Waves in composites and metamaterials/Mie theory and Bloch theorem

From Wikiversity

[edit] Scattering of radiation from a sphere

Recall the sphere shown in Figure 1. We set up our coordinate system such that the origin is at the center of the sphere. The sphere has a magnetic permeability of $μ$ and a permittivity $ε$ . The medium outside the sphere has a permittivity $ε 0$ and a permeability $μ 0$ . The electric field is oriented parallel to the $x 1$ axis and the $x 2$ axis points out of the plane of the paper.

Figure 1. Scattering of radiation from a sphere.

Also recall that

$\mu = \mu_0 ~;~~ \epsilon = \epsilon_r ~\epsilon_0 = n^2 ~\epsilon_0$

where $ε r$ is the relative permittivity of the material inside the sphere and that the incident plane wave is given by

$\mathbf{E}_i = e^{ikx_3}~\mathbf{e}_1$

where $\mathbf{e}_1$ is the unit vector in the $x 1$ direction.

The most widely used superpotentials are the electric and magnetic Hertz vector potentials $\boldsymbol{\Pi}_e$ and $\boldsymbol{\Pi}_m$ (also known as polarization potentials).

In the last lecture we discussed the Hertz vector potentials and that the $\mathbf{E}$ and $\mathbf{H}$ fields can be expressed as

$\text{(1)} \qquad \begin{align} \mathbf{E} & = \boldsymbol{\nabla} \times \boldsymbol{\nabla}\times{\boldsymbol{\Pi}_e} - \mu~\boldsymbol{\nabla} \times \frac{\partial \boldsymbol{\Pi}_m}{\partial t} \\ \mathbf{H} & = \boldsymbol{\nabla} \times \boldsymbol{\nabla}\times{\boldsymbol{\Pi}_m} + \epsilon~\boldsymbol{\nabla} \times \frac{\partial \boldsymbol{\Pi}_e}{\partial t} ~. \end{align}$

For spherically symmetric time harmonic problems, such as we find in the problem of scattering of EM waves by a sphere, we stated that an important class of Hertz vector potentials are the Debye potentials of the form

$\boldsymbol{\Pi}_e = u~\mathbf{r} ~;~~ \boldsymbol{\Pi}_m = v~\mathbf{r} \qquad \text{where} \quad \mathbf{r} \equiv (x_1, x_2, x_3) ~.$

Let the time harmonic fields be given by

$\mathbf{E} = \widehat{\mathbf{E}}~e^{-i\omega t} ~;~~ \mathbf{H} = \widehat{\mathbf{H}}~e^{-i\omega t} ~;~~ u = \hat{u}~e^{-i\omega t} ~;~~ v = \hat{v}~e^{-i\omega t} ~.$

Plugging these into (1) and dropping the hats gives the Maxwell equations at fixed frequency:

$\begin{align} \mathbf{E} & = \boldsymbol{\nabla} \times \boldsymbol{\nabla}\times{(u~\mathbf{r})} + i\omega\mu~\boldsymbol{\nabla} \times (v~\mathbf{r}) \\ \mathbf{H} & = \boldsymbol{\nabla} \times \boldsymbol{\nabla}\times{(v~\mathbf{r})} - i\omega\epsilon~\boldsymbol{\nabla} \times (u~\mathbf{r}) ~. \end{align}$

Recall that the Debye potentials satisfy the homogeneous wave equations

$\text{(2)} \qquad (\nabla^2 + k^2) u = 0 \qquad \text{and} \qquad (\nabla^2 + k^2) v = 0 ~.$

To deal with the problem of scattering by a sphere, let us split the potentials $u$ and $v$ (outside the sphere) into incident and scattered fields:^[1]

$u = u_i + u_s ~;~~ v = v_i + v_s$

where the subscript $i$ indicates an incident field and the subscript $s$ indicates a scattered field.

Inside the sphere, the potentials are denoted by

$u = u_r ~;~~ v = v_r$

where the subscript $r$ indicates a refracted + reflected field.

