Abstract

Accurate determination of laser intensity is of fundamental importance to study various phenomena in intense laser-atom/molecule interactions. We theoretically demonstrate a scheme to measure laser intensity by examining the holographic structure originating from the interference between the direct and near-forward rescattering electrons in strong-field tunneling ionization. By adding a weak second-harmonic field with polarization orthogonal to the strong fundamental driving field, the interference pattern oscillates with the changing relative phases of the two-color fields. Interestingly, the amplitude of this oscillation in the photoelectron momentum spectrum depends on the parallel momentum. With the quantum-orbit analysis, we show that the amplitude of the oscillation minimizes when the time difference between the recollision and ionization of near-forward rescattering electron is half cycle of the fundamental driving field. This enables us to measure accurately the laser intensity by seeking the minimum of the oscillation amplitude. Moreover, we show that this minimum can be determined without scanning the relative phases, instead, by just monitoring the interference patterns for two relative phases. This facilitates the application of our scheme in experiment.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Tunneling ionization of atoms and molecules exposed to the strong laser fields is a fundamental process in intense laser-matter interactions [1–4]. The tunneled electron wave packet further accelerated by the oscillating electric field of laser pulse may return back and rescatter or recombine with its parent ion, resulting in various nonlinear phenomena [5], such as high-order above-threshold ionization [6–8], high-order harmonic generation [9–12] and nonsequential double ionization [13–19]. Exploring the application of these nonlinear processes consists the main contents of the attosecond science. In the past decades, these rescattering processes have been extensively used for attosecond pulses generation [20–23], molecular orbit tomography [24–26] and probing electronic and nuclear dynamics [27–31]. For these applications, deep insights into the details of the processes are necessary and it has drew plenty attentions. For example, the ionization time and rescattering time of the tunneled electron wave packet have been determined with attosecond accuracy [32]. In these strong-field processes, the laser intensity is an important parameter. Uncertainty in the laser intensity could lead to misunderstanding on the underlying dynamics [33]. Therefore, accurate determination of the laser intensity is essential for fully understanding the strong-field processes and for further performing their applications. However, this is a longstanding and challenging task [34–36].

Traditionally, the laser intensity could be estimated by measuring the pulse energy, the spatial and temporal profiles. This method usually gives an uncertainty as large as 50% [37]. In strong-field ionization, various in situ methods have been proposed to measure laser intensity. For example, it has been shown that the laser intensity could be calibrated by comparing the experimentally measured ionization probability with these from the theories [38]. This method requires the accurate theory on calculating ionization probability. The laser intensity has also been usually estimated from the 2Up or 10Up (Up is the ponderomotive potential of the laser field) cutoff in the photoelectron energy spectrum [39]. However, in the actual experiments such cutoffs are usually not clear. Moreover, in the multiphoton ionization range, such cutoffs do not exist. Another method of calibration of laser intensity is based on the ac-stark shift of the ionization potential of the target [40–42]. By checking the shift of the above-threshold ionization (ATI) peaks in the energy spectrum, the laser intensity can be estimated on the accuracy of a few percents. While the existence of the resonant ionization pathway in which the ATI peaks are laser intensity-independent, impedes its application [43]. Moreover, in the tunneling region for the mid-infrared laser fields, the ATI peaks are not visible and thus a reliable calibration of laser intensity is prevented. For the elliptically and circularly polarized laser pulses, the laser intensity can be determined from the center position of the photoelectron momentum distribution (PEMD) or the width of transverse momentum distribution [33, 44]. However, this method sensitively depends on either the adiabatic or nonadiabatic ionization picture used in calibration.

In this work, we introduce another novel in situ method based on the holographic pattern of the photoelectron momentum distribution (PEMD) in strong-field tunneling ionization to determine the laser intensity precisely. This holographic pattern originates from the interference of the direct electron (which reaches the detector directly after tunneling ionization) and the near-forward rescattering electron in strong-field tunneling ionization. This pattern has been experimentally observed several years ago [45] and it has been expected to be an effective tool in probing atomic and molecular structure and their ultrafast dynamics. In recent years, its application has been widely explored [46–52]. For example, it has been employed to extract the tunneling exit point [53] and the phase structure of the electron wave packet in molecular tunneling ionization [54,55]. Very recently, we have shown that with this photoelectron holographic interference the phase of the scattering amplitude can be extracted [56,57]. More fantastically, we have demonstrated that this holographic interference is capable of measuring the attosecond charge migration in molecules where unprecedented attosecond and picometer spatio temporal resolutions could be achieved [58]. Here, we explore its application in determining the laser intensity.

We analyze the holographic interference pattern in the PEMD obtained by solving the time-dependent Schrödinger equation (TDSE). When a weak perturbative second harmonic (SH) with polarization orthogonal to the fundamental driving field is added, the holographic interference varies with relative phases of this orthogonal two-color (OTC) fields. We monitor the positions of interference minima. The results show that the positions of these minima oscillate in the transverse direction (perpendicular to the polarization of the fundamental field) periodically with the relative phases of the OTC fields. The amplitude of the oscillation depends on the parallel momentum (parallel to the laser polarization of the fundamental field) and minimizes at a certain parallel momentum. Employing the quantum-orbit analysis, we demonstrate how the amplitude depends on the parallel momentum and show that it minimizes when the traveling time (the time between recollision and tunneling ionization of the electron) is half cycle of the fundamental field. This establishes the relation between the parallel momentum where the amplitude minimizes and the traveling time and thus the tunneling ionization time. With this relationship, the intensity of the fundamental driving field is determined by seeking the minimum of the oscillation amplitude of the holographic interference. Moreover, we demonstrate that this minimum can be determined by analyzing the holographic interference in the OTC fields at only two relative phases. This facilitates the application of our method in experiment.

