- #1

Barry Johnston

## Homework Statement

Gaussian beam of radius R_i and beam width w_i, The beam is reflected off a mirror with a radius of curvature R = R_i and the reflectivity of this mirror is given as rho(r) = rho_0*exp(-r^2/a^2), where r is the radial distance from the center of the mirror and a is a constant. Derive an expression for the modified beam width after reflection. Assumed that we are in air, ideal conditions, and we are talking about the fundamental TEM(0,0) mode.

## Homework Equations

Electric field (propogation along z) E(x,y,z) = E_0*(w_0/w(z))*exp(-j*(k*z-nu(z)-(k*r^2/2*R(z)))*exp(-r^2/w(z)^2)

where k = 2*Pi*n/Lambda

Radius of curvature, R(z) = z*(1+z_0^2/z^2)

phase parameter, nu(z) = tan^-1(z/z_0)

Rayleigh range, z_0 = Pi*n*w_0^2/Lambda

beam width, w(z) = w_0*(1+z^2/z_0^2)^(0.5)

w_0 = initial beam width

## The Attempt at a Solution

I have attempted to solve this problem by taking the square root of the reflectivity to get the Fresnel reflection coefficient. I then multiplied this by the electric field and tried to solve for a new initial beam waist so that I could plug this into the beam width express w(z). The initial beam waist is found by setting z = 0 and solving for the real part of the exponential exponent being equal to -1 (where r = w_0 in this case). I know that since the reflectivity of the mirror is equal to a Gaussian, we would expect the reflected beam width to decrease since we are taking a Gaussian of a Gaussian beam. So far I get nowhere near the results expected, I would appreciate any help to put me in the right direction, thanks!