Waves in composites and metamaterials/Bloch waves and the quasistatic limit

From Wikiversity
Jump to navigation Jump to search

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.

Bloch Theorem[edit | edit source]

In the previous lecture we showed that Maxwell's equations at fixed frequency can be formulated in terms of the fields and as [1]

Equations (1) suggest that we should look for solutions and in the space of divergence-free fields such that

where the operator is given by

If the medium is such that the permittivity and the permeability are periodic, i.e.,

where is a lattice vector (see Figure 1) then the operator has the same periodicity as the medium.

Figure 1. Lattice vector in a periodic medium.

Also recall the translation operator defined as

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

[2] The translation operator is unitary, i.e.,

This means that the adjoint operator is equal to the inverse operator .

The translation operator also commutes, i.e.,

[3] Also, since and commute, the operators and must also commute. This implies that

Hence the eigenstates of and the eigenstates of lie in the same space. Therefore, any solution can be expressed in fields which are simultaneously eigenstates of all the , i.e., these eigenstates have the property

Since , we have

So it suffices to know when where the 's are the primitive vectors of the lattice, i.e.,

Let us assume that

for a suitable choice of .

Then for any lattice vector

we have


Define a vector

where the vectors are the reciprocal lattice vectors satisfying



Therefore, we have

Plugging this expression into (4), we get


Equation (5) is called the Bloch condition.

In summary, the solutions to the electromagnetic equations in a periodic medium can be expressed in Bloch waves where each Bloch wave is a time harmonic solution to the electromagnetic equations which in addition satisfies the Bloch condition for all lattice vectors and for some appropriate choice of .

Note that for any vector , the Bloch condition implies that

Therefore the quantity

is periodic.

Quasistatic Limit[edit | edit source]

Let us now consider the solution of Maxwell's equation in periodic media in the quasistatic limit. [4] Consider the periodic medium shown in Figure 2. The lattice spacing is .

Figure 2. Periodic medium with and spaces.


These are periodic functions, i.e.,

where are the primitive lattice vectors. We may also write these periodicity conditions as

Similarly, define

Then Maxwell's equations can be written as

Let us look for Bloch wave solutions of the form

where have the same periodicity as and , i.e.,

From the constitutive relations, we get

Recall that, for periodic media, Maxwell's equations may be expressed as

Here is an eigenvalue of . However, depends on via and . {\bf Bloch wave solutions do not exists unless takes one of a discrete set of values.}

Let these discrete values be

where the superscript labels the solution branches.

Let us see what the Bloch wave solutions reduce to as . Following standard multiple scale analysis, let us assume that the periodic complex fields have the expansions

Let us also assume that the dependence of on and has an expansion of the form

Plugging (8) and (7) into (6) gives


Then, for a vector field , using the chain rule we get

Using definitions (10) in (9) and collecting terms of order gives

These are the solutions in the quasistatic limit. Also, from the constitutive equations

Similarly, collecting terms of order 1 from the expanded Maxwell's equations (9) we get

Since are periodic, this implies that

where is the volume average over the unit cell. So a necessary condition that equations (12) have a solution is that

Note that the second pair of (14) implies the first pair.

Footnotes[edit | edit source]

  1. The following discussion is based on Ashcroft76 (p. 133-139).
  2. We can see that the two operators commute by working out the operations. Thus,
  3. We can see that the translation operator commutes by working out the operations. Thus,
  4. The following discussion is based on Milton02

References[edit | edit source]

  • N. W. Ashcroft and N. D. Mermin. Solid State Physics. Saunders, New York, 1976.
  • G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.