A critical survey of approximate scattering wave theories from random rough surfaces

Coherent reflection recovery in scattering from the ocean surface using the frequency-difference autoproducta)

The coherence of rough sea-surface-scattered acoustic fields decreases with increasing frequency. The frequency-difference autoproduct, a quadratic product of acoustic fields at nearby frequencies, mimics a genuine field at the difference frequency. In rough-surface scattering, the autoproduct's lower effective frequency decreases the apparent surface roughness, restoring coherent reflection. Herein, the recovery of coherent reflection in sea surface scattering via the frequency-difference autoproduct is examined for data collected off the coast of New Jersey during the Shallow Water '06 (SW06) experiment. An acoustic source at depth 40 m and receiver at depth 24.3 m and range 200 m interrogated 160 independent realizations of the ocean surface. The root mean square surface height h was 0.167 m, and broadcast frequencies were 14-20 kHz, so that 2.5 ≤kh cos θ≤ 3.7 for acoustic wavenumber k and incidence angle θ. Measured autoproducts, constructed from scattered constituent fields, show significant coherent reflection at sufficiently low difference frequencies. Theoretical results, using the Kirchhoff approximation and a non-analytic surface autocorrelation function, agree with experimental findings. The match is improved using a numerical strategy, exploiting the relationship between autoproduct-based coherence recovery, the ocean-surface autocorrelation function, and the ocean-surface height spectrum. Error bars computed from Monte Carlo scattering simulations support the validity of the measured coherence recovery.

Testing the theoreticians

The Reflections series takes a look back on historical articles from The Journal of the Acoustical Society of America that have had a significant impact on the science and practice of acoustics.

The visual appearances of disordered optical metasurfaces

Nanostructured materials have recently emerged as a promising approach for material appearance design. Research has mainly focused on creating structural colours by wave interference, leaving aside other important aspects that constitute the visual appearance of an object, such as the respective weight of specular and diffuse reflectances, object macroscopic shape, illumination and viewing conditions. Here we report the potential of disordered optical metasurfaces to harness visual appearance. We develop a multiscale modelling platform for the predictive rendering of macroscopic objects covered by metasurfaces in realistic settings, and show how nanoscale resonances and mesoscale interferences can be used to spectrally and angularly shape reflected light and thus create unusual visual effects at the macroscale. We validate this property with realistic synthetic images of macroscopic objects and centimetre-scale samples observable with the naked eye. This framework opens new perspectives in many branches of fine and applied visual arts.

Asymptotic Modeling of Three-Dimensional Radar Backscattering from Oil Slicks on Sea Surfaces

This paper presents new results of a simulation of radar backscatter from oil slick areas on a real three-dimensional sea surface, based on a physical hydrodynamic model of surface wave damping in the presence of oil films, the local equilibrium model (MLB). To solve this problem, the modelling was carried out by using the first-order small-slope approximation (SSA1) model. It presents the advantage of having a very good compromise between rapidity and accuracy of the calculation. The choice of the model is justified by solving the two-dimensional problem with several asymptotic methods and further comparing the results with a rigorous numerical method, based on the Method of Moments (MoM). Two approaches called “thin-layer” (TL) and “classical” were used to deal with the double layer (air/oil/sea) problem. The TL approach assumes that this double-layer problem can be seen locally as a Fabry–Pérot interferometer, which implies that the Kirchhoff-tangent plane approximation (KA) is valid. The classical approach consists in neglecting the presence of the oil layer for dealing with electromagnetic backscattering, which is valid for very thin oil films compared to the electromagnetic (EM) wavelength. It is shown that these two approaches have rather complementary validity domains: The TL approach is always valid for small observation angles, which makes it suitable for near nadir sensors such as altimeters, whereas the classical approach is valid for moderate observation angles, which makes it suitable for most satellite applications. The 3D modelling results are compared with C-band and X-band measured data (CSK experiment and OOW NOFO experiment) in VV polarization. The calculation takes into account that the oil film on the sea surface is mainly in an emulsion state. The results highlighted the relevance of the MLB hydrodynamic model, as well as the SSA1 EM model combined wit the classical approach, for quantifying NRCS in seas contaminated with marine oil or surfactants. The agreement is indeed very good in the X-band range.

Physics-Based Forward Modeling of Ocean Surface Swell Effects on SMAP L1-C NRCS Observations

This paper examines the impact of ocean surface swell waves on near-coastal L-band high-resolution synthetic aperture radar (SAR) data collected using the National Aeronautics and Space Administration's (NASA) Soil Moisture Active/Passive (SMAP) radar at 40° incidence angle. The two-scale model and a more efficient off-nadir approximation of the second-order small-slope-approximation are used for co- and cross-polarized backscatter normalized radar cross-section (NRCS) predictions of the ocean surface, respectively. Backscatter NRCS predictions are modeled using a combined wind and swell model where wind-driven surface roughness is characterized using the Durden-Vesecky directional spectrum, while swell effects are represented through their contribution to the long wave slope variance (mean-square slopes, or MSS). The swell-only MSS is numerically computed based on a model defined using the JONSWAP spectrum with parameters calculated using the National Data Buoy Center and Wave Watch III data. The backscatter NRCS model is further refined to include fetch-limited and low-wind corrections. The results show an improved agreement between modeled and observed HH-polarized backscatter NRCS when swell effects are included and indicate a relatively larger swell impact on L-band compared to higher radar frequencies. Preliminary investigations into the potential swell retrieval capabilities in the form of excess MSS are encouraging, however further refinements are required to make broadly applicable conclusions.

