Principal frequency of an ultrashort laser pulse

We introduce an alternative definition of the main frequency of an ultrashort laser pulse—the principal frequency ${\omega }_P$. This parameter is complementary to the most accepted and widely used carrier frequency ${\omega }_0$. Given the fact that these ultrashort pulses, also known as transients, have a temporal width comprising only a few cycles of the carrier wave, corresponding to a spectral bandwidth $\mathrm{\Delta }\omega$ covering several octaves, ${\omega }_P$ describes, in a more precise way, the dynamics driven by these sources. We present examples where, for instance, ${\omega }_P$ is able to correctly predict the high-order harmonic cutoff independent of the carrier envelope phase. This is confirmed by solving the time-dependent Schrödinger equation in reduced dimensions, supplemented with the time-analysis of the quantum spectra, where it is possible to observe how the subcycle electron dynamics is better described using ${\omega }_P$. The concept of ${\omega }_P$, however, can be applied to a large variety of scenarios, not only within the strong-field physics domain.

Figure 1

(a)–(c) Spectral power $S\left(\omega \right)$ of the fields $E_R\left(\omega \right), E_{2R}\left(\omega \right)$, and $E_G\left(\omega \right)$ for $E_0=1$. All the cases are centered at the same frequency, ${\omega }_1=2\pi$. The spectral bandwidth of both $E_R\left(\omega \right)$ and $E_G\left(\omega \right)$ is $\mathrm{\Delta }\omega =\pi$; meanwhile, for $E_{2R}\left(\omega \right)$ each “subspectrum” has a bandwidth $\mathrm{\Delta }\omega =\pi /2$. In (b) $\delta \omega =\pi /2$. The blue solid (red dashed) line represents the ${\omega }_P$ (${\omega }_0$) frequency (see the text for details). (d)–(f) Normalized time-dependent fields $E_R\left(t\right), E_{2R}\left(t\right)$, and $E_G\left(t\right)$. Thick blue (thin red) line defines a cosine(sine)-like pulse. In (e) the dashed (dotted) line corresponds to the temporal distance between two consecutive zeros (maximum and minimum) of the field for the $sin(cos$)-like pulses, i.e., $T_0/2$ and $T_P/2$, respectively (see the text for details).

Figure 2

Principal period $T_P$ as a function of the pulse bandwidth $\mathrm{\Delta }\omega$ for (a) $E_R\left(t\right)$, (b) $E_{2R}\left(t\right)$, and (c) $E_G\left(t\right)$. In all the cases: dotted violet line, $T_0=2\pi /{\omega }_0$ [note that for the case of $E_G\left(t\right)$ pulses, ${\omega }_0$ depends on $\mathrm{\Delta }\omega$]; dashed green line, $T_P$ computed from $T_P=2\pi /{\omega }_P$; thick blue (thin red) solid line, $T_{\mathrm{COS}}$ ($T_{\mathrm{SIN}}$) extracted for the time-dependent cosine(sine)-like pulses (see the text for details).

Figure 3

Cosine-like (a) and sine-like (b) $E_R\left(t\right)$ pulses for different bandwidths $\mathrm{\Delta }\omega$. The dotted red, dashed green, and solid blue lines correspond to pulses with a bandwidth of $\mathrm{\Delta }\omega =\pi , \mathrm{\Delta }\omega =2\pi$, and $\mathrm{\Delta }\omega =3\pi$ in units of ${\omega }_0$, which we take corresponding to a laser wavelength of ${\lambda }_0=2000$ nm, respectively. The labels I–III denote the ionization and recombination regions (see the text for more details). HHG spectra for the cosine-like (c) and sine-like (d) pulses plotted in panels (a) and (b), respectively. The vertical lines correspond to the different HHG cutoffs (see the text for more details).

Figure 4

Time-frequency analysis extracted from the 1D-TDSE HHG spectra. Superimposed we plot the classically computed rescattering energies of electrons as a function of the ionization time, in white dots, and recombination time, in black dots, for the laser pulses of Fig. 3 [panels (a)–(c)] and Fig. 3 [panels (d)–(f)]. In panels (a)–(c) the solid red (dashed green) arrow corresponds to the electron trajectory that contributes to the more (less) energetic HHG cutoff. Meanwhile, in panels (d)–(f) the green arrow corresponds to the electron trajectory that contributes to the single HHG cutoff (see the text for more details). The ionization potential, $I_p$, is indicated by a black arrow.

Figure 5

Cosine-like (a) and sine-like (b) $E_{2R}\left(t\right)$ pulses for different values of $\mathrm{\Delta }\omega$ and $\delta \omega$. The dotted red (solid blue) line corresponds to pulses with $\mathrm{\Delta }\omega =0.1$ and $\delta \omega =\pi$ ($\mathrm{\Delta }\omega =\pi /2$ and $\delta \omega =\pi /2$). The labels I–III denote the ionization and recombination regions (see the text for more details). HHG spectra for the cosine-like (c) and sine-like (d) pulses plotted in panels (a) and (b), respectively. The vertical lines correspond to the different HHG cutoffs (see the text for more details).

Figure 6

Time-frequency analysis extracted from the 1D-TDSE HHG spectra. Superimposed we plot the classically computed rescattering energies of electrons as a function of the ionization time, in white dots, and recombination time, in black dots, for the laser pulses of Fig. 5 [panels (a) and (b)] and Fig. 5 [panels (c) and (d)]. In all the panels the green arrow corresponds to the electron trajectory that contributes to the single HHG cutoff (see the text for more details). The ionization potential, $I_p$, is indicated by a black arrow.