A.2 - The WKB approximation

Summary

In Chapter 3 we considered OPAs in which the parametric gain coefficient g is a constant along the fiber. This allowed us to obtain closed-form solutions for the gain in several important situations. In practice, however, fibers generally have properties that may cause g to vary as a function of z. Examples of these properties are: fiber loss, which causes the pump power to drop exponentially; non-uniform dispersion, which causes Δβ to fluctuate; birefringence (fixed or randomly varying), which introduces a complex evolution of SOPs. Under these circumstances the constant-g solutions are no longer applicable.

In several areas of physics that involve wave propagation in media with slowly varying properties, one often uses approximate solutions derived by making use of the Wentzel–Kramers–Brillouin (WKB) approximation, also referred to as the phase-integral method [1, 2]. This method was originally introduced in quantum mechanics. In optics, it has been used extensively to study propagation in multimode fibers [3]. It can lead to closed-form solutions if the properties vary in a simple fashion, such as linearly. If the variations are not simple, one may still be able to use the method to obtain some useful expressions involving the average of g along the fiber.

Karlsson first applied the WKB method to fiber OPAs, in the context of modulation instability (MI) [4]. Here we present a slightly different version, starting from the basic OPA equations.