Multichromatic Polarization-Controlled Pulse Sequences for Coherent Control of Multiphoton Ionization

In this review, we report on recent progress in the generation and application of multichromatic polarization-tailored pulse sequences for the coherent control of multiphoton ionization (MPI) dynamics and present unpublished experimental results that complement our previous findings. Specifically, we utilize single-color, bichromatic, and trichromatic polarization-controlled pulse sequences generated by spectral amplitude, phase and polarization modulation of a carrier-envelope phase (CEP)-stable white light supercontinuum for MPI. The analysis of the number of ionization pathways and the number of distinct final free electron states shows that both increase significantly, but scale differently with the number of absorbed photons and the number of pulses in the sequence. In our experiments, ultrafast polarization shaping is combined with high-resolution photoelectron tomography to generate, control, and reconstruct three-dimensional photoelectron momentum distributions from atomic and molecular MPI. We discuss the use of polarization-controlled single-color and bichromatic pulse sequences in perturbative and non-perturbative coherent control of coupled electron-nuclear dynamics in molecules, atomic spin-orbit wave packet dynamics and the directional photoemission from atoms and chiral molecules. We compare the coherent control of CEP-insensitive intraband multipath interference in the MPI with a fixed number of photons with CEP-sensitive interband multipath interference in the ionization with a different number of photons. The generation and control of free electron vortices with even-numbered rotational symmetry by MPI with single-color pulse sequences is contrasted with the bichromatic control of CEP-sensitive electron vortices with odd-numbered rotational symmetry. To illustrate the potential of multichromatic pulse sequences for coherent control, we present a trichromatic scheme for shaper-based quantum state holography.

The use of advanced polarization-shaped pulses for the generation of free electron wave packets by MPI has necessitated 3D detection to characterize the full 3D photoelectron momentum distribution (PMD). While time-offlight techniques enabled the kinetic energy-resolved detection of photoelectrons from ultrafast MPI dynamics [81,82], their angular distribution became available through the use of velocity map imaging (VMI) techniques [83][84][85]. Currently, the most sophisticated detection method is based on COLTRIMS [86,87]. Cylindrically symmetric PMDs can be retrieved from the measured 2D projections using the Abelinversion technique [88]. In order to image non-cylindrically symmetric PMDs, we have introduced a VMI-based photoelectron tomography technique [89]. By now, photoelectron tomography has become an established method with versatile applications including 3D imaging of PMDs from atomic strong-field ionization [90], the reconstruction of molecular PMDs in the laboratory [91] and molecular [92] frame, the retrieval of circularly polarized XUV fields from HHG [93] or the reconstruction of PMDs from strong-field photodetachment of negatively charged ions [94]. Applications of photoelectron tomography have ranged from the generation and characterization of designer electron wave packets [95,96] and multiphoton photoelectron circular dichroism (PECD) measurements [97] to the extraction of transition matrix elements from tomographic data [98]. More recently, we have used photoelectron tomography to study free electron angular momentum wave packets [99], spin-orbit [100] and Rydberg wave packet dynamics [101] and photoelectron vortices with even [102][103][104] and odd [73] rotational symmetries. An overview of our recent photoelectron tomography studies of coherent control of atomic MPI can be found in [105,106].
In this contribution, we review the application of shapergenerated multichromatic polarization-controlled fields for coherent control of atomic and molecular MPI. We study the quantum dynamics of atomic and molecular model systems induced by absorption of N photons from M pulses, focussing on the new opportunities offered by polarization-tailored multichromatic fields. The paper is organized as follows: In Section 2, we introduce the time-dependent laser electric field and summarize the theoretical methods that we have used to reproduce and analyze our experimental results. Details of our experimental setup for generating polarization-controlled multichromatic pulse sequences by spectral amplitude and phase modulation of a CEP-stable supercontinuum and the VMI-based photoelectron tomography technique are given in Section 3. In Section 4 we present our results on coherent control of MPI with multichromatic pulse trains ordered by the type of pulse sequence we have used in the experiment. We start with experiments using single-color linearly polarized pulse sequences (Section 4.1) for perturbative and nonperturbative control of atomic and molecular MPI. By analyzing the 3D PMD, we rationalize the observed photoelectron spectra in an experiment on the interference of ultrashort free electron wave packets [11]. We review the generation of free electron vortices (FEVs) with c 4 , c 6 and c 8 rotational symmetry using circularly polarized pulse sequences and present previously unpublished results on the vortex formation. The article's focus on intraband and interband coherent control scenarios using linearly and circularly polarized bichromatic pulse sequences is elaborated in Section 4.2. Examples include the control of frequency mixing contributions in the MPI with orthogonal linearly polarized (OLP) and CRCP bichromatic fields with applications to the observation of spin-orbit wave packets. In addition, we present the control of directional photoemission in the MPI of atoms and chiral molecules by bichromatic parallel linearly polarized (PLP) fields or few-cycle circularly polarized fields, and the use of cycloidal bichromatic fields to generate and manipulate 3D PMDs with c 1 and c 7 rotational symmetry. Finally, results on coherent control by trichromatic pulse sequences are presented in Section 4.3. The phase-sensitive combination of intra-and interband interferences is used for trichromatic shaper-based quantum state holography. We conclude with a summary and an outlook in Section 5.

THEORY
The MPI of atoms and molecules has been studied extensively both experimentally and theoretically. The fundamentals are documented in a vast body of literature, see for example [107][108][109][110][111][112]. In this section, we present the notation to describe multichromatic pulse sequences (Section 2.1) and summarize the theoretical concepts based on either nonperturbative description of the atomic and molecular neutral dynamics perturbatively coupled to the continuum (Section 2.2) or fully perturbative intraband and interband interference in MPI. The theoretical description is specifically adapted to reproduce the experimental results and discuss the relevant control aspects.

Ultrashort Polarization-Controlled Multichromatic Pulse Sequences
We start by describing the electric field of a pulse sequence consisting of M pulses, each of which with a different polarization state, e.g. linear, circular or in general elliptic. E j (t − τ j ) represents the complex valued timedelayed (τ j ) envelope of the jth pulse with a pulseduration of Δt, which contains the effects due to higherorder spectral phase functions [113]. Here, Δt describes the full width at half maximum of the laser intensity. Taking into account the carrier oscillation with the central angular frequency ω j , the relative phase φ j , and the common CEP φ ce , the scalar field of the jth pulse reads E j (t) E j (t − τ j )e −i(ωjt+φ j +φ ce ) . (1) The coefficients c −1 j and c +1 j in Eq. 2 describe the weights of the respective right-handed circularly polarized (RCP) (−1) and lefthanded circularly polarized (LCP) (+1) components For example, c j (c −1 j , c +1 j ) (1, 0) denotes RCP light while c j (1, 1)/ 2 √ represents horizontally linearly polarized light. Using the Jones vectors e −1 1 2 √ (e x − ie y ) for RCP and e +1 − 1 2 √ (e x + ie y ) for LCP pulses, we obtain the time-dependent vectorial electric field of the jth pulse The total field is obtained by superposition of all M pulses in the sequence The physical field is given by Re[E(t)]. Schematic representations of single-color (M 2, ω 1 ω 2 ), bichromatic (M 2, ω 1 ≠ ω 2 ) and trichromatic (M 3, ω i ≠ ω j ) pulse sequences with different polarizations are depicted in Figures  4, 7, 12, respectively.

Simulation of Multiphoton Ionization
Many theoretical concepts and numerical methods are available to describe and simulate the quantum dynamics of atoms and molecules induced by the interaction with ultrashort laser pulses [4,6,114]. In the past two decades, we have used different methods to model ultrafast dynamics driven by shaped femtosecond pulses to model our experiments and interpret our observations. In parallel with our experimental studies, we have presented the corresponding theoretical descriptions and numerical simulations of the multilevel coherent control of atomic [115,116] and molecular [117,118] excitation. We have devised a resonant strong-field control mechanism based on the manipulation of dressed state populations and energies [119] and, in this context, developed a formalism to model photoelectron spectra from resonance-enhanced multiphoton ionization (REMPI) accounting for non-perturbative population dynamics and AC Stark shifts [118,120]. In [48,103], we provided a description for the calculation of sculpted 3D free electron wave packets from MPI, including the energy and angular distribution. Recently, we employed ab initio methods to numerically solve the time-dependent Schrödinger equation (TDSE) for a 2D model atom interacting with a polarizationtailored bichromatic femtosecond laser pulse sequence [121]. The model was applied to reproduce and analyze our experimental results reported in [73,103] taking into account phase fluctuations in the CEP and the Gouy phase.

Non-Perturbative Neutral Dynamics Coupled to the Continuum
Resonant strong-field control of MPI dynamics is based on the non-perturbative manipulation of resonant bound state dynamics and the simultaneous perturbative mapping of those dynamics into the ionization continuum by absorption of additional photons. To model this type of non-perturbative REMPI processes we adapt the concept from Meier et al. [122] and divide the calculation into two steps. In the first step, we calculate the non-perturbative neutral dynamics of the time-dependent amplitudes c i (t) by solving the corresponding TDSE numerically for a multistate quantum system, including all relevant bound states. This procedure yields the time-dependent bound state populations |c i (t)| 2 . In the second step, we compute the photoelectron kinetic energy spectrum using time-dependent perturbation theory by inserting the amplitudes determined in the first step. This strategy is illustrated in Figure 1A for a generic atomic system and in (B) for the potassium molecule.

