# In situ measurement and control of the laser-induced strong field ionization
phenomena

Noam Shlomo, Eugene Frumker<EMAIL_ADDRESS>Department of Physics, Ben-
Gurion University of the Negev, Beer-Sheva 84105, Israel

###### Abstract

Accurately characterizing the intensity and duration of strong-field
femtosecond pulses within the interaction volume is crucial for attosecond
science. However, this remains a major bottleneck, limiting accuracy of the
strong-field, and in particular, high harmonic generation experiments. We
present a novel scheme for the in situ measurement and control of spatially
resolved strong-field femtosecond pulse intensity and duration within the
interaction focal region. Our approach combines conjugate focal imaging with
in situ ion measurements using gas densities pertinent to attosecond science
experiments. Independent measurements in helium and argon, accompanied by a
fitting to a strong field ionization dynamic model, yield accurate and
consistent results across a wide range of gas densities and underscores the
significance of double ionization, as well as barrier suppression ionization.
Direct spatially resolved characterization of the driving laser is a critical
step towards resolving the averaging problem in the interaction volume, paving
the way for more accurate and reliable attosecond experiments.

##  I Introduction

Ultrafast laser-induced strong field phenomena play a pivotal role across a
broad spectrum of fundamental areas within modern physics. For instance, the
first step in high harmonic generation (HHG) [1, 2], a cornerstone of
attosecond science, relies on strong field-driven tunnel ionization.
Similarly, powerful techniques like ’attoclock’ employ this phenomenon to
probe intricate details of quantum tunnelling [3, 4]. Strong field induced
Coulomb explosion, fragmentation, tunneling and multiphoton ionization in the
velocity map imaging (VMI) [5] and cold target recoil ion momentum
spectroscopy (COLTRIMS) [6] experiments are used to study variety of most
fundamental physical and chemical processes in atoms and molecules [7, 8]. In
the wake-field acceleration [9, 10] experiments the strong laser field allows
to accelerate particles with a record gradients enabling construction of high
performance accelerators of much smaller size than conventional ones or even
at the photonic chip scale [11, 12, 13].

Accurate measurement, optimization and control of laser intensity distribution
and of the pulse duration in the interaction region are critically important
for most of the strong field-driven experiments.

Upon initial examination, the problem may appear straightforward: to directly
measure or estimate fundamental laser pulse characteristics, such as energy,
width, and duration, and subsequently deduce the peak intensity. However, it
has long been recognized and experimentally validated [14, 15, 2] that such
seemingly simple approach usually results in unacceptably large errors,
frequently exceeding 50%, which can significantly impact the interpretation of
experimental results. Several factors can contribute to such errors, including
imperfect laser beam quality, aberrations in the focusing optics, distortion
of ultrafast pulses due to spatial chirp (e.g., caused by misalignment of the
compressor grating), pulse dispersion in the generation medium, and more.

Furthermore, high-power laser systems often are prone to temporal fluctuations
in pulse intensity and variations in their spatial and temporal profiles,
compounding the issue due to the highly nonlinear nature of strong-field
phenomena. Therefore, a reliable in situ approach to measure and control these
parameters in the focal volume becomes critically important for all areas
involving strong-field laser physics.

To address this challenge, various methodologies have been reported.
Typically, intricate, specialized, and costly techniques have been employed to
achieve this objective, including mass spectrometry [16], velocity map imaging
(VMI) [17], and cold target recoil-ion momentum spectroscopy (COLTRIMS) [15].
Furthermore, the utilization of these approaches necessitates very low gas
densities, generally incompatible with HHG experiments, and often extremely
low transverse temperatures, which further complicate the experimental setup,
requiring larger system dimensions.

For high gas density measurements, particularly critical in strong-field
physics, such as attosecond science, the estimation of laser-peak intensity
has often relied on the high harmonic generation (HHG) cutoff [18, 19].
However, in practice, determining the cutoff is nontrivial, as contrary to an
idealized and clearly defined cutoff, a series of low-intensity harmonics
frequently appear in the experimentally measured HHG spectrum ([20, 2]). As a
result, determining the cutoff harmonic order is somewhat subjective.
Furthermore, it has been demonstrated that macroscopic effects, arising from
averaging and phase matching, can distort the single-atom HHG cutoff response
[21], leading to inaccurate intensity estimates and a lack of access to pulse
duration estimation within the interaction region.

In this study we introduce a novel in situ measurement and control approach
that solves this long-standing critical problem, opening the door to
unprecedented accuracy in strong-field, in particular, attosecond science
experiments. By conducting independent measurements in helium (He), the gas
with a highest known ionization potential, $I_{p}=24.6$ eV, and argon (Ar),
$I_{p}=15.76$ eV, [22] with exact the same beam parameters in the focal
volume, we have identified indelible and meaningful signatures of double
ionization, and barrier-suppression ionization (BSI).

Furthermore, we realized a dynamic theoretical model that incorporates single
ionization, sequential double ionization (SDI), non-sequential double
ionization (NSDI), non-sequential excitation (NSE), recollision-induced
excitation with subsequent ionization (RESI), and BSI phenomena for strong
field ionization. This model demonstrates excellent agreement and consistency
with our experimental results. Importantly, while demonstrated here using
argon as a test case, this approach seamlessly extends to experiments with
other atomic gases, including neon (Ne), krypton (Kr), and xenon (Xe).

Figure 1:  Schematic of the experimental setup. The ion detector mesh
$\textrm{ID}^{\textrm{-}}$ is connected to the negative voltage of the high
voltage power supply ($V_{PS}$) through the shunt resistor ($R_{1}=1M\Omega$).
The output is monitored by an oscilloscope (OSC) connected to the circuit via
a DC decoupling capacitor ($C_{1}=100pF$) and a resistor ($R_{2}=10M\Omega$).
The input capacitance of the scope (15 pf) and input impedance ($1M\Omega$)
are not shown in the figure but are accounted for in the calculations.

##  II Main

The schematic of our experimental setup is shown in Fig. 1. A linearly
polarized laser pulse from a Ti:sapphire amplifier ( centered around
$\lambda=800$ nm) was focused by a spherical f=500 mm focusing mirror M2
through the vacuum chamber window (W) onto a gas jet (GJ). To maintain precise
control over the pulse energy, a motorized $\lambda/2$ waveplate (RWP) and a
thin-film polarizer (TFP) were employed. The focal plane on the gas jet was
monitored via an image-conjugated microscope, which comprised a beam-picking
mirror (FM), an objective lens (OL), and a CCD imager.

