Strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies

We perform a detailed semianalytical study of the characteristics of strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies. We show that the photoelectron spectra depend to a great extent on the absolute phases of the field components and on the duration of the pulse. By using the saddle-point method, we show that this dependence can be exploited to control the electron dynamics in the laser field. Besides the photoelectron spectra, we also investigate the spectra of high-order harmonics. Again, we find a strong dependence on the absolute phases and the duration of the pulse. We present the spectra using a false color scale in the absolute phase-harmonic energy plane, and show that particular regions in this plane can be assessed using the simple man's model founded on the classical solution of the Newton equation of motion. Finally, we investigate the symmetry properties of the photoelectron and high-order harmonic spectra with respect to the transformation which includes the change of the absolute phase, and compare them with those valid for a long driving pulse with a flat envelope.

Figure 1

Logarithm of the differential ionization probability as a function of the photoelectron energy of Ar for HATI induced by the $\omega \text{−}2\omega$ linearly polarized few-cycle pulse with the component intensity $E_1^2=E_2^2={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, fundamental carrier wavelength of 800 nm, and the values of the absolute phases as indicated in the panel. The total pulse duration and the emission angle are ${\tau }_p=4T$ and ${\theta }_e=0^{\circ }$, respectively.

Figure 2

Logarithm of the differential ionization probability as a function of the photoelectron energy of Ar for HATI induced by the $\omega \text{−}2\omega$ linearly polarized few-cycle pulse which consists of $n_p$ optical cycles, as indicated in the upper left panel, and for the values of the absolute phase ${\phi }_2$ as indicated in the panels. Other driving-field parameters and the emission angle are the same as in Fig. 1.

Figure 3

Saddle-point partial contributions to the differential ionization probability for the driving pulse with ${\phi }_2={120}^{\circ }$ and the same other parameters as in Fig. 1. The pair which is dominant in the high-energy part of the spectrum is denoted by $h$, while the pair whose contribution is significant in the medium part of the spectrum is denoted by $m_1$.

Figure 4

Complex saddle-point solutions for the high-energy (left panel) and medium-energy (right panel) pair for the driving pulse with the absolute phase ${\phi }_2={180}^{\circ }$ (black lines; largest $t_{0\text{R}}$), ${\phi }_2={220}^{\circ }$ (red lines; medium $t_{0\text{R}}$), and ${\phi }_2={260}^{\circ }$ (green lines; smallest $t_{0\text{R}}$), and the same other parameters as in Fig. 1.

Figure 5

Photoelectron trajectories for the convergent solutions of the $h$ [blue and green (upper) lines] and $m_1$ [magenta and orange (lower) lines] pairs of the saddle-point solutions for the absolute phase as indicated in the panel. The photoelectron energy is $E_{\mathbf{p}}=7.5U_{p0}$. Other pulse parameters are the same as in Fig. 1. Inset: The electric field for the two values of the phase as indicated in the panel, and with the relevant real part of the ionization times denoted by dots.

Figure 6

Logarithm of the differential ionization probability as a function of the absolute phase ${\phi }_1$ and the photoelectron energy for the driving pulse with ${\phi }_2=0^{\circ }$ and the same other parameters as in Fig. 1.

Figure 7

Logarithm of the differential ionization probability as a function of the absolute phase ${\phi }_1$ and the photoelectron energy for the driving field with $n_p=3$ (a) and $n_p=5$ (b) optical cycles per pulse, and for the long pulse with a flat envelope (c). The absolute phase is ${\phi }_2=0^{\circ }$ and the other parameters are as in Fig. 1.

Figure 8

Logarithm of the differential ionization probability as a function of the photoelectron energy calculated by numerical integration (black solid lines) and by using the saddle-point method [red (gray) solid lines], together with the contributions of different saddle-point solutions. The spectra calculated using the saddle-point method do not include the contributions of the direct electrons, while they are included in the spectra calculated by the numerical integration. The values of the absolute phases and the number of optical cycles per pulse are denoted in the panels. Other parameters are the same as in Fig. 1.

Figure 9

Logarithm of the harmonic intensity as a function of the harmonic energy for the Ar atom exposed to the $\omega \text{−}2\omega$ linearly polarized field which consists of three optical cycles per pulse and for the values of the absolute phases ${\phi }_2=0^{\circ }$ and ${\phi }_1$ as indicated in the panels. The intensity of the driving-field components is $E_1^2=E_2^2={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, while the fundamental carrier wavelength is 1600 nm.

Figure 10

Logarithm of the harmonic intensity $I$ as a function of the absolute phase ${\phi }_1$ and the harmonic energy. The intensity of the driving-field components is $E_1^2=E_2^2={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, while the fundamental carrier wavelength and the absolute phase ${\phi }_2$ are 800 nm and $0^{\circ }$, respectively. The number of optical cycles per pulse is indicated above the panels.

Figure 11

Logarithm of the harmonic intensity $I$ as a function of the absolute phase ${\phi }_2$ and the harmonic energy. The intensity of the driving-field components is $E_1^2=E_2^2={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, while the fundamental carrier wavelength and the absolute phase ${\phi }_1$ are 900 nm and $0^{\circ }$, respectively. The number of optical cycles per pulse is $n_p=3$. The black curve, obtained using the one-dimensional simple man's model, determines the optimal harmonic intensity.

Figure 12

Logarithm of the harmonic intensity as a function of the absolute phase ${\phi }_1$ and the harmonic energy (${\phi }_2=0^{\circ }$) for the Ar atom exposed to the $\omega \text{−}2\omega$ long driving field with the component intensity $E_1^2=E_2^2={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and the fundamental wavelength of 1600 nm.