1. Introduction

Recent progresses in ultrafast laser technologies have enabled us to explore many interesting research fields in strong-field physics such as high-harmonic and attosecond pulse generation [1] and their applications to time-resolved measurement and the control of electron dynamics [2]. Because of their spatial and temporal coherence, high harmonics are unique short-wavelength light sources to extend laser-based nonlinear optics to the extreme ultraviolet and soft x-ray regions and to attosecond time scales. The physics behind HHG is well described by the three-step model [3, 4] where an electron (i) first tunnels through the Coulomb barrier, which is distorted by a strong electric field, then (ii) moves as a free electron in the field, and finally, (iii) recombines with a parent atom releasing a photon of the energy equal to the sum of the kinetic energy of an electron and the ionization potential. All three steps give opportunities to control high harmonics. Especially the timing of tunnel ionization and the field acceleration in the continuum mostly determine the yield, the chirp, and the cut-off energy of high harmonics. Since all the fundamental processes occur on sub-cycle time scale, it is essential to control the shape of field oscillation on attosecond time scales. In this viewpoint, we examined the possibility to control the ionization process, the electron motion in the field, and, therefore, the high harmonics by the use of a phase-locked two-color laser field. There are many controllable parameters in the phase-locked two-color field such as a relative phase, their intensities, and polarizations. In this experiment we monitor high-harmonic spectra by changing a relative phase between the fundamental and second harmonic (SH) of the two-color field, E(t), defined throughout this paper as

where E _f and E _SH express the amplitudes of the fundamental and SH, respectively. ω is a fundamental angular frequency and ϕ is the relative phase between the fundamental and SH. A semiclassical trajectory of an electron motion in a monochromatic field repeats every half cycle of the field as shown in Fig. 1(a). On the other hand, the second harmonic field added to the fundamental field results in the appearance of two major trajectories, which repeat every one period of the fundamental field as shown in Fig. 1(b). Figure 1(b) shows two major branches and we call a branch with the higher-energy cut off as an upper branch and a branch with the lower-energy cut off as a lower branch [5]. Such deformation of the trajectory and the increase of the ionization probability due to the additional SH cause variety of unique phenomena. These phenomena include relative-phase-averaged two-color experiments such as the enhancement of the high-harmonic yield [6, 7] and the cut-off extension of high harmonics [8]. The use of relative-phase-locked two-color field results in experiments such as the relative-phase- dependent enhancement of the high-harmonic yield [9, 10], the generation of continuum spectrum in the extreme ultraviolet [11, 12], the generation and spectral shaping of attosecond pulse trains [13, 14], and the generation of extreme ultraviolet continuum aiming isolated attosecond pulses by the use of a sub-10 fs two-color laser pulse [15]. Despite many experiments, so far, researchers focus on phenomena originated in the lower branch trajectory although the effects of the upper branch were implicitly involved. To understand as well as to utilize a phase-locked two-color field for HHG and attosecond pulse generation, the first step is to resolve the contribution of two major trajectories, especially the upper branch. In this paper, we have demonstrated the control of the quantum paths in a two-color field and, therefore, the separation of the contribution of two trajectories to the cut offs. A dominant mechanism is found to be the ionization process gating the yield of an electron when trajectories start. Experimental results are confirmed by the numerical calculation based on the solution of the time-dependent one-dimensional Schrödinger equation.

 

Fig. 1. (a). Electron trajectories in a monochromatic field. (b). Electron trajectories in a bichromatic field. Black and blue lines show the kinetic energy of a return electron as a function of ionization and recombination times. Solid and dotted lines represent a short and a long trajectories of each branch, respectively. A red line shows an electric field. The amplitude ratio of the SH and fundamental (E _SH/E _f) is 0.25 as in the experiment. X-axis is plotted in units of a fundamental (800 nm) light period.

