Nonlinear Optics of Intense Attosecond Light Pulses

The interaction of an intense light pulse of "subatomic" duration with a system of multiple discrete quantum states is analyzed. The nonperturbative character of the response to the pulse field leading to an efficient conversion into high order harmonics is predicted. The spatial-temporal evolution of the field is shown to obey a generalized nonlinear wave equation of the double-sine-Gordon type. In addition to the solitary wave structures, it predicts a nontrivial regime of pulse amplification accompanied by extreme temporal self-contraction of the amplified field.

Figure 1

(a) The interaction of an ASP of “subatomic” duration with optical transitions is close to the field interaction with a quasidegenerate system. (b) Assuming inversion symmetry in the system, all states can be divided into two groups according to their parity.

Figure 2

Solitary wave structures in an absorbing medium ($q+2p>0$). (a) 1) $\tilde{p}=1/3$, $\tilde{q}=1/3$; 2) $\tilde{p}=0.47$, $\tilde{q}=0.06$; (b) 1) $\tilde{p}=-0.01$, $\tilde{q}=1.02$; 2) $\tilde{p}=-5.0$, $\tilde{q}=11.0$; (c) 1) $\tilde{p}=1.12$, $\tilde{q}=-1.24$; 2) $\tilde{p}=0.52$, $\tilde{q}=-0.04$. $E(t)$ in units of $E_{\mathrm{NL}}=\hslash /(D{\tau }_p)$, $\tilde{p}=p/(q+2p)$, $\tilde{q}=p/(q+2p)$.

Figure 3

The temporal structure of the ASP-field $E(z,t)$ in an amplifying medium ($q+2p<0$) at different propagation lengths $Z$. Also shown is the energy $\Delta W(z,t)$ radiated by the system. The time in units of the field period $T_0=2\pi /{\omega }_0$ at $Z=0$; the field in units of $E_{\mathrm{NL}}=\hslash /2D{T}_0$, the length $z$ in units of $Z_{\mathrm{NL}}=\hslash c{D}^2/(8\pi N{T}_0|{\overrightarrow{d}}_{l_0}{|}^2|q+2p|)$; $p/|q+2p|=-0.6$; $q/|q+2p|=2.2$. For $T_0=30\mathrm{As}$, $D\approx 10d_B$, $|q+2p|\approx \mathrm{Ry}(d_B{)}^2$, $N={10}^{19}{\mathrm{cm}}^{-3}$, the units: $E_{\mathrm{NL}}=4.5\times {10}^8\mathrm{V}/\mathrm{cm}$, $Z_{\mathrm{NL}}={10}^{-2}\mathrm{cm}$.