⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 113201 Chirp Compensation for Generating Ultrashort Attosecond Pulses with 800-nm Few-Cycle Pulses Li Wang (王力)1,2, Xiaowei Wang (王小伟)1,2*, Fan Xiao (肖凡)1,2, Jiacan Wang (王家灿)1,2, Wenkai Tao (陶文凯)1,2, Dongwen Zhang (张栋文)1,2, and Zengxiu Zhao (赵增秀)1,2* Affiliations 1Department of Physics, National University of Defense Technology, Changsha 410073, China 2Hunan Key Laboratory of Extreme Matter and Applications, National University of Defense Technology, Changsha 410073, China Received 19 September 2023; accepted manuscript online 1 November 2023; published online 13 November 2023 *Corresponding authors. Email: xiaowei.wang@nudt.edu.cn; zhaozengxiu@nudt.edu.cn Citation Text: Wang L, Wang X W, Xiao F et al. 2023 Chin. Phys. Lett. 40 113201    Abstract We show that it is feasible to generate sub-40-attosecond pulses with near-infrared few-cycle pulses centered at 800 nm. With proper gating technique, super-broadband continuum spectrum extending from 50 eV to above 200 eV can be obtained, and the intrinsic atto-chirp can be satisfactorily compensated with C filter, producing isolated attosecond pulses with duration of 33 as. According to the wavelength scaling law of high-order harmonic generation, the proposed scheme is of great significance to develop high-flux ultrashort attosecond sources.
cpl-40-11-113201-fig1.png
cpl-40-11-113201-fig2.png
cpl-40-11-113201-fig3.png
cpl-40-11-113201-fig4.png
 DOI:10.1088/0256-307X/40/11/113201 © 2023 Chinese Physics Society Article Text As the basis of attosecond science and technology growing up in the last two decades,[1-4] the generation of isolated attosecond pulses (IAPs) has been rightfully drawing a great deal of attention.[5-7] Improving the flux and shortening the pulse duration of IAPs are the two key tasks in the development of attosecond light sources. Although high-flux attosecond pulses are produced with free electron lasers, IAPs gated from high-order harmonic (HH) generation (HHG)[8,9] in strong laser fields have reached a much wider audience as its affordability and table-top compactness. Therefore, proposals for juggling the conversion efficiency and pulse duration in HHG process are desired. According to the energy-time uncertainty principle, the time-bandwidth product for Gaussian pulses satisfies $\Delta E \tau \geq1.83$ eV$\cdot$fs,[10] where $\Delta E$ and $\tau$ are the full width at half maximum (FWHM) of the spectral intensity and temporal profile, respectively. It suggests that ultrashort attosecond pulses shorter than 40 as require spectral bandwidth wider than 45.8 eV in FWHM, i.e., $\sim$ $100$ eV in full width. The cut-off energy $E_{\max}$ of HHG from atoms/molecules with ionization potential of $I_{\rm p}$ depends on both the peak intensity $I_0$ and central wavelength $\lambda_0$ as $(E_{\max}-I_{\rm p})\propto I_0\lambda_0^2$. The cut-off energy is proportional to the square of central wavelength, and the peak intensity is limited by the ionization, therefore strong laser pulses in mid-infrared (MIR) region are developed[11-14] to serve the purpose of broadband soft x-ray continuum producing[15,16] and ultrashort attosecond pulse generation.[17] The shortest IAP pulse that has been reported so far is generated with MIR pulses centered at 1.8 µm.[18] However, the conversion efficiency of HHG from a single atom proves to be scaling unfavorably with the laser wavelength as $\lambda^{-\mu}$ with $\mu=5$–6 for near-infrared (NIR) and MIR laser pulses,[19,20] which suggests 2–3 orders of magnitude dropping in conversion efficiency if the wavelength is tuned to be 1800 nm from 800 nm. On the other hand, in spite of higher conversion efficiency, IAP generation with NIR pulses suffers from lower photon energy limited by the saturation intensity. However, it is possible to breakthrough the limitation of saturation effect by generating IAP/HHG with ions whose ionization threshold is much higher than neutral atoms. It has been shown that the cut-off energy can be extended to above 500 eV with multiply charged argon ions subjected to 2.5 PW/cm$^2$ laser pulses centered at 800 nm,[21] and with 1.45 µm driving pulses the cut-off energy can even go up to 5.2 keV[22] from krypton atoms. Even if only moderately strong laser pulses, whose peak intensity are not strong enough to produce a significant amount of ions, are available, the bandwidth of NIR-driven IAPs from neutral atoms is still appreciable. The shortest IAP based on Ti:sapphire amplifiers is 67 as,[5] but their widest experimental bandwidth was larger than 100 eV. High energy photons must be removed via phase-matching process due to lack of appropriate dispersion compensation materials. Isolated extreme ultraviolet (XUV) continua supporting 32 as ultrashort pulses was generated with few-cycle NIR pulses.