The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$ , $S\subset \mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 \in \mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $\omega$ , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.

An infrared laser propagating through a semiconductor waveguide, embedded with a density ripple of wave number q, resonantly excites third harmonic radiation. The phase matching is achieved when q equals difference between third harmonic wave number and three times the wave number of the laser. The excited third harmonic is in higher order radial Eigen mode than the pump laser. The third harmonic efficiency increases with electron concentration and decreases with the frequency of the laser.