## Physics Friday 19

Consider a material of a refractive index linearly increasing with depth from the surface: we have n=1+ky, y>0, for some positive constant k, with y being depth from the surface. What will be the trajectory of a light ray entering the surface y=0 at an angle θ>0 from the surface normal.

Let x be the direction along the surface, with the x component of the initial direction being positive for θ>0. We choose our origin to be the point the ray meets the surface.
Using Fermat’s Principle, we know that the optical path length must be extremal.

For a path from (0,0) to (xf,yf), xf≥0, with a path described by , we obtain an optical path length of:


We want to find the extrema, so we use the calculus of variations, and the Euler-Lagrange equation:
with , we have


Now:
,
,
,

So:





This is a second-order non-linear differential equation, with no explicit dependence on the independent variable x. So, we make the transformation ,  (see case II in this Monday Math post), and get:

This is separable:




This, in turn, is separable:


Rearranging,
,
and using y(0)=0, we find:


And so


Now, we note that . We have

so

Thus:



And plugging this back into our equation for y(x), we get:


And thus the path of a light ray within the material is a segment of a catenary (see also Physics Friday 6).