[23] Furthermore, the cut-off energy can even reach carbon K-edge[24,25] with NIR pulses, as comparable to the case with MIR pulses.[17] Thus it is feasible to generate ultrashort attosecond pulses with 800-nm drivers in the sense of bandwidth. The other key factor for generating ultrashort attosecond pulses, i.e., atto-chirp compensation, becomes more important and needs extensive investigations. In this Letter, we propose a route to generate sub-40 as IAPs with NIR pulses. It is shown that ultrabroadband continuum in the 50–250 eV spectral range can be typically produced with intense NIR pulses centered at 800 nm. Moreover, the intrinsic atto-chirp can be well compensated with C foil, resulting in near Fourier transform ultrashort IAPs. With short pulse duration and high conversion efficiency, NIR-driven IAPs based on HHG are promising table-top attosecond light sources, which are essential to give a boost to attosecond science and technology.
cpl-40-11-113201-fig1.png
Fig. 1. Intrinsic GDD of IAP under different peak intensities of driving pulses centered at 800 nm (a) and 1800 nm (b). As is well-known, the cut-off energy of long-wavelength driving pulses is much larger that of the short-wavelength driving pulses, however, super-broadband continuum extending from 50 eV to above 200 eV can be generated with 800 nm driving pulses when peak intensity is higher than 1 PW/cm$^2$. Moreover, for given photon energy, the GDD for shorter wavelength is smaller, which is favorable for ultrashort attosecond pulse generation.
The intrinsic chirp of IAPs can be predicted with the three-step model, which states that the HH photons are emitted during the recombination of the returning electrons liberated and energized by the strong laser fields, with photon energy being equal to the summation of ionization potential and kinetic energy of the returning electrons. For short trajectory electrons, the earlier arrived electrons have lower kinetic energy. Therefore, short trajectory HHs are born with positive group delay dispersion (GDD), which makes the attosecond pulses longer than their Fourier transform limited pulse width. By examining the classic trajectory of free electrons, the GDD of IAP is calculated for 800 nm and 1800 nm pulses for different laser intensities, as shown in Figs. 1(a) and 1(b), respectively. It is shown that for certain peak intensity the cut-off energy of 1800 nm pulses is much larger than that of 800 nm pulses, however, the cut-off energy of 800 nm pulses can still go up to about 300 eV for 1.5 PW/cm$^2$. The GDD of the two cases has very similar behaviors. Firstly, for a given laser intensity, the GDD for low and high (near cut-off) energy photons is huge, and is quite flat and small for the central part of IAP bandwidth. Secondly, the overall GDD drops as peak intensity increases. Lastly, the GDD is positive and on the order of hundreds to thousands of square attoseconds. Figure 1(a) shows that the IAP generated with 800 nm pulses can cover a spectral range wider than 50–250 eV when peak intensity is higher than 1 PW/cm$^2$, which is quite sufficient for generation of 40 as IAPs. The challenge is the atto-chirp compensation. The positive intrinsic atto-chirp is usually compensated with solid foils or gas medium with negative material dispersion. In the pioneer research, the dispersion of various materials are investigated in the XUV region,[26] and near Fourier transform limited pulses can be obtained with Xe[26] or Sn[27,28] as the dispersion compensation material for the spectral window of 150–300 eV. Hydrogen plasma[29] and neutral hydrogen[30] are proposed to be capable of compensating the atto-chirp in 300–500 eV and 500–1000 eV, respectively, which benefits the IAP generation with MIR drivers. With 800 nm driving pulses, the shortest IAP reported so far was 67 as,[5] and Zr foil was employed as the compensation material. The IAP spectra must be tailored to be narrower to match the compensation window of Zr foil although the experimental bandwidth can support IAPs shorter than 50 as. Therefore, it is desired to find materials with negative GDD and acceptable transmissivity in the range of 50–250 eV to compress the IAPs based on 800 nm drivers.
