Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie

Modally Resolved Fabry-Perot Experiment with Semiconductor Waveguides

Based on the interaction between different spatial modes, semiconductor Bragg-reflection waveguides provide a highly functional platform for non-linear optics. Therefore, the control and engineering of the properties of each spatial mode is essential. Despite the multimodeness of our waveguide, the well-established Fabry-Perot technique for recording fringes in the optical transmission spectrum can successfully be employed for a detailed linear optical characterization when combined with Fourier analysis. A prerequisite for the modal sensitivity is a finely resolved transmission spectrum that is recorded over a broad frequency band. Our results highlight how the features of different spatial modes, such as their loss characteristics and dispersion properties, can be separated from each other allowing their comparison. The mode-resolved measurements are important for optimizing the performance of such multimode waveguides by tailoring the properties of their spatial modes.Comment: 8 pages, 7 figure

Multilayer optical learning networks

A new approach to learning in a multilayer optical neural network based on holographically interconnected nonlinear devices is presented. The proposed network can learn the interconnections that form a distributed representation of a desired pattern transformation operation. The interconnections are formed in an adaptive and self-aligning fashioias volume holographic gratings in photorefractive crystals. Parallel arrays of globally space-integrated inner products diffracted by the interconnecting hologram illuminate arrays of nonlinear Fabry-Perot etalons for fast thresholding of the transformed patterns. A phase conjugated reference wave interferes with a backward propagating error signal to form holographic interference patterns which are time integrated in the volume of a photorefractive crystal to modify slowly and learn the appropriate self-aligning interconnections. This multilayer system performs an approximate implementation of the backpropagation learning procedure in a massively parallel high-speed nonlinear optical network

Exploring, tailoring, and traversing the solution landscape of a phase-shaped CARS process

Pulse shaping techniques are used to improve the selectivity of broadband CARS experiments, and to reject the overwhelming background. Knowledge about the fitness landscape and the capability of tailoring it is crucial for both fundamental insight and performing an efficient optimization of phase shapes. We use an evolutionary algorithm to find the optimal spectral phase of the broadband pump and probe beams in a background-suppressed shaped CARS process. We then investigate the shapes, symmetries, and topologies of the landscape contour lines around the optimal solution and also around the point corresponding to zero phase. We demonstrate the significance of the employed phase bases in achieving convex contour lines, suppressed local optima, and high optimization fitness with a few (and even a single) optimization parameter

Modeling the partially coherent behavior of few-mode far-infrared grating spectrometers

Modelling ultra-low-noise far-infrared grating spectrometers has become crucial for the next generation of far-infrared space observatories. Conventional techniques are awkward to apply because of the partially coherent form of the incident spectral field, and the few-mode response of the optics and detectors. We present a modal technique for modelling the behaviour of spectrometers, which allows for the propagation and detection of partially coherent fields, and the inclusion of straylight radiated by warm internal surfaces. We illustrate the technique by modelling the behaviour of the Long Wavelength Band of the proposed SAFARI instrument on the well-studied SPICA mission.Comment: This paper is submitted to Journal Optical Society of America A. When accepted, the paper can be found here: https://opg.optica.org/josaa/home.cf

Digital signal processing for fiber-optic communication systems

As the available bandwidth of optical fibers has been almost fully exploited, Digital Signal Processing (DSP) comes to rescue and is a critical technology underpinning the next generation advanced fiber-optic systems. Literally, it contributes two principal enforcements with respect to information communication. One is the implementation of spectrally-efficient modulation schemes, and the other is the guarantee of the recovery of information from the spectrally-efficient optical signals after channel transmission. The dissertation is dedicated to DSP techniques for the advanced fiber-optic systems. It consists of two main research topics. The first topic is about Fast-orthogonal frequency-division multiplexing (OFDM) — a variant OFDM scheme whose subcarrier spacing is half of that of conventional OFDM. The second one is about Fresnel transform with the derivation of an interesting discrete Fresnel transform (DFnT), and the proposal of orthogonal chirp-division multiplexing (OCDM), which is fundamentally underlain by the Fresnel transform. In the first part, equalization and signal recovery problems result from the halved subcarrier spacing in both double-sideband (DSB) and single-sideband (SSB) modulated Fast-OFDM systems are studied, respectively. By exploiting the relation between the multiplexing kernels of Fast-OFDM systems and Fourier transform, equalization algorithms are proposed for respective Fast-OFDM systems for information recovery. Detailed analysis is also provided. With the proposed algorithms, the DSB Fast-OFDM was experimentally implemented by intensity-modulation and direct detection in the conventional 1.55-μm and the emerging 2-μm fiber-optic systems, and the SSB Fast-OFDM was first implemented in coherent fiber-optic system with a spectral efficiency of 6 bit/s/Hz at 36 Gbps, for the first time. In the second part, Fresnel transform from optical Fresnel diffraction is studied. The discrete Fresnel transform (DFnT) is derived, as an interesting transformation that would be potentially useful for DSP. Its properties are proved. One of the attractive properties, the convolution-preservation property states that the DFnT of a circular convolution of two sequences is equal to the DFnT of either one convolving with the other. One application of DFnT is practically utilized in the proposal of OCDM. In the OCDM system, a large number of orthogonal chirped waveforms are multiplexed for high-speed communication, achieving the maximum spectral efficiency of chirp spread spectrum systems, in the same way as OFDM attains the maximum spectral efficiency of frequency-division multiplexing. Owing to the unique time-frequency properties of chirped waveforms, OCDM outperforms OFDM and single-carrier systems, and is more resilient against the noise effect, especially, when time-domain and frequency-domain distortions are severe. Experiments were carried out to validate the feasibility and advantages of the proposed OCDM systems

