Control of photoelectron momentum distributions by bichromatic polarization-shaped laser fields

Bichromatic polarization-shaped femtosecond laser pulses are used to control three-dimensional photoelectron momentum distributions (3D-EDs) from resonance enhanced multi-photon ionization of potassium atoms. The light fields consisting of two spectral bands with different ellipticity are produced using an ultrafast polarization pulse shaper equipped with a custom polarizer in the Fourier plane. The tomographically reconstructed 3D-EDs from ionization with counterrotating circularly or orthogonal linearly polarized bichromatic laser pulses show different angular momentum superposition states at four distinct photoelectron energies. The analysis of the measured 3D-EDs reveals that the underlying physical mechanism is based on the interplay of ionization pathway selection via quantum mechanical selection rules for optical transitions and intrapulse frequency mixing of the spectral bands with different ellipticity.


Introduction
Femtosecond polarization pulse shaping [1][2][3][4][5] is an established experimental technique which provides access to the spatial nature of the light-matter interaction by control of the instantaneous ellipticity of the laser pulse. Control by polarization-shaped femtosecond laser pulses has been demonstrated in numerous experiments including coherent manipulation of two-photon absorption in atomic rubidium [6] and optimization of the ion yield in the multi-photon ionization of potassium dimers [7]. In addition, in [8] the extraction of a full set of ionization matrix elements from multi-photon ionization with polarization-shaped femtosecond laser pulses was demonstrated. Recently, ultrashort bichromatic laser fields with individual polarization state of each color have emerged as a powerful tool for coherent control of ultrafast electron dynamics in diverse applications comprising strong-field ionization of atoms [9][10][11][12][13], charge localization in ultrafast photochemical reactions [14,15], and phase control of electric currents in semiconductors [16,17] and metals [18]. In particular, in highorder harmonic generation (HHG) and HHG spectroscopy, polarization-shaped bichromatic fields are routinely employed to steer electron wave packets from tunneling ionization and control the rescattering process. The use of linearly polarized bichromatic driving fields permits the generation of even harmonics [19][20][21] and temporal shaping of the attosecond pulse train [20,[22][23][24][25]. The use of circularly polarized bichromatic driving fields leads to the emission of elliptically polarized harmonics, as proposed in [19,26,27] and demonstrated only recently [28][29][30]. In addition, bicircular HHG has been proposed as a technique to generate attosecond magnetic field pulses [31] and to extract spectroscopic information on atomic and molecular symmetries [32][33][34].
In this contribution, we demonstrate control of three-dimensional photoelectron momentum distributions (3D-EDs) from atomic resonance enhanced multi-photon ionization (REMPI) in the perturbative regime using bichromatic polarization-shaped femtosecond laser fields. Specifically, we study 1+2 REMPI of potassium atoms with a sequence of counterrotating circularly polarized (CNR-CP) and orthogonal linearly polarized (O-LP) bichromatic femtosecond laser pulses. The use of bichromatic polarization-shaped pulses disentangles