cpl-40-11-113201-fig2.png
Fig. 2. Transmissivity (a) and GDD (b) for 200-nm-thick C foil (circle), Ti foil (upward triangle), Sn foil (square) and Zr foil (downward triangle). The phase compensation range of Zr filter is below 150 eV, whereas Sn filter is usually used for high photon energy IAPs with upper working limit beyond 300 eV. However, it does not work for low energy photons below 140 eV. Ti filter can further extend the lower limit down to about 80 eV in spite of smaller transmissivity. In contrast, C filter is the best choice in terms of both transmission and GDD for the spectral window of 50–230 eV.
For the considered wavelength range which is longer compared to the atomic dimensions, the atoms inside the elementary substance or compound can be considered to scatter as dipoles with complex atomic scattering factor $\tilde{f}$, therefore, the complex refractive index in XUV region can be evaluated as[31] \begin{align} \tilde{n}(\lambda) = 1-\frac{r_0}{2\pi}\lambda^2\sum_{q}{n_q \tilde{f}_q}, \tag {1} \end{align} where $r_0$ is classic electron radius, and $n_q$ is the number of type $q$ atoms per unit volume. Then the GDD introduced by a medium with thickness $L$ can be obtained as follows: \begin{align} G_{\rm dd}(\omega) = \frac{L}{c_0}\Big(2\frac{dn}{d\omega}+\omega\frac{d^2n}{d\omega^2}\Big), \tag {2} \end{align} where $n$ is real part of $\tilde{n}$, and $c_0$ is the speed of light in vacuum. The transmission of the XUV light is determined by the imaginary part of $\tilde{n}$, i.e., the extinction coefficient $\kappa$. In Fig. 2, we show the optical properties of several 200-nm-thick candidates that have adequate transmission in the interested spectral window. The Zr filter, which has been widely used for IAP dispersion compensation with 800-nm driver, has decent transmission, however, its phase compensation range is below 150 eV. This is why the spectrum has to be narrowed via phase-mismatch in previous experiment.[5] The shortest IAP (43 as)[18] that has been demonstrated so far was produced with Zr filter, and its highest photon energy is around 150 eV as well. For high photon energy IAPs, Sn filter is usually used,[17,32] since its upper limit of negative GDD can go beyond 300 eV. However, it does not work for low energy photons below 140 eV. Titanium (Ti) filter can further extend the lower limit down to about 80 eV in spite of smaller transmissivity. In contrast, C filter is the best choice in terms of both transmission and GDD for the spectral window of 50–230 eV. To demonstrate the ability of C filter as a qualified compensation material, we simulate an IAP produced with polarization gating (PG) technique[33,34] by numerically solving the three-dimensional time-dependent Schrödinger equation (TDSE). The PG scheme is realized by partially overlapping two co-propagating counter-rotating circularly polarized pulses, such that a linearly polarized electric field with width being half the laser cycle is synthesized in the overlap region. Since HHG is suppressed when laser ellipticity is higher than 0.2,[35-37] HHG occurs only in the linear portion of the PG field, resulting in an IAP. In our simulation, only the electric field inside the PG gate window is considered. The TDSE was solved in the length gauge: \begin{align} i\frac{\partial}{\partial t}\varPsi (\boldsymbol{r},t)=\Big[-\frac{1}{2}\nabla ^2 - V(\boldsymbol{r}) - E(t)\boldsymbol{r}\Big]\varPsi (\boldsymbol{r},t), \tag {3} \end{align} where $\varPsi (\boldsymbol{r},t)$ is time-dependent wave function, $V(\boldsymbol{r})$ is atomic