Scattering of an electromagnetic plane wave by a homogeneous sphere made of an orthorhombic dielectric&#x2013;magnetic material

A. D. Ulfat Jafri; Akhlesh Lakhtakia

doi:10.1364/JOSAA.31.000089

1. INTRODUCTION

The homogeneous sphere is ubiquitous in electromagnetic-scattering literature, whether the sphere is electrically small [1–3] or electrically huge [4,5] or anywhere in between in size [6,7]. Exact analytical solutions are known for scattering by isotropic dielectric [8–10], isotropic dielectric–magnetic [11,12], and bi-isotropic spheres [13], because vector spherical wavefunctions are available in closed form for these materials. However, similarly analytical solutions for scattering by anisotropic spheres remain unknown, because the relevant vector spherical wavefunctions in closed form have not been found.

Recently, however, closed-form vector spherical wavefunctions were found for orthorhombic dielectric–magnetic material with gyrotropic-like magnetoelectric properties [14], thereby extending our options in solving scattering problems analytically. For the simplest application of the newly found wavefunctions, we decided to calculate scattering by a homogeneous sphere of an orthorhombic dielectric–magnetic material described by the frequency-domain constitutive relations

\begin{matrix} D (r, ω) = ϵ_{0} ϵ_{r} (ω) A̳ (ω) \cdot A̳ (ω) \cdot E (r, ω) \\ B (r, ω) = μ_{0} μ_{r} (ω) A̳ (ω) \cdot A̳ (ω) \cdot H (r, ω) \end{matrix}},

where

ϵ_{0}

and

μ_{0}

are the permittivity and the permeability of free space, respectively; the scalars

ϵ_{r} (ω)

and

μ_{r} (ω)

are complex-valued scalar functions of the angular frequency

ω

; the diagonal dyadic,

A̳ (ω) = α_{x}^{- 1} (ω) \hat{x} \hat{x} + α_{y}^{- 1} (ω) \hat{y} \hat{y} + \hat{z} \hat{z},

contains the real-valued scalars

α_{x} (ω) > 0

and

α_{y} (ω) > 0

that are also frequency dependent; and the Cartesian unit vectors are written as

\hat{x}

,

\hat{y}

, and

\hat{z}

.

Belonging to the magnetic point group $D_{2 h}$ or $D_{2 h}$ [15, p. 68], the chosen material is anisotropic but pathologically unirefringent because its permeability dyadic is a scalar multiple of its permittivity dyadic [16]. Suppose that a plane wave propagates in an arbitrary direction in this material with wavenumber $k_{c}$ . Then, $k_{c}$ is bounded by either (i) $k$ and $k α_{x}$ or (ii) $k$ and $k α_{y}$ , where $k = ω \sqrt{ϵ_{0} μ_{0}} \sqrt{ϵ_{r}} \sqrt{μ_{r}} {(α_{x} α_{y})}^{- 1}$ . In naturally occurring materials, both $| α_{x}^{2} - 1 |$ and $| α_{y}^{2} - 1 |$ are expected be very small ( $≲ 0.1$ ), based on available data for solid dielectric crystals [17]. However, both of these quantities could be larger for manufactured composite materials. An example is a composite material made by dispersing a mixture of electrically small springs that are aligned along any one of the three Cartesian axes, provided that every spring has a coaligned enantiomorph present in close proximity. Although simple techniques for making composite materials containing aligned springs [18] have been around for at least a century, nano/microscale additive manufacturing techniques [19] should play a major role in the very near future toward engineering complex materials. Furthermore, suspensions of ferromagnetic grains in liquid crystals [20] are also candidate materials.

Although the closed-form vector spherical wavefunctions applicable for the chosen material form a complete set [14,21], these functions are orthogonal not on a unit sphere described solely by the angular variables $θ$ and $ϕ$ in $r$ space, but on a unit sphere in a transformed space. Therefore, scattering by this sphere cannot be described in terms of analogs of the “Mie coefficients” found in modern treatments of scattering by homogeneous isotropic spheres [11,12]. Instead, a transition matrix (commonly called the “T matrix”) is needed to describe scattering by a homogeneous sphere made of the chosen material. The derivation of the T matrix for general nonspherical scatterers being available [14], we provide essential expressions and final results in Section 2. Section 3 presents numerical results to explicate the effects of anisotropy on the scattering of an incident plane wave. Special attention is paid to the scattering cross section, extinction cross section, and differential cross section in relation to (i) the size parameter and the anisotropy of the sphere as well as (ii) the polarization state and the direction of propagation of the incident plane wave. The time dependency $\exp (- i ω t)$ is present but suppressed, $k_{0} = ω \sqrt{ϵ_{0} μ_{0}}$ is the free-space wavenumber, and $η_{0} = \sqrt{μ_{0} / ϵ_{0}}$ is the intrinsic impedance of free space. The asterisk denotes the complex conjugate.

2. THEORY

Let a homogeneous sphere of radius $a$ be located with its center at the origin of the spherical coordinate system $(r, θ, ϕ)$ . The sphere is made of a material that obeys the constitutive relations stated via Eqs. (1) and (2). The region $r > a$ is vacuous.

A. Incident Electromagnetic Field

With the assumption that the source of the incident field lies in the region $r > b > a$ , the incident electric field can be expressed as

E_{inc} (r) = \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} {D_{m n} [A_{s m n}^{(1)} M_{s m n}^{(1)} (k_{0} r) + B_{s m n}^{(1)} N_{s m n}^{(1)} (k_{0} r)]}, r < b,

and the incident magnetic field as

H_{inc} (r) = \frac{k_{0}}{i ω μ_{0}} \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} {D_{m n} [A_{s m n}^{(1)} N_{s m n}^{(1)} (k_{0} r) + B_{s m n}^{(1)} M_{s m n}^{(1)} (k_{0} r)]}, r < b .

The vector spherical wavefunctions for free space used in these two equations are defined as [11,12]

\begin{array}{l} M_{e m n}^{(1)} (k_{0} r) = \nabla \times [r j_{n} (k_{0} r) P_{n}^{m} (\cos θ) \cos (m ϕ)] \\ M_{o m n}^{(1)} (k_{0} r) = \nabla \times [r j_{n} (k_{0} r) P_{n}^{m} (\cos θ) \sin (m ϕ)] \\ N_{s m n}^{(1)} (k_{0} r) = k_{0}^{- 1} \nabla \times M_{s m n}^{(1)} (k_{0} r), s \in {e, o} \end{array}}, m \in {0, 1, 2, \dots, n}, n \in {1, 2, 3, \dots}

with

j_{n} (\cdot)

denoting the spherical Bessel function of order

n

, and

P_{n}^{m} (\cdot)

the associated Legendre function of order

n

and degree

m

. The normalization factor

D_{m n} = (2 - δ_{m 0}) \frac{(2 n + 1) (n - m)!}{4 n (n + 1) (n + m)!}

employs the Kronecker delta

δ_{p q}

.

