arXiv subjects

Austin McDaniel

Publications and source records attributed to Austin McDaniel.

9 recordsLinked to original sources

arXiv2024

A vector Helmholtz electromagnetic wave propagator for inhomogeneous media

The vector electric-field Helmholtz equation, containing cross-polarization terms, is factored to produce both pseudo-differential and exponential operator forms of a three-dimensional, one-way, vector, wave equation for propagation through inhomogeneous media. From this operator factorization we develop a high-order approximate, vector Helmholtz propagator that correctly handles forward-arc, high-angle scattering and diffraction from inhomogeneities at all resolved length scales, and seamlessly includes evanescent waves. A rational approximation/partial fraction decomposition of the exponential operator converts the propagator into a moderate number of large, sparse, linear solves whose results are summed together at each step to advance the electric field in space. We use a new AAA-Lawson rational interpolant for this approximation, rather than the more common Pad\'{e} expansions that have appeared in the seismic and ocean acoustics literature previously. GMRES is used to solve these large systems. A direct-solve, free space propagation method proves to be an effective preconditioner for GMRES, but can also serve as a standalone propagator in homogeneous media. Scalar computational examples shown include plane wave diffraction by a circular aperture and Gaussian beam propagation through sine-product and homogeneous refractive index fields. The sine-product example compares its results to that of paraxial propagation through the same media, and demonstrates the substantial differences between these propagator paradigms when the scale of the inhomogeneities is of the order of the fundamental wavelength in the Helmholtz equation. We also examine the convergence of the homogeneous media beam results to fields generated by Clenshaw-Curtis evaluation of the first Rayleigh-Sommerfeld integral for the same initial conditions.

physics.comp-ph↗

arXiv2022

Dynamic Mode Decomposition for Aero-Optic Wavefront Characterization

Aero-optical beam control relies on the development of low-latency forecasting techniques to quickly predict wavefronts aberrated by the Turbulent Boundary Layer (TBL) around an airborne optical system, and its study applies to a multi-domain need from astronomy to microscopy for high-fidelity laser propagation. We leverage the forecasting capabilities of the Dynamic Mode Decomposition (DMD) -- an equation-free, data-driven method for identifying coherent flow structures and their associated spatiotemporal dynamics -- in order to estimate future state wavefront phase aberrations to feed into an adaptive optic (AO) control loop. We specifically leverage the optimized DMD (opt-DMD) algorithm on a subset of the Airborne Aero-Optics Laboratory Transonic (AAOL-T) experimental dataset, characterizing aberrated wavefront dynamics for 23 beam propagation directions via the spatiotemporal decomposition underlying DMD. Critically, we show that opt-DMD produces an optimally de-biased eigenvalue spectrum with imaginary eigenvalues, allowing for arbitrarily long forecasting to produce a robust future-state prediction, while exact DMD loses structural information due to modal decay rates.

physics.flu-dyn↗

arXiv2019

A small delay and correlation time limit of stochastic differential delay equations with state-dependent colored noise

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein-Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

math-ph↗

arXiv2018

Coupling of paraxial and white-noise approximations of the Helmholtz equation in randomly layered media

We study the simultaneous paraxial and white-noise limit of the Helmholtz equation in randomly layered media where the refractive index fluctuations are in the direction of propagation. We consider the regime in which the wavelength is of the same order as the correlation length of the random fluctuations of the refractive index. We show that this simultaneous limit can be taken in this regime by introducing into the equation an arbitrarily small regularization parameter. The corresponding paraxial white-noise approximation that we derive is different from that of the previously studied high-frequency regime. Since the correlation length of the refractive index fluctuations due to atmospheric turbulence varies substantially, our results are relevant for numerous different propagation scenarios including microwave and radiowave propagation through various regions of the atmosphere.

math.NA↗

arXiv2018

Stochastic differential equation model for spontaneous emission and carrier noise in semiconductor lasers

