Transient Cherenkov radiation from an inhomogeneous string excited by an ultrashort laser pulse at superluminal velocity

The optical response of a one-dimensional string made of dipoles with a periodically varying density excited by a spot of light moving along the string at superluminal (subluminal) velocity is studied theoretically. The Cherenkov radiation in such a system is rather unusual, possessing both transient and resonant character. We show that, under certain conditions, in addition to the resonant Cherenkov peak, another Doppler-like frequency appears in the radiation spectrum. Both linear (small-signal) and nonlinear regimes as well as different string topologies are considered.

Figure 1

(a) Excitation of a string at superluminal velocity. 1, a short spectrally broadband laser pulse source; 2,3, lenses; 4, the pump pulse. The intersection of the plane pulse and the medium moves along the string at the velocity $V=c/sin\beta >c$. (b) The observation geometry of the string of length $Z_m$. The observer is placed far away from the string or in the focus of a lens $L$ which collects the radiation of the string parallel to the $z$ axis. (c) The source must produce a significantly broadband and flat spectrum [58, 59, 60, 61, 62, 63, 64, 65, 66], which includes also the resonance frequency ${\omega }_0$ of the dipoles forming the string.

Figure 2

Time dependence of the field $E\left(t\right)$ according to Eq. (14) (a),(c) and its spectral intensity $I\left(\omega \right)$ (b),(d) normalized to their maximal values vs normalized time $t/T_0$ and frequency $\omega /{\omega }_0$, for $V/c=2$, $Z_m/{\Lambda }_z=9.55$, ${\Lambda }_z/{\lambda }_0=5$, and ${\omega }_0/\gamma =22.22$ for the observation angle $\phi =0$ (a),(b) and $\phi ={60}^{\circ }$ (c),(d); the latter corresponds to the Cherenkov emission angle.

Figure 3

Dependence of the spectral intensity $I\left(\omega \right)$ of the string response according to Eq. (14) on the observation angle $\phi$ (a), the excitation velocity $V$ (b), and the string density modulation period ${\Lambda }_z$ (c). The other parameters coincide with those given in Figs. 2 and 2. The spectral intensity is presented on a logarithmic scale.

Figure 4

Time dependence (a) of the string response field $E\left(t\right)$ according to Eq. (14), and (b) of its spectral intensity $I\left(\omega \right)$ normalized to the maximum value vs normalized time $t/T_0$ and frequency $\omega /{\omega }_0$, for $V/c=0.7$, $Z_m/{\Lambda }_z=9.55$, ${\Lambda }_z/{\lambda }_0=5$, and ${\omega }_0/\gamma =22.22$ and observation angle $\phi =0$.

Figure 5

Dependence of the spectral intensity $I\left(\omega \right)$ of the string response according to Eq. (14) on the observation angle $\phi$ (a) and the string density modulation period ${\Lambda }_z$ (b). The other parameters coincide with the ones in Fig. 4. Note the logarithmic scale in the plot.

Figure 6

Circular geometry of the string. The source of a short pulse with a broad spectrum (c) is located in the center of the circle and quickly rotates. The crossing of the pulse and medium (yellow dot) moves at the velocity $v$ along the string (black circle). As in the previous case, the string is made of dipoles characterized by the resonance frequency ${\omega }_0$ and the dipole number density is modulated along the string periodically with the angular period ${\Lambda }_{\varphi }$.

Figure 7

(a) Time dependence of the electric field $E\left(t\right)$ excited by the string and (b) the corresponding intensity spectrum $I\left(\omega \right)$ in the center of the circle for the circular scheme depicted in Fig. 6 and the parameters $V/c=3.75$, ${\Lambda }_{\varphi }R/{\lambda }_0=2$, ${\omega }_0/\gamma =22.2$, and ${\omega }_0/{\Omega }_2=0.53$.

Figure 8

(a) Dependence of the radiation spectrum on the normalized propagation speed $V$ of the excitation (a) and the radius of the circle $R$ (b). Other parameters are as in Fig. 7.

Figure 9

(a) Time dependence of the string response field $E\left(t\right)$ according to Eq. (24). (b) S line: its spectral intensity $I\left(\omega \right)$ normalized to the maximum values vs normalized time $t/T_0$ and frequency $\omega /{\omega }_0$, for the parameters of Fig. 2 and total pulse area $\Phi =\pi /2$, ${\Omega }_R=0.07{\omega }_0$, ${\tau }_p=2T_0$; dashed line: spectrum of the excitation pulse.

Figure 10

Dependence of the radiation spectrum on the pulse area $\Phi$. Other parameters are as in Fig. 9.