Waves in composites and metamaterials/Mie theory and Bloch theorem

The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.

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 $\mu$ and a permittivity $ϵ$. The medium outside the sphere has a permittivity ${ϵ}_0$ and a permeability ${\mu }_0$. The electric field is oriented parallel to the $x_1$ axis and the $x_2$ axis points out of the plane of the paper.

File:Lec15Fig1.jpg Figure 1. Scattering of radiation from a sphere.

Also recall that

$\mu ={\mu }_0;ϵ={ϵ}_r{ϵ}_0=n^2{ϵ}_0$

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

${𝐄}_i=e^{ik{x}_3}{𝐞}_1$

where ${𝐞}_1$ is the unit vector in the $x_1$ direction.

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

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

$\text{(1)}\begin{array}{cc}𝐄& =\mathrm{\nabla }\times \mathrm{\nabla }\times {\mathrm{\Pi }}_e-\mu \mathrm{\nabla }\times \frac{\partial {\mathrm{\Pi }}_m}{\partial t}\\ 𝐇& =\mathrm{\nabla }\times \mathrm{\nabla }\times {\mathrm{\Pi }}_m+ϵ\mathrm{\nabla }\times \frac{\partial {\mathrm{\Pi }}_e}{\partial t}.\end{array}$

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

${\mathrm{\Pi }}_e=u𝐫;{\mathrm{\Pi }}_m=v𝐫\text{where}𝐫\equiv (x_1,x_2,x_3).$

Let the time harmonic fields be given by

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

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

$\begin{array}{cc}𝐄& =\mathrm{\nabla }\times \mathrm{\nabla }\times (u𝐫)+i\omega \mu \mathrm{\nabla }\times (v𝐫)\\ 𝐇& =\mathrm{\nabla }\times \mathrm{\nabla }\times (v𝐫)-i\omega ϵ\mathrm{\nabla }\times (u𝐫).\end{array}$

Recall that the Debye potentials satisfy the homogeneous wave equations

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

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:

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

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.,

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

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

${𝐄}_i=e^{ik{x}_3}{𝐞}_1$

have the expression

$\begin{array}{cc}r{u}_i& =\frac{1}{k^2}\sum _{l=1}^{\mathrm{\infty }}\frac{i^{l-1}(2l+1)}{l(l+1)}{\psi }_l(kr)P_l^1(\cos \theta )\cos \varphi \\ r{v}_i& =\frac{1}{\eta k^2}\sum _{l=1}^{\mathrm{\infty }}\frac{i^{l-1}(2l+1)}{l(l+1)}{\psi }_l(kr)P_l^1(\cos \theta )\sin \varphi \end{array}$

where

${\psi }_l(\rho )=\sqrt{\frac{\pi r}{2}}{J}_{l+1/2}(\rho );\eta =\sqrt{\frac{{\mu }_0}{{ϵ}_0}}.$

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

$\frac{d}{dx}[(1-x^2)\frac{d{P}_l^1}{dx}]+[l(l+1)-\frac{1}{1-x^2}]P_l^1=0$

and $J_{\nu }(\rho )$ are the Bessel functions which solve

$\frac{d^2{J}_{\nu }}{d{\rho }^2}+\frac{1}{\rho }\frac{d{J}_{\nu }}{d\rho }+[1-\frac{{\nu }^2}{{\rho }^2}]J_{\nu }=0.$

The functions $\psi (\rho )$ are chosen such that

${\psi }_{\nu }(\rho )=\sqrt{\frac{\pi r}{2}}{J}_{\nu }(\rho )$

is regular at the origin.

The scattered fields have a similar expansion

$\begin{array}{cc}r{u}_s& =\frac{-1}{k^2}\sum _{l=1}^{\mathrm{\infty }}\frac{i^{l-1}(2l+1)}{l(l+1)}{a}_l{\zeta }_l(kr)P_l^1(\cos \theta )\cos \varphi \\ r{v}_s& =\frac{-1}{\eta k^2}\sum _{l=1}^{\mathrm{\infty }}\frac{i^{l-1}(2l+1)}{l(l+1)}{b}_l{\zeta }_l(kr)P_l^1(\cos \theta )\sin \varphi \end{array}$

where

${\zeta }_l(\rho )=\sqrt{\frac{\pi r}{2}}{H}_{l+1/2}^{(1)}(\rho )$

and $H_{\nu }^{(1)}(\rho )$ 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

$\begin{array}{cc}r{u}_r& =\frac{1}{k^2{n}^2}\sum _{l=1}^{\mathrm{\infty }}\frac{i^{l-1}(2l+1)}{l(l+1)}{c}_l{\psi }_l(knr)P_l^1(\cos \theta )\cos \varphi \\ r{v}_r& =\frac{1}{\eta k^2{n}^2}\sum _{l=1}^{\mathrm{\infty }}\frac{i^{l-1}(2l+1)}{l(l+1)}{b}_l{\zeta }_l(knr)P_l^1(\cos \theta )\sin \varphi \end{array}$

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 $E_{\theta },E_{\varphi },H_{\theta },H_{\varphi }$ (tangential components of $𝐄$ and $𝐇$) are continuous across the surface of the sphere at $r=a$, it is sufficient that

$n^{2}u,\frac{\partial }{\partial r}(ru);\frac{\partial }{\partial r}(rv)$

are continuous.

