Use of circular cylinders as surrogates for hexagonal pristine ice crystals in scattering calculations at infrared wavelengths

We investigate the errors associated with the use of circular cylinders as surrogates for hexagonal columns in computing the optical properties of pristine ice crystals at infrared (8-12-microm) wavelengths. The equivalent circular cylinders are specified in terms of volume (V), projected area (A), and volume-to-area ratio that are equal to those of the hexagonal columns. We use the T-matrix method to compute the optical properties of the equivalent circular cylinders. We apply the finite-difference time-domain method to compute the optical properties of hexagonal ice columns smaller than 40 microm. For hexagonal columns larger than 40 microm we employ an improved geometric optics method and a stretched scattering potential technique developed in previous studies to calculate the phase function and the extinction (or absorption) efficiency, respectively. The differences between the results for circular cylinders and hexagonal columns are of the order of a few percent. Thus it is quite reasonable to use a circular cylinder geometry as a surrogate for pristine hexagonal ice columns for scattering calculations at infrared (8-12-microm) wavelengths. Although the pristine ice crystals can be approximated as circular cylinders in scattering calculations at infrared wavelengths, it is shown that optical properties of individual aggregates cannot be well approximated by those of individual finite columns or cylinders.     

Scattering of a gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results

We present numerical results concerning the properties of the electromagnetic field scattered by an infinite circular cylinder illuminated by a circular Gaussian beam. The cylinder is arbitrarily located and arbitrarily oriented with respect to the illuminating Gaussian beam. Numerical evaluations are provided within the framework of a rigorous electromagnetic theory, the generalized Lorenz-Mie theory, for infinite cylinders. This theory provides new insights that could not be obtained from older formulations, i.e., geometrical optics and plane-wave scattering. In particular, some emphasis is laid on the waveguiding effect and on the rainbow phenomenon whose fine structure is hardly predictable by use of geometrical optics.     

Local cross-sections and energy efficiencies in the multiple electromagnetic/optical scattering by two perfect electrically-conducting cylindrical particles

Exact mathematical expressions for the intrinsic electromagnetic (EM) cross–sections (i.e. extinction, scattering and absorption) for a pair of perfectly conducting circular cylinders in a homogeneous non–absorptive medium are derived. The multipole expansion method in cylindrical coordinates and the translational addition theorem, applicable to any range of frequencies or particle sizes are used. An effective EM field, incident on the probed cylinder is defined first, which includes the initial and re-scattered field from the second cylinder. It is used jointly with the scattered field to derive the mathematical expressions for the intrinsic/local cross–sections. Numerical computations for the intrinsic extinction (or scattering) energy efficiencies per unit-length for a pair of conducting circular cylinders with different radii in a homogeneous medium are considered. The results computed a priori can be useful in the full characterization of a multiple scattering system of many particles, in conjunction with experimental data for the extrinsic cross–sections.     

On the generation of water waves by cylindrical porous wave maker

Summary By the direct application of Havelock's expansion theorem and exploitation of various properties of Bessel functions, the problem of generation of cylindrical surface waves, in the case of water of infinite depth, in the presence of an impermeable circular cylinder surrounded by a co-axial permeable cylinder immersed vertically in the fluid region, is investigated. The wave motion is generated due to the simple harmonic motion in the radial direction of the (i) inner impermeable cylinder and (ii) co-axial permeable cylinder, when one of the two cylinders is kept fixed. As an application, the problem of scattering of water waves is analyzed when both the cylinders are kept fixed.     

Trapping and Near-Trapping by Arrays of Cylinders in Waves

In this paper, some recent developments and new results concerning the trapping of waves by arrays of vertical circular cylinders is presented. In particular, the cases are examined when there is a circular arrangement of cylinders and both finite and infinite periodic linear arrays of identical cylinders. Only for the infinite array is there pure trapping of waves – known as Rayleigh–Bloch or edge waves – which, for particular dominant wavenumbers, reduce to the well-known trapped-mode solutions for a cylinder between two parallel walls having either Neumann or Dirichlet conditions upon them. This latter case is considered separately and some new results are presented. In the circular array and finite linear array the concept of near-trapping is introduced where large resonant motions are found to occur at certain frequencies of the incident wave field. In the case of the finite linear array, these near-trapping frequencies are related to the Rayleigh–Bloch trapped-wave frequencies for the infinite array. Finally, the case when there are two or more lines of cylinders in the linear array is examined.     

