# Tunneling ionization in ultrashort laser pulses: edge-effect and remedy

M. Klaiber<EMAIL_ADDRESS>Q. Z. Lv<EMAIL_ADDRESS>K. Z.
Hatsagortsyan  C. H. Keitel  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

Tunneling ionization of an atom in ultrashort laser pulses is considered. When
the driving laser pulse is switched-on and -off with a steep slope, the
photoelectron momentum distribution (PMD) shows an edge-effect because of the
photoelectron diffraction by the time-slit of the pulse. The trivial
diffraction pattern of the edge effect consisting of fast oscillations in the
PMD disguises in the deep nonadiabatic regime the physically more interesting
features in the spectrum which originate from the photoelectron dynamics. We
point out the precise conditions how to avoid this scenario experimentally and
if unavoidable in theory we put forward an efficient method to remove the
edge-effect in the PMD. This allows to highlight the nonadiabatic dynamical
features of the PMD, which is indispensable for their further investigation in
complex computationally demanding scenarios. The method is firstly
demonstrated on a one-dimensional problem, and further applied in three-
dimensions for the attoclock. The method is validated by a comparison of
analytical results via the strong-field approximation with numerical solutions
of the time-dependent Schrödinger equation.

##  I Introduction

Modern state-of-the-art laser techniques allow for full control over the wave
form of a laser pulse, and in particular, the generation of few-cycle strong
laser pulses [1, 2, 3, 4]. Such few-cycle laser pulses of sufficient strength
are an efficient tool in attoscience [5, 6, 7]. They have been employed for
the generation of isolated attosecond pulses via high-order harmonic
generation (HHG) [8, 9, 10, 11], for molecular imaging and laser induced
electron diffraction [12, 13, 14, 15], as well as for the time-resolved study
of strong-field phenomena, such as nonsequential double ionization [16, 17,
18, 19, 20, 21, 22, 23] and dissociative ionization [24, 25]. The theoretical
description of strong-field phenomena in few-cycle pulses within the strong
field approximation (SFA) is outlined in Ref. [26].

In ultrashort laser pulses an abrupt switch-on and -off of the laser pulse can
induce a diffraction effect of the photoelectrons by the time slit of the
pulse due to the pulse edges, the so-called edge-effect. The edge-effect is
exhibited as oscillations in the photoelectron momentum distribution (PMD),
additionally to the dynamical features of PMD, and it disappears in the case
of a smooth laser pulse. The edge-effect distorts the most important dynamical
physical signal in strong-field ionization and for this reason one tries to
avoid or separate it. The distortion is especially conspicuous at low laser
intensities when the ionization signal is weak, but just in this deeply
nonadiabatic regime the dynamical features of PMD are nontrivial. We underline
that there are observed unexplained features in PMD in elliptically polarized
laser fields in the weak field regime [27, 28], and the edge-effect hinders
their analysis.

In an experiment the role of edge-effects could be diminished using more and
more smoother laser pulses. In a theoretical description via numerical
solution of time-dependent Schrödinger equation (TDSE), as well as within SFA,
different forms of laser pulses with a smooth switch-on and -off are employed.
The most simple description of a short $N$-cycle laser pulse is via a
$\cos^{2}$-envelope: $f(t)=\cos^{2}(\omega t/N)$, with the laser frequency
$\omega$, see e.g. [26, 29]. Smoother pulses are obtained via
$\cos^{n}$-envelopes with $n=4$ or larger (in this case one needs to take into
account the change of the effective frequency of the laser field). A better
description is obtained with the use of a Gaussian pulse with a long tail
[30], which however requires rather time consuming computationally expensive
calculations.

In this paper we put forward a simple method to separate the edge-effect and
single-out the PMD dynamical signal in SFA calculations as well as in the
numerical solution of TDSE, while using laser pulses with no-smooth switching.
The method (U-contour method) mimics the saddle-point integration, however,
without explicit finding and classification of all relevant saddle-points for
the given PMD. We demonstrate the method in a one-dimensional (1D) model of
tunneling ionization in half-cycle pulses of $\cos^{2}$ and truncated-Gaussian
form, and confirm its accuracy in comparison with the numerical TDSE solution.
Finally, we apply the method in a 3D example of the attoclock. The U-contour
method in 3D has a clear advantage with respect to the saddle-point
integration, as the latter would require the calculation of a large data set
of saddle-points.

