Citation: Zhou, Y.; Quan, W.; Zhao,

M.; Wang, Z.; Wang, M.; Chen, S.;

Chen, J.; Liu, X. Improved

Carrier-Envelope Phase Determination

Method for Few-Cycle Laser Pulses

Using High-Order Above-Threshold

Ionization. Photonics 2022, 9, 528.

https://doi.org/10.3390/

photonics9080528

Academic Editors: Bonggu Shim and

Xiaohui Gao

Received: 10 July 2022

Accepted: 27 July 2022

Published: 29 July 2022

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photonicshv

Article

Improved Carrier-Envelope Phase Determination Method forFew-Cycle Laser Pulses Using High-OrderAbove-Threshold IonizationYu Zhou 1,2 , Wei Quan 1,2,* , Meng Zhao 1,2, Zhiqiang Wang 1, Minghui Wang 1,2, Sijin Cheng 1,2, Jing Chen 3,4

and Xiaojun Liu 1,2

1 State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physicsand Mathematics, Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences, Wuhan 430071, China; zhouyu93@126.com (Y.Z.);zhaomeng191@mails.ucas.ac.cn (M.Z.); 19834426740@163.com (Z.W.); wmh@apm.ac.cn (M.W.);sijin_cheng2021@163.com (S.C.); xjliu@wipm.ac.cn (X.L.)

2 University of Chinese Academy of Sciences, Beijing 100049, China3 Key Laboratory of High Energy Density Physics Simulation, Center for Applied Physics and Technology,

Peking University, Beijing 100084, China; chen_jing@iapcm.ac.cn4 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China* Correspondence: charlywing@wipm.ac.cn

Abstract: Recent studies indicate that the stereo-ATI carrier-envelope phase meter (CEPM) is aneffective method to determine the carrier-envelope phase (CEP) of each and every single few-cyclelaser pulse. In this method, a two-dimensional parametric asymmetry plot (PAP), which can beobtained with the measured data in two short time-of-flight intervals, is applied to extract the CEP.Thus, part of the data containing useful CEP information is discarded in the PAP method. In thiswork, an improved method was developed to effectively exploit most of the experimental data. Bythis method, we achieve a CEP precision of 57 mrad over the entire 2π range for 5.0 fs laser pulses.

Keywords: high-order above-threshold ionization; few-cycle pulses; strong-field physics; carrier-envelope phase

1. Introduction

The technological advances in ultrafast lasers have permitted researchers to study theionization dynamics of matter with laser pulses comprising only a few oscillation opticalcycles (see, e.g., [1–8]). The electric field of a few-cycle laser pulse can be described byE(τ) = f (τ)E0cos(ωτ + φ), where f (τ) is the envelope, E0 is the electric field amplitude,ω is the carrier angular frequency, and φ is the carrier-envelope phase (CEP). If the pulseduration is close to an optical cycle, the electric field strength can be altered conspicuouslyby a controlled CEP. Indeed, the CEP effect has been shown to play a prominent role incontrolling generation of attosecond pulses [2,9], ionization of atoms [6,10–12], ionizationand dissociation of molecules [3,13] and clusters [14], strong-field interaction with nano-tips [15], and other nano-scale systems [16].

Methods for measuring the CEP of a few-cycle laser pulse have been well devel-oped based on a series of techniques, such as f − 2 f interferometry [17,18], frequencyresolved optical grating employing photoionization [19], dispersive Fourier transform [20],angle-resolved photoelectron spectra of noble gas atoms in circularly polarized few-cyclepulses [21], etc. Recently, with the technique of f − 2 f interferometry, an excellent CEPstability with a jitter of only 82 mrad (root-mean-square) [18] was achieved. Based onthis achievement, recent work [22] on above-threshold ionization (ATI) of atomic cesiumshows that the directional asymmetry in the energy spectra of backscattered electronsoscillates three times, rather than once, due to interference between quantum orbits as

