Silicon photonics devices for integrated analog signal processing and sampling

Silicon photonics offers the possibility of a reduction in size weight and power for many optical systems, and could open up the ability to build optical systems with complexities that would otherwise be impossible to achieve. Silicon photonics is an emerging technology that has already been inserted into commercial communication products. This technology has also been applied to analog signal processing applications. MIT Lincoln Laboratory in collaboration with groups at MIT has developed a toolkit of silicon photonic devices with a focus on the needs of analog systems. This toolkit includes low-loss waveguides, a high-speed modulator, ring resonator based filter bank, and all-silicon photodiodes. The components are integrated together for a hybrid photonic and electronic analog-to-digital converter. The development and performance of these devices will be discussed. Additionally, the linear performance of these devices, which is important for analog systems, is also investigated

A Spectral Graph Uncertainty Principle

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure

Preasymptotic Convergence of Randomized Kaczmarz Method

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the preasymptotic convergence behavior of the randomized Kaczmarz method, and show that the low-frequency error (with respect to the right singular vectors) decays faster during first iterations than the high-frequency error. Under the assumption that the inverse solution is smooth (e.g., sourcewise representation), the result explains the fast empirical convergence behavior, thereby shedding new insights into the excellent performance of the randomized Kaczmarz method in practice. Further, we propose a simple strategy to stabilize the asymptotic convergence of the iteration by means of variance reduction. We provide extensive numerical experiments to confirm the analysis and to elucidate the behavior of the algorithms.Comment: 20 page

Integrated mode-locked lasers in a CMOS-compatible silicon photonic platform

CLEO: Science and Innovations 2015 San Jose, California United States 10–15 May 2015 ISBN: 978-1-55752-968-8 From the session: Silicon Photonic Systems (SM2I)The final version is available from the publisher via the DOI in this record.Integrated components necessary for a mode-locked laser are demonstrated on a platform that allows for monolithic integration with active silicon photonics and CMOS circuitry. CW lasing and Q-switched mode-locking are observed in the full structures.This work was supported under the DARPA E-PHI project, grant no. HR0011-12-2-0007

Problems in Signal Processing and Inference on Graphs

Modern datasets are often massive due to the sharp decrease in the cost of collecting and storing data. Many are endowed with relational structure modeled by a graph, an object comprising a set of points and a set of pairwise connections between them. A ``signal on a graph'' has elements related to each other through a graph---it could model, for example, measurements from a sensor network. In this dissertation we study several problems in signal processing and inference on graphs. We begin by introducing an analogue to Heisenberg's time-frequency uncertainty principle for signals on graphs. We use spectral graph theory and the standard extension of Fourier analysis to graphs. Our spectral graph uncertainty principle makes precise the notion that a highly localized signal on a graph must have a broad spectrum, and vice versa. Next, we consider the problem of detecting a random walk on a graph from noisy observations. We characterize the performance of the optimal detector through the (type-II) error exponent, borrowing techniques from statistical physics to develop a lower bound exhibiting a phase transition. Strong performance is only guaranteed when the signal to noise ratio exceeds twice the random walk's entropy rate. Monte Carlo simulations show that the lower bound is quite close to the true exponent. Next, we introduce a technique for inferring the source of an epidemic from observations at a few nodes. We develop a Monte Carlo technique to simulate the infection process, and use statistics computed from these simulations to approximate the likelihood, which we then maximize to locate the source. We further introduce a logistic autoregressive model (ALARM), a simple model for binary processes on graphs that can still capture a variety of behavior. We demonstrate its simplicity by showing how to easily infer the underlying graph structure from measurements; a technique versatile enough that it can work under model mismatch. Finally, we introduce the exact formula for the error of the randomized Kaczmarz algorithm, a linear system solver for sparse systems, which often arise in graph theory. This is important because, as we show, existing performance bounds are quite loose.Engineering and Applied Sciences - Engineering Science

Strategies for limiting interference and interception of free space optical communications

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 167-168).Free space optical systems provide an attractive solution to communication needs that require inexpensive, easily deployable links capable of high data rate transmissions. A major challenge of free space optical communication is ensuring the integrity and confidentiality of the transmitted information. In an optical wireless network with close-in users, communication between two users could interfere with another link in the network. Such systems are also susceptible to eavesdropping, especially inside the main lobe of the transmitted beam. In this thesis, we propose a method of controlling the direction of energy propagation from an optical transmitter to maximize the power received by the remote terminal of a link while limiting the power received in a broadly defined region within the main lobe of the transmission. We consider specifically an optical transmitter comprised of an array of apertures with controllable amplitudes and phases, and we approximate the intended suppression region with a finite number of points. We assume the total transmitted power is held fixed. Via iterative numerical methods, we solve a nonlinear optimization problem for the weight vector that maximizes the intensity at the receiver while limiting the intensity at the specified suppression points to below some fraction of the intensity at the receiver. For a linear aperture array, we show that without overly limiting the power to the intended receiver, it is possible to suppress the signal intensity in a 1 beamwidth region located 0.2 beamwidths from the intended user down to one tenth of the intensity at the intended receiver. For a two dimensional array, we show that we can similarly suppress a !!!!! beamwidth region as close as 0.2333 beamwidths to the intended receiver. We further show that by increasing the number of suppression points used to approximate the suppression region, we can suppress a region much closer to the receiver, but at the cost of significantly lowering the intensity at the receiver. We also observe a tradeoff between the size of the suppression region and our ability to limit the signal intensity throughout the entire region. We show that our ability to successfully suppress power to the required region is limited by both our available transmit power and our uncertainty of the position of the eavesdropper or network user.by Manishika P. Agaskar.M. Eng

Randomized Kaczmarz algorithms: Exact MSE analysis and optimal sampling probabilities

The Kaczmarz method, or the algebraic reconstruction technique (ART), is a popular method for solving large-scale overdetermined systems of equations. Recently, Strohmer et al. proposed the randomized Kaczmarz algorithm, an improvement that guarantees exponential convergence to the solution. This has spurred much interest in the algorithm and its extensions. We provide in this paper an exact formula for the mean squared error (MSE) in the value reconstructed by the algorithm. We also compute the exponential decay rate of the MSE, which we call the “annealed” error exponent. We show that the typical performance of the algorithm is far better than the average performance. We define the “quenched” error exponent to characterize the typical performance. This is far harder to computethan the annealed error exponent, but we provide an approximation that matches empirical results. We also explore optimizing the algorithm’s row-selection probabilities to speed up the algorithm’s convergence.Engineering and Applied Science

A one femtojoule athermal silicon modulator

Silicon photonics has emerged as the leading candidate for implementing ultralow power wavelength division multiplexed communication networks in high-performance computers, yet current components (lasers, modulators, filters, and detectors) consume too much power for the femtojouleclass links that will ultimately be required. Here, we propose, demonstrate, and characterize the first modulator to achieve simultaneous high-speed (25-Gb/s), low voltage (0.5VPP) and efficient 1-fJ/bit error-free operation while maintaining athermal operation. Both the low energy and athermal operation were enabled by a record free-carrier accumulation/depletion response obtained in a vertical p-n junction device that at 250-pm/V (30-GHz/V) is up to ten times larger than prior demonstrations. Over a 7.5{\deg}C temperature range, the massive electro-optic response was used to compensate for thermal drift without increasing energy consumption and over a 10{\deg}C temperature range, increasing energy consumption by only 2-fJ/bit. The results represent a new paradigm in modulator development, one where thermal compensation is achieved electro-optically.Comment: 23 pages, 5 figure