Measurement of the Spatial Shape of Photons

UNIVERSITAT POLITÈCNICA
DE CATALUNYA
Measurement of the Spatial
Shape of Photons
Memory presented by
Noelia González Rodrı́guez
Thesis supervised by
J.P Torres
Dpto. Teoria del Senyal i Comunicacions
Universitat Politècnica de Catalunya
Institut de Ciències Fotòniques
Barcelona, 2009
A mis padres
A mi hermana, Jordi y Albert
A Giu
Agradecimientos
Este conjunto de hojas agrupado bajo el nombre de Tesis, es seguramente
a la vista del lector, una recopilación del trabajo llevado a cabo durante
mis casi 5 años de doctorado. Siendo ası́ espero que su contenido no sea
excesivamente tedioso y pueda aportar algo positivo a aquel que lo lea.
A mis ojos es la última página de un libro, leer la última lı́nea, llegar
a la palabra Fin. Parar la mirada sobre la palabra Fin, sin mirarla, mientras mis ojos miran las imágenes contenidas en esta historia. Posar la vista
sobre la palabra Fin, mirándola ahora, siendo consciente de que el próximo
movimiento es cerrar el libro.
Esta reflexión podrı́a parecer triste, pero no lo es, más bien todo lo contrario. Porque después de cerrar un libro abrimos otro, con la excitación
y el entusiasmo de adentrarnos en algo nuevo, y con la experiencia de lo
aprendido en el libro anterior.
Ası́ pues, cierro este libro, pero no sin antes dedicar unas lı́neas de
agradecimiento a aquellos que me han ayudado y que, en este momentos,
aparecen en mi mente.
Gracias a...
Juan, por darme la oportunidad de trabajar en su grupo.
Jordi Mompart por su gran humanidad en este mundo que a veces parece
carecer de ella. Por motivarme y alentarme en mi último año de carrera y
en el inicio de mi doctorado.
Gabi. A él quiero darle no las gracias, sino un número infinito de gracias. Por
haber sido un compañero de trabajo excelente y mejor amigo. Por hacerme
la vida en el laboratorio más fácil con su compañı́a, por su eterna paciencia,
por todo lo que me ha enseñado, por su alegrı́a que se contagia. Gracias
Gabi, no cambies nunca.
Al resto de mi grupo, especialmente a Xiaojuan Shi, por ser una estupenda
compañera de despacho para compartir penas y alegrı́as.
A Roser, que realizó la parte experimental del SPI y me cedió sus datos para
completar mi Tesis.
A Mery, Laia, Manuela, Olga y Laura, que han sufrido mi despiste y des-
organización y siempre me han ayudado en toda la burocracia, trámites y
preparación de presentaciones.
A Mafi, Daniel, Flavia, Leandro, Milene, Nico, por ser los mejores compañeros
de cervecitas, cenas, conciertos y fiestas varias (siempre a tope).
A Sybille, por ser tan dulce y intentar ayudarme siempre.
A Toni, por su elegancia.
A mis ”flamenquitas” . Mi amistad con ellas empezó al mismo tiempo que
mi doctorado y han sido un punto de equilibrio muy importante. Me encanta
hablar con ellas que viven en el otro extremo del mundo cientı́fico.
A Nuria, Manel y Piconcito, amigos que siempre están ahı́, aunque los años
pasen.
A mis padres y mi hermana, que siempre me han apoyado a pesar de llevar
una forma de vida tan diferente de la suya. Creo que estos años han sido
una etapa de aprendizaje, tanto para ellos como para mı́.
El pensamiento más fuerte y presente para Giu, por descubrirme y ayudarme
en la investigación más difı́cil, la de uno mismo.
Contents
1 Introduction.
1
1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Some Basic concepts. . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Quantum States. . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Bipartite quantum systems. . . . . . . . . . . . . . . . 4
1.4 Implementation of quantum entangled states with photons. . . 5
1.4.1 Generation of quantum entangled states by Spontaneous Parametric Down conversion (SPDC). . . . . . . 6
1.4.2 Quantum orbital angular momentum (OAM) of photons. 9
1.5 Atomic medium as quantum memory . . . . . . . . . . . . . . 12
1.5.1 Electromagnetically induced transparency (EIT) . . . . 12
2 How a Dove prism transforms the orbital angular momentum
of a light beam.
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 ABCD law for a Dove prism . . . . . . . . . . . . . . . . . . .
2.2.1 Theoretical development . . . . . . . . . . . . . . . . .
2.2.2 Experimental validation of the ABCD law. . . . . . . .
2.3 Ellipticity induced by a Dove prism . . . . . . . . . . . . . . .
2.3.1 Theoretical development . . . . . . . . . . . . . . . . .
2.3.2 Experimental observation of the ellipticity . . . . . . .
2.4 OAM transformation rule of the Dove
prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Characterization of the spatial shape
spiral phase interferometry
3.1 Introduction . . . . . . . . . . . . . .
3.2 Basics of spiral phase filtering. . . . .
3.2.1 Isotropic spiral phase filtering.
vii
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of optical beams with
29
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viii
3.3
3.4
3.5
3.6
3.7
3.8
CONTENTS
3.2.2 Non-isotropic phase filtering. . . . . . . . . . . . . . .
Spiral phase interferometry revisited. . . . . . . . . . . . . .
3.3.1 Mathematical development. . . . . . . . . . . . . . .
3.3.2 SPI problems. . . . . . . . . . . . . . . . . . . . . . .
Solutions for the original SPI method. New implementation.
3.4.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Processing. . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 Reconstruction. . . . . . . . . . . . . . . . . . . . . .
Numerical simulations for the new implementation . . . . . .
Experimental set-up . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Optical beam generation. . . . . . . . . . . . . . . . .
3.6.2 Optical Fourier Transform 1 (OFT1) . . . . . . . . .
3.6.3 Optical filtering . . . . . . . . . . . . . . . . . . . . .
3.6.4 Imaging System 1 . . . . . . . . . . . . . . . . . . . .
3.6.5 Optical Fourier Transform 2 (OFT2) + Imaging System 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental results. . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Filter centered at the optical axes. . . . . . . . . . .
3.7.2 Filter displaced with respect to the optical axes. . . .
3.7.3 Other experimental results. . . . . . . . . . . . . . .
Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Measurement of the spatial Wigner function of paired photons
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Wigner function . . . . . . . . . . . . . . . . . . . . . . .
4.3 The source of spatially entangled photons. . . . . . . . . . . .
4.4 Remote preparation of a pure state . . . . . . . . . . . . . . .
4.4.1 Projection onto a Gaussian state . . . . . . . . . . . .
4.4.2 Projection onto superpositions of Hermite-Gaussian and
Gaussian states . . . . . . . . . . . . . . . . . . . . . .
4.5 Wigner function of the signal photon and the amount of spatial
entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Experimental set-up for measuring the Wigner function of the
signal photon. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Entangled photons source. . . . . . . . . . . . . . . . .
4.6.2 Idler operation. . . . . . . . . . . . . . . . . . . . . . .
4.6.3 Sagnac interferometer for the signal photon. . . . . . .
4.6.4 Coincidence Counting. . . . . . . . . . . . . . . . . . .
4.7 A more realistic analysis: propagation of the fields from the
crystal to the detectors. . . . . . . . . . . . . . . . . . . . . .
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CONTENTS
ix
4.8 Experimental results and discussion. . . . . . . . . . . . . . . 80
4.9 Wigner function of the two-photon state . . . . . . . . . . . . 82
4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Dynamics of saturated Bragg diffraction in a stored light
grating in cold atoms.
86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Some Concepts about EIT and LS. . . . . . . . . . . . . . . . 87
5.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Grating formation and storage . . . . . . . . . . . . . . 99
5.3.2 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.3 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.1 Atomic source. . . . . . . . . . . . . . . . . . . . . . . 104
5.4.2 Incident laser beams. . . . . . . . . . . . . . . . . . . . 106
5.4.3 Detection. . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5 Experimental results and discussion. . . . . . . . . . . . . . . 108
5.5.1 Diffracted signal D for different storage times. . . . . . 108
5.5.2 Diffracted signal D for different reading beam intensities.110
5.5.3 Diffracted signal D for different writing beam intensities.113
5.5.4 Energy retrieved saturation. . . . . . . . . . . . . . . . 113
5.6 Collapse and revival of the stored light grating. . . . . . . . . 114
5.7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Conclusions.
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter 1
Introduction.
Because this is the first chapter of this thesis, our purpose here is twofold.
Firstly, to expose the motivation of the work that will be presented below,
and secondly, to introduce some fundamental concepts that are going to be
extensively used. The aim is not to give a very detailed explanation, but to
provide the tools (if they are needed) in order to make the reading of this
work easier and more understandable.
1.1
Motivation.
Quantum Optics Information Technology (QOIT) is a new branch of Science and Technology that has emerged in the last few years. QOIT offers a
qualitative new view about what it is and how we can manipulate information. Importantly, two technologies (communications and computing) that
crucially shape our present world, and surely will strongly contribute to the
tailoring our future world, are fundamentally based on information. Therefore, a central topic in the field of Quantum Information is study of the
properties of quantum states and their potentiality when they are used in
information-related applications [11].
Photons and atoms are the key physical objects. By exploiting their
quantum properties QOIT holds the promise of computational capabilities
beyond those of any classical computer, it promises absolutely secure communication, and it offers the possibility to implement new applications, such as,
e.g., quantum enhanced clock synchronization and positioning sensors, as well
as ultra high resolution quantum imaging. So that, the use of quantum states
implemented by photons and atoms might guide the elucidation of proof-ofprinciple capacity-increased quantum information processing schemes.
Of special interest is the generation and application of entanglement. En1
2
Introduction.
tanglement of identical particles is one of the most genuine features of the
quantum world, and it forms the core of quantum cryptography, computing,
and teleportation [11–16]. To date research has focused on quantum states
belonging to two-dimensional Hilbert spaces, or qubits. Most quantum information applications use the polarization of photons, or polarization entanglement between paired photons as the quantum resource. Photons can
be described as a superpositions of two orthogonal polarizations: vertical
and horizontal, or alternatively, right-hand or left-hand circular polarization.
Photons entangled in polarization have been used to demonstrate quantum
teleportation, quantum dense coding, and entanglement swapping [13].
However, multidimensional entangled states, or qudits, provide higher dimensional alphabets, thus enhancing the potential of quantum techniques.
For example, by using qudits the security of quantum key distribution cryptography can be improved [17], and the efficiency of quantum communication
protocols can be enhanced [18]. Therefore, the challenge is the implementation of the d -channel. Quantum systems entangled in continuous variables
have been receiving increasing attention from the scientific community. The
most widely explored continuous variable system is based on the quadratures
of the electrical field of photons.
The transverse spatial shape of photons (i.e transverse position and momentum degree of freedom) is another of such continuous variable systems. It
has already been shown that pairs of photons can be entangled in their spatial properties [120]. Up to now, the entanglement in transverse momentum
has been used to test some quantum protocols in a finite number of dimensions [9,121]. On the other hand, due to its ease of control, the spatial degree
of photons is prone to be used for testing some of the new physical predictions that continuous variables quantum systems can offer [122]. Following
this line, the central topic of this Thesis is the characterization and measurement of the spatial shape of photons. Different techniques to completely
characterize and implement quantum states using the spatial properties of
photons has been developed and they will be presented in the following chapters.
The realization of many basic concepts in quantum information and computation science, not only requires the production and transport of quantum
states, if not the capability of storage and retrieval them, and is here where
atoms are a key element. The coherent and reversible storage of photons
states in matter is an outstanding problem. In general photons are difficult
to localize and store, for this reason matter (e.g., spins) will likely serve as
quantum memory elements. Therefore the challenge is to develop a technique
for coherent transfer of quantum information carried by light to atoms and
viceversa. In other words it is necessary to have a quantum memory that
1.2 Overview.
3
is capable to storing and releasing quantum states. Such a device needs to
be entirely coherent, and in order to achieve a unidirectional transfer (from
field to atoms or viceversa), an explicit time-dependent control mechanism is
required. The work presented in the last chapter of this Thesis is dedicated
to this aim.
1.2
Overview.
Below we summarize the contents of this Thesis:
Chapter 2: How a Dove prism transforms the orbital angular
momentum of a light beam. It is generally assumed that a light beam
with orbital angular momentum (OAM) per photon of l, is transformed,
when traversing a Dove prism, into a light beam with OAM per photon
of −l. In this chapter, we show theoretically and experimentally that this
OAM transformation rule does not apply for highly focused light beams. This
result should be taken into account when designing classical and quantum
algorithms that make use of Dove prims to manipulate the OAM of light.
Chapter 3: Characterization of optical beams with spiral phase
interferometry. In this chapter we study both theoretically and experimentally a method to characterize the amplitude and phase of a paraxial optical
beam. The method is based on the spiral phase interferometry method, recently proposed. We theoretically analyze how to adapt the original proposal
to deal with the special characteristics of finite optical beams. Finally, we
compare a series of numerical and experimental results to show the advantages and limitations of our proposal.
Chapter 4: Measurement of the spatial Wigner function of
paired photons. Here we analyze some important characteristics of the
spatial Wigner function of entangled photon pairs. We show that the Wigner
description of quantum states that live in the infinitive-dimensional spacemomentum degree of freedom proves to be particularly useful. We propose a
experimental configuration that can be used to retrieve the Wigner function
of paired photons entangled in the spatial degree of freedom. In particular, it allows the full characterization of the paired photons emitted from a
spontaneous parametric down-conversion (SPDC) source.
Chapter 5: Dynamics of saturated Bragg diffraction in a stored
light grating in cold atoms. We report on a detailed investigation of the
dynamics and the saturation of a light grating stored in a sample of cold
cesium atoms. We employ Bragg diffraction to retrieve the stored optical
information impressed into the atomic coherence by the incident light fields.
4
Introduction.
The diffracted efficiency is studied as a function of the intensities of both
writing and reading laser beams. A theoretical model is developed to predict
the temporal pulse shape of the retrieved signal and compares reasonably
well with the observed results.
1.3
1.3.1
Some Basic concepts.
Quantum States.
In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which ”prepares” this quantum state. The mathematical
function that contains the full description about the state generated is the
density matrix ρ̂. ρ̂ is Hermitian and normalized, so that T r ρ̂ = 1. An important case corresponds to pure states, which can be written as ρ = |ΨΨ|,
where |Ψ is a vector in a Hilbert space. Pure states fulfill T rρ2 = 1, while
all other states, called mixed states, fulfill T rρ2 < 1.
1.3.2
Bipartite quantum systems.
Let’s to consider one quantum system, described by ρ̂, which consists of two
subsystems, 1 and 2. We can make experiments on each of the subsystems
independently, or experiments that involve the measurements of correlations
between the outcomes of the two subsystems.
In the case that we observe only quantities referring to one subsystem,
we should make use of the reduced density operator
ρ̂1 = tr2 {ρ̂}
(1.1)
where the tr2 should be calculated with respect to the unobserved degrees of
freedom. The so-constructed reduced operator ρ̂1 obeys all the requirements
for a physically meaningful density operator- it is normalized, Hermitian,
and nonnegative because the total density operator meets this criteria. Consequently, we can regard ρ̂1 as describing the quantum state of the reduced
system.
The parts of a composite system are in general mixed states. Note that
even if the total system is in a pure state, the reduced system might be statistically mixed. This feature relies on the entanglement of the subsystems (and
hence it can be used as measure for entanglement [5], [6]). It is not possible
observe all aspects of an entangled system by considering the subsystems
1.4 Implementation of quantum entangled states with photons. 5
only. The lack of knowledge about the partner object causes statistical uncertainty in the state of the subsystem, explaining why the reduced system
may be in a mixed state.
If the whole systems is separable, we can write
ρ=
an ρ̂n1 ⊗ ρ̂n2
(1.2)
n
where the symbol ⊗ denotes the tensor product, and ρ̂i denotes a density
matrix describing subsystem i. The state space of the total system is the
tensor product of the subspaces. Quantum states that can not be written as
Eq. (1.2) are called entangled states.
An example of this type of states for a bipartite system is
|Φ =
|H1 ⊗ |V 2 − |V 1 ⊗ |H2
√
2
(1.3)
which can not be expressed as |ψ1 1 ⊗ |ψ2 2 for any |ψ1 1 , |ψ2 2 . H and V
refer to horizontal and vertical polarizations of a photons. An important
characteristic of entangled states is that the measurements of correlations
between the subsystems show very peculiar features, which cannot be explained classically. The global system is not any more a mere composition of
its parts, because the subsystems are correlated, even when both parts are
spatially separated.. This correlation may bridge space and time, showing
the potential nonlocality of quantum mechanics, as expressed for instance
in the Einstein-Podolsky-Rosen paradox [2] and in Bell’s inequalities [3, 4].
Entangled states are a fundamental key in quantum computation, quantum
communication and quantum cryptography.
1.4
Implementation of quantum entangled states
with photons.
Entangled states are a key element to perform quantum computation, quantum communication and quantum cryptography. Indeed, one of the aims in
quantum physics is the experimental implementation of this type of states.
Good candidates to implement such interesting kind of quantum states, the
entangled states, are paired photons. In the following we are going to explain
how to generate entangled states where photons are correlated in polarization, spatial shape or the orbital angular momentum, and in the frequency
content. In section 1.4.1 we present the spontaneous parametric down conversion (SPDC) process, used to generate the entangled photons whereas
6
Introduction.
in section 1.4.2 we are going to center our attention in the orbital angular
momentum (OAM) of photons as a degree of freedom.
1.4.1
Generation of quantum entangled states by Spontaneous Parametric Down conversion (SPDC).
The most widely used method for the generation of pairs of entangled photons
is spontaneous parametric down conversion (SPDC). In SPDC, a beam of
radiation, called the pump, is incident on a birefringent crystal. The pump
is intense enough so that nonlinear effects lead to the conversion of pump
photons into pairs of correlated photons. The lower-frequency generated
photons can be entangled in polarization or spin angular momentum [19], in
frequency [20], and in orbital angular momentum [21, 22] as will be shown
with more detail below. The down conversion process is said to be of type
I or type II, depending on wether the photons in the pair have parallel or
orthogonal polarizations. For frequency degenerate photon pairs, in the type
I case, paired photons emerge from the crystal forming an unique cone, as it
is showed in Fig. 1.1(a). In the type II case, showed in Fig. 1.1(b), emerging
photons form two differentiate cones. The photons in a pair may come out
in different directions (non collinear configuration) or they may come out
in the same direction (collinear configuration). The frequency and direction
of the photons is determined by the orientation of the crystal. Figure 1.2
show some examples of the down conversion cones measured experimentally.
It may be stressed that if other frequencies are taken into account, then
Fig. 1.1 becomes more complicated, with different frequencies emerging at
different angles.
(a)
(b)
Figure 1.1: Schematic examples of SPDC process.(a) Type I: paired photons
emerge from the crystal forming an unique cone. (b) Type II: emerging photons
form two differentiate cones.
1.4 Implementation of quantum entangled states with photons. 7
Figure 1.2: Example of SPDC cones measured experimentally: (a) and (b) correspond to collinear configuration, where photons in a pair came out in the same
direction; (c) and (d) show the non collinear configuration where the photons in a
pair come out in different directions.
The effective Hamiltonian.
In the interaction picture, the effective Hamiltonian for the optical parametric
process in a crystal pumped by a laser beam is
HI (t) = ε0
χ(2) Êp+ (r, t)Ês− (r, t)Êi− (r, t) + hc
(1.4)
V
where the integral is performed over the interaction volume V , Êj+ (r, t) and
Êj− (r, t) are the positive and negative-frequency components of the electric
field operator associated to the signal (j=s), idler (j=i) and pump (j=p)
†
photons, related by Êj+ (r, t) = Êj− (r, t) and h.c stands for the Hermitian
conjugate. The positive frequency Êj+ (r, t) is given by
+
Êj (x, z, t) ∝ dωj dqj âj (ωj , qj ) exp (ikj z + iqj [x + z] − iωj t) (1.5)
where âj (ωj , qj ) is the photon annihilation operator for the j-th polarized
mode with transverse wave number qj = (qjx , qji ) and frequency ωj , kj =
(ωj nj /c)2 − |qj | is the longitudinal wave number inside the crystal, nj
is the refractive index at the central angular frequency, is the Poynting
vector walk-off and x = (x, y) is the position in the transverse plane. Since
spontaneous parametric down-conversion is a very inefficient process, the
pump field must be relatively intense. Accordingly, the electric-field operator
for the pump beam, Êp+ (r, t) may be replaced by the classical field
Ep (x, z, t) = dωp dqp E0 (ωp , qp ) exp ikp z + iqp [x + z] − iωp t (1.6)
where E0 (ωp , qp ) is the classical amplitude of the pump beam.
8
Introduction.
The two photon quantum state.
The two-photon quantum state |Ψ at the output of the nonlinear crystal to
first order in perturbation theory reads
τ
|Ψ = |0, 0 − (i/)
dtHI (t)|0, 0
(1.7)
0
where |0, 0 is the vacuum state, and τ the interaction time. The time integral gives 2πδ (ωs + ωi − ωp ) which is the steady-state or frequency phasematching condition. In the absence of walk-off, the integral in the z direction
over the length L of the crystal gives
W (Δk L/2) = sinc (Δk L/2) exp (isk L/2)
(1.8)
where Δk determines the spectral width of the two-photon state and is expressed as
Δk = kp (ωs + ωi, qs + qi ) − ks (ωs , qs ) − ki (ωi , qi )
(1.9)
sk = kp (ωs + ωi , qs + qi ) + ks (ωs , qs ) + ki (ωi, qi ) .
(1.10)
and
We obtain that the two-photon state is given by
|Ψ = |0, 0 + dωs dωi dqs dqi Φ (ωs , ωi , qs , qi ) â†s (ωs , qs ) â†i (ωi, qi ) |0, 0
(1.11)
where we have defined Φ (ωs , ωi, qs , qi ) as
Φ (ωs , ωi, qs , qi ) = E0 (ωs + ωi , qs + qi ) W (Δk L/2) .
(1.12)
If L is infinite, then the integral over the length of the crystal becomes a δ
function. In this case, phase-matching conditions
ωs + ωi = ωp
ks + ki = kp êz
(1.13)
both hold and the phase matching is said to be perfect. Here kj = (qj , kj ) is
the total wave number. The phase-matching conditions arise from the fact
that the SPDC process is a coherent process in which all parts of the crystal
contribute in phase. For finite L the wave vector phase matching condition is
relaxed so that |Δk | may vary over an interval of order 1/L. The state |Ψ is a
linear superposition of the vacuum state and a state containing two photons.
In general, the second term is much smaller than the first by 5 − 6 orders of
1.4 Implementation of quantum entangled states with photons. 9
magnitude. Higher-order terms containing four, six, etc. photons are negligible for continuous pumping with power below 1 W. The two-photon part
of the state is an entangled state in frequency, wave number and polarization. In frequency space, the entanglement is a result of the phase-matching
conditions, which implies that the detection of a photon at frequency ω requires the other photon to have the frequency ωp − ω. This is the origin of
interesting experiments aimed at illuminating the Einstein-Podolsky-Rosen
paradox. The state is also entangled with respect to the wave vector k since
the function Φ defined in Eq. (1.12) cannot be written as a product of a
function of ks times a function of ki . In the general case, the wave number
entanglement has implications for the spatial correlations (orbital angular
momentum correlations) as we will see below. Finally the two-photon state
in Eq. (1.11) is entangled in both k and spin degrees of freedom since the
polarization vectors are themselves functions of k because they must satisfy êj · k = 0. In the collinear case, where the crystal is oriented so that
all the beams (pump, signal and idler) have the same direction, the spatial
and temporal correlations are indistinguishable. The lack of entanglement
in polarization is a special feature of the collinear case and follows from the
symmetry of the state with respect to the frequency and wave number.
1.4.2
Quantum orbital angular momentum (OAM) of
photons.
As we have seen above, spontaneous parametric down conversion is a reliable source for the generation of entangled photon pairs. Pairs of downconverted photons entangled in polarization, or spin angular momentum,
were used, e.g., in the demonstration of quantum teleportation [51], and in
the recent realization of a quantum universal NOT gate [14]. However, the
down-converted photons can also be entangled in orbital angular momentum
(OAM) [21], which belongs to an infinite dimensional Hilbert space and thus
allows encoding qudits with arbitrary d [28]. The quantum state of a photon
is described by a mode function Ψ. Any paraxial mode function expressed in
cilindrical spatial coordinates, i.e Ψ = Ψ (ρ, φ, z) with an arbitrary amplitude
profile can be expanded into Laguerre Gaussian (LG) modes,
Ψ (ρ, φ, z) =
∞ ∞
l=−∞ p=0
LGlp (ρ, φ, z) .
(1.14)
10
Introduction.
(a)
(b)
(c)
Imax
0
(e)
(d)
(f)
p
-p
Figure 1.3: Some examples of Laguerre Gaussian modes. The amplitude is
showed in the top arrow whereas the phase is showed below: (a) amplitude and
(d) phase of a LG01 mode; (b) amplitude and (e) phase of a LG20 mode, were we
can observe the absence of spiral phase due to the exp(im ) contribution; (c) and
(f) amplitude and phase of a LG21 mode respectively.
The normalized Laguerre Gaussian mode at its beam waist (z = 0) is given
in cylindrical coordinates by
√ |l|
2
2ρ
1
2p!
2ρ
l
|l|
exp −ρ2 /w02 exp (ilφ)
Lp
LGp (ρ, φ, z = 0) =
2
π (|l| + p)! w0
w0
w0
(1.15)
|l|
where Lp (ρ) are the associated Laguerre polynomials,
L|l|
p
(ρ) =
p
m=0
(−1)m
(|l| + p)!
ρm
(p − m)! (|l| + m)!m!
(1.16)
w0 is the beam width, p is the number of nonaxial radial nodes of the mode,
and the index l, referred to as the winding number, describes the helical
structure of the wave front around a phase dislocation. Some examples of
LG modes ares showed in Fig. 1.3.
When the mode function is a pure LG mode with winding number l, the
quantum sate is an eigenstate of the OAM operator with eigenvalue l [78].
State vectors, which are not represented by a pure LG mode, correspond to
photons in a superposition state, with the weights of the quantum superposition dictated by the contribution of the lth angular harmonics.
1.4 Implementation of quantum entangled states with photons. 11
Coming back to the paired photons produced in the SPDC process, the mode
function which describes the two-photon state at the output of the no nonlinear crystal given by Eq. (1.11) can also be expressed in the spatial-temporal
space. Without taking into account the temporal part and the vacuum contribution, the mode function reads [53]
(1.17)
|Ψ = dr⊥ Φ (⊥) â†s (⊥) â†i (⊥) |0, 0
where r⊥ is the radial coordinate in the transverse x − y plane. Here Φ (⊥) is
the spatial distribution of the pump beam at the input faced of the crystal.
A photon state described by a pure LG mode can be written as
(1.18)
|lp = dr⊥ LGlp (r⊥ ) ↠(⊥) |0
Using I = lp |lplp|, once can express the quantum state |Ψ using the
eigenstates of the orbital angular momentum operator as
|Ψ =
Cpl11,l,p22 |l1 , p1 ; l2 , p2 (1.19)
l1 ,p1 l2 ,p2
where (l1 , p1 ) correspond to the signal mode and (l2 , p2 ) correspond to the
idler mode. The expression of the probability amplitude Cpl11,l,p22 is given by
[54, 55]
∗ ∗
l1 ,l2
Cp1 ,p2 ∼ dr⊥ Φ (r⊥ ) LGlp11 (r⊥ ) LGlp22 (r⊥ ) .
(1.20)
The pump beam Φ (r⊥ ) can be expanded into spiral harmonics to get
Φ (ρ, φ) =
∞
al (ρ) exp (ilφ) ,
(1.21)
l=−∞
therefore, by inserting Eq. (1.21) in Eq. (1.20) it can be seen that the
quantum probability amplitude Cpl11,l,p22 depends only on the radial profile of
the (l1 + l2 )th angular harmonic of the pump beam. Thus, such harmonic
content of the pump beam translates to the complex probability amplitude of
the quantum state with l1 +l2 = m resulting in the entanglement between the
signal and idler photon. The weights of the quantum superposition are given
by Ppl11,l,p22 = |Cpl11,l,p22 |2 /η, which gives the value of the joint detection probability
for finding one photon in the signal mode (l1 , p1 ) and one two-photon in the
idler mode (l2 , p2 ). The two-photon state produced in the down-conversion
process is a coherent superposition of an infinite number of states of the form
|l1 , p1 ; l2 , p2 , so that using eigenstates with index l = 0, ..., d − 1, produces
qudits of arbitrary d.
12
1.5
Introduction.
Atomic medium as quantum memory
In the previous sections we have seen that photons are ideal candidates to
carry quantum information. They offer the possibility of implementing quantum states in many degrees of freedom, moreover they are fast, robust and
readily available. However, the capability of storage and retrieval the information that they carry, i.e. a quantum memory, is also crucial in quantum
information and computation science.
The conceptually simplest approach to a quantum memory is to ”store”
the state of a single photon in an individual atom. This approach involves
a coherent absorption and emission of single photons by single atoms. However, the single-atom absorption cross sections is very small, which makes
such a process very inefficient. Cavity QED provides a solution for this problem [56]- [58]. Placing an atom in a high-Q resonator effectively enhances
its cross-section by the number of photon round trips during the ring-down
time and thus makes an effective transfer possible. Raman adiabatic-passage
techniques [59,60], with time-dependent external control fields can be used to
implement a directed but reversible transfer of quantum state of a photon to
the atoms (i.e., coherent absorption). However, despite the enormous experimental progress in this field [61] it is technically very challenging to achieve
the necessary strong-coupling highly susceptible regime. Furthermore, the
single-atom system is by construction highly susceptible to the loss of atoms
and the speed of operations is limited by the large Q factor.
On the other hand a photon can be absorbed with unit probability in an
optically thick ensemble of atoms. Nevertheless it has been shown that such
absorption of light leads to a partial mapping of this quantum properties to
atomic ensembles [62–64]. As a consequence of dissipation these methods do
not allow to reversibly store the quantum state on the level of individual photon wave packets (single qubits). Alternatively, light storage (LS) has been
demonstrated to be a reliable technique to trap, store, and release excitations
carried by light pulses in atomic mediums, and to date several experimental
observations of these effects were realized in different systems [65–68, 141].
This light storage technique is based on the phenomenon of ultraslow light
group velocity [70–72], which is made possible by electromagnetically induced
transparency (EIT).
1.5.1
Electromagnetically induced transparency (EIT)
Electromagnetically induced transparency (EIT), termed in this way by Harris and co-workers (Harris et al., 1990, consist in modifying the optical response of an atomic medium by means of laser-induced coherence of atomic
1.5 Atomic medium as quantum memory
D2
D1
G31 G32
13
3
wc
wp
2
1
Figure 1.4: Generic system for EIT: lambda-type scheme with signal field of
frequency ωp and coupling field of frequency ωc . Δ1 = ω31 − ωp and Δ2 = ω32 − ωc
denote detunings form atomic resonances and Λjk radiative decays rates from state
|j to state |k
states. The generic system for EIT consist of a Λ-type three-level system
driven by a coherent coupling field, how it can be seen in Fig. 1.4. As a
result of the laser-induced coherence, the different excitation pathways that
control the optical response interfere. The absorption and refraction (linear
susceptibility), can in this way be eliminated at the resonance frequency of
a transition. The importance of EIT stems from the fact that it gives rise
to greatly enhanced nonlinear susceptibility in the spectral region of induced
transparency of the medium and is associated with step dispersion. A more
detailed explanation about the EIT process and how it allows LS is given in
chapter 5.
Chapter 2
How a Dove prism transforms
the orbital angular momentum
of a light beam.
2.1
Introduction.
The Dove prism is a very well known tool in optics. It is a type of reflective
prism which is used to invert an image. Dove prisms are shaped from a
truncated right-angle prism. A beam of light entering one of the sloped faces
of the prism undergoes total internal reflection from the inside of the longest
(bottom) face and emerges from the opposite sloped face. Images passing
through the prism are flipped, and because only one reflection takes place,
the image is inverted but not laterally transposed.
Figure 2.1: Dove prism
14
2.1 Introduction.
15
y
YD
y
x
y
XD
x
q
x
y
2q
x’
YD
XD
x
Figure 2.2: Example of how a Dove prism works. If the coordinate axis of the image
coincides with the optical axis of the Dove prism it acts as an image flipper (top figure).
If the Dove prism is rotated an angle θ about its optical axis, images passing through the
prism are flipped and rotated an angle 2θ (bottom image). The phase is rotated 2θ also.
Dove prisms show an interesting property when they are rotated along
their longitudinal axis. The transmitted image rotates at twice the rate of the
prism. This property means they can rotate a beam of light by an arbitrary
angle, making them useful as beam rotators, which have applications in fields
such as interferometry, astronomy, and pattern recognition [87]. Moreover,
this makes the OAM of a light beam to change its sign, which has turned
Dove prims into a key element in some recent classical and quantum optics
implementations that make use of the OAM of light as a resource.
A control-NOT gate, which has recently been implemented using polarization and transverse spatial modes [88], it makes use of a Dove prism
located in one of the arms of an interferometer, where the spatial profile of
the light beam (or photon) is properly rotated. Dove prisms are key elements
of an interferometric method for measuring the orbital angular momentum
of single photons [89], as well as of a scheme that allows the measurement of
the orbital angular momentum content of a superposition of LG beams [90].
Recently, another interferometric method has been proposed for measuring
the amount of spatial entanglement that exist between certain entangled
paired photons generated in parametric down conversion [91]. A scheme to
generate arbitrary coherent superpositions of OAM states in Bose-Einstein
condensates makes use of Dove prisms to change the handedness of light [92].
When a light beam with a well defined OAM per photon of l, i.e., with
16
How a Dove prism transforms...
spatial shape in cylindrical coordinates
Ain = A0 (ρ) exp (ilϕ)
(2.1)
traverses a Dove prim, it is generally assumed that the output beam has a
well defined OAM per photon of −l, i.e., with spatial shape
Aout = A0 (ρ) exp (−ilϕ) exp (−2ilθ)
(2.2)
where θ is the angle of rotation of the Dove prism. Notice that the Dove
prism introduces also a phase of the form 2lθ. If the Dove prism rotates
continuously, the angle of rotation θ changes with time, and therefore the
phase shift 2lθ(t) depends on time. This effect makes possible the observation
of the rotational frequency shift of light beams [93], which can be used to
measure the OAM content of a light beam [45]. The rotation of a Dove
prism can also introduce time-dependent polarization changes into the light
beam [94], since the polarization also depends on the specific configuration
used.
How can we derive the change of sign of the OAM (l → −l) of the
input beam, and the introduction of a new phase (2lθ), after the beam has
traversed the Dove prism? To see this point, let us rewrite the initial beam
in normalized cartesian coordinates as
Ain (x, y) = Ã0 (x, y) (x + iy)l exp − x2 + y 2 .
(2.3)
The Dove prism, characterized by the coordinate axis (xD , yD ), is rotated an
angle θ respect to the coordinate axes (x, y). We can thus write
x = xD cos θ + yD sin θ,
y = −xD sin θ + yD cos θ.
(2.4)
The input beam can be written in the new coordinates as
Ain (xD , yD ) = Ã0 (xD , yD ) [(cos θ − i sin θ)(xD + iyD )]l .
(2.5)
The Dove prims performs the transformation xD → xD , yD → −yD . When
we rewrite the output beam back to the original axis (x, y), making use of
xD = x cos θ − y sin θ,
yD = x sin θ + y cos θ
(2.6)
we obtain
Aout (x, y) = Ã0 (x, y) (cos 2θ − i sin 2θ)l (x − iy)l .
(2.7)
2.1 Introduction.
17
a)
input plane
(x1 , y1)
Y
Z
X
output plane
(x2 , y2)
(x’’ , y’’)
(x’ , y’)
iy
oy
?
b)
iy’
X
Y
Z
ix
ix’
ox
L
Figure 2.3: Geometrical configuration of a Dove prism. (a) Lateral view (yz−plane)
and (b) Top view (xz−plane). Solid and dashed lines represent the typical path of two
optical rays.
Finally, taking into account some well-known trigonometric relationships
(exp(iθ) = cos θ + i sin θ), we obtain that
Aout (x, y) = Ã0 (x, y) (x − iy)l exp (−il2θ)
(2.8)
which it is the expression that we were looking for.
In this section we will show theoretically and experimentally that the
OAM transformation rule l ⇒ −l, derived above, it is not valid for highly
focused light beams, since Dove prisms inherently introduce astigmatism, and
therefore further OAM changes. The transformation of the light beam due
to the Dove prims can not be written, for highly focused beams, as just the
introducion of a π phase in one transverse coordinate (y → −y). Light beams
with a well defined value of the OAM per photon, after traversing the Dove
prism, are transformed into a superposition of states with well defined OAM.
The violation of the rule l ⇒ −l, turns out to be more severe for highly
focused light beams. This result should be taken into account when designing
classical and quantum algorithms that make use of Dove prims to manipulate
the OAM of light. We will provide a quantitative study of the properties of
the Dove prism, by making use of the geometrical optics properties of the
Dove prism, and we will verify experimentally the validity of our theoretical
results in a series of experiments with a commercially available Dove prism.
