Davood Mardani

Efficient modal analysis using compressive optical interferometry.

Interferometry is routinely used for spectral or modal analysis of optical signals. By posing interferometric modal analysis as a sparse recovery problem, we show that compressive sampling helps exploit the sparsity of typical optical signals in modal space and reduces the number of required measurements. Instead of collecting evenly spaced interferometric samples at the Nyquist rate followed by a Fourier transform as is common practice, we show that random sampling at sub-Nyquist rates followed by a sparse reconstruction algorithm suffices. We demonstrate our approach, which we call compressive interferometry, numerically in the context of modal analysis of spatial beams using a generalized interferometric configuration. Compressive interferometry applies to widely used optical modal sets and is robust with respect to noise, thus holding promise to enhance real-time processing in optical imaging and communications.

Basis-neutral Hilbert-space analyzers

Interferometry is one of the central organizing principles of optics. Key to interferometry is the concept of optical delay, which facilitates spectral analysis in terms of time-harmonics. In contrast, when analyzing a beam in a Hilbert space spanned by spatial modes – a critical task for spatial-mode multiplexing and quantum communication – basis-specific principles are invoked that are altogether distinct from that of ‘delay’. Here, we extend the traditional concept of temporal delay to the spatial domain, thereby enabling the analysis of a beam in an arbitrary spatial-mode basis – exemplified using Hermite-Gaussian and radial Laguerre-Gaussian modes. Such generalized delays correspond to optical implementations of fractional transforms; for example, the fractional Hankel transform is the generalized delay associated with the space of Laguerre-Gaussian modes, and an interferometer incorporating such a ‘delay’ obtains modal weights in the associated Hilbert space. By implementing an inherently stable, reconfigurable spatial-light-modulator-based polarization-interferometer, we have constructed a ‘Hilbert-space analyzer’ capable of projecting optical beams onto any modal basis.

Adaptive sequential compressive detection

Sparsity is at the heart of numerous applications dealing with multidimensional phenomena with low-information content. The primary question that this work investigates is whether, and how much, further compressive gains could be achieved if the goal of the inference task does not require exact reconstruction of the underlying signal. In particular, if the goal is to detect the existence of a sparse signal in noise, it is shown that the number of measurements can be reduced. In contrast to prior work, which considered non-adaptive strategies, a sequential adaptive approach for compressed signal detection is proposed. The key insight is that the decision can be made as soon as a stopping criterion is met during sequential reconstructions. Two sources of performance gains are studied, namely, compressive gains due to adaptation, and computational gains via recursive sparse reconstruction algorithms that fuse newly acquired measurements and previous signal estimates.

Signal reconstruction from interferometric measurements under sensing constraints

This paper develops a unifying framework for signal reconstruction from interferometric measurements that is broadly applicable to various applications of interferometry. In this framework, the problem of signal reconstruction in interferometry amounts to one of basis analysis. Its applicability is shown to extend beyond conventional temporal interferometry, which leverages the relative delay between the two arms of an interferometer, to arbitrary degrees of freedom of the input signal. This allows for reconstruction of signals supported in other domains (e.g., spatial) with no modification to the underlying structure except for replacing the standard temporal delay with a generalized delay, that is, a practically realizable unitary transformation for which the basis elements are eigenfunctions. Under the proposed model, the interferometric measurements are shown to be linear in the basis coefficients, thereby enabling efficient and fast recovery of the desired information. While the corresponding linear transformation has only a limited number of degrees of freedom set by the structure of the interferometer giving rise to a highly constrained sensing structure, we show that the problem of signal recovery from such measurements can still be carried out compressively. This signifies significant reduction in sample complexity without introducing any additional randomization as is typically done in prior work leveraging compressive sensing techniques. We provide performance guarantees under constrained sensing by proving that the transformation satisfies sufficient conditions for successful reconstruction of sparse signals using concentration arguments. We showcase the effectiveness of the proposed approach using simulation results, as well as actual experimental results in the context of optical modal analysis of spatial beams.

On sparse recovery with Structured Noise under sensing constraints

This paper considers sparse signal recovery under sensing constraints originating from the limitations of practical data acquisition systems. Such limitations introduce non-linearities in the underlying measurement model. We first develop a more accurate measurement model with structured noise representing a known non-linear function of the sparse signal obtained by leveraging side information about the physical sampling structure. Then, we devise two iterative denoising algorithms, namely, Orthogonal Matching Pursuit with Structured Noise (OMPSN), and Subspace Pursuit with Structured Noise (SPSN) that are shown to enhance the quality of sparse recovery in presence of physical constraints by iteratively estimating and eliminating the non-linear term from the measurements. Numerical and simulation results demonstrate that the proposed algorithms outperform standard algorithms in detecting the support and estimating the sparse vector.

Sparse reconstruction under sensing constraints: A controlled approach

This paper considers a controlled approach to sparse reconstruction under sensing constraints that have been largely ignored in related work on compressive sensing and sparse recovery. The first constraint stems from the reduced number of degrees of freedom of actual information gathering systems, which imposes specific structures on the sensing matrix departing from the conventional random ensembles. The second limitation originates from the unknown statistical model of the corrupting noise. A controlled sensing approach is proposed to guide the collection of informative measurements given the constrained sensing structure. In the presence of additive noise with unknown statistics, the proposed approach is shown to yield stable recovery and dispenses with the usual de-noising requirements. In addition, a sequential implementation with a stopping rule is proposed, thereby reducing the sample complexity for a target performance in reconstruction.