Recovery of coherent reflection from rough-surface scattered acoustic fields via the frequency-difference autoproduct

The acoustic field reflected from a random rough surface loses coherence with the incident field in the Kirchhoff approximation as kh cos θ increases, where k is the incident field wavenumber, h is the root mean square roughness height, and θ is the incidence angle. Thus, for fixed rough-surface properties and incidence angle, a reflected field at lower wavenumber should retain more coherence. Recent results suggest that the frequency-difference autoproduct formed from complex acoustic field amplitudes at two nearby frequencies can recover acoustic information at the difference of those frequencies even when the difference frequency is below the recorded field's bandwidth. Herein analytical, computational, and experimental results are presented for the extent to which the frequency-difference autoproduct recovers coherence from randomly rough-surface-scattered constituent fields that have lost coherence. The analytical results, developed from the Kirchhoff approximation and formal ensemble averaging over randomly rough surfaces with Gaussian height distributions and Gaussian correlation functions, indicate that the coherence of the rough-surface-reflected frequency-difference autoproduct depends on the surface correlation length and Δkh cos θ, where Δk is the difference of the autoproduct's constituent field wavenumbers. These results compare favorably with Monte Carlo simulations of rough surface scattering, and with laboratory experiments involving long surface correlation lengths where 1  ≤kh cos θ≤ 3.

Ka-Band Radar Cross-Section of Breaking Wind Waves

The effective normalized radar cross section (NRCS) of breaking waves, σwb, is empirically derived based on joint synchronized Ka-band radar and video records of the sea surface from a research tower. The σwb is a key parameter that, along with the breaker footprint fraction, Q, defines the contribution of non-polarized backscattering, NP =σwbQ, to the total sea surface NRCS. Combined with the right representation of the regular Bragg and specular backscattering components, the NP component is fundamental to model and interpret sea surface radar measurements. As the first step, the difference between NRCS values for breaking and non-breaking conditions is scaled with the optically-observed Q and compared with the geometric optics model of breaker backscattering. Optically-derived Q might not be optimal to represent the effect of breaking waves on the radar measurements. Alternatively, we rely on the breaking crest length that is firmly detected by the video technique and the empirically estimated breaker decay (inverse wavelength) scale in the direction of breaking wave propagation. A simplified model of breaker NRCS is then proposed using the geometric optics approach. This semi-analytical model parameterizes the along-wave breaker decay from reported breaker roughness spectra, obtained in laboratory experiments with mechanically-generated breakers. These proposed empirical breaker NRCS estimates agree satisfactorily with observations.

A Sensitivity Analysis on the Spectral Signatures of Low-Backscattering Sea Areas in Sentinel-1 SAR Images

Satellite synthetic aperture radar (SAR) is a unique tool to collect measurements over sea surface but the physical interpretation of such data is not always straightforward. Among the different sea targets of interest, low-backscattering areas are often associated to marine oil pollution even if several physical phenomena may also result in low-backscattering patches at sea. In this study, the effects of low-backscattering areas of anthropogenic and natural origin on the azimuth autocorrelation function (AACF) are analyzed using VV-polarized SAR measurements. Two objective metrics are introduced to quantify the deviation of the AACF evaluated over low-backscattering areas with reference to slick-free sea surface. Experiments, undertaken on six Sentinel-1 SAR scenes, collected in Interferometric Wide Swath VV+VH imaging mode over large low-backscattering areas of different origin under low-to-moderate wind conditions (speed ≤ 7 m/s), spanning a wide range of incidence angles (from about 30° up to 46°), demonstrated that the AACF evaluated within low-backscattering sea areas remarkably deviates from the slick-free sea surface one and the largest deviation is observed over oil slicks.

Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances

This paper proposes a brief review of acoustic wave scattering models from rough surfaces. This review is intended to provide an up-to-date survey of the analytical approximate or semi-analytical methods that are encountered in acoustic scattering from random rough surfaces. Thus, this review focuses only on the scattering of acoustic waves and does not deal with the transmission through a rough interface of waves within a solid material. The main used approximations are classified here into two types: the two historical approximations (Kirchhoff approximation and the perturbation theory) and some sound propagation models more suitable for grazing observation angles on rough surfaces, such as the small slope approximation, the integral equation method and the parabolic equation. The use of the existing approximations in the scientific literature and their validity are highlighted. Rough surfaces with Gaussian height distribution are usually considered in the models hypotheses. Rather few comparisons between models and measurements have been found in the literature. Some new criteria have been recently determined for the validity of the Kirchhoff approximation, which is one of the most used models, owing to its implementation simplicity.

Two domain decomposition methods, SDIM and CBFM, for scattering from a two-dimensional perfectly conducting rough surface: comparison and parametric study

This paper focuses on the two domain decomposition methods, the subdomain decomposition iterative method (SDIM) and the characteristics basis function method (CBFM), combined with adaptive cross approximation (ACA) to compute the normalized radar cross section (NRCS) from a perfectly conducting two-dimensional (2D) randomly rough surface. The 3D electromagnetic problem is solved from the electric field integral equation discretized by the Galerkin method of moments with the Rao-Wilton-Glisson basis functions. In addition, a parametric study versus the number of blocks, the number of overlapping edges, the thresholds of recompressed ACA (RACA; ACA combined with two QR decompositions and truncated by a SVD procedure, also named ACA-SVD or ACA-TSVD), and the parameters inherent to the CBFM is investigated. The complexity of the two methods is also addressed.