18
2.2
2.2.1
How a Dove prism transforms...
ABCD law for a Dove prism
Theoretical development
In Fig. 2.3 we present the basic geometrical configuration of a Dove prism, by
showing a typical optical ray tracing. We will find the relationship between
the output position (x2 , y2 ) and angle (ox , oy ) of a ray, and the input position
(x1 , y1 ) and angle (ix , iy ) by making use of the laws of geometrical optics. In
order to do so, one follows the rays trajectories in the figure. This is done in
three steps. Firstly, we propagate the ray from the input plane to the input
face of the prism (x1 , y1 ) → (x , y ). Secondly, we let the ray traverse the
Dove prism (x , y ) → (x , y ), and finally, we calculate the ray trajectory
from the output face of the prism to the output plane (x , y ) → (x2 , y2 ).
The first and last steps are straightforward free-space propagations, which in
our case just means finding the crossings in the three dimensional space of a
straight line with a plane.
On the other hand, the middle step is divided into refraction from air to
glass at the input face of the prism, reflection of the ray at the floor of the
prism, and another refraction from glass to air. The final result relates the
position and angle of the ray at the input plane (x1 , y1 ; ix , iy ), with those at
the output plane (x2 , y2 ; ox , oy ) in the following way [95]
x2 = x1 + L
tan(ix ) + tan(α) tan(α + iy ) tan(ix )
,
1 + tan(α) tan(α + iy )
ox = ix ;
y2 = L
tan(α) − tan(iy )
1 + tan(α) tan(α + iy )
oy = −iy .
− y1 ,
(2.9)
In these formulas we use the refraction angles inside the crystal
−1
ix , iy = sin (sin(ix )/n), sin−1 (sin(π/2 − α − iy )/n ,
(2.10)
which are shown in Fig. 2.3.
Next, we perform a Taylor expansion to first order in the angles of these
equation, since we consider the paraxial approximation regime. The result
2.2 ABCD law for a Dove prism
of these approximations are
19
L
h0
1
+
x2 = x1 +
1−
ix
n tan α
n
ox = ix
η
1
+
y2 = (h0 − y1 ) − h0
iy
n tan α
oy = −iy
(2.11)
where L is the length of the base of the Dove prism, n is the refractive index
of the material, α is the base angle, and
−1
1
−1 cos α
+
h0 = L tan α + sin
n
tan α
cos α cos α 2 −1/2
h0 sin α
η=
cos−2 α + sin−1
1−
.(2.12)
L
n
n
A few comments are now in order. First, one notes that, although in the full
equations, the output positions of a ray depend on all input angles (ix , iy ),
in the linearized equations the two transverse dimensions are completely
decoupled. This allows a simplification of the ABCD law, which otherwise
would become a larger matrix [99]. Nevertheless, this simplification is only
valid within the paraxial approximation, i.e. to first order in the incoming
angles.
Secondly, Eqs. (2.9) show that the magnitudes of the angles are not
changed in the process. This is due to the fact that the input and output
media are the same (air). The change in sign of the angle in the vertical
direction is due to the reflection of one ray at the floor of the prism.
Finally, we would like to mention the physical meaning of the parameter
h0 , which is explicitly written in Eq. (2.12). In order to clarify the following
discussion we are going to introduce a slight variation in the formula for y2
and we rewrite in the form
1
h0
h0
η
+
y2 −
(2.13)
= − y1 −
− h0
iy
2
2
n tan α
It can be easily checked from Eq. (2.13), that in the case of incidence angle
parallel to the base of the Dove prism (iy = 0), h0 /2 is exactly the position
where the Dove prim has no effect over the ray (y2 = y1 = h0 /2).
The set of equations (2.11) can be directly cast into the ABCD matrix
form
x2
Ax Bx
x1
=
,
(2.14)
ox
C x Dx
ix
20
How a Dove prism transforms...
L2
L1
He-Ne laser
M1
f1
y
z
L4
L3
CCD
f2
x
M2
f3
f4
f3
f4
Imaging system
Figure 2.4: Sketch of the setup for checking the validity of Eqs. (2.11) and (2.12).
Lenses L1 and L2 with focal lens f1 = 200 mm and f 2 = 200 mm, are used in order
to shape the laser beam conveniently at the input of the Dove prism, having a width of
560μm. Mirrors M1 and M2 control respectively the angle and the position of the beam at
the input plane. Lenses L3 and L4 with focal lens f3 = 35 mm and f4 = 75 mm, work as
an image system of the output plane. Their focal length is conveniently chosen in order to
fit it on a CCD camera.
and the corresponding one for the vertical direction. As mentioned above,
due to the decoupling of the dimensions, we have one ABCD matrix for every
distinct direction. In order to use the ABCD matrix to calculate the effect
of an optical system in a Gaussian beam, we make use of the complex radius
of curvature [99] q = (z − z0 ) − iλ/(πw02 ), where z is the actual longitudinal
position of the beam, z0 the position of the beam waist of the beam, λ the
wavelength of the light and w0 the beam width at the waist position. The
beam can have a different complex radius of curvature for each dimension
(qx , qy ). The transformation through an optical system gives
q¯i =
Ai + Bi qi
Ci + Di qi
(2.15)
with i ∈ {x, y}, for each dimension. We can write it in this simple way,
because Eqs. (2.11) are decoupled for the two transverse directions.
2.2.2
Experimental validation of the ABCD law.
To check the validity of Eqs. (2.11) and (2.12) we have performed a series
of experiments with a commercially available Dove prism (Thorlabs) making
use of the scheme showed in Fig. 2.4. The Dove prism parameters are
L = 63mm, α = 45o and n = 1.51. We use a CW He-Ne laser (wavelength
2.3 Ellipticity induced by a Dove prism
21
(a)
(b)
displacement (mm)
800
y-plane
x-plane
400
0
-400
-800
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
-0.03
angle (rad)
-0.02
-0.01
0.00
0.01
0.02
0.03
angle (rad)
Figure 2.5: Location of the center of the light beam at the output plane. (a) The
angle in the x-plane (ix ) is changed. (b) The angle in the y-plane (iy ) is changed. Dots:
experimental results. Solid line: theoretical results.
633nm). The output beam of the laser is conveniently shaped so that at the
input plane of the Dove prism, the beam width is w0 560μm. The beam is
directed to the Dove prism by means of two mirrors to accurately control the
angle and position of the beam at the input plane. The beam at the output
plane of the system is demagnified to fit on a CCD camera (from Lumenera
company) with an appropriate imaging system. We have took the image of
the beam at the output plane when the input beam propagates with different
angles at the input plane of the Dove prism for both coordinates (x, y).
Figure 2.5(a) shows the position of the center of the beam at the output
plane when the input beam, centered at (x1 = 0, y1 = 0), propagates with
different angles (ix ) at the input plane of the Dove prism. Similarly, Fig.
2.5(b) corresponds to the case of changing the angle iy . We can see how the
experimentally measured values agree well with the theoretical predictions
as given by Eqs. (2.11) and (2.12).
2.3
2.3.1
Ellipticity induced by a Dove prism
Theoretical development
From the ABCD matrix derived in the previous section, it is possible to
calculate the effect of the Dove prism on the beam width and on the location
of the beam waist of an optical beam [99]. The important point here is that,
apart from the well known image inversion in the y direction, Eqs. (2.11),
combined with Eq. (2.15) also show that the Dove prism modifies the beam
waist position of the beam, (for the input beam, we have zx and zy , with
22
How a Dove prism transforms...
L1
L2
He-Ne laser
M1
razor-edge
system
L4
L3
y
z
x
M2
f3
Power meter
f4
f3
f4
Imaging system
Figure 2.6: Sketch of the setup used to measure the beam width of each transverse
coordinate at the output plane. In this case we use different values of the focal length f2
in order to obtain different beam widths at the input plane. A razor-edge system before de
CCD camera is used to measure the beam width in the two transverse dimensions at the
output plane
zx = zy ), differently in both transverse dimensions.
The new beam waist positions (z̄x and z̄y ) read
L
h0
1
z̄x = zx +
−
1−
n tan α
n
η
1
+
.
z̄y = zy + h0
n tan α
(2.16)
The appearance of two different beam waist positions for each transverse
dimension induce astigmatism in the output beam, and therefore, changes in
the OAM content of the output beam [100]. Generally speaking, any optical
device that introduces different optical path lengths for rays propagating
in different transverse planes, should produce changes in the orbital angular
momentum content of the output light beam. For the case of a Dove prims, as
considered here, the difference between ray propagation in the two transverse
dimensions is only noticeable for highly focused beam.
After traversing the Dove prism, the width of the light beam at the
output plane is given by the well known formula for LG beams w̄x,y =
1/2
, where w0 is the width of the beam at the input plane
w0 1 + (z̄x,y /z0 )2
and z0 is the corresponding Rayleigh range.
2.3.2
Experimental observation of the ellipticity
In order to measure change of the beam width for the two transverse coordinates (x and y) after traversing the dove prism (i.e there are different beam
2.3 Ellipticity induced by a Dove prism
23
1.2
0.8
180
width (mm)
ellipticity
1.0
0.6
165
150
135
120
105
0
90
180
270
360
rotation angle(degrees)
0.4
0
150
300
450
600
beam width(mm)
Figure 2.7: Ellipticity of the output beam at the output plane, after traversing the Dove
prism. Filled circles: Experimental results with the Dove prism. Triangles: experimental
results when the Dove prism is removed. The solid and dashed lines are the theoretical
results, as explained in the text. The dashed line corresponds to the theoretical value of the
ellipticity (e = 1) when the Dove prism is removed. Inset: Filled circles: x-axis; Empty
circles: y-axis. Input beam waist: w0 50μm.
waist for each coordinate), we have made use of the experimental set-up
showed in Fig. 2.6. The beam width at the input plane is changed with a series of lenses, but keeping always the beam waist position at the input plane.
For the measurement of the beam widths in both transverse dimensions, we
have used a razor-edge measurement technique. A razor blade was mounted
onto two motorized translation stages (horizontal and vertical), which allows
the displacement of the razor blade over the transverse plane of the laser
beam in both orthogonal directions. The potency of the laser beam is measured after the motorized razor for each position of the razor blade, which at
the starting position allows the total pass of the beam, whereas at the end
position blocks the beam totally. The curve obtained after processing the
measured data gives a very good approximation of the beam width.
Figure 2.7 shows the experimentally measured ellipticity at the output
plane of an input gaussian beam, after traversing the Dove prim. For the
sake of comparison, we have also measured the ellipticity of the output beam
when the Dove prism was removed, which is also shown in Fig. 2.7. The
theoretical curve shown in Figure 2.7 corresponds to e = (w̄x /w̄y )2 . The
inset of Fig. 2.7 shows how the output elliptical beam rotates when the Dove
prism rotates.
How a Dove prism transforms...
transverse coordinate
transverse coordinate
24
(a)
(b)
(c)
(d)
300
150
0
−150
−300
300
150
0
−150
−300
−300
−150
0
150
300 −300
transverse coordinate
−150
0
150
300
transverse coordinate
Figure 2.8: Spatial light intensity measured at the output plane, with the Dove prism
removed (a) and (c), and with the Dove prism, (b) and (d). (a) and (b): w0 = 560μm,
(c) and (d) w0 = 50μm. All dimensions are in μm.
Figure 2.8 shows two typical spatial shape measurements obtained at the
output plane, when the Dove prism is present or when it is removed. The
input beam is a vortex beam with winding number m = 2 that we have
generated using a computer generated phase hologram which is located after
the He-Ne laser. Figure 2.9 shows an example of this type of holograms.
The vortex beam is shaped with two different beam widths using the proper
lenses. For very large beam widths, (a) and (b), the astigmatism induced by
the Dove prism is not relevant, contrary to the case of highly focused beams,
as shown in (c) and (d).
2.4
OAM transformation rule of the Dove
prism
The astigmatism induced by the Dove prism will transform the OAM of
the output beam differently from the expected transformation l ⇒ −l. Let
us consider that the input beam (at the input plane) writes Ain (ρ, ϕ) ∝
ρl exp (−ρ2 /w02 ) exp (ilϕ), which corresponds to a LG beam with winding
number l and radial index p = 0. The OAM of a light beam is related to
2.4 OAM transformation rule of the Dove
prism
25
Figure 2.9: Example of generated phase hologram.
the azimuthal index l, while it does not change for light beams with different
index p. The light beam at the output plane writes
x
y
−i
Aout (ρ, ϕ) = N
w¯x
w̄y
2
2
ky
kx
+i
× exp i
2R̄x
2R̄y
l
x2
y2
exp − 2 − 2
w̄x w̄y
(2.17)
where k is the wavenumber, N is the normalizationfactor and the wavefront
radius of curvature reads R̄x,y = z̄x,y 1 + (z0 /z̄x,y )2 .
Due to the astigmatism induced by the Dove prism, the output beam is no
longer a pure spiral harmonic with winding number −l, but a superposition
of spiral harmonics that can be written as [79]
Aout (ρ, ϕ) =
1 am (ρ) exp (imϕ)
(2π)1/2 m
(2.18)
where am (ρ) is the distribution function associated to the spiral harmonic
with winding 2number m. The weight of the m-harmonic is given by Cm =
ρdρ|am (ρ) | , thus in order to know the contribution of different harmonics
the
at the output plane of the Dove prism we have to find
out am (ρ). Because
spiral harmonics are orthogonal, i.e, they satisfy dϕ exp {i (m − m ) ϕ} =
2πδm,m , the distribution function am (ρ) can be easily calculated and reads
1
am (ρ) =
(2.19)
dϕAout (ρ, ϕ) exp (−imϕ) .
(2π)2
We thus obtain that the weights of the OAM superposition {Cm } that de-
26
How a Dove prism transforms...
weight
(a)
(b)
1
1
0.5
0.5
0
−10
−5
0
5
10
0
−10
−5
weight
(c)
1
0.5
0.5
−5
0
5
10
5
10
(d)
1
0
−10
0
5
mode number
10
0
−10
−5
0
mode number
Figure 2.10: OAM decomposition of the output beam. (a) Input beam width w0 = 20μm,
winding number l = 0; (b) w0 = 100μm, l = 0; (c) w0 = 20μm, l = 2; (d) w0 = 100μm,
l = 2.
scribes the light beam, after traversing the Dove prism, are given by [101]
1
1
1
2l+1
2
Cm =
dρ exp −ρ
+
ρ
2l−2 l!w̄x w̄y
w̄x2 w̄y2
2
k l−k
l 1
1
1
1
l
−k
−
+
J(l+m)/2−k (s) (2.20)
i
×
k
w̄x w̄y
w̄x w̄y
k=0
when (l + m)/2 is an integer and Cm = 0 otherwise. In the formula above
Jm is the Bessel function of the first kind and order m, that for integer m
can be defined as:
1
Jm (z) =
dτ exp −i (mτ − z sin τ )
(2.21)
2π
and the parameter s reads
ρ2 1
1
1
1
kρ2
+i
.
−
−
s=
4
2 w̄x2 w̄y2
R̄x R̄y
(2.22)
Figure 2.10 show how the Dove prism transforms the OAM decomposition of an input beam, by showing the distribution {Cm } for different cases.
2.4 OAM transformation rule of the Dove
prism
27
weight of the central mode
1
0.5
0
20
40
60
80
100
beam waist (μm)
Figure 2.11: Weight of the central mode of the output beam. Solid line: weight of the
mode m = 0, for an input gaussian beam (l = 0). Dashed line: weight of the m = −1
mode, for an input l = 1 vortex beam.
Figures 2.10(a) and (b) show the OAM decomposition of the output beam
for a gaussian input beam, and Figs.2.10 (c) and (d) shows the corresponding
OAM decomposition for a l = 1 vortex input beam. The width of the input
beam is 20μm for (a) and (c) and 100μm for (b) and (d).
In Figs. 2.10(b) and (d), the OAM decomposition of the output beam
shows a single line, so in this case the Dove prism transforms the OAM of the
light beam from l to −l, as initially expected. For highly focused light beams,
such as it is the case of Figs. 2.10(a) and (c), the Dove prism transform a
pure LG beam into a superposition of spiral harmonics with different OAM
index as we expected.
In order to quantify the validity of the rule l ⇒ −l to describe the OAM
related behavior of the Dove prism, we plot in Fig. 2.11 the weight of the
central mode (to be defined below) as a function of the waist of the input
beam (beam waist). The central mode corresponds to m = 0 for the case of
an input gaussian beam, and m = −1 for the case of a l = 1 input vortex
beam. Generally speaking, a Dove prism performs the OAM transformation
l ⇒ {Cm }
(2.23)
where the decomposition Cm is determined by Eq. (2.20). For highly focused
light beams, the OAM decomposition shows many modes. For larger beam
widths values, the usual transformation l ⇒ −l holds. From Fig. 2.11, we
notice that, for a given value of the input beam width, the weight of the
central mode of the OAM superposition is smaller for the case of the input
vortex beam than for the gaussian beam.
28
2.5
How a Dove prism transforms...
Conclusions
In this chapter we have shown from a theoretical and experimental point of
view that the OAM transformation rule l ⇒ −l, believed to be correct for
Dove prisms, is not valid for highly focused light beams, since Dove prisms
inherently introduce astigmatism, and therefore further OAM changes. We
have analyzed how light beams with a well defined value of the OAM per
photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule l ⇒ −l,
turns out to be more important for highly focused light beams. Dove prisms
are being extensively used in many physical settings that make use of the
OAM of light [88–92]. In view of the results presented here, the use of Dove
prisms with highly focused beams could require the use of some compensating
schemes, such as appropriate combinations of cylindrical lenses.
Chapter 3
Characterization of the spatial
shape of optical beams with
spiral phase interferometry
3.1
Introduction
Here we study both theoretically and experimentally a method to characterize the amplitude and phase of a paraxial optical beam. The method is
based on a technique called spiral phase interferometry. The phase of an
optical beam is a property which is usually only accessible using interferometric methods, where a reference beam with well defined amplitude and
phase is used to retrieve the phase properties of the desired beam. Interferometric methods usually demand very stable set-ups, which can be difficult
to implement in every day applications.
As an alternative to those interferometric systems, some self-referenced
techniques have been proposed and implemented [110, 112, 115]. In the last
few years, one of those techniques, the so called spiral phase interferometry
(SPI) [112, 115], is gaining increasing interest due to its simple implementation. It has already been successfully used in high resolution microscopy
applications [111, 113, 114]. In principle, this technique could be used for
characterizing an unknown optical beam, whose transversal structure could
be used to codify information. In particular, it could be very interesting in
applications where the orbital angular momentum of an optical beam has to
be measured. We expose here some of the problems that these methods has
to overcome, and a possible way to solve them, by using a slight variation of
the usual SPI. We analyze how to adapt the original proposal to deal with
the special characteristics of finite optical beams. We have experimentally
29
30
Characterization of the spatial...
implemented such a system, and showed experimentally the advantages and
limitations of our proposal.
3.2
Basics of spiral phase filtering.
The spiral phase transform, which is also known as the Rieszt transform,
vortex transform, or two-dimensional isotropic Hilbert-transform its originally a purely numerical tool used, for example, in the analysis of fringes in
interferometers [102,103], or as a tool for the analysis of speckle patterns [?].
However, there are also applications where the spiral phase transform is performed using real optical methods, by introducing a spiral phase filter in
the Fourier plane of a imaging setup. The spiral phase filter can be constructed with an on-axis element, a so-called spiral phase plate [104] or by
making use of the diffraction of a specially designed off-axis hologram [116].
Fig. (3.1) shows an example of both cases. These phase filters can also be
constructed using high resolution spatial light modulators which can acts as
two-dimensional arrays of individually addressable pixels.
The sketch of a 4f-system for implementing a spatial Fourier filter is shown
in Fig. 3.2. A first lens (L1) generates a Fourier transform of the object plane
in the focal plane, where the spiral phase plate is located. The design of the
spiral phase plate is shown below. The zero-order Fourier component of
the field in the object plane focuses in the center of the phase plate. The
other orders focus at different positions of the spiral phase plate. The spiral
phase plate adds a phase offset to each off-axis beam. A second lens (L2)
placed at a focal distance behind the spiral phase plate performs a reverse
Fourier transform and creates the output image in its focal plane. There,
the zero-order component of the incident light field is again a plane wave,
superposing coherently with the remaining light field. This remaining light
field now carries a spatially dependent phase-offset with respect to the input
field.
3.2.1
Isotropic spiral phase filtering.
Typically, the spiral phase transformation is mathematically defined as a
multiplication of the Fourier transform of an input field with a vortex phase
profile, i.e with exp(iφ), where φ is the polar angle in a plane transverse to
the light propagation measured from the center of the spiral phase plate.
Fig. 3.3 (a) shows the example of an hologram which performs an isotropic
spiral phase transformation. We point out that this mathematical definition
excludes information about the central point of the spiral phase element,
3.2 Basics of spiral phase filtering.
31
(a)
(b)
Figure 3.1: (a) Spiral phase plate, grey-values correspond to phase values between 0 and
2π. (b) Off-axis hologram.
Object
plane
Fourier
plane
L1
f1
f1
Output
plane
L2
f2
f2
Figure 3.2: Sketch of a 4f-system used to implement a spatial Fourier filter. Go to the
text for more detail.
32
Characterization of the spatial...
(a)
(b)
Figure 3.3: (a)Isotropic spiral filter. (b) No-isotropic Spiral filter with R = 0.
where a phase singularity exists. However, if the spiral transformation is
implemented using a real spiral phase element, this central position is very
important, since it coincides with the zero-order Fourier component of the
input image, which usually contains the largest fraction of the total intensity.
In practise, real spiral phase elements have a central point with well-defined
amplitude and phase instead of a singularity. Only in the case where the
central region (or pixel) with a size of the order of the zero-order component
of the input field has no transmission, the resulting spiral phase transform is
really isotropic, i.e. our vortex phase profile is well described by exp(iφ). In
this cases the output image is isotropically edge enhanced.
3.2.2
Non-isotropic phase filtering.
In the case of non-isotropic phase filtering the central region of the spiral
phase filter acts as a transmissive phase shifter. In order to perform the
spiral phase transformation the optical Fourier transformed input beam is
multiplied with:
exp(iθ(x, y)) if x2 + y 2 > R2
(3.1)
eiα
if x2 + y 2 ≤ R2
where θ(x, y) = arctan xy is the azimuthal angle, R is a typically small radius
which separates the two regions of the filter and α is a constant phase. Figure
3.3(b) shows typical filter with this shape.
For this kind of spiral phase filters, the rotational symmetry of the spiral
phase filter is broken, and it is therefore called a non-isotropic phase filter.
If such a non-isotropic phase filter is used as a spatial Fourier filter, then the
output image shows a relief-like shadow profile [114]. This effect is due to the
interferometric superposition of the two output fields: one generated by the
central region of the phase filter, and the other one generated by the rest of
3.3 Spiral phase interferometry revisited.
33
the filter. The zero-order Fourier component of the input field, focussing in
the center of the spiral phase plate, is transformed into a plane wave by the
reverse Fourier transform performed by the following lens, interfering with
the output field. Thus, a spiral phase plate with a transmissive center acts
effectively as a self-referenced (or common-path) interferometer [107–109],
using the unmodulated zero-order component of the original input field as a
reference wave.
This system is usually more stable than conventional interferometric setups. Moreover, changing the phase of the central pixel of the spiral phase
plate (or rotating the whole spiral phase plate respect to the central pixel) results in a corresponding rotation of the apparent shadow direction. Recently,
it has been shown that this rotating shadow effect can be used to reconstruct
the exact phase and amplitude transmission of a complex sample by post
processing a series of at least three shadow images, recorded at evenly distributed shadow rotation angles in an interval between 0 and 2π [111]. The
feature that a field can be uniquely reconstructed is based on the fact that
there is no information loss in non-isotropically spiral phase filtered field due
to not erasing the information of the zero-order Fourier component of the
filtered input field. This missing information would not just result in an
insignificant offset, but in a strong corruption of the output field, since the
plane wave offset coherently superposes with the remaining image field, leading to an amplification or suppression of different components. It makes the
spiral phase transform with non-isotropic phase filtering a reversible operation, using for example a second spiral phase transform with a complimentary
spiral phase element (consisting of complex conjugate phase pixels).
3.3
3.3.1
Spiral phase interferometry revisited.
Mathematical development.
In this section we present a more detailed explanation of how a complex field
can be completely retrieved using the SPI technique proposed in [111]. This
method can offer nice results in some contexts like microscopy [114, 115],
and for phase-only modulated (constant amplitude) input beans [112, 115]
but can be problematic in some cases, especially if we want to extend it to
arbitrary beams. Our aim is to find a way to solve these problems and to
extend this method for reconstructing any complex beam.
Let us start with an initial paraxial beam with scalar amplitude Ein (xin , yin ).
In the same way that is showed in the experimental set-up sketched in Fig.
3.2 the input field is optically Fourier transformed with a 2f system, but now
34
Characterization of the spatial...
is multiplied by a set of spiral phase filters Hi=1,2,3 created by a SLM with
the following shape:
Φ = exp[iθ(x, y)] if x2 + y 2 > R2
Hk (x, y) =
(3.2)
eiαk
if x2 + y 2 ≤ R2
where θ(x, y) = arctan xy is the azimuthal angle, R is a typically small radius,
which separates the two regions of the filter and αk = (2π/3)k is a constant
phase which is different for every spiral phase filter. After performing another
Fourier transform with the second 2f system, the result of this spiral filtering
can be derived by a convolution of the initial field with the Fourier transform
of the filter:
(k)
Eout (xout , yout) = F {F {Ein (xin , yin )} Hk (x , y )}
= F {F {Ein (xin , yin )}} ⊗ F {Hk (x , y )}
∝ Ein ⊗ H̃k (xout , yout )
(3.3)
where F {g} represents the optical Fourier transform of the function g,
(xin , yin ) are the transverse coordinates at the input plane, (xout , yout ) represent those at the output plane (CCD camera), (x , y ) represent the plane
where the filtering takes place (after the first optical Fourier transform) and
H̃k is the Fourier transform of the filter. Another equivalent expression of
the output field is:
(k)
Eout (xout , yout ) = A(xout , yout ) + exp(iαk )B(xout , yout )
(3.4)
where the functions A(xout , yout ) and B(xout , yout ) represent the two parts of
the beam traversing the two zones of spiral phase filter, the central transmissive one for x2 + y2 > R2 and the rest of the filter with x2 + y2 < R2
respectively. For the sake of simplicity, we have normalized the transverse
coordinates so that we do not take into account the trivial magnification
factors and inversions due to the set of lenses chosen.
The analytical expressions for A(xout , yout ) and B(xout , yout) are more easily expressed in cylindrical coordinates and read:
∞
2π
A(rout , θout ) =
r dr
Ẽin (r , θ ) exp(iθ ) exp(irout r cos(θout − θ ))dθ
R
0
R
2π
Ẽin (r , θ ) exp(irout r cos(θout − θ ))dθ . (3.5)
B(rout , θout ) =
r dr 0
0
where Ẽin (r , θ ) = F {Ein } is the Fourier transform of Ein . The original
decoding method [111] is based on the measurement of the intensity distri(k)
(k)
bution of a series of three images Iout = |Eout |2 for k = 1, 2, 3. Assuming
3.3 Spiral phase interferometry revisited.
35
that B(xout , yout ) is nearly a constant, and with the calculation of these two
quantities:
Ic
1 (k) −iαk
=
Iout e
= A∗ (xout , yout )B(xout , yout )
3 k=1
Itot
1 (k)
=
Iout = |A(xout , yout )|2 + |B(xout , yout)|2
3 k=1
3
3
(3.6)
one can retrieve completely the initial field. To see it more clearly let’s
start assuming that the central part of the spiral phase filter works perfectly
and only transmits the zero-order Fourier component (including the complex phase) of the input field. In this case the functions A(xout , yout ) and
B(xout , yout ) can be expressed as
A(xout , yout ) = F {F {(Ein (xin , yin ) − Ein0 )}} ⊗ F {Φ(x , y )}
∝ [(Ein − Ein0 ) ⊗ Φ̃](xout , yout )
B(xout , yout ) = F {F {Ein0 }} ∝ Ein0
(3.7)
where Ein0 is the (still unknown) constant corresponding to the zero-order
Fourier component and Φ̃ is the Fourier transform of Φ. Combining Eqs.
(3.6) and (3.7), the expressions of Ic and Itot reads [111]
∗
Ic = [(Ein (xout , yout ) − Ein0 ) ⊗ Φ̃(xout , yout )]Ein
0
Itot = | (Ein (xout , yout ) − Ein0 ) ⊗ Φ̃(xout , yout )|2 + |Ein0 |2
(3.8)
Since we have assumed Ein0 = |Ein0 | exp iθin0 to be a constant, the convolution in Eq. (3.8) can be reversed by numerically performing the deconvolution
with the inverse function Φ̃−1 which corresponds to a numerical spiral-back
transformation. The initial field is then retrieved by:
|Ein0 (xout , yout )| exp {i (θin (xout , yout ) − θin0 )} =
(Ic ⊗ Φ̃−1 + |Ein0 |2 )
(3.9)
|Ein0 |
where Ein0 (xout , yout) has been split into its absolute value and its phase.
Therefore, if the intensity |Ein0 |2 of the constant zero-order Fourier component of the input image is known, it is possible to reconstruct the initial
field up to a phase offset θin0 , which corresponds to the spatially constant
phase of the zero-order Fourier component. Combining the expressions for
Ic and Itot given in Eq. (3.8), |Ein0 |2 can be calculated using:
|Ein0 |4 − Itot |Ein0 |2 + |Ic |2 = 0
(3.10)
36
Characterization of the spatial...
which has two possible solutions:
1
|Ein0 |2 = Itot ±
2
2
Itot
− 4|Ic2|
(3.11)
The positive or negative solutions are chosen depending on the experimental
context. Positive one applies for pure amplitude samples, and for samples
with a sufficiently small phase modulation, which is the case of imaging in
microscopy. On the other hand, the negative solution can be used for samples
with a deep phase modulation (of the order of π or larger), or with a high
spatial frequency.
3.3.2
SPI problems.
As we have pointed before, this method has delivered nice results in the fields
of microscopy [111, 114] and for phase modulated constant amplitude input
beams [111,115] but unfortunately it doesn’t work properly if we try to extend
it to more general beams. The main restriction of the protocol explained
before is the assumption that the function B(xout , yout ) is a constant. This
is a necessary condition in order to perform the deconvolution process in Eq.
(3.8), which allows to find easily an expression to retrieve the initial field. In
fact this condition is rarely satisfied for different reasons.
Firstly, as already stated in the original paper, there is a limit on how
small R can be. The ideal situation is to make R tend to zero so that
the function B(x, y) is nearly a plane wave. However, making R small can
create some problems. The smaller R, the smaller the amplitude of B, which
therefore reduces the amplitude of Ic . The natural limit for R is then given
by the noise in the recording apparatus and also depends on the shape of
the input beam. In typical optical beams, one needs R to be a significant
fraction of the input beam, to be able to overcome the noise of a standard
CCD camera. In most cases, this invalidates the approximation of B(x, y)
being a plane wave.
Secondly, even in the case that B(x, y) is close to a plane wave after
making R bigger in order to overcome the CCD noise, the algorithm to
retrieve B(x, y) not always gives a clear solution, i.e. we are not able to
separate easily the solutions obtained with the positive or negative signs.
This is the case, for example, of beams carrying orbital angular momentum.
Fig. 3.4 shows an example where this problem appears. Figures 3.4 (a) and
3.4 (b) present the intensity and phase of a beam with a vortex of charge
1, respectively, and 3.4 (c) and 3.4 (d) show the two possible solutions of
B(x, y).
3.4 Solutions for the original SPI method. New implementation.
37
(b)
(a)
(c)
Imax
p
0
-p
(d)
Imax
p
0
-p
Figure 3.4: Initial field with a vortex of charge 1 displaced simulated numerically, (a)
intensity and (b) phase. The two possible possible solutions of function B(x, y) calculated
numerically can be seen in the bottom arrow, (c) positive solution an (d) negative solution.
In order to see clearly that at some points we are not able to separate the
positive and negative solution, in Fig. 3.5 we have highlighted a transversal
plane (indicated by the black line in left figures) of both solutions together.
One of the solutions is bigger in the largest part of the interval (the positive
solution). If we focus our attention in the area marked with a circle, we can
see that both solutions are equal in some points. Moreover, it seems that
the two solutions exchange their role, i.e the positive solution becomes the
negative one and vice versa.
In order to overcome these problems, we have devised a slight variation
of the original spiral phase contrast method which we have successfully experimentally tested.
3.4
Solutions for the original SPI method.
New implementation.
In this section we present some useful variations of the method, with the
aim of overcoming the problems mentioned in the previous section. The
modifications have been introduced in order to make the reconstruction of
the initial beam easier and valid for more general optical beams. Fig.3.6
shows a diagram of the new scheme proposed. This new protocol can be
38
Characterization of the spatial...
Figure 3.5: Example of a region where the two possible solutions of B(x, y) are not
clearly separated. The right part of the figure represents both solution in one dimension,
along the plane marked in the left part of the figure. The solution with the positive sign is
represented by a green line, whereas the solution with the negative sign is represented by
the blue one. Conflictive points are inside the circle.
divided in three parts (pre-processing, processing and reconstruction), which
are explained in more detail below.
3.4.1
Pre-processing
To start with, we perform a pre-processing: we make an image of the Fourier
transformed input beam, which allows us to find a proper point (xR , yR )
where to center our filter. We look for a spot in the beam with a local
maximum of intensity. With this simple pre-processing, we obtain several
advantages: First, the Ic field will be maximized for a fixed radius R. Second, we also avoid zeros of the input beam which will make Ic close to zero.
Finally, the amplitude of the beam within the circle of radius R in the filter
is rather constant, which will allow us to make some simplifications to retrieve the input beam information. In our implementation we performed this
step visually, but it can be easily automatized with proper image processing
algorithms.
3.4 Solutions for the original SPI method. New implementation.
39
Figure 3.6: Block diagram of our system. (a) represents the optical processing, OFT
mean optical Fourier transform, and (b) represents the numerical post-processing to recover
the field, FFT and IFFT are the fast Fourier transform and its inverse, and sqrt{} is the
square root operation. More details are given in the text.
40
3.4.2
Characterization of the spatial...
Processing.
After the pre-processing, we continue with the processing which is illustrated
in Fig. 3.6 (a). First, we obtain the three filtered images as in the original
spiral phase contrast method and we also record an additional image where
the filter has been completely removed, i.e. we take an image of the in(0)
tensity Iout = |Ein (xout , yout )|2 . This image will be identical to the input
beam, except for the trivial rescaling and inversion due to the optical Fourier
transforming processes.
3.4.3
Reconstruction.
The reconstruction of Ein , which is shown in Fig. 3.6 (b) is rather simple
and reads:
y
Φ = arg{F {F {Ic(xout , yout )}e−i arctan x }} − (xR x + yR y)
0
Erec =
Iout
exp(iΦ)
(3.12)
Since we have made the pre-processing part of our protocol, we know that
we are using the proper spot to place our filter. Because of the radius R of
our spot is big enough to assure that Ic overcomes the CCD noise, we can’t
approximate B(x, y) by a constant, but we can safely assure that is very close
to diffraction image of a circular aperture:
2
2
J1 R (x − δx ) + (y − δy )
exp(i(xR x + yR y))
(3.13)
B(x, y) R (x − δx )2 + (y − δy )2
except for a trivial rescaling due to the optical Fourier transform. To obtain
this expression one has to assume a flat amplitude across the circle R and a
linear change in the phase. These approximations are based on the fact that
we have chosen the right amplitude spot in the beam and that the radius R
is small enough so that we can approximate any changes in the phase to first
order.
This change in the phase is responsible for the displacement of the diffraction image: (δx , δy ). As we will see, this typically small displacement of
the image does not affect our reconstruction. Finally, the added phase
(xR x + yR y) in B(x, y) is due to the displacement of the filter (if it’s needed)
and is corrected in the reconstruction given by Eq. (3.12). Note that even
under this approximation, the phase of B(x, y) presents some radial phase
singularities, i.e. there are π phase jumps at some radial positions, given by
the Airy function. Although these singularities could be properly taken into
3.5 Numerical simulations for the new implementation
41
account, usually we do not have to deal with them as they are out of the area
of interest. Taking into account Eqs. (3.6) and (3.12), we observe that the
reconstructed phase is actually the result of applying a mean filter of size R
to the Fourier transformed beam. Finally we point out that the errors in the
reconstruction of the intensity, which was made using the fourth image of the
(0)
intensity, Iout = |Ein (xout , yout )|2 , are only due to the noise of the imaging
and recording systems.
3.5
Numerical simulations for the new implementation
In Fig. 3.7 we present a numerical example of how the reconstruction process
works. Our input field consists of a beam with some phase singularities.