The ion detector [23, 19, 24] is composed of a stainless steel grounded plate
($\textrm{ID}^{\textrm{G}}$) connected mechanically and galvanically to the
body of the pulsed valve , and the metal mesh ($\textrm{ID}^{\textrm{-}}$)
attached to the ($\textrm{ID}^{\textrm{G}}$) with dielectric rods and
connected to the negative high-voltage, $V_{PS}$ via simple passive electronic
circuit shown in Fig. 1. In this experiment, we employed a pulsed valve
(Parker series 9 solenoid valve) that is widely recognized for its versatility
and widespread use in numerous strong-field experiments. It is important to
note that our approach is not limited to this specific valve and can
seamlessly adapt to other gas jet sources as well. We applied a voltage of
$V_{PS}=-500V$ and set the distance between the grounded plate and the mesh to
d=80 mm. The inclusion of the mesh instead of solid component is instrumental
in minimizing gas flow resistance, thereby enabling efficient evacuation of
the residual gas by a vacuum turbomolecular pump located beneath the vacuum
chamber. The pulsed gas valve was installed on a micron-precision (XYZ) vacuum
positioner, ensuring that the laser beam intersected the jet at the center,
approximately $100\mu\textrm{m}$ below the injection flange. This intentional
positioning was chosen for optimal conditions for attosecond science
experiments (HHG), facilitating in situ measurements of HHG [19, 2]. This
simple and straightforward construction not only enhances the versatility of
the ion detector but also simplifies its deployment across a wide range of
experimental configurations. Consequently, our approach can be readily applied
in a diverse array of strong-field experiments.

Figure 2:  (a) Example of measured (dotted blue curve) vs. calculated (solid
red curve) ion signal temporal evolution for argon jet. The laser pulse energy
was 0.6mJ, and the gas backpressure was 7.5 Bar. The best fitted parameters
were found $R_{0}=13.27M\Omega$ and $\tau=30.74\mu s$ with goodness of fit
$R^{2}=0.98$. (b) Total charge (integrated current, modeled in LTspice and
shown by the dotted blue curve) vs. measured ion signal. The goodness of
linear fit (solid red line) $R^{2}=0.999$.

When the laser pulse interacts with the gas jet, the gas molecules and atoms
undergo ionization within the pulse’s relatively short duration, typically on
the order of tens of femtoseconds, through the strong-field ionization
mechanisms. As a result, the gas medium’s resistance drops, enabling current
to flow between the ground plate and the mesh, recorded as the voltage
evolution on an oscilloscope as illustrated in Fig. 1. Utilizing a freely
available professional analog electronic circuit simulator LTspice, we
performed the calculations to determine the transient response of the circuit,
and fit it to the experimentally measured voltage trace, with a typical result
shown in Fig. 2 (a).

The laser ionized gas was modeled as a time-dependent resistor
$R_{ID}(t)=R_{ID}(t_{0})\exp[(t-t_{0})/\tau_{ID}]$ in parallel with a
capacitor, representing the ion detector geometry. Transmission lines with
defined impedances and lengths were included to account for cable effects. The
initial resistance $R_{ID}(t_{0})$, ionizing pulse arrival time - $t_{0}$, and
the decay constant $\tau_{ID}$ were fit to experimentally measured data.

Note, that the minor ”bump” observed 1 millisecond after the primary pulse
results from the ionization of residual gas induced by a successive laser
pulse from our 1 kHz amplified laser system. We systematically varied the
laser intensity by adjusting the Rotating Waveplate (RWP), Fig. 1, and
recorded the voltage trace for each intensity setting. For each measured
voltage trace, we used the LTspice to compute the total charge, which
corresponds to the integral of the current between the ground plate and the
mesh, and confirm a linear dependence between the ion signal and the total
charge as shown in Fig. 2 (b). It’s worth noting that the term ”ion signal”
can refer to either the maximum voltage measured on the oscilloscope trace or
the integral of the voltage trace (as used in our analysis), both result in
the linear dependence between the ion signal and the total charge.

Figure 3:  Measured Intensity distribution of the beam at the focal plane. The
inner black and the outer red lines represent the $\gamma=1$ contours for
helium and argon gases respectively, emblematically illustrating the regions
of tunneling (inner part) and multiphoton (outer part) regimes. The intensity
was calibrated using energy per pulse of 0.4mJ and $\tau_{FWHM}=43$ fs. The
spatial intensity distribution in the focal plane was imaged using a
microscope as illustrated in Fig. 1.

The measured spatial intensity distribution of the beam in the focus is shown
in the Fig 3. The input beam has been apertured down to 9.25 mm before the
focusing mirror (M2) with an iris to improve the beam quality and to extend
effective Raleigh range in the focus. By scanning the image-conjugated
microscope (depicted as a dashed box in Fig. 1 ) we validated that the beam
width and shape don’t change within the measured 0.4mm extension (FWHM) of the
gas jet.

We now turn our focus to the fundamental processes underlying strong-field
ionization phenomena, which play a pivotal role in our measurements. Single-
electron ionization of neutral atoms or ions represents the most fundamental
aspect of strong-field ionization and has been extensively studied for several
decades [25, 26, 27, 3, 4]. Keldysh’s seminal contribution [25] laid the
foundation for our understanding of strong-field ionization in the so-called
’transparency region’, where the ionization potential, $I_{p}$, exceeds the
energy of the incident laser field, $\hbar\omega_{L}$.

The nature of the ionization process is characterized by the adiabaticity or
Keldysh parameter, denoted as $\gamma$. This parameter is defined as
$\gamma=2\pi\frac{T_{\textrm{tunnel}}}{T_{L}}$, where
$T_{L}=\frac{2\pi}{\omega_{L}}$ represents the laser field’s period, and
$T_{\textrm{tunnel}}$ is is referred to as the ”tunneling time.” The tunneling
time [25] is defined as a ratio of twice the length of the potential barrier,
$\textrm{L}_{\textrm{barrier}}$ divided by a classical velocity,
$\textrm{v}_{\textrm{e}}$, of an electron moving under the barrier. Using the
virial theorem - $\textrm{v}_{\textrm{e}}=\sqrt{2I_{p}/m_{e}}$, where $m_{e}$
is the electron’s mass. Assuming a point-like potential for electron-nuclear
interaction and approximating the total potential with a triangular barrier
solely created by the laser field, one gets
$\textrm{L}_{\textrm{barrier}}=I_{p}/{eE_{L}}$. Consequently, for the
tunneling time we get
$T_{\textrm{tunnel}}=\frac{2\textrm{L}_{\textrm{barrier}}}{{\textrm{v}}_{\textrm{e}}}=\frac{\sqrt{2I_{p}m_{e}}}{eE_{L}}$,
and for the Keldysh parameter we obtain:

