Need help with a differential equation

[origin]

Hi, I haven't studied diff. eqs. much, so I need some help. I decided to use trig & calc to solve for a light beam in a medium with a continuously changing index of refraction described by n(x,y). The light vector is L, and dL/dt is the change in the light ray's direction. I assumed |L| is always 1. Here's the result I got:

dL/dt = (n(x,y) / n(x+dn(x,y)/dx,y+dn(x,y)/dy)) * |grad(n(x,y)) x L| * (k x grad(n(x,y))) - L

This is derived from simple vector subtraction & shows that L changes parallel to the virtual optical surface (k x grad(n(x,y))) as one would expect from snell's law. By the way, I've assumed it's just a 2D problem for simplicity. I need help changing this to a function of just L & t (getting rid of dL/dt). Can anyone help?

 

[RainmanTime]

Hmmmm, interesting stuff. I'm not much into optics, but I think I can see the physical basis behind the equation. Just one question to make sure I understand everything: k is intended to represent the unit vector normal to the x-y plane which contains the material with index of refraction n(x,y), correct?

I'll have to dig out my diff eq book when I get home tonight, because the grad implies you've got partials in there and PDEs are always a bit more tricky than ODEs.

RMT

 

[origin]

That's right. k = i x j

I'll also need to be able to trace the path of the light ray. So if r(t) is the position of the photon at time t, then dr/dt = L.

 

[origin]

I don't know the specific form of the final (solved) equation; however, it can be seen that the photon is "accelerated" because of the 2nd derivative in dr/dt. So I found that the light behaves similarly to a particle in an energy well/hill. A force is exerted on it which is antiparallel to the gradient at its current location, pushing it toward the well or away from the hill. I will have to learn about differential equations before I can solve this explicitly, but I think i can make some simple simulations now.