# Interference in $\omega-2\omega$ atomic ionization within the strong-field
approximation: Beyond the perturbative regime

Diego G. Arbó1,2, Sebastián D. López${}^{1},$ 1 Institute for Astronomy and
Space Physics - IAFE (CONICET-UBA), CC 67, Suc. 28, C1428ZAA, Buenos Aires,
Argentina.  2Universidad de Buenos Aires, Facultad de Ciencias Exactas y
Naturales y Ciclo Básico Común, Buenos Aires, Argentina.

###### Abstract

We analyze interference processes in atomic ionization induced by a two-color
laser with fundamental frequency $\omega$ and its second harmonic $2\omega$.
The interplay between inter- and intracycle interference processes give rise
to multiphoton peaks which can be named as main or ATI peaks and sidebands, in
analogy to the well-known RABBIT (reconstruction of attosecond harmonic
beating by interference of two-photon transitions). We use the saddle point
approximation (SPA) to extract the complex ionization times of the interfering
electron trajectories. Changing the relative phase between the two colors, the
doubly differential momentum distribution of emitted electrons can be
controlled. We study the dependence of the electron emission as a function of
the relative phase between the $\omega$ and $2\omega$ fields within the strong
field approximation (SFA) but beyond the perturbative regime. We focus on the
extraction of the phase delays accounting the electron forward emission in the
direction of the polarized electric fields. We characterize the time delays in
the emission of electrons for visible frequency of the pump and its first
harmonic as a probe [Ti:Sapphire laser ($800$ nm) together with the first
harmonic ($400$ nm)] for a typical $\omega-2\omega$ configuration for argon
ionization. We find excellent agreement between our SPA results and the
corresponding SFA (without any further approximation) and also with previous
perturbative theories.

###### pacs:

32.80.Rm,32.80.Fb,03.65.Sq

##  I Introduction

Photoionization is defined as the detachment of one or multiple electrons from
a system, such as an atomic or molecular ion, a cluster, or a solid, due to
the electromagnetic force exerted by external radiation. Two different
pictures has been widely used to depict the photoionization process. On one
hand, the exiting electron tunnels through the barrier formed by the parent
structure and the external field when the time that the electron travels
through the barrier is longer than the time variation of the external field
(situation known as the tunneling regime). And on the other hand, when the
variation of the external field is very rapid, tunneling is not possible and
photoionization proceeds through the absorption of one or several photons
allowing the target gain energy up to the continuum near threshold or even
higher (situation known as the multiphoton regime) Keldysh64 , Keldysh65 ,
Bergues07 . First experiments used rather weak lasers, for which, the
ionization processes were deep in the multiphoton regime Agostini79 . Theories
accompanying these experiments were firstly perturbative Faisal73a , Faisal73b
but, as irradiance of laser beams grew, these perturbation theories became
obsolete and a new paradigm was necessary Corkum94 , Lewenstein95 , Ivanov95 .
Based on rather simple pictures of the photoionization processes like the
simpleman’s model (SMM) or the strong field approximation (SFA), interference
structures in photoelectron spectra have been identified as a diffraction
pattern from a time grating composed of inter- and intracycle interferences
Bivona08 , Arbo10a , Arbo10b , Arbo12 . Whereas the intercycle interferences
give rise to multiphoton peaks, intracycle interferences lead to a modulation
of the ATI spectrum offering information on the subcycle ionization dynamics
Xie12 , Arbo10a , Arbo14a .

Metrology of atomic processes became accessible through pump-probe techniques
such as attosecond streaking Itatani02 , Goulielmakis04 , Goulielmakis07  and
RABBIT (reconstruction of attosecond harmonic beating by interference of two-
photon transitions) Veniard95 , Veniard96 , Paul01 . These techniques involve
lasers of at least two very different frequencies: Electrons are emitted due
to the absorption of an XUV field and probed by a moderately intense field in
the near-infrared region of the electromagnetic spectrum. In this way, it was
possible to measure phase shifts compatible with attosecond delays for noble
gas atoms Schultze10 , Klunder11 , Guenot12 , Guenot14 . For attosecond
streaking, the oscillating probe field moves classically the electron
previously freed by the pump field, producing gains and losses of its final
kinetic energy Itatani02 , Fuchs20 . In RABBIT, ionization is given by two
consecutive high harmonics fields of odd parity followed by absorption or
emission of a photon with the fundamental frequency. By analyzing the
interference between these two ionization paths, the intrinsic phase shifts in
above threshold ionization (ATI) spectra could be extracted by means of
second-order perturbative calculations Dahlstrom13  or by analyzing the
asymptotic behavior of the scattered electron wave packet from numerically
exact solutions of the time-dependent Schrödinger equation (TDSE) Nagele12 ,
Pazourek12 , Kheifets13 , Feist14 , Su13 , Pazourek15 , Boll16 .

Strong-field ionization by laser fields with commensurate frequencies and
well-defined relative phase permits the tune and control of the emission
process Schumacher94 , Arbo15 , Ehlotzky01 , Xie12 , Arbo14a . ATI by laser
pulses using the fundamental component and one of its harmonics were
investigated Ehlotzky01 , Schumacher94 , Muller90  and applied for controlled
ionization Thompson97 , Sheehy95 , Ohmura04 , dichroism Fifirig03 , Cionga03 ,
orientation of molecules De09 , and control of interference fringes in the
electron momentum distribution Xie12 , Arbo14a . The temporal shape of the
two-color field is determined by the intensities of the two components and
their relative phase. Coherent phase control refers to the manipulations of
some physical processes through the relative phase Ehlotzky01 . The concept of
phase shifts and time delays in RABBIT has been extended by Zipp et al. Zipp14
for two-color ($\omega-2\omega$) lasers with controlled relative phase. Very
recently, we have theoretically explored the extraction of $\omega-2\omega$
phase delays by means of the ab initio solution of the TDSE and also through
the development of a perturbation theory Lopez21 .

In this work, we developed a non-perturbation theory of the electronic
photoemission process in atomic argon due to a two-color ($\omega-2\omega$)
linearly polarized short laser pulse in the multiphoton regime. In Sec. II, we
introduce the general theory based on the saddle point approximation (SPA) to
calculate the ionization time of each interfering electron trajectory, firstly
analyzing one-color ionization and then the $\omega-2\omega$ setup. Different
interference structures of the doubly differential momentum distribution are
analyzed. We focus on the extraction of the phase shifts using directional
emission in the forward direction. We show that our SPA results are in
excellent agreement with the results of the SFA (without any further
approximation) and also the perturbation theories in the literature
Bertolino21 , Lopez21 . We make our final remarks in Sec. III. Atomic units
($e=\hbar=m_{e}=1$ a.u.) are used throughout unless stated otherwise.

