= Ray in an exponential-speed medium
{title2=$e^{\lambda y}=\cos(\lambda x)/\cos(\lambda a)$}
For positive $\lambda$ and light speed $c(y)=c_0e^{\lambda y}$, the <Fermat principle> uses the time integrand $c_0^{-1}e^{-\lambda y}\sqrt{1+y'^2}$. The <Beltrami identity> sets $e^{-\lambda y}/\sqrt{1+y'^2}$ 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 $2/(c_0\lambda)$.
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