Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam

Huayong Zhang; Zhixiang Huang; Yuan Shi

doi:10.1364/OE.21.015645

Optics Express
Vol. 21,
Issue 13,
pp. 15645-15653
(2013)
•https://doi.org/10.1364/OE.21.015645

Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam

Huayong Zhang, Zhixiang Huang, and Yuan Shi

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Huayong Zhang, Zhixiang Huang, and Yuan Shi, "Internal and near-surface electromagnetic fields for a uniaxial anisotropic cylinder illuminated with a Gaussian beam," Opt. Express 21, 15645-15653 (2013)

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About this Article
History
- Original Manuscript: May 13, 2013
- Revised Manuscript: June 12, 2013
- Manuscript Accepted: June 14, 2013
- Published: June 21, 2013
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 8, Iss. 8

Abstract

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.

1. Introduction

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) [1], coupled dipole approximation method [2], integral equation technique [3], and frequency domain finite difference method (FDFD) [4], have been employed in the analysis of the scattering problems. By using the Fourier transform and vector wave eigenfunctions expansions [5], analytical solutions have been developed to the plane wave scattering by an anisotropic circular cylinder [6], uniaxial anisotropic sphere [7], 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 [16], to attack the problem of interaction between an arbitrary shaped beam and an infinite cylinder. In [17], we have obtained the expansion of an incident Gaussian beam (a focused TEM₀₀ mode laser beam) in terms of the CVWFs, and subsequently in [18] 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. Formulation

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 $r_{0}$ is attached to the Cartesian coordinate system $O x y z$ . An incident Gaussian beam propagates in free space and along the positive $z^{'}$ axis in the $x O z$ plane, with the middle of its beam waist located at origin $O^{'}$ on the $z^{'}$ axis. Origin $O$ has a coordinate $z_{0}$ on the axis $O^{'} z^{'}$ (on-axis case), and the angle made by the axis $O^{'} z^{'}$ with the axis $O z$ is $β$ . In this paper, the time-dependent part of the electromagnetic fields is assumed to be $\exp (- i ω t)$ .

Fig. 1 Uniaxial anisotropic cylinder illuminated by an on-axis incident Gaussian beam.

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We have obtained an expansion in [17] 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 $O x y z$ , in the following form

E^{i} = E_{0} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [I_{m, T E} (ζ) m_{m λ}^{(1)} (h) + I_{m, T M} (ζ) n_{m λ}^{(1)} (h)] e^{i h z} d ζ,

H^{i} = - i E_{0} \frac{k_{0}}{ω μ_{0}} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [I_{m, T E} (ζ) n_{m λ}^{(1)} (h) + I_{m, T M} (ζ) m_{m λ}^{(1)} (h)] e^{i h z} d ζ,

where

λ = k_{0} \sin ζ

h = k_{0} \cos ζ

k_{0}

and

μ_{0}

are the free-space wave number and permeability, respectively, and

I_{n, T E}^{m}

I_{n, T M}^{m}

are the beam shape coefficients.

For a TE polarized Gaussian beam, the coefficients $I_{n, T E}^{m}$ , $I_{n, T M}^{m}$ are

\begin{matrix} I_{m, T E} = \frac{{(- i)}^{m + 1}}{k_{0}} \sum_{n = | m |}^{\infty} \frac{(n - m)!}{(n + m)!} \frac{2 n + 1}{2 n (n + 1)} \\ \times g_{n} [m^{2} \frac{P_{n}^{m} (\cos β)}{\sin β} \frac{P_{n}^{m} (\cos ζ)}{\sin ζ} + \frac{d P_{n}^{m} (\cos β)}{d β} \frac{d P_{n}^{m} (\cos ζ)}{d ζ}], \end{matrix}

\begin{matrix} I_{m, T M} = \frac{{(- i)}^{m + 1}}{k_{0}} m \sum_{n = | m |}^{\infty} \frac{(n - m)!}{(n + m)!} \frac{2 n + 1}{2 n (n + 1)} \\ \times g_{n} [\frac{P_{n}^{m} (\cos β)}{\sin β} \frac{d P_{n}^{m} (\cos ζ)}{d ζ} + \frac{d P_{n}^{m} (\cos β)}{d β} \frac{P_{n}^{m} (\cos ζ)}{\sin ζ}], \end{matrix}

where

g_{n}

are the Gaussian beam shape coefficients in spherical coordinates, and for the Davis-Barton model of the Gaussian beam [19], can be described by the localized approximation as [20, 21]

g_{n} = \frac{1}{1 + 2 i s z_{0} / w_{0}} \exp (i k z_{0}) \exp [\frac{- s^{2} {(n + 1 / 2)}^{2}}{1 + 2 i s z_{0} / w_{0}}],

where

s = 1 / (k_{0} w_{0})

, and

w_{0}

is the beam waist radius.

