Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 88 / 3 / c / Solution

With constant permeability, the electric equation reduces to $({\partial }_x^2+{\partial }_z^2+k^2(z))E_y=0$. Insert the oblique-wave dependence $E_y=e_y(z)e^{-i{k}_0\sin {\theta }_{0}x}$ and set $s=z-z_0$. The inverse-square refractive-index profile givesSubstituting $e_y=\sqrt{|s|}w(\beta |s|)$ gives the Bessel equation of order $\nu$. On either side of the profile singularity, when ${\beta }^2>0$,Equivalently use Hankel functions for an incoming/outgoing wave basis. If ${\beta }^2<0$, write $\gamma =\sqrt{-{\beta }^2}$ and replace $J_{\nu },Y_{\nu }$ by modified Bessel functions $I_{\nu },K_{\nu }$ with argument $\gamma |s|$. For $\beta =0$, the solutions are $|s{|}^{1/2+\nu }$ and $|s{|}^{1/2-\nu }$, with a logarithmic second solution in the repeated-root case $\nu =0$. The constants on each side require incident-wave and boundary or matching data; the supplied singular profile alone does not choose them.