##  II Non-perturbative strong-field approximation

In general, ionization of an atomic system by a linearly polarized laser pulse
can be considered in the single-active-electron approximation. The TDSE then
reads

$i\frac{\partial}{\partial
t}\left|\psi(t)\right\rangle=\left[H_{0}+H_{\text{int}}(t)\right]\left|\psi(t)\right\rangle,$
(1)

where $H_{0}=\vec{p}^{2}/2+V(r)$ is the time-independent atomic Hamiltonian,
whose first term corresponds to the electron kinetic energy and its second
term to the electron-core Coulomb interaction. In Eq. (1), $H_{\text{int}}(t)$
corresponds to the interaction hamiltonian between the atomic system and the
external radiation field. Because of the presence of the external laser field,
the electron initially bound in an atomic state $|\phi_{i}\rangle$ can either
remain in the same state, be excited to another atomic bound state, or be
emitted to a final continuum state $|\phi_{f}\rangle$ with final momentum
$\vec{k}$ and energy $E=k^{2}/2$. In the latter case, we call the process
photoionization and the transition amplitude within the time-dependent
distorted wave theory in the prior form is expressed as Macri03 , Arbo08a

$T_{\mathrm{if}}=-i\int_{-\infty}^{+\infty}dt\,\langle\chi_{f}^{-}(\vec{r},t)|H_{\text{int}}(\vec{r},t)|\phi_{i}(\vec{r},t)\rangle,$
(2)

where $\phi_{i}(\vec{r},t)=\varphi_{i}(\vec{r})\,e^{iI_{p}t}$ is the initial
atomic state with ionization potential $I_{p}$ and $\chi_{f}^{-}(\vec{r},t)$
is the distorted final state. Eq. (2) is exact as far as the final channel
$\chi_{f}^{-}(\vec{r},t)$ is the exact solution of Eq. (1), within the dipole
approximation. Throughout this paper, we will be considering linearly
polarized laser fields (in the $\hat{z}$ direction).

Several degrees of approximation have been considered in the literature to
solve Eq. (2). The widest known one is the strong field approximation (SFA),
which neglects the Coulomb distortion (in the final channel) produced on the
ejected-electron state due to its interaction with the residual ion and
discard the influence of the laser field in the initial ground state and the
depletion of the ground state Lewenstein94 , Lewenstein95 . The SFA consists
in approximating the distorted final state with the solution of the TDSE for a
free electron in an electromagnetic field, namely, a Volkov function Volkov35
, i.e., $\chi_{f}^{-}(\vec{r},t)=\chi_{f}^{V}(\vec{r},t)$, where

$\displaystyle\chi_{f}^{V}(\vec{r},t)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3/2}}\exp\\{i[\vec{k}+\vec{A}(t)]\cdot\vec{r}\\}$
(3) $\displaystyle\times$
$\displaystyle\exp\left\\{\frac{i}{2}\int_{t}^{\infty}[\vec{k}+\vec{A}(t^{\prime})]^{2}dt^{\prime}\right\\}$

the vector potential due to the total external field is defined as
$\vec{A}(t)=-\int_{-\infty}^{t}dt^{\prime}\vec{F}(t^{\prime}),$ and
$\vec{F}(t)$ denotes the external laser field. The final Volkov function in
Eq. (3) is calculated within the length gauge, i.e.,
$H_{\text{int}}(\vec{r},t)=\vec{F}(t)\cdot\vec{r}$.

Therefore, $T$-matrix in Eq. (2) can be written as

$T_{\mathrm{if}}=\int_{-\infty}^{+\infty}\,\ell(t)\ e^{iS(t)}\,\,dt,$ (4)

where

$\displaystyle\ell(t)$ $\displaystyle=$
$\displaystyle-i\vec{F}(t)\cdot\vec{d}\left[\vec{k}+\vec{A}(t)\right]$
$\displaystyle\mathrm{and}$ $\displaystyle S(t)$ $\displaystyle=$
$\displaystyle-\int_{t}^{\infty}dt^{\prime}\left\\{\frac{\left[\vec{k}+\vec{A}(t^{\prime})\right]^{2}}{2}+I_{p}\right\\}$
(5)

with the dipole transition moment defined as
$\vec{d}(\vec{v})=(2\pi)^{-3/2}\langle
e^{i\vec{v}\cdot\vec{r}}|\vec{r}|\varphi_{i}(\vec{r})\rangle$, and $S(t)$ is
the Volkov action.

We assume that the pump field is composed of $2N$ optical cycles each of
duration $T=2\pi/\omega$. Then,

$\displaystyle T_{\mathrm{if}}$ $\displaystyle=$
$\displaystyle\int_{0}^{NT}\,\ell(t)e^{iS(t)}\,\,dt$ (6) $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}\int_{jT}^{(j+1)T}\ell(t)e^{iS(t)}dt.$

We consider now a general electric field (and vector potential) with a smooth
envelope with a central flat-top region where both $\vec{F}(t)$ and
$\vec{A}(t)$ are oscillating with period $T$. From Eq. (5), it is
straightforward to realize that $S(t)-at$ is a time-oscillating function with
the same period of the laser field and vector potential,

$S(t+jT)=S(t)+ajT,$ (7)

where

$a=\frac{k^{2}}{2}+I_{p}+U_{p},$ (8)

and $U_{p}=\int_{t}^{t+T}dt^{\prime}A(t^{\prime})^{2}.$ In light of the
periodicity properties of the action in Eq. (7) and that $\ell(t+jT)=\ell(t)$,
the transition matrix $T_{\mathrm{if}}$ in Eq. (4) can be written in terms of
the contribution of the first fundamental cycle or unit cell Arbo12 ,
DellaPicca20 ,

$\displaystyle T_{\mathrm{if}}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}\int_{jT}^{(j+1)T}\ell(t+jT)e^{iS(t+jT)}dt$ (9)
$\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}e^{iajT}\int_{0}^{T}\ell(t)e^{iS(t)}dt$
$\displaystyle=$
$\displaystyle\frac{\sin{(aTN/2)}}{\sin{(aT/2)}}\,e^{(iaT(2N-1)/2)}I(\vec{k}).$

From the absolute value of the transition matrix we can extract probabilistic
information, like the doubly differential momentum distribution or the angle
resolved photoelectron spectrum. Because of the azimuthal symmetry, the
electron distribution can be expressed in terms of only two physical
magnitudes, i.e., the final electron momentum parallel $k_{z}$ and transversal
$k_{\rho}$ to the field polarization direction or, alternatively, the final
kinetic energy $E$ and the final polar emission angle $\theta$:

$\left|T_{\mathrm{if}}\right|^{2}=\frac{dP}{2\pi
k_{\rho}dk_{\rho}dk_{z}}=\frac{dP}{2\pi\sqrt{2E}\ dE\ d(\cos\theta)}.$ (10)

The factor $I(\vec{k})=\int_{0}^{T}\ell(t)e^{iS(t^{\prime})}dt^{\prime}$ in
Eq. (9) corresponds to the contribution into one optical cycle of the $\omega$
field and $|I(\vec{k})|^{2}$ is known in the literature as the _intracycle_
contribution to the ionization probability Arbo08a , Arbo08b , DellaPicca20 .
Thus, the photoelectron spectrum (PES) can be expressed as a product of the
_intracycle_ factor $|I(\vec{k})|^{2}$ and the _intercycle_ factor
$\left(\sin{(aTN/2)}/\sin{(aT/2)}\right)^{2}$, being the latter the result of
the phase interference arising from the $N$ different optical cycles of the
field Arbo10a , Arbo10b , Arbo12 . We want to point out that Eq. (9) is a mere
consequence of the periodicity of the transition matrix with no further
approximations, except for a flat-top pulse DellaPicca20 .

Finite maxima are reached at the zeroes of the denominator of the intercycle
factor $\left(\sin{(aTN/2)/}\sin{(aT/2)}\right)^{2}$, i.e., the energy values
satisfying $aT/2=n\pi$, since the numerator also cancels out at these points.
Such maxima are recognized as the multiphoton peaks of the PES. They occur
when

$E_{n}=n\omega-I_{p}-U_{p},$ (11)

where we have used Eq. (8). In fact, when $N\rightarrow\infty$, the intercycle
factor becomes a series of delta functions, i.e., $\sum_{n}\delta(E-E_{n})$,
satisfying the conservation of energy. Instead, for finite pulse durations
$\tau=NT$ (composed of $N$ cycles), each multiphoton peak has a width $\Delta
E\sim 2\pi/NT$, fulfilling the uncertainty relation $\Delta E\tau\sim 2\pi$.