Intensity and phase are represented in (a) and (b), respectively. The order
of the phase singularities can be easily identified in (b), where we observe
that the beam presents one single charged vortex and another second order
vortex (where the phase twists twice around the singularity). Both vortices
are separated by some distance. The pre-processing is illustrated in (c) and
(d). (c) is the Fourier transform of the initial beam and the white spot
represents the maximum of intensity, where we locate the filters (one of the
filters is shown in (d)). The 3 different images obtained after the filtering,
and the last Fourier transform during the processing are displayed in the
panels (e), (f) and (g). Finally, in the last panels of the figure, (h) and (i),
we present the numerically reconstruction of the beam using Eqs. (3.12),
which are to be compared with panels (a) and (b).
3.6
Experimental set-up
We implement the experimental setup sketched in Fig. 3.8. To help following
how the system works, we divide the figure in five parts: optical beam generation; optical Fourier transform 1 (OFT1); optical filtering; Imaging system
1; optical Fourier transform 2 (OFT2) + Imaging system 2. We explain
below with more detail each part.
3.6.1
Optical beam generation.
To generate the initial optical beam we used the system showed in Fig. 3.9.
Our source of light was a 810nm diode laser which was coupled to a single
mode fiber to obtain a pure Gaussian spatial mode. The light from the output
42
Characterization of the spatial...
Imax
0
Imax
(a)
(b)
(c)
(d)
p
-p
p
-p
0
(e)
Imax
0
Imax
0
(g)
(f)
(h)
(i)
p
-p
Figure 3.7: Numerical example of the reconstruction system. (a) Intensity of the input
field. (b) Phase of the input field. (c) Amplitude of the Fourier transform of the input
beam. The white dot indicates the position of the center of the filter. (d) One of the filters
used, the white dot indicates the center of the filter. (e), (f ), (g) Output intensities of the
system, corresponding to the different filters used. (h) Intensity of the recovered field. (i)
Phase of the recovered field.
3.6 Experimental set-up
Initial beam generator
43
OFT 1
OFT 2
Imaging
System 1
Imaging
System 2
Figure 3.8: Sketch of the experimental setup. A computer generated phase hologram is
illuminated with a collimated diode laser light to produce a Laguerre-Gaussian-like beam
in the object plane, using the lens L1 and an iris (to select the first order of diffraction).
Then L2 makes the Fourier transform of the object and puts it on the SLM surface, where
the filters are displayed. After the filtering, we make the Fourier transform again with L3
and we rescale the image to fit the CCD chip with the imaging system 1. With the imaging
system 2 we make an image of the SLM on the CCD to find a proper point where to center
the filters.
44
Characterization of the spatial...
Figure 3.9: Photograph of the optical beam generation composed by the laser, the hologram, the lens and the pinhole used. The distances between the different optical elements
are indicated.
of the optical fiber was collimated and illuminated a computer-generated
phase hologram. As the holograms we use are custom made, we can produce
the appropriate hologram to modulate the optical beam in the desired way. In
the example in Fig. 3.8 we present a simple fork-like dislocation [116]. This
kind of holograms are well known to produce superpositions of LaguerreGaussian beams [117, 118].
To control the position of the beam with respect to the dislocation of the
hologram, we have used a micrometric support which allows us to move the
hologram with high precision. 50 mm after the hologram, we put a lens L1,
with a focal length f 1 = 50mm, so that 50mm after the lens, we obtain the
Fourier transform of the correspondent field at the hologram position. At
this point we use an iris in order to select the first order of diffraction from
all orders produced by the hologram. A numerical example of the intensity
shape of the resulting beam at this point, called the object plane, can be
seen in Fig. 3.8.
3.6.2
Optical Fourier Transform 1 (OFT1)
The next step is Fourier transform the object onto the surface of the spatial light modulator (SLM), trying to use the maximum part of the area
of the modulator. The modulator’s area is 20mm × 20mm, so we have to
choose properly the lens L2 in the 2f -system to perform the optical Fourier
transform.
In Fig. 3.10 we show two different possibilities that we have tried during
our experiment. The difference between the two paths is the central lens:
3.6 Experimental set-up
45
Figure 3.10: Photography of the 2f -system used to perform the first optical Fourier
transform (OFT1) of the initial field. Both possible trajectories tested to fit the beam
in the SLM screen have been indicated as well as the optical elements and the distances
between them.
L2A and L2B with focal lens f 2A = 150mm and f 2B = 300mm respectively.
The distance from the object plane to both lenses, and from the lenses to the
SLM correspond to the the focal length for each lens, so that they perform
the Fourier transform of the object plane at the SLM plane. Both paths don’t
work simultaneously: if we want to use lens L2B , we have to use mirrors M1
and M4 to guide the beam through the lens. On the other hand, if we want
to make use of lens L2A , the two mirrors have to be removed of the setup.
Considering lens L1 and L2 as a telescope, the magnification given by the
two possibilities reads:
f2A,B
|mA,B | =
(3.14)
f1
In our case mA = 3 and mB = 6. The diameter of the beam, measured at
the hologram position, is 2mm. In order to take profit of the maximum area
of the SLM for all the experiments, we have used the lens L2B , obtaining a
beam of 12mm of diameter at the surface of the modulator. Just before the
SLM we use a beam splitter (BS), so that the beam reflected by the SLM
changes its trajectory to mirror M5, in order to continue with the next step
of our implementation.
3.6.3
Optical filtering
To realize the optical filtering we use the spatial light modulator (SLM)
showed in Fig. 3.11, which allows us to filter the incident field with the
46
Characterization of the spatial...
Figure 3.11: Photography of the spatial light modulator used in the experiment.
3 filters needed in a sequential way. The SLM can implement in real time
the 3 filters, working in phase mode, therefore affecting only the phase of
the field. We reconstruct the holograms with the SLM using the calculation
of the spiral phase profile, and adding a certain phase in a small circle of
radius R variable (depending on the visibility). Each filter has a different
relative phase in the center. Finally, we superpose a grating to the filters so
that we generate different orders of diffraction. The SLM is connected to a
computer to control the different filters using a Lab View program which has
been designed to implement the 3 filters in a comfortable and fast way. For
each filter we take an image with the CCD camera. These images are the
necessary intensities to make the digital post-processing. In order to record
the initial field without filtering we only have to construct an hologram of
zero order with R = 0, taking into account the rescaling due to the Fourier
transforms.
3.6.4
Imaging System 1
With two flip mirrors (dashed lines in the figure), we could choose to direct
the light from the SLM to a CCD camera either through an imaging system
(imaging system 2) or with a Fourier transforming system (lens L3) re-scaled
(with lens L4), so that the resulting image fits the CCD chip. We direct the
beam to the Imaging System 1 to perform the pre-processing of our protocol.
We scan the shape of the beam in the SLM, looking for maxima of intensity.
In this way, we choose the position of the circular area of radius R in the
filters created by the SLM. The amplitude of the beam within that circle is
rather constant, which will allow us to make some simplifications to retrieve
the input beam information, as we have explained above.
3.6 Experimental set-up
47
(b)
(a)
Figure 3.12: Photography of the 2f -system used to perform the second optical Fourier
transform (OFT2) where optical path from the spatial light modulator(SLM) to the CCD
camera can be observed. The optical elements used and the distances which separate them
are indicated as well.
3.6.5
Optical Fourier Transform 2 (OFT2) + Imaging
System 2
Once we found a suitable zone, we switched to the Fourier set-up with lens L3
that we can see in Fig. 3.12 (a). This set-up Fourier transform the product of
the Fourier transformed initial object with the filter in the SLM. The beam
goes through L3 after reflecting in mirror M5. The focal length of L3 is
250mm, so that if the distance from the SLM to L3 is 250mm, at the same
distance after the lens (i.e the focal length) we obtain the Fourier transform
of the beam in the SLM surface. In this point, as shown in Fig.3.12 (b),
we put a pinhole to select the convenient order of diffraction from all the
orders produced by the filter in the SLM (the first order). This is the field
that we want to record, but it’s too much focussed to be recorded with the
CCD camera. In order to make it bigger we have used the Imaging System
2 implemented by lens L4 which focal length is 100mm, as we can see in
Fig.3.12 (b). The equation correspondent to thin lenses is given by:
1
1
1
−
=
z1 z2
f
(3.15)
We have put the lens 130mm after the pinhole plane (i.e z1 = 130mm) so
that the image of this plane is formed at z2 = 433mm after the lens, position
in which we have situated the CCD camera. The magnification of the beam
reads:
z2
|m| = | |
(3.16)
z1
48
Characterization of the spatial...
Figure 3.13: Photography of the CCD camera and the density filters used in the experiment. The filters are used to attenuate the beam in order to protect the sensor from a
excess of intensity.
The beam is magnified by a factor m = 4.33 in the CCD plane, taking profit
of a 25% of the camera’s chip.
Now we can take the four images needed for the protocol. The first one
was taken with the blank filter in the SLM (order zero and R = 0) thus we just
retrieved the intensity pattern of the object plane. The three other images
were taken with three different filters in the SLM as explained previously.
Each filter consisted on a fork-like pattern (similar to that in the hologram
of Fig. 3.8), but the position of the dislocation was covered with a circle of
variable radius (depending on the visibility conditions). Every filter had a
different relative phase in the circle.
We have used a CCD from Lumenera, which can be seen in Fig. 3.13. It
can been observed that we have used intensity filters before the sensor of the
camera because an excess of intensity could damage it. The images recorded
allows us to recover the initial optical beam in intensity and phase by means
the numeric post-processing explained in previous sections.
3.7
Experimental results.
In this section we present some of the experimental results that have been
obtained implementing the protocol proposed. As we have commented previously, there are two possible positions for the filters depending on the type
of initial beam that we want to characterize. If the initial beam presents a
maximum of intensity in its center, i.e, in its optical axis, then the filters also
are centered in the optical axis and (xR , yR ) = (0, 0). On the other hand,
if the initial beam presents a minimum or a zero of intensity in its optical
axis, then the filters have to be displaced to the position (xR , yR ) in such a
way that the center of the filters matches with a maximum of intensity in the
3.7 Experimental results.
(a)
Imax
49
(b)
0
Imax
(c)
0
Imax
0
Figure 3.14: The three intensities recorded experimentally with the CCD camera. (a),
(b) and (c) correspond with different angles αi .
initial beam. Below we explain the reconstruction of optical beams for both
cases. First we show cases where the center of the beam presents a relative
maximum of intensity. Next we present the retrieval of a Laguerre-Gaussian
mode whose intensity vanish in its center. Finally some other experimental
examples are shown.
3.7.1
Filter centered at the optical axes.
In this section we present the process that we have realized in order to reconstruct the phase and intensity of a complex beam which consist of four
phase dislocations, four spatial vortexes forming a square. The collimated
beam goes through a custom-made hologram designed to generate our initial
field in the object plane. Because the intensity of this field is different from
zero and quite constant in its center, we can center our filters in the optical
axes, so that (xR , yR ) = (0, 0) (the correction term in the phase vanishes) and
B(x, y) satisfies the needed approximations to simplify the reconstruction of
the incident field. Following our protocol we take the four images needed:
(0)
firs, the intensity corresponding to the beam in the object plane Iout , and
(k)
afterwards the resulting images after the three filters Iout . Fig 3.14 shows the
three intensities measured experimentally and used below to calculate the
compensated intensity Ic ,showed in Fig. 3.15, using the digital processing.
The final result obtained with the method previously described can be
seen in Fig. 3.16 In the upper row of the figure we present, for the sake of
comparison, a numerical calculation of the beam we expected. In the calculation we used our knowledge of both the incoming beam and the hologram
that we used. Observe that the reconstruction phase follows remarkably well
the expected features. The reconstructed phase is rather noisy far from the
center of the beam, where the method is prone to give worse results as the
noise of the camera is of the same order as the recovered signal. Note also
that from the phase measurements we can observe that the beam has a small
50
Characterization of the spatial...
(a)
Imax
(b)
p
-p
0
Figure 3.15: Compensated intensity Ic obtained by digital processing with the experimental intensities recorded previously. (a) intensity and (b) phase.
(a)
(b)
Imax
-p
0
(c)
p
(d)
Imax
p
0
-p
Figure 3.16: Characterization of a Gaussian beam with four phase singularities. Intensity (a) and phase (b) of the expected beam calculated numerically. Intensity (c) and
phase (c) experimentally reconstructed. In (a) and (c) can be observed the four nulls of
intensity due to the four phase singularities.
divergence, which can be observed from the curvature of the iso-phase lines.
This is an indication that the laser beam was not perfectly collimated in the
object plane. Finally, from the intensity measurements a small ellipticity in
the beam can be observed. This is probably due to some inhomogeneities of
the SLM and is in agreement with other series of measurements not shown
here.
3.7.2
Filter displaced with respect to the optical axes.
In this section the initial beam that we want to characterize has a zero of
intensity in its center. More specifically, we have a beam where we have introduced a vortex of charge 2 (similar to a LG02 ) using a computer-generated
hologram. In order to retrieve this field, we displaced the filter to (xR , yR
3.7 Experimental results.
51
(b)
(a)
p
Imax
p
0
(c)
Imax
0
-p
(d)
p
-p
Figure 3.17: Characterization of a Laguerre-Gaussian beam with charge 2. (a) and
(b) show the numerical simulations for the expected intensity and phase respectively. (c)
Intensity and (d) phase of the experimental reconstruction without the compensation of the
phase introduced by the displacement of the filter (xR , yR ).
with respect to the optical axes. After recording the four images needed, we
proceed with the digital post-processing process in the same way that in the
previous case. Figure 3.17 (a) and 3.17 (b) show the intensity and phase that
we expect to recover, Fig.3.17 (c) and Fig. 3.17 (d) show the experimental
results. It can be observed that the phase of the experimental retrieved field
presents a lineal phase which doesn’t appear in the expected one. This additional phase correspond with the displacement of the filter (xR , yR ) which
has not been compensated in the phase reconstruction.
In order to compensate the phase introduced by the filter, we have to
determine its position respect to the optical axes ,(xR , yR ). To do it we
remove the hologram which generates the initial field, and we represent in
the SLM screen a phase vortex at the position of the filter. In this way
we obtain a gaussian beam which traverses the OFT1 system, resulting in
another gaussian beam, which its multiplied by a spiral phase and traverses
the the OFT2 system, which makes the optical Fourier transform of this
product how shows Fig.3.18 If we calculate the distance between the center
of the beam and the point where the intensity vanishes we can determine the
filter displacement in number of pixels, (Δx, Δy). The initial field after the
displacement correction is given by:
Δx
Δy
Erec = Erec exp i
x+
y
(3.17)
Nx
Ny
where Nx and Ny are the CCD number of pixels and normalize the displace-
52
Characterization of the spatial...
Dx
Imax
Dy
0
Figure 3.18: Beam obtained after the reflection of a Gaussian beam known in the spatial
light modulator (SLM) with a phase vortex in the position of the filter used in the previous
reconstruction. This beam is used to determine the position of the filter respect to the
optical axes (Δx, Δy).
(a)
(b)
Imax
p
0
-p
Figure 3.19: Characterization of a Laguerre-Gaussian beam with charge 2. (c) Intensity and (d) phase of the experimental reconstruction with the compensation of the phase
introduced by the displacement of the filter (xR , yR ).
ment of the filter.
Fig. 3.19 presents the obtained results after the displacement correction.
It can be observed that now, the phase in Fig. 3.19 (b) is very similar to the
expected in Fig.3.17 (b).
For the sake of comparison we can study the result of the recovered field
without displacing the filter respect to the optical axis. In this case the area
of the beam where the intensity is a minima is bigger than the circular area
of constant phase that we put in the center of the filter. In this context
B(x, y) can not be approximated by (3.13) so that our expression for the
reconstruction are not valid, thus it’s impossible to recover the total beam
how is showed in Fig. 3.20
3.7.3
Other experimental results.
In this section we present other examples in which our protocol has been
used to reconstruct the initial field, from a very simple case, the Gaussian
beam, to more complicated ones which phases indetermination.
3.7 Experimental results.
53
(a)
(b)
Imax
p
0
-p
Figure 3.20: Characterization of a Laguerre-Gaussian beam with charge 2 without performing the displacement of the filter respect to the optical axes. (a) Intensity and (b)
phase of the experimental reconstruction.
(b)
(a)
Imax
-p
0
(c)
Imax
0
p
(d)
p
-p
Figure 3.21: Characterization of a Gaussian beam. Upper arrow, intensity (a) and
phase (b) of the beam expected numerically calculated. Bottom arrow, intensity (c) and
phase (d) of the experimentally reconstructed beam.
Gaussian beam.
The Gaussian beam presents a maximum of intensity in its center, so in this
cases is not necessary to displace the filter with respect to the optical axes.
Note that in this a case the original protocol proposed in [114] should works
properly. Fig. 3.21 presents the obtained results. In the upper row, (a) and
(b) are the numerical calculations taking into account the shape of the initial
beam which presents a radial phase. The lower row, (c) and (d), show the
experimental results. We can see that the reconstructed beam is not totally
symmetric which can be due to some misalignment in the set-up.
54
Characterization of the spatial...
(a)
(c)
Imax
0
Imax
0
(b)
p
-p
(d)
p
-p
Figure 3.22: Characterization of a Gaussian beam with a phase jump. Intensity (a)
and (b) phase of the numerical simulations for the expected beam with a phase jump of
0.8π. (c) and (d), experimental reconstruction of intensity and phase respectively.
Beam with a phase jump.
In this case, the initial beam has a phase jump of π that is generated using
a hologram with the corresponding phase jump. Figures 3.22 (a) and 3.22
(b) show the intensity and phase of the initial beam numerically calculated.
Figures 3.22 (c) and 3.22 (d) the experimental reconstructions for both of
them. In the reconstruction we can see a phase jump of approximately π
as it is expected, where the unexpected variation can be produced by the
hologram itself.
Beam with vortex of first order.
This is an example of the reconstruction of a beam similar to a LG01 . This
case is similar to the explained in the section before where the reconstruction
of a LG02 was presented. The intensity of the initial field vanishes in its center
so that the filter has to be displaced respect to the optical axes. Figure 3.23
show the obtained results. In the upper arrow are presented the expected
field that has been calculated numerically like in the cases before, taking into
account the characteristics of the hologram used and the shape of the gaussian
beams used to generate the initial field. The bottom row show the results
obtained experimentally. We can see that the phase in Fig. 3.23(a) matches
with the expected once, whereas the intensity in Fig. 3.23(b) presents some
differences that can be done to the no perfect collimation of the beam.
In order to compare the reconstruction of the initial field with and with-
3.7 Experimental results.
(a)
55
Imax
(b)
0
(c)
Imax
p
-p
(d)
p
-p
0
Figure 3.23: Characterization of a beam with a vortex of first order displacing the
filter respect to the optical axes. The compensation in the reconstruction of the phase has
been performed.At the upper row, (a) and (b) show the numerical intensity and phase of
the expected field respectively. At the bottom, (c) and (d) show the intensity and phase
characterized experimentally.
out displacing the filter, the protocol was performed for both situations. In
Figure. 3.24 we can see the intensity and phases recovered without displacing
the filter, which means that the compensated intensity Ic does not overcomes
the CCD noise and that the approximations taking into account in our protocol are not valid. It is clear that the reconstruction is far from the expected
results.
(a)
Imax
0
(b)
p
-p
Figure 3.24: Characterization of a beam with a vortex of first order without displacing the filter respect to the optical axes. Numerical simulations of the intensity (a) and
phase(b) of the expected beam. Intensity (c) and phase (d) recovered experimentally.
56
Characterization of the spatial...
(a)
(c)
Imax
0
Imax
(b)
p
(d)
p
-p
0
-p
Figure 3.25: Characterization of a Gaussian beam with a vortex of first order displaced
respect to the optical axes. The protocol has been performed without displacing the filter
from the center of the beam. Intensity (a) and phase (b) calculated numerically. Intensity
(c) and phase (b) retrieved experimentally.
Gaussian beam with a vortex of first order displaced.
As a result of displacing the hologram used in the previous example with
respect to the optical axes, the initial field is a vortex of first order displaced.
In this context, the area where the intensity vanishes it is not located in the
center anymore, so that our protocol works properly without displacing the
filter. Figure 3.25 shows that the experimental results are very similar to the
calculated ones.
Beam with vortex of second order displaced.
This example is analogous to the last one, but now using a second order
hologram instead of a first order hologram. Without displacing the filter
respect to the optical axes the reconstructed field showed in Fig. 3.26 (c)
and 3.26 (d) is near from the expected results represented in Fig. 3.26 (a)
and 3.26 (b) except for some difference in the intensity, that like in previous
examples, can be done to misalignment of the beam.
Beam with three vortexes of first order.
Finally, we present the reconstruction of a initial field with three vortexes of
first order without displacing the filter. In this case the protocol seems not
work so properly than in the other examples. In Figure 3.27 (b) and 3.27
3.8 Conclusions.
57
(a)
(c)
Imax
0
Imax
0
(b)
(d)
p
-p
p
-p
Figure 3.26: Characterization of a Gaussian beam with a vortex of first order displaced
respect to the optical axes. The protocol has been performed without displacing the filter
from the center of the beam. Intensity (a) and phase (b) calculated numerically. Intensity
(c) and phase (b) recovered performing the protocol experimentally.
(b) can be seen that the level of noise in the reconstruction, amplitude and
phase respectively, is bigger than in the other reconstructions, making them
quite different from the intensity and phase expected showed in Fig. 3.27 (c)
and 3.27 (d) respectively.
3.8
Conclusions.
In this chapter we have presented, from a theoretical and experimental point
of view, a method to measure the amplitude an phase of Laguerre-Gaussian
like beams. This method is based on a small variation of the spiral phase interferometry technique, which allows to avoid some technical problems that
can be found in the reconstruction of finite sized beams and beams with
phase singularities. A few examples of the use of our technique for the characterization of complex beams has been presented. From this examples we
can conclude that this new method avoid the problems that present the spiral phase interferometry technique proposed in [114] when extrapolating it
to general beams, but presenting some limitations as well. The limitations
found in the reconstructions presented are basically due to the measurement devices and the numerical processing. The aberrations produced by
the misalignment in the optical system and the imperfections of the holograms produced manually are an important source of errors. Regarding to
the experimental devices, both the spatial light modulator and the CCD cam-
58
Characterization of the spatial...
(a)
(c)
Imax
0
Imax
0
(b)
(d)
p
-p
p
-p
Figure 3.27: Characterization of a Gaussian beam with three phase singularities. (a)
Intensity and (b) phase of the expected beam calculated numerically.Intensity (c) and phase
(d) reconstructed experimentally without displacing the filter from the optical axes.
era present a limited resolution thus affecting the final reconstruction. The
errors in the numerical processing are introduced principally by the measurement of the displacement of the filter, which couldn’t be exact in some cases.
Nevertheless, avoiding this technical problems, the method presented here is
a powerful tool in many areas where the spatial structure of light is used.
Chapter 4
Measurement of the spatial
Wigner function of paired
photons
4.1
Introduction
It is known than one suitable representation of the quantum state of a system is the Wigner function [123]. The Wigner function formalism is fully
equivalent to the density matrix representation, thus providing all the accessible information of the system to the observer. Moreover, Wigner functions
are specially useful for describing continuous variables. In particular, it has
been used for describing the quadratures of the electrical field of coherent
and squeezed states [124], and of single photon states [125].
Therefore, we have chosen the Wigner function formulism to study the
properties of entangled photons in the transverse momentum degree of freedom. The description of the spatial transverse modes of an optical field
in terms of the Wigner function can be found in [135, 136]. In particular,
these results can be directly applied to describe the transverse spatial shape
of bipartite entangled photons generated in Spontaneous Parametric DownConversion (SPDC) processes. In section 4.2 we revise the main properties
of the Wigner function that we will use, in section 4.3 we describe a typical quantum state that describes the momentum of photons generated in a
SPDC process.
For the sake of clarity, we analyze the spatial Wigner function in three
different situations: Firstly, we consider the case where one of the photons
is projected onto a specific pure state, obtaining the Wigner function of the
remaining photon (section 4.4). We also consider the case when one simply
59
60
Measurement fo the spatial Wigner...
disregard one of the photons, the other being, in principle, in a mixed state
(section 4.5). In section 4.6 we propose an experimental scheme for measuring
the spatial properties of the signal photon in the cases presented before. A
more realistic analysis from the experimental point of view is presented in 4.7
and the experimental results obtained are presented in section 4.8. Finally
we propose an experimental scheme for measuring the spatial properties of
the whole entangled state, and analyze the properties of the Wigner function
of the whole system in section 4.9.
4.2
The Wigner function
The Wigner function associated with a quantum state |Ψ can be expressed
as the expectation value of the operator Π̂rq [126]
1
Ψ|Π̂rq |Ψ
π2
(4.1)
dr0 exp (−2iqr0 ) |r − r0 r + r0 | =
(4.2)
W (r, q) =
where Π̂rq is defined as
Π̂rq =
dq0 exp (−2irq0 ) |q + q0 q − q0 |
The operator Π̂rq performs a reflection about the phase-space point (r, q) and
is thus the parity operator about that point. In the case of a “two particle”
state the Wigner function can be expressed as [127]
W (r1 , q1 , r2 , q2 ) =
1
Ψ1,2 |Π̂1,2 |Ψ1,2 π4
(4.3)
where Π̂1,2 is the product of the two displaced operators, each one acting
over one particle:
Π̂1,2 = Π̂r1 ,q1 ⊗ Π̂r2 ,q2 .
(4.4)
The Wigner function
corresponding to a generally mixed state with density matrix ρ =
λi |Ψi Ψi| can be written, making use of Eq. (4.3), as
W (r, q) = T r(ρΠ̂rq ). An interesting property of the Wigner function, easily obtained from Eqs. (1.1) and (1.2), is that the
Wigner function
of a
probabilistic mixture of density matrices, i.e. ρ =
λi ρi is W =
λi Wi .
Notwithstanding, this is not the case for a linear superposition of pure states,
whose Wigner function does not result in a linear superposition of the corresponding Wigner functions.
4.3 The source of spatially entangled photons.
4.3
61
The source of spatially entangled photons.
In the following we will consider the important case of a two-photon state,
whose quantum state can be written as
|Ψ = dqs dqi Ψ (qs , qi ) a†s (qs ) a†i (qi ) |0s |0i
(4.5)
where qs = (qs,x , qs,y ) and qi = (qi,x , qi,y ) are the corresponding transverse
momenta of the signal and idler photons, respectively. a†s (qs ) is the creation
operator of a signal photon with transverse momentum qs , and similarly for
the idler photon.
In the rest of the work presented here, we will confine ourselves to the
case where the probability amplitude Ψ writes
|AB|1/2
(4.6)
exp −A|qs + qi |2 − B|qs − qi |2 ,
π
where A and B are two possibly complex constants that will allow us to analyze different types of momentum correlations among the photons. Ψ (qs , qi )
is normalized, so that dqs dqi |Ψ (qs , qi ) |2 = 1. The type of states described
by Eq. (4.6) is ubiquitous when describing quantum systems of continuous
variables. In the case of momentum correlated photons, the state given by
Eq. (4.6) is a very good approximation for describing paired photons entangled in the momentum degree of freedom [128, 129].
In particular, this kind of states can be produced when a second order
nonlinear crystal is illuminated by a quasi-monochromatic pump beam in a
Gaussian mode in order to produce frequency downconverted waves. The
downconverted waves should be generated in a collinear configuration (all
interacting waves propagate along the same direction), and the Poynting
vector walk-off should be negligible. One way of achieving such conditions
is the use of noncritical type-II quasi-phase-matched nonlinear crystals. If
such a crystal geometry is chosen, and under the approximation that the
refractive indices of pump, signal and idler are nearly equal, Eq. (4.6) is a
good approximation to the state of the photons at the output face of the
crystal. The values obtained of the two constants, A and B, are [129]
wp2
1
σ0
1 L
A=
−
+i
(4.7)
4 1 + wp4 /σ02
4 kp0 1 + σ02 /wp4
Ψ (qs , qi ) =
B=
L
αL
+ i 0,
0
4kp
4kp
(4.8)
62
Measurement fo the spatial Wigner...
where L is the length of the nonlinear crystal, α is a fitting constant to
approximate the phase matching functions sinc function by a Gaussian function (in our case we use α = 0.455), wp and σ0 are the pump beam width
and radius of curvature considered at the center of the crystal, respectively,
kp0 = ωp np /c, and ωp and np are the corresponding angular frecuency and
refractive index.
From Eq. (4.6) it can be readily checked that the momentum correlations
in each of the two transverse coordinates are completely independent, i.e.
there is no cross-correlation between direction x and direction y. Therefore,
in the rest of the section we will drop all vector quantities, and focus on just
one of the transverse dimensions of the photons.
4.4
Remote preparation of a pure state
The two-photon state described by Eq. (4.6) is pure. After projecting the
idler photon in a pure state, the signal photon will also remain in a pure
state. The specific spatial shape of the signal photon will depend on a) the
two-photon momentum correlations of signal and idler and b) the spatial
shape of the mode onto which the idler photon is projected. In this sense,
we talk about remote preparation of pure states and the goal of this section
is analyze from a theoretical point of view how the Wigner function of the
signal photon looks like.
After projecting the idler photon, the Wigner function of the signal photon can be expressed as
WΦ,s (xs , qs ) =
1
Ψs,i|PΦ,i ⊗ Π̂xs ,qs |Ψs,i
π2
where PΦ,i = |ΦΦ|i projects the idler photon onto the state
|Φ = dqΦ(q)a†i (q) |0i .
(4.9)
(4.10)
We will analyze this remote preparation procedure with two different cases:
the projection onto a gaussian mode, and projections onto coherent superpositions of Gaussian and Hermite-Gaussian modes.
4.4.1
Projection onto a Gaussian state
We describe the projection of the idler photon into a Gaussian state by the
projector PG = |GG| where |G, in the transverse momentum space, is
4.4 Remote preparation of a pure state
given by
|G = NG
1 q2
dq exp −
|q
4 μx
63
(4.11)
NG is a normalization constant and μx is the complex beam width in real
space given by
1
k
μx = 2 + i .
(4.12)
w0
2R
w0 is the beam waist, k is the longitudinal idler wavevector and R is the
radius of curvature of the beam. Making use of Eq. (4.9), we find that the
Wigner function for the signal photon reads
(xs − hI qs )2
1
2
WG,s (xs , qs ) = exp −hR qs exp −
(4.13)
π
hR
where hG = (hR + ihI )/2 is defined as
hG = (A + B) −
(B − A)2
.
B + A + 1/ (4μ∗x )
(4.14)
Since the biphoton function given by Eq. (4.6) describes perfect correlations
in orbital angular momentum (OAM) between the signal and idler photons
(ms + mi = 0, with ms,i being the OAM index of the signal (idler) beams)
[130], Eq. (4.13) correspond to the Wigner function of a pure Gaussian state.
It can be easily demonstrated if we use Eq. (4.1) to calculate the Wigner
function corresponding to the pure Gaussian state given by Eq. (4.11). The
obtained Wigner function reads:
1
1
1
1
2
2
exp −
WG (x, q) =
)qs + xs ] .
q exp −
[(
4π|μx |
2μx
2μx
2μx
(4.15)
Eq. (4.13) clearly corresponds with the Wigner of a pure Gaussian state with
a complex beam width in real space μx = 1/4hG . Thus, the signal photon is
prepared remotely in a Gaussian state whose parameters depend on the idler
photon projection.
In Figure 4.1 we show a few examples of the remote preparation of Gaussian states as given by Eq. (4.13) and (4.14). Note the different features of
the Wigner function. For example, tilting of the Wigner function appears
whenever the state presents curvature, i.e. when either A, B or μx have nonvanishing imaginary parts. In Figures 4.1(a) and (b) the Wigner function it
is not tilted because the chosen parameters make A, B and μx purely real (L
small and Ri and Rp tending to infinity). Also, note that the two transverse
64
Measurement fo the spatial Wigner...
widths of the elliptical Wigner function, in the qs and (xs − hI qs ) directions,
are inversely related as shown in Eq. (4.13). We will see below that this only
happens when the represented state is pure and can be clearly observed by
comparing Figs. 4.1(a) and (b).
Also, the idler Gaussian width only enters Eq. (4.13) through hG . From
Eq. (4.14) it is readily observed that when A = B, the spatial shape of the
signal photon is independent of the coincident idler projection. This could
have already been anticipated since Eq. (4.6) represents a separable state
whenever A = B [129]. In Fig. 4.2 we plot two cases with different pumps
but fixed idler projections. In particular, Fig. 4.2(a) shows the especial case
when A = B.
4.4.2
Projection onto superpositions of Hermite-Gaussian
and Gaussian states
Now we consider a multimode situation, i.e., the idler photon is projected
onto a coherent superposition of two modes: a first order Hermite-Gaussian
mode and a Gaussian mode. The idler photon projector into a coherent
superposition can be written as PHG = |HGHG|, where
1 q2
|q,
(4.16)
|HG = NHG dq(2q1 + iq) exp −
4 μx
q1 = x1 μx gives the relative amplitude between the pure Gaussian and the
pure Hermite-Gaussian modes being x1 the position of the null appearing in
the field in real space. When μx is purely real, then q1 = x1 /w02. The Wigner
function corresponding to the quantum state of the signal, after projecting
the idler onto the mode given by Eq. (4.16) reads
WHG,s (xs , qs ) = NHG,s CH,s (xs , qs ) WG,s (xs , qs )
(4.17)
where
|g0|2
|g0|2
2
(h
q
+
x
)
−
−
I
s
s
h2R
2hR
(q1∗ g0 )
(hI qs + xs ) + 4|q1 |2 ,
4(q1∗g0 )qs − 4
hR
CH,s (xs , qs ) = |g0 |2 qs2 +
(4.18)
WG,s is defined by Eq. (4.13) and g0 writes
g0 =
B−A
.
B + A + 1/ (4μ∗x )
(4.19)
4.4 Remote preparation of a pure state
−3
x 10
(a)
65
(b)
0.33
−1
q(μm )
−5
0
5
0
−3
x 10
(c)
(d)
0.33
−1
q(μm )
−5
0
5
0
−2000
0
x(μm)
2000−2000
0
2000
x(μm)
Figure 4.1: Examples of remote preparation of a single photon in a spatial Gaussian state. The pump is a Gaussian mode, and the idler is projected onto a
Gaussian state. (a) and (b) show two Wigner functions with no curvature of the
idler mode (Ri → ∞), and different values of the beam width of the idler photon
wi : (a) wi = 0.4 mm, (b) wi = 1 mm. (c) and (d) correspond to the same beam
width of idler photon, wi = 1 mm, but different values of the radius of curvature ,
(c) Ri = 1 m, (d) Ri = 200 m. In all cases: beam width of the pump beam wp = 1
mm; no curvature of the pump beam (Rp → ∞); Length of the nonlinear crystal
L = 5 mm.
66
Measurement fo the spatial Wigner...
−3
(a)
x 10
0.37
−5
−1
q(μm )
−0.05
(b)
0
0
0.05
5
−200
0
x(μm)
200
−2000
0
0
2000
x(μm)
Figure 4.2: Effect of the pump beam on the remote preparation of a Gaussian
state (a)no curvature of the pump beam (Rp → ∞), wp = 38.23μ m and L = 50
mm satisfying A = B; (b) Rp = 2m, wp = 1 mm, no curvature of the idler mode
(Ri → ∞), wi = 0.5 mm, L = 5 mm.
The Wigner function corresponding to the pure state given by Eq. (4.16)
can be calculated using Eq. (4.1) and reads
WHG (x, q) = NHG CH (x, q) WG (x, q)
(4.20)
where
1
1
((1/2μx )q + x)2 −
−
(1/2μx )
2(1/2μx )
(q1 )
((1/2μx )q + x) + 4|q1 |2 .
4(q1 )q − 4
(1/2μx )
CH (x, q) = 2q 2 +
(4.21)
Comparing Eqs. (4.17) and (4.18) with Eqs. (4.20) and (4.21) it is clear
that the Wigner function of the signal photon corresponds with the Wigner
function of a superposition of pure Gaussian and Hermite-Gaussian states,
i.e the signal photon has been remotely prepared in a superposition of pure
Gaussian and Hermite-Gaussian states. Equation (4.17) is identical to Eq.
(4.20) when μx = 1/4hG and g0 . Notice again that when A = B, g0 = 0
and the Wigner function of the signal photon corresponds to a Gaussian one
independently of idler photon, i.e. the two photons are in a separable state.
Thus, if in this situation we set q1 = 0, i.e. we project the idler into a
pure Hermite-Gaussian mode, the probability for a signal-idler coincidence
is exactly zero, as expected [130]. Finally, notice that the superposition of a
Gaussian and a Hermite-Gaussian state does not result on the sum of their
respective Wigner functions.
4.5 Wigner function of the signal photon and the amount of
spatial entanglement
67
In Fig. 4.3 we show some examples of the Wigner function for the signal photon remotely prepared in a superposition of Gaussian and HermiteGaussian state with different values of q1. When q1 is purely real, changing
its value results in a displacement of the singularity in the x-coordinate, as
it can be appreciated in Figs. (a) and (b). Note that the superposition of a
Gaussian and a Hermite-Gaussian state does not result on the sum of their
respective Wigner functions.
