TASK

Nonlinear optics

In Lecture 11, we analyzed nondegenerate optical parametric ampli cation (NOPA) under the undepleted-pump approximation and assuming no linear loss  for all lightwaves involved. In practice, however, the linear losses could be significant.

Let Γs and Γi be the linear loss coefficients per unit length for the signal and the idler waves, respectively, write down the equations of motion for the signal and idler waves undergoing NOPA, assuming undepleted pump and phase matching (10 pts).

When there is no idler input, solve the NOPA dynamics for Γi =0 and the limiting cases of (i) Γs being very small (meaning the loss occurs at a speed much lower than the nonlinear process does); (ii) Γs is predominantly large.  Based on your results, give a rough estimate on how large Γs can be for the signal’s parametric gain to be greater than 1 (20 pts).

In class, we have shown that the parametric gains for the signal wave, g, and the idler wave, g′, satisfy g′ =g−1, when the two waves are degenerate in frequency and are not subject to any linear loss. Does this relation still hold for Γs ≠0? Why?

When there is no idler input, solve the NOPA dynamics for Γs =0 and in the limit of Γi being predominantly large. In this case, can a large parametric gain be obtained for the signal? [Hint, by working out this problem, you have sailed into the wonderland of quantum Zeno effect. For an example of quantum Zeno effect in nonlinear optics, refer to Phys. Rev. A 82, 063826 (2010)] (20 points)

Problem 2 (40 pts):

In lecture 14, we studied propagation of a monochromatic wave in an isotropic nonlinear medium, where we derived the nonlinear refractive indices for left and right circularly polarizations.

(a) Derive nonlinear refractive indices for the two orthogonal linear polarizations .

(b) From your result, will a linearly polarized input wave experience rotation of polarization? Why? How about an initially elliptically polarized wave?