potential, and $E(t)$ is the driving laser field in the PG gate with central wavelength of 800 nm. For Ne atoms, the atomic potential we adopted[38] predicts a ground state energy of $-0.7928$ a.u., which agrees with the actual value of $-0.7923$ a.u. quite well. The evolution of the wave function in time was solved with splitting operator method,[39,40] and the time-dependent dipole moment is obtained by the following equation: \begin{align} d(t)=\int \varPsi^*(\boldsymbol{r},t)\boldsymbol{r}\varPsi (\boldsymbol{r},t)\,d\boldsymbol{r}. \tag {4} \end{align}
cpl-40-11-113201-fig3.png
Fig. 3. Simulated IAP spectrum by numerically solving the three-dimensional TDSE. (a) The time-frequency analysis of the calculated dipole. It confirms that IAPs are generated from short trajectory returning electrons, and the IAPs have positive intrinsic chirp. (b) Spectrum (filled area) and spectral phase (solid line) in the range of 50–250 eV obtained by the Fourier transform of $d(t)$. The plateau ends at about 200 eV, and the cut-off region stretches further into 250 eV. Although the spectral phases accumulated during the electron excursion seems linear, the high-order terms need to be properly compensated to yield short IAP.
In the simulation, the dipole is calculated with peak laser intensify of 1.5 PW/cm$^2$ within the half-cycle linear polarization window gated by PG, so that the cut-off energy can go above 250 eV. Only short trajectory contribution is kept in the calculated dipole, and HHG from long trajectory is restrained by setting appropriate absorption boundaries. This can be done experimentally with proper phase matching conditions. To confirm the absence of long trajectory and the generation of IAP instead of attosecond pulse train, the time-frequency analysis of the dipole is performed with spectrogram method, as shown in Fig. 3(a). The spectrogram shows that XUV photons with energy larger than 50 eV are emitted at a specific moment within only one half laser cycle. In addition, higher energy photons are born later than lower energy photons, i.e., the emission is positively chirped. Therefore, the calculated continuum spectra are indeed radiated from the recombination of short trajectory electrons to the ionic core. In Fig. 3(b), we show the spectrum (filled area) and spectral phase (solid line) in the range of 50–250 eV obtained by the Fourier transform of $d(t)$. The plateau ends at about 200 eV, and the cut-off region stretches further into 250 eV. The spectral phases accumulated during the electron excursion are nearly linear according to the figure, although there exists high-order terms suggested by the time-frequency analysis. This is due to the fact that the first order phase term is dominant. However, the high-order terms need to be properly compensated to yield short IAP.
cpl-40-11-113201-fig4.png
Fig. 4. Intrinsic phase compensation with C filter. (a) The high-order IAP intrinsic phase $\phi_{\rm in}$, the material phase introduced by 600 nm C foil $\phi_{\rm fo}$ and their summation $\phi_{\scriptscriptstyle{\rm IAP}}$. The transmitted spectrum is obtained by multiplying the simulated IAP spectrum and the transmissivity of C foil. (b) The temporal profile of compensated IAP, the FWHM is $33 \pm 1$ as, which is the shorter than the shortest IAP experimentally characterized so far. The uncertainty of the IAP width obtained by assuming the thickness of C foil is subjected to have 20 nm error. There are some asymmetric structures in the temporal profile, which is caused by the uncompensated higher-order phase terms.
