Nonlinear Optics
Homework 4
due Tuesday, 13 Feb 2007

Problem 1: 
Boyd, problem 2.9

Problem 2:
Boyd, problem 2.19

Problem 3:
In third-harmonic microscopy, the harmonic light generated is not really generated at the  interface, but it is localized near there because of the role of phase matching. In this problem, we will consider the longitudinal spatial resolution of this kind of microscopy. Here are the parameters: 
- We focus a Gaussian beam with a vacuum wavelength of 1 micron at f/1
- Suppose we are looking at a fused silica slide in air. If you can find Sellmeier equations for water (let me know), that would be better. You may assume that the thickness of the fused silica slide is much greater than the Rayleigh range of the focus. Be careful in how to introduce the refractive index of the slide into the equations for the focal spot size and the Rayleigh range.    
Calculate the signal vs. z-position as the focal spot is moved in the z-direction. The FWHM (full-width at half-maximum) is effectively the impulse response of the imaging system in the longitudinal direction.

dispersion for fused silica

as1 =  .6961663;
as2 =  .4079426;
as3 =  .8974794;
bs1 =  .0684043^2;
bs2 =  .1162414^2;
bs3 = 9.896161^2;
ns[λ_] := √(1 + ((as1 (λ^2))/((λ^2) - bs1)) + ((as2 (λ^2))/((λ^2) - bs2)) + ((as3 (λ^2))/((λ^2) - bs3)));

Problem 4: 
Consider wave mixing in a square cross-section hollow waveguide. 
a. write down or derive an expression for the propagation constant and the eigenmodes for a gas-filled waveguide. Allow the pressure to be a variable. 
b. derive a phase matching condition for third-harmonic generation. Assuming the fundamental is in a single, well-defined mode, to what modes is it possible to obtain phase matching in the 3rd harmonic? 
c. Calculate an expression for the mode overlap factor for the modes that can be phasematched. Calculate the overlap integral for the best output mode, and compare it with the next best one. 
d. Consider a 4-wave mixing experiment, where the second harmonic is mixed with the fundamental to produce the 3rd harmonic: (Subscript[ω, 3]) = 2  (Subscript[ω, 2]) - (Subscript[ω, 1]). You should see that you can phase match if all beams are in the lowest order spatial mode. Calculate the mode overlap factor in this case.

Argon dispersion equations (good down to 110 nm)
wavelength in microns

dnAr[λ_] := 
        2*120.625 (10^(-6)) ((14.0/(87.892 - (((λ))^(-2)))) + (14.0/(91.011 - (((λ))^(-2)))) + 
              (430.63/(217.6 - (((λ))^(-2)))));
nAr[λ_, pr_] = 1 + pr dnAr[λ];