Multichromatic supercontinuum polarization shaping applied to photoelectron holography

We present a supercontinuum pulse shaping method specifically designed for the generation of polarization-tailored multichromatic femtosecond laser fields. By combining a 4f-polarization pulse shaper with a custom-made polarizer kit, we independently modulate different spectral bands of a white-light supercontinuum in amplitude, phase, and polarization. The scheme is highly modular, scalable to any number of bands supported by the input spectrum and aims at the physically motivated design of specific laser fields tailored to the relevant transitions and dynamics of the quantum system. The power and versatility of the scheme is showcased in three different scenarios based on atomic multiphoton ionization employing polarization-tailored trichromatic pulse sequences. (1) We demonstrate the creation of an fx(x2−3y2) -type free electron wave packet and reconstruct its 3D momentum distribution by a waveplate-free shaper-based photoelectron tomography technique. (2) We utilize the holographic properties of the wave packet to study time-resolved and phase-sensitive ultrafast dynamics of bound states. (3) We investigate the field-free time-evolution of a photoelectron wave packet in the continuum. Our results on an atomic model system demonstrate the great potential of the modular shaping scheme for a wide range of applications on more complex quantum systems, including polyatomic molecules, nanoscopic structures and solids.

In this contribution, we combine multichromatic polarization pulse shaping with photoelectron holography for the time-resolved investigation of quantum phases in atomic resonance-enhanced multiphoton ionization (REMPI) induced by the bound-state dynamics and by the time-evolution in the photoionization continuum.To this end, we present a pulse-shaping methodology specifically designed for the generation of polarization-tailored multichromatic femtosecond laser fields.The concept is based on the independent phase-and amplitude-modulation of a white-light supercontinuum (WLS), ideally suited due to its ultrabroad spectral range.The multichromatic spectral amplitude profile is sculptured from the WLS via amplitude-modulation using a 4f -polarization pulse shaper equipped with a dual-layer liquid crystal spatial light modulator (LC-SLM).Instead of the standard polarizer usually employed for this purpose, we introduce a polarizer kit containing a set of polarizing elements with various widths and transmission axes specifically adaptable to the application at hand.The polarizer kit develops the specific setups used previously [12,78] into a fully generalized scheme.The optical elements arranged in the Fourier plane form a structured polarization mask.The generalized scheme provides full polarization control of each spectral band in the multichromatic field and enables the generation of pulse sequences that could not be generated with our previous setups.The presented scheme is highly modular and readily scalable to any number of spectral bands supported by the input spectrum.In addition, the spectral phase of each band can be shaped separately, rendering the scheme very versatile for applications in spectroscopy and coherent control.For example, adjusting the constant relative phase between different bands permits to control Ramsey-type interference phenomena [68].Systematic variation of the linear phase of each band, i.e. the time-delay between the corresponding pulses, enables the implementation of polarization-sensitive pump-probe [69] and multidimensional spectroscopy [20] schemes.Non-linear spectral phase functions, e.g.linear chirps [70], sinusoidal phase functions [71], discontinuous step-functions [72,73] and arbitrary combinations thereof, are provided for the coherent control of quantum dynamics.In general, combining independent phase-and amplitude-modulation with modular and individual polarization control of multiple spectral bands allows for the physically motivated design of laser fields specifically tailored to the relevant transitions and dynamics of the quantum system under study.
We showcase the power of our scheme by investigating unusual photoelectron wave packets with even-numbered rotational symmetry created by atomic multiphoton ionization (MPI) with polarization-tailored trichromatic pulse sequences.Specifically, we demonstrate the creation of an f x(x 2 −3y 2 ) -type free electron wave packet.This wave packet is characterized by a real-valued angular momentum distribution commonly used in chemistry [74].Its bound-state analog-the atomic f x(x 2 −3y 2 ) -orbital-plays a prominent role in the coordination chemistry of lanthanide and actinide complexes [75][76][77].The wave packet is described by a coherent superposition of the angular momentum eigenstates |f, m = −3⟩ and |f, m = +3⟩.The corresponding electron density forms a spherical standing wave with c 6 rotational symmetry.A different type of c 6 -symmetric wave packet, that is a six-armed free electron vortex [65], was demonstrated in [78] using counter-rotating CP (CRCP) single-color pulse sequences.In that single-color experiment, the temporal separation of the pulses by a time-delay τ was required in order to maintain the rotational symmetry of the wave packet.For τ → 0, the CRCP sequence evolves into a single LP pulse opening up new ionization pathways by the mixing of photons from both pulses.Because in the single-color case, all pathways interfere in the same energy window, the shape and the symmetry of the created photoelectron wave packet is altered when the pulses overlap in time.As a result, it is impossible to create the c 6 -symmetric f x(x 2 −3y 2 ) -type wave packet using single-color techniques [78].However, here we show that using multichromatic fields, the interfering frequency mixing pathways are energetically disentangled from the relevant pathways constituting the sculptured wave packet, preserving its symmetry at τ = 0.
