⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 114203 Intensity-Dependent Dipole Phase in High-Order Harmonic Interferometry Li Wang (王力), Fan Xiao (肖凡), Pan Song (宋盼), Wenkai Tao (陶文凯), Xu Sun (孙旭), Jiacan Wang (王家灿), Zhigang Zheng (郑志刚), Jing Zhao (赵晶), Xiaowei Wang (王小伟)*, and Zengxiu Zhao (赵增秀)* Affiliations Department of Physics, National University of Defense Technology, Changsha 410073, China Received 27 August 2023; accepted manuscript online 9 October 2023; published online 13 November 2023 *Corresponding authors. Email: xiaowei.wang@nudt.edu.cn; zhaozengxiu@nudt.edu.cn Citation Text: Wang L, Xiao F, Song P et al. 2023 Chin. Phys. Lett. 40 114203    Abstract High-order harmonics are ideal probes to resolve the attosecond dynamics of strong-field recollision processes. An easy-to-implement phase mask is utilized to covert the Gaussian beam to TEM01 transverse electromagnetic mode, allowing the realization of two-source interferometry of high-order harmonics. We experimentally measure the intensity dependence of dipole phase directly with high-order harmonic interferometry, in which the driving laser intensity can be precisely adjusted. The classical electron excursion simulations reproduce the experimental findings quite well, demonstrating that Coulomb potential plays subtle roles on movement of electrons for harmonics near the ionization threshold. This work is of great importance for precision measurements of ultrafast dynamics in strong-field physics.
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 DOI:10.1088/0256-307X/40/11/114203 © 2023 Chinese Physics Society Article Text As coherent extreme ultra-violet (XUV) light sources, high-order harmonics (HH) radiated from extreme nonlinear optical process in strong laser fields form the basis of attosecond science. High-order harmonic generation (HHG) enables production of ultra-broadband XUV spectra with highest photon energy of up to thousands of keV,[1,2] the generation of single attosecond pulses with pulse durations down to tens of attoseconds,[3-6] and the investigation of electron and nuclear ultrafast dynamics in matter on timescales ranging from femtoseconds to attoseconds.[7-15] Although the strong-field approximation (SFA), in which the liberated electron interacts with strong laser fields, the role of Coulomb potential (CP) needs to be examined for near-threshold harmonics with near-zero electron kinetic energy. One of the examples of the significance of CP in strong-field physics is the theoretical study of sub-cycle ionization dynamics,[16] in which the SFA fails to describe the dynamics even qualitatively. For liberated electrons, the excursion time is found to be overestimated if ionic potential is neglected,[17] which implies that the spectral phase of HH cannot be accurately calculated with the SFA. In particular, when near-threshold harmonics are investigated, the short trajectory dynamics exhibits a strong correlation with the ionic structure,[18] and thus holds the potential to resolve the dynamics of atomic excited states and molecular orbitals. Precise measurements of the long-discussed ionization delay time as well as recombination times can be improved by tens of attoseconds when long-range Coulomb interaction is taken into account.[19] Subtle but vital time differences in electron dynamics were also observed with XUV interferometry recently, which was attributed to strong laser–Coulomb coupling.[20] It was later demonstrated that the slowdown time of outgoing electrons by CP is about 35 as and measurable by using a streaking technique with complex time trajectory analysis.[21] With state-of-the-art attosecond precision measurements, abundant ultrafast electron dynamics can be more deeply and precisely investigated. Among the rich structural and temporal information encoded in HH, the spectral phase indicates the electron excursion dynamics, which is of great importance to understand strong-field recollision processes, such as nonsequential double ionization, frustrated tunneling ionization, and ultrashort attosecond pulse generation. While spectral intensity is readily obtainable experimentally, the measurement of spectral phase requires complex correlation techniques, such as RABBITT,[22,23] attosecond streak cameras,[24] XFROG,[25] and the recently developed all-optical characterization method.[26,27] These methods are either time-consuming or heavily dependent on further assumptions, putting all kinds of limitations on the HH phase determination. On the other hand, direct phase measurements with HH interference, based on Young's double-slit interferometry,[28] have been successfully used for atomic and solid dipole phase measurements,[29,30] the demonstration of phase-locked HHG,[31] Gouy phase detection,[32] molecular multi-electron dynamic tomography,[28] and ultrafast charge transfer.