2. Theoretical method

2.1. Numerically solving the time dependent Schrödinger equation

The PEMD was obtained by numerically solving the two-dimensional TDSE with single-active-electron (SAE) approximation (atomic unites are used unless otherwise stated),

iΨ(r,t)t=H(r,t)Ψ(r,t),
where Ψ(r, t) represents wave function of system and H(r, t) is Hamiltonian
H(r,t)=122+VC(r)+VL(r,t),
where VC(r) = −(x2 + y2 + a)−1/2 is the effective soft-core potential with soft-core parameter a = 0.92 which yields the ionization potential of Xe (Ip = 0.446 a.u.). VL(r, t) = r · F(t) represents the interaction of electron with the laser field. r = (x,y) denotes the position coordinate of electron. F(t) is the electric field of the OTC pulses, which is written as
F(t)=f(t)[Fx(t)e^x+Fy(t)e^y]=f(t)[F1cos(ωt)e^x+F1cos(2ωt+Φ)e^y].
Here, f(t) is pulse envelope. It has a trapezoidal shape, rising linearly during one cycle, then keeping constant for three cycles and decreasing linearly during the last one cycle of the fundamental pulse. The fundamental driving pulse Fx(t) is polarized along the x axis with amplitude F1 and the SH field Fy(t) is polarized along y axis with amplitude F2. In our calculations, the wavelength of the driving field is 1600 nm and the SH field is 800 nm. Φ is the relative phase of the two-color fields.

The split-operator spectral method on a Cartesian grid is used to solve the TDSE [59]. The initial wave function is obtained with imaginary-time propagation method [60]. The whole space is split into the inner (0 − Rs) and outer (RsRmax) region smoothly by the splitting function Fs(Rs) at the time τ [61]:

Ψ(τ)=Ψ(τ)[1Fs(Rs)]+Ψ(τ)Fs(Rs)=Ψ1(τ)+Ψ2(τ).
Here, Fs(Rs) = 1/[1 + e−(rRs)/Δ], Δ is the width of crossover region and Rc represents the boundary of the inner space [62]. Ψ1(τ) is the wave function in the inner space which is numerically propagated under the full Hamiltonian, and Ψ2(τ) represents the wave function in the outer space which is analytically propagated under the Volkov Hamiltonian [61,63]. Specially, at the time τ during the propagation, the wave packet Ψ2(τ) is firstly transformed into momentum space C(p, τ), and then it is propagated from time τ to the end of laser pulse by
Ψ2(,τ)=C¯(p,τ)eipτ2πd2p,
with C¯(p,τ)=eiτ12[p+A(τ)]2dτ C(p, τ). Here, p is the electron final momentum and A(t)=0tF(t)dt is the vector potential of the OTC fields. Therefore, we obtain the final momentum distribution related to the sum of the wave function in momentum space at τ,
dP(p)dEdθ=2E|τC¯(p,τ)|2,
where E = p2/2 is electron energy and θ is the the angle between p and the direction of laser polarization. In our calculations, we choose time step δτ = 0.05 a.u. At the end of the pulse, the wave function is propagated for one additional optical cycle to make sure the “slow” electrons reach the boundary Rs [64]. The boundary Rs is set to be 200 a.u. and Δ = 8 a.u.

2.2. Quantum-orbit analysis

In this work, we focused on the holographic pattern in the PEMDs. This holographic pattern results from the interference of the direct and near-forward rescattering electrons, and it can be written as

|M|2=|Md|2+|Mr|2+2|Md||Mr|cos(Δφ),
where |Md| and |Mr| are the amplitudes of the direct and rescattering electron wave packets, respectively. Δφ is the phase difference between the direct and rescattering electrons. Following [45], this phase difference can be calculated with the saddle-point approximation, which provides us the quantum orbit [65] to analysis the holographic interference. In the OTC fields Δφ can be written as
Δφ=φrφd=12tidtr[py+Ay(t)]2dt+12tidtr[px+Ax(t)]2dt12tirtr[ky+Ay(t)]2dt12tirtr[kx+Ax(t)]2dt+Ip(tirtid).
Here, Ip is the ionization potential of Xe. tir is the ionization time of rescattering electron and tid is the ionization time of direct electron. tr is the recollision time of the rescattering electron. px and py are the parallel and transverse momenta, respectively. Ax(t) and Ay(t) are the vector potential of the fundamental field and SH field, respectively. kx and ky are the components of canonical momentum k of electrons before rescattering in the horizontal and vertical direction, respectively. The first and third terms account for the phase acquired for the motion of direct and rescattering electrons in the direction of SH field polarization. The second and forth terms represent the phase obtained from the motion of the direct and rescattering electrons parallel to the fundamental laser polarization. The fifth term is caused by the different ionization time.

In this study, the SH field is very weak (with the intensity of about 1% of the fundamental field), and thus the ionization time and the recollision time can be approximately determined from the saddle-point equation in the fundamental driving field [32]. The evolution of the electron after tunneling ionization is affected by the SH field. Then, the saddle-point equation for the direct electron is

12[p+Ax(tid)]2+Ip=0,
and the saddle-point equations for the rescattering electron are
12[kx+Ax(tir)]2+Ip=0,
12[kx+Ax(tr)]2=12[p+Ax(tr)]2,
trtir[kx+Ax(t)]dt=0,
trtir[ky+Ay(t)]dt=0,
where Eqs. (9) and (10) stand for the energy conservation of direct and rescattering electron at tunneling ionization, and Eq. (11) indicates the energy conservation during rescattering. Equations (12) and (13) present the return condition.

3. Result and discussion

Figure 1(a) shows the PEMD obtained by solving TDSE of Xe in 1600-nm single-color laser field. Figures 1(b)–1(d) display the PEMDs in the OTC laser fields combined a 1600-nm fundamental component and a much weaker 800-nm field, where the relative phases are Φ = 0.5π, π and 1.5π, respectively. The intensity of the 1600-nm laser field is 1.5 × 1014 W/cm2. The intensity of the 800-nm field is 1 × 1012 W/cm2 and it can be considered as a weak perturbation. px and py are respectively the electron momentum along polarization directions of the 1600-nm and 800-nm fields. The nearly horizontal fringes are clearly visible in the PEMDs. These fringes are the holographic pattern originating from the interference of the direct and near-forward rescattering electrons. For the single-color field, this interference structure is exactly symmetric about py = 0. However, for the OTC fields the fringes are distorted. The fringes shift along py direction and this shift depends on the parallel momentum px. For example, at the relative phase Φ = 0.5π, the interference minima shift up at px = 1.0 a.u. and shift down at px = 1.9 a.u., as shown in Fig. 1(b). The shift of the fringes changes with relative phase. For example, the shift of the fringes is reversed for Φ = 1.5π, as compared to Φ = 0.5π.

 figure: Fig. 1

Fig. 1 (a) The TDSE results of PEMD for strong-field tunneling ionization of Xe in a single-color laser field with wavelength 1600 nm. (b)–(d) The PEMDs in the OTC laser fields with relative phases Φ = 0.5π, 1π and 1.5π, respectively. The OTC laser fields are combined by a 1600-nm fundamental pulse and weak SH pulse. The 1600-nm field is polarized along the x axis with intensity 1.5 × 1014 W/cm2 and the 800-nm field is polarized along y axis with intensity 1 × 1012 W/cm2. A sequence of PEMDs in the OTC laser fields for varying relative phases are shown in Visualization 1.