Black paints covered with multidielectrics: light absorbers

Black paints are commonly used to provide broadband light absorbers in high-precision optics. We show how multidielectric coatings improve the performances of these absorbers. The coated rough paints still exhibit a quasi-lambertian diffuse reflection, but this scattering pattern can be reduced by several orders of magnitude, which strongly enhances absorption. Predictions are based on an exact electromagnetic theory of light scattering from arbitrary rough multilayers. Results are also compared to useful approximate theories.

Mineral Oil Slicks Identification Using Dual Co-polarized Radarsat-2 and TerraSAR-X SAR Imagery

This study is devoted to a generalization of C-band Radarsat-2 and X-band TerraSAR-X synthetic aperture radar (SAR) data in the form of a diagram serving to easily identify mineral oil slicks (crude oil and emulsions) and separate them from the other oil slicks. The diagram is based on the multi-polarization parameter called Resonant to Non-resonant signal Damping (RND) introduced by Ivonin et al. in 2016, which is related to the ratio between damping within the slick of the short waves and wave breakings. SAR images acquired in the North Sea during oil-on-water exercises in 2011–2012 containing three types of oil spills (crude oil, emulsion, and plant oil) were used. The analysis was performed under moderate sea conditions (wind speeds of 2–6 m/s and sea wave heights of less than 2 m), the incidence angles of 27°–49°, and the signal-to-noise ratio (SNR) of −3 to 11 dB within slicks. On the diagram plane, created by the RND parameter and the Bragg wave number, the mineral oil samples form a well-outlined zone, called a mineral oil zone. For C-band data, the plant oil samples were clearly distinguished from the mineral oils in the diagram. Determination of the confidence level for the detection of mineral oils versus plant oil was proposed using the mineral oil zone boundaries. The mineral oil data with SNR within slicks better than 2 dB lay within this zone with a confidence level better than 65%. The plant oil data with the same SNR lay outside this zone with a confidence level of better than 80%. For mineral oil with SNR of −3 dB, the confidence level is 55%.

Polarization-coded material classification in automotive LIDAR aiming at safer autonomous driving implementations

LIDAR sensors are one of the key enabling technologies for the wide acceptance of autonomous driving implementations. Target identification is a requisite in image processing, informing decision making in complex scenarios. The polarization from the backscattered signal provides an unambiguous signature for common metallic car paints and can serve as one-point measurement for target classification. This provides additional redundant information for sensor fusion and greatly alleviates hardware requirements for intensive morphological image processing. Industry decision makers should consider polarization-coded LIDAR implementations. Governmental policy makers should consider maximizing the potential for polarization-coded material classification by enforcing appropriate regulatory legislation. Both initiatives will contribute to faster (safer, cheaper, and more widely available) advanced driver-assistance systems and autonomous functions. Polarization-coded material classification in automotive applications stems from the characteristic signature of the source of LIDAR backscattering: specular components preserve the degree of polarization while diffuse contributions are predominantly depolarizing.

Inverse scattering for reflection intensity phase microscopy

Reflection phase imaging provides label-free, high-resolution characterization of biological samples, typically using interferometric-based techniques. Here, we investigate reflection phase microscopy from intensity-only measurements under diverse illumination. We evaluate the forward and inverse scattering model based on the first Born approximation for imaging scattering objects above a glass slide. Under this design, the measured field combines linear forward-scattering and height-dependent nonlinear back-scattering from the object that complicates object phase recovery. Using only the forward-scattering, we derive a linear inverse scattering model and evaluate this model's validity range in simulation and experiment using a standard reflection microscope modified with a programmable light source. Our method provides enhanced contrast of thin, weakly scattering samples that complement transmission techniques. This model provides a promising development for creating simplified intensity-based reflection quantitative phase imaging systems easily adoptable for biological research.

Surface-roughness measurement based on scalar field correlation with active millimeter-wave imaging system

Active millimeter-wave (MMW) imaging is of interest because it has played an important role in the area of personnel surveillance over the past decades. Through reconstructing reflectivity, potential threats can be recognized based on shape. Besides reflectivity, diverse physical characteristics should be explored so that supplementary information can be used to assist recognition. This paper presents a surface-roughness measurement method using holographic data that has already been acquired during security checks. Based on scalar diffraction theory and speckle metrology, a simple mathematical relation between the correlation of holographic fields and surface roughness has been derived. Consequently, another kind of information, that is, the surface-roughness estimate, can be used to help differentiate similarly shaped articles. Results of simulations and laboratory experiments have shown its validation and potential in application to MMW imaging systems.

Linking Remote Sensing and Geodiversity and Their Traits Relevant to Biodiversity—Part I: Soil Characteristics

In the face of rapid global change it is imperative to preserve geodiversity for the overall conservation of biodiversity. Geodiversity is important for understanding complex biogeochemical and physical processes and is directly and indirectly linked to biodiversity on all scales of ecosystem organization. Despite the great importance of geodiversity, there is a lack of suitable monitoring methods. Compared to conventional in-situ techniques, remote sensing (RS) techniques provide a pathway towards cost-effective, increasingly more available, comprehensive, and repeatable, as well as standardized monitoring of continuous geodiversity on the local to global scale. This paper gives an overview of the state-of-the-art approaches for monitoring soil characteristics and soil moisture with unmanned aerial vehicles (UAV) and air- and spaceborne remote sensing techniques. Initially, the definitions for geodiversity along with its five essential characteristics are provided, with an explanation for the latter. Then, the approaches of spectral traits (ST) and spectral trait variations (STV) to record geodiversity using RS are defined. LiDAR (light detection and ranging), thermal and microwave sensors, multispectral, and hyperspectral RS technologies to monitor soil characteristics and soil moisture are also presented. Furthermore, the paper discusses current and future satellite-borne sensors and missions as well as existing data products. Due to the prospects and limitations of the characteristics of different RS sensors, only specific geotraits and geodiversity characteristics can be recorded. The paper provides an overview of those geotraits.