$\gamma=2\pi\frac{\sqrt{2I_{p}m_{e}}}{eE_{L}T_{L}}.$ (1)

For a given laser frequency, a weaker laser field, $E_{L}$, results in a
longer potential barrier length, $\textrm{L}_{\textrm{barrier}}$, leading to
an increase in the Keldysh parameter, $\gamma$. In the limit of $\gamma\gg 1$
or $E_{L}\ll\sqrt{2I_{p}m_{e}\omega_{L}}/e$, the laser field undergoes
multiple oscillations during the tunneling time, defining the multiphoton
regime. In this regime, the ionization rate, denoted as $W_{i}$, exhibits a
power-law dependence on the electric field, a well-known feature in
perturbative non-linear optics [28]: $W_{i}\propto E_{L}^{2K}$, with
$K=\lfloor I_{p}/\omega_{L}+1\rfloor$ representing the minimum number of
absorbed photons required to conserve energy, known as multiquantumness.
Conversely, in the limit of $\gamma\ll 1$ or
$E_{L}\gg\sqrt{2I_{p}m_{e}}\omega_{L}/e$, the driving laser field experiences
minimal change during the tunneling time, defining the well-known tunnelling
regime with ionization rate for hydrogen atom given in Gaussian units [29] as
$W_{i}=\frac{4m_{e}^{3}e^{9}}{E_{L}\hbar^{7}}\exp[-2E_{a}/(3E_{L})]$, where
$E_{a}={m_{e}}^{2}e^{5}/\hbar^{4}$ \- electrical field experienced by an
electron situated at the Bohr radius $a_{0}=\frac{\hbar^{2}}{m_{e}e^{2}}$ in
the hydrogen atom.

It is essential to emphasize that in our experiment, as illustrated by the
measured profile in Figure 3, the intensity varies across the beam so that at
the central region of the beam, ionization may predominantly occurs in the
tunneling regime, while as we move away from the center, the field strength
gradually decreases, transitioning towards the multiphoton ionization regime.
This intermediate regime is particularly rich and interesting as it contains
signatures of both tunnelling and multiphoton ionization mechanisms. Helium’s
higher ionization potential restricts the dominant tunneling ionization to a
smaller area compared to argon.

Shortly after seminal Keldysh work, Perelomov, Popov and Terentev in their
work [26, 30, 31] addressed the effect of Coulomb interaction on ionization
rate. They obtained Eq. 4 for the optical cycle-averaged ionization rate
$W_{PPT}(E_{L},\omega_{L})$ for the linear polarized driving laser field (from
now on and throughout the rest of the paper, unless otherwise explicitly
stated, we will use atomic units ($\hbar=e=m_{e}=1$).

The refinement of strong field ionization models along with the experimental
effort for its validation is an ongoing active field of research [32, 27, 33,
34, 35, 36, 37, 38, 39, 40, 16, 41]. It has been experimentally shown that
ionization rate predicted by the PPT model is excellent for atoms up to the
Keldysh parameter $\gamma\sim 3-4$ [38, 16] and even valid for small molecules
[39]. Another widely used ionization model is the Ammosov, Delone, Krainov
(ADK) that can be obtained from PPT model in the limit of $\gamma\rightarrow
0$. Main advantage of the ADK is its relative simplicity. However, it has been
shown experimentally that validity of the ADK formula is limited to the
tunneling regime with Keldish parameter $\gamma\leq 0.5$ [42]. In this work,
we employ the PPT model for calculating the cycle averaged ionization rate -
$W_{01}$ Eq. 4 including the BSI correction Eq. 18.

After a neutral atom is ionized by a strong laser, with a probability rate
$W_{01}$, resulting in the formation of a singly charged ion, there exists a
probability that the remaining ion will undergo further ionization in the
laser field, ultimately yielding a doubly charged ion. This phenomenon is
referred to as sequential double ionization. The probability rate, denoted as
$W_{12}$, can be calculated using the same PPT ionization Eq. 4, but with
consideration of the ionization potential, $I_{p}$, for the singly charged
ion, which is higher than that of the neutral atom. Additionally, the residual
charge is equal to $Z=2$. Consequently, these factors lead to a reduced
ionization rate when compared to single-electron ionization.

When an electron gets ejected from the parent ion, there is a probability that
the electron wavepacket, driven by the oscillating laser field in the
continuum, will re-collide with the parent ion. There are four possible
outcomes of this process. Firstly, the re-colliding electron may experience
elastic scattering, or it may contribute to the HHG [1]. These processes do
not contribute to ionization in the medium after the laser pulse is gone. The
recolliding electron wavepacket may also cause either the ionization of the
parent ion, which is referenced as non-sequential double ionization (NSDI),
with a rate of $W_{NSDI}(E(t))$ or ion’s excitation to a higher energy state,
denoted as $1^{*}$, with a rate $W_{NSE}(E(t))$, and its subsequent ionization
(RESI) [43, 44, 45], with a rate $W_{1^{*}2}(E(t))$ . This $W_{1^{*}2}$
ionization rate is calculated again using the PPT ionization model with
ionization potential of singly charged ion in the excited state.

To account for NSDI and the RESI processes in ionization dynamics, we
implement a semiclassical model for calculations of the $W_{NSDI}(E(t))$ and
$W_{NSE}(E(t))$ rates [45]. In the nutshell, we calculate the probability of
ionization - $P(t_{i})$ in the time window $[t_{i}\;t_{i}+dt]$ within an
optical cycle of period - T, $P(t_{i})=W(E(t_{i}))\bar{P}(0\leq t\leq
t_{i})dt=W(E(t_{i}))\exp{[-\int_{0}^{t_{i}}W(E(t^{\prime}))dt^{\prime}]}dt$,
where $W(E(t_{i}))$ is an instantaneous ionization rate Eq. 19 and
$\bar{P}(0\leq t\leq
t_{i})=\exp{[-\int_{0}^{t_{i}}W(E(t^{\prime}))dt^{\prime}]}$ \- is the
probability that no ionization occurred in the $[0\;t_{i}]$ time window. For
each ionization time, we calculate the classical trajectory of electron within
an oscillating field $E(t)=A\cos{(\omega t)}$, the return time - $t_{r}$ and
return energy $E_{k}(t_{r})$. Note, that in semiclassical picture, upon
ionization the electron wavepacket emerges from the tunnel barrier with a
momentum coordinates, $p_{\perp}$, perpendicular to classical trajectory
direction (in the atomic units) [46, 47]:

$\psi(p_{\perp})=\psi(0)\exp{\left(-\frac{p_{\perp}^{2}T_{\textrm{tunnel}}}{2}\right)}$
(2)

This wavepacket spreads as it propagates in the continuum during the
$t_{r}-t_{i}$ time window. Upon recollision we calculate the overlap of the
recolliding wavepacket with the respective energy, $E_{k}(t_{r})$ \- dependent
cross-sections: $\sigma_{NSDI}$ -cross section of electron impact ionization
of ion, and $\sigma_{NSE}$-cross section of electron impact excitation of ion
[48, 49]. We determine the probability of the process of interest by examining
the overlap, as derived in the section ”Non-sequential double ionization” of
the Methods. The rates $W_{NSDI}$ and $W_{NSE}$ are found by normalizing the
respective probabilities by half the period of the driving laser field.