Photonics 2022, 9, 528. https://doi.org/10.3390/photonics9080528 https://www.mdpi.com/journal/photonics

Photonics 2022, 9, 528 2 of 10

the CEP is changed from 0 to 2π. In addition to the aforementioned work, since 2009,much attention has been paid to a CEP determination method by stereographic detectionof the high-order ATI spectra of noble gas atoms [23], which is also dubbed stereo-ATI CEPmeter (CEPM) [24]. The CEPM can be applied to perform real-time, every-single-shot CEPmeasurement with high precision [23,24] for a laser system with a high repetition rate, upto 100 kHz [25], and a high power, up to multi-terawatt [26]. If the measured CEP can berecorded in a computer along with the experimental data for each laser pulse, the CEPdependent experimental data can be extracted by the offline analysis. This technique isdubbed phase tagging [23,24], which has been applied to study the CEP effects for non-sequential [5,27] and sequential [28] double ionization and ionization and dissociation ofH2 [29,30], and to develop the techniques of attosecond streaking [31] and refractive indexdispersion measurement [32], etc.

Due to the strong CEP dependence of the involved high-order nonlinear processes,high precision of CEPM measurement can be achieved. The physical principle of CEPMrelies on the radical CEP dependent left–right asymmetries of angle-resolved high-orderATI yields [33–35], and the CEP dependence of this asymmetry can be stronger by anorder of magnitude than that of direct ATI [23]. In 2011, a precision of 113 mrad overthe entire 2π range was reported [24] for a typical CEPM. Moreover, it was documentedthat a better precision could be achieved for a shorter pulse duration [36]. In contrast,for a long acquisition time, drift of laser power (or intensity) will affect the precision ofCEPM [37], which could be remedied by periodic recalibration. Recent progress indicatesthat a technique of intensity tagging [38,39] can be applied to reduce the influence ofintensity fluctuations and increase the sensitivity of phase tagging. Up until now, as far aswe know, the best precision of CEPM is 82 mrad for 3.5 fs laser pulses [25].

In this work, we propose a method for precisely determining the CEP of each andevery single few-cycle laser pulse. Our method relies on a measurement of the left–rightasymmetry of photoelectron kinetic energy/time-of-flight (TOF) distribution. To extractthe CEP, our numerical procedure can exploit most of the TOF distribution, rather thanonly two asymmetric parameters that can be obtained by the data in two short TOFintervals. Technically, we have made several improvements of our homemade phase meterto optimize the signal-to-noise ratio (SNR), the intensity, and phase distribution in thevicinity of the focus. Moreover, the intensity tagging technique has also been applied toincrease the CEP precision. According to the analysis, we have achieved a CEP precision of57 mrad over the entire 2π range for 5.0 fs laser pulses.

This paper is organized as follows. Our experimental setup is described in the follow-ing section. The methods of data processing and numerical solution of the time-dependentSchrödinger equation (TDSE) are briefly described in Sections 3 and 4. In Section 5, the ex-perimental and theoretical results are discussed. Finally, our conclusion is given in Section 6.Atomic units are used throughout unless otherwise indicated.

2. Experimental Setup

We have made several improvements of our homemade stereo-ATI photoelectronspectrometer, which has already been described in an earlier paper [40] and is schemati-cally illustrated in Figure 1a. A commercial Ti:sapphire femtosecond laser system (FEM-TOPOWER compact PRO CE-Phase HP/HR) can generate laser pulses with a duration ofapproximately 30 fs and a center wavelength of 800 nm. The laser beam is focused into a1.1 m long, 300-micron diameter hollow-core fiber filled with 2.5 bar Ne. Spectral distri-bution of the beam is broadened in the hollow-core fiber. Subsequently, the laser pulsesare compressed by a set of chirped mirrors and a pair of fused silica wedges. The pulseduration of the achieved few-cycle laser pulses is 5.0 fs, which is reformulated by an auto-correlator (FEMTOMETERTM), and the central wavelength of the few-cycle laser pulse is780 nm. With a beam splitter, a small portion of the few-cycle laser beam is directed to anddetected by a fast silicon photo-diode (PD) to perform the intensity tagging, which has

Photonics 2022, 9, 528 3 of 10

reduced the single pulse energy uncertainty from 0.8% to approximately 0.2%, and, at thesame time, to trigger a PXI system equipped with a pair of fast digitizers.

