Nonlinear Optics
Homework 3
due Tuesday, 6 Feb 2007

Problem 1: 
Following the calculations done for second harmonic generation shown in the notebook mixing solutions.nb, numerically solve for the intensity vs. propagation length for sum frequency mixing. 
a) First start wih the nonlinear coupled equations 2.2.11, 2.2.13, and 2.2.14 and redefine the fields scaled to the total intensity Itot = I1 + I2 + I3, along the lines of what is shown in 2.6.13, but expressing the fields in terms of (Subscript[a, i]) instead of (Subscript[u, i]) (e^(i (Subscript[ϕ, i])))(see my notes). Unlike the case for SHG, there will be some residual frequency dependence to the scaling factor. If you choose your normalization factor to include the frequency (Subscript[ω, 3]), then the expressions for (Subscript[E, 1]) and (Subscript[E, 2]) will have an extra factor of e.g.  (Subscript[ω, 1])/(Subscript[ω, 3]). 

b) Construct the numerical solution using NDSolve[ ] as shown in my example. Find input conditions to generate plots like Figures 2.4.2, 2.4.3 and 2.5.2. Note that the case of difference frequency mixing is also solved here - it is just a matter of what input waves are there. Show a couple examples for DFM, for example illusrating an OPA (strong pump at (Subscript[ω, 3]) and a weak seed at (Subscript[ω, 1]). For sum-frequency generation, determine conditions that allow complete conversion of the two inputs to (Subscript[ω, 3]) without any back conversion.

Problem 2:
Boyd problem 2.1 (If you didn't do it already). "Optimum focusing" means that the length of the crystal equals the confocal parameter (2 x the Rayleigh range) of the lowest frequency beam.

Problem 3:
One of the most common nonlinear crystals is KDP (potassium dihydrogen phosphate). The Sellmeier equations for KDP are given below. 
a. Calculate the variation of ne with angle θ to the z-axis. 
b. Calculate the optimum phase matching angle for doubling 1000nm light with Type I and Type II. What is the shortest input wavelength that can be phasematched?
c. For both types, plot the harmonic yield (intensity) as a function of incident angle for a few thicknesses. This shows the angular acceptance. 
d. For both types, plot the harmonic yield as a function of input wavelength for three thicknesses. This shows the phase matching bandwidth. 
e. Finally, for type I phase matching, calculate the group delay for the fundamental and the harmonic for 1mm thickness. How thin must the crystal be to make the group delay difference less than 100fs?

KDP sellmeier: wavelength in microns

indexSellmaier[lambda_, aA_, bB_, cC_, dD_, 
        eE_] := (Sqrt[aA + ((bB (lambda^2))/((lambda^2) - cC)) + (dD/((lambda^2) - eE))])

aoKDP = 2.259276;
boKDP = 13.00522;
coKDP = 400;
doKDP = 0.01008956;
eoKDP = 0.012942625;
aeKDP = 2.132668;
beKDP = 3.2279924;
ceKDP = 400;
deKDP = 0.008637494;
eeKDP = 0.012281043;

no[lambda_] = 
        indexSellmaier[lambda, aoKDP, boKDP, coKDP, doKDP, 
          eoKDP];
ne[lambda_] = 
        indexSellmaier[lambda, aeKDP, beKDP, ceKDP, deKDP, 
          eeKDP];