Attosecond pulse formation via switching of resonant interaction by tunnel ionization

We derive an analytical solution uncovering the origin of few-cycle attosecond pulse formation from vacuum-ultraviolet (VUV) radiation in an atomic gas simultaneously irradiated by a moderately strong infrared (IR) laser field, which does not perturb atoms in the ground state, but induces rapid quasistatic ionization from the excited states [Polovinkin et al., Opt. Lett. 36, 2296 (2011)]. The derived solution shows that the pulses are produced due to periodic switching of the resonant interaction between the incident VUV radiation and the atoms: turning it off near the crests of the IR-field strength and switching it back on near the IR-field zero crossings. We extend the method originally proposed by Polovinkin  et al. [Opt. Lett. 36, 2296 (2011)] to non-hydrogen-like media and show that the pulses can be produced from resonant VUV radiation in a variety of atomic gases. The pulses are nearly bandwidth limited without external adjustment of phases of the generated sidebands. Proximity of the carrier frequency of the produced pulses to intra-atomic resonances may allow their efficient utilization for nondestructive steering of ultrafast dynamics of the bound electrons. The experimental possibilities for attosecond pulse formation from 58.4 nm VUV radiation in helium and from 73.6 nm VUV radiation in neon dressed by the 3.9 μm laser field, as well as from 122 nm VUV radiation in atomic hydrogen dressed by $\mathrm{C}{\mathrm{O}}_2$-laser field are discussed.

Figure 1

Sketch of the theoretical model used for derivation of the analytical solution. The incident HF radiation with frequency ω selects the lower, $|1\rangle$, and upper, $|2\rangle$, states of the resonant atomic transition, possessing the energy ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$, respectively. The LF field with frequency Ω rapidly ionizes atoms from the excited state $|2\rangle$ leading to periodic broadening of the corresponding energy level twice within the LF-field cycle. At the same time, the ground atomic state $|1\rangle$ remains unaffected by the LF field. The linewidth of the resonant transition $|1\rangle \leftrightarrow |2\rangle$ takes the minimum value $2{\overline{\gamma }}_{min}$ during the time intervals $\Delta t_{\mathrm{zero}}$ near zero crossings of the LF field and the maximum value $2{\overline{\gamma }}_{max}$ during the time intervals $\pi /\Omega -\Delta t_{\mathrm{zero}}$ near the LF-field crests.

Figure 2

Time dependence of radiation, resonantly scattered by the atoms of helium under the simultaneous action of 58.4 nm VUV radiation, exciting $1s\leftrightarrow 2p$ atomic transition, and $3.9\mu \mathrm{m}$ IR radiation with intensity $I_{\mathrm{IR}}=1.5\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, rapidly ionizing atoms from the $2p$ state twice within the IR-field cycle. The resonant scattering is confined to the time intervals $\Delta t_{\mathrm{zero}}$ near zero crossings of the IR field. For the rest of the time the resonant scattering is suppressed due to the huge broadening of the resonant transition line (see Fig. 1). The black bold dashed line corresponds to the radiation envelope, calculated analytically [(15),(16)]. The rapidly oscillating blue solid line corresponds to absolute value of the radiation strength, calculated numerically within the three-level model [22], taking into account the nonadiabatic coupling between $2p$ and $2s$ states of helium, as well as instantaneous quadratic Stark shift of the corresponding energy levels and tunnel ionization from them.

Figure 3

Fourier transform of the slowly varying envelope of the resonantly scattered radiation plotted in Fig. 2. The result of analytical calculation for the spectral amplitudes of the scattered radiation on the basis of Eqs. (18) and (19) is shown by cyan squares. Amplitudes of the spectral components calculated numerically within the three-level model [22] (see caption to Fig. 2) before and after spectral filtering (see the text) are shown by red solid and bold lavender dashed lines, respectively (these lines almost overlap except for the three central spectral components). Transmission of the spectral filter, which is zero at the resonance frequency (zero frequency of the envelope) and unity far away from it, is shown by black dash-dotted line. The result of analytical calculation for the spectral phases of the scattered radiation using Eqs. (18) and (19) is shown by red stars. The numerically calculated [22] phases of the spectral components at the multiples of the doubled IR-field frequency, ±2Ω,±4Ω,… are marked by blue circles.

