We present a method for detecting indirect, off-axis light radiated by a laser beam propagating in the atmosphere, in the presence of (primarily solar) background. Unlike in most existent methods, where laser light detection is based on its monochromaticity or intensity, the proposed approach uses a high degree of temporal coherence of laser radiation as the discriminating factor against potentially strong but very low-coherence background. The method also relies on intensity interferometry, rather than amplitude interferometry approaches frequently found in the literature. The analytic developments of this paper revolve around the evaluation of two quantities-the intensity correlation signal and its fluctuations (or noise)-envisaged as measures of the proposed coherence signature, designed to apply to both stationary and pulsed radiation. Reliable evaluation of the noise, due to strong statistical fluctuations of the high-temporal-coherence scattered field, is essential, as well as challenging, because of the presence of higher, up to the fourth order, moments of the measured optical intensities. We calculated full effects of statistical fluctuations of the laser- and background-related radiation and established the optimal detector parameters maximizing the obtained signal-to-noise value. We show that the signal-to-noise ratio may be on the order of 10 for a single recorded pulse and, in practically relevant case of a train of pulses, increases as the square-root of the number of pulses in the sequence. While the proposed approach has direct applications in the development of laser warning systems, closely related techniques should be applicable in active imaging, particularly in LiDAR systems operating in the presence of strong background.

We have recently proposed (E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, Waves in Random and Complex Media, 33, 1319-1345, 2023 and Journal of the Optical Society of America A, submitted, 2025) a method for detection of off-axis radiation of a (stationary or pulsed) laser beam propagating in the atmosphere, in the presence of potentially strong solar background. The proposed approach uses a high degree of temporal coherence of laser radiation as the discriminating factor against very low-coherence background.

We construct a coupled set of integral equations for metasurface sheets, following from non-local, derivative boundary conditions. Such integral equations are applicable to infinite as well as finite size metasurface sheets with a boundary, having a discontinuity in the material properties. We present expressions for matrix elements calculated with suitably modified Rao-Wilton-Glisson basis functions accounting for material discontinuity at sheet boundary edges and the resulting edge behavior of the solutions. We discuss representative applications of our approach for planar sheet problems.

We describe some aspects of a method for detecting indirect laser light (the weak off-axis or back-scattered radiation of a laser beam scattering in the atmosphere or from a rough surface) in the presence of low coherence (primarily solar) background. Unlike in most existent methods, which detect laser light based on its monochromaticity or intensity, the proposed approach uses a high degree of temporal coherence as the distinguishing property of laser radiation. It also utilizes intensity interferometry, which, in our case, amounts to the measurement of correlations of time-averaged photocurrents generated by two photodetectors located within the correlation range. Those correlations are proportional to the radiation's co-herence time, hence are sizable for the scattered laser light and negligible for the background. However, the low-coherence background does contribute to the noise (understood as statistical fluctuations of the current correlation); additional large noise components come from intensity fluctuations of the scattered laser radiation itself. We carry out a rigorous calculation of the signal and noise, the latter involving high - up to eighth - moments of the scattered field. The derived expressions allow us to determine the preferred range of the detector parameters, maximizing the signal-to-noise ratio (which can be sizable even for detection of a single pulse) and minimizing its sensitivity to the background.

We describe an approach to detecting off-axis radiation of laser beams propagating in scattering media, especially in the atmosphere, in the presence of background (solar) radiation. The method relies on a generalization of the conventional intensity interferometry (II) theory to scenarios involving coexisting sources of relatively long (laser radiation) and much shorter (background) coherence times. In such circumstances, the high coherence of the laser light allows its discrimination against even much stronger, but low-coherence, background. We propose a simple detection system consisting of a small array of photodetectors (e.g. photodiodes) and estimate the ratio of the background-to-laser irradiances at which the laser radiation is expected to be detectable.

We consider two approaches of evaluating matrix elements of electromagnetic volume integral equations.