Atomic Multiphoton Ionization
To simulate our experiments, we have modeled atomic systems by multilevel schemes involving two [120,[123][124][125], three [48,116] or more [115] bound states. As an example, we consider a twolevel atom, consisting of a ground state |1〉 and an excited state |2〉 separated in energy by Zω 21 Zω 2 −Zω 1 and coupled by the transition dipole moment μ 21 . The non-perturbative dynamics of the atom interacting with an intense near-resnant femtosecond laser pulse E(t) E(t)e −iω 0 t , where ω 0 is the laser central frequency and E(t) is the generally complex-valued temporal envelope, is governed by the TDSE. Using both, the dipole approximation and rotating wave approximation, the interaction picture TDSE for the time-dependent ground and excited state amplitudes c 1 (t) and c 2 (t) reads [5,119,126].
Herein, δ ω 0 −ω 21 is the detuning of the laser with respect to the atomic resonance. The TDSE in Eq. 5 is solved numerically on a discrete temporal grid using short-time propagator methods [4]. This procedure yields the time evolution of the state vector c(t) [c 1 (t), c 2 (t)] T . With the knowledge of the excited state amplitude c 2 (t), we calculate the amplitudes c(ω k ) of the free electrons, released with a kinetic energy of Zω k by perturbative N -photon ionization from the excited state, by N -th order timedependent perturbation theory [115,123,127] The notation was adapted from Dudovich et al. [128]. The multiphoton transition dipole moment μ (N ) for the boundfree transition is in general complex-valued and depends on the atomic structure. In the narrow energy windows considered here, we assume an unstructured continuum, i.e., treat the coupling as energy-independent. This numerical method was applied to reproduce the experimental photoelectron spectra obtained in our studies of basic physical mechanisms of resonant non-perturbative coherent control of atomic model systems using PLP single-color double pulse sequences [123,124], multipulse sequences [117,120,129,130], chirped pulses [115,125,130] and shaped pulses from spectral step-phase modulation [116,127] (cf. Section 4.1.1.2).

Molecular Multiphoton Ionization
Molecular systems are modeled in the framework of the Born-Oppenheimer approximation by a set of N bound electronic states |ϕ n 〉 characterized by the adiabatic potential energy curves V n (R). For simplicity, we restrict ourselves to the diatomic case, where the nuclear coordinate R denotes the internuclear separation of the atoms and describes the vibration of the molecule. In the electronic basis, the TDSE for the state vector ψ(R, t) (ψ 1 (R, t), ‥, ψ n (r, t), ‥, ψ N ) T reads [131] iZ The time-dependent probability amplitudes ψ n (R, t) of the electronic states describe the nuclear wave packets in the electronic potentials V n (R). Making use of the dipole approximation, the Born-Oppenheimer Hamiltonian is given by [132] H with the reduced mass m r of the molecule. Explicit expressions of the Hamiltonian matrix are provided in [118,132]. The diagonal matrix V̂(R) contains the potentials V n (R). These potentials, as well as the R-dependent transition dipole matrix elements contained in μ(R), were supplied by C. Meier and F. Spiegelman (see [118] for more information) and were Scheme for perturbative control of intraband [73] and interband [105] MPI with polarization-shaped pulses along with multiphoton spectra. (D) Model potential for ab initio simulations of MPI with cycloidal laser pulses [121].
recently confirmed by Petersen et al. [133]. The TDSE in Eq. 7 is solved numerically by iterative propagation of the state vector on a temporal grid employing a Fourier-based split-operator method [134]. The time evolution of ψ(R, t) yields the coupled electron-nuclear (vibronic) wave packet dynamics in the bound molecular system. In a theoretical study, we demonstrated efficient ultrafast switching between different electronic target states in potassium dimers ( Figure 1B) by resonant non-perturbative control of the vibronic dynamics in the X 1 Σ g ground and A 1 Σ u states using an intense PLP single-color pulse sequence [118,131] (cf. Section 4.1.1.2).
To calculate the photoelectron contribution from a bound electronic state V n (R), produced by perturbative one-photon ionization with a probe pulse E pr (t) E pr (t)e -iωprt , we pursue two alternative strategies. The first approach makes use of the semi-classical difference potential analysis technique by Mulliken [81,135,136]. The combination of energy conservation and the Franck-Condon principle provides a mapping R → ω k ω k (R) of the internuclear distance R to the photoelectron excess energy Zω k , which is mediated by the difference potential ΔV n (R) V ion (R)− V n (R). Here, V ion (R) is the Born-Oppenheimer potential of the ionic state. If this mapping is invertible, requiring that ΔV n (R) is strictly monotonous throughout the ionization R-window, then the photoelectron contribution from state n can be written as If ΔV n (R) is non-monotonous within the ionization window, i.e. ΔV n ′(R) vanishes locally, we decompose the wave packet into a set of Gaussian functions which are locally mapped into Gaussianshaped photoelectron distributions with a narrow, but finite spectral width in the order of the experimental energy resolution. The second approach is based on a perturbative quantum mechanical treatment of the ionization process. Consider a transition from the bound vibrational eigenstate ] to an ionic vibrational eigenstate ]′ with the transition frequency ω ]′] ω ]′ −ω ] . For a fixed spectral component ω of the probe pulse spectrum Epr(ω) F [E pr (t)](ω), the photoelectron excess energy is given by ω k ω−ω ]′] . The photoelectron contribution from the nth electronic state is determined by the Franck-Condon factor |〈]′|]〉| 2 , the final population of the bound vibrational states |〈]|ψ ∞ n 〉| 2 |〈]|ψ n 〉 t → ∞ | 2 and the spectral intensity of the probe pulse: The quantum mechanical description was used to reproduce and analyze the experimental results in [132,137]. Using intense PLP single-color multipulse sequences followed by an ionizing probe pulse with a different color, we were able to demonstrate the resonant non-pertubative control scheme described above experimentally [131,132,137].

Perturbative Intraband and Interband Interference in Multiphoton Ionization
In this section we consider perturbative atomic MPI with a pulse train consisting of M pulses. A schematic representation of the available intraband pathways for absorption of N 2 photons is shown in Figure 2 for (A) single-color, (B) bichromatic, and (C) trichromatic ionization scenarios. The specific scheme for bichromatic intraband MPI reported in [99] and interband MPI by (3ω:4ω) pulse sequences reported in [73] is presented in Figure 1C. To describe the general intraband case, we extend the perturbative description of non-resonant bichromatic MPI reported in [73,105,106] to photoionization with N photons from a multichromatic CEP-stable sequence of M pulses. To this end we consider absorption of r j RCP and l j LCP photons from the jth polarization components E +1 j (t) and E −1 j (t) introduced in Eq. 2, respectively. Representing the total number of absorbed RCP and LCP photons by r and l, the order of the MPI process is N r + l. We introduce (l, r) (l 1 , . . . , l M , r 1 , . . . , r M ) as a shorthand notation for the tuples of absorbed photon numbers to characterize the relevant temporal multiphoton field which is written down explicitly for a tricromatic field (M 3) Taking the Fourier transform of the temporal multiphoton field E (l,r) (t), we obtain the multiphoton spectrum S (l,r) (ω) to describe the absorption of (l, r) photons Taking into consideration the selection rules for absorption of RCP (Δm −1) and LCP (Δm + 1) photons and ignoring for simplicity Δl −1, the dipole moments μ (l,r) , the multiplicity of a specific N -photon ionization pathway given by the multinomial coefficients (N , (l, r)), the kinetic energy Zω and ionization potential Zω IP as well as the angular distribution Y N ,l−r (θ, ϕ) for the absorption of N l + r photons, we obtain the photoelectron wave function where we sum over all indices (l, r) such that l + r N . The α k describe the complex-valued coefficients of the partial wave functions ψ l,m k including the multiphoton transition dipole moments and the electric field amplitudes and phases. In the discussion of the different control scenarios below, we use a shorthand notation based on Eq. 15 including the amplitudes and phases of α k relevant to the control scenario. In addition, since we focus on the field-induced variation of the wave function, the intrinsic atomic phases and radial amplitudes of the dipole couplings are generally omitted. The geometric part of the dipole couplings, given by the Wigner 3-j symbols, is written explicitly whenever the amplitudes of the participating partial wave functions cannot be manipulated individually by the electric field amplitudes. Otherwise, the field amplitudes are generally adapted in the experiment to optimize the interference contrast. The optical phases are written explicitly whenever they are used as control parameters to manipulate the resulting photoelectron wave packet. A more comprehensive theoretical description is given for example in [107][108][109]138]. Control of multiphoton processes by multichromatic pulse sequences is based on the interference of multiple pathways leading to the same final state. As we increase the number of pulses, both the number of available quantum pathways and the number of final states increase significantly, opening up more versatile options for control.
An overview of the numbers of angular momentum states (C l ), free electron energy states (C e ), pathways (C p ) and distinguishable pathways (C d ) for ionization with N 1, . . . , 4 and M 1, 2, 3 pulses is presented in Table 1.
In addition to the above discussed intraband interferences characterized by absorption of N photons, in multichromatic scenarios further interferences arise when pathways with a different number of absorbed photons N 1 and N 2 interfere. In this case the wave function is a superposition of both contributions ψ(ε, θ, ϕ) ψ N 1 (ε, θ, ϕ) + ψ N 2 (ε, θ, ϕ).
If both pathways address overlapping final continuum states, i.e. the energy gained by absorption of N 1 photons with Zω 1 agrees within the bandwidth with the energy delivered by N 2 photons with Zω 2 , as shown in Figure 1C for a (3ω:4ω) bichromatic field, CEP-sensitive interband interferences occur. Bichromatic fields with commensurable frequencies ω 2 ω 1 · N 1 /N 2 satisfy the above condition ε ZN 1 ω 1 ZN 2 ω 2 . Examples for CEPcontrol of interband interferences in the MPI by linearly and circularly polarized bichromatic pulse sequences are presented in Section 4.2.1.1 and Section 4.2.1.2.

