Nonlinear Optics
Homework 3
due Friday, 1 Feb 2013

Problem 1: practice with polarization
A real waveplate that is made of a single piece of quartz (Δn=(Subscript[n, o])-(Subscript[n, e])=0.009) can be thick enough that the designer must aim for the desired phase retardation plus m*(2 π), where m is an integer. A zero-order waveplate is where m = 0. A 10th order half waveplate would have a total phase shift (ω/c)Δn L = 20.5 π.

a) if the design wavelength is 800nm, how thick should a zero-order half-wave plate be? 
b) what order would the half waveplate be if the thickness is approximately 1mm?
c) A way to make a zero-order half waveplate is to combine two waveplates with the optic axis of the second plate rotated (90^o) to the first. If each half of the waveplate is approximately 1mm, calculate the exact thicknesses of the two halves of a zero order half waveplate.

Problem 2:
A nonlinear crystal used for doubling the 3000nm output of an optical parametric amplifier (OPA) is (Subscript[LiIO, 3]) (lithium iodate).

a. Using the Phase_match_KDP.nb notebook as a template, code the dispersion equations for (Subscript[LiIO, 3]) and calculate the variation of (Subscript[n, e]) with angle θ to the z-axis. The Sellmeier equations for (Subscript[LiIO, 3]) are given in the datasheet posted online.

b. Calculate the optimum phase matching angle for doubling 3000nm light with Type I and Type II. If we changed the input wavelength to something shorter, what is the shortest input wavelengths that can be phasematched for the two types of phase matching?

c. To show the angular acceptance of the crystal, plot the normalized harmonic yield (intensity) as a function of incident angle for three thicknesses (this is the (sinc^2)(Δk L/2) function. Put all three thicknesses on the same plot. Make a plot like this for type I, type II (input wavelength 3μm) and type III for the input wavelength that is at the extreme allowed by type I.

d. To show the phase matching bandwidth, plot the normalized harmonic yield as a function of input wavelength for the same three thicknesses. Use an incident angle corresponding to optimum phase matching at 3μm.

Problem 3
Working with the d-matrix for BBO to calculate the effective nonlinearity.

BBO has a 3m symmetry, with the Kleinman symmetry valid. The contracted notation for the d-matrix can be written as:

d3m={
{0,0,0,0,d31,-d22},
{-d22,d22,0,d31,0,0},
{d31,d31,d33,0,0,0}};
Print["d matrix 3m: ",MatrixForm[d3m]];

in contracted notation, there is only one matrix, and the two electric fields are combined into a 6 element vector:
These combinations for sum frequency mixing are shown in eqn 1.5.27

E1E2Vector[E1_,E2_]:={(Subscript[E1, [[1]]]) (Subscript[E2, [[1]]]),(Subscript[E1, [[2]]]) (Subscript[E2, [[2]]]),(Subscript[E1, [[3]]]) (Subscript[E2, [[3]]]),
(Subscript[E1, [[2]]]) (Subscript[E2, [[3]]])+(Subscript[E1, [[3]]]) (Subscript[E2, [[2]]]),(Subscript[E1, [[1]]]) (Subscript[E2, [[3]]])+(Subscript[E1, [[3]]]) (Subscript[E2, [[1]]]),(Subscript[E1, [[1]]]) (Subscript[E2, [[2]]])+(Subscript[E1, [[2]]]) (Subscript[E2, [[1]]])};
E1={E1x,E1y,E1z};
E2={E2x,E2y,E2z};
E1E2General=E1E2Vector[E1,E2];
Print["general form for E1E2 vector: ",MatrixForm[E1E2General]];

The nonlinear polarization is the matrix product of the d-matrix with this 6 element E1E2 vector.

Print["general form for output polarization vector: ",MatrixForm[d3m.E1E2General]];

Example: if the input field is polarized only along the x direction, we get NL polarization along y and z.

E1={A1x,0,0};
E2={A2x,0,0};
E1E2=E1E2Vector[E1,E2];
Print[MatrixForm[d3m],".",MatrixForm[E1E2]," = ",MatrixForm[d3m.E1E2]];

Interpretation: in this case, if E1 and E2 were the same field, a second harmonic signal would be generated along the y and z axes. Note that specifying the input polarization direction does not fully specify the wave direction. So if the wave direction (k) is in the y direction, the second harmonic would be generated polarized in the z direction.

a) For the cases where the input field polarization is along y, calculate the NL polarization vector. Do the same for input along z and the case with input A1x and A2y.

b) For what crystal orientations should Type I phase matched generation of SH be possible? Recall that BBO is negative uniaxial.

Problem 4: 
The commonly-used nonlinear crystal BBO (beta- barium borate) is a uniaxial crystal with nonlinear coefficients (Subscript[d, 31]) = 0.16 pm/V and (Subscript[d, 22]) = 2.2 pm/V. The effective nonlinear coefficients as a function of the direction of the fundamental beam for this crystal are given in equation 1.5.30. The angle θ is fixed according to phase matching, but we are free to choose whatever value of ϕ that will optimize the magnitude of the nonlinearity |(Subscript[d, eff])|.

a) For an input wavelength of 800nm, the type I phase matching angle is θ = 29 deg. Plot (Subscript[d, eff])vs ϕ and determine the value of ϕ that gives the largest magnitude of (Subscript[d, eff]) (sign is unimportant).

b) Do the same for type II, where the angle is θ = 42.3 deg. Which phase matching type gives a higher nonlinearity?