14.6.1 General Considerations

We first demonstrate that TM and TE waves are not transversal, i.e. they do possess longitudinal components. In the cases of TE and TM waves the expressions for E _? and B _? in Eqs. (14.20) and (14.21) also simplify. Setting

we obtain from Eqs. (14.20) and (14.21) for ? ² ? 0:

TM: B _z = 0 everywhere,

TE: E _z = 0 everywhere,

TM: Inserting in the second of Eqs. (14.67) the expression for ? _? E _z from the first of Eqs. (14.67), we obtain

TE: Inserting similarly in Eqs. (14.68) the expression of the equation for ? _? B _z into the other, we obtain

We conclude: E _z, B _z ? 0, because from Eqs. (14.67) for TM

and from Eqs. (14.68) for TE

Thus the components E _z (TM), B _z (TE) are not both zero; i.e. the TM and TE waves are not transversal waves, they possess longitudinal components E _z and B _z.

We saw previously (see Eqs. (14.30)), that E( x, y), B( x, y) satisfy the equations

(i.e. for each of the three components of E and B). We have therefore in particular for the z-components

TM:

TE:

Equations (14.72) and (14.73) together with their boundary conditions define an eigenvalue problem for the determination...