Theory
A theoretical description of photoelectron distributions from multi-photon atomic ionization with polarization-shaped laser pulses has been presented in the context of designer electron wave packets [41] and highly multiplexed coherent quantum metrology [8]. Here, we consider REMPI of atoms using polarization- connects the s 4 ground state to a continuum with an angular momentum of l=3, i.e. an f-type continuum. The resulting free electron wave packet is composed of the angular momentum eigenstates f m , j | ñ with  In general, the a m j ( ) e are determined by the shape of the ionizing laser pulse [39,43] and describe the kinetic energy distribution of each angular momentum state f m , j | ñ. In the weak-field limit, these amplitudes can be calculated using third order time-dependent perturbation theory and are given by the coherent superposition of all ionization pathways that lead from the ground state s 4 , 0 | ñ to the target states f m , j | ñ. For a complete description, the p 4 resonance needs to be taken into account [8,[44][45][46]. In order to present a simple and intuitive physical picture however, we restrict the following analytical description to non-resonant multi-photon ionization. For ionization pathways proceeding exactly via the p 4 resonance, e.g. pathway figure 1(b), an additional resonant term arises. Generally, the interference between resonant and non-resonant contributions leads to an additional modulation of the photoelectron energy distribution [44] which is not captured in the analytical model presented below (but is considered in the numerical simulations). Some effects due to the resonance observed in the experimental data are discussed in sections 4.1 and 4.2. In the non-resonant case, each individual ionization pathway depicted in figure 1 maps the third order optical spectrum of the driving laser pulse onto the continuum [47], taking into account the helicity of the photons absorbed in each transition. We describe the polarization-shaped laser pulse by the (negative frequency) analytic signal of its electric field E t ( ) , where e L R are the spherical basis vectors.
Then, for example, the contribution from the ionization pathway s being the sum of the kinetic excess energy ε and the atomic ionization potential IP. The prefactor R s p d f ) is the Fourier transform of the corresponding temporal field E t q ( ) and '⊗' denotes the convolution. In general, by associating q with the photon angular momentum, i.e. q=1 for L-CP and q 1 =for R-CP light, the total amplitude a m j ( ) e in equation (1) can be expressed as m q q q q q q q q q q q q m q q q 0 The sum runs over all helicity combinations q q q + + = , that is all three-photon ionization pathways connecting the ground state s 4 , 0 | ñ with a given continuum state f m , j | ñ. Equation (4) highlights the interplay between the dipole selection rules represented by the CGC products and frequency mixing reflected by the third order optical spectra.
So far, the discussion is valid for any polarization-shaped laser field. Now, we specifically consider a bichromatic laser pulse consisting of two, e.g., Gaussian-shaped spectral bands with central frequencies r w (red band) and b r w w > (blue band) and individual ellipticities. In general, frequency mixing of both colors gives rise to four contributions to E q q q  Table 2 gives an overview of all coefficients a nj in the O-LP case. Finally, we specialize our discussion on a polarization-shaped bichromatic pulse with two spectral bands of identical shape described by a common shape function g˜( ) w , such that being complex-valued amplitudes. Then, the convolution in equation (3) is readily performed yielding a third order spectrum of the form E c g q q q n n q q q n 3 0 The coefficients c n q q q , and describe the total amplitudes of the four energy contributions resulting from constructive or destructive interference of all frequency mixing pathways with the same helicity combination q q q is the third order optical spectrum of the spectral shape function. In the CNR-CP case, the weights of all contributions in the third order spectrum are equal (see table 1(b)). In the O-LP case, however, the contributions at 1 e and 2 e are reduced to 1/3 for the m 1 j =  states due to partial destructive interference of the three individual pathways contributing to the same final state (see table 2(b)). Inserting this result into equation (4) and introducing the energy distribution function  , the photoelectron wave packet within the nth energy band reads (see equation (1) ) g e e and an angular part given by the product of CGCs along the corresponding ionization pathway. Due to the unique mapping between helicity and color in the CNR-CP case, each energy window is associated with only one state f m , j | ñ, leading to a complete energetic disentanglement of the target states. The weights a nj are given by the products of the coefficients c n q q q