4.5
Wigner function of the signal photon and
the amount of spatial entanglement
In this section we consider the spatial properties of the signal photon alone,
i.e. regardless of the state of the idler photon. The Wigner function of the
signal photon alone can be calculated from the two-photon state given by
Eq. (4.6) using
W (xs , qs ) =
1
T r[Ψs,i|Iˆi ⊗ Π̂xs ,qs |Ψs,i]
π2
(4.22)
which corresponds to calculating the Wigner function of the signal photon
alone, applying the identity and tracing out the idler photon. The Wigner
function obtained reads
−[xs − 2HI qs ]2
1 HR − G
2
exp −2(HR − G)qs exp
Ws (xs , qs ) =
π HR + G
2(HR + G)
(4.23)
where H and G are defined as
H = HR + iHI = (A + B) −
G=
(A − B)2
2(A + B)
|A − B|2
2(A + B)
(4.24)
(4.25)
Note that in this case, the purity of the quantum state that describes the
signal photon, which can be easily calculated once the Wigner function has
been measured, can be used to determine the degree of entanglement of the
initial two-photon state. The
purity of the signal photon can be calculated
as P = Px2 , where Px = 2π dxs dqs Ws (xs , qs )2 and reads
Px =
HR − G
HR + G
1/2
.
(4.26)
68
Measurement fo the spatial Wigner...
−3
q(μm−1)
x 10
(a)
(b)
0.29
−5
0
5
−0.34
−3
q(μm−1)
x 10
(c)
(d)
0.48
−5
0
5
−2.9
−2000
0
x(μm)
2000 −2000
0
2000
x(μm)
Figure 4.3: Examples of remote preparation of a single photon in a spatial superposition of a Gaussian and a Hermite Gaussian state for different values of q1 .
The pump is a Gaussian mode, whereas the idler photon is projected onto a given
superposition of the states. (a) and (b) show two cases where q1 is purely real,
whereas in (c) and (d) q1 is purely imaginary. (a) q1 = 0, (b) q1 = 2 × 10−4 μm,
(c) q1 = i 10−8 μm, (d) q1 = i 2 × 10−8 μm. In all cases: Beam width of the idler
and pump: wi = wp = 1 mm; Curvature of the idler photon and the pump beam
Ri = Rp = 100 m; Crystal length L = 5 mm.
4.6 Experimental set-up for measuring the Wigner function of
the signal photon.
69
From Eq. (4.23) and Eq. (4.26) it can be seen than the purity of the signal
photon is directely related to the product of the two transverse widths of the
elliptical Wigner function, in the qs and (xs − HI qs ) directions respectively,
satisfying the constrain
P = Δqs Δ(xs −HI qs ) ≤ 1.
(4.27)
Some comments are in order: first, notice again that if A = B, G = 0 and the
two-photon state is separable. In this case the Wigner function of the signal
photon correspond to the Wigner function of a Gaussian state with the same
conditions, so that the state of the signal photon is pure. Also, note that in
this situation the two transverse widths of the Wigner function are inversely
related. Therefore, as it is expected, the purity of the signal state is P = 1
and we can conclude that the transverse widths of the Wigner function of a
pure Gaussian state are inversely related.
Some numerical examples of the Wigner function given by Eq. (4.23) are
shown in Figs. 4.4(a) and 4.4(c). For Fig. 4.4(c) we have chosen wp and
L to satisfy the condition A = B. As it is expected, in this case the twophoton state, which is separable, gives the Wigner function of a Gaussian
state with equal conditions. This is clear if we compare Fig. 4.4(c) with Fig.
4.2(a) which shows the same Wigner function. Also, note that comparing
Fig. 4.4(a) with Fig. 4.1(b), it is clearly seen how the purity of the former
case is less than one, meaning some degree of spatial incoherence.
4.6
Experimental set-up for measuring the
Wigner function of the signal photon.
The Wigner function of the signal photon described in the previous sections
can be measured experimentally using an extension of the method proposed
in [131] where they measured the Wigner function of a quantum state of
light at a single photon level using a parity-inverting Sagnac interferometer.
Figure 4.5 shows the scheme proposed to measure the Wigner function of
the signal from the entangled two-photon pair. The signal photon is sent
through a parity-inverting Sagnac interferometer whereas the appropriate
optical elements are used to perform the different operations over the idler
photon. From the measurement of the coincidences rates at both output
ports, signal and idler, we can extract the Wigner function of the signal
photon. In the following we explain with more detail each of the parts that
compose the set-up: entangled photons source, Sagnac interferometer for the
signal photon, idler photon operation and coincidence counting.
70
Measurement fo the spatial Wigner...
(a)
(b)
0.0077
−1
q(μm )
−0.2
0
0.2
0
−2000
0
2000−2000
0
−3
(c)
x 10
2000
(d)
0.32
−0.05
−1
q(μm )
−5
0
0
5
0.05
−200
0
0
x(μm)
200 −2000
0
2000
x(μm)
Figure 4.4: Examples of the Wigner function of the single photon alone, i.e
without considering the idler photon. The pump is a Gaussian beam. (a) and
(c) show two cases where the signal photon is analyzed at the output face of the
nonlinear crystal. In (b) and (d) we have considered the propagation of the signal
photon using a lens with focal lenght f mm located L1 after the output face of the
nonlinear crystal, and at L2 from the entrance of the interferometer. For (a) and
(b) wp = 1 mm and L = 5 mm. For (c) and (d) wp = 38.23μ m and L = 50 mm
(satisfying the condition A=B. In (b) f = 150 mm, L1 = 300 mm and L2 = 300
mm, in (d) f = 200 mm, L1 = 175 mm and L2 = 200 mm. No curvature of the
pump beam: Rp → ∞ for all cases.
4.6 Experimental set-up for measuring the Wigner function of
the signal photon.
71
Coincidence
Counting
HWP’s
Path 1
BS
L
Path 2
Detector 1
M0
MS
q
Dx
Pump
PPKTP
Gaussian
projection
signal
PBS
L
HG
projection
Idler
Without
projection
Detector 2
Figure 4.5: Sketch of the set-up proposed for the retrieval of the Wigner function of a remotely prepared photon. Pairs of momentum entangled photons are
generated from a nonlinear crystal illuminated by a quasi monochromatic gaussian
pump beam in a collinear configuration. The signal and idler photon follow different paths after traversing a PBS. The idler photon is detected by Detector2 after
been collected using the appropriate optical elements depending on the remote
preparation that we want to perform. The signal photon is detected by Detector1
after passing throw the Sagnac interferometer and collected with a multimode
fiber. Using coincident logical detection, we measure the Wigner function of the
signal photon.
72
Measurement fo the spatial Wigner...
signal
Diode Laser
405 nm
Spatial
Filtering
HWP
fp
PPKTP
DM
BPF
PBS
Idler
Figure 4.6: Scheme of the set-up used to generate the two entangled photons (go
to the text for more detail).
4.6.1
Entangled photons source.
A more detailed scheme of the photon source is showed in Fig. 4.6. In order
to generate the two spatially entangled photons we pump, with a current and
temperature stabilized 405 nm diode laser, a periodically poled KTP crystal
(ppKTP). The collimated pump beam first passes through a spatial filter in
order to improve the quality of beam profile. The beam polarization is later
adjusted with a half wave plate (HWP) so that the pump beam is polarized
in the correct direction to achieve the phase matching of the waves. The
beam is focused at the center of the crystal with a lens with focal length fp .
The crystal, with a longitude of L = 5 mm, is cut so that it allows the
generation of two orthogonally polarized photons at 810nm in a configuration
where all the waves propagate along one of the optical axis, thus avoiding
spatial walk-off of the beams. Phase matching of the waves to produce the
nonlinear effect is guaranteed by the poling of the crystal, which had a period
of around 10μm. In this situation, the spatial quantum state of the two
photons can be well approximated by Eqs. (4.6), (4.7) and (4.8).
As the pump beam and the down converted photons are propagating in
the same direction, we use a dichroic mirror (DM) after the nonlinear crystal
which reflects the blue light and transmits the down converted light. Later,
a band pass filter is mounted as the mirror does not have 100% reflectivity
at 405 nm so we need further filtering of the remained blue light. The signal
and idler beam are then spatially separated with a polarizing beam splitter
(PBS).
4.6.2
Idler operation.
The idler photon traverse a 2f system. In the middle of the optical path
from the crystal to the coupler a lens is mounted with a focal length distance
equal to the half of the distance. The down converted photons are coupled
into an optical fiber to guide the photon to a Single Photon Detector (SPD)
that depending on the experiment will be either a single mode fiber (SMF)
or a standard multi mode fiber (MMF).
4.6 Experimental set-up for measuring the Wigner function of
the signal photon.
73
L
signal
Coupler
SMF
Idler
PBS
f
Detector 2
f
Figure 4.7: Sketch of the basic set-up used to project the idler photon onto a
Gaussian state.
signal
Slab
SMF
idler
PBS
f
L
f
Coupler
Detector 2
Figure 4.8: Scheme of the set-up used to project the idler photon onto a superposition of Gaussian and Hermite-Gaussian states.
Projection onto a Gaussian state.
First, let us describe the basic set-up used to project the idler photon onto a
Gaussian mode in order to remotely prepare the signal photon in a Gaussian
state. In this way, we collect the idler photon with a single mode fiber (SMF)
and detect it as showed in Fig. 4.7.
Projection onto superpositions of Gaussian and Hermit-Gaussian
states
The projection onto a superposition of Gaussian and Hermit-Gaussian states
in order to prepare the signal photon into a similar superposition, can be
done by introducing a mode transformation of the idler spatial profile before
the projection operation implemented by a SMF. The mode transformation
together with the projection onto a Gaussian state can be seen as a projection
onto a HG state. As it is sketched in Fig. 4.8 the HG projection was possible
by introducing a phase jump in the idler spatial profile before the SMF fiber.
The phase jump was produced with a silica slab, and was monitorized with
an aligning diode and a CCD camera.
74
Measurement fo the spatial Wigner...
L
signal
Coupler
MMF
Idler
PBS
f
f
Detector 2
Figure 4.9: Schematic set-up used to regard the idler photon.
Measurement of the Wigner function of the signal photon regardless the idler photon
As it is showed in Fig. 4.9 the measurement of the Wigner function of the
signal photon alone, without taking into account the idler photon, can be
done collecting the idler photon with a multimode fiber (MMF) in order to
avoid spatial filtering.
4.6.3
Sagnac interferometer for the signal photon.
As it is showed in Fig. 4.10, the signal photon is directed with a set of
mirrors and lenses to our Wigner measurement device, a three dimensional
Sagnac interferometer with a top mirror configuration. We have used a lens
of focal length f = 250 mm located at a distance L1 = 250 mm after the
nonlinear crystal and L2 = 500 mm from the mirror Ms . M0 and Ms are
mirrors mounted in Gimbal mounts allowing us to align the beam inside the
interferometer with micro metric precision.
Figure 4.11 illustrates with more detail how the Sagnac interferometer
works. After mirror Ms the field is split by a 50 : 50 beam splitter into
clockwise and counter-clockwise propagation beams. As the two beams pass
through the interferometer, they are directed out of the table by three mirrors (”top-mirror configuration”). The angle between all beam propagation
directions in the interferometer are 90◦ . This arrangement rotates the wave
fronts of the two beams by ±90◦ for the counter-clockwise and clockwise directions, respectively, and inverts them along the horizonal giving the transformations E(x, y) → E(±x, ±y). The opposition of the rotations can be
understood using a Berry’s phase argument [132]. This effectively performs
a two-dimensional parity operation on one of the wave fronts while leaving
the other unchanged. The beams are recombined at the beam splitter. The
symmetry point of the interferometer is given by the geometry of the system and it is experimentally assessed. A motorized mirror mounted over a
4.6 Experimental set-up for measuring the Wigner function of
the signal photon.
75
HWP’s
Path 1
BS
L
Path 2
Detector 1
M0
MS
q
Dx
L2
signal
L
Pump
PPKTP
L1
Idler
PBS
Figure 4.10: Sketch of the Sagnac interferometer used for measuring the Wigner
function of the signal photon. The wave-front transformations along the clockwise
path are depicted. The external steering mirror Ms tilts and translates.
Parity-inverting
Sagnac-interferometer
Detector
BS
PHOTON
SOURCE
Ms
Figure 4.11: Sketch of the Sagnac interferometer used for measuring the Wigner
function of the signal photon. The wave-front transformations along the clockwise
path are depicted. The external steering mirror Ms tilts and translates.
76
Measurement fo the spatial Wigner...
tangent-arm rotation stage, which is simultaneously mounted over a micro
translation stage, can displace and tilt the input signal photon at will.
The output of the interferometer is collected by an imaging system into a
200μm core MMF, which then carries the signal to the signal photon detector
(SPD). We used narrow interference filters, to avoid momentum-frequency
correlations, and to filter external noise.
Then,under these conditions, the probability of one signal photon to be
detected (Ps ) is
1 1
Ps = + α
2 2
+∞
−∞
ρs (x − xs , x + xs ) exp (i2qs x) =
1 γ
+ W̃ (xs , qs ) (4.28)
2 2
where ρs (x, x ) is the density matrix describing the state of the photon at the
motorized mirror and W̃ (xs , qs ) is proportional to the corresponding Wigner
function. The parameters xs and qs are controlled with the displacement
and the tilt of the motorized mirror, respectively. The factor γ gives the
visibility of our Wigner function measurement with respect to the constant
background. It is critical to observe weak signals or Wigner functions with
a low amplitude, where we need that |γ| 1. In particular, the parameter γ
depends on the transmission (T ) and refraction (R) of the beam splitter and
on the polarization of the input photon.
In the general case, the combined effect of the out of plane geometry
and the polarization dependent mirrors and beam splitter, makes the two
counter-propagating signals to acquire different polarizations, thus lowering
the visibility. In order to control this problem we modified the original implementation of the interferometer by introducing a set of three wave plates
in one of the arms of the interferometer how can be seen in Fig. 4.10. These
plates can produce any transformation in the polarization and so we could
compensate the effect of the interferometer. With the help of a polarimeter and an aligning diode, we set the plates in the right position, so that
|γ| RT , i.e. does not depends any more on the polarization. The sign of γ
depended on the exact transformation performed, due to the PancharatnamBerry effect [133]. In our case, the interference was constructive.
4.6.4
Coincidence Counting.
The detection of a single photon is a conversion of incoming light into a
readable electrical signal. To date the best choice for single photon detection
is the use of avalanche photo diodes (APD) which work in Geiger mode. The
detection of the coincidence counts consist of three parts: coupling, detection
and processing from TTL signal to coincidence logical module.
4.6 Experimental set-up for measuring the Wigner function of
the signal photon.
77
Coupling.
The coupling of the down converted photons to the couplers has been done
by the use of an aspheric lens (in our case f = 11 mm) mounted on a XY
translator which focuses the light to a fiber (multi mode for the signal photon
and single mode or multi mode, depending on the case, for the signal photon).
The fiber is mounted on a Z translation stage for optimizing the coupling.
The entire coupling system was mounted on another XY translation stage
to enable us to align the system with respect to the down converted light.
Detection.
The light signal from the fibers is then transferred to a single counting module (based on an avalanche photo diode), a product of Perkin Elmer which
detects single photons in the wavelength range of 400 nm to 1060 nm. The
efficiency for the wavelength we need to detect, 810 nm is about 60%. The
dark counts coming from the detectors which vary depending on the lab
conditions were numerically extracted from the data files.
Signal processing.
The last part of the coincidence detection scheme is processing the transistortransistor logic (TTL) signal coming from the counting modules to a coincidence logic module which in a determines which counts coming from the
two detectors correspond to a correlated photon pair. As there is strong time
correlation between down converted photons coming from the same generation process, it is possible to ”filter out” the incoming photons and take into
account only the ones which arrive to the coincidence logic into the short period of time called coincidence window. If the coincidence logic is triggered
by a photon arriving from the first channel, then a coincidence count will be
produced if a photon arrives from the second detector in the next 10ns.
As there are always small differences in the optical paths of the down
converted photons due to different distances from the output of the crystal
to the detectors, different length of the optical fibers and differences in the
length of the cables delivering the TTL signal, our coincidence logic has
the option of adjusting the time delay between the two lines. There is a
certain probability of producing a coincidence count (accidental count) from
two photons which are not generated together. We should take into account
these when reading the experimental data. The number of accidental counts
is given by
A = ns ∗ ni ∗ Δt
(4.29)
78
Measurement fo the spatial Wigner...
where Δt is the coincidence window time and ns and ni are the count rates
for the signal and idler photons. It may be seen that the accidental counts
increase quadratically with the number of photons arriving to the detectors.
Finally the TTL signal produced from the coincidence logic is processed with
a LabView program.
Under our experimental conditions, the coincidences rates from the both
output ports, signal and idler, will have the following shape:
1
2
nc ∝ ns + ns + 2γ W̃s (xs , qs ) ni
(4.30)
where ni is the count rate of the idler photon, and njs = αj ns , with j = 1, 2,
are proportional to the counts corresponding to the two counter propagating
directions of the signal photon in the interferometer (see Fig. 4.5). Here, ns
is the photon flux rate of the incoming signal beam and α1 = R2 (α2 = T 2 )
depends on the reflectivity r (transmissivity t) of the beam-splitter. On the
other hand, γ = RT and W̃s (xs , qs ) is proportional to the Wigner function of
the signal photon. Therefore, the proper Wigner function can be extracted
from the measurement of the coincidences of the two outputs, after subtracting the background and properly normalizing the result.
4.7
A more realistic analysis: propagation of
the fields from the crystal to the detectors.
In order to properly compare the results obtained with these experimental
set-ups, with our previous theoretical results, an important element should
be considered: the effects on the Wigner function of the propagation of the
fields from the output face of the nonlinear crystal to the detectors. In
the situations described so far, we have shown the Wigner function at the
output face of the crystal. Let’s now consider a more realistic situation, in
which both photons propagate from the output face of the crystal to the
interferometer, maybe traversing a lens in their way. The idler propagation
can be properly accounted for by using the correct values of q1 and μx . Thus,
we are left with the propagation of the signal. In the case of the remote
preparation of the signal photon (section 4.4), the effect of the propagation
is trivial. The overall shape of the Wigner function would still be described
by Eqs. (4.13) and (4.20), but with different values for the characterizing
parameters.
The most interesting case is the one described in Sec. 4.5, where the state
of the idler photon is disregarded. If we use a lens of focal length f located
4.7 A more realistic analysis: propagation of the fields from the
crystal to the detectors.
79
a distance L1 after the output face of the nonlinear crystal, and located at
a distance L2 from the Sagnac interferometer(Ms ), the state of the signal
photon at the output face of the nonlinear crystal is transformed as:
|Ψs,i = Ûs ⊗ Iˆi |Ψs,i
(4.31)
Us is an unitary transformation over the signal photon, given by the product
of the transformations due to the free propagation and the effect of the lenses:
Ûs = ÛL2 Ûf ÛL1
ÛLn and Ûf , in transverse momentum space, have the form
Ln 2
q|ÛLn |q = exp −i q δ(q − q ) n = 1, 2
2k
1
k 2
q|Ûf |q =
dx exp [i(q − q )x] exp −i x
2π
2f
(4.32)
(4.33)
(4.34)
We take into account this more realistic scenario to calculate the Wigner
function of the signal photon alone. Making use of the new |Ψs,i state in
Eq. (4.22), the new expression for the Wigner function is
HR − G
[xs − 2HI qs ]2
1
2
exp −2(HR − G)qs exp −
Ws (qs , xs ) = √
,
π HI + G
2(HR + G)
(4.35)
where
2
f
|G1 |2
(4.36)
G =
2k
(2HR1 )
2 f
1
(G1 )2
H = HR + iHI =
+
− iT2 (4.37)
2k
A + B − iT1
2HR1
B−A
(4.38)
G1 = G1R + iG1I =
A + B − iT1
(B − A)2
H 1 = HR1 + iHI1 = A + B −
(4.39)
A + B − iT1
f − L1
(4.40)
T1 =
2k
f − L2
(4.41)
T2 =
2k
Let us to note that Eqs. (4.35) and (4.23) have the same mathematical
structure.
80
Measurement fo the spatial Wigner...
In Figs. 4.4(b) and (d) we show two examples of how the consideration of
the propagation of the signal photon affects the shape of the Wigner function
retrieved. Figures 4.4 (a) and (c) show the Wigner function at the output
face of the nonlinear crystal, without considering the propagation effects (as
described in section 4.5). Figures 4.4(b) and (c) plot similar cases, but now
taking into account the propagation from the output face of the nonlinear
crystal to the entrance of the Sagnac interferometer, showing clearly how
the shape of the Wigner function of the signal photon changes. However, it
should be noted that this transformation preserves the purity of the signal
photon, as no filtering occurs in the process. This can be mathematically
shown by inserting the expressions for H and G in Eq. (4.26). Note that
Figs. 4.4(d) is exactly the same than Fig. 4.1(b), although they represent two
very different experimental situations. In the case of Fig. 4.1(b), the signal
photon results in a coherent Gaussian state with a plane wavefront by means
of projecting the idler photon onto a given state. In Fig. 4.4(d), the signal
photon results on exactly the same state but without any post-selection.
4.8
Experimental results and discussion.
Projection onto a Gaussian state.
In the first set of experiments, we project the idler photon state into a Gaussian mode, using a SMF. The pump beam is focused onto the crystal with
wp 26μm using a lens of fp = 400 mm. We have chosen this value of wp
following previous results in our group, they conclude that the two photon
state is highly entangled for this case. We record the coincident signal photons after it has traversed the interferometer. Finally, after subtracting the
coincidences background, we obtain an interference pattern which is proportional to the Wigner function, as given by Eq. (4.30). In this situation the
state of the signal photon coincident with the detected idler should be a pure
Gaussian state, with a phase curvature dependent on the propagation of the
signal and idler photons to the motorized mirror and the SMF respectively.
In Fig. 4.14(a) it can be observed the Wigner function obtained with such
a measurement procedure. The contour plots have been obtained using Eq.
(4.15) with the best fitted curvatures and widths. It was very easy as well to
remotely prepare a displaced Gaussian state, just by moving the idler photon
coupler. The resulting Wigner function can be observed in Fig. 4.14(b).
Projection onto a superposition of Gaussian and HermitGaussian states.
4.8 Experimental results and discussion.
81
(a)
(b)
2000
x(μm)
1500
1000
500
0
−500
−1000
2
4
6
8
q(μm−1)
10
12
2
4
−3
x 10
6
8
q(μm−1)
10
12
−3
x 10
Figure 4.12: Experimental remote preparation of a single photon in a spatial
Gaussian state. The pump is a Gaussian mode, and the idler is projected onto
a Gaussian state using a single mode fiber (SMF). (a) and (b) show two Wigner
functions for different positions of the idler coupler: in (b) the coupler has been displaced 4 mm respect to the position in (a). The contour plots have been obtained
using Eq. (4.15) with the best fitted curvatures and widths.
In order to further test the remote state preparation capabilities we then projected the idler photon into a Hermite-Gaussian state. The HG projection
was done by introducing a phase jump in the idler spatial profile before the
SMF fiber. As in the previous case the pump beam is again focused onto
the crystal with wp 26μm. The resulting Wigner function is plotted in
Fig. 4.13. The contour plots have been obtained using Eq. (4.20) with the
best fitted curvatures and widths. One can observe the phase jump and the
curvature of the field.
Mixed states and entanglement.
In Fig. 4.14 we show the two different Wigner functions corresponding to the
states of the signal photon when the pump beam was focused onto the crystal
with wp 26μm (4.14 (a)) and with wp 103μm (4.14 (b)). Following the
previous work of our group, the two photon state is almost a separable state
in the first situation (wp 26μm) whereas it is highly entangled in the second
one (wp 103μm). The contour plots have been obtained using Eq. (4.35)
with the best fitted curvatures and widths. It can be observed that the signal
to noise ratio, i.e, the visibility, decreases in the case (b) respect to the case
(a). This effect can be understood in two consistent ways. In the first place,
as the signal photon is more spatially incoherent, the interference visibility
has to decrease. On the other hand, as the Wigner function x and p widths
82
Measurement fo the spatial Wigner...
2000
1500
x(μm)
1000
500
0
−500
−1000
2
4
6
q(μm−1)
8
10
12
−3
x 10
Figure 4.13: Experimental remote preparation of a single photon in a spatial
superposition of a Gaussian and a Hermite Gaussian state. The pump is a Gaussian
mode, whereas the idler photon is projected onto a given superposition of the states
by introducing a phase jump in the idler spatial profile before the SMF fiber. The
contour plots have been obtained using Eq. (4.20) with the best fitted curvatures
and widths.
should no longer Fourier transform related, the Wigner function spreads out
and the local amplitude is lower.
4.9
Wigner function of the two-photon state
Finally, in this section we analyze the properties of the Wigner function of
the whole biphoton system. The Wigner function of the two-photon state
can be calculated using the more general Eq. (4.3). It can be readily shown
that at the output face of the crystal it has the form
1
exp −2(A)(qs − qi )2 − 2(B)(qs + qi )2
Ws,i(qs , xs , qi , , xi ) =
π
−[2(B − A)(qi − qs ) + xs ]2
× exp
(A + B)
(2(B − A)(1 − ν1 )(qs − qi ) + xi − ν1 xs )2
× exp −
2(HR − G)
(4.42)
where
(B − A)
(4.43)
(A + B)
As was pointed out in [131, 137] this Wigner function could be measured
experimentally using an extension of the procedure to measure the Wigner
ν1 =
4.10 Conclusions
5
83
5
(b)
x 10
(a)
x 10
3
3
2.5
2
2
x(μm)
1
1.5
0
1
0.5
−1
0
−2
−0.5
−3
5000
10000
−1
q(μm )
15000
−1
0
1
2
q(μm−1)
3
4
x 10
Figure 4.14: Experimental measurements of the Wigner function of the single
photon alone, i.e without considering the idler photon which is collected using a
multi mode fiber. The pump is a Gaussian beam focused onto the crystal to: (a)
wp 26μm, (b) wp 103μm
function of the signal alone. In this case, instead of collecting the idler photon
with a multimode fiber, it will pass through another Sagnac interferometer
similar to the one used for the signal photon, as it is sketched in Fig. 4.15.
Under these experimental conditions, the coincidence rates from the output
ports of both Sagnac interferometers will have the following shape:
Rc ∝ Is1 (Ii1 + Ii2 ) + Is2 (Ii1 + Ii2 )
+(Is1 + Is2 )Wi (xi , qi ) + (Ii1 + Ii2 )Ws (xs , qs )
+Ws,i(xs , qs , xi , qi )
(4.44)
where Ijn , with n = 1, 2 and j = s, i, are the counts corresponding to a
signal or idler photon which travels through either path of the interferometer. Wj (xj , qj ) is the Wigner function of the signal (idler) photon alone
regardless of the state of the idler (signal) photon, as described in Section
V. Ws,i(xs , qs , xi , qi ) is the Wigner function of the two-photon state at the
output face of the crystal and can be extracted from the coincidences of the
two outputs, after subtracting the background elements.
4.10
Conclusions
In this work we have presented some theoretical and experimental results
regarding the spatial Wigner function of entangled paired photons. We have
84
Measurement fo the spatial Wigner...
Coincidence
Counting
D1
BS
D2
Path 1
Path 2
BS
Dx
q
Path 2
q
M0
MS
MS
Path 1
Dx
signal
Pump
PPKTP
PBS
Idler
M0
Figure 4.15: Sketch of the set-up for measuring the Wigner function of the spatial
state of two photons produced from an SPDC source. In this case, both signal and
idler photon are sent to a Sagnac interferometer and detected by Detector1 and
Detector2 respectively after been collected by a multimode fiber. Using coincident
logical detection and after substracting the coincident background, we obtain the
spatial Wigner function corresponding to the two-photon state.
4.10 Conclusions
85
studied the remote preparation of one of the photons of the pair, showing
explicit results (analytical expressions and experimental measurements) for
the photon being in a Gaussian beam and in a superposition of two modes.
Also, we have analyzed and measured the important case of the Wigner
function of only one of the two photons of the pair, which shows that one
photon of the pair is in a statistical mixture of modes.
More importantly we could experimentally estimate the total spatial entanglement of a pair of photons. The experimental results showed still need
further scrutiny they fit qualitatively with the theoretical results. Thus, this
work is an important first step in the total characterization of entangled
states. A possible way in order to improve the signal to noise ratio could be
detecting the second output port of the Sagnac interferometer with the help
of an optical isolator.
Finally, we propose an experiment to completely measure the spatial state
of the two photons, by using a pair of Sagnac interferometers. We have given
analytical expressions for the results one would expect in this case.
Chapter 5
Dynamics of saturated Bragg
diffraction in a stored light
grating in cold atoms.
5.1
Introduction
The storage of light information in an atomic ensemble is a well understood
phenomenon which has a promising prospect for applications both in classical
and quantum information processing. The light storage (LS) phenomenon
allows us to obtain later information about a previously stored light pulse,
as well as to manipulate the stored information. How we have seen, LS can
be described as being due to the creation of a spatially dependent ground
state coherence that contains information on the amplitude and phase of a
light pulse, and which survives after the switching-off of the incident light.
This work was realized in the Federal University of Pernambuco (UFPE)
in collaboration with the Optics and Atomic Physics group, led by professor Jose Wellington Tabosa. Here we present a theoretical and experimental
investigation on the dynamics of the grating stored in an EIT medium associated with a degenerate two-level system. The dependence of the stored
light grating with the intensities of the incident writing and reading beams
is investigated. Bragg diffraction of the stored grating is employed to probe
its dynamics under different experimental conditions. After giving some concepts about EIT and LS in section 5.2, in section 5.3 we present the theoretical model developed. The experimental setup used is described in section
5.4 an the experimental results obtained are showed in sections 5.5 and 5.6.
The demonstration of the reversible storage and manipulation of the spatial light phase structure stored into the atomic ensemble, and its extension
86
5.2 Some Concepts about EIT and LS.
87
to include beams carrying orbital angular momentum, would be of great importance to demonstrate the capability of quantum information encoded in a
higher dimensional state space [143, 144]. Moreover, the storage of this light
grating opens up the possibility to investigate the generation of correlated
photons pairs in a previously coherently prepared atomic ensemble [146].
5.2
Some Concepts about EIT and LS.
EIT characteristics.
The optical properties of atomic and molecular gases are fundamentally tied
to their intrinsic energy-level structure. The linear response of an atom to
resonant light is described by the first-order susceptibility χ(1) . The imaginary part of this susceptibility Im[χ(1) ] determines the dissipation of the field
by the atomic gas (absorption), while the real part Re[χ(1) ] determines the
refractive index (dispersion). The form of Im[χ(1) ] at a dipole-allowed transition as a function of frequency is that a Lorentzian function with a width set
by the damping. The refractive index Re[χ(1) ] follows the familiar dispersion
profile, with anomalous dispersion (decrease in Re[χ(1) ] with field frequency)
in the central part of the absorption profile within the linewidth.
Figure 5.1 illustrates both the conventional form of the absorption and
dispersion curves (dashed line) and the modified form that results from EIT
(solid line) as a function of the signal field detuning from resonance. Figure
5.2 shows the corresponding third order nonlinear susceptibility. It can be
observed that Im[χ(1) ] undergoes destructive interference in the region of
resonance, i.e., the coherently driven medium is transparent to the prove
field. The fact that transparency of the sample is attained at resonance is
not in itself of great importance, as the same degree of transparency can
be obtained simply by tuning sufficiently away from resonance. What is
important is that in the same spectral region where there is a high degree of
of transmission the nonlinear response χ(3) displays constructive interference,
i.e., its value at resonance is larger than expected from a sum of two split
Lorentzian lines.
Furthermore, the dispersion variation in the vicinity of the resonance differs markedly from the steep anomalous dispersion familiar at an undressed
resonance. Instead, there is a normal dispersion in a region of low absorption, the steepness of which is controlled by the coupling-laser strength (i.e
very steep for low values of the drive laser coupling). Thus despite the transparency the transmitted laser pulse can still experience strong dispersive
and nonlinear effects. It is most significant that the refractive index passes
88
Dynamics of saturated Bragg....
Im [ c ]
1
0
-3
0
3
0
3
Re [ c ]
0.4
-0.4
-3
(wp-w31)/g31
Figure 5.1: Susceptibility as a function of the frequency ωp of the applied field relative to the atomic resonance frequency ω31 , for a radiatively broadened two-level
system with radiative width γ31 (dashed line) and an EIT system with resonant
coupling field (solid line): top, imaginary part of χ(1) characterizing absorption;
bottom, real part of χ(1) determining the refractive properties of the medium.
|c |
(3)
1
0
-2
0
2
(wp-w31)/g31
Figure 5.2: Absolute value of nonlinear susceptibility for sum-frequency generation |χ(3) |
as a function of ωp , in arbitrary units.
5.2 Some Concepts about EIT and LS.
89
Dressed states
Bare states
a
3
(+)
a
G
G
Dressing
2
Signal
(-)
(+)
G(-)
Signal
1
1
Figure 5.3: Interference generated by coherent coupling: left, coherent coupling
of a metaestable state |2 to an excited state |3 by the coupling laser generates
(right) interference os excitation pathways through the doublet of dressed states
|a± (Autler-Townes doublet) provided the decay out of state |2 is negligible
compared to that of state |3
.
through the vacuum value and the dispersion is steep and linear exactly
where absorption is small. This gives rise to effects such as ultraslow group
velocities, longitudinal pulse compression, and light storage (LS). Furthermore through the destructive (constructive) interference in Im[χ(1) ] (Im[χ(3) ])
and the elimination of the effect of resonance upon the refractive index, the
conditions for efficient nonlinear mixing are met.
Intuitive understanding of EIT.
Boller et al. (1991), in discussing the first experimental observation of EIT
in Sr vapor, pointed out that there are two physically informative ways that
we can view EIT: considering the bare or the dressed atomic states showed
in Fig. 5.3. The bare atomic states are the eigenstates of the Hamiltonian
of the atom alone Ĥ0 whereas, the dresses atomic states are the eigenstates
of the hamiltonian Ĥ = Ĥ0 + V̂ , where the interaction of the atoms with the
applied fields, described by V̂ , is considered.
In the dressed atomic states case we use the picture that arises from
the work of Imamoglu and Harris (1989), in which the dressed states can
be viewed as simply comprising two closely spaced resonances effectively
decaying to the same continuum (Boller et al., 1991; Zhang et al., 1995). If
the probe field is tuned exactly to the zero-field resonance frequency, then the
contributions to the linear susceptibility due to the two resonances, which are
90
Dynamics of saturated Bragg....
equally spaced but with opposite signs of detuning, will be equal and opposite
and thus lead to the cancelation of the response at this frequency due to the
interference of de decay channels. In the alternative and equivalent picture,
considering the bare states, EIT can be seen as arising through different
pathways between the bare states. The effect of the fields is to transfer a
small but finite amplitude into state |2. The amplitude for |3, which is
assumed to be the only decaying state and thus the only way to absorption,
is thus driven by two routes- directly via the |1 → |3 pathway, or indirectly
via the |1 → |3 → |2 → |3 pathway (or higher order variants). Because
the coupling field is much more intense than the probe, this indirect pathways
has a probability amplitude that is in fact of equal magnitude to the direct
way, but for resonant fields it is of opposite sign.
Quantitative description of EIT: Bloch equations of the Λ-system.
The essential features of EIT and many of its applications can be quantitatively described using a semiclassical analysis. We shall assume a continuous
wave (cw) classical fields interacting with a single atom that can be modeled
as the Λ-system showed in Fig. 1.4. We consider an ensemble of identical
atoms whose dynamics can be described by taking into account only three
of its eigenstates. In the absence of electromagnetic fields, all atoms are assumed to be in the lowest energy state |1. State |2 has the same parity as
|1 and is assumed to have a very long coherence time. The highest-energy
state |3 is of opposite parity and has nonzero electric dipole coupling to
both |1 and |2. A (near) resonant nonperturbative electromagnetic field of
frequency ωc , termed the coupling field, is applied on the |2-|3 transition.
A probe field ωp is applied on the |1-|3 transition. EIT is primarily concerned with the modification of the linear and nonlinear optical properties of
this-typically perturbative-probe field. The coupling and probe field can be
expressed as:
c = Ec (r, t)ei(kc ·r−ωc t)
E
p = Ep (r, t)ei(kp ·r−ωp t)
E
(5.1)
where Ec (r, t) and Ep (r, t) specify the transversal modes and temporal envelopes of coupling and probe fields with wave vector kc and kp respectively.
The time dependent hamiltonian that describes the system is
Ĥ(t) = Ĥ0 + V̂ (t)
(5.2)
where Ĥ0 is the free atom hamiltonian and reads
Ĥ0 = ω1 |11| + ω2 |22| + ω3 |33|
(5.3)
5.2 Some Concepts about EIT and LS.
91
being ωk for k = 1, 2, 3 the energy correspondent to each level energetic in
Fig. 1.4 respect to a reference level, and
V̂ (t) = −d · E
(5.4)
c is the
=E
p + E
where d is the transition electronic dipole moment and E
total electric field interacting with the atom. Considering our Λ-system, the
electronic dipole moment has the form
d = d3,1 |31| + d3,2 |32| + h.c
(5.5)
where dα,β is the atomic dipole associated with the transition α → β. Thus,
the interaction hamiltonian reads
p (t)|31| − d3,2 · E
c (t)|32| + h.c.
