Time-Dependent Theory of Electron Emission Perpendicular to Laser Polarization for Reconstruction of Attosecond Harmonic Beating by Interference of Multiphoton Transitions

Abstract

We present a time-dependent nonperturbative theory of the reconstruction of attosecond beating by interference of multiphoton transitions (RABBIT) for photoelectron emission from hydrogen atoms in the transverse direction relative to the laser polarization axis. Extending our recent semiclassical strong-field approximation (SFA) model developed for parallel emission, we deduce analytical expressions for the transition amplitudes and demonstrate that the photoelectron probability distribution can be factorized into interhalf- and intrahalfcycle interference contributions, the latter modulating the intercycle pattern responsible for sideband formation. We identify the intrahalfcycle interference arising from trajectories released within the same half cycle as the mechanism governing attosecond phase delays in the perpendicular geometry. Our results reveal the suppression of even-order sidebands due to destructive interhalfcycle interference, leading to a characteristic spacing between adjacent peaks that doubles the standard spacing observed along the polarization axis. Comparisons with numerical calculations of the SFA and the ab initio solution of the time-dependent Schrödinger equation confirm the accuracy of the semiclassical description. This work provides a unified framework for understanding quantum interferences in attosecond chronoscopy, bridging the cases of parallel and perpendicular electron emission in RABBIT-like protocols.

1. Introduction

Attosecond science has opened the way to probe and control electronic dynamics in matter on their natural timescales. Pump–probe techniques combining attosecond extreme-ultraviolet (XUV) pulses with near-infrared (NIR) or visible laser fields provide access to phase and timing information encoded in photoelectron wave packets. Among them, attosecond streaking [1,2,3] and the reconstruction of attosecond beating by interference of two-photon transitions (RABBIT) [4,5] have become the keystone of attosecond chronoscopy of photoionization processes in atoms [4,6,7,8,9,10], molecules [11,12,13,14,15,16,17], and solids [3,18,19,20,21]. While streaking can be interpreted classically as an energy shift induced by the probe field [22,23,24,25,26,27], RABBIT is understood as a quantum interferometer arising from multiple indistinguishable paths towards one final continuum state [8,22].

Most theoretical descriptions of RABBIT fall back on perturbative treatments of the probe field, which assume that only two-photon processes dominate and that the probe field intensity is weak enough not to alter the ionization dynamics [28,29,30]. Nevertheless, the continuum–continuum transitions triggered by the probe are fundamental to the measurement process and cannot be ignored. Approaches relying on the strong-field approximation (SFA) [24,31,32,33,34,35,36,37] offer a robust nonperturbative framework that enables the study of RABBIT outside the perturbative regime, allowing the inclusion of multiple probe photons and allowing the identification of different interfering electron trajectories responsible for the observed phase delays. In our recent work [29], we developed a semiclassical time-dependent nonperturbative model within the SFA to describe electron emission in a RABBIT-like scheme along the polarization direction. We demonstrated that the photoelectron probability can be factored into intracycle and intercycle contributions, and we identified intracycle interference as the main mechanism governing the phase delays.

Anisotropy of photoemission delays was first discovered by Keller’s group [38], and subsequent angle-resolved studies of laser-assisted photoionization emission (LAPE) have revealed rich interference structures beyond the polarization axis [39]. Fano’s propensity rule was identified to explain the $\pi$ phase change of RABBIT in perpendicular emission as occurring due to different angular shapes following absorption or emission of IR photons [40,41]. In particular, photoelectron emission in the perpendicular direction exhibits a distinct interference pattern [42]. This leads to a natural factorization of the spectrum into intrahalfcycle and interhalfcycle contributions, which modulate the well-known intercycle interference responsible for sideband formation. Unlike the case of parallel emission [30], perpendicular emission shows destructive interference between half cycles for the absorption and emission of even numbers of NIR photons, resulting in the suppression of certain sidebands and the double of the characteristic energy spacing between neighboring peaks [42,43,44,45]. These features highlight the crucial role of symmetry and emission geometry to determine the structure of the photoelectron spectrum (PES).

In this work, we generalize the nonperturbative, time-dependent SFA-based RABBIT framework introduced in [29] to describe photoelectron emission in the direction orthogonal to the laser polarization axis. We demonstrate that our approach provides a unified semiclassical framework for understanding perpendicular emission in RABBIT-like protocols. We obtain an analytical factorization of the photoelectron momentum distribution into intrahalfcycle and interhalfcycle interference contributions. We also demonstrate that the intrahalfcycle interference is directly responsible for the attosecond phase delays. This work thus bridges previous studies on parallel [29] and perpendicular [42] emission, offering new insights into the role of quantum interferences in attosecond chronoscopy.

The structure of the paper is as follows. Section 2 outlines the semiclassical model (SCM) employed to compute photoelectron spectra for emission perpendicular to the laser polarization within the RABBIT scheme. Section 3 presents and analyzes the results, comparing the SCM predictions with those obtained from the SFA and from ab initio calculations based on the time-dependent Schrödinger equation (TDSE) [46,47,48]. Section 4 contains the concluding remarks. Unless noted otherwise, all quantities in this work are expressed in atomic units.