For a TM polarized Gaussian beam, the corresponding expansions can be obtained only by replacing $I_{n, T E}^{m}$ in Eqs. (1) and (2) by $i I_{n, T M}^{m}$ , and $I_{n, T M}^{m}$ by $i I_{n, T E}^{m}$ .

It is worth mentioning that the coefficients $I_{n, T E}^{m}$ , $I_{n, T M}^{m}$ 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 [22].

By following Eqs. (1) and (2), an appropriate expansion of the scattered fields in terms of the CVWFs can be written as

E^{s} = E_{0} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [α_{m} (ζ) m_{m λ}^{(3)} + β_{m} (ζ) n_{m λ}^{(3)}] e^{i h z} d ζ,

H^{s} = - i E_{0} \frac{k_{0}}{ω μ_{0}} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [α_{m} (ζ) n_{m λ}^{(3)} + β_{m} (ζ) m_{m λ}^{(3)}] e^{i h z} d ζ,

where

α_{m} (ζ)

β_{m} (ζ)

are the unknown expansion coefficients to be determined.

Let’s consider that a uniaxial anisotropic medium is characterized by a permittivity tensor $\bar{ε} = \hat{x} \hat{x} ε_{t} + \hat{y} \hat{y} ε_{t} + \hat{z} \hat{z} ε_{z}$ in the system $O x y z$ and a scalar permeability $μ = μ_{0}$ .

From Appendix, the electromagnetic fields within the uniaxial anisotropic cylinder can be expanded in terms of the CVWFs (refer to Appendix), as follows:

E^{w} = E_{0} \sum_{q = 1}^{2} \sum_{m = - \infty}^{\infty} \int_{0}^{π} F_{m q} (ζ) [α_{q}^{e} (ζ) m_{m λ_{q}}^{(1)} + β_{q}^{e} (ζ) n_{m λ_{q}}^{(1)} + γ_{q}^{e} (ζ) l_{m λ_{q}}^{(1)}] e^{i h_{q} z} d ζ,

H^{w} = - i E_{0} \sum_{q = 1}^{2} \sum_{m = - \infty}^{\infty} \int_{0}^{π} \frac{k_{q}}{ω μ_{0}} F_{m q} (ζ) [β_{q}^{e} (ζ) (ζ) m_{m λ_{q}}^{(1)} + α_{q}^{e} (ζ) n_{m λ_{q}}^{(1)}] e^{i h_{q} z} d ζ,

where

h_{1} = h_{2} = h = k_{0} \cos ζ,

a_{1}^{2} = ω^{2} ε_{t} μ_{0}, a_{2}^{2} = ω^{2} ε_{z} μ_{0},

k_{1} = a_{1}, k_{2} = \frac{1}{a_{1}} \sqrt{a_{1}^{2} a_{2}^{2} + (a_{1}^{2} - a_{2}^{2}) k_{0}^{2} \cos^{2} ζ},

λ_{1} = \sqrt{a_{1}^{2} - k_{0}^{2} \cos^{2} ζ}, λ_{2} = k_{2} \sin θ_{k} = \frac{a_{2}}{a_{1}} \sqrt{a_{1}^{2} - k_{0}^{2} \cos^{2} ζ},

α_{1}^{e} (ζ) = 1, β_{1}^{e} (ζ) = γ_{1}^{e} (ζ) = α_{2}^{e} (ζ) = 0,

β_{2}^{e} (ζ) = - i \frac{a_{1}^{2} a_{2}}{\sqrt{(a_{1}^{2} - k_{0}^{2} \cos^{2} ζ) [a_{1}^{2} a_{2}^{2} + (a_{1}^{2} - a_{2}^{2}) k_{0}^{2} \cos^{2} ζ]}},

γ_{2}^{e} (ζ) = - \frac{a_{1}^{2} - a_{2}^{2}}{a_{1}^{2}} \frac{a_{1} a_{2} k_{0} \cos ζ \sqrt{a_{1}^{2} - k_{0}^{2} \cos^{2} ζ}}{a_{1}^{2} a_{2}^{2} + (a_{1}^{2} - a_{2}^{2}) k_{0}^{2} \cos^{2} ζ},

2.2 On-axis Gaussian beam scattering by a uniaxial anisotropic cylinder

The unknown expansion coefficients $α_{m} (ζ)$ , $β_{m} (ζ)$ in Eqs. (6) and (7) as well as $F_{m 1} (ζ)$ , $F_{m 2} (ζ)$ in Eqs. (8) and (9) can be determined by using the following boundary conditions

\begin{matrix} \begin{matrix} E_{ϕ}^{i} + E_{ϕ}^{s} = E_{ϕ}^{w}, & E_{z}^{i} + E_{z}^{s} = E_{z}^{w} \end{matrix} \\ \begin{matrix} H_{ϕ}^{i} + H_{ϕ}^{s} = H_{ϕ}^{w}, & H_{z}^{i} + H_{z}^{s} = H_{z}^{w} \end{matrix} \end{matrix}} \begin{matrix} \begin{matrix} a t & r = r_{0} \end{matrix} \end{matrix},

