On the nature of the Sommerfeld–Brillouin forerunners (or precursors)

Abstract

We present a brief overview of Sommerfeld’s forerunner signal, which occurs when a monochromatic plane-wave (frequency $\omega =\omega _{\text {s}})$ suddenly arrives, at time $t=0$ and at normal incidence, at the surface of a dispersive dielectric medium of refractive index $n\left( \omega \right)$. Deep inside the dielectric host at a distance $z_0$ from the surface, no signal arrives until $t=z_0/c$, where c is the speed of light in vacuum. Immediately after this point in time, however, a weak but extremely high frequency signal is observed at $z=z_0$. This so-called Sommerfeld forerunner (or precursor) is highly chirped, meaning that its frequency, which is much greater than $\omega _{\text {s}}$ immediately after $t=z_0/c$, declines rapidly with the passage of time. The incident light with its characteristic frequency $\omega _{\text {s}}$ eventually arrives at $t\cong z_0 /v_{\text {g}}$, where $v_{\text {g}}$ is the group velocity of the incident light inside the host medium—it is being assumed here that $\omega _{\text {s}}$ is outside the anomalous dispersion region of the host. Brillouin has identified a second forerunner that occupies the interval between the end of the Sommerfeld forerunner at $t\cong n\left( 0 \right) z_0 /c$ and the beginning of the steady signal (i.e., that which has the incident frequency $\omega _{\text {s}}$) at $t=z_0 /v_{\text {g}}$. This second forerunner, which is also weak and chirped, having a frequency that is well below $\omega _{\text {s}}$ at first, then grows rapidly in time to reach $\omega _{\text {s}}$, is commonly referred to as the Brillouin forerunner (or precursor). Given that the incident wave has a sudden start at $t=0$, its frequency spectrum spans the entire range of frequencies from $-\infty$ to $\infty$. Consequently, the high-frequency first forerunner cannot be considered a superoscillation, nor can the low-frequency second forerunner be regarded as a suboscillation. The goal of the present paper is to extend the Sommerfeld–Brillouin theory of precursors to bandlimited incident signals, in an effort to determine the conditions under which these precursors would continue to exist, and to answer the question as to whether or not such precursors, upon arising from a bandlimited incident signal, constitute super- or suboscillations.

Appendices

Appendix A

At the location of the right-hand saddle, that is, at $\omega _{\mathrm{saddle}} \cong \left[ {\omega _{\text {p}} /\sqrt{2\left( {\tau -1} \right) }\,} \right] -\hbox {i}\gamma$, assuming that $\tau \gtrsim 1$, the exponent of the integrand in Eq. (1) can be approximated as follows:

(A1)

The real part of the above expression is the exponent of the damping factor introduced by Brillouin [1, 5] as a correction to Sommerfeld’s precursor amplitude—since Sommerfeld’s original derivation required that $\gamma$ be set to zero. The imaginary part of the exponent in Eq. (A1) eventually becomes the chirped frequency of oscillations that appears in Eq. (10).

Brillouin’s damping factor is precisely what one would obtain by allowing the frequency content of the incident spectrum at $\omega =\omega _{\mathrm{spc}} =\omega _{\text {p}} /\sqrt{2\left( {\tau -1} \right) }$ to propagate directly to the observation point at $z=z_0$. To see this, note that the real part of the exponent of the propagation factor at $\omega =\omega _{\mathrm{spc}}$ is given by

$$\begin{aligned} {\text {Re}}\left[ {{\text {i}}\zeta \omega _{{{\text {spc}}}} n\left( {\omega _{{{\text {spc}}}} } \right) } \right]= & {} - \zeta \omega _{{{\text {spc}}}} n^{\prime \prime }\left( {\omega _{{{\text {spc}}}} } \right) = - \zeta \omega _{{{\text {spc}}}} ~{\text {Im}}\left( {\sqrt{1 + \frac{{\omega _{p}^{2} }}{{\omega _{r}^{2} ~ - ~\omega _{{{\text {spc}}}}^{2} ~ - ~{\text {i}}\gamma \omega _{{{\text {spc}}}} }}} } \right) \nonumber \\\cong & {} -\zeta \omega _{\mathrm{spc}} \hbox { Im}\left[ {1-\frac{\omega _{\text {p}}^2 }{2\left( {\omega _{\mathrm{spc}}^2 \,+\,\hbox {i}\gamma \omega _{\mathrm{spc}} } \right) }} \right] =\frac{1}{2}\zeta \,\hbox {Im}\left( {\frac{\omega _{\text {p}}^2 }{\omega _{\mathrm{spc}} \,+\,\hbox {i}\gamma }} \right) \nonumber \\\cong & {} \frac{1}{2}\zeta \,\hbox {Im}\left[ {\frac{\omega _{\text {p}}^2 }{\omega _{\mathrm{spc}} \,}\left( {1-\frac{\hbox {i}\gamma }{\omega _{\mathrm{spc}} }} \right) } \right] =-\frac{1}{2}\gamma \zeta \left( {\omega _{\text {p}} /\omega _{\mathrm{spc}} } \right) ^{2}\nonumber \\= & {} -\gamma \zeta \left( {\tau -1} \right) =-\gamma \left( {t-z_0 /c} \right) . \end{aligned}$$