We report new developments in the analytical evaluation of the near-field contribution to the matrix elements of the electric and magnetic field operators for planar conducting structures embedded in a layered medium. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions supported on parallel interfaces in the medium. Our method is an extension of the approach described in [1] of representing a Green function as a two-dimensional Laplacian of an auxiliary function. Such Laplacian representations can be obtained for the asymptotic forms of the Green functions, which are being subtracted in order to regularize the behavior of the Sommerfeld-type integrals. Matrix elements resulting from these asymptotic forms, given originally as quadruple surface integrals with singular integrands, are then reduced to double contour integrals over the perimeters of the surface elements, involving simple closed-form non-singular auxiliary functions.

We describe a concept of a system for detecting laser radiation in the presence of other (background) radiation sources. Unlike most existent laser-warning methods, which detect laser beams based on their intensity and monochromaticity, the proposed approach uses a high degree of coherence as the distinguishing property of laser radiation. Further, in contrast to previously considered coherence detection systems, based on amplitude interferometry (AI), we propose to utilize intensity interferometry (II), pioneered by Hanbury Brown and Twiss and originally applied to thermal radiation sources, but here generalized to coexisting high- and low-coherence radiation. In this contribution we concentrate on a possible application of the proposed system in detection of off-axis laser beam radiation scattered on atmospheric medium particles (water droplets and aerosols, especially dust). A particular design, which may operate in the visible and infrared regions, utilizes a small array of photodetectors (such as PIN photodiodes) and electronic correlator circuits identifying correlations in photocurrent fluctuations of neighboring detectors. It is shown that the normalized cross-correlation coefficients will be significant, even in the presence of a strong incoherent (e.g., solar radiation) background, provided the coherent radiation degeneracy parameter (number of photoelectrons generated by the detector during the coherence-time interval) exceeds the ratio of the background to signal intensities. This condition, which should not be difficult to achieve in realistic situations, ensures that the coherent signal is larger than the shot noise due to the background.

A short pulse propagating through a medium consisting of randomly distributed scatterers, large compared to the wavelength, is expected to develop an "early-time diffusion" (ETD) behavior: a sharply rising structure in the time-resolved intensity, immediately following the coherent (ballistic) component. Since the ETD signal is attenuated at a rate substantially lower than the coherent wave, it offers a possibility of application in imaging through diverse scattering media, such as atmospheric obscurants (clouds, fog, mist), dust, aerosols, fuel sprays, or biological tissues. We describe here a two-way (reflection) imaging scenario utilizing the ETD phenomenon, and propose a specific image formation technique. We evaluate, by using the radiative transport theory, the resulting point-spread function (PSF) characterizing the image resolution. We show that the directly formed image has an angular resolution comparable to the width of the forward peak in the ensemble-averaged scattering cross section of the medium constituents. Subsequently, we show that, through the application of a regularized deconvolution technique enhancing higher Fourier components of the PSF, the resolution can be further significantly improved-at least by a factor of ${\sim}4$ for a medium layer of optical thickness of the order of 20. Such an improvement can be reached even if the noise level is a few orders of magnitude higher than the coherent (ballistic) image component.

We consider the method of evaluating matrix elements of electromagnetic volume integral equations with the help of suitably constructed Laplacian-type representations of singular kernels (Green functions) appearing in electromagnetic volume and surface integral equations. The method consists of representing the singular kernels (Green functions) in terms of a generalized Laplacian operator acting on some auxiliary functions. Such a representations allow us, by using Gauss divergence theorem, to convert volumetric and surface integrals representing matrix elements to integrals always involving only non-singular integrands. The task of of finding particular Laplacian representation of different kernels amounts to solving appropriate ordinary or partial inhomogeneous differential equations. We apply the Laplacian method to analytic evaluation of matrix elements of the volume integral equation by employing two different Laplacian representations of the integral equation kernels. The first representation allows us to reduce 6-dimensional volume integrals over tetrahedra to 2-dimensional surface integrals over tetrahedra faces. The second representation is subsequently used reducing surface integrals over tetrahedra faces to pairs of of 1-dimensional line integrals over tetrahedra edges. We also derive analytic expressions for the resulting line integrals given in terms of elementary functions.