The intracycle amplitude
$I(\vec{k})=\int_{0}^{T}\ell(t)e^{iS(t^{\prime})}dt^{\prime}$ in Eq. (9) can
be calculated either numerically (SFA) or within the saddle point
approximation (SPA). In the latter, the intracycle amplitude can be regarded
as a superposition of all electron trajectories within any optical cycle or
unit cell with final momentum $\vec{k}$

$I(\vec{k})\simeq\sum_{\beta}\ell(t_{\beta})\frac{e^{iS(t_{\beta})}}{\left|\ddot{S}(t_{\beta})\right|^{1/2}},$
(12)

each starting at a complex ionization times $t_{\beta}$ fulfilling the saddle
equation $\dot{S}(t_{\beta})=0$ (where the dot denotes the time derivative),
i.e.,

$\frac{\left[\vec{k}+\vec{A}(t_{\beta})\right]^{2}}{2}+I_{p}=0.$ (13)

In general, solutions of Eq. (13) come in pairs
($t_{\beta},t_{\beta}^{\ast}$), where the star means complex conjugate. From
each couple, we select only the solution with positive imaginary part to avoid
spurious exponential growth of probabilities and only keep exponential decays
when calculating
$\exp\left[iS(t_{\beta})\right]=\exp\left\\{i\operatorname{Re}\left[S(t_{\beta})\right]\right\\}\exp\left\\{-\operatorname{Im}\left[S(t_{\beta})\right]\right\\}$
in Eq. (12). The SMM considers real ionization times by neglecting $I_{p}$ and
the perpendicular momentum reducing Eq. (13) to $k_{z}+A(t_{\beta})=0.$

###  II.1 One color photoionization

For the case of atomic photoionization by a one color field
$\vec{F}(t)=Ff(t)\cos(\omega t+\phi)\ \hat{z}$ with $f(t)$ a smooth function
between $0$ and $1$ mimicking the pulse envelope, $\hat{z}$ the polarization
direction, and $F$ the field strength, the action can be calculated from Eq.
(5) as

$S_{0}(t)=at+b\cos(\omega t)+c\sin(2\omega t),$ (14)

where $a$ is given by Eq. (8), $b=F/\omega^{2}\hat{z}\cdot\vec{k},$
$c=-U_{p}/2\omega$, and the ponderomotive energy is $U_{p}=(F/(2\omega))^{2}$.
We denote $S_{0}$ the one-color action just to distinguish it from the two-
color action in the next subsection.

From Eq. (13), two ionization times can be analytically calculated with the
following expressions

$\displaystyle t_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\omega}\sin^{-1}\left[\frac{\omega}{F}\left(k_{z}+i\sqrt{2I_{p}+k_{\rho}^{2}}\right)\right],$
$\displaystyle t_{2}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\omega}-t_{1}^{\ast},$ (15)

for $k_{z}\geq 0$ and,

$\displaystyle t_{1}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\omega}+\frac{1}{\omega}\sin^{-1}\left[\frac{\omega}{F}\left(k_{z}+i\sqrt{2I_{p}+k_{\rho}^{2}}\right)\right],$
$\displaystyle t_{2}$ $\displaystyle=$
$\displaystyle\frac{3\pi}{\omega}-t_{1}^{\ast},$ (16)

for $k_{z}\leq 0.$

We show the real part of $t_{1}$ and $t_{2}$ in Fig. 1a and the imaginary
parts in Fig. 1b for an electric field of strength $F=0.0469$ a.u. and
frequency $\omega=0.114,$ for the special case that $k_{\bot}=0$ (forward and
backward emission). The SMM times are drawn in dash line and were calculated
by including a tiny ionization potential $I_{p}=10^{-6}.$ The shaded region of
longitudinal momentum $|k_{z}|\leq 2\sqrt{U_{p}}$ corresponds to the classical
accessible region for the electron according to the SMM. Inside this region
$\operatorname{Im}(t_{1})=\operatorname{Im}(t_{2})=0$ while outside,
$\operatorname{Im}(t_{1})=\operatorname{Im}(t_{2})>0.$ In turn, for the SPA,
we have used an ionization potential $I_{p}=0.58$ a.u. (corresponding to the
ground state of atomic argon).

Figure 1: Complex saddle times $t_{1}$ and $t_{2}$ as a function of the
longitudinal momentum $k_{z}$ for $k_{\bot}=0$ for one-color ionization. In
solid (dash) lines the results of the SPA (SMM, including an ionization
potential of $10^{-6}$ to generate the imaginary part of the saddle time). (a)
In black $\operatorname{Re}(t_{1})$ and in red $\operatorname{Re}(t_{2})$. (b)
In solid (dash) lines the results of the SPA (SMM)
$\operatorname{Im}(t_{1})=\operatorname{Im}(t_{2})$. The yellow region in the
$k_{z}$ domain corresponds to the allowed classical region in the SMM. In the
classical-forbiden region, the imaginary part of the SMM saddle times
increases.

In Fig. 2a we show the intercycle factor
$\left(\sin{(aTN/2)/}\sin{(aT/2)}\right)^{2}$ with $N=4$ as a function of the
longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$ (whose
magnitude is equal to $k_{\rho}.$ Isotropic rings are observed at radii
$k_{n}=\sqrt{2E_{n}},$ where $E_{n}$ are given by Eq. (11), corresponding to
the absorption of $n$ photons. The intracycle factor is displayed in Fig. 2d
exhibiting a strong angular dependence. The total doubly differential momentum
distribution is proportional to the multiplication of the intercycle and
intracycle factors and is displayed in Fig 2f. In this paper we omit in the
calculations the factor $l(t)$ containing the atomic dipole moment just to
focus on interference aspects of the photoionization processes.

Figure 2: Doubly differential momentum distribution as a function of the
longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$ for
one-color ($2\omega$) ionization. (a) Intercycle factor considering
$4\pi/\omega$ periodicity generating ATI and SB rings, (b) Intercycle factor
considering $2\pi/\omega$ periodicity generating only ATI rings (of double
width), (c) Multiplication of (a) and (b), the ATI rings are visible but the
SB rings disappeared since they coincide with the minima of the distribution
in (b). (d) corresponds to the intracycle interference, (e) to the
multiplication of distributions in (d) and (b), and in (f) the total momentum
distribution is displayed (multiplication of (c) and (d)).

###  II.2 Interference in ($\omega-2\omega$) photoionization

The main goal of this work is to extend the well-known interference structures
of electron photoemission in one-color atomic photoionization to the case of
two colors, where one main frequency (harmonic) doubles the other
(fundamental), i.e., $\omega-2\omega$ photoionization emission, within the
SFA. If the fundamental intensity is very low compared to the intensity of its
second harmonic, some connection between $\omega-2\omega$ ionization and
RABBIT can be speculated and analyzed showing some similarities and some
differences Zipp14 , Bertolino21 , Lopez21 . In the RABBIT jargon for even $n$
in Eq. (11), the energy maxima are named ATI peaks and for odd $n$ they are
named sidebands, but although in our context that denomination of the
multiphoton peaks is arbitrary, we will maintain it for clarity. In general
$n=n_{2\omega}+$ $n_{\omega}$ denotes the absorption (emission) of a
$n_{2\omega}$ number of $2\omega$ photons and absorption (emission) and a
$n_{\omega}$ number of $\omega$ photons for positive (negative) $n_{2\omega}$
and $n_{\omega}$ values. We consider the two-color electric field of the form

Figure 3: Vector potential (red) and electric field (black) along the
polarization axis as a function of time. In solid line the two-color version
and in dash line the one-color version (switching off the $\omega$ terms in
Eqs. (17) and (18). (a) for $\phi=0,$ (b) for $\phi=\pi/2,$ (c) for
$\phi=\pi,$ and (d) for $\phi=3\pi/2.$ Vector potentials and electric fields
are normalized with respect to their respective one-color version.