Polarization Aberrations in Astronomical Telescopes: The Point Spread Function

Detailed knowledge of the image of the point spread function (PSF) is necessary to optimize astronomical coronagraph masks and to understand potential sources of errors in astrometric measurements. The PSF for astronomical telescopes and instruments depends not only on geometric aberrations and scalar wave diffraction but also on those wavefront errors introduced by the physical optics and the polarization properties of reflecting and transmitting surfaces within the optical system. These vector wave aberrations, called polarization aberrations, result from two sources: (1) the mirror coatings necessary to make the highly reflecting mirror surfaces, and (2) the optical prescription with its inevitable non-normal incidence of rays on reflecting surfaces. The purpose of this article is to characterize the importance of polarization aberrations, to describe the analytical tools to calculate the PSF image, and to provide the background to understand how astronomical image data may be affected. To show the order of magnitude of the effects of polarization aberrations on astronomical images, a generic astronomical telescope configuration is analyzed here by modeling a fast Cassegrain telescope followed by a single 90° deviation fold mirror. All mirrors in this example use bare aluminum reflective coatings and the illumination wavelength is 800 nm. Our findings for this example telescope are: (1) The image plane irradiance distribution is the linear superposition of four PSF images: one for each of the two orthogonal polarizations and one for each of two cross-coupled polarization terms. (2) The PSF image is brighter by 9% for one polarization component compared to its orthogonal state. (3) The PSF images for two orthogonal linearly polarization components are shifted with respect to each other, causing the PSF image for unpolarized point sources to become slightly elongated (elliptical) with a centroid separation of about 0.6 mas. This is important for both astrometry and coronagraph applications. (4) Part of the aberration is a polarization-dependent astigmatism, with a magnitude of 22 milliwaves, which enlarges the PSF image. (5) The orthogonally polarized components of unpolarized sources contain different wavefront aberrations, which differ by approximately 32 milliwaves. This implies that a wavefront correction system cannot optimally correct the aberrations for all polarizations simultaneously. (6) The polarization aberrations couple small parts of each polarization component of the light (∼10^(-4)) into the orthogonal polarization where these components cause highly distorted secondary, or “ghost” PSF images. (7) The radius of the spatial extent of the 90% encircled energy of these two ghost PSF image is twice as large as the radius of the Airy diffraction pattern. Coronagraphs for terrestrial exoplanet science are expected to image objects 10^(-10), or 6 orders of magnitude less than the intensity of the instrument-induced “ghost” PSF image, which will interfere with exoplanet measurements. A polarization aberration expansion which approximates the Jones pupil of the example telescope in six polarization terms is presented in the appendix. Individual terms can be associated with particular polarization defects. The dependence of these terms on angles of incidence, numerical aperture, and the Taylor series representation of the Fresnel equations lead to algebraic relations between these parameters and the scaling of the polarization aberrations. These “design rules” applicable to the example telescope are collected in § 5. Currently, exoplanet coronagraph masks are designed and optimized for scalar diffraction in optical systems. Radiation from the “ghost” PSF image leaks around currently designed image plane masks. Here, we show a vector-wave or polarization optimization is recommended. These effects follow from a natural description of the optical system in terms of the Jones matrices associated with each ray path of interest. The importance of these effects varies by orders of magnitude between different optical systems, depending on the optical design and coatings selected. Some of these effects can be calibrated while others are more problematic. Polarization aberration mitigation methods and technologies to minimize these effects are discussed. These effects have important implications for high-contrast imaging, coronagraphy, and astrometry with their stringent PSF image symmetry and scattered light requirements

Hybrid propagation physics for the design and modeling of astronomical observatories: a coronagraphic example

