# Analytical calculation of a reflected light ray in glass fibre

I am trying to analytically calculate the propagation of a light ray in a glass fiber. While the physics of it is clear to me, I am not sure how describe arbitrary geometries of the fiber and then calculate analytically the intersection point and angle of the intersection. From this I would need to define a new, reflected and shifted ray. Then again calculate the intersection etc. .

Here I had attempted an idea using analytical geometry, where I would describe the ray as a line, the fiber as a curve with a cylindrical surface. I was told there might be easier solutions to my problem. I would appreciate your help and ideas to this. Thanks!

Given a curve $$\gamma:\quad [a,b]\to{\mathbb R}^3,\qquad t\mapsto{\bf x}(t)$$ it is easy to produce a parametric representation of a tube $S$ of radius $\rho>0$ with "soul" $\gamma$. But there is no "analytical solution" of your light ray problem. The entry of the tube is a disc of radius $\rho$, and you can specify the exact entry point of the ray on this disc, as well as its initial direction. But already the first point of reflection of this ray has to be found numerically, not only as a point in ${\mathbb R}^3$, but also as a pair of parameter values $(t,\phi)$. You will then be able to describe (numerically) the next leg of the ray, until it again hits $S$ at some point farther down, etcetera.