Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.7 WKB METHODS

This still requiresk 0 n′(x) to be small (compared with, say,k^20 n^2 (x)), but is some

improvement (not least in complexity!) on (25.44) and gives some measure of the

conditions under which the solution might be a suitable approximation.

The integral in equation (25.45) embodies what is sometimes referred to as the

phase memoryapproach; it expresses the notion that the phase of the wave-like

solution is the cumulative effect of changes it undergoes as it passes through the

medium. If the medium were uniform the overall change would be proportional

tonx, as in (25.43); the extent to which it is not uniform is reflected in the amount

by which the integral differs fromnx.

The condition for solution (25.45) to be a reasonable approximation can be

written asn′k 0 −^1 n^2 or, in words, the change innover anx-range ofk− 01 should

be small compared withn^2. For light in an optical medium, this means that the

refractive indexn, which is of the order of unity, must change very little over a

distance of a few wavelengths.

For some purposes the above approximation is adequate, but for others further

refinement is needed. This comes from considering solutions that are still wave-

like but have amplitudes, as well as phases, that vary with position. These are the

WKB solutions developed and studied in the next three subsections.

25.7.2 Constructing the WKB solutions

Having formulated the notion of phase memory, we now construct the WKB

solutions of the general equation (25.42), in whichf(x) can now be both position-

dependent and complex. As we have already seen, it is the possibility of a complex

phase that permits the existence of wave-like solutions with varying amplitudes.

Sincen(x) is calculated as the square root off(x), there is an ambiguity in its

overall sign. In physical applications this is normally resolved unambiguously by

considerations such as the inevitable increase in entropy of the system, but, so far

as dealing with purely mathematical questions is concerned, the ambiguity must

be borne in mind.

The process we adopt is an iterative one based on the assumption that the

second derivative of the complex phase with respect toxis very small and can

be approximated at each stage of the iteration. So we start with equation (25.42)

and look for a solution of the form

y(x)=Aexp[iφ(x)], (25.46)

whereAis a constant. When this is substituted into (25.42) the equation becomes
[

−

(

dφ

dx

) 2

+i

d^2 φ

dx^2

+k 02 n^2 (x)

]

y(x)=0. (25.47)

Setting the quantity in square brackets to zero produces a non-linear equation for