[College Electrical Engineering/Physics] Waves & Lens

[DerpyDalek]

Homework Statement

Show that when a plane wave is transmitted through a thin lens of focal length f in a direction parallel to the axis of the lens, it is converted into a paraboloidal wave (the Fresnel approximation of a spherical wave) centered about a point at a distance f from the lens.

Homework Equations

U(r) = (A_o/z)exp(-jkz)exp[-jk(x^2 + y^2)/2z]

The Attempt at a Solution

I am having trouble determining how to best set up this problem, i understand visually why this occurs as a plane wave contacts a thin lens and transforms but am having problems seeing the math to "prove" that this is what occurs.

Currently looking into how the lens affects the speed of light as it passes through each point of the lens at the front of the wave

 

[rude man]

Use 2 dimensions to start. You should be able to show by analytic geometry that the locus of off-axis rays impinging on the focal plane is a parabola. Then extend the argument to 3 dimensions.