We present a new stochastic differential equation model for the spontaneous emission noise and carrier noise in semiconductor lasers. The correlations between these two types of noise have often been neglected in recent studies of the effects of the noise on the laser dynamics. However, the classic results of Henry show that the intensity noise and the carrier noise are strongly negatively correlated. Our model demonstrates how to properly account for these correlations since the corresponding diffusion coefficients agree exactly with those derived by Henry. We show that in fact in the correct model the spontaneous emission noise and the carrier noise are driven by the same Wiener processes. Furthermore, we demonstrate that the nonzero correlation time of the physical noise affects the mean dynamics of both the electric field amplitude and the carrier number. We show that these are systematic corrections that can be described by additional drift terms in the model.

physics.app-ph↗

arXiv2016

An SDE approximation for stochastic differential delay equations with state-dependent colored noise

We consider a general multidimensional stochastic differential delay equation (SDDE) with state-dependent colored noises. We approximate it by a stochastic differential equation (SDE) system and calculate its limit as the time delays and the correlation times of the noises go to zero. The main result is proven using a theorem about convergence of stochastic integrals by Kurtz and Protter. It formalizes and extends a result that has been obtained in the analysis of a noisy electrical circuit with delayed state-dependent noise, and may be used as a working SDE approximation of an SDDE modeling a real system where noises are correlated in time and whose response to noise sources depends on the system's state at a previous time.

math.PR↗

arXiv2015

Engineering sensorial delay to control phototaxis and emergent collective behaviors

Collective motions emerging from the interaction of autonomous mobile individuals play a key role in many phenomena, from the growth of bacterial colonies to the coordination of robotic swarms. For these collective behaviours to take hold, the individuals must be able to emit, sense and react to signals. When dealing with simple organisms and robots, these signals are necessarily very elementary, e.g. a cell might signal its presence by releasing chemicals and a robot by shining light. An additional challenge arises because the motion of the individuals is often noisy, e.g. the orientation of cells can be altered by Brownian motion and that of robots by an uneven terrain. Therefore, the emphasis is on achieving complex and tunable behaviors from simple autonomous agents communicating with each other in robust ways. Here, we show that the delay between sensing and reacting to a signal can determine the individual and collective long-term behavior of autonomous agents whose motion is intrinsically noisy. We experimentally demonstrate that the collective behaviour of a group of phototactic robots capable of emitting a radially decaying light field can be tuned from segregation to aggregation and clustering by controlling the delay with which they change their propulsion speed in response to the light intensity they measure. We track this transition to the underlying dynamics of this system, in particular, to the ratio between the robots' sensorial delay time and the characteristic time of the robots' random reorientation. Supported by numerics, we discuss how the same mechanism can be applied to control active agents, e.g. airborne drones, moving in a three-dimensional space.

physics.bio-ph↗

arXiv2014

The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.

math-ph↗

arXiv2012

Dynamical Effects of Multiplicative Feedback on a Noisy System

Intrinsically noisy mechanisms drive most physical, biological and economic phenomena, from stock pricing to phenotypic variability. Frequently, the system's state influences the driving noise intensity, as, for example, the actual value of a commodity may alter its volatility or the concentration of gene products may regulate their expression. All these phenomena are often modeled using stochastic differential equations (SDEs). However, an SDE is not sufficient to fully describe a noisy system with a multiplicative feedback, because it can be interpreted according to various conventions -- in particular, Itô calculus and Stratonovitch calculus --, each of which leads to a qualitatively different solution. Which convention to adopt must be determined case by case on the basis of the available experimental data; for example, the SDE describing electrical circuits driven by a noise are known to obey Stratonovich calculus. Once such an SDE-convention pair is determined, it c an be employed to predict the system's behavior under new conditions. Here, we experimentally demonstrate that the convention for a given physical system may actually vary under varying operational conditions. We show that, under certain conditions, a noisy electric circuit shifts to obey Itô calculus, which may dramatically alter the system's long term stability. We track such Stratonovich-to-Itô transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We briefly discuss ramifications of our conclusions for biology and economics: the possibility of similar transitions and their dramatic consequences should be recognized and accounted for where SDEs are employed to predict the evolution of complex phenomena.

cond-mat.stat-mech↗