2. Theory and Methods

We study the ionization of an atom interacting with an external laser field in the single-active-electron (SAE) approximation. The TDSE reads

$$i{\displaystyle \frac{\partial }{\partial t}}\left|\psi \left(t\right)\right.〉=\left[H_0+H_{\mathrm{int}}\left(t\right)\right]\left|\psi \left(t\right)\right.〉,$$

(1)

where $H_0={\overrightarrow{p}}^2/2+V\left(r\right)$ denotes the field-free atomic Hamiltonian, with the first term representing the electron’s kinetic energy and the second describing its Coulomb interaction with the ionic core. The laser–atom coupling, referred as the interaction Hamiltonian $H_{\mathrm{int}}$, on the right-hand side of Equation (1) produces the promotion of the initially bound electron in the atomic state $|{\varphi }_i\rangle$ to the continuum final state $|{\varphi }_f\rangle$ with energy $E=k^2/2$ and final momentum $\overrightarrow{k}$. The photoelectron momentum spectrum can then be evaluated as

$${\displaystyle \frac{dP}{d\overrightarrow{k}}}={\left|T_{if}\right|}^2$$

(2)

where $T_{if}$ denotes the transition amplitude ${\varphi }_i\to {\varphi }_f$.

In time-dependent distorted-wave theory, the prior-form transition amplitude is given by [33,49]

$$T_{if}=-i{\int }_{-\infty }^{+\infty }dt\langle {\chi }_f^{-}(\overrightarrow{r},t)|H_{\mathrm{int}}(\overrightarrow{r},t)|{\varphi }_i(\overrightarrow{r},t)\rangle .$$

(3)

Here, ${\varphi }_i(\overrightarrow{r},t)={\phi }_i\left(\overrightarrow{r}\right)e^{i{I}_{p}t}$ corresponds to the initial atomic state characterized by the ionization potential $I_p$, while ${\chi }_f^{-}(\overrightarrow{r},t)$ is the final-state wave function. Within the SFA framework, the Coulomb influence of the residual ion on the emitted electron is ignored, so the final state reduces to the Volkov solution [50], ${\chi }_f^{-}={\chi }_f^V$, describing a free electron interacting solely with the electromagnetic field. For simplicity, in this work, we consider photoionization of a hydrogen atom due to a short XUV pulse train produced by a pair of consecutive odd high-order harmonics and supported by a near-infrared (NIR) driving field at the fundamental frequency $\omega$. The approach developed here can be straightforwardly extended to other atomic systems within the SAE framework. In our setup, both the HH fields and the NIR laser are linearly polarized along the same axis, $\overrightarrow{F}\left(t\right)=F\left(t\right),\widehat{z}$, where

$$\begin{array}{ccc}\hfill F\left(t\right)=f_0\left(t\right)F_{0}cos(\omega t+\varphi )& +& f_{m-1}\left(t\right)F_{m-1}cos\left[\left(2m-1\right)\omega t+{\varphi }_{m-1}\right]\hfill \\ & +& f_m\left(t\right)F_{m}cos\left[\left(2m+1\right)\omega t+{\varphi }_m\right],\hfill \end{array}$$

(4)

where $f_0\left(t\right)$, $f_{m-1}\left(t\right)$, and $f_m\left(t\right)$ are functions between 0 and 1 that mimic the corresponding pulse envelopes, and $F_0, F_{m-1},$ and $F_m$ are the field strengths of the NIR and HH lasers, respectively. Each harmonic takes on a phase ${\varphi }_{m-1}$ and ${\varphi }_m$, while the NIR takes a phase $\varphi$. In the case of a long pulse that is smoothly switched on and off, the vector potential $\overrightarrow{A}\left(t\right)=-{\int }_{-\infty }^{t}d{t}^{\prime }\overrightarrow{F}\left(t^{\prime }\right)$ takes the following form in its central portion:

$$\overrightarrow{A}\left(t\right)\simeq -{\displaystyle \frac{f_0\left(t\right)}{\omega }}{F}_{0}sin(\omega t+\varphi )\widehat{z},$$

(5)

where we have supposed that the harmonic intensity of this component is much smaller than that of the fundamental field, satisfying $F_m/(2m+1)\ll F_0$ for large harmonic orders, i.e., $m\gg 1$. Therefore, the vector potential of the two HHs can be neglected.

Our study is limited to photoelectron momenta in the direction orthogonal to the field’s polarization axis ($\overrightarrow{k}=k\widehat{\rho }$ in cylindrical coordinates). We analyze similarities and differences with those obtained for emission along the polarization axis ($\overrightarrow{k}=k\widehat{z}$), a configuration recently investigated in [29]. As we use the length gauge in the present work, i.e., $H_{\mathrm{int}}(\overrightarrow{r},t)=\overrightarrow{r}·\overrightarrow{F}\left(t\right)$, then from Equation (4), the expression for $T_{if}$ in Equation (3) is decomposable into parts corresponding to the different driving fields, i.e., $T=T_0+T_{m-1}+T_m$, with

$$T_m=F_m{\int }_{-\infty }^{+\infty }{\ell }_m\left(t\right)e^{i\left[S_m\left(t\right)-{\varphi }_m\right]}dt,$$

(6)

where

$${\ell }_m\left(t\right)=-{\displaystyle \frac{i}{2}}{f}_m\left(t\right)\widehat{z}·\overrightarrow{d}\left[\overrightarrow{k}+\overrightarrow{A}\left(t\right)\right],$$

(7)

and

$$S_m\left(t\right)=-{\int }_t^{\infty }d{t}^{\prime }\left\{{\displaystyle \frac{{\left[\overrightarrow{k}+\overrightarrow{A}\left(t^{\prime }\right)\right]}^2}{2}}+I_p-(2m+1)\omega \right\},$$

