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 | edit source]
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.
Figure 1. Scattering of radiation from a sphere.
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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
![{\displaystyle {\cfrac {d}{dx}}\left[(1-x^{2})~{\cfrac {dP_{l}^{1}}{dx}}\right]+\left[l(l+1)-{\cfrac {1}{1-x^{2}}}\right]~P_{l}^{1}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/943e4bfd4e0c873ca3286e2429b223bf089402bc)
and
are the Bessel functions which solve
![{\displaystyle {\cfrac {d^{2}J_{\nu }}{d\rho ^{2}}}+{\cfrac {1}{\rho }}~{\cfrac {dJ_{\nu }}{d\rho }}+\left[1-{\cfrac {\nu ^{2}}{\rho ^{2}}}\right]~J_{\nu }=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5cb1218980ddca5091433146e56a13f8698529)
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
![{\displaystyle {\begin{aligned}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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8d2763a911e24b43ada04f3f154896a24842da)
where

Note that the tangential components of
fall off as
while
the radial component falls off as
.
Periodic Media and Bloch's Theorem[edit | edit source]
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.,

Figure 2. Lattice vector in a periodic medium.
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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.
- ↑ This discussion is based on Ishimaru78. Please
consult that text and the reference cited therein for further details.
- 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.