Nonlinear Optics
Homework 3
due Friday, 1 Feb 2013

Problem 1:
A nonlinear crystal used for doubling the 3000nm output of an optical parametric amplifier (OPA) is (Subscript[LiIO, 3]) (lithium iodate).

a. Using the Phase_match_code.nb notebook as a template, code the dispersion equations for (Subscript[LiIO, 3]) and calculate the variation of (Subscript[n, e]) with angle θ to the z-axis. The Sellmeier equations for (Subscript[LiIO, 3]) are given in the datasheet posted online.

b. Calculate the optimum phase matching angle for doubling 3000nm light for Type I and Type II. If we changed the input wavelength to something shorter, what is the shortest input wavelengths that can be phasematched for the two types of phase matching?

c. To show the angular acceptance of the crystal, plot the normalized harmonic yield (intensity) as a function of incident angle for three thicknesses (this is the (sinc^2)(Δk L/2) function. Put all three thicknesses on the same plot. Make a plot like this for type I, type II (input wavelength 3μm) and type III for the input wavelength that is at the extreme allowed by type I.

d. To show the phase matching bandwidth, plot the normalized harmonic yield as a function of input wavelength for the same three thicknesses. Use an incident angle corresponding to optimum phase matching at 3μm.

Problem 2:
Boyd problem 2.1. See section 2.6 for a derivation that treats the case of upconversion. For the analytic calculation of the quantum efficiency, if you assume Δk=0, you will get an expression for the QE in a simple form. 

Assumptions for this problem:
*   let "optimum focusing" mean 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). 
*   let Δk=0, and assume that we don’t need to account for focusing phase mismatch.
*   Proustite is (Subscript[Ag, 3])(Subscript[AsS, 3]) (silver arsenic sulfide), use (Subscript[d, eff])=(Subscript[d, 22]) from Table 1.5.15. Also use n(λ=10um) = 2.75 and n(λ=0.635um) = 3.0. 
*   With these assumptions, I get a QE = 1.2% in my calculation.

Problem 3: 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 4: 
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.)