Application of the sinc basis moment method to the reconstruction of infinite circular cylinders

A solution of the ultrasonic scattering and inverse scattering problem has been obtained by solving the inhomogeneous Helmholtz wave equation by the sinc basis moment method. In this numerical study, the algorithm of S.A. Johnson and M.L. Tracy (1983) has been applied to the reconstruction of an infinite circular cylinder that is subject to an incident cylindrical wave of ultrasound and is surrounded by a homogeneous coupling medium. For weak scattering cylinders, successful reconstructions have been obtained using the known exact solution for the scattered field as the input data for the algorithm. A detailed discussion of sampling requirements for this algorithm is presented, and the threshold derived correlates well with results of a numerical study of variation of the sampling density. Effects of varying object contrast, object size, grid size, sampling density, and method of iteration are investigated. Because the algorithm is slow, optimization of computation is described.     

Finite-difference time-domain solution of light scattering by an infinite dielectric column immersed in an absorbing medium

The two-dimensional (2-D) finite-difference time-domain (FDTD) method is applied to calculate light scattering and absorption by an arbitrarily shaped infinite column embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. The single-scattering properties of the infinite column embedded in the absorbing medium, including scattering phase functions and extinction and absorption efficiencies, are derived by use of an area integration of the internal field. An exact solution for light scattering and absorption by a circular cylinder in an absorbing medium is used to examine the accuracy of the 2-D UPML FDTD code. With use of a cell size of 1/120 incident wavelength in the FDTD calculations, the errors in the extinction and absorption efficiencies and asymmetry factors from the 2-D UPML FDTD are generally smaller than approximately 0.1%. The errors in the scattering phase functions are typically smaller than approximately 4%. With the 2-D UPML FDTD technique, light scattering and absorption by long noncircular columns embedded in absorbing media can be accurately solved.     

Single scattering of light by circular cylinders

We consider two topics pertaining to light scattering by circular cylinders. (A) Scattering properties of cylinders with increasing aspect ratio are studied. It is shown that the solution for finite cylinders does not converge to the solution for infinitely long cylinders if the aspect ratio increases. This is due to differences in the treatment of diffraction for finite and infinite cylinders. (B) Finite cylinders have sharp edges, so their scattering properties differ from those of spheroids having the same aspect ratio. To illustrate these differences we present scattering matrix elements of cylinders and spheroids for a large set of aspect ratios. To handle the large amount of data, the scattering matrix elements as functions of aspect ratio and scattering angle are presented in so-called three-dimensional figures.     

Scattering of light from arbitrarily oriented finite cylinders

An iterative approach to the scattering of light from a finite dielectric cylinder first developed by Shifrin and extended by Acquista is applied to cases where the phase shift is <2, and the cylinder is arbitrarily oriented. It is found that the first 2 orders of the iteration converge to within 1% when the aspect ratio (length/diameter) of the cylinder is as small as 20. The results are compared to the exact theory for infinite cylinders, and the effects of finite size are calculated and discussed.     

On the uniqueness of some helical flows of a second grade fluid

Summary The uniqueness of some helical flows of a second grade fluid, between two infinite circular cylinders, is proved. Initially, the fluid is at rest and flow is produced by the motion of the cylinders. Finally, the special case of a flow in a circular cylinder is considered.     

Vibration of infinite piezoelectric cylinders with anisotropic properties using cylindrical finite elements

Natural frequencies for free vibration of infinite piezoelectric cylinders are computed using finite elements that are formulated in cylindrical coordinates. The finite-element method is used to model the cross-section of the cylinder in r, theta coordinates using circular sectors. Material constants that are functions of theta are allowed to vary in each circular sector and are computed using standard tensor transformations. The accuracy of the finite-element formulation is verified using previous results for isotropic cylinders and axisymmetric piezoelectric cylinders. New results are tabulated for frequencies of free vibration of solid and hollow piezoelectric cylinders of LiNbO(3) of crystal class 3m. Displacements for typical mode shapes are illustrated graphically.     