We consider new developments in the analytical evaluation of the near-field contribution to the matrix elements of the electric and magnetic field operators for planar conducting structures embedded in a layered medium. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions supported on parallel interfaces in the layered medium. The method uses suitably constructed representations of the mixed potential formulation integral kernels in terms of two-dimensional Laplacian of auxiliary functions. Such Laplacian representations can be obtained for the asymptotic forms of the Green functions, which are being subtracted in order to regularize the behavior of the Sommerfeld-type integrals. Matrix elements resulting from these asymptotic forms, given originally as quadruple surface integrals with singular integrands, are then reduced to double contour integrals over the perimeters of the surface elements, involving simple closed-form non-singular auxiliary functions M. The new developments include: · Derivation of relations between elements of the asymptotic dyadic Green functions and the kernels of the mixed-potential representation of the fields. · Inclusion of additional terms introduced in [1], which improve convergence of the Sommerfeld integrals. These additional kernel components, related to half-line source potentials, were not included in our previous paper [1]; they constitute non-leading asymptotic contributions to the mixed-potential kernels K Φ and K Ψ . · Construction of two additional auxiliary functions needed to represent the above-mentioned additional terms. The resultant auxiliary functions are expressed as integrals of the previously obtained [1] functions for the leading asymptotic kernel terms. · Construction of simplified analytical expressions for the matrix elements of the asymptotic parts of the pertinent dyadic Green functions. The asymptotic matrix elements, given in terms of quadruple surface integrals with singular integrands, are subsequently converted, by using suitably constructed Laplacian representations of the Green function, to double contour integrals over the perimeters of the surface elements, with simple, non-singular, smoothly varying integrands. The line integrals can be either evaluated analytically or by means of low order numerical quadratures. We discuss the relative merits of the direct numerical and analytic evaluation if these line integrals.

The radiative transport theory predicts that a short pulse propagating through a random medium consisting of discrete scatterers of sizes large compared to the wavelength develops to develop an "early-time diffusion" (ETD) component: a sharply rising structure in the time-resolved intensity, immediately following the coherent (ballistic) signal, but attenuated at a rate substantially lower than the coherent attenuation. This phenomenon offers a possibility of application in imaging through obscuring (e.g., atmospheric) media. We describe here an imaging scenario utilizing the ETD signal, evaluate the resulting point-spread function characterizing the image resolution, and show how that resolution can be significantly improved by means of regularized deconvolution techniques.

We consider an exact evaluation of the asymptotic, small distance contribution to the matrix elements of the planarly layered-medium Green function. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions with supports located on the medium interfaces and the asymptotic contribution is defined as that involving all matrix elements between pairs of basis functions supported on either the same interface or on a pair of interfaces of a single homogeneous material layer. The asymptotic matrix elements, given in terms of quadruple surface integrals with singular integrands, are subsequently converted, by using suitably constructed Laplacian representations of the Green function, to double contour integrals over the perimeters of the surface elements, with simple, nonsingular, smoothly varying integrands. The line integrals can be either evaluated analytically (currently resulting in rather lengthy expressions involving elementary functions), or by means of low order numerical quadratures.

We consider extensions and selected applications of the recently proposed method of evaluating Galerkin matrix elements of electromagnetic volume and surface integral equations with the help of suitably constructed Laplacian-type representations of singular kernels (Green functions) in terms of non-singular auxiliary functions.

We present an approach to calculation of matrix elements for arbitrarily oriented planar surface elements, based on reduction of the surface integrals to line integrals over perimeters of the elements. The described method is a generalization of our previous developments applicable to parallel geometry elements; it utilizes a representation of the Helmholtz-equation Green function in terms of an auxiliary function acted upon by a differential operator involving tangential gradients in two different planes. Integration by parts allows then reduction of surface- to line integrals with nonsingular integrands. The latter integrals can be either evaluated as standard quadratures or expressed analytically, in every order of expansion in the wave number k, in closed form. The method can be applied to all operators arising in Maxwell equations and in similar problems in acoustics and elastodynamics, offering significant advantages in accuracy and the computational cost.