PDLB

PH PS

(b)(a)

W HWP

PS

ECEC

Right-MCPLeft-MCP

Power Divider

BS

0.50 0.75 1.000.0

0.5

1.0 VOAP VSM

Vol

ume

(arb

.uni

t)

I/I0

0.0

0.2

0.4

LPOAP Mirror

DfOAP DfSM

Df (r

ad)

Figure 1. (a) Schematic view of the experimental setup. Laser beam (LB) with polarization alongthe horizontal direction is introduced into the sample chamber by a off-axis parabolic (OAP) mirror.The photoelectrons produced at the laser focus fly towards left and right MCP detectors through twopinholes (PH). The signals recorded by the MCP detectors are extracted by two homemade electroniccircuits (EC). The photoelectron signals are sent to a PXI system (PS) equipped with a pair of fastdigitizers. By a beam splitter (BS), a small portion of laser is directed and detected by a photo-diode(PD). The PD output is further split evenly by a power divider to trigger the PS and perform intensitytagging. A pair of fused silica wedges (W) are applied to compensate the residual positive chirp ofthe laser beam. A combination of an achromatic broadband half-wave plate (HWP) and a commercial(Edmund #36-652) linear polarization (LP) is applied to alter the laser intensity. See text for moredetails. (b) The intensity dependence of the CEP range (∆φ) and volume (V) of each iso-intensityshell of few-cycle laser pulses focused by an OAP mirror and a spherical mirror (SM), respectively.

The main portion of the few-cycle laser beam is focused into the sample chamberfilled with Xe by an off-axis parabolic (OAP) mirror with an effective focal length of6 inches (152.4 mm). We replace the spherical mirror (SM), which is employed by astandard CEPM [23] and our earlier work [40], by an OAP mirror because, in a typicalexperimental configuration of a CEPM (see, e.g., the design of light path in [40,41]), it isimpossible for the laser beam to be aligned perfectly perpendicular to the SM. Moreover,the deviation could be significant and cannot be conveniently determined. However,the OAP mirror employed here can provide an intensity distribution closer to a standardGaussian distribution. Note that, in contrast to the well-understood intensity distribution,as documented, the spatially dependent CEP distribution of focused few-cycle pulses is verycomplex [42–45]. In Figure 1b, based on numerical calculations, we present the intensitydependence of the CEP range (∆φ) and volume (V) of each iso-intensity shell in the vicinityof the focus of few-cycle laser pulses in the cases of an OAP mirror and an SM (incidenceangle of 4.5◦) in a typical experimental configuration of CEPM, respectively. It can be foundthat, if an SM was applied, the volume of each iso-intensity shell would be larger for theintensities around the peak, which is relevant to comprehend our experimental resultsbelow. In contrast, in the case of an OAP mirror, the CEP range of each iso-intensity shellis significantly smaller in the whole intensity range explored. This would be helpful tooptimize the CEP precision of our CEPM.

Instead of a pair of thin vertical slits, which are usually employed in a standardCEPM [23], two pinholes with a diameter of 0.7 mm, which are smaller than those usedin [40], are employed for the photoelectrons to leave the sample chamber and fly towards thetwo micro-channel plate (MCP) detectors on both sides of the photoelectron spectrometer(left-MCP and right-MCP). The smaller pinholes can support a higher sample pressure(2.0 Pa) in the interaction region, and, in turn, improve the SNR. The low-energy electronswith kinetic energy of Ek < 42 eV are discriminated by an electrostatic repeller installedbefore each MCP detector. The laser intensity is determined by a comparison between themeasured CEP-averaged photoelectron kinetic energy spectra (PES) and TDSE calculationswith focal averaging effect included.