For the creation of the f x(x 2 −3y 2 ) -type angular momentum wave packet, we utilize the trichromatic photoelectron holography scheme described in [43].The scheme is based on the interference of a probe photoelectron wave packet, created by (2+1) REMPI of potassium atoms via the 3d-state, with a reference wave packet, created by direct three-photon ionization from the ground state.The resulting photoelectron hologram is detected using velocity map imaging (VMI) techniques [79].To show the versatility of our shaping scheme, we manipulate the structured wave packet in three different scenarios.(1) We introduce a shaper-based photoelectron tomography technique to reconstruct the three-dimensional (3D) photoelectron momentum distribution (PMD).In contrast to the established photoelectron tomography technique [80,81], where a half wave plate (HWP) is used to rotate the entire laser field, we use optical phases to rotate the hologram [82] and compare the results of both techniques.(2) We utilize the hologram for the time-resolved and phase-sensitive investigation of ultrafast photoionization dynamics in the bound atomic system.After its excitation to the 3d-state, the electron accumulates the field-free time-evolution phase which rotates the 3D PMD analogously to the optical phase.(3) We image the time-evolution of the reference wave packet after its release to the photoionization continuum.The propagation of the reference wave packet in the continuum induces an angular shearing of the 3D PMD which twists the hologram into a six-arm free electron vortex similar to our previous observations in the single-color case [78].
The paper is organized as follows, first in section 2 we introduce the generalized shaping scheme.Section 3 describes the physical system and the excitation scheme for the photoelectron holography.The results in section 4 are subdivided into three parts.In section 4.1 the optical control of the created photoelectron hologram is demonstrated and used to perform a shaper-based photoelectron tomography.The following sections 4.2 and 4.3 investigate ionization phases induced by the bound-state and continuum dynamics, which manifest in the photoelectron hologram as rotations and azimuthal shearing, respectively.We end the article in section 5 with a conclusion and outlook.

Multichromatic polarization pulse shaping
In this section, we introduce the optical setup of the multichromatic polarization shaping scheme and provide a mathematical description of the shaped output pulses based on the Jones-matrix formalism.The primary light source is an actively CEP-stabilized multipass chirped pulse amplifier system (FEMTOLASERS RAINBOW 500, CEP4 MODULE, FEMTOPOWER HR CEP, 3 kHz) which provides near-infrared pulses with a pulse duration of 20 fs at a central wavelength of 790 nm with a pulse energy of 1 mJ.Using a neon-filled hollow-core fiber (absolute pressure 2.0 bar, inner diameter: 250 µm), we generate a WLS ranging from 450 nm to 950 nm.The multichromatic amplitude profile is sculptured from the WLS employing a home-built polarization pulse shaper in 4f -geometry equipped with custom polarizer segments (CODIXX colorPol) from the polarizer kit.The segments are mounted behind the dual-layer LC-SLM (JENOPTIK SLM-640D) in the Fourier-plane of the 4f -setup, forming a spectrally resolved composite polarization mask.The spatial resolution of the polarizer mask is only limited by the minimum manufacturable width of the polarizer segments and ultimately by the pixel width of the spatial light modulator used.The spectral phase can also be controlled individually for each pixel so that, e.g.different time-delays can be set for each spectral band.In combination, the LC-SLM and the polarization mask allow the generation of tailored multichromatic fields controlled in amplitude, phase and polarization state of each spectral band.An alternative approach for multichromatic pulse shaping is based on dielectric metasurfaces in the Fourier-plane [83].In this case, the phase and polarization masks are implemented by nanostructures.This approach offers a higher spatial resolution but provides less flexibility, since the mask is not directly programmable as is the case for LC-SLMs.Our shaping scheme is illustrated in figure 1.The left inset to (a) shows a top view of the entire shaper setup.The input WLS is shown as white line and the shaped multichromatic output field is depicted as petrol-blue line.At the shaper output, a superachromatic quarter wave plate (QWP; Λ 4 ) is used to convert LP field components into CP fields.The shaped output fields are focused into a VMI spectrometer, where the experiment, i.e. the interaction with the sample, takes place.Using the VMI spectrometer, we detect two-dimensional projections of the created photoelectron wave packets with energy and angular resolution.An additional HWP (Λ 2 ) serves to rotate the entire laser field about its propagation direction, to implement photoelectron tomography [80] and reconstruct the 3D PMD.
