Simple asymptotic forms for Sommerfeld and Brillouin precursors

This article mainly deals with the propagation of step-modulated light pulses in a dense Lorentz medium at distances such that the medium is opaque in a broad spectral region including the carrier frequency. The transmitted field is then reduced to the celebrated precursors of Sommerfeld and Brillouin, far apart from each other. We obtain simple analytical expressions of the first (Sommerfeld) precursor, whose shape only depends on the order of the initial discontinuity of the incident field and whose amplitude rapidly decreases with this order (rise-time effects). We show that, in a strictly asymptotic limit, the second (Brillouin) precursor is entirely determined by the frequency dependence of the medium attenuation and has a Gaussian or Gaussian-derivative shape. We point out that this result applies to the precursor directly observed in a Debye medium at decimetric wavelengths. When attenuation and group-delay dispersion both contribute to its formation, we establish a more general expression of the Brillouin precursor, containing the previous one (dominant-attenuation limit) and that obtained by Brillouin (dominant-dispersion limit) as particular cases. We finally study the propagation of square or Gaussian pulses, and we determine the pulse parameters optimizing the Brillouin precursor. Obtained by standard Laplace-Fourier procedures, our results are explicit and contrast in their simplicity those derived by the uniform saddle-point methods, from which it is difficult to retrieve our asymptotic forms.

Figure 1

Amplitude transmission $\left|H\right(z,\omega |$ of the medium as a function of the frequency modulus $\left|\omega \right|$ (logarithmic scale). Parameters (units of ${\omega }_0$) : ${\omega }_p=1.11$, $\gamma =0.0707$ and $\xi =8.31\times {10}^5$ for the curve (a) corresponding to the Brillouin choice ($z={10}^{-2}$ m). The curves (b), (c),(d) and (e) are obtained for propagation distances (and thus $\xi$) respectively 10, 100, 1 000 and 10 000 times smaller.

Figure 2

Sommerfeld precursor originated by the incident field $\cos \left({\omega }_{c}t\right)u_H\left(t\right)$. The solid (dashed) line is the exact numerical solution (the approximate analytic solution). Parameters (units of ${\omega }_0$) are ${\omega }_c=1$, ${\omega }_p=1.11$, $\gamma =0.0707$, and $\xi =831$. The inset is an enlargement of the tail of the precursor.

Figure 3

Sommerfeld precursor originated by the canonical incident field $\sin \left({\omega }_{c}t\right)u_H\left(t\right)$. The solid (dashed) line is the exact numerical solution (the approximate analytic solution). Parameters are as in Fig. 2.

Figure 4

Sommerfeld precursor originated by the incident field $e(0,t)=\tanh \left(rt\right)\sin \left({\omega }_{c}t\right)u_H\left(t\right)$. The solid (dashed) line is the exact numerical solution (the approximate analytic solution) obtained for $r={\omega }_0$. Other parameters are as in Fig. 2.

Figure 5

Brillouin precursor obtained under the Brillouin conditions, namely, for ${\omega }_c=0.1$, ${\omega }_p=1.11$, $\gamma =0.0707$, and $\xi =8.31\times {10}^5$ (units of ${\omega }_0$). For these parameters, ${\omega }_0{t}_B\approx 6.654\times {10}^5$ and $\beta \approx 1.78\times {10}^{-3}{\omega }_0=1.78\times {10}^{-2}{\omega }_c$. Solid lines (dots) are the exact numerical solutions (the analytic solutions). Curve (a) is the precursor obtained with the canonical incident field $\sin \left({\omega }_{c}t\right)u_H\left(t\right)$. The precursor of curve (b) is originated by the incident field $e(0,t)=\sin \left({\omega }_{c}t\right)[1+\text{erf}\left(rt\right)]/2$ for $r={\omega }_c/2\sqrt{2}$. The inset shows the Sommerfeld precursor magnified by a factor ${10}^6$ obtained with the conditions of curve (a). It fully vanishes under the conditions of curve (b).

Figure 6

Brillouin precursor obtained with the incident fields $(1-e^{-rt})\cos \left({\omega }_{c}t\right)u_H\left(t\right)$ for (a) $r\to \infty$, (b) $r=65{\omega }_c$, and (c) $r=20{\omega }_c$ (solid lines). Other parameters are as in Fig. 5. The dots correspond to the analytical solutions given by Eq. (27) or by the combination of this equation with Eq. (26).

Figure 7

Brillouin precursor and main field obtained for ${\omega }_c\approx 3.84\beta$ as a function of $\beta (t-t_B)$. The solid line, the dots, and the dashed line are, respectively, the exact numerical solution, the analytical solution given by Eq. (29), and its approximate form given by Eq. (30). The inset shows the Brillouin precursor obtained for ${\omega }_c\approx 7.67\beta$. The two analytical solutions are indistinguishable in this case, and the amplitude of the main field is negligible.

