Parabolic light rays in a reciprocal-square-root speed profile ($y=y_0-\frac{k(x-x_c{)}^2}{4(1-k{y}_0)}$)

For speed of light $c(y)=c_0/\sqrt{1-ky}$, where $c_0>0$ and $1-ky>0$, the travel-time integrand in Fermat's principle is $c_0^{-1}\sqrt{(1-ky)(1+y^{\prime 2})}$. For nonzero $k$, the Beltrami identity giveswhere $y_0$ is the turning height. The corresponding stationary rays are the parabolasFor equal-height endpoints $(\pm x_0,0)$ with $x_0>0$, symmetry gives $x_c=0$ and $(k{x}_0{)}^2=4k{y}_0(1-k{y}_0)$. Real turning heights therefore require $|k|x_0\le 1$, not merely a one-sided condition on signed $k$. When the inequality is strict, the two algebraic values are $y_0=[1\pm \sqrt{1-k^2{x}_0^2}]/(2k)$; only rays remaining inside the specified optical layer are admissible. Equality merges the two branches. At $k=0$, the medium has constant speed of light and the ray joining these endpoints is the straight horizontal segment; the formulas containing $1/k$ must be replaced by that limit.