The coefficients $A_{s m n}^{(1)}$ and $B_{s m n}^{(1)}$ are presumed to be known. Suppose that the incident electromagnetic field is that of a plane wave with a propagation vector

k_{inc} = k_{0} (\hat{x} \sin θ_{inc} \cos ϕ_{inc} + \hat{y} \sin θ_{inc} \sin ϕ_{inc} + \hat{z} \cos θ_{inc})

with

θ_{inc} \in [0, π]

and

ϕ_{inc} \in [0, 2 π)

. If the incident electric field is written as

E_{inc} (r) = e_{inc} \exp (i k_{inc} \cdot r)

with

e_{inc} \cdot k_{inc} = 0

, then the incident magnetic field is

H_{inc} (r) = \frac{k_{inc} \times e_{inc}}{ω μ_{0}} \exp (i k_{inc} \cdot r) .

The identity [22, p. 1866]

e_{inc} \exp (i k_{inc} \cdot r) = 4 \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} {i^{n} D_{n m} \sqrt{n (n + 1)} [C_{s m n} (θ_{inc}, ϕ_{inc}) M_{s m n}^{(1)} (k_{0} r) - i B_{s m n} (θ_{inc}, ϕ_{inc}) N_{s m n}^{(1)} (k_{0} r)]},

where [22, p. 1899]

\begin{array}{l} B_{e m n} (θ, ϕ) = \frac{1}{\sqrt{n (n + 1)}} [\frac{d P_{n}^{m} (\cos θ)}{d θ} \cos (m ϕ) \hat{θ} - m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \sin (m ϕ) \hat{ϕ}] \\ B_{o m n} (θ, ϕ) = \frac{1}{\sqrt{n (n + 1)}} [\frac{d P_{n}^{m} (\cos θ)}{d θ} \sin (m ϕ) \hat{θ} + m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \cos (m ϕ) \hat{ϕ}] \\ C_{e m n} (θ, ϕ) = - \frac{1}{\sqrt{n (n + 1)}} [m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \sin (m ϕ) \hat{θ} + \frac{d P_{n}^{m} (\cos θ)}{d θ} \cos (m ϕ) \hat{ϕ}] \\ C_{o m n} (θ, ϕ) = \frac{1}{\sqrt{n (n + 1)}} [m \frac{P_{n}^{m} (\cos θ)}{\sin θ} \cos (m ϕ) \hat{θ} - \frac{d P_{n}^{m} (\cos θ)}{d θ} \sin (m ϕ) \hat{ϕ}] \end{array}},

then yields

\begin{array}{l} A_{s m n}^{(1)} = 4 i^{n} \sqrt{n (n + 1)} e_{inc} \cdot C_{s m n} (θ_{inc}, ϕ_{inc}) \\ B_{s m n}^{(1)} = 4 i^{n - 1} \sqrt{n (n + 1)} e_{inc} \cdot B_{s m n} (θ_{inc}, ϕ_{inc}) \end{array}} .

B. Scattered Electromagnetic Field

With the assumption that the field scattered by the sphere does not interact with (and modify) the source of the incident field, we can state the scattered electric and magnetic fields as follows:

E_{sca} (r) = \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} {D_{m n} [A_{s m n}^{(3)} M_{s m n}^{(3)} (k_{0} r) + B_{s m n}^{(3)} N_{s m n}^{(3)} (k_{0} r)]}, r > a,

H_{sca} (r) = \frac{k_{0}}{i ω μ_{0}} \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} {D_{m n} [A_{s m n}^{(3)} N_{s m n}^{(3)} (k_{0} r) + B_{s m n}^{(3)} M_{s m n}^{(3)} (k_{0} r)]}, r > a,

In these expressions [11,12],

\begin{array}{l} M_{e m n}^{(3)} (k_{0} r) = \nabla \times [r h_{n}^{(1)} (k_{0} r) P_{n}^{m} (\cos θ) \cos (m ϕ)] \\ M_{o m n}^{(3)} (k_{0} r) = \nabla \times [r h_{n}^{(1)} (k_{0} r) P_{n}^{m} (\cos θ) \sin (m ϕ)] \\ N_{s m n}^{(3)} (k_{0} r) = k_{0}^{- 1} \nabla \times M_{s m n}^{(3)} (k_{0} r), s \in {e, o} \end{array}}, m \in {0, 1, 2, \dots, n}, n \in {1, 2, 3, \dots},

where

h_{n}^{(1)} (\cdot)

is the spherical Hankel function of the first kind and order

n

, but the coefficients

A_{s m n}^{(3)}

and

B_{s m n}^{(3)}

have to be determined by the solution of a boundary-value problem [14].

In the far zone, the relation

\lim_{k_{0} r \to \infty} [k_{0} r \exp (- i k_{0} r) h_{n}^{(1)} (k_{0} r)] = {(- i)}^{n + 1}

leads to

\begin{array}{l} \lim_{k_{0} r \to \infty} [k_{0} r \exp (- i k_{0} r) M_{s m n}^{(3)} (k_{0} r)] = {(- i)}^{n + 1} \sqrt{n (n + 1)} C_{s m n} (θ, ϕ) \\ \lim_{k_{0} r \to \infty} [k_{0} r \exp (- i k_{0} r) N_{s m n}^{(3)} (k_{0} r)] = {(- i)}^{n} \sqrt{n (n + 1)} B_{s m n} (θ, ϕ) \end{array}} .

Accordingly, in the far zone the scattered electric field may be approximated as

E_{sca} (r) \approx F_{sca} (θ, ϕ) \frac{\exp (i k_{0} r)}{r}

and the scattered magnetic field as

H_{sca} (r) \approx η_{0}^{- 1} \hat{r} \times F_{sca} (θ, ϕ) \frac{\exp (i k_{0} r)}{r},

where

\hat{r} = r / r

and the vector far-field scattering amplitude

F_{sca} (θ, ϕ) = k_{0}^{- 1} \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} {{(- i)}^{n} D_{m n} \sqrt{n (n + 1)} [- i A_{s m n}^{(3)} C_{s m n} (θ, ϕ) + B_{s m n}^{(3)} B_{s m n} (θ, ϕ)]} .

