1. Introduction

Intense few-cycle optical pulses with a pulse energy around 400 µJ and duration of 3.5 fs (measured at the full-width at half-maximum (FWHM) of their temporal intensity profile) have been generated very recently [1]. Considering a wavelength of 720 nm, an optical cylce has a duration of 2.4 fs, i.e., the pulses consist of very few optical cycles only. One of the most remarkable advantages is that it is possible to tailor the electric field of such pulses by controlling their phase of the carrier frequency with respect to the envelope, the so-called carrier-envelope phase (CEP). Few-cycle pulses have mainly been used for generating coherent x-rays [2] and attosecond pulses [3, 4]. However, the generation of attosecond pulses requires phase-stabilized driving pulses and precise control of the driving electric field waveform [5]. Therefore, the ability to precisely measure and stabilize the CEP is of great importance. The experimental evidence of phase effects has been obtained in the left-right asymmetry of above-threshold ionization (ATI) yield [6], high-order harmonics generation (HHG) [7] and nonsequential double ionization (NSDI) [8].

Conversely, this strong dependence can be used to measure the CEP. The left-right asymmetry of high-order ATI yield has been investigated theoretically [9, 10] and experimentally used to measure and stabilize the absolute phase of linearly polarized few-cycle pulses [11]. These strong-field processes can be described by the rescattering model [12] qualitatively. In this model, the rescattering photoelectrons play a important role in these processes. The kinetic energy spectrum of high-energy photoelectrons also can be used to measure the CEP. For fewcycle pulses, the rescattering electrons bursting in the time interval closest to the pulse maximum would dominate the high-energy part of the corresponding spectrum and their highest energy (cutoff energy) is very sensitive to the CEP (see Fig. 1), especially for high laser intensities. In this paper, we propose the cutoff energy of rescattering photoelectrons, whose spectrum is obtained in a quantum-mechanical calculation, as an accurate tool for CEP measurement.

2. Results and discussions

The electric field of a few-cycle laser pulse can be written as E(t,φ)=e⃗_xsin²(πt/T_p)cos(ωt+φ), where e⃗_x, sin²(πt/T_p), ω, φ and T_p denote the axis of polarization, envelope, frequency, CEP and total duration of the pulse, respectively. φ=0 and φ=π/2 correspond to a cosine-like pulse and a - sine-like pulse respectively, as shown in Fig. 1. The photoelectron spectra are obtained by numerically solving the 1D time-dependent Schrödinger equation (TDSE). The total wave function is split as Ψ=Ψ[1-F_s(x_c)]+ΨF_s(x_c)=Ψ_I+Ψ_II. Here, F_s(x_c)=1/(1+e ^{-(|x|-x_c)/Δ}) is a split function [13] that separates the whole space into the inner (|x|<x_c) and outer (x_c<|x|<x_max) regions smoothly. Δ represents the width of the crossover region.Ψ_I represents the wave function in the inner region. In this region, the wave function is propagated as prescribed by the TDSE for an electron in a potential V(x) driven by an electric field E(t) polarized along the x axis,

 

Fig. 1. Electric field (blue solid line) and envelope (cyan solid line) of a 800 nm four-cycle laser pulse with an intensity of I=2×10¹⁵ W/cm² for the CEPs 0 (left panel) and π/2 (right panel). The drift momentum of high-energy photoelectrons as a function of the ionization time t ₀ is shown in black (emission to the left, negative direction along the x axis) and red (emission to the right, positive direction along the x axis) dotted lines, calculated with a classical model. The ionization probability (green dotted line) is also given by the quasistatic tunneling rate with arb. units as a function of t ₀ when the rescattering photoelectrons are ionized. The photoelectrons ionized in the time intervals labeled with 1 and 2 would dominate the high-energy parts of respective spectra.