(A2)

The equivalence of the damping factor obtained by Brillouin’s saddle-point approximation of the integral in Eq. (1), and that obtained by direct propagation of the incident spectral content at $\omega =\omega _{\mathrm{spc}}$ (albeit at the corresponding group velocity $v_{\text {g}} \left( {\omega _{\mathrm{spc}} } \right) )$, is further confirmation that the Sommerfeld precursor is not a manifestation of superoscillatory behavior, but rather a systematic temporal arrangement of the frequencies that are already present in the incident waveform.

Appendix B

The derivative with respect to $\omega$ of the refractive index $n\left( \omega \right)$ of Eq. (2) is evaluated as follows:

$$\begin{aligned} \dot{n}\left( \omega \right)= & {} \frac{{{\text {d}}n\left( \omega \right) }}{{{\text {d}}\omega }} = \frac{{\text {d}}}{{{\text {d}}\omega }}\left[ {{{\left( {\omega - {\omega _a}} \right) }^{\frac{1}{2}} }{{\left( {\omega + \omega _a^*} \right) }^{\frac{1}{2}} }{{\left( {\omega - {\omega _b}} \right) }^{ -\frac{1}{2}}}{{\left( {\omega + \omega _b^*} \right) }^{-\frac{1}{2}}}} \right] \nonumber \\= & {} \frac{1}{2}\left( {\omega -\omega _a } \right) ^{-\frac{1}{2}}\left( {\omega +\omega _a^*} \right) ^{-\frac{1}{2}}\left( {\omega -\omega _b } \right) ^{-\frac{1}{2}}\left( {\omega +\omega _b^*} \right) ^{-\frac{1}{2}}\nonumber \\&\times \left[ {\left( {\omega +\omega _a^*} \right) +\left( {\omega -\omega _a } \right) -\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) \left( {\omega -\omega _b } \right) ^{-1}-\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) \left( {\omega +\omega _b^*} \right) ^{-1}} \right] \nonumber \\= & {} \frac{1}{2}\left[ {\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) \left( {\omega -\omega _b } \right) \left( {\omega +\omega _b^*} \right) } \right] ^{-\frac{1}{2}}\nonumber \\&\times \left[ {2\omega +\omega _a^*-\omega _a -\frac{\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) }{\left( {\omega -\omega _b } \right) \left( {\omega +\omega _b^*} \right) }\left( {2\omega +\omega _b^*-\omega _b } \right) } \right] \nonumber \\= & {} \left( {\omega +\frac{1}{2}\hbox {i}\gamma } \right) \left[ {1-\frac{\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) }{\left( {\omega -\omega _b } \right) \left( {\omega +\omega _b^*} \right) }} \right] \left[ {\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) \left( {\omega -\omega _b } \right) \left( {\omega +\omega _b^*} \right) } \right] ^{-\frac{1}{2}}\nonumber \\= & {} \omega _{\text {p}}^2 \left( {\omega +\frac{1}{2}\hbox {i}\gamma } \right) \left( {\omega -\omega _a } \right) ^{-\frac{1}{2}}\left( {\omega +\omega _a^*} \right) ^{-\frac{1}{2}}\left( {\omega -\omega _b } \right) ^{-3/2}\left( {\omega +\omega _b^*} \right) ^{-3/2}\nonumber \\= & {} \frac{\omega _{\text {p}}^2 \left( {\omega +\frac{1}{2}\hbox {i}\gamma } \right) n\left( \omega \right) }{\left( {\omega -\omega _a } \right) \left( {\omega +\omega _a^*} \right) \left( {\omega -\omega _b } \right) \left( {\omega +\omega _b^*} \right) }\quad . \end{aligned}$$

(B1)

Note that, if $\omega$ happens to be on the imaginary axis, $n\left( \omega \right)$ will be purely real, but $\dot{n}\left( \omega \right)$ will be purely imaginary, due to the fact that the denominator of $\hbox {d}n\left( \omega \right) /\hbox {d}\omega$ will be imaginary.