The right part of figure 1(a) shows a magnification of the Fourier-plane of the pulse shaper.The incident WLS is p-polarized and described by the scalar positive frequency spectrum Ẽ+ in (ω).Using the Jones-matrix formalism, the corresponding vector field reads Ẽ+ in (ω) = Ẽ+ in (ω) (0, 1) T .After passing all optical elements, the modulated laser field Ẽ+ mod (ω) is described by the optical cascade where M(ω) is the Jones-matrix of the dual-layer LC-SLM, P N (ω) denotes the Jones-matrix of the polarization mask, consisting of N polarizer segments, and the Λ n (n = 2, 4) are the Jones-matrices of the QWP and the HWP with their respective rotation angles α q and α h .The modulator matrix M with optical axes oriented at α = ±45 • is given specifically by being the rotation matrix [84].The spectral phase functions φ A (ω) and φ B (ω) are defined piece-wise, encoding the computer-controlled phase masks of each spectral band for the SLM A and the SLM B, respectively (see [84] for a more detailed description).Prominent examples are constant phases [12], time-delays implemented by linear phase functions [78], linear chirps generated by quadratic phase-modulation [70] or multipulse sequences resulting from sinusoidal phase-modulation [71].Similarly, the composite polarizer mask P N (ω) is described piece-wise as where the P j (j = 1, . . ., N) are the Jones-matrices of the individual polarizer segments and the ω j are the boundary frequencies of the corresponding spectral bands.Although polarizer segments can be fabricated for any transmission direction, here we demonstrate the use of the following set ) , where the first two elements correspond to an s-and a p-polarizer and the last two elements correspond to a combination of a p-polarizer with a polarizer whose fast axis is oriented along +45 • and −45 • respectively, resulting in diagonal polarization (d ± ).The position and width of each polarizer segment can be chosen arbitrarily, allowing for spectrally resolved multichromatic polarization shaping.The subsequent QWP Λ 4 set to α q = ± π 4 converts the s-and p-LP light into CP light but leaves the d ± -LP light unaltered (and vice versa for α q = 0).Finally, the HWP Λ 2 (α h ) rotates the polarization profile by an angle of 2α h about the laser propagation direction.The two wave plates Λ n (α) are described by the Jones-matrices [85] Λ with n = 2 for the HWP and n = 4 for the QWP.
To illustrate the versatility of the scheme, four prototypical examples of polarization-tailored trichromatic laser fields are shown in figures 1(b)-(e).The corresponding polarizer and QWP configurations are given in the bottom row of each panel.Panel (b) shows a sequence of three temporally separated colors with the linear polarizations p, d and s to demonstrate individual polarization control of each spectral band.This type of pulse sequence is useful, e.g. for polarization-sensitive two-dimensional spectroscopy applications.Panel (c) depicts a propeller-shaped pulse with c 7 -symmetry followed by a diagonal LP pulse.The propeller-pulse is generated by application of the QWP to an orthogonal LP bichromatic (3ω:4ω) field (green and blue bands), converting its polarization state to CRCP.The diagonal LP polarized red band passes the QWP unaffected.This type of pulse sequence is ideally suited for multicolor pump-probe studies utilizing a polarization-tailored pump and a well-characterized probe pulse.Panel (d) shows a pulse sequence consisting of a corkscrew-shaped pulse followed by a short CP pulse.The corkscrew-pulse is generated by the interference of two narrow spectral bands centered in the red edge of the WLS and with opposite circularity (bichromatic CRCP).For optimal time resolution the ultrashort CP pulse is comparatively broadband covering the central and blue part of the WLS.In atomic MPI, it was shown that such corkscrew-pulses disentangle different angular momentum states |l, m⟩ energetically in the ionization continuum [86] allowing for the background-free imaging of ultrafast bound-state dynamics [39].Superimposing the different angular momentum photoelectron wave packets by an additional broadband reference wave packet, created by the CP pulse, enables the holographic, i.e. phase-sensitive detection of the dynamics [43].The last panel (e) shows a generic version of the pulse sequence used in the photoelectron holography experiments presented here.It consists of a time-delayed LCP bichromatic pump-probe sequence (red and blue bands) combined with an RCP reference pulse (green band) which overlaps temporally with the probe pulse.In ascending order of the central frequencies, this sequence is referred to as LRL-CP sequence.A more detailed discussion of this type of sequence is given in the following section 3.