By virtue of the field expansions, the above boundary conditions can be written as

\begin{array}{l} ξ \frac{d}{d ξ} J_{m} (ξ) I_{m, T E} + \frac{h m}{k_{0}} J_{m} (ξ) I_{m, T M} + ξ \frac{d}{d ξ} H_{m}^{(1)} (ξ) α_{m} (ζ) + \frac{h m}{k_{0}} H_{m}^{(1)} (ξ) β_{m} (ζ) \\ = F_{m 1} (ζ) ξ_{1} \frac{d}{d ξ_{1}} J_{m} (ξ_{1}) + F_{m 2} (ζ) β_{2}^{e} (ζ) \frac{h m}{k_{2}} J_{m} (ξ_{2}) - F_{m 2} (ζ) γ_{2}^{e} (ζ) i m J_{m} (ξ_{2}), \end{array}

ξ^{2} [J_{m} (ξ) I_{m, T M} + H_{m}^{(1)} (ξ) β_{m} (ζ)] = F_{m 2} (ζ) \frac{k_{0}}{k_{2}} ξ_{2}^{2} J_{m} (ξ_{2}) [β_{2}^{e} (ζ) + γ_{2}^{e} (ζ) \frac{i h k_{2}}{λ_{2}^{2}}],

\begin{array}{l} \frac{h m}{k_{0}} J_{m} (ξ) I_{m, T E} + ξ \frac{d}{d ξ} J_{m} (ξ) I_{m, T M} + \frac{h m}{k_{0}} H_{m}^{(1)} (ξ) α_{m} (ζ) + ξ \frac{d}{d ξ} H_{m}^{(1)} (ξ) β_{m} (ζ) \\ = \frac{h m}{k_{0}} F_{m 1} (ζ) J_{m} (ξ_{1}) + \frac{k_{2}}{k_{0}} F_{m 2} (ζ) β_{2}^{e} (ζ) ξ_{2} \frac{d}{d ξ_{2}} J_{m} (ξ_{2}), \end{array}

ξ^{2} [J_{m} (ξ) I_{m, T E} + H_{m}^{(1)} (ξ) α_{m} (ζ)] = F_{m 1} (ζ) ξ_{1}^{2} J_{m} (ξ_{1}),

where

ξ = λ r_{0}

ξ_{1} = λ_{1} r_{0}

, and

ξ_{2} = λ_{2} r_{0}

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

{| E^{w} / E_{0} |}^{2} = {({| E_{r}^{w} |}^{2} + {| E_{ϕ}^{w} |}^{2} + {| E_{z}^{w} |}^{2}) / | E_{0} |}^{2},

and

{| (E^{i} + E^{s}) / E_{0} |}^{2} = {({| E_{r}^{i} + E_{r}^{s} |}^{2} + {| E_{ϕ}^{i} + E_{ϕ}^{s} |}^{2} + {| E_{z}^{i} + E_{z}^{s} |}^{2}) / | E_{0} |}^{2},

The subscripts $r$ , $ϕ$ and $z$ in Eqs. (22) and (23) denote the $r$ , $ϕ$ and $z$ components of the electric fields. By taking Eq. (22) as an example, the $r$ , $ϕ$ and $z$ components of the internal electric field $E^{w}$ are expressed as

\begin{matrix} E_{r}^{w} = E_{0} \sum_{m = - \infty}^{\infty} e^{i m ϕ} \int_{0}^{π} {F_{m 1} (ζ) i \frac{m}{r} J_{m} (λ_{1} r) \\ + F_{m 2} (ζ) \frac{\partial}{\partial r} J_{m} (λ_{2} r) [β_{2}^{e} (ζ) i \frac{h}{k_{2}} + γ_{2}^{e} (ζ)]} e^{i h z} d ζ \end{matrix}

\begin{matrix} E_{φ}^{w} = E_{0} \sum_{m = - \infty}^{\infty} e^{i m ϕ} \int_{0}^{π} {- F_{m 1} (ζ) \frac{\partial}{\partial r} J_{m} (λ_{1} r) \\ + F_{m 2} (ζ) \frac{m}{r} J_{m} (λ_{2} r) [- β_{2}^{e} (ζ) \frac{h}{k_{2}} + i γ_{2}^{e} (ζ)]} e^{i h z} d ζ \end{matrix}

E_{z}^{w} = E_{0} \sum_{m = - \infty}^{\infty} e^{i m ϕ} \int_{0}^{π} F_{m 2} (ζ) J_{m} (λ_{2} r) [β_{2}^{e} (ζ) \frac{λ_{2}^{2}}{k_{2}} + i h γ_{2}^{e} (ζ)] e^{i h z} d ζ

In a similar way, from Eqs. (1) and (6) the $r$ , $ϕ$ and $z$ components of the electric fields $E^{i}$ and $E^{s}$ can also be obtained.