The foregoing expressions can be used to verify the satisfaction of the Sommerfeld radiation conditions [23]

\begin{array}{l} \lim_{k_{0} r \to \infty} | \frac{\partial}{\partial r} E_{sca} (r) - i k_{0} E_{sca} (r) | \leq O (r^{- 2}) \\ \lim_{k_{0} r \to \infty} | \frac{\partial}{\partial r} H_{sca} (r) - i k_{0} H_{sca} (r) | \leq O (r^{- 2}) \end{array}}

as well as of the Silver–Müller radiation conditions [24]

\begin{array}{l} \lim_{k_{0} r \to \infty} | E_{sca} (r) + η_{0} \hat{r} \times H_{sca} (r) | \leq O (r^{- 2}) \\ \lim_{k_{0} r \to \infty} | η_{0} H_{sca} (r) - \hat{r} \times E_{sca} (r) | \leq O (r^{- 2}) \end{array}}

by Eqs. (13) and (14).

C. Internal Electromagnetic Field

The electric and magnetic fields excited inside the sphere are represented by [14]

E_{exc} (r) = \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} [b_{s m n} M_{s m n} (r) + c_{s m n} N_{s m n} (r)], r < a,

H_{exc} (r) = \frac{k_{0}}{i ω μ_{0}} \sqrt{\frac{ϵ_{r}}{μ_{r}}} \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} [b_{s m n} N_{s m n} (r) + c_{s m n} M_{s m n} (r)], r < a,

where the coefficients

b_{s m n}

and

c_{s m n}

are not known.

The vector spherical wavefunctions appearing in these expressions are given as

M_{s m n} (r) = \frac{J_{n} (k r)}{f_{1} (ϕ)} {A̳}^{- 1} \cdot {\hat{r} [\frac{f_{4} (ϕ) - f_{1}^{2} (ϕ)}{f_{2} (θ, ϕ)} \sin θ \cos θ Q_{s m n} (θ, ϕ) - (α_{x} - α_{y}) \sin θ \sin ϕ \cos ϕ R_{s m n} (θ, ϕ)] + \hat{θ} [\frac{f_{4} (ϕ) \cos^{2} θ + f_{1}^{2} (ϕ) \sin^{2} θ}{f_{2} (θ, ϕ)} Q_{s m n} (θ, ϕ) - (α_{x} - α_{y}) \cos θ \sin ϕ \cos ϕ R_{s m n} (θ, ϕ)] + \hat{ϕ} [- \frac{α_{x} - α_{y}}{f_{2} (θ, ϕ)} \cos θ \sin ϕ \cos ϕ Q_{s m n} (θ, ϕ) - f_{4} (ϕ) R_{s m n} (θ, ϕ)]}

and

N_{s m n} (r) = {A̳}^{- 1} \cdot (\hat{r} {\frac{J_{n} (k r)}{k r} [\frac{\cos^{2} θ + f_{4} (ϕ) \sin^{2} θ}{f_{2}^{2} (θ, ϕ)}] P_{s m n} (θ, ϕ) + \frac{K_{n} (k r)}{f_{1} (ϕ)} [\frac{f_{4} (ϕ) - f_{1}^{2} (ϕ)}{f_{2} (θ, ϕ)} \sin θ \cos θ R_{s m n} (θ, ϕ) + (α_{x} - α_{y}) \sin θ \sin ϕ \cos ϕ Q_{s m n} (θ, ϕ)]} + \hat{θ} {\frac{J_{n} (k r)}{k r} [\frac{f_{4} (ϕ) - 1}{f_{2}^{2} (θ, ϕ)} \sin θ \cos θ] P_{s m n} (θ, ϕ) + \frac{K_{n} (k r)}{f_{1} (ϕ)} [\frac{f_{4} (ϕ) \cos^{2} θ + f_{1}^{2} (ϕ) \sin^{2} θ}{f_{2} (θ, ϕ)} R_{s m n} (θ, ϕ) + (α_{x} - α_{y}) \cos θ \sin ϕ \cos ϕ Q_{s m n} (θ, ϕ)]} + \hat{ϕ} {- \frac{J_{n} (k r)}{k r} [\frac{α_{x} - α_{y}}{f_{2}^{2} (θ, ϕ)} \sin θ \sin ϕ \cos ϕ] P_{s m n} (θ, ϕ) + \frac{K_{n} (k r)}{f_{1} (ϕ)} [- \frac{α_{x} - α_{y}}{f_{2} (θ, ϕ)} \cos θ \sin ϕ \cos ϕ R_{s m n} (θ, ϕ) + f_{4} (ϕ) Q_{s m n} (θ, ϕ)]}),

where

J_{n} (k r) = j_{n} [k r f_{2} (θ, ϕ)],

K_{n} (k r) = \frac{n + 1}{k r f_{2} (θ, ϕ)} J_{n} (k r) - J_{n + 1} (k r),

P_{s m n} (θ, ϕ) = n (n + 1) P_{n}^{m} [\frac{\cos θ}{f_{2} (θ, ϕ)}] V_{s m} (ϕ),

Q_{s m n} (θ, ϕ) = m P_{n}^{m} [\frac{\cos θ}{f_{2} (θ, ϕ)}] \frac{f_{2} (θ, ϕ)}{f_{1} (ϕ) \sin θ} U_{s m} (ϕ),

R_{s m n} (θ, ϕ) = \frac{1}{f_{1} (ϕ) \sin θ} {(n - m + 1) f_{2} (θ, ϕ) P_{n + 1}^{m} [\frac{\cos θ}{f_{2} (θ, ϕ)}] - (n + 1) \cos θ P_{n}^{m} [\frac{\cos θ}{f_{2} (θ, ϕ)}]} V_{s m} (ϕ),

U_{s m} (ϕ) = {\begin{matrix} - \sin [m f_{3} (ϕ)] \\ \cos [m f_{3} (ϕ)] \end{matrix}}, s = {\begin{matrix} e \\ o \end{matrix},

V_{s m} (ϕ) = {\begin{matrix} \cos [m f_{3} (ϕ)] \\ \sin [m f_{3} (ϕ)] \end{matrix}}, s = {\begin{matrix} e \\ o \end{matrix},

f_{1} (ϕ) = + {(α_{x}^{2} \cos^{2} ϕ + α_{y}^{2} \sin^{2} ϕ)}^{1 / 2},

f_{2} (θ, ϕ) = + {[f_{1}^{2} (ϕ) \sin^{2} θ + \cos^{2} θ]}^{1 / 2},

f_{3} (ϕ) = \tan^{- 1} (\frac{α_{y}}{α_{x}} \tan ϕ),

f_{4} (ϕ) = α_{x} \cos^{2} ϕ + α_{y} \sin^{2} ϕ .

The angle

f_{3} (ϕ)

must lie in the same quadrant as its argument. Let us also point out that

M_{s m n} (r)

and

N_{s m n} (r)

are provided here in closed form and do not require the solution of any eigenvalue problem [25,26].

