THE SCATTERING PROBLEM FOR CYLINDRICAL PARTICLES
Leading to: Limiting cases of the general Mie theory
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:
In the case of randomly polarized incident radiation, the efficiency factors and the asymmetry factor are determined as follows:
Coefficients ak, bk are given by the following equations:
Jk is the Bessel function, and Hk(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:
The scattering function for randomly polarized incident radiation is written as
If radiation illuminates a cylinder along the normal to the axis, akE = bkH = 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
The following designations are used in Eqs. (8):
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,
In the simplest case of a homogeneous cylinder at normal incidence, one can write the following equations instead of (10)-(13):
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]:
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
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, Redding, CT, and New York, 1996.
Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.
Gurwich, I., Shiloah, N., and Kleiman, M., The recursive algorithm for electromagnetic scattering by tilted infinite circular multilayered cylinder, J. Quant. Spectr. Radiat. Transfer, vol. 63, no. 2-6, pp. 217-229, 1999.
Gurvich, I., Shiloah, N., and Kleiman, M., Calculations of the Mie scattering coefficients for multilayered particles with large size parameters, J. Quant. Spectr. Radiat. Transfer, vol. 70, no. 4-6, pp. 433-440, 2001.
Hau-Riege, S. P., Extending the size-parameter range for plane-wave light scattering from infinite homogeneous circular cylinders, Appl. Opt., vol. 45, no. 6, pp. 1219-1224, 2006.
Lee, S. C., Radiative transfer through a fibrous medium: Allowance for fiber orientation, J. Quant. Spectr. Radiat. Transfer, vol. 36, no. 3, pp. 253-263, 1986.
Lee, S. C., Radiation heat-transfer model for fibers oriented parallel to diffuse boundaries, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 303-308, 1988.
Swathi, P. S. and Tong, T. W., A new algorithm for computing the scattering coefficients of highly absorbing cylinders, J. Quant. Spectrosc. Radiat. Transfer, vol. 40, no. 4, pp. 525-530, 1988.