Anti-scattering effect analyzed with an exact theory of light scattering from rough multilayers

First-order theories of light scattering previously revealed the existence of anti-scattering effects in optical multilayers. Here we present an exact electromagnetic theory that is able to complete the scattering analysis when first-order scattering is cancelled. The theory is valid for arbitrary rough multilayers.

C-curve feature of complex permittivity estimation based on multi-polarization measurements in passive millimeter-wave sensing

Passive millimeter-wave (PMMW) sensing has been used for several close-range applications such as scene monitoring and security check. Aiming to sense man-made or natural dielectrics, we report the estimation characteristics of complex permittivity by multi-polarization measurements. According to the polarization brightness temperature (TB) model, the complex permittivities of dielectric materials are estimated by using the TB data of multiple linear polarizations. By Monte Carlo simulations, we found that the estimated complex permittivities can form a special cluster. All estimation values of each dielectric spread around a special nonlinear curve (called the "C-curve"). Then the "C-curve" distribution characteristics of different parameter conditions such as polarization number, incident angle, roughness, and measurement noise, have been analyzed. The 94 GHz multi-polarization measurements have been conducted to prove the existence of "C-curve" feature. The results suggest that this invariant feature can be possibly used for material classification and image segmentation in PMMW sensing.

Diffuse scattering due to stochastic disturbances of 1D-gratings on the example of line edge roughness

Diffuse scattering of optical one-dimensional gratings becomes increasingly critical as it constrains the performance, e.g., of grating spectrometers. In particular, stochastic disturbances of the ideal grating structure provoke straylight. In this paper, the straylight spectrum of stochastically disturbed gratings is examined. First, a 1D-method is presented that allows to calculate 2D-diffuse scattering of arbitrarily polarized light originating from stochastic disturbances of the grating geometry on the basis of standard optical simulation tools. Within the scope of this method an enormous reduction of computational effort is achieved compared to the full 2D-simulation approach, i.e., the computation time can be reduced by several orders of magnitude. Hence, the method also allows to address even large period gratings that are not possible to calculate within a full 2D-approach. In analogy to scattering theories for surface roughness the method relies on typical characteristics of straylight originating from small disturbances, that the angle resolved scattering (ARS) can be separated into a product of the power spectral density describing the 2D stochastic process and additional factors depending on the undisturbed 1D grating structure. In a second part, an analytical model within Fourier optics utilizing thin element approximation (TEA) describing the wide angle scattering of lamellar gratings disturbed by line edge roughness (LER) for TE-polarized light is derived and verified by applying the 1D-simulation method. For shallow gratings, we find an excellent agreement between simulation and TEA over the whole transmission half space. In addition, this model allows a descriptive understanding of the underlying physical effects and, accordingly, the influence of relevant parameters (grating geometry, refractive indices, illumination) onto the scattering spectra is discussed. Further, it is shown that LER-scattering can be described within a modified Rayleigh-Rice-ARS usually found within the frame of surface roughness.

Photon momentum change of quasi-smooth solar sails

The solar photon sail (SPS) allows space missions without propellant that would otherwise not be feasible. Thrust models frequently used in the literature for the calculation of trajectories often underestimate the effect that the surface roughness has on SPS dynamics. A small variation of the thrust vector can induce a large modification of sail flight. In this work, the variation of the photon momentum vector (PMV) is computed as resulting from the incident Sun radiation, taking into account the absorbed and reflected photons. The momentum resulting from diffuse light has been modeled by using vectorial scattering theories in the limit of a quasi-smooth sail where the first-order of Rayleigh-Rice can be applied. In particular, the momentum change resulting from diffuse radiation causes a PMV reduction as well as a deviation of its direction from what is foreseen in the case of an ideally smooth sail.

Validity of the Kirchhoff approximation for the scattering of electromagnetic waves from dielectric, doubly periodic surfaces

Accuracy of Kirchhoff approximation (KA) for rough-surface electromagnetic wave scattering is studied by comparison with accurate numerical solutions in the context of three-dimensional dielectric surfaces. The Kirchhoff tangent plane approximation is examined without resorting to the principle of stationary phase. In particular, it is shown that this additional assumption leads to zero cross-polarized backscattered power, but not the tangent plane approximation itself. Extensive numerical results in the case of a bisinusoidal surface are presented for a wide range of problem parameters: height-to-period, wavelength, incidence angles, and dielectric constants. In particular, this paper shows that the range of validity inherent in the KA includes surfaces whose curvature is not only much smaller, but also comparable to the incident wavelength, with errors smaller than 5% in total reflectivity, thus presenting a detailed and reliable source for the validity of the KA in a three-dimensional fully polarimetric formulation.

Light scattering by periodic rough surfaces: equivalent jump conditions

We present an interface model based on two-scale homogenization to predict the coherent scattering of light by a periodic rough interface between air and a dielectric. Contrary to previous approaches where the roughnesses are replaced by a layer filled with an equivalent medium, our modeling yields effective jump conditions applying across the region containing the roughnesses. The validity of the model is inspected by comparison with direct numerics and with experimental measurements on an air/silicium rough interface near the Brewster angle. It is shown that the interface model reproduces accurately the shift in the Brewster phenomenon without any adjustable parameter, which is of practical importance in retrieval methods to get thickness or filling fraction with reliable physical values.