Counterrotating circular polarization
The scheme for three-photon ionization with a CNR-CP bichromatic femtosecond laser pulse is depicted in figure 1(b). The spectrum of the ionizing laser field shown in figure 2 has two Gaussian-shaped bands corresponding to two pulses with different central frequencies: If the two pulses coincide in time, the superposition of both colors produces a corkscrew-type temporal polarization profile, as illustrated in figure 2. For larger spectral separation and commensurable center frequencies the polarization profile changes into a characteristic propeller shape [26,36,37]. Here, due to the small spectral separation, the propeller leaves are very narrow, resulting in quasilinear instantaneous polarization slowly rotating with the beating frequency 2 . Absorption of three red L-CP photons leads to selective excitation of the f , 3 | ñ free electron wave packet with a toroidal momentum distribution centered at a kinetic energy of 3 IP pathway creates a torus-shaped free electron wave packet at . Because the electron wave packets from absorption of three red L-CP photons ) do not overlap energetically, the resulting photoelectron distribution is a 'torus within a torus'-in contrast to the electron vortex observed by single color REMPI [42]. In addition, if the red and blue By multiplication of the respective transition amplitudes and coherent addition of all contributions, the relative weight of this ionization pathway is determined to be a 3 5 11 = (see table 1(a)). Indeed, in both frequency mixing cases the respective three ionization pathways end in a single continuum state, i.e. state f , 1 | ñ at 1 e and state f , 1 | -ñat 2 e . Ionization with bichromatic CNR-CP fields creates a unique mapping of the angular momentum states f m , j | ñ to the respective kinetic energy. Therefore, the matrix to describe the state vector at a given kinetic energy of the free electron wave packet , , n ( ) y e q f in terms of the angular momentum states f m , j | ñ described by equation (6) is diagonal for ionization with CNR-CP pulses (see table 1(a)).

Orthogonal linear polarization
The scheme for excitation and ionization with O-LP bichromatic femtosecond laser pulses is shown in figure 1(c).

Experimental
In our experiment, we combine bichromatic polarization pulse shaping with photoelectron imaging in order to create and detect photoelectron wave packets from 1+2 REMPI of potassium atoms with CNR-CP and O-LP bichromatic laser pulses. Using tomographic techniques, the full 3D-ED is reconstructed from a series of 2D projections of the wave packets measured under various different angles.

Bichromatic polarization pulse shaping
Recently, we introduced an optical common-path scheme based on a f 4 polarization pulse shaper for the generation of polarization-shaped bichromatic laser pulses [36,37]. The shaper, equipped with a dual-layer liquid crystal spatial light modulator (LC-SLM; Jenoptik, SLM-S640d) for independent amplitude and phase modulation, is employed to sculpture a bichromatic amplitude profile from the spectrum of a 20 fs, 790 nm input pulse provided by a femtosecond laser system (Femtolasers Femtopower HR 3 kHz CEP amplifier seeded by a Rainbow 500 oscillator). A custom composite polarizer consisting of two parts with orthogonal transmission axes (sand p-polarized) is mounted in the Fourier plane of the f 4 setup to enable independent spectral amplitude and phase modulation of two orthogonally polarized spectral bands. Application of the phase functions with 0 w denoting the center frequency of the input spectral amplitude Ẽ( ) w , to the LC displays A and B of the LC-SLM yields O-LP bichromatic fields with individually adjustable amplitude profiles 1 2 ( ) w  , phase modulation functions 1 2 ( ) j w , and polarization states (sor p-polarized) of both colors. Conversion to CNR-CP bichromatic fields is achieved using a quarter wave plate at the shaper output with optical axis aligned at±45°with respect to the x-axis (see figure 2). The inset to figure 2 shows a measured bichromatic spectrum, as used in the experiments, in front of the input spectrum shown as gray-shaded background. For the bichromatic spectrum we used two Gaussian-shaped amplitude profiles . The amplitude ratio of roughly 2:1 was chosen to match the one-color photoelectron yields of the resonant and the red-detuned pulse (see also section 4). In order to introduce a time delay τ between the two colors with the pulse shaper, we use linear spectral phases (8) and (9). By applying the linear phases with respect to the center frequencies r w and b w , only the temporal envelopes of both colors are shifted in time while the respective carriers remain fixed, leaving the relative phase between the colors unaltered. To ensure bandwidth-limited bichromatic output pulses, we compensate the residual spectral phase of the input pulse using the pulse shaper and an evolutionary algorithm to adaptively optimize the second harmonic generation in a β-barium borate crystal [36,48,49].