V̂ (t) = −d3,1 · E
(5.6)
Defining the Rabi frequencies associated with the coupling and probe fields
as
id3,1 Ep (r, t)ei(kp ·r)
Ωp =
id3,2 Ec (r, t)ei(kc ·r)
Ωc =
(5.7)
V̂ (t) can be rewritten in the form:
V̂ (t) = −
Ωp (t)e−iωp t + Ωc (t)eiωc t + h.c
2
(5.8)
92
Dynamics of saturated Bragg....
The Bloch equations which describes the dynamics of this laser-driven atomic
system are given by:
i
dρ33
= − 3|[Ĥ, ρ̂]|3 − Γ33 ρ33
dt
i
dρ11
= − 1|[Ĥ, ρ̂]|1 + Γ11 ρ33
dt
i
dρ22
= − 2|[Ĥ, ρ̂]|2 + Γ22 ρ33
dt
dρ31
i
= − 3|[Ĥ, ρ̂]|1 − Γ31 ρ31
dt
dρ32
i
= − 3|[Ĥ, ρ̂]|2 − Γ32 ρ32
dt
i
dρ21
= − 2|[Ĥ, ρ̂]|1 − γρ21
(5.9)
dt
The spontaneous relaxation rates are indicated by Γ31 , Γ32 and Γ33 for coherence and population decays, respectively. Γ11 and Γ22 indicates the rate
at which the ρ33 population decays into the populations ρ11 and ρ22 . Finally,
the ground-state-coherence decay rate is introduced by γ. For convenience
we define Γ3 = Γ33 + Γ31 + Γ32 , γ31 = Γ3 + Γ11 , γ32 = Γ3 + Γ22 and γ21 = γ.
The macroscopic polarization generated in the atomic medium by the applied fields is of primary interest, since it acts as a source term in Marxwell’s
equations and determines the electromagnetic field dynamics. The expectation value of the macroscopic atomic polarization is
P (t) =
N
diαβ i=1
V
(5.10)
where N indicates the number of atoms in the medium. If we assume that
all N atoms contained in the volume V couple identically to the electromagnetically fields then
N P (t) =
(5.11)
d31 ρ31 e−iω31 t + d32 ρ32 e−iω32 t + c.c
V
Assuming d31 = d31 ẑ and d21 = d21 ẑ, we let = N/V and obtain Pz (t) = P (t)
as
P (t) = d31 ρ31 e−iω31 t + d32 ρ32 e−iω32 t + c.c
(5.12)
5.2 Some Concepts about EIT and LS.
93
We now focus on the perturbative regime in the probe field and evaluate the
off-diagonal density-matrix elements ρ31 (t), ρ32 (t) and ρ12 (t) to obtain P (t),
or, equivalently, the linear susceptibility χ(1) (−ωp , ωp ). Taking ρ11 1 and
using a rotating frame to eliminate fast exponential time dependencies, we
find
ρ32 =
iΩc eiΔ1 t
ρ12 ,
γ32 + i2Δ2
ρ12 =
iΩc eiΔ2 t
ρ13 ,
γ21 + i2 (Δ2 − Δ1 )
ρ31 =
iΩp eiΔ1 t
iΩc eiΔ2 t
+
γ31 + i2Δ1 γ31 + i2Δ1
ρ21 .
(5.13)
where Δ1 = ω31 − ωp , Δ2 = ω32 − ωc denote the single-photon detuning from
the resonant frequency ωαβ for the transition α → β, and δ = Δ1 − Δ2 as the
two-photon detuning. Keeping track of the terms that oscillate with e−iωp t ,
we obtain
2
4δ (|Ωc |2 − 4δΔ1 ) − 4Δ1 γ21
|μ13 |2 (1)
χ (−ωp , ωp ) =
0 ||Ωc |2 + (γ31 + i2Δ1 ) (γ21 + i2δ)|
8δ 2 γ31 + 2γ21 (|Ωc |2 + γ21 γ31 )
|μ13 |2 +i
0 ||Ωc |2 + (γ31 + i2Δ1 ) (γ21 + i2δ)|
(5.14)
The linear susceptibility given in Eq. (5.14) contains many of the important features of EIT detailled explained in [73]. For our purpose we are going
to center our attention in the slow and ultraslow light effect.
Linear response:Slow and ultraslow light.
We are going to consider the properties associated with the linear response
of an EIT medium to the probe field Ep . From Eq. (5.14)The most characteristic feature of the real part of the susceptibility spectrum is a linear
dependence on the frequency close to the two-photon resonance δ = 0. For
a negligible decay of the |1 − |2 coherence one finds
2Γ31 δ
Re χ(1) = η 2 + O(δ 2 )
Ωc
(5.15)
where η = (3/4π 2 )λ3 is the normalized density and λ is the transition
wavelength in vacuum. Since the linear dispersion dn/dωp of the refractive
94
Dynamics of saturated Bragg....
index n = 1 + Re[χ] is positive, EIT is associated with a reduction of the
group velocity according to
vgr ≡
dωp
c
|δ=0 =
dkp
n + ωp (dn/dωp )
(5.16)
which was first pointed out by Harris et al. (1992). At the same time, the
index of refraction on an ideal three-level medium is unity and thus the phase
velocity of the probe field is just the vacuum speed of light:
vph ≡
ωp
c
|δ=0 = = c
kp
n
(5.17)
An important property of the EIT system is that the second-order term in
Eq. (5.15) vanishes exactly if there is also single photon resonance Δ1 = 0 of
the probe field. As a consequence there is no group velocity dispersion, i.e.,
no wave-packet spreading. Using Eq. (5.15) yields at two-photon resonance
vgr =
c
1 + ngr
(5.18)
with ngr = σc ΓΩ312 , where ηk = σ was used being σ = 3λ2 /2π the absorption
c
cross section of an atom and the atom number density. The reduced group
velocity gives rise to a group delay in a medium of length L:
1
Γ31
1
Lngr
= σc 2
−
(5.19)
τd = L
=
vgr
c
c
Ωc
Due to the vanishing imaginary part of the susceptibility, i.e., perfect transparency, at δ = 0, relatively high atom densities and low intensities of the
coupling field Ic ∼ Ω2c can be used. Thus the group index ngr can be rather
large compared to unity, and extremely small group velocities are possible.
The lossless slowdown of a light pulse in a medium is associated with a number of important effects. When a pulse enters such a medium, it becomes
spatially compressed in the propagation direction by the ratio of group velocity to the speed of light outside the medium (Harris and Hau, 1999). This
compression emerges because when the pulse enters the sample its front end
propagates much more slowly than its back end. At the same time, however, the electrical-field strength remains the same. The reverse happens
when the pulse leaves the sample. Spatial compression from a kilometer to
a submilimiter scale has been observed by Hau et al. (1999).
Although in the absence of losses the time-integrated photon flux through
any plane inside the medium is constant, the total number of probe photons
5.2 Some Concepts about EIT and LS.
95
inside the medium is reduced by a factor vgr /c due to spatial compression.
Thus photons or electromagnetic energy must be temporarily stored in the
combined system of atoms and coupling field. The notion of a group velocity
of light is still used even for vgr c, where only a tiny fraction of the original
pulse energy remains electromagnetic.
Now, we can consider the slow-light propagation from the point of view
of the atoms. From this perspective, before the probe pulse interacts with
three-level atoms, a cw coupling field puts all atoms into state |1 by optical
pumping. When the front end of the probe pulse arrives at an atom, the dark
state makes a small rotation from state |1 to a superposition between |1
and |2. In this process energy is taken out of the probe pulse and transferred
into the atoms and the coupling field. Thus energy is returned to the probe
pulse at its back end. The excursion of the superposition state between |1
and |2, called dark sate, away from state |1 and hence the characteristic
time of the adiabatic return process depends ont the strength of the coupling
field. The weaker coupling field, the larger the excursion and thus the larger
the pulse delay.
The slowing down of light has a number of important applications. A
reduction of the group velocity of photons leads to an enhanced interaction
time in a nonlinear medium, which is important in enhancing the efficiency
of nonlinear processes (Harris and Hau, 1999; Lukin Yelin, and Flieschhaure,
2000; Lukin and Imamoglu, 2001).
Moreover, making use of the substantial pulse deformation at the boundary of an EIT medium is also of interest for the storage of information contained in long pulses, which in this way can be compressed to very small
spatial volume. But in this case we should consider the limitations that EIT
presents. A convenient figure of merit for this is not the achievable group
velocity itself, but the ratio of achievable delay time τd of a pulse in an EIT
medium to its pulse length τp . One upper limit for the delay time is given
by probe absorption due to the finite lifetime of the dark resonance. Furthermore, for a pulsed probe field, i.e., for a probe field with a finite spectral
width, the absorption of the nonresonant frequency components is nonzero
even under ideal conditions of an infinitely long-lived dark state. A more
detailed discussion about this upper bound for the delay time τd can be
found in [73]. This discussion shows that is not possible to bring a pulse to
a complete stop by using EIT with a stationary coupling field, so that is not
possible to store the information carried by the pulse in the atomic medium.
96
Dynamics of saturated Bragg....
”Stopping of light” using dynamic EIT.
As discussed before it is not possible to bring a light pulse to a complete stop
with stationary EIT. Nevertheless, as was shown by Fleischhauer and Lukin
(2000) it is possible to achieve this goal by changing the group velocity in
time. In the following we will explain with more detail the ”stopping” of
light and this potential applications.
At this point a word of caution is needed, however. The expression ”stooping of light” should be no be taken literally. As mentioned before, the reduction of the propagation velocity of light in a lossless, passive medium is
always associated with a temporary transfer of its energy to the medium.
In the extreme limit of zero velocity relative to the stationary medium no
electromagnetic excitation is left at all. Nevertheless, the notion of vanishing
group velocity of light has here the same justification as the notion of a group
velocity in the case of ultra slow pulse propagation, in which likewise only a
tiny fraction of the original excitation remains in the form of photons.
A key conceptual advance occurred when it was realized by Fleischhauer
and Lukin (2000) that the stopping of light should be possible by reducing
adiabatically the group velocity to zero in time using the EIT scheme. This
can be achieved, for example, by reducing the Rabi frequency of the drive
field. Later, this conditions of adiabaticity have been analyzed by Matsko,
Rosotvtsev, Kocharovskaya, et al. (2001) and by Fleischhauer and Lukin
(2001). The resulting limitations on the rate of change of the coupling field
are rather weak. Furthermore, if the coupling field and thus the group velocity is already very small, even an instantaneous switch off would lead only
to a loss of the very small electromagnetic component of the spin excitation.
By increasing the strength of the coupling field leads to a ”reacceleration”
of the light pulse. In this sense we talk about ”storage” and ”retrieval” of
light. Figure 5.4 shows an illustrative picture of how the LS is produced
using dynamics EIT.
As it was first proposed, light storage (LS) in an EIT medium, can be
described in terms of a mixed two component light-matter exitation, called
dark state polariton (DSP), where each component of the excitation can be
externally controlled [74]. In this picture, when the probe pulse enters the
EIT medium, DSP is formed, and have no sense to talk about the propagation
of the probe pulse inside the medium if not to talk about the propagation
of the DSP state. Decreasing the Rabi frequency of the drive field leads to
a deceleration of the polariton to a full stop. At the same time its character
changes from that of electromagnetic field to that of a pure spin excitation
and all properties of the original light pulse are coherently transferred to
the atomic system in this process controlled by the coupling field. To date
5.2 Some Concepts about EIT and LS.
97
Figure 5.4: Illustrative picture about the LS process. On top arrow is showed the
conventional EIT process whereas bottom arrow show the dynamic EIT producing
light sotorage. In dynamic EIT (bottom arrow) the coupling (or control) field is
switched off thus ”stoping” the prove pulse. After a time τ the prove pulse is
retrieved by means of switching on the coupling field.
several experimental observations of these effects were realized in different
systems [65–68, 141].
Alternatively, the LS process can also be described as being due to the
creation of a spatially dependent ground states coherence (coherent grating)
that contains, respectively, the information on the amplitude and phase of
a light pulse and which survives after the switching off of the incident light.
Using this simpler picture, it was recently demonstrated the storage of a
polarization light grating into an atomic coherence via a backward four-wave
mixing configuration [75].
Light Storage (LS) and quantum memories for photons.
The most important potential application of (LS) is certainly in the field
of quantum information. In this process, the quantum states of photons
are transferred to collective excitations of the medium, from the point of the
DSP picture, or to the spatially dependent ground states coherence, from the
coherent grating picture. In this way the EIT medium can acts as a quantum
memory for photons: we are able to obtain later information on a previously
stored light pulse, as well as to manipulate the stored information. Compared
98
Dynamics of saturated Bragg....
(a)
(b)
(c)
Figure 5.5: (a) Simplified Zeeman level scheme, showing the coupling and the
propagation directions of the grating writing beams (W and W ) and (b) the coupling and the propagation direction of the reading (R) and diffracted (D) beams.
The beams W and W make a small angle θ and are circularly polarized with
opposite handedness, while the beam R is counterpropagating to the beam W and
have a circular polarization opposite to this beam. The diffracted beam is detected
in a direction opposite to the beam W .(c) The switching time sequence for the
writing and reading beams.
to other proposals for quantum memories, LS using the EIT-based system
is capable of storing individual photon wave packets with high fidelity and
without the need for a strongly coupling resonator. Finally, comment that
other schemes have also been recently employed to store spatial structures
(images) in atomic vapors [77, 142]. For instance, a light vortex was stored
in a hot vapor for hundreds of microseconds [142].
5.3
Theoretical model
We consider an ensemble of cold atoms excited by three different fields: two
writing (W and W ) and one reading (R) laser pulses. The atomic ensemble
5.3 Theoretical model
99
can be well approximated by a set of degenerate two-level atoms, with a
ground-state manifold composed of two degenerate states (|1a and |1b) and
the excited-state manifold having a single state (|2). As illustrated in Fig.
5.5, the ground-state degeneracy corresponds to the Zeemam degeneracy of
atomic cesium in the experiment. In this way, the different atomic levels
are connected by fields of different polarizations with respect to the atom.
We consider fields W and R having σ̂ − polarization and field W having
σ̂ + polarization. W and R excite then the transition 1b → 2, and W the
transition 1a → 2.
The fields W and W propagate in different directions, with a small angle
θ between them. The R field is counter-propagating with respect to W . The
signal we want to model corresponds to the diffraction of the R field in the
spatial grating formed by fields W and W . In the case of cw excitation of
the ensemble, this signal corresponds to the well-know conjugated signal in
four-wave mixing (FWM) processes [145]. Here we call it the D field (see
Fig.5.5(b)).
We use this FWM configuration to store and later retrieve a coherence
grating written in the atomic ensemble. In order to address this coherence
storage process, we use a specific time sequence for the pulsed excitation of
the ensemble showed in Fig.5.5 (c). First we prepare the sample by exciting
it with the two, long writing pulses. In this writing process, the goal is
to leave the system in its stationary state. Then we turn off the writing
beams, and wait a certain amount of time, the storage time, before turning
the reading pulse on. The information about this reading pulse stays on
also for a long time, enough to extract the whole stored grating from the
ensemble. A field-D pulse is then generated during the read process. In
the following theoretical analysis, we want to model and study this field-D
generation process in detail, considering the three-level-atom approximation
discussed above.
5.3.1
Grating formation and storage
W , propagating in
We consider an atom excited by two writing beams: one, E
W . The fields
W , forming an angle θ with E
the z direction and the other, E
+
−
W and E
W have orthogonal circular polarizations σ̂ and σ̂ , respectively.
E
We consider small enough angles so that we can assume, to a good approxi W as being σ̂ − on the same state basis in which
mation, the polarization of E
W is σ̂ + . We can then write
E
W = EW (r)ei(kW z−ωW t) σ̂ + ,
E
(5.20a)
W = EW (r)ei(kW ·r−ωW t) σ̂ − ,
E
(5.20b)
100
Dynamics of saturated Bragg....
where EW (r) and EW (r) represent the transverse modes of each field. We
assumed both of them having constant intensities. The frequencies of the
fields are ωW and ωW , and their wavevectors are kW ẑ and kW , respectively.
The energy difference between fundamental and excited levels is ωe . The
system Hamiltonian can be written as
Ĥ(t) = Ĥ0 + V̂ (t) ,
(5.21)
Ĥ0 = ωe |22|
(5.22)
where
is the Hamiltonian for the free atom and
W (t) |21a|
V̂ (t) = − d2,1a · E
W (t) |21b| + h.c.
− d2,1b · E
(5.23)
is the interaction hamiltonian. Defining the Rabi frequencies
id2,1a EW (r)eikW z
,
id2,1b EW (r)eikW ·r
,
ΩW (r) =
ΩW (r) =
(5.24a)
(5.24b)
and assuming the resonance condition ωW = ωW = ωe , the whole set of
Bloch equations which describe the evolution of the atomic system, in the
rotating-wave approximation, becomes
dρ22
dt
dρ1a,1a
dt
dρ1b,1b
dt
dσ1a,2
dt
dσ1b,2
dt
dρ1a,1b
dt
= [ΩW σ1a,2 + ΩW σ1b,2 + c.c.] − Γ22 ρ22 ,
(5.25a)
= [−ΩW σ1a,2 + c.c.] + Γ1a,1a ρ22 ,
(5.25b)
= [−ΩW σ1b,2 + c.c.] + Γ1b,1b ρ22 ,
(5.25c)
= −Ω∗W (ρ22 − ρ1a,1a ) + Ω∗W ρ1a,1b − Γ12 σ1a,2 ,
(5.25d)
= −Ω∗W (ρ22 − ρ1b,1b ) + Ω∗W ρ1b,1a − Γ12 σ1b,2 ,
(5.25e)
= −Ω∗W σ2,1b − ΩW σ1a,2 − γρ1a,1b ,
(5.25f)
with σ1a,2 = ρ1a,2 e−iωW t and σ1b,2 = ρ1b,2 e−iωW t . The spontaneous relaxation
rates are indicated by Γ12 and Γ22 , for the coherence and population decays,
respectively. Γ1a,1a and Γ1b,1b indicate the rates at which the ρ22 population
5.3 Theoretical model
101
decays into the populations ρ1a,1a and ρ1b,1b , respectively. For simplicity, in
these equations and in the following, we omit the spatial dependence of the
Rabi frequencies. The ground-state-coherence decay rate γ is introduced to
take into account, in an effective way, the decay induced by any residual
magnetic fields. Such decay is usually a result of inhomogeneous broadening in the ensemble of atoms, each subject to a slightly different magnetic
field [147]. For the signal we are treating here, however, this simple model
which consider the same decay constant for the whole ensemble is enough to
obtain a good description with the experimental data.
After a sufficiently long time, the system reaches a steady situation in
which dρkl /dt = 0, for all ρkl density-matrix elements. The steady-state
coherence (i.e the coherence grating) ρe1a,1b between the two ground state
levels is then given by
Γ1a,1a |ΩW |2 + Γ1b,1b |ΩW |2
e
Ω∗W ΩW ,
ρ1a,1b = −
(5.26)
A
with
A = Γ1a,1a |ΩW |2 + Γ1b,1b |ΩW |2 γΓ12 + |ΩW |2 + |ΩW |2 + 6γ|ΩW |2 |ΩW |2
(5.27)
We are particularly interested in the situation where γ is very small when
compared to any other frequency in the system, since this corresponds to our
experimental condition. In this limit, note then that the above expression
simplifies to
Ω∗W ΩW ρe1a,1b = −
.
(5.28)
|ΩW |2 + |ΩW |2
W and E
W are turned off, the coherences in the system
Once the fields E
evolve according to their respective decay times. Since γ << Γ12 , after a time
ts >> 1/Γ12 the stored coherences in the sample can be well approximated
by
s
σ1a,2
(ts ) = 0 ,
(5.29a)
s
σ1b,2
(ts ) = 0 ,
(5.29b)
ρs1a,1b (ts )
5.3.2
=
ρe1a,1b e−γts
.
(5.29c)
Reading
The stored coherence grating can be extracted from the sample using a σ̂ − R counter-propagating with respect to E
W :
polarized third field E
R = ER (r)ei(−kR z−ωR t) σ̂ − ,
E
(5.30)
102
Dynamics of saturated Bragg....
with ER , kR , and ωR represent the transverse mode, wavevector, and fre R . If we assume the condition ωe = ωR holds, and define
quency of field E
the Rabi frequency
id2,1b ER (r)e−ikR z
ΩR (r) =
,
(5.31)
the relevant Bloch equations describing the reading process become
dσ1a,2
= Ω∗R ρ1a,1b − Γ12 σ1a,2 ,
dt
dρ1a,1b
= −ΩR σ1a,2 − γρ1a,1b ,
dt
(5.32a)
(5.32b)
with σ1a,2 = ρ1a,2 e−iωR t . Note that the equations for σ1a,2 and ρ1a,1b are
actually de-coupled from the rest of the system of Bloch equations.
D that is phase conjugated to
We are interested in calculating the field E
W . This field is generated by the medium in the transient excitation of
E
the σ1a,2 coherence, corresponding to the extraction of the stored coherence
grating. Using the stored state as initial conditions, the solution of the above
equations for σ1a,2 (t) are
σ1a,2 (t) =
Ω∗R ρs1a,1b (ts )e−γ1 t senh (γ2 t)
,
γ2
(5.33)
with
γ1 =
γ2 =
Γ12 + γ
,
2
(5.34a)
(Γ12 − γ)2 − 4|ΩR |2
2
,
(5.34b)
The single-atom polarization vector p2,1a on the 2 → 1a transition is then
given by
p2,1a (r, t) = d2,1a σ2,1a (r, t)e−iωe t .
(5.35)
5.3.3
Signal
D of the D field coming from the diffraction of E
R (see
The electric field E
Fig.5.5 (b)) is a result of the constructive interference of the emission of all
atoms in the −kW direction. If we neglect interaction between atoms and
D in the k direction can be
propagation effects of the D field, the value of E
obtained by the superposition of all atomic contributions on that direction:
1
ED (k, t) =
(5.36)
η(r)p2,1a (r, t)e−ik·r d3r ,
3/2
4π0 (2π)
5.3 Theoretical model
103
where η(r) represents the atomic density at r, 0 is the vacuum permittivity,
and the integration runs over the whole ensemble volume. Approximating the
fields W , W , and R as plane waves, we can neglect the spatial dependence
on EW , EW , and ER . In this case, we can write
IW ikW z
ΩW
=i
e
,
(5.37a)
Γ12
2Isa
ΩW IW ikW ·r
=i
e
,
(5.37b)
Γ12
2I
sb
ΩR
IR −ikR z
=i
e
,
(5.37c)
Γ12
2Isb
with IW , IW , and IR being the intensities of the W , W , and R fields,
respectively. Isa and Isb are the saturation intensities of the 1a → 2 and
1b → 2 transitions, defined according to [148].
Since kR − kW = 0, Eq. (5.36) can be written as
D (k, t) =
E
id2,1a |ρs1a,1b |fR (t)e−iωe t
4π0 (2π)3/2
η(r)e−i(k+kW )·r d3r ,
√
IW IW e−γts
=
Isb
Isa
IW
+ IW Isa
Isb
representing the modulus of the stored ground-state coherence, and
IR e−γ1 t senh (γ2 t)
fR (t) =
2Isb
γ2 /Γ12
(5.38)
with
|ρs1a,1b |
(5.39)
(5.40)
a function describing the temporal profile of the D-field pulse. Note that
fR (t) is a function of the read field parameters only.
If we approximate the distribution of atoms as having a gaussian profile
with the same rms width L in all three direction, we can write
η(r) =
N
−
r ·
r/2L2
e
,
(2πL2 )3/2
(5.41)
where N is the total number of atoms in the cloud. Using this expression for
η(r), Eq. (5.38) becomes
D (k, t) =
E
id2,1a N|ρs1a,1b |fR (t)e−iωe t −|k+k |2 L2 /2
W
e
,
4π0 (2π)3/2
(5.42)
104
Dynamics of saturated Bragg....
which explicitly shows that the emission of the D-field occurs in the −kW direction mainly with a spread in vector space, of the order of the inverse of
the atomic-distribution spatial width, L−1 .
The detection apparatus can be arranged to collect all light of the D-field.
The detection of the field is performed with a fast detector compared
to the time variation of fR (t), the signal Sfast (t) is then proportional to the
integration of the intensity of light over all k:
D (k, t)|2 d3k ,
(5.43)
Sfast (t) = A |E
where A is a proportionality constant. From Eq. (5.42), we see that such
detected signal is given by
Sfast (t) = A |ρs1a,1b |2 |fR (t)|2 ,
(5.44)
with A being a different proportionality constant.
Another important quantity that can be directly derived from Sfast (t)
is the total energy, UD , extracted in mode D. Note that, in light-storage
measurements, the goal is usually to extract as much information and energy
as possible from the coherence grating [149]. From the expressions derived
above we have
∞
Sfast (t)dt
UD =
0
2 A |ρs1a,1b |2
IR /2Isb
=
γ
Γ12
(1 + Γ12 )( 2IIRsb +
5.4
5.4.1
γ
)
Γ12
.
(5.45)
Experimental setup.
Atomic source.
As indicated in Figs. 5.5(a) and 5.5(b) the experiment was performed using a degenerate two-level system. This system corresponds in the experiment to the cycling transition 6S1/2 (F = 3) ↔ 6P3/2 (F = 2) of the
cesium D2 line showed in Fig. 5.6. The cesium atoms were previously
cooled in a magneto optical trap (MOT) operating in the closed transition
6S1/2 (F = 4) ↔ 6P3/2 (F = 5) with a repumping beam resonant with the
open transition 6S1/2 (F = 3) ↔ 6P3/2 (F = 3). The temperature of the
MOT was estimated approximately to be around 1 mK. The beams which
generate the MOT come from a Ti:saphire laser. This beams, with a power of
80 mW and a diameter of 2.0 cm are 12 MHz out of resonance with the trap
5.4 Experimental setup.
105
6P3/2
wT
wRP wW’ wD
wW wR
wW
wD
wW’
wR
6S1/2
Figure 5.6: Left: D2 line energy levels. ωT and ωRP are the frequency of the trap
and repumping fields respectively, ωW and ωW are the frequency of the writing
beams and ωR , ωD the frequency of the reading and diffracted beam. Right: the
relevant Zeeman levels coupled by writing and reading fields.
transition. To prepare the atoms into the state 6S1/2 (F = 3), we switch off
the repumping beam for a period of about 1 ms to allow optical pumping by
the trapping beams via non resonant excitation to the excited state F = 4.
The correspondent lasers which generate the trap and repumping bean are
showed in Fig. 5.7.
After optical pumping, the optical density of the sample of cold atoms in
the F = 3 ground state is approximately equal to 3 for appropriate MOT
parameters. In order to compensate for spurious magnetic fields, three independent pairs of Helmholtz coils with adjustable currents are placed around
(a)
(b)
Trap laser
Repumping laser
Figure 5.7: (a) Ti:Saphire laser; (b) repumping laser.
106
Dynamics of saturated Bragg....
Figure 5.8: Magneto optical trap (MOT) used to cool the cesium atoms. The
atomic cell is in the center. Three independent pairs of Helmholtz coils are placed
around the MOT to compensate the magnetic field in that region.
the MOT and regulated in order to cancel the magnetic field in that region,
how is showed in Fig. 5.8. This cancelation is optimized by narrowing the
EIT peak according to [75].
5.4.2
Incident laser beams.
A simplified experimental setup is depicted in Fig. 5.9 showing how the
different incident laser beams indicated in Fig. 5.5 are generated. All the
incident laser beams indicated in Fig. 5.5 are provided by an external cavity
diode laser which is locked to the F = 3 ↔ F = 2 transition. Figure 5.10
shows the controls and the external cavity which generate the incident laser
beams of the experiment.
The grating writing beams (W and W ) have the same frequency and
both of them pass through a pair of acousto-optical modulators, showed
in Fig. 5.11, (AOM) with one of them operating in double passage. The
(AOM), also called a Bragg cell, uses the acousto-optic effect and shift the
frequency of light using sound waves (usually at radio-frequency). After
passing through them, W and W fields can have their frequency scanned
around the F = 3 ↔ F = 2 transition. The two AOM’s also allow us
to control their intensity. These two beams are circularly polarized with
opposite handedness and are incident in the MOT forming a small angle
θ ≈ 60,mrad which leads to a polarization grating with a spatial period
λ
given by Λ = 2sin(θ/2)
, where λ is the light wavelength. The reading beam R
is circularly polarized opposite the writing beam W and also passes through
another pair of AOM’s which does not change its frequency but allow us to
control its intensity.
5.4 Experimental setup.
107
Figure 5.9: Simplified experimental scheme. The diode laser is locked to the
F = 3 ↔ F = 2 transition. AOM, acousto-optical modulator; BS, beam splitter.
(a)
(b)
Figure 5.10: controls and external cavity of the laser which produce the incident
fields.
108
Dynamics of saturated Bragg....
Figure 5.11: Acousto optic modulators used in the experiment.
5.4.3
Detection.
The detection of the diffracted beam D was performed using a photodiode.
The time constant of the detector is ≤ 0.5μs, which limits the raising time
of the experimentally retrieved pulse as we will see below. Finally, the analogic signal is digitalized and recorded by the oscilloscope connected to the
detector.
5.5
Experimental results and discussion.
Employing the time sequence shown in Fig. 5.5(c) we have investigated
the light grating storage dynamics through the observation of delayed Bragg
diffraction of the reading beam R in the Zeeman coherence grating induced
by the writing beams W and W . The writing and reading pulses are trigged
by the switching off of the repumping laser which also triggers the turn off
of the MOT quadrupole magnetic field. The spurious magnetic fields are
compensated adjusting the current of the three independent pairs of coils
situated around the MOT.
5.5.1
Diffracted signal D for different storage times.
In this section we study the Bragg diffracted beam as a function of the storage
time of the Zeeman coherence grating. In Fig. 5.12 we show the cw-FWM
and the Bragg diffracted signal which is retrieved from the stored Zeeman
coherence grating for different storage times. We have experimentally verified that the polarization of the diffracted beam, both for the steady state
Diffracted Signal Intensity, D (arb. units.)
5.5 Experimental results and discussion.
109
switching off
of W and W'
storage time, ts
-6
-4
-2
0
2
4
6
8
10
Time ( s)
Figure 5.12: Bragg diffraction signal retrieved from the stored grating for different storage times
cw-FWM signal (real time Bragg diffraction) and for the retrieved signal,
is always opposite to the polarization of the reading beam as schematically
depicted in Fig. 5.5(b). We have been able to observe the diffracted signal
up to a time of 10 μs. This maximum storage time is very sensitive to the
compensation of the residual magnetic field. It is interesting to note that for
short storage times the retrieved signal peak intensity is much larger than
the corresponding cw-FWM signal. This effect is related to the simultaneous presence of the writing and reading beams in the cw regime, where
the reading beam contributes to decrease the contrast of the coherence grating induced by the writing beams. The decay of the peak intensity of the
diffracted pulse, normalized by its steady state value is presented in Fig.
5.13. The exponential decay behavior is evidenced by the exponential fitting
(solid curve). For the data presented in Fig. 5.13, the intensities of the
writing beams W and W are approximately equal to 5.0 mW/cm2 and 1.5
mW/cm2 respectively, while the intensity of the reading beam R is about 8.0
mW/cm2 . From the measurement presented in Fig. 3, we obtain a decay
time of the order of 2.9 μs, which corresponds to the Zeeman ground state
coherence decay. It is worth noticing that we have experimentally verified
that the measured coherence time does not depend on the intensity of either
the writing and the reading beams.
Dynamics of saturated Bragg....
Diffracted Maximum Intensity (arb. units)
110
Experiments
fitting
~exp(-t/tc)
t-1
c
g 2p=0.05 MHz
Decay time tc=2.9ms
PR = 209 mW
PW` = 32 mW
PW = 105 mW
0
1
(W0)W,W`= 1,60 mm
(W0)R = 1,80 mm
2
3
4
5
6
7
8
Time (ms)
Figure 5.13: Normalized Bragg diffraction peak signal for different storage times.
The solid curve corresponds to a fit with an exponential function.
5.5.2
Diffracted signal D for different reading beam
intensities.
In this section we have measured the temporal pulse shape of the retrieved
signal for different reading beam intensities when fixing the storage time of
the Zeeman coherent grating to approximately 1μs, which means that the
reading beam is turn off 1μs after turning off the writing beans, W and W .
Left column in Fig. 5.14 show the experimental results obtained for three different values of the reading beam intensity. We note that the experimentally
retrieved pulse raising time is limited by the time constant of the detector.
As we have discussed previously the coherently prepared atomic system couples to the reading beam to transiently generate the diffracted pulse signal.
The temporal width of the generated pulse decreases for increasing reading
beam intensity, a direct consequence of the effect of the increased dumping
of the Zeeman ground state coherence caused by spontaneous emission induced by the reading beam in the process of mapping the stored Zeeman
coherence into the optical coherence. By sake of comparison, in the right
column of Fig. 5.14 we show the corresponding retrieved pulse obtained using the previously developed theory, assuming Isb the saturation intensity
for the 6S1/2 (F = 3, mF = +3) → 6P3/2 (F = 2, mF = +2) transition. We
have used an adjustable parameter of the order of a ≈ 0.02 to re-scale all
the measured reading beam intensities (i. e. IR → aIR ), which accounts for
the uncertainty in the determination of the experimental value of the Rabi
Diffracted Signal Intensity (arb. units)
5.5 Experimental results and discussion.
Experiment
Theory
WRW
/G2 /12G=
2 0.18
= 0.015
IR » 8.5 mW/cm2
R
22
WR/2G122=0.14
IR » 5.0 mW/cm2
WR/ G22= 0.009
2
=0.11
WW
/ 12G222
= 0.005
R/G
IR » 2.9 mW/cm2
0.000000 0.000001 0.000002 0.000003
111
R
0
1
Time ( s)
2
3
4
Time ( s)
Figure 5.14: Retrieved-pulse temporal shape corresponding to different intensities of the reading beam, for a fixed storage time. The writing beam intensities
are 7.0 mW/cm2 and 1.0 mW/cm2 respectively. As described in the text, for
comparison with the theory, the measured reading beam intensities needs to be
re-scaled by a factor a ≈ 0.02.
frequency associated with the reading beam.
In order to show more clearly the dependence of the retrieved pulse with
the intensity of the read beam, we plot in Fig. 5.15 plot the measured pulse
width (full width half maximum, FWHM) for different reading beam intensities. In these measurements, for each value of the intensity of the reading
beam , we have recorded three curves of the retrieved pulse which allows
us to estimate the corresponding error bars. The solid line curve in Fig.
5.15 corresponds to a calculation of the pulse width using the previously developed theoretical model. In this calculation we have used γ/Γ12 ≈ 0.014
in order to obtain the best agreement with the experiment. Note that this
value is of the same order of the experimentally measured decay rate, obtained from the data shown in Fig. 5.13, and estimated as γ/Γ22 ≈ 0.02,
with Γ22 /2π = 2Γ12 /2π = 5.2 MHz.
We show in Fig. 5.16 the retrieved pulse energy. The corresponding
solid curve is a theoretical fitting obtained using Eq. (5.3.3) with the same
adjustable parameter a. The agreement between theory and experiment is
qualitatively satisfactory, taking into account the simplifications of the theoretical model, which consider a single three-level system and do not take into
acount for the manifold Zeeman degeneracy.
112
Dynamics of saturated Bragg....
1,6
FHWM, 'W (Ps)
1,4
1,2
1,0
0,8
0,6
0
2
4
6
8
10
2
Intensity of R (mW/cm )
Figure 5.15: Measurement of the temporal width (Full Width at Half Maximum)
of the retrieved pulse for different intensities of the reading beam, obtained in
similar experimental conditions as in Fig. 5.13. The empty-circles-connected curve
is a theoretical fitting using the model described in the text.
250
200
J/* Retrieved peak intensity (arb. units)
Retrieved pulse energy (arb. units)
Experiment
Theory
150
100
50
25
B
pico
20
15
10
5
0
0
2
4
6
8
2
Reading beam intensity (mW/cm )
10
0
0
2
4
6
8
10
2
Reading beam intensity (mW/cm )
Figure 5.16: Retrieved pulse energy for different intensities of the reading beam,
obtained on similar experimental conditions as in Fig. 4(a-c). Inset: The corresponding variation of the peak intensity of the retrieved pulse. The solid and the
empty-circles-connected curves are theoretical fittings employing the model described in the text with the same intensity adjust parameter used to fit the pulse
width in Fig. 5.