Photonics 2022, 9, 528 4 of 10

3. The Procedure to Extract the Absolute Carrier-Envelope Phase

With the procedure of the standard CEPM [23,40,41], one can choose two energyintervals (low and high), whose CEP dependence is offset by a constant phase of Φ,to achieve two corresponding asymmetry parameters, αLow and αHigh. The asymmetryparameter is defined by α(φ, Ek) = (YL(φ, Ek)−YR(φ, Ek))/(YL(φ, Ek)+YR(φ, Ek)) , whereYL(φ, Ek) and YR(φ, Ek) are the yields of photoelectrons emitted along the left and rightdirections (the opposing directions parallel to the laser polarization) with an energy ofEk at a CEP of φ. The two asymmetry parameters can be plotted against each other in atwo-dimensional parametric asymmetry plot (PAP), where a one-to-one correspondencebetween the polar angle (θ) and the CEP (φ) can be established. To achieve a good precisionover the entire 2π range, the polar angle (θ) is supposed to vary approximately linearlywith rising CEP (φ), which means that the pattern of PAP distribution should be close to anannulus [24,41]. Thus, we have to make sure Φ = π/2 and, at the same time, the oscillationamplitude of αLow has to equal to that of αHigh.

Experimentally, it is not always convenient to fulfill the two conditions noted above. Ithas been shown that the pattern of PAP distribution is sensitive to the pulse duration [46],laser intensity [38,39], and phase-volume effect [45]. The physical origin can be com-prehended by the results in [40], where it was shown that, with rising Ek, the oscillationamplitude of the asymmetry parameter shows a rapid rising trend until a maximum aroundthe cutoff of ATI spectra, and a step structure appears at a relatively lower energy if an SMis employed in the CEPM for a 5.0 fs laser pulse. However, to ensure Φ = π/2, the energyintervals of αHigh and αLow have to be carefully chosen and, usually, the difference betweenthe centers of the two intervals could be approximately 15 eV (see, e.g., [23]). It is easy tounderstand that one cannot obtain equal oscillation amplitudes of αHigh and αLow and, atthe same time, maintain Φ = π/2 for a pure rising trend of the oscillation amplitude of theasymmetry parameter. As demonstrated in [40], the step structure will give rise to moreslowly varying oscillation amplitudes, which means the two conditions noted in the lastparagraph can be approximately fulfilled. Nevertheless, in some cases, the PAP distributionis still different from a perfect annulus [23,38,39,46].

Here, we propose a dedicated data processing procedure that can exploit most ofthe effective experimental data to extract CEP. To this end, it is necessary to obtain theasymmetry of photoelectron kinetic energy or TOF spectra [23,24]. In Figure 2a, we presentthe typical photoelectron TOF spectra measured by the left and right detectors at a seriesof CEPs. In Figure 2b, we present the asymmetry of the photoelectron TOF spectra thatare obtained by calculating the asymmetry parameter α(φ, t), where t indicates TOF of thephotoelectron, with the data in Figure 2a. As documented [23,24], the CEP dependence ofthe asymmetry of the photoelectron yields at a chosen kinetic energy or TOF is a sine/cosinecurve. Thus, instead of choosing two energy/TOF intervals (low and high) [23,41], wedecided to fit the αt(φ) at each TOF with the measured CEP dependent photoelectron yieldsin a wide TOF range, with a function of Atcos(φ− Φt), to increase the precision of theextracted CEP.

