OPTICS NONLINEAR (PHYC/ECE 568)
Fall 2008 - Instructor: M. Sheik-Bahae University of New Mexico
Homework #3, Due Thursday, Oct. 16
Problem 1. SHG in KDP: (4 points) a. Calculate the type-I phase matching angle for SHG in KDP using 1.06 m output of a Nd:YAG laser. b. For a beam radius w0=500 m, calculate the aperture length defined as la= w0 / where is the Poynting vector walk-off angle. Obtain the aperture length for w0=20 m and discuss the role of additional limitations that may be imposed due to diffraction of the beam.
Problem 2. SHG Bandwidth: (4 points) a. Calculate the bandwidth associated with a phase-matched SHG process in terms of the group velocities vg(1) and vg(21). In the low-depletion approximation, this corresponds to the width of the Sinc2 function which is taken to be (kL)=2 with L denoting the length of the nonlinear crystal. Hint: Use the first-order term in the Taylor series expansion of k(). b. Discuss how your results in (a) explains the limitation on the SHG-efficiency when ultrashort laser pulses are used. Problem 3. What about the fundamental wave? (10 points) Consider the case of a phase-matchable SHG process; but instead of being concerned about the secondharmonic beam (at 2), we would like to determine the fate of the transmitted fundamental field at (see also problem 2.20 in Boyd, 3rd ed.). (a) Start with the coupled amplitude equations (i.e. Eqns. 2.7.10-11 in Boyd). Eliminate A2 to obtain the following nonlinear differential equation for A1:
d 2 A1 dA + ik 1 + 2 A1 2 | A1 / A1 (0) |2 1 = 0 2 d z dz
where 2=4d212|A1(0)|2/(c2n1n2). Now (b) make the low-depletion approximation by setting |A1|2=|A1(0)|2 in the above equation. Solve for A1 for a propagation length L. (Hint: You need a second initial condition that is obtained from E2(0)=0). (c) Taking A1=|A1|ei, plot |A1|/|A1(0)| and versus kL (from - 4 to 4) for 2L2=0.1, 0.2, and 0.4. Discuss your results (i.e. sign reversal vs. k, etc.)
The above process (i.e. the intensity-dependent phase variation of the fundamental wave) has been termed (2):(2) cascading nonlinearity. It mimics a third order (3) process where (3)eff(2)(;2,-)(2)(2;,) is effectively a cascade of two second order effects. The cascading nonlinearity has generated some interest for applications requiring large (3) effects (i.e. optical switching, spatial solitons, and, in general, processes requiring an n2-type nonlinearity). See Sheik-Bahae and Hasselbeck (OSA Handbook, Chapter 17).
Problem 4. Cascading for THG (Third-Harmonic Generation) in KDP. (2 points...