3.1

Generation and phase-matched HHG and FWM

For a far negative delay (-200 fs), the driving beam arrives before the control beam, and we obtain only the HHG spectrum driven by the 800 nm beam. Figure 2(a) shows the odd-harmonic spectrum which exhibits a spectral comb structure optimized around the phase-matching condition for harmonic H21 (32.55 eV). The full spectral linewidth of each harmonic is optimized around 500 meV.

Figure 2:

HHG spectrum with 49 Torr of argon. The intensity is normalized to the maximum in each spectrum. (a) HHG spectrum for far negative delay -200 fs, (b) FWM spectrum for phase-matched zero delay 0 fs for cases with/without the presence of 560 nm corresponding to the color spectrum (green/red).

The physics underlying the signature spectrum of HHG has been successfully explained using the semi-classical three-step model¹⁷, the semi-analytical quantum mechanics model¹⁸ and the fully quantum theory of extreme nonlinear optics¹⁹. In order to obtain an odd-harmonic spectrum, all the emitted photons have to add constructively together at the moment the electrons recollide back to the parent ion. Calling the fundamental driving laser frequency ω₈₀₀, the phase-matching for the q^th harmonic (ω_q=qω₈₀₀) must satisfy the wave-vector mismatch between the driving field and the harmonic field:

The dispersion factor Δk_neutral arises from the neutral gas density, while Δk_plasma is due to the free electron cloud from the ionic gas which is unavoidable during the HHG process. To control these two factors, we optimize the effective focus intensity of the driving field to around 2x10¹⁴ W/cm² to maintain a low ionization rate around a critical value of 3%^{20, 21}. In this way, we can maintain a balanced contribution of positive and negative dispersion of the neutral gas and free-electron cloud within the interaction length. Moreover, the effect of free charges on the propagation of the driving field and the XUV field is negligible at this moderate ionization rate²².

In addition, the optimal aperture size of iris I4 can be finely adjusted to around 2 to 3 mm, which corresponds to a variable Rayleigh length of approximately 10 to 20 mm when used with a lens L3 of focal length 30 cm. Thus, the Rayleigh length is much longer than the effective interaction length (2 mm to 5 mm), so that the contribution of the intensity-dependent harmonic dipole phase shift Δk_dipole and the geometric-configuration Gouy phase shift Δk_geom can be neglected^{22, 23}. The effective interaction length (2 mm to 5 mm), i.e., the displacement of focusing point toward the exit pinhole inside the gas cell, for phase-matched HHG has been demonstrated in our previous studies²⁴ by the observation of the HHG intensity varying quadratically with gas pressure and interaction length.

For a phase-matched zero delay (0 fs), the control field overlaps with the driving field, and additional frequencies showing four-wave mixing peaks are generated on both sides of each harmonic as shown in Figure 2(b). The enrichment of the spectrum signal can be understood as a perturbative cascaded wave-mixing process¹⁶. In our experiment, we name as follows: wave-vectors k₈₀₀ and k_XUV for the 800 nm beam and the q^th harmonic, respectively, k_Ctrl for the control field for which we can change frequency, and k_MIX for a new frequency generated by the wave-mixing process. Since the coherence property of the driving field is transferred to the XUV pulse while the contribution of the neutral and plasma dispersion is small for the XUV pulse²², the phase-matching condition for the generation of the wave-mixing field within the coherence length of the q^th harmonic is Δk = k_MIX - (k_XUV ± (k₈₀₀ - k_Ctrl) ≈ 0. The frequency of the mixing field is determined by the law of parity, momentum, and energy conservation in the perturbative wave-mixing regime ^{15, 25-28}.

As a result, the sum and difference frequency mixing frequencies on both sides of the q^th harmonic in the XUV region are generated in our experiment. The factor m = 1, 2 corresponds to the third and fifth-order nonlinear wave-mixing process, which is displayed as two peaks on each side of the main XUV harmonic. The difference energy of 0.67 eV corresponds to the difference frequency Δω = ω₈₀₀ - ω_Ctrl.

From our previous studies the effective interaction length for the FWM process is identified as about 2 mm before the exit pinhole inside the gas cell²⁸. Within this effective interaction length, the mixing field intensity scales quadratically with gas pressure and interaction length. Plus, the phase effect acting on all component fields is negligible since the ionization rate of the argon gas is kept well below 3% in our dispersive interactive medium²².

3.2

Studies of multi-color FWM in XUV region

In this experiment we focus on the influence of the multi-color fields on the modulation of the spectrum without considering complicated energy transitions between resonance states on the background of the continuum of states. Accordingly, we focus on the perturbation nonlinear wave-mixing process with two different FWM processes with and without the presence of the additional field 560 nm around harmonic H21.

To ensure the reliability of the data, we monitor the spectrum variation based on the standard odd-harmonic spectrum of HHG. Figure 3(c) shows the sharp feature of HHG peaks which is maintained in our experiment for a far-negative and positive delay. This demonstrates the stability of the driving laser and the HHG generation process before and after the perturbation of the control field.

Figure 3:

Intensity of XUV radiation versus time-delay at harmonic H21 and its sum and difference frequency, (a) FWM of three input beams of H21, 800 nm and 1400 nm, (b) FWM of four input beams of H21, 800 nm, 1400 nm and 560 nm, (c) variation of main harmonic spectrum at H19, H21, H23 for far-negative and positive delay.