Trichromatic MPI holography
To demonstrate the capabilities of the multichromatic polarization shaping scheme, we apply it to the creation of an unusual f x(x 2 −3y 2 ) -type free electron wave packet in the MPI of potassium atoms.To this end, we employ the shaper-based quantum state holography (SQuaSH) method developed in [43] and consider the resonant three-photon ionization of potassium atoms by a polarization-shaped trichromatic pulse sequence.The characteristic feature of the f x(x 2 −3y 2 ) -type wave packet is its c 6 rotational symmetry which, so far, has been realized in the shape of free electron vortices created using time-delayed CRCP single-color pulse sequences [65,78].In the single-color case, however, the field evolves into LP for temporally overlapping subpulses, entailing the creation of an |f, 0⟩-type photoelectron wave packet which is no longer c 6 -symmetric.In contrast, the trichromatic approach allows us to generalize the vortex-scheme towards overlapping pulses while maintaining the c 6 -symmetry.In the following, we describe the perturbative MPI of potassium atoms by polarization-shaped trichromatic laser fields and discuss the resulting photoelectron wave packets.
The excitation scheme for the interaction of potassium with a trichromatic LRL-CP sequence, as used in the experiment, is shown in figure 2(a).The LCP red band serves as a pump pulse to excite the |3d, 2⟩ state via two σ + -transitions.The 3d-population is mapped into an |f, 3⟩-type photoelectron continuum by the LCP blue band acting as a probe pulse.The probe wave packet is superimposed by an |f, −3⟩-type reference wave packet created via direct three-photon ionization of the 4s ground state by the RCP green band, hence termed reference pulse.When both partial wave packets are created simultaneously (τ ref = 0), the interference of both partial wave packets gives rise to the f x(x 2 −3y 2 ) -type photoelectron wave packet illustrated in the top inset to figure 2(a).Its c 6 rotational symmetry is determined by the difference between the magnetic quantum numbers of the partial wave packets.The labels 'probe' and 'reference' indicate that the structured wave packet is a photoelectron hologram.In our second and third scenario (see sections 4.2 and 4.3), we utilize the holographic properties of the superposition wave packet to determine the spectral phase of the partial wave packets and infer details of the MPI dynamics [43,82].
To highlight the role of the polarization in the MPI scheme and the options associated with it, we briefly describe two other polarization combinations, using different polarization masks.The corresponding ionization schemes are depicted in figures 2(b) and (c).Switching the circularity of the red pump pulse to RCP, as shown in (b), creates a superposition of continuum states with different orbital angular momentum, specifically, the |f, −1⟩-and the |p, −1⟩-type continuum.The angular part of the probe wave packet is hence described by |f, −1⟩ + a p (ε) |p, −1⟩, where the generally energy-dependent complex-valued parameter a p (ε) accounts for the different ionization amplitudes and phases of the two continua [41,82].Recently, we demonstrated the retrieval of the phase of a p by photoelectron holography [82].The difference between the magnetic quantum numbers of the interfering partial wave packets is 2 in this case, yielding the c 2 -symmetric hologram shown in figure 2(b).Switching the circularity of the reference pulse in addition, as shown in (c), changes the symmetry of the hologram once again.Now, the LCP reference pulse creates an |f, 3⟩-type reference wave packet.Because the magnetic quantum number difference is 4 in this case, the resulting hologram is c 4 -symmetric [82].A distinct advantage of the trichromatic schemes over the single-color schemes is based on the energetic disentanglement of the relevant MPI pathways from other frequency-mixing pathways and also the single-color pathways.The energetic separation of the target wave packet from the other contributions allows the unperturbed creation and observation of the photoelectron hologram.
