2.1

Excitation of evanescent waves by a subwavelength slit

Consider an infinitely long slit of width a confined to the plane ( , ?) and directed along the axis (see Fig. P2.1.1).

Figure P2.1.1: Infinite slit of width a.

The boundary conditions in the plane of a slit z = 0 are as follows. Inside the slit, the magnetic field H( y, z = 0) is parallel to the slit and uniform H ₀ exp( ?i ?t), while outside the slit, the magnetic field is zero.

1. Expanding the solution of Maxwell s equations in the half space z ? 0 in terms of the outgoing and evanescent plane waves, find a complete solution of the problem satisfying the above-mentioned boundary conditions.

2. Assuming that the slit is subwavelength, a ? ( ? = 2 ?c/ ?), find the dependence of the evanescent field H _evanescent( y = 0 , z) for z ~ ?.

2.2

Zero derivative of the dispersion relation at the edge of a Brillouin zone

Consider a system exhibiting discrete translational symmetry along the axis with spatial period a. In this case, as demonstrated in Section 2.4.5 (see the subsection on 1D discrete translational symmetry) the general form of a solution is given by:

Moreover, time-reversal symmetry, ?( k _z) = ?( ? k _z), limits the choice of the distinct wave vectors...