[10] These developments have inspired the use of optical interferometry to detect the phase variation of HH with laser intensity. In this work, interferences between two HH sources are utilized to directly measure the phase variation of HH from argon atoms with respect to the driving laser intensity. We develop an easy-to-implement phase mask to covert the Gaussian beam to TEM01 transverse electromagnetic mode, allowing relative intensity adjustment to investigate intensity-dependent HH phase. Our experimental results corroborate the semiclassical three step model under the SFA. Moreover, CP is found to play marginal roles on the HH phase even for the near threshold harmonics. In the experiments, HH was generated from argon atoms subjected to strong laser fields formed by focusing 30 fs laser pulses centered at 793 nm with repetition rate of 1 kHz and pulse energy up to 1.6 mJ, as illustrated in Fig. 1(a). The input pulse energy is adjustable with an ultrafast half-wave plate and a broadband polarizer. To create two HH sources for interferometry measurements, a phase mask consisting of two fused silica plates (FSPs) was placed in the beam path to divide the laser beam into two parts. The two FSPs, which act as a 0–$\pi$ phase plate,[33,34] split the wavefront into two parts and impose additional spectral phases. By tuning the tilt angle of one of the FSPs, the phase difference introduced by the material dispersion can be set to $\pi$. This alignment results in the destructive interference of the two beams at the focal center, leading to the formation of a dark region and the emergence of two distinct focal spots, as shown in Fig. 1(b). In comparison with the 0–$\pi$ phase plate, our method is easy to implement and applicable for pulses with different wavelengths. The intensity integrals along the $x$ and $y$ axes are shown in Fig. 1(c), which manifests that the full width at half maximum (FWHM) of each focus is about 75 µm and the distance between the two focuses is 110 µm. Since the two FSPs pick almost equal pulse energy from the incident beam, the peak intensities of the two focuses are fairly identical.
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Fig. 1. Schematic of HH interferometry. (a) A phase mask consisting of two fused silica plates employed to divide the 30 fs, 1 kHz pulses into two parts, and their interference at the focus producing two separate focal spots by tuning the tilt angle of one of the two plates. The interference fringes between the two HH sources are detected by a home-made spectrometer. (b) The intensity distribution of the two focal spots on the focal plane. (c) The intensity integrals along $x$ and $y$ axes, which manifests that the full width at half maximum of each focus is about 75 µm and the distance between the two focuses is 110 µm.
To probe the intensity dependence of the HH phase, the laser intensity of one of the two focal spots (probe HH source) needs to be tunable, while the laser intensity of the other source (reference HH source) is kept to be constant. An iris was inserted in one of the beam paths to allow relative pulse energy adjustments. By changing the aperture of the iris, the intensity of both foci changes accordingly due to the interference. We then adjusted the total input pulse energy by rotating the half-wave plate to hold the intensity of the reference beam. This was accomplished by monitoring the foci intensity with a charge-coupled device camera. Figure 2 shows seven different probe intensities from 51 TW/cm$^2$ to 71 TW/cm$^2$ while keeping the reference intensity as 143 TW/cm$^2$. Although the beam diameter is adjusted with an iris, the beam size variation was tiny (from 4.23 mm to 5 mm). It can be seen from Fig. 2 that during the intensity tuning, the focal spot size and the gap between the two foci hardly changed. The Gouy phase change due to the focusing geometry is estimated to be only 0.15 rad, which can be ignored since it is much smaller than the dipole phase change introduced by the laser intensity change.
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Fig. 2. Intensity adjustment of one of the two foci. By tuning the iris aperture and the input pulse energy, seven different probe intensities from 51 TW/cm$^2$ to 71 TW/cm$^2$ are obtained, while the reference intensity is kept at 143 TW/cm$^2$. The focal spot size and the gap between the two foci do not change significantly.
An argon gas nozzle with a back pressure of 2 bar was employed to generate HH, which was observed using a homemade XUV spectrometer[35] with spatial and spectral resolving power. Due to the short propagation length of HH in the gas jet, the macroscopic effect is considered to be insignificant and the measured HH signal is considered as a direct indication of the single atom response. In contrast to conventional HHG experiments, there are two coherent HH sources, which interfere with each other, and fringes show up in the spectrum of each order HH. An interferential spectrum example is shown in Fig. 3(a), in which odd harmonics in 11–21st orders are detected. Interference fringes are apparent for each order of HH along the spatial (vertical) axis. The detailed spatial profiles of the intensity of all the observed HHs are plotted in Fig. 3(b). The spatial interval between adjacent fringes is inversely proportional to the distance between the two laser foci and to the wavelength of HH. Thus, higher-order HHs have denser fringes. Additionally, the phase of the fringes indicates the phase difference of the two HH sources. Therefore, electron excursion dynamics in the strong field can be investigated by observing the shifting of the interference fringes under different laser intensities.
