Nonlinear Optics
Homework 1
due Thursday, 26 Jan 2017

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 1.5
- to estimate the order of magnitude of the contributions, use the scaling arguments discussed in the reading. Assume there are no resonances that would affect the order of magnitude.

Problem 3: Boyd 1.12
This problem is similar to an induced grating experiment (see intro slides).
Make a sketch of the directions of the beams that are driven by the nonlinear polarization

Problem 4: 
The commonly-used nonlinear crystal BBO (beta- barium borate) is a uniaxial crystal with nonlinear coefficients (Subscript[d, 31]) = 0.16 pm/V and (Subscript[d, 22]) = 2.2 pm/V. The effective nonlinear coefficients as a function of the direction of the fundamental beam for this crystal are given in equation 1.5.30. The angle θ is fixed according to phase matching, but we are free to choose whatever value of ϕ that will optimize the magnitude of the nonlinearity |(Subscript[d, eff])|. 

For an input wavelength of 800nm, the type I phase matching angle is θ = 29 deg and for type II, the angle is θ = 42.3 deg. 
- Plot (Subscript[d, eff])vs ϕ for both cases, and determine the optimum value.    
- Which phase matching type gives a higher nonlinearity?

Problem 5: 
We can numerically solve differential equations without a perturbation approximation. Use NDSolve[ ] in Mathematica (or another differential equation solver if you're using another program) to solve for the displacement of the classical nonlinear oscillator (equation 1.4.1). In Mathematica, use the help on NDSolve for examples of how to program it. 
It is actually quite easy:

a. Make plot of x(t) for a = 0, a=0.03 and a = 0.3. Explain the origin of the transients you see at early times. 
Other parameters: ω0=3; ω=1; γ=0.1; c=1;

b. The solutions for a > 0 are not that different for the linear case. Subtract the linear solution function from the nonlinear solution for the two cases, and plot the two results in a region of time after the transients have died down (say after t>50). Discuss the difference between the small and large nonlinearity cases.