# 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 edit

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 and a permittivity . The medium outside the sphere has a permittivity and a permeability . The electric field is oriented parallel to the axis and the axis points out of the plane of the paper.

Also recall that

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

where is the unit vector in the direction.

The most widely used ** superpotentials** are the electric and magnetic
** Hertz vector potentials** and (also known as
polarization potentials).

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

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

Let the time harmonic fields be given by

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

Recall that the Debye potentials satisfy the homogeneous wave equations

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

where the subscript indicates an incident field and the subscript indicates a scattered field.

Inside the sphere, the potentials are denoted by

where the subscript indicates a refracted + reflected field.

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

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

have the expression

where

Here are the Legendre polynomials which solve

and are the Bessel functions which solve

The functions are chosen such that

is regular at the origin.

The scattered fields have a similar expansion

where

and 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

To find the constants we need to apply continuity conditions across the boundary of the sphere.

To ensure that (tangential components of and ) are continuous across the surface of the sphere at , it is sufficient that

are continuous.

Applying these conditions, we get

where

The scattered field , far from the sphere are given by

where

where

Note that the tangential components of fall off as while the radial component falls off as .

## Periodic Media and Bloch's Theorem edit

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 and the permeability are periodic. Recall that, at fixed frequency, the Maxwell equations are

Also recall the constitutive relations

Plugging (4) into (3), we get

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

where the operator is given by

Since and are periodic, the operator has the same periodicity as the medium.

Clearly, equation (6) represents an eigenvalue problem where is an eigenvalue of and is the corresponding eigenvector.

Let define a translation operator which, when acting upon a pair of the fields shifts the argument by a vector , where is taken to be a lattice vector (see Figure~2), i.e.,

Periodicity of the medium implies that commutes with , i.e.,

Note that , like , maps divergence-free fields to divergence-free fields.

Now, consider the space of field pairs which are divergence-free and which are in the null space of , i.e., they satisfy

This subspace is closed under the action of which is unitary, i.e.,

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

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

The Bloch condition will be discussed in the next lecture.

## Footnotes edit

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

## References edit

- 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.