The Sommerfeld precursor

Let us consider the situation immediately after the arrival of the signal; i.e., when $s$ is small and positive. Let us start from Eq. (4.102), which can be written in the form

$$f(x,t) = \frac{1}{\tau}\int_C {\rm e}^{{\rm i}\,([k-\omega/c]x-\omega s)} \frac{d\omega}{\omega^2 -(2\pi/\tau)^2}.$$ (740)

We can deform the original path of integration $C$ into a large semi-circle of radius $R$ in the upper half-plane, plus the segments of the real axis, as shown in Fig. 9. Because of the denominator $\omega^2-(2\pi/\tau)^2$, the integrand tends to zero as $1/\omega^2$ on the real axis. We may add the path in the lower half plane which is shown as a dotted line in the figure, for if the radius of the semi-circular portion of this lower path is increased to infinity, the integrand vanishes exponentially because $s>0$. Therefore, we may replace our original path of integration by the entire circle S. Thus,

$$f(x,t) = \frac{1}{\tau}\oint_S{\rm e}^{{\rm i}\,([k-\omega/c]x-\omega s)} \frac{d\omega}{\omega^2 -(2\pi/\tau)^2}$$ (741)

in the limit that the radius of the circle $R$ tends to infinity.

Figure 9: Sketch of the integration contour used to evaluate Eq. (4.107)

The dispersion relation (4.86) yields

$$k -\frac{\omega}{c} \simeq \frac{\omega}{c} \left(\sqrt{1-\f... ...ga^2}} - 1\right) \simeq - \frac{\omega_p^{~2}}{2\,c\,\omega}$$ (742)

in the limit $\vert\omega\vert \rightarrow\infty$. Using the abbreviation

$$\xi = \frac{\omega_p^{~2}}{2\,c}\, x,$$ (743)

and henceforth neglecting $2\pi/\tau$ with respect to $\omega$, we obtain from Eq. (4.107)

$$f(x,t) = f_1(\xi,t) \simeq\frac{1}{\tau} \oint_S \exp\left[\... ...i}{\omega} -\omega\, s\right)\right]\frac{d\omega} {\omega^2}.$$ (744)

This expression can also be written

$$f_1(\xi,t) = \frac{1}{\tau} \oint_S \exp\left[\,-{\rm i}\, \... ...a \sqrt{\frac{s}{\xi}}\right)\right] \frac{d\omega}{\omega^2}.$$ (745)

Let

$$\omega \sqrt{\frac{s}{\xi}} = {\rm e}^{{\rm i}\,u}.$$ (746)

It follows that

$$\frac{d\omega}{\omega} = {\rm i} \,du,$$ (747)

giving

$$\frac{d\omega}{\omega^2} = {\rm i} \sqrt{\frac{s}{\xi}} \,{\rm e}^{-{\rm i} \,u} \,du.$$ (748)

Substituting the angular variable $u$ for $\omega$ as the integration variable in Eq. (4.111) yields

$$f_1(\xi, t) = \frac{{\rm i}}{\tau} \sqrt{\frac{s}{\xi}}\int_... ...(-2{\rm i} \sqrt{\xi s}\, \cos u)\,{\rm e}^{-{\rm i}\,u}\, du.$$ (749)

Here, we have taken $\sqrt{\xi/s}$ as the radius of the circular integration path in the $\omega$-plane. This is indeed a large radius, since $s\ll 1$. From symmetry, Eq. (4.115) simplifies to

$$f_1(\xi, t) = \frac{{\rm i}}{\tau} \sqrt{\frac{s}{\xi}}\int_0^{2\pi} \exp(-2{\rm i} \sqrt{\xi s}\, \cos u)\,\cos u\, du.$$ (750)

The following mathematical identity is very well-known¹¹

$$J_n(z) = \frac{{\rm i}^{-n}}{2\pi}\int_0^{2\pi} {\rm e}^{{\rm i}\,z\,\cos\theta} \cos(n\theta)\,d\theta,$$ (751)

where $J_n(z)$ is Bessel function of order $n$. It follows from Eq. (4.115) that

$$f_1(\xi, t) = \frac{2\pi}{\tau} \sqrt{\frac{s}{\xi}} \,J_1(2\sqrt{\xi s}).$$ (752)

Here, we have made use of the fact that $J_1(-z) = - J_1(z)$.

Figure 10: The Bessel function $J_1(z)$

The properties of Bessel functions are well-known and are listed in many standard references on mathematical functions (see, for instance, Abramowitz and Stegun). In the small argument limit $z\ll 1$ we find that

$$J_1(z) = \frac{z}{2} + O(z^3).$$ (753)

On the other hand, in the large argument limit $z\gg 1$ we obtain

$$J_1(z) = \sqrt{\frac{2}{\pi z}} \cos(z- 3\pi/4) + O(z^{-3/2}).$$ (754)

The behaviour of $J_1(z)$ is further illustrated in Fig. 10.

We are now in a position to make some quantitative statements regarding the signal which first arrives at depth $x$ in the dispersive medium. This signal propagates at the velocity of light in vacuum and is called the Sommerfeld precursor. The first important point to note is that the amplitude of the Sommerfeld precursor is very small compared to that of the incident wave (whose amplitude is normalized to unity). We can easily see this because in deriving Eq. (4.118) we assumed that $\vert\omega\vert=\sqrt{\xi/s}\gg 2\pi/\tau$ on the circular integration path $S$. Since the magnitude of $J_1$ is always less than, or of order, unity, it is clear that $\vert f_1\vert\ll 1$. This is a comforting result, since in a naive treatment of wave propagation through a dielectric medium the wave front propagates at the group velocity $v_g$ (which is usually less than $c$) and, therefore, no signal should reach depth $x$ in the medium before time $x/v_g$. We are finding that there is, in fact, a precursor which arrives at $t=x/c$, but that this signal is fairly small. Note from Eq. (4.109) that $\xi$ is proportional to $x$. Clearly, the amplitude of the Sommerfeld precursor decreases like one over the distance traveled by the wave front through the dispersive medium (since $J_1$ attains its maximum value when $s\sim 1/\xi$). Thus, the Sommerfeld precursor is likely to become undetectable after the wave has traveled a long distance through the medium.

Figure 11: The Sommerfeld precursor

Equation (4.118) can be written

$$f_1(\xi, t) = \frac{\pi}{\xi\,\tau} \,g(s/s_0),$$ (755)

where $s_0 = 1/4\,\xi$, and

$$g(z) = \sqrt{z} \,J_1(\sqrt{z}).$$ (756)

The normalized Sommerfeld precursor $g(z)$ is shown in Fig. 11. It can be seen that both the amplitude and the oscillation period of the precursor gradually increase. The roots of $J_1(z)$ [i.e., the solutions of $J_1(z)=0$] are spaced at distances of approximately $\pi$ apart. Thus, the time interval for the $m$th half period of the precursor is approximately given by

$$\Delta t_m \sim \frac{m \pi^2}{2\xi}.$$ (757)

Note that the initial period of oscillation,

$$\Delta t_0 \sim \frac{ \pi^2}{2\xi},$$ (758)

is extremely small compared to the incident period $\tau$. Moreover, the initial period of oscillation is completely independent of the frequency of the incident wave. In fact, $\Delta t_0$ depends only on the depth $x$ and on the dispersive power of the medium. The period decreases with increasing distance $x$ traveled by the wave front though the medium. So, when visible radiation is incident on some dispersive medium it is quite possible for the first signal detected well inside the medium to lie in the X-ray region of the electromagnetic spectrum.