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To show the shift of the interference fringes more clearly, we employ the procedure introduced in [56] to extract the phase difference Δφ between the direct and rescattering electrons from the PEMDs. In Fig. 2(a), we present the extracted interference term cos(Δφ) as a function of transverse momentum at px = 1.9 a.u. It is shown that for Φ = 0.5π the line shifts toward left and for Φ = 1.5π it shifts toward right, as compared to the single-color field. To reveal this shift quantitatively, we seek the position of the first minimum of the interference term at py > 0 in the OTC laser fields and calculate its shift Δpy relative to the position in the single-color field. In Fig. 2(b), we present Δpy as a function of relative phase. It is shown that Δpy oscillates with relative phase. In Fig. 2(b), we have traced Δpy at different parallel momenta px. It is interesting that the behaviors of Δpy depend on the parallel momentum px. For example, Δpy maximizes at Φ = 1.5π for px = 1.9 a.u. and it maximizes at Φ = 2.0π for px = 1.0 a.u. Additionally, the amplitudes of oscillation of these lines also depend on the parallel momentum px. These behaviors are more clearly seen in Fig. 2(c), where we display Δpy as a function of relative phase Φ for parallel momentum px ranging from 0.75 a.u. to 1.95 a.u. The solid black line marks the maximum of Δpy. It is obvious that the corresponding relative phase where Δpy maximizes changes gradually with parallel momentum.

 figure: Fig. 2

Fig. 2 (a) The interference term cos(Δφ) as a function of py at px = 1.9 a.u. The blue dashed line represents the result extracted from the PEMD of single-color field. The yellow and green solid lines are the results extracted from the PEMDs in the OTC laser fields with relative phases Φ = 0.5π and 1.5π, respectively. The black dashed lines indicate the position of the first minimum of the interference term at py > 0. (b) The shift Δpy of the first minimum of the interference term at py > 0 as a function of relative phase at px = 1.0 a.u. (the yellow solid line), 1.3 a.u. (the green solid line), 1.6 a.u. (the purple solid line) and 1.9 a.u (the blue solid line). (c) The shift Δpy extracted from the TDSE results as a function of relative phase for px ranging from 0.75 a.u. to 1.95 a.u. (d) The shift Δpy obtained from Eq. (8) as a function of relative phase for px ranging from 0.75 a.u. to 1.95 a.u. The black solid lines in (c) and (d) indicate the maximum of the shift Δpy.

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We also calculate the oscillation Δpy of the position of the first minimum with Eq. (8) as shown in Fig. 2(d). Equation (8) is based on the strong-field approximation. It should be mentioned that Eq. (8) does not take into account the phase induced by the interaction between the parent ion and the rescattering electron (i.e., the phase of the scattering amplitude [56]), and thus its prediction of the holographic interference fringes deviates from the TDSE calculation and experimental data [66]. In order to correctly describe the holographic interferences, we should add a phase accounting for this interaction to Eq. (8) [56]. In our calculations, the orthogonal component is a weak perturbation, and thus this phase in the OTC laser fields is the same as that of the single-color field. Therefore, when considering the shift of the fringes induced by the OTC laser fields with respect to the single-color laser field, this phase does not matter. Thus, we can expect that the shift of the fringes can be accurately dealt with Eq. (8). This is confirmed by comparing Figs. 2(c) and 2(d), where the excellent agreement indicates that Eq. (8) is accurate in describing the shift of the fringes induced by the weak perturbation.

The periodic oscillation in Figs. 2(b)–2(d) introduces us to fit the relative phase dependence of Δpy with

Δpy=Pmcos(ΦΦm),
where the quantity Pm describes the amplitude of this oscillation and the quantity Φm shows at which phase the shift Δpy maximizes. Technically, Pm and Φm are determined by Fourier transforming Δpy with respect to Φ for each px [67,68]. In Figs. 3(a) and 3(b), we display the obtained Pm and Φm respectively as a function of px for the data from TDSE results in Fig. 2(c). It is shown that the amplitude Pm minimizes at px = 1.3 a.u. The obtained Φm depends on parallel momentum and varies from 2.0π to 1.4π when px changes from 0.75 a.u. to 1.95 a.u. For comparison, the results extracted from Fig. 2(d) are also shown in Fig. 3. Both quantities agree with the TDSE results excellently. In the following, we will focus on the amplitude Pm, which enables us to determine the laser intensity accurately.

 figure: Fig. 3

Fig. 3 (a) The amplitude Pm as a function of px. The solid line and dashed line represent the TDSE results extracted from Fig. 2(c) and the results obtained from Fig. 2(d), respectively. The black dashed line indicates the position where Pm gets minimum. (b) The same as (a) but for the quantity Φm.