The zeroth and first orders of the phase, which do not change the pulse shape, are removed and the remaining high-order terms of $\phi_{\rm in}$ are plotted in Fig. 4(a). The concave upward curve is a good indication of positive chirp. The material phase of 600 nm C foil $\phi_{\rm fo}$ is calculated from the refractive index described by Eq. (1). The $\phi_{\rm fo}$ curve is concave downward, and thus compensates the intrinsic phase $\phi_{\rm in}$, as indicated by their summation shown as $\phi_{\scriptscriptstyle{\rm IAP}}$ in Fig. 4(a). The total phase curve is near zero over a large wavelength range, and deviates from zero in the low and high energy parts of the entire spectrum. In addition to the phase, the spectrum intensity is also altered by the non-uniform transmissivity. The transmitted spectrum is shown by the filled line. With the spectral intensity and spectral phase in hand, it is straightforward to obtain the temporal intensity profile, as shown in Fig. 4(b). The FWHM of the IAP is $33 \pm 1$ as, which is the shorter than the shortest IAP experimentally characterized so far. The uncertainty of the pulse duration is introduced by the C foil thickness variation ($\pm 20$ nm), since we do not employ any optimization algorithm to find optimum thickness for the shortest IAP pulse. There are some asymmetric structures in the temporal profile, which is caused by the uncompensated higher-order phase terms. Besides the single atom response taken into accounted in this work, the macroscopic propagation or phase matching effect are also important in practice. The possible issues associated with macroscopic effects are the bandwidth narrowing and dispersion introducing in the isolated attosecond pulse generation. However, it does not necessarily play a part. Thin gas jet with propagation length as short as a few hundreds of micrometers can be used to eliminate the propagation effect. Loose focusing geometry with Rayleigh length larger than tens of millimeters can make Gouy phase effect unimportant. Or, low gas pressure can hold the cut-off photons and introduce neglectable dispersion while employing tight focusing to favor the short trajectory. Moreover, the carrier-envelope phase of the driving laser, which determines the electric waveform, affects both the cut-off energy and atto-chirp, so it should be stabilized to make sure every IAP short. In conclusion, we have shown that ultrabroadband XUV spectrum can be generated with PG technique from neon atoms subjected to intense few-cycle 800 nm pulses. The intrinsic chirp can be compensated with 600 nm C foil, resulting in an ultrashort IAP with FWHM of 33 as, which is shorter than the IAPs generated with 1800 nm drivers. The work relaxes the requirement of long-wavelength drivers, which is mainly produced with optical parametric amplifiers with efficiency less than 30%, to generate IAPs with pulse duration down to 40 as. In addition, short wavelength drivers result in much higher conversion efficiency according to the wavelength scaling law of HHG, which may help to enable high-flux IAP generation and attosecond nonlinear optics. Acknowledgments. 