113
Experiment
Theory
200
J/* 150
Retrieved peak intensity (arb. units)
Retrieved pulse energy (arb. units)
5.5 Experimental results and discussion.
100
50
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
2
Writing beam intensity, W (mW/cm )
0
0
1
2
3
4
5
2
6
7
Writing beam intensity, W (mW/cm )
Figure 5.17: Variation of the maximum peak intensity of the retrieved pulse
for different intensities of the grating writing beam W. For these data, the corresponding intensities of the writing (W ) and reading (R) beams were fixed at 1.0
mW/cm2 and 9.0 mW/cm2 , respectively. The solid curves are again theoretical
fittings using the model described in the text. We have used the same intensity
adjust parameter, a ≈ 1.9, in both curves.
5.5.3
Diffracted signal D for different writing beam intensities.
We also have measured the variation of the diffracted signal as a function
of the intensity of one of the writing beams (i.e. the beam W ). The results for the corresponding pulse energy are shown in Fig. 5.17. For these
measurements, the intensities of the writing beam W and the reading beam
were respectively 1.0 mW/cm2 and 9.0 mW/cm2 . The solid curve in Fig.
5.17 corresponds to a theoretical fitting with the calculated retrieved pulse
energy given by Eq. 5.45, assuming that Isa is the saturation intensity of
the 6S1/2 (F = 3, mF = +1) → 6P3/2 (F = 2, mF = +2) transition. Again,
to take into account for the uncertainty in the experimental value of the
Rabi frequency associated with the writing beams W and W , we have used
another adjustable parameter, which in the present case is of the order of
a ≈ 1.9, so we re-scale the intensity ratio measured between these beams
(i.e., IW /IW → a IW /IW ).
5.5.4
Energy retrieved saturation.
From Fig. 5.16 and Fig. 5.17 it can be observed that the amount of energy that can be retrieved from the medium clearly saturates as a function
114
Dynamics of saturated Bragg....
(b)
"Without Magnetic Field"
switching off
Diffracted Signal Intensity, D (arb. units)
Diffracted Signal Intensity, D (arb. units)
(a)
of W and W`
-4
0
4 Time ( s) 8
12
B ~ 0.2 Gauss
switching off
of W and W
-4
0
4
Time ( s)
8
12
Figure 5.18: (a) Diffracted signal D for different storage time in the absence of
applied magnetic field; (b) Diffracted signal for the same sequence of storage time
used in (a) but now applying a transversal magnetic field of order of 0.5 G.
of writing and reading beam intensities. In particular, this shows that for
fixed writing beams intensities, there is a maximum amount of energy that
can be retrieved from the stored coherence. As observed, the peak of the
retrieved pulse saturates more strongly with the writing beam intensity as
when compared with the saturation induced by the reading beam. The saturation induced by the read beam is related mainly to the total retrieved
energy. The pulse peak, however, can increase much further with the read
power, since it is closely related also with the speed of the reading process.
On the other hand, the increase of the writing beam intensity will saturate
the Zeeman coherence grating, therefore reducing its contrast. This effect
has a strong influence on the Bragg diffraction efficiency, affecting equally
the total retrieved energy and the pulse peak.
5.6
Collapse and revival of the stored light
grating.
Finally we present some experimental results on the observation of collapse
and revival of the stored grating due to its evolution in an external magnetic
field, which is associated with the Larmor precession of the induced coherent
grating [150, 151]. This coherent grating, which consist entirely of a spin
wave, precesses about the external magnetic field axis revealing a series of
collapses and revivals. Bragg diffraction is employed to probe its dynamics
under different experimental conditions. In Fig. 5.18 we show the diffracted
5.7 Conclusions.
115
signal for different storage time when a transversal magnetic field of order
of 0.5 G is applied. Clearly we can observe the collapse and revival of the
stored coherence grating. Under appropriate conditions, we have observed
up to three revival events after the switching off of the grating writing beams.
5.7
Conclusions.
In this work we have investigated from the theoretical and experimental point
of view, the storage of a spatial light polarization grating into the Zeeman
ground states coherence of cold cesium atoms. We have performed systematic measurements in order to reveal the saturation behavior of the retrieved
signal as a function of the intensities of the writing and reading beams. The
developed simple theoretical model accounts reasonably well for the observed
results and in particular for the measured pulse temporal shape. We consider
that our results are of considerable importance for a better understanding
of the coherent memory for multidimensional state spaces. Finally, we also
have observed the coherent evolution of the stored grating in the presence
of an applied magnetic field, which shows collapses and the revivals of the
stored coherence grating, associated with the Larmor precession of the induced grating around the applied magnetic field. This effect strongly support the possibility of manipulating more complex spatial information stored
into an atomic medium. Further investigation on this effect was done by
the group of the UFPE later. The results were recently published in [152]
where they report on the storage of orbital angular momentum of light using
the same scheme presented in this chapter. In the same way, they employ
Bragg diffraction to retrieve the stored optical information impressed into
the atomic coherence by the incident light fields. Moreover, the stored information was manipulated by an applied magnetic field and they were able
to observe collapses and revivals due to the rotation of the stored atomic
Zeeman coherence for times longer than 15μs.
Chapter 6
Conclusions.
After presenting in detail the work realized in this Thesis, it is time to summarize the goals that have been achieved and also to comment the perspectives
following the developed work.
Quantum systems entanglement in continuous variables has been receiving increasing attention from the scientific community. The transverse spatial
shape of photons (i.e transverse position and momentum degree of freedom)
is such a continuous variable systems, which due to its ease of control is prone
to be used for testing some of the new physical predictions that continuous
variables quantum systems can offer, as the improvement of the security of
quantum key distribution cryptography and the efficiency enhancement of
quantum communication protocols.
Following this line, the main goal of this Thesis has been the characterization and measurement of the spatial shape of photons. Different techniques
to completely characterize and implement quantum states using the spatial
properties of photons has been developed and presented. Specifically, it has
been the aim of Chapter 2, Chapter 3, and Chapter 4, where we have shown
theoretical analysis and experimental verifications.
Another outstanding problem in Quantum Optics Information Technology (QOIT) is the coherent and reversible storage of photons states in matter,
i.e. the necessity of a quantum memory. This topic has been studied in Chapter 5, where theoretical and experimental results has been showed. Below we
summarize the more important contributions of this Thesis:
Demonstration about the non-validity of the OAM transformation rule l ⇒ −l for highly focused light beams: We have shown from
a theoretical and experimental point of view that the OAM transformation
rule l ⇒ −l is not valid for highly focused light beams, since Dove prisms
inherently introduce astigmatism, and therefore further OAM changes. We
116
117
have analyzed how light beams with a well defined value of the OAM per
photon, after traversing the Dove prism, are transformed into a superposition of states with well defined OAM. The violation of the rule l ⇒ −l,
turns out to be more important for highly focused light beams. We have
provided a quantitative study of the properties of the Dove prism, and we
have verified experimentally the validity of the theoretical results in a series
of experiments with a commercially available Dove prism (Chapter 2).
Presentation of a new method to measure the amplitude and
phase of Laguerre-Gaussian like beams: A method to measure the amplitude an phase of Laguerre-Gaussian like beams have been presented, from
a theoretical and experimental point of view. This method is based on a
small variation of the spiral phase interferometry technique, which allows to
avoid some technical problems that can be found in the reconstruction of
finite sized beams and beams with phase singularities. A few examples of
the use of our technique for the characterization of complex beams has been
presented (Chapter 3).
Description of the transverse spatial shape of bipartite entangled photons generated in Spontaneous Parametric Down Conversion (SPDC) using the Wigner function formulism: We have shown
some theoretical and experimental results regarding the spatial Wigner function of entangled paired photons generated in SPDC. Concretely, we have
studied the remote preparation of one of the photons of the pair, for the photon being in a Gaussian beam and in a superposition of two modes. Also,
we have analyzed and measured the case of the Wigner function of only one
of the two photons of the pair, which shows that one photon of the pair is
in a statistical mixture of modes. Finally, an experiment to completely measure the spatial state of the two photons was proposed, and also analytical
expressions for the results one would expect in this case were given.
Investigation about the storage of spatial light polarization grating into the Zeeman ground states of cold cesium atoms: We have
investigated theoretically and experimentally the dynamics of light grating
stored in electromagnetically induced transparent (EIT) medium associated
with a degenerate two-level system. The dependence of the stored light
grating with the intensities of the incident and reading beams has been investigated. Bragg diffraction into the stored grating has been employed to
probe the dynamics under different conditions showing reasonable results in
agreement with the theoretical model developed. Moreover, we also have
observed the coherent evolution of the stored grating in the presence of an
118
Conclusions.
applied magnetic field, which allows collapses and revivals of the stored grating, which is associated with the Larmor precession of the induced grating
around the applied magnetic field.
As it can be observed from the anterior relation, a remarkable point of
this Thesis is that each theoretical prediction has been tested experimentally. We think that the results presented here are an important step in
order to explore the quantum capabilities of the transverse spatial shape of
photons and can encourage to go further in its capabilities. Firstly, the characterization and measurement of the transverse momentum phase space of
entangled photons it’s a crucial point in order to test some of the new physical predictions that continuous variables quantum systems can offer, such as
quantum computation with continuous variables and another interesting protocols. Moreover, the investigation on the dynamics of light grating stored in
an EIT medium is of considerable importance for a better understanding of
the coherent memory for multidimensional state spaces. Thus, a further and
more detailed investigation of the topics presented here should be of special
interest for the scientific community.
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How a Dove prism transforms the orbital
angular momentum of a light beam
N. González1 , G. Molina-Terriza1 , and J. P. Torres1,2
ICFO-Institut de Ciencies Fotoniques1 , and Department of Signal Theory and
Communications2 , Universitat Politecnica de Catalunya, Mediterranean Technology Park,
08860 Castelldefels (Barcelona), Spain
[email protected]
Abstract: It is generally assumed that a light beam with orbital angular
momentum (OAM) per photon of l h̄, is transformed, when traversing a
Dove prism, into a light beam with OAM per photon of −l h̄. In this paper,
we show theoretically and experimentally that this OAM transformation
rule does not apply for highly focused light beams. This result should be
taken into account when designing classical and quantum algorithms that
make use of Dove prims to manipulate the OAM of light.
© 2006 Optical Society of America
OCIS codes: (080.0080) Geometrical optics; (230.5480) prisms; (090.1970) Holography,diffractive optics
References and links
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and
the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45 8185 (1992).
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Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. 88, 013601 (2002).
3. Graham Gibson, Johannes Courtial, Miles J. Padgett, Mikhail Vasnetsov, Valeriy Pasko, Stephen M. Barnett, and
Sonja Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,”
Opt. Express 12, 5448 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5448
4. R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, “The use of orbital angular momentum of light
beams for super-high density optical data storage,” OSA Annual Meeting, (Optical Society of America, 2004)
paper FTuG14.
5. Lluis Torner, Juan P. Torres, and Silvia Carrasco, “Digital spiral imaging,” Opt. Express 13, 873 (2005).
6. A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,”
Nature 412, 313 (2001).
7. A. Vaziri, G. Weihs and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum
communication,” Phys. Rev. Lett. 89, 240401 (2002).
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94, 040501 (2005).
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Photon Pairs,” Phys. Rev. Lett. 95, 260501 (2005).
10. M. Born and E. Wolf, Principles of Optics, Pergamon Press, 1993.
11. A. N. de Oliveira, S. P. Walborn and C H Monken, “Implementing the Deutsch algorithm with polarization and
transverse spatial modes,” J. Opt. B: Quantum Semiclass. Opt. 7, 288–292 (2005).
12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88, 257901 (2002).
13. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,”
Phys. Rev. Lett. 96, 113901 (2006).
14. W. Chan, J.P. Torres, and J.H. Eberly, “Entanglement Migration of Biphotons in Spontaneous Parametric Downconversion,” in CLEO/QELS 2006 Technical Digest (Optical Society of America, Long Beach, California, May
2006.
15. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95 173601 (2005).
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Received 11 July 2006; revised 8 September 2006; accepted 12 September 2006
2 October 2006 / Vol. 14, No. 20 / OPTICS EXPRESS 9093
16. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light
Beam,” Phys. Rev. Lett. 81, 4828 (1998).
17. M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46, 175–179 (1999)
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(2004).
23. I. S. Gradshteyn and I. M. Ryzhik, Tables of series, integrals and products, Academic Press, 1980. We make use
of some useful properties of series of Bessel functions in chapters 8-9 about Special functions.
1.
Introduction
Light possesses orbital angular momentum (OAM), which is associated with the amplitude and
phase of its transverse spatial profile [1]. A light beam with an azimuthal phase dependence of
the type exp (il ϕ ), carries an OAM per photon of l h̄. In general, in the paraxial approximation,
light beams can be represented as superpositions of Laguerre-Gaussian (LG) beams, or alternatively, as superposition of spiral harmonics. The weights of the superposition determine the
corresponding angular momentum content of the light beam [2].
The OAM of light is receiving increasing attention as a resource, in classical and quantum
optics, since the OAM exists in an inherently multidimensional space. For instance, information
can be encoded into higher dimensional OAM-alphabets for its use in free space communications systems [3], and in high density optical storage in compact disks [4]. Generally speaking,
the use of the OAM of light might represent a new strategy for optical imaging [5].
In quantum optics, the OAM of single and paired photons is used as a quantum resource
that allows to increase the dimensionality of the working Hilbert space [2, 6], which can be
used to implement new quantum applications. Illustrative examples include the violation of
Bell inequalities with qutrits [7], the implementation of the quantum coin tossing protocol [8],
and the generation of a quantum state in a highly multidimensional state [9].
The Dove prism is a very well known tool in optics. It acts as an image flipper in one transverse dimension, while leaving unchanged the image in the other transverse dimension. This
characteristics, which makes it very useful in certain optical instruments [10], makes the OAM
of a light beam to change. This property has turned Dove prims into a key element in some
recent classical and quantum optics implementations that make use of the OAM of light as a
resource.
A control-NOT gate, which has recently been implemented using polarization and transverse
spatial modes [11], it makes use of a Dove prism located in one of the arms of an interferometer,
where the spatial profile of the light beam (or photon) is properly rotated. Dove prisms are key
elements of an interferometric method for measuring the orbital angular momentum of single
photons [12], as well as of a scheme that allows the measurement of the orbital angular momentum content of a superposition of LG beams [13]. Recently, another interferometric method has
been proposed for measuring the amount of spatial entanglement that exists between certain
entangled paired photons generated in parametric down conversion [14]. A scheme to generate
arbitrary coherent superpositions of OAM states in Bose-Einstein condensates makes use of
Dove prism to change the handedness of light [15].
When a light beam with a well defined OAM per photon of l h̄, i.e., with spatial shape in
cylindrical coordinates at the beam waist Ain = A0 (ρ ) exp (il ϕ ), traverse a Dove prim, it is
generally assumed that the output beam has a well defined OAM per photon of −l h̄, i.e., with
spatial shape Aout = A0 (ρ ) exp (−il ϕ ) exp (−il γ ), where γ /2 is the angle of rotation of the Dove
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prism. The time dependence of the angle of rotation, and therefore the phase shift l γ , makes
possible the observation of the rotational frequency shift of light beams [16]. The rotated Dove
prism can also introduce polarization changes into the light beam [17].
Generally speaking, the polarization and spatial properties of light beams can not be considered separately [18]. For instance, highly focused light beams of fixed linear polarization do
not exist [19]. Notwithstanding, within the paraxial regime, both contributions can be measured
and manipulated separately [20].
In this paper we will show theoretically and experimentally that the OAM transformation rule
l h̄ ⇒ −l h̄ is not valid for highly focused light beams, since Dove prisms inherently introduce
astigmatism, and therefore further OAM changes. Light beams with a well defined value of
the OAM per photon, after traversing the Dove prism, are transformed into a superposition of
states with well defined OAM. The violation of the rule l h̄ ⇒ −l h̄, turns out to be more severe
for highly focused light beams. We will provide a quantitative study of the properties of the
Dove prism, by making use of the geometrical optics properties of the Dove prism, and we will
verify experimentally the validity of our theoretical results in a series of experiments with a
commercially available Dove prism.
a)
input plane
(x1 , y1)
Y
Z
X
output plane
(x2 , y2)
(x’’ , y’’)
(x’ , y’)
iy
oy
?
b)
iy’
X
Y
Z
ix
ix’
ox
L
Fig. 1. Geometrical configuration of a Dove prism. (a) Lateral view (yz−plane) and (b) Top
view (xz−plane). Solid and dashed lines represent the typical path of two optical rays.
2.
ABCD law for a Dove prism
In Fig. 1, we present the basic geometrical configuration of a Dove prism, by showing a typical
optical ray tracing. By making use of the laws of geometrical optics, one finds that the relationship between the output position (x2 ,y2 ) and angle (ox ,oy ) of a ray, and the input position (x1 ,y1 )
and angle (ix ,iy ) are given by (see appendix)
1
h0
L
1−
+
ix
x2 = x1 +
n tan α
n
ox = ix
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y2 = (h0 − y1 ) − h0
1
η
+
n tan α
iy
oy = −iy
(1)
where L is the length of the base of the Dove prism, n is the refractive index of the material, α
is the base angle, and
−1
1
+
h0 = L tan α + sin
n
tan α
−1/2
cos α o
n
cos α 2
h0 sin α
η=
1−
cos−2 α + sin−1
L
n
n
h
−1
cos α i
(2)
In order to derive Eqs. (1), we have made use of the paraxial approximation, so we have
only kept first order terms in the angles of the optical rays. Inspection of Eqs. (1) show that the
propagation of rays through the Dove prism is described by two decoupled ABCD matrices,
one for each transverse dimension. We can analyze ray behaviour in each transverse coordinate
separately and independently, using the appropriate ABCD matrix [21].
We have performed a series of experiments with a commercially available Dove prism (Thorlabs) to check the validity of Eqs. (1) and (2). The Dove prism parameters are L = 63mm,
α = 45o and n = 1.51. We use a CW He-Ne laser (wavelength 633nm). The output beam of
the laser is conveniently shaped so that at the input plane of the Dove prism, the beam width is
w0 ≃ 560µ m. The beam is directed to the Dove prism by means of two mirrors to accurately
control the angle and position of the beam at the input plane. The beam at the output plane of
the system is demagnified to fit on a CCD camera with an appropriate imaging system.
(a)
(b)
displacement (mm)
800
y-plane
x-plane
400
0
-400
-800
-0.03
-0.02
-0.01
0.00
0.01
0.02
angle (rad)
0.03
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
angle (rad)
Fig. 2. Location of the center of the light beam at the output plane. (a) The angle in the
x-plane (ix ) is changed. (b) The angle in the y-plane (iy ) is changed. Dots: experimental
results. Solid line: theoretical results.
Figure 2(a) shows the position of the center of the beam at the output plane when the input
beam, centered at (x1 = 0, y1 = 0), propagates with different angles (ix ) at the input plane of
the Dove prism. Similarly, Fig. 2(b) corresponds to the case of changing the angle iy . The
experimentally measured values agree well with the theoretical predictions as given by Eqs. (1)
and (2).
3.
Ellipticity induced by a Dove prism
From the ABCD matrix derived in the previous section, it is possible to calculate the effect
of the Dove prism on the width and the waist position of an optical beam [21]. The important
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point here is that, apart from the well known image inversion in the y direction, Eqs. (1) also
show that the Dove prism modifies the beam waist position of the beam, (zx and zy , zx = zy ),
differently in both transverse dimensions. The new beam waist positions (z̄x and z̄y ) read
1
L
h0
1−
z̄x = zx +
−
n tan α
n
η
1
z̄y = zy + h0
(3)
+
n tan α
The appearance of two different beam waist positions for each transverse dimension induce
astigmatism in the output beam, and therefore, changes in the OAM content of the output beam
[22]. Generally speaking, any optical device that introduces different optical path lengths for
rays propagating in different transverse planes, should produce changes in the orbital angular
momentum content of the output light beam. For the case of a Dove prism, as considered here,
the difference between ray propagation in the two transverse dimensions is only noticeable for
highly focused beam.
After traversing the Dove prism, the width of the light beam at the output plane is given by
h
i1/2
, where w0 is the width of
the well known formula for LG beams w̄x,y = w0 1 + (z̄x,y /z0 )2
the beam at the input plane and z0 is the corresponding Rayleigh range.
1.2
0.8
180
width (mm)
ellipticity
1.0
0.6
165
150
135
120
105
0
90
180
270
360
rotation angle(degrees)
0.4
0
150
300
450
600
beam width(mm)
Fig. 3. Ellipticity of the output beam at the output plane, after traversing the Dove prism.
Filled circles: Experimental results with the Dove prism. Triangles: experimental results
when the Dove prism is removed. The solid and dashed lines are the theoretical results, as
explained in the text. The dashed line corresponds to the theoretical value of the ellipticity
(e = 1) when the Dove prism is removed. Inset: Filled circles: x-axis; Empty circles: y-axis.
Input beam waist: w0 ≃ 50µ m.
Figure 3 shows the experimentally measured ellipticity at the output plane of an input gaussian beam, after traversing the Dove prim. The beam width at the input plane is changed with a
series of lenses, but keeping the beam waist position at the input plane. For the measurement of
the beam widths in both transverse dimensions, we have used a razor-edge measurement technique, for the two orthogonal directions. For the sake of comparison, we have also measured
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transverse coordinate
transverse coordinate
the ellipticity of the output beam when the Dove prism was removed, which is also shown in
Fig. 3. The theoretical curve shown in Fig. 3 corresponds to e = (w̄x /w̄y )2 . The inset of Fig. 3
shows how the output elliptical beam rotates when the Dove prism rotates.
(a)
(b)
(c)
(d)
300
150
0
−150
−300
300
150
0
−150
−300
−300
−150
0
150
300 −300
transverse coordinate
−150
0
150
300
transverse coordinate
Fig. 4. Spatial light intensity measured at the output plane, with the Dove prism removed
(a) and (c), and with the Dove prism, (b) and (d). (a) and (b): w0 = 560µ m, (c) and (d)
w0 = 50µ m. All dimensions are in µ m.
Figure 4 shows two typical spatial shape measurements obtained at the output plane, when
the Dove prism is present or when it is removed. The input beam is a vortex beam with winding
number m = 2, with two different beam widths. For very large beam widths, (a) and (b), the
astigmatism induced by the Dove prism is not relevant, contrary to the case of highly focused
beams, as shown in (c) and (d).
4.
OAM transformation rule of the Dove prism
The astigmatism induced by the Dove prism will transform the OAM of the output beam differently from the expected transformation l ⇒ −l.
Let us consider that the input beam (at the
input plane) writes Ain (ρ , ϕ ) ∝ ρ l exp −ρ 2 /w20 exp (il ϕ ), which corresponds to a LG beam
with winding number l and radial index p = 0. The OAM of a light beam is related to the azimuthal index l, while it does not change for light beams with different index p. From Eqs. (1),
the normalized beam at the output plane writes
!
2
x
y l
x2
y2
kx
ky2
+i
exp − 2 − 2 exp i
(4)
Aout (ρ , ϕ ) = N
+i
exp (−il ϕ )
w̄x
w̄y
w̄x w̄y
2R̄x
2R̄y
where k is the wavenumber,
N is
h
i the normalization factor and the wavefront radius of curvature
2
reads R̄x,y = z̄x,y 1 + (z0 /z̄x,y ) .
Due to the astigmatism induced by the Dove prism, the output beam is no longer a pure spiral
harmonic with winding number −l, but a superposition of spiral harmonics that can be written
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weight
(a)
(b)
1
1
0.5
0.5
0
−10
−5
0
5
10
0
−10
−5
0
weight
(c)
1
1
0.5
0.5
0
−10
−5
0
5
10
5
10
(d)
5
10
0
−10
mode number
−5
0
mode number
Fig. 5. OAM decomposition of the output beam. (a) Input beam width w0 = 20µ m, winding
number l = 0; (b) w0 = 100µ m, l = 0; (c) w0 = 20µ m, l = 1; (d) w0 = 100µ m, l = 1.
weight of the central mode
1
0.5
0
20
40
60
80
100
beam waist (µm)
Fig. 6. Weight of the central mode of the output beam. Solid line: weight of the mode m = 0,
for an input gaussian beam (l = 0). Dashed line: weight of the m = −1 mode, for an input
l = 1 vortex beam.
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as [2]
Aout (ρ , ϕ ) =
R
1
∑ am (ρ ) exp (imϕ )
(2π )1/2 m
(5)
where amR(ρ ) = 1/(2π )1/2 d ϕ Aout (ρ , ϕ ) exp (−imϕ ). The weight of the m-harmonic is given
by Cm = ρ d ρ |am (ρ ) |2 . We thus obtain [23] that the weights of the OAM superposition {Cm }
that describes the light beam, after traversing the Dove prism, is given by
!#
"
Z
1
1
1
Cm =
+
ρ 2l+1 d ρ exp −ρ 2
w̄2x w̄2y
2l−2 l!w̄x w̄y
2
k l−k
l
1
1
1
1
l
×∑
−
+
i−k
J(l+m)/2−k (s)
(6)
k=0 k
w̄x w̄y
w̄x w̄y
when (l + m)/2 is an integer and Cm = 0 otherwise. In the formula above Jm is the Bessel
function of the first kind and order m, and the parameter s reads
!
1
1
ρ2 1
kρ 2 1
−
+i
−
(7)
s=
4
2 w̄2x w̄2y
R̄x R̄y
Figure 5(a) and (b) shows the OAM decomposition of the output beam for a gaussian input
beam, and Figs. 5(c) and (d) shows the corresponding OAM decomposition for a l = 1 vortex
input beam. In all cases, the OAM decomposition of the output beam is centered at −l.
In Figs. 5(b) and (d), the OAM decomposition of the output beam shows a single line, so in
this case Dove prism transforms the OAM of the light beam from l to −l. For highly focused
light beams, such as it is the case of Figs. 5(a) and (c), the Dove prism transform a pure LG
beam into a superposition of spiral harmonics with different OAM index.
In order to quantify the validity of the rule l ⇒ −l to describe the OAM related behaviour
of the Dove prism, Fig. 6 shows the weight of the central mode, which corresponds to m = 0
for the case of an input gaussian beam, and m = −1 for the case of a l = 1 input vortex beam.
Generally speaking, a Dove prism performs the OAM transformation
l ⇒ {Cm }
(8)
where the decomposition Cm is determined by Eq. (6). For highly focused light beams, the OAM
decomposition shows many modes. For larger beam widths values, the usual transformation
l ⇒ −l holds. From Fig. 6, we notice that, for a given value of the input beam width, the weight
of the central mode of the OAM superposition is smaller for the case of the input vortex beam
than for the gaussian beam.
5.
Conclusions
We have demonstrated theoretically and experimentally that a highly focused light beam with a
well defined value of the OAM per photon is transformed into a OAM superposition state when
traversing a Dove prism, due to the introduction of astigmatism into the light beam propagation.
Dove prisms are being extensively used in many physical settings that make use of the OAM
of light [11, 12, 13, 14, 15]. In view of the results presented here, the use of Dove prisms with
highly focused beams could require the use of some compensating schemes, such as appropriate
combinations of cylindrical lenses.
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6.
Appendix: Derivation of the ABCD matrix for a Dove prism
In this section we will derive Eqs. (1),(2) and (3), making use of the scheme shown in Fig. 1. In
order to do so, one follows the rays trajectories in the figure. This is done in three steps. Firstly,
we propagate the ray from the input plane to the input face of the prism (x1 , y1 ) → (x′ , y′ ).
Secondly, we let the ray traverse the Dove prism (x′ , y′ ) → (x′′ , y′′ ), and finally, we calculate the
ray trajectory from the output face of the prism to the output plane (x′′ , y′′ ) → (x2 , y2 ). The first
and last steps are straightforward free-space propagations, which in our case just means finding
the crossings in the three dimensional space of a straight line with a plane.
On the other hand, the middle step is divided into refraction from air to glass at the input face
of the prism, reflection of the ray at the floor of the prism, and another refraction from glass to
air.
The final result relates the position and angle of the ray at the input plane (x1 , y1 ; ix , iy ), with
those at the output plane (x2 , y2 ; ox , oy ) in the following way
tan(ix ) + tan(α ) tan(α + i′y ) tan(i′x )
,
1 + tan(α ) tan(α + i′y )
!
tan(α ) − tan(iy )
− y1 ,
1 + tan(α ) tan(α + i′y )
x2 = x1 + L
y2 = L
ox = ix ,
oy = −iy .
(9)
In these formulas we use the refraction angles inside the crystal (i′x , i′y ) =
(arcsin(sin(ix )/n), arcsin(sin(π /2 − α − iy )/n), which are shown in Fig. 1. Next, we
perform a Taylor expansion to first order in the angles of these equation, since we consider the
paraxial approximation regime. The result of this approximation are Eqs.(1) and (2), which we
repeat here to ease the following discussion
1
L
h0
x2 = x1 +
1−
+
ix
n tan α
n
ox = ix
h0
1
h0
η
iy
= − y1 −
− h0
+
y2 −
2
2
n tan α
oy = −iy
(10)
One can note a slight variation in the formula for y2 , in order to clarify the following discussion.
A few comments are now in order. First, one notes that, although in the full equations, the
output positions of a ray depend on all input angles ((ix , iy )), in the linearized equations the two
transverse dimensions are completely decoupled. This allows a simplification for the ABCD
law, which otherwise would become a larger matrix [21]. Nevertheless, this simplification is
only valid within the paraxial approximation, i.e. to first order in the incoming angles.
Secondly, Eqs.(9) show that the magnitudes of the angles are not changed in the process.
This is due to the fact that the input and output media are the same (air). The change in sign of
the angle in the vertical direction is due to the reflection of one ray at the floor of the prism.
Finally, we would like to mention the physical meaning of the parameter h0 , which is explicitly written in Eq.(2). It can be easily checked from the equation for y2 , that in the case
of incidence angle parallel to the base of the Dove prism (iy = 0), h0 /2 is exactly the position
where the Dove prim has no effect over the ray (y2 = y1 = h0 /2).
The set of equations (10) can be directly cast into the ABCD matrix form
x2
Ax Bx
x1
=
,
(11)
ox
Cx Dx
ix
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and the corresponding one for the vertical direction. As mentioned above, due to the decoupling
of the dimensions, we have one ABCD matrix for every distinct direction.
In order to use the ABCD matrix to calculate the effect of an optical system to a Gaussian
beam, we have to introduce the complex radius of curvature [21] q = (z − z0 ) − iλ /(π w20 ),
where z is the actual longitudinal position of the beam, z0 the position of the beam waist of the
beam, λ the wavelength of the light and w0 the beam width at the waist position. The beam can
have a different complex radius of curvature for each dimension (qx , qy ). The transformation
through an optical system gives
Ai + Bi qi
,
(12)
q̄i =
Ci + Di qi
with i ∈ {x, y}, for each dimension. We can write it in this simple way, because Eqs.(10) are
decoupled for the two transversal directions.
Acknowledgments
This work was supported by the Grant No. FIS2004-03556 from the Government of Spain; by
the Generalitat de Catalunya, and by the European Commission under the Integrated Project
Qubit Applications (QAP) funded by the IST directorate as Contract No. 015848. GMT acknowledges support from the Government of Spain through a Ramon y Cajal fellowship.
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Characterization of optical beams with
spiral phase interferometry
Roser Juanola-Parramon 1, Noelia Gonzalez 1
and Gabriel Molina-Terriza1,2
1 ICFO-Institut
de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels
(Barcelona), Spain
2 ICREA - Institució Catalana de Recerca i Estudis Avançats, 08010, Barcelona, Spain
[email protected]
Abstract: In this paper we study both theoretically and experimentally a
method to characterize the amplitude and phase of a paraxial optical beam.
The method is based on the spiral phase interferometry technique, recently
proposed. We theoretically analyze how to adapt the original proposal
to deal with the special characteristics of finite optical beams. Finally,
we compare a series of numerical and experimental results to show the
advantages and limitations of our proposal.
© 2008 Optical Society of America
OCIS codes: (070.6110) Fourier optics and signal processing: Spatial filtering; (090.1970)
Holography : Diffractive optics
References and links
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photon,” Phys. Rev. A 74, 053809 (2006).
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quantum state of light at the single-photon level,” Opt. Lett. 30, 3365–3367 (2005).
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(2005).
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Express 13, 689–694 (2005).
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1. Introduction
The phase of a paraxial optical beam is used in many applications, including holography, optical metrology, characterization of the optical properties of materials, etc. [1] More recently,
the transversal phase of an optical beam has acquired great importance in the field of singular optics [2], where phase singularities of the optical field are used in many applications as
micromachining [3], microfluidics [4], data storage [5], new imaging schemes [6, 7] or new astronomical instruments [8]. Also, the transversal spatial structure, both phase and amplitude, of
an optical beam defines its orbital angular momentum content [9, 10] which can be transferred
to material particles using optical tweezers [11], used to directly control the state of atomic
ensembles [12, 13, 14] or in quantum information applications [15].
Nevertheless, the phase of an optical beam is a property which is usually only accessible
using interferometric methods, where a reference beam with well defined amplitude and phase
is used to retrieve the properties of the desired beam. Interferometric methods usually demand
very stable set-ups, which can be difficult to implement in every day applications. As an alternative to those interferometric systems, some self-referenced techniques have been proposed
and implemented [16, 17, 18]. Recently one of those techniques, the so called spiral phase
interferometry (SPI) [17, 18], is gaining interest in the optical community due to its simple implementation. It has already been successfully used in high resolution microscopy applications
[19, 20, 21].
In principle, this technique could be used for characterizing an unknown optical beam, which
could be of interest in the above mentioned applications where the transversal structure of an
optical beam is used to codify information. In particular it could be very interesting in applications where the orbital angular momentum of an optical beam has to be measured. In this article
we expose some of the problems to fulfil this program and a possible way to solve it, by using a
slight variation of the usual SPI. We have experimentally implemented such a system and here
we present a few examples of the results we obtained with complex beams.
2. Spiral phase contrast revisited
The spiral phase contrast method is based on the convolution of a given image with a spiral filter.
To be more specific let us start with an initial paraxial beam with scalar amplitude E in (xin , yin ).
In an experimental set-up, the input field would be optically Fourier transformed with a 2 f
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system and multiplied by a set of phase masks H k=1,2,3 with the following shape:
exp(iθ (x , y )) if x 2 + y 2 > R2
Hk (x , y ) =
e iα k
if x 2 + y 2 ≤ R2
(1)
where θ (x , y ) = arctan yx is the azimuthal angle, R is a typically small radius, which separates
the two regions of the filter and α k = 23π k is a constant phase which is different for every mask
in the set.
After performing another optical Fourier transform, the resulting field is an interference between the two different parts of the beams, traversing the two zones of the mask. As it is nicely
explained in ref. [19], when R tends to zero, the output field is the self-referenced interference
of the input beam with a plane wave, with a well defined phase. Different masks in the set
provide different phases of the plane wave, which in the end allows to reconstruct the initial
beam’s intensity and phase.
This method has delivered nice results in the fields of microscopy [19, 21] and for phase
modulated constant amplitude input beams [17, 18]. Our aim in this paper is to find a way to
extend those results to arbitrary beams. This extension can be problematic in some cases and
we will provide some solutions, which we have successfully experimentally tested. Let us start
by writing the intensity of the output beam as recorded by the CCD camera:
2
(k)
Iout (xout , yout ) ∝ F F {Ein (xin , yin )} Hk (x , y ) =
|A(xout , yout ) + exp(iαk )B(xout , yout )|2
(2)
where (xin , yin ) represent the transversal coordinates of the input plane, (x out , yout ) those of the
output plane (CCD camera), and (x , y ) the coordinates of the plane where the filtering takes
place (after the first optical Fourier transform). For the sake of simplicity we have normalized
the transversal coordinates so that we do not take into account the trivial magnification factors
and inversions due to the set of lenses chosen. F {g} represents the optical Fourier transform
of the function g and the functions A(x out , yout ) and B(xout , yout ) are more easily expressed in
cylindrical coordinates:
A(rout , θout )
=
B(rout , θout )
=
∞
R
R
0
r dr
r dr
2π
0
2π
0
Ẽin (r , θ ) exp(iθ ) exp(irout r cos(θout − θ ))d θ Ẽin (r , θ ) exp(irout r cos(θout − θ ))d θ .
(3)
here, Ẽin (r , θ ) = F {Ein } is the Fourier transform of E in .
The original decoding method [19] is based on the calculation of these two quantities:
Ic
=
1 3 (k) −iαk
∑ Iout e = A∗ (xout , yout )B(xout , yout )
3 k=1
Itot
=
1 3 (k)
∑ Iout = |A(xout , yout )|2 + |B(xout , yout )|2
3 k=1
(4)
Assuming that B(xout , yout ) is sufficiently close to a plane wave, one can retrieve the separate information of A(x out , yout ) and B(xout , yout ). Unfortunately for general optical beams, this
method runs into some problems. First, as already stated in the original paper, there is a limit
on how small we can make R. The smaller R is, the smaller the amplitude B is, which reduces
the amplitude of Ic . The natural limit for R is then given by the noise in the recording apparatus
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and also depends on the shape of the input beam. In typical optical beams, one needs R to be
a significant fraction of the input beam, to be able to overcome the noise in a standard CCD
camera. This invalidates the approximation of B(x, y) as a plane wave. Second, even in the case
that B(x, y) is close to a plane wave, the original algorithm to retrieve B(x, y) does not give a
single solution when the input beams contain points with zero intensity, as would be the case
for beams carrying orbital angular momentum.
