Laser intensity determination using nonadiabatic tunneling ionization of atoms in close-to-circularly polarized laser fields

Wei Quan; MingHu Yuan; ShaoGang Yu; SongPo Xu; YongJu Chen; YanLan Wang; RenPing Sun; ZhiLei Xiao; Cheng Gong; LinQiang Hua; XuanYang Lai; XiaoJun Liu; Jing Chen

doi:10.1364/OE.24.023248

1. Introduction

Precise determination of laser intensity is crucial to understand the ultrafast dynamics of atoms and molecules subject to ultrashort strong laser field. There are two kinds of experimental procedures employed ubiquitously to determine the laser intensity: (i) Intensity calibration based on geometric measurements [1,2], which is operative if the focus geometry and the time evolution function of the laser field are known accurately; (ii) The procedure based on the measurements of ultrafast ionization phenomena, such as Freeman resonances [3,4], intensity scaling of high-order above-threshold ionization (HATI) cutoff [5,6], intensity dependence of single or double charged ion yields [7,8] etc. The latter procedure can be performed with very high accuracy if the ionization process in question can be well reproduced theoretically. Considering that averaging over the intensity distribution of the laser focus will significantly change the result, it is paramount to employ the focal averaging during the calculation. Therefore, even with the procedure (ii), the knowledge of the focal averaging function is still indispensable.

Recently, much attention has been paid to a laser intensity calibration procedure based on measurements of photoelectron/photoion momentum distribution from ionization by circularly polarized laser fields [9,10]. As is documented, this procedure is accurate and reliable, where the measurements can be well reproduced with the theoretical methods [11,12]. Moreover, this procedure can be applied with both velocity map imaging (VMI) and cold target recoil ion momentum spectroscopy (COLTRIMS), which are the two of most popular imaging spectrometers. Nevertheless, to perform this procedure, a premise has to be fulfilled, which is the accurate determination of the monotonous correspondence between the instantaneous laser intensity, $I$ , and the measured momenta of the photoelectron or photoion, $p$ . Based on the simple-man model [13,14], the function of,

I (p) = \frac{1 + ε^{2}}{ε^{2}} {(p ω)}^{2}

is adopted widely [9,11,12], where

ω

is the laser frequency and

ε

the ellipticity of the laser field. Atomic units (a.u.) are used unless otherwise stated.

As long as the over-the-barrier ionization is insignificant, Eq. (1) is justified for the adiabatic (quasi-static) tunneling dynamics featured by the Keldysh parameter $γ < < 1$ [15], where $γ = \sqrt{I_{p} / 2 U_{p}}$ , $I_{p}$ is the ionization potential and $U_{p}$ the ponderomotive energy ( $U_{p} = I / 4 ω^{2}$ ). The documented experimental works employed this intensity calibration procedure [11,12] are performed mainly in the regime of $γ$ much less than 1, when adiabatic tunneling ionization dominates and the nonadiabatic effect can be neglected. However, equipped with the popular Ti: Sapphire laser beam with wavelength of 800 nm, a lot of experiments for noble gas atoms, in essence, are performed at the regime of $γ \approx 1$ , when the laser field evolves so fast that the electron cannot adiabatically follows the change of the barrier. Recently, plenty of papers [10,16–18] point out that in this regime Eq. (1) doesn’t hold any more and the nonadiabatic effect becomes significant. It is documented [16,19] that the impact of nonadiabatic effect on the laser intensity calibration results could be appreciable under typical experimental conditions. Therefore, it is necessary to include the nonadiabatic effect when determining the intensity with the typical experimental parameters employed by most experiments.

Based on the pioneering works [9,11,12], in this paper, we exploit an improved procedure to calibrate the laser intensity using nonadiabatic tunneling ionization of several noble gas atoms, where $γ$ is slightly larger than 1. We deliberately choose this regime to highlight the paramount role of nonadiabatic effect when tunneling picture is still valid. In contrast to the circularly polarized field employed in the preceding papers, we demonstrate that our procedure is viable even for close-to-circularly polarized laser field, which is less demanding for the experimental techniques, especially for the broadband few-cycle laser fields. Furthermore, our work confirms the paramount role of Coulomb potential in the ultrafast ionization dynamics. To test the accuracy of the calibrated intensity, we performthe measurements of HATI for Ar subject to linearly polarized laser fields, which are further compared with the numerical calculation of the time-dependent Schrödinger equation (TDSE) in the single-electron approximation.