At zero delay, we have a maximum population depletion of the main harmonic (H21). Figure 3(a) shows the single-color FWM process around harmonic H21 (H21, 800 nm, and 1400 nm), in which we have the energy of the sum frequency E_sum = E_H21 + (E₈₀₀ - E₁₄₀₀) = 33.22 eV and the difference frequency E_diff = E_H21 - (E₈₀₀ - E₁₄₀₀) = 31.88 eV. The phase-matched window for this case falls within the range of 100 fs. Figure 3(b) shows the two-color FWM process around harmonic H21 (H21, 800 nm, 1400 nm, and 560 nm). We observe similar peaks for the sum and difference frequencies but with a different intensity modulation along the time-delay axis. With the additional field 560 nm, the FWM process occurs with a smaller phase-matched window but with higher efficiency for the population transfer from H21 to first-order mixing frequencies.

To understand the population oscillation among different radiation states within this cascaded wave-mixing process, we interpret the phenomena as follows. Within our single gas cell interaction chamber, the effective interaction length of the HHG process is about 2-5 mm while the generation of the FWM is about 2 mm before the exit pinhole²⁸. Therefore, the cascaded wave-mixing process can be analyzed in two processes. First, the intense driving field of k₈₀₀ initiates the HHG process in the argon gas medium. The result is a coherent constructive interference of the emitted XUV radiation which exhibits high temporal and spatial coherence, inherited from k₈₀₀. Therefore, the synthesized phase-locked XUV pulse and the 800 nm pulse create a dipole moment and give rise a macroscopic polarization, i.e., addressing the electron wave-packet in a highly excited state. Consequently, if there is no additional effect taking place in this process, the induced electric field of the highly excited atomic gas will cancel the initial field and cause an absorption process. This process will exhibit a characteristic dephasing time which can be recorded by attosecond transient absorption spectroscopy²⁹. Second, along the propagation length of k_XUV and k₈₀₀, we introduce a perturbation of the control field k_Ctrl with moderate intensity and parallel polarization to alter the dynamics of the prepared electron wave-packet. For the phase-matching condition, all component fields combine to induce a third-order and fifth-order nonlinear response in the interactive medium. The new generated mixing frequencies enrich our spectrum in the XUV region and this observation indicates the correlation physics with perturbation nonlinear optics theory^{15, 16}.

When we introduce a time-delay in the control beam, this will result in a time-varying cascaded wave-mixing process which creates a population oscillation among different radiation states at different time-delays as illustrated in Figure 3 and Figure 4. The dynamical process in this nonlinear interaction regime can be extracted to find the correlation of the time-delay dependence between the intrinsic oscillating dipole moment of the medium and the external manipulating electric field. We note that in our recent studies a frequency shift of the FWM field was identified when we analyzed the coupling state of the free electron wave-packet, in which we have a superposition of the continuum and a discrete real excited state of atomic krypton around harmonic H17 (26.3 eV)³⁰ and atomic argon around harmonic H19 (29.3 eV)³¹.

Figure 4:

Cascaded FWM spectrum versus time-delay around harmonic H21 (32.55 eV). The sum and difference frequencies appear on the right and left side of H21 corresponding to energies of 33.22 eV and 32.88 eV. (a) FWM without the presence of 560 nm, (b) FWM with the presence of 560 nm, (c) energy diagram for FWM process around H21.

In the present experiment, we focus our analysis on the spectrum around harmonic H21 which lies outside the resonance frequency of argon. From this, we consider the influence of the multi-color fields on the modulation of the XUV spectrum without concern about the resonance of single electron excitation states participating in the dynamics of our system. The energy diagram as illustrated in Figure 4(c) explains the reason for the maximum population transfer in the case of two-color FWM.

The figure illustrates two FWM processes. Interestingly, when we combine the two-color fields of k₁₄₀₀ and k₅₆₀ in the FWM process, the sum-frequency generation process in the case of FWM with 1400 nm ends up at the same energy level as with the difference-frequency generation of FWM with 560 nm. Similarly, the difference-frequency generation of the FWM with 1400 nm ends up at the same energy level as the sum-frequency generation of the FWM with 560 nm.

The above result demonstrates the very high efficiency of the sum and difference frequency generation processes. The contour Figure 4(b) illustrates the wave-mixing process and shows a clear depletion of the main harmonic H21 at 0 fs in the case of two-color FWM. This phenomenon is in contrast with the single-color FWM result in Figure 4(a) where we observe an unstable intensity oscillation and a sideband oscillation on the sum-frequency side. This result demonstrates the power of the multi-color field FWM process which not only amplifies the new frequency generation process but also produces a higher flux of coherent electron wave-packets. Such a result will be significant for time-resolved studies of electron wave-packet dynamics in highly excited states and in the above-threshold ionization regime. Additionally, the generated wave-packets are background-free and are produced by a perturbative nonlinear wave-mixing process which does not require a dipole-allowed transition to reach the radiation states. As a result of the influence of H21 on the atomic system in this experiment, we acknowledge that the XUV radiation of H21 (32.55 eV) lies above the first ionization threshold of argon (15.76 eV) and this could lead to either direct ionization or autoionization to the continuum of high Rydberg states in the one-electron excitation system. The higher frequency generation can even exceed the second ionization threshold and become involved in even more complex situations with two or more electron excitation processes.