 Award ID(s):
 2010046
 NSFPAR ID:
 10475953
 Publisher / Repository:
 SIAM MMS
 Date Published:
 Journal Name:
 Multiscale Modeling & Simulation
 Volume:
 21
 Issue:
 1
 ISSN:
 15403459
 Page Range / eLocation ID:
 80 to 118
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may or may not be integrable. We focus attention mostly on the nonintegrable case, which corresponds to a random perturbation with longrange correlations, that is, relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a highfrequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation, and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schr\"odingertype equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications, such as imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence. We show explicitly that the commonly used approximations for the decorrelation of spatial and spectral components are appropriate for the Kolmogorov model but fail for models with longrange correlations.more » « less

A theory for the characterization of the fourthorder moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian–Schell model is used for the partially coherent random source. The whitenoise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the ItôSchrödinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourthorder moment equations are considered. The general fourthorder moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of the intensity covariance function. The form of the intensity covariance function derives from the solution of the transport equation for the Wigner distribution associated with the secondorder wave moment. The fourthorder moment results for polarized waves are used in an application for imaging of partially coherent sources.more » « less

null (Ed.)Abstract We outline and interpret a recently developed theory of impedance matching or reflectionless excitation of arbitrary finite photonic structures in any dimension. The theory includes both the case of guided wave and freespace excitation. It describes the necessary and sufficient conditions for perfectly reflectionless excitation to be possible and specifies how many physical parameters must be tuned to achieve this. In the absence of geometric symmetries, such as parity and timereversal, the product of parity and timereversal, or rotational symmetry, the tuning of at least one structural parameter will be necessary to achieve reflectionless excitation. The theory employs a recently identified set of complex frequency solutions of the Maxwell equations as a starting point, which are defined by having zero reflection into a chosen set of input channels, and which are referred to as Rzeros. Tuning is generically necessary in order to move an Rzero to the real frequency axis, where it becomes a physical steadystate impedancematched solution, which we refer to as a reflectionless scattering mode (RSM). In addition, except in singlechannel systems, the RSM corresponds to a particular input wavefront, and any other wavefront will generally not be reflectionless. It is useful to consider the theory as representing a generalization of the concept of critical coupling of a resonator, but it holds in arbitrary dimension, for arbitrary number of channels, and even when resonances are not spectrally isolated. In a structure with parity and timereversal symmetry (a real dielectric function) or with parity–time symmetry, generically a subset of the Rzeros has real frequencies, and reflectionless states exist at discrete frequencies without tuning. However, they do not exist within every spectral range, as they do in the special case of the Fabry–Pérot or twomirror resonator, due to a spontaneous symmetrybreaking phenomenon when two RSMs meet. Such symmetrybreaking transitions correspond to a new kind of exceptional point, only recently identified, at which the shape of the reflection and transmission resonance lineshape is flattened. Numerical examples of RSMs are given for onedimensional multimirror cavities, a twodimensional multiwaveguide junction, and a multimode waveguide functioning as a perfect mode converter. Two solution methods to find Rzeros and RSMs are discussed. The first one is a straightforward generalization of the complex scaling or perfectly matched layer method and is applicable in a number of important cases; the second one involves a modespecific boundary matching method that has only recently been demonstrated and can be applied to all geometries for which the theory is valid, including free space and multimode waveguide problems of the type solved here.more » « less

Dynamic and steadystate aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium through all channels, the Wigner time delay, which is the sum of dwell times in all channels coupled to the medium, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among fundamental wave properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We begin with the demonstrations that the transmission time in random 1D media is equal to the density of states even in the presence of ultrastrong absorption and that its ensemble average is independent of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the transmission time remains equal to the density of states and is independent of the scattering strength in unitary systems but falls with absorption to a degree that increases with the strengths of absorption and scattering, and the number of channels coupled to the medium. We show that the relationships between key propagation variables in nonHermitian systems can be understood in terms of the singularities of the phase of the determinant of the transmission matrix. The poles of the transmission matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the transmission zeros are topological: in unitary systems they occur only singly on the real axis or as conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation of the sample has a squareroot singularity at a zero point at which two single zeros and a complex pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and zeros. The sum over poles is the density of states with an average that is independent of scattering and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of channels in ways that make it possible to control ultranarrow spectral features in transmission and transmission time. We show that the field, including the contribution of the still coherent incident wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell time even in the presence of loss or gain.more » « less

Superresolution optical sensing is of critical importance in science and technology and has required prior information about an imaging system or obtrusive nearfield probing. Additionally, coherent imaging and sensing in heavily scattering media such as biological tissue has been challenging, and practical approaches have either been restricted to measuring the field transmission of a single point source, or to where the medium is thin. We present the concept of farsubwavelength spatial sensing with relative object motion in speckle as a means to coherently sense through heavy scatter. Experimental results demonstrate the ability to distinguish nominally identical objects with nanometerscale translation while hidden in randomly scattering media, without the need for precise or known location and with imprecise replacement. The theory and supportive illustrations presented provide the basis for superresolution sensing and the possibility of virtually unlimited spatial resolution, including through thick, heavily scattering media with relative motion of an object in a structured field. This work provides enabling opportunities for material inspection, security, and biological sensing.