Propagation of waves in a medium varying transverse to the direction of propagation

For an inhomogeneous dielectric in which the dielectric constant ε depends on
x
only, it has been shown that it is possible to consider modes propagating in the
z
direction having
H_{x} = 0
or
E_{x} = 0
such that other field components can be expressed in terms of E
x
and H
x
. The usual TE and TM modes are obtained in the special case of no
y
dependence of the fields. Explicit field patterns and the propagation constants have been obtained for a square-law medium.