Complex amplitude reflectance of a spherical mirror

[semc]

Homework Statement

Prove the complex amplitude reflectance of a spherical mirror is given as exp[-jk(x²+y²)/R]

Homework Equations

Transmittance of a spherical mirror is also exp[jk(x²+y²)/2f]

The Attempt at a Solution

I have totally no idea how to go about doing this. Can I just say that the reflectance is the same as the transmittance just that the wave changes the direction of propagation?

 

[Jan Jenko]

I realize this is late but here it goes:

See how much phase you accumulate, relative to the plane wave that travels along the optical axis and bounces off a planar mirror.

If you are looking at the plane wave ##\rho = \sqrt{x^2+y^2}## away from the axis in the plane ##z=0## (mirror centre is at ##z=R## ), then to get to the mirror you need to travel additional distance ##d##. Denote ##z_0## the z at which the ray intersects with the mirror at a given ##\rho##.

$$d = z_0 = R - \sqrt{R^2 - \rho^2} = R - R\sqrt{1-(\frac \rho R)^2}$$

Assuming ##\rho## is small (we are close to the axis) compared to R, we can write
$$\sqrt{1-(\frac \rho R)^2} = 1-\frac {\rho^2} {2R^2}$$
and so
$$d = \frac {\rho^2} {2R}$$

Because we traverse that distance twice, the phase gained is ##k*2d = k \frac {\rho^2} {R}## and your complex reflectance is ##e^{ik \frac {\rho^2} {R}}##