When laser field approaches or exceeds the critical value of
$E_{BSI}=I_{p}^{2}/(4Z)$, the ionization potential $I_{p}$ reaches or
surpasses the potential barrier created by the combination of Coulomb
attraction to the atomic core with charge Z and the instantaneous laser field.
In this so-called barrier-suppression ionization (BSI) region, it is well-
known that strong-field ionization models [31, 27] overestimate the ionization
rate [42, 50]. To account for this discrepancy, we use a correction factor
[51] for ionization rate calculations Eq. 18, as detailed in the section
”Barrier-suppression ionization correction” of the Methods.

Taking into account all the aforementioned processes, we derive the following
set of coupled equations that describe the ionization dynamics:

$\displaystyle\frac{dN_{0}(\vec{r},t)}{dt}$
$\displaystyle=-W_{01}(E(\vec{r},t))N_{0}(t)-W_{NSDI}(E(\vec{r},t))N_{0}(\vec{r},t)-W_{NSE}(E(\vec{r},t))N_{0}(\vec{r},t)$
$\displaystyle\frac{dN_{1}(\vec{r},t)}{dt}$
$\displaystyle=W_{01}(E(\vec{r},t))N_{0}(t)-W_{12}(E(\vec{r},t))N_{1}(\vec{r},t)$
$\displaystyle\frac{dN^{*}_{1}(\vec{r},t)}{dt}$
$\displaystyle=W_{NSE}(E(\vec{r},t))N_{0}(\vec{r},t)-W_{1^{*}2}(E(\vec{r},t))N_{1}^{*}(\vec{r},t)$
$\displaystyle\frac{dN_{2}(\vec{r},t)}{dt}$
$\displaystyle=W_{NSDI}(E(\vec{r},t))N_{0}(t)+W_{12}(E(\vec{r},t))N_{1}(t)+W_{1^{*}2}(\vec{r},t)N_{1}^{*}(\vec{r},t)$
(3)

where $N_{0}(\vec{r},t)$, $N_{1}(\vec{r},t)$, $N^{*}_{1}(\vec{r},t)$, and
$N_{2}(\vec{r},t)$ are densities of neutral, singly ionized, singly ionized in
the excited state and doubly ionized atoms/molecules respectively.

Figure 4:  Normalized Ion density overlaid with the beam intensity
distribution in the interaction volume, shown within the measured and
tomographically reconstructed normalized argon gas density in the pulsed jet
(back pressure 7.5 bar, laser pulse energy 0.5 mJ). Gas jet is visualized as
cut-outs within isosurfaces of constant gas density.

To comprehensively elucidate the ionization dynamics within the focal volume
in both space and time, we solve the system of Eqs. (3) taking into account
measured intensity cross-section of the laser beam in the focus as shown in
Fig. 3, as well as the change of gas jet density along the laser propagation
direction within the jet. The gas density distribution, $N_{0}(\vec{r},t=0)$
within the jet has been measured using the laser induced strong-field
ionization gas jet tomography [24] by scanning the laser beam across the slice
of interest and measuring the ion signal, as depicted in Fig. 4. The spatial
intensity distribution of the laser beam couples into the ionization dynamics
equations via spatial dependence of respective ionization rates,
$W(E(\vec{r},t))$.

Figure 5:  Ionization dynamics in space and time in the focal volume for argon
(a,b,c) and helium (d,e,f,g). The temporal evolution of the spatial ion
distributions is visualized in the Supplementary movie [?file-link?]. This
movie provides further insights into the dynamics observed in the figure.

Figure 5 illustrates the typical ionization dynamics for helium and argon at
the focus of a femtosecond laser pulse. For this analysis, we assume a typical
for a femtosecond laser $f^{2}(t)=\textrm{sech}^{2}(t/\tau)$ temporal
intensity pulse profile [52] , with a FWHM pulse duration of 43 fs and pulse
energy of 2 mJ. The rationale for choosing these pulse parameters will become
evident from the discussion below. Note that for pulse durations in this
range, we can safely assume space-time invariance [53, 54, 55], i.e.
$E(\vec{r}_{\bot},t)=E_{peak}\varepsilon(\vec{r}_{\bot})f(t)$, where
$E_{peak}$ \- electric field peak amplitude,
$\varepsilon(\vec{r}_{\bot})=\sqrt{I_{N}(\vec{r}_{\bot})}$, with
$I_{N}(\vec{r}_{\bot})$ \- measured (as depicted in Fig. 3) normalized focused
beam intensity distribution.

Figure 5 (a) depicts the time evolution of the laser intensity, the density of
the neutral helium atoms, and the density of the singly ionized He atoms at
the center of the driving laser beam. It is noteworthy that due to the very
high ionization potential of helium, double ionization is negligible for the
considered laser pulse, and ionization of neutral atoms with the $W_{01}$ rate
is the dominant process. Consequently, for helium, the sum of neutrals and
singly ionized atoms’ density remains constant. It is evident that ionization
becomes significant before the peak of the laser pulse. In this example,
approximately $20\%$ of the initial helium gas gets ionized, $~{}10$ fs before
the pulse reaches its peak intensity (at this point in time, the intensity
reaches $~{}85\%$ of its peak). After the laser pulse has passed,
approximately $80\%$ of the gas at the beam’s center has been ionized. This
creates a distinct ’hole-burning’ effect in the spatial distribution of
neutral atoms, as shown in Fig. 5 (b), complemented by the generation of
spatially ”molehill”-like localized singly ionized helium atoms, whose
normalized density is visualized in Fig. 5 (c).

Figure 5 (d) presents the time evolution of the ionization process for argon
at the beam’s center, driven by the same laser pulse used in the helium
example. Notably, while helium reached $70\%$ ionization only past the peak
intensity of the laser pulse on its falling slope, argon achieves this same
level by the time the femtosecond pulse reaches just $40\%$ of its peak
intensity on the leading edge of the pulse. This significant difference
clearly arises from argon’s lower ionization potential compared to helium.

These observations underscores the fact that strong-field ionization exhibits
high sensitivity not only to the peak laser intensity but also to the temporal
evolution of the driving pulse, particularly within the time window around the
peak where the majority of the gas undergoes ionization.