In this contribution, we focus on the creation of the c 6 -symmetric f x(x 2 −3y 2 ) -type wave packet shown in figure 2(a).Experimentally, we generate a trichromatic LRL-CP laser field with central wavelengths of λ r = 927 nm, λ g = 835 nm and λ b = 702 nm from the WLS.The measured power spectral density of the trichromatic field is shown in figure 2(d).The residual spectral phase of the WLS is compensated by applying an evolutionary optimization procedure to maximize the second harmonic generation (SHG) in a β-barium borate crystal (EKSMA OPTICS, cutting angle 29.2 • , 5 µm thickness) [12].For this purpose, a broadband p-polarizer covering the entire WLS is inserted into the Fourier-plane.Subsequently, the p-polarizer is replaced by the tailored polarization mask and the time-zero of all three pulses is fine-tuned by optimizing the SHG signal again, this time applying only linear spectral phases to the pump and the reference pulse.The time-delay τ pr = 0 of the probe pulse is defined as time-zero.The optimized SHG-spectrum, shown in figure 2(e), displays all relevant single-color and frequency mixing contributions.The signals labeled by (i), (iv) and (vi) represent the single-color contributions of the blue, green and red component, respectively.The signals (ii), (iii) and (v) result from blue-red, blue-green and green-red frequency mixing which is highly sensitive to the temporal overlap of the colors.From this starting point, we varied the time-delay of the red pump pulse by τ pu and the green reference pulse by τ ref relative to the probe pulse.
In multichromatic photoelectron holography, the relevant photoelectron wave function is a superposition of a probe-and a reference wave packet.Considering three-photon ionization by a trichromatic LRL-CP pulse sequence, the pump and the probe pulse drive σ + -transitions whereas the reference pulse drives σ − -transitions.Therefore, according to the selection rules for optical dipole transitions, both MPI pathways couple the ground state exclusively to f -type continua.Unlike the experiment reported in [82], no contribution from the p-type continuum needs to be considered here which ensures the purity of the f x(x 2 −3y 2 ) -type wave packet.In spherical coordinates and in energy representation, the photoelectron wave function is expressed as [82] with the spherical harmonics Y 3,±3 (ϑ, ϕ) and the detunings ϖ ref (ε) = 3ω ref − ω IP − ε h and ϖ pu = 2ω pu −ω 3d .The radial parts R ref (ε) and R pr (ε) of the two partial wave packets are determined by the transition moments and the optical spectra of reference and probe pulse, respectively.The maximum interference contrast is achieved by experimentally adjusting the spectral widths and relative amplitudes of the different spectral bands such that the radial parts match, R ref (ε) ≈ R pr (ε).The relative phase between both partial wave packets is determined by the phase difference where α h is the rotation angle of the HWP, as well as the bound-state and the free time-evolution phases ϖ pu τ pu and ϖ ref (ε)τ ref , respectively.The last term in equation ( 7) arises because a HWP rotation by α h rotates the entire pulse by 2α h .Since both wave packets are created by absorption of three photons of opposite circularity, the total phase difference acquired is In addition, we note that-although different colors are involved-the carrier-envelope phase (CEP) does not enter equation ( 7), i.e. the hologram described by equation ( 6) is not CEP-sensitive.This is due to the same number of photons absorbed in the creation of both partial wave packets [63].Assuming matched radial parts, the photoelectron density describing the PMD measured in the experiment is given by [82] |Ψ The density is cosine-modulated in ϕ-direction with an angular frequency determined by the difference of the magnetic quantum numbers of the two partial wave packets, i.e. 6 in this case.Therefore, the photoelectron hologram in equation ( 8) exhibits a six-fold rotational symmetry in the polarization plane.
The density also reveals that the phase introduced by the HWP and the relative optical phases between the colors rotate the hologram by an angle of The option to rotate the wave packet by variation of the optical phases is utilized experimentally for the shaper-based, HWP-free tomographic reconstruction of the 3D PMD in section 4.1.In addition, the hologram in equation ( 8) is rotated by the energy-dependent angle (10) which arises due to the field-free time-evolution of the electron wave function in between the three pulses.
The first term proportional to the pump pulse delay τ pu results from the time-evolution of the electron in the excited 3d-state (bound system).This energy-independent phase induces a rotation of the entire hologram analogous to the optical phases in equation ( 9).In contrast, the second term proportional to the reference pulse time-delay τ ref describes an energy-dependent phase induced by the free time-evolution of either the reference (τ ref < 0) or the probe (τ ref > 0) wave packet in the photoelectron continuum, depending on which is created first.The effect is a differential rotation, i.e. azimuthal shearing of the hologram, twisting the PMD into the shape of an Archimedean spiral.We compare both the bound and the free wave packet dynamics by performing time-resolved measurements of the hologram as a function of the time-delays τ pu and τ ref .
The experimental results are presented in sections 4.2 and 4.3.

Results and discussion
We show the capabilities of our multichromatic shaping scheme in three prototype scenarios based on the creation of a f x(x 2 −3y 2 ) -type photoelectron wave packet using trichromatic LRL-CP sequences.First, we demonstrate a HWP-free shaper-based tomographic reconstruction of the 3D PMD.Second, we utilize the holographic properties of the structured wave packet for the time-resolved and phase-sensitive investigation of bound-state MPI dynamics.Finally, we holographically investigate the free dynamics of the reference wave packet in the ionization continuum.

