Central frequency of few-cycle laser pulses in strong-field processes

We analyze the role of the difference between the central frequencies of the spectral distributions of the vector potential and the electric field of a short laser pulse. The frequency shift arises when the electric field is determined as the derivative of the vector potential to ensure that both quantities vanish at the beginning and end of the pulse. We derive an analytical estimate of the frequency shift and show how it affects various light induced processes, such as excitation, ionization, and high-harmonic generation. Since observables depend on the frequency spectrum of the electric field, the shift should be taken into account when setting the central frequency of the vector potential to avoid potential misinterpretation of numerical results for processes induced by few-cycle pulses.

Figure 1

Temporal (left) and spectral (right) distributions of vector potential (solid lines) and electric field (dashed line) for pulses with FWHM of 1 cycle (top), and 3 cycles (bottom) at central frequency ${\omega }_A=1.0$ a.u. of the vector potential. Also shown is the Gaussian envelope of the vector potential.

Figure 2

Ratio ${\omega }_E/{\omega }_A$ as a function of the normalized number of cycles $N$ defined in Eq. (5). The numerical results were calculated by maximizing $\tilde{E}\left(\omega \right)$ for Gaussian (dotted line), sech (dashed-dotted line), and ${sin}^4$ (dashed line) envelopes and are compared with the simple analytic estimate (solid line) given in Eq. (4). The inset reveals a slight dependence on envelope shape, which can be attributed to the correction term in Eq. (3).

Figure 3

Photoelectron spectra $P\left(k\right)$ as function of photoelectron momentum obtained for interaction of hydrogen atom with laser pulses at central frequencies ${\omega }_A=2$ (dashed line) and ${\omega }_E=2$ (solid line) and duration of (a) 10 cycles and (b) 2 cycles FWHM.

Figure 4

Population in $2p$ state following one-photon excitation of hydrogen atom with a laser pulse as a function of ${\omega }_A$ (left) and ${\omega }_E$ (right) for different pulse lengths at peak intensity ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. Each line represents results obtained for a fixed pulse duration in terms of ${\tau }_1=405$ as. The results from time-dependent Schrödinger equation (TDSE) calculations (a),(b) and predictions within first-order perturbation theory (PT) (c),(d) are in excellent agreement. The vertical line marks the energy difference between $2p$ and initial $1s$ state. The green dots indicate the maximum excited-state population for each pulse duration.

Figure 5

Numerical results for population in $2s$ state following two-photon excitation with ${\tau }_2=811$ as. The vertical line marks half of the energy gap between $1s$ and $2s$ representing the resonance condition for the two-photon process. Other parameters are the same as in Fig. 4.

Figure 6

HHG spectrum at driver wavelength 730 nm (${\omega }_{\text{central}}=0.0625$ a.u.) vs number of cycles $N_{\text{FWHM}}$. In the upper plot the central frequency ${\omega }_A={\omega }_{\text{central}}$, while in the lower panel ${\omega }_E={\omega }_{\text{central}}$. The vertical white dashed lines mark field-free transition energies between excited states and the ground state, while the green solid lines mark the harmonic energies $n_p{\omega }_E$ with respect to the central frequency of the electric field.