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

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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.

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

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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.

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

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

References

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  • 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.