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.

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




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




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




The scattered field  ,   far from the sphere are given by






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.,

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 edit

  1. 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.