αt(φ) = Atcos(φ−Φt). (1)

Note that, in the above equation, for each laser pulse, φ is a constant. In contrast, consideringthat the CEP is not actively stabilized during the measurement, the CEPs of our laser pulsescould be assumed to be quasi-uniformly distributed over the range of 0 ≤ φ < 2π. Thus,we can make an estimation of the amplitude of At and the phase shift Φt based on thestatistical distribution of αt(φ) at each TOF before fitting. This procedure can significantlyincrease the fitting precision and efficiency. The amplitude of At can be estimated by

At =√

2 ·

√∑i=N

i=1 (αt(φ))2i

N, (2)

Photonics 2022, 9, 528 5 of 10

where N is the number of sampling pulses involved. Additionally, to make an estimationof Φt, a simple transformation of Equation (1) can be obtained,

βt =αt(φ)

At= cos(φ−Φt). (3)

Then we calculate the derivative of the above equation with respect to t,

dβt

dt= sin(φ−Φt)

dΦt

dt. (4)

With the above equation and considering that Φt decreases with rising t [40], we get

dΦt

dt= −

∣∣∣∣∣∣dβtdt√

1− β2t

∣∣∣∣∣∣. (5)

After an integration of the above equation and averaging over all the pulses, we get

Φt = −∫ tn

t0

∑i=Ni=1

∣∣∣∣ dβidt√

1−β2i

∣∣∣∣N

dt + Φ0, (6)

where Φ0 is an arbitrary offset, indicating that only the relative CEP can be obtained if thereis no more information available. With Equations (2) and (6) and the experimental data inthe TOF range of high SNR (from 40.5 ns to 48.0 ns here (see Figure 2a,b)), we can make anestimate of At and Φt. Next, we put the initially estimated At and Φt into Equation (1) andfit the measured αt with the experimental data to achieve the initial φ. The obtained initialφ is further inserted into Equation (1) to fit the measured data again to achieve the moreaccurate At and Φt. The fitting procedure can be iterated until all the extracted parameters,including φ, At, and Φt, no longer change. Note that the determination of absolute CEP isfinally achieved by a comparison with TDSE calculations. The obtained photoelectron TOF(kinetic energy) dependent At(Ae) and Φt(Φe) are presented in Figure 2c,d). Additionally,the CEP (φ) and photoelectron kinetic energy (Ek) dependence of α(φ, Ek) distributions arepresented in Figure 2e.

40 45 500.0

0.5

1.0(a)

Yie

lds (

arb.

unit)

TOF (ns)

YL

YR

40 45 50-1.0

-0.5

0.0

0.5

1.0

aLowaHigh

(f)

(e)

(d)(b)

(c)

a

TOF (ns)

a0

1

2

f (p

)

-1 0 1

Expt.

40 45 500.0

0.5

1.0

1.5 a

At

TOF (ns)

At

50 750.0

0.5

1.0

1.5

Ae

Photoelectron kinetic energy (eV)

AeExpt.

AeTDSE50 750

1

2 TDSE

f (p

)

Photoelectron kinetic energy (eV)

-1.0

-0.5

0.0

0.5 Ft

Ft (

p)

-1.0

-0.5

0.0

0.5 Fe Expt.

Fe TDSE

Fe (

p)

Figure 2. (a) Typical time-of-flight (TOF) photoelectron distributions measured by the left and rightdetectors of CEPM at a series of CEPs. (b) The TOF-dependence of the left-right asymmetry (α(φ, t))

Photonics 2022, 9, 528 6 of 10

at a series of CEPs. (c) The extracted TOF-dependent asymmetry amplitude (At) and phase shift(Φt) with our data processing procedure. (d) A comparison between the extracted energy-dependentasymmetry amplitude (Ae) and phase shift (Φe) and the TDSE calculations. We also present themeasured (e) and calculated (f) 2-dimensional α(Ek, φ) distributions. See text for more details.

