The diagram on the right depicts scattering of a plane wave by a uniform flat slab. The frequency is ω and the wave vector in the x-direction is k. The (free-space) speed of light in the air is c=c

The field u satisfies the Helmholtz equation (∂

The region between the cones ω=c

Pairs (k,ω) below the cone ω=c

Let us now impose an artificial period of 2π in the x-direction and write each Helmholtz field u(x,z) = v(z)e

u(x,z) = w(x,z)e

Now, for each k in [-1/2,1/2), there corresponds a sequence of guided modes, whose frequencies are indicated by the red dispersion relations that have been "folded over" into this zone. Those that fall on the blue zone corresponding to scattering states can be thought of as frequencies of guided modes that are "embedded" in the continuous spectrum of extended states.

For a fixed value of k, the frequencies of these guided modes are eigenvalues embedded in the continuous spectrum of the Helmholtz operator restricted to pseudo-periodic fields in the infinite strip (0,2π)x(-∞,∞) with pseudo-periodic factor e

If we now impose a genuine periodicity upon the slab, the solutions are no longer separable, and the periodic factor w(x,z) in the Bloch fields attains all Fourier harmonics,

w(x,z) = Σ

in other words, the Bloch wave number k is only determined up to an additive integer, so we take it to lie in the first Brillouin zone. To the left and right of the slab, the functions v

v

(with different coefficients to the left and right) in which (ω/c)

Since the Bloch fields typically contain all Fourier harmonics, those with (k,ω) in the blue region above the cone ω=c

If the slab is symmetric about a horizontal line, then, for k=0, solutions can be decomposed into symmetric and antisymmetric parts. If, in addition, there is only one propagating harmonic (meaning that (k,ω) is in the blue diamond shape bounded by the cutoff cone and the dotted lines above it), then this propagating harmonic, which is symmetric, vanishes for antisymmetric solutions. Thus, if anitsymmetric solutions exist, they will be guided modes. In fact, one can prove that they do exist. They correspond to points on the complex dispersion relation where ω happens to be real; their frequencies are indicated by solid red dots on the graph. Since a perturbation of k results in a small imaginary part of ω, the guide mode is seen to disappear, and we call it

This is an instance of the disppearance of an eigenvalue embedded in the continuous spectrum of the Helmholtz operator with pseudo-periodic boundary conditions in the strip (0,2π)x(-∞,∞), where the parameter of perturbation is the pseudo-periodic factor e

Numerical calculations of scattering by the yellow structure consisting of an infinite row of rods (seen in cross-section) demonstrate that a nonrobust guided mode produces anomalous transmission behavior. The structure admits a nonrobust guided mode at k = 0 and ω = ω

In the lower right, one period of the intensity of the guided mode is shown, and, below that, a field scattered by an incident wave from the left at a small nonzero value of k and a frequency near ω

The scattering problem can be expressed in terms of values and normal derivative of the source and total (or scattered) fields on the air-slab interface in one period cell (in this case, over the boundary of one circlular cross section of a rod). These boundary data are called the

This is what the boundary-integral equations look like. The exterior (radiating) and interior Green functions are denoted by G