The structure of the paper is the following. In Sec. II the SFA model is
introduced,and the edge-effect is described. The conditions for the appearance
of the effect are discussed in Sec. III. The U-contour method for separation
of the edge-effect is introduced in Sec. IV, and its performance is tested in
comparison with numerical solutions of TDSE. The application of the U-contour
method for the analysis of the edge-effect in 3D case of attoclock is
presented in Sec. V, and the conclusion is given in Sec. VI.

Figure 1: PMD with a field $E(t)=-E_{0}\cos^{2}(\omega t)$: (a) $E_{0}=0.1$,
(b) $E_{0}=0.05$, (c) $E_{0}=0.025$, $\omega=0.05$ a.u., $\kappa=1$ a.u., the
field is truncated at $\omega t_{i}=-\pi/2$ and $\omega t_{f}=\pi/2$, (orange,
dashed) via SFA with the edge-effect, (blue, solid) via SFA with the edge-
effect subtracted, (dotted, green) via numerical solution of TDSE.

Figure 2: PMD with a field $E(t)=-E_{0}\exp[-(\omega t)^{2}]$: (left column)
via first-order SFA Eq. (2); (right column) via numerical solution of TDSE;
(a,c) $E_{0}=0.1$, (b,d)) $E_{0}=0.05$; (dashed, orange) short pulse with the
truncation points of the Gaussian at $\omega t_{i}=-2$ and $\omega t_{f}=2$,
(dotted, green) long pulse with $\omega t_{i}=-8$ and $\omega t_{f}=8$, (blue,
solid) the edge-effect subtracted, (red, dot-dashed) in (b) is SFA calculation
with SPA; $\omega=0.05$ a.u., $\kappa=1$ a.u.

Figure 3: PMD with a field $E(t)=-E_{0}\exp[-(\omega t)^{2}]$, $E_{0}=0.025$:
(a) via first-order SFA, (b) via numerical solution of TDSE; (dashed, orange)
short pulse with the truncation points of the Gaussian at $\omega t_{i}=-2$
and $\omega t_{f}=2$, (dotted, green) long pulse with $\omega t_{i}=-8$ and
$\omega t_{f}=8$, (blue, solid) SFA with the edge-effect subtracted, (dot-
dashed, red) via Patchkowsii’s truncated Gaussian [30] with parameters $\omega
t_{1}=2$, and $\omega t_{2}=2.5$; $\omega=0.05$ a.u., $\kappa=1$ a.u. Figure
4: Determination of the threshold field strength in $\cos^{n}$-pulses, below
which the edge-effect contaminates the strong-field ionization PMD. The edge-
effect is induced by the high-energy component of the field with
$\Omega\gtrsim I_{p}$, the probability of photoionization $W_{\Omega}$ via
such an one high-energy photon absorption: (blue) in $\cos^{2}$-pulse,
(orange) in $\cos^{4}$-pulse, (green) in $\cos^{6}$-pulse; (red) the PPT-
probability $W_{SFI}$. The edge-effect will be visible below the field value
when $W_{\Omega}=W_{SFI}$. Figure 5:  Contours of the time-integration in Eqs.
(5) and (17): (red, horizontal) the original contour along the real time-axis
from the pulse outset $t_{i}$ to the end $t_{f}$, (green, dashed) the saddle-
point contour, (red, solid) the proposed U-contour to remove the pulse edge-
effect. The laser pulse form is illustrated via the blue, dashed line.  Figure
6:  The amplitude of the integrand along the vertical contour, exactly (blue,
solid), and approximated by the expansion of Eq. (10) (orange, dashed).

##  II Edge-effect and theoretical description

We consider ionization of an electron bound in an atomic potential $V(r)$, in
a laser pulse with electric field $\mathbf{E}(t)$. The asymptotic momentum
distribution,

$w(p)=|m(p)|^{2},$ (1)

is determined by the SFA direct ionization amplitude [31]:

$\displaystyle m(p)=-i\int dt\langle\psi_{p}^{V}(t)|H_{i}(t)|\phi(t)\rangle,$
(2)

where $\phi(\mathbf{r},t)$ is the bound state wave function,
$\psi^{V}_{p}(\mathbf{r},t)$ the Volkov wave function [32],
$H_{i}(t)=\mathbf{r}\cdot\mathbf{E}(t)$ the electron interaction Hamiltonian
with the laser field. Atomic units are used throughout. The integrals in the
amplitudes of Eqs. (2) are calculated in two ways, fully numerically and with
the saddle-point approximation (SPA) analytically.

