## THE SCATTERING PROBLEM FOR CYLINDRICAL PARTICLES

**Following from: **
Radiative properties of single particles and fibers: the hypothesis of independent scattering and the Mie theory

**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 *a*_{k}, *b*_{k} are given by the following equations:

where

*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:

The scattering function for randomly polarized incident radiation is written as

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, *T*_{3} = *T*_{4} = 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

where

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,

where

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]:

or

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**

#### REFERENCES

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.

#### Referencias

- 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.