Let us require that these potentials satisfy wave equations of the form given in (2), i.e.,

$\begin{align} (\nabla^2 + k^2) u_i &= 0 & \qquad \text{and} \qquad & (\nabla^2 + k^2) v_i = 0\\ (\nabla^2 + k^2) u_s &= 0 & \qquad \text{and} \qquad & (\nabla^2 + k^2) v_s = 0\\ (\nabla^2 + k^2 n^2) u_r &= 0 & \qquad \text{and} \qquad & (\nabla^2 + k^2 n^2) v_r = 0 ~. \end{align}$

Since each of these satisfies a scalar wave equation, we can express each potential in terms of spherical harmonics.

In particular, the Debye potentials associated with the incident field

$\mathbf{E}_i = e^{ikx_3}~\mathbf{e}_1$

have the expression

${ \begin{align} r u_i & = \cfrac{1}{k^2} \sum_{l=1}^{\infty} \cfrac{i^{l-1} (2l+1)}{l(l+1)}~\psi_l(k r)~P_l^1(\cos\theta)~\cos\phi \\ r v_i & = \cfrac{1}{\eta k^2} \sum_{l=1}^{\infty} \cfrac{i^{l-1} (2l+1)}{l(l+1)}~\psi_l(k r)~P_l^1(\cos\theta)~\sin\phi \end{align} }$

where

$\psi_l(\rho) = \sqrt{\cfrac{\pi~r}{2}}~J_{l+1/2}(\rho) ~;~~ \eta = \sqrt{\cfrac{\mu_0}{\epsilon_0}} ~.$

Here $P_l^1(x)$ are the Legendre polynomials which solve

$\cfrac{d }{d x}\left[ (1- x^2)~\cfrac{d P^1_l}{d x}\right] + \left[l(l+1) - \cfrac{1}{1 - x^2}\right]~P^1_l = 0$

and $J ν (ρ)$ are the Bessel functions which solve

$\cfrac{d^2 J_\nu}{d \rho^2} + \cfrac{1}{\rho}~\cfrac{d J_\nu}{d \rho} + \left[1 - \cfrac{\nu^2}{\rho^2}\right]~J_\nu = 0 ~.$

The functions $ψ(ρ)$ are chosen such that

$\psi_\nu(\rho) = \sqrt{\cfrac{\pi~r}{2}}~J_\nu(\rho)$

is regular at the origin.

The scattered fields have a similar expansion

${ \begin{align} r u_s & = \cfrac{-1}{k^2} \sum_{l=1}^{\infty} \cfrac{i^{l-1} (2l+1)}{l(l+1)}~a_l~\zeta_l(k r)~P_l^1(\cos\theta)~ \cos\phi \\ r v_s & = \cfrac{-1}{\eta k^2} \sum_{l=1}^{\infty} \cfrac{i^{l-1} (2l+1)}{l(l+1)}~b_l~\zeta_l(k r)~P_l^1(\cos\theta)~ \sin\phi \end{align} }$

where

$\zeta_l(\rho) = \sqrt{\cfrac{\pi~r}{2}}~H^{(1)}_{l+1/2}(\rho)$

and $H^{(1)}_\nu(\rho)$ is one of the Hankel functions solving the same equation as the Bessel function but decaying at infinity.

Inside the sphere, the expansion of the fields takes the form

${ \begin{align} r u_r & = \cfrac{1}{k^2 n^2} \sum_{l=1}^{\infty} \cfrac{i^{l-1} (2l+1)}{l(l+1)}~c_l~\psi_l(k n r)~P_l^1(\cos\theta)~ \cos\phi \\ r v_r & = \cfrac{1}{\eta k^2 n^2} \sum_{l=1}^{\infty} \cfrac{i^{l-1} (2l+1)}{l(l+1)}~b_l~\zeta_l(k n r)~P_l^1(\cos\theta)~ \sin\phi \end{align} }$

To find the constants $a l, b l, c l, d l$ we need to apply continuity conditions across the boundary of the sphere.

To ensure that $E θ, E φ, H θ, H φ$ (tangential components of $\mathbf{E}$ and $\mathbf{H}$ ) are continuous across the surface of the sphere at $r = a$ , it is sufficient that

$n^2 u ~,~~ \frac{\partial }{\partial r}(r u) ~;~~ \frac{\partial }{\partial r}(r v)$

are continuous.