Applying these conditions, we get

$\begin{array}{cc}{a}_l& =\frac{{\psi }_l(\alpha )\psi {\text{'}}_l(\beta )-n{\psi }_l(\beta )\psi {\text{'}}_l(\alpha )}{{\zeta }_l(\alpha )\psi {\text{'}}_l(\beta )-n{\psi }_l(\beta )\zeta {\text{'}}_l(\alpha )}\\ b_l& =\frac{n{\psi }_l(\alpha )\psi {\text{'}}_l(\beta )-{\psi }_l(\beta )\psi {\text{'}}_l(\alpha )}{n{\zeta }_l(\alpha )\psi {\text{'}}_l(\beta )-{\psi }_l(\beta )\zeta {\text{'}}_l(\alpha )}\end{array}$

where

$\alpha =ka;\beta =kna.$

The scattered field $E_{\theta }$, $E_{\varphi }$ far from the sphere are given by

$\begin{array}{cc}{E}_{\theta }& =\frac{i{e}^{ikr}}{kr}{S}_1(\theta )\cos \varphi \\ E_{\varphi }& =-\frac{i{e}^{ikr}}{kr}{S}_1(\theta )\sin \varphi \end{array}$

where

$\begin{array}{cc}{S}_1(\theta )& =\sum _{l=1}^{\mathrm{\infty }}\frac{2l+1}{l(l+1)}[a_l{\pi }_l(\cos \theta )+b_l{\tau }_l(\cos \theta )]\\ S_2(\theta )& =\sum _{l=1}^{\mathrm{\infty }}\frac{2l+1}{l(l+1)}[a_l{\tau }_l(\cos \theta )+b_l{\pi }_l(\cos \theta )]\end{array}$

where

${\pi }_l(\cos \theta )=\frac{P_l^1(\cos \theta )}{\sin \theta };{\tau }_l(\cos \theta )=\frac{d}{d\theta }{P}_l^1(\cos \theta ).$

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

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 $ϵ(𝐱)$ and the permeability $\mu (𝐱)$ are periodic. Recall that, at fixed frequency, the Maxwell equations are

$\text{(3)}\mathrm{\nabla }\cdot (ϵ𝐄)=0;\mathrm{\nabla }\cdot (\mu 𝐇)=0;\mathrm{\nabla }\times 𝐄+i\omega \mu 𝐇=0;\mathrm{\nabla }\times 𝐇-i\omega ϵ𝐄=0.$

Also recall the constitutive relations

$\text{(4)}𝐃=ϵ𝐄;𝐁=\mu 𝐇.$

Plugging (4) into (3), we get

$\text{(5)}\mathrm{\nabla }\cdot 𝐃=0;\mathrm{\nabla }\cdot 𝐁=0;i\mathrm{\nabla }\times \left(\frac{𝐃}{ϵ}\right)=\omega 𝐁;-i\mathrm{\nabla }\times \left(\frac{𝐁}{\mu }\right)=\omega 𝐃.$

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

$\text{(6)}ℒ\left[\begin{array}{c}𝐃\\ 𝐁\end{array}\right]=\omega \left[\begin{array}{c}𝐃\\ 𝐁\end{array}\right]$

where the operator $ℒ$ is given by

$\text{(7)}ℒ:=\left[\begin{array}{cc}0& -i\mathrm{\nabla }\times {\mu }^{-1}\\ i\mathrm{\nabla }\times {ϵ}^{-1}& 0\end{array}\right].$

Since $ϵ$ and $\mu$ are periodic, the operator $ℒ$ has the same periodicity as the medium.

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

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

${𝒯}_R\left[\begin{array}{c}𝐃(𝐱)\\ 𝐁(𝐱)\end{array}\right]=\left[\begin{array}{c}𝐃(𝐱+𝑹v)\\ 𝐁(𝐱+𝑹v)\end{array}\right].$

File:Lec16Fig1.jpg Figure 2. Lattice vector in a periodic medium.

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

${𝒯}_Rℒ=ℒ{𝒯}_R.$

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

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

$(ℒ-\omega 1)\left[\begin{array}{c}𝐃\\ 𝐁\end{array}\right]=0.$

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

${𝒯}_R{𝒯}_R^T={𝒯}_R{𝒯}_{-R}=1.$

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

${𝒯}_R{𝒯}_{R^{\prime }}={𝒯}_{R^{\prime }}{𝒯}_R={𝒯}_{R+R^{\prime }}.$

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

${𝒯}_R\left[\begin{array}{c}𝐃(𝐱)\\ 𝐁(𝐱)\end{array}\right]=c(𝑹v)\left[\begin{array}{c}𝐃(𝐱)\\ 𝐁(𝐱)\end{array}\right].$

The Bloch condition will be discussed in the next lecture.

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.