We illustrate the edge-effect on a 1D problem of ionization of an electron
bound in a 1D zero-range potential $V(x)=-\kappa\delta(x)$, in a half-cycle
laser pulse with electric field $E(t)=-E_{0}\cos^{2}(\omega t)$, with the
field vanishing at $t<t_{i}$ and $t>t_{f}$. Here $\omega=0.05$ a.u.,
$\kappa=\sqrt{2I_{p}}=1$ a.u., $I_{p}$ is the ionization potential,
$\gamma=\tilde{\omega}\kappa/E_{0}$ the Keldysh parameter, with the effective
frequency $\tilde{\omega}\equiv\sqrt{2}\omega$ related to the $\cos^{2}$-pulse
(the effective frequency is defined as
$\tilde{\omega}=\sqrt{-E^{\prime\prime}(0)/E(0)}$ at the field maximum $t=0$).
We calculate PMD for different laser fields, using SFA amplitude of Eq. (2),
with the bound state wave function
$\phi(x,t)=\sqrt{\kappa}\exp(-\kappa|x|+i\kappa^{2}/2t)$. The results are
presented in Figs. 1-3.

In strong fields, the PMD is a smooth function of the asymptotic momentum, see
the case of $E_{0}=0.1$ a.u. for $\cos^{2}$-pulse in Fig. 1(a), and for
truncated-Gaussian pulses in Fig. 2(a,c). In contrast, at weak fields PMD
appears to be superimposed by the diffraction pattern due to the time-slit of
the pulse edges, see $E_{0}=0.05$ a.u. and $E_{0}=0.025$ a.u. in Fig. 1(b) and
(c) for $\cos^{2}$-pulse, and Fig. 2(b,d) for Gaussian pulses, respectively
(orange-dashed lines in figures correspond to SFA, and green-dotted lines to
the TDSE numerical solutions). This effect is large in weak fields, when the
ionization dynamical signal is weak, and strongly dependent on the pulse
shape. In fact, in a Gaussian pulse $E(t)=-E_{0}\exp[-(\omega t)^{2}]$ of the
same effective frequency ($\omega=0.05$) as in $\cos^{2}$ one, the edge-effect
gradually decreases with increasing of the Gaussian truncation. In particular,
the edge-effect vanishes, i.e. oscillations in PMD disappear, if a rather
large truncation time is applied, see the green-dotted lines in Figs. 2(b,d),
and 3 corresponding to $\omega(t_{f}-t_{i})=16$. However, the edge-effect
persists at smaller truncation time at the same field strength and the same
frequency, see the dashed lines in Fig. 2(b,d) corresponding to
$\omega(t_{f}-t_{i})=4$. While the use of Patchkovskii’s smooth-truncated-
Gaussian pulse of Ref. [30] decreases the edge-effect, see red-dot-dashed line
in Fig. 3, however, at weak fields the boundary terms still contribute and
contaminate the physical PMD.

In Figs. 1-3, we provide PMD via SFA, as well as via the numerical solution of
TDSE. For strong fields $E_{0}=0.1$ and $E_{0}=0.05$, the SFA results are in
close accordance with the numerical ones in any pulse. Deviations mainly
originate from the Stark-shift that is not accounted in SFA, yielding slightly
overestimated ionization probabilities. In weak fields, $E_{0}=0.025$ the
results are still in accordance with long Gaussian pulses, but in short
truncated-Gaussian pulses the edge effects are different in SFA and in TDSE,
concurring only qualitatively.

##  III Conditions for the edge-effect appearance

Generally, the smoother the switching-on and -off of the laser pulse, the less
pronounced are the edge-effects. However, for a given smooth pulse shape there
is a threshold intensity, below which the edge-effects again show up. This is
illustrated for $\cos^{n}$-type pulses in Fig. 4. We estimate the condition
for the edge-effect appearance as follows. The switching-on/-off of the laser
pulse results in appearance of high-energy components in the field spectrum.
The edge-effect is induced by the high-energy component of the field with
$\Omega\gtrsim I_{p}$, available in the spectrum of the pulse. We characterize
the edge-effect by the probability of photoionization via absorption of such a
high-energy photon [33]:

$W_{\Omega}\sim\left(\frac{E_{\Omega}a}{\Omega}\right)^{2},$ (3)

with the field strength $E_{\Omega}\sim E_{0}(\omega/\Omega)^{n}$ of the high-
frequency component $\Omega$, and the typical atomic length $a=1/\kappa$. The
edge-effect will be visible if this probability is comparable with (or larger
than) the strong-field ionization probability [34] due to the monochromatic
field of the effective frequency of the pulse:

