Extended Boundary Condition Method Research Articles

Issues of analytical continuation of the field interior to the target surface and a more general T-matrix for submerged fluid targets

The T-matrix or extended boundary condition method developed by P. Waterman for wave scattering from classical targets and extended by a host of researchers is a marvelous generalization of normal mode solutions allowed for separable geometries to extend to nonseparable geometries by means of mode coupling. The method is based on Huygens’ principle and partial wave expansions of physical quantities. In general the expansions don’t obey orthogonality and one must take care in the way expansions are made. One must never take expansions of a quantity and then represent expansions of their differential forms in terms of the partial wave differential forms using the same expansion coefficients lest one violate a mathematical principle (the Rayleigh hypothesis). One may circumvent any issues without resort to analytical continuation of interior solutions by including all solutions subject to interior and exterior Greens functions. This leads to matrix relations (constraining matrices) between the expansion terms which when used lead to new forms of the fluid T-matrix, for example. The new form differs from Waterman’s T-matrix for the case when the constraining matrices are not symmetric. Otherwise the original expression by Waterman is valid. These issues are discussed and examples are presented.

Calculation of cutoff frequency of arbitrary weak guidance optical waveguides by extended boundary condition method

The extended boundary condition method is applied to both circular and elliptical regions for calculating cutoff frequencies Vc of weakly guiding optical waveguides, which consist of a uniform cladding and a core with arbitrary shape and index profile. The frequency shift formula method is explained by means of the extended boundary condition method on circular regions. Comparing the method with analytic and other rigorous methods, we show that it is indeed a powerful tool. Numerical results confirm that the mode designation and arrangement order in elliptical and rectangular waveguides are, in general, dependent on the aspect ratio.

Single-expansion EBCM computations for osculating spheres

We show that the standard, single-expansion extended boundary condition method provides convergent scattering results for osculating dielectric spheres and discuss the implications of this result.

Atmospheric effect on microwave polarimetric passive remote sensing of ocean surfaces

A theoretical emission model of combined ocean surface and atmosphere is presented to predict the microwave emissivity of the ocean. The modeled ocean surface is one‐dimensional with a random rough profile. The electromagnetic scattering from the surface is calculated based on the extended boundary condition method. Realizations of rough surfaces are created using Monte Carlo simulations. The bistatic scattering coefficients are computed from the ensemble average. The millimeter‐wave propagation model is used to evaluate the absorption of microwave radiation at all height levels in the atmosphere. An expression for the observed brightness temperatures is derived by solving the radiative transfer equations. The radiative transfer model results show a good agreement with the measured data from the 1995 NASA Jet Propulsion Laboratory WIND radiometer (WINDRAD) campaign. An approximate model is provided to estimate the atmospheric effect on the ocean brightness temperatures based on the overall atmospheric attenuation. The approximate model also compares well with the WINDRAD data. Further comparisons are made between the approximate formula and the radiative transfer results on the ratio of the third Stokes parameter in the atmosphere to the one in free space by varying the atmospheric conditions, surface roughness, and radiation frequencies. The approximate formula shows its usefulness for the prediction of the ocean brightness temperatures.

Light Scattering Simulation and Measurement of Monodisperse Spheroids using a Phase Doppler Anemometer

Mathematical tools are provided for the computation of the scattered field produced by non-spherical particles moving through the measurement volume of a phase Doppler anemometer. The phase distribution of a spheroid with random orientation is computed by using the rigorous extended boundary condition method and the ray theory. In a phase Doppler experiment the spheroid parameters are obtained by fitting the measured phase distribution with the simulated phase distribution. The numerical simulations are supported by experimental results on monodisperse spheroids.

Light Scattering Simulation and Measurement of Monodisperse Spheroids using a Phase Doppler Anemometer

Mathematical tools are provided for the computation of the scattered field produced by non-spherical particles moving through the measurement volume of a phase Doppler anemometer. The phase distribution of a spheroid with random orientation is computed by using the rigorous extended boundary condition method and the ray theory. In a phase Doppler experiment the spheroid parameters are obtained by fitting the measured phase distribution with the simulated phase distribution. The numerical simulations are supported by experimental results on monodisperse spheroids.

Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates

A method other than the extended-boundary-condition method (EBCM) to compute the T matrix for electromagnetic scattering is presented. The separation-of-variables method (SVM) is used to solve the electromagnetic scattering problem for a spheroidal particle and to derive its T matrix in spheroidal coordinates. A transformation is developed for transforming the T matrix in spheroidal coordinates into the corresponding T matrix in spherical coordinates. The T matrix so obtained can be used for analytical calculation of the optical properties of ensembles of randomly oriented spheroids of arbitrary shape by use of an existing method to average over orientational angles. The optical properties obtained with the SVM and the EBCM are compared for different test cases. For mildly aspherical particles the two methods yield indistinguishable results. Small differences appear for highly aspherical particles. The new approach can be used to compute optical properties for arbitrary values of the aspect ratio. To test the accuracy of the expansion coefficients of the spheroidal functions for arbitrary arguments, a new testing method based on the completeness relation of the spheroidal functions is developed.

Scattering analysis of conducting axisymmetic particles using the extended boundary condition method with discrete sources

Scattering analysis of conducting axisymmetic particles using the extended boundary condition method with discrete sources

Scattering analysis of conducting axisymmetic particles using the extended boundary condition method with discrete source

In this paper the extended boundary condition method with discrete sources (DS) is introduced for the electromagnetic scattering problem of a conducting axisymmetric particle. The approximate solution is constructed as a linear combination of “Mie-potentials” with different origins. In order to take advantage of the rotational symmetry the DS are distributed on an auxiliary curve in the plane of the generatrix. The complete system of vector functions have an explicit exp (jmϕ-dependence which lead to a decoupling of the scattering problem over each azimuthal mode. It is shown that with the method discussed herein, the computational efficiency can be improved in comparison to that which use DS distributed on auxiliary surfaces.

Generalization of the separation of variables method for non-spherical scattering on dielectric objects

During the last years, the T-matrix approach based on the Extended Boundary Condition Method (EBCM) has become a powerful technique for solving the problem of plane wave scattering by non-spherical particles. It can be applied to different geometries and to large size parameters. For certain scattering configurations this method is able to bridge the gap between the resonance region and the geometric optics approximation. Despite these advantages it still remains a difficult method for practitioners. It is accompanied by loss of simplicity in concept and execution caused by the formulation in terms of integrals and related problems like the formulation of the non-local EBC, and the open question if it suffers from a restricted range of applicability of the Rayleigh hypothesis. In this paper, a new method will be presented which offers a much simpler way to derive a T-matrix without running into these problems. This is achieved by a generalization of the Separation of Variables Method (SVM) which uses the differential formulation of the scattering process. In this way, we are able to remove the mistake that SVM can be applied only if the boundary surface of the scatterer coincides with a constant coordinate line. Additional advantages of this approach are the possibility to consolidate different methods into a common mathematical body, thus allowing a better estimation of the accuracy of different numerical techniques, and the possibility to find an answer to the problem of the Rayleigh hypothesis. To formulate this new approach the method of lines (MoL) is used as a starting point. The MoL is a mathematical tool for solving partial differential equations, and it has been applied very successfully in guided wave theories. It also forms the mathematical basis of the Discretized Mie-formalism (DMF) which has been recently developed for plane wave scattering. With this contribution, there will be discussed interesting new aspects of the MoL.

Experimental size determination of spheroidal particles via the T-matrix method

Light scattering (LS) properties have been used for determining the size and shape in colloidal suspensions of hematite particles in water. The system has been approximated by a suspension of randomly oriented, monodisperse, spheroidal particles with an equivalent-volume-diameter d eq and an axial ratio, or eccentricity, ε= a/ b ( b=revolution axis), in the single-scattering mode. Relative intensity measurements have been taken at several angles, and the results have been compared with theoretical models obtained by combining Waterman’s Extended Boundary Condition Method (EBCM) together with Mishchenko’s analytical averaging scheme for randomly oriented particles. Both full and depolarized LS have been measured and fitted to theoretical curves. While both methods give good agreement on particle size, shape is not always characterized unequivocally with full LS. The problem is overcome by using depolarized LS, which, for spherical particles, is zero; fit between experimental depolarized LS and theoretical predictions give the correct value of eccentricity, as well as a more accurate value of d eq. This is an experimental validation that depolarized LS can be useful for determining size and shape of real particle systems.