(8)

where the dipole matrix element is given by $\overrightarrow{d}\left(\overrightarrow{v}\right)={\left(2\pi \right)}^{-3/2}\langle e^{i\overrightarrow{v}·\overrightarrow{r}}|\overrightarrow{r}|{\phi }_i\left(\overrightarrow{r}\right)\rangle$, and $S_m\left(t\right)$ denotes the generalized classical action associated with the harmonic frequency $(2m+1)\omega$, i.e., HH$(2m+1)$. The vector potential [Equation (5)] is only used in the action in the exponent, while the full field [Equation (4)] including all terms is used in the electric field in the exponential prefactor. With a suitable selection of NIR and XUV parameters, the energy region associated with the LAPE processes driven by HH$(2m+1)$ and HH$(2m-1)$ remains well separated from the energy range corresponding to ionization by the NIR field alone. Therefore, we can neglect the contribution of NIR ionization $T_0$ with respect to the contribution of the XUV pulse train. We justify this assumption in the next section. Equation (6) employs the rotating-wave approximation [29,30,39,42], treating the high-frequency ionization amplitude associated with HH$(2m+1)$ as a single-photon absorption event with energy $(2m+1)\omega$. In the flat-top interval of the vector potential in Equation (5)—that is, when $f_0\left(t\right)=1$—the temporal integral in Equation (8) can be solved analytically, i.e.,

$$S_m\left(t\right)=a_{m}t-{\displaystyle \frac{U_p}{2\omega }}sin(2\omega t+2\varphi ),$$

(9)

where $a_m=k^2/2+I_p+U_p-(2m+1)\omega$ and $U_p={(F_0/2\omega )}^2$ define the electron ponderomotive energy under the fundamental-frequency NIR electric field.

The RABBIT methodology can be interpreted as a specific realization of LAPE in which a sequence of high-harmonic components initiates ionization. To streamline the discussion, we focus on the situation where only two contiguous odd harmonics—HH$(2m-1)$ and HH$(2m+1)$—with frequencies $(2m-1)\omega$ and $(2m+1)\omega$ interact with the atom. After ionization, the electron may undergo continuum–continuum transitions driven by the fundamental field of frequency $\omega$. In this framework, the transition probability from the initial bound state to the continuum is described as the coherent sum of the contributions associated with each harmonic, i.e.,

$${\left|T\right|}^2={\left|T_{m-1}+T_m\right|}^2,$$

(10)

where $T_m$ is defined as in Equation (6). The case of a single contributing HH has already been studied, showing that the phase $\varphi$ is not important, affecting the photoelectron spectrum only through boundary conditions of the short XUV pulse [30,39,42]. Instead, in the RABBIT protocol, the phases associated with the two transition amplitudes, $T_{m-1}$ and $T_m$, in Equation (10) become relevant. In the RABBIT scheme, the introduction of multiple high-frequency components in the pump field—two in the present case—removes the time-translation invariance that exists when ionization is driven by a single harmonic. As a result, one can define a reference time for the phase of the probe field [29]. Consequently, the ionization probability [Equation (10)] is sensitive to the relative phases $\varphi$, ${\varphi }_{m-1}$, and ${\varphi }_m$ in Equation (4).

Assuming that the constant portion of the NIR envelope field begins at $t=0$, the square of the vector potential in Equation (5) is periodic every half cycle, i.e., $A^2(t+j\pi /\omega )=A^2\left(t\right)$, or equivalently, $\overrightarrow{A}(t+j\pi /\omega )={(-1)}^j\overrightarrow{A}\left(t\right),$ with j being any positive integer number, as well as the square of the dipole moment, i.e., $d^2\left[\overrightarrow{k}+\overrightarrow{A}(t+j\pi /\omega )\right]=d^2\left[\overrightarrow{k}+\overrightarrow{A}\left(t\right)\right],$ or equivalently, $\overrightarrow{d}\left[\overrightarrow{k}+\overrightarrow{A}(t+j\pi /\omega )\right]={(-1)}^j\overrightarrow{d}\left[\overrightarrow{k}+\overrightarrow{A}\left(t\right)\right]$ (see Appendix A). We also find that the second term on the right-hand side of Equation (9) exhibits temporal oscillations with a period $\pi /\omega$, which is half the period of the driving field, namely,

$$S_m(t+j{\displaystyle \frac{\pi }{\omega }})=S_m\left(t\right)+{\displaystyle \frac{j\pi a_m}{\omega }}.$$

(11)

The integration domain of Equation (6) extends to the non-zero values of the envelope $f_m\left(t\right),$ for simplicity $\tau =2\pi N/\omega$, with N any real positive number. However, we specify our analysis for half-integer values of $N.$ Thus, we can rewrite the matrix element $T_m$ due to the contribution of HH($2m+1$) in Equation (6) as the addition of N optical laser cycles of the probe field, namely

$$\begin{array}{ccc}\hfill T_m& =& F_m{\int }_0^{2\pi N/\omega }{\ell }_m\left(t\right)e^{i\left[S_m\left(t\right)-{\varphi }_m\right]}dt\hfill \\ & =& F_m{e}^{-i{\varphi }_m}\sum _{j=0}^{N-1}{\int }_{2\pi j/\omega }^{2\pi (j+1)/\omega }{\ell }_m\left(t\right)e^{i{S}_m\left(t\right)}dt.\hfill \end{array}$$

(12)

By applying the change of variables $t=t^{\prime }+j,2\pi /\omega$, the time integral in the second line of Equation (12) is shifted by j cycles of the probe field. Taking into account the periodic behavior of ${\ell }_m\left(t\right)$, which follows from the periodicity of the dipole term when $f_m\left(t\right)=1$, the transition amplitude can then be factorized as [51,52]

$$\begin{array}{ccc}\hfill T_m& =& F_m{e}^{-i{\varphi }_m}\sum _{j=0}^{N-1}{e}^{i2\pi a_{m}j/\omega }{\int }_0^{2\pi /\omega }{\ell }_m\left(t^{\prime }\right)e^{i{S}_m\left(t^{\prime }\right)}d{t}^{\prime }\hfill \\ & =& F_m{e}^{-i{\varphi }_m}{\displaystyle \frac{sin{(a}_m\pi N/\omega )}{sin{(a}_m\pi /\omega )}}{e}^{i\pi a_m(N-1)/\omega }{I}_m\left(k\right),\hfill \end{array}$$

(13)

where the factor

$$I_m\left(k\right)={\int }_0^{2\pi /\omega }{\ell }_m\left(t^{\prime }\right)e^{i{S}_m\left(t^{\prime }\right)}d{t}^{\prime }$$

(14)

in Equation (13) accounts for the contribution of HH$(2m+1)$ ionization produced during a single cycle of the fundamental field.

