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## Homework Statement

Consider a cylindrical piece of material of a constant thickness

*d*and radius

*r*=

*a*, but made out of a material with position-dependent index of refraction. How must the index of refraction vary with radial distance

*r*such that the cylindrical piece act as a lens with a focal length of

*f*. Assume

*d*<<

*a*, i.e., a thin lens.

## The Attempt at a Solution

Ok, first I remind myself that light follows the path, where the optical path length is stationary. So I write

[tex]

\Gamma = \int {n\left( {r,\phi ,z} \right)} \,rdrd\phi dz.

[/tex]

I can integrate dz and dφ out, since we only want to look at the r-dependence. Now, I need to find dGamma/dr = 0, but that just gives me n(r)=0, which is clearly wrong. Can you give me a hint?

Best,

Niles.

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