Another major consequence of argon’s lower ionization potential compared to
helium is that double ionization plays a significant roles in the ionization
dynamics of argon. This is in striking contrast to helium, where the impact of
these processes on ionization was negligibly small. Initially, as the laser
pulse intensity increases, strong-field single ionization begins to dominate
the ionization dynamics. This phenomenon leads to an increase in the density
$N_{1}(t)$ of singly ionized $Ar^{+1}$ ions, naturally accompanied by a
decrease in the density of neutral atoms ($N_{0}(t)$), as illustrated in
Figure 5 (d). With increasing density of $Ar^{+1}$ ions and laser intensity,
double ionization gain prominence, leading to double ionization of $Ar^{+1}$
ions (depicted as the dotted black curve labeled $Ar^{+2}$ in the figure).

This phenomenon depletes the transient population of singly ionized $Ar^{+1}$
ions at the beam’s center. Notably, the duration of the transient $Ar^{+1}$
population at the beam’s center is only 19 fs (FWHM), significantly shorter
than the 43 fs laser pulse duration. While the high laser intensity at the
center drives double ionization, it weakens with increasing distance further
away from the center, leading to single ionization there as the dominant
process.

This spatial dependence results in the characteristic ”molerun”-like
distribution of $Ar^{+1}$ ion density across the focal plane once the laser
pulse has passed, as illustrated in Figure 5 (f). This phenomenon is
accompanied by a localized ”molehill” peak of $Ar^{+2}$ ions (Figure 5 (g))
and a significantly more depleted ”hole-burning” compared to helium (Figure 5
(e)).

Figure 6:  Measured ion signal vs. laser pulse energy. (a) In helium (b) In
helium normalized (c) In argon (d) In argon normalized.

We now turn to the experimental results of strong field ionization
measurements as a function of laser pulse energy. Figure 6 (a) and (c) shows
the measured ion signal for helium and argon, respectively, at various back
pressures in the pulsed valve, ranging from 1.5 bar to 7.5bar. As expected,
increasing back pressure leads to a higher gas density in the focal volume of
the gas jet, consequently enhancing the ion signal for a given gas.
Additionally, at the same pulse energy, the argon signal surpasses that of
helium due to the difference in their ionization potentials.

To gain deeper insights, we normalized the same measurements as shown in Figs.
6(b) and (d). Surprisingly, our experiment reveals that the normalized
ionization curves for both helium and argon exhibit no discernible dependence
on the gas density within the jet. Notably, apart from a single outlier point
at 4.5 bar for argon, all ionization curves remain strikingly close within the
expected range of random measurement error. We attribute this lone outlier
among the ten measurements in both helium and argon to an anomaly in the
driving laser stability and exclude it from further consideration.

It’s important to note that after the laser pulse is over, a multitude of
recombination processes may occur within the ionized plasma, such as three-
body recombination, radiative recombination, and others [56]. Accompanied by
space charge effects such as Debye shielding [57] and ambipolar diffusion
[56], which disrupts the charge current and separation of ions and electrons
within the laser-induced plasma subjected to the electric field of an ion
detector, these recombination processes can decrease the fraction of ions and
electrons reaching the detector plates and consequently diminish the measured
ion signal.

One might anticipate a pronounced, generally nonlinear influence of gas
density on the measured ion signal due to the interplay of these processes,
potentially altering the ionization curve shapes. However, our experimental
results clearly indicate that it is not the case - the normalized ionization
curves exhibit remarkable resilience and remain independent of gas density in
the jet within the studied range.

Next, considering that the ion signal is proportional to the total ionized
charge as illustrated in Fig. 2 (b), we calculate and fit the theoretically
calculated ion yield, $Y_{ions}$, to the measured normalized ion signals shown
in Figs. 6 (b) and (d) for every back pressure for both helium and argon. The
ion yield is determined by the integral of ion charge density,
$\rho_{ions}(\vec{r})$ (as depicted in Fig. 4), over the focal volume:
$Y_{ions}=\iiint_{V_{focal}}\rho_{ions}(\vec{r})dV$. The ion charge density
after the ionizing laser pulse has passed is given by
$\rho_{ions}(\vec{r})=\left(N_{1}(\vec{r}\right)+N^{*}_{1}(\vec{r}))+2N_{2}(\vec{r})$,
where $N_{1}(\vec{r})$, $N^{*}_{1}(\vec{r})$ and $N_{2}(\vec{r})$ represent
singly and doubly ionized densities calculated at the end of the pulse by
solving the set of Eqs. 3, as has been described.

Note that we measure both - the total energy of the pulse as well as spatial
intensity distribution in the focal volume as shown in Fig. 3. Therefore, for
a specific pulse shape, the only free parameter in the fitting procedure is
the pulse duration, $\tau_{FWHM}$, or equivalently, the peak intensity
$I_{peak}$. More details about the fitting procedure are provided in the
Methods.

Figure 7:  The strong field ionization model fitted to the measured ionization
curves at 7.5 bar back-pressure in (a) helium and (b) argon. For helium, the
extracted pulse duration is $\tau_{FWHM}=(41.7\pm 0.5)$ fs with a goodness of
fit $R^{2}=0.997$. For argon the extracted pulse duration is
$\tau_{FWHM}=(43.3\pm 1.6)$ fs with a goodness of fit $R^{2}=0.989$. Both
intensity and $\gamma$ were calculated using $\tau_{FWHM}$.

Figure 7 (a) shows a typical result of this fitting procedure in helium at a
back-pressure of 7.5 bar. The optimal fit yields a full width at half maximum
pulse duration of $\tau_{FWHM}=(41.7\pm 0.5)$ fs. The upper horizontal axis in
the figure displays the pulse peak intensity corresponding to this pulse
duration.

It’s noteworthy that increasing the ionizing pulse energy generates a plasma
with higher density and larger volume within the gas jet by the time the
driving laser pulse ends. This, in turn, can potentially amplify recombination
and space-charge effects, as previously discussed, and subsequently diminish
the measured ion signal at higher intensities. In the presented version, our
theoretical model does not incorporate these macroscopic recombination or
space-charge phenomena. Therefore, one might anticipate a systematic error
that increases at higher intensities, potentially distorting the shape of the
measured signal and causing a deviation from the theoretically calculated
curve.