2. Experimental setup

The laser system employed in the experiment is a 5 kHz femtosecond Ti: Sapphire laser system (Femtopower Compact Pro Amplifier, FEMTOLASERS Produktions GmbH), which can produce ultrashort laser pulse with duration of around 25 fs at 800 nm and output energy up to 0.8 mJ. The pulse shape is near Gaussian distribution in both temporal and spatial domains. The laser pulse energy is precisely controlled with the combination of a λ/2 plate and a thin film polarizer. When necessary, an optional extra λ/4 plate is applied to change the polarization of the laser electric field from linear to close-to-circular before the laser beam is directed into a COLTRIMS [20], where the laser beam with diameter of around 8 mm is focused by a concave mirror with the focal length of 75 mm. The Rayleigh length is estimated to be around 20 μm. The noble gas atoms are ionized inside the COLTRIMS, where the momenta of photoelectrons and photoions are measured coincidently. A uniform electric field (2.2 V/cm for the case of linearly polarized laser field and 13.8 V/cm for the close-to-circularly polarized one) is applied to accelerate the photoelectrons and photoions flying towards Microchannel Plates (MCP) detectors, where the delay line anodes (HEX75 and DLD80, RoentDek Handels GmbH, for electron and ion respectively) are equipped. The distance from the laser focus to the ion detector is around 10 cm and the one for electron is around 16 cm. There is no field-free region in our spectrometer. The 3 dimensional momenta of both fragments can be retrieved from their impact positions and time-of-flights which can be resolved by the detectors. A uniform magnetic field (2.3 Gauss for the case of linearly polarized laser field and 7.8 Gauss for the close-to-circularly polarized one) generated by a pair of Helmholtz coils can confine the photoelectron cyclotron radius in the plane perpendicular to the electric field to maintain the 4π solid angle measurement. Weak magnetic fields produced by two pairs of auxiliary coils, which are aligned perpendicularly to each other, can be adjusted carefully to compensate the residual earth magnetic field. A controllable slit, which can be tuned with four micrometer drivers outside the vacuum chamber, is employed to reduce the intersection of supersonic beam and therefore the production rate of photofragments if necessary. Great care has been taken to ensure that the beam will go through the center of the focus. During the measurement, the count rate of the electron/ion is kept around 0.2/0.04 per laser shot to avoid false coincidence.

3. Simulation procedure

The numerical calculation procedure employed in this paper is developed on the basis of a well-verified adiabatic semiclassical numerical method [21,22]. In this semiclassical model, the bound electron is firstly tunnel ionized in the combined laser electric field and atomic Coulomb field. In the case of elliptically polarization [22], the electric field of the laser is rotating in the x-z plane. At each tunneling moment $t_{0}$ , the $\vec{s}$ axis is rotated to be parallel to the instantaneous electric field. The initial tunnel exit, the initial transverse velocity and the weight of the electron trajectory are calculated in the rotated coordinates according to the tunneling model the same as in the linearly polarized field and then projected to the x-y-z coordinates. After ionization, the evolution of the free electron is governed by the 3 dimensional Newton’s equations of motion and the Coulomb interaction between electron and parent ionic core is fully considered.

