Nonlinear Optics
Homework 1
due Thursday, 17 Jan 2012

Problem 1: 
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:

solution=NDSolve[{x''[t]+2 γ x'[t]+((ω0)^2)x[t]+a ((x[t])^2)==-Cos[ω t],x[0]==0,x'[0]== 0},x,{t,0,tMax}];
xSolution[t_]=x[t]/.solution;

The equation shown here is the equation of motion for an anharmonic oscillator (compare to eqn 1.4.1 in Boyd). Here for convenience, we have set e = 1, m = 1, (Subscript[E, 0])=1/2. Notice that the (Subscript[E, 0])=1/2 in the driving term makes it be consistent with the definition in equation 1.4.4. 
 
When you define the function name, you can choose different names so you can work with the solutions later on, for example to subtract the linear solution from the nonlinear solution.

a. First consider the situation where a = 0 (no nonlinearity). 
Use ω0 = 3; ω = 1; γ = 0.1;
This makes the process non-resonant at both ω and 2ω. 
Make plots of x(t) from t = 0 to t = 100 for two driving terms:
  E(t) = Cos[ω t]
  E(t) = Sin[ω t]
You should notice transients you see at early times. Explain the origin of these transients and why they go away. (For your own understanding, you might play with the initial and system conditions.)

b. Now make plots of x(t) for a=0.03 and a = 0.3 for the Cos[ω t] driving term. 
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.

c. In the perturbative approach, we assume that the solution can be written as a sum of the solution for the linear equation (x^((1)))(t) and higher order terms. Use the perturbative solutions in Boyd and make a plot along with the numeric solutions to compare them.