Figures (2) and (3), sharing the same colorbar, show the normalized internal and near-surface field intensity distributions ${| E^{w} / E_{0} |}^{2}$ and ${| (E^{i} + E^{s}) / E_{0} |}^{2}$ in the $x O z$ plane for a uniaxial anisotropic cylinder, illuminated by a TM and TE polarized Gaussian beam, respectively. The radius of the cylinder $r_{0}$ and the beam waist radius $w_{0}$ are assumed to be five and two times the wavelength of the incident Gaussian beam, and $a_{1} = \sqrt{3} k_{0}$ , $a_{2} = \sqrt{2} k_{0}$ , $β = π / 4$ , $z_{0} = 0$ .

Fig. 2 ${| E^{w} / E_{0} |}^{2}$ and ${| (E^{i} + E^{s}) / E_{0} |}^{2}$ for a uniaxial anisotropic cylinder illuminated by a TM polarized Gaussian beam.

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Fig. 3 Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.

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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.

4. Conclusion

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.

Appendix

In [7], the eigen plane wave spectrum representation of the electric field in uniaxial anisotropic media is given by the Fourier transform, as follows:

E^{w} = E_{0} \sum_{q = 1}^{2} \int k_{q}^{2} \sin θ_{k} d θ_{k} \int_{0}^{2 π} F_{q}^{e} (θ_{k}, ϕ_{k}) f_{q} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r} d ϕ_{k},

where the wave numbers

k_{q}

and electric field eigenvectors

F_{q}^{e} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r}

(

q = 1, 2

) are

k_{1} = a_{1}, k_{2} = a_{1} a_{2} \sqrt{\frac{1}{a_{1}^{2} \sin^{2} θ_{k} + a_{2}^{2} \cos^{2} θ_{k}}},

F_{q}^{e} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r} = (F_{q x}^{e} \hat{x} + F_{q y}^{e} \hat{y} + F_{q z}^{e} \hat{z}) e^{i k_{q} \cdot r},

F_{q x}^{e} = {\begin{matrix} - \sin ϕ_{k} \begin{matrix} , & q = 1 \end{matrix} \\ - \frac{a_{2}^{2}}{a_{1}^{2}} \frac{\cos θ_{k}}{\sin θ_{k}} \cos ϕ_{k} \begin{matrix} , & q = 2 \end{matrix} \end{matrix}

F_{q y}^{e} = {\begin{matrix} \cos ϕ_{k} \begin{matrix} , & q = 1 \end{matrix} \\ - \frac{a_{2}^{2}}{a_{1}^{2}} \frac{\cos θ_{k}}{\sin θ_{k}} \sin ϕ_{k} \begin{matrix} , & q = 2 \end{matrix} \end{matrix}

F_{q z}^{e} = {\begin{matrix} 0 \begin{matrix} , & q = 1 \end{matrix} \\ 1 \begin{matrix} , & q = 2 \end{matrix} \end{matrix}

The unknown angular spectrum amplitude $f_{q} (θ_{k}, ϕ_{k})$ in Eq. (27) is periodic with respect to $ϕ_{k}$ , so we use a Fourier expansion for the $f_{q} (θ_{k}, ϕ_{k})$ as

f_{q} (θ_{k}, ϕ_{k}) = \sum_{n = - \infty}^{\infty} {G^{'}}_{n q} (θ_{k}) e^{i n ϕ_{k}},

A substitution of Eq. (33) into Eq. (27) leads to

E^{w} = E_{0} \sum_{q = 1}^{2} \sum_{n = - \infty}^{\infty} \int {G^{'}}_{n q} (θ_{k}) k_{q}^{2} \sin θ_{k} d θ_{k} \int_{0}^{2 π} F_{q}^{e} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r} e^{i n ϕ_{k}} d ϕ_{k},

An arbitrary plane electromagnetic wave of unit amplitude can be represented in a convergent series of the CVWFs, as follows [5, 6]:

\hat{x} e^{i k \cdot r} = \sum_{m = - \infty}^{\infty} (a_{m}^{x} m_{m λ}^{(1)} + b_{m}^{x} n_{m λ}^{(1)} + c_{m}^{x} l_{m λ}^{(1)}) e^{i h z},

\hat{y} e^{i k \cdot r} = \sum_{m = - \infty}^{\infty} (a_{m}^{y} m_{m λ}^{(1)} + b_{m}^{y} n_{m λ}^{(1)} + c_{m}^{y} l_{m λ}^{(1)}) e^{i h z},