2D Ab Initio Simulation of Atomic Multiphoton Ionization in Cycloidal Laser Fields
Another efficient method to simulate atomic photoionization dynamics by polarization-shaped ultrashort laser fields is based on solving of the TDSE numerically for a 2D model system aligned in the laser polarization plane. In [121], we employed the 2D model potential shown in Figure 1D V with the error function erf and a soft-core parameter a, and solved the TDSE for the time-dependent Hamiltonian on a 2D grid using the Fourier-based split-operator method [134]. Specifically, atomic MPI with cycloidal, i.e. COCP and CRCP, femtosecond laser pulse sequences was investigated. Employing single-color pulse sequences, we were able to reproduce selected recent experimental results on the generation of photoelectron vortices [102,103] and to analyze the transient vortex formation dynamics in different physical pictures, including the bound state population dynamics, the time evolution of the free electron wave function and its asymptotic topological charge. Employing cycloidal bichromatic fields, we reproduced the experimental results on the generation of free electron wave packets with odd rotational symmetries [73]. We analyzed the final state wave function with respect to the relative orientation between field and PMD and extracted the azimuthal probability density current to validate the physical discussions in the experimental studies [73,105,106]. Furthermore, the simulations allowed us to examine the influence of experimental CEP and Gouy phase fluctuations on the measured PMD. Eventually, we applied the technique to investigate photoelectron vortices theoretically in so far unexplored intensity regimes.

EXPERIMENT
In this section, we introduce our experimental scheme for the coherent control of ultrafast MPI dynamics in atoms and molecules using polarization-tailored single-color or multichromatic fields. In the experimental studies, we combine advanced supercontinuum pulse shaping techniques with the highly differential detection of photoelectrons using a VMI spectrometer. The experimental setup is portrayed in Figure 3. CEP-stable polarization-shaped laser pulse sequences are generated using a phase-stabilized white light polarization shaper in 4f geometry. Photoelectron wave packets, released by the interaction of the shaped pulses with the atomic or molecular sample in the VMI spectrometer, are observed with energy and angular resolution. By rotating the pulses about the laser beam axis using a wave plate, the VMI spectrometer is turned into a photoelectron tomography apparatus, allowing us to retrieve the full 3D PMD. In the following, we give an overview of white light polarization shaping in Section 3.1 and photoelectron tomography in Section 3.2. For a more detailed description of both techniques, we refer to [106].

Supercontinuum Polarization Pulse Shaping
Various experimental schemes for the generation of multicolor laser fields have been reported, including 4f pulse shapers adapted to an octave-spanning WLS [61,64] and extensions to optical waveform synthesizers [139,140] as well as four-wave mixing schemes [141] and molecular modulation [142][143][144]. Our method to generate multichromatic polarization-tailored pulse sequences is based on 4f-polarization pulse shaping of a white light supercontinuum. The primary light source of our experiments is an actively CEP-stabilized multipass chirped pulse amplifier system (FEMTOLASERS Rainbow 500, CEP 4 module, Femtopower HR 3 kHz) providing 20 fs infrared pulses centered at 790 nm with an energy of 1.0 mJ. These pulses are used to seed a neon-filled hollow-core fiber (absolute pressure ∼ 2.0 bar) generating an over-octave spanning WLS which is amplitude-and phase-modulated in the spectral domain using a home-built 4f polarization pulse shaper [48,50,51,145]. While traditional 4f pulse shaping allows for either independent amplitude and phase modulation or phase and polarization shaping [51,145], recently we introduced a pulse shaping scheme specifically adapted for the independent shaping of amplitude, phase and polarization of bichromatic fields [65,106,146]. The bichromatic white light polarization shaper is shown on the lefthand side of Figure 3. Bichromatic amplitude profiles are sculptured from the WLS via combined amplitude and phase modulation using a traditional polarization shaping setup consisting of a 640 pixel dual-layer liquid crystal spatial light modulator (LC-SLM) in the Fourier plane of a folded 4f setup with holographic transmission gratings at the in-and output. A customized composite polarizer mounted directly behind the LC-SLM enables independent polarization control of both spectral bands (colors). Along the spectral axis, the polarizer is divided into two parts, each of which can be chosen in s-or p-configuration, resulting in the generation of either bichromatic PLP (both colors s-or p-polarized) or OLP 1 | Overview of the total number of possible pathways for ionization with a given number of photons (N ) from (M) pulses. Shown are the numbers of angular momentum states (C l ), free electron energy states (C e ), pathways (C p ) and distinguishable pathways (C d ).
Frontiers in Physics | www.frontiersin.org August 2021 | Volume 9 | Article 675258 (one color s-, the other one p-polarized) fields. The bichromatic pulse shaping approach allows us to control the center frequency ratio, spectral bandwidths and amplitude profiles of both spectral bands and, moreover, independently permits the application of arbitrary phase modulation functions. By use of a superachromatic quarter wave plate (λ/4) at the shaper output, we generate circularly polarized single-color or bichromatic pulse sequences. An additional half wave plate (λ/2) is used to rotate the polarization-shaped laser pulses about the propagation axis, which is the basis for the photoelectron tomography technique discussed in Section 3.2. To ensure the CEP-stability of the shaped pulses, we use the shaper to generate an additional (ω: 2ω) field which is split off the main beam by a dichroic mirror and sent to an f-2f interferometer. The interferometer feeds the active control loop of the amplifier, which stabilizes the CEP of the shaped pulses before they are focused into the VMI spectrometer. The root mean squared CEP stability after the shaper was measured over 11 h and is better than 215 mrad [101]. Finally, dispersion control and pulse characterization are implemented in situ, i.e. in the interaction region of the VMI spectrometer, by using the shaper for adaptive optimization of the highly nonlinear MPI of xenon atoms and to perform shaper-based crosscorrelation measurements [146].

Photoelectron Tomography
In order to study the full 3D PMD caused by the interaction of atoms and molecules with polarization-tailored ultrashort laser pulses, we employ the VMI technique [83,103] for the energyand angle-resolved measurement of photoelectron wave packets.
The VMI spectrometer is shown on the righthand side of Figure 3. The tailored laser pulses are focused via a broadband silver mirror with a focal length of 250 mm into the interaction region of the VMI spectrometer. The background pressure is typically in the order of 5 × 10 −7 mbar. All experiments were carried out in the gas phase. Alkali atomic samples are supplied by dispenser sources (working pressure: ∼10 −7 mbar), whereas rare gases or molecular samples are injected using an effusive gas-inlet (working pressure: ∼10 −6 mbar). An electrostatic lens setup consisting of a repeller, an extractor and an Einzellens, projects the 3D PMD created by MPI of the sample by the incident laser pulses onto a chevron-type microchannel plate detector stacked with a phosphor screen. The resulting 2D image of the projected PMD is recorded by a charge-coupled device camera. PMDs created by MPI with either linearly or circularly polarized laser pulses are, in general, cylindrically symmetric and can therefore be reconstructed from a single 2D image via Abel inversion, e.g. using the pBASEX algorithm [88]. In contrast, PMDs created by polarization-shaped pulses typically exhibit no such symmetry.
For the retrieval of 3D PMDs with arbitrary shape, we developed a tomographic reconstruction technique [89] based on the rotation of the laser pulse-and hence the PMD-using a superachromatic λ/2 wave plate. After acquiring numerous 2D projections of the PMD under various rotation angles, we retrieve the 3D PMD by . Multichromatic amplitude, phase and polarization tailoring of an input WLS is implemented by a specifically adapted polarization shaper in folded 4f geometry. The WLS input beam is spectrally dispersed at the input grating (G), collimated at the cylindrical mirror (CM) and spectrally modulated in the Fourier plane by a dual-layer LC-SLM and a custom composite polarizer (CP). Active CEP-stabilization is realized by an f-2f interferometer feeding the control loop of the laser system. A superachromatic λ/4 wave plate at the output allows to generate circularly polarized pulse sequences. Using an additional λ/2 wave plate, the shaped laser pulse is rotated about the beam axis, which is the basis for VMI-based photoelectron tomography. The VMI spectrometer on the right consists of an electrostatic lens system projecting the PMD onto an microchannel plate detector attached to a phosphor screen. Images of the resulting 2D projections are recorded with a charge-coupled device camera. Exemplarily, a CRCP (3ω:4ω) output pulse is depicted in magenta along with the tomographically reconstructed 3D PMD created by MPI of potassium atoms.
Frontiers in Physics | www.frontiersin.org August 2021 | Volume 9 | Article 675258 application of tomographic techniques such as the Fourier slice algorithm [89,147,148] or the backprojection algorithm [90,147,149]. An advantage of the Fourier slice method is that the angular discretization inherent to the measurement appears only in the Fourier domain representation, whereas the retrieved real-space PMD is angularly smooth. For a typical tomographic measurement, we use about 45 different orientations of the λ/2 wave plate ranging from −90°to 86°. Energy calibration of the reconstructed 3D PMD is performed by taking 151 azimuthal 2D slices through the 3D data cube and convert the radial momentum distributions into corresponding energy distributions. We estimate an energy-resolution of better than 80 meV for photoelectrons with a kinetic energy of about 1 eV. Thus, our VMI-based photoelectron tomography technique is well suited to retrieve the highly structured 3D PMDs generally created by MPI with multichromatic polarization-shaped laser pulse sequences. Similar tomographic techniques have been applied, e.g., for the imaging of molecular orbitals [148], the discrimination of chiral molecules via the multiphoton PECD [150] and the time-resolved imaging of ultrafast laser-matter interactions inside transparent media [149].