Comparison of computational scattering methods

There are various methods to compute electromagnetic scattering by arbitrarily shaped particles. The aim of this article is merely to give a short introduction to three very different types of methods and have a look at the applicabilities and shortcomings of each. At first some comments are made. • to the Discrete Dipole Approximation (DDA), alias Coupled Dipole Method (CDM), as a special form of the Volume Integral Equation Method (VIE); • to the Finite Difference Time Domain (FDTD) and • to the Extended Boundary Condition Method (EBCM). As an example the results and the parameters of different codes for a cube are compared to give just a hint of the computational demands.

Light scattering from a particle on or near a surface

The problem of light scattering by a particle on or near a surface is treated using the extended boundary condition method's solution for scattering by a particle in a homogeneous medium and the integral representation of spherical vector wave functions over plane waves for the calculation of the reflection of the scattered field by the surface. An approximate solution is also given by assuming that the scattered field, reflecting off the surface and interacting with the particle, is incident upon the surface at near-normal incidence. The range of validity of the approximate method is checked from a numerical point of view

Formulations of the extended boundary condition method for three-dimensional scattering using the method of discrete sources

Formulations of the extended boundary condition method for three-dimensional scattering using the method of discrete sources

Formulations of the extended boundary condition method for three-dimensional scattering using the method of discrete sources

Novel formulations of the extended boundary condition method for three-dimensional scattering problems are derived by using a system of magnetic and electric dipoles, a system of ‘Mie potentials’ and a system of lowest-order multipoles as complete systems of functions. The key step in these approaches is to use discrete sources located on auxiliary supports to guarantee the null-field condition for the total electric field and to generate the surface current densities.

Comparison between various formulations of the extended boundary condition method

The accuracy of various formulations of the extended boundary condition method (EBCM) for solving the scattering problem of a cone-sphere particle is investigated. We compare the standard EBCM, the multiple miltipole EBCM, the lowest-order multipole EBCM and the EBCM with sub-domain bases. For this pusposes we choose the residual of the total electric field on spherical surfaces with shifted origins as error criterion. A hybrid method consisting of a combination of sub-boundary bases and global-boundary bases for the surface current density representation is also discussed.

Extended boundary condition method with multipole sources located in the complex plane

A novel formulation for improving the numerical stability of the extended boundary condition method for highly elongated and flattened dielectric objects is presented. The key step in this approach is to approximate the surface currents by using lowest-order multipoles located in the complex plane. The accuracy of the proposed method for strongly deformed particles is verified by comparing the numerical results with those derived by using other approaches.

Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions

Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions

Formulations of the extended boundary condition method for incident gaussian beams using multiple-multipole expansions

Novel formulations for improving the numerical stability of the extended boundary conditions method and extending its application to highly elongated dielectric objects illuminated by a Gaussian beam are described. The derived formulations are obtained as special cases of a general approach, which considers arbitrary complete system functions on the particle surface. The surface currents are approximated by multiple spherical expansions or, for a fixed azimuthal mode, by the lowest-order multipoles located inside the particle surface. Results illustrating the application of the procedures developed are presented.

Light Scattering by Homogeneous Axisymmetric Particles for PDA Calculations to Measure Both Axes of Spheroidal Particles

Mathematical tools are provided for the computation of the scattered field produced by a non‐spherical particle moving through the measurement volume of a phase Doppler anemometer. The Doppler signal is modeled by using the rigorous extended boundary condition method and an approximate ray theory. The region of applicability of ray theory is discussed. In particular, a pure refraction model is considered for large particles. It is shown that under some circumstances the phase Doppler technique may be used for sizing of spheroidal particles.