However, our results unequivocally demonstrate the opposite - the normalized
measured ionization curves exhibit remarkable agreement with our theoretical
model, characterized by a parametric fit error of $\pm 0.5$ fs ($95\%$
confidence interval) and goodness of fit of $R^{2}=0.997$, even at high pulse
energies and gas back-pressure. In principle, this pulse duration uncertainty
encompass both statistical errors and systematic errors. By repeating the
measurements we find a statistical errors to be within $~{}1$ fs range for
single-pulse energy scan across 150 points spanning the energy range. To
isolate systematic errors arising from theoretical model imperfections, we
analyzed and presented in Figs. 7 data averaged over 100 ion signal
measurements for each pulse energy, effectively reducing statistical error to
0.1 fs.

Only at low pulse energies ($<~{}0.8$ mJ) and correspondingly very low
ionization yield do we measure ionization yield in helium higher than
predicted by the theory. This is attributed to ionization of the residual gas
impurities in helium, though not visible on the scale of the plot and having
negligible effect on the fitting, as these parasitic contributions are very
low compared to the dynamic range of the measured ion signal. Given the high
ionization potential of helium, at low intensities, residual gases such as
nitrogen, oxygen, and water vapor mainly contribute to the ionization signal,
surpassing the minor contribution from ionization of helium itself.

Figure 7(b) depicts a representative result of the analogous fitting procedure
in argon at a the same back-pressure of 7.5 bar. The best fit results in a
pulse duration of $\tau_{FWHM}=(43.3\pm 1.6)$ fs. In contrast to helium, where
single-electron ionization dominates across the measured intensity range due
to its high ionization potential, double ionization phenomena comes into play
in argon. Our calculations show that SDI plays a major role, followed by NSDI,
while RESI contribution was negligible in the relevant intensity range.
Additionally, BSI has been found to play significant role as well. This
complex interplay also explains why, while the fitting results for argon
remain good with a pulse duration fit error of $\pm 1.6$ fs and $R^{2}=0.989$,
they fall slightly short of the exceptional goodness of fit achieved for
helium. Remarkably, despite this intricate interplay of ionization processes
in the model, we obtained a pulse duration in argon that closely matches that
measured in helium.

It is crucial to include these strong field ionization phenomena in the
computations; neglecting them leads to an inconsistent pulse duration in argon
compared to the helium result, indicating a deficiency in the comprehensive
ionization model. For instance, not accounting for sequential double
ionization in argon results in a pulse duration of 74 fs, a stark $>70\%$
deviation from the helium ($~{}42$ fs) or well-modeled argon ($~{}43$ fs)
result. Omitting BSI only in argon leads to 58 fs, still very significant 38%
deviation.

At the same time, we validated that neglecting SDI (or any double ionization
process) in helium ionization calculations made, as expected, no difference to
the pulse duration result. Note that neglecting NSDI doubles the systematic
error in argon compared to helium, but this effect is significantly less
important than the SDI contribution. This is because the contribution of
$W_{12}$ dominates over $W_{NSDI}$ in argon already at the peak intensity of
$5\times 10^{14}$ $\textrm{W}/\textrm{cm}^{2}$.

Figure 8:  Pulse duration $\tau_{FWHM}$ extracted by fitting the theoretical
model to the measured ionization curves for helium (red crosses) and argon
(blue squares). The error bars indicate a parametric fit error ($95\%$
confidence interval).

Figure 8 summarizes the pulse durations extracted for both helium and argon
across various back pressures. Notably, the average pulse duration across all
gases and pressures is $\tau_{FWHM}=43.1\pm 1.6$ fs.

As each data point in the plot is shown after averaging reducing statistical
error to 0.1 fs, the error bars indicate the pulse duration parametric fit
error. The average value for pulse duration for the measurements in He is
$42.3\pm 1.5$ fs and in argon $44\pm 1$ fs. Thus we find that each of the
systematic errors due to imperfections in theoretical model, the variations in
the back pressure and usage of different gases doesn’t exceed 2 fs or is less
than $5\%$ of measured pulse duration. The main source of statistical error in
our set-up was caused by beam pointing stability issues - this can be further
improved, for instance, by implementing active beam stabilization techniques
[58]. However, for most practical purposes, the demonstrated statistical error
(approximately 1 fs), even without averaging, is already satisfactory and
comparable to the systematic errors.

The pulse duration at the output of our regenerative amplifier is 30 fs;
however, in situ measurements using our scheme reveal a pulse length $~{}40\%$
longer. Since peak intensity is inversely proportional to pulse duration,
relying solely on the output pulse duration would result in a significant
error in estimating the peak intensity within the interaction region. This
once again underscores the importance of in situ measurements for both pulse
duration and peak intensity, especially in strong field and attosecond science
experiments.

To assess the impact of temporal pulse shape on our results, we considered
both the typical $\textrm{sech}^{2}$ and Gaussian profiles used to
characterize femtosecond laser outputs. Additionally, we modeled the influence
of dispersion-induced pulse distortions, by propagating 30 fs pulse through 50
mm fused silica. Notably, the estimated laser peak intensity remained
remarkably robust, showing a maximum variation of only 3% across different
pulse shapes and distortions.

It is important to stress that the measured ion signal depends not only on the
peak intensity, but also on the pulse duration. As the strong field ionization
is highly nonlinear process, the shape around the peak of the pulse becomes
particularly important. Interestingly, for $\textrm{sech}^{2}$ , Gaussian, and
dispersion-distorted pulses with the same FWHM, the pulse intensity varies by
no more than $2\%$ of its peak within the upper half of the intensity profile
- where the most ions signal is generated. However, the main variation in
intensity between these pulses occurs around their off-peak regions, where
intensity is low and impact on the strong field ionization is minor. This
further underscores an important advantage of our technique for strong-field
ionization experiments: it is inherently sensitive to the pulse characteristic
most crucial for the process, namely the intensity profile around the peak,
where the majority of strong field ionization phenomena take place.

##  III Conclusion

In conclusion, the accuracy of strong-field experiments, particularly in
attosecond science, should no longer be constrained by uncertainties in the
femtosecond laser beam characteristics within the interaction region.
Spatially resolved, in situ intensity measurements of the driving laser,
combined with the measurement of spectrally resolved HHG wavefronts [20], pave
the way for resolving a long-standing issue in attosecond science experiments:
the spatial averaged measurements across the focal volume [59]. This approach
holds the potential to significantly enhance experimental precision, for
instance, in imaging orbitals [18, 60], tracing molecular dynamics [61],
identifying and time resolving tunneling wave packets [62], following Auger
decay [63], and probing polar molecules with high harmonic spectroscopy [2]\-
enabling access to a single-molecule response in a strong laser field [59].

