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deedsy

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## Homework Statement

A wave travels in a stratified medium whose index of refraction is a function of the coordinate y. Show that the angle ##\theta## between a ray and the y-axis obeys the following law:

## \frac{d\theta}{ds} = \frac{-(dn/dy) sin(\theta)}{n} ## , where the distance s is measured along the ray.

Using this result, you can verify the

*ray equation*##\frac{d}{ds} (nt) = \nabla n## , where t is a unit vector tangent to the ray at a point where the index of refraction is n.

Hint: Select your y-axis along ##\nabla n## and your x-axis in the plane of incidence.

## Homework Equations

Snells Law --> ##n_1 sin(\theta_1) = n_2 sin(\theta_2)##

## The Attempt at a Solution

I need an equation that relates ##\theta, s, y, ##and ##n(y) ##, which shouldn't be hard but I can't seem to figure it out.

I attached a simple figure of how I am setting this up.

From the figure, s and y are related by

## cos(\theta) = \frac{dy}{ds} ##

##\theta = \cos^{-1}(\frac{dy}{ds})##

n and ##\theta## are related by Snell's Law, which for this problem, can be applied to a large number of thin y-axis layers with varying refractive indexes

## n_1 sin(\theta_1) = n_2 sin(\theta_2) = n_3 sin(\theta_3) = A##, where A is just a constant

##n = \frac{A}{\sin(\theta)}##

## \frac{dn}{d\theta} = \frac{-A cos(\theta)}{sin^2(\theta)}##

With these expressions for ## dn, dy, ds, d\theta##, I've just been manipulating these expressions trying to match what the answer should be, but I'm not making any progress. Am I on the right track trying to solve this using only the geometry of the problem and Snell's law, or is there another way someone can reccomend? I've seen the ray equation derived by using phase differences and fermat's principle, but those didn't yield what the first part of this question is asking for.