D. Solution of Boundary–Value Problem

Application of the standard boundary conditions,

\begin{array}{l} \hat{θ} \cdot E_{exc} (r) = \hat{θ} \cdot [E_{inc} (r) + E_{sca} (r)] \\ \hat{ϕ} \cdot E_{exc} (r) = \hat{ϕ} \cdot [E_{inc} (r) + E_{sca} (r)] \\ \hat{θ} \cdot H_{exc} (r) = \hat{θ} \cdot [H_{inc} (r) + H_{sca} (r)] \\ \hat{ϕ} \cdot H_{exc} (r) = \hat{ϕ} \cdot [H_{inc} (r) + H_{sca} (r)] \end{array}}, r = a,

across the closed surface

r = a

as well as the orthogonalities of the functions

B_{s m n} (θ, ϕ)

and

C_{s m n} (θ, ϕ)

on the unit sphere leads to the following simultaneous algebraic equations for every combination of

j \in {1, 3}

,

s \in {e, o}

,

n \in {1, 2, 3, \dots}

, and

m \in {0, 1, 2, \dots, n}

[14]:

A_{s m n}^{(1)} = \sum_{s^{'} \in {e, o}} \sum_{n^{'} = 1}^{\infty} \sum_{m^{'} = 0}^{n^{'}} [I_{s m n, s^{'} m^{'} n^{'}}^{(1)} b_{s^{'} m^{'} n^{'}} + J_{s m n, s^{'} m^{'} n^{'}}^{(1)} c_{s^{'} m^{'} n^{'}}],

B_{s m n}^{(1)} = \sum_{s^{'} \in {e, o}} \sum_{n^{'} = 1}^{\infty} \sum_{m^{'} = 0}^{n^{'}} [K_{s m n, s^{'} m^{'} n^{'}}^{(1)} b_{s^{'} m^{'} n^{'}} + L_{s m n, s^{'} m^{'} n^{'}}^{(1)} c_{s^{'} m^{'} n^{'}}],

A_{s m n}^{(3)} = - \sum_{s^{'} \in {e, o}} \sum_{n^{'} = 1}^{\infty} \sum_{m^{'} = 0}^{n^{'}} [I_{s m n, s^{'} m^{'} n^{'}}^{(3)} b_{s^{'} m^{'} n^{'}} + J_{s m n, s^{'} m^{'} n^{'}}^{(3)} c_{s^{'} m^{'} n^{'}}],

B_{s m n}^{(3)} = - \sum_{s^{'} \in {e, o}} \sum_{n^{'} = 1}^{\infty} \sum_{m^{'} = 0}^{n^{'}} [K_{s m n, s^{'} m^{'} n^{'}}^{(3)} b_{s^{'} m^{'} n^{'}} + L_{s m n, s^{'} m^{'} n^{'}}^{(3)} c_{s^{'} m^{'} n^{'}}] .

In these equations, the quantities

I_{s m n, s^{'} m^{'} n^{'}}^{(1)}

, etc., are the integrals

I_{s m n, s^{'} m^{'} n^{'}}^{(j)} = - \frac{i {(k_{0} a)}^{2}}{π} \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \sin θ {N_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times M_{s^{'} m^{'} n^{'}} (a \hat{r})] + \sqrt{\frac{ϵ_{r}}{μ_{r}}} M_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times N_{s^{'} m^{'} n^{'}} (a \hat{r})]},

J_{s m n, s^{'} m^{'} n^{'}}^{(j)} = - \frac{i {(k_{0} a)}^{2}}{π} \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \sin θ {N_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times N_{s^{'} m^{'} n^{'}} (a \hat{r})] + \sqrt{\frac{ϵ_{r}}{μ_{r}}} M_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times M_{s^{'} m^{'} n^{'}} (a \hat{r})]},

K_{s m n, s^{'} m^{'} n^{'}}^{(j)} = - \frac{i {(k_{0} a)}^{2}}{π} \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \sin θ {M_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times M_{s^{'} m^{'} n^{'}} (a \hat{r})] + \sqrt{\frac{ϵ_{r}}{μ_{r}}} N_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times N_{s^{'} m^{'} n^{'}} (a \hat{r})]},

L_{s m n, s^{'} m^{'} n^{'}}^{(j)} = - \frac{i {(k_{0} a)}^{2}}{π} \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \sin θ {M_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times N_{s^{'} m^{'} n^{'}} (a \hat{r})] + \sqrt{\frac{ϵ_{r}}{μ_{r}}} N_{s m n}^{(ℓ)} (k_{0} a \hat{r}) \cdot [\hat{r} \times M_{s^{'} m^{'} n^{'}} (a \hat{r})]},

where

j \in {1, 3}

,

ℓ = j + 2 (\mod 4) \in {3, 1}

, and

\hat{r} = (\hat{x} \cos ϕ + \hat{y} \sin ϕ) \sin θ + \hat{z} \cos θ

is the unit radial vector. Let us note here that the radial components of

M_{s^{'} m^{'} n^{'}} (a \hat{r})

and

N_{s^{'} m^{'} n^{'}} (a \hat{r})

do not show up in the foregoing integrals.

After truncating the summations over $n^{'} \in {1, 2, 3, \dots}$ to $n^{'} \in {1, 2, 3, \dots, N}$ and restricting $n \in {1, 2, 3, \dots}$ to $n \in {1, 2, 3, \dots, N}$ similarly, Eqs. (39)–(42) can be written symbolically as the matrix equations

[\begin{matrix} A^{(1)} \\ B^{(1)} \end{matrix}] = [Y^{(1)}] [\begin{matrix} b \\ c \end{matrix}], [\begin{matrix} A^{(3)} \\ B^{(3)} \end{matrix}] = - [Y^{(3)}] [\begin{matrix} b \\ c \end{matrix}],

whence

[\begin{matrix} A^{(3)} \\ B^{(3)} \end{matrix}] = - [Y^{(3)}] {[Y^{(1)}]}^{- 1} [\begin{matrix} A^{(1)} \\ B^{(1)} \end{matrix}] = - [T] [\begin{matrix} A^{(1)} \\ B^{(1)} \end{matrix}] .

In this equation,

[T] = - [Y^{(3)}] {[Y^{(1)}]}^{- 1}

is the T matrix of the chosen sphere embedded in free space. It has to be evaluated for increasing values of

N

until the various cross sections converge to preset tolerances.

E. Uniaxial Dielectric–Magnetic Sphere

When the sphere is made of a uniaxial dielectric–magnetic material, i.e., $α_{x} = α_{y}$ , then the integrals (43)–(46) show symmetries with respect to the indices $s$ and $m$ as follows:

I_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto δ_{s s^{'}} (δ_{m m^{'}} + δ_{m, m^{'} - 1} + δ_{m, m^{'} + 1}),

J_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto (1 - δ_{s s^{'}}) (δ_{m m^{'}} + δ_{m, m^{'} - 1} + δ_{m, m^{'} + 1}),

K_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto (1 - δ_{s s^{'}}) (δ_{m m^{'}} + δ_{m, m^{'} - 1} + δ_{m, m^{'} + 1}),

L_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto δ_{s s^{'}} (δ_{m m^{'}} + δ_{m, m^{'} - 1} + δ_{m, m^{'} + 1}) .