RESULTS AND DISCUSSION
In this section, we discuss our experimental results on photoelectron tomography of 3D PMDs obtained by atomic and molecular MPI using multichromatic polarizationcontrolled pulse sequences. We present our findings in the order of the type of pulse sequence: We start with experiments on MPI with single-color pulse sequences in Section 4.1 and focus on various coherent control scenarios by bichromatic pulse sequences in Section 4.2. New results on trichromatic pulse sequences are presented in Section 4.3.

Single-Color Pulse Sequences
Single-color pulses consist of a single connected spectral band with bandwidth Δω centered around a carrier frequency ω 1 . A set of single-color pulse sequences for different polarizations is depicted in Figure 4. These pulses are the starting point of our discussion.

Single-Color Linearly Polarized Pulse Sequences
Single-color PLP pulse sequences with variable time-delay τ, as shown in Figure 4A, are the standard tool in pump-probe experiments [151]. Even in the overlapping region τ ≈ 0, such pulses remain linearly polarized. In contrast, OLP pulse sequences, as shown in Figure 4B, additionally address the spatiotemporal aspect of the light-matter interaction due to their time-dependent polarization. Their application ranges from the phase-control of currents in semiconductors [152] to the steering of molecular rotation [153,154]. Crossed polarizations demand a vectorial description of the field and thus provide access to the angular properties of the interaction energy −μ · E. Depending on the relative phase, OLP pulses are either linearly, elliptically or circularly polarized at τ 0.

Perturbative Multiphoton Ionization
In [11], Wollenhaupt et al. have used phase-locked pairs of linearly polarized single-color pulses to study the interference of ultrashort free electron wave packets generated by perturbative MPI of potassium atoms. By variation of the time delay in the PLP sequence, the interference pattern in the photoelectron kinetic energy spectrum from one-photon ionization of the excited 5p state was controlled. The observation of interferences in the photoelectron spectrum demonstrated that the coherence properties of the ultrashort laser pulses were transferred to the electrons. Using OLP sequences, i.e. crossed polarizations, no such interference fringes have been found in the threshold electrons. However, the quantum interferences reappeared at twice the modulation frequency in the above-threshold ionization (ATI).
To rationalize the experimental findings reported in [11], we apply the general formalism discussed in Section 2.2.2 to describe one-and two-photon ionization of a pre-excited atom with single-color PLP and OLP pulse sequences and analyze the full 3D PMD. Figure 5A shows a schematic energy level diagram of potassium atoms for the excitation of the 5p state and subsequent photoionization at the threshold (THR) and the ATI. The description in the spherical basis provides a consistent picture for excitation and ionization with both PLP and OLP sequences. Simulated PMDs at the ionization threshold and the ATI are depicted in Figure 5B. The left column shows the PMDs from ionization with either a single pulse or a PLP pulse sequence whose polarizations are parallel (p) to the initial blue excitation pulse. Ionization with a single orthogonal (o) polarized pulse or an OLP pulse sequence are depicted in the right column. To calculate the modulation of the interference in the photoelectron spectra, we first consider the wave function in the excited 5p state prepared by the initial, linearly polarized blue (405 nm) laser pulse. In the spherical basis, the p state is a superposition of the two ψ 1,±1 states Taking into account the transition dipole moments determined by the Wigner 3-j symbols, the absorption of another parallel or orthogonal polarized photon from the 5p state yields the wave functions for the photoelectrons at the ionization threshold By calculating the electron density ρ(ε, θ, ϕ) |Ψ(ε, θ, ϕ)| 2 , we see that ionization with a parallel polarized field yields the expected d-type electron density ρ p THR plotted on the left side of Figure 5B, whereas orthogonal ionization of the aligned 5p state gives rise to the d x,y -type state due to the cancellation of the ψ 2,0 state via destructive quantum interference. The generation of such a d x,ytype state in the continuum, exhibiting a c 4 rotational symmetry, has been experimentally demonstrated using a two-color OLP sequence to ionize potassium atoms [105]. Ionization with a single-color PLP sequence, i.e. two phase locked time-delayed parallel polarized pulses, leads to the superposition state characterized by the interference structures in the electron density ρ PLP THR . By integration over the angular coordinates θ and ϕ, we obtain the energy-resolved photoelectron spectrum yielding the fully cosinusoidally modulated spectrum in agreement with the experimental results [11]. In contrast, in the OLP case, the superposition of two different wave functions results in the highly patterned electron density ρ OLP THR displayed on the right side of Figure 5B. Again, by angular integration we find that in the OLP case the interference in the photoelectron spectrum cancels out as reported in [11]. Consequently, the OLP-photoelectron spectrum at the ionization threshold is unstructured and independent of the time delay In the next step, we apply the same procedure to analyze the PMDs at the ATI. To this end, we write down the corresponding wave functions and densities and calculate the photoelectron spectra to rationalize the experimental findings in the ATI spectrum. Absorption of another parallel (orthogonal) linearly polarized photon from the threshold states given by Eq. 22 yields the wave functions in the ATI The corresponding densities ρ p ATI and ρ o ATI are shown in the upper row of Figure 5B. Assuming the two pulses in the sequence are fully separated (τ ≫Δt), i.e. the temporal overlap vanishes, there are no contributions from paths describing sequential ionization from different pulses, and hence the wave functions for PLP and OLP sequences read Again, in the PLP case, the PMDs given by ρ PLP ATI , and hence the corresponding photoelectron spectra in the ATI are fully modulated by the delay-dependent interferences Using the above method to calculate the PMDs from two-photon ionization with an OLP sequence, given by ρ OLP ATI , and the respective photoelectron spectra, we obtain In contrast to the unmodulated photoelectron spectra from ionization with OLP pulses at the threshold (Eq. 26), the spectra in the ATI are indeed modulated-albeit more weakly-in the OLP case in agreement with the experimental work.
In another polarization-sensitive pump-probe experiment, single-color PLP and OLP pulse sequences have been used to control the perturbative MPI of potassium molecules [52]. It was demonstrated that the ionization efficiency is higher for OLP pulses than for PLP pulses. This result was rationalized by the orthogonality of transition dipole moments encountered along the major MPI pathways in the potassium dimer. The experiment, which included the first demonstration of control of molecular dynamics via ultrafast polarization shaping, showed that tailored polarization-shaped pulses are better suited than the corresponding linearly polarized fields to manipulate the spatiotemporal evolution of molecular wave functions.

Non-perturbative Resonance-Enhanced Multiphoton Ionization
The basic concept behind the non-perturbative control of REMPI processes is depicted in Figure 1A. The scheme is based on the selective population of dressed states (SPODS), recently reviewed in [119], by a sequence of PLP single-color femtosecond laser pulses. A relatively weak prepulse with a pulse area of π/2 [5] with respect to the resonant transition-typically between the ground and first excited state-is followed by an intense main pulse. Initially, the prepulse excites an electronic wave packet of maximum coherence, i.e., an efficient charge oscillation described by a time-dependent induced dipole moment μ i (t). After a time delay τ, the electric field E(t) of the main pulse couples to the induced charge oscillation. By suitable adaption of the optical phase to the electric dipole dynamics, the interaction energy −μ i (t) · E(t) is maximized (minimized), equivalent to the selective population of the upper (lower) dressed state in the strongly coupled resonant sub-system. This energy shift (resonant AC-Stark shift) opens up new multiphoton routes to higher-lying (lower-lying) target states which are inaccessible by perturbative excitation as shown in Figure 1A.
Resonant non-perturbative control via SPODS was first reported on 1 + 2 REMPI of potassium atoms [123]. Using interferometric double pulse sequences, we demonstrated efficient ultrafast switching between different ionization target channels by controlling the dressed state populations of the resonant 4s−4p transition via the relative optical phase. The interpretation of the scheme in a dressed state picture, was presented in [124]. On the same system, we also implemented SPODS with PLP single-color multipulse trains generated by sinusoidal spectral phase modulation [120]. In that work, dressed state control by all sine-parameters was demonstrated.
Subsequently, we applied SPODS to the non-perturbative control of coupled electron-nuclear dynamics in molecules. In the molecular case, the charge dynamics induced by the prepulse is altered due to the coupling between electronic and nuclear degrees of freedom. For example, the amplitude of the dipole oscillation depends on the overlap of vibrational wave packets launched in the resonant electronic states. Also, as the vibrational wave packets propagate, the electronic resonance frequency varies changing the eigenfrequency of the dipole and introducing an additional phase drift which the main pulse needs to adapt to. Our proof-of-principle studies were performed on the potassium dimer serving as a molecular prototype system. The corresponding excitation scheme is depicted in Figure 1B. The prepulse resonantly couples the X 1 Σ + g ground state and the A 1 Σ + u excited state, giving rise to two light-induced potentials (i.e. molecular dressed states) and launching the vibronic dynamics. By selective population of a single light-induced potential through the main pulse, population is transferred efficiently to a predefined target state out of a manifold of high-lying but bound electronic states. In a first theoretical study, we demonstrated ultrafast efficient (∼ 80% population transfer) switching between the 2 1 Π g and 5 1 Σ + g state using PLP single-color double pulse sequences [118]. The scenario was demonstrated experimentally employing singlecolor multipulse trains from sine-modulation [137].

