Nonlinear Optics
Homework 6
due Friday, 29 March 2013

Problem 1: 
Nonlinear ellipse rotation for intensity contrast enhancement. 

The following sequence of polarization optics can be used to suppress low intensity wings of a pulse while transmitting high intensity. The input polarization is linear in the x direction. A quarter waveplate at a rotation angle (Subscript[ϕ, 1]) is used to change to elliptical polarization. The beam then passes through a cell filled with carbon disulfide ((Subscript[CS, 2])) where nonlinear propagation rotates the polarization ellipse by an amount proportional to the intensity. A second quarter waveplate is place at the cell exit with an orientation -(Subscript[ϕ, 1]) so that at low intensity, the light returns to the original x polarization. Finally, a polarizer is placed at the end so that it transmits y-polarized light. At low intensity nothing gets through, but at higher intensity the polarization rotates enough to transmit. 

a. Use the matrices defined in the polarization_tools.nb notebook to construct the linear terms for the sequence that operates on the input polarization. Define a function transm[ϕ1_]=((Abs[vout[[2]]])^2) that represents the output intensity as a function of the rotation angles of the quarter wave plates. 

b. Now use the expressions in Boyd (4.2.28 with θ defined through 4.2.26) to define a matrix (which is a function of intensity through 4.2.24) that represents the operation of the nonlinear ellipse rotation. 

c. Plot the transmission through the polarizer vs input intensity and determine what intensity is required for complete transmission. 

d. If the pulse is preceded by some ASE that is four orders of magnitude less intense, what is the ratio of the output intensity at the peak to the transmitted ASE? (The primary limitation turns out to be the quality of the waveplates.)

Problem 2:
Boyd problem 4.6 Solve the simpler case where (Subscript[D, 1])=(Subscript[D, 2]). In this case, you should find that the temperature profile (and therefore the induced refractive index profile) is parabolic. This parabolic index profile acts as a lens. Calculate the phase shift after propagation through the rod, then compare your expression to that of a thin lens,
 ϕ(r)=-k (r^2)/2 f, thereby calculating the effective focal length of the thermally induced lens.

For typical parameters, use the parameters for fused silica (Table 4.5.1):

cm=0.01m;W=1;Kelvin=1;
Qin=10W/(cm^3);
κ=1. W/(m Kelvin);
a=1 cm;
len=1cm;
dndT=1.2 (10^(-5))/Kelvin;

Problem 3:
Boyd problem 4.7, parts (a) and (b).

Problem 4:
Boyd, problem 4.8
Here you assume that the intensity dependence of the beam is unaffected by the nonlinearity. This means you don’t account for self-focusing and you use the Gaussian beam formula for the variation of the beam size with position.

Problem 5: 
Consider wave mixing in a square cross-section gas-filled hollow waveguide. 

Assuming the field goes to zero at the boundaries of the waveguide, the propagation constant and the eigenmodes for a gas-filled waveguide are calculated as follows: 

From the vector relationship for the wavevector components: 
((Subscript[k, z])^2)=((Subscript[k, 0])^2)(n^2)-((Subscript[k, x])^2)-((Subscript[k, y])^2)

where (Subscript[k, 0])=(ω/c) and n is the refractive index of the gas:
(n^2)=1+p dn , where dn > 0
and
(Subscript[k, x]) = (Subscript[m, x]) π/L
(Subscript[k, y]) = (Subscript[m, y]) π/L
where (Subscript[m, x])=(Subscript[m, y])=integers ≥1. 
The transverse mode profiles follow the functions: 
E(x,y)∝(Subscript[a, 0])Sin[(((Subscript[m, x])π x)/L)]Sin[(((Subscript[m, y])π y)/L)]
These functions are not normalized. 

This results in:
(Subscript[k, z])=β=(Sqrt[((Subscript[k, 0])^2)(1+p dn)-((((Subscript[m, x]) π/L))^2)-((((Subscript[m, y]) π/L))^2)])

Note that the deviation from the vacuum wavenumber is very small. We can expand the square root out: 
β=(Subscript[k, 0])(Sqrt[1+p dn-((((Subscript[m, x]) π/(Subscript[k, 0]) L))^2)-((((Subscript[m, y]) π/(Subscript[k, 0]) L))^2)])
= (Subscript[k, 0])(1+p dn/2-(1/2)((((π/((Subscript[k, 0]) L))))^2)(((Subscript[m, x])^2)+((Subscript[m, y])^2)))

a. Using this information, derive a phase matching condition for third-harmonic generation. Assuming the fundamental is in a single, well-defined mode, to what modes can the phase matching be satisfied for the 3rd harmonic? Note that the pressure is variable, p > 0, and can be tuned to satisfy the phase matching condition.

b. Now consider a 4-wave mixing experiment, where the second harmonic is mixed with the fundamental to produce the 3rd harmonic: (Subscript[ω, 3])=2(Subscript[ω, 2])-(Subscript[ω, 1]). You should see that it is possible to achieve phase matching if all beams are in the lowest order spatial mode. 

It is not necessary to evaluate the phase-matching pressure, but if you are interested, an expression for the dispersion of argon is below.

Argon dispersion equations (good down to 110 nm)
wavelength in microns

dnAr[λ_]:=2*120.625 (10^(-6)) ((14.0/(87.892-(((λ))^(-2))))+(14.0/(91.011-(((λ))^(-2))))+(430.63/(217.6-(((λ))^(-2)))));
nAr[λ_,pr_]=1+pr dnAr[λ];