\hat{z} e^{i k \cdot r} = \sum_{m = - \infty}^{\infty} (a_{m}^{z} m_{m λ}^{(1)} + b_{m}^{z} n_{m λ}^{(1)} + c_{m}^{z} l_{m λ}^{(1)}) e^{i h z},

where the expansion coefficients are determined by

[\begin{matrix} a_{m}^{x} & b_{m}^{x} & c_{m}^{x} \end{matrix}] = \frac{i^{m - 1} e^{- i m ϕ_{k}}}{k} [\begin{matrix} \frac{1}{\sin θ_{k}} \sin ϕ_{k} & i \frac{\cos θ_{k}}{\sin θ_{k}} \cos ϕ_{k} & \sin θ_{k} \cos ϕ_{k} \end{matrix}],

[\begin{matrix} a_{m}^{y} & b_{m}^{y} & c_{m}^{y} \end{matrix}] = \frac{i^{m - 1} e^{- i m ϕ_{k}}}{k} [\begin{matrix} - \frac{1}{\sin θ_{k}} \cos ϕ_{k} & - i \frac{\cos θ_{k}}{\sin θ_{k}} \sin ϕ_{k} & \sin θ_{k} \sin ϕ_{k} \end{matrix}],

[\begin{matrix} a_{m}^{z} & b_{m}^{z} & c_{m}^{z} \end{matrix}] = \frac{i^{m - 1} e^{- i m ϕ_{k}}}{k} [\begin{matrix} 0 & i & \cos θ_{k} \end{matrix}] .

Substituting Eqs. (35)-(37) into Eq. (34) and considering that $\int_{0}^{2 π} e^{i (n - m) ϕ_{k}} d ϕ_{k}$ equals $2 π$ when $m = n$ and $0$ when $m \neq n$ , we end up with

E^{w} = E_{0} \sum_{q = 1}^{2} \sum_{m = - \infty}^{\infty} \int G_{m q} (θ_{k}) [A_{q}^{e} (θ_{k}) m_{m λ_{q}}^{(1)} + B_{q}^{e} (θ_{k}) n_{m λ_{q}}^{(1)} + C_{q}^{e} (θ_{k}) l_{m λ_{q}}^{(1)}] e^{i h_{q} z} d θ_{k},

where

G_{m q} (θ_{k}) = 2 π i^{m + 1} {G^{'}}_{m q} (θ_{k}) k_{q}

λ_{q} = k_{q} \sin θ_{k}

h_{q} = k_{q} \cos θ_{k}

, and

A_{1}^{e} (θ_{k}) = 1, B_{1}^{e} (θ_{k}) = C_{1}^{e} (θ_{k}) = A_{2}^{e} (θ_{k}) = 0,

B_{2}^{e} (θ_{k}) = - i \frac{a_{1}^{2} \sin^{2} θ_{k} + a_{2}^{2} \cos^{2} θ_{k}}{a_{1}^{2} \sin θ_{k}},

C_{2}^{e} (θ_{k}) = - \frac{a_{1}^{2} - a_{2}^{2}}{a_{1}^{2}} \sin θ_{k} \cos θ_{k},

Due to the phase continuity over the cylinder surface when using the boundary conditions, we have $h_{q} = k_{q} \cos θ_{k} = h = k_{0} \cos ζ$ ( $q = 1, 2$ ). Then, Eq. (41) is transformed into Eq. (8), where $A_{1}^{e} (θ_{k}) = α_{1}^{e} (ζ)$ , $B_{1}^{e} (θ_{k}) = β_{1}^{e} (ζ)$ , $C_{1}^{e} (θ_{k}) = γ_{1}^{e} (ζ)$ , $A_{2}^{e} (θ_{k}) = α_{2}^{e} (ζ)$ , $B_{2}^{e} (θ_{k}) = β_{2}^{e} (ζ)$ , $C_{2}^{e} (θ_{k}) = γ_{2}^{e} (ζ)$ , and $G_{m q} (θ_{k}) d θ_{k} = F_{m q} (ζ) d ζ$ .

Acknowledgments

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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Fig. 1 Uniaxial anisotropic cylinder illuminated by an on-axis incident Gaussian beam.

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Fig. 2

{| E^{w} / E_{0} |}^{2}

and

{| (E^{i} + E^{s}) / E_{0} |}^{2}

for a uniaxial anisotropic cylinder illuminated by a TM polarized Gaussian beam.

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Fig. 3 Same model as in Fig. 2 but illuminated by a TE polarized Gaussian beam.

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Equations (44)

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E^{i} = E_{0} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [I_{m, T E} (ζ) m_{m λ}^{(1)} (h) + I_{m, T M} (ζ) n_{m λ}^{(1)} (h)] e^{i h z} d ζ,

