Optical limiting of short laser pulses

The dynamics of pulse propagation accompanied by harmonic generation, stimulated Raman scattering, amplified spontaneous emission, and superfluorescence is studied near the two-photon resonance. We explore the optical limiting of intense and short laser pulses. The numerical solutions of the coupled Bloch and Maxwell’s equations for the $4,4^{\prime }$-bis(dimethylamino) stilbene molecule are compared with the two-photon area theorem. It is shown that the area theorem explains qualitatively the major dynamical properties of pulse propagation even if the propagation is accompanied by the generation of new fields. In agreement with the area theorem, we see that the conventional dependence of the transmittance on the propagation depth is not valid for intense pulses.

Figure 1

The $4,4^{\prime }$-bis(dimethylamino) stilbene molecule.

Figure 2

(Color online) Scheme of transitions.

Figure 3

Graphical analysis of the area theorem (5). $n=1,2,\dots$.

Figure 4

Characteristic lengths $L_0$ [Eq. (9)], $L$ [Eq. (3)], and $z_c$ [Eq. (6)] of the optical limiting. The conventional length of TPA $\left(L_0\right)$ and the TPA length $\left(L\right)$ in the region of enhanced optical limiting [panel (a)]. The coordinate $z_c$ [panel (b)]. $\tau =100\text{fs}$. Peak intensity of the incident pulse is calculated as $I_0=c{ϵ}_0{E}_0^2∕2$.

Figure 5

Temporal shape of the pulse $F\left(t,z\right)$ (left panels) and its Fourier transform $F\left(\nu ,z\right)$ (right panels) for different initial intensities $I_0$ and propagation depths $z$. $\tau =2.5\text{fs}$. $\omega =2.135\text{eV}$ $\left(\delta =0\right)$. $I_0=4.54\times {10}^{12}\text{W}∕{\text{cm}}^2$ corresponds to $\theta \left(0\right)=3\pi$.

Figure 6

Output area of the pulse $\theta \left(z\right)$ versus input pulse area $\theta \left(0\right)$. The curves are calculated using the area theorem (5) except curves $\tau =2.5\text{fs}$ and $\tau =7.5\text{fs}$ which are computed using the “TPA area” [Eq. (3)], $\theta \left(z\right)=\alpha {\int }_{-\infty }^{\infty }I\left(t,z\right)dt$, of the total field. The intensity of the total field $I\left(t,z\right)=FH$ is obtained from strict numerical solution of the coupled Bloch and Maxwell’s equations. $L=1.1646\mu \text{m}$.

Figure 7

Transmittance $T\left(z\right)$ [Eq. (7)] versus $z$ for different input areas of the pulse, $\theta \left(0\right)=0.05\pi ,\dots$. The transmittance is calculated using the area theorem (5). $L=1.1646\mu \text{m}$.

Figure 8

Transmittance $T\left(z\right)$ [Eq. (14)] versus $z$ for different input areas of the pulse. Simulations are based on strict numerical solutions of Eqs. (13) which take into account harmonic, SRS, and ASE fields (see Fig. 5). $\delta =0$. Transmittances marked by WF limit are computed using Eq. (8). $\tau =2.5\text{fs}$. $I_0=830\text{GW}∕{\text{cm}}^2$ for $\theta \left(0\right)=0.55\pi$.