Free Electron Vortices With Circularly Polarized Pulse Sequences
In this section we consider the creation of free electron wave packets by REMPI of potassium atoms with single-color circularly polarized pulse sequences (Figures 4C,D). First, in Section 4.1.2.1, we discuss the formation of a FEV by perturbative MPI at the ionization threshold with c 6 rotational symmetry and in the ATI with c 8 rotational symmetry. In addition, we study the evolution of the c 6 FEV into an angular momentum eigenstate as the time-delay in the sequence vanishes. Then, non-perturbative REMPI resulting in a FEV with c 4 rotational symmetry is examined in Section 4.1.2.2. We show, how the wave function of the latter FEV is reconstructed using a holographic method.

Perturbative Resonance-Enhanced Multiphoton Ionization
Recently, the generation of FEVs by photoionization with two time-delayed CRCP attosecond laser pulses has been theoretically predicted and termed "an unusual kind of Ramsey interference" [155]. FEVs with c |m1−m2| rotational symmetry arise from the interference of two free electron angular momentum states ψ l,m with different magnetic quantum numbers m 1 and m 2 . In general, the wave function Ψ FEV of this superposition state reads Note that, in the single-color case l 1 l 2 . Motivated by the helical interference structures in the electron density, Starace and coworkers [155] coined the term "electron vortex" for this type of PMD. In general, this notion of an electron vortex is not identical with a vortex state defined by its non-vanishing azimuthal probability density current j. The probability density current of the vortex state defined in Eq. 31 has been derived in [106] and reads where e ε and e ϕ denote the respective unit vectors in energy and azimuthal direction. A detailed discussion of the probability density current of vortex states can be found in [105,106,156]. Interestingly, PMDs exhibiting simultaneously helical interference structures and a non-vanishing azimuthal probability density current (Eq. 32) have been experimentally demonstrated by bichromatic MPI using time-delayed CRCP pulses [73] and analyzed by TDSE simulations [121]. FEVs have recently attracted much attention, both theoretically [155,[157][158][159][160][161][162][163][164][165][166][167][168][169][170][171][172][173][174] and experimentally [73,[102][103][104]. In this section we discuss our experimental results on the creation and manipulation of FEVs from femtosecond atomic MPI using shaper-generated single-color femtosecond CRCP pulse sequences ( Figure 4D). By combining ultrafast polarization pulse shaping with photoelectron tomography, Pengel et al. [102] were the first to experimentally demonstrate FEVs from REMPI of potassium atoms. Initially, it was shown that perturbative REMPI with a single-color CRCP sequence gives rise to the six-armed Archimedean spiral-shaped FEV depicted in Figure 6B. The corresponding wave function produces a vortex-shaped electron density distribution. In the same work, it was shown that changing from MPI with a timedelayed single-color CRCP sequence to the corresponding singlecolor COCP sequence ( Figure 4C) results in the wave function corresponding to a toroidal PMD with spectral interference fringes in the radial (energy) direction spaced by h/τ. Subsequently, Pengel et al. [103] reported on the formation of FEVs from polarizationshaped supercontinua and on the generation of an eight-armed FEV in the ATI with the wave function The c 8 rotationally symmetric FEV in the ATI is shown in Figure 6C. The time-dependent electric field of single-color CRCP sequences shown in Figure 4D reveals that, as the time delay vanishes, the sequence evolves into a linearly polarized field whose orientation is determined by the relative phase between the LCP and RCP components. In Figure 6D we present results from tomographic reconstruction of the PMDs as τ → 0. The initial c 6 rotationally symmetric FEV (τ 20 fs) evolves into a distorted vortex (14 fs) with more and more pronounced lobes in the vicinity of the x-axis at 10 fs and eventually turns into the rotated angular momentum state R x (π/2)ψ 3,0 at τ 0. Partially overlapping LCP and RCP pulses resulting in the non-vanishing product [E −1 1 (t)] r1 · [E +1 1 (t)] l1 give rise to the additional pathways with alternating circularity, e.g. LCP → RCP → LCP and RCP → LCP → RCP, shown in Figure 1C with light arrows for a bichromatic MPI scenario. These pathways populate the free electron angular momentum states ψ 3,−1 and ψ 3,1 in addition, leading to the superposition of four ψ 3,m states with m ∈ (−3, −1, 1, 3). MPI with the horizontally polarized pulse at τ 0 results in the rotated angular momentum state R x (π/2)ψ 3,0 which is a superposition of the four ψ 3,m states according to where the amplitudes have been taken from Eq. 27. This result can be checked by decomposition of the rotated state into the unrotated basis R x (π/2)ψ 3,0 (θ, ϕ) m −3,−1,1,3

Non-Perturbative Resonance-Enhanced Multiphoton Ionization
So far, we discussed c 6 and c 8 rotationally symmetric FEVs generated by perturbative three-and four-photon ionization, respectively, with the intermediate resonant 4p state being only weakly populated and not contributing significantly to the vortex formation. In contrast, it was demonstrated that FEVs with c 4 rotational symmetry can be generated by non-perturbative REMPI [102]. By using CRCP pulse sequences with a pulse area of π [5], the 4s state was fully depleted and the 4p state fully populated. Hence, ionization with the second pulse started from the excited 4p state resulting in a wave function characterized by the superposition of angular momentum states with different values of |m| The tomographic reconstruction of the corresponding PMD depicted in Figure 6A

Bichromatic Pulse Sequences
In this section, we discuss a number of applications of bichromatic polarization-shaped pulse sequences for femtosecond spectroscopy and coherent control of ultrafast quantum dynamics [105,106]. Bichromatic fields are characterized by two spectral bands with individual bandwidths of Δω 1 and Δω 2 centered around two different carrier frequencies ω 1 and ω 2 , as detailed in [99]. The spectral band with the lower (higher) carrier frequency is typically referred to as red (blue) component. Our shaper-based approach for the generation of bichromatic fields offers independent control over the amplitude, phase and polarization profile of the field [65,146]. Shaper-generated polarization-tailored bichromatic fields are therefore very versatile, not only regarding their use in bichromatic coherent control schemes but also in terms of their application in timeresolved studies of ultrafast dynamics. For example, employing a linear spectral phase function (ω−ω i ) · τ allows us to introduce a variable time delay τ between the two colors to generate bichromatic pulse sequences for polarization-sensitive twocolor pump-probe experiments [177]. Several examples of polarization-controlled bichromatic pulse sequences, with both  Figure 7, including the prototypical PLP, OLP, COCP and CRCP fields. The bottom row illustrates the same sequences for τ 0. In contrast to the single-color case, temporally overlapping bichromatic fields occur in a plethora of shapes and symmetries, depending on the frequency ratio, polarization, amplitude and relative phase of the two colors. For example, in the OLP case shown in Figure 7B, the bichromatic polarization profile exhibits a Lissajous-type shape, while in the COCP and CRCP cases, shown in (C) and (D), the polarization profiles are cycloidally shaped. The applications of polarization-tailored bichromatic fields are as versatile as their shapes. For example, PLP bichromatic fields were applied to control plasmonenhanced photoemission from silver clusters [178] and strongfield photoemission from gold nanotips [79]. OLP bichromatic fields with commensurable frequencies, characterized by Lissajous-type polarization profiles which exhibit a timevarying optical chirality, have been used to investigate subcycle variations in the PECD of chiral molecules [179,180]. Commensurable CRCP bichromatic fields exhibit unique propeller shapes with distinct rotational symmetry. Currently, such fields play a prominent role in HHG [67, 68, 70-72, 181, 182], strong-field ionization [158,[183][184][185][186], and were employed for the time-resolved probing of molecular chirality [187]. In the perturbative regime, coherent control of bichromatic MPI is governed by the manipulation of intra-and interband interferences [105] (Section 2.2.2). In the following, we discuss examples for the bichromatic control of MPI in atoms (Section 4.2.1) and molecules (Section 4.2.2).

Atomic Multiphoton Ionization
Atomic model systems interacting with multichromatic pulse sequences are ideally suited to elucidate the physical mechanisms underlying the coherent control. In this section, we study two mechanisms of coherent control of multipath interference in the atomic MPI by polarization-tailored bichromatic fields. We compare the CEP-insensitive intraband MPI with N photons to the CEP-sensitive interband MPI with different numbers N 1 ≠ N 2 of photons. Different bichromatic control scenarios are implemented using linearly (Section 4.2.1.1) and circularly (Section 4.2.1.2) polarized bichromatic fields.