We consider an approach allowing conversion of surface integrals (over planar surface elements) to line integrals with nonsingular integrand, in evaluating matrix elements of Helmholtz-equation Green function and its derivatives, in particular the vector Green function in electromagnetics. A general procedure is outlined consisting of finding suitable auxiliary functions applicable to particular kernel operators. Explicit analytical for expressions for the resulting line integrals can be obtained for arbitrary frequency and are provided in the static limit. The accuracy and the computational cost of proposed technique for its analytical and numerical line integral versions is compared on representative examples involving different surface elements geometrical configurations.

A novel procedure is presented for the evaluation of integrals involving singular Green function and Rao-Wilton-Glisson basis functions with arbitrary mutual non-planar geometrical configuration which appear in surface integral equations representation of Maxwell equations. The proposed procedure constitutes a generalization of our previously reported result valid for planar geometries. The method employs a suitably constructed representation of the Helmholtz equation Green function in terms of an differential operator acting on an auxiliary function which allows one to reduce four-dimensional surface integrals with singular integrands to line integrals over triangle edges with regular integrands. Advantages of our approach include simplicity and high accuracy at a computational cost considerably lower than for previously considered methods, such as the singularity subtraction technique.

A novel procedure is presented for the evaluation of integrals involving singular Green function and Rao-Wilton-Glisson basis functions with arbitrary mutual non-planar geometrical configuration which appear in surface integral equations representation of Maxwell equations. The proposed procedure constitutes a generalization of our previously reported result valid for planar geometries. The method employs a suitably constructed representation of the Helmholtz equation Green function in terms of an differential operator acting on an auxiliary function which allows one to reduce four-dimensional surface integrals with singular integrands to line integrals over triangle edges with regular integrands. Advantages of our approach include simplicity and high accuracy at a computational cost considerably lower than for previously considered methods, such as the singularity subtraction technique.

This presentation describes an ongoing work by the authors on the solution of the time-dependent radiative transfer equation (RTE) and its application to propagation of short pulses through dilute scattering media (in particular, atmospheric obscurants, such as clouds, fog, or aerosols). It concentrates on exploitation of the “early-time diffusion” phenomenon arising for media in which scatterers are significantly larger than the pulse wavelength. The early time diffusion signature is a sharply rising structure in the time-resolved intensity, immediately following the ballistic (coherent) signal; its rise time is, typically, orders of magnitude shorter than that of the usual “late-time” diffusion and its decay with the propagation distance is significantly slower than for the coherent intensity contribution. Our previous analysis of the early-time diffusion pertained to an infinite random medium. Here we present its generalization to the case of a laterally infinite slab of a finite thickness, and concentrate on an imaging scenario in which the transmitter (collocated with the receiver) and the observed object are located on the opposite sides of the slab. Compared to the infinite medium, the slab geometry offers 1. a possibility of a more direct comparison with typical experimental setups; 2. more favorable circumstances for reducing the effect of back-scattered light as a background for the signal reflected from the object; and 3. an opportunity of exploiting the shower-curtain effect. A general formulation of the problem will be illustrated by preliminary computational results, suggesting improved prospects of applicability of early-time diffusion in remote sensing through atmospheric obscurants, such as clouds, fog, or aerosols.

We present an approach allowing conversion of surface integrals (over planar surface elements) to line integrals, in evaluating matrix elements of the derivatives of the Helmholtz-equation Green function, in particular the vector Green function in electromagnetics. A general procedure is outlined and explicit expressions are provided in the static limit. In the latter case, the accuracy and the computational cost of the proposed technique are compared to those of a more conventional approach based on evaluating surface integrals of a known closed-form potential.