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Fig. 3. (a) Interference fringes of the two HH sources. The 11–21st order harmonics are detected. Interference fringes are apparent for each order of HH along the spatial (vertical) axis. (b) The spatial profiles of each HH. The spatial interval between adjacent fringes is inversely proportional to the distance between the two laser foci and to the wavelength of HH, so higher-order harmonics have denser fringes.
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Fig. 4. Shift of interference fringes versus laser intensity for 11th (a), 13th (b), 15th (c) and 17th (d) HHs. The fringes for 11th, 13th and 15th harmonics show clear interferences with discrete crests and troughs for all the intensities, while the modulation of the interference fringes for 17th HH with laser intensity lower than 55 TW/cm$^2$ is vague. It is consistent with the cutoff energy law of HHG.
According to the well-accepted semi-classical three-step model,[36] HHs are emitted right at the moment that the returning electron recollides with the parent ion with energy being the summation of the ionization potential and electron kinetic energy. As earlier returning electrons have lower/higher kinetic energy for short/long trajectory, intrinsic chirp exists in HHs and the phases of HHs carry rich information about the electron releasing, excursion and recombination. The driving laser intensity for the reference HH source is tuned to be 143 TW/cm$^2$, while the laser intensity for the probe source was varied from 51 TW/cm$^2$ to 71 TW/cm$^2$ as addressed before to generate HH covering barely above the threshold to well above the threshold spectral range. The corresponding fringes under different laser intensities for 11th, 13th, 15th and 17th HHs are illustrated in Figs. 4(a), 4(b), 4(c), and 4(d), respectively, in which only the most prominent interference features within divergence angles from $-2$ to 2 mrad are exhibited. As is expected, the fringes shift (upwards) as driving laser intensity increases. The fringes for lower order HHs (11th, 13th, and 15th) show clear interferences with discrete crests and troughs for all the intensities, while the modulation of the interference fringes for 17th HH with laser intensity lower than 55 TW/cm$^2$ is vague. The interference is only present when both the probe and reference sources have the same spectral component. Since the reference source contains HHs up to 27th order due to high laser intensity, the turning point of the spectral interferences implies that the 17th HH from the probe source is generated with laser intensity higher than 55 TW/cm$^2$. It is in agreement with the cutoff energy law of HHG, i.e., $\omega_{\max}=I_{\rm p}+3.17U_{\rm p}$, with $I_{\rm p}$ being the field-free ionization potential of the argon atoms and $U_{\rm p}$ being the ponderomotive energy. For 17th HH with photon energy of 26.8 eV, the required driving laser intensity is 59 TW/cm$^2$, which is in excellent agreement with experimental observations. Therefore, the interference onset intensity for a certain order of HH provides a novel approach to accurately calibrate the intensity of laser pulses. In spite of the intensity calibration, more valuable information regarding electron dynamics is encoded in the shifting of the fringes. As the reference source is kept to be unchanged, the spatial movement $\Delta L$ of the interference pattern reflects the phase change $\Delta\varphi$ of HH radiated from probe source, which satisfies \begin{align} \Delta \varphi(I) = 2\pi\frac{\Delta L(I)}{L_0}, \tag {1} \end{align} where $L_0$ is spatial oscillation period and $I$ is the laser peak intensity. Note that the absolute phase of each harmonic is not obtainable, only the intensity-related phase change can be decoded. The phase difference caused by gas density imparity between the two foci, for example, can be excluded by the interference with the same reference HH source for different driving intensities of probe source. As shown in Fig. 4, the phase changes $\Delta \varphi(I)$ for 11th (square dots), 13th (circle dots) and 15th (triangle dots) harmonics are extracted by performing Fourier analysis for the interference fringes. To examine the role of CP on electron motion, the classical trajectory method with and without CP is adopted to calculate the return time, as a function of laser intensity, of electrons responsible for the emission of different order harmonics. The laser pulse is assumed to have Gaussian temporal profile with an FWHM of 30 fs and a central wavelength of 793 nm, which is consistent with the experimental conditions. Electron motion is simulated using Newton's equations (atomic units are used throughout unless stated otherwise): \begin{align} {\boldsymbol F}(t)=-{\boldsymbol E}(t)-U'(x), \tag {2} \end{align} where ${\boldsymbol F}$ is the force imposed on the electron, $\boldsymbol {E}$ is the laser electric field strength and $U$ is the soft CP with the following form: \begin{align} U(x) = -\frac{1}{\sqrt{x^2+\big(\frac{1}{U_0}\big)^2}}, \tag {3} \end{align} where $U_0=0.7$ a.u. (19 eV) and it is close to the argon ionization energy of 15.6 eV. To reasonably investigate the low-order harmonics, classical over-the-barrier escape[37] needs to be included with initial conditions: \begin{align} x(t_0)=0,~~p(t_0)=\pm \sqrt{2(E_i+U_0)}, \tag {4} \end{align} and $E_i$ is set to zero.