Thus, all four of these integrals vanish if

m

and

m^{'}

differ by more than unity. Furthermore,

I_{e m n, o m^{'} n^{'}}^{(j)} = I_{o m n, e m^{'} n^{'}}^{(j)} = L_{e m n, o m^{'} n^{'}}^{(j)} = L_{o m n, e m^{'} n^{'}}^{(j)} = 0

and

J_{e m n, e m^{'} n^{'}}^{(j)} = J_{o m n, o m^{'} n^{'}}^{(j)} = K_{e m n, e m^{'} n^{'}}^{(j)} = K_{o m n, o m^{'} n^{'}}^{(j)} = 0

. These symmetries simplify the storage of and reduce the time to calculate the matrices

[Y^{(j)}]

defined in Eqs. (47).

F. Isotropic Dielectric–Magnetic Sphere

When $α_{x} = α_{y} = 1$ , the symmetries (49)–(52) expand further to

I_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto δ_{s s^{'}} δ_{m m^{'}} δ_{n n^{'}},

J_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto (1 - δ_{s s^{'}}) δ_{m m^{'}} δ_{n n^{'}},

K_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto (1 - δ_{s s^{'}}) δ_{m m^{'}} δ_{n n^{'}},

L_{s m n, s^{'} m^{'} n^{'}}^{(j)} \propto δ_{s s^{'}} δ_{m m^{'}} δ_{n n^{'}},

and these integrals can then be obtained analytically because the simplifications

M_{s m n} (r) = M_{s m n}^{(1)} (k r)

and

N_{s m n} (r) = N_{s m n}^{(1)} (k r)

emerge. We have verified that the usual Lorenz–Mie solution [12] for an isotropic dielectric–magnetic sphere embedded in free space is recovered.

G. Scattering Cross Sections and Efficiencies

The plane wave scattering response is usually quantified through the differential scattering cross section [27, Section 8.8]

σ_{D} (θ, ϕ) = 4 π \frac{F_{sca} (θ, ϕ) \cdot F_{sca}^{*} (θ, ϕ)}{e_{inc} \cdot e_{inc}^{*}}

and the total scattering cross section

σ_{sca} = \frac{1}{e_{inc} \cdot e_{inc}^{*}} \int_{ϕ = 0}^{2 π} \int_{θ = 0}^{π} [F_{sca} (θ, ϕ) \cdot F_{sca}^{*} (θ, ϕ)] \sin θ d θ d ϕ,

= \frac{1}{e_{inc} \cdot e_{inc}^{*}} \frac{π}{k_{0}^{2}} \sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} [D_{m n} ({| A_{s m n}^{(3)} |}^{2} + {| B_{s m n}^{(3)} |}^{2})] .

The differential scattering efficiency is defined as

Q_{D} = σ_{D} / π a^{2}

, and the total scattering efficiency as

Q_{sca} = σ_{sca} / π a^{2}

.

H. Extinction and Absorption Cross Sections and Efficiencies

In order to quantify the removal of energy from the incident plane wave through both absorption and scattering, the extinction cross section

σ_{ext} = - \frac{4 π}{k_{0}} Im [\frac{F_{sca} (θ_{inc}, ϕ_{inc}) \cdot e_{inc}^{*}}{e_{inc} \cdot e_{inc}^{*}}]

is calculated. The extinction efficiency is defined as

Q_{ext} = σ_{ext} / π a^{2}

.

The excess of $σ_{ext}$ over $σ_{sca}$ is the absorption cross section $σ_{abs}$ , which is null valued if the sphere is made of a nondissipative material (i.e., both $ϵ_{r}$ and $μ_{r}$ are both positive and real). When $σ_{abs}$ is normalized by $π a^{2}$ , the result is the absorption efficiency $Q_{abs}$ .

3. NUMERICAL RESULTS AND DISCUSSION

In order to compute the T matrix as well as the differential, total scattering, and extinction cross sections, we set up a Mathematica program. For that purpose, we made the replacement

\sum_{s \in {e, o}} \sum_{n = 1}^{\infty} \sum_{m = 0}^{n} \to \sum_{s \in {e, o}} \sum_{n = 1}^{N} \sum_{m = 0}^{n}

for the representation of all fields. The value of

N

would be incremented in steps of unity as the T matrix and the backscattering efficiency

Q_{b} = σ_{D} (π + θ_{inc}, π + ϕ_{inc}) / π a^{2}

were calculated for fixed

{k_{0} a, ϵ_{r}, μ_{r}, α_{x}, α_{y}, θ_{inc}, ϕ_{inc}}

. The procedure would be terminated once the backscattering cross section

σ_{b} = Q_{b} π a^{2}

had converged within a preset tolerance of 0.1%. In general, the larger the deviation of

A̳

from

I̳

and/or the larger the deviation of

| ϵ_{r} μ_{r} |

from unity, the greater the adequate value of

N

for converged results. For all results presented here,

N \leq 11

. The program was implemented on a laptop computer.

For partial validation of the Mathematica program, $α_{x} = α_{y} = 1$ was set therein so that the scattering sphere was taken to be made of an isotropic dielectric–magnetic material. First, the forward-scattering efficiency,

Q_{f} = σ_{D} (θ_{inc}, ϕ_{inc}) / π a^{2},

the backscattering efficiency, and the total scattering efficiency were checked for an isotropic dielectric sphere (

μ_{r} = 1

) against data computed using the Lorenz–Mie solution and published in standard works [12,28]. Second, a Mathematica program was written to implement the Lorenz–Mie solution for an isotropic dielectric–magnetic sphere (

ϵ_{r} \neq 1

,

μ_{r} \neq 1

) and was used to check the results yielded by the program written for the anisotropic dielectric–magnetic sphere. The latter program satisfied both tests.

Additional tests for validation involved symmetry. For example, for a uniaxial dielectric–magnetic sphere ( $α_{x} = α_{y} \neq 1$ ), we found that the total scattering cross section does not change if (i) $E_{inc} = \hat{x} \exp (i k_{0} z)$ is replaced by $E_{inc} = \hat{y} \exp (i k_{0} z)$ , (ii) $E_{inc} = \hat{z} \exp (i k_{0} x)$ is replaced by $E_{inc} = \hat{z} \exp (i k_{0} y)$ , or (iii) $E_{inc} = \hat{x} \exp (i k_{0} y)$ is replaced by $E_{inc} = \hat{y} \exp (i k_{0} x)$ . Similarly, we verified that $σ_{D} (θ, 0)$ for $E_{inc} = \hat{x} \exp (i k_{0} z)$ remains the same as $σ_{D} (θ, π / 2)$ for $E_{inc} = \hat{y} \exp (i k_{0} z)$ , provided that the values of $α_{x}$ and $α_{y}$ are interchanged.