$\vec{F}(t)=f(t)\left[F_{2\omega}\cos\left(2\omega
t\right)+F_{\omega}\cos(\omega t+\phi)\right]\ \hat{z},$ (17)

with $\phi$ the relative phase of the second harmonic with respect to the
fundamental laser field, $f(t)$ is a smooth function between $0$ and $1$
mimicking the pulse envelope, $\hat{z}$ is the polarization direction of both
fields, and $F_{2\omega}$ and $F_{\omega}$ are the field strengths of the
second harmonic and fundamental frequency, respectively. For a long pulse with
adiabatic switch on and off the vector potential can be written in its central
part $f(t)\simeq 1$), as

$\vec{A}(t)=-f(t)\left[\frac{F_{2\omega}}{2\omega}\sin(2\omega
t)+\frac{F_{\omega}}{\omega}\sin(\omega t+\phi)\right]\hat{z},$ (18)

giving rise to a periodicity property of the vector potential and the electric
field, i.e., $\vec{A}(t)=\vec{A}(t+2j\pi/\omega)$ and
$\vec{F}(t)=\vec{F}(t+2j\pi/\omega),$ with $j$ any integer number provided
that $f(t+2j\pi/\omega)=1.$ For our calculations we use the same parameters as
in Ref. Zipp14  with $F_{2\omega}=0.0469$ a.u. ($I_{2\omega}=8\times 10^{13}$
W/cm2) and $F_{\omega}=0.00332$ a.u. ($I_{\omega}=4\times 10^{11}$ W/cm2). In
Fig. 3 we show the electric field in black and the vector potential in red as
a function of time. We have normalized both fields to the one-color case
($F_{\omega}=0$) displayed in dotted lines. As the one-color fields show
obviously invariant under changes of the relative phase, there are small but
appreciable changes in the two-color fields. For example, the one-color fields
are $\pi/\omega$-periodic, whereas the two-color fields are
$2\pi/\omega$-periodic. As shown for the one-color case Arbo10a , Arbo10b ,
the correct way to choose the unit cell corresponding to one optical cycle of
the field is from a root of the vector potential. For the one-color case this
corresponds to, for example, $t=0,$ whereas this value changes for the two-
color case as a function of the relative phase $\phi.$ For example,
considering the roots of the vector potential closest to the origin as the
left border of the unit cells, they are determined by
$t\in\left[0,2\pi/\omega\right]$ ($\phi=0$ in Fig. 3a),
$t\in\left[-1.243,-1.243+2\pi/\omega\right]$ (in a.u. for $\phi=\pi/2$ in Fig
3b), $t\in\left[0,2\pi/\omega\right]$ ($\phi=\pi$ in Fig. 3c), and
$t\in\left[1.243,1.243+2\pi/\omega\right]$ (in a.u. for $\phi=3/2$ in Fig.
3d).

Under the assumption of adiabatic switch on and off, the action in Eq. (5) can
be analytically calculated (in the central region where $f(t)=1$) as

$\displaystyle S(t)$ $\displaystyle=$ $\displaystyle at+b\cos(2\omega
t)+c\sin(4\omega t)+d\cos(\omega t+\phi)$ $\displaystyle+e\sin(\omega
t-\phi)+f\sin(2\omega t+2\phi)+g\sin(3\omega t+\phi),$

where $a$ is given by Eq. (8), and

$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{F_{2\omega}}{4\omega^{2}}\hat{z}\cdot\vec{k},$ (20)
$\displaystyle c$ $\displaystyle=$ $\displaystyle-\frac{U_{p,2}}{4\omega},$
$\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{F_{\omega}}{\omega^{2}}\hat{z}\cdot\vec{k},$
$\displaystyle e$ $\displaystyle=$
$\displaystyle\frac{F_{2\omega}F_{\omega}}{4\omega^{3}},$ $\displaystyle f$
$\displaystyle=$ $\displaystyle-\frac{U_{p,1}}{2\omega},$ $\displaystyle g$
$\displaystyle=$ $\displaystyle-\frac{F_{2\omega}F_{\omega}}{12\omega^{3}},$

and
$U_{p}=U_{p,2}+U_{p,1}=(F_{2\omega}/(4\omega))^{2}+(F_{\omega}/(2\omega))^{2}$
defines the ponderomotive energy as the addition of the individual
ponderomotive energies of each color. We we have dropped diverging terms in
Eq. (II.2) since only the accumulated action, computed as differences of
phases, is relevant. The two-color action of Eq. (II.2) reduces to the one-
color action of Eq. (14) when either $F_{\omega}=0$ (since $d=e=f=g=0$) or
$F_{2\omega}=0$ (since $b=c=e=0$).

For the case of an $\omega-2\omega$ field with weak probe $\omega$ field
compared to the harmonic $2\omega$ field, there are four physical solutions of
Eq. (13) per optical cycle (in the unit cell), i.e., $t_{1}$, $t_{2}$,
$t_{3},$ and $t_{4}$. We have solved all the saddle times from Eq. (13)
numerically by separating their real and imaginary parts obtaining two coupled
equations:

$\displaystyle\frac{F_{2\omega}}{2\omega}\sin\left(2\omega\operatorname{Re}t_{\beta}\right)\cosh(2\omega\operatorname{Im}t_{\beta})$
(21)
$\displaystyle+\frac{F_{\omega}}{\omega}\sin\left(\omega\operatorname{Re}t_{\beta}+\phi\right)\cosh\left(\omega\operatorname{Im}t_{\beta}\right)$
$\displaystyle=$ $\displaystyle k_{z}$
$\displaystyle\frac{F_{2\omega}}{2\omega}\cos\left(2\omega\operatorname{Re}t_{\beta}\right)\sinh(2\omega\operatorname{Im}t_{\beta})$
(22)
$\displaystyle+\frac{F_{\omega}}{\omega}\cos\left(\omega\operatorname{Re}t_{\beta}+\phi\right)\sinh\left(\omega\operatorname{Im}t_{\beta}\right)$
$\displaystyle=$ $\displaystyle i\sqrt{2I_{p}+k_{\rho}^{2}},$

with $\beta=1,2,3,$ and $4.$ By neglecting either the first or the second
terms of Eq. (II.2) and summing the two equations we recover the saddle times
in Eqs. (16) and (15) for the one-color case.

Figure 4:  Intrahalfcycle factors $\cos^{2}\left(\Delta S_{12}/2\right)$ in
(a) and $\cos^{2}\left(\Delta S_{34}/2\right)$ in (b).

