= 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 $E_y=e_y(z)e^{-ik_0\sin\theta_0x}$ and set $s=z-z_0$. The <inverse-square refractive-index profile> gives
$$
e_y''+\left[\beta^2-\frac{\nu^2-1/4}{s^2}\right]e_y=0,
\qquad \beta^2=k_0^2(a^2-\sin^2\theta_0),
\qquad \nu^2=\frac14+k_0^2b^2.
$$
Substituting $e_y=\sqrt{|s|}\,w(\beta|s|)$ gives the <Bessel equation> of order $\nu$. On either side of the profile singularity, when $\beta^2>0$,
$$
\boxed{e_y(z)=\sqrt{|z-z_0|}\left[A J_\nu(\beta|z-z_0|)+B Y_\nu(\beta|z-z_0|)\right].}
$$
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.
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