4. Numerical Solution of the Time-Dependent Schrödinger Equation (TDSE)

To determine the absolute CEP, calculations based on numerical solutions of TDSEwere performed and compared with the experimental data. The TDSE of Xe subject to astrong laser field is written as

i∂ψ(r, τ)

∂τ=

[−1

2∇2 + Vm(r) + iA(τ) · 5

]ψ(r, τ) (7)

where A(τ) = −∫ τ−∞ E(τ′)dτ′ is the laser field’s vector potential, and the model potential

is given by [47,48]

Vm(r) = −1 + (a1e−a2r + a3re−a4r + a5e−a6r)

r. (8)

The parameters in the above equation, for a Xe atom with an initial 5p state orientedalong the laser polarization, can be found in Ref [48]. The laser vector potential employedin the calculation is A(τ) = − E0

ω cos2 ωτ2n sin(ωτ + φ), where E0, ω = 0.0584 a.u., n = 5.27,

and φ are the peak amplitude, the central angular frequency, number of optical cycles andCEP of the laser field, respectively. These parameters are chosen according to the experi-mental configurations. The maximum angular momentum quantum number employed inthe calculation is lmax = 85. The TDSE is solved numerically using the QPROP code [49].To compare with the experimental data, we have calculated PES at a series of CEPs andintensities, which are varied in the steps of 0.0625π and 0.1× 1013 W/cm2, respectively.The peak laser intensity is 1.2× 1014 W/cm2 in our TDSE calculations. These PES are focal av-eraged for both intensity [50,51] and CEP [42–45] according to the experimental configurations.

5. Results and Discussion

To validate our method, we compared the measured CEP (φ) and photoelectron kineticenergy (Ek) dependence of α(φ, Ek) distributions (Figure 2e) with the TDSE calculations(Figure 2f). The agreement between the experimental data and the focal-averaged calcula-tions are good. Moreover, in Figure 2d, the Ek dependence of asymmetry amplitude Ae(Ek)and phase Φe(Ek) extracted from the experimental data can be well reproduced by theTDSE calculations. Note that the step structure, which is obvious in an earlier work [40],disappears because of the altered intensity dependence of the volume of each iso-intensityshell in the vicinity of the focus of few-cycle laser pulses in our experimental setup (seeFigure 1b).

Our experimental data were also processed with the standard CEPM procedure [23,24].Two energy intervals of (61.3∼69.5 eV) and (47.0∼52.0 eV) (see the shaded areas in Figure 2b)were chosen to obtain the high (αHigh) and low (αLow) asymmetry parameters because ofgood SNR in these intervals and the smallest CEP uncertainty that can be eventually obtained.The achieved PAP distribution is presented in Figure 3a, where a potato-like distributioncan be found. The deviation of the PAP distribution from a perfect annulus can be mainlyattributed to the residual difference between the oscillation amplitudes of αHigh and αLow (seeFigure 2d). This difference can become smaller if an SM is employed [40], because a stepstructure will appear and diminish the difference to a large extent. In our case, the applicationof an OAP enhanced the focusing averaging effect significantly (see Figure 1b) and eliminatedthe influence of the step structure.

Photonics 2022, 9, 528 7 of 10

YL, 1.5pYL, 1.5p

Yie

lds (

arb.

unit)

TOF (ns)

Figure 3. (a) The PAP obtained with the typical CEPM procedure with the data given in Figure 2 andTDSE calculations. (b) The polar angle (θ) as a function of CEP extracted from the PAP in (a). The grayshaded area indicates the uncertainties of the experimental data. (c) The photoelectron TOF spectrameasured by the left (YL) and right (YR) detectors over a small CEP range of φ = 1.5π ± 5 mraddetermined by our procedure for 50 pulses, which are randomly chosen from 1033 pulses for bettervisualization of the jitter, and the averaged curve of YL and YR. (d) A comparison of the CEPdependence of measurement precision (δφ) of our procedure and the standard method employingPAP. See text for more details.