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2. Experimental setup

An experimental procedure is summarized in Fig. 2(a). 20-fs, 10-mJ pulses at a 1-kHz repetition rate from a Ti:sapphire chirped-pulse amplification system [16] were used for the two-color experiment. The output pulses from the laser system were sent to a two-color Michelson interferometer shown in Fig. 2(b) to produce the relative-phase-locked two-color pulses with the pulse energies of 1.64 mJ at 800 nm and 0.24 mJ at 400 nm, respectively. The relative delay was controlled by the feedback stage (FS-1020-PX, Sigma Tech) with a precision of 20 nm. The polarizations of both fundamental and SH beams were set to be linear and parallel with each other. The fundamental and SH beams were spatially filtered independently by irises to have the diameters of 12 mm and 7 mm at the exit of the interferometer, respectively. This helped us to maximize the beam spatial overlap between two fields at the focus. A harmonic generation and detection setup is depicted in Fig. 2(c). The two-color pulses from the interferometer were focused into a 1-kHz pulsed neon (Ne) jet by a concave mirror with a focal length of 1000 mm. The detailed information of HHG scheme with a pulsed gas jet is described elsewhere [17, 18]. The peak intensity of the fundamental field at the interaction region was estimated to be 3.7×10¹⁴W/cm² from the cut-off energy of high harmonics, when only the fundamental pulses were used. The peak intensity of the SH field can be derived to be 2.4×10 ¹³W/cm² assuming the fundamental peak intensity above. Assuming these intensities, the ionization probability of Ne can be calculated based on the ADK model [19] to be from 4 to 9 % depending on the relative phase between two fields. This low level of ionization suppresses the effect of the plasma formed in the interaction region. Generated harmonics were spectrally resolved by an x-ray spectrometer (SXR-II-1, Hettrick Scientific), passed through two 150-nm-thick Zr filters (Luxel), and were detected by a back-illuminated CCD camera (Princeton instruments). We also measured Ne radiation at 585 nm (2p₁ - 1s₂ (Paschen notation), 3p ^′[1/2]₀-3s ^′[1/2]⁰ ₁ (Racah notation)) [20, 21] to monitor a Ne⁺ yield using a spectrometer (USB4000, Ocean Optics). Because Ne+ created by the strong laser field at the focus recombines with an electron and the resulting neutral Ne of an excited state decays to its ground state via the 2p _n - 1s_m (Paschen) paths, the radiation intensity is considered to be proportional to a Ne⁺ yield.

 

Fig. 2. (a). Experimental procedure. (b). Two-color Michelson interferometer, BBO, 150µm-thick β -BaB₂O₄; DM, dichroic-mirror; WP, zero-order half-wave plate at 800 nm. (c). Harmonic generation and detection setup, OMA, optical multichannel analyzer (USB4000, Ocean Optics); SXR-II-1, x-ray spectrometer (Hettrick Scientific); TMP, turbomolecular pump.

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3. Results and discussion

The fundamental pulses without the SH pulses were used to generate high harmonics having the cut-off energy of 92 eV, which gives the estimation of the peak intensity. Mixing of the SH with the fundamental resulted in the ten-times enhancement of the harmonic intensity and also the appearance of even harmonics. The intensities of even harmonics became equal to those of odd harmonics. Only the SH did not have enough intensity to generate any observable harmonics in the spectral region of interest. Series of harmonic spectra depicted in Fig. 3(1) were obtained at every 20-nm delay (corresponding to the shift of the phase, ϕ, by π/10 rad.) with an exposure time of 3 seconds (3000 laser shots). Typical harmonic spectra shown in Fig. 3(3) were measured at three phases indicated in Fig. 3(1). In Fig. 3(3), the black line (A) has the highest cut-off energy but has the lower intensity in the plateau region. On the contrary, the green line (C) has the lowest cut-off energy but has the higher intensity than the black line (A). In the middle of the two phases (A) and (C), the red line (B) has a two-step feature in the observed spectrum. This two-step spectrum indicates that the upper and lower branches contribute to the weak spectral component with the higher cut-off energy and the intense one with the lower cut-off energy, respectively. With this relative phase, assuming the ponderomotive potential, U _p, of 22.1 eV and the Ne ionization potential, I _p, of 21.6 eV, the cut-off energies of the upper and lower branches can be evaluated from the classic trajectory as 3.70U _p+I _p=103.4 eV and 2.72U _p+I _p=81.7 eV, respectively. These values well correspond to the experimental cut-off energies of the two-step spectrum. To confirm the experimental results, the time-dependent one-dimensional Schrödinger equation is numerically integrated to calculate the dipole response of a single electron in a model atom using the same parameters as those used in the experiment. The results of the numerical study are shown in Fig. 3(2), where the dipole intensity is plotted as a function of the relative phase and the corresponding photon energy of the high harmonics. Figure 3(4) shows the harmonic spectra at three phases selected from Fig. 3(2). As can be seen, the experimental and numerical results are in good agreement in the appearance of the two step structure where the lower-energy cut off around 80 eV has the higher intensity than the higher-energy cut off around 100 eV. The underlying mechanism behind these observations can be understood by the semiclassical electron trajectories shown in Fig. 4. We pick up the same phases, 0.3π, 0.5π, and 0.8π rad. as those in Fig. 3. In all phases, the relative phase does not affect the upper and lower branches so much, in spite of the slight change of trajectories’ shapes and cut-off energies. Meanwhile, the electric field amplitude at the time of ionization varies by about 10 %. This small change of the field intensity, however, influences the ionization probability drastically since tunnel ionization is the extremely nonlinear process [19]. Therefore, we conclude that the ionization gating is the major mechanism for the selection of quantum trajectories. In Fig. 4(a), the upper branch is dominant more than the lower branch in the harmonic spectrum because the electric field at the time of ionization of the upper branch is stronger than the lower branch. As an opposite case, in Fig. 4(c), although the semiclassical trajectories are almost the same as those in Fig. 4(a), the lower cut off harmonics are now strongly enhanced since the field amplitude at the time of ionization for the lower branch is higher, resulting in the intense radiation with the lower cut-off energy. In Fig. 4(b), in the middle of two extreme cases, both branches can be seen as the clear indication of a two step structure in the high harmonic spectrum. These explanations are well consistent with the experimental observations and the numerical results. The selection of a quantum path is a novel method to change the characteristics of the high harmonics, e.g., its intensity, the pulse duration and the chirp of generated harmonics, and the timing between high-harmonic pulses in the train. This control is also effective for the generation of an isolated attosecond pulse using a sub-10-fs laser pulse. The requirement of the driver pulse duration for the isolation of harmonics can be relaxed by carefully choosing the relative phase and, therefore, a specific branch (either a lower or upper branch) to suppress neighboring satellite harmonic pulses. We also investigate the relative-phase dependence of the harmonic intensity in two energy regions (upper branch dominant (>88 eV) and lower branch dominant regions (<88 eV)), and the ionization probability of Ne. Measured and calculated traces are plotted in Figs. 5(a) and (b), respectively. In both figures, black, blue, and orange lines are the harmonic intensities integrated in the higher-energy region (>88 eV), and in the lower-energy region (<88 eV) and Ne recombination emission intensity, respectively. The ionization probability of Ne enables us to determine the absolute value of the relative phase between two fields because the ionization peaks at a relative phase equal to 0. Therefore simultaneous measurement of the high-harmonic changes and ion yield is a useful way to obtain the precise value of the relative phase and investigate the relative-phase dependence of strong field phenomena. Past experiments have only estimated the relative phase indirectly by comparing experimental data with simulation results. The high-harmonic intensity of the upper branch is almost suppressed to be zero because of the drastic ionization gating. To the contrary, the highharmonic intensity in the low energy region shows a periodic modulation and has a minimum at a specific delay. However, even at a minimum point, the harmonic intensity has an offset from zero because of the contribution of low-energy harmonics from the strong upper branch. None of the peaks of all three lines does not coincide with each other at all. This result indicates that, by choosing a proper relative phase in a two-color HHG, one can efficiently extend the cut-off and enhance the high-energy harmonics without fully ionizing Ne. This scheme can be an useful method for the cut-off extension and the enhancement of the harmonics, which is usually limited by the ionization of a gaseous medium. In other words, in contrast to a monochromatic field, two-color fields can control the electron produced at the maximum field amplitudes to recombine efficiently with a parent atom of the maximum kinetic energy.

