The method of stationary phase

Equation (4.102) can be written in the form

$$f(x,t) = \int_C {\rm e}^{{\rm i} \,\phi(\omega)} F(\omega) \,d\omega$$ (759)

where

$$F(\omega) = \frac{1}{\tau}\frac{1}{\omega^2-(2\pi/\tau)^2},$$ (760)

and

$$\phi(\omega) = k(\omega)\, x - \omega t.$$ (761)

It is clear that $F(\omega)$ is a relatively slowly varying function of $\omega$ (except in the immediate vicinity of the singular points $\omega=\pm 2\pi/\tau$), whereas the phase $\phi(\omega)$ is generally large and rapidly varying. The rapid oscillations of $\exp(\,{\rm i}\,\phi)$ over most of the range of integration means that the integrand averages to almost zero. Exceptions to this cancellation rule occur only when $\phi(\omega)$ is stationary; i.e., when $\phi(\omega)$ has an extremum. The integral can therefore be estimated by finding places where $\phi(\omega)$ has a vanishing derivative, evaluating (approximately) the integral in the neighbourhood of each of these points, and summing the contributions. This procedure is called the method of stationary phase.

Suppose that $\phi(\omega)$ has a vanishing first derivative at $\omega=\omega_s$. In the neighbourhood of this point, $\phi(\omega)$ can be expanded as a Taylor series,

$$\phi(\omega) = \phi_s + \frac{1}{2} \phi''_s(\omega-\omega_s)^2+\cdots.$$ (762)

Here, the subscript $s$ is used to indicate $\phi$ or its second derivative evaluated at $\omega=\omega_s$. Since $F(\omega)$ is slowly varying, the contribution to the integral from this stationary phase point is approximately

$$f_s \simeq F(\omega_s) {\em e}^{{\rm i}\,\phi_s} \int_{\inft... ...ty} {\rm e}^{({\rm i}/2)\phi_s''(\omega-\omega_s)^2}\,d\omega.$$ (763)

It is tacitly assumed that the stationary point lies on the real axis in $\omega$-space, so that locally the integral along the contour $C$ is an integral along the real axis in the direction of decreasing $\omega$. The above expression can be written in the form

$$f_s\simeq -F(\omega_s) \,{\rm e}^{{\rm i}\,\phi_s} \sqrt{\fr... ...ty \left[ \cos(\pi t^2/2)+{\rm i}\,\sin(\pi t^2/2)\right]\,dt,$$ (764)

where

$$\frac{\pi}{2}\, t^2 = \frac{1}{2}\, \phi_s'' \,(\omega-\omega_s)^2.$$ (765)

The integrals in the above expression are, Fresnel integrals¹² and can be shown to take the values

$$\int_0^\infty\cos(\pi t^2/2)\,dt = \int_0^\infty\sin(\pi t^2/2)\,dt =\frac{1}{2}.$$ (766)

It follows that

$$f_s \simeq - \sqrt{\frac{2\pi\,{\rm i}}{\phi_s''}} \, F(\omega_s) \,{\rm e}^{\,{\rm i}\,\phi_s}.$$ (767)

It is easily seen that the arc length (in $\omega$-space) of the integration contour which makes a significant contribution to $f_s$ is of order $\Delta \omega/\omega_s \sim 1/\sqrt{k(\omega_s)\, x}$. Thus, the arc length is relatively short provided that the wavelength of the signal is much less than the distance propagated through the dispersive medium. If there is more than one point of stationary phase in the range of integration then the integral is approximated as a sum of terms like the above.

Integrals of the form (4.125) can be calculated exactly using the method of steepest decent.¹³ The stationary phase approximation (4.133) agrees with the leading term of the method of steepest decent (which is far more difficult to implement than the method of stationary phase) provided that $\phi(\omega)$ is real (i.e., provided that the stationary point lies on the real axis). If $\phi$ is complex, however, the stationary phase method can yield erroneous results. This suggests that the stationary phase method is likely to break down when the extremum point $\omega=\omega_s$ approaches any poles or branch cuts in the $\omega$-plane (see Fig. 8).