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 edit

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

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

  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

  • 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.