When the XUV pulse comprises an integer number of optical cycles N of the fundamental field, the factors $exp[i\pi a_m(N-1)/\omega ]$ and $sin{(a}_m\pi N/\omega )/sin{(a}_m\pi /\omega )$ do not depend on the harmonic index m. Therefore, they can be factorized in Equation (10), leading to

$${\left|T\right|}^2=\underset{\mathtt{intercycle}}{\underbrace{{\left|{\displaystyle \frac{sin{(a}_m\pi N/\omega )}{sin{(a}_m\pi /\omega )}}\right|}^2}}\underset{\mathtt{intracycle}}{\underbrace{{\left|F_{m-1}{e}^{-i{\varphi }_{m-1}}{I}_{m-1}\left(k\right)+F_m{e}^{-i{\varphi }_m}{I}_m\left(k\right)\right|}^2}}.$$

(15)

We manage to split the total transition probability in the RABBIT protocol into the intercycle interference factor [first factor in Equation (15)] and the intracyle interference factor [second factor in Equation (15)] [29,33,52,53]. The condition for the denominator of the intercycle factor to vanish, $a\ast m\pi /\omega =n\pi$, yields the energy values

$$E_{2m\pm 1+n}=\left(2m\pm 1+n\right)\omega -I_p-U_p,$$

(16)

associated with absorption of a single photon from HH$(2m\pm 1)$, with a frequency of $(2m\pm 1)\omega$, together with the additional absorption or emission of (|n|) fundamental photons at frequency $\omega$ for $n>0$ or $n<0$, respectively. The resulting maxima at the energies given by Equation (16) correspond to the SBs in the PES of the HH$\left(2m\pm 1\right)$ in the presence of the fundamental laser [29,30,39,42], and they are also the SBs formed by the laser-dressed train of pulses composed by HH$(2m-1)$ and HH$(2m+1)$. In fact, when the number of cycles of the NIR $N\to \infty$, the intercycle factor satisfies the energy conservation. When the pump pulse has a finite duration $\tau$ (approximately $2\pi N/\omega$), the sidebands are no longer infinitely sharp but instead exhibit a width $\mathrm{\Delta }E$∼$\omega /N$. This satisfies the uncertainty condition $\mathrm{\Delta }E,\tau \sim 2\pi$, with $\tau$ representing the pulse duration. In Figure 1, we have depicted the corresponding SBs as a function of the intensity of the NIR laser, which results in straight lines of negative slope $-U_p/I$. One can observe that their energies are straight lines, since the ponderomotive energy $U_p$ increases proportionally with the NIR laser intensity. In Figure 1 we can observe the SBs’ values given by Equation (16) for a combination of values of m and n.

Alternatively, we can regard the intracycle factor $I_m\left(k\right)$ in Equation (14) as the separate contribution of the two half cycles

$$\begin{array}{ccc}\hfill I_m\left(k\right)& =& {\int }_0^{{\displaystyle \frac{\pi }{\omega }}}{\ell }_m\left(t^{\prime }\right)e^{i{S}_m\left(t^{\prime }\right)}d{t}^{\prime }+{\int }_{{\displaystyle \frac{\pi }{\omega }}}^{{\displaystyle \frac{2\pi }{\omega }}}{\ell }_m\left(t^{\prime }\right)e^{i{S}_m\left(t^{\prime }\right)}d{t}^{\prime }\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& =& 2i{e}^{i{\displaystyle \frac{\pi a_m}{2\omega }}}sin\left[{\displaystyle \frac{\pi a_m}{2\omega }}\right]J_m\left(k\right)\hfill \end{array}$$

(18)

where in the second term of Equation (17) the substitution $t^{\prime \prime }=t^{\prime }-\pi /\omega$ has been performed, and the periodicity properties of ${\ell }_m\left(t\right)$ and $S_m\left(t\right)$ have been used to arrive at Equation (18). The factor $sin\left[\pi a_m/\left(2\omega \right)\right]$ in Equation (18) corresponds to the interference between the ionization during the first and second half cycles of only one optical cycle of the NIR laser and its square, known as the interhalfcycle interference pattern, which cancels at $a_m=n\omega$ with $\left|n\right|=0,2,4,\dots$ As a consequence, LAPE channels corresponding to the absorption or emission of an even number of NIR photons are suppressed or, equivalently, $\left|n\right|$ must be odd in Equation (16). Therefore, SBs with even numbers of exchanged photons in Figure 1, shown in dotted–dashed light blue (gray), become frustrated due to the interhalfcycle interference. This result is not surprising because the model assumes an initial 1 s state. In a partial wave development of the transition amplitude, the corresponding odd spherical harmonics (reached by an even number of NIR photons and one XUV photon) have zero values for perpendicular direction, i.e., $\theta =\pi /2$, while the even spherical harmonics (reached by an odd number of NIR photon and one XUV photon) are non-zero in the perpendicular direction [42,43,44,45]. Conversely, our SCM shows no need for any partial wave development. The last factor in Equation (18) corresponds to the ionization amplitude due to HH$(2m+1)$ occurring during only one half-cycle pulse of the NIR, defined as

$$J_m\left(k\right)={\int }_0^{{\displaystyle \frac{\pi }{\omega }}}\ell \left(t^{\prime }\right)e^{i{S}_m\left(t^{\prime }\right)}d{t}^{\prime }.$$