$W_{SFI}\sim\frac{E_{a}}{E_{s}}\exp\left\\{-\frac{2I_{p}}{\tilde{\omega}}\left[\left(1+\frac{1}{2\gamma^{2}}\right)\operatorname{arcsinh}\gamma-\frac{\sqrt{1+\gamma^{2}}}{2\gamma}\right]\right\\}.$
(4)

where $E_{s}=E_{0}\sqrt{1+\gamma^{2}}$ is the field value at the time saddle-
point. Thus, the condition of the onset of the edge-effect is
$W_{\Omega}\gtrsim W_{SFI}$. In Fig. 4 $W_{\Omega}=W_{SFI}$ corresponds to the
crossing point of the red line representing $W_{SFI}$ with the corresponding
one-photon probabilities $W_{\Omega}$ for different pulses. Thus, the edge-
effect will be visible in the corresponding pulses with the field strength
below the crossing points. The smoother the laser pulse, the smaller will be
the laser intensity below which the edge-effect will emerge. The edge-effect
appears at low laser intensities, when the tunneling ionization signal is weak
and becomes comparable with the diffraction signal. This effect hinders the
understanding of nonadiabatic tunneling at large Keldysh parameters , as it
conceals specific nonadiabatic features in PMD.

##  IV Separation of the edge-effect

In this section we put forward a method for separation of the edge effect and
singling-out the dynamical features of PMD at given laser parameters. In the
total PMD with the edge-effect, the dynamical signal is superimposed by the
trivial diffraction pattern due to the time-slit of the pulse. Meanwhile, the
dynamical signal is most interesting physically because it provides
information on the nonadiabatic dynamics of the photoelectron in a weak field
regime. As an example we refer to structures inside the attoclock ring in a
weak elliptically polarized laser field [28], which also could be related to
the unexplained large attoclock offset angles in the multiphoton regime [27].

###  IV.1 U-contour method

Before introducing the method for separation of the edge-effects, let us to
note that the edge-effect can be avoided in the calculation of PMD within SFA,
when using SPA for the time-integration, e.g. see red-dashed line in Fig. 2(b)
[35]. However, in 3D cases and for a large range of PMD, e.g. in the
attoclock, see Sec. V, it is a cumbersome procedure to find all saddle-points
of the full PMD. Moreover, there still remains the question how to remove the
edge-effect for TDSE. In the latter the only possibility is to use a Gaussian
pulse with a very large truncation time, which requires extensive
computational resources.

Here we propose a simple method for the calculation of the edge-effect-free
PMD. The method mimics the saddle-point time-integration method for the
ionization amplitude. In the first-order SFA the integrand $m(t)$ of the
ionization amplitude

$m(p)=\int_{t_{i}}^{t_{f}}dt\,m(t),$ (5)

has the form

$\displaystyle m(t)={\cal
C}\frac{(\mathbf{p}+\mathbf{A}(t))\cdot\mathbf{E}(t)}{\left[\left(\mathbf{p}+\mathbf{A}(t)\right)^{2}+\kappa^{2}\right]^{2}}\exp[-iS(t)+i\kappa^{2}/2t],$
(6)

with the classical action in the laser field
$S(t)=\int^{t_{f}}_{t}ds(\mathbf{p}+\mathbf{A}(s))^{2}/2$, and the constant
${\cal C}_{1D}=-2i\sqrt{\frac{2}{\pi}}\kappa^{3/2}$ for the 1D case, and
${\cal C}_{3D}=1/(\sqrt{2\pi}\kappa){\cal C}_{1D}$ for the 3D case. In the
original Eq. (5) the time-integration runs along the real time-axis from the
onset of the pulse $t_{i}$ to the end $t_{f}$, see Fig. 5. When one applies
SPA, the original contour is deformed to the steepest-descent contour of the
saddle-point. Generally, one has to find all saddle-points corresponding to
the given asymptotic momentum via an appropriate deformation of the initial
contour of the time-integration. As is shown in Fig. 5, the integral along the
steepest-descent contour equals to that along the U-contour (red in Fig. 5).
Thus, the edge-free PMD can be obtained adding two integrals along the
vertical contour to the main integral along the real axis (from $t_{i}$ to
$t_{f}$). This can be done analytically within SFA, as well as numerically in
TDSE solution.

Further, we note that the U-contour can also be used to calculate half-cycle
resolved ionization probabilities in long sinusoidal fields via truncating the
field at the beginning and the end of the half-cycle of interest.

