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Beginning with an eigenmode H that minimizes U f , we rewrite the numerator of (20) using (11), (8), and (9), and we rewrite the denominator using (17) and (8). The result is: U f (H) = = (∇ × E, ∇ × E) (E, ε(r)E) d3 r |∇ × E(r)|2 . 13 Although we derived (23) by starting with an eigenmode H and rewriting the (minimized) energy functional in terms of E, it can be shown (using the E eigenproblem of the next section) that (23) is also a valid variational theorem: the lowest-frequency eigenmode is given by the E field that minimizes (23), subject to ∇ · εE = 0.

2002). In these special cases, the Maxwell equations can be written in a Schrödinger-like form. ] On the other hand, if ε does not depend on a particular coordinate, then that particular dimension of the problem is always separable, as we will see in the section Continuous Translational Symmetry of chapter 3. tex 24 CHAPTER 2 Further Reading A particularly lucid undergraduate text on electromagnetism is Griffiths (1989). A more advanced and complete treatment of the macroscopic Maxwell equations, including a derivation from their microscopic counterparts, is contained in Jackson (1998).

In the limit of larger and larger |k |, one obtains more and more guided bands, and eventually one approaches the ray-optics limit of totally internally reflected rays with a continuum of angles θ > θc . Discrete Translational Symmetry Photonic crystals, like traditional crystals of atoms or molecules, do not have continuous translational symmetry. Instead, they have discrete translational symmetry. That is, they are not invariant under translations of any distance, but rather, only distances that are a multiple of some fixed step length.

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