Photoelectron imaging tomography
The 3D-EDs are measured using photoelectron imaging spectroscopy in combination with a tomographic reconstruction technique described in [39,41]. As sketched in figure 2, the laser is focused by a 250 mm lens into the interaction region of a VMI spectrometer to interact with potassium vapor (pressure 5 10 mbar Photoelectron wave packets released by the laser-atom interaction are imaged onto a multi-channel plate detector in chevron configuration stacked with a phosphor screen. About two events per laser pulse were detected on the screen. Images of the screen are recorded by a CCD camera with an exposure time of 250 ms. Each projected electron momentum distribution (PED) was recorded by accumulation of 400 images. The energy resolution of the PEDs is better than 80 meV at 1 eV. In general, 3D-EDs from photoionization with polarization-shaped laser pulses exhibit no rotational symmetry, precluding a reconstruction by Abel inversion. By rotating the incident laser pulses using a 2 l wave plate, we measured PEDs under 31 angles from 0°to 90°.
From the recorded 2D PEDs the 3D-EDs were reconstructed using the Fourier slice algorithm [50]. The Fourierbased algorithm described in [39] is specifically adapted to the reconstruction in spherical coordinates and was found to be better suited than the back projection algorithm.

Results and discussion
In this section, we present measured free electron wave packets from REMPI of potassium atoms by CNR-CP (section 4.1) and O-LP (section 4.2) bichromatic pulses. , with m e denoting the electron mass and n e being the central energy of the nth energy window as introduced in section 2. For best visibility, the weaker p 1and p 2 -contribution were enhanced by factors of 4.7 and 3.1, respectively. The right half of each section shows the enhanced data, while the original reconstructed data is displayed in the left half for comparison. Contributions from one-color multi-photon ionization by the red and blue components are visible at momenta p 0 (red arrow) and p 3 (blue arrow). The circularly symmetric signals in the x-y-plane and the crescent-shaped signals in both the x-z-and the y-z-plane indicate a toroidal shape of the corresponding wave packets, in accordance with the photoelectron states f , 3 0 | | y ñ µ ñand f , 3 3 | | y ñ µ -ñpredicted by equation (1) and table 1. In addition, two mixing terms, corresponding to the absorption of either two red and one blue or one red and two blue photons, are visible around p 1 (purple arrow) and p 2 (magenta arrow). While their contribution is likewise circularly symmetric in the x-y-plane, their angular distribution in the x-z-and the y-z-plane exhibits three distinct lobes in the upper and lower hemisphere, reflecting the spherical harmonics Y ,

