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 $c(y)=c_0/\sqrt{1-ky}$, with $1-ky>0$, the elapsed-time integrand is $\sqrt{(1-ky)(1+(y^{\prime }{)}^2)}/c_0$. The Beltrami identity gives $(1-ky)/(1+(y^{\prime }{)}^2)=1-k{y}_0$, where $y_0$ is the turning height. Integration yields the displayed parabola. If the endpoints are $(\pm x_0,0)$, symmetry gives $x_c=0$ and $(k{x}_0{)}^2=4k{y}_0(1-k{y}_0)$. Hence $k{x}_0\le 1$, and the two algebraic turning heights are $[1\pm \sqrt{1-k^2{x}_0^2}]/(2k)$. Only heights lying within the specified optical layer are admissible.