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Given the agreement in Fig. 3, we reveal the origin of the minimum in Pm using Eq. (8). For the near-forward rescattering electrons, we have pxkx through Eq. (11). Then it can be easily proved that the second, forth and fifth term in the right side of Eq. (8) can approximately cancel each other. In fact, this point has been confirmed by our numerical calculations (not shown here). Then the first and third terms in Eq. (8) are the dominant terms accounting for the phase difference [45]. With this in mind, the phase difference between the direct and rescattering electrons is simplified as

Δφ12titr{[py+Ay(t)]2[ky+Ay(t)]2}dt.
Here, the superscript in ti has been omitted because tirtid. From Eq. (13), we have ky=titrAy(t)dt/(trti). Obviously, ky = 0 for the single-color field. From Eq. (15), we can obtain the transverse momentum py of the first minimum in the single-color and OTC fields. Then the shift of the holographic interference fringes in the OTC fields with respect to that of the single-color is
Δpy=pyOTCpyS=titrAy(t)dttrti=F22ωtitrsin(2ωt+Φ)dttrti.
Here, pyOTC and pyS are the momenta of the first minimum of the holographic fringes in the OTC fields and the single-color fundamental field, respectively. Equation (16) shows that the shift of the holographic interference fringes in the OTC fields with respect to the fundamental field equals to ky (ky=titrAy(t)dttrti). This indicates that Δpy depends on the relative phase, and also on the parallel momentum through ti and tr.

In Fig. 4(a), we show Δpy calculated with Eq. (16) as a function of relative phase Φ for each px. This results are consistent with that shown in Fig. 2(c), confirming the validation of the approximation made in Eq. (16). In Fig. 4(b), we present the amplitude Pm [Eq. (16)] as a function of px at various fundamental laser intensities I = 0.75I0, I0, 1.5I0 and 2.0I0 (I0 = 1.0 × 1014 W/cm2), while the intensity of the SH field is the same (1.0 ×1012 W/cm2). As a comparison, Pm from TDSE results are shown by solid lines. Clearly, Pm minimized at different px. Specifically, the minimum of Pm moves towards larger px as the intensity increases.

 figure: Fig. 4

Fig. 4 (a) Δpy calculated by Eq. (16) as a function of relative phase for px ranging from 0.75 a.u. to 1.95 a.u. The laser intensity is 1.5I0 (I0 = 1.0 × 1014 W/cm2). The black solid line indicates the maximum of Δpy. (b) The amplitude Pm of the oscillating Δpy as a function of px at laser intensities I = 0.75I0 (the blue lines), 1.0I0 (the green lines), 1.5I0 (the yellow lines) and 2.0I0 (the red lines), respectively. The solid lines represent the TDSE results and dashed lines stand for the results calculated by Eq. (16). (c) Left axis (the blue solid line): Pm as a function of traveling time Δt. Here T1 is the period of the fundamental field. Right axis (the green solid line): the parallel momentum px as a function of the traveling time Δt. A1 is the amplitude of the vector potential of the fundamental field. The laser intensity is 1.5I0. The black dashed lines stand for the position where the amplitude Pm minimizes. (d) The same as (b) but with px scaled by A1.

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Equation (16) indicates that Δpy is always zero when traveling time Δt = trti equals to the period of the SH field, i.e., half cycle of the fundamental field, Δt = 0.5T1, where T1 is the period of the fundamental driving field. It means that Pm minimizes at px where the corresponding traveling time of the rescattering electrons is half cycle of the fundamental field. In Fig. 4(c), we plot Pm as a function of traveling time. Clearly, the minimum locates at traveling time Δt = 0.5T1. Since the traveling time is an injective function of the parallel momentum for the near-forward rescattering electron [the green solid line in Fig. 4(c)], one can obtain that the corresponding parallel momentum for Δt = 0.5T1 is 0.6A1 (A1 is the amplitude of the vector potential of the fundamental driving field). Thus, by seeking the position px where the amplitude Pm minimizes, A1 of the driving field can be determined. Then the laser intensity is measured. It should be noted that the minimum of Pm is not zero. This is due to the fact that the ionization time in Eqs. (9)(11) is a complex number. In Fig. 4(d), we display the amplitude Pm for different laser intensities. The data here are the same as those presented in Fig. 4(b) but with px scaled by the vector potential A1 of the fundamental driving field. It is shown that the position of minimum always locates around 0.6A1. It means that the laser intensity can be determined with the relation px = 0.6A1. Thus, it provides a feasible way to measure the laser intensity.

Closer inspection of Fig. 4(d) shows that the position of minimum changes a little with laser intensity. This results from the fact that the ionization and rescattering time vary with laser intensity, as displayed by Eqs. (9)(11). The corresponding traveling time changes and thus the position of the minimum in Pm varies with laser intensity. Therefore, if one aims to determine the laser intensity more accurately, it is necessary to establish the map where the minimum of Pm locates as a function of laser intensity. In Fig. 5(a), we show the traveling time at different px as a function of the laser intensity of fundamental field. The solid line indicates the traveling time of 0.5T1, i.e, the position where the minimum of Pm should be located. With this map, the intensity of the driving field can be accurately determined. As a comparison, we show the obtained minimum position of Pm from the TDSE calculations at different laser intensities. The results are shown in Fig. 5(b). They are well located on the solid line predicted by Fig. 5(a). It confirms that the laser intensity can be preciously determined by seeking the minimum of Pm and comparing it with the solid line in Fig. 5(a).

 figure: Fig. 5

Fig. 5 (a) Δt as a function of px at different laser intensities ranging from 0.5I0 to 3.0I0. The black solid line represents Δt = 0.5T1. The vertical axis in (a) is scaled by A1. (b) The black solid line shows the parallel momentum corresponding to Δt = 0.5T1, as marked in (a). The red circles stand for the TDSE results of the position where Pm minimizes.

