Within the generalized Lorenz-Mie theory (GLMT) framework, an analytical solution to the scattering by a uniaxial anisotropic cylinder, for oblique incidence of an on-axis Gaussian beam, is constructed by expanding the incident Gaussian beam, scattered fields as well as internal fields in terms of appropriate cylindrical vector wave functions (CVWFs). The unknown expansion coefficients are determined by virtue of the boundary conditions. For a localized beam model, numerical results are provided for the normalized internal and near-surface field intensity distributions, and the scattering characteristics are discussed concisely.
©2013 Optical Society of America
There are many natural and artificial anisotropic materials, having found a wide variety of applications in optical signal processing, optimum design of optical fibers, radar cross section controlling, and microwave device fabrication, etc. One of the basic problems to investigate the interaction between electromagnetic waves and anisotropic media is to characterize the scattering properties of anisotropic objects. Some numerical methods, such as the method of moments (MOM) , coupled dipole approximation method , integral equation technique , and frequency domain finite difference method (FDFD) , have been employed in the analysis of the scattering problems. By using the Fourier transform and vector wave eigenfunctions expansions , analytical solutions have been developed to the plane wave scattering by an anisotropic circular cylinder , uniaxial anisotropic sphere , anisotropic coated sphere [8, 9], and aggregate of uniaxial anisotropic spheres [10, 11]. Wu et al. studied the scattered and internal fields of a uniaxial anisotropic sphere illuminated by an on-axis, off-axis, and arbitrarily oriented Gaussian beam based on the GLMT for spheres [12, 13]. Gouesbet et al. published the theory of distributions [14, 15], and later on Lock employed the angular spectrum of plane waves model , to attack the problem of interaction between an arbitrary shaped beam and an infinite cylinder. In , we have obtained the expansion of an incident Gaussian beam (a focused TEM00 mode laser beam) in terms of the CVWFs, and subsequently in  investigated the Gaussian beam scattering by a homogeneous dielectric cylinder in the framework of the GLMT for cylinders. As an extension of our previous works, this paper is devoted to the case of a uniaxial anisotropic cylinder.
The paper is organized as follows. Section 2 provides the theoretical procedure for the determination of the scattered and internal fields of a uniaxial anisotropic cylinder illuminated by a Gaussian beam. In Section 3, the normalized internal and near-surface field intensity distributions are displayed. Section 4 is the conclusion.
2.1 Expansions of Gaussian beam, scattered fields and internal fields in cylindrical coordinates
As shown in Fig. 1, an infinite uniaxial anisotropic cylinder of radius is attached to the Cartesian coordinate system . An incident Gaussian beam propagates in free space and along the positive axis in the plane, with the middle of its beam waist located at origin on the axis. Origin has a coordinate on the axis (on-axis case), and the angle made by the axis with the axis is . In this paper, the time-dependent part of the electromagnetic fields is assumed to be .
We have obtained an expansion in  of the electromagnetic fields of an incident Gaussian beam (a focused TEM00 mode laser beam) in terms of the CVWFs with respect to the system , in the following form
For a TE polarized Gaussian beam, the coefficients , are19], can be described by the localized approximation as [20, 21]
It is worth mentioning that the coefficients , in Eqs. (3) and (4) are expressed in terms of beam shape coefficients in spherical coordinates, obviously an extrinsic method. Gouesbet et al. introduced a cylindrical localized approximation to evaluate the coefficients, in which an intrinsic method is used .
Let’s consider that a uniaxial anisotropic medium is characterized by a permittivity tensor in the system and a scalar permeability .
From Appendix, the electromagnetic fields within the uniaxial anisotropic cylinder can be expanded in terms of the CVWFs (refer to Appendix), as follows:
2.2 On-axis Gaussian beam scattering by a uniaxial anisotropic cylinder
From the system consisting of Eqs. (18)-(21), the expansion coefficients of the scattered and internal fields can be determined. By substituting them into Eqs. (6)-(9), the solution to the scattering of an on-axis Gaussian beam by a uniaxial anisotropic cylinder is obtained.
3. Numerical results
In this paper, we will focus our attention on the normalized internal and near-surface field intensity distributions, which are described, respectively, by
Figures (2) and (3), sharing the same colorbar, show the normalized internal and near-surface field intensity distributions and in the plane for a uniaxial anisotropic cylinder, illuminated by a TM and TE polarized Gaussian beam, respectively. The radius of the cylinder and the beam waist radius are assumed to be five and two times the wavelength of the incident Gaussian beam, and , , , .
From Figs. 2 and 3 we can see that a TM polarized Gaussian beam is reflected less than a TE polarized one, so it is reasonable to conclude that in the same case the transmitted power is larger for the former. Another noticeable difference lies in the fact that the middle of the transmitted beam for a TM polarized Gaussian beam is lower than that for a TE polarized one, demonstrating that the former is refracted more strongly, which dose not appear for a dielectric isotropic cylinder.
An approach to compute the on-axis Gaussian beam scattering by a uniaxial anisotropic cylinder is presented. Numerical results of the normalized internal and near-surface field intensity distributions show that, compared with a TE polarized Gaussian beam, a TM polarized one is reflected less and refracted more strongly. As a result, this study provides an exact analytical model for interpretation of Gaussian beam scattering phenomena for a uniaxial anisotropic cylinder.
In , the eigen plane wave spectrum representation of the electric field in uniaxial anisotropic media is given by the Fourier transform, as follows:
The unknown angular spectrum amplitude in Eq. (27) is periodic with respect to , so we use a Fourier expansion for the as
The authors would acknowledge the support by the NSFC (Nos.60931002, 61101064, 61201122), Distinguished Natural Science Foundation (No.1108085J01), Universities Natural Science Foundation of Anhui Province (Nos.KJ2011A002, KJ2011A242, KJ2012A013), DFMEC (No.20123401110009) and NCET (No.NCET-12-0596) of China.
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