Shaper-based tomography of an f x(x 2 −3y 2 ) -type photoelectron wave packet
Photoelectron tomography relies on the detection of various 2D projections of the 3D PMD under different angles [80].This is achieved either by rotating the PMD while keeping the detector fixed or by rotating the detector around the PMD.For the reconstruction of photoelectron wave packets, it is generally more convenient to rotate the object than rotating the detection apparatus.Typically, a HWP is used to rotate the entire laser field-and hence the PMD-about the laser propagation direction [80,81].In contrast to this established approach, referred to as HWP-based tomography, here we demonstrate a HWP-free approach, where the f x(x 2 −3y 2 ) -type wave packet is rotated utilizing its sensitivity to the relative optical phases [82] which we control by the pulse shaper.Considering equations ( 8) and ( 9), each of the relative optical phases individually rotates the PMD.Consequently, it is sufficient to vary one of these phases, thereby rotating exclusively the corresponding color rather than the entire pulse.Here, we chose to rotate the red pump pulse by variation of φ pu .For the reconstruction of the c 6 -symmetric wave packet, a rotation about 60 • is, in principle, sufficient.According to equation ( 9), this rotation is realized by varying φ pu from 0 to π.As a check, we varied φ pu in 30 steps from 0 to 2π thus rotating the wave packet over 120 • .The remaining relative optical phases φ pr and φ ref were set to zero and the HWP was kept fix at α h = 0 • For selected pump phases of φ pu = 0, 1 3 π, 2 3 π, π, we additionally performed HWP-based tomographies of the 3D PMD.These reconstructions are shown in figure 3(a) together with their projections along the Cartesian axes.The c 6 Fourier component of all PMDs was enhanced by a factor of 4 [78].The rotation of the PMD is best discernible in the z-projections, but is also visible in the variations of the x-and y-projections.All reconstructed PMDs display a slight asymmetry in x-direction, clearly seen in the z-projections.This asymmetry is ascribed to imperfections of the HWP which induce systematic variations of the laser focus geometry during the HWP-rotation.Figure 3(b) depicts the azimuthal (ϕ) angular distributions (blue lines) of the wave packets.The rotation angles indicated as red circular segments are calculated according to equation (9).The calculated angles match the measured photoelectron angular distributions very well, demonstrating the high fidelity of the rotation induced via the optical phase.Figure 3(c) shows the reconstructed 3D PMDs obtained by HWP-based tomography (top frame) in comparison to the shaper-based tomography result (bottom frame).Both reconstructions are in very good agreement.The main difference is, that the shaper-based reconstruction does not exhibit the above-mentioned asymmetry in x-direction, since it is not affected by imperfections of the HWP.
Overall, the shaper-based tomography technique is a powerful alternative to the established HWP-based method.In some cases, such as the scenario presented here, it is not even necessary to rotate the entire laser field in order to rotate the PMD.In general, however, the field needs to be rotated as a whole, which is feasible using the pulse shaper provided sufficient modulation capabilities are available [87,88].

Time-resolved holography of bound-state dynamics
Next, we demonstrate the capabilities of the shaping scheme for the phase-sensitive detection of ultrafast ionization dynamics in the bound system.For this purpose, we consider the f x(x 2 −3y 2 ) -type wave packet as a hologram.By variation of the pump time-delay τ pu , we perform a pump-probe study of the electron's field-free time-evolution in the 3d-state.In the absence of an external field, the excited electron accumulates the time-evolution phase which is mapped into the phase of the probe wave packet.Therefore, measuring the density of the probe wave packet as a function of τ pu is not sufficient to reveal the 3d time-evolution.Only by the superposition of the probe with the reference wave packet, the 3d-phase is mapped into the density of the hologram where it manifests itself in the azimuthal rotation of the c 6 -symmetric interference pattern.