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As demonstrated above, to determine the position where Pm minimizes, one needs to scan the relative phases of the OTC fields to obtain the PEMDs. Experimentally, this is a very tedious task. In the following, we demonstrate that the position where Pm minimizes can be determined by just analyzing the PEMDs at two relative phases separated by 0.5π. According to Eq. (14), the shift Δpy at px for the relative phases Φ and Φ + 0.5π have the relation:

Δpy(Φ;px)2+Δpy(Φ+0.5π;px)2=Pm2(px).
Thus, by extracting the shift of holographic structure at two relative phases, the amplitude Pm of the oscillating Δpy can be determined. In Fig. 6(a) we show the the shift Δpy calculated by Eq. (16) for Φ = 0.1π and Φ = 0.6π. The amplitude Pm obtained through Eq. (17) is shown by the open circles. There is a clear minimum at px = 1.3 a.u. In Fig. 6(b), the data Pm at another pair of relative phases Φ = 1.3π and Φ = 1.8π are shown. The obtained Pm is the same as those in Fig. 6(a). The minimum also locates at px = 1.3 a.u. Thus, by analyzing the holographic pattern for an arbitrary pair of relative phases Φ and Φ + 0.5π, the minimum position of Pm can be determined. Then, the laser intensity can be measured. Figure 6(c) displays the determined laser intensities through this two-relative-phase measurement. The data are well located around the accurate value shown by the solid line, confirming the validity and accuracy of our method.

 figure: Fig. 6

Fig. 6 (a) Δpy calculated by Eq. (16) as a function of px at relative phases Φ = 0.1π (the green solid line) and 0.6π (the purple solid line). The red circles represent the amplitude Pm obtained by Eq. (17). The yellow dashed line is the same as that in Fig. 4(b). Here the laser intensity of fundamental field is 1.5I0. (b) The same as (a) but for another pair of relative phases Φ = 1.3π and Φ = 1.8π. (c) The black solid line is the same as that in Fig. 5(a). The red circles stand for the position where the Pm extracted from the TDSE results by Eq. (17) minimizes.

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Our scheme of determining the laser intensity is based on seeking the minimum of Pm. This minimum is still observable when the laser focal volume effect is taken into account, as shown in Fig. 7 where a Gaussian beam profile is considered. Thus our method can be used in real experiment. Note that in our scheme, the measured laser intensity is not the peak intensity of the laser pulse, but the effective intensity which is most relevant for the strong-field phenomena. In Fig. 7, the peak intensity we used is 2.25 ×1014 W/cm2. From the minimum of Pm, we can obtained that the effective laser intensity for the ionization signal is 1.13 ×1014 W/cm2. In the results above, a pulse shape with flat top is used in our calculation. For the realistic Gaussian-shape pulses (not shown here), the minimum in Pm is also clear, and thus our scheme of determining laser intensity is applicable.

 figure: Fig. 7

Fig. 7 The amplitude Pm as a function of the parallel momentum px. Here laser focal volume effect has been taken into account by considering a Gaussian beam profile. The peak laser intensity is 2.25 ×1014 W/cm2. The vertical dashed line indicates the minimum of Pm.

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4. Conclusion

In summary, we have investigated the holographic pattern in strong-field tunneling ionization in OTC fields. Our results show that the positions of the interference minima oscillate with the relative phase. The amplitude of this oscillation depends on the parallel momentum. With the quantum-orbit analysis based on saddle-point approximation, we showed that the amplitude minimizes when the traveling time of the rescattering electron is half cycle of the fundamental driving field. This predicates that the minimum for oscillation amplitude locates at the parallel momentum of 0.6A1, which is confirmed by our TDSE calculations. Therefore, scanning the relative phases of OTC fields and seeking the minimum of the oscillation amplitude of the interference pattern, the vector potential and thus the intensity of the fundamental driving field can be accurately determined. Moreover, we demonstrated that the amplitude of the oscillations as a function of parallel momentum can be determined by just analyzing the PEMDs at two relative phases separated by 0.5π. Thus the laser intensity can be measured by just monitoring the PEMDs at two relative phases.

Funding

National Natural Science Foundation of China (11622431, 61475055, 11604108, 11627809); Program for HUST Academic Frontier Youth Team.

Acknowledgments

Numerical simulations presented in this paper were carried out using the High Performance Computing Center experimental testbed in SCTS/CGCL (see http://grid.hust.edu.cn/hpcc).

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20. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. D. Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314(5798), 443–446 (2006). [CrossRef]   [PubMed]  

21. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef]   [PubMed]  

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23. J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017). [CrossRef]   [PubMed]  

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25. C. Zhai, X. Zhang, X. Zhu, L. He, Y. Zhang, B. Wang, Q. Zhang, P. Lan, and P. Lu, “Single-shot molecular orbit tomography with orthogonal two-color fields,” Opt. Express 26(3), 2775–2784 (2018). [CrossRef]   [PubMed]  

26. H. Yuan, L. He, F. Wang, B. Wang, X. Zhu, P. Lan, and P. Lu, “Tomography of asymmetric molecular orbits with a one-color inhomogeneous field,” Opt. Lett. 43(4), 931–934 (2018). [CrossRef]   [PubMed]  

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32. D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485(7398), 343 (2012). [CrossRef]   [PubMed]  

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2018 (9)

Y. Chen, Y. Zhou, Y. Li, M. Li, P. Lan, and P. Lu, “Rabi oscillation in few-photon double ionization through doubly excited states,” Phys. Rev. A 97(1), 013428 (2018).
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L. Li, P. Lan, L. He, X. Zhu, J. Chen, and P. Lu, “Scaling law of high harmonic generation in the framework of photon channels,” Phys. Rev. Lett. 120(22), 223203 (2018).
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S. Luo, X. Ma, H. Xie, M. Li, Y. Zhou, W. Cao, and P. Lu, “Controlling nonsequential double ionization of Ne with parallel-polarized two-color laser pulse,” Opt. Express 26(10), 13666–13676 (2018).
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C. Zhai, X. Zhang, X. Zhu, L. He, Y. Zhang, B. Wang, Q. Zhang, P. Lan, and P. Lu, “Single-shot molecular orbit tomography with orthogonal two-color fields,” Opt. Express 26(3), 2775–2784 (2018).
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H. Yuan, L. He, F. Wang, B. Wang, X. Zhu, P. Lan, and P. Lu, “Tomography of asymmetric molecular orbits with a one-color inhomogeneous field,” Opt. Lett. 43(4), 931–934 (2018).
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L. He, Q. Zhang, P. Lan, W. Cao, X. Zhu, C. Zhai, F. Wang, W. Shi, M. Li, X. Bian, P. Lu, and A. D. Bandrauk, “Monitoring ultrafast vibrational dynamic of isotopic molecules with frequency modulation of high-order harmonics,” Nat. Commun. 9(1), 1108 (2018).
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M. Li, W. Jiang, H. Xie, S. Luo, Y. Zhou, and P. Lu, “Strong-field photoelectron holography of atoms by bicircular two-color laser pulses,” Phys. Rev. A 97(2), 023415 (2018).
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M. He, Y. Li, Y. Zhou, M. Li, W. Cao, and P. Lu, “Direct visualization of valence electron motion using strong-field photoelectron holography,” Phys. Rev. Lett. 120(13), 133204 (2018).
[Crossref] [PubMed]