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where A(t)=-∫^t ₀ E(t′)dt′. (Atomic units are used throughout unless otherwise stated.) We choose He atom because of its large ionization potential (0.9 a.u.) and a soft-Coulomb potential $V\left(x\right)=-\frac{1}{\sqrt{0.48+x^2}}$ is used to reproduce its ground state energy. This model is suitable to describe qualitatively many features of ionization which usually occurs near the ground-state geometry. The initial wave function is the field-free ground state, which is obtained by propagation in imaginary time by means of the split-operator method [14]. Ψ_II stands for the wave function in the outer region where the electron is far away from the nucleus. Therefore, the interaction between core and electron is neglected. The TDSE then becomes

and the propagation of the wave function is simply accomplished by multiplications in momentum space. After the end of the pulse, the whole wave function is propagated without field for an additional time in order to collect all electrons with high energies above 3 U_p, where U_p=E ² ₀/(4ω ²) is the ponderomotive energy. The ATI spectra are calculated according to the formula $\frac{\mathit{dP}\left(E\right)}{\mathit{dE}}=\frac{1}{\mid p\mid }{\mid {\mathrm{\Psi }}_{\mathrm{II}}(p,t)\mid }^2,$ , where E=p ²/2. In a 1D model, the multiple rescattering and overall rescattering probabilities are overestimated compared to a 3D model. However, this effect should be small in the case of a few-cycle laser pulse. In present simulation, we choose x_max=400 a.u., x_c=250 a.u. and Δ=15 a.u. Varying x_c from 200 to 300 a.u., the final results are insensitive to the choice of x_c in this range.

 

Fig. 2. High-energy photoelectron spectra (log scale) of He as a function of the CEP, calculated with the same laser parameters as in Fig. 1. Left panel for photoelectrons emitted to the left and right panel for photoelectrons emitted to the right. Both panels show the same information because a phase shift of π mirrors the pulse field in space and thus only reverses the roles of the left and right.

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Figure 2 shows the resulting high-energy photoelectron spectra as a function of the CEP for a 800 nm four-cycle (3.9 fs FWHM) laser pulse with an intensity I=2×10¹⁵ W/cm². The cutoff energy of the photoelectron spectrum is very sensitive to the CEP. For the photoelectrons emitted to the left, the cutoff energy increases from ~450 to ~1200 eV as the CEP increases from -0.5π to π. While for the photoelectrons emitted to the right, the cutoff energy also increases from ~450 to ~1200 eV as the CEP increases from 0.5π to 2π. Changing the CEP by π only interchanges the roles of the left and right. In addition, the spectra exhibit clear interference fringes. This pattern may be due to the interference of the long and short orbits in quantum theory [15]. The interference fringe separation is much larger than the laser photon energy and becomes larger with the photoelectron energy increasing. The number of interference fringes increases as the cutoff energy increases. For fewer-cycle pulses, the cutoff energy is more sensitive to the CEP. For a three-cycle (2.9 fs FWHM) laser pulse, the cutoff energy varies ranging from ~450 to ~1200 eV for a CEP variation of π, as shown in Fig. 3. More importantly, the relation between the cutoff energy and the CEP is approximately linear. This simple relation can be used to measure and stabilize the CEP precisely. One can measure the CEP from 0 to π using the cutoff energy of the spectrum of the left and from π to 2π using that of the right.

Firstly, one need to determine the maximal and minimal energies of an interference fringe in the cutoff region of the left spectra as the CEP is scanned over (0,π), or in the cutoff region of the right spectra as the CEP is scanned over (π,2π). Then, one can precisely measure the variation of the CEP by accurately measuring the shift of this interference fringe. Figure 4 shows the photoelectron spectra in the cutoff region for different CEPs. The spectra in Fig. 4(a) also exhibit interference fringes with a small separation close to the laser photon energy (1.55 eV). This is due to the interference of photoelectrons bursting in two time intervals apart with one optical cycle. The ionization probability is as the same order of magnitude in one time interval as that in the other time interval.

We only consider the large-scale interference fringes. As shown in Fig. 4(d), the interference dips closest to the cutoff energies occur at energies about 685.2, 688, 690.5 and 693 eV for CEPs φ=0, 0.005π, 0.01π and 0.015π, respectively. This implies that an average cutoff energy variation of 2.5 eV corresponds to a CEP variation of 0.005π. For CEPs φ=0.5, 0.505π, 0.51π and 0.515π, the interference dips closest to the cutoff energies occur at energies about 935.6, 938, 940.3 and 942.6 eV, respectively. An average cutoff energy variation of 2.3 eV corresponds to a CEP variation of 0.005π. As the CEP increases further, the linear relation between the cutoff energy and the CEP changes very slightly. For CEPs φ=0.9π, 0.905π, 0.91π and 0.915π, the interference dips closest to the cutoff energies occur at energies about 1112, 1114, 1116 and 1118 eV, respectively. For a CEP variation of 0.005π, the variation of the cutoff energy diminishes to 2 eV. Overall, the cutoff energy variation is about 2.3 eV corresponding to a CEP variation of 0.005π for laser parameters employed in Fig. 1. For higher laser intensities, the cutoff energy is more sensitive to the CEP.

