Nonlinear Optics
Homework 2
due Tuesday, 30 Jan 2007

Problem 1: 
Following the calculations shown at the end of the crystal symmetries.nb notebook, calculate deff[ θ, ϕ] for type I phase matching for the crystal KDP (see Table 1.5.3 and fig 1.5.2). You may also find the posted Midwinter reference useful.

Problem 2:
The impulse response for a classical damped harmonic oscillator is 
h[t] = Exp[ - γ t] Sin[ ω0s t] for t > 0, and h[t] = 0 for t<0
where ω0s is a slightly shifted resonance frequency defined by 
ω0s = (Sqrt[ω(0^2) - (γ^2)])

Use transform theorems to calculate the corresponding transfer function to show that you recover what is expected from the linear solution described in Chapter 1. You may check your result by doing the transform in Mathematica, but I want the work to be done analytically.

Problem 3:
This calculation illustrates the response of the system to a pulsed input. 
Using the sample program convolution demo.nb as guide to the use of the numerical convolution routine ListConvolve, perform the following convolution numerically: 
fOut[ t ] = h[ t ] ⊗ fIn[ t ], 
where h[ t ] is defined above in problem 2, and fIn[t] = Exp[-(t^2)/(tp^2)] Cos[ω t]

Plot the input and output functions for the off resonant case and the nearly resonant case. 
Use the following conditions: 
ω0s = 10	resonance frequency
γ = 0.1		damping coefficient
ω = 7		input driving frequency

Calculate the output for two different pulse widths (tw = 1 and 2). 

Discuss the differences. Your discussion should reflect your understanding in both time and frequency domain.

Problem 4: 
In the previous problem, the response shown is that of the individual induced dipole in the system. When a wave propagates through a material made up of these dipoles, the waveform itself is changed according to: 

Eout[ ω ] = Exp[ i ω n[ω] L/c] Ein[ ω ]

In this respect, Hsys[ ω ] = Exp[ i ω n[ω] L/c] can be considered a transfer function. 
This transfer function is different from that in problems 2 and 3, though they are related since n[ ω ] = \(TraditionalForm\`\@\(1\  + \ 4\ π\ \(\(χ\^\((1)\)\)(ω)\)\)\). 
For convenience (so we don't have to worry about dimensions), you may define the index as 
n[ ω ] = 1  +  2 π dens/Dres(ω), where Dres(ω) is the resonant denominator, and you can use dens = 0.02 for the number density weight. 
Then in applying the propagation phase, ω n[ω] L/c, you set c = 1 and L = 50. 

Feel free to change these numbers around to see what happens, but include plots for these numbers so that I can more easily compare yours to mine. 

Using the same input pulse as in the previous problem, calculate the output pulse in two different ways: 
a) Calculate Eout[ ω ] through the product form shown above, then do a numeric Fourier transform to calculate Eout[ t ].  

b) Calculate the impulse response hsys[ t ], then numerically do the convolution as in problem 3. 

For help with the numeric Fourier transforms, look at the FFT demo.nb notebook. 
You can also see examples in the non-perturbative response.nb notebook.

Problem 5: 
Boyd problem 2.1