J. Tan, Y. Li, Y. Zhou, M. He, Y. Chen, M. Li, and P. Lu, “Identifying the contributions of multiple-returning recollision orbits in strong-field above-threshold ionization,” Opt. Quant. Electron. 50(2), 57 (2018).
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2017 (4)

M. He, Y. Zhou, Y. Li, M. Li, and P. Lu, “Revealing the target structure information encoded in strong-field photoelectron hologram,” Opt. Quant. Electron. 49(6), 232 (2017).
[Crossref]

S. G. Walt, N. B. Ram, M. Atala, N. I. S. Shilovski, A. Conta, D. Baykusheva, M. Lein, and H. J. Wörner, “Dynamics of valence-shell electrons and nuclei probed by strong-field holography and rescattering,” Nat. Commun. 8, 15651 (2017).
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T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. A. Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver,” Opt. Express 25(22), 27506–27518 (2017).
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J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8(1), 186 (2017).
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2016 (8)

B. Wolter, M. G. Pullen, A. T. Le, M. Baudisch, K. D. Dier, A. Senftleben, M. Hemmer, C. D. Schröter, J. Ullrich, T. Pfeifer, R. Moshammer, S. Gräfe, O. Vendrell, C. D. Lin, and J. Biegert, “Ultrafast electron diffraction imaging of bond breaking in di-ionized acetylene,” Science 354(6310), 308–312 (2016).
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M. Haertelt, X. Bian, M. Spanner, A. Staudte, and P. B. Corkum, “Probing Molecular Dynamics by Laser-Induced Backscattering Holography,” Phys. Rev. Lett. 116(13), 133001 (2016).
[Crossref] [PubMed]

M. He, Y. Li, Y. Zhou, M. Li, and P. Lu, “Temporal and spatial manipulation of the recolliding wave packet in strong-field photoelectron holography,” Phys. Rev. A 93(3), 033406 (2016).
[Crossref]

Y. Li, Y. Zhou, M. He, M. Li, and P. Lu, “Identifying backward-rescattering photoelectron hologram with orthogonal two-color laser fields,” Opt. Express 24(21), 23697–23706 (2016).
[Crossref] [PubMed]

S. Zhao, A. T. Le, C. Jin, X. Wang, and C. D. Lin, “Analytical model for calibrating laser intensity in strong-field-ionization experiment,” Phys. Rev. A 93(2), 023413 (2016).
[Crossref]

C. Hofmann, T. Zimmermann, A. Zielinski, and A. S. Landsman, “Non-adiabatic imprints on the electron wave packet in strong field ionization with circular polarization,” New J. Phys. 18(4), 043001 (2016).
[Crossref]

M. Liu, M. Li, C. Wu, Q. Gong, A. Staudte, and Y. Liu, “Phase structure of strong-field tunneling wave packet from molecules,” Phys. Rev. Lett. 116(16), 163004 (2016).
[Crossref] [PubMed]

Y. Zhou, O. I. Tolstikhin, and T. Morishita, “Near-forward rescattering photoelectron holography in strong-field ionization: extraction of the phase of the scattering amplitude,” Phys. Rev. Lett. 116(17), 173001 (2016).
[Crossref] [PubMed]

2015 (4)

S. Skruszewicz, J. Tiggesbäumker, K. H. M. Broes, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

P. He, N. Takemoto, and F. He, “Photoelectron momentum distribution of atomic and molecular systems in strong circularly or elliptically polarized laser field,” Phys. Rev. A 91(6), 063413 (2015).
[Crossref]

Y. Shao, M. Li, M. Liu, X. Sun, X. Xie, P. Wang, Y. Deng, C. Wu, Q. Gong, and Y. Liu, “Isolating resonant excitation from above-threshold ionization,” Phys. Rev. A 92(1), 013415 (2015).
[Crossref]

M. Li, P. Zhang, S. Luo, Y. Zhou, Q. Zhang, P. Lan, and P. Lu, “Selective enhancement of resonant multiphoton ionization with strong laser fields,” Phys. Rev. A 92(6), 063404 (2015).
[Crossref]

2014 (1)

M. Meckel, A. Staudte, S. Patchkovskii, D. M. Villeneuve, P. B. Corkum, R. Dörner, and M. Spanner, “Signatures of the continuum electron phase in molecular strong-field photoelectron holography,” Nat. Phys. 10(8), 594 (2014).
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2013 (1)

R. Boge, C. Cirelli, A. S. Landsman, S. Heuser, A. Ludwig, J. Maurer, M. Weger, L. Gallmann, and U. Keller, “Probing nonadiabatic effects in strong-field tunnel ionization,” Phys. Rev. Lett. 111(10), 103003 (2013).
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2012 (8)

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485(7398), 343 (2012).
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C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro, and C. D. Lin, “Imaging ultrafast molecular dynamics with laser-induced electron diffraction,” Nature 483(7388), 194 (2012).
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Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical simulations including electron correlations for sequential double ionization,” Phys. Rev. Lett. 109(5), 053004 (2012).
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W. Becker, X. Liu, P. J. Ho, and J. H. Eberly, “Theories of photoelectron correlation in laser-driven multiple atomic ionization,” Rev. Mod. Phys. 84(3), 1011 (2012).
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Y. Huismans, A. Gijsbertsen, A. S. Smolkowska, J. H. Jungmann, A. Rouzée, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. Meer, M. Yu. Ivanov, T. M. Yan, D. Bauer, O. Smirnova, and M. J. J. Vrakking, “Scaling laws for photoelectron holography in the midinfrared wavelength regime,” Phys. Rev. Lett. 109(1), 013002 (2012).
[Crossref] [PubMed]