For diffraction-limited optical systems an accurate physical optics model is necessary to properly evaluate instrument performance. Astronomical observatories outfitted with coronagraphs for direct exoplanet imaging require physical optics models to simulate the effects of misalignment and diffraction. Accurate knowledge of the observatory's PSF is integral for the design of high-contrast imaging instruments and simulation of astrophysical observations. The state of the art is to model the misalignment, ray aberration, and diffraction across multiple software packages, which complicates the design process. Gaussian Beamlet Decomposition (GBD) is a ray-based method of diffraction calculation that has been widely implemented in commercial optical design software. By performing the coherent calculation with data from the ray model of the observatory, the ray aberration errors can be fed directly into the physical optics model of the coronagraph, enabling a more integrated model of the observatory. We develop a formal algorithm for the transfer-matrix method of GBD, and evaluate it against analytical results and a traditional physical optics model to assess the suitability of GBD for high-contrast imaging simulations. Our GBD simulations of the observatory PSF, when compared to the analytical Airy function, have a sum-normalized RMS difference of ~10^-6. These fields are then propagated through a Fraunhofer model of a exoplanet imaging coronagraph where the mean residual numerical contrast is 4x10^-11, with a maximum near the inner working angle at 5x10^-9. These results show considerable promise for the future development of GBD as a viable propagation technique in high-contrast imaging. We developed this algorithm in an open-source software package and outlined a path for its continued development to increase the fidelity and flexibility of diffraction simulations using GBD.Comment: 58 pages, 15 figures, preprint version for article in press. Accepted to SPIE's Journal of Astronomical Telescopes, Instruments, and Systems on October 23 202

Fresnel diffractive imagery arrays : ground-based validation and space applications

L'imageur de Fresnel est un instrument focalisation diffractive qui est développé pour l'observation astrophysique depuis l'espace. Il est a longue focale, et de ce fait requiert le vol en formation. Pendant ma thèse, le concept a été étudié avec des prototypes au sol et validé en mesurant la qualité des images obtenues et les fonctionnalités de l'instrument. Depuis 2007, un prototype diffractif d'imageur de Fresnel de génération 2 a été connu, il a été mis au point en 2009, et fonctionne en parallèle sur un tube de lunette long de 18 mètres. Plusieurs grilles de Fresnel carrées de 200x200 mm ont été construites, testées au sol par l'observation de cibles astrophysiques, et améliorées pour tendre vers la qualité requise pour des applications depuis l'espace avec des grilles grande taille 4m ou plus. Des observations sont faites à Nice depuis juin 2009 avec un prototype de 200x200 mm d'ouverture, et 18 m de focale. Elles montrent que le concept fonctionne en conditions réelles. Les données acquises et les images obtenues démontrent la fonctionnalité et la qualité de cet imageur de Fresnel. La première partie de la thèse présente le principe de focalisation diffractive utilisé dans l'imageur de Fresnel, la formation des images, et le détail de sa conception : les caractéristiques des différents composants optiques du système et leurs performances nominales. La deuxième partie décrit les essais faits pour valider le concept et pour démontrer la formation des images à haute dynamique, les résultats d'observations, et l'optimisation du prototype au sol. La troisième et dernière partie propose une mission spatiale, et sa configuration pour l'observation depuis l'espace. Une cible telle qu'un disque d'accrétion autour d'une étoile jeune est prise comme cas d'étude pour la mission spatiale d'un imageur de Fresnel avec une ouverture de 20X20 m, ouvrant de nouveaux domaines de l'astrophysique dans l'ultraviolet.The Fresnel Array is a diffractive focalization instrument developed to be applied for space observation in astrophysics. It implies long-focal length focusing, thus, formation flying in space. During my thesis, the concept has been tested with ground-based prototypes and validated by measuring the image quality and instruments functionalities. Since 2007, a second generation Fresnel Diffractive imager prototype has been built, and operates on a 18 meter long refractor tube. 200x200 mm square Fresnel arrays have been constructed, tested with ground-based observation of astrophysical targets, and upgraded to demonstrate that they meet the quality for space applications at large array size : 4m. Observations are made in Nice since June 2009. They prove that the concept is working in real conditions, The collected data and the consequent images demonstrate the functionality and quality of Fresnel diffractive Imager. The first part of this thesis introduces the principle of Fresnel diffractive Imaging and its design. The specifications of the components in the system present the nominal performance. The second part describes the tests made to validate the instrument concept and demonstrate the high dynamic range imaging, the observation results, and the optimization of the ground-based prototype. The last part proposes a space mission, providing the space observations configurations. A target such as an accretion disc around a young star is shown as a study case for a Fresnel space mission with 20x20 meter arrays, opening new domains of astrophysics in the ultra violet