Appendix C

For large values of $\omega$, Eq. (35) can be solved with the aid of elementary approximation methods, as follows:

$$\begin{aligned}&\omega \dot{n}\left( \omega \right) + n\left( \omega \right) = \tau \nonumber \\&\quad \rightarrow \frac{\omega _{\text {p}}^2 \left( {\omega \,+\,\hbox {i}\gamma /2} \right) \omega }{\left( {\omega -\omega _a } \right) ^{\frac{1}{2}}\left( {\omega +\omega _a^*} \right) ^{\frac{1}{2}}\left( {\omega -\omega _b } \right) ^{3/2}\left( {\omega +\omega _b^*} \right) ^{3/2}}+\sqrt{1+\frac{\omega _{\text {p}}^2 }{\omega _{\text {r}}^2 \,-\,\omega ^{2}\,-\,\hbox {i}\gamma \omega }}=\tau \nonumber \\&\quad \rightarrow \frac{\left( {\omega _{\text {p}} /\omega } \right) ^{2}\left( {1\,+\,\hbox {i}\gamma /2\omega } \right) }{\left( {1-\omega _a /\omega } \right) ^{\frac{1}{2}}\left( {1+\omega _a^*/\omega } \right) ^{\frac{1}{2}}\left( {1-\omega _b /\omega } \right) ^{3/2}\left( {1+\omega _b^*/\omega } \right) ^{3/2}}+1-\frac{\left( {\omega _{\text {p}} /\omega } \right) ^{2}}{2\left[ {1+\left( {\hbox {i}\gamma /\omega } \right) -\left( {\omega _{\text {r}} /\omega } \right) ^{2}} \right] }\cong \tau \nonumber \\&\quad \rightarrow \left( {\omega _{\text {p}} /\omega } \right) ^{2}\left( {1+\hbox {i}\gamma /2\omega } \right) \left( {1+\omega _a /2\omega } \right) \left( {1-\omega _a^*/2\omega } \right) \left( {1+3\omega _b /2\omega } \right) \left( {1-3\omega _b^*/2\omega } \right) \nonumber \\&\qquad -\frac{1}{2}\left( {\omega _{\text {p}} /\omega } \right) ^{2}\left[ {1-\left( {\hbox {i}\gamma /\omega } \right) +\left( {\omega _{\text {r}} /\omega } \right) ^{2}} \right] \cong \tau -1\nonumber \\&\quad \rightarrow \left( {\omega _{\text {p}} /\omega } \right) ^{2}\left[ {1+\left( {\hbox {i}\gamma /2\omega } \right) + \mathop {\underbrace{\left( {\omega _a -\omega _a^*} \right) }}\limits _{-\mathrm{i}\upgamma } /2\omega +3\mathop {\underbrace{\left( {\omega _b -\omega _b^*} \right) }}\limits _{-\mathrm{i}\upgamma } /2\omega -\frac{1}{2}+\frac{1}{2}\left( {\hbox {i}\gamma /\omega } \right) } \right] \cong \tau -1\nonumber \\&\quad \rightarrow \left( {\omega _{\text {p}} /\omega } \right) ^{2}\left[ {\frac{1}{2}-\hbox {i}\left( {\gamma /\omega } \right) } \right] \cong \tau -1\nonumber \\&\quad \rightarrow \omega /\omega _{\text {p}} \cong \pm \sqrt{1-\hbox {i}\left( {2\gamma /\omega } \right) }/\sqrt{2\left( {\tau -1} \right) }\nonumber \\&\quad \rightarrow \omega \cong \pm \frac{\omega _{\text {p}} }{\sqrt{2\left( {\tau \,-\,1} \right) }}\left[ {1-\hbox {i}\left( {\gamma /\omega } \right) } \right] \qquad \quad \rightarrow \quad \qquad \omega _{\mathrm{saddle}} \cong \pm \frac{\omega _{\text {p}} }{\sqrt{2\left( {\tau \,-\,1} \right) }}-\hbox {i}\gamma . \end{aligned}$$