As documented [16–18,23], the nonadiabatic effect in elliptically polarized laser field will influence the ionization rate, the tunneling exit coordinate, the initial transverse momentum distribution, and the delay of the appearance of the photoelectron in the continuum (tunneling delay). To include the nonadiabatic effect, modifications have been made to these aspects of the semiclassical method, except for the tunneling delay, the effect of which has been circumvented in our calculation because consensus has not been achieved on how long the tunneling delay precisely is [24]. Fortunately, the effect of this delay has been comprehended very well, which may give rise to an angular offset of the 2 dimensional momentum distribution in the plane of polarization [25]. To remove the angular offset induced by the tunneling delay, we deliberately rotate the experimental and theoretical momentum distributions to a well-defined angle, where the two yield maximums in each 2 dimensional momentum distribution perfectly overlap with each other in the distributions in horizontal direction (denoted as u, see Fig. 1(b)), and symmetric about $p = 0$ in the vertical direction (denoted as v, see Fig. 1(b)). Note that this angular offset is also sensitive to the Coulomb potential. As shown in Fig. 1(a), the blue dashed line, which goes through the centers of the two maximums of the 2 dimensional momentum distribution, forms an angle with the red solid line $p_{z} = 0$ in the plane of ( $p_{z}$ , $p_{x}$ ). In Fig. 1(b), the blue dashed line is defined as $p_{u} = 0$ . In this way, the unresolved problem of tunneling delay can be avoided in our numerical simulations.

Fig. 1 Illustration of the procedure to determine the directions of “u” and “v”. (a) The calculated 2 dimensional momentum distribution of Kr subject to linearly polarized laser field with intensity of 1.9 × 10¹⁴ W/cm² and wavelength of 800 nm in the plane of ( $p_{z}$ , $p_{x}$ ); (b) The same data in the plane of ( $p_{u}$ , $p_{v}$ ). The solid red line is drawn according to the condition of $p_{z} = 0$ . The dashed blue line, which goes through the centers of the two yield maximums of 2 dimensional momentum distribution, is defined as $p_{u} = 0$ in (b).

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The details of our calculation procedure are presented in the following. The laser field can be described with

E (t) = F (t) (\cos (ω t + ϕ) \vec{z} + ε \sin (ω t + ϕ) \vec{x})

where

F (t) = E_{0} \cos^{2} (t / τ_{p})

if the condition of

- τ_{p} π / 2 \leq t \leq τ_{p} π / 2

is fulfilled, otherwise

F (t) = 0

.

ω

is the laser angular frequency,

ϕ

the carrier envelope phase (CEP), which is randomly distributed, and

τ_{p}

is 13 optical periods. The evolution of the tunneled electrons are traced by launching a set of trajectories at

t_{0}

, which are randomly chosen within the interval of [

- τ_{p} π / 2

,

τ_{p} π / 2

]. The ionization rate is described by nonadiabatic tunneling ionization theory [26],

W (γ (t), p_{| |}, p_{⊥}, t) = \exp (- \frac{2 I_{p}}{ω} f (γ (t), p_{| |}, p_{⊥}, t))

where,

f (γ (t), p_{| |}, p_{⊥}, t) = (1 + \frac{1}{2 γ {(t)}^{2}} + \frac{p {(t)}^{2}}{2 I_{p}}) arc \cos α - \sqrt{α^{2} - 1} (\frac{β}{γ (t)} \sqrt{\frac{2}{I_{p}}} (p_{| |} - A (t)) + \frac{α [1 - 2 β^{2}]}{2 γ {(t)}^{2}})

α (β) = \frac{γ (t)}{2} (\sqrt{\frac{p {(t)}^{2}}{2 I_{p}} + \frac{2}{γ (t)} \frac{(p_{| |} - A (t))}{\sqrt{2 I_{p}}} + \frac{1}{γ {(t)}^{2}} + 1}) + (-) \sqrt{\frac{p {(t)}^{2}}{2 I_{p}} - \frac{2}{γ (t)} \frac{(p_{| |} - A (t))}{\sqrt{2 I_{p}}} + \frac{1}{γ {(t)}^{2}} + 1})

p {(t)}^{2} = {(p_{| |} - A (t))}^{2} + p_{⊥}^{2}

p_{| |}

indicates the momentum parallel to the instantaneous laser electric field,

p_{⊥}

the one perpendicular to the polarization plane,

γ (t) = ω \sqrt{2 I_{p}} / E (t)