Finally, we also verified that the absorption cross section $σ_{abs}$ is null valued if the sphere is made of a nondissipative material.

A. Rayleigh Scattering

Scattering by an electrically small object [6, Section 6.4] is called Rayleigh scattering. Not only is it studied to explain the electromagnetic response of particulate materials [29,30], but it is also used to design composite materials [31,32].

A long-wavelength approximation is very useful to obtain closed-form analytical results for scattering by homogeneous and electrically small objects [33]. Accordingly, scattering by an electrically small sphere described by the constitutive relations (1) is equivalent to radiation jointly by an electric dipole moment

p_{eqvt} = 4 π a^{3} ϵ_{0} (ϵ_{r} A̳ \cdot A̳ - I̳) \cdot {(ϵ_{r} A̳ \cdot A̳ + 2 I̳)}^{- 1} \cdot e_{inc}

and a magnetic dipole moment

m_{eqvt} = 4 π a^{3} μ_{0} (μ_{r} A̳ \cdot A̳ - I̳) \cdot {(μ_{r} A̳ \cdot A̳ + 2 I̳)}^{- 1} \cdot (\frac{k_{inc} \times e_{inc}}{ω μ_{0}}),

both located at the centroid of the sphere, with

I̳

denoting the identity dyadic. Therefore, the Rayleigh estimate of the vector far-field scattering amplitude is

F_{sca}^{Rayleigh} (\hat{r}) = - \frac{ω^{2} μ_{0}}{4 π} [\hat{r} \times (\hat{r} \times p_{eqvt}) + \sqrt{\frac{ϵ_{0}}{μ_{0}}} \hat{r} \times m_{eqvt}] = - k_{0}^{2} a^{3} (\hat{r} \times I̳) \cdot [(\hat{r} \times I̳) \cdot (ϵ_{r} A̳ \cdot A̳ - I̳) \cdot {(ϵ_{r} A̳ \cdot A̳ + 2 I̳)}^{- 1} + (μ_{r} A̳ \cdot A̳ - I̳) \cdot {(μ_{r} A̳ \cdot A̳ + 2 I̳)}^{- 1} \cdot ({\hat{k}}_{inc} \times I̳)] \cdot e_{inc},

where

{\hat{k}}_{inc} = k_{inc} / k_{0}

. The cross sections

σ_{D}^{Rayleigh} (θ, ϕ)

,

σ_{sca}^{Rayleigh}

, and

σ_{ext}^{Rayleigh}

can be calculated thereafter by using Eq. (66) in Eqs. (57), (58), and (60), respectively. These expressions can be expected to hold for [6, Section 10.6.1]

\begin{array}{l} k_{0} a \leq π / 5 \\ k_{0} a \sqrt{ϵ_{r}} \sqrt{μ_{r}} \max {α_{x}^{- 1}, α_{y}^{- 1}, 1} \leq π / 5 \end{array}},

i.e., when the radius

a

is equal to or less than a tenth of the smallest wavelength in the ambient medium (vacuum) and the scattering material.

We computed the differential scattering, total scattering, and extinction efficiencies, exactly using Eq. (20) and approximately using Eq. (66), of an electrically small uniaxial dielectric–magnetic sphere ( $ϵ_{r} = 4$ , $μ_{r} = 1.1$ , $α_{x} = α_{y} = 1.2$ ) illuminated by a plane wave with $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ . No distinction can be seen in the curves of $Q_{sca}$ and $Q_{sca}^{Rayleigh}$ versus $k_{0} a \in (0, 0.5]$ presented in Fig. 1(a). The same curves also hold for $E_{inc} = \hat{y} \exp (i k_{0} z) V m^{- 1}$ . Although the long-wavelength approximation is expected to be valid for $k_{0} a \leq 0.3$ according to the inequalities (67), the Rayleigh expressions hold well in Fig. 1(a) for $k_{0} a$ as high as 0.5.

Fig. 1. Total scattering efficiency—computed exactly using Eq. (20) and approximately using Eq. (66)—as a function of $k_{0} a$ , when $ϵ_{r} = 4$ and $μ_{r} = 1.1$ . (a) $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ and $α_{x} = α_{y} = 1.2$ . (b) $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ .

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For Fig. 1(a), the sphere is transversely isotropic with respect to the direction of the incident plane wave, because $α_{x} = α_{y}$ and ${\hat{k}}_{inc} ‖ \hat{z}$ . By setting $α_{x}$ and $α_{y}$ unequal without any other changes, the condition of transverse isotropy can be changed. Figure 1(b) presents $Q_{sca}$ and $Q_{sca}^{Rayleigh}$ versus $k_{0} a \in (0, 0.5]$ for an electrically small biaxial dielectric–magnetic sphere ( $ϵ_{r} = 4$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , $α_{y} = 1.2$ ) illuminated by a plane wave with $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ . Clearly now, the long-wavelength approximation is highly satisfactory for $k_{0} a \leq 0.3$ , as predicted by the inequalities (67), but the discrepancies between the exact and the approximate results begin to grow as $k_{0} a$ increases further.

The biaxiality of the electrically small dielectric–magnetic sphere in Fig. 1(b) also enhances the roles of the directions of both $e_{inc}$ and $k_{inc}$ . Comparing the plots of $Q_{sca}$ versus $k_{0} a \in (0, 0.5]$ presented in

• Figure 1(b) for $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ ,
• Figure 2(a) for $E_{inc} = \hat{x} \exp (i k_{0} y) V m^{- 1}$ , and
• Figure 2(b) for $E_{inc} = \hat{z} \exp (i k_{0} y) V m^{- 1}$

for the same sphere (

ϵ_{r} = 4

,

μ_{r} = 1.1

,

α_{x} = 1.1

,

α_{y} = 1.2

), we see that the total scattering efficiency is significantly affected by the directions of both

e_{inc}

and

k_{inc}

for

k_{0} a

as low as the upper limit predicted by the inequalities (67).

Although we presented data in Subsection 3.A on only the total scattering efficiencies of electrically small spheres here, the conclusions drawn on the adequacy of the Rayleigh expression (66) therefrom are also applicable to the differential scattering and extinction efficiencies.

Fig. 2. Total scattering efficiency—computed exactly using Eq. (20) and approximately using Eq. (66)—as a function of $k_{0} a$ , when $ϵ_{r} = 4$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ . (a) $E_{inc} = \hat{x} \exp (i k_{0} y) V m^{- 1}$ , (b) $E_{inc} = \hat{z} \exp (i k_{0} y) V m^{- 1}$ .