Fig. 1. Block diagram of our system. (a) represents the optical processing, OFT mean optical Fourier transform, and (b) represents the numerical post-processing to recover the field,
FFT and IFFT are the fast Fourier transform and its inverse, and sqrt{} is the square root
operation. More details are given in the text.
In order to overcome these problems we have devised a slight variation of the original spiral
phase contrast method. Our implementation can be seen in Fig. 1. To start with, we perform a
pre-processing: we make an image of the Fourier transformed input beam, which allows us to
find a proper point (x R , yR ) where to center our filter. We look for a spot in the beam with a local
maximum of intensity. With this simple pre-processing, we obtain several advantages: First, the
Ic field will be maximized for a fixed radius R. Second, we also avoid zeros of the input beam
which will make Ic close to zero. Finally, the amplitude of the beam within the circle of radius
R in the filter is rather constant which will allow us to make some simplifications to retrieve the
input beam information. In our implementation we performed this step visually, but it can be
easily automatized with proper image processing algorithms. After the spot has been chosen,
we proceed in the following way. First, we obtain the three filtered images as in the original
spiral phase contrast method and we also record an additional image where the filter has been
(0)
completely removed, i.e. we take an image of the intensity I out = |Ein (xout , yout )|2 , see Fig. 1(a).
This image will be identical to the input beam, except for the trivial rescaling and inversion due
to the optical Fourier transforming processes.
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The reconstruction of E in , which is shown in Fig. 1(b) is rather simple and reads:
Φ
Erec
y
= arg{F {F {Ic (xout , yout )}e−i arctan x }} − (xR x + yR y)
0 exp(iΦ)
=
Iout
(5)
where the errors in the reconstruction of the intensity are only due to the noise of the imaging
and recording systems. On the other hand, the reconstruction of the phase provides a good
approximation, as we will show below. When using the proper spot to place our filter, we
assure that the function B(x out , yout ) is very close to diffraction of a circular aperture:
J1 R (x − δx )2 + (y − δy)2
B(x, y) exp(i(xR x + yR y))
(6)
R (x − δx )2 + (y − δy)2
except for trivial rescaling due to the optical Fourier transform. To obtain this expression one
has to assume a flat amplitude across the circle R and a linear change in the phase. These approximations are based on the fact that we have chosen the right amplitude spot in the beam and
that the radius R is small enough so that we can approximate any changes in the phase to first
order. This change in the phase is responsible of the displacement of the diffraction function:
(δx , δy ). As we will see, this typically small displacement of the function does not affect our
Fig. 2. Numerical example of the reconstruction system. (a) Intensity of the input field. (b)
Phase of the input field. (c) Amplitude of the Fourier transform of the input beam. The
white dot indicates the position of the center of the filter. (d) One of the filters used, the
white dot indicates the center of the filter. (e), (f), (g) Output intensities of the system,
corresponding to the different filters used. (h) Intensity of the recovered field. (i) Phase of
the recovered field.
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reconstruction. Finally, the added phase (x R x + yR y) in B(x, y) is due to the displacement of the
filter and is corrected in the reconstruction Eq. (5). Note that even under this approximation,
the phase of B(x, y) presents some radial phase singularities, i.e. there are π phase jumps at
some radial positions, given by the Airy function. Although these singularities could be properly taken into account, usually we do not have to deal with them as they are out of the area of
interest. Finally, taking into account Eq. (4) and (5) we observe that the reconstructed phase is
actually the result of applying a mean filter of size R to the Fourier transformed beam.
In Fig. 2 we present a numerical example of how the reconstruction works. Our input field
consists in a beam with some phase singularities. The order of the phase singularities can be
easily identified in Fig. 2(b), where we observe that the beam presents one single charged
vortex and another second order vortex (where the phase twists twice around the singularity).
Both vortices are separated by some distance. Figure 2(c) is the Fourier transform of this beam
and the white spot represents the intensity maximum where we position the filters (one of the
filters is shown in Fig. 2(d)). The 3 different images obtained after the filtering and the last
Fourier transform are displayed in the panels (e), (f) and (g). Finally, in the last panels of the
figure, (h) and (i), we present the numerically reconstructed beam with Eqs. (5), which are to
be compared with panels (a) and (b).
3. Experimental set-up
Fig. 3. Sketch of the experimental setup. A computer generated phase hologram is illuminated with a collimated diode laser light to produce a Laguerre-Gaussian-like beam in the
object plane, using the lens L1 and an iris (to select the first order of diffraction). Then L2
makes the Fourier transform of the object and puts it on the SLM surface, where the filters are displayed. After the filtering, we make the Fourier transform again with L3 and we
rescale the image to fit the CCD chip with the imaging system 1. With the imaging system
2 we make an image of the SLM on the CCD to find a proper point where to center the
filters.
The experimental setup of our system is sketched in Fig. 3. Our source of light was a 810nm
diode laser which was coupled to a single mode fiber to obtain a pure Gaussian spatial mode.
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The light from the output of the optical fiber was collimated and illuminated a computer generated phase hologram. As the holograms we use are custom made, we can produce the appropriate hologram to modulate the optical beam in the desired way. In the example in Fig. 3 we
present a simple fork-like dislocation [22]. This kind of holograms are well known to produce
superpositions of Laguerre-Gaussian beams [23, 24]. A numerical example of the intensity
shape of the resulting beam can be seen in Fig. 3 in the object plane. In this case, the hologram
dislocation is shifted from the center of the illuminating Gaussian beam. Then the output from
the hologram is a superposition of a Gaussian beam and a Laguerre-Gaussian beam [23]. We
use then lens L1 and an iris to select the first order of diffraction from the hologram.
With lens L2 we Fourier transform the object onto the surface of an spatial light modulator
(SLM). The SLM allows us to display on real time the different filters needed for the protocol.
Our SLM was set to work in phase mode (only affecting the phase of the incoming beam). L1
and L2 were chosen to magnify the beam to take advantage of the SLM surface.
Finally, with two flip mirrors (dashed lines in the figure) we could choose to direct the light
from the SLM to a CCD camera either through an imaging system (imaging system 2) or with
a Fourier transforming system (lens L3) rescaled (with lens L4) so that the resulting image fits
the CCD chip.
First, we scanned with the imaging set-up the shape of the beam in the SLM, looking for
maxima of intensity. Once we found a suitable zone, we switched to the Fourier set-up and took
the four images needed for the protocol. The first one was taken with a blank filter in the SLM,
thus we just retrieved the intensity pattern of the object plane. The three other images were
taken with three different filters in the SLM as explained previously. Each filter consisted on a
fork-like pattern (similar to that in the hologram of Fig. 3), but the position of the dislocation
was covered with a circle of variable radius (depending on the visibility conditions). Every filter
had a different relative phase in the circle.
Fig. 4. Characterization of a Gaussian beam with four embedded phase singularities. Upper
row, simulation of the expected field. Lower row, experimental reconstruction. (a) and (c)
Intensity pattern with the four zeroes associated with the four phase dislocations. (b) and
(d) Phase pattern.
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In Fig. 4 we present an example of the experimental results we obtained with our system.
The initial vortex beam traversed a hologram consisting on four phase dislocations, forming a
square. In the upper row of the figure we present, for the sake of comparison, a numerical calculation of the beam we expected. In the calculation we used our knowledge of both the incoming
beam and the hologram we used. Observe that the reconstruction phase follows remarkably
well the expected features. The reconstructed phase is rather noisy far from the center of the
beam, where the method is prone to give worse results as the noise of the camera is of the same
order as the recovered signal. Note also that from the phase measurements we can observe that
the beam has a small divergence, which can be observed from the curvature of the iso-phase
lines. This is an indication that the laser beam was not perfectly collimated in the object plane.
Finally, from the intensity measurements a small ellipticity in the beam can be observed. This
is probably due to some inhomogeneities of the SLM and is in agreement with other series of
measurements not shown here.
Fig. 5. Characterization of a Gaussian beam with a phase jump. Upper row, simulation of
the expected field using a 0.8π phase jump. Lower row, experimental reconstruction. (a)
and (c) Intensity pattern. (b) and (d) Phase pattern.
Another example of the reconstruction process can be found in Fig. 5. Here, the hologram
that we used consisted in a simple phase jump. The reconstruction shows that the jump was
actually of approximately 0.8π , a value consistent with the design of our hologram. Note again
the small curvature of the beam as in the previous case.
In conclusion, we have presented here a method to measure the amplitude and phase of
Laguerre-Gaussian-like beams. This method is based on a small variation of the spiral phase
interferometry technique. Our method avoids some technical problems that can be found in the
reconstruction of finite sized beams and beams with phase singularities. We have shown a few
examples of the use of our technique for the characterization of complex beams.
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31 March 2008 / Vol. 16, No. 7 / OPTICS EXPRESS 4478
PHYSICAL REVIEW A 80, 043804 共2009兲
Properties of the spatial Wigner function of entangled photon pairs
Noelia Gonzalez,1,* Gabriel Molina-Terriza,1,2 and Juan P. Torres1,3
1
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, Castelldefels, 08860 Barcelona, Spain
2
ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain
3
Departament of Signal Theory and Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain
共Received 18 May 2009; published 5 October 2009兲
In this paper we analyze some important characteristics of the spatial Wigner function of entangled photon
pairs. We show that the Wigner description of quantum states that live in the infinite-dimensional spacemomentum degree of freedom proves to be particularly useful. We propose an experimental configuration that
can be used to retrieve the Wigner function of paired photons entangled in the spatial degree of freedom. In
particular, it allows the full characterization of the paired photons emitted from a spontaneous parametric
down-conversion source.
DOI: 10.1103/PhysRevA.80.043804
PACS number共s兲: 42.50.Dv
I. INTRODUCTION
The efficient and reliable generation, control and detection of specific states of quantum systems is at the roots of
the field of quantum information 关1兴. One important property
of quantum systems composed of several parties is that they
can become entangled, i.e., the quantum state of the whole
system cannot be described by a set of separable quantum
states for each party of the ensemble. Entangled states have
found many applications in quantum computing, quantum
teleportation, and are at the core of certain cryptographic
protocols 关1–3兴.
Quantum systems which present entanglement in continuous variables have been receiving increasing attention from
the scientific community due to their potentiality. The most
widely explored continuous variable system is based on the
quadratures of the electrical field of photons. The transverse
spatial shape of photons 共i.e., transverse position and momentum degree of freedom兲 is another of such continuous
variable systems. It has already been shown that pairs of
photons can be entangled in their spatial properties 关4兴. Up to
now, the entanglement in transverse momentum has been
used to test some quantum protocols in a finite number of
dimensions 关5–8兴. On the other hand, due to its ease of control, the spatial degree of photons is prone to be used for
testing some of the new physical predictions that continuous
variables quantum systems can offer 关9兴.
One suitable representation of the quantum state of a system is the Wigner function 关10,11兴. The Wigner-function formulism is fully equivalent to the density matrix representation, thus providing all the accessible information of the
system to the observer. Wigner functions are especially useful for describing continuous variables. In particular, it has
been used for describing the quadratures of the electrical
field with coherent and squeezed states 关12兴 or single photon
states 关13兴.
Here we analyze the structure of the phase space of entangled photons in the transverse momentum degree of freedom using the Wigner-function formulism. The description
*[email protected]
1050-2947/2009/80共4兲/043804共8兲
of the spatial transverse modes of an optical field in terms of
the Wigner function can be found in 关14,15兴. In this paper we
present some results which can be directly applied to describe the transverse spatial shape of bipartite entangled photons generated in spontaneous parametric down-conversion
共SPDC兲 processes. Thus, in Sec. II we revise the essential
properties of the Wigner function that we will use along the
paper and in Sec. III we describe the typical quantum state of
the momentum of photons generated in a SPDC process. For
the sake of clarity, we analyze the spatial Wigner function in
three different situations: First, we consider the case where
one of the photons is projected onto a specific pure state,
obtaining the Wigner function of the remaining photon 共Sec.
IV兲. We also consider the case when one simply disregard
one of the photons, the other being, in principle, in a mixed
state 共Sec. V兲. Finally we propose an experimental scheme
for measuring the spatial properties of the whole entangled
state, and analyze the properties of the Wigner function of
the whole system 共Sec. VI兲.
II. WIGNER FUNCTION
The Wigner function associated with a quantum state 兩⌿典
can be expressed as the expectation value of the operator ⌸̂rq
关16兴,
W共r,q兲 =
1
具⌿兩⌸̂rq兩⌿典,
␲2
共1兲
where ⌸̂rq is defined as
⌸̂rq =
=
冕
冕
dr0 exp共− 2iq · r0兲兩r − r0典具r + r0兩
dq0 exp共− 2ir · q0兲兩q + q0典具q − q0兩.
共2兲
The operator ⌸̂rq performs a reflection about the phase-space
point 共r , q兲 and is thus the parity operator about that point. In
the case of a “two particle” state the Wigner function can be
expressed as 关17兴
043804-1
©2009 The American Physical Society
PHYSICAL REVIEW A 80, 043804 共2009兲
GONZALEZ, MOLINA-TERRIZA, AND TORRES
W共r1,q1,r2,q2兲 =
1
具⌿1,2兩⌸̂1,2兩⌿1,2典,
␲4
共3兲
A=
where ⌸̂1,2 is the product of the two displaced operators,
each one acting over one particle,
⌸̂1,2 = ⌸̂r1,q1 丢 ⌸̂r2,q2 .
共4兲
The Wigner function corresponding to a generally mixed
state with density matrix ␳ = 兺␭i兩⌿i典具⌿i兩 can be written,
making use of Eq. 共3兲, as W共r , q兲 = Tr共␳⌸̂rq兲. An interesting
property of the Wigner function, easily obtained from Eqs.
共1兲 and 共2兲, is that the Wigner function of a probabilistic
mixtures of density matrices, i.e., ␳ = 兺␭i␳i is W = 兺␭iWi.
Notwithstanding, this is not the case for a linear superposition of pure states, whose Wigner function does not result in
a linear superposition of the corresponding Wigner functions.
III. SOURCE OF SPATIALLY ENTANGLED PHOTONS
In the following we will consider the important case of a
two-photon state, whose quantum state can be written as
兩⌿典 =
冕
dqsdqi⌿共qs,qi兲as†共qs兲a†i 共qi兲兩0典s兩0典i ,
共5兲
where qs = 共qs,x , qs,y兲 and qi = 共qi,x , qi,y兲 are the corresponding
transverse momenta of the signal and idler photons, respectively. as†共qs兲 is the creation operator of a signal photon with
transverse momentum qs, and similarly for the idler photon.
In the rest of the paper we will confine ourselves to the
case where the probability amplitude ⌿ writes
⌿共qs,qi兲 =
兩AB兩1/2
exp共− A兩qs + qi兩2 − B兩qs − qi兩2兲,
␲
共6兲
where A and B are two possibly complex constants that will
allow us to analyze different types of momentum correlations
among the photons. The quantum state given by Eq. 共6兲 is
normalized, so that 兰dqs , dqi兩⌿共qs , qi兲兩2 = 1. The type of
states described by Eq. 共6兲 is ubiquitous when describing
quantum systems of continuous variables. In the case of momentum correlated photons, the state given by Eq. 共6兲 is a
very good approximation for describing paired photons entangled in the momentum degree of freedom 关18,19兴.
In particular, this kind of states can be produced when a
second-order nonlinear crystal is illuminated by a quasimonochromatic pump beam in a Gaussian mode in order to
produce frequency downconverted waves. The downconverted waves should be generated in a collinear configuration
共all interacting waves propagate along the same direction兲,
and the Poynting vector walkoff should be negligible. One
way of achieving such conditions is the use of noncritical
type-II quasi-phase-matched nonlinear crystals. If such a
crystal geometry is chosen, and under the approximation that
the refractive indices of pump, signal and idler are nearly
equal, Eq. 共6兲 is a good approximation to the state of the
photons at the output face of the crystal. The values obtained
of the two constants, A and B, are 关19兴
冉
冊 冉
冊
w2p
1 L
1
␴0
+i
−
,
4 k0p 1 + ␴20/w4p
4 1 + w4p/␴20
B=
L
␣L
,
0 +i
4k p 4k0p
共7兲
where L is the length of the nonlinear crystal, ␣ is a fitting
constant to approximate the phase matching functions sinc
function by a Gaussian function 共in our case we use ␣
= 0.455兲, w p and ␴0 are the pump beam width and radius of
curvature considered at the center of the crystal, respectively,
k0p = ␻ pn p / c, and ␻ p and n p are the corresponding angular
frequency and refractive index.
From Eq. 共6兲 it can be readily checked that the momentum correlations in each of the two transverse coordinates are
completely independent, i.e., there is no cross-correlation between direction x and direction y. Therefore, in the rest of the
paper we will drop all vector quantities, and focus on just
one of the transverse dimensions of the photons.
IV. REMOTE PREPARATION OF A PURE STATE
The two-photon state described by Eq. 共6兲 is pure. After
projecting the idler photon in a pure state, the signal photon
will also remain in a pure state. The specific spatial shape of
the signal photon will depend on 共a兲 the two-photon momentum correlations of signal and idler and 共b兲 the spatial shape
of the mode onto which the idler photon is projected. In this
sense, we talk about remote preparation of pure states. Theoretically, whenever Eq. 共6兲 represents a nonseparable state,
one could achieve any remote preparation of a given spatial
state. This is due to the fact that the two-photon state admits
a Schmidt decomposition onto an infinite series of spatial
modes, each one of them with a finite, but ever decreasing
amplitude. However, in any experimental implementation the
limits of control would be given by the precision in the
preparation of states, the noise in the generation of the photons and the fundamental limits given by the paraxial approximation.
After projecting the idler photon, the Wigner function of
the signal photon can be expressed as
W⌽,s共xs,qs兲 =
1
具⌿s,i兩P⌽,i 丢 ⌸̂xs,qs兩⌿s,i典,
␲2
共8兲
where P⌽,i = 兩⌽典具⌽兩i projects the idler photon onto the state
兩⌽典 = 兰dq⌽共q兲a†i 共q兲兩0典i. We will exemplify this remote preparation procedure with two different cases: the projection onto
a Gaussian mode, and projections onto coherent superpositions of Gaussian and Hermite-Gaussian 共HG兲 modes.
A. Projection onto a Gaussian state
We describe the projection of the idler photon into a
Gaussian state by the projector PG = 兩G典具G兩 where 兩G典, in the
transverse momentum space, is given by
兩G典 = NG
冕
冉 冊
dq exp −
1 q2
兩q典,
4 ␮x
共9兲
NG is a normalization constant and ␮x is the complex beam
width in real space given by
043804-2
PHYSICAL REVIEW A 80, 043804 共2009兲
PROPERTIES OF THE SPATIAL WIGNER FUNCTION OF…
−3
x 10
(a)
(b)
(a)
0.33
−0.05
q(µm−1)
q(µm−1)
−5
0
5
−3
x 10
(c)
(d)
0
0
5
−200
0.33
0.37
−5
0.05
0
(b)
−3
x 10
0
x(µm)
200
−2000
0
x(µm)
2000
0
FIG. 2. 共Color online兲 Effect of the pump beam on the remote
preparation of a Gaussian state 共a兲no curvature of the pump beam
共R p → ⬁兲, wp = 38.23 ␮m and L = 50 mm satisfying A = B; 共b兲
R p = 2m, w p = 1 mm, no curvature of the idler mode 共Ri → ⬁兲,
wi = 0.5 mm, L = 5 mm.
q(µm−1)
−5
0
5
−2000
0
2000−2000
0
x(µm)
2000
0
x(µm)
FIG. 1. 共Color online兲 Examples of remote preparation of a
single photon in a spatial Gaussian state. The pump is a Gaussian
mode, and the idler is projected onto a Gaussian state. 共a兲 and 共b兲
show two Wigner functions with no curvature of the idler mode
共Ri → ⬁兲, and different values of the beam width of the idler photon
wi: 共a兲 wi = 0.4 mm, 共b兲 wi = 1 mm. 共c兲 and 共d兲 correspond to the
same beam width of idler photon, wi = 1 mm, but different values of
the radius of curvature, 共c兲 Ri = 1 m, 共d兲 Ri = 200 m. In all cases:
beam width of the pump beam w p = 1 mm; no curvature of the
pump beam 共R p → ⬁兲; Length of the nonlinear crystal L = 5 mm.
␮x =
1
w20
+i
k
.
2R
冉
冊
1
共xs − hIqs兲2
exp共− hRqs2兲exp −
, 共11兲
␲
hR
where hG = 共hR + ihI兲 / 2 is defined as
hG = 共A + B兲 −
共B − A兲2
B + A + 1/共4␮ⴱx 兲
B. Projection onto superpositions of Hermite-Gaussian
and Gaussian states
共10兲
w0 is the beam waist, k is the longitudinal idler wave vector
and R is the radius of curvature of the beam. Making use of
Eq. 共8兲, we find that the Wigner function for the signal photon reads
WG,s共xs,qs兲 =
infinitive兲. Also, note that the two transverse widths of the
elliptical Wigner function, in the qs and 共xs − hIqs兲 directions,
are inversely related as shown in Eq. 共11兲. We will see below
that this only happens when the represented state is pure and
can be clearly observed by comparing Figs. 1共a兲 and 1共b兲.
Also, the idler Gaussian width only enters Eq. 共11兲
through hG. From Eq. 共12兲 it is readily observed that when
A = B, the spatial shape of the signal photon is independent of
the coincident idler projection. This could have already been
anticipated since Eq. 共6兲 represents a separable state whenever A = B 关19兴. In Fig. 2 we plot two cases with different
pumps but fixed idler projections. In particular, Fig. 2共a兲
shows the especial case when A = B.
.
共12兲
Since the biphoton function given by Eq. 共6兲 describes perfect correlations in orbital angular momentum 共OAM兲 between the signal and idler photons 关ms + mi = 0, with ms,i being the OAM index of the signal 共idler兲 beams兴 关20兴, Eq. 共11兲
correspond to the Wigner function of a pure Gaussian state.
In Fig. 1 we show a few examples of the remote preparation of Gaussian states as given by Eq. 共11兲 and 共12兲. Note
the different features of the Wigner function. For example,
tilting of the Wigner function appears whenever the state
presents curvature, i.e., when either A, B, or ␮x have nonvanishing imaginary parts. In Figs. 1共a兲 and 1共b兲 the Wigner
function it is not tilted because the chosen parameters make
A, B, and ␮x purely real 共L small and Ri and R p tending to
Now we consider a multimode situation, i.e., the idler
photon is projected onto a coherent superposition of two
modes: a first order Hermite-Gaussian 共HG兲 mode and a
Gaussian mode. The idler photon projector into a coherent
superposition can be written as PHG = 兩HG典具HG兩, where
兩HG典 = NHG
冕
冉 冊
dq共2q1 + iq兲exp −
1 q2
兩q典,
4 ␮x
共13兲
where q1 = x1␮x gives the relative amplitude between the pure
Gaussian and the pure Hermite-Gaussian modes being x1 the
position of the null appearing in the field in real space. When
␮x is purely real, then q1 = x1 / w20. The Wigner function corresponding to the quantum state of the signal, after projecting the idler onto the mode given by Eq. 共13兲 reads
WHG,s共xs,qs兲 = NHGCH,s共xs,qs兲WG,s共xs,qs兲,
共14兲
where
CH,s共xs,qs兲 = 兩g0兩2qs2 +
−4
兩g0兩2
hR2
共hIqs + xs兲2 −
兩g0兩2
− 4Im共qⴱ1g0兲qs
2hR
Re共qⴱ1g0兲
共hIqs + xs兲 + 4兩q1兩2 ,
hR
WG,s is defined by Eq. 共11兲 and g0 writes
043804-3
共15兲
PHYSICAL REVIEW A 80, 043804 共2009兲
GONZALEZ, MOLINA-TERRIZA, AND TORRES
−3
(a)
(b)
0.29
Ws共xs,qs兲 =
−5
−1
q(µm )
x 10
1
␲
冑
HR − G
exp兵− 2共HR − G兲qs2其
HR + G
再
⫻exp
0
5
− 关xs − 2HIqs兴2
2共HR + G兲
共18兲
where H and G are defined as
−0.34
−3
x 10
(c)
(d)
H = HR + iHI = 共A + B兲 −
0.48
共A − B兲2
,
2Re共A + B兲
共19兲
−5
−1
q(µm )
冎
0
G=
兩A − B兩2
.
2Re共A + B兲
共20兲
5
−2000
0
2000 −2000
x(µm)
0
2000
−2.9
x(µm)
FIG. 3. 共Color online兲 Examples of remote preparation of a
single photon in a spatial superposition of a Gaussian and a Hermite
Gaussian state for different values of q1. The pump is a Gaussian
mode, whereas the idler photon is projected onto a given superposition of the states. 共a兲 and 共b兲 show two cases where q1 is purely
real, whereas in 共c兲 and 共d兲 q1 is purely imaginary. 共a兲 q1 = 0, 共b兲
q1 = 2 ⫻ 10−4 ␮m, 共c兲 q1 = i10−8 ␮m, 共d兲 q1 = i2 ⫻ 10−8 ␮m. In all
cases: Beam width of the idler and pump: wi = w p = 1 mm; curvature
of the idler photon and the pump beam Ri = R p = 100 m; Crystal
length L = 5 mm.
g0 =
B−A
B + A + 1/共4␮ⴱx 兲
.
V. WIGNER FUNCTION OF THE SIGNAL PHOTON
AND THE AMOUNT OF SPATIAL ENTANGLEMENT
In this section we consider the spatial properties of the
signal photon alone, i.e., regardless of the state of the idler
photon. The Wigner function of the signal photon alone can
be calculated from the two-photon state given by Eq. 共6兲
using
1
Tr关具⌿s,i兩Îi 丢 ⌸̂xs,qs兩⌿s,i典兴,
␲2
Px =
共17兲
which corresponds to calculating the Wigner function of the
signal photon alone, applying the identity and tracing out the
idler photon. The Wigner function obtained reads
冉
HR − G
HR + G
冊
1/2
.
共21兲
From Eqs. 共18兲 and 共21兲 it can be seen than the purity of
the signal photon is directly related to the product of the two
transverse widths of the elliptical Wigner function, in the qs
and 共xs − 2HIqs兲 directions, respectively, satisfying the constrain
共16兲
Notice again that when A = B, g0 = 0 and two-photon state is
separable. In this situation the Wigner function of the signal
photon corresponds to a Gaussian one. When the two photons are in a separable state, if we set q1 = 0, i.e., we project
the idler into a pure Hermite-Gaussian mode, the probability
for a signal-idler coincidence is exactly zero 关20兴. Some examples of the shapes described by Eq. 共14兲 are shown in Fig.
3. Note that the superposition of a Gaussian and a HermiteGaussian state does not result on the sum of their respective
Wigner functions.
W共xs,qs兲 =
Note that in this case, the purity of the quantum state that
describes the signal photon, which can be easily calculated
once the Wigner function has been measured, can be used to
determine the degree of entanglement of the initial twophoton state. The purity of the signal photon can be calculated as P = P2x , where Px = 2␲兰dxsdqsWs共xs , qs兲2 and reads
Px = ⌬qs⌬共xs−2HIqs兲 ⱕ 1.
共22兲
Some comments are in order: first, notice again that if A
= B, G = 0 and the two-photon state is separable. In this case
the Wigner function of the signal photon corresponds to the
Wigner function of a Gaussian state with the same conditions, so that the state of the signal photon is pure. Note that
in this situation the two transverse widths of the Wigner
function are inversely related and as it is expected the purity
of the signal estate given by Eq. 共22兲 is Px = 1. Finally, we
can quantify the degree of entanglement of the initial state
with the von Neumann entropy Sx, which in this case depends only with the purity of the traced state in the following
way 关21,22兴:
Sx共Px兲 =
冉 冊 冉 冊
1 − Px
1 + Px
2Px
ln
− ln
.
2Px
1 − Px
1 + Px
共23兲
Some numerical examples of the Wigner function given
by Eq. 共18兲 are shown in Figs. 4共a兲 and 4共c兲. For Fig. 4共c兲 we
have chosen w p and L to satisfy the condition A = B. As it is
expected, in this case the two-photon state, which is separable, gives the Wigner function of a Gaussian state with
equal conditions. It is clear if we compare Fig. 4共c兲 with Fig.
2共a兲 which shows the same Wigner function. Also, note that
comparing Fig. 4共a兲 with Fig. 1共b兲 it is clearly seen how the
purity of the former case is less than one.
043804-4
PHYSICAL REVIEW A 80, 043804 共2009兲
PROPERTIES OF THE SPATIAL WIGNER FUNCTION OF…
(a)
(b)
0.0077
q(µm−1)
−0.2
Coincidence
Counting
0
Path 1
BS
0.2
−2000
0
2000−2000
−3
(c)
q(µm−1)
−0.05
0
x 10
2000
(d)
0
Path 2
Detector 1
0.32
−5
0
M0
MS
x
0
Pump
PPKTP
Gaussian
projection
signal
PBS
HG
projection
Idler
5
0.05
−200
0
x(µm)
200 −2000
Detector 2
0
2000
Without
projection
0
x(µm)
FIG. 4. 共Color online兲 Examples of the Wigner function of the
single photon alone, i.e., without considering the idler photon. The
pump is a Gaussian beam. 共a兲 and 共c兲 show two cases where the
signal photon is analyzed at the output face of the nonlinear crystal.
In 共b兲 and 共d兲 we have considered the propagation of the signal
photon using a lens with focal length f mm located L1 after the
output face of the nonlinear crystal, and at L2 from the entrance of
the interferometer. For 共a兲 and 共b兲 w p = 1 mm and L = 5 mm. For 共c兲
and 共d兲 w p = 38.23 ␮m and L = 50 mm 共satisfying the condition A
= B. In 共b兲 f = 150 mm, L1 = 300 mm and L2 = 300 mm, in 共d兲 f
= 200 mm, L1 = 175 mm and L2 = 200 mm. No curvature of the
pump beam: R p → ⬁ for all cases.
VI. EXPERIMENTAL PROPOSALS FOR MEASURING
THE WIGNER FUNCTION
A. Measurement of the Wigner function of the signal photon
The Wigner function of the signal photon described in the
previous sections can be measured experimentally using an
extension of the method proposed in 关23兴. Figure 5 shows the
scheme proposed to measure the Wigner function of the signal from the entangled two-photon pair. The Wigner function
of the signal photon is measured by means of a threedimensional 共3D兲 Sagnac interferometer. The detailed description of this device can be found in 关23兴. At the output of
the 3D Sagnac interferometer, the two beams which counterpropagate, produce an interference pattern which is then sent
to a detector. The role of the out of the plane reflections
共round mirror in the figure兲 is to spatially invert one of the
interfering beams with respect to the other, in order to produce the parity operation described in Eq. 共2兲. On the other
hand, the input beam is properly displaced and tilted to complete the operator in Eq. 共2兲. Under these experimental conditions, the coincidences rates from the both output ports,
signal and idler, will have the following shape:
Rc ⬀ 关Is1 + Is2 + 2␥W̃s共xs,qs兲兴Ii ,
共24兲
where Ii is the detected idler photon flux, Isn = ␣nIs, with n
= 1 , 2, are proportional to the counts corresponding to the
FIG. 5. 共Color online兲 Sketch of the setup proposed for the
retrieval of the Wigner function of a remotely prepared photon.
Pairs of momentum entangled photons are generated from a nonlinear crystal illuminated by a quasimonochromatic Gaussian pump
beam in a collinear configuration. The signal and idler photon follow different paths after traversing a PBS. The idler photon is detected by Detector2 after been collected using the appropriate optical elements depending on the remote preparation that we want to
perform. The signal photon is detected by Detector1 after passing
through a Sagnac interferometer, and collected with a multimode
fiber. Using coincident logical detection, we measure the Wigner
function of the signal photon.
two counterpropagating directions of the signal photon in the
interferometer. Here, Is is the photon flux rate of the incoming signal beam and ␣1 = R2 共␣2 = T2兲 depends on the reflectivity, R 共transmissivity T兲 of the beam-splitter. On the other
hand, ␥ = RT and W̃s共xs , qs兲 is proportional to the Wigner
function of the signal photon. Therefore, the proper Wigner
function can be extracted from the measurement of the coincidences of the two outputs, after subtracting the background
and properly normalizing the result.
First, let us describe the basic setup for measuring the
quantum state of the signal once the idler has been projected
onto a Gaussian state 共Sec. IV A兲. In this case, we collect the
idler photon with a single mode fiber 共SMF兲 and detect it,
thus effectively projecting the idler onto a Gaussian state. We
record the coincident signal photons after it has traversed the
interferometer. Finally, after subtracting the coincidences
background, we obtain an interference pattern which is proportional to the Wigner function, as given by Eq. 共11兲. Note
that the value of ␮x in Eq. 共9兲 is the one of the idler beam at
the output face of the crystal, thus one has to take into account the propagation from the fiber to the crystal.
The projection in a superposition of Gaussian and
Hermite-Gaussian states 关Sec. IV B兴 can be done in a similar
way, but now introducing a mode transformation of the idler
spatial profile before the projection operation implemented
by the SMF. The mode transformation together with the projection onto a Gaussian state can be seen as a projection onto
a HG state. Again, the proper values of q1 and ␮x in Eq. 共13兲
043804-5
PHYSICAL REVIEW A 80, 043804 共2009兲
GONZALEZ, MOLINA-TERRIZA, AND TORRES
are those taken at the output face of the crystal. Although the
propagation does not change the weight of the modes in the
superposition, it might change the relative phases of the
complex amplitude.
Finally, the measurement of the Wigner function of the
signal photon alone, without taking into account the idler
photon, can be done collecting the idler photon with a multimode fiber 共MMF兲 in order to avoid spatial filtering. In this
case, the propagation of the idler mode from the crystal to
the detector is not relevant, as the idler state is traced out.
In order to properly compare the results obtained with
these experimental setups, with our previous theoretical results, an important element should be considered: the effects
on the Wigner function of the propagation of the fields from
the output face of the nonlinear crystal to the detectors. In
the situations described so far, we have shown the Wigner
function at the output face of the crystal. Let us now consider
a more realistic situation, in which both photons propagate
from the output face of the crystal to the interferometer,
maybe traversing a lens in their way. As we have already
mentioned, the idler propagation can be properly accounted
for by using the correct values of q1 and ␮x. Thus, we are left
with only the propagation of the signal. In the case of the
remote preparation of the signal photon 共Sec. IV兲, the effect
of the propagation is trivial. The overall shape of the Wigner
function would still be described by Eqs. 共12兲 and 共15兲, but
with different values for the characterizing parameters.
The most interesting case is the one described in Sec. V,
where the state of the idler photon is disregarded. If we use a
lens of focal length f located a distance L1 after the output
face of the nonlinear crystal, and located at a distance L2
from the Sagnac interferometer 共M s兲, the state of the signal
photon at the output face of the nonlinear crystal is transformed as
⬘ 典 = Ûs 丢 Îi兩⌿s,i典,
兩⌿s,i
共25兲
Us is an unitary transformation over the signal photon, given
by the product of the transformations due to the free propagation and the effect of the lenses,
共26兲
Ûs = ÛL2Û f ÛL1 ,
ÛLn and Û f , in transverse momentum space, have the form
冉
具q兩ÛLn兩q⬘典 = exp − i
具q兩Û f 兩q⬘典 =
1
2␲
冕
冊
Ln 2
q ␦共q − q⬘兲n = 1,2,
2k
冉
dx exp关i共q − q⬘兲x兴exp − i
共27兲
冊
k 2
x .
2f
共28兲
We take into account this more realistic scenario to calculate
the Wigner function of the signal photon alone. Making use
of the new 兩⌿s,i
⬘ 典 state in Eq. 共17兲, the new expression for the
Wigner function is
Ws⬘共qs,xs兲 =
1
冑␲
冑
HR⬘ − G⬘
HI⬘ + G⬘
再
⫻exp −
exp关− 2共HR⬘ − G⬘兲qs2兴
关xs − 2HI⬘qs兴2
2共HR⬘ + G⬘兲
冎
共29兲
,
where
G⬘ =
H⬘ = HR⬘ + iHI⬘ =
冉 冊
f
2k
冉 冊冉
f
2k
2
2
兩G1兩2
共2HR1 兲
共30兲
,
冊
1
共G1兲2
+
− iT2 ,
A + B − iT1 2HR1
共31兲
G1 = GR1 + iGI1 =
B−A
,
A + B − iT1
H1 = HR1 + iHI1 = A + B −
共B − A兲2
,
A + B − iT1
共32兲
共33兲
T1 =
f − L1
,
2k
共34兲
T2 =
f − L2
.
2k
共35兲
Let us to note that Eqs. 共29兲 and 共18兲 have the same mathematical structure.
