Diffractive optics applied to free-space optical interconnects

The optimum design of free-space optical interconnection systems utilizing diffractive optics is determined from a practical engineering standpoint for systems ranging from space invariant to fully space variant. System volume is calculated in terms of parameters such as the f-number of the diffractive lens, the wavelength of light, and also the total number, size, and separation of the optical sources and detectors. Performance issues such as interconnection complexity, diffraction efficiency, and signal-tonoise ratio are discussed. Diffractive optics fabricated by electron-beam direct-write techniques are used to provide experimental results for both shuffle-exchange and twin-butterfly free-space optical interconnects.     

Influence of geometry on polychromatic speckle contrast

Understanding speckle behavior is very important in speckle metrology application. The contrast of a polychromatic speckle depends not only on surface roughness and the coherence length of a light source, as shown in previous works, but also on optical geometry. We applied the Fresnel approach of diffraction theory for the free-space geometry and derived a simple analytical relationship between contrast, coherence length, size of illuminated spot, and distances between source, object, and observation plane. The effect of contrast reduction is found to be significant for low-coherence light sources.     

Monte Carlo modeling of spatial coherence: free-space diffraction

We present a Monte Carlo method for propagating partially coherent fields through complex deterministic optical systems. A Gaussian copula is used to synthesize a random source with an arbitrary spatial coherence function. Physical optics and Monte Carlo predictions of the first- and second-order statistics of the field are shown for coherent and partially coherent sources for free-space propagation, imaging using a binary Fresnel zone plate, and propagation through a limiting aperture. Excellent agreement between the physical optics and Monte Carlo predictions is demonstrated in all cases. Convergence criteria are presented for judging the quality of the Monte Carlo predictions.     

Pupil Effect in a GRIN Lens with Gaussian Illumination

Abstract

Using the propagator method, the quasi-geometric transmittance function and complex amplitude distribution produced by a GRIN lens are evaluated when it is illuminated by a Gaussian beam and the pupil effect is taken into account. The effective pupil is defined and derived by both quasi-geometrical optics and the exact Fresnel diffraction, which it is evaluated in a Rubinowicz representation.     

Fresnel diffraction as an aperture edge integral

Abstract

The standard Fresnel diffraction wave field from an aperture in a plane screen involves an integral over the aperture area. Here it is reduced to an aperture edge integral instead. The reduction really amounts to performing one integral but can be considered as the reduction of the ‘Maggi-Rubinovics’ edge formula from Kirchhoff optics to paraxial (i.e. Fresnel) optics.     

Far zone spectral shifts in light propagation from the exit face of a few-mode fibre waveguide guiding the frequency-dependent linearly polarized modes

Abstract

Far zone spectral shifts, including free space propagation of light from a planar secondary, partially coherent source represented by the exit face of a few-mode fibre waveguide, are investigated in the paraxial regime by using frequency-dependent linearly polarized (LP) modes guided by the fibre waveguide. The origin of the far zone blue or red spectral shift of individual frequency-dependent LP modes, as well as of their superposition at the particular observation point, is clarified. In particular, the dependence of the wavelength shifts of spectra on the position of the observation point in the far zone plane and the primary source linewidth is studied for the two-mode, weakly guiding, step-index fibre waveguide.     

Jones matrix treatment for polarization fourier optics

Abstract

In this work we introduce the use of a Jones matrix method to evaluate the far-field diffraction produced by spatially variant polarization elements. We extend the scalar Fourier optics theory to a vectorial theory by the use of the Jones matrix formalism. With this method it is possible to analyse the diffraction pattern and the local state of polarization in the Fraunhofer approximation by means of the usual Jones matrix calculus.     

Free-space communication through turbulence: a comparison of plane-wave and orbital-angular-momentum encodings

Abstract

Free-space communication allows one to use spatial mode encoding, which is susceptible to the effects of diffraction and turbulence. Here, we discuss the optimum communication modes of a system while taking such effects into account. We construct a free-space communication system that encodes information onto the plane-wave (PW) modes of light. We study the performance of this system in the presence of atmospheric turbulence, and compare it with previous results for a system employing orbital-angular-momentum (OAM) encoding. We are able to show that the PW basis is the preferred basis set for communication through atmospheric turbulence for a system with a large Fresnel number product. This study has important implications for high-dimensional quantum key distribution systems.     

Rigorous coupled wave analysis applied to transmission efficiency of diffractive beam array relays for free-space optical interconnects

Parallel high-density free-space optical interconnects typically relay multiple channels in an array configuration; thus, they require good uniformity across their aperture for optimum performance. Rigorous coupled wave analysis is used to determine the throughput off-axis diffraction efficiency for Fresnel lenses within a diffractive imaging relay. The rigorous results are compared with scalar theory and show a significant nonuniformity not predicted by scalar theory. However, the polarization sensitivity is found to be negligible for the f-numbers considered (f/2.9 to f/10.2). These results are supported by experiment.     

Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation

Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.     

Computation considerations and fast algorithms for calculating the diffraction integral

Abstract

One of the most basic optical ‘components’ is free-space propagation. A common approximation used when calculating the resultant field distribution after propagation is the Fresnel integral. This integral can be evaluated in two ways: directly or by using the angular spectrum. In this paper, we estimate the regions in which each mode of evaluation is preferable according to computing efficiency and accuracy considerations. A fast numerical algorithm is introduced for each region. The result is relevant also for the evaluation of the Rayleigh-Sommerfeld diffraction formula.     