 

Fig. 3. (1) Measured high-harmonic spectra as a function of a relative phase. Vertical axis, photon energy; horizontal axis, relative phase, ϕ. A, B, and C indicate measurement points of the spectra shown in Fig. 3(3). (2) Calculated high-harmonic spectra as a function of a relative phase. Vertical axis, photon energy; horizontal axis, relative phase, ϕ. D, E, and F indicate measurement points of the spectra shown in Fig. 3(4). (3) Measured harmonic spectra at three phases. (4) Calculated harmonic spectra at three phases.

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Fig. 4. Relative-phase dependent electron trajectories with the relative phase of 0.3π (a), 0.5π (b), and 0.8π rad. (c). Black and blue lines (solid line: short trajectory, dotted line: long trajectory) represent the kinetic energy of a return electron as a function of ionization and recombination times in units of fundamental (800-nm) light periods, respectively. A red line shows a two-color field.

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Fig. 5. (a). Measured high-harmonic intensities and ion yield. Black line, harmonic intensity integrated in the higher-energy region (>88 eV, upper branch); blue line, harmonic intensity in the lower-energy region (<88 eV, lower branch); orange line, Ne recombination emission intensity. (b). Calculated high-harmonic intensities and ion yield. Black line, harmonic intensity integrated in the higher-energy region (>88 eV, upper branch); blue line, harmonic intensity in the lower-energy region (<88 eV, lower branch); orange line, calculated Ne ionization probability.

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4. Conclusion

We have produced high harmonics with a phase-locked two-color field and observed the phase dependence of the spectral features near the cut off (70–110 eV). To our knowledge, it is the first clear observation of the two-step structure around high-harmonic cut off and the separation of the upper and lower branches’ contribution to high harmonics. We found that the mechanism of these phenomena is mainly attributed to the tunnel ionization gating of the quantum path. We have also demonstrated the relative-phase-dependent measurement of the high-harmonic intensity in the different energy regions. We simultaneously monitored the degree of ionization using a simple spectrometer. This is a useful method to determine the absolute value of the relative phase of two fields in a two-color HHG experiments. The experimental results are well reproduced in the simulation results. The selection of the quantum path is a key point for producing an isolated attosecond pulse by the use of a sub-10 fs fundamental pulse. The addition of the second harmonic to the fundamental field double the periodicity of the harmonics generation process. Therefore, the requirement of the pulse duration for the isolated attosecond pulse generation can be relaxed by using a two-color field driver and by the choice of a proper relative phase.