H^{i} = - i E_{0} \frac{k_{0}}{ω μ_{0}} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [I_{m, T E} (ζ) n_{m λ}^{(1)} (h) + I_{m, T M} (ζ) m_{m λ}^{(1)} (h)] e^{i h z} d ζ,

\begin{matrix} I_{m, T E} = \frac{{(- i)}^{m + 1}}{k_{0}} \sum_{n = | m |}^{\infty} \frac{(n - m)!}{(n + m)!} \frac{2 n + 1}{2 n (n + 1)} \\ \times g_{n} [m^{2} \frac{P_{n}^{m} (\cos β)}{\sin β} \frac{P_{n}^{m} (\cos ζ)}{\sin ζ} + \frac{d P_{n}^{m} (\cos β)}{d β} \frac{d P_{n}^{m} (\cos ζ)}{d ζ}], \end{matrix}

\begin{matrix} I_{m, T M} = \frac{{(- i)}^{m + 1}}{k_{0}} m \sum_{n = | m |}^{\infty} \frac{(n - m)!}{(n + m)!} \frac{2 n + 1}{2 n (n + 1)} \\ \times g_{n} [\frac{P_{n}^{m} (\cos β)}{\sin β} \frac{d P_{n}^{m} (\cos ζ)}{d ζ} + \frac{d P_{n}^{m} (\cos β)}{d β} \frac{P_{n}^{m} (\cos ζ)}{\sin ζ}], \end{matrix}

g_{n} = \frac{1}{1 + 2 i s z_{0} / w_{0}} \exp (i k z_{0}) \exp [\frac{- s^{2} {(n + 1 / 2)}^{2}}{1 + 2 i s z_{0} / w_{0}}],

E^{s} = E_{0} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [α_{m} (ζ) m_{m λ}^{(3)} + β_{m} (ζ) n_{m λ}^{(3)}] e^{i h z} d ζ,

H^{s} = - i E_{0} \frac{k_{0}}{ω μ_{0}} \sum_{m = - \infty}^{\infty} \int_{0}^{π} [α_{m} (ζ) n_{m λ}^{(3)} + β_{m} (ζ) m_{m λ}^{(3)}] e^{i h z} d ζ,

E^{w} = E_{0} \sum_{q = 1}^{2} \sum_{m = - \infty}^{\infty} \int_{0}^{π} F_{m q} (ζ) [α_{q}^{e} (ζ) m_{m λ_{q}}^{(1)} + β_{q}^{e} (ζ) n_{m λ_{q}}^{(1)} + γ_{q}^{e} (ζ) l_{m λ_{q}}^{(1)}] e^{i h_{q} z} d ζ,

H^{w} = - i E_{0} \sum_{q = 1}^{2} \sum_{m = - \infty}^{\infty} \int_{0}^{π} \frac{k_{q}}{ω μ_{0}} F_{m q} (ζ) [β_{q}^{e} (ζ) (ζ) m_{m λ_{q}}^{(1)} + α_{q}^{e} (ζ) n_{m λ_{q}}^{(1)}] e^{i h_{q} z} d ζ,

h_{1} = h_{2} = h = k_{0} \cos ζ,

a_{1}^{2} = ω^{2} ε_{t} μ_{0}, a_{2}^{2} = ω^{2} ε_{z} μ_{0},

k_{1} = a_{1}, k_{2} = \frac{1}{a_{1}} \sqrt{a_{1}^{2} a_{2}^{2} + (a_{1}^{2} - a_{2}^{2}) k_{0}^{2} \cos^{2} ζ},

λ_{1} = \sqrt{a_{1}^{2} - k_{0}^{2} \cos^{2} ζ}, λ_{2} = k_{2} \sin θ_{k} = \frac{a_{2}}{a_{1}} \sqrt{a_{1}^{2} - k_{0}^{2} \cos^{2} ζ},

α_{1}^{e} (ζ) = 1, β_{1}^{e} (ζ) = γ_{1}^{e} (ζ) = α_{2}^{e} (ζ) = 0,

β_{2}^{e} (ζ) = - i \frac{a_{1}^{2} a_{2}}{\sqrt{(a_{1}^{2} - k_{0}^{2} \cos^{2} ζ) [a_{1}^{2} a_{2}^{2} + (a_{1}^{2} - a_{2}^{2}) k_{0}^{2} \cos^{2} ζ]}},

γ_{2}^{e} (ζ) = - \frac{a_{1}^{2} - a_{2}^{2}}{a_{1}^{2}} \frac{a_{1} a_{2} k_{0} \cos ζ \sqrt{a_{1}^{2} - k_{0}^{2} \cos^{2} ζ}}{a_{1}^{2} a_{2}^{2} + (a_{1}^{2} - a_{2}^{2}) k_{0}^{2} \cos^{2} ζ},

\begin{matrix} \begin{matrix} E_{ϕ}^{i} + E_{ϕ}^{s} = E_{ϕ}^{w}, & E_{z}^{i} + E_{z}^{s} = E_{z}^{w} \end{matrix} \\ \begin{matrix} H_{ϕ}^{i} + H_{ϕ}^{s} = H_{ϕ}^{w}, & H_{z}^{i} + H_{z}^{s} = H_{z}^{w} \end{matrix} \end{matrix}} \begin{matrix} \begin{matrix} a t & r = r_{0} \end{matrix} \end{matrix},

\begin{array}{l} ξ \frac{d}{d ξ} J_{m} (ξ) I_{m, T E} + \frac{h m}{k_{0}} J_{m} (ξ) I_{m, T M} + ξ \frac{d}{d ξ} H_{m}^{(1)} (ξ) α_{m} (ζ) + \frac{h m}{k_{0}} H_{m}^{(1)} (ξ) β_{m} (ζ) \\ = F_{m 1} (ζ) ξ_{1} \frac{d}{d ξ_{1}} J_{m} (ξ_{1}) + F_{m 2} (ζ) β_{2}^{e} (ζ) \frac{h m}{k_{2}} J_{m} (ξ_{2}) - F_{m 2} (ζ) γ_{2}^{e} (ζ) i m J_{m} (ξ_{2}), \end{array}

