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

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

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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

 

or,

 

Define a vector

 

where the vectors   are the reciprocal lattice vectors satisfying

 

Then,

 

or,

 

Therefore, we have

 

Plugging this expression into (4), we get

 

or,

 

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

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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.

Define

 

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

 

Define

 

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

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  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

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  • 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.