Helium’s exceptionally high $I_{p}$ allows the single-electron PPT model to
adequately describe its ionization within the relevant intensity range.
However, most other molecules and even noble gas atoms possess significantly
lower $I_{p}$ \- similar or even lower than argon. To comprehensively
elucidate the ionization dynamics of such systems, accounting for double
ionization and BSI becomes crucial. Frequently, ion measurements have been
interpreted solely through single-electron strong-field ionization models,
considering only peak intensity ionization rates; our findings clearly
illustrate the inadequacy of this approach.

Furthermore, the implications of our findings extend beyond improving the
fidelity of strong-field and attosecond science experiments. The observed
robustness to pressure variations and remarkable agreement between our
measurements and the single-atom ionization model for helium suggest that ion
detection approach can serve as a valuable experimental tool for validating
and refining theoretical models of strong-field ionization for more complex
atoms and molecules.

##  IV Methods

###  IV.1 Strong-field ionization rate PPT

In this section, we explicitly summarize the formulas for the optical cycled
averaged ionization rate, employing the Perelomov, Popov, Terentev (PPT)
formulas [26], which are used in this work with certain corrections:

$W_{mPPT}(E_{L},\omega_{L})=C_{n^{*}l^{*}}^{2}f_{lm}I_{p}\sqrt{\frac{6}{\pi}}\left(\frac{2E_{0}}{E_{L}}\right)^{2n^{*}-|m|-3/2}\\\
\times(1+\gamma^{2})^{|m|/2+3/4}A_{m}(\omega_{L},\gamma)\\\
\times\exp{\left(-\frac{2E_{0}}{3E_{L}}g(\gamma)\right)},$ (4)

where:

$E_{L}$ \- laser field

$\omega_{L}$ \- laser cycle frequency

$n^{*}$ \- effective principal quantum number:

$n^{*}=\frac{Z}{\sqrt{2I_{p}}},$ (5)

$l^{*}$ \- effective orbital quantum number:

$l^{*}=n^{*}-1,$ (6)

$E_{0}$ \- the characteristic atomic electric field:

$E_{0}=\left(2I_{p}\right)^{\frac{3}{2}},$ (7)

$C_{n^{*}l^{*}}^{2}=\frac{2^{2n^{*}}}{n^{*}\Gamma(n^{*}+l^{*}+1)\Gamma(n^{*}-l^{*})},$
(8)

$f_{lm}=\frac{(2l+1)(l+|m|)!}{2^{|m|}|m|!(l-|m|)!},$ (9)

$A_{m}(\omega_{L},\gamma)=\frac{4}{\sqrt{3\pi}|m|!}\frac{\gamma^{2}}{1+\gamma^{2}}\sum_{k\geq
k_{min}}^{+\infty}e^{-\alpha(\gamma)(k-k_{min})}\xi_{m}\left(\sqrt{\beta(\gamma)(k-k_{min})}\right)$
(10)

$\alpha(\gamma)=2\left[\textrm{arcsinh}(\gamma)-\frac{\gamma}{\sqrt{1+\gamma^{2}}}\right],$
(11)

$\beta(\gamma)=\frac{2\gamma}{\sqrt{1+\gamma^{2}}},$ (12)

$\xi_{m}(x)=\frac{x^{2|m|+1}}{2}\int^{1}_{0}\frac{e^{-x^{2}t}t^{|m|}}{\sqrt{1-t}}\,dt,$
(13)

$g(\gamma)=\frac{3}{2\gamma}\left[\left(1+\frac{1}{2\gamma^{2}}\right)\textrm{arcsinh}(\gamma)-\frac{\sqrt{1+\gamma^{2}}}{2\gamma}\right],$
(14)

where k - is the number of Above threshold ionization photons (ATI) [22],
starting with the minimum number of ATI photons, $k_{min}$ (this formula for
clearity is written in the SI units):

$k_{min}=\lceil{\frac{I_{p}+U_{p}}{\hbar\omega_{L}}}\rceil=\lceil{\frac{I_{p}}{\hbar\omega}\left({1+\frac{1}{2\gamma^{2}}}\right)}\rceil;U_{p}=\frac{e^{2}E_{L}^{2}}{4m_{e}\omega_{L}^{2}},$
(15)

Here, $U_{p}$ \- is a ponderomotive energy, which is the cycle-averaged
kinetic energy of the electron motion in the field. Note, that $k_{min}$ is
rounded up to the nearest integer here, whereas in the original PPT paper
[26], $k_{min}$ was not rounded, which leads to unphysical results.

The simplified cycled averaged ADK formula for the ionization rate,
$W_{ADK}(E_{L},\omega_{L})$, is often used [27] (can be obtained from
$W_{ADK}(E_{L},\omega_{L})$ in the limit $\gamma\rightarrow 0$ as shown in
[26] Eq. 59):

$W_{mADK}(E_{L},\omega_{L})=C_{n^{*}l^{*}}^{2}f_{lm}I_{p}\sqrt{\frac{6}{\pi}}\left(\frac{2E_{0}}{E_{L}}\right)^{2n^{*}-|m|-3/2}\times\exp{\left(-\frac{2E_{0}}{3E_{L}}\right)},$
(16)

We explicitly emphasize the use of the full $W_{PPT}$ cycle averaged
ionization rate Eq. 4, omitting the ADK approximation, for single electron
ionization in this work.

For argon gas with an orbital angular momentum quantum number ($l=1$), we
calculated the ionization rate by averaging over all possible values of its
magnetic quantum number ($m$), representing the projection of the angular
momentum along the direction of the applied field. This methodology follows
the approach outlined by Larochelle et al. in their investigation of the
Coulomb effect in multiphoton ionization of rare-gas atoms [38] and is
consistent with the averaging procedure detailed in equation (61) of PPT model
[26].

$W_{PPT}(E_{L},\omega_{L})=\frac{1}{2l+1}\sum_{m=-l}^{l}W_{mPPT}(E_{L},\omega_{L})$
(17)

###  IV.2 Barrier-suppression ionization correction

The BSI correction to the ionization rate, described in the Eq. 18 [51], has
been implemented. The relevant parameter values $a_{1},a_{2},a_{3}$ are
provided in Table 1. It has been demonstrated that this modified ionization
rate formula aligns well with $E_{L}\leq 4.5E_{BSI}$, falling within the range
the peak laser field used in this work.

$W(E_{L},\omega_{L})=W_{PPT}(E_{L},\omega_{L})\exp\left(a_{1}\frac{E_{L}^{2}}{E_{BSI}^{2}}+a_{2}\frac{E_{L}}{E_{BSI}}+a_{3}\right)$
(18)

(Note, that we corrected a typo in the the original paper [51] , where a minus
sign appears in the Eq. (8).)