Radiative transfer calculations in multilayer systems with smooth or rough interfaces

Abstract

A very general discrete-ordinate method is presented for radiative transfer calculations in plane-parallel systems with partially reflecting interfaces. It is the natural extension of existing treatments for atmospheres (like DISORT) to account for changes of refractive index (Fresnel interfaces). The system under study is made of several dielectric layers, each layer containing spherical particles which produce anisotropic scattering (Mie scatterers). Incident light can be either collimated at normal or oblique incidence, or diffuse. A reflecting substrate can be added to the system. A procedure can be used to calculate the scattered fluxes at arbitrary angles, from the fluxes firstly obtained at a fixed set of polar angles. An efficient matrix formalism allows one to consider various boundary conditions: smooth or rough (scattering) interfaces. Surface scattering is introduced through a combination of Kirchhoff's expressions (for the specular reflection attenuation) and a geometrical optics model (for the scattering lobe). Comparisons with previous models and examples of simulations are presented. The main limitation of the method seems to be the approximate account for the polarization in this scalar treatment.     

Achromatic optical fourier transformer with planar-integrated free-space optics

We address the problem of achromatization of an optical system for the realization of planar-integrated, free-space optics. In particular we demonstrate an integrated optical Fourier transformation module that was achromatized for the visible spectrum by means of a diffractive lens doublet. The optical system design is studied by using the parabolic approximation of the scalar diffraction theory, including terms related to astigmatism. Based on the method of ABCD ray matrices, the optical specifications of the lens doublet are derived and the chromatic correction effect is quantified. For experimental confirmation the diffraction patterns of various grating structures are evaluated.     

Synthesis of three-dimensional light fields with binary spatial light modulators

Computation of a binary spatial light modulator (SLM) pattern that generates a desired light field is a challenging quantization problem for which several algorithms have been proposed, mainly for far-field or Fourier plane reconstructions. We study this problem assuming that the desired light field is synthesized within a volumetric region in the non-far-field range after free space propagation from the SLM plane. We use Fresnel and Rayleigh-Sommerfeld scalar diffraction theories for propagation of light. We show that, when the desired field is confined to a sufficiently narrow region of space, the ideal gray-level complex-valued SLM pattern generating it becomes sufficiently low pass (oversampled) so it can be successfully halftoned into a binary SLM pattern by solving two decoupled real-valued constrained halftoning problems. Our simulation results indicate that, when the synthesis region is considered, the binary SLM is indistinguishable from a lower resolution full complex gray-level SLM. In our approach, free space propagation related computations are done only once at the beginning, and the rest of the computation time is spent on carrying out standard image halftoning.     

Spatial coherence wavelets

Abstract

It is shown that the cross-spectral density at a plane in the Fresnel—Fraunhofer domain can be expressed as a certain diffraction pattern, which is generated by the superposition of second-order spatial coherence wavelets that emerge from the aperture. The amplitude of each coherence wavelet exhibits units of power density (average energy) and the power spectrum at the far zone plane will be the summation of the amplitudes of such wavelets. Thus, the spatial coherence wavelet constitutes a vehicle for both correlation and energy transport in free space. Some simulation results are discussed to illustrate these ideas.     

Localized fringes from identical twin image sources

Abstract

Twin coherent monochromatic point sources of light produce Young's optical interference fringes throughout space. Often in interferometry, there are effectively identical twin sources (partially reflected images of a single source by plane surfaces). They are small, but are far from being point sources. The consequence is long known: the fringes are ‘localized’, being visible in a certain region only. Here the isometry between the image sources (their identical shape and size) is exploited to deduce the three-dimensional shape of this region and the form of the fringes within it by means of wave optics. In the short wavelength limit the fringes are localized in a definite tube about a certain curve in space, given in cylindrical polar coordinates (r, φ, z) by z = Z tan φ/ tan φ, r = R, where (R, ± φ, ±, Z) locate the sources (and describe their mutual orientation). Also the fringe sheets within this tube differ from the hyperboloids of Young's fringes, being instead nested pieces of concentric cylindrical surfaces.     

Fraunhofer and Fresnel field distributions from the LP01 mode in a micro-sized hollow optical fibre

Abstract

The electric field distribution of a LP₀₁ mode and properties of the evanescent-wave field in a micro-sized hollow optical fibre under the weakly guiding approximation are analysed. The far- and near-field distributions of the output beam from the LP₀₁ mode and its propagation characteristics in free space are calculated numerically from Fraunhofer and Fresnel diffraction theory. We also derive an analytical expression of the far-field distribution of the LP₀₁ mode and discuss its applicable conditions. Our study shows that the output beam of the LP₀₁ mode in the hollow fibre is a dark hollow divergent beam whose near-field divergent angle is slightly smaller than the far-field divergent angle. The dark spot size (DSS) of the beam in the near field is about equal to the beam radius r ₀, whereas the DSS in the far field is smaller than its beam radius. We analyse the dependences of the far-field divergent angle of the output beam from the LP₀₁ mode on various fibre parameters and briefly discuss potential applications of the dark hollow beam in atomic physics and atomic optics.     

Numerical calculation of a converging vector electromagnetic wave diffracted by an aperture using Borgnis potentials. II. Application to the study of focal shift

Focal shift of the converging spherical wavefront light diffracted by a circular aperture is numerically studied with the method of calculating the vector diffractive field by using Borgnis potentials given in Part I [J. Opt. Soc. Am. A23, 872 (2006)]. The quantitative dependence of the focal shift on the geometric parameters is discussed. The focal shift is mainly determined by the Fresnel number (N(f)) on the geometric focusing plane of the converging light, and an empirical formula between the fractional focal shift and the Fresnel number is deduced for N(f)<2. The focal shift of the same geometry is also studied on the basis of the scalar Rayleigh theory of diffraction, and its comparison with and difference from the result of our method are presented.