The solution of the infinitesimal propagation equation for atmospheric propagation of single-photon and entangled quantum states, represented in terms of Laguerre-Gauss modes, which is a discrete orbital angular momentum (OAM) basis, is compared with numerical simulations for the propagation of optical fields that carry OAM in atmospheric turbulence. The numerical simulations are performed using the multi-phase screen model based on the Kolmogorov theory of turbulence. The comparison was done under various turbulence conditions and propagation distances to allow comparison under both weak and strong scintillation conditions. The results show that there is an agreement between the infinitesimal propagation equation and the numerical simulations. Also, we note that in the limit of weak scintillation both methods, the infinitesimal propagation equation and numerical simulations, agree with the predictions of single-phase screen model.
Combining the multiple degrees of freedom of photons has become topical in quantum communication and information
processes. This provides advantages such as increasing the amount of information that is be packed into
a photon or probing the wave-particle nature of light through path-polarisation entanglement. Here we present
two experiments that show the advantages of using hybrid entanglement between orbital angular moment (OAM)
and polarisation. Firstly, we present results where high dimensional quantum key distribution is demonstrated
with spatial modes that have non-separable polarisation-OAM DOF called vector modes. Secondly, we show
that through OAM-polarisation entanglement, the traditional which-way experiment can be performed without
using the traditional physical path interference approach.
High-dimensional encoding using higher degrees of freedom has become topical in quantum communication protocols. When taking advantage of entanglement correlations, the state space can be made even larger. Here, we exploit the entanglement between two dimensional space and polarization qubits, to realize a four-dimensional quantum key distribution protocol. This is achieved by using entangled states as a basis, analogous to the Bell basis, rather than typically encoding information on individual qubits. The encoding and decoding in the required complementary bases is achieved by manipulating the Pancharatnam-Berry phase with a single optical element: a q-plate. Our scheme shows a transmission fidelity of 0.98 and secret key rate of 0.9 bits per photon. While the use of only static elements is preferable, we show that the low secret key rate is a consequence of the filter based detection of the modes, rather than our choice of encoding modes.
Considering the quantum state produced in type I spontaneous parametric down-conversion with collinear, degenerate signal and idler beams, and a Gaussian pump, we show that the azimuthal Schmidt number in the Laguerre-Gaussian (LG) basis increases when the radial indices of the LG modes detected in the signal and idler beams are different. These observations are confirmed by the good agreement between theoretical and experimental results. The theoretical results are obtained by deriving expressions for the probability amplitude to detect LG modes with any combination of azimuthal and radial indices in a down-converted photonic quantum state.
Bessel-Gaussian (BG) modes possess unique characteristics that have been exploited in the classical world and which may also offer advantages over other modes in the quantum regime. The easily adjustable radial scale of BG modes provides a more favourable basis of orbital angular momentum (OAM) entanglement over Laguerre- Gaussian (LG) modes, where the radial dependence is often ignored. We demonstrate high-dimensional entanglement with the BG modes and show a higher fidelity than the LG modes. We use the reconstruction property of BG modes to recover the degree of entanglement of our quantum state after encountering an obstruction. By moving the obstruction along the path of propagation of the entangled photon pairs, we quantitatively show a increase in the degree of entanglement as the obstruction was moved beyond that minimum distance.
We propose a 2-dimensional method for Bessel Gaussian beam azimuthal and radial decomposition using digital holograms. We illustrate the reconstruction of a Bessel Gaussian beam after encountering an obstruction. From the measured decomposition we show the reconstruction of the amplitude, phase and azimuthal index of the field with high degree of accuracy.
We present a simple way of simulating Spontaneous parametric down-conversion (SPDC) by modulating a classical laser beam with two spatial light modulators (SLM) through a back projection setup. This system has the advantage of having very high photon count rates, it can simulate a large range of pump beam profiles simply by modifying the hologram on the SLM, and it can be easily converted to a SPDC setup by simply changing only two of its components without the need to perform realignment. This setup can be used to give an indication whether a SPDC experiment will be feasible in a very short amount of time.
Optical vortices are always created or annihilated in pairs with opposite topological charges. However, the presence of such a vortex dipole does not directly indicate whether they are associated with a creation or an annihilation event. Here we propose a method to distinguish between vortex dipoles that have just been created and those that are about to be annihilated. We use first and second order transverse derivatives of the optical field to construct a quantity that reveals the nature of the dipoles. Numerical examples are provided as demonstration of the method.