Linearly Polarized Bichromatic Pulse Sequences
The two prototypes of linearly polarized bichromatic pulse sequences, PLP and OLP fields, are illustrated in Figures  7A,B, respectively. To illustrate the difference between interband and intraband interferences in optical phase-control, we discuss below the results from bichromatic MPI of alkali atoms using CEP-stable commensurable PLP (3ω:4ω) fields [73] on the one hand and incommensurable OLP bichromatic fields [99,100] on the other hand. We start by investigating phasesensitive interband interferences arising from the superposition of continuum states with opposite parity [73,188]. MPI of sodium atoms with a PLP (3ω:4ω) field creates an f-type photoelectron wave packet (ψ 3,0 ) via 3-photon ionization by the blue pulse (ω 2 ) and a g-type wave packet (ψ 4,0 ) via 4photon ionization in the continuum by the red pulse (ω 1 ). Both wave packets overlap at a kinetic energy of ε 4Zω 1 3Zω 2 resulting in interband interferences. Because the difference between the number of photons absorbed by each partial wave is odd, the resulting photoelectron wave function is a superposition of two states with opposite parity As illustrated in Figure 8A, this wave function exhibits a directional asymmetry along the laser polarization axis (yaxis), which is sensitive to the CEP. To study the CEP- dependence of the asymmetry, we measured 2D projections of the PMD for φ ce 0 and φ ce π. The recorded projections were Abelinverted using the pBASEX algorithm [88]. The retrieved 2D sections of the PMD through the (x, y) polarization plane are shown in the left and middle frames of Figure 8A. The sections display a pronounced left/right asymmetry which is inverted by switching the CEP from 0 to π. To quantify the asymmetry, we derived the asymmetry contrast [105,189] directly from the measured projections. The result shown in the right frame of Figure 8A reveals a CEP-induced directional asymmetry of up to ±28%. A similar scheme for the control of directional photoemission from xenon atoms by bichromatic MPI with phase-modulated PLP (7ω:8ω) fields was recently reported in [189,190]. In that contribution, the bichromatic pulses were specifically tailored to induce phase-controlled 7-vs. 8-photon interband interference in the 5P 3/2 ionization continuum of xenon. An alternative strategy to steer the directionality of ultrafast electronic processes in atoms [188,189,191,192], molecules [193][194][195] or solids [196][197][198][199][200][201] is based on the use of CEP-stable few-cycle femtosecond laser pulses. In this case, however, the interpretation of the interference mechanism is generally complicated by the multitude of MPI pathways connecting the ground state to a specific energy target state. Using shaper-generated commensurable bichromatic fields tailored to the MPI process, has enabled us to excite specific asymmetric target states in the continuum by two-path interband interference, fully controllable via the CEP and the relative phase of the two colors. In contrast to the few-cycle approach, we were thus able to localize phase-controlled asymmetries in a preselected photoelectron energy window [189]. To demonstrate the concept of intraband interference, we employed temporally overlapping (τ < Δt) OLP bichromatic fields with incommensurable frequencies for 3-photon ionization of potassium atoms. While both the blue and the red component produce individual single-color signals centered at ε 0 3Zω 1 −Zω IP and ε 3 3Zω 2 −Zω IP , third-order intrapulse frequency mixing leads to two additional contributions centered at ε 1 2Zω 1 + Zω 2 − Zω IP and ε 2 Zω 1 + 2Zω 2 − Zω IP , which are energetically disentangled from the single-color signals (cf. Eq. 15) [99,100]. The measured and tomographically reconstructed 3D PMD is shown in the main frame of Figure 8B. The different above mentioned contributions are color-coded. Since each of the frequency mixing pathways is composed of three photons-albeit with different combinations of red and blue photons-the accumulated optical phase in each target state is identical, rendering the resulting intraband interference of all target states CEP-insensitive. Each of the photoelectron angular distributions within a given energy interval is determined by a specific coherent superposition of angular momentum states as described in Eq. 15. For example, the two single-color signals are described by rotated f-type wave packets perpendicularly to one another along the respective laser polarization axis, i.e. the x-and y-axis for the OLP pulse shown in Figure 7B. The two inner partial wave packets in the energy interval around ε 1 (magenta) and ε 2 (purple) resulting from intraband interference read analogously to Eq. 27, where the upper and lower signs in the sum correspond to ε 1 and ε 2 , respectively. Both reconstructed partial wave packets are shown separately in Figure 8C. They are also aligned perpendicularly with respect to one another and exhibit a pseudo c 6 rotational symmetry. Using (ω:2ω) bichromatic OLP fields, we recently demonstrated the generation of a c 4 rotationally symmetric PMD in the 2-photon ionization of potassium atoms [105] similar to the d x,y state shown in Figure 5B. Again, the mechanism was based on the cancellation of specific quantum pathways by intraband interference. Due to the energetic disentanglement of the different photoelectron contributions Frontiers in Physics | www.frontiersin.org August 2021 | Volume 9 | Article 675258 from intraband interference, this scheme is suitable for pumpprobe experiments featuring background-free detection of ultrafast dynamics. For example, employing shaper-generated linearly polarized bichromatic pump-probe pulse sequences, we investigated Rydberg [177] and spin-orbit [100] wave packet dynamics. The full potential of polarization-shaped bichromatic fields to generate and manipulate angular momentum superposition states via frequency mixing has been studied in [99]. Very recently, we introduced a shaperbased quantum state holography (SQuaSH) method by combining intra-and interband interferences using commensurable CEP-stable bichromatic fields for phasesensitive pump-probe measurements [101].

Circularly Polarized Bichromatic Pulse Sequences
Circularly polarized bichromatic fields, with their cycloidal polarization profiles ( Figures 7C,D), offer fascinating possibilities for the generation and control of electronic angular momentum wave packets with unusual physical properties, whether in the neutral system as in the case of spin-orbit wave packets (SOWPs) or in the ionization continuum as in the case of free electron wave packets. Similar to the linearly polarized case discussed in Section 4.2.1.1, the mechanism to control the spectrum, the symmetry and the rotation of the PMD is based on the manipulation of intraband and interband interference by the polarization state of the sequence (CRCP or COCP) and the optical phases. We illustrate the basic concepts with two examples. In the first example, we use bichromatic CRCP pulse sequences with incommensurable frequencies for MPI of potassium atoms and make use of the energetic disentanglement of the angular momentum states in the continuum. This ionization scheme is utilized for background-free observation of SOWP dynamics in the bound state. The second example deals with the creation and coherent control of FEVs with unusual rotational symmetry by interband interference in the MPI of sodium atoms with CEPstable commensurable cycloidal bichromatic pulse sequences.
Recently, we demonstrated intraband frequency mixing by shaper-generated CRCP bichromatic fields in the three-photon ionization of potassium [99]. Using temporally overlapping (τ < Δt) sequences of two colors with incommensurable frequencies, we showed that in the CRCP case, frequency mixing results in a complete energetic disentanglement of the different angular momentum target states in the ionization continuum. The reconstructed 3D PMD is depicted in Figure 9A. The two singlecolor contributions in the inner-(red) and outermost (blue) energy channels correspond to the torus-shaped partial waves ψ 3,±3 . The two frequency mixing contributions in the intermediate energy channels ε 1 and ε 2 correspond to the partial waves displayed in the insets. This result shows that in the CRCP case, bichromatic frequency mixing provides a unique mapping between the number of absorbed red and blue photons and the different continuum target states, enabling the selective excitation of individual angular momentum states which otherwise (e.g. in the single-color case) overlap inseparably in the energy spectrum (Eq. 36). As pointed out in Section 4.2.1.1, if the two colors are spectrally disjoint, the frequency mixing signals are insensitive to the optical phases implying that frequency mixing signals are inherently robust with respect to phase fluctuations of the CEP, the relative phase between the colors or the Gouy-phase. Moreover, the process of frequency mixing is very efficient due to the large number of pathways leading to the same final state (see Figure 2). These properties, i.e. efficiency, robustness and the disentanglement of target states, make frequency mixing a well-suited test-bed for bichromatic pumpprobe studies. The potential of shaper-based bichromatic pumpprobe spectroscopy was recently demonstrated on the examples of time-and angular-resolved measurement [177] and holographic observation [101] of ultrafast Rydberg dynamics and the time-resolved 3D imaging of ultrafast SOWP dynamics in potassium atoms [100]. In the latter experiment, the LCP blue pulse (pump) was tuned on resonance with the two 4p fine structure states 4p 1/2 and 4p 3/2 . These states are separated energetically by Δε 7 meV which corresponds to a SOWP oscillation period of T h/Δε 580 fs. The bandwidth of the pump was chosen sufficiently large to excite both states coherently and, thus, launch the SOWP in the neutral system. After a variable time delay τ, introduced experimentally by linear spectral phase modulation via the shaper, the RCP red pulse (probe) ionized the atom from the 4p states in a two-photon process, mapping the SOWP into the ionization continuum as depicted in Figure 10A. More specifically, the electron dynamics in the neutral atom were mapped into the ε 1 -channel, where-due to the energetic disentanglement from other contributions-the time evolution of the SOWP could be observed without any interfering background. By tomographic reconstruction of the PMD as a function of time delay τ, we obtained 3D images of the spatiotemporal SOWP dynamics. The images are shown in Figure 10B. Initially, at τ 0, the partial wave from the ε 1 -channel displays ψ 3,−1 symmetry. This symmetry maps the torus-shaped ψ 1,1 state (4p (m 1)) in the neutral system, which is consistent with the impulsive excitation of the 4p state by the circularly polarized pump pulse. At half-period τ ≈ T/2, the symmetry of the photoelectron angular distribution in the ε 1 -channel evolves into that of a ψ 3,−2 state, mapping the dumbbell-shaped ψ 1,0 state [4p (m 0)]. The time evolution of the SOWP between these two states is driven by the spin-orbit interaction which, in a semiclassical picture, is interpreted as the precession of the coupled spin and orbital angular momenta around the total angular momentum. After a full period T, the distribution in the ε 1channel returns to ψ 3,−1 symmetry indicating the completion of the first SOWP oscillation cycle. These findings, along with additional results obtained for OLP bichromatic pump-probe sequences [99,100], showcase the capabilities of bichromatic pump-probe spectroscopy, based on shaper-generated polarization-shaped tailored bichromatic pulse sequences in combination with photoelectron tomography, for the background-free 3D imaging of ultrafast spatiotemporal quantum dynamics.
Next, we discuss the application of circularly polarized bichromatic pulse sequences with commensurable frequencies for the coherent control of interband N 1 vs. N 2 MPI processes. When the different colored pulses overlap in time, i.e. at τ 0, the resulting field exhibits a cycloidal polarization profile with an S opt -fold rotational symmetry ( Figures 7C,D) described by Here, the upper and lower sign correspond to CRCP and COCP pulses, respectively, and gcd denotes the greatest common divisor. For the sake of clarity, we discuss below the case of gcd(N 1 , N 2 ) 1. The use of commensurable frequency cycloidal fields enables the creation of FEVs with unusual rotational symmetry by controlling the multipath interference in N 1 -vs. N 2 -photon ionization. The resulting FEV exhibits an S FEV -fold rotational symmetry with S FEV N 1 ± N 2 , reflecting the field symmetry from Eq. 42. Due to the different photonic orders of the contributing ionization pathways, the resulting interband interference is sensitive to the optical phases (cf. Eq. 39). Both, the relative phases φ 1 and φ 2 of the two colors and the CEP φ ce of the field induce an azimuthal rotation of the FEV, i.e. in the polarization plane, by different angles and in different directions. The general expression for the total rotation angle induced by the optical phases reads [73]  with ΔN N 2 − N 1 . Again, the upper and lower sign correspond to the CRCP and the COCP case, respectively. This relation first implies that for the observation of interband interference in the experiment, the stabilization of the CEP is crucial, otherwise the interference pattern is rotationally averaged out [73]. Apparently, the response of the quantum system to fluctuations of the CEP is different for interband interference compared to intraband frequency mixing (Section 4.2.1.1), which is robust with respect to optical phase variations. The second important implication of Eq. 43 is that the orientation of the FEV in polarization plane is controllable by all optical phases, albeit to different extents. Recently, we have experimentally demonstrated the creation and manipulation of 1-and 7-fold rotationally symmetric FEVs by controlled interband interference in the 3-vs. 4-photon ionization of sodium atoms using CEP-stable (3ω: 4ω) circularly polarized pulses [73]. Similar to the linear case (cf. Eq. 39), the photoelectron wave function in the circular case is described by the coherent superposition of a partial wave ψ 4,4 , created by absorption of four LCP photons from the red pulse, and a partial wave ψ 3,±3 created by absorption of three LCP (+: COCP case) or RCP (−: CRCP case) photons from the blue pulse The phase term Δφ 4φ 1 − 3φ 2 + φ ce describes the optically induced relative phase between both partial waves. The phase term ετ 1 /Z accounts for an additional energy-dependent propagation phase accumulated if the red pulse is timedelayed by τ 1 relative to the blue pulse. The FEV described by Ψ 1 , created by COCP ionization, exhibits a single azimuthal lobe reminiscent of a crescent, while the FEV with the wave function Ψ 7 , created by CRCP ionization, displays seven lobes in the polarization plane. The corresponding tomographically reconstructed 3D PMDs, measured at the parameters Δφ 0 and τ 1 0, are shown in Figures 9B,C, respectively. To demonstrate the rotational control of both FEVs, we varied the CEP from φ ce 0 to φ ce π, analogously to the directional control discussed in Section 4.2.1.1 (Figure 8). The resulting rotation of the PMD is depicted in Figures  10C,D, indicated by the green and blue angle segments, respectively. While the c 1 FEV is rotated by α COCP π, the c 7 FEV is rotated by an angle of α CRCP π 7 , in agreement with Eq. 43. In addition, we demonstrated the generation of a vortex-shaped PMD with c 7 rotational symmetry by introducing a time-delay of τ 1 −20 fs between the two colors in the CRCP pulse sequence. The reconstructed corresponding 3D PMD of the c 7 rotationally symmetric FEV, shown in Figure 10E, exhibits the characteristic helical interference structure, causing the vortex-like tilt of the lobes. The results on the FEVs with odd rotational symmetry created by MPI with circularly polarized commensurable bichromatic fields demonstrate our ability to manipulate the properties of the PMD by precise control on multipath interband interference. The rotational symmetry of FEVs is determined by the selection of the MPI pathways via the polarization state (COCP or CRCP), while its azimuthal orientation is manipulated by the optical phases including the relative phase and the CEP.