A rheological model for immersed corrugated elastic plates

The influence of surface imperfections on the propagation of guided waves in an immersed elastic plate can be interpreted by means of a rheological model. The corrugated surface is modeled by a very thin interface, similar to a Jones spring model, which replaces the continuity boundary conditions at the liquid - corrugated solid-plate interface. As the surrounding liquid is considered to be perfect, only one complex stiffness is used for the model of Jones. The selection of the plate guided mode and the test frequency are motivated by the detectability and non-interference with other modes. The spring stiffness is obtained by a best fit procedure, between the analytical solution and the results obtained by the finite elements method (FEM). One way ensuring the agreement of the two approaches, rheological and FEM, is to consider angular resonances provided by the transmission coefficients. Small changes in the parameters of the roughness keep the positions of the angular resonances of the plate practically unchanged, while at the same time large variations of the half width of the transmission coefficient curve is observed. The effect of corrugation parameters on the guided modes in the plate can be predicted by using the rheological model with the deduced spring complex stiffness.

The scattering of torsional guided waves from Gaussian rough surfaces in pipework

In older sections of industrial pipework there are often regions of general corrosion that typically have a Gaussian thickness distribution. During guided wave inspection this corrosion causes an increase in the background noise and a significant attenuation of the inspection wave. These effects are investigated in this paper through finite element modelling of the interaction of torsional guided waves with rough surfaces in pipes. Pipes of different diameter and rough surface profile are modelled and it is found that the attenuation of waves is explained by significant mode conversion and scattering within the rough surface. This mode conversion is greatest when the non-axisymmetric modes to which energy is scattered are close to the cutoff frequency or when the ratio of surface correlation length to wavelength is around 0.2-0.25. Mode conversion increases with increasing surface roughness and is a strong function of frequency-diameter product, with larger pipes causing more mode conversion. When this mode conversion occurs the energy is lost mostly to those waves with a displacement profile closest to the original torsional inspection wave. Resulting attenuation of the inspection signal can be severe; for example a mean wall thickness loss of 28% can cause 2.7 dB/m attenuation in a pulse-echo configuration.

Reflection of diffuse light from dielectric one-dimensional rough surfaces

We study the reflection of diffuse light from 1D randomly rough dielectric interfaces. Results for the reflectance under diffuse illumination are obtained by rigorous numerical simulations and then contrasted with those obtained for flat surfaces. We also explore the possibility of using perturbation theories and conclude that they are limited for this type of study. Numerical techniques based on Kirchhoff approximation and reduced Rayleigh equations yield better results. We find that, depending on the refractive index contrast and nature of the irregularities, the roughness can increase or decrease the diffuse reflectance of the surface.

Efficient polarimetric BRDF model

The purpose of the present manuscript is to present a polarimetric bidirectional reflectance distribution function (BRDF) model suitable for hyperspectral and polarimetric signature modelling. The model is based on a further development of a previously published four-parameter model that has been generalized in order to account for different types of surface structures (generalized Gaussian distribution). A generalization of the Lambertian diffuse model is presented. The pBRDF-functions are normalized using numerical integration. Using directional-hemispherical reflectance (DHR) measurements, three of the four basic parameters can be determined for any wavelength. This simplifies considerably the development of multispectral polarimetric BRDF applications. The scattering parameter has to be determined from at least one BRDF measurement. The model deals with linear polarized radiation; and in similarity with e.g. the facet model depolarization is not included. The model is very general and can inherently model extreme surfaces such as mirrors and Lambertian surfaces. The complex mixture of sources is described by the sum of two basic models, a generalized Gaussian/Fresnel model and a generalized Lambertian model. Although the physics inspired model has some ad hoc features, the predictive power of the model is impressive over a wide range of angles and scattering magnitudes. The model has been applied successfully to painted surfaces, both dull and glossy and also on metallic bead blasted surfaces. The simple and efficient model should be attractive for polarimetric simulations and polarimetric remote sensing.

Comparison of microfacet BRDF model to modified Beckmann-Kirchhoff BRDF model for rough and smooth surfaces

A popular class of BRDF models is the microfacet models, where geometric optics is assumed. In contrast, more complex physical optics models may more accurately predict the BRDF, but the calculation is more resource intensive. These seemingly disparate approaches are compared in detail for the rough and smooth surface approximations of the modified Beckmann-Kirchhoff BRDF model, assuming Gaussian surface statistics. An approximation relating standard Fresnel reflection with the semi-rough surface polarization term, Q, is presented for unpolarized light. For rough surfaces, the angular dependence of direction cosine space is shown to be identical to the angular dependence in the microfacet distribution function. For polished surfaces, the same comparison shows a breakdown in the microfacet models. Similarities and differences between microfacet BRDF models and the modified Beckmann-Kirchhoff model are identified. The rationale for the original Beckmann-Kirchhoff F(bk)(2) geometric term relative to both microfacet models and generalized Harvey-Shack model is presented. A modification to the geometric F(bk)(2) term in original Beckmann-Kirchhoff BRDF theory is proposed.

Measurement of roughness based on the Talbot effect in reflection from rough surfaces

In the present work, the Talbot effect of a square grating is analyzed when light is reflected from a rough surface. It is shown theoretically that the scattered light intensity in the Fresnel diffraction limit depends on statistical properties of the rough surface, the angle of incidence of the light, the grating period, and a geometric coefficient, related to the ratio of distance of the rough surface and the observation plane from the grating. At Talbot distances of the grating, the surface height difference function, in terms of multiplication of the Talbot number, the grating period, and the geometric coefficient is the modulation transfer function (MTF) of the scattering in reflection from the rough surface. If the argument of the height difference function is larger than twice the surface correlation length, the height difference function is constant for different spatial frequencies. Therefore, the square wave is reproduced with smaller contrast. The surface roughness can be obtained by measuring the contrast at different incident angles. It is also shown that the contrast measurements in both reflection and transmission, provide the refractive index of transparent samples with a rough surface. In experimental studies, the roughness of three metal standard rough surfaces are determined at different angles of incidence. Also, the refractive index of a sheet glass with a rough surface is obtained. The results are quite consistent.