###  IV.2 Calculation of the time-integral along the vertical contour

Figure 7: PMD in the attoclock via first-order SFA: (a) PMD with the edge
effect included, (b) PMD with the edge effect subtracted; The laser field is
$\mathbf{E}(t)=-E_{0}\cos(\omega t/6)^{2}(\cos(\omega t),\epsilon\sin(\omega
t))$, $\omega=0.062$ a.u., $\kappa=1.345$ a.u., $E_{0}=0.04$ a.u.,
$\epsilon=0.87$ a.u. the field truncation is at $\omega t_{i}=-3\pi/\omega$
and $\omega t_{f}=3\pi/\omega$.

The calculation of the time-integrals along the vertical contours in Fig. 5,
$C_{i}$ and $C_{f}$, is facilitated by the fact that the integrand is
exponentially suppressed at large imaginary times, see Fig. 6, and only the
beginning of the contour close to the real axis gives the main contribution to
the integral. In order to account for the edge-effect analytically, we
approximate the prefactor and the exponential of the integrand $m(t)$ near the
truncation points $t_{i}$ and $t_{f}$:

$\displaystyle m_{C_{i}}(t)$ $\displaystyle\approx$ $\displaystyle{\cal
C}\sum_{n}\frac{(p+A(t_{i}))A^{(n+1)}(t_{i})(t-t_{i})^{n}}{n!\left((p+A(t_{i}))^{2}+\kappa^{2}\right)^{2}}$
(7) $\displaystyle\times$
$\displaystyle\exp\left[-iS(t_{i})+i\kappa^{2}/2t_{i}\right.$ $\displaystyle+$
$\displaystyle\left.i(\kappa^{2}/2+(p+A(t_{i}))^{2}/2)(t-t_{i})\right]$
$\displaystyle m_{C_{f}}(t)$ $\displaystyle\approx$ $\displaystyle{\cal
C}\sum_{n}\frac{(p+A(t_{f}))A^{(n+1)}(t_{f})(t-t_{f})^{n}}{n!\left((p+A(t_{f}))^{2}+\kappa^{2}\right)^{2}}$
(8) $\displaystyle\times$
$\displaystyle\exp\left[-iS(t_{f})+i\kappa^{2}/2t_{f}\right.$ $\displaystyle+$
$\displaystyle\left.i(\kappa^{2}/2+(p+A(t_{f}))^{2}/2)(t-t_{f})\right],$

where the summation over $n$ begins from the first non-vanishing derivative of
the function $A^{\prime}(t)$ up to the next third orders at weak fields and
terms of the order of $A^{\prime}(t_{i})^{2}$ or $A^{\prime}(t_{f})^{2}$ are
neglected. These approximated integrands can now be integrated along the
steepest descent contour at the truncation points. Since
$\kappa^{2}/2+(p+A(t_{i}))^{2}/2$ and $\kappa^{2}/2+(p+A(t_{f}))^{2}/2$ are
real numbers, the contour is vertically aligned in the complex plane starting
at $t_{i}$ or $t_{f}$, respectively. The integration yields

$\displaystyle m_{C_{i}}(p)$ $\displaystyle=$ $\displaystyle{\cal
C}\sum_{n}2^{n+1}\exp[-iS(t_{i})+i\kappa^{2}/2t_{i})]$ (9)
$\displaystyle\times$
$\displaystyle\frac{(p+A(t_{i}))A^{n+1}(t_{i})}{((p+A(t_{i}))+\kappa^{2})^{n+1}}$
$\displaystyle m_{C_{f}}(p)$ $\displaystyle=$ $\displaystyle{\cal
C}\sum_{n}2^{n+1}\exp[-iS(t_{f})+i\kappa^{2}/2t_{f})]$ (10)
$\displaystyle\times$
$\displaystyle\frac{(p+A(t_{f}))A^{n+1}(t_{f})}{((p+A(t_{f}))+\kappa^{2})^{n+1}}.$

The approximated integrand function of Eq. (10) is shown in Fig. 6. It
coincides with the analytical one. Thus, using expressions of Eqs. (10), (10)
the contribution of the vertical contours $C_{i}$, $C_{f}$ can be subtracted
analytically, which corresponds to the subtraction of the edge-effect.