ξ^{2} [J_{m} (ξ) I_{m, T M} + H_{m}^{(1)} (ξ) β_{m} (ζ)] = F_{m 2} (ζ) \frac{k_{0}}{k_{2}} ξ_{2}^{2} J_{m} (ξ_{2}) [β_{2}^{e} (ζ) + γ_{2}^{e} (ζ) \frac{i h k_{2}}{λ_{2}^{2}}],

\begin{array}{l} \frac{h m}{k_{0}} J_{m} (ξ) I_{m, T E} + ξ \frac{d}{d ξ} J_{m} (ξ) I_{m, T M} + \frac{h m}{k_{0}} H_{m}^{(1)} (ξ) α_{m} (ζ) + ξ \frac{d}{d ξ} H_{m}^{(1)} (ξ) β_{m} (ζ) \\ = \frac{h m}{k_{0}} F_{m 1} (ζ) J_{m} (ξ_{1}) + \frac{k_{2}}{k_{0}} F_{m 2} (ζ) β_{2}^{e} (ζ) ξ_{2} \frac{d}{d ξ_{2}} J_{m} (ξ_{2}), \end{array}

ξ^{2} [J_{m} (ξ) I_{m, T E} + H_{m}^{(1)} (ξ) α_{m} (ζ)] = F_{m 1} (ζ) ξ_{1}^{2} J_{m} (ξ_{1}),

{| E^{w} / E_{0} |}^{2} = {({| E_{r}^{w} |}^{2} + {| E_{ϕ}^{w} |}^{2} + {| E_{z}^{w} |}^{2}) / | E_{0} |}^{2},

{| (E^{i} + E^{s}) / E_{0} |}^{2} = {({| E_{r}^{i} + E_{r}^{s} |}^{2} + {| E_{ϕ}^{i} + E_{ϕ}^{s} |}^{2} + {| E_{z}^{i} + E_{z}^{s} |}^{2}) / | E_{0} |}^{2},

\begin{matrix} E_{r}^{w} = E_{0} \sum_{m = - \infty}^{\infty} e^{i m ϕ} \int_{0}^{π} {F_{m 1} (ζ) i \frac{m}{r} J_{m} (λ_{1} r) \\ + F_{m 2} (ζ) \frac{\partial}{\partial r} J_{m} (λ_{2} r) [β_{2}^{e} (ζ) i \frac{h}{k_{2}} + γ_{2}^{e} (ζ)]} e^{i h z} d ζ \end{matrix}

\begin{matrix} E_{φ}^{w} = E_{0} \sum_{m = - \infty}^{\infty} e^{i m ϕ} \int_{0}^{π} {- F_{m 1} (ζ) \frac{\partial}{\partial r} J_{m} (λ_{1} r) \\ + F_{m 2} (ζ) \frac{m}{r} J_{m} (λ_{2} r) [- β_{2}^{e} (ζ) \frac{h}{k_{2}} + i γ_{2}^{e} (ζ)]} e^{i h z} d ζ \end{matrix}

E_{z}^{w} = E_{0} \sum_{m = - \infty}^{\infty} e^{i m ϕ} \int_{0}^{π} F_{m 2} (ζ) J_{m} (λ_{2} r) [β_{2}^{e} (ζ) \frac{λ_{2}^{2}}{k_{2}} + i h γ_{2}^{e} (ζ)] e^{i h z} d ζ

E^{w} = E_{0} \sum_{q = 1}^{2} \int k_{q}^{2} \sin θ_{k} d θ_{k} \int_{0}^{2 π} F_{q}^{e} (θ_{k}, ϕ_{k}) f_{q} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r} d ϕ_{k},

k_{1} = a_{1}, k_{2} = a_{1} a_{2} \sqrt{\frac{1}{a_{1}^{2} \sin^{2} θ_{k} + a_{2}^{2} \cos^{2} θ_{k}}},

F_{q}^{e} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r} = (F_{q x}^{e} \hat{x} + F_{q y}^{e} \hat{y} + F_{q z}^{e} \hat{z}) e^{i k_{q} \cdot r},