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Fig. 5. (a) The calculated intensity-dependent dipole phase change without CP (dashed lines) and with CP (solid lines), as well as experimentally measured dipole phase with HH interferometry (dots), for 11th (square), 13th (circle) and 15th (triangle) HH. (b) Electron return time calculated for 11–21th odd-order harmonics versus the driving laser intensity without CP (dashed lines) and with CP (solid lines).
When the electron trajectories have been calculated for all light intensities, we can then select those trajectories that are consistent with the experimental HHG energies and extract their return times as a function of light intensity, which can be expressed as $T_{\rm r}(q,I)$, where $q$ and $I$ represent the harmonic order and laser intensity, respectively. For the returned electron corresponding to any order harmonic, the phase difference can be expressed as \begin{align} \Delta \varphi_q(I)=\frac{T_{\rm r}(q,I) - T_{\rm r}(q,I_0)}{T_q}, \tag {5} \end{align} with $I_0$ and $T_q$ are the reference laser intensity and period of the $q$th harmonic. The unit of the $\Delta \varphi_q(I)$ is $2\pi$. The calculated intensity-dependent dipole phase changes without CP (dashed lines) and with CP (solid lines) are compared with the experimental data, as shown in Fig. 5(a). The data for 17th HH are not shown since the interferences occur only for part of the scanned intensities as stated above. It is demonstrated that both the theoretical models reproduce the experimental points fairly well, which suggests that CP plays a generally insignificant role in the movement of electrons, especially for harmonics well above the ionization threshold. However, for harmonics near the threshold such as the 11th harmonic, the calculation including CP yields slightly better conformity with the measured phase change. For 15th HH, there exists obvious discrepancy between simulation and experimental data, since the classic three-step simulation overestimates the electron excursion time for the dominate short trajectory electrons, especially for higher-order harmonics.[38] Regarding the electron excursion dynamics in the electron rescattering process, the electron return time for 11–21th odd order harmonics versus the driving laser intensity are calculated, as presented in Fig. 5(b), with CP (solid lines) and without CP (dashed lines). For HH well above the ionization threshold, the classical calculations under the SFA overestimate the travel time for the liberated electron. The electron is speeded up by the CP by about 10 as; whereas for electrons with low kinetic energy, whose recombination with the ionic core radiates at near-threshold harmonics (11th), CP slows down the electrons by about 35 as, which is in accordance with a previous study.[21] However, the time difference is too marginal to be measured by the proposed high-order harmonic interferometry, which validates the SFA model. In conclusion, the interferometry of HHs is realized with an easy-to-implement phase mask and it is employed to experimentally measure the phase change of different-order HHs produced by argon gas at different laser intensities. The experimental observations are in excellent agreement with the classical analysis. We show that the on/off status of the interference fringes for certain order HH can be used to perform precise laser intensity calibrations. Furthermore, it is demonstrated that CP plays an insignificant role in the movement of electrons for harmonics above the ionization threshold. The “double-slit” HH interferometry powered by the coherent nature of HHs provides a quick and reliable means to reveal the intensity-dependent phase change of HHs. This work shows the great potential of HH interferometry in measurements of ultrafast strong-field dynamics. Acknowledgements. This work was supported by the National Key Research and Development Program of China (Grant No. 2019YFA0307703), and the National Natural Science Foundation of China (Grant Nos. 12234020 and 11974426). 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