1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex amplitude reflectance of a spherical mirror

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove the complex amplitude reflectance of a spherical mirror is given as exp[-jk(x2+y2)/R]


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


    3. 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?
     
  2. jcsd
  3. Apr 12, 2017 #2
    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}}##
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Complex amplitude reflectance of a spherical mirror
Loading...