= Catenary light ray in a linear refractive index
{title2=$x(y)=-1/\beta+(c/\beta)\cosh[\beta(y-y_0)/c]$}
For height $x$ and horizontal position $y$, <Fermat principle> in a positive index profile $n(x)=1+\beta x$, $\beta>0$, conserves $n(x)y'/\sqrt{1+(y')^2}$ on a graph $y(x)$. For positive magnitude $c$ of the horizontal optical momentum, integration gives the displayed <catenary>. Its minimum height is $x_0=(c-1)/\beta$, at $y=y_0$. Thus $c>1$ produces a smooth turning ray wholly above a road at $x=0$. The two graph branches have opposite signs of $dy/dx$; $x(y)$ continues smoothly through their common horizontal tangent. For a signed negative graph constant, use $|c|$ in the prefactor; $c=0$ gives a vertical ray separately.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-q9-mirage-ray.png]
{title=A catenary light ray turns smoothly above a hot road}
{description=Dimensionless coordinates $X=\beta x$ and $Y=\beta(y-y_0)$ show the case $c=1.4$. The <refractive index> increases with height, and the <light ray> turns at $X=c-1>0$.}
Back to article page