(19)

Taking into account Equations (17) and (19), the transition probability of LAPE corresponding to HH$(2m+1)$ becomes

$$\begin{array}{ccc}\hfill T_m& =& 2i{e}^{i\pi a_m(N-1/2)/\omega }{F}_m{e}^{-i{\varphi }_m}{\displaystyle \frac{sin{(a}_m\pi N/\omega )}{sin{(a}_m\pi /\omega )}}sin\left[\pi a_m/\left(2\omega \right)\right]J_m\left(k\right)\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& =& i{e}^{i\pi a_m(N-1/2)/\omega }{F}_m{e}^{-i{\varphi }_m}{\displaystyle \frac{sin{(a}_m\pi N/\omega )}{cos\left[a_m\pi /\left(2\omega \right)\right]}}{J}_m\left(k\right).\hfill \end{array}$$

(21)

It is easy to show that when the duration of both laser fields of HH$(2m-1)$ and HH$(2m+1)$ are an integer number of half cycles of the NIR laser, i.e., $\tau =2\pi N/\omega$ with $2N=1,2,3,\dots$, the factor $exp[i\pi a_{m-1}(N-1/2)/\omega ]sin{(a}_{m-1}\pi N/\omega )/cos\left[a_{m-1}\pi /\left(2\omega \right)\right]$ is independent of the HH index $m,$ which means that it can be factorized in Equation (10) as

$${\left|T\right|}^2=\underset{\mathrm{interhalfcycle}}{\underbrace{{\left({\displaystyle \frac{sin{(a}_m\pi N/\omega )}{cos\left[a_m\pi /\left(2\omega \right)\right]}}\right)}^2}}\underset{\mathrm{intrahalfcycle}}{\underbrace{{\left|F_{m-1}{e}^{-i{\varphi }_{m-1}}{J}_{m-1}\left(k\right)+F_m{e}^{-i{\varphi }_m}{J}_m\left(k\right)\right|}^2}}.$$

(22)

Equation (22) indicates that the photoelectron yield factorizes into an interhalfcycle factor—capturing interference between emissions from different half cycles—and an intrahalfcycle factor that describes interference processes occurring within a single half cycle. So far, it is worth mentioning that factorizations into interhalf- and intrahalfcycle [Equation (22)] or inter- and intracycle interferences [Equation (15)] are direct consequences of the time symmetries of the problem and rely only on the SFA.

We can compute $J_m\left(k\right)$ either numerically, whose result we call SFA, or analytically, by using the saddle-point approximation, which is the cornerstone of SCM; namely

$$J_m\left(k\right)=\sum _{\alpha =1}^2{g}_m(k,t^{\left(\alpha \right)})exp\left[i{S}_m\left(t^{\left(\alpha \right)}\right)+i{\displaystyle \frac{\pi }{4}}\mathrm{sgn}\left[{\ddot{S}}_m\left(t^{\left(\alpha \right)}\right)\right]\right],$$

(23)

whose saddle times $t^{\left(\alpha \right)}$ and weighting factor $g_m(k,t^{\left(\alpha \right)})$ are found in Appendix B. In Figure 1 we plot the classical allowed regions for the LAPE ionization from HH29 and HH31, observing an overlap between them, which conforms to the classical allowed region in our non-perturbative SCM for RABBIT (see Appendix B). The minimum value of the classical allowed region for RABBIT corresponds to $U_P=\omega$, which, for $\omega =0.05$ a.u., corresponds to $I_{min}=1.7\times {10}^{13}$ W/cm², which is shown in Figure 1 with the vertical left dotted line.

Using the accumulated action $\mathrm{\Delta }{S}_m=S_m\left(t^{\left(2\right)}\right)-S_m\left(t^{\left(1\right)}\right)$ and the average action ${\overline{S}}_m=[S_m\left(t^{\left(1\right)}\right)+S_m\left(t^{\left(2\right)}\right)]/2$, the expression for $J_m\left(k\right)$ becomes

$$J_m\left(k\right)=2g_m\left(k\right)e^{i{\overline{S}}_m}{\gamma }_m\left(k\right),$$

(24)

where the factor ${\gamma }_m\left(k\right)$ is given by

$${\gamma }_m\left(k\right)=\left\{\begin{array}{cc}cos\left({\displaystyle \frac{\mathrm{\Delta }{S}_m}{2}}-{\displaystyle \frac{\pi }{4}}\right),\hfill & \mathrm{if}A\left(t^{\left(1\right)}\right)A\left(t^{\left(2\right)}\right)\ge 0,\hfill \\ sin\left({\displaystyle \frac{\mathrm{\Delta }{S}_m}{2}}-{\displaystyle \frac{\pi }{4}}\right),\hfill & \mathrm{if}A\left(t^{\left(1\right)}\right)A\left(t^{\left(2\right)}\right)<0.\hfill \end{array}\right.$$

(25)

Summing up, within the SCM, the RABBIT transition probability to the continuum can be written as

$${\left|T\right|}^2=\underset{\mathrm{interhalfcycle}}{\underbrace{{\left|{\displaystyle \frac{sin\left(\frac{\pi a_{m}N}{\omega }\right)}{cos\left(\frac{{\pi a}_m}{2\omega }\right)}}\right|}^2}}\underset{\mathrm{intrahalfcycle}}{\underbrace{{\left|F_{m-1}{e}^{-i{\varphi }_{m-1}}{g}_{m-1}\left(k\right)e^{i{\overline{S}}_{m-1}}{\gamma }_{m-1}\left(k\right)+F_m{e}^{-i{\varphi }_m}{g}_m\left(k\right){\gamma }_m\left(k\right)e^{i{\overline{S}}_m}\right|}^2}},$$