Counterrotating circular polarization
The angular-integrated plots showing the photoelectron distribution as a function of the momentum in a given plane and the angular-independent momentum distribution are displayed in figure 3(b). It is clearly seen that the 2D distributions in (a) allow to discern features which are unresolved in the angular-independent momentum distribution. Even higher differential information is obtained by integration over the angular segments between [-30°..30°] and [60°..120°] in the planar momentum distribution. The resulting differential photoelectron spectra, displayed in the respective insets, highlight the contributions at p 0 and p 3 (in [-30°..30°]) as well as p 1 and p 2 (in [60°..120°]).
Next, we introduce a time delay between the two pulses in the CNR-CP sequence. Separating both colors in time provides an additional check to discriminate different contributions in the photoelectron spectrum. At a time delay of 300 fs t = -(left column of figure 3(a)) the red component precedes the blue. In this case, only the p 0 -and p 3 -contribution are observed in the photoelectron spectrum, confirming the one-color character of these signals. In contrast, for reversed pulse ordering at 300 fs t = (right column), the term around p 1 persists. Since the blue band is resonant with the potassium p s 4 4 ¬ transition, excitation by the blue prepulse generates population in the p 4 state. After time τ the excited system is probed by the red postpulse in a non-resonant twophoton ionization step. Similar to the 0 t = case, the corresponding wave packet detected around p 1 exhibits circular symmetry in the x-y-plane. However, the contributions in the x-z-and y-z-plane reveal that the symmetry of the wave packet has changed from three to two lobes in each hemisphere. This delay-dependent shape of the wave packet's angular distribution is due to the time evolution of a SOWP launched by coherent excitation of the two fine structure components p 4 1 2 and p 4 3 2 . The chosen time delay roughly corresponds to half the SOWP oscillation period of 578 fs [51,52]. Note that the SOWP dynamics are captured exclusively by the p 1 -contribution. Both one-color ionization pathways are insensitive to the time delay and the p 2 -mixing term vanishes with the temporal separation of the two colors. This example of REMPI with bichromatic CNR-CP pulses shows that the energetic disentanglement of the different ionization pathways reveals neutral system dynamics which may be concealed otherwise due to overlapping contributions. A detailed investigation of the SOWP dynamics utilizing shaper-generated polarization-tailored bichromatic pulses in a two-color pumpprobe experiment will be presented in a forthcoming publication.
The 3D-EDs presented in figure 4 highlight the excellent agreement of the measured and simulated photoelectron angular distributions for CNR-CP ionization. Separating the photoelectron distributions in the spherical shells containing narrow energy intervals around p 0 to p 3 (insets to figure 4)-much like peeling an onion-permits selective inspection of all four angular momentum states f m , j | ñ.

Orthogonal linear polarizations
The reconstructed 3D-ED created by ionization with O-LP pulses at 0 fs t = is shown in figure 6. Sections through the 3D distributions at 0 fs t = and 300 fs t =  are depicted in figure 5(a). Weak contributions were enhanced for better visibility using enhancement factors up to 3.9. At 0 fs t = (middle column) we observe contributions from all four frequency mixing pathways. The outermost contribution at p 3 , resulting from onecolor ionization by the blue pulse, exhibits an f , 0 | ñ-type shape in the x-y-plane (upper row). Due to the ppolarization of the blue band, the two main lobes of the signal are aligned horizontally along the y-axis.
Consequently, the signal vanishes in the x-z-plane (middle row) being a nodal plane of the 'f-wave'. The same symmetry is observed for the innermost contribution at p 0 resulting from one-color ionization by the red pulse. However, due to the s-polarization of the red band the corresponding photoelectron signal is aligned vertically along the x-axis and has-due to the rotational symmetry about the x-axis-similar contributions in both the x-y-and the x-z-planes. The symmetry of the two mixing terms is distinctly different. The first mixing term at p 1 exhibits six lobes in the x-y-plane distributed equidistantly in angular intervals of 60°, with the two weakest lobes aligned in opposite directions along the y-axis. The same structure, albeit less pronounced, appears in the x-z-plane, indicating a preferential alignment of the wave packet in the laser polarization plane (x-y-plane). This observation is in agreement with the generic wave packet shown in the inset to figure 1(c) ( 1 e -frame) and the discussion in section 2.2, where the six-lobe structure was rationalized to arise from a coherent superposition dominated by the states f , 3 |  ñ. A similar structure is observed for the p 2 -mixing term. In this case however, the two weak lobes in the x-y-plane are aligned along the x-axis. In the x-z-plane only two crescent-shaped signals are observed. Both observations suggest a 90°rotation of the p 2 -wave packet relative to the p 1 -wave packet about the laser propagation direction (z-axis), as discussed in section 2.2.
By introducing a time delay of 300 fs t = -(left column) only the one-color contributions around p 0 and p 3 remain, while the two mixing terms vanish from the photoelectron spectrum. For the reversed ordering of colors (right column), a pronounced signal is observed around p 2 resulting from resonant excitation and time-delayed probing of the atom. As in the CNR-CP case discussed in the previous section, the angular distribution of the p 1 -wave packet evolves in time due to the spin-orbit interaction in the excited p 4 state. As a result, the contributions in the x-y-and the x-z-plane exchange their roles between 0 t = and 300 fs, indicating a 90°r otation of the wave packet about the x-axis. This observation shows again that subtle details of the neutral dynamics are revealed by bichromatic REMPI due to the background-free detection of the relevant two-color signal.
The 'fwithin an f-wave' along with the two frequency mixing contributions discussed in section 2.2 are clearly observed in the experimental 3D-ED and the simulation shown in figure 6. By isolating the photoelectron distributions belonging to different energy shells around p 0 to p 3 , we compare simulated and measured contributions individually (see insets to figure 6) to find excellent agreement, although the equatorial node in the reconstructed p 2 -wave packet is not fully resolved due to the energetic overlap with the neighboring lobe of the p 3 -wave packet.