Molecular Multiphoton Ionization
Two-color femtosecond pump-probe experiments are ideally suited to study ultrafast dynamics in molecules driven by femtosecond laser pulses [151]. In the following Section 4.2.2.1 and Section 4.2.2.2, we discuss the application of this concept to the observation of non-Born-Oppenheimer dynamics in sodium dimers and the mapping of non-perturbative control of concerted electron-nuclear dynamics in potassium dimers. In Section 4.2.2.3, we present new results demonstrating axial and lateral asymmetries in the photoemission from chiral molecules induced by MPI with CEP-stable circularly polarized few-cycle pulses from a WLS.

Non-adiabatic Molecular Dynamics
In [202], we used a PLP two-color pump-probe sequence to study the effect of adiabatic (non-Born-Oppenheimer) dynamics in the 2 1 Σ + u double-minimum state of the sodium dimer. The sequence consisted of a 340 nm pump pulse, to launch a vibrational wave packet in the 2 1 Σ + u state by single photon excitation, and a timedelayed 265 nm probe pulse to interrogate the vibrating molecule by single photon ionization. Time-and energy-resolved photoelectron spectra were measured in order to map the time evolution of the nuclear wave packet along the vibrational coordinate R. The sodium double-minimum state arises from an avoided crossing of diabatic states [203]. Therefore, the electronic structure changes rapidly as a function of R suggesting a strong dependence of the ionization probability on the internuclear distance. This reasoning was supported by theoretical predictions in [204,205]. In the experiment, the nuclear wave packets were used as local probes to determine R-dependent ionization probabilities. By comparison of the measured photoelectron spectra to numerical wave packet simulations a linear R-dependence of the dipole moment μ n→i (R) for the neutral-to-ionic transition was retrieved. The experiment demonstrated that PLP two-color pump-probe sequences combined with time-resolved photoelectron spectroscopy is a powerful technique for the observation and unraveling of ultrafast photochemical dynamics in molecules.

Non-perturbative Electron-Nuclear Dynamics
In the experiments reported in [132,137], we applied a two-color pump-probe scheme to demonstrate efficient ultrafast switching between bound electronic target states in the potassium dimer, based on a detailed understanding of the underlying strong-field control mechanism. Intense femtosecond pulse sequences, shaped with attosecond precision [50], were used to switch the neutral excitation selectively between the 2 1 Π g and the 5 1 Σ + g state (cf. Section 4.1.1.2). Employing a time-delayed 570 nm probe pulse, generated by an optical parametric amplifier and combined interferometrically with the pump pulse, the final population of the target states was mapped into the energy-resolved photoelectron spectrum via one-photon ionization. By careful choice of the probe wavelength, the target state signals were tuned to an almost background-free energy window within the crowded photoelectron spectrum. In the measurement of non-perturbative bound state dynamics, signatures of strong-field interaction are typically washed out by volume averaging over the intensity distribution of the pump beam focus [132]. One way to counteract this effect is the utilization of a non-linear probe step. Due to the non-linear decrease of efficiency towards the focal edges, the probe volume is effectively reduced [119,120]. In this experiment however, dealing with a linear probe step, the probe beam was expanded to be focused more tightly into the interaction region than the pump. By this means, the probe mapped only the central part of the interaction region with an approximately homogeneous pump intensity distribution. Both the energetic disentanglement of the two-color probe signals from the broad single-color background and the minimization of the focal intensity averaging were key to the interference-free observation of the non-perturbative population dynamics induced in the target states by the intense tailored pump pulse sequence. The results demonstrated our ability to efficiently steer the coupled electron-nuclear dynamics of a molecule into different preselected target channels by controlling the delicate interplay between the quantum phase of the induced charge dynamics and the optical phase of the driving laser field.

Asymmetries in the Photoemission From Chiral Molecules
Polarization-shaped bichromatic fields are particularly suitable to chiral applications. Recent examples comprise the control of optical chirality [206], fundamental investigations on the PECD of chiral molecules [179,180] and time-resolved studies of molecular chirality [187,207]. Motivated by our experimental findings on the CEP-control of lateral asymmetries in the atomic MPI of sodium atoms by (3ω: 4ω) [73] and xenon atoms by (7ω: 8ω) [189] bichromatic PLP pulses (cf. Section 4.2.1.1), we investigated asymmetries in the PMD from MPI of the chiral fenchone molecule using CEP-stable few-cycle pulses. The manipulation of lateral asymmetries in chiral molecules is particularly interesting because these molecules also exhibit a pronounced axial, i.e., forward/backward asymmetry in the photoelectron emission due to the PECD [208]. Recent measurements of the PECD of (-)-fenchone molecules from MPI with circularly polarized 400 nm pulses, depicted in Figure 11A, reproduce our previous results [97,150]. Displayed is the antisymmetric part of the PMD with respect to the polarization plane, showing the well-known asymmetries of up to ±12%. Using a CEP-stable circularly polarized bandwidth-limited white light pulse centered at 800 nm, we observed CEP-sensitive lateral, i.e., left/right asymmetries in the photoelectron emission of (-)-fenchone. In Figure 11B we plot the difference between the PMDs measured with a CEP of φ ce 0 and φ ce π. This representation yields the antisymmetric part of the spectrum with respect to the propagation direction (zdirection), revealing pronounced lateral asymmetries of up to ±15%. The CEP-dependence of this asymmetry is shown in Figure 11C in the form of an energy-resolved asymmetry map obtained by integrating the spectra over the angular coordinate in one half-plane. These results demonstrate that MPI of chiral molecules with circularly polarized few-cycle pulses can be used to control the axial asymmetry via the PECD and, moreover, the lateral asymmetry via the CEP. Very recently, fully 3D polarization-tailored laser fields, generated by superposition of two polarization-controlled multichromatic laser pulses propagating non-collinearly in different directions, have been proposed as novel tools for the investigation of chiral light-matter interactions [209][210][211][212]. These so-called superchiral light fields [213] promise an unprecedented enantio-sensitivity which makes them highly attractive for chiral discrimination applications.