Rigorous speckle simulation using surface integral equations and higher order boundary element method

The scattering of electromagnetic waves from rough surfaces has been actively studied for more than a century now because of its involvement in vast application areas. In the past two decades, great advances have been made by incorporating multiple scattering effects into analytical approaches. However, no model can yet be applied to surfaces with arbitrary roughness. It is also very difficult to study the cross-polarization, shadowing, or multiple scattering effects. In order to study more fundamentally the interaction of polarized light with more general rough surfaces of general media, we have developed a rigorous numerical simulator to calculate the resulting speckle fields. The full Maxwell equations were solved using surface integral equations combined with a boundary element method. The rough surface was discretized by higher order quadrilateral edge elements. The effective tangential electric and magnetic fields in each element in terms of 10 edges were first solved. The scattered electric and magnetic fields everywhere in space were then calculated correspondingly. One of the great advantages of such a simulator is that both the near and far fields can be calculated directly. Preliminary results of different kinds of metallic structures are presented, by which the advantages of the method are demonstrated.

Scattering of near normal incidence SH waves by sinusoidal and rough surfaces in 3-D: comparison to the scalar wave approximation

The challenge of accurately simulating how incident scalar waves interact with rough boundaries has made it an important area of research within many scientific disciplines. Conventional methods, which in the majority of cases focus only on scattering in two dimensions, often suffer from long simulation times or reduced accuracy, neglecting phenomena such as multiple scattering and surface self-shadowing. A simulation based on the scalar wave distributed point source method (DPSM) is presented as an alternative which is computationally more efficient than fully meshed numerical methods while obtaining greater accuracy than approximate analytical techniques. Comparison is made to simulated results obtained using the finite element method for a sinusoidally periodic surface where scattering only occurs in two dimensions, showing very good agreement (<0.2 dB). In addition to two-dimensional scattering, comparison to experimental results is also carried out for scattering in three dimensions when the surface has a Gaussian roughness distribution. Results indicate that for two-dimensional scattering and for rough surfaces with a correlation length equal to the incident wavelength (λ) and a root mean square height less than 0.2λ, the scalar wave approximation predicts reflected pulse shape change and envelope amplitudes generally to within 1 dB. Comparison between transducers within a three-element array also illustrate the sensitivity pulse amplitude can have to sensor position above a rough surface, differing by as much as 17 dB with a positional change of just 1.25λ.

Scattering by complex inhomogeneous objects: a first-order reciprocity method

The scattering by slightly inhomogeneous objects has been studied by a first-order method and reciprocity theorem. The scattering calculation reported in this manuscript is based on a simple computation of the field in a defectless structure at different incidence angles. The numerical results have been compared to those given by an exact calculation. It is shown that the method enables to handle complex structures with an affordable computational burden. A major advantage of the method is its ability to treat different defects without recomputing the field, i.e, the main part of the computation time. In addition, for defects in periodic structures, the field computation can be limited to a single period thus leading to an important decrease of the computational time and required memory. This method is believed to provide significant advantages for the engineering of optical devices.

Numerical solutions of the Rayleigh equations for the scattering of light from a two-dimensional randomly rough perfectly conducting surface

We present rigorous, nonperturbative, purely numerical solutions of the Rayleigh equations for the scattering of p- and s-polarized light from a two-dimensional randomly rough perfectly conducting surface. The solutions are used to calculate the reflectivity of the surface, the mean differential reflection coefficients, and the full angular distribution of the intensity of the scattered field. These results are compared with previously published rigorous numerical solutions of the Stratton-Chu equations, and very good agreement is found.

Determination of the rough interface parameters using the self-imaging effect

In this work, a linear grating is used to project a periodic light intensity distribution on a rough interface, and the near field transmitted light scattering is studied. It is shown theoretically that the intensity in the Fresnel regime depends on statistical properties of the rough interface and the light intensity period. The self-image contrast exponentially depends on the interface height-height correlation function. The correlation is obtained in terms of multiplication of the self-image number and the period of the light intensity distribution. Therefore, the roughness and the correlation length of the interface can be obtained by determining the contrast of the self-images when the light intensity period is smaller than the interface correlation length. For periods longer than twice the correlation length, the contrast measurements only provide the interface roughness. In experimental studies, the roughness of interfaces is determined by square gratings with periods much longer than the correlation lengths. The rough interfaces are prepared by roughening sheet glass by powders of different grit numbers. The results for different gratings and light wavelengths are quite consistent.

A fast and high-order accurate surface perturbation method for nanoplasmonic simulations: basic concepts, analytic continuation and applications

In this paper we demonstrate that rigorous high-order perturbation of surfaces (HOPS) methods coupled with analytic continuation mechanisms are particularly well-suited for the assessment and design of nanoscale devices (e.g., biosensors) that operate based on surface plasmon resonances generated through the interaction of light with a periodic (metallic) grating. In this connection we explain that the characteristics of the latter are perfectly aligned with the optimal domain of applicability of HOPS schemes, as these procedures can be shown to be the methods of choice for low to moderate wavelengths of radiation and grating roughness that is representable by a few (e.g., tens of) Fourier coefficients. We argue that, in this context, the method can, for instance, produce full and precise reflectivity maps in computational times that are orders of magnitude faster than those of alternative numerical schemes (e.g., the popular "C-method," finite differences, integral equations or finite elements). In this initial study we concentrate on the description of the basic principles that underlie the solution scheme, including those that relate to analytic continuation procedures. Within this framework, we explain how, in spite of conventional wisdom to the contrary, the resulting perturbative techniques can provide a most valuable tool for practical investigations in plasmonics. We demonstrate this with some examples that have been previously discussed in the literature (including treatments of the reflectivity and band gap structure of some simple geometries) and extend this to demonstrate the wider applicability of the proposed approach.