Figure 8

Brillouin precursor obtained in the simple asymptotic limit with the canonical incident field $\sin \left({\omega }_{c}t\right)u_H\left(t\right)$. Parameters (units of ${\omega }_0$) are ${\omega }_c=1$, ${\omega }_p=1.11$, $\gamma =0.0707$, and $\xi =831$, leading to ${\omega }_0{t}_B\approx 665.4$, $b\approx 8.44\times {10}^{-2}{\omega }_0$, and $\beta \approx 5.64\times {10}^{-2}{\omega }_0$. The solid line, the dots, and the dashed line are, respectively, the exact numerical solution, the analytical solution given by Eq. (34), and the Gaussian that would be obtained in the dominant-attenuation approximation. The conditions are those of Fig. 3. The corresponding Sommerfeld precursor magnified by a factor 1000 is given in the inset for reference.

Figure 9

Comparison of the Brillouin precursor obtained outside the asymptotic limit (solid line) with the analytical forms given by Eqs. (34) (dots) and (36) (dashed line). Parameters (units of ${\omega }_0$) are ${\omega }_c=1$, ${\omega }_p=1.11$, $\gamma =0.0707$, and $\xi =83.1$, leading to ${\omega }_0{t}_B\approx 66.54$, $b\approx 0.182{\omega }_0$, and $\beta \approx 0.178{\omega }_0$. The inset shows the corresponding Sommerfeld precursor (solid line) compared to the analytic form given by Eq. (21) (dashed line).

Figure 10

Brillouin precursor in the dispersion-dominant limit. The solid line (dots) is the exact numerical solution (the analytical solution). Parameters (units of ${\omega }_0$) are ${\omega }_c=0.836$, ${\omega }_p=1.11$, $\gamma ={10}^{-4}$, and $\xi =2.95\times {10}^4$, leading to ${\omega }_0{t}_B\approx 2.3641\times {10}^4$, $b\approx 0.0257{\omega }_0$, and $\beta \approx 0.252{\omega }_0$. The carrier frequency ${\omega }_c$ is at the lower boundary of the opacity region [$\alpha \left({\omega }_c\right)z\approx 20$].

Figure 11

Comparison of the Brillouin precursors $e_B^{\prime }(z,t)$ generated by an incident square-modulated field of duration (a) $T=T_c/2$ and (b) $T=T_c$. The parameters are those of Fig. 8. The solid and dashed lines are the exact numerical solutions, indiscernible from the analytical solutions given by Eq. (37). The dots are the approximate solutions (a) $2e_B(z,t-T_c/4)$ and (b) $T_c{\dot{e}}_B(z,t-T_c/2)$. As expected the precursor amplitude for $T=T_c/2$ is twice that attained with a step-modulated field (see Fig. 8), whereas that attained for $T=T_c$ is much smaller. The inset shows the corresponding incident fields.

Figure 12

Brillouin precursors generated by the incident fields of Gaussian envelope (a) $e^{-{(t/T)}^2}\cos \left({\omega }_{c}t\right)$ with $T=\sqrt{2}/{\omega }_c$ and (b) $e^{-{(t/T)}^2}\sin \left({\omega }_{c}t\right)$ with $T=\sqrt{6}/{\omega }_c$. The parameters are those of Fig. 8. In both cases, the pulse duration has been chosen in order to maximize the precursor amplitude (see text). The solid and dashed lines are the exact numerical solutions, whereas the dots are the analytical solutions obtained in the short-pulse approximation [Eqs. (45) and (46)], indiscernible from those obtained without approximation [Eq. (43)]. The inset shows the corresponding incident fields. Numerical calculations show that the Sommerfeld precursors generated by these fields have negligible amplitudes, $5.6\times {10}^{-7}$ for (a) and $1.15\times {10}^{-10}$ for (b).

Figure 13

Brillouin precursor generated by a chirped incident pulse $e^{-{(t/T)}^2}\cos ({\omega }_{c}t+{\chi }^2{t}^2)$, with $T=4\sqrt{2}/{\omega }_c$ and $\chi ={\omega }_c/4$. The other parameters are the same as those of Figs. 8 and 12. The solid line, the dots, and the dashed line are respectively the exact numerical solution, the analytic solution derived from Eq. (43), and the approximate analytic solution of Eq. (48), obtained in the short-pulse approximation. The inset shows the incident pulse. The corresponding Sommerfeld precursor has fully negligible amplitude ($9\times {10}^{-11}$).