(26)

where the average action fulfills ${\overline{S}}_m=-n(\varphi -f_m)$, with $f_m$ being a constant multiple of $\pi /2$. When electrons are emitted perpendicularly, $f_m$ is not affected by changes in laser intensity, whereas in the parallel case it varies with field strength [29]. In Figure 2a we plot the interhalfcycle interference factor with the solid orange line. We observe that even values of exchanged NIR photons n are precluded, meaning that we can name even SBs as frustrated SBs. This characteristic applies not only to the RABBIT protocol, but also to LAPE and photoionization in the perpendicular direction [42,43,44,45]. In Figure 2b, through its modulation of the intracycle pattern, the intrahalfcycle interference factor ultimately governs the modulation of the total ionization probability. The small peaks at $E\simeq 0.77$ and $0.8$ correspond to the incipient SB with $n=0$, which is suppressed by interhalfcycle interference. In Figure 2a we plot the intercycle factor as a function of the photoelectron energy for $N=2$, showing the SB energy peaks of width $\mathrm{\Delta }E$ and positions given by Equation (16). In Figure 2b, the intracycle interference pattern for $N=1$ and the total pattern for $N=2$ are shown using red and black lines. One can observe that the intracycle pattern for $N=1$ modulates the corresponding PES to $N=2.$ This result was already observed for forward emission in the RABBIT protocol [29]. In the following, we point out the differences between forward and perpendicular emission.

We now examine the phase of the probability amplitude corresponding to each SB for energies sufficiently close to a sideband [Equation (16)]. In this limit, where $a_m\approx n\omega$, the total amplitude in Equation (26) can be expressed as

$$|T^{\left(n\right)}{|}^2=4N^2{\left|F_{m-1}{g}_{m-1}^{(n+2)}{e}^{-i{\varphi }_{m-1}}{e}^{-i\varphi }{\tilde{\gamma }}_{m-1}^{(n+2)}+F_m{g}_m^{\left(n\right)}{e}^{-i{\varphi }_m}{e}^{i\varphi }{\tilde{\gamma }}_m^{\left(n\right)}\right|}^2$$

(27)

where we have defined ${\tilde{\gamma }}_m^{\left(n\right)}=e^{in{f}_m}{\gamma }_m^{\left(n\right)}$, and $g_m^{\left(n\right)}=g_m\left(k\right)$ corresponds to energies $k^2/2=E_{2m+1+n}$ in Equation (16).

Taking into account Equation (27), the transition probability to a specific SBn can be written as

$${\left|T^{\left(n\right)}\right|}^2=A+Bcos\left(2\varphi +\delta \right).$$

(28)

Here, A and B quantify the strengths of each interfering pathway, and $\delta$ is the total phase delay (see Appendix B of [29]):

$$\delta =\underset{\mathrm{HH}}{\underbrace{{\varphi }_m-{\varphi }_{m-1}}}+\underset{\mathrm{atomic}}{\underbrace{arg\left[{\tilde{\gamma }}_m^{\left(n\right)}\right]-arg\left[{\tilde{\gamma }}_{m-1}^{(n+2)}\right]}}.$$

(29)

The first contribution to $\delta$ stems from the relative group delays of the two neighboring harmonics driving ionization, whereas the second arises from atomic phase shifts encoded in the SFA dipole transition elements. Within our semiclassical RABBIT model, each sideband is produced by interference between two multi-path contributions, rather than by a single pair of two-photon channels as predicted by lowest-order perturbation theory [4,6,8,54,55].

3. Results and Discussion

To test the general result that the ionization probability for photoelectrons emitted perpendicular to the polarization axis of the XUV pulse train and the NIR laser can be factorized in two distinct ways—either into intracycle and intercycle interferences [Equation (15)] or into intrahalf- and interhalfcycle interferences [Equation (26)]—we compare the photoelectron spectrum ${\displaystyle \frac{dP}{dE}}=2\pi \sqrt{2E},{\left|T_{if}\right|}^2$ as obtained from the SCM, the SFA, and ab initio TDSE simulations. The NIR and XUV frequencies are taken as $\omega =0.05$ a.u. and ${\omega }_{2m+1}=(2m+1)\omega$, respectively, with $m=14$ and 15. We consider the NIR laser duration to be $8\pi /\omega$, corresponding to four optical cycles, and three different durations of the XUV pulse train, $\tau =\pi /\omega$, $2\pi /\omega$, and $4\pi /\omega$, corresponding to $N=1/2,1$, and 2 optical cycles of the NIR laser, all of them starting right after the second cycle of the NIR laser, saving the first cycle for a linear ramp on and the fourth cycle for a linear ramp off. The envelopes $f_m\left(t\right)$ are defined in the same way: a unit-amplitude flat-top with one-cycle linear ramps on and off.