The scattering of water waves by an array of circular cylinders in a channel

The full linear problem of the scattering of water waves by an array of N bottom-mounted vertical circular cylinders situated in a channel of constant depth and width is solved using the method of multipoles. A simple formula is derived for the velocity potential in the vicinity of a cylinder, and in particular on the cylinder surfaces, which allows hydrodynamic quantities such as forces to be easily evaluated. The simplicity of the solution makes the evaluation of quantities of interest straightforward and extensive results are given. An approximate solution for the forces on the cylinders, based on the assumption that the wavelength of the incident wave is long compared with the cylinder radii, is also given, and this is compared with results from the exact linear solution.     

The trapping of surface waves by multiple submerged horizontal cylinders

The existence of edge waves, or trapped modes, travelling above a single long horizontal submerged cylinder is well established in the linearised theory of water waves. In the present paper, the possibility of wave-trapping by multiple submerged horizontal circular cylinders is considered. The trapped mode solutions are constructed by means of a multipole approach combined with an addition formula for Bessel functions and requires finding the non-trivial solutions of a real infinite system of algebraic equations. The case of a single submerged cylinder is returned to briefly, where results for symmetric trapped modes are reproduced and new numerical results for antisymmetric modes are presented. A large range of results are also presented for multiple cylinders.     

Anomalous diffraction theory for arbitrarily oriented finite circular cylinders and comparison with exact T-matrix results

A general method is developed to formulate extinction and absorption efficiency for nonspherical particles at arbitrary and random orientations by use of anomalous diffraction theory (ADT). An ADT for finite circular cylinders is evaluated as an example. Existing ADT's for infinite cylinders at arbitrary orientations and for finite cylinders at the normal incidence are shown to be special cases of the new formulation. ADT solutions for finite cylinders are shown to approach the rigorous T-matrix results when the refractive indices approach unity. The importance of some physical processes that are neglected in the ADT approximation are evaluated by comparisons between ADT and rigorous calculations for different particle geometries. For spheres, van de Hulst's ADT and Mie theory are used, whereas the ADT that we present and T-matrix calculations are used for cylinders of different diameter-to-length ratios. The results show that the differences in extinction between ADT and exact solutions generally decrease with nonsphericity. A similar decrease occurs for absorption at wavelengths of relatively strong absorption. The influence of complex refractive index is evaluated. Our results suggest that ADT may provide a useful approximation in parameterization and remote sensing of cirrus clouds in the Christiansen bands where the real part of the refractive index approaches unity and/or where relative absorption is strong.     

Electromagnetic scattering by an infinite cylinder of material or metamaterial coating eccentrically a dielectric cylinder

The electromagnetic scattering by an infinite cylinder of dielectric material or metamaterial, coating eccentrically another infinite dielectric cylinder, is treated in this work. The problem is solved using classical separation of variables techniques. No use is made of the translational addition theorem. For small eccentricities h = d/a(? 1), where d is the distance between the axes of the cylinders and a the radius of the outer cylinder, we use instead the cosine and the sine laws to satisfy the boundary conditions at the surface of the outer cylinder. Keeping terms up to the order h2 we finally obtain exact, closed-form expressions for the expansion coefficients g(1) and g(2) in the relation S(h) = S(0)[1 + g(1)h + g(2)h2 + O(h3)], giving the scattered field and the scattering cross sections of the problem, where S(0) corresponds to the coaxial geometry, with h = 0 (d = 0). Both polarizations are considered for normal incidence. Numerical results are given for various values of the parameters, corresponding to materials or metamaterials. Our method is an alternative of the one using the translational addition theorem in the case of small eccentricities h.     

Discrete-dipole approximation for periodic targets: theory and tests

The discrete-dipole approximation (DDA) is a powerful method for calculating absorption and scattering by targets that have sizes smaller than or comparable to the wavelength of the incident radiation. The DDA can be extended to targets that are singly or doubly periodic. We generalize the scattering amplitude matrix and the 4 x 4 Mueller matrix to describe scattering by singly and doubly periodic targets and show how these matrices can be calculated using the DDA. The accuracy of DDA calculations using the open-source code DDSCAT is demonstrated by comparison with exact results for infinite cylinders and infinite slabs. A method for using the DDA solution to obtain fields within and near the target is presented, with results shown for infinite slabs.     