Bessel-Gaussian (BG) modes possess unique characteristics that have been exploited in the classical world and which may also o er advantages over other modes in the quantum regime. We use the reconstruction property of BG modes to recover the degree of entanglement of our quantum state after encountering an obstruction. For BG modes, there exists a minimum distance behind an obstruction before reconstruction of the mode occurs. By moving the obstruction along the path of propagation of the entangled photon pairs, we quantitatively show a increase in the degree of entanglement as the obstruction moved beyond that minimum distance.
A quantum walk is the quantum analog of the classical random walks. Despite their simple structure they form a universal platform to implement any algorithm of quantum computation. However, it is very hard to realize quantum walks with a sufficient number of iterations in quantum systems due to their sensitivity to environmental influences and subsequent loss of coherence. Here we present a scalable implementation scheme for one-dimensional quantum walks for arbitrary number of steps using the orbital angular momentum modes of classical light beams. Furthermore, we show that using the same setup with a minor adjustment we can also realize electric quantum walks.
The use of Higher-dimensional entangled systems have been proved to signi cantly improve many quantum in- formation tasks. For instance, it has been shown that the use of higher-dimensional entangled systems provides a higher information capacity and an increased security in quantum cryptography. The orbital angular momentum (OAM) state of light is a potential candidate for the implementation of higher-dimensional entangled systems and has thus been considered for free-space quantum communication. However, atmospheric turbulence severely affects the OAM state of photons. In this work, we study the evolution of the OAM entanglement between two qutrits (three-dimensional quantum systems) in atmospheric turbulence both numerically and experimentally. The qutrits are photons entangled in their orbital angular momentum (OAM) states generated by spontaneous parametric down conversion. We propagate one of the photons through turbulence while leaving the other undis- turbed. To compare our results with previous work, we simulate the turbulent atmosphere with a single phase screen based on the Kolmogorov theory of turbulence and we use the tangle to quantify the amount of entangle- ment between the two qutrits. We compare our results with the evolution of OAM entanglement between two qubits.
The study of optical vortices in stochastic optical fields involves various quantities, including the vortex density and topological charge density, that are defined in terms of local expectation values of distributions of optical vortices. For stochastic optical fields that are inhomogeneous or not normally distributed, these local quantities often have nontrivial transient evolution as a function of propagation distance. The field of stochastic singular optics strive, among other things, to understand this dynamics. Here we review the tools and challenges of stochastic singular optics and provide some details of recent progress in this field.
In this work we will present two techniques for the measurement of superimposed higher-order Bessel beams. In the first technique we will outline a simple approach using only a spatial light modulator and a Fourier transforming lens to decompose the OAM spectrum of an optical field. We test this approach on symmetric and non-symmetric superpositions of non-diffracting higher-order Bessel beams. Our second procedure consists of two refractive optical elements which perform a Cartesian to log-polar coordinate transformation, translating helically phased beams into a transverse phase gradient. By introducing two cylindrical lenses we can focus each of the azimuthal modes associated with each Bessel beam to a different lateral position in the Fourier plane, while separating the radial wave-vectors in the image-plane.
The OAM or spiral bandwidth indicates the dimensionality of an entangled state that is produced by the spontaneous parametric down-conversion process. Normally this bandwidth is determined by modulating the signal and idler beams with helical phase functions with opposite azimuthal indices on the spatial light modulators in the signal and idler beams, respectively. We added an additional binary Bessel function to the helical phase, thereby specifying the radial dependence of the mode to be Bessel-Gaussian (BG) modes. This comes down to a post selection process, which is known to have the ability to increase entanglement. The result is a modification to the shape of the OAM spectrum, which leads to a higher dimensionality for the quantum states. We perform analytical calculations to show that the bandwidths obtained by measuring in the BG modal basis are larger than those for the LG modes. These theoretical predictions are confirmed by experimental measurements of the bandwidths for LG modes and for BG modes with different transverse scales.
