Nonlinear Optics
Homework 2
due Thursday, 24 Jan 2013

Problem 1: Boyd 1.5
- to estimate the order of magnitude of the contributions, use the scaling arguments discussed in the reading. Assume there are no resonances for this part.

Problem 2: Boyd 1.12
This problem is similar to an induced grating experiment (see intro slides). The keys are: 
- to recognize that although the input waves are all at the same frequency ω, they have different vector directions, k. 
- since one of the beams is “weak”, you can ignore any terms that are higher order than 1 in the weak field. 
Make a sketch of the directions of the beams that are driven by the nonlinear polarization

Problem 3: numerical solution of the coupled wave equations for second harmonic generation
Equations 2.7.10 and 2.7.11 in Boyd are the coupled equations for the evolution of the amplitudes for the fundamental ((Subscript[A, 1])) and the second harmonic ((Subscript[A, 2])). In this problem you will use NDSolve to calculate the evolution of the fields in the crystal.

a) Show that you can write the equations in a simpler form:
	(∂_z) (Subscript[A, 1])=i κ (Subscript[A, 2])(SuperStar[(Subscript[A, 1])])(e^(-i Δk z))
	(∂_z) (Subscript[A, 2])=i κ ((Subscript[A, 1])^2)(e^(+i Δk z))
Here in the coefficient κ we are making the assumption that (Subscript[n, 1])≈(Subscript[n, 2])=n. 
Define κ. 
What are the dimensions of κ?

b) Use NDSolve[ ] to numerically solve for both fields. For the purposes of looking at the functional form of the evolution of the fields, we can set κ = 1. 
- For Δk = 0, show that you can get a plot resembling Figure 2.7.3, where the conversion saturates.

c) Make a plot that shows that for short propagation lengths (no saturation), the non-depleted pump approximation is valid.

d) Make a plot resembling Figure 2.7.4, where you are choosing different values of Δk.