Mathematical Methods for Physics and Engineering - Matematica.NET

25.7 WKB METHODSThis still requires k 0 n ′ (x) to be small (compared with, say, k0 2n2 (x)), but is someimprovement (not least in complexity!) on (25.44) and gives some measure of theconditions under which the solution might be a suitable approximation.The integral in equation (25.45) embodies what is sometimes referred to as thephase memory approach; it expresses the notion that the phase of the wave-likesolution is the cumulative effect of changes it undergoes as it passes through themedium. If the medium were uniform the overall change would be proportionalto nx, as in (25.43); the extent to which it is not uniform is reflected in the amountby which the integral differs from nx.The condition for solution (25.45) to be a reasonable approximation can bewritten as n ′ k0 −1 ≪ n 2 or, in words, the change in n over an x-range of k0 −1 shouldbe small compared with n 2 . For light in an optical medium, this means that therefractive index n, which is of the order of unity, must change very little over adistance of a few wavelengths.For some purposes the above approximation is adequate, but for others furtherrefinement is needed. This comes from considering solutions that are still wavelikebut have amplitudes, as well as phases, that vary with position. These are theWKB solutions developed and studied in the next three subsections.25.7.2 Constructing the WKB solutionsHaving formulated the notion of phase memory, we now construct the WKBsolutions of the general equation (25.42), in which f(x) can now be both positiondependentand complex. As we have already seen, it is the possibility of a complexphase that permits the existence of wave-like solutions with varying amplitudes.Since n(x) is calculated as the square root of f(x), there is an ambiguity in itsoverall sign. In physical applications this is normally resolved unambiguously byconsiderations such as the inevitable increase in entropy of the system, but, so faras dealing with purely mathematical questions is concerned, the ambiguity mustbe borne in mind.The process we adopt is an iterative one based on the assumption that thesecond derivative of the complex phase with respect to x is very small and canbe approximated at each stage of the iteration. So we start with equation (25.42)and look for a solution of the formy(x) =A exp[ iφ(x)], (25.46)where A is a constant. When this is substituted into (25.42) the equation becomes[ ( ) ]2 dφ− + i d2 φdx dx 2 + k2 0n 2 (x) y(x) =0. (25.47)Setting the quantity in square brackets to zero produces a non-linear equation for897