In the experiment, the pump pulse was slightly blue-detuned with respect to the 4s →→ 3d resonance.The pump time-delay was varied from the initial value τ pu = −275 fs in five steps of 100 fs to −775 fs, successively shifting the pump to earlier times while keeping probe and reference synchronized at τ pr = τ ref = 0.For each step, we performed a HWP-based tomography of the 3D PMD.The experimental results are presented on the left-hand side of figure 4. Two selected reconstructions of the PMD for τ pu = −475 fs and −675 fs are shown along the left time axis.The 3d-induced azimuthal rotation is clearly visible and also highlighted by red arrows, whose angles are determined via Fourier analysis.From the rotation angles, we extract the phases ϖ pu τ pu which are depicted in the top left inset as a function of the pump time-delay.As expected from equation (10), we find a linear dependence of the phase on τ pu .Using a linear fit model, we determine the slope of the linear function as ϖ pu = 7(1) mrad fs −1 which, according to the relation ϖ pu = 2ω pu − ω 3d , corresponds to a blue-detuning of the pump pulse of about 1.4(2) nm, i.e. a center wavelength of the pump pulse of 927.3(2) nm.This result is consistent with the measured spectrum in figure 2(d), validating the fidelity of the phase retrieval via the shaper-based photoelectron holography scheme.
The time-resolved holography of prototypical bound-state dynamics shows that shaper-generated polarization-tailored multichromatic pulse sequences are powerful and versatile tools for the real-time imaging of ultrafast dynamics including the retrieval of quantum-mechanical phase information.The technique is also applicable to more complex quantum systems with higher densities of states, where the excitation of bound-state wave packets is mapped into richer, energy-dependent rotational dynamics combined with the amplitude modulation of the hologram.

Time-resolved holography of continuum dynamics
So far, we demonstrated the creation of the f x(x 2 −3y 2 ) -type wave packet and controlled its rotation by constant optical phases and the time-delay between the pump and the probe pulse.In our third scenario, we highlight the different physical roles of the three colors in the trichromatic pulse sequence.We show that the probe-reference time-delay has a different effect on the wave packet, i.e. it alters the shape of the sculptured wave packet while maintaining its symmetry.To do so, we fixed the pump time-delay to τ pu = −275 fs and decreased the reference time-delay in two steps from τ ref = 0 fs to −30 fs and −50 fs.For each step, we performed a HWP-based tomography of the PMD.The experimental results are presented on the right-hand side of figure 4. We observe that the f x(x 2 −3y 2 ) -type wave packet is twisted into a six-arm photoelectron vortex with counterclockwise sense of rotation [78].Since the probe no longer coincides with the reference pulse, the reference wave packet is created before the probe wave packet and, as described in section 3, accumulates an additional phase of −ϖ ref (ε)τ ref in the continuum.Due to the linear energy-dependence, this free time-evolution phase induces a differential rotation, i.e. shearing of the photoelectron wave packet in azimuthal direction yielding a spiral-shaped PMD.The slope of the spiral arms is inverse proportional to τ ref [78] tending towards infinity as τ ref → 0 (in this limit, the vortex-shaped wave packet converges towards the f x(x 2 −3y 2 ) -type wave packet).As a consequence, the absolute slope is larger and the photoelectron spiral is less twisted for τ ref = −30 fs than for −50 fs.To analyze the azimuthal shearing angle in more detail, we integrated the reconstructed PMDs over the polar angle ϑ.The resulting 2D representations of the wave packet are shown in the top right inset to figure 4. The green curves indicate the theoretical shearing angle of the lobes as predicted by equation (10).The slopes are calculated according to the relation ε(ϕ) = 6h/τ ref • ϕ [78].The theoretical slopes match the experimental results very well.
Our results show that the different physical mechanisms at play in our scenario have very different effects on the photoelectron wave packet.The shaper-generated trichromatic pulse sequences provide access to all of these mechanisms and enable us to explore them independently.This underscores the power and versatility of the multichromatic shaping scheme for the rational design of ultrashort multichromatic pulse sequences specifically tailored to the application at hand.

Conclusion and outlook
We presented a supercontinuum pulse shaping scheme specifically designed for the generation of polarization-tailored multichromatic femtosecond laser fields and applied the scheme to the creation of unusual photoelectron wave packets.The shaping scheme combines a WLS polarization pulse shaper with a custom-made polarizer kit consisting of various polarizing elements with different widths and transmission axes.The elements are arranged modularly along the spectral axis to form a structured polarization mask which, when inserted in the Fourier plane of the shaper, provides independent control over the amplitude, phase, and polarization of different WLS bands.