From Eq. (12), the intracycle amplitude stemming from the electron
trajectories with released times $t_{\beta}$ ($\beta=1,2,3,$ and $4$) is
proportional to

$\sum_{\beta=1}^{4}e^{iS(t_{\beta})}=e^{i\overline{S}_{1,2}}\cos\left[\frac{\Delta
S_{1,2}}{2}\right]+e^{i\overline{S}_{3,4}}\cos\left[\frac{\Delta
S_{3,4}}{2}\right]$ (23)

where we have omitted the prefactors of each of the terms corresponding to the
electron trajectories departing at $t_{1},t_{2},t_{3},$ and $t_{4},$ to
highlight the interference patterns. In Eq. (23)
$\overline{S}_{i,j}=\left[S(t_{i})+S(t_{j})\right]/2$ is the average action
between $t_{i}$ and $t_{j}$ and $\Delta S_{i,j}=S(t_{j})-S(t_{i})$ is the
accumulated action between $t_{i}$ and $t_{j}.$ The accumulated actions
$\Delta S_{1,2}$ and $\Delta S_{3,4}$ in the last equation contribute to the
intrahalfcycle interference of the first and second half cycles. In Fig. 4 we
compare the intrahalfcycle factors of the first and second half cycles. They
are similar but not equal since the vector potential in the first half cycle
differs from the second half cycle. As the probe field is weak compared to the
pump field, the intrahalfcycle distributions of Fig. 4 are also similar to the
one color case exhibited in Fig. 2d. Taking the zeroth-order perturbation in
the probe field $\left(\Delta S_{0}\right)_{1,2}=\left(\Delta
S_{0}\right)_{3,4}\equiv\Delta S_{0},$ where $\Delta S_{0}$ denotes the one-
color accumulated action in Eq. (14) . Therefore, the probability,calculated
as the square of the absolute value of the coherent addition of the four
different terms in Eq. (23), can be written as

$\displaystyle\left|\sum_{\beta=1}^{4}e^{iS(t_{\beta})}\right|^{2}$
$\displaystyle\simeq$
$\displaystyle\left|e^{i\overline{S}_{1,2}}+e^{i\overline{S}_{3,4}}\right|^{2}\cos^{2}\left[\frac{\Delta
S_{0}}{2}\right]$ (24) $\displaystyle\simeq$ $\displaystyle
4\underset{\mathrm{interhalfcycle}}{\ \underbrace{\cos^{2}\left(\frac{\Delta
S}{2}\right)}}\underset{\mathrm{intrahalfcycle}}{\underbrace{\cos^{2}\left[\frac{\Delta
S_{0}}{2}\right]}},$

where $\Delta S=$ $\overline{S}_{3,4}-\overline{S}_{1,2}.$ To get Eq. (24) we
have considered the periodicity of $S_{0}(t).$ Eq. (24) shows that the
intracycle factor $\left|I(\vec{k})\right|^{2}$ can be approximately splitted
as two factors: (i) the intrahalfcycle interference pattern
$\cos^{2}\left[\Delta S_{0}/2\right]$ stemming from the interference of the
two electron trajectories released during half optical cycle of the $\omega$
field (or within one optical cycle of the $2\omega$ field) and (ii) the
interhalfcycle interference between the contribution of the two half cycles of
the $\omega$ field (or between the two optical cycle of the $2\omega$ field).

If we go back to one-color ionization, i.e., $S(t)=S_{0}(t),$ which fulfills
the periodicity property $S_{0}(t+jT/2)=S_{0}(t)+ajT/2.$ Then,
$\overline{S}_{3,4}=\overline{S}_{1,2}+aT/2,$ and $\Delta S=\Delta
S_{0}=aT/2=a\pi/\omega=(E+I_{p}+U_{p})\pi/\omega.$ Replacing the energy for
its value at the multiphoton peaks in Eq. (11), then $\Delta S_{0}=n\pi.$
Therefore, the intracycle factor $\cos\left(\Delta S_{0}/2\right)$ in Eq. (24)
becomes equal to $\pm 1$ for even $n$ (constructive interference for the ATI
peaks) and $0$ for odd $n$ (destructive interference for the sidebands). It
can be observed that the odd intercycle rings in Fig 2a coincide with the
minima of the intracycle factor in Fig. 2b. Thus, the sidebands are not formed
in Fig. 2c and Fig. 2f, which simply means that sidebands are only present
when an $\omega$ field is applied.

Figure 5: Complex saddle times $t_{1}$, $t_{2}$, $t_{3}$, and $t_{4}$ as a
function of the longitudinal momentum $k_{z}$ for $k_{\bot}=0$ for
$\omega-2\omega$ ionization. In solid (dash) lines the results of the SPA real
(imaginary) parts for different relative phases $\phi=0$, $\phi=\pi/2$,
$\phi=\pi$, and $\phi=3\pi/2$. The variation of both real and imaginary parts
of the saddle times with the relative phase is very small and cannot be
discerned in this figure. Figure 6: Close up of the real parts of the saddle
times of Fig. 4 for different relative phases (a) $\phi=0$, (b) $\phi=\pi/2$,
(c) $\phi=\pi$, and (d) $\phi=3\pi/2$.

More generally and beyond the last approximation, complex ionization (saddle)
times depend on the longitudinal and transverse momenta, similar to the one
color case. For $\omega-2\omega$ ionization, Fig. 5 displays the real parts of
the saddle times $t_{1}$ (in black), $t_{2}$ (in green), $t_{3}$ (in cyan) and
$t_{4}$ (in yellow) as a function of the longitudinal momentum $k_{z}$ for
$k_{\bot}=0$ and for relative phases $\phi=0,\pi/2,\pi,$ and $3\pi/2.$ The
same is displayed for the imaginary parts of the ionization times. For each
value of $k_{z},$ the different solutions of Eqs. (II.2) are very similar and
cannot be distinguished in the figure. In Figs. 6 we show that the apparent
degeneracy with respect of the relative phase is not such (close up of Fig.
4). Whereas for $k_{z}\geq 0$, $\operatorname{Re}(t_{1})$ are the same for
$\phi=0$ and $\pi$ and is zero for $k_{z}=0,$ it has smaller values for
$\phi=\pi/2$ and higher values for $\phi=3\pi/2$ being the difference of about
$1$ a.u.. The degeneracy of $\operatorname{Re}(t_{1})$ between $\phi=0$ and
$\pi$ is removed for $k_{z}<0$ at expenses of a new degeneracy between
$\phi=\pi/2$ and $3\pi/2.$ In Fig 7b, we show that $\operatorname{Re}(t_{2})$
lying in the second quarter cycle are the same for $\phi=\pi/2$ and $3\pi/2$
for $k_{z}\geq 0\ $whereas it has smaller values for $\phi=\pi$ and higher
values for $\phi=0$. The degeneracy of $\operatorname{Re}(t_{2})$ between
$\phi=\pi/2$ and $3\pi/2$ is removed for $k_{z}<0$ at expenses of a new
degeneracy between $\phi=0$ and $\pi.$ $\operatorname{Re}(t_{3})$ lying in the
third quarter cycle is shown in Figs. 6c. The degeneracy is the same as for
$\operatorname{Re}(t_{1})$ whereas it has smaller values for $\phi=3\pi/2$ and
higher values for $\phi=\pi/2$ for $k_{z}\geq 0$ and the inverse for
$k_{z}<0$. Finally, the same is observed for $\operatorname{Re}(t_{4})$ with
respect to $\operatorname{Re}(t_{2})$ in Figs. 6d. We do not analyze the
imaginary part of the ionization times in detail since it is not relevant for
the interference patterns.

Figure 7: Doubly differential momentum distribution as a function of the
longitudinal $k_{z}$ and perpendicular momenta $k_{\bot}$ for the
$\omega-2\omega$ ionization within the SPA with relative phase $\phi=0$. (a)
Intercycle factor, (b) interhalfcycle factor, (c) multiplication of (a) and
(b), (d) intrahalfcycle factor, (e) intracycle factor [multiplication of
distributions in (d) and (b)], and (f) the total momentum distribution
[multiplication of (c) and (d)]. All distributions are normalized.