F_{q x}^{e} = {\begin{matrix} - \sin ϕ_{k} \begin{matrix} , & q = 1 \end{matrix} \\ - \frac{a_{2}^{2}}{a_{1}^{2}} \frac{\cos θ_{k}}{\sin θ_{k}} \cos ϕ_{k} \begin{matrix} , & q = 2 \end{matrix} \end{matrix}

F_{q y}^{e} = {\begin{matrix} \cos ϕ_{k} \begin{matrix} , & q = 1 \end{matrix} \\ - \frac{a_{2}^{2}}{a_{1}^{2}} \frac{\cos θ_{k}}{\sin θ_{k}} \sin ϕ_{k} \begin{matrix} , & q = 2 \end{matrix} \end{matrix}

F_{q z}^{e} = {\begin{matrix} 0 \begin{matrix} , & q = 1 \end{matrix} \\ 1 \begin{matrix} , & q = 2 \end{matrix} \end{matrix}

f_{q} (θ_{k}, ϕ_{k}) = \sum_{n = - \infty}^{\infty} {G^{'}}_{n q} (θ_{k}) e^{i n ϕ_{k}},

E^{w} = E_{0} \sum_{q = 1}^{2} \sum_{n = - \infty}^{\infty} \int {G^{'}}_{n q} (θ_{k}) k_{q}^{2} \sin θ_{k} d θ_{k} \int_{0}^{2 π} F_{q}^{e} (θ_{k}, ϕ_{k}) e^{i k_{q} \cdot r} e^{i n ϕ_{k}} d ϕ_{k},

\hat{x} e^{i k \cdot r} = \sum_{m = - \infty}^{\infty} (a_{m}^{x} m_{m λ}^{(1)} + b_{m}^{x} n_{m λ}^{(1)} + c_{m}^{x} l_{m λ}^{(1)}) e^{i h z},

\hat{y} e^{i k \cdot r} = \sum_{m = - \infty}^{\infty} (a_{m}^{y} m_{m λ}^{(1)} + b_{m}^{y} n_{m λ}^{(1)} + c_{m}^{y} l_{m λ}^{(1)}) e^{i h z},

\hat{z} e^{i k \cdot r} = \sum_{m = - \infty}^{\infty} (a_{m}^{z} m_{m λ}^{(1)} + b_{m}^{z} n_{m λ}^{(1)} + c_{m}^{z} l_{m λ}^{(1)}) e^{i h z},

[\begin{matrix} a_{m}^{x} & b_{m}^{x} & c_{m}^{x} \end{matrix}] = \frac{i^{m - 1} e^{- i m ϕ_{k}}}{k} [\begin{matrix} \frac{1}{\sin θ_{k}} \sin ϕ_{k} & i \frac{\cos θ_{k}}{\sin θ_{k}} \cos ϕ_{k} & \sin θ_{k} \cos ϕ_{k} \end{matrix}],

[\begin{matrix} a_{m}^{y} & b_{m}^{y} & c_{m}^{y} \end{matrix}] = \frac{i^{m - 1} e^{- i m ϕ_{k}}}{k} [\begin{matrix} - \frac{1}{\sin θ_{k}} \cos ϕ_{k} & - i \frac{\cos θ_{k}}{\sin θ_{k}} \sin ϕ_{k} & \sin θ_{k} \sin ϕ_{k} \end{matrix}],

[\begin{matrix} a_{m}^{z} & b_{m}^{z} & c_{m}^{z} \end{matrix}] = \frac{i^{m - 1} e^{- i m ϕ_{k}}}{k} [\begin{matrix} 0 & i & \cos θ_{k} \end{matrix}] .

E^{w} = E_{0} \sum_{q = 1}^{2} \sum_{m = - \infty}^{\infty} \int G_{m q} (θ_{k}) [A_{q}^{e} (θ_{k}) m_{m λ_{q}}^{(1)} + B_{q}^{e} (θ_{k}) n_{m λ_{q}}^{(1)} + C_{q}^{e} (θ_{k}) l_{m λ_{q}}^{(1)}] e^{i h_{q} z} d θ_{k},

A_{1}^{e} (θ_{k}) = 1, B_{1}^{e} (θ_{k}) = C_{1}^{e} (θ_{k}) = A_{2}^{e} (θ_{k}) = 0,

B_{2}^{e} (θ_{k}) = - i \frac{a_{1}^{2} \sin^{2} θ_{k} + a_{2}^{2} \cos^{2} θ_{k}}{a_{1}^{2} \sin θ_{k}},

C_{2}^{e} (θ_{k}) = - \frac{a_{1}^{2} - a_{2}^{2}}{a_{1}^{2}} \sin θ_{k} \cos θ_{k},

Abstract

1. Introduction

2. Formulation

2.1 Expansions of Gaussian beam, scattered fields and internal fields in cylindrical coordinates

2.2 On-axis Gaussian beam scattering by a uniaxial anisotropic cylinder

3. Numerical results

4. Conclusion

Appendix

Acknowledgments

References and links

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Figures (3)

Equations (44)

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