On the Convergence of Numerical Computations for Both Exact and Approximate Solutions for Electromagnetic Scattering by Nonspherical Dielectric Particles.

We summarize the size parameter range of the applicability of four light-scattering computational methods for nonspherical dielectric particles. These methods include two exact methods - the extended boundary condition method (EBCM) and the invariant imbedding T-matrix method (II-TM) and two approximate approaches - the physical-geometric optics method (PGOM) and the improved geometric optics method (IGOM). For spheroids, the single-scattering properties computed by EBCM and II-TM agree for size parameters up to 150, and the comparison gives us confidence in using IITM as a benchmark for size parameters up to 150 for other geometries (e.g., hexagonal columns) because the applicability of II-TM with respect to particle shape is generic, as demonstrated in our previous studies involving a complex aggregate. This study demonstrates the convergence of the exact II-TM and approximate PGOM solutions for the complete set of single-scattering properties of a nonspherical shape other than spheroids and circular cylinders with particle sizes of ~ 48λ(size parameter ~150), specifically a hexagonal column with a length size parameter of kL = 300 where k = 2π/λ and L is the column length. IGOM is also quite accurate except near the exact 180°backscattering direction. This study demonstrates that a synergetic combination of the numerically-exact II-TM and the approximate PGOM can seamlessly cover the entire size parameter range of practical interest. To demonstrate the applicability of the approach, we compute the optical properties of dust particles and demonstrate a downstream application to the retrieval of dust aerosol optical thickness and effective particle size from satellite polarimetric observations.

On Babinet's principle and diffraction associated with an arbitrary particle.

Babinet's principle is widely used to compute the diffraction by a particle. However, the diffraction by a 3-D object is not totally the same as that simulated with Babinet's principle. This Letter uses a surface integral equation to exactly formulate the diffraction by an arbitrary particle and illustrate the condition for the applicability of Babinet's principle. The present results may serve to close the debate on the diffraction formalism.

Physical-geometric optics method for large size faceted particles.

A new physical-geometric optics method is developed to compute the single-scattering properties of faceted particles. It incorporates a general absorption vector to accurately account for inhomogeneous wave effects, and subsequently yields the relevant analytical formulas effective and computationally efficient for absorptive scattering particles. A bundle of rays incident on a certain facet can be traced as a single beam. For a beam incident on multiple facets, a systematic beam-splitting technique based on computer graphics is used to split the original beam into several sub-beams so that each sub-beam is incident only on an individual facet. The new beam-splitting technique significantly reduces the computational burden. The present physical-geometric optics method can be generalized to arbitrary faceted particles with either convex or concave shapes and with a homogeneous or an inhomogeneous (e.g., a particle with a core) composition. The single-scattering properties of irregular convex homogeneous and inhomogeneous hexahedra are simulated and compared to their counterparts from two other methods including a numerically rigorous method.

Validation of quasi-invariant ice cloud radiative quantities with MODIS satellite-based cloud property retrievals.

Similarity relations applied to ice cloud radiance calculations are theoretically analyzed and numerically validated. If τ(1-ϖ) and τ(1-ϖg) are conserved where τ is optical thickness, ϖ the single-scattering albedo, and g the asymmetry factor, it is possible that substantially different phase functions may give rise to similar radiances in both conservative and non-conservative scattering cases, particularly in the case of large optical thicknesses. In addition to theoretical analysis, this study uses operational ice cloud optical thickness retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS) Level 2 Collection 5 (C5) and Collection 6 (C6) cloud property products to verify radiative similarity relations. It is found that, if the MODIS C5 and C6 ice cloud optical thickness values are multiplied by their respective (1-ϖg) factors, the resultant products referred to as the effective optical thicknesses become similar with their ratio values around unity. Furthermore, the ratios of the C5 and C6 ice cloud effective optical thicknesses display an angular variation pattern similar to that of the corresponding ice cloud phase function ratios. The MODIS C5 and C6 values of ice cloud similarity parameter, defined as [(1-ϖ)/(1-ϖg)]1/2, also tend to be similar.

Polarimetric imaging and retrieval of target polarization characteristics in underwater environment.