By using digital holograms, we present a simple technique for performing a complete azimuthal decomposition of an
arbitrary laser mode. The match-filter, used to perform the azimuthal decomposition, is bounded by an annular ring,
allowing us to conduct a scale-independent decomposition on our selected mode. This technique therefore requires no
prior knowledge of the mode structure, the mode phases, or the amplitude distribution. A basis comprising of the angular
harmonics is used to express the spatial distribution of the selected mode in terms of spatially dependant coefficients. We
use this to infer directly from the measured weightings of the azimuthally decomposed modes and their phase-delay
measurements, the intensity of the selected field, its phase, and its orbital angular momentum (OAM) density. We
illustrate the concept by executing a full decomposition of two examples: a superposition of two Bessel beams, with
relative phase differences, and an off-axis vortex mode. We show a reconstruction of the amplitude, phase and OAM
density of these fields with a high degree of accuracy.
A procedure to efficiently sort orbital angular momentum (OAM) states of light, by performing a Cartesian to log-polar
coordinate transformation which translates helically phased beams into a transverse phase gradient, currently exists1. We
implement this mode transformer, which comprises of two custom refractive optical elements2, to efficiently sort Bessel
beams carrying OAM. Introducing two cylindrical lenses, allows the focusing of each of the input OAM Bessel states to
a different lateral position in the Fourier plane and separates the radial wave-vectors in the image-plane. We demonstrate
the concept by separating over forty OAM states and radial wave-vectors.
We experimentally generated superpositions of higher-order Bessel beams that possess no global orbital angular momentum (OAM), yet exhibit an angular rotation in their intensity profile as the field propagates. The digital holograms encoded on a spatial light modulator (SLM), used for generating such fields, consist of two annular rings of unequal radial wave-vectors where each ring is encoded with an azimuthal mode of equal order but opposite charge. We present experimentally measured angular rotation rates for some example superposition fields, which are shown to be in good agreement with that predicted theoretically. Introducing a second SLM and a Fourier transforming lens, we demonstrate a simple approach to perform an azimuthal decomposition of our generated optical fields. Bounding the match-filter to an annular ring, of varying radius, we are able to perform a scale-independent azimuthal decomposition of our initial field. From the measured weightings of the azimuthally decomposed modes we show reconstruction of the cross-sectional intensity profile and OAM density of our initial field.
Phase-only spatial light modulators are now ubiquitous tools in modern optics laboratories, and are often used to
generate so-called structured light. In this work we outline the use of a phase-only spatial light modulator to achieve full complex amplitude modulation of the light, i.e., in amplitude and phase. We outline the theoretical concept, and then illustrate its use with the example of the laser beam shaping of Gaussian beams into flat-top beams. We quantify the performance of this approach for the creation of such fields, and compare the results to conventional lossless approaches to flat-top beam generation.
The orbital angular momentum (OAM) state of light can potentially be used to implement higher dimensional
entangled systems for quantum communication. Unfortunately, optical fibers in use today support only modes
with zero OAM values. Free-space quantum communication is an alternative to traditional way of communicating
through optical fibers. However the refractive index fluctuation of the atmosphere gives rise to random phase
aberrations on a propagating optical beam. To transmit quantum information successfully through a free-space
optical channel, one needs to understand how atmospheric turbulence influences quantum entanglement. Here,
we present a numerical study of the evolution of quantum entanglement between a pair of qubits. The qubits
consist of photons entangled in the OAM basis. The photons propagate in a turbulent atmosphere modeled by a
series of consecutive phase screens based on the Kolmogorov theory of turbulence. Maximally entangled initial
states are considered, and the concurrence is used as a measure of entanglement. We show how the evolution
of entanglement is influenced by various parameters such as the beam waist, the strength of the turbulence and
the wavelength of the beam. We restricted our analysis to the OAM values l = ±1 and we compared our results
to previous work.