S. Xu, X. Sun, B. Zeng, W. Chu, J. Zhao, W. Liu, Y. Cheng, Z. Xu, and S. L. Chin, “Simple method of measuring laser peak intensity inside femtosecond laser filament in air,” Opt. Express 20(1), 299–307 (2012).
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D. D. Hickstein, P. Ranitovic, S. Witte, X. Tong, Y. Huismans, P. Arpin, X. Zhou, K. E. Keister, C. W. Hogle, B. Zhang, C. Ding, P. Johnsson, N. Toshima, M. J. J. Vrakking, M. M. Murnane, and H. C. Kapteyn, “Direct visualization of laser-driven electron multiple scattering and tunneling distance in strong-field ionization,” Phys. Rev. Lett. 109(7), 073004 (2012).
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X. Bian and A. D. Bandrauk, “Attosecond Time-Resolved Imaging of Molecular Structure by Photoelectron Holography,” Phys. Rev. Lett. 108(26), 263003 (2012).
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2011 (5)

C. Smeenk, J. Z. Salvail, L. Arissian, P. B. Corkum, C. T. Hebeisen, and A. Staudte, “Precise in-situ measurement of laser pulse intensity using strong field ionization,” Opt. Express 19(10), 9336–9344 (2011).
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Y. Huismans, A. Rouzée, A. Gijsbertsen, J. H. Jungmann, A. S. Smolkowska, P. S. W. M. Logman, F. Lépine, C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker, G. Berden, B. Redlich, A. F. G. Meer, H. G. Muller, W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov, O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J. Vrakking, “Time-resolved holography with photoelectrons,” Science 331(6013), 61–64 (2011).
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T. Marchenko, Y. Huismans, K. J. Schafer, and M. J. J. Vrakking, “Criteria for the observation of strong-field photoelectron holography,” Phys. Rev. A 84(5), 053427 (2011).
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Y. Zhou, C. Huang, and P. Lu, “Coulomb-tail effect of electron-electron interaction on nonsequential double ionizaiton,” Phys. Rev. A 84(2), 023405 (2011).
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Y. Zhou, C. Huang, A. Tong, Q. Liao, and P. Lu, “Correlated electron dynamics in nonsequential double ionization by orthogonal two-color laser pulses,” Opt. Express 19(3), 2301–2308 (2011).
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2010 (2)

H. Kang, W. Quan, Y. Wang, Z. Lin, M. Wu, H. Liu, X. Liu, B. Wang, H. Liu, Y. Gu, X. Jia, J. Liu, J. Chen, and Y. Cheng, “Structure effect in angle-resolved high-order above-threshold ionization of molecules,” Phys. Rev. Lett. 104(20), 203001 (2010).
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Y. Zhou, Q. Liao, and P. Lu, “Asymmetric electron energy sharing in strong-field double ionization of helium,” Phys. Rev. A 82(5), 053402 (2010).
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2009 (1)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163 (2009).
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2008 (1)

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008).
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2007 (1)

X. Tong, S. Watahiki, K. Hino, and N. Toshima, “Numerical observation of the rescattering wave packet in laser-atom interaction,” Phys. Rev. Lett. 99(9), 093001 (2007).
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2006 (4)

D. B. Milošević, G. G. Paulus, D. Bauer, and W. Becker, “Above-threshold ionization by few-cycle pulses,” J. Phys. B 39(14), R203 (2006).
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X. Tong, K. Hino, and N. Toshima, “Phase-dependent atomic ionization in few-cycle intense laser fields,” Phys. Rev. A 74(3), 031405 (2006).
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S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilǎ, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing Proton Dynamics in Molecules on an Attosecond Time Scale,” Science 312(5772), 424–427 (2006).
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G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. D. Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314(5798), 443–446 (2006).
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2004 (3)

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867 (2004).
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A. S. Alnaser, X. Tong, T. Osipov, S. Voss, C. M. Maharjan, B. Shan, Z. Chang, and C. L. Cocke, “Laser peak intensity calibration using recoil-ion momentum imaging,” Phys. Rev. A 70(2), 023413 (2004).
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V. L. B. Jesus, B. Feuerstein, K. Zrost, D. Fischer, A. Rudenko, F. Afaneh, C. D. Schröter, R. Moshammer, and J. Ullrich, “Atomic structure dependence of nonsequential double ionization of He, Ne and Ar in strong laser pulses,” J. Phys. B 37(8) L161 (2004).
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2002 (2)

H. Niikura, F. Légaré, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417(6892), 917 (2002).
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W. Becker, F. Grasbon, R. Kopold, D. B. Milošević, G. G. paulus, and H. Walther, “Above-threshold ionization: from classical features to quantum effect,” Adv. At. Mol. Opt. Phys. 48, 35–98 (2002).
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2001 (1)

P. Salières, B. Carré, L. L. Déroff, F. Grasbon, G. G. Paulus, H. Walther, R. Kopold, W. Becker, D. B. Milošević, A. Sanpera, and M. Lewenstein, “Feynman’s path-integral approach for intense-laser-atom interactions,” Science 292(5518), 902–905 (2001).
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2000 (1)

T. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emission in multiphoton double ionization,” Nature 405(6787), 658 (2000).
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1999 (1)

E. Constant, D. Garzella, P. Breger, E. Mével, C. Dorrer, C. L. Blanc, F. Salin, and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Model and experiment,” Phys. Rev. Lett. 82(8), 1668 (1999).
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1998 (1)

S. Chelkowski, C. Foisy, and A. D. Bandrauk, “Electron-nuclear dynamics of multiphoton H2+ dissociative ionization intense laser fields,” Phys. Rev. A 57(2), 1176 (1998).
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1997 (1)

M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389 (1997).
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1994 (2)

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Supplementary Material (1)

NameDescription
» Visualization 1       This video shows a sequence of PEMDs in the OTC laser fields for varying relative phases.