In Figs. 7 we show the SPA doubly differential photoelectron momentum
distribution as a function of the longitudinal momentum $k_{z}$ and the
perpendicular momentum $k_{\bot}$ for zero relative phase ($\phi=0$) between
the two colors. The intercycle interference pattern for $N=4$ displays in
Figs. 7a a set of multiphoton (ATI and sidebands) peaks. The number of minima
between consecutive multiphoton rings is $N-1=3.$ This factor is practically
the same as the one-color intercycle factor in Fig. 2a with an almost
imperceptible difference stemming from the inclusion of the ponderomotive
energy of the $\omega$ field, which is $U_{p,1}=8.5\times 10^{-4}$ ($2\%$ of
$U_{p,2}$). The intercycle factor is also independent of the electron emission
angle and the relative phase $\phi.$ The interhalfycle factor in Figs. 7b also
consists in a set of concentric rings, but the isotropy is lost and the rings
appear slightly stretched along the longitudinal momentum. Therefore, the
minima of the intracycle rings do not perfectly match with the sidebands of
the intercycle factor in Figs. 7a and thus, they survive when one multiply the
inter- and intracycle factors as shown in Figs. 7c, unlike the one-color case.
In Figs. 7d we show the intrahalfcycle factor calculated as the intrahalfcycle
of the one color case, i.e., $\cos^{2}\left[\left(\Delta
S_{0}\right)/2\right]$. Therefore, the intrahalfcycle pattern is independent
of the relative phase $\phi.$ The intracycle pattern (multiplication of the
interhalf- and intrahalfcycle patterns) is shown in Figs. 7e. The quasi-
isotropic intracycle factor appears modulated by the highly angle-dependent
intrahalfcycle pattern (or viceversa). The total emission pattern is the
multiplication of the inter- (Figs. 7a) and intracycle (Figs. 7d) (see Fig.
7f). ATI peaks and sidebands of Figs. 7c modulated by the intracycle
interference pattern are observed in Figs. 7d.

Figure 8: Doubly differential momentum distribution as a function of the
longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$ for
the $\omega-2\omega$ ionization within the SPA with relative phase
$\phi=\pi/2$. (a) interhalfcycle factor, (b) multiplication of the intercycle
factor in Fig 7a and the interhalfcycle factor in Figs. 8a, (c) intracycle
factor (multiplication of the intrahalfcycle factor in Figs. 7d and the
interhalfcycle of Figs. 8a), and (d) total distribution calculated as the
multiplication of the intercycle factor in Figs. 7a and the intracycle factor
in Figs. 8c. All distributions are normalized.

The doubly differential photoelectron momentum distribution as a function of
the longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$
for relative phase $\phi=\pi/2$ is shown in Figs. 8. The interhalfcycle factor
in Figs. 8a also consists in a set of concentric rings with a discontinuity
for $k_{z}=0$ since the the vector potential is not antisymmetric (with
respect to the middle of the unit cell) as in the case of $\phi=0$ (see Fig.
3b). Such discontinuities are an artifact of the SPA and also appears for
laser assisted photoionization emission (two colors with one frequency much
higher than the other) Gramajo17 , Gramajo18 . Again, the minima of the
interhalfcycle rings do not match with the sidebands of the intercycle factor
in Figs. 7a and thus, they survive when one multiply the inter- and
interhalfcycle factors (see Fig. 8b). The intracycle factor in Fig 8c inherits
the discontinuity of the interhalfcycle factor of Figs. 8a. The total emission
pattern is the multiplication of the inter- (Figs. 7a) and intracycle (Figs.
8c) patterns (see Fig. 8d).

Figure 9: Doubly differential momentum distribution as a function of the
longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$ for
the $\omega-2\omega$ ionization within the SPA with relative phase
$\phi=\pi$.(a) interhalfcycle factor, (b) multiplication of the intercycle
factor in Fig 7a and the interhalfcycle factor in Figs. 9a, (c) intracycle
factor (multiplication of the intrahalfcycle factor in Figs. 7d and the
interhalfcycle of Figs. 9a), and (d) total distribution calculated as the
multiplication of the intercycle factor in Figs. 7a and the intracycle factor
in Figs. 9c. All distributions are normalized.

Despite the case of $\phi=\pi/2,$ for $\phi=\pi$ the two half cycles have the
same duration and the interhalfcycle distribution is continuous as displayed
in Figs. 9a (as for $\phi=0$). The multiplication of the intercycle factor of
Figs. 7a and the interhalfcycle factor of Figs. 9a is displayed in Figs. 9b
exhibiting all ATIs and sidebands. In Figs. 9c the intracycle pattern is
displayed. The total momentum distribution for $\phi=\pi$ is shown in Figs.
9d. For $\phi=3\pi/2$ the first half cycle of the vector potential is shorter
than its second half cycle, thus, bigger intracycle rings yield for positive
$k_{z},$ as shown in Fig. 10a, instead of for negative $k_{z}$ in the case for
$\phi=\pi/2.$ The multiplication of the intercycle factor of Figs. 7a and the
interhalfcycle factor of Figs. 10a is displayed in Fig. 10b, which shows ATIs
and sidebands. In Fig. 10c the intracycle interference pattern exhibits again
a discontinuity like the case of $\phi=\pi/2$ in Figs. 8c. The total momentum
distribution for $\phi=3\pi/2$ is shown in Fig. 10d.

Figure 10: Doubly differential momentum distribution as a function of the
longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$ for
the $\omega-2\omega$ ionization within the SPA with relative phase
$\phi=3\pi/2$. (a) interhalfcycle factor, (b) multiplication of the intercycle
factor in Fig 7a and the interhalfcycle factor in Fig. 10a, (c) intracycle
factor (multiplication of the intrahalfcycle factor in Figs. 7d and the
interhalfcycle of Fig. 10a), and (d) total distribution calculated as the
multiplication of the intercycle factor in Figs. 7a and the intracycle factor
in Fig. 10c. All distributions are normalized.

In order to test the validity of the SPA, we perform the time integral in Eq.
(4) numerically within the SFA Macri98 , Arbo08a , Kazansky06 . For the sake
of simplicity, we model the atomic argon as a hydrogen-like atom with
effective charge $Z_{\mathrm{eff}}=\sqrt{2n^{2}I_{P}}$, where $n$ is the
principal quantum number of the initial state, in this case $n=3$ and the
initial orbital quantum number is $l=1$ ($p$-state) Belkic79 . This effective
charge ensures the ionization potential to be taken into account properly and,
consequently, the intercycle fringes in the electron spectra to be situated at
the energy values of Eq. (11). We consider an electric field with a ramp on
and ramp off of duration $2\pi/\omega$ each and a flat-top region of duration
$4\pi/\omega$. In Fig. 12 we show the doubly differential momentum
distribution for relative phase $\phi=0$ in (a), $\phi=\pi/2$ in (b),
$\phi=\pi$ in (c), and $\phi=3\pi/2$ in (d). In order to highlight the
interference patterns we have multiplied the momentum distribution by
$\exp(10E)$ and plotted in logarithmic scale to neutralize the exponential
decay of the SFA as a function of the energy. The intercycle interference
pattern appeared as concentric rings situated at $\sqrt{2E_{n}},$ whereas the
intracycle interference pattern does it with the shape of waning and waxing
moons (depending on the sign of $k_{z})$. Whereas distributions for $\phi=0$
and $\pi$ in Figs 12a and 12c exhibit forward-backward symmetry, the momentum
distribution for $\phi=\pi/2$ results in a small asymmetry enhancing forward
emission and, contrary, the momentum distribution for $\phi=3\pi/2$ results in
a small asymmetry enhancing backward emission. The agreement between the SPA
distributions in Figs. 8f, 9f, 10f, and 11f and the SFA distributions in Figs.
11 is very good. Some differences for the angular distribution near threshold
stem from the effect of the dipole matrix element (from a $p$-state to the
continuum) in the SFA, which has been disregarded within the SPA. We have
checked this performing calculations for a hydrogenic atom from a fictitious
$1s$ state with $I_{p}=0.58$ (not shown).