Conclusion
In this paper, we presented the first application of shaper-based polarization-tailored bichromatic femtosecond laser fields to the generation of controlled 3D free electron wave packets. Specifically, we reported on control of 3D-EDs from REMPI of potassium atoms by CNR-CP and O-LP bichromatic laser pulses. In the experiment, bichromatic laser fields consisting of two spectral bands with different ellipticity have been produced using an ultrafast polarization pulse shaper equipped with a custom polarizer in the Fourier plane. Photoelectron momentum images from ionization with CNR-CP and O-LP bichromatic laser pulses have been measured employing a VMI spectrometer. The 3D-EDs have been reconstructed from numerous VMI images using a tomographic algorithm. Our results showed that bichromatic 1+2 REMPI produces photoelectron wave packets with kinetic energies within four distinct energy bands. Both outermost 3D-EDs, at maximum and minimum kinetic energy, originate from one-color ionization respectively, whereas the contributions in between are created by frequency mixing of pulses with different ellipticity. By analyzing the pathways for multiphoton ionization with bichromatic CNR-CP and O-LP pulses, we could accurately characterize the angular quantum states of the observed free electron wave packets. We found that ionization with bichromatic CNR-CP fields creates a unique mapping of the angular momentum states f m , j | ñ to the respective kinetic energy bands, permitting selective detection of all four angular momentum states f m , j | ñ. The two one-color contributions create a 'torus within a torus' and two additional wave packets are generated by frequency mixing. In contrast, bichromatic O-LP laser pulses created four different superposition states, including the 'fwithin an f-wave' and two uncommon angular momentum superposition states. By introducing a time delay between the two pulses of different color and ellipticity, the non-resonant frequency mixing contribution vanished, whereas the resonant part persisted when the resonant excitation pulse preceded the off-resonant ionization pulse. Besides the discrimination of resonant and non-resonant ionization pathways, bichromatic excitation enabled backgroundfree detection of a SOWP in one specific photoelectron kinetic energy band.
In this contribution, REMPI of atoms with bichromatic CNR-CP and O-LP pulses served as a model system to demonstrate the extraction of detailed information on neutral dynamics and control of multi-photon ionization pathways. Our analysis of the 3D-EDs showed that the physical mechanism of control by bichromatic Note that the node of the measured p 2 -wave packet (purple) in the y-z-plane is not resolved due to the energetic overlap with the intense main-lobe of the p 3 -wave packet (blue).
polarization-shaped fields is based on the interplay of photoionization pathway selection by the ellipticity of the pulses and intrapulse frequency mixing of the spectral bands with different ellipticity.
In general, shaper-based creation of bichromatic polarization-tailored fields is a powerful experimental technique to study and manipulate the dynamics of quantum systems because it combines the advantages of polarization shaping, i.e. control of the 3D nature of the light-matter interaction, with the spectroscopic characteristics of bichromatic control. Currently we perform two-color polarization-sensitive pump-probe experiments using bichromatic polarization-shaped pulses to investigate spin-orbit and Rydberg wave packet dynamics and to perform time-resolved studies of the photoelectron circular dichroism of chiral molecules.