Polarized light fields contain more information than simple irradiance and such capabilities provide an advanced tool for underwater imaging. The concept of the beam spread function (BSF) for analysis of scalar underwater imaging was extended to a polarized BSF which considers polarization. The following studies of the polarized BSF in an underwater environment through Monte Carlo simulations and experiments led to a simplified underwater polarimetric imaging model. With the knowledge acquired in the analysis of the polarimetric imaging formation process of a manmade underwater target with known polarization properties, a method to extract the inherent optical properties of the water and to retrieve polarization characteristics of the target was explored. The proposed method for retrieval of underwater target polarization characteristics should contribute to future efforts to reveal the underlying mechanism of polarization camouflage possessed by marine animals and finally to generalize guidelines for creating engineered surfaces capable of similar polarization camouflage abilities in an underwater environment.

Open-ocean fish reveal an omnidirectional solution to camouflage in polarized environments.

Despite appearing featureless to our eyes, the open ocean is a highly variable environment for polarization-sensitive viewers. Dynamic visual backgrounds coupled with predator encounters from all possible directions make this habitat one of the most challenging for camouflage. We tested open-ocean crypsis in nature by collecting more than 1500 videopolarimetry measurements from live fish from distinct habitats under a variety of viewing conditions. Open-ocean fish species exhibited camouflage that was superior to that of both nearshore fish and mirrorlike surfaces, with significantly higher crypsis at angles associated with predator detection and pursuit. Histological measurements revealed that specific arrangements of reflective guanine platelets in the fish's skin produce angle-dependent polarization modifications for polarocrypsis in the open ocean, suggesting a mechanism for natural selection to shape reflectance properties in this complex environment.

Broadband and polarization reflectors in the lookdown, Selene vomer.

Predator evasion in the open ocean is difficult because there are no objects to hide behind. The silvery surface of fish plays an important role in open water camouflage. Various models have been proposed to account for the broadband reflectance by the fish skin that involve one-dimensional variations in the arrangement of guanine crystal reflectors, yet the three-dimensional organization of these guanine platelets have not been well characterized. Here, we report the three-dimensional organization and the optical properties of integumentary guanine platelets in a silvery marine fish, the lookdown (Selene vomer). Our structural analysis and computational modelling show that stacks of guanine platelets with random yaw angles in the fish skin produce broadband reflectance via colour mixing. Optical axes of the guanine platelets and the collagen layer are aligned closely and provide bulk birefringence properties that influence the polarization reflectance by the skin. These data demonstrate how the lookdown preserves or alters polarization states at different incident polarization angles. These optical properties resulted from the organization of these guanine platelets and the collagen layer may have implications for open ocean camouflage in varying light fields.

Inherent optical properties of the coccolithophore: Emiliania huxleyi.

A realistic nonspherical model for Emiliania huxleyi (EHUX) is built, based on electron micrographs of coccolithophore cells. The Inherent Optical Properties (IOP) of the EHUX are then calculated numerically by using the discrete dipole approximation. The coccolithophore model includes a near-spherical core with the refractive index of 1.04 + m(i)j, and a carbonate shell formed by smaller coccoliths with refractive index of 1.2 + m(i)j, where m(i) = 0 or 0.01 and j(2) = -1. The reported IOP are the Mueller scattering matrix, backscattering probability, and depolarization ratio. Our calculation shows that the Mueller matrices of coccolithophores show different angular dependence from those of coccoliths.

Mueller matrix holographic method for small particle characterization: theory and numerical studies.

Holographic imaging has proved to be useful for spherical particle characterization, including the retrieval of particle size, refractive index, and 3D location. In this method, the interference pattern of the incident and scattered light fields is recorded by a camera and compared with the relevant Lorenz-Mie solutions. However, the method is limited to spherical particles, and the complete polarized scattering components have not been studied. This work extends the Mueller matrix formalism for the scattered light to describe the interference light field, and proposes a Mueller matrix holography method, through which complete polarization information can be obtained. The mathematical formalism of the holographic Mueller matrix is derived, and numerical examples of birefringent spheres are provided. The Mueller matrix holography method may provide a better opportunity than conventional methods to study anisotropic particles.

Genesis and evolution of polarization of light in the ocean [invited].

