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

[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 $e^{ik \frac {\rho^2} {R}}$