PHGN 585 Take-home midterm
cascaded (χ^((2))) for controlled (Subscript[n, 2]) experiments

When a doubling crystal is used in a non-phasematched orientation, the input beam can experience a nonlinear phase shift that is like what would result from a (χ^((3))) medium. Except here, the effective (Subscript[n, 2]) can be controlled both in magnitude and sign. In this take-home midterm, you will derive how this works and come up with an experimental design for working with this effect. This is an expanded version of problem 2.20 in Boyd. 
You can use any resources except working with others. If you make use of any sources, cite them. You can, of course, use Mathematica, but any code must be well commented and presented in a way that others can follow what you are doing. For example, you may need to manually rearrange equation output to make them more presentable.

1. Equation for the evolution of the fundamental

Starting from equations 
((d (Subscript[A, 1]))/dz)=((2 i ((Subscript[ω, 1])^2)(Subscript[d, eff]))/((Subscript[k, 1])(c^2)))(Subscript[A, 2])(SuperStar[(Subscript[A, 1])])(e^(-i Δk z))		(2.7.10)
((d (Subscript[A, 2]))/dz)=((i ((Subscript[ω, 2])^2)(Subscript[d, eff]))/((Subscript[k, 2])(c^2)))((Subscript[A, 1])^2)(e^(+i Δk z))		(2.7.11),
eliminate (Subscript[A, 2]) from the equations to show that 
(((d^2)(Subscript[A, 1]))/(d (z^2)))+i Δk ((d (Subscript[A, 1]))/dz)-(Γ^2)(1-2|(Subscript[A, 1])/(Subscript[A, 10])((|)^2))(Subscript[A, 1])=0, 
where Γ=|(Subscript[A, 10])|((( 2(Subscript[ω, 1])(Subscript[d, eff]))/(c (Sqrt[(Subscript[n, 1])(Subscript[n, 2])]))))

At z=0, (Subscript[A, 2])=0, so (Subscript[A, 10])(SuperStar[(Subscript[A, 10])]) is proportional to the total input power. 
Hint: you will need to use the Manley-Rowe relations for photon conservation. 
Also: factors of 2 can be hard to get right here, so don’t worry too much if you’re off a bit.

2. Non-depleted approximation: cascaded (χ^((2)))

If the phase mismatch is sufficiently large, the amplitude of the fundamental is not changed, so (((Subscript[A, 1])(SuperStar[(Subscript[A, 1])]))/((Subscript[A, 10])(SuperStar[(Subscript[A, 10])])))=1.

a. Slowly-varying envelope approximation (SVEA)

Use the SVEA to justify dropping the second derivative, and thereby show that the input beam propagates by acquiring a nonlinear phase.

b. calculate an expression for the effective (Subscript[n, 2]), and describe how one can adjust its magnitude and sign.

c. Under the non-depleted approximation, solve for (Subscript[A, 1])[z] without using the SVEA. Under what conditions can you recover the solution found in part (a)?

3. numerical solutions

By numerically solving (NDSolve[ ]) the original equation (what you derived in problem 1), make plots of the intensity (((Abs[A1])^2)) and the phase Arg[A1] as a function of (Γ^2) for a fixed crystal length of L=1. Do this for values of Δk = 6, 20, and 50. (Note that the products Δk L and Γ L are dimensionless).  Choose the initial amplitude (Subscript[A, 10])=1.