Nonlinear Optics
Homework 4
due Friday, 8 Feb 2013

Problem 1: Boyd 1.4
- The first part comes directly from evaluating the Kroneker deltas in the equation. 
- For the explanation, consider how the nonlinear restoring force changes when driven in different directions.

Problem 2:
Boyd problem 2.1. For this problem, assume that "optimum focusing" means that the length of the crystal equals the confocal parameter (2 x the Rayleigh range) of the pump beam at 0.635μm. (See the discussion at the end of 2.10). You don't need to account for focusing phase mismatch (we'll treat this later). Section 2.6 treats the case of upconversion. Proustite is (Subscript[Ag, 3])(Subscript[AsS, 3]), and you’ll find the refractive index data on refractiveindex.info. The data there only goes out to about 4.5μm, so you can use that index for 10μm. I get about 1.2% in my calculation.

Problem 3: Quasi phase matching
Boyd problem 2.9. I did this by calculating the Fourier series for the periodic function of the domains, then following the steps of the derivation in 2.4.

Problem 4: Numerical solutions of quasi phase matching
Plot the growth of signal for second-harmonic generation in a quasi-phasematched crystal in which the (Subscript[d, eff]) is modulated sinusoidally. Make plots for two cases, 1st order and 3rd order quasi-phase matching.

Problem 5: 
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 2.2.10, 2.2.12a, and 2.2.12b and redefine the fields scaled to the total intensity (Subscript[I, tot])=(Subscript[I, 1])+(Subscript[I, 2])+(Subscript[I, 3]), along the lines of what is shown in 2.7.13. It is easier to express the fields in terms of a single complex variable (Subscript[a, i]) instead of (Subscript[u, i])(e^(i (Subscript[ϕ, i])))(which is the treatment in the book). Unlike the case for SHG, there will be some residual frequency dependence to the scaling factor. Choose your normalization factor to include the frequency (Subscript[ω, 3]), then the expressions to normalize (Subscript[A, 1]) and (Subscript[A, 2]) will have an extra factor of  (Subscript[ω, 1])/(Subscript[ω, 3]) and (Subscript[ω, 2])/(Subscript[ω, 3]), respectively. Note that eqn 2.7.19 should have the (Subscript[ϵ, 0])c term in the numerator of the square root.

b) Construct the numerical solution using NDSolve[ ] as shown in my example. 
Find input conditions to generate plots like Figures 2.6.2, 2.6.3 and 2.8.2.

c) Note that the case of difference frequency mixing (DFM) is also solved here - it is just a matter of what input waves are there. Show an example for DFM, illustrating an OPA (strong pump at (Subscript[ω, 3]) and a weak seed at (Subscript[ω, 1]). (This is an extension of the plot 2.8.2 that includes saturation.)

d) For sum-frequency generation, determine input conditions that allow complete conversion of the two inputs to (Subscript[ω, 3]) without any back conversion. (Hint: consider the Manley-Rowe relations.)