An optics problem - elliptical lens and plane waves

[pergradus]

Homework Statement

I must show that a point source placed at one of the focii will produce a plane wave (or vise versa) when the light is refracted through the interface. This problem is really kicking my ***, I've spent at least three hours trying to figure out different ways to do it...

Homework Equations

I tried writing the OPL such that y satisfies the equation of an ellipse centered at the origin. I apply Fermat's theorem, differentiate with respect to x and set this equal to zero.

The result should satisfy the initial condition that n₂ > than n₁ but I seem to get the opposite.

The Attempt at a Solution

See attached picture!

 

[pergradus]

I just figured out my mistake. The OPL should just be proportional to the distance traveled times the refractive index, not 1/refractive index. The letter will give 1/time rather than just time which is what I'm looking for.

I didn't get any help but I managed to follow the train of logic on my own

 

[Phruizler]

Pergradus did you ever solve this problem and if so how