Cylinders, scattering infinite

In Chapter 8 we shall derive the field scattered by an infinite cylinder of arbitrary radius and refractive index we shall also consider scattering by a finite cylinder in the diffraction theory approximation. Although the finite cylinder scattering problem is not exactly soluble, we can obtain analytical expressions for the amplitude scattering matrix elements in the Rayleigh-Gan s approximation. 

There are many naturally occurring particles, such as some viruses and asbestos fibers, which are best represented as cylinders long compared with their diameter. Therefore, in this section we shall construct the exact solution to the problem of absorption and scattering by an infinitely long right circular cylinder and examine some of the properties of this solution. 

Figure 8.4 Wave front and wave normals of light scattered by an infinite cylinder.

In Chapter 3 we derived a general expression for the amplitude scattering matrix for an arbitrary particle. An unstated assumption underlying that derivation is that the particle is confined within a bounded region, a condition that is not satisfied by an infinite cylinder. Nevertheless, we can express the field scattered by such a cylinder in a concise form by resolving the incident and scattered fields into components parallel and perpendicular to planes determined by the cylinder axis (ez) and the appropriate wave normals (see Fig. 8.3). That is, we write the incident field... 

Although we have repeatedly referred to an infinite cylinder, it is clear that no such cylinder exists except as an idealization. So what we really have in mind is a cylinder long compared with its diameter. Later in this section we shall try to acquire some insight into how long a cylinder must be before it is effectively infinite by considering scattering in the diffraction theory approximation. 

Figure 8.7 Scattering cross section per unit particle volume for normally incident light polarized parallel (---) and perpendicular (...) to the axis of an infinite cylinder in air.

Equation (8.43) provides us with an approximate criterion, subject to the limitations of diffraction theory, for when a finite cylinder may be regarded as effectively infinite if R > 10, say, there will be comparatively little light scattered in directions other than those in a plane perpendicular to the cylinder axis. The greater is R, the more the scattered light is concentrated in this plane in the limit of indefinitely large R, no light is scattered in directions other than in this plane. We may show this as follows. The phase function may be written in the form p(0, ) = G(0, )F(0, ), where... 

We need consider only scattering directions in the plane = tt/2 (or < > = 377/2) because p vanishes outside this plane we also have 0 = 0 when = 77/2 and 0 = - 6 when = 377/2, where 0 = 0 is the forward direction. Thus, we may take the phase function for scattering by an infinite cylinder in the diffraction theory approximation to be... 

Figure 8.10 Phase function for scattering of unpolarized light by an infinite cylinder. The arrows indicate minima according to diffraction theory.

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

An infinite right circular cylinder is another particle shape for which the scattering problem is exactly soluble (Section 8.4), although it might be thought that such cylinders are so unphysical as to be totally irrelevant to real... 

Bohren, C. F., and V. Timbrell, 1979. Computer programs for calculating scattering and absorption by normally illuminated infinite cylinders, Project No. 38, Institute of Occupational and Environmental Health, Montreal. 

Analysis of low angle X-ray scattering data gives a value for the radius of gyration at infinite dilution of 18.3 A (254)- The scattering curve can best be explained on the basis of a cylinder or prolate ellipsoid... 

The motivation for this work is the potential use of chiral (and maybe biisotropic) cylinders as rod antennas and scatterers. Accordingly, Lakhtakia investigated the boundary value problem relevant to the scattering of an incident oblique plane EM wave by an infinitely long homogeneous biisotropic cylinder. This medium is described by the so-called Fedorov representation through the monochromatic frequency-domain constitutive relations ... 

Recently, Lee and Pilon (2013) demonstrated that the absorption and scattering cross-sections per unit length of randomly oriented linear chains of spheres, representative of filamentous cyanobacteria (Fig. ID), can be approximated as those of randomly oriented infinitely long cylinders with equivalent volume per unit length. Then, for linear chains of monodisperse cells of diameter d, the diameter 4, v of the volume-equivalent infinitely... 

The number of scattering problems that can be solved analytically is severly limited by the inseparability of the vector wave equation in all but a very few coordinate systems. In the majority of cases various approximate methods have to be used. An excellent review of the analytic results for perfectly conducting bodies has been given by BOWMAN et al. [4.291. These include circular, elliptic, parabolic, and hyperbolic cylinders the wedge, the half plane, and other geometries. For infinite dielectric circular cylinders, see the review in KERKER [4.2]. 

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