In Figure 3b, we depict the CEP (φ) dependence of the polar angle (θ), which wasextracted from the measured PAP. Apparently, this relationship is a complex combinationof an oscillation and a linearly rising trend, instead of a simple line. As expected, the curveextracted from the experimental data can be well reproduced by the TDSE calculations.The grey shaded area indicates the uncertainties of the measured CEP by the standardCEPM procedure.

A Monte Carlo algorithm, which was proposed in Ref. [24], was employed to deter-mine the uncertainties of the CEPs, which are measured by our procedure. As shown inFigure 3c, the reference spectra in both directions (indicated by YL and YR in Figure 3c)can be obtained by averaging the measured photoelectron TOF spectra over a small CEPrange of ±5 mrad. The statistical error due to electron counting and intensity fluctuationsis numerically simulated by the Monte Carlo algorithm. In this way, the uncertainties ofCEP determined by our procedure are obtained and shown in Figure 3d.

To demonstrate the merit of our procedure, the uncertainties of CEPs determined byour procedure and also those of the standard CEPM are compared in Figure 3d. As shownin this panel, the CEP uncertainty of our method is significantly smaller than that of themethod employing PAP distribution. Moreover, The CEP precision of the two methods,which is determined by the maximum of CEP uncertainties over the entire 2π range in eachcase, is 57 mrad and 156 mrad, respectively.

To reveal the performance of the intensity tagging, the CEP dependent precision inthe case of including all the experiment data is presented in Figure 3d. The correspondingintensity uncertainty is 0.8% for a typical data acquiring time of approximately 4 min.As shown in Figure 3d, the increase of intensity uncertainty from approximately 0.2% to0.8% gives rise to slightly larger uncertainties of CEP in the case of a standard CEPM proce-dure and almost identical CEP precision for our numerical procedure. This result indicatesthat our procedure will be less sensitive to the laser intensity fluctuation. Although the

Photonics 2022, 9, 528 8 of 10

advantage of the intensity tagging is not significant in this experiment, its performancecould be more prominent for the experiment with a longer data acquiring time (e.g., tens ofhours) because of possible larger intensity fluctuation.

6. Conclusions

In this work, we demonstrate that, with an improved carrier-envelope phase deter-mination method for few-cycle laser pulses, a CEP precision of 57 mrad over the entire2π range can be achieved for 5.0 fs laser pulses. In contrast, the CEP uncertainty becomes156 mrad if the traditional method employing PAP distribution is applied for the samedata. The improvement relies mainly on a dedicated data processing procedure that caneffectively exploit most of the experimental data to extract the CEP. Moreover, we havemade several technical improvements of our homemade phase meter to optimize the SNR,the intensity, and phase distribution in the vicinity of the focus. The intensity taggingtechnique was also applied to reduce the intensity uncertainty. A comparison between theexperimental data and TDSE calculations indicates that our procedure is robust and reliable.

Author Contributions: Methodology, Y.Z. and W.Q.; validation, Y.Z., W.Q., M.W., M.Z., Z.W.,and S.C.; data curation, Y.Z.; formal analysis, Y.Z. and W.Q.; writing—original draft preparation, Y.Z.;writing—review and editing, W.Q., X.L., and J.C.; visualization, Y.Z.; supervision, W.Q. and X.L.;funding acquisition, W.Q. and X.L. All authors have read and agreed to the published version ofthe manuscript.

Funding: This work is supported by the National Key Research and Development Program of China(Grant No. 2019YFA0307700), the National Natural Science Foundation of China (Nos. 11834015,11974383 and 12121004), the Science and Technology Department of Hubei Province (Nos. 2019CFA035and 2020CFA029).

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable

Data Availability Statement: Not applicable.

Acknowledgments: We thank Xu Y. Q., Sun H. Y., Wan C. Z., Wang Y., and Chen Q. F. for technicalsupport of laser system, electronic devices, and vacuum system.

Conflicts of Interest: The authors declare no conflict of interest.

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