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Figures (7)

Fig. 1
Fig. 1 (a) The TDSE results of PEMD for strong-field tunneling ionization of Xe in a single-color laser field with wavelength 1600 nm. (b)–(d) The PEMDs in the OTC laser fields with relative phases Φ = 0.5π, 1π and 1.5π, respectively. The OTC laser fields are combined by a 1600-nm fundamental pulse and weak SH pulse. The 1600-nm field is polarized along the x axis with intensity 1.5 × 1014 W/cm2 and the 800-nm field is polarized along y axis with intensity 1 × 1012 W/cm2. A sequence of PEMDs in the OTC laser fields for varying relative phases are shown in Visualization 1.
Fig. 2
Fig. 2 (a) The interference term cos(Δφ) as a function of py at px = 1.9 a.u. The blue dashed line represents the result extracted from the PEMD of single-color field. The yellow and green solid lines are the results extracted from the PEMDs in the OTC laser fields with relative phases Φ = 0.5π and 1.5π, respectively. The black dashed lines indicate the position of the first minimum of the interference term at py > 0. (b) The shift Δpy of the first minimum of the interference term at py > 0 as a function of relative phase at px = 1.0 a.u. (the yellow solid line), 1.3 a.u. (the green solid line), 1.6 a.u. (the purple solid line) and 1.9 a.u (the blue solid line). (c) The shift Δpy extracted from the TDSE results as a function of relative phase for px ranging from 0.75 a.u. to 1.95 a.u. (d) The shift Δpy obtained from Eq. (8) as a function of relative phase for px ranging from 0.75 a.u. to 1.95 a.u. The black solid lines in (c) and (d) indicate the maximum of the shift Δpy.
Fig. 3
Fig. 3 (a) The amplitude Pm as a function of px. The solid line and dashed line represent the TDSE results extracted from Fig. 2(c) and the results obtained from Fig. 2(d), respectively. The black dashed line indicates the position where Pm gets minimum. (b) The same as (a) but for the quantity Φm.
Fig. 4
Fig. 4 (a) Δpy calculated by Eq. (16) as a function of relative phase for px ranging from 0.75 a.u. to 1.95 a.u. The laser intensity is 1.5I0 (I0 = 1.0 × 1014 W/cm2). The black solid line indicates the maximum of Δpy. (b) The amplitude Pm of the oscillating Δpy as a function of px at laser intensities I = 0.75I0 (the blue lines), 1.0I0 (the green lines), 1.5I0 (the yellow lines) and 2.0I0 (the red lines), respectively. The solid lines represent the TDSE results and dashed lines stand for the results calculated by Eq. (16). (c) Left axis (the blue solid line): Pm as a function of traveling time Δt. Here T1 is the period of the fundamental field. Right axis (the green solid line): the parallel momentum px as a function of the traveling time Δt. A1 is the amplitude of the vector potential of the fundamental field. The laser intensity is 1.5I0. The black dashed lines stand for the position where the amplitude Pm minimizes. (d) The same as (b) but with px scaled by A1.
Fig. 5
Fig. 5 (a) Δt as a function of px at different laser intensities ranging from 0.5I0 to 3.0I0. The black solid line represents Δt = 0.5T1. The vertical axis in (a) is scaled by A1. (b) The black solid line shows the parallel momentum corresponding to Δt = 0.5T1, as marked in (a). The red circles stand for the TDSE results of the position where Pm minimizes.
Fig. 6
Fig. 6 (a) Δpy calculated by Eq. (16) as a function of px at relative phases Φ = 0.1π (the green solid line) and 0.6π (the purple solid line). The red circles represent the amplitude Pm obtained by Eq. (17). The yellow dashed line is the same as that in Fig. 4(b). Here the laser intensity of fundamental field is 1.5I0. (b) The same as (a) but for another pair of relative phases Φ = 1.3π and Φ = 1.8π. (c) The black solid line is the same as that in Fig. 5(a). The red circles stand for the position where the Pm extracted from the TDSE results by Eq. (17) minimizes.
Fig. 7
Fig. 7 The amplitude Pm as a function of the parallel momentum px. Here laser focal volume effect has been taken into account by considering a Gaussian beam profile. The peak laser intensity is 2.25 ×1014 W/cm2. The vertical dashed line indicates the minimum of Pm.

Equations (17)

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i Ψ ( r , t ) t = H ( r , t ) Ψ ( r , t ) ,
H ( r , t ) = 1 2 2 + V C ( r ) + V L ( r , t ) ,
F ( t ) = f ( t ) [ F x ( t ) e ^ x + F y ( t ) e ^ y ] = f ( t ) [ F 1 cos ( ω t ) e ^ x + F 1 cos ( 2 ω t + Φ ) e ^ y ] .
Ψ ( τ ) = Ψ ( τ ) [ 1 F s ( R s ) ] + Ψ ( τ ) F s ( R s ) = Ψ 1 ( τ ) + Ψ 2 ( τ ) .
Ψ 2 ( , τ ) = C ¯ ( p , τ ) e i p τ 2 π d 2 p ,
d P ( p ) d E d θ = 2 E | τ C ¯ ( p , τ ) | 2 ,
| M | 2 = | M d | 2 + | M r | 2 + 2 | M d | | M r | cos ( Δ φ ) ,
Δ φ = φ r φ d = 1 2 t i d t r [ p y + A y ( t ) ] 2 d t + 1 2 t i d t r [ p x + A x ( t ) ] 2 d t 1 2 t i r t r [ k y + A y ( t ) ] 2 d t 1 2 t i r t r [ k x + A x ( t ) ] 2 d t + I p ( t i r t i d ) .
1 2 [ p + A x ( t i d ) ] 2 + I p = 0 ,
1 2 [ k x + A x ( t i r ) ] 2 + I p = 0 ,
1 2 [ k x + A x ( t r ) ] 2 = 1 2 [ p + A x ( t r ) ] 2 ,
t r t i r [ k x + A x ( t ) ] d t = 0 ,
t r t i r [ k y + A y ( t ) ] d t = 0 ,
Δ p y = P m cos ( Φ Φ m ) ,
Δ φ 1 2 t i t r { [ p y + A y ( t ) ] 2 [ k y + A y ( t ) ] 2 } d t .
Δ p y = p y O T C p y S = t i t r A y ( t ) d t t r t i = F 2 2 ω t i t r sin ( 2 ω t + Φ ) d t t r t i .
Δ p y ( Φ ; p x ) 2 + Δ p y ( Φ + 0.5 π ; p x ) 2 = P m 2 ( p x ) .

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