Define in the charge-free interior of a wave guide the electric vector potential G by E = ? G. Similarly we have B = ? A. Find the wave equations of the vector potentials and sketch qualitatively the moduli of TE and TM modes of the two-dimensional fields.

Solution: In the charge-free space ( ? = 0, j = 0) we can define an electric vector potential G by E = ? G, so that ? E = 0. Maxwell's curl equations are here

From these we deduce

Moreover

and with " curl curl = grad div ? div grad" we obtain

and

We choose the gauge fixing conditions

Then

and a similar equation holds for A. TE modes are defined by E _z = 0. With G = (0,0, ?), it follows that

It may be noted that the potential G has in this case only a z-component (compare with B in Sec. 8.4.). As an example we take, with m, n = 0,1,..., as in Eq. (14.87),

Lines of constant electric field strength are given by E ² = const., i.e.

The curves given by this equation are ellipses in the ( x, y)-plane, as one can see by selecting specific modes. One...