the time dependent Keldysh parameter and

A (t) = - \int E (t) d t

the vector potential of the laser field. Note that here we use

A (t)

to replace

A_{| |} (t)

for simplicity and the influence induced by this replacement is negligible when the ellipticity is close to 1 (According to our calculation, the deviation of ionization rate induced by this replacement is around 0.01% for Kr subject to the laser field with intensity of 2.0 × 10¹⁴ W/cm² and ellipticity of 0.98. Deviation larger than 25% is possible, if the ellipticity is lower than 0.8). As documented [10,16,17], nonadiabatic effect can induce a momentum offset and a broadened momentum distribution in the transverse direction at the tunneling exit. Here, we explicitly insert one exponential term to simulate the transverse momentum distribution. The final nonadiabatic ionization rate reads,

R (γ (t), p_{| |}, p_{⊥}, t) = W (γ (t), p_{| |}, p_{⊥}, t) \exp (- \frac{{(p_{p} - p_{0})}^{2}}{{(C \times p_{0})}^{2}})

where,

p_{0}

is the transverse momentum shift,

p_{p}

denotes the transverse momentum inside the polarization plane, which is a random real number in the range of [−3

p_{0}

, 5

p_{0}

], and

C

is a parameter to adjust the width of initial transverse momentum distribution. The value of

C

, in the range of [1.1,1.5], is chosen to match measured momentum distributions and the documented theoretical calculations [16,27]. According to the imaginary time method [28,29], the non-zero velocity shift can be calculated with,

p_{0} = \frac{\sqrt{(1 + ε^{2}) E_{0}^{2} - E {(t)}^{2}}}{ω} (\frac{\sin h (τ_{0})}{τ_{0}} - 1)

where

τ_{0}

is determined from,

\sin h^{2} τ_{0} (1 - ε^{2} {(\coth τ_{0} - \frac{1}{τ_{0}})}^{2}) = γ^{2}

The tunneling exit with nonadiabatic effect employed can be calculated with,

s_{0} = \frac{E_{0}}{ω^{2}} (\cos h τ_{0} - 1) \vec{s}

\vec{s}

indicates the instantaneous laser electric field direction. The focal averages of tight and weak focusing have been carried out in the calculations. As discussed in the section of experimental setup, we have applied a controllable slit to reduce the intersection of the supersonic beam in the direction of laser beam prorogation if necessary. When the slit is fully open, the width of the supersonic beam is around 1 mm, which is larger than the Rayleigh range of the laser beam focus. Therefore, the tight-focusing approximation is applicable [12,30] and the yield from a given iso-intensity shell in the laser focus is,

\frac{d N}{d I} = (1 - e^{- \int_{- \infty}^{\infty} R (I) d t}) \int_{- \infty}^{\infty} \frac{- π z_{0} ω_{0}^{2}}{3} \frac{1}{I^{5 / 2}} (I_{0} F (t) + 2 I) {(I_{0} F (t) - I)}^{1 / 2} d t

where

z_{0}

is the Rayleigh range,

ω_{0}

the 1/e beam radius in intensity and

I_{0}

the peak intensity. If the counts are too large for coincidence measurements, we can reduce the intersection of the supersonic beam with the two micrometer drivers outside the vacuum chamber. In this way, the measurements at higher intensity can be performed coincidently with COLTRIMS. In the meantime, the tight-focusing approximation doesn’t hold any more because the width of the supersonic beam (around 1 μm in our case) is much less than the Rayleigh range of the focus (around 20 μm in our case). Therefore the weak-focusing approximation can be employed [30], where the Gaussian temporal distribution is assumed,

\frac{d N}{d I} = π τ_{p} Δ z ω_{0}^{2} \frac{R (I)}{I} \sqrt{\ln (\frac{I_{0}}{I})}

where

Δ z

is the width of the supersonic beam and

τ_{p}

the pulse duration.

