Nonlinear Optics

1. (50 points) The equations of motion describing nonlinear coupling of three plain waves in a lossless χ(2) medium are given in the slowly-varying envelope approximation as
(1)
(2)
(3)
where ωj and nj are the angular frequency and refractive index, respectively, of jth wave (j = 1,2,3). K is the effective second-order susceptibility. c is the speed of light in vacuum. ∆k is the phase mismatch per unit length.

(a). What kind of nonlinear processes do those dynamical equations describe? (10 points)

(b). Use those equations of motion to show that the following quantities are constants of motion (30 points)
(4)
(5)
(6)
(7)

The above equations are known as the Manly-Rowe relations.

(c). Give a physical explanation as to why Manly-Rowe relations hold in lossless media (10).

2. (50 points) Equations (1)-(3) are for a lossless χ(2) medium. Now consider a χ(2) media with linear propagation loss for the ω3 wave only. The equations of motion then become
(8)
(9)
, (10)
where Γ is the loss coefficient per unit length.

(a) Are W, M1,M2 still constants of motion? How about M3? Show your mathematical derivations.
(20)

(b) What are the physical explanations of your results above? (15)
(c) Now consider a similar medium but with linear loss for the ω1 wave only. Do any of W, M1, M2, M3 remain constants of motion? Why? (15)