Surface and bulk scattering by magnetic and dielectric inhomogeneities: a first-order method

The scattering of a magnetodielectric multilayer has been studied by a first-order method. The model reported in this manuscript relies on the equivalence between heterogeneities of the medium and fictitious electric and magnetic sources. Types of inhomogeneities considered are roughness and bulk inhomogeneities and concern both permittivity and permeability. The numerical results are compared to those given in previous papers for optical scattering. It is shown in the microwave spectra that angle-resolved scattering allows identification of the scattering origins (permittivity or permeability spectra). The cases of isotropic films and metamaterials are presented and discussed.

An eigenvalue correction due to scattering by a rough wall of an acoustic waveguide

In this paper a derivation of the attenuation factor in a waveguide with stochastic walls is presented. The perturbation method and Fourier analysis are employed to derive asymptotically consistent boundary-value problems at each asymptotic order. The derived approximation predicts the attenuation of the propagating mode in a rough waveguide through a correction to the eigenvalue corresponding to smooth walls. The proposed approach can be used to derive results that are consistent with those obtained by Bass et al. [IEEE Trans. Antennas Propag. 22, 278-288 (1974)]. The novelty of the method is that it does not involve the integral Dyson-type equation and, as a result, the large number of statistical moments included in the equation in the form of the mass operator of the volume scattering theory. The derived eigenvalue correction is described by the correlation function of the randomly rough surface. The averaged solution in the plane wave regime is approximated by the exponential function dependent on the derived eigenvalue correction. The approximations are compared with numerical results obtained using the finite element method (FEM). An approach to retrieve the correct deviation in roughness height and correlation length from multiple numerical realizations of the stochastic surface is proposed to account for the oversampling of the rough surface occurring in the FEM meshing procedure.

Physical optics solution for the scattering of a partially-coherent wave from a statistically rough material surface

The scattering of a partially-coherent wave from a statistically rough material surface is investigated via derivation of the scattered field cross-spectral density function. Two forms of the cross-spectral density are derived using the physical optics approximation. The first is applicable to smooth-to-moderately rough surfaces and is a complicated expression of source and surface parameters. Physical insight is gleaned from its analytical form and presented in this work. The second form of the cross-spectral density function is applicable to very rough surfaces and is remarkably physical. Its form is discussed at length and closed-form expressions are derived for the angular spectral degree of coherence and spectral density radii. Furthermore, it is found that, under certain circumstances, the cross-spectral density function maintains a Gaussian Schell-model form. This is consistent with published results applicable only in the paraxial regime. Lastly, the closed-form cross-spectral density functions derived here are rigorously validated with scatterometer measurements and full-wave electromagnetic and physical optics simulations. Good agreement is noted between the analytical predictions and the measured and simulated results.

Optimizing and characterizing grating efficiency for a soft X-ray emission spectrometer

The efficiency of soft X-ray diffraction gratings is studied using measurements and calculations based on the differential method with the S-matrix propagation algorithm. New open-source software is introduced for efficiency modelling that accounts for arbitrary groove profiles, such as those based on atomic force microscopy (AFM) measurements; the software also exploits multi-core processors and high-performance computing resources for faster calculations. Insights from these calculations, including a new principle of optimal incidence angle, are used to design a soft X-ray emission spectrometer with high efficiency and high resolution for the REIXS beamline at the Canadian Light Source: a theoretical grating efficiency above 10% and resolving power E/ΔE > 2500 over the energy range from 100 eV to 1000 eV are achieved. The design also exploits an efficiency peak in the third diffraction order to provide a high-resolution mode offering E/ΔE > 14000 at 280 eV, and E/ΔE > 10000 at 710 eV, with theoretical grating efficiencies from 2% to 5%. The manufactured gratings are characterized using AFM measurements of the grooves and diffractometer measurements of the efficiency as a function of wavelength. The measured and theoretical efficiency spectra are compared, and the discrepancies are explained by accounting for real-world effects: groove geometry errors, oxidation and surface roughness. A curve-fitting process is used to invert the calculations to predict grating parameters that match the calculated and measured efficiency spectra; the predicted blaze angles are found to agree closely with the AFM estimates, and a method of characterizing grating parameters that are difficult or impossible to measure directly is suggested.

Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles

Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena resulting from random phase variations in the reflected or transmitted wavefront. Rayleigh-Rice, Beckmann-Kirchhoff. or Harvey-Shack surface scatter theories are commonly used to predict surface scatter effects. Smooth-surface and/or paraxial approximations have severely limited the range of applicability of each of the above theoretical treatments. A recent linear systems formulation of nonparaxial scalar diffraction theory applied to surface scatter phenomena resulted first in an empirically modified Beckmann-Kirchhoff surface scatter model, then a generalized Harvey-Shack theory that produces accurate results for rougher surfaces than the Rayleigh-Rice theory and for larger incident and scattered angles than the classical Beckmann-Kirchhoff and the original Harvey-Shack theories. These new developments simplify the analysis and understanding of nonintuitive scattering behavior from rough surfaces illuminated at arbitrary incident angles.