4. Results

We have measured the photoelectron momentum distributions for Ar, Kr and Xe subject to close-to-circularly polarized laser field with the ellipticity of 0.98 at 800 nm, where the laser powers are adjusted to be low enough to fulfill the requirements, for coincident measurements, to the photoelectron counts. Since the ionization thresholds of the three noble gas atoms are different significantly, the intensities selected will distribute in a large interval. To study the ionization dynamics at higher intensity for one certain kind of atoms, e.g. Kr, we can reduce the width of the controllable slit to reduce the number of the atoms in the focus with the two micrometer drivers outside the vacuum chamber. Then the photoelectron counts are kept still low enough for the coincident measurements even if the laser intensity is increased. As discussed in the section of simulation procedure, we have rotated the 2 dimensional photoelectron momentum distributions to a well-defined angle, where the two new axes are denoted as u and v, to remove the rotation effect of tunneling delay and the Coulomb potential.

In Fig. 2, the experimental measurements are compared with the numerical calculation results. The measurements are shown with blue open squares. A prominent double-hump structure can be identified for each sample in the direction of v, while a flat-top single-hump structure in the direction of u. This is different from the case of circularly polarized laser field, where double-hump structure appears in any direction. This difference can be comprehended because the deviation of the ellipticity from 1 leads to two significant maximums of photoelectron yields due to the exponential dependence of the ionization rate on laser intensity, which, in turn, induces the two different structures in the direction of u and v. The numerical simulations, pictured with red thick lines, can reproduce the measurements very well. Considering that there are always some small deviations of ellipticity from 1 for circularly polarized laser field in experiments, the agreements of the measured and calculated momentum distributions in both directions (u and v) make the simulations for close-to-circularly polarization more reliable than those for circularly polarization, where usually the comparisons of momentum distributions in only one direction is performed.

Fig. 2 Measured and calculated photoelectron momentum distributions for noble gas atoms subject to close-to-circularly polarized laser fields with ellipticity of 0.98. (a) Kr subject to the laser field with peak intensity of 1.9 × 10¹⁴W/cm² in the direction of v; (b) same as (a), but in the direction of u. (c) Xe subject to the laser field with peak intensity of 6.5 × 10¹³ W/cm² in the direction of v; (d) same as (c), but in the direction of u. The blue open square dots indicate the measured data, the red thick lines illustrate the calculation results with the nonadiabatic model employed in the paper.

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To shed more light on the role of Coulomb potential and nonadiabatic effect, we deliberately remove the Coulomb potential and the nonadiabatic effect in the calculations, as shown in Fig. 3. As we can see, the influences of both Coulomb potential and nonadiabatic effect are significant. Without the Coulomb potential, as shown in Fig. 3(b), the momentum distribution of photoelectrons becomes wider. This result can be understood with the deceleration by the Coulomb potential during the passage of photoelectrons from their tunnel exits to the detector. The influence of Coulomb potential is indispensable and the inclusion of this influence will make our numerical simulations more accurate. In contrast, without the initial transverse momentum offsets induced by the nonadiabatic effect, as shown in Fig. 3(c), the two peaks of the double-hump structure in the momentum distribution of photoelectrons become much closer, which is a direct consequence of removing initial velocity offset pointing away from the Coulomb center. Apparently, these two effects will compensate each other and the effect of the initial transverse momentum offset is stronger. If the broadened initial transverse momentum distribution induced by the nonadiabatic effect is removed further, as shown in Fig. 3(d), the width of each hump becomes very small, which leads to larger deviation from the measurement.

Fig. 3 Calculated momentum distributions in the direction of v for Kr subject to close-to-circularly polarized laser fields with ellipticity of 0.98 at 1.9 × 10¹⁴ W/cm² and 800 nm with the numerical model employed all the influences induced by the nonadiabatic effect and Coulomb potential (a), all but the Coulomb potential (b), all but the offset of the initial transverse momentum induced by nonadiabtic effect (c), all but the offset of the initial transverse momentum and the broadened initial transverse momentum distribution induced by nonadiabtic effect (d).

