Nonlinear optics take-home midterm

Revision 1: 7am Saturday. Adjusted wording for problem 4. 
Revision 2: 7am Sunday: Reworded intro, problems 3 and 4.
Revision 3: 7 am Monday (last one) Primarily clarified problem 7.

Suppose we have a way to align a molecule such as gaseous CS2 (experimentally, this would be done with an auxiliary laser). We want to do third harmonic generation of an input wave at frequency ω. The problems on this test have to do with various aspects of this possible experiment.

1. The molecules are ellipsoidal in shape, with one long axis (y) and two symmetric short axes (x and z). The molecules in the gas do not interact with each other. Explain why we cannot frequency double (2  (Subscript[ω, 1]) = (Subscript[ω, 2])) in this medium.

2. Even though the rotational response is greater for a molecule, it is not likely that we can use that contribution to the nonlinearity for this mixing process. Why would this be true? (Short answer)

3. We set up a model equation for the motion of a bound electron as (compare to 1.4.36): 
\(TraditionalForm\`r\&¨\  + 
          2\ γ\ r\& . \  + \ \(ω\_x0\^2\)(
            x\ x\&^\  + z\ z\&^) + \ \(ω\_y0\^2\) 
            y\ y\&^ - \(b(r·r)\)\ r\  = \(-e\)\ \(\(\(E(t)\)/
            m\)\(\ \)\)\), 
where r and E are vectors. Explicitly r can be expanded out to give:
\(TraditionalForm\`\(x\&^\)(
            x\&¨\  + 
              2\ γ\ x\& . \  + \ \(ω\_x0\^2\) 
                x\  - \(b(r·r)\)\ x)\  + \(y\&^\)(
            y\&¨\  + 
              2\ γ\ y\& . \  + \ \(ω\_y0\^2\) 
                y\  - \(b(r·r)\)\ y) + \(z\&^\)(
            z\&¨\  + 
              2\ γ\ z\& . \  + \ \(ω\_x0\^2\) 
                z\  - \(b(r·r)\)\ z) = \(-e\)\ \(\(\(E(t)\)/
            m\)\(\ \)\)\)

Initially, we'll consider the molecules to be aligned to the y-direction, which is the long axis, with (Subscript[ω, y0]) <  (Subscript[ω, x0]). Note that when the molecules are aligned the medium behaves like a uniaxial birefringent crystal. 
 
In this part you will calculate the linear and 3rd order responses using perturbation techniques: write x and y in separate expansions as in 1.4.6. The response for z will be the same as for x.

a. Aligning the molecules has essentially made the gas birefringent. Show that the two values of (χ^((1)))are 
\(TraditionalForm\`\(χ\_11\^\((1)\)\)(ω) = \ \(N\ e\^2/m\)\/\(\(D\_x\)(ω)\)\) and
\(TraditionalForm\`\(χ\_22\^\((1)\)\)(ω) = \ \(N\ e\^2/m\)\/\(\(D\_y\)(ω)\)\)
where (Subscript[D, x])(ω) = ((Subscript[ω, x0])^2) - (ω^2) - 
          2 i ω γ, with a corresponding definition for (Subscript[D, y])(ω).

b. Calculate expresssions for two of the values of (χ^((3))): ((Subscript[χ, 1221])^((3)))((Subscript[ω, 3]) = 
          3  (Subscript[ω, 1])) and ((Subscript[χ, 2222])^((3)))((Subscript[ω, 3]) = 
          3  (Subscript[ω, 1])). If you need other values later, you may deduce the pattern to write them down. What is one symmetry rule that dictates that a large number of these tensor elements are zero?

4. To get good efficiency, the tripling process must be phase matched. Angle-tuning is possible by adjusting the direction of the aligning field in the y-z plane. Relative to the molecule coordinates, this means the beam direction is in the y-z plane, with the E-vector in a plane normal to the k-vector. Type I and Type II phase matching techniques are developed for second-order mixing. Describe in detail how the input polarization should be oriented to achieve phase matching using the linear, birefringent refractive index, making clear which direction the output polarization will be.
- To help visualize the process, use the sample input parameters of ω=1, (Subscript[ω, x0]) = 10, (Subscript[ω, y0]) = 8, γ= 0.1 and plot the relevant linear ordinary and extraordinary refractive indices vs ω.
- Write down a phase matching condition that explicitly shows which linear refractive indices are used in which terms, with expressions for those refractive indices. I am not asking you to numerically solve for the aligning field angle, though you need to show through your plot that it is possible to find such an angle.
- Assume that the process is not absorbing for any of the relevant frequency combinations. 
- For simplicity, assume the value of (χ^((3))) appropriate to the process is not affected by adjustment of the aligning field. (Recall that (Subscript[d, eff])is affected by the incident angle in general - I'm saying we can neglect this effect here.) 
- To what maximum harmonic order could you achieve phase matching?

5. If we were to use focused beams instead of plane waves (tight-focusing limit), what effect would that have on the phase-matching criterion?  (short answer)

6. Suppose we could alternate our aligning field either along x or y so that molecules were periodically aligned either along the short or long axes. Write down an expression for what period would be required to achieve first-order quasi-phasematching. (short answer)

7. Estimate the importance of the nonlinear refractive index (B-integral) for this process: 
- By scaling the nonlinear equations for the growth of the harmonic field, find the scale length for the conversion to the third harmonic (along the lines of 2.6.19). 
- Also estimate the distance for the B-integral to build up to a value of one. To do this calculation, use the value of ((Subscript[χ, 2222])^((3))) as a representative value to estimate (Subscript[n, 2]). 
- Estimate the ratio of these distances, and comment on the feasibility of converting to third harmonic without substantial B-integral.