Scattering problem for cylindrical particles

Let us consider the absorption and scattering of a plane electromagnetic wave by an arbitrary oriented homogeneous infinite circular cylinder. The geometrical scheme of the problem is shown in Fig. 1. It is sufficient to consider two polarization modes: “”, i.e, polarization with magnetic field vector perpendicular to vectors and _z, and “”, i.e., polarization with electric vector perpendicular to vectors and _z. The scattered radiation propagates along the conical surface with the axis OZ and the angle π -2α. The front of scattered wave at normal incidence (α = 0) is a cylindrical surface with the axis OZ.

Figure 1. Scheme of the scattering problem for an infinite cylinder: 1--“E” polarization, 2--“H” polarization

In the case of a homogeneous cylindrical particle, the efficiency factors of scattering and extinction, as well as the asymmetry factor of scattering, depend on the angle of incidence α, complex index of refraction m, and diffraction parameter x. The following expressions are known:

(1a)

(1b)

(1c)

(1d)

(1e)

In the case of randomly polarized incident radiation, the efficiency factors and the asymmetry factor are determined as follows:

(2a)

(2b)

Coefficients a_k, b_k are given by the following equations:

(3a)

(3b)

where

(4a)

(4b)

(4c)

(4d)

J_k is the Bessel function, and H_k⁽²⁾ is the Hankel function of the second kind. The amplitude matrix components for arbitrary oriented cylinder are expressed by the Mie coefficients and azimuth in the following manner:

(5a)

(5b)

The scattering function for randomly polarized incident radiation is written as

(6)

If radiation illuminates a cylinder along the normal to the axis, a_k^E = b_k^H = 0 and the above equations are considerably simplified. Particularly, T₃ = T₄ = 0 in this case.

Optical properties of hollow or two-layer cylinders can also be calculated by Eqs. (1), (2), (5), and (6), but x should be replaced by x'' defined by the external radius. In this case, the expressions for the Mie coefficients are considerably more complex. In general case, we have

(7a)

(7b)

(7c)

where

(8a)

(8b)

(8c)

The following designations are used in Eqs. (8):

(9a)

(9b)

(9c)

where x' is the diffraction parameter defined by the core or cavity radius, and m', m'' are the complex refractive indices of the core and the shell.

At normal incidence of radiation, considerably more simple equations take place,

(10a)

(10b)

(10c)

(11)

where

(12)

(13)

(14)

In the simplest case of a homogeneous cylinder at normal incidence, one can write the following equations instead of (10)-(13):

(15a)

(15b)

(15c)

(16)

Note that Eqs. (10)-(14) are similar to the analogous equations for two-layer spherical particles and Eqs. (15)-(16) to the equations for homogeneous spherical particles.

We do not consider the problem of reliable calculations of special functions for cylinders by using recursion relations. This was discussed in the book by Dombrovsky (1996) and in some other special publications. A reader can find useful information on this subject both for homogeneous and multilayered cylinders in papers by Swathi and Tong (1988), Gurwich et al. (1999, 2001), and Hau-Riege (2006). An additional bibliography on this subject can be found in the recent monograph by Dombrovsky and Baillis (2010).

One often comes across disperse systems of particles randomly oriented in space (isotropic system) or in parallel planes (transversely isotropic system, i.e., a system of isotropic layers of fibers). In these cases, it is convenient to introduce the efficiency factors averaged over orientations. For randomly polarized radiation, this can be written as follows [see original papers by Lee (1986, 1988) for further details]:

(17)

or

(18)

Equation (17) is referred to an isotropic disperse system, and Eq. (18) to a transversally isotropic disperse system. In the last case, the average values depend on the angle of illumination θ (see the scheme in Fig. 2).

Figure 2. Scheme of the problem for transversally isotropic composition of fibers: θ is the incidence angle for the plane of fibers P, α is the incidence angle for a single fiber f, and ψ is the angle between the plane of incidence I and the normal plane N