 

Fig. 3. High-energy photoelectron spectra (log scale) of He as a function of the CEP, calculated with the same laser parameters as in Fig. 2 except for the total duration of the few-cycle pulse, which is three cycles. Left panel for photoelectrons emitted to the left and right panel for photoelectron emitted to the right.

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Fig. 4. Higher-energy parts of photoelectron spectra from left panel of Fig. 2 for different CEPs. The black, green, blue and red lines correspond to: panels (a) φ=0, 0.005π, 0.01π, 0.015π; panels (b) φ=0.5π, 0.505π, 0.51π, 0.515π; panels (c) φ=0.9π, 0.905π, 0.91π, 0.915π, respectively. The spectra in right panels are enlarged forms of the spectra near cutoff energies in respective left panels.

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Photoelectron spectra with clear interference fringes with pacing close to the photon energy have been measured for argon with 6 fs 760 nm laser pulses for intensity 1×10 ¹⁴ W/cm² in the attosecond double-slit experiment [16]. In our approach, note that only a fraction of the spectrum in the cutoff region is required to be measured. The space dependence of the electric field in a real laser pulse has little influence on the measurement of the interference structure in the cutoff region experimentally, since high-energy electrons can be generated only in the vicinity of the beam axis. Therefore, it is expected that the interference fringes of the calculated spectra could be observed in experiments. Photoelectron spectra are measured by the stereo-ATI spectrometer experimentally. The resolution of an electron spectrometer, i.e., the ability of distinguishing electrons of very small kinetic energy differences, determines the minimum observable shift of interference fringes in experiment. The minimum detectable energy difference of an electron spectrometer used in reference [6] is less than 1 eV in the energy range of 0-100 eV. In order to increase significantly the count rate in the ATI spectrum in the cutoff region and the energy resolution for high-energetic photoelectrons, one can mount two additional metal grids in front of the time-of-flight tubes. While the first is grounded, the second is kept at some adjustable negative voltage. As a consequence, only high energetic photoelectrons can overcome the created electric potential barrier, whereas low energetic photoelectrons are repelled. By adjusting the voltage, only photoelectrons in the cutoff region are allowed to enter detectors. In addition, the energies of these photoelectrons are very low when they enter detectors. This allows increasing significantly the count rate in the cutoff region and the higher energy resolution for high-energetic photoelectrons. Note that the distance between the two grids is kept small in order to avoid influence on the photoelectrons’ time-of-flight and subsequent artifacts in the spectra. Considering a typical resolution of 0.5 eV, it is expected that a CEP measurement precision of 4 mrad can be reached for laser parameters employed in Fig. 2.

In our method, the intensity range can be chosen as (1–4)×10¹⁵ W/cm². The ionization potentials of the target must be large. Moreover, the single ionization yields of He are two orders of magnitude higher than the double ionization yields of He in the high intensity range of [17]. We present other two methods for a comparison with our method. The CEP measurement precision with the asymmetry of ATI yield achieves 100 mrad for 6.5 fs 20 µJ pulses [18]. The precision with the THz system is 700 mrad for 8 fs 66 µJ pulses and a factor of four improvement can be achieved with 6.5 fs instead of 8 fs pulses [19]. The precisions with the two methods can be improved only by several times for 3.9 fs instead of 6.5 fs pulses, which is much lower than that with our method.

3. Conclusion

In conclusion, we present a new method of measuring the CEP of few-cycle pulses accurately using the strong phase dependence of the cutoff energy of rescattering photoelectrons bursting in very short time intervals close to the maximum of the pulse envelope. By accurately measuring the shift of the interference fringes in cutoff region, we can measure the slight variation of the CEP. Furthermore, only a small part of the spectrum in the cutoff region is required to measure in our scheme. Comparing with other two methods used in references [15, 16], our scheme can improve the CEP measurement precision significantly.