Stochastic vortex fields are found in laser speckle, in scintillated beams propagating through a turbulent atmosphere,
in images of holograms produced by Iterative Fourier Transform methods and in the beams produced
by certain diffractive optical elements, to name but a few. Apart from the vortex fields found in laser speckle,
the properties and dynamics of stochastic vortex fields are largely unexplored. Stochastic vortex fields with
non-equilibrium initial conditions exhibit a surprisingly rich phenomenology in their subsequent evolution during
free-space propagation. Currently there does not exist a general theory that can predict this behavior and
only limited progress has thus far been made in its understanding. Curves of the evolution of optical vortex
distributions during free-space propagation that are obtained from numerical simulations, will be presented. A
variety of different stochastic vortex fields are used as input to these simulations, including vortex fields that
are homogeneous in their vortex distributions, as well as inhomogeneous vortex fields where, for example, the
topological charge densities vary sinusoidally along one or two dimensions. Some aspects of the dynamics of
stochastic vortex fields have been uncovered with the aid of these numerical simulations. For example, the
numerical results demonstrate that stochastic vortex fields contain both diffusion and drift motions that are
driven by local and global variations in amplitude and phase. The mechanisms for these will be explained. The
results also provide evidence that global variations in amplitude and phase are caused by variations in the vortex
distributions, giving rise to feedback mechanisms and nonlinear behavior.
Orbital angular momentum (OAM) entangled bi-photons are a resource for the higher dimensional implementation
of quantum cryptography, which allows secure communication over various channels. In the case where
free-space is used as communication channel the initial OAM entangled bi-photon loses some or even all of its
entanglement because of the scintillation that it experiences while propagating through the turbulence in the
atmosphere. This decoherence of OAM entanglement has so far only been studied for the case of weak turbulence.
Unfortunately, it is the more challenging strong turbulence scenario that is relevant for the practical
implementation of free-space quantum communication through the atmosphere. Using an approach that differs
from previous approaches, we derive a master equation for the evolution of an OAM entangled bi-photon during
propagation through turbulence. However, in our approach the equation contains a derivative with respect to the
propagation distance instead of time. The principle is to consider the propagation over an infinitesimal distance
of OAM basis states through a random medium. This approach allows one to include, not only the effect of
turbulence of arbitrary strength, but also the effect of the inner and outer scale of the turbulence, as represented
by the Tartarskii and von Karman spectra. The resulting expression can predict the rates of decoherence for
arbitrary initial OAM entangled states and can be used to calculate the concurrence, which measures the amount
of entanglement, as a function of propagation distance for different initial entangled OAM states.
Polynomial Gaussian beams, which are laser beams with Gaussian envelops and complex bivariate polynomial
prefactors, provide us with a tractable means to investigate the evolution of optical vortices. A formalism for
the propagation of such beams allows one to determine how the coefficients of the polynomial transform during
propagation. This formalism is used to proof that global topological charge is conserved, provided that the
Gaussian envelope of the beam is rotationally symmetric. For astigmatic Gaussian beams the global topological
charge is not conserved and can change during propagation in steps of 2 when one of the optical vortices undergoes
topological charge inversion. The global topological charge is bounded by the order of the polynomial prefactor.
One can also investigate the behavior of vortices in random vortex fields by modelling them as polynomial
Gaussian beams. The phase functions that exist in the vicinity of the annihilation of a vortex dipole are similar,
regardless of the type of beam in which the vortices exist. One can therefore use polynomial Gaussian beams to
find a way to force vortices in random vortex fields to annihilate. The number of optical vortices that can exist
in a polynomial Gaussian beam depends on the reducibility of the polynomial prefactor. During propagation
the reducibility of the prefactor is generally destroyed. However, if the morphologies of the vortices of a fully
reducible prefactor are all the same, the reducibility is maintained during propagation. The results obtained
from the analyses of polynomial Gaussian beams are confirmed by numerical simulations.
A photonic crystal can behave like a medium with a negative effective refractive index within certain frequency
ranges. As a result, under these conditions, a photonic crystal slab acts as a Veselago lens, which produces
images via inverse propagation. To have low reflection the effective index is usually designed to be −1. However,
this implies that the lens is half as thick as the distance between the object and its image. We investigate the
possibility of varying the refractive index along the propagation direction from −1 at the interfaces to an effective
index with a smaller magnitude in the bulk of the medium in order to shorten the Veselago lens while keeping the
reflections from the interfaces low. The hole radius in the photonic crystal is varied over the photonic crystal to
produce a varying effective refractive index. We present finite-difference time-domain simulations of a Veselago
lens with a varying index, demonstrating that Veselago lenses can be shortened in this way.
The phase velocity in photonic crystals is analyzed with the aid of a finite-difference time-domain simulation.
