Ultimate capabilities for few-cycle pulse formation via resonant interaction of XUV radiation with IR-field-dressed atoms

We perform an ab initio study of the ultimate capabilities and limits of applicability of the method for few-cycle pulse formation via the resonant interaction of extreme ultraviolet (XUV) radiation with atoms dressed by a moderately strong infrared (IR) laser field proposed in two earlier works [Y. V. Radeonychev et al., Phys. Rev. Lett. 105, 183902 (2010) and V. A. Polovinkin et al., Opt. Lett. 36, 2296 (2011)]. Taking into account all the multiphoton processes in the systems under consideration on the basis of numerical solution of the three-dimensional time-dependent Schrödinger equation (TDSE) in the single-active-electron approximation, we show the possibilities to produce 1.1-fs pulses from 124.6-nm XUV radiation via the linear Stark effect in atomic hydrogen, as well as 500-as pulses from 58.4-nm XUV radiation via excited-state ionization in helium. We derive a generalized analytical solution, which takes into account the interplay between sub-laser-cycle Stark effect and excited-state ionization and allows us to analyze the results of TDSE calculations. We found that the ultimate intensity of the IR field suitable for few-cycle pulse formation via the linear Stark effect or excited-state ionization is limited by the threshold for atomic ionization from the resonant excited state or the ground state, respectively. We show that the pulses with shorter duration can be produced in the medium of ions with higher values of the ionization potential.

Figure 1

Time dependencies of a probability for a hydrogen atom initially excited into the state $|2\rangle$ or $|3\rangle$ to remain nonionized by the monochromatic IR field of intensity $I_{\mathrm{IR}}=2.5\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{IR}}=8\mu \mathrm{m}$. The lower red curves correspond to the state $|2\rangle$, while the upper blue curves characterize the state $|3\rangle$. The solid curves are the results of ab initio solution of the TDSE, whereas the dashed curves represent the approximation (10b).

Figure 2

(a) Time dependence of intensity $I\sim |{\tilde{E}}_{\mathrm{XUV}}{|}^2$ of XUV radiation at the exit of an optically thin medium of atomic hydrogen simultaneously irradiated by the $\mathrm{C}{\mathrm{O}}_2$-laser field with intensity $I_{\mathrm{IR}}=1.4\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{IR}}=10.65\mu \mathrm{m}$ and the XUV radiation with intensity $I_{\mathrm{XUV}}=2.2\times {10}^8\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{XUV}}=122.2\mathrm{nm}$. The dimensionless parameters in analytical solution are $P_{\omega }^{\left(1\right)}=4.45, P_{\omega }^{\left(2\right)}=0.23, P_{\gamma }^{\left(1\right)}=0.05, P_{\gamma }^{\left(2\right)}=0.015$, and ${\overline{\gamma }}_{\mathrm{tr}}=0.04\mathrm{\Omega }$. The dashed red curve and the solid green curve correspond to the analytical solutions (11) and (13), respectively. The rapidly oscillating dash-dotted blue curve shows the numerical solution of the TDSE for the squared value of the XUV field strength, $|E_{\mathrm{XUV}}{|}^2$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). The results provided by the generalized analytical solution (12) for the amplitudes and phases of the spectral components are shown by red squares and blue circles, while the predictions of the simplified analytical theory (13) for the spectral amplitudes and phases are plotted by black asterisks and filled green circles. The lavender curve and cyan crosses show the ab initio solution of the TDSE for the amplitudes of the spectral components, as well as their phases at the combinational frequencies, $\omega ={\omega }_0+n\mathrm{\Omega }, n=0,\pm 1,\pm 2,...$, respectively. The resonant spectral component, $\omega ={\omega }_0$, of the output XUV radiation is attenuated to the level of the generated sidebands.

Figure 3

(a) Same as Fig. 2, but for IR field with intensity $I_{\mathrm{IR}}=2.5\times {10}^{12}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{IR}}=8\mu \mathrm{m}$, and XUV radiation with intensity $I_{\mathrm{XUV}}=1.6\times {10}^9\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{XUV}}=122.6\mathrm{nm}$. The dimensionless parameters in analytical solutions are $P_{\omega }^{\left(1\right)}=4.45, P_{\omega }^{\left(2\right)}=0.32, P_{\gamma }^{\left(1\right)}=0.21, P_{\gamma }^{\left(2\right)}=0.074$, and ${\overline{\gamma }}_{\mathrm{tr}}=0.19\mathrm{\Omega }$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 2.

Figure 4

(a) Same as Fig. 2, but for IR field with intensity $I_{\mathrm{IR}}={10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{IR}}=4\mu \mathrm{m}$, and XUV radiation with intensity $I_{\mathrm{XUV}}={10}^9\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{XUV}}=124.6\mathrm{nm}$. The dimensionless parameters in analytical solutions are $P_{\omega }^{\left(1\right)}=4.45, P_{\omega }^{\left(2\right)}=0.23, P_{\gamma }^{\left(1\right)}=0.8, P_{\gamma }^{\left(2\right)}=0.6$, and ${\overline{\gamma }}_{\mathrm{tr}}=1.3\mathrm{\Omega }$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 2.

Figure 5

(a) Same as Fig. 2, but for IR field with intensity $I_{\mathrm{IR}}=4\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{IR}}=2\mu \mathrm{m}$, and XUV radiation with intensity $I_{\mathrm{XUV}}=4\times {10}^9\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{XUV}}=133.4\mathrm{nm}$. The dimensionless parameters in analytical solutions are $P_{\omega }^{\left(1\right)}=4.45, P_{\omega }^{\left(2\right)}=0.5, P_{\gamma }^{\left(1\right)}=1.9, P_{\gamma }^{\left(2\right)}=1.8$, and ${\overline{\gamma }}_{\mathrm{tr}}=3.8\mathrm{\Omega }$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 2.