Figure 3 presents the SFA and TDSE results obtained using the same XUV and NIR pulse parameters as in Figure 2. We consider $F_{m-1}=F_m=0.1$ a.u. and two values of the NIR field amplitude: $F_0=0.03$ a.u. $(I\simeq 3.16\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2)$ for panels (a) and (c), and $F_0=0.04$ a.u. ($I\simeq 5.62\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$) for panels (b) and (d). In all cases, the relative phases satisfy ${\varphi }_{m-1}={\varphi }_m$ and $\varphi =0$. The agreement between the PES along the perpendicular direction calculated with SFA and TDSE is remarkable in RABBIT. This fact was already emphasized in Ref. [43] for the case of photoionization by a quasi-monochromatic NIR laser, where the two cases of perpendicular and parallel emission were compared and the reasons why the SFA works as well as it does for the parallel case were pointed out, and in Refs. [30,42] for the case of LAPE with only one XUV frequency. This complements their comparison with the SCM results in Figure 2b. Several conclusions can be drawn. First, The insignificant role played by the Coulomb potential of the remaining ion in shaping the electron yield is confirmed by the almost indistinguishable SFA and TDSE spectra. A more detailed analysis of Coulomb effects in RABBIT spectra will be presented in a forthcoming work. Second, the factorization of perpendicular photoelectron emission into either intra- and intercycle interference patterns [Equation (15)] or intra- and interhalfcycle interference patterns [Equation (26)] is validated not only in the SCM and SFA spectra but also in the full TDSE results. Figure 3 shows that the intrahalfcycle interference pattern—obtained from the energy distribution generated by an XUV pulse of duration $\tau =\pi /\omega$ (half a laser cycle)—modulates the intracycle interference pattern, which is obtained from the energy distribution corresponding to an XUV pulse with a duration of one full laser cycle, i.e., $\tau =2\pi /\omega$, as predicted in Equation (26). In the same way, the intracycle interference factor modulates the SBs in the energy distribution for a longer XUV pulse, i.e., $\tau =4\pi /\omega$. When the XUV pulse duration involves several periods of the laser, i.e., $\tau =4\pi /\omega$, the positions of the SBs in the SFA and TDSE calculations in Figure 4 agree with the SCM expressed in Equation (16) for an odd number $\left|n\right|$ of exchanged NIR photons, as described in the previous section. Conversely, SBs with an even number $\left|n\right|$ of exchanged NIR photons are suppressed by the destructive interference between XUV emission during the first and second half cycles of all the cycles of the NIR laser involved. The photoelectron spectra obtained from the SFA and TDSE calculations extend beyond the classical upper-limit energies predicted by the model, $E_{29}^{+}=0.95$ a.u. and $E_{31}^{+}=1.05$ a.u., and beyond the lower limits $E_{29}^{-}=0.77$ a.u. ($0.63$ a.u.) and $E_{31}^{-}=0.87$ a.u. ($0.73$ a.u.) for $I\simeq 3.16\times {10}^{13}$ W/cm² ($I\simeq 5.62\times {10}^{13}$ W/cm²) and $F_0=0.03$ a.u. ($0.04$ a.u.).

We further examine how the perpendicular PES varies with NIR laser intensity, following the schematic diagram in Figure 1. Figure 4 presents the energy distributions obtained using the SCM (a, d, g), the SFA (b, e, h), and the TDSE (c, f, i) for NIR laser intensities up to ${10}^{14}$ W/cm² ($F_0=0.053$ a.u.). The analysis includes different XUV pulse durations, $\tau =\pi /\omega$, $2\pi /\omega$, and $4\pi /\omega$. Figure 2 and Figure 3 correspond to cuts of Figure 4 at $I\simeq 3.16\times {10}^{13}$ W/cm² and $I\simeq 5.62\times {10}^{13}$ W/cm². By definition, the classical boundaries $E_{29}^{+}$ and $E_{31}^{-}$, shown as dotted lines, delimit the SCM spectra in Figure 4a,d,g. For $\tau =\pi /\omega$ (first row), the intrahalfcycle interference stripes exhibit a negative slope. For $\tau =2\pi /\omega$ (second row), the intrahalfcycle interference patterns are flanked by nodes which arise at the zeros of the interhalfcycle factor appearing in Equation (26). i.e., $4{sin}^2\frac{\pi a_m}{2\omega }$ for $N=2$, which vanishes for $a_m=2n\omega$, i.e., for all even $\left|n\right|$ number of exchanged NIR photons. The slope of maxima (visible SBs) and minima (frustrated SBs) is $7.13\times {10}^{-4}/{\omega }^2,$ where the intensity units are in W/cm² and the difference in energy between consecutive maxima (as well as minima) is $2\omega$. This result is inherent to all emissions in the perpendicular direction to the polarization of the laser fields, not only for RABBIT and LAPE but also for one-color ionization [43,44,45]. For $\tau =4\pi /\omega$ (third column), the sidebands become thinner due to the time–energy uncertainty relation $\mathrm{\Delta }E\mathrm{\Delta }\tau \approx \hslash$, which reflects destructive intercycle interference for energies far from conservation values. The SCM successfully recovers the negative-slope intrahalf- and intracycle interference stripes observed in the SFA and TDSE spectra, but only within the classical energy bounds $E_{31}^{-}$ and $E_{29}^{+}$ (see Appendix B). The quantum SFA and TDSE spectra naturally spread beyond these limits because of quantum diffusion. We have checked the convergence of our TDSE code and found that it deteriorates abruptly for intensities greater than $I=8\times {10}^{13}$ W/cm². For $I={10}^{14}$ W/cm² (not shown), the loss of the continuum wave function relative to the ionization probability is as high as 14% due to absorption in the numerical box, whereas for $I=8\times {10}^{13}$ W/cm², it was 7%. For $I=7\times {10}^{13}$ W/cm², it was 2%, whereas for or $I=6\times {10}^{13}$ W/cm², it is as low as 0.24%.

