Gaussian beam spherical mirror reflection question

[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!

 

[anathema]

Thank you for your interesting question. I would like to approach this problem by first understanding the physical concept behind it. In this case, we are dealing with the reflection of a Gaussian beam off a curved mirror. The reflectivity of the mirror is given as a Gaussian function, which means that the reflected beam will also have a Gaussian shape. The key here is to understand how the curved mirror affects the shape of the reflected beam.

To solve this problem, we can use the concept of Gaussian optics, which describes the propagation of Gaussian beams through optical systems. The key parameters in Gaussian optics are the beam waist (w0), the Rayleigh range (z0), and the radius of curvature (R). These parameters are related to each other through the equations given in the homework section.

Now, to derive an expression for the modified beam width after reflection, we need to consider the effect of the curved mirror on the beam waist and the Rayleigh range. Since the mirror has the same radius of curvature as the incident beam, the reflected beam will have the same Rayleigh range as the incident beam. However, the beam waist will be modified due to the reflection off the curved mirror.

To find the new beam waist, we can use the principle of conservation of energy. The reflected beam must have the same energy as the incident beam, which means that the product of the electric field and the beam width must be constant. Using this principle and the equations given in the homework section, we can derive an expression for the modified beam width after reflection.

I hope this helps you in solving the problem. If you have any further questions, please do not hesitate to ask. Good luck!