In Figs. 4共b兲 and 4共d兲 we show two examples of how the
consideration of the propagation of the signal photon affects
the shape of the Wigner function retrieved. Figures 4共a兲 and
4共c兲 show the Wigner function at the output face of the nonlinear crystal, without considering the propagation effects 共as
described in Sec. V兲. Figures 4共b兲 and 4共c兲 plot similar cases,
but now taking into account the propagation from the output
face of the nonlinear crystal to the entrance of the Sagnac
interferometer, showing clearly how the shape of the Wigner
function of the signal photon changes. However, it should be
noted that this transformation preserves the purity of the signal photon, as no filtering occurs in the process. This can be
mathematically shown by inserting the expressions for H⬘
and G⬘ in Eq. 共21兲. Note that Fig. 4共d兲 is exactly the same
than Fig. 1共b兲, although they represent two very different
experimental situations. In the case of Fig. 1共b兲 the signal
photon results in a coherent Gaussian state with a plane wave
front by means of projecting the idler photon onto a given
state. In Fig. 4共d兲, the signal photon results on exactly the
same state but without any postselection.
B. Wigner function of the two-photon state
The Wigner function of the two-photon state can be calculated using the more general Eq. 共3兲. It can be readily
shown that at the output face of the crystal it has the form
043804-6
PHYSICAL REVIEW A 80, 043804 共2009兲
PROPERTIES OF THE SPATIAL WIGNER FUNCTION OF…
Ws,i共qs,xs,qi,xi兲
=
Coincidence
Counting
1
exp兵− 2Re共A兲共qs − qi兲2 − 2Re共B兲共qs + qi兲2其
␲
再
再
⫻exp
− 关2Im共B − A兲共qi − qs兲 + xs兴
Re共A + B兲
2
冎
D1
BS
关2Im共B − A兲共1 − ␯1兲共qs − qi兲 + xi − ␯1xs兴2
⫻exp −
2共HR − G兲
冎
D2
Path 1
Path 2
BS
x
共36兲
Path 2
MS
MS
x
Path 1
signal
where
Im共B − A兲
␯1 =
.
Re共A + B兲
Pump
共37兲
This Wigner function can be measured experimentally using an extension of the procedure to measure the Wigner
function of the signal alone. In this case, instead of collecting
the idler photon with a multimode fiber, it will pass through
another Sagnac interferometer similar to the one used for the
signal photon, as it is sketched in 6. Under these experimental conditions, the coincidence rates from the output ports of
both Sagnac interferometers will have the following shape:
Rc ⬀ Is1共I1i + I2i 兲 + Is2共I1i + I2i 兲 + 共Is1 + Is2兲Wi共xi,qi兲
+ 共I1i + I2i 兲Ws共xs,qs兲 + Ws,i共xs,qs,xi,qi兲,
共38兲
where Inj , with n = 1 , 2 and j = s , i, are the counts corresponding to a signal or idler photon which travels through either
path of the interferometer. W j共x j , q j兲 is the Wigner function
of the signal 共idler兲 photon alone regardless of the state of
the idler 共signal兲 photon, as described in Sec. V.
Ws,i共xs , qs , xi , qi兲 is the Wigner function of the two-photon
state at the output face of the crystal and can be extracted
from the coincidences of the two outputs, after subtracting
the background elements.
PPKTP
PBS
Idler
FIG. 6. 共Color online兲 Sketch of the setup for measuring the
Wigner function of the spatial state of two photons produced from
an SPDC source. In this case, both signal and idler photon are sent
to a Sagnac interferometer and detected by Detector1 and Detector2, respectively, after been collected by a multimode fiber. Using
coincident logical detection and after subtracting the coincident
background, we obtain the spatial Wigner function corresponding to
the two-photon state.
Wigner function of only one of the two photons of the pair,
which shows that one photon of the pair is in a statistical
mixture of modes. This case can be particularly important to
estimate the amount of spatial entanglement of the source.
Finally, we propose an experiment to completely measure the
spatial state of the two photons, by using a pair of Sagnac
interferometers. We have given analytical expressions for the
results one would expect in this case.
All these analytical studies show that the spatial degree of
freedom of photons can be used to explore the physics of
continuous variable systems. Also, it will allow us to further
control the spatial state of the photons.
VII. CONCLUSIONS
ACKNOWLEDGMENTS
In this paper we have presented some useful results regarding the spatial Wigner function of entangled paired photons. We have studied the remote preparation of one of the
photons of the pair, showing explicit results for the photon
being in a Gaussian beam and in a superposition of two
modes. Also, we have studied the important case of the
This Work was supported by the European Commission
共Qubit Applications, Contract No. 015848兲 and by the Government of Spain 关Consolider Ingenio 2010 共QOIT兲
CSD2006-00019 and FIS2007-60179兴.
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S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev.
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043804-8
PHYSICAL REVIEW A 78, 023811 共2008兲
Dynamics of Bragg diffraction in a stored light grating in cold atoms
D. Moretti,1 N. Gonzalez,2 D. Felinto,1 and J. W. R. Tabosa1
1
Departamento de Física, Universidade Federal de Pernambuco, Cidade Universitária, 50670-901 Recife, PE, Brazil
2
Institut de Ciències Fotòniques, Barcelona, Spain
共Received 15 April 2008; published 7 August 2008兲
We report on a detailed investigation of the dynamics and the saturation of a light grating stored in a sample
of cold cesium atoms. We employ Bragg diffraction to retrieve the stored optical information impressed into
the atomic coherence by the incident light fields. The diffracted efficiency is studied as a function of the
intensities of both writing and reading laser beams. A theoretical model is developed to predict the temporal
pulse shape of the retrieved signal and compares reasonably well with the observed results.
DOI: 10.1103/PhysRevA.78.023811
PACS number共s兲: 42.50.Gy, 32.80.Qk, 42.65.Hw
I. INTRODUCTION
II. THEORETICAL MODEL
The storage of light information in an atomic ensemble is
a well understood phenomenon which has a promising prospect for application both in classical and quantuminformation processing 关1兴. The light storage 共LS兲 phenomenon allows us to obtain later information on a previously
stored light pulse, as well as to manipulate the stored information. As it was first proposed, LS in an electromagnetically induced transparency 共EIT兲 medium 关2,3兴 is described
in terms of a mixed two component light-matter excitation,
called dark state polariton 共DSP兲, where each component of
the excitation can be externally controlled 关4兴. To date, several experimental observations of these effects were realized
in different systems 关5–10兴.
Alternatively, the LS process can also be described as
being due to the creation of a spatially dependent ground
states coherence that contains, respectively, the information
on the amplitude and phase of a light pulse and which survives after the switching off of the incident light. Using this
simpler picture, we have recently demonstrated the storage
of a polarization light grating into an atomic coherence via a
backward four-wave mixing configuration 关11兴. Other
schemes have also been recently employed to store spatial
structures 共images兲 in atomic vapors 关12–15兴. For instance, a
light vortex was stored in a hot vapor for hundreds of microseconds 关12兴.
In this work we present experimental and theoretical investigation on the dynamics of light grating stored in an EIT
medium associated with a degenerate two-level system. The
dependence of the stored light grating with the intensities of
the incident writing and reading beams is investigated. Bragg
diffraction into the stored grating is employed to probe its
dynamics under different experimental conditions. The demonstration of the reversible storage and the manipulation of
the spatial light phase structure stored into the atomic ensemble, and its extension to include beams carrying orbital
angular momentum, would be of great importance to demonstrate the manipulation of quantum information encoded in a
higher dimensional state space 关16,17兴. Moreover, the storage of this light grating opens up the possibility to investigate the generation of correlated photons pairs in a previously coherently prepared atomic ensemble 关18兴.
We consider an ensemble of cold atoms excited by three
different fields: Two writing 共W and W⬘兲 and one reading 共R兲
laser pulses. The atomic ensemble can be well approximated
1050-2947/2008/78共2兲/023811共8兲
(a)
(b)
2
2
6P3/2, F=2
W, σ+
W´, σ−
D, σ+
R, σ−
1a
1b
1a
1b
6S1/2, F=3
MOT
MOT
W
W´
θ
R
D
(c)
W, W’
R
“ON”
“OFF”
“OFF”
“ON”
0
Time
FIG. 1. 共a兲 Simplified Zeeman level scheme, showing the coupling and the propagation directions of the grating writing beams
共W and W⬘兲 and 共b兲 the coupling and the propagation direction of
the reading 共R兲 and diffracted 共D兲 beams. The beams W and W⬘
make a small angle ␪ and are circularly polarized with opposite
handedness, while the beam R is counterpropagating to the beam W
and have, with respect to the atom, a circular polarization opposite
to this beam. The diffracted beam is detected in a direction opposite
to the beam W⬘. 共c兲 The switching time sequence for the writing
and reading beams.
023811-1
©2008 The American Physical Society
PHYSICAL REVIEW A 78, 023811 共2008兲
MORETTI et al.
by a set of degenerate two-level atoms, with a ground-state
manifold composed of two degenerate states 共兩1a典 and 兩1b典兲
and the excited-state manifold having a single state 共兩2典兲. As
illustrated in Fig. 1, the ground-state degeneracy corresponds
to the Zeemam degeneracy of atomic cesium in the experiment. In this way, the different levels are connected by fields
of different polarizations with respect to the atom. We consider fields W⬘ and R having ␴ˆ − polarization, with respect to
the atomic transitions, and field W having ␴ˆ + polarization.
W⬘ and R excite then the transition 1b → 2, and W excites the
transition 1a → 2.
The fields W and W⬘ propagate in different directions,
corresponding to a small angle ␪ between them. The R field
is counterpropagating with respect to W. The signal we want
to model corresponds to the diffraction of the R field in the
spatial grating formed by fields W and W⬘. In the case of
continuous wave 共cw兲 excitation of the ensemble, this signal
corresponds to the well-known conjugated signal in fourwave mixing 共FWM兲 processes 关19兴. Here we call it D field
关see Fig. 1共b兲兴.
We use this FWM configuration to store and later retrieve
a coherence grating written in the atomic ensemble. In order
to address this coherence storage process, we use a specific
time sequence for the pulsed excitation of the ensemble, see
Fig. 1共c兲. First we prepare the sample by exciting it with the
two, long writing pulses. In this writing process, the goal is
to leave the system in its stationary state. Then we turn off
the writing beams, and wait a certain amount of time, the
storage time, before turning the reading pulse on. This reading pulse stays on also for a long time, enough to extract the
whole stored grating from the ensemble. A field-D pulse is
then generated during the read process. In the following theoretical analysis, we want to model and study this field-D
generation process in detail, considering the three-level-atom
approximation discussed above.
Ĥ共t兲 = Ĥ0 + V̂共t兲,
共2兲
Ĥ0 = ប␻e兩2典具2兩
共3兲
where
is the Hamiltonian for the free atom and
ជ 共t兲兩2典具1a兩 − dជ
ជ
V̂共t兲 = − dជ 2,1a · E
W
2,1b · EW⬘共t兲兩2典具1b兩 + H.c.
共4兲
is the interaction Hamiltonian. Defining the Rabi frequencies
⍀W共rជ兲 =
id2,1aEW共rជ兲eikWz
ប
共5a兲
,
ជ
⍀W⬘共rជ兲 =
id2,1bEW⬘共rជ兲eikW⬘·rជ
ប
,
共5b兲
and assuming the resonance condition ␻W = ␻W⬘ = ␻e, the
whole set of Bloch equations for the system, in the rotatingwave approximation, becomes
d␳22
= 共⍀W␴1a,2 + ⍀W⬘␴1b,2 + c.c.兲 − ⌫22␳22 ,
dt
共6a兲
d␳1a,1a
⬘ ␳22 ,
= 共− ⍀W␴1a,2 + c.c.兲 + ⌫1a,1a
dt
共6b兲
d␳1b,1b
⬘ ␳22 ,
= 共− ⍀W⬘␴1b,2 + c.c.兲 + ⌫1b,1b
dt
共6c兲
d␴1a,2
*
* 共␳ − ␳
= − ⍀W
22
1a,1a兲 + ⍀W ␳1a,1b − ⌫12␴1a,2 ,
⬘
dt
共6d兲
A. Grating formation and storage
ជ ,
Consider an atom excited by two writing beams: One, E
W
ជ , forming an
propagating in the z direction and the other, E
W⬘
ជ . The fields Eជ and Eជ have orthogonal
angle ␪ with E
W
W
W⬘
circular polarizations ␴ˆ + and ␴ˆ −, respectively. We consider
small enough angles so that we can assume, to a good apជ as being ␴ˆ − on the same
proximation, the polarization of E
W⬘
+
state basis in which Eជ W is ␴ˆ . We can then write
ជ = E 共rជ兲ei共kWz−␻Wt兲␴ˆ + ,
E
W
W
共1a兲
ជ
Eជ W⬘ = EW⬘共rជ兲ei共kW⬘·rជ−␻W⬘t兲␴ˆ − ,
共1b兲
where EW共rជ兲 and EW⬘共rជ兲 represent the transversal modes of
each field, and we assumed both of them having constant
intensities. The frequencies of the fields are ␻W and ␻W⬘, and
their wave vectors are kWẑ and kជ W⬘, respectively. The energy
difference between fundamental and excited levels is ប␻e.
The system Hamiltonian can then be written as
d␴1b,2
* 共␳ − ␳
*
= − ⍀W
1b,1b兲 + ⍀W␳1b,1a − ⌫12␴1b,2 ,
⬘ 22
dt
共6e兲
d␳1a,1b
*␴
= − ⍀W
2,1b − ⍀W⬘␴1a,2 − ␥␳1a,1b ,
dt
共6f兲
with ␴1a,2 = ␳1a,2e−i␻Wt and ␴1b,2 = ␳1b,2e−i␻W⬘t. The spontaneous relaxation rates are indicated by ⌫12 and ⌫22, for the
and
coherence and population decays, respectively. ⌫1a,1a
⬘
indicate the rates at which the ␳22 population decays
⌫1b,1b
⬘
into the populations ␳1a,1a and ␳1b,1b, respectively. For simplicity, in these equations and in the following, we omit the
spatial dependence of the Rabi frequencies. The groundstate-coherence decay rate ␥ is introduced to take into account, in an effective way, the decay induced by residual
magnetic fields. Such decay is usually a result of inhomogeneous broadening in the ensemble of atoms, each subject to a
slightly different magnetic field 关21兴. For the signal we are
treating here, however, this simple model considering the
same decay constant for the whole ensemble is already
023811-2
PHYSICAL REVIEW A 78, 023811 共2008兲
DYNAMICS OF BRAGG DIFFRACTION IN A STORED…
enough to obtain a good comparison with the experimental
data probing the coherence decay.
After a sufficiently long time, the system reaches a steady
situation in which d␳kl / dt = 0, for all ␳kl density-matrix elee
between the two
ments. The steady-state coherence ␳1a,1b
ground-state levels is then given by
e
␳1a,1b
=−
⬘ 兩⍀W⬘兩2 + ⌫1b,1b
⬘ 兩⍀W兩2兲
共⌫1a,1a
A
*⍀ ,
⍀W
W⬘
共7兲
d␳1a,1b
= − ⍀R␴1a,2 − ␥␳1a,1b ,
dt
with ␴1a,2 = ␳1a,2e−i␻Rt. Note that the equations for ␴1a,2 and
␳1a,1b are actually decoupled from the rest of the system of
Bloch equations.
Eliminating ␳1a,1b in this last set of equations, we obtain
the following second-order differential equation for ␴1a,2,
d2␴1a,2
d␴1a,2
+ 共␥ + ⌫12兲
+ 共␥⌫12 + 兩⍀R兩2兲␴1a,2 = 0.
dt2
dt
with
共14兲
⬘ 兩⍀W⬘兩2 + ⌫1b,1b
⬘ 兩⍀W兩2兲共␥⌫12 + 兩⍀W兩2 + 兩⍀W⬘兩2兲
A = 共⌫1a,1a
+ 6␥兩⍀W⬘兩2兩⍀W兩2 .
共8兲
We are particularly interested in the situation where ␥ is very
small when compared to any other frequency in the system,
since this corresponds to our experimental condition. In this
limit, note then that the above expression simplifies to
e
␳1a,1b
=−
*⍀
⍀W
W⬘
兩⍀W兩2 + 兩⍀W⬘兩2
.
共9兲
ជ and Eជ are turned off, the coherences
Once the fields E
W
W⬘
in the system evolve according to their respective decay
times. Since ␥ Ⰶ ⌫12, after a time ts Ⰷ 1 / ⌫12 the stored coherences in the sample can be well approximated by
s
␴1a,2
共ts兲 = 0,
共10a兲
s
␴1b,2
共ts兲 = 0,
共10b兲
s
e
␳1a,1b
共ts兲 = ␳1a,1b
e −␥ts .
共10c兲
This is the same equation describing a damped harmonic
oscillator subject to a step-function excitation. When comparing to the experimental data, we will find later that our
experimental conditions correspond, formally, to the overdamped regime of a damped harmonic oscillator.
ជ that is phase
We are interested in calculating the field E
D
ជ
conjugated to EW⬘. This field is generated by the medium in
the transient excitation of the ␴1a,2 coherence, corresponding
to the extraction of the stored coherence grating. Using the
stored state as initial conditions, the solution of the above
equations for ␴1a,2共t兲 is
␴1a,2共t兲 =
s
共ts兲e−␥1t sinh共␥2t兲
⍀R*␳1a,1b
␥2
␥1 =
␥2 =
⌫12 + ␥
,
2
冑共⌫12 − ␥兲2 − 4兩⍀R兩2
2
共16a兲
共16b兲
.
pជ 2,1a共rជ,t兲 = dជ 2,1a␴2,1a共rជ,t兲e−i␻et .
ជ = E 共rជ兲ei共−kRz−␻Rt兲␴ˆ − ,
E
R
R
C. Signal
共11兲
with ER, kR, and ␻R representing the transversal mode, wave
ជ . After similar
vector, and frequency, respectively, of field E
R
considerations as for the grating-formation process, including the resonance condition ␻e = ␻R, and the analogous definition of a third Rabi frequency
ប
共15兲
with
The stored coherence grating can be extracted from the
ជ counterpropagatsample using a ␴ˆ −-polarized third field E
R
ជ
ing with respect to EW,
id2,1bER共rជ兲e−ikRz
,
The single-atom polarization vector pជ 2,1a on the 2 → 1a transition is then given by
B. Reading
⍀R共rជ兲 =
共13b兲
,
共12兲
ជ for the D field coming from the
The electric field E
D
ជ
diffraction of ER on the sample coherence grating 关see Fig.
1共b兲兴 is a result of the constructive interference of the emission of all atoms in the −kជ W⬘ direction. If we neglect interaction between atoms and propagation effects on the D field,
for simplicity and since we deal only with relatively low
ជ in the
densities in the experiments of Sec. III, the value of E
D
kជ direction can be obtained by the superposition of all atomic
contributions on that direction,
the relevant Bloch equations describing the reading process
become
d␴1a,2
= ⍀R*␳1a,1b − ⌫12␴1a,2 ,
dt
共13a兲
共17兲
ជ 共kជ ,t兲 =
E
D
1
4␲⑀0共2␲兲3/2
冕
ជ
␩共rជ兲pជ 2,1a共rជ,t兲e−ik·rជd3rជ , 共18兲
where ␩共rជ兲 represents the atomic density at rជ, ⑀0 is the
vacuum permittivity, and the integration runs over the whole
023811-3
PHYSICAL REVIEW A 78, 023811 共2008兲
MORETTI et al.
ensemble volume. Approximating the fields W, W⬘, and R as
plane waves, we can neglect the spatial dependence on EW,
EW⬘, and ER, respectively. In this case, we can write
⍀W
=i
⌫12
⍀ W⬘
⌫12
冑
冑
冑
=i
⍀R
=i
⌫12
IW ik z
e W,
2Isa
I W⬘
2Isb
e
ikជ W⬘·rជ
冑
冑
冑
W
M
λ/4
λ/4
R
AOM-2
FIG. 2. Simplified experimental scheme. The diode laser is
locked to the F = 3 ↔ F⬘ = 2 transition. AOM, acousto-optical modulator 共AOM-1,2: 200 MHz; AOM-3: 100 MHz兲; PBS, polarizing
beam splitter; BS, beam splitter; M, mirror.
Sfast共t兲 = A
Isa
Isb
共21兲
IR e−␥1t sinh共␥2t兲
␥2/⌫12
2Isb
共22兲
N
2
e−rជ·rជ/2L ,
共2␲L2兲3/2
s
兩f R共t兲e−i␻et −兩kជ + kជ 兩2L2/2
idជ 2,1aN兩␳1a,1b
W⬘
Eជ D共kជ ,t兲 =
e
,
4␲⑀0共2␲兲3/2
共24兲
which explicitly shows that the emission of the D-field occurs in the −kជ W⬘ direction only with a spread in vector space,
on each direction, of the order of the inverse of the atomicdistribution spatial width, L−1.
The detection apparatus can be arranged to collect all
light in the D-field mode. In this case, and if the detection of
the field is performed with a fast detector compared to the
time variation of f R共t兲, the signal Sfast共t兲 is then proportional
to the integration of the intensity of light in field D over all kជ ,
ជ 共kជ ,t兲兩2d3kជ ,
兩E
D
s
兩2兩f R共t兲兩2 ,
Sfast共t兲 = A⬘兩␳1a,1b
共25兲
共26兲
with A⬘ a different proportionality constant.
Another important quantity that can be directly derived
from Sfast共t兲 is the total energy, UD, extracted in mode D.
Note that, in light-storage measurements, the goal is usually
to extract as much information and energy as possible from
the coherence grating 关23兴. From the expressions derived
above we have then
UD =
冕
⬁
0
Sfast共t兲dt =
s
兩2
2A⬘兩␳1a,1b
⌫12
冉
IR/2Isb
.
IR
␥
␥
+
1+
⌫12 2Isb ⌫12
冊冉
冊
共27兲
III. EXPERIMENTAL RESULTS AND DISCUSSIONS
共23兲
where N is the total number of atoms in the cloud. Using this
expression for ␩共rជ兲, Eq. 共20兲 becomes
冕
where A is a proportionality constant. From Eq. 共24兲, we see
that such detected signal is given by
a function describing the temporal profile of the D-field
pulse. Note that f R共t兲 is a function of the read field parameters only.
If we approximate the distribution of atoms as having a
Gaussian profile with the same rms width L in all three directions, we can write
␩共rជ兲 =
PBS
M
ជ ជ
representing the modulus of the stored ground-state coherence, and
f R共t兲 =
AOM-1
␩共rជ兲e−i共k+kW⬘兲·rជd3rជ ,
Isb
+ I W⬘
Isa
λ/2
BS
冑IWIW⬘e−␥ts
IW
MOT
W’
共19c兲
with
=
D
BS
共19b兲
,
共20兲
s
兩
兩␳1a,1b
BS
AOM-3
LASER
with IW, IW⬘, and IR the intensities of the W, W⬘, and R fields,
respectively. Isa and Isb are the saturation intensities of the
1a → 2 and 1b → 2 transitions, respectively, defined according to Ref. 关22兴.
Since kR − kW = 0, Eq. 共18兲 can be written as
冕
Detector
共19a兲
IR −ik z
e R,
2Isb
s
兩f R共t兲e−i␻et
idជ 2,1a兩␳1a,1b
Eជ D共kជ ,t兲 =
4␲⑀0共2␲兲3/2
M
As indicated in Figs. 1共a兲 and 1共b兲 the experiment was
performed using a degenerate two-level system. This system
corresponds in the experiment to the cycling transition
6S1/2共F = 3兲 ↔ 6P3/2共F⬘ = 2兲 of the cesium D2 line. The cesium atoms were previously cooled in a MOT operating in
the closed transition 6S1/2共F = 4兲 ↔ 6P3/2共F⬘ = 5兲 with a repumping beam resonant with the open transition 6S1/2共F
= 3兲 ↔ 6P3/2共F⬘ = 3兲. To prepare the atoms in the state
6S1/2共F = 3兲, we switch off the repumping beam for a period
of about 1 ms to allow optical pumping by the trapping
beams via nonresonant excitation to the excited state F⬘ = 4.
After optical pumping, the optical density of the sample of
cold atoms in the F = 3 ground state is approximately equal to
3 for appropriate MOT parameters.
A simplified experimental setup is depicted in Fig. 2,
which shows how the different incident laser beams indi-
023811-4
PHYSICAL REVIEW A 78, 023811 共2008兲
Retrieved peak Intensity (arb. units)
Diffracted Signal Intensity (arb. units.)
DYNAMICS OF BRAGG DIFFRACTION IN A STORED…
cw-FWM
Reading beam "off"
Writing beams "off"
1.6
1.2
Decay time τ = 2.9 µs
0.8
0.4
0.0
0
-6
-4
-2
0
2
4
6
8
2
4
6
8
Time (µs)
10
Time (µs)
FIG. 3. Bragg diffraction signal retrieved from the stored grating
for different storage times.
FIG. 4. Normalized Bragg diffraction peak signal for different
storage times. The solid curve corresponds to a fit with an exponential function.
cated in Fig. 1 are generated. All the incident beams are
provided by an external cavity diode laser which is locked to
the F = 3 ↔ F⬘ = 2 transition. The grating writing beams 共W
and W⬘兲 have the same frequency. After passing through a
pair of acousto-optical modulators 共AOM兲 with one of them
operating in double passage, they can have their frequency
scanned around the F = 3 ↔ F⬘ = 2 transition. The two AOM’s
also allow us to control their intensity and timing, with
switching on or off times smaller than 100 ns. These two
beams are circularly polarized with opposite handedness and
are incident in the MOT forming a small angle ␪ ⬇ 60 mrad,
which leads to a polarization grating with a spatial period
given by ⌳ = 2 sin共␭␪/2兲 , where ␭ is the light wavelength. The
reading beam R is circularly polarized opposite to the writing
beam W and also passes through another pair of AOM’s
which does not change its frequency but allows us to control
its intensity.
Employing the time sequence shown in Fig. 1共c兲 we have
investigated the light grating storage dynamics through the
observation of delayed Bragg diffraction of the reading beam
R in the Zeeman coherence grating induced by the writing
beams W and W⬘. The writing and reading pulses are triggered to the switching off of the repumping laser which also
triggers the turn off of the MOT quadrupole magnetic field.
In order to compensate for spurious magnetic fields, three
independent pairs of Helmholtz coils with adjustable currents
are placed around the MOT and regulated in order to cancel
the magnetic field in that region. This canceling is optimized
by narrowing the EIT peak according to 关11兴, and also by
maximizing the storage time.
In Fig. 3 we show the cw-FWM and the Bragg diffracted
signal which is retrieved from the stored Zeeman coherence
grating for different storage times. We have experimentally
verified that the polarization of the diffracted beam, both for
the steady-state cw-FWM signal 共real time Bragg diffraction兲
and for the retrieved signal, is always opposite to the polarization of the reading beam as schematically depicted in Fig.
1共b兲. We have been able to observe the diffracted signal up to
a time of 10 ␮s. This maximum storage time is very sensitive to the compensation of the residual magnetic field. It is
interesting to note that for short storage times the retrieved
signal peak intensity is much larger than the corresponding
cw-FWM signal. This effect is related to the simultaneous
presence of the writing and reading beams in the cw regime,
where the reading beam contributes to decrease the contrast
of the coherence grating induced by the writing beams. The
decay of the peak intensity of the diffracted pulse, normalized by its steady-state value 共cw-FWM signal兲 is presented
in Fig. 4. The exponential decay behavior is evidenced by the
exponential fitting 共solid curve兲. For the data presented in
Fig. 4, the intensities of the writing beams W and W⬘ are
approximately equal to 5.0 mW/ cm2 and 1.5 mW/ cm2 respectively, while the intensity of the reading beam R is about
8.0 mW/ cm2. From the measurement presented in Fig. 3, we
obtain a decay time of the order of 2.9 ␮s, which corresponds to the Zeeman ground-state coherence decay. We
have experimentally verified that the measured coherence
time does not depend on the intensity of either the writing or
the reading beams.
It is worth mentioning that the grating storage we have
observed in this experiment is only possible if cold atoms are
employed. The induced grating will be completely washed
out by the atomic motion for a thermal atomic vapor. Although for the small angle used in our experiment the effect
of the atomic motion could be neglected, for a much larger
angle, i.e., small grating period, even the very small velocities associated with cold atoms would effect the grating decay time 关20兴.
For a fixed storage time of approximately 1 ␮s, we have
also measured the temporal pulse shape of the retrieved signal for different reading beam intensities and the results are
shown in Figs. 5共a兲–5共c兲 for three different values of the
reading beam intensity. We note that the experimentally retrieved pulse raising time is limited by the time constant of
the detector 共艋0.5 ␮s兲. As we have discussed previously the
coherently prepared atomic system couples to the reading
beam to transiently generate the diffracted pulse signal. The
temporal width of the generated pulse decreases for increasing reading beam intensity, a direct consequence of the effect
of the increased dumping of the Zeeman ground-state coherence caused by spontaneous emission induced by the reading
beam in the process of mapping the stored Zeeman coher-
023811-5
PHYSICAL REVIEW A 78, 023811 共2008兲
Theory
3
2
(a)
1
(d)
1
2
IR= 8.5 mW/cm
ΩR/ Γ12= 0.18
0
2
0
2
(b)
1
(e)
1
ΩR/ Γ12= 0.14
2
IR= 4.9 mW/cm
0
0
1
1
(c)
(f)
ΩR/ Γ12= 0.11
2
IR= 2.9 mW/cm
0
0
1
2
0
4 0
3
1
2
3
4
Time (µs)
Time (µs)
ence into the optical coherence. In Figs. 5共d兲–5共f兲 we show
the corresponding retrieved pulse obtained using the previously developed theory, assuming Isb the saturation intensity
for the 6S1/2共F = 3 , mF = + 3兲 → 6P3/2共F = 2 , mF = + 2兲 transition. We have used an adjustable parameter of the order of
a ⬇ 0.02 to rescale all of the theoretical reading beam intensities 共i.e., IR → aIR兲, which accounts for the uncertainty in
the determination of the effective experimental value of the
Rabi frequency associated with the reading beam.
More systematically, in Fig. 6 we plot the measured pulse
width 关full width at half-maximum 共FWHM兲兴 for different
reading beam intensities. In these measurements, for each
value of the reading beam intensity, we have recorded three
curves of the retrieved pulse which allows us to estimate the
corresponding error bars. The solid curve in Fig. 6 corresponds to the calculation of the pulse temporal width using
1.6
FWHM, ∆τ (µs)
1.4
Experiment
Theory
1.0
0.8
0.6
0
2
4
6
8
Experiment
Theory
20
3.0
15
10
5
2
10
Reading beam intensity (mW/cm )
FIG. 6. Measurement of the temporal width 共full width at halfmaximum兲 of the retrieved pulse for different intensities of the
reading beam, obtained in similar experimental conditions as in
Figs. 5共a兲–5共c兲. The solid curve is a theoretical fitting using the
model described in the text.
2.5
2.0
1.5
1.0
Experiment
Theory
0.5
0.0
0
2
4
6
8
2
10
Reading beam intensity (mW/cm )
0
0
FIG. 5. Experimental 共right-hand side兲 and calculated 共left-hand
side兲 retrieved-pulse temporal shape corresponding to different intensities of the reading beam, for a fixed storage time. The writing
beam intensities, W and W⬘, are 7.0 mW/ cm2 and 1.0 mW/ cm2,
respectively. As described in the text, for comparison between
theory and experiment, all of the theoretical reading beam intensities need to be rescaled by a factor a ⬇ 0.02.
1.2
25
Retrieved peak intensity
Experiment
2
Retrieved pulse energy (arb. units)
Diffracted Signal Intensity (arb. units.)
MORETTI et al.
2
4
6
8
10
Reading beam intensity (mW/cm2)
FIG. 7. Retrieved pulse energy for different intensities of the
reading beam, obtained on similar experimental conditions as in
Figs. 5共a兲–5共c兲. Inset; the corresponding variation of the peak intensity of the retrieved pulse. The solid curves are theoretical fittings
employing the model described in the text with the same intensity
adjustment parameter used to fit the pulse width in Fig. 6.
the signal shape function given by Eq. 共26兲. In this calculation we have used ␥ / ⌫12 ⬇ 0.014 in order to obtain the best
agreement with the experiment. Note that this value is of the
same order of the experimentally measured decay rate, obtained from the different set of data shown in Fig. 4, and
estimated as ␥ / ⌫12 ⬇ 0.02, with 2⌫12 / 2␲ = ⌫22 / 2␲
= 5.2 MHz. From the same set of data as Fig. 6, we show in
Fig. 7 the retrieved pulse energy, obtained by time integration of the measured pulse intensity. The corresponding solid
curve is a theoretical fitting obtained using Eq. 共27兲 with the
same adjustable parameter a.
Despite the smallness of the fitting parameter a, we consider that the agreement between theory and experiment is
qualitatively satisfactory, owning to the simplification of the
theoretical model, which uses a single three-level system and
does not account for the many-fold Zeeman degeneracy. Indeed, in this three-level model we have considered the reading beam interacting with the transition having the smallest
saturation intensity 共largest Clebsh-Gordan coefficient兲,
while for the intensities of the writing beams used in the
experiment the atomic system will not be pumped into the
highest magnetic sublevel and ground-state coherence involving different pairs of Zeeman sublevels can actually exist. This will lead to the interaction of the reading beam with
transitions having smaller saturation intensities and, as a consequence, to a smaller effective reading beam intensity,
which is simulated by the adjustable parameter a. Another
possible source of such disagreement is associated with the
difficulty of optimizing alignment and the position of the
beams around the center of the MOT.
We also have measured the variation of the diffracted signal as a function of the intensity of one of the grating writing
beams 共i.e., the beam W兲 and the results for the corresponding pulse energy are shown in Fig. 8. For these measurements, the intensities of the grating writing beam W⬘ and the
reading beam were, respectively, equal to 1.0 mW/ cm2 and
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DYNAMICS OF BRAGG DIFFRACTION IN A STORED…
the peak intensity of the retrieved pulse with the corresponding intensity, as shown in the insets of Figs. 7 and 8. As
observed, the maximum peak of the retrieved pulse saturates
more strongly with the writing beam intensity as compared
with the saturation induced by the reading beam. As one
should expect, the saturation induced by the reading beam is
related mainly to the total retrieved energy. The pulse peak,
however, can increase much further with the reading power,
since it is closely related also with the speed of the reading
process. On the other hand, the increase of the writing beam
intensity will saturate the Zeeman coherence grating, therefore reducing its contrast. This effect has a strong influence
on the Bragg diffraction efficiency, affecting equally the total
retrieved energy and the pulse peak.
Experiment
Theory
20
15
Retrieved peak intensity
Retrieved pulse energy (arb. units)
25
10
5
2.5
2.0
1.5
Experiment
Theory
1.0
0.5
0.0
0
1
2
3
4
5
6
7
Writing beam intensity, W (mW/cm2)
0
0
1
2
3
4
Writing beam intensity, W
5
6
7
(mW/cm2)
FIG. 8. Variation of the retrieved pulse energy for different intensities of the grating writing beam W. For these data, the corresponding intensities of the writing 共W⬘兲 and reading 共R兲 beams
were fixed at 1.0 mW/ cm2 and 9.0 mW/ cm2, respectively. Inset,
variation of the corresponding peak intensity. The solid curves are
again theoretical fittings using the model described in the text. We
have used the same intensity adjustment parameter, a⬘ ⬇ 1.9, in both
curves.
9.0 mW/ cm2. The solid curve in Fig. 8 corresponds to a
theoretical fitting with the calculated retrieved pulse energy
given by Eq. 共27兲, assuming that Isa is the saturation intensity
of the 6S1/2共F = 3 , mF = + 1兲 → 6P3/2共F = 2 , mF = + 2兲 transition, with Isa = 15Isb according to the ratio between the corresponding Clebsch-Gordan coefficient. Again, to account
for the uncertainty in the experimental value of the Rabi
frequency associated with the writing beams W and W⬘, we
have used another adjustable parameter, which in the present
case is of the order of a⬘ ⬇ 1.9, to rescale the theoretical
intensity ratio between these beams 共i.e., IW / IW⬘ → a⬘IW / IW⬘兲.
As can be observed from Figs. 7 and 8 the amount of
energy that can be retrieved from the medium clearly saturates with the writing and reading beam intensities. In particular, this shows that for fixed writing beam intensities,
there is a maximum amount of energy that can be retrieved
from the stored coherence. However, it is worth mentioning
the different saturation behavior observed for the variation of
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IV. SUMMARY
We have investigated, both theoretical and experimentally, the storage of a spatial light polarization grating into
the Zeeman ground-state coherence of cold cesium atoms.
Systematic measurements were performed to reveal the saturation behavior of the retrieved signal as a function of the
intensities of the writing and reading beams. The developed
simple theoretical model accounts reasonably well for the
observed results and in particular for the measured pulse
temporal shape. We consider our results are of considerable
importance for a better understanding of the coherent
memory for multidimensional state spaces. Finally, we
would like to mention that we also have observed the coherent evolution of the stored grating in the presence of an applied magnetic field, which shows collapses and the revivals
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Further investigation on this effect is currently under way
and will be presented elsewhere.
ACKNOWLEDGMENTS
We gratefully acknowledge Marcos Aurelio for his technical assistance during the experiment. This work was supported by the Brazilian Agencies CNPq/PRONEX, CNPq/
Institute Milênio, and FINEP.
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