Applying these conditions, we get

${ \begin{align} a_l & = \cfrac{\psi_l(\alpha)~\psi'_l(\beta) - n~\psi_l(\beta)~\psi'_l(\alpha)} {\zeta_l(\alpha)~\psi'_l(\beta) - n~\psi_l(\beta)~\zeta'_l(\alpha)} \\ b_l & = \cfrac{n~\psi_l(\alpha)~\psi'_l(\beta) - \psi_l(\beta)~\psi'_l(\alpha)} {n~\zeta_l(\alpha)~\psi'_l(\beta) - \psi_l(\beta)~\zeta'_l(\alpha)} \end{align} }$

where

$\alpha = k~a ~;~~ \beta = k~n~a ~.$

The scattered field $E θ$ , $E φ$ far from the sphere are given by

$\begin{align} E_\theta & = \cfrac{i~e^{i k r}}{k r}~S_1(\theta)~\cos\phi \\ E_\phi & = -\cfrac{i~e^{i k r}}{k r}~S_1(\theta)~\sin\phi \end{align}$

where

$\begin{align} S_1(\theta) & = \sum_{l=1}^\infty \cfrac{2l+1}{l(l+1)} \left[a_l~\pi_l(\cos\theta) + b_l~\tau_l(\cos\theta)\right] \\ S_2(\theta) & = \sum_{l=1}^\infty \cfrac{2l+1}{l(l+1)} \left[a_l~\tau_l(\cos\theta) + b_l~\pi_l(\cos\theta)\right] \end{align}$

where

$\pi_l(\cos\theta) = \cfrac{P^1_l(\cos\theta)}{\sin\theta} ~;~~ \tau_l(\cos\theta) = \cfrac{d }{d \theta}~P^1_l(\cos\theta) ~.$

Note that the tangential components of $\mathbf{E}$ fall off as $1 / r$ while the radial component falls off as $1 / r 2$ .

[edit] Periodic Media and Bloch's Theorem

The following discussion is based on Ashcroft76 (p. 133-139). For a more detailed mathematical treatment see Kuchment93.

Suppose that the medium is such that the permittivity $\epsilon(\mathbf{x})$ and the permeability $\mu(\mathbf{x})$ are periodic. Recall that, at fixed frequency, the Maxwell equations are

$\text{(3)} \qquad \boldsymbol{\nabla} \cdot (\epsilon~\mathbf{E}) = 0 ~;~~ \boldsymbol{\nabla} \cdot (\mu~\mathbf{H}) = 0 ~;~~ \boldsymbol{\nabla} \times \mathbf{E} + i\omega\mu~\mathbf{H} = 0 ~;~~ \boldsymbol{\nabla} \times \mathbf{H} - i\omega\epsilon~\mathbf{E} = 0 ~.$

Also recall the constitutive relations

$\text{(4)} \qquad \mathbf{D} = \epsilon~\mathbf{E} ~;~~ \mathbf{B} = \mu~\mathbf{H} ~.$

Plugging (4) into (3), we get

$\text{(5)} \qquad \boldsymbol{\nabla} \cdot \mathbf{D} = 0 ~;~~ \boldsymbol{\nabla} \cdot \mathbf{B} = 0 ~;~~ i~\boldsymbol{\nabla} \times \left(\cfrac{\mathbf{D}}{\epsilon}\right) = \omega~\mathbf{B} ~;~~ -i~\boldsymbol{\nabla} \times \left(\cfrac{\mathbf{B}}{\mu}\right) = \omega~\mathbf{D} ~.$

Equations (5) suggest that we should look for solutions $\mathbf{D}$ and $\mathbf{B}$ in the space of divergence-free fields such that

$\text{(6)} \qquad \mathcal{L}\begin{bmatrix} \mathbf{D} \\ \mathbf{B} \end{bmatrix} = \omega~\begin{bmatrix} \mathbf{D} \\ \mathbf{B} \end{bmatrix}$

where the operator $\mathcal{L}$ is given by

$\text{(7)} \qquad \mathcal{L} := \begin{bmatrix} 0 & -i~\boldsymbol{\nabla} \times \mu^{-1} \\ i~\boldsymbol{\nabla} \times \epsilon^{-1} & 0 \end{bmatrix} ~.$

Since $ε$ and $μ$ are periodic, the operator $\mathcal{L}$ has the same periodicity as the medium.