Modeling of light scattering in different regimes of surface roughness

The light scattering of rough metallic surfaces with roughness levels ranging from a few to several hundred nanometers is modeled and compared to experimental data. Different modeling approaches such as the classical Rayleigh-Rice vector perturbation theory and the new Generalized Harvey-Shack theory are used and critically assessed with respect to ranges of validity, accuracy, and practicability. Based on theoretical calculations and comparisons with Rigorous Coupled Wave Analysis for sinusoidal phase gratings, it is demonstrated that the approximate scatter models yield surprisingly accurate results and at the same time provide insight into light scattering phenomena. For stochastically rough metal surfaces, the predicted angles resolved scattering is compared to experimental results at 325 nm, 532 nm, and 1064 nm. In addition, the possibilities of retrieving roughness information from measured scattering data for different roughness regimes are discussed.

Investigation on wide-band scattering of a 2-D target above 1-D randomly rough surface by FDTD method

Finite-difference time-domain (FDTD) algorithm with a pulse wave excitation is used to investigate the wide-band composite scattering from a two-dimensional(2-D) infinitely long target with arbitrary cross section located above a one-dimensional(1-D) randomly rough surface. The FDTD calculation is performed with a pulse wave incidence, and the 2-D representative time-domain scattered field in the far zone is obtained directly by extrapolating the currently calculated data on the output boundary. Then the 2-D wide-band scattering result is acquired by transforming the representative time-domain field to the frequency domain with a Fourier transform. Taking the composite scattering of an infinitely long cylinder above rough surface as an example, the wide-band response in the far zone by FDTD with the pulsed excitation is computed and it shows a good agreement with the numerical result by FDTD with the sinusoidal illumination. Finally, the normalized radar cross section (NRCS) from a 2-D target above 1-D rough surface versus the incident frequency, and the representative scattered fields in the far zone versus the time are analyzed in detail.

Statistical study of radiation loss from planar optical waveguides: the curvilinear coordinate method and the small perturbation method

This article presents an original method for the theoretical analysis of the intensity radiated by a dielectric waveguide with rough walls. The method is based on Maxwell's equations under their covariant form written in nonorthogonal coordinate systems adapted to the geometry of the waveguide. The solutions are found by using a perturbation method starting from a guide with smooth walls. The statistical characteristics of the radiant intensity, the mean value, and the probability density function are analytically determined.

Neural network emulation of the integral equation model with multiple scattering

The Integral Equation Model with multiple scattering (IEMM) represents a well-established method that provides a theoretical framework for the scattering of electromagnetic waves from rough surfaces. A critical aspect is the long computational time required to run such a complex model. To deal with this problem, a neural network technique is proposed in this work. In particular, we have adopted neural networks to reproduce the backscattering coefficients predicted by IEMM at L- and C-bands, thus making reference to presently operative satellite radar sensors, i.e., that aboard ERS-2, ASAR on board ENVISAT (C-band), and PALSAR aboard ALOS (L-band). The neural network-based model has been designed for radar observations of both flat and tilted surfaces, in order to make it applicable for hilly terrains too. The assessment of the proposed approach has been carried out by comparing neural network-derived backscattering coefficients with IEMM-derived ones. Different databases with respect to those employed to train the networks have been used for this purpose. The outcomes seem to prove the feasibility of relying on a neural network approach to efficiently and reliably approximate an electromagnetic model of surface scattering.

Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity

In this work it is shown theoretically and examined experimentally that the measurement of coherently transmitted or reflected monochromatic light intensity from a randomly rough interface as a function of incident angle provides the height distribution on the interface. It is also shown that the spectrum of coherently transmitted or reflected light from a rough interface is modified and the modified spectrum yields the height distribution. The experimental results obtained by applying both methods, in transmission and reflection, on rough surfaces prepared by roughening the sheet glasses by powders of different grain sizes are quite consistent. In addition, the effect of the surrounding medium's refractive index on the roughness measurement is studied by immersing the samples into liquids of different refractive indices. Also, the application range and limitations of the introduced methods are discussed.

Analytical fitting model for rough-surface BRDF

A physics-based model is developed for rough surface BRDF, taking into account angles of incidence and scattering, effective index, surface autocovariance, and correlation length. Shadowing is introduced on surface correlation length and reflectance. Separate terms are included for surface scatter, bulk scatter and retroreflection. Using the FindFit function in Mathematica, the functional form is fitted to BRDF measurements over a wide range of incident angles. The model has fourteen fitting parameters; once these are fixed, the model accurately describes scattering data over two orders of magnitude in BRDF without further adjustment. The resulting analytical model is convenient for numerical computations.

On the Soil Roughness Parameterization Problem in Soil Moisture Retrieval of Bare Surfaces from Synthetic Aperture Radar

Synthetic Aperture Radar has shown its large potential for retrieving soil moisture maps at regional scales. However, since the backscattered signal is determined by several surface characteristics, the retrieval of soil moisture is an ill-posed problem when using single configuration imagery. Unless accurate surface roughness parameter values are available, retrieving soil moisture from radar backscatter usually provides inaccurate estimates. The characterization of soil roughness is not fully understood, and a large range of roughness parameter values can be obtained for the same surface when different measurement methodologies are used. In this paper, a literature review is made that summarizes the problems encountered when parameterizing soil roughness as well as the reported impact of the errors made on the retrieved soil moisture. A number of suggestions were made for resolving issues in roughness parameterization and studying the impact of these roughness problems on the soil moisture retrieval accuracy and scale.