The radiative transfer of sunlight through the deep oceans of the world is a complex and only partially solved environmental optical problem. Empirically, in situ systematic measurements of key parameters such as polarization of deep open seawater have been very sparse in recent decades. Although we have the necessary equation of transfer to solve this complex problem, until it can be solved explicitly, only approximations and partial analytic solutions are possible in addition to some successful computer modeling. Further complexity is added by the diversity of researchers' interests from academic to international policy making, as well as the ineffective communication between the different disciplines concerned, ranging from mathematics to endangered species. As a result, isolated focused pockets of good data and theory have been developed in recent decades without the needed breadth of understanding. This present review intends to bring together some visual biology and optical physics in order to understand the role of polarization in navigation, communication, and identification of marine animals as well as a possible tool for remotely sensing underwater objects.

Effect of surrounding inhomogeneities on whispering gallery modes in spherical resonators.

The Amsterdam discrete dipole approximation (ADDA) is used to study the effects of an inhomogeneous refractive index in the surrounding medium of a microspherical resonator on the quality and position of the whispering gallery modes (WGMs). The model consists of a polystyrene microsphere with a refractive index, n, of 1.587 surrounded by water (n=1.333) and an inhomogeneity (n=1.5) on top of the microsphere. The effect of the area of the inhomogeneity on the WGMs is modeled using the ADDA code and compared with Lorenz-Mie code. WGMs of various quantum dot embedded microspheres mounted on atomic force microscope cantilevers are experimentally measured and shown to be consistent with the model.

Benthic effects on the polarization of light in shallow waters.

Measurements of the upwelling polarized radiance in relatively shallow waters of varying depths and benthic conditions are compared to simulations, revealing the depolarizing nature of the seafloor. The simulations, executed with the software package RayXP, are solutions to the vector radiative transfer equation, which depends on the incident light field and three types of parameters: inherent optical properties, the scattering matrix, and the benthic reflectance. These were measured directly or calculated from measurements with additional assumptions. Specifically, the Lambertian model used to simulate benthic reflectances is something of a simplification of reality, but the bottoms used in this study are found to be crucial for accurate simulations of polarization. Comparisons of simulations with and without bottom contributions show that only the former corroborate measurements of the Stokes components and the degree of linear polarization (DoLP) collected by the polarimeter developed at the City College of New York. Because this polarimeter is multiangular and hyperspectral, errors can be computed point-wise over a large range of scattering angles and wavelengths. Trends also become apparent. DoLP is highly sensitive to the benthic reflectance and to the incident wavelength, peaking in the red band, but the angle of linear polarization is almost spectrally constant and independent of the bottom. These results can thus facilitate the detection of benthic materials as well as future studies of camouflage by benthic biota; to hide underwater successfully, animals must reflect light just as depolarized as that reflected by benthic materials.

Backscattering properties of small layered plates: a model for iridosomes.

The backscattering properties of small layered plates are studied for various size parameter values with respect to the plate thickness, plate aspect ratio, number of layers, incident direction, and polarization states of the incident light. The results are compared with the analytical results for semi-infinite plates. The phase functions and the corresponding backscattering efficiencies of the small plates are computed with the Discrete Dipole Approximation method. The angular width of the reflection peak is found to depend on both the size parameter and the aspect ratio. The criteria for using the reflectance of semi-infinite plates to approximate the backscattering efficiencies of finite size plates are quantified with respect to the number of layers, incident angle, and polarization state.

Measurements and simulations of polarization states of underwater light in clear oceanic waters.

Polarization states of the underwater light field were measured by a hyperspectral and multiangular polarimeter and a video polarimeter under various atmospheric, surface, and water conditions, as well as solar and viewing geometries, in clear oceanic waters near Port Aransas, Texas. Some of the first comprehensive comparisons were made between the measured polarized light, including the degree and angle of linear polarization and linear Stokes parameters (Q and U), and those from Monte Carlo simulations that used concurrently measured water inherent optical properties and particle volume scattering functions as input. For selected wavelengths in the visible spectrum, measured and model-simulated polarization characteristics were found to be consistent in most cases. Measured degree and angle of linear polarization are found to be largely determined by an in-water single-scattering model. Model simulations suggest that the degree of linear polarization (DoLP) at horizontal viewing directions is highly dependent on the viewing azimuth angle for a low solar elevation. This implies that animals can use the DoLP signal for orientation.

Observation of non-principal plane neutral points in the in-water upwelling polarized light field.