So far, we have considered transverse electron emission for a fixed phase $\varphi =0$. To examine how the intracycle interference structure evolves with the relative phase, we analyze the photoelectron spectra as $\varphi$ is varied from 0 to $2\pi$. Figure 5a shows the intrahalfcycle interference pattern ($N=1/2$) calculated within the SCM (for $\tau =\pi /\omega$). We mark the two discontinuities for HH29 and HH31 at energy values $E_{\mathrm{disc},m}=\frac{1}{2}\left[v_{0,m}^2-A^2(\varphi /\omega )\right]$ (see Appendix B and [42]. For $\varphi =0$, the discontinuity lies at $E_{\mathrm{disc},m}=v_{0,m}^2/2$, which coincides with the upper classical boundary. As $\varphi$ changes, the discontinuity tracks the squared vector potential with $\pi$-periodicity, in contrast with the $2\pi$-periodicity characteristic of parallel emission [29,30,42]. For $\varphi =0,\pi ,2\pi$, the discontinuity occurs at $E_{2m+1}^{+}=v_{0,m}^2/2$, while for $\varphi =\pi /2,3\pi /2$, it shifts to $E_{2m+1}^{-}=v_{0,m}^2/2-2U_p$. The intracycle interference ($N=1$) and total spectra ($N=2$) shown in Figure 5d,e display jumps in the probability distribution at the discontinuities. For $\tau =2\pi /\omega$, the SCM spectrum (Figure 5d) shows horizontal lines corresponding to intracycle interference, i.e., interference between the first and second half cycles, consistent with the factor ${sin}^2(\pi a_m/2\omega )$ in Equation (18). For $\tau =4\pi /\omega$ (Figure 5g), the SCM spectrum shows intercycle interference modulated by the intracycle pattern, but the discontinuities at the SBs [Equation (16)] are absent. As the SBs narrow, the discontinuities blur. The SFA and TDSE results further extend the classical limits, exhibiting more SBs, as seen in second and third columns of Figure 5, and agree closely with the SCM within the classical boundaries, validating the time-dependent semiclassical picture. All SCM spectrograms in Figure 5 show a good qualitative agreement with the quantum SFA and TDSE counterparts. However, SCM discrepancies arise, like classical boundaries, their concomitant caustics inherent to all semiclassical theories, and discontinuities (see Appendix B). These three types of discrepancies are evident in the first row of Figure 5 for a half cycle duration of the XUV pulse and exist irrespective of the weight used in the theory. However, as the XUV pulse duration increases, sidebands become thinner, which conveniently masks the mentioned SCM discrepancies, improving the similarities between the SCM and quantum results.

Finally, to scrutinize the phase delays of the sidebands as functions of the NIR intensity, Figure 6 shows the perpendicular photoelectron spectra of hydrogen for SB32, SB30, and SB28 within the SCM and SFA. The SCM reproduces all the main features of the SFA, except for SB32, which is out of the applicability region. At low NIR intensities, the distributions for SB30 and SB28 maximize at $\varphi =0$ and $\pi$, indicating that both sidebands are in phase, i.e., ${\delta }_{30}={\delta }_{28}=0$ [see Equation (28)]. Conversely, SB32 peaks at $\varphi =\pi /2$ and $3\pi /2$ for low intensity, with an ensuing phase delay ${\delta }_{32}=\pi .$

The maxima of SB28 remain invariant with probe intensity (at least up to the studied intensity of ${10}^{14}$ W/cm²), while SB32 and SB30 exhibit behavior transitions. SB32 peaks at $\varphi =0,\pi$ for low intensities ($I\lesssim 3.16\times {10}^{13}$ W/cm²), giving ${\delta }_{32}=0$; at $\varphi =\pi /2,3\pi /2$ for $3.16\times {10}^{13}$ W/cm² $\lesssim I\lesssim 8.9\times {10}^{13}$ W/cm², with ${\delta }_{32}=\pi$; and again at $\varphi =0,\pi$ for higher intensities ($I\gtrsim 8.9\times {10}^{13}$ W/cm²), leading to ${\delta }_{32}=0$. We observe that the phase delays are always $\delta =0$ or $\pi$ depending on the relative sign of the two contributing terms ${\tilde{\gamma }}_{m-1}^{(n+2)}$ and ${\tilde{\gamma }}_m^{\left(n\right)}$, as expected from Equations (28) and (29) under the strong-field approximation. From Equations (28) and (29) we have demonstrated that sidebands have a 0 or $\pi$ phase within the SFA, which has also been shown by L. B. Madsen [56], and in $\omega -2\omega$ above-threshold ionization [57], where the properties of the generalized Bessel functions can change sign as the intensity of the NIR laser field is increased. In addition, this reflects the common “rule of thumb” for phase delays predicted within the perturbative RABBIT framework [40,55,58]. The redistribution of population in the continuum to multiple sidebands has been addressed, for example, in the three-SB setup by Bharti et al. in Refs. [59,60]. The SCM could be extended to the non-classical region by using theories that are more sophisticated, for example, the uniform saddle-point approximation [61]; however, this is out of the scope of the present work.

4. Conclusions

We have developed a time-dependent nonperturbative theory of the RABBIT protocol for photoelectron emission perpendicular to the laser polarization axis. Building on our previous work for parallel emission, we generalized the semiclassical strong-field approximation by deriving analytical expressions that allow for a transparent interpretation of the phase-delay information. Our analysis shows that the perpendicular emission spectra can be factored into intrahalf- and interhalfcycle contributions or, alternatively, as intra- and intercycle contributions. In particular, we identified intrahalfcycle interference between trajectories released within the same half cycle as the main mechanism responsible for attosecond phase delays in this geometry. A distinctive feature of the perpendicular direction is the suppression of even-order sidebands, leading to a sideband spacing of $2\omega$, in sharp contrast to the spacing $\omega$ observed along the polarization axis. Comparisons with full SFA and the ab initio TDSE simulations confirm the validity of our nonperturbative semiclassical approach. The present theory thus provides a unified framework for describing attosecond chronoscopy in both parallel and perpendicular emission geometries within the strong-field approximation, where the influence of the Coulomb interaction can be neglected. Future work may extend this analysis to angle-resolved photoemission, to polarization-resolved pump–probe schemes, and to the inclusion of Coulomb effects, paving the way toward a more complete understanding of attosecond electron dynamics.