Ray in an exponential-speed medium ($e^{\lambda y}=\cos (\lambda x)/\cos (\lambda a)$)

For positive $\lambda$ and light speed $c(y)=c_0{e}^{\lambda y}$, the Fermat principle uses the time integrand $c_0^{-1}{e}^{-\lambda y}\sqrt{1+y^{\prime 2}}$. The Beltrami identity sets $e^{-\lambda y}/\sqrt{1+y^{\prime 2}}$ constant. Solving this first integral gives the displayed symmetric ray between $(\pm a,0)$ when $0<\lambda a<\pi /2$. Its travel time is $2\sin (\lambda a)/(c_0\lambda )$. The ray's central height diverges as $\lambda a\uparrow \pi /2$, while its travel time tends to $2/(c_0\lambda )$.