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In Fig. 4, with filled red squares, we depict the calibrated laser peak intensities, which are determined with the nonadiabatic numerical procedure discussed above, with respect to the laser powers measured during the experiments. A linear fit, depicted with red solid line, is performed, where the standard error for the slope is around 3% and the intercept is −2.4 × 10¹²W/cm². This intercept corresponds to the laser power of −0.6 mW, which can be taken as the residual uncompensated readings from the power meter. In the meantime, we also perform the intensity calibration with the adiabatic procedure discussed in [12], which is supposed to be valid only if nonadiabatic effect is negligible. Due to the absence of nonadiabatic effect, the adiabatic calculation results couldn’t match the measurements very well (as shown in Fig. 3(d)), which gives rise to larger error bars in Fig. 4. Moreover, as expected, the calibrated intensities, indicated with open blue circles, deviate larger from the linear fit. The standard error for the slope is more than 19%. The intercept is 1.1 × 10¹³ W/cm², the absolute value of which is much larger than that in the case of nonadiabatic model. Apparently, the laser intensities determined with our nonadiabatic model are lower than those with the adiabatic model. This result can be understood with the compensation between the influence of the offset of the initial transverse momentum from the nonadiabatic effect and that of the Coulomb potential, where the latter is weaker, as discussed above.

Fig. 4 The calibrated intensity versus laser power. (a) Laser intensity calibration performed with the nonadiabatic model employed in this work; (b) Linear fit of (a); (c) Intensity calibration performed with the adiabatic model from the pioneer work [12]; (d) Linear fit of (c).

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To test if the intensities calibrated with the nonadiabatic procedure are more reliable, we further perform the measurements for Ar subject to linearly polarized laser fields at 800 nm (see Fig. 5(a)) and compare the HATI photoelectron spectra with the numerical calculation of TDSE with the single-active-electron approximation. The details of the numerical integration of TDSE have been described in [31,32], where the accuracy of the calculation is well verified. Essentials of this TDSE method will be given in the following. In the method, the Schrödinger equation is written as,

i \frac{\partial}{\partial t} ψ (r, t) = (- \frac{\nabla^{2}}{2} + V^{(a)} (r) + V^{(F)} (r, t)) ψ (r, t)

where

V^{(a)}

represents the spherically symmetric three-dimensional long-range model potential of atomic system [33] and

V^{(F)}

is the laser-atom interaction. To solve the equation, we expand the time-dependent wave function in spherical coordinates as

ψ (r, t) = \sum_{l = 0}^{l_{\max}} \sum_{m = - l}^{l} (1 / r) χ_{_{l, m}} (r, t) Y_{_{l, m}} (θ, ϕ)

,where

χ_{l, m} (r, t)

is the reduced radial wave function, and

Y_{l, m} (θ, ϕ)

is the spherical harmonics. The second-order split-operator scheme is employed to propagate the wave function. At the end of the propagation, we can obtain the energy spectrum by projecting the final wave function

ψ (r, t = T)

onto the continuum field-free wave function. Calculations have been performed at a series of laser intensities to include the influence of focal average with Eq. (11). The envelope of laser field is a cos-square function with twelve optical cycles, which is long enough to avoid the carrier envelope phase effect [34].

Fig. 5 The measured and calculated HATI photoelectron spectra for Ar subject to linearly polarized laser field. According to the nonadiabatic fit function of laser intensity versus power, the calibrated laser intensity for the measurement is (a) 1.0 × 10¹⁴ W/cm². In the meantime, the intensities employed in the TDSE calculations are (b) 1.0 × 10¹⁴W/cm² and (c) 1.2 × 10¹⁴W/cm².