Figure 6

(a) Time dependence of intensity of XUV radiation at the exit of an optically thin medium of helium irradiated by the IR field with intensity $I_{\mathrm{IR}}=1.5\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{IR}}=4\mu \mathrm{m}$, and the XUV radiation with intensity $I_{\mathrm{XUV}}={10}^{11}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ and wavelength ${\lambda }_{\mathrm{XUV}}=58.4\mathrm{nm}$. The incident spectral component of XUV radiation is suppressed. The dashed red curve and the solid green curve correspond to the harmonic and steplike analytical solutions (17), and (20, 21, 22, 23), respectively. The dimensionless parameters in analytical calculations are $P_{\omega }^{\left(2\right)}=15, P_{\gamma }^{\left(2\right)}=11.5, {\overline{\gamma }}_{min}/\phantom{{\overline{\gamma }}_{min}\mathrm{\Omega }}\mathrm{\Omega }=0.056, {\overline{\gamma }}_{max}/\phantom{{\overline{\gamma }}_{max}\mathrm{\Omega }}\mathrm{\Omega }=2.7$, and $\mathrm{\Omega }\mathrm{\Delta }{t}_{\mathrm{zero}}=0.2\pi$. The rapidly oscillating dash-dotted blue curve is the numerical solution of the TDSE for the squared value of the XUV field strength, $|E_{\mathrm{XUV}}{|}^2$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Red squares and blue circles are the amplitudes and phases of the spectral components calculated analytically via the harmonic analytical solution (18). The corresponding results of the steplike analytical solution (20, 21, 22, 23) are plotted by black asterisks and green filled circles. The lavender curve and cyan crosses show the ab initio solution of the TDSE for the amplitudes of the spectral components, as well as their phases at the combinational frequencies, $\omega ={\omega }_0+n\mathrm{\Omega }, n=0,\pm 1,\pm 2,...$, respectively. The upper frequency limit corresponds to the ionization potential of helium.

Figure 7

(a) Same as Fig. 6, but for IR field with wavelength ${\lambda }_{\mathrm{IR}}=2\mu \mathrm{m}$. The dimensionless parameters in the analytical solutions are $P_{\omega }^{\left(2\right)}=4.0, P_{\gamma }^{\left(2\right)}=4.6, {\overline{\gamma }}_{min}/\phantom{{\overline{\gamma }}_{min}\mathrm{\Omega }}\mathrm{\Omega }=0.14, {\overline{\gamma }}_{max}/\phantom{{\overline{\gamma }}_{max}\mathrm{\Omega }}\mathrm{\Omega }=1.5$, and $\mathrm{\Omega }\mathrm{\Delta }{t}_{\mathrm{zero}}=0.25\pi$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 6.

Figure 8

(a) Same as Fig. 6, but for IR field with wavelength ${\lambda }_{\mathrm{IR}}=1\mu \mathrm{m}$. The dimensionless parameters in the analytical solutions are $P_{\omega }^{\left(2\right)}=1.0, P_{\gamma }^{\left(2\right)}=1.2, {\overline{\gamma }}_{min}/\phantom{{\overline{\gamma }}_{min}\mathrm{\Omega }}\mathrm{\Omega }=0.22, {\overline{\gamma }}_{max}/\phantom{{\overline{\gamma }}_{max}\mathrm{\Omega }}\mathrm{\Omega }=0.88$, and $\mathrm{\Omega }\mathrm{\Delta }{t}_{\mathrm{zero}}=0.4\pi$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 6.

Figure 9

(a) Same as Fig. 6, but for IR field with intensity $I_{\mathrm{IR}}=4\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The dimensionless parameters in the analytical solutions are $P_{\omega }^{\left(2\right)}=25, P_{\gamma }^{\left(2\right)}=20.7, {\overline{\gamma }}_{min}/\phantom{{\overline{\gamma }}_{min}\mathrm{\Omega }}\mathrm{\Omega }=0.24, {\overline{\gamma }}_{max}/\phantom{{\overline{\gamma }}_{max}\mathrm{\Omega }}\mathrm{\Omega }=6.2$, and $\mathrm{\Omega }\mathrm{\Delta }{t}_{\mathrm{zero}}=0.16\pi$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 6. (i) Cyan crosses and (ii) dark blue pluses show the results of the TDSE solution for the phases of spectral components at (i) the combinational frequencies, $\omega ={\omega }_0+n\mathrm{\Omega }, n=0,\pm 1,\pm 2,...$, and (ii) the high-order harmonics of the IR field, $\omega =(2k+1)\mathrm{\Omega }, k=1,2,...$, respectively.

Figure 10

(a) Same as Fig. 6, but for IR field with intensity $I_{\mathrm{IR}}=8\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. The dimensionless parameters in the analytical solutions are $P_{\omega }^{\left(2\right)}=50, P_{\gamma }^{\left(2\right)}=51, {\overline{\gamma }}_{min}=0.66\mathrm{\Omega }, {\overline{\gamma }}_{max}=11.2\mathrm{\Omega }$, and $\mathrm{\Delta }{t}_{\mathrm{zero}}=0.1\pi /\phantom{\pi \mathrm{\Omega }}\mathrm{\Omega }$. (b) Fourier transform of the output XUV radiation corresponding to the time dependence in (a). Designations are the same as in Fig. 9.