Figure 4

Intensity of the output VUV radiation, propagated through an optically thin medium of helium, simultaneously irradiated by the resonant 58.4 nm VUV radiation and the strong $3.9\mu \mathrm{m}$ IR field (see caption to Fig. 2). The output VUV radiation results from coherent summation of the resonantly scattered radiation (Fig. 2) with the incident one. The blue solid line represents the result of analytical calculation for the envelope of the output VUV intensity. The rapidly oscillating dashed cyan curve corresponds to the square of the normalized VUV radiation strength, calculated numerically within the model [22]. The presented calculation imply approximation of an infinitely thin medium, so that the depth of intensity dips in Fig. 4 is much smaller than unity.

Figure 5

Intensity of the attosecond pulses produced from the output VUV radiation (see Fig. 4) via suppression of the resonant components of its spectrum according to Fig. 3. The lavender solid line represents the result of analytical calculation for the envelope of the pulses. The rapidly oscillating dashed red curve corresponds to the square of the normalized VUV radiation strength, calculated numerically within the model [22]. The pulse duration is ${\tau }_{\mathrm{pulse}}=420$ as; the pulse repetition period equals half-cycle of the IR field, $T=6.5\mathrm{fs}$.

Figure 6

Time dependence of intensity of the attosecond pulse train, produced from 73.6 nm VUV radiation in neon, dressed by $3.9\mu \mathrm{m}$ IR field with intensity $I_{\mathrm{IR}}=5\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, via the resonant interaction with $2s^{2}2p^6\leftrightarrow 2s^{2}2p^5\left({}^2{P^0}_{1/2}\right)3s$ atomic transition and suppression of the resonant component of the output VUV spectrum. The solid lavender curve represents the analytical solution for the envelope of the pulses. The rapidly oscillating blue curve corresponds to the square of the normalized VUV radiation strength, calculated numerically within the eight-level model [23], which takes into account transitions between the ground atomic state $2s^{2}2p^6$ and the excited states $2s^{2}2p^{5}3s$ and $2s^{2}2p^{5}3p$, coupled to each other by the VUV radiation and the IR field. The latter also induces time-dependent ionization from all the excited atomic states. The pulse duration is ${\tau }_{\mathrm{pulse}}=460$ as; the pulse repetition period equals half-cycle of the IR field, $T=6.5\mathrm{fs}$.

Figure 7

Intensity of the ultrashort pulses, produced from 122 nm VUV radiation due to the resonant interaction with $1s\leftrightarrow 2p$ transition of atomic hydrogen, dressed by $10.65\mu \mathrm{m}\mathrm{C}{\mathrm{O}}_2$-laser field with intensity $I_{\mathrm{C}{\mathrm{O}}_2}=2.2\times {10}^{13}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$, and suppression of the resonant component of the output VUV spectrum. The solid black curve represents the analytical solution for the envelope of the pulses. The rapidly oscillating lavender curve corresponds to the square of the normalized VUV radiation strength, calculated numerically within the three-level model [16], which takes into account both tunnel ionization from the excited atomic states $|2\rangle =\left(|2s\rangle +|2p\rangle \right)/\sqrt{2}$ and $|3\rangle =\left(|2\mathrm{s}\rangle -|2\mathrm{p}\rangle \right)/\sqrt{2}$, and Stark shift of the corresponding energy levels. The pulse duration is ${\tau }_{\mathrm{pulse}}=1.1\mathrm{fs}$; the pulse repetition period equals half-cycle of the $\mathrm{C}{\mathrm{O}}_2$-laser field, $T=17.8\mathrm{fs}$.