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B. Differential Scattering Efficiency

In order to present representative results for differential scattering efficiency, we fixed ${\hat{k}}_{inc} ‖ \hat{z}$ , $ϵ_{r} = 4$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ . Plots of $Q_{D} (θ, 0 °)$ and $Q_{D} (θ, 90 °)$ are presented in Figs. 3 and 4 for $k_{0} a \in {0.5, 2.5, 4.5}$ and either ${\hat{e}}_{inc} ‖ \hat{x}$ or ${\hat{e}}_{inc} ‖ \hat{y}$ , where ${\hat{e}}_{inc} = e_{inc} / | e_{inc} |$ . A very notable feature of these plots is the appearance of multiple lobes as the size parameter $k_{0} a$ increases. This happens in any scattering plane, as exemplified by the plots of $Q_{D}$ versus $θ$ in the $ϕ = 0 °$ and $ϕ = 90 °$ planes. However, when the electrical size is small, lobes may not appear in certain scattering planes—e.g., in the $ϕ = 90 °$ plane in Fig. 3(b) and the $ϕ = 0 °$ plane in Fig. 4(a). Even though the sphere is made of an anisotropic material lacking transverse isotropy with respect to the direction of propagation of the incident plane wave, the sphere appears to be isotropic just because it is electrically small. As the size parameter increases, anisotropy asserts its presence.

Fig. 3. Differential scattering efficiency as a function of $θ$ , when $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ , $ϵ_{r} = 4$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ . The dashed red line is for $k_{0} a = 0.5$ , the solid blue line is for $k_{0} a = 2.5$ , and the dashed-and-dotted black line is for $k_{0} a = 4.5$ . (a) $ϕ = 0 °$ , (b) $ϕ = 90 °$ .

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Fig. 4. Same as Fig. 3, except that $E_{inc} = \hat{y} \exp (i k_{0} z) V m^{- 1}$ .

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With the size parameter fixed at $k_{0} a = 4.5$ , we also compared the effects of uniaxiality and biaxiality. Figures 5 and 6 show plots of $Q_{D} (θ, 0 °)$ and $Q_{D} (θ, 90 °)$ , when ${\hat{k}}_{inc} ‖ \hat{z}$ , either ${\hat{e}}_{inc} ‖ \hat{x}$ or ${\hat{e}}_{inc} ‖ \hat{y}$ , $ϵ_{r} = 4$ , and $μ_{r} = 1.1$ . The sphere is transversely isotropic with respect to the direction of propagation of the incident plane wave if the material is uniaxial ( $α_{x} = α_{y} = 1.1$ ), but not if it is biaxial ( $α_{x} = 1.1$ , $α_{y} = 1.2$ ). The curve of $Q_{D} (θ, 0 °)$ versus $θ$ for ${\hat{e}}_{inc} ‖ \hat{x}$ in Fig. 5(a) is identical to that of $Q_{D} (θ, 90 °)$ versus $θ$ for ${\hat{e}}_{inc} ‖ \hat{y}$ in Fig. 6(b), and the curve of $Q_{D} (θ, 90 °)$ versus $θ$ for ${\hat{e}}_{inc} ‖ \hat{x}$ in Fig. 5(b) is identical to that of $Q_{D} (θ, 0 °)$ versus $θ$ for ${\hat{e}}_{inc} ‖ \hat{y}$ in Fig. 6(a), for the uniaxial sphere. This symmetry is not evident for the biaxial sphere.

Fig. 5. Differential scattering efficiency as a function of $θ$ , when $E_{inc} = \hat{x} \exp (i k_{0} z) V m^{- 1}$ , $k_{0} a = 4.5$ , $ϵ_{r} = 4$ , and $μ_{r} = 1.1$ . The red dashed line is for $α_{x} = α_{y} = 1.1$ , and the solid blue line is for $α_{x} = 1.1$ and $α_{y} = 1.2$ . (a) $ϕ = 0 °$ , (b) $ϕ = 90 °$ .

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Fig. 6. Same as Fig. 5, except that $E_{inc} = \hat{y} \exp (i k_{0} z) V m^{- 1}$ .

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C. Total Scattering and Extinction Efficiencies

The total scattering and extinction efficiencies calculated using Eqs. (58) and (60), respectively, are shown in Figs. 7–9 as functions of $k_{0} a$ for six combinations of the directions of $k_{inc}$ and $e_{inc}$ . The sphere was taken to be dissipative and biaxial: $ϵ_{r} = 4 (1 + i 0.1)$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ .

Fig. 7. Extinction, total scattering, absorption, backscattering, and forward-scattering efficiencies as functions of $k_{0} a$ , when ${\hat{k}}_{inc} ‖ \hat{z}$ , $ϵ_{r} = 4 (1 + i 0.1)$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ . (a) ${\hat{e}}_{inc} ‖ \hat{x}$ , (b) ${\hat{e}}_{inc} ‖ \hat{y}$ .

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Fig. 8. Same as Fig. 7, except that ${\hat{k}}_{inc} ‖ \hat{x}$ . (a) ${\hat{e}}_{inc} ‖ \hat{y}$ , (b) ${\hat{e}}_{inc} ‖ \hat{z}$ .

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Fig. 9. Same as Fig. 7, except that ${\hat{k}}_{inc} ‖ \hat{y}$ . (a) ${\hat{e}}_{inc} ‖ \hat{x}$ , (b) ${\hat{e}}_{inc} ‖ \hat{z}$ .

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Because of dissipation, $Q_{ext} > Q_{sca}$ for all $k_{0} a > 0$ in these figures. Furthermore, $Q_{ext}$ tracks $Q_{sca}$ so well that the absorption efficiency $Q_{abs}$ rises steadily with $k_{0} a \in [0, 2]$ and becomes a weak function of $k_{0} a \in [2, 4.5]$ , regardless of the directions of $k_{inc}$ and $e_{inc}$ .

All six panels in Figs. 7–9 clearly indicate that $Q_{ext}$ and $Q_{sca}$ are maximum at some $k_{0} a \in [2.2, 3.2]$ . The plots suggest that both $Q_{ext}$ and $Q_{sca}$ would settle down to show a weak dependence on $k_{0} a$ as that parameter increases further—just like an isotropic sphere [12, Section 4.4]—but calculations could not be carried out on a laptop computer for $k_{0} a > 4.5$ in a reasonable period of time. This limitation arises from the functions $M_{s m n} (r)$ and $N_{s m n} (r)$ not displaying orthogonality properties on the surface of any sphere centered at the origin when $A̳ \neq I̳$ [14]. We conjectured from Figs. 7–9 that the highest peaks of $Q_{ext}$ and $Q_{sca}$ versus $k_{0} a$ are obtained when the direction of propagation of the incident plane wave is such that the scalar ${\hat{k}}_{inc} \cdot A̳ \cdot {\hat{k}}_{inc}$ has a minimum value. This conjecture was verified by several calculations for different values of $α_{x} \in (1, 1.5]$ and $α_{y} \in (1, 1.5]$ .