Optical extinction by closely spaced parallel cylinders inside a finite dielectric slab

The formulation for the extinction and scattering cross sections of closely spaced parallel infinite cylinders in a dielectric medium of finite thickness is presented. We consider the general case of dissimilar refractive indices for the half-spaces on both sides of the slab, and the diameter and refractive index of each cylinder can be different. The formulation accounts for the coherent scattering between the cylinders and scattering of the multiply reflected internal waves inside the slab. Discontinuity in the refractive index across the dielectric slab interfaces results in boundary reflections that modify the angular distribution of the scattered intensity in both forward and backward directions. The extinction cross section, which is derived by a formal application of the optical theorem, is shown to consist of both a forward and a backward component. The general solution is applied to obtain the formulas for the cases of cylinders in front of a reflecting plane, cylinders inside a semi-infinite dielectric medium, and cylinders in free space.     

Split off-specular reflection and surface scattering from woven materials

We measured radiance distributions for black lining cloth and copper gauze using the convenient technique of wrapping the materials around a circular cylinder, irradiating it with a parallel light source and collecting the scattered radiance by a digital camera. One family of parallel threads (weave or weft) was parallel to the cylinder generator. The most salient features for such glossy plane weaves are a splitting up of the reflection peak due to the wavy variations in local slopes of the threads around the cylinders and a surface scattering lobe due to the threads that run along the cylinder. These scattering characteristics are quite different from the (off-)specular peaks and lobes that were found before for random rough specular surfaces. The split off-specular reflection is due to the regular structures in our samples of man-made materials. We derived simple approximations for these reflectance characteristics using geometrical optics.     

Scattering of light by polydisperse, randomly oriented, finite circular cylinders

We use the T-matrix method, as described by Mishchenko [Appl. Opt. 32, 4652 (1993)], to compute rigorously light scattering by finite circular cylinders in random orientation. First we discuss numerical aspects of T -matrix computations specific for finite cylinders and present results of benchmark computations for a simple cylinder model. Then we report results of extensive computations for polydisperse, randomly oriented cylinders with a refractive index of 1.53 + 0.008i, diameter-to-length ratios of 1/2, 1/1.4, 1, 1.4, and 2, and effective size parameters ranging from 0 to 25. These computations parallel our recent study of light scattering by polydisperse, randomly oriented spheroids and are used to compare scattering properties of the two classes of simple convex particles. Despite the significant difference in shape between the two particle types (entirely smooth surface for spheroids and sharp rectangular edges for cylinders), the comparison shows rather small differences in the integral photometric characteristics (total optical cross sections, single-scattering albedo, and asymmetry parameter of the phase function) and the phase function. The general patterns of the other elements of the scattering matrix for cylinders and aspect-ratio-equivalent spheroids are also qualitatively similar, although noticeable quantitative differences can be found in some particular cases. In general, cylinders demonstrate much less shape dependence of the elements of the scattering matrix than do spheroids. Our computations show that, like spheroids and bispheres, cylinders with surface-equivalent radii smaller than a wavelength can strongly depolarize backscattered light, thus suggesting that backscattering depolarization for nonspherical particles cannot be universally explained by using only geometric-optics considerations.     

Light scattering by a coated infinite cylinder in an absorbing medium

The scattering formulation for a coated infinite cylinder in an absorbing medium is presented in this paper. The cylinder is subjected to an arbitrarily polarized plane wave propagating in a general direction at the cylinder. The refractive index and magnetic permeability of the host medium, as well as those for the core and coating of the cylinder, can be real or complex. The scattering and extinction efficiencies and the scattering amplitudes are derived for both the near field and the far field. As the medium is absorbing, the "true" extinction and scattering efficiencies are derived based on the radiative energy outflow at the surface of the cylinder. The radiative efficiencies in the far field are denoted as "apparent" properties because they include absorption by the intervening medium. The influence of the refractive index and permeability of the host medium on the scattering properties of a coated cylinder is illustrated by numerical examples.