The results of the simulation is used to display an animation of the phase evolution of an electromagnetic wave
propagating through a photonic crystal slab. The phase velocity is indicated by the direction of motion of the
phase inside the photonic crystal. This motion is found to be in the direction of propagation even though the
eective refractive index under the operating conditions is negative. This can be explained by the location of the
dominant values of the Fourier coeffcient function in the k-space representation of fields in the photonic crystal.
Optical vortices are ubiquitous in coherent optical beams -- they appear naturally in speckle fields and can also be excited artificially with, for instance, diffractive optical elements. An understanding of the propagation of such vortices would be useful for applications of this phenomenon. A method is provided to compute the trajectories of optical vortices in complicated scenarios. The Gaussian beam in which the vortices are embedded is expressed in terms of paraxial modes. The positions of the vortices as a function of the propagation distance can then be computed analytically. The case of a vortex dipole is analyzed and shown to undergo annihilation and revival of the pair under certain conditions. Expressions are provided for the trajectories of a canonically launched vortex dipole. The analytical predictions are compared with numerical simulations.
We present an optical system for the polar formatting of data in a spotlight mode SAR. This system is implemented with only one holographic optical element (HOE). Previously such a HOE could not be produced because the phase of the required transmission function of the HOE does not obey the continuity condition which is prerequisite for the conventional implementation of such optical transforms. Here we show how a HOE can be produced to perform the complete polar formatting transform by incorporating branch point phase singularities in the transmission function of the HOE. The computation of the transmission function is shown and numerically computed diffraction patterns obtained from this HOE are also shown.
Optical correlation is sensitive to out-of-plane rotations of the object in the input image. Such an out-of-plane rotation causes the correlation peak to degrade. Here, the degradation of this correlation peak is investigated and a model is proposed to explain its behavior for arbitrary convex objects. First an analytical expression is derived for the peak degradation caused by out-of-plane rotations of flat objects, which is equivalent to a scaling in one dimension. This analytical result compares favorably with experimental results obtained from computer simulation. The 1-D theoretical model is then generalized to explain the behavior of the correlation peak for out-of-plane rotations of arbitrary convex objects. Experimental results for the latter case are also provided.
Diffractive lenses are generally quite sensitive to changes in wavelength. An expression for the spectra of thin diffractive lenses is derived from the Fresnel diffraction integral. Spectra computed with this expression are compared with spectra obtained from computed diffraction patterns. Based on a new definition for the bandwidth of a thin diffractive lens, an expression for this bandwidth is derived.
All applications for diffractive optical elements can be seen as some type of linear optical transform implementation. Methods for the implementation of arbitrary linear optical transformations are discussed. Necessary and sufficient conditions for the implementation of linear optical transformations are considered. These are derived from the properties of a linear optical system. A taxonomy of linear optical transformations is provided. Point transforms are one of the groups of linear optical transformations. These are considered in more detail. The necessary and sufficient conditions naturally leads to the discrete phase technique of implementation. This technique comprise the weighted summation of localized diffraction gratings (or Fresnel lenses in the case of a lensless implementation). As an example of an implementation of this technique the Hough transform is considered. This well known transform is used for the processing of two dimensional images. The conventional Hough transform maps lines in an input image to points in a two dimensional output plane. The cartesian coordinates of the points in the output plane denote the orientations and locations of the lines in the input image.
Holography and optical processing often require coherent input beams with intensity distributions that are different from the usual Gaussian distribution of a laser. Aspherical lenses are used to perform general beam shaping in order to obtain the required intensity distributions. Computer-generated holograms (CGHs) can also be used for beam shaping. In this paper the computational procedure for CGHs, which implements a rotationally symmetric transformation, is supplied. The computational procedure consists of two steps. In the first step a transformation equation is computed from the two known intensity distributions by using an integral equation. This transformation equation will transform the available intensity distribution into the required intensity distribution. It is substituted into a differential equation from which a phase function is cornputed, and is then encoded as a CGH. Two CGHs are required to perform the transformation. The first one diffracts the light to form the desired intensity distribution at a given distance behind the CGH. At that position the second CGH is placed to cancel the diffraction in order to retain the desired intensity distribution. The resulting intensity distribution remains unchanged from then on except for the effect of the diffraction of the distribution itself. General intensity distributions have been implemented with CGHs produced with this technique, and the results are presented in this paper.
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