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). References Attosecond Technology and ScienceAttosecond metrology: from electron capture to future signal processingThe birth of attosecond physics and its coming of ageReal-Time Observation of Electron-Hole Coherence Induced by Strong-Field IonizationTailoring a 67 attosecond pulse through advantageous phase-mismatchGeneration of 88 as Isolated Attosecond Pulses with Double Optical Gating*Strong-field coherent control of isolated attosecond pulse generationCoherent Control of High Harmonic Generation Driven by Metal Nanotip PhotoemissionMultiband Dynamics of Extended Harmonic Generation in Solids under Ultraviolet InjectionGeneration of Intense Sub-10 fs Pulses at 385nmHigh-efficiency optical parametric chirped-pulse amplifier in BiB_3O_6 for generation of 3 mJ, two-cycle, carrier-envelope-phase-stable pulses at 17 μmGeneration of 1 kHz, 23 mJ, 88 fs, 25 μm pulses from a Cr2+ :ZnSe chirped pulse amplifierGeneration of few-cycle multi-millijoule 2.5 μm pulses from a single-stage Cr2+:ZnSe amplifierEfficient generation of mid-infrared few-cycle pulses by the intrapulse difference-frequency generation in YCOBBright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond LasersGeneration of bright isolated attosecond soft X-ray pulses driven by multicycle midinfrared lasersCrossref OpenURL ResolverStreaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driverScaling laws for high-order-harmonic generation with midinfrared laser pulsesWavelength scaling laws for high-order harmonic yield from atoms driven by mid- and long-wave infrared laser fieldsEnhanced High Harmonic Generation from Multiply Ionized Argon above 500 eV through Laser Pulse Self-CompressionHigh-order harmonic generation in an x-ray range from laser-induced multivalent ions of noble gasSub-50-as isolated extreme ultraviolet continua generated by 1.6-cycle near-infrared pulse combined with double optical gating schemeExtreme ultraviolet supercontinua supporting pulse durations of less than one atomic unit of timeAttosecond pulse generation in carbon K-edge region (284 eV) with sub-250 μ J driving laser using generalized double optical gating methodAttosecond-chirp compensation with material dispersion to produce near transform-limited attosecond pulsesSingle sub − 50 − attosecond pulse generation from chirp-compensated harmonic radiation using material dispersionCompression of harmonic pulses by using material dispersionAttosecond chirp compensation in water window by plasma dispersionCompensating chirp of attosecond X-ray pulses by a neutral hydrogen gasX-Ray Interactions: Photoabsorption, Scattering, Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92Monitoring Argon L-Shell Auger Decay Using 250-eV Attosecond X-ray PulsesSubfemtosecond pulsesControlling attosecond electron dynamics by phase-stabilized polarization gatingSingle attosecond pulse and xuv supercontinuum in the high-order harmonic plateauGeneration of attosecond pulses with ellipticity-modulated fundamentalDependence of high-order-harmonic-generation yield on driving-laser ellipticitySubcycle laser control and quantum interferences in attosecond photoabsorption of neonSplit-operator spectral method for solving the time-dependent Schrödinger equation in spherical coordinatesTheoretical study of multiple high-order harmonic generation by intense ultrashort pulsed laser fields: A new generalized pseudospectral time-dependent method
[1] Sansone G, Calegari F, and Nisoli M 2012 IEEE J. Sel. Top. Quantum Electron. 18 507
[2] Krausz F and Stockman M I 2014 Nat. Photonics 8 205
[3] Krausz F 2016 Phys. Scr. 91 063011
[4] Zhao J, Liu J, Wang X, Yuan J, and Zhao Z 2022 Chin. Phys. Lett. 39 123201
[5] Zhao K, Zhang Q, Chini M, Wu Y, Wang X, and Chang Z 2012 Opt. Lett. 37 3891