Neutral points are specific directions in the light field where the three Stokes parameters Q, U, V, and thus the degree of polarization simultaneously go to zero. We have made the first measurement of non-principal-plane neutral points in the upwelling light field in natural waters. These neutral points are located at approximately 40°- 80° nadir angle and between 120° - 160° azimuth to the sun which is well off of the principal plane. Calculations show that the neutral point positions are very sensitive to the balance in the incident light between the partially polarized skylight and the direct solar beam.

Simulation of the optical properties of plate aggregates for application to the remote sensing of cirrus clouds.

In regions of deep tropical convection, ice particles often undergo aggregation and form complex chains. To investigate the effect of the representation of aggregates on electromagnetic scattering calculations, we developed an algorithm to efficiently specify the geometries of aggregates and to compute some of their geometric parameters, such as the projected area. Based on in situ observations, ice aggregates are defined as clusters of hexagonal plates with a chainlike overall shape, which may have smooth or roughened surfaces. An aggregate representation is developed with 10 ensemble members, each consisting of between 4-12 hexagonal plates. The scattering properties of an individual aggregate ice particle are computed using either the discrete dipole approximation or an improved geometric optics method, depending upon the size parameters. Subsequently, the aggregate properties are averaged over all geometries. The scattering properties of the aggregate representation closely agree with those computed from 1000 different aggregate geometries. As a result, the aggregate representation provides an accurate and computationally efficient way to represent all aggregates occurring within ice clouds. Furthermore, the aggregate representation can be used to study the influence of these complex ice particles on the satellite-based remote sensing of ice clouds. The computed cloud reflectances for aggregates are different from those associated with randomly oriented individual hexagonal plates. When aggregates are neglected, simulated cloud reflectances are generally lower at visible and shortwave-infrared wavelengths, resulting in smaller effective particle sizes but larger optical thicknesses.

Edge-effect contribution to the extinction of light by dielectric disks and cylindrical particles.

The extinction efficiency factor associated with the scattering of a plane electromagnetic wave impinging on a basal face of a dielectric disk or a cylindrical particle is investigated by employing the physical-geometric optics hybrid (PGOH) method and the discrete-dipole approximation (DDA) method. It is found that the derived extinction efficiency factor from the PGOH is a function of the thickness of the disk, or the length of the cylinder, and the refractive index, but is independent of the diameter and shape of the cross section of the basal face of the particle. Furthermore, the oscillations of the extinction efficiency factor versus the thickness or length of the particle do not diminish if the particle is not absorptive. The values of the extinction efficiency factor simulated from the DDA method are quite different from those computed from the PGOH, although the size parameter of the particle is in the commonly recognized geometric optics regime. To explain the difference, the concept of the edge effect associated with the tunneling rays in the semiclassical scattering theory is generalized from the case of spherical particles to that of nonspherical particles based on the localization principle. Accordingly, the edge-effect contribution can be distinguished and removed from the extinction cross section calculation by the DDA method. The remaining part of the extinction cross section, associated with the interference between the transmitted rays and incident rays, agrees well with the results computed from the PGOH, and the agreement illustrates the presence of the edge effect in the case of nonspherical particles with surfaces that have no curvature along the incident direction. It is found that the asymptotic extinction efficiency factor may not necessarily converge to 2, but it depends on the specific physical processes of the interference between diffracted and transmitted light and of the edge effect.

Platform effects on optical variability and prediction of underwater visibility.

We present hydrographic and optical data collected concurrently from two different platforms, the R/P FLoating Instrument Platform and the R/V Kilo Moana, located about 2km apart in the Santa Barbara Channel in California. We show that optical variability between the two platforms was due primarily to platform effects, specifically the breakdown of stratification from mixing by the hull of R/P FLIP. Modeled underwater radiance distribution differed by as much as 50% between the two platforms during stratified conditions. We determine that the observed optical variability resulted in up to 57% differences in predicted horizontal visibility of a black target.

Equivalent path lengths in an integrating cavity: comment.

The equivalent absorption path length in an integrating cavity is examined. In an otherwise excellent paper, Tranchart et al. [Appl. Opt. 35, 7070 (1996)] made an important error in obtaining the expressions for the equivalent path length in an integrating cavity. This error has been propagated through several other publications in the literature. Since the equivalent path length is the sine qua non for obtaining an accurate absorption coefficient when using an integrating cavity, it is our intent here to give the correct formulas to prevent further errors when extracting absorption coefficients.