###  IV.3 Edge-effect subtraction in the numerical solution of TDSE

The PMD according to numerical solution of TDSE can be written as:

$w(p)=|m(p)|^{2}=|\langle{\psi_{p}^{V}(x,t_{f})}|U(t_{f},t_{i})|{\phi(x,t_{i})}\rangle|^{2},$
(11)

where the time evolution operator $U(t_{f},t_{i})$ can be obtained through the
normal Schrödinger equation

$U(t_{f},t_{i})=\mathcal{T}\exp\left[-i\int_{t_{i}}^{t_{f}}dt\,H(t)\right],$
(12)

with $H(t)=\hat{\mathbf{p}}^{2}/2m+\mathbf{r}\cdot\mathbf{E}(t)+V(\mathbf{r})$
and $\mathcal{T}$ being the time ordering operator, or based on the Dyson
equation

$U(t_{f},t_{i})=U_{0}(t_{f},t_{i})-i\int_{t_{i}}^{t_{f}}dt\,U(t_{f},t)H_{i}(t)U_{0}(t,t_{i}),$
(13)

with $H_{i}(t)=\mathbf{r}\cdot\mathbf{E}(t)$ being the interaction Hamiltonian
and $U_{0}(t_{2},t_{1})$ being the field free time evolution operator.

The ionization amplitude $m(p)$ along the real axis is calculated as

$m_{C_{r}}(p)=\langle{\psi_{p}^{V}(t_{f})}|\mathcal{T}\exp\left[-i\int_{t_{i}}^{t_{f}}dt\,H(t)\right]|{\phi(t_{i})}\rangle$
(14)

employing the traditional time-splitting operator method. Along the vertical
contours ($C_{i}$ and $C_{f}$), on the other hand, the amplitude is obtained
through

$\displaystyle
m_{C_{i}}(p)=-i\int_{C_{i}}dt\,\langle\psi_{p}^{V}(t_{f})|U(t_{f},t)H_{i}(t)U_{0}(t,t_{i})|\phi(t_{i})\rangle,$
(15) $\displaystyle
m_{C_{f}}(p)=-i\int_{C_{f}}dt\,\langle\psi_{p}^{V}(t_{f})|U(t_{f},t)H_{i}(t)U_{0}(t,t_{i})|\phi(t_{i})\rangle.$
(16)

Here $U(t_{f},t)$ is the exact time evolution operator in the numerical
simulation rather than the time evolution operator in the laser pulse only,
which is used in the SFA calculations.

Finally, the total PMD is calculated as

$w(p)=|m_{C_{r}}(p)+m_{C_{i}}(p)+m_{C_{f}}(p)|^{2}.$ (17)

##  V Attoclock

The proposed U-contour method proves very efficient for the calculation of PMD
of the attoclock at weak laser intensities. We calculate PMD via 3D first-
order SFA in the attoclock case with an elliptically polarized laser field

$\mathbf{E}(t)=-E_{0}\cos(\omega t/6)^{2}\left[\cos(\omega
t),\epsilon\sin(\omega t)\right],$ (18)

with $\omega=0.062$ a.u., $\kappa=1.345$ a.u., $E_{0}=0.04$ a.u.,
$\epsilon=0.87$ a.u. and truncation at $\omega t_{i}=-3\pi/\omega$ and $\omega
t_{f}=3\pi/\omega$. The time integral in Eqs. (5)-(6) is calculated
numerically and the edge terms are subtracted analytically as shown above.

The corresponding momentum distribution is shown in Fig. 7. Note that in a
long Gaussian pulse the PMD coincides with the edge-effect-free result of the
U-contour method. One can see that the edge-effects significantly disturb PMD,
even changing the topology of the distribution. Usually the Coulomb field
effect increases the ionization signal, and makes the edge-effect relevant at
smaller field strength than in the short-range potential.

##  VI Conclusion

We have developed a new method to remove the edge-effect of the laser pulse
because of the diffraction from the time slit created by the edges, in PMD of
tunnel-ionized electrons. The method consists of replacing the original time-
integral in the ionization amplitude along the real time axis with the, so-
called, U-contour, adding two integrals along the imaginary time axis,
starting at the time edges of the laser pulse. The method can be applied
analytically for SFA, as well as for the numerical solution of TDSE. The edge-
effect adds a trivial diffraction patterns originated from the time-edges of
the laser pulse, which disappear when using more smooth laser pulses (long-
truncated-Gaussian pulse) of the same frequency and intensity. The edge-effect
hides the physical structures in PMD due to nonadiabatic processes in weak
laser fields that underlines the important application of the proposed method
to reveal the dynamical signal of strong-field ionization in the deep
nonadiabatic regime.

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