[6] Wang X W, Wang L, Xiao F, Zhang D W, Lue Z, Yuan J, Zhao Z, Lü Z H, Yuan J M, and Zhao Z X 2020 Chin. Phys. Lett. 37 023201
[7] Yang Y D, Mainz R E, Rossi G M, Scheiba F, Silva-Toledo M A, Keathley P D, Cirmi G, and Kärtner F X 2021 Nat. Commun. 12 6641
[8] Zhang H D, Liu X W, Jin F C, Zhu M, Yang S D, Dong W H, Song X H, and Yang W F 2021 Chin. Phys. Lett. 38 063201
[9] Lang Y, Peng Z Y, and Zhao Z X 2022 Chin. Phys. Lett. 39 114201
[10] Xiao F, Fan X, Wang L, Zhang D, Wu J, Wang X, and Zhao Z 2020 Chin. Phys. Lett. 37 114202
[11] Yin Y C, Li J, Ren X M, Zhao K, Wu Y, Cunningham E, and Chang Z H 2016 Opt. Lett. 41 1142
[12] Ren X M, Mach L H, Yin Y C, Wang Y, and Chang Z H 2018 Opt. Lett. 43 3381
[13] Wu Y, Zhou F, Larsen E W, Zhuang F, Yin Y, and Chang Z 2020 Sci. Rep. 10 7775
[14] Gu X B, Liu J S, Yuan P, Tu X H, Zhang D F, Wang J, Xie G Q, and Ma J Q 2022 Opt. Lett. 47 5244
[15] Popmintchev T, Chen M C, Popmintchev D, Arpin P, Brown S, Ališauskas S, Andriukaitis G, Balčiunas T, Mücke O D, Pugzlys A, Baltuška A, Shim B, Schrauth S E, Gaeta A, Hernández-García C, Plaja L, Becker A, Jaron-Becker A, Murnane M M, and Kapteyn H C 2012 Science 336 1287
[16] Chen M C, Mancuso C, Hernández-García C, Dollar F, Galloway B, Popmintchev D, Huang P C, Walker B, Plaja L, Jaroń-Becker A A, Becker A, Murnane M M, Kapteyn H C, and Popmintchev T 2014 Proc. Natl. Acad. Sci. USA 111 E2361
[17] Li J, Ren X M, Yin Y C, Zhao K, Chew A, Cheng Y, Cunningham E, Wang Y, Hu S Y, Wu Y, Chini M, and Chang Z H 2017 Nat. Commun. 8 186
[18] Gaumnitz T, Jain A, Pertot Y, Huppert M, Jordan I, Ardana-Lamas F, and Wörner H J 2017 Opt. Express 25 27506
[19] Frolov M V, Manakov N L, Xiong W H, Peng L Y, Burgdörfer J, and Starace A F 2015 Phys. Rev. A 92 023409
[20] Emelina A S, Emelin M Y, and Ryabikin M Y 2019 J. Opt. Soc. Am. B 36 3236
[21] Arpin P, Popmintchev T, Wagner N L, Lytle A L, Cohen O, Kapteyn H C, and Murnane M M 2009 Phys. Rev. Lett. 103 143901
[22] Gao J x, Wu J q, Lou Z Y, Yang F, Qian J Y, Peng Y J, Leng Y X, Zheng Y H, Zeng Z N, and Li R X 2022 Optica 9 1003
[23] Oguri K, Mashiko H, Ogawa T, Hanada Y, Nakano H, and Gotoh H 2018 Appl. Phys. Lett. 112 181105
[24] Mashiko H, Gilbertson S, Chini M, Feng X, Yun C, Wang H, Khan S D, Chen S, and Chang Z 2009 Opt. Lett. 34 3337
[25] Mashiko H, Oguri K, Sogawa T, Mashiko H, Oguri K, and Sogawa T 2009 Appl. Phys. Lett. 102 171111
[26] Ko D H, Kim K T, and Nam C H 2012 J. Phys. B 45 074015
[27] Kim K T, Kim C M, Baik M G, Umesh G, and Nam C H 2004 Phys. Rev. A 69 051805
[28] Kim K T, Kim C M, Baik M G, Umesh G, and Nam C H 2004 Appl. Phys. B 79 563
[29] Chang Z H 2018 Opt. Express 26 33238
[30] Chang Z H 2019 OSA Continuum 2 314
[31] Henke B L, Gullikson E M, and Davis J C 1993 At. Data Nucl. Data Tables 54 181
[32] Han S, Zhao K, and Chang Z 2022 Sensors 22 7513
[33] Corkum P B, Burnett N H, and Ivanov M Y 1994 Opt. Lett. 19 1870
[34] Sola I J, Mével E, Elouga L, Constant E, Strelkov V, Poletto L, Villoresi P, Benedetti E, Caumes J P, Stagira S, Vozzi C, Sansone G, and Nisoli M 2006 Nat. Phys. 2 319
[35] Chang Z H 2004 Phys. Rev. A 70 043802
[36] Strelkov V, Zair A, Tcherbakoff O, Lopez-Martens R, Cormier E, Mevel E, and Constant E 2004 Appl. Phys. B 78 879
[37] Möller M, Cheng Y, Khan S D, Zhao B, Zhao K, Chini M, Paulus G G, and Chang Z 2012 Phys. Rev. A 86 011401
[38] Wang X W, Chini M, Cheng Y, Wu Y, Tong X M, and Chang Z H 2013 Phys. Rev. A 87 063413
[39] Hermann M R and Fleck J A J 1988 Phys. Rev. A 38 6000
[40] Tong X M and Chu S I 1997 Chem. Phys. 217 119