Figure 11: Doubly differential momentum distribution as a function of the
longitudinal momentum $k_{z}$ and the perpendicular momentum $k_{\bot}$ for
the $\omega-2\omega$ ionization within the SFA with relative phase $\phi=0$ in
(a), $\phi=\pi/2$ in (b), $\phi=\pi$ in (c), and $\phi=3\pi/2$ in (d). In
order to highlight the interference patterns we have multiplied the momentum
distribution by $\exp(10E)$ and plotted in logarithmic scale.

###  II.3 Phase delays in $\omega-2\omega$ ionization

In order to get a simple close form, and considering that ionization takes
place at times near the extremes of the electric field [Eq. (17)], at zeroth-
order perturbation, these ionization times are $t_{j}=(j-1)\pi/(2\omega)$ with
$j=1,2,3,$ and $4$. After a bit of algebra the accumulated action becomes

$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle
n\pi+\left(-d-e+g\right)\left(\cos\phi-\sin\phi\right)$ $\displaystyle=$
$\displaystyle n\pi+2\chi\cos\left(\phi+\pi/4\right),$

where
$\chi=\left(-d-e+g\right)/\sqrt{2}=-F_{\omega}\left(k_{z}/\omega^{2}+F_{2\omega}/(3\omega^{3})\right)/\sqrt{2}.$

Figure 12: SPA intracycle interference for ATI peaks and sidebands and their
respective perturbative prediction both given by Eqs. (26) and (27) as a
function of relative phases $\phi$ (lower axis) and $\phi^{\prime}$ (upper
axis) for $\chi=0.5$. ATI maximizes at $\phi=0.25\pi$ and $1.25\pi$
($\phi^{\prime}=3\pi$ and $\pi$), whereas sidebands maximizes at
$\phi=0.75\pi$ and $1.75\pi$ ($\phi^{\prime}=2\pi$ and $0$)

For an ATI peak, $n$ is even then replacing Eq. (II.3) into Eq. (24) we
demonstrate that the intracycle interference probability factor
$\cos^{2}\left(\frac{\Delta S}{2}\right)$ is equal to

$\cos^{2}[\chi\cos\left(\phi+\pi/4\right)]\simeq
1-\frac{\chi^{2}}{2}+\frac{\chi^{2}}{2}\cos\left[2(\phi+3\pi/4)\right]+O(\chi^{4}),$
(26)

where in the last term we have performed a series expansion in terms of the
perturbation parameter $\chi.$ This shows that the phase delay of an ATI peak
is $\phi_{0}=-3\pi/4$. In Fig. 12 we display the analytical expression
$\cos^{2}[\chi\cos\left(\phi+\pi/4\right)]$ and its first order perturbative
approximation $1-\chi^{2}/2+\chi^{2}\cos\left[2(\phi+3\pi/4)\right]/2$ for the
value $\chi=0.5$ which reproduces the SFA value in Zipp14  and the
perturbative theory in Lopez21 .

Instead, for any sideband, $n$ is odd and thus the intracycle interference
probability factor $\cos^{2}\left(\frac{\Delta S}{2}\right)$ can be written as

$\sin^{2}[\chi\cos\left(\phi+\pi/4\right)]\simeq\frac{\chi^{2}}{2}+\frac{\chi^{2}}{2}\cos\left[2(\phi+\pi/4)\right]+O(\chi^{4}),$
(27)

where in the last term we have performed a series expansion in terms of the
peturbation parameter $\chi.$ This shows that the phase delay of any sideband
is $\phi_{0}=-\pi/4,$ or equivalently
$\phi_{0}^{\prime}=-2\left(\phi_{0}+\pi/4\right)=0.$ In Fig. 12 we display the
analytical expression $\sin^{2}[\chi\cos\left(\phi+\pi/4\right)]$ and its
first order perturbative approximation
$1+\chi^{2}\cos\left[2(\phi+\pi/4)\right]/2$ for the value $\chi=0.5$ which
reproduces the SFA in Zipp14  and the perturbative theory in Lopez21 . The
variation of the probabilities of the ATI peaks and sidebands as a function of
the relative phase $\phi$ is small if $\chi$ is small, or equivalently, if the
probe field is weak. It is worth to notice from Eqs. (26) and (27) that the
probability of ATI peaks and sidebands as a function of the relative phase
$\phi$ have opposite phases (phase difference of $\pi$), as expected. The fact
that the addition of the intracycle pattern of ATI and sidebands is unity
assures the conservation of probability. We obtain similar results to the non-
perturbative theory in Ref. Boll16  for RABBIT. Especially, Eqs. (26) and (27)
are similar to Eqs. (14) and (15) in Boll16 . The factor $\chi\simeq-\alpha
k_{z}\left(1+F_{2\omega}/(3\omega
k_{z})\right)/\sqrt{2}=-\vec{\alpha}_{\omega}\cdot\vec{k}\left(1+\vec{\alpha}_{2\omega}\cdot\vec{k}/(3E_{z})\right)/\sqrt{2},$
where $\vec{\alpha}_{\omega}=\vec{F}_{\omega}/\omega^{2},$
$\vec{\alpha}_{2\omega}=\vec{F}_{2\omega}/\left(2\omega\right)^{2}$ represents
the quiver vector for the two different $\omega$ and $2\omega$ fields and
$E_{z}=k_{z}^{2}/2$. Therefore, it may be thought that the interhalfcycle
interference pattern stem from two point sources

$\displaystyle\cos^{2}[\chi\cos\left(\phi+\pi/4\right)]$ $\displaystyle=$
$\displaystyle\cos^{2}\left[\vec{k}\cdot\left(\vec{R}_{+}-\vec{R}_{-}\right)/2\right]\qquad\text{{(ATI)}}$
$\displaystyle\sin^{2}[\chi\cos\left(\phi+\pi/4\right)]$ $\displaystyle=$
$\displaystyle\sin^{2}\left[\vec{k}\cdot\left(\vec{R}_{+}-\vec{R}_{-}\right)/2\right]\qquad\text{{(SB)}}$

at
$\vec{R}_{+(-)}=\pm\vec{\alpha}_{\omega}\left(1+\vec{\alpha}_{2\omega}\cdot\vec{k}/(3E_{z})\right)\cos\left(\phi+\pi/4\right)/\sqrt{2}\hat{z}$,
similar to a diatomic molecule aligned along the polarization axis. For the
case of the ATIs, these two point sources emit in phase and constructuve
interference is produced at perpendicular emission; instead, for sidebands,
the two point sources emit in counterphase, leading to partial destructive
interference in the perpendicular direction since in this case
$\chi=-F_{\omega}F_{2\omega}/(3\sqrt{2}\omega^{3})$ and not zero as for the
emission from a diatomic molecule. In Figs. 7c, 8c, 9c, and 10c we observe a
minimum as a function of the angle, whereas ATIs exhibit maxima values.