In contrast to the clear maximums of both $Q_{ext}$ and $Q_{sca}$ versus $k_{0} a$ in Figs. 7–9 the absorption efficiency $Q_{abs}$ —despite some undulations—shows a generally increasing trend with $k_{0} a$ for six combinations of the directions of $k_{inc}$ and $e_{inc}$ . In all six panels in these figures, $Q_{sca}$ first increases slower than $Q_{ext}$ and then decreases faster than $Q_{ext}$ , as $k_{0} a$ increases.

D. Backscattering and Forward-Scattering Efficiencies

The backscattering and forward-scattering efficiencies computed using Eqs. (62) and (63), respectively, are also shown in Figs. 7–9 as functions of $k_{0} a$ for six combinations of the directions of $k_{inc}$ and $e_{inc}$ , when $ϵ_{r} = 4 (1 + i 0.1)$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ .

We observe that $Q_{f}$ keeps on generally increasing as the size parameter $k_{0} a$ increases up to a limit and then exhibits undulations with further increases of $k_{0} a$ . The undulations are most pronounced for the smallest value of ${\hat{k}}_{inc} \cdot A̳ \cdot {\hat{k}}_{inc}$ .

In contrast to $Q_{f}$ , the plots of $Q_{b}$ versus $k_{0} a$ are oscillatory regardless of the directions of $k_{inc}$ and $e_{inc}$ . Monostatic detection of the sphere would be easier at frequencies where $Q_{b}$ peaks, but it will be difficult at frequencies for which $Q_{b}$ reduces to a minimum, which can be substantially smaller than the adjacent peak values. In this characteristic, a sphere made of the chosen material responds similarly to a sphere made of an isotropic achiral material [28, p. 246]. But the anisotropy of the chosen material shows its effect too. Higher $Q_{b}$ peaks are favored by the highest value of ${\hat{e}}_{inc} \cdot A̳ \cdot {\hat{e}}_{inc}$ , as can be concluded from Fig. 10. Even more importantly, the backscattering efficiency of a sphere of a specific size parameter can be halved from its maximum value simply by reorienting the sphere with respect to the directions of the wave vector and the electric field of the incident plane wave. Thus, certain orientations of the sphere make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection.

Fig. 10. Backscattering efficiency as a function of $k_{0} a$ , when $ϵ_{r} = 4 (1 + i 0.1)$ , $μ_{r} = 1.1$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ , for six different combinations of ${\hat{e}}_{inc}$ and ${\hat{k}}_{inc}$ .

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E. Impedance Match

When $ϵ_{r} = μ_{r}$ , the chosen material can be said to be impedance-matched to the ambient medium (i.e., free space). If $A̳ = I̳$ additionally, the sphere would be made of an isotropic dielectric–magnetic material, and then $Q_{b} \equiv 0$ by virtue of a theorem of Weston [34]. Figure 11 shows plots of $Q_{b}$ versus $k_{0} a$ for six different combinations of ${\hat{k}}_{inc}$ and ${\hat{e}}_{inc}$ , when $ϵ_{r} = μ_{r} = 2 (1 + i 0.1)$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ . Whereas $Q_{b}$ is independent of ${\hat{e}}_{inc}$ in Fig. 11, it does depend on ${\hat{k}}_{inc}$ . Comparing with Fig. 10, we conclude that the independence of $Q_{b}$ from ${\hat{e}}_{inc}$ is due to the impedance match with the ambient free space.

Fig. 11. Same as Fig. 10, except that $ϵ_{r} = μ_{r} = 2 (1 + i 0.1)$ , $α_{x} = 1.1$ , and $α_{y} = 1.2$ .

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Even though the values of $k / k_{0}$ are quite similar in Figs. 10 and 11, a significant consequence of impedance match to the ambient free space is a tenfold decrease in the backscattering efficiency. That $Q_{b}$ did not actually drop to zero is because anisotropy makes the chosen sphere noninvariant under a rotation by $π / 2$ about an axis parallel to the direction of incidence, a condition required by Weston’s theorem [34] for null backscattering.

4. CONCLUDING REMARKS

Electromagnetic scattering by an orthorhombic dielectric–magnetic sphere whose permeability dyadic is a scalar multiple of its permittivity dyadic was formulated exactly in terms of a transition matrix, simplifying a recent development wherein closed-form vector spherical wavefunctions were formulated to capture anisotropy. Calculating the total scattering, extinction, differential scattering, backscattering, and forward-scattering efficiencies, we examined the evolving role of anisotropy as the size parameter increases.

Multiple lobes appear in the plots of the differential scattering efficiency in any scattering plane, as the size parameter increases. However, lobes may not appear in certain scattering planes when the sphere is electrically small in size. As the size parameter increases, the total scattering, extinction, and forward-scattering efficiencies exhibit a prominent maximum each, but the absorption efficiency generally increases with weak undulations. Certain orientations of the sphere with respect to the directions of propagation and the electric field of the incident plane wave make it highly susceptible to detection in a monostatic configuration, whereas other orientations make it much less vulnerable to detection. Finally, the backscattering efficiency can be reduced significantly, if the sphere is impedance-matched to the ambient free space.

ACKNOWLEDGMENTS

A. D. U. J. acknowledges the support of the Higher Education Commission of Pakistan for sponsoring a six-month visit to the Pennsylvania State University, where this work was done. A. L. is grateful to the Charles Godfrey Binder Endowment at the Pennsylvania State University for ongoing support of his research.

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Scattering of an electromagnetic plane wave by a homogeneous sphere made of an orthorhombic dielectric–magnetic material

Abstract

Corrections

1. INTRODUCTION

2. THEORY

A. Incident Electromagnetic Field

B. Scattered Electromagnetic Field

C. Internal Electromagnetic Field

D. Solution of Boundary–Value Problem

E. Uniaxial Dielectric–Magnetic Sphere

F. Isotropic Dielectric–Magnetic Sphere

G. Scattering Cross Sections and Efficiencies

H. Extinction and Absorption Cross Sections and Efficiencies

3. NUMERICAL RESULTS AND DISCUSSION

A. Rayleigh Scattering

B. Differential Scattering Efficiency

C. Total Scattering and Extinction Efficiencies

D. Backscattering and Forward-Scattering Efficiencies

E. Impedance Match

4. CONCLUDING REMARKS

ACKNOWLEDGMENTS

REFERENCES

Cited By

Figures (11)

Equations (67)

Journal of the Optical Society of America A