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As shown in Fig. 5, the measured HATI spectra are compared with the results calculated with the numerical procedure of TDSE. The measured laser power for the experimental data in Fig. 5 is 27.5( $\pm$ 1.0) mW. According to the nonadiabatic fit function of laser intensity versus power presented in Fig. 4, the corresponding intensity is 1.0 × 10¹⁴W/cm². In contrast, the intensity determined according to the adiabatic fit function is 1.2 × 10¹⁴W/cm². We present the TDSE results at both intensities in Fig. 5. As we can see, the TDSE result at 1.0 × 10¹⁴W/cm² (see Fig. 5(b)) is clearly closer to the measurement, especially in the high energy region around 60 eV, which is the cutoff energy of the high energy plateau in the HATI spectra predicted by the simple-man model [13,14,35] at this intensity. Considering the sensitive dependence of the cutoff energy on the laser intensity, the coincidence between the measurement and calculation confirms the validity of our intensity calibration procedure based on the nonadiabatic tunneling ionization. In contrast, the deviation of the calculated HATI spectra at 1.2 × 10¹⁴ W/cm² (see Fig. 5(c)) from the measurement is significant, which means that the adiabatic intensity procedure may lead to a large deviation when nonadiabatic effect is nonnegligible. This comparison demonstrates that our intensity calibration procedure based on the nonadiabatic numerical calculation is accurate and robust.

The uncertainty of our intensity calibration procedure is estimated to be around 10%. The intensity calibration uncertainty induced by the comparison between the numerical simulations and measurements followed by the linear fit procedure is less than 4%. The uncertainty of laser power measured by the power meter is around 3%. The other experimental factors, such as laser ellipticity, alignment etc., will contribute less than 3%. Therefore, the total accuracy of the intensity determination is less than 10%.

5. Discussion

In the regime of $γ < < 1$ , where the tunneling process is adiabatic, Eq. (1) establishes a simple correspondence between the laser instantaneous intensity and the measured photoelectron momentum. Based on this relationship, the absolute values of the momenta of the photofragments can be taken as the direct measurements of the laser intensities at the time of ionization and the local laser intensity for each ionization event can be determined easily [9,11,12]. As a matter of fact, a prerequisite of this procedure is that quantum mechanical uncertainty of the momentum in the polarization plane is small enough. However, this prerequisite is justified until the nonadiabatic effect plays a significant role, when the width of the initial transverse momentum distribution is broadened significantly [16,28]. And there are large uncertainties of the correspondence between the laser instantaneous intensities and the final photoelectron momenta. Therefore, the simple correspondence between the laser instantaneous intensity and the measured photoelectron momentum cannot be applied when the nonadiabatic effect is significant.

As demonstrated above, the comparisons of measurements and calculations in this work can be performed even if the simple correspondence between the laser instantaneous intensity and the measured photoelectron momentum doesn’t hold anymore. Therefore, the coincident measurement technique, which is applied in this work, is not always necessary when the nonadiabatic intensity calibration procedure described here is employed. In fact, the other spectrometers (e.g. VMI), which can acquire reliable 2 dimensional momentum distributions of noble gas atoms without applying the coincident technique, can also be employed in the nonadiabatic intensity calibration procedure discussed here. In brief, our procedure is a general and reliable procedure to calibrate the laser intensity with the nonadiabatic effect included.

6. Summary

In summary, we have determined the laser intensities using nonadiabatic tunneling ionization of atoms in close-to-circularly polarized laser fields. The measurements can be well reproduced with the numerical calculation, where the nonadiabatic effect and Coulomb potential influence have been included. The determined laser intensity has been confirmed by the comparison of the measured HATI spectrum in the linearly polarized laser field with the TDSE calculation result. Out work demonstrates that the nonadiabatic effect is indispensable to calibrate the laser intensity in the regime of $γ \approx 1$ . The uncertainty in our intensity calibration procedure is estimated to be around 10%.

Funding

National Basic Research Program of China (No. 2013CB922201); National Natural Science Foundation of China (Nos. 11274050, 11204356, 11334009, 11374329, 11425414 and 11474321) ; Youth Innovation Promotion Association of CAS (2011242).

Acknowledgment

We would like to thank Prof. XueBin Bian for helpful discussions with the details of the numerical calculation of TDSE.

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Laser intensity determination using nonadiabatic tunneling ionization of atoms in close-to-circularly polarized laser fields

Abstract

1. Introduction

2. Experimental setup

3. Simulation procedure

4. Results

5. Discussion

6. Summary

Funding

Acknowledgment

References and links

Cited By

Figures (5)

Equations (13)

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