Clearly, equation (6) represents an eigenvalue problem where $ω$ is an eigenvalue of $\mathcal{L}$ and $[\mathbf{D},\mathbf{B}]$ is the corresponding eigenvector.

Let $\mathcal{T}_R$ define a translation operator which, when acting upon a pair of the fields $[\mathbf{D}, \mathbf{B}]$ shifts the argument by a vector $\boldsymbol{R}v$ , where $\boldsymbol{R}v$ is taken to be a lattice vector (see Figure~2), i.e.,

$\mathcal{T}_R\begin{bmatrix} \mathbf{D}(\mathbf{x}) \\ \mathbf{B}(\mathbf{x}) \end{bmatrix} = \begin{bmatrix} \mathbf{D}(\mathbf{x}+\boldsymbol{R}v) \\ \mathbf{B}(\mathbf{x}+\boldsymbol{R}v) \end{bmatrix} ~.$

Figure 2. Lattice vector in a periodic medium.

Periodicity of the medium implies that $\mathcal{T}_R$ commutes with $\mathcal{L}$ , i.e.,

$\mathcal{T}_R~\mathcal{L} = \mathcal{L}~\mathcal{T}_R ~.$

Note that $\mathcal{T}_R$ , like $\mathcal{L}$ , maps divergence-free fields to divergence-free fields.

Now, consider the space of field pairs $\mathbf{D}, \mathbf{B}$ which are divergence-free and which are in the null space of $\mathcal{L} - \omega~\boldsymbol{1}$ , i.e., they satisfy

$(\mathcal{L} - \omega~\boldsymbol{1})\begin{bmatrix}\mathbf{D} \\ \mathbf{B} \end{bmatrix} = \boldsymbol{0} ~.$

This subspace is closed under the action of $\mathcal{T}_R$ which is unitary, i.e.,

$\mathcal{T}_R~\mathcal{T}_R^T = \mathcal{T}_R~\mathcal{T}_{-R} = \boldsymbol{1} ~.$

Also, the translation operator commutes, i.e.,

$\mathcal{T}_R~\mathcal{T}_{R'} = \mathcal{T}_{R'}~\mathcal{T}_R = \mathcal{T}_{R+R'} ~.$

Therefore, any solution can be expressed in fields which are simultaneously eigenstates of all the $\mathcal{T}_R$ . These eigenstates have the property

$\mathcal{T}_R\begin{bmatrix} \mathbf{D}(\mathbf{x}) \\ \mathbf{B}(\mathbf{x}) \end{bmatrix} = c(\boldsymbol{R}v)~\begin{bmatrix} \mathbf{D}(\mathbf{x}) \\ \mathbf{B}(\mathbf{x}) \end{bmatrix} ~.$

The Bloch condition will be discussed in the next lecture.

[edit] Footnotes

↑ This discussion is based on Ishimaru78. Please consult that text and the reference cited therein for further details.

[edit] References

N. W. Ashcroft and N. D. Mermin. Solid State Physics. Saunders, New York, 1976.
A. Ishimaru. Wave Propagation and Scattering in Random Media. Academic Press, New York, 1978.
P. Kuchment. Floquet Theory For Partial Differential Equations. Birkhauser Verlag, Basel, 1993.

[0] This discussion is based on Ishimaru78. Please consult that text and the reference cited therein for further details.

[1]

Waves in composites and metamaterials/Mie theory and Bloch theorem

From Wikiversity

Contents

[edit] Scattering of radiation from a sphere

[edit] Periodic Media and Bloch's Theorem

[edit] Footnotes

[edit] References

Views

Personal tools

Search

Navigation

Community

Toolbox