Figure 13: Energy distribution in the forward direction as a function of the
relative phase $\phi$ for the $\omega-2\omega$ ionization within the SPA. (a)
Intercycle factor, (b) interhalfcycle factor, (c) multiplication of (a) and
(b), (d) intrahalfcycle factor, (e) intracycle factor [multiplication of
distributions in (b) and (d)], and (f) the total momentum distribution
[multiplication of (c) and (d)]. All distributions are normalized.

In Fig. 13 we show the energy spectrum in the forward direction as a function
of the relative phase $\phi$ between the two colors within the SPA. In Fig
13a, we show that the intercycle factor is independent $\phi$. All multiphoton
peaks (ATIs and sidebands) are present in the intercycle factor in Fig. 13a
with separation of one $\omega$ photon energy. The interhalfcycle factor can
be observed as a $2\pi$-periodic function in Fig. 12b. The separation between
interhalfcycle maxima corresponds to a $2\omega$ photon energy and the
amplitude of the oscillation increases with energy, since the accumulated
action $\Delta S$ in Eq. (II.3) increases with $k_{z}=\sqrt{2E}$ through the
factor $\chi$. The interplay between the inter- and interhalfcycle
interferences is plotted in Fig. 13c where both ATI peaks and sidebands arise.
From Eq. (11), we see that the first multiphoton peak just above threshold
($E_{n}=0.0038$ a.u.) corresponds to $n=11$ and, as it is an odd number, it is
a sideband with $\phi_{0}=3\pi/4$ and $7\pi/4$ (as all sidebands). In turn,
ATI peaks maximize at $\phi_{0}=\pi/4$ and $5\pi/4.$ This confirms our
prediction of Eqs. (26) and (27) shown in Fig. 12.

Figure 14: Energy distribution in the forward direction as a function of the
relative phase $\phi$ for the $\omega-2\omega$ ionization within the SFA. In
order to highlight the interference patterns we have multiplied the momentum
distribution by $\exp(15E)$

We have also calculated the SFA forward emission spectrum as a function of the
relative phase $\phi$ (see Fig. 14). In order to highlight the interference
patterns we have multiplied the momentum distribution by $\exp(15E)$ to
neutralize the exponential decay of the SFA as a function of the energy. One
observe a very good agreement between SFA (Fig. 14) and SPA results (Figs. 13c
and 13f). We think that the small deviations of the SFA from the SPA stem from
the inclusion of the starting and ending ramps. It can be observed a
$\phi$-independent modulation with a minimum about $0.2$ a.u. similar to the
intrahalfcycle interference in Fig. 13d, 13e, and 13f. Besides,there is an
anomaly in the alternation of ATIs and sidebands at energy close to the
intrahalfcycle minimum at $E\sim 0.25$ . This phenomenon could be due to the
cooper minimum stemming from the $3p$ initial state of the argon atom.

Figure 15: Asymmetry parameter as a function of the energy $E$ and relative
phase $\phi$ in (a) for the SPA and (d) for the SFA. The horizontal dash lines
indicate the energy positions of ATIs (grey) and sidebands (white). In (b) the
asymmetry parameter for the first ATI (at $E=0.061)$ is plotted for the SPA
and the SFA. In (d) the asymmetry parameter for the second sideband (at
$E=0.118)$ is plotted for the SPA and the SFA.

Another important quantity to map out ionization phases in the
$\omega-2\omega$ protocol is the forward-backward
($\theta\leftrightarrow\pi-\theta$) asymmetry of the photoelectron emission
probability

$A(E,\phi)=\frac{\frac{dP}{dE}(\theta=0,\phi)-\frac{dP}{dE}(\theta=\pi,\phi)}{\frac{dP}{dE}(\theta=0,\phi)+\frac{dP}{dE}(\theta=\pi,\phi)},$
(29)

where the forward (backward) emission spectra $\frac{dP}{dE}(\theta=0,\phi)$
($\frac{dP}{dE}(\theta=\pi,\phi)$) are defined in Eq. (10). In Fig. 29a we
show the SPA asymmetry parameter $A(E,\phi)$ as a function of the final
electron kinetic energy $E$ and the relative phase $\phi.$ The energy
positions of ATIs are marked with a horizontal grey dash line whereas the
position of sidebands are marked with a white dahline. In Fig. 29d we show the
corresponding asymmetry parameter calculated within the SFA. At first shight,
there are significant differences between the SPA and SFA results, however,
when one inspect on the asymmetry at the position of the first ATI (in Fig.
29b) and second sideband (in Fig. 29c), similar oscillatory behaviors are
found, maximizing the ATIs at $\phi=0.25\pi$ and $1.25\pi$ and thee sidebands
at $\phi=0.75\pi$ and $1.75\pi.$

For a close comparison between our SPA and SFA results with the perturbative
theory developed in Ref. Zipp14  accompanying an experiment for the ionization
of atomic argon by a $\omega-2\omega$ and also our recent perturbative theory
developed in Ref. Lopez21 , we perform the transformation
$t=t^{\prime}+\phi^{\prime}/(2\omega)-\pi/(4\omega)$ and
$\phi=-\phi^{\prime}/2-\pi/4$ in Eq. (17), becoming the electric field in Eq.
(30) equivalent to the following expression [see Eq. (1) of Ref. Zipp14  only
differing in a factor 2 for the definition of the frequencies, and Eq. (1) of
Ref. Lopez21 ,

$\vec{F}(t^{\prime})=f(t^{\prime})\left[F_{2\omega}\sin\left(2\omega
t^{\prime}+\phi^{\prime}\right)+F_{\omega}\sin(\omega
t^{\prime})\right]\hat{z},$ (30)

where we have supposed that the envelope $f(t)$ remains invariant due to its
smoothness as a function of time. In Eq. (30) $\phi^{\prime}$ is the relative
phase of the second harmonic with respect to the fundamental laser field.
Figs. 5, 13c, 13f, and 15 show that the ATIs maximize at $\phi_{0}=\pi/4$ and
$5\pi/4,$ which is equivalent to
$\phi_{0}^{\prime}=-2\left(\phi_{0}+\pi/4\right)=\pi,$ and $3\pi,$ whereas the
sidebands maximize at $\phi_{0}=3\pi/4$ and $7\pi/4,$ which is equivalent to
$\phi_{0}^{\prime}=-2\left(\phi_{0}+\pi/4\right)=0,$ and $2\pi$ (modulo
$2\pi$). Therefore, there are an agreement not only between our SFA and SPA
calculations but also with our own perturbation theory Lopez21  and the
perturbation theory in Ref. Zipp14 .

##  III Conclusions

We have developed a non-perturbative strong field theory for the atomic
ionization by a linearly polarized $\omega-2\omega$ laser pulse. We have
derived the formation of sidebands as a result of the interplay between inter-
and interhalfcycle interference patterns stemming from the effect of a
$\omega$ field with respect to a stronger $2\omega$ component. We have
individualized both interhalf- and intrahalfcycle interferences conforming the
intracycle interference pattern. We show that phase delays calculated within
our SPA agree not only with our SFA calculations but also with previous
perturbation theories Zipp14 , Lopez21  extending their validity to stronger
pulses.

## Acknowledgements

This work was supported by CONICET PIP0386, PICT-2016-0296 PICT-2017-2945 and
PICT-2016-3029 of ANPCyT (Argentina). D.G.A especially thanks S. Eckart, M.
Dalhström, and M. Bertolino for fruitfull discussions.

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