The versatility of the shaping scheme was demonstrated in three photoelectron holography scenarios based on the resonant MPI of potassium atoms with polarization-tailored trichromatic pulse sequences.First, we presented the creation of an f x(x 2 −3y 2 ) -type free electron wave packet-an unusual angular momentum superposition state with c 6 rotational symmetry which cannot be created by single-color techniques.By variation of the relative optical phase between the colors, we coherently controlled the rotation of the wave packet and performed a fully shaper-based tomographic reconstruction of the 3D PMD without rotating optical elements.In the second and third study, we utilized the structured wave packet as a holographic probe of the photoionization dynamics.By independent variation of the time-delays between the three colors, we mapped out the laser-excited bound-state dynamics and explored the dynamics of a partial photoelectron wave packet in the continuum after ionization.Each type of dynamics imprinted distinct signatures on the photoelectron hologram: While the bound dynamics induced a rotation of the 3D PMD, similar to the optical phase, the continuum dynamics twisted the 3D PMD into a free electron vortex.
Beyond the variation of time-delays, i.e. linear spectral phases, our shaping scheme allows to apply arbitrary spectral phase functions to the different colors.The ability to generate sequences of shaped pulses with tailored spectral, temporal and polarization profiles opens up perspectives for coherent control spectroscopy [89].Here, a spectroscopic signal is measured as a function of arbitrary control parameters to determine not only the structure and dynamics of a quantum system but also the physical mechanisms underlying the light-induced processes.A simple realization of this concept is the recording of 2D maps of a molecular signal as a function of the second-(chirp) and third-order dispersion parameter [90].More complex implementations include sinusoidal spectral phase functions [71,89,91,92] and discrete spectral phase steps [72,92].Our shaping scheme is ideally suited for this application, combining the advantages of ultrashort multichromatic fields with the complete repertoire of ultrafast pulse shaping techniques for the physically motivated design of tailored pulse sequences specifically adapted to the quantum system under study.Currently we are working on a shaping setup based on three displays to disentangle the amplitude modulation from the phase and polarization control.By this means, no individual polarizer segments are required, improving the modularity and flexibility of the pulse shaping setup even further.

Figure 1 .
Figure 1.Modular multichromatic polarization pulse-shaping setup.(a) Schematic of the polarization pulse shaper (left) and magnification of the Fourier-plane (right) with the two layers A and B of the LC-SLM with an exemplary configuration of three polarization segments P1, P2 and P3 taken from the polarizer kit and mounted behind the LC-SLM.The structure polarization mask enables individual polarization control and independent amplitude-and phase-modulation of three spectral bands of the input WLS.The bottom panels (b)-(e) show a selection of pulse sequences generated by the multichromatic polarization shaping scheme for different applications (see main text) to highlight the versatility of the approach.

Figure 2 .
Figure 2. Photoelectron holography schemes based on the REMPI of potassium atoms interacting perturbatively with polarization-tailored trichromatic pulse sequences.The top insets display partial photoelectron wave packets (generic) created along the different MPI pathways together with the total photoelectron wave packet arising from the coherent superposition.(a) shows the scheme for an LRL-CP sequence as used in the experiment.(b) and (c) show schemes for RRL-and RLL-CP sequences, respectively, to highlight the role of the polarization for the created photoelectron hologram.(d) and (e) depict the measured fundamental and SHG spectrum of the trichromatic field used on the experiment for temporally overlapping colors, i.e. τpu = τ ref = 0 fs.Single-color contributions to the SHG spectrum are labeled by (i), (iv) and (vi), frequency-mixing contributions are labeled by (ii), (iii) and (v).

Figure 3 .
Figure 3. Tomographic reconstruction of the c6-symmetric f x(x 2 −3y 2 ) -type photoelectron wave packet.(a) Measured HWP-based reconstructions of the 3D PMD for different relative phases of the pump pulse.The rotation of the PMD by φpu is clearly visible.(b) Corresponding azimuthal photoelectron angular distributions (blue) with theoretical rotation angles calculated using equation (9).(c) Comparison of the HWP-based (top) and the shaper-based (bottom) tomography result.

Figure 4 .
Figure 4. Time-resolved holography of bound-state (left branch) and continuum (right branch) dynamics.Left: by variation of the pump-probe time-delay, the time-evolution of the electron in the excited 3d-state is mapped into the hologram.The time-evolution phase acquired in the bound atomic system manifests itself in a rotation of the 3D PMD.The top left inset shows the measured rotation angles as a function of τpu (black circles).The result of the linear fit (red line) is in good accordance with equation (10).Right: by variation of the probe-reference time-delay, the time-evolution of the reference photoelectron wave packet in the continuum is mapped into the hologram.The time-evolution phase acquired in the continuum is energy-dependent.It induces an azimuthal shearing of the hologram yielding a vortex-shaped PMD.The top right inset shows the PMD in polar representation for different values of τ ref .The shearing angles calculated according to equation (10) (green lines) match the experimental results very well.