Table 1: Relevant values of parameters for BSI correction for our experiment [51]. Parameter |  | He |  | Ar |  | $\mathrm{Ar^{+}}$
---|---|---|---|---|---|---
$a_{1}$ |  | 0.13550 |  | 0.16178 |  | 0.30441
$a_{2}$ |  | -0.86210 |  | -1.50441 |  | -2.70461
$a_{3}$ |  | 0.021562 |  | 0.32127 |  | 1.40821

Note that the BSI formula we use is a further development of the correction
proposed by Tong et al. [64]. It has been noted that their original correction
for the tunneling ionization fits well for $E_{L}<2.5E_{BSI}$. However, for
larger values of $E_{L}$, the ionization rate decreases for argon, which is
inconsistent with the expected physical behavior.

###  IV.3 Non-sequential double ionization

Non-Sequential Double Ionization (NSDI) [1, 65] is a process in which a singly
charged ion undergoes secondary ionization due to recollision with the
electron released earlier in the same laser cycle.

The initial step of non-sequential double ionization (NSDI) involves the
ionization of an atom within the laser cycle. Consequently, the instantaneous
ionization rate [26] is used:

$W(E(t))=I_{p}C_{n^{*}l^{*}}^{2}f_{lm}\left(\frac{2E_{0}}{E(t)}\right)^{2n^{*}-|m|-1}\exp\left\\{-\frac{2E_{0}}{3E(t)}\right\\},$
(19)

where time dependency comes from the field, $E(t)=A\cos(\omega t)$, where $A$
is the amplitude of the specific cycle under the slowly varying envelope
approximation.

As discussed in the Main section of the manuscript, we calculate the
probability of ionization - $P(t_{i})$ in the time window $[t_{i}\;t_{i}+dt]$
within an optical cycle of period - T.

Electrons that do return to their parent ion possess a probability of either
ionization or excitation, determined by the electron’s kinetic energy and the
respective energy dependent cross-section for ionization/excitation by
collision ($\sigma_{NSDI/NSE}$) [48, 49]. This probability is computed as an
overlap between the electron’s wavepacket spread in the laser field and the
corresponding cross-section. The calculations are outlined as follows:

$\displaystyle P_{c}^{NSDI/NSE}(t_{i})$
$\displaystyle=2\pi\int_{0}^{R_{c}^{NSDI/NSE}(E_{k}(t_{i}))}|\Psi(r_{\perp})|^{2}r_{\perp}dr_{\perp}$
$\displaystyle=2\pi
N\int_{0}^{R_{c}^{NSDI/NSE}(E_{k}(t_{i}))}\exp\left(-\frac{r_{\perp}^{2}}{2\Delta
r_{\perp}^{2}(t_{i})}\right)r_{\perp}dr_{\perp}$
$\displaystyle=1-\exp\left(-\frac{\left(R_{c}^{NSDI/NSE}(E_{k}(t_{i}))\right)^{2}}{2\Delta
r_{\perp}^{2}(t_{i})}\right)$ (20)

Where,

$N$ \- normalizing factor,

$N=\frac{1}{2\pi\Delta r_{\perp}^{2}}.$ (21)

$\Delta r_{\perp}$ \- the electron wavepacket spread,

$\Delta r_{\perp}(t_{i})=\Delta
p_{\perp}(t_{r}-t_{i})=\frac{1}{\sqrt{2T_{tunnel}}}(t_{r}-t_{i})=\frac{\sqrt{E(t_{i})}}{(8I_{p})^{1/4}}(t_{r}-t_{i}).$
(22)

$R_{c}$ \- collision radius,

$R_{c}^{NSDI/NSE}=\sqrt{\frac{\sigma_{NSDI/NSE}(E_{k}(t_{i}))}{\pi}}.$ (23)

Finally, the NSDI rate and NSE rate are,

$W_{NSDI/NSE}=\frac{\sum_{t_{i}}P(t_{i})P_{c}^{NSDI/NSE}(t_{i})}{\frac{T}{2}}.$
(24)

If the electron, following its collision with the parent ion, excites rather
than ionizes the ion, the ion itself may undergo ionization from the laser
field while in its excited state. This phenomenon is termed recollision-
induced excitation with subsequent ionization (RESI) [43, 44, 45]. From the
ion’s excited state, its ionization rate is calculated by Eq. 18, with the
proper values for the ion’s $I_{p}$ and $Z=2$.

###  IV.4 Procedure for pulse duration extraction by fitting to the ionization
curve

In our experiment, we measure the laser’s pulse energy (PE), the normalized
spatial intensity distribution in the interaction region $(I_{N}(r_{\perp})$
as illustrated in Fig. 3, and the ion signal as a function of PE (ionization
curve).

The PE is related to the laser field amplitude $E(\vec{r}_{\bot},t)$ by:

$E(\vec{r}_{\bot},t)=E_{peak}\sqrt{I_{N}(\vec{r}_{\bot})}f(t)$ (25)

where $E_{P}eak$ denotes the laser’s pulse peak field. The pulse peak
intensity ($I_{peak}$) can be expressed as:

$I_{peak}=c\varepsilon_{0}E^{2}_{peak}/2$ (26)

We then relate the peak intensity ($I_{peak}$) to the pulse energy (PE) using
the equation:

$I_{peak}=\frac{\mathrm{PE}}{\iint_{0}^{\infty}I_{N}(\vec{r}_{\perp})d^{2}\vec{r}_{\perp}\int_{-\infty}^{\infty}f^{2}(t)dt}.$
(27)

Here, $c$ \- speed of light in the vacuum, $\varepsilon_{0}$ \- vacuum
permittivity. Note that
$\int_{-\infty}^{\infty}f^{2}(t)dt=C_{S}\times\tau_{FWHM}$, where $C_{S}$ is a
constant depending on the normalized temporal pulse envelope $f(t)$ (e.g. for
$f(t)=\text{sech}\left(2\textrm{arcsech}(\sqrt{0.5})t/\tau_{FWHM}\right)$,
$C_{S}=1/{\textrm{arcosh}\left(\sqrt{2}\right)}$. Therefore, the pulse
duration $\tau_{FWHM}$ is the only free fitting parameter.

For each $\tau_{FWHM}$, we solve the set of ionization dynamics rate equations
and calculate the total ion signal $Y_{ions}$ (see details in the main paper).
We use the Matlab Curve Fitting Toolbox to fit the ionization curve to the
calculated $Y_{ions}$ and determine the optimal $\tau_{FWHM}$, the fitting
error, and the goodness of fit $R^{2}$. Subsequently, we use Eq. 27 to extract
the $I_{peak}$.

###  IV.5 Acknowledgements

Acknowledgements: E.F. acknowledges the support of Israel Science Foundation
(ISF) grant 2855/21. N.S. acknowledges financial support from the Kreitman Hi-
tech student fellowship.

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