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## Homework Statement

An optical fiber is one way to guide light efficiently from one point to an other. It is currently used for data communication: it offers low loss and very high bandwidth, ideal for the requirements of the internet. Generally, we can describe an optical fiber as a medium with an index of refraction

*n*which depends only on the distance from its axis for example

*O*. If we assume an incoming beam in the plane

_{x}*O*at start crossing

_{xy}*O*with an angle

_{x}*θ*,

_{0}show that, by solving the Euler-Lagrange equations, the equation for the beam path can be written as:

##n \sin(i) = a##

with

*a*a constant to determine and

*i*the angle that the ray makes with the normal. You will have to keep some approximation on the angle

## Homework Equations

Euler lagrange equations:

##\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial x'}-\frac{\partial\mathcal{L}}{\partial x}=0 ##

##\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial y'}-\frac{\partial\mathcal{L}}{\partial y}=0 ##

##\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial z'}-\frac{\partial\mathcal{L}}{\partial z}=0 ##

## The Attempt at a Solution

I've tried rewriting the Euler-Lagrange equations using the Principle of Fermat, that light will travel the path between A and b that is shortest, in order to get a differential equation:

##\frac{d}{ds}\left(n\frac{d\vec{r}}{ds}\right)={\vec\nabla}n##

with

*s*the distance travelled by the light and

*r*the position in space.

From this point I did not get any further on relating this equations to this particular beam path length in an optical fiber proposed, especially on relating it to the angle

*i.*