(C1)

As t rises beyond $z_0 /c$, the parameter $\tau$ goes above 1.0, and the right and left saddles at $\omega _{\mathrm{saddle}}$ of Eq. (C1) move rather swiftly from $\pm \infty -\hbox {i}\gamma$ to $\sim \pm 10\omega _{\text {p}} -\hbox {i}\gamma$ (at $\tau =1.005)$, to $\sim \pm 5\omega _{\text {p}} -\hbox {i}\gamma$ (at $\tau =1.02)$, and then to $\sim \pm 3\omega _{\text {p}} -\hbox {i}\gamma$ (at $\tau =1.055)$. Needless to say, as $\tau$ drifts further and further beyond 1.0, the approximations that lead to Eq. (C1) make the above estimate of $\omega _{\mathrm{saddle}}$ less and less trustworthy.

Appendix D

The saddle points on the imaginary axis can be found by substituting $\hbox {i}{\omega }''$ for $\omega$, then writing

$$\begin{aligned} n\left( {{\text {i}}\omega ''} \right)= & {} \left( {1 + \frac{{\omega _{\text {p}}^2}}{{{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2}}} \right) ^{\frac{1}{2}}. \end{aligned}$$

(D1)

$$\begin{aligned} \dot{n}\left( {{\text {i}}\omega ''} \right)= & {} \frac{{{\text {i}}\omega _{\text {p}}^2\left( {2\omega '' + \gamma } \right) }}{{2{{\left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2} \right) }^2}}}{\left( {1 + \frac{{\omega _{\text {p}}^2}}{{{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2}}} \right) ^{ -\frac{1}{2}}}. \end{aligned}$$

(D2)

$$\begin{aligned} n\left( {{\text {i}}\omega ''} \right) + {\text {i}}\omega ''\dot{n}\left( {{\text {i}}\omega ''} \right)= & {} {\left( {1 + \frac{{\omega _{\text {p}}^2}}{{{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2}}} \right) ^{\frac{1}{2}} } - \frac{{\omega _{\text {p}}^2\left( {2\omega '' + \gamma } \right) \omega ''}}{{2{{\left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2} \right) }^2}}}{\left( {1 + \frac{{\omega _{\text {p}}^2}}{{{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2}}} \right) ^{ -\frac{1}{2}}} = \tau .\nonumber \\ \end{aligned}$$

(D3)

For any given value of $\tau$, Eq. (D3) must be solved numerically to reveal the location of the saddle point(s) on the ${\omega }''$-axis. To better understand the nature of these solutions, we equate the derivative with respect to ${\omega }''$ of the left-hand side of Eq. (D3) to zero, arriving at

$$\begin{aligned} \frac{{{\text {d}}\left[ {n\left( {{\text {i}}\omega ''} \right) + {\text {i}}\omega ''\dot{n}\left( {{\text {i}}\omega ''} \right) } \right] }}{{{\text {d}}\omega ''}}= & {} \omega _{\text {p}}^2{\left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2} \right) ^{ - 5/2}}{\left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2 + \omega _{\text {p}}^2} \right) ^{ - 3/2}}\nonumber \\&\times \left[ {\omega ''{{\left( {2\omega '' + \gamma } \right) }^2}\left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2 + \frac{3}{4} \omega _{\text {p}}^2} \right) } \right. \nonumber \\&\left. { - \left( {3\omega '' + \gamma } \right) \left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2} \right) \left( {{\omega ^{'' 2}} + \gamma \omega '' + \omega _{\text {r}}^2 + \omega _{\text {p}}^2} \right) } \right] = 0. \end{aligned}$$

(D4)

For typical values of the parameter set $\left( {\omega _{\text {p}} ,\,\omega _{\text {r}} ,\gamma } \right)$, the $5{\mathrm{th}}$ order polynomial equation appearing in Eq. (D4) has five solutions of which only three are real-valued. These represent the locations of a single maximum and two nearby minima of the function on the left-hand side of Eq. (D3). The location of the maximum, inferred from Eq. (D4), is $\omega '' \cong - \gamma /3$. An examination of Eqs. (D3) and (D4) reveals the general profile of $n\left( {{\text {i}}\omega ''} \right) + {\text {i}}\omega ''\dot{n}\left( {{\text {i}}\omega ''} \right)$ as depicted in Fig. 12.