The W.K.B. solutions as asymptotic series

We have seen that the W.K.B. solution

$$E_y = n^{-1/2} \,\exp\left(\pm {\rm i}\, k \int^z n\,dz\right)$$ (905)

is an approximate solution of the differential equation

$$\frac{d^2 E_y}{dz^2} + k^2n^2(z)\,E_y = 0$$ (906)

in the limit where the typical wavelength, $2\pi/n k$, is much smaller than the typical variation length-scale of the refractive index. But, what sort of approximation is involved in writing this solution?

It is convenient to define the scaled variable

$$\hat{z} = \frac{z}{L},$$ (907)

where $L$ is the typical variation length-scale of the refractive index, $n(z)$. Equation (4.254) can then be written

$$w'' + h^2 \,q\, w = 0,$$ (908)

where $w(\hat{z},h)\equiv E_y(L \,\hat{z})$, $q(\hat{z}) \equiv n^2(L\, \hat{z})$, $'\equiv d/d\hat{z}$, and $h= kL$. Note that, in general, $q(\hat{z})$, $q'(\hat{z})$, $q''(\hat{z})$, etc. are $O(1)$ quantities. The non-dimensional constant $h$ is of order the ratio of the variation length-scale of the refractive index to the wavelength. Let us seek the solutions to Eq. (4.256) in the limit $h\gg 1$.

We can write

$$w(\hat{z}, h) = \exp\left[\,{\rm i}\,h\,\phi(\hat{z}, h)\right].$$ (909)

Equation (4.256) transforms to

$$\frac{\rm i}{h} \,\phi'' - (\phi')^2 + q = 0.$$ (910)

Expanding in powers of $1/h$, we obtain

$$\phi' = \pm q^{1/2} + \frac{\rm i}{4 h} \frac{q'}{q} + O\left(\frac{1}{h^2}\right),$$ (911)

which yields

$$w(\hat{z}, h) = q^{-1/4} \,\exp\left( \pm {\rm i}\, h\int^{\... ...}} q\,d\hat{z}\right)\left[1+O\left(\frac{1}{h}\right)\right].$$ (912)

Of course, we immediately recognize this expression as a W.K.B. solution.

Suppose that we keep expanding in powers of $1/h$ in Eq. (4.259). The appropriate generalization of Eq. (4.260) is a series solution of the form

$$w(\hat{z}, h) = q^{-1/4} \,\exp\left( \pm {\rm i}\, h\int^{\... ...ght)\left[1+\sum_{p=1}^\infty \frac{A_p(\hat{z})}{h^p}\right].$$ (913)

This is, in fact, an asymptotic series in $h$. We can now appreciate that a W.K.B. solution is just a highly truncated asymptotic series in $h$, in which only the first term in the series is retained.

But, why is it so important that we recognize that W.K.B. solutions are highly truncated asymptotic series? The point is that the W.K.B. method was initially rather controversial after it was popularized in the 1920s. A lot of people thought that the method was completely wrong. Let us try to understand what the problem was. Suppose that we have never heard of an asymptotic series. Looking at Eq. (4.261), we would imagine that the expression in square brackets is a power law expansion in $1/h$. The W.K.B. approximation involves neglecting all terms in this expansion except the first. This sounds fine, as long as $h$ is much greater than unity. But, surely, to be mathematically rigorous, we have to check that the sum of all of the terms in the expansion which we are neglecting is small compared to the first term? However, if we attempt this we discover, much to our consternation, that the expansion is divergent. In other words, the sum of all of the neglected terms is infinite! Thus, if we interpret Eq. (4.261) as a conventional power law expansion in $1/h$, the W.K.B. method is clearly nonsense: the W.K.B. solution is the first approximation to infinity. However, once we appreciate that Eq. (4.261) is actually an asymptotic series in $h$, the fact that the series diverges becomes irrelevant. If we retain the first $n$ terms in the series, the series approximates the exact solution of Eq. (4.261) with an intrinsic (fractional) error which is of order $1/h^n$ (i.e., the first neglected term in the series). The error is minimized at a particular value of $h$. As the number of terms in the series is increased, the intrinsic error decreases, and the value of $h$ at which the error is minimized increases. In particular, we can see that there is an intrinsic error associated with a W.K.B. solution which is of order $1/h$ times the solution.

It is amusing to note that if Eq. (4.261) were not a divergent series then it would be impossible to obtain total reflection of the W.K.B. solutions at the point $q=0$. As we shall discover, the reflection is directly associated with the fact that the expansion (4.261) exhibits a Stokes phenomenon. It is, of course, impossible for a convergent power series expansion to exhibit a Stokes phenomenon.