Multichromatic Pulse Sequences
A natural extension of bichromatic white-light polarization pulse shaping (cf. Section 3.1) is to generate multichromatic polarization-controlled pulse sequences. As a first step towards more complex pulses, we consider a trichromatic application. Trichromatic fields offer enhanced possibilities for ultrafast applications such as multicolor spectroscopy [214], strong-field physics [215,216], the generation of terahertz radiation [217,218] and HHG [219]. Further examples of trichromatic pulse sequences are illustrated in Figure 12. Each sequence consists of a temporally overlapping polarization-tailored bichromatic pulse which is either preceded by a circularly polarized [(A) and (B)] or followed by a linearly polarized [(C) and (D)] single pulse of a third color. The use of an additional color allows us to combine the bichromatic mechanism of intraband interference, e.g. for the background-free observation of ultrafast dynamics, with phasesensitive interband interferences for the coherent control of the underlying multiphoton processes. For example, an auxiliary pulse can be used to provide a separate reference wave packet for an experimental implementation of wave packet holography. This extension opens up a manifold of coherent control scenarios ranging from multichromatic SQuaSH via the creation and manipulation of multiple FEVs to multichannel quantumphase clocks [101,220]. However, trichromatic MPI is not only an extension of bichromatic MPI but also changes the physical picture, since the number of available pathways and the number of target states scales differently with the number of pulses, as discussed in Section 2.2.2. So far, we have distinguished between intraband and interband interference in bichromatic MPI scenarios. Phase-sensitive interband interference was characterized by interfering pathways involving the absorption of a different number of photons per color, i.e. N 1 ≠ N 2 , resulting in the condition ε ZN 1 ω 1 ZN 2 ω 2 . If more than two colors are present this condition can be generalized into involving two distinct pathways, e.g., with three pairwise different numbers of photons N 1 ≠ N 2 ≠ N 3 . Building on the results of self-referenced bichromatic SQuaSH in [101], we devise a trichromatic scenario to study the interference of bichromatic 2 + 1 REMPI for the detection of a background free pump-probe signal with direct three photon ionization delivering a reference signal for wave packet holography. According to Eq. 45, the three frequencies are adjusted so that both contributions yield photoelectrons in the same energy window. In this particular scenario, we choose N 1 + N 2 N 3 , such that the CEPdependence in the final state cancels. Here we present previously unpublished data from SQuaSH on 2 + 1 REMPI of potassium atoms using a trichromatic PLP pulse sequence. As shown in the excitation scheme in Figure 13A a photoelectron wave packet is created by resonant two-photon excitation of the 3d state by the red pump and subsequent one-photon ionization by the blue probe pulse. This wave packet interferes with the reference wave packet originating from offresonant 3-photon ionization of the ground state by the green reference pulse to generate a phase-sensitive photoelectron hologram. We expect the hologram to be sensitive to the relative phases between the pump, the probe and the reference, but, because the same number of photons are absorbed in each pathway, the interference pattern should not depend on the CEP. To investigate this scheme experimentally, we apply spectral amplitude modulation of the WLS (cf. Figure 13B) to generate a tailored PLP trichromatic pulse sequence composed of a λ pu 928 nm pump pulse (red band), followed by a λ pr 722 nm probe (blue band) and λ ref 848 nm reference pulse (green band). The time delays between the pulses are introduced via additional linear spectral phase modulation of the pump and probe band with τ pu −215 fs and τ pr −15 fs relative to the reference pulse. The wave function of the resulting photoelectron hologram reads where c pu , c pr and c ref denote the complex valued amplitudes including delay-dependent phases and φ pu , φ pr and φ ref the relative optical phases corresponding to the pump-the probeand the reference pulse, respectively. A tomographically reconstructed 3D PMD is depicted in Figure 13C. To evaluate the phase-dependence of the hologram, we measure the modulation of the integral cross-section in the SQuaSH channel of the photoelectron spectrum by variation of the relative optical phase φ ref ∈ [−π, π]. To this end, the measured PMDs are energycalibrated [221] and integrated over the angular coordinates to yield the photoelectron spectra. In Figures 13D,E the resulting energy-resolved photoelectron spectra I (ε; φ ref ) with ε ∈ [0.01, 0.1] eV are plotted as a function of the relative phase of the reference pulse. The insets in the top frame show the measured projections for constructive (φ ref −1.64 rad) and destructive interference (−0.77 rad). In agreement with Eq. 46, the interference signal shows a 2π 3 -periodic oscillation because the reference pulse ionizes the system by absorption of three photons. We invert the interference structure ( Figure 13E) by introducing a phase of φ pu π 2 to the pump pulse (Eq. 46), because the pump pulse (A) Circularly polarized single pulse followed by a CRCP pulse sequence, (B) Circularly polarized single pulse followed by an OLP pulse sequence, (C) CRCP pulse sequence followed by a linearly polarized (s) single pulse and (D) COCP pulse sequence followed by a linearly polarized single pulse tilted by 45°. In all figures the single pulse is separated by a delay τ > 2Δt from the coinciding pulse sequences.
Frontiers in Physics | www.frontiersin.org August 2021 | Volume 9 | Article 675258 contributes two photons to the REMPI process. The cosinusoidal oscillation is confirmed by showing sections through both interference patterns together with cosine-fits in Figure 13F.

SUMMARY AND OUTLOOK
In this review, we have reported on coherent control of MPI dynamics of atomic and molecular model systems using fully coherent polarization-tailored single-color, bichromatic and trichromatic pulse sequences. Building on previous experiments with single-color pulse sequences, we have shown that the advanced polarization shaping of CEP-stable supercontinua enables the generation of unprecedented bichromatic laser fields such as propeller-shaped CRCP pulses or Lissajous-like OLP pulses. Even more flexibility is achieved in trichromatic polarization shaping by the many possible combinations of linearly or circularly polarized pulses, resulting in an increasing number of final states and interfering pathways. These new pulse shapes enable the targeted use of CEP-insensitive intraband and CEP-sensitive interband multipath interference for quantum control. The experimental results on atomic and molecular MPI by multichromatic pulse sequences reviewed here, serve to illustrate the new options for coherent control. We have summarized the theoretical methods used to reproduce and analyze our experimental results and presented a powerful experimental technique combining supercontinuum polarization shaping and VMI-based high-resolution photoelectron tomography to reconstruct the full 3D PMD. The presented applications of singlecolor pulse sequences included perturbative and non-perturbative control of atomic and molecular MPI. We have analyzed the 3D PMD from a previous experiment on the interference of free electron wave packets created by PLP sequences and reported on free electron vortices with even-numbered rotational symmetry originating from MPI with single-color CRCP sequences. We have reviewed results on bichromatic coherent control by intraband and interband multipath interference on various examples including control of the directional photoemission in the MPI of atoms and chiral molecules by PLP and COCP fields, the use of COCP and CRCP cycloidal fields to generate and manipulate free electron vortices with odd-numbered rotational symmetry, and spin-orbit wave packets controlled by PLP pulses. Finally, we have introduced trichromatic shaper-based quantum state holography using a trichromatic PLP pulse sequence by adding a reference pulse to a bichromatic pump-probe sequence. As an example for the new options of trichromatic pulse shaping, we have devised an MPI scheme based on a polarization-tailored trichromatic CRCP pulse sequence which results in a CEPinsensitive c 6 rotationally symmetric inner vortex surrounded by a CEP-sensitive crescent-shaped wave-packet ( Figure 14). Extensions of our shaping scheme towards the generation of independently polarization-controlled multichromatic multipulse sequences are currently explored in our labs.
Shaper-based multi-color 2D spectroscopy using fully coherent multichromatic CEP-stable pulse sequences is another emerging application of our technique. Taking advantage of the shaper's capability to additionally introduce higher order spectral phase modulation, the trichromatic scheme can be further developed into a powerful tool for coherent control spectroscopy by using sequences of tailored pulses, thus generalizing the concept of 2D spectroscopy. In addition, the tunability of the shaper-generated polarization-controlled multicolor fields makes them attractive for applications in quantum information [222], quantum metrology [109,110,223] and ultrafast nanotechnology [79,178,214,224,225]. In all the experiments reported above, the tailored polarization profile was confined to the (x, y)-plane perpendicular to the propagation direction of the pulse. Building on recent theoretical proposals [209][210][211][212]226], we envision the next challenge in experimental coherent control to be the generation and tailoring of full 3D light fields by superimposing two polarization-shaped laser pulses propagating non-collinearly in different directions.

AUTHOR CONTRIBUTIONS
KE and MW developed the topic and idea, collected the data, and prepared the original manuscript. MW coordinated this review and managed the project. KE, LE, TB, and MW contributed to the conception and development of the narrative. KE, TB, and MW were responsible for data analysis and simulations. LE contributed to the drafting of the manuscript and prepared the illustrations. All authors contributed to manuscript revision, read, and approved the submitted version.