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

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: a red L-CP laser pulse ${\tilde{E}}_{L}^{-}(\omega )$ centered at ${\omega }_{r}$ and a blue R-CP pulse ${\tilde{E}}_{R}^{-}(\omega )$ centered at ${\omega }_{b}$. 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 quasi-linear instantaneous polarization slowly rotating with the beating frequency $({\omega }_{b}-{\omega }_{r})/2$. Absorption of three red L-CP photons leads to selective excitation of the $| f,3\rangle$ free electron wave packet with a toroidal momentum distribution centered at a kinetic energy of ${\varepsilon }_{0}=3{\hslash }{\omega }_{r}-{\rm{IP}}$ via the $| 4s,0\rangle \mathop{\longrightarrow }\limits^{L}| 4p,1\rangle \mathop{\longrightarrow }\limits^{L}| d,2\rangle \mathop{\longrightarrow }\limits^{L}| f,3\rangle$ ionization pathway. Analogously, ionization through the $| 4s,0\rangle \mathop{\longrightarrow }\limits^{R}| 4p,-1\rangle \mathop{\longrightarrow }\limits^{R}| d,-2\rangle \mathop{\longrightarrow }\limits^{R}| f,-3\rangle$ pathway creates a torus-shaped free electron wave packet at ${\varepsilon }_{3}=3{\hslash }{\omega }_{b}-{\rm{IP}}$. Because the electron wave packets from absorption of three red L-CP photons ($| f,3\rangle$ at ${\varepsilon }_{0}=3{\hslash }{\omega }_{r}-{\rm{IP}}$) and three blue R-CP photons ($| f,-3\rangle$ at ${\varepsilon }_{3}=3{\hslash }{\omega }_{b}-{\rm{IP}}$) 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 pulses coincide in time, further pathways for ionization are available by frequency mixing. The states at the two intermediate kinetic energies ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ are each connected to the ground state via three ionization pathways with different combinations of red L-CP and blue R-CP photons. For example, all three possible pathways for photoionization by absorption of one blue R-CP photon and two red L-CP photons via the pathways $| 4s,0\rangle \mathop{\longrightarrow }\limits^{R}| 4p,-1\rangle \mathop{\longrightarrow }\limits^{L}| d,0\rangle \mathop{\longrightarrow }\limits^{L}| f,1\rangle$, $| 4s,0\rangle \mathop{\longrightarrow }\limits^{L}| 4p,1\rangle \mathop{\longrightarrow }\limits^{R}| d,0\rangle \mathop{\longrightarrow }\limits^{L}| f,1\rangle$ and $| 4s,0\rangle \mathop{\longrightarrow }\limits^{L}| 4p,1\rangle \mathop{\longrightarrow }\limits^{L}| d,2\rangle \mathop{\longrightarrow }\limits^{R}| f,1\rangle$ end in the $| f,1\rangle$ continuum state at ${\varepsilon }_{1}=2{\hslash }{\omega }_{r}+{\hslash }{\omega }_{b}-{\rm{IP}}$. By multiplication of the respective transition amplitudes and coherent addition of all contributions, the relative weight of this ionization pathway is determined to be ${a}_{11}=\sqrt{3/5}$ (see table 1(a)). Indeed, in both frequency mixing cases the respective three ionization pathways end in a single continuum state, i.e. state $| f,1\rangle$ at ${\varepsilon }_{1}$ and state $| f,-1\rangle$ at ${\varepsilon }_{2}$. Ionization with bichromatic CNR-CP fields creates a unique mapping of the angular momentum states $| f,{m}_{j}\rangle$ to the respective kinetic energy. Therefore, the matrix to describe the state vector at a given kinetic energy of the free electron wave packet ${\psi }_{n}(\varepsilon ,\theta ,\phi )$ in terms of the angular momentum states $| f,{m}_{j}\rangle$ described by equation (6) is diagonal for ionization with CNR-CP pulses (see table 1(a)).

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The scheme for excitation and ionization with O-LP bichromatic femtosecond laser pulses is shown in figure 1(c). In the spherical basis both linearly polarized pulses, i.e. the red s- and the blue p-polarized pulse, are represented by superpositions of an R-CP and an L-CP field. The relative optical phase of $\phi =\pi$ in the R-CP component of the red pulse is introduced to describe the 90° spatial rotation of the red pulse with respect to the blue. This optical phase is imparted on all ${\rm{\Delta }}{m}=-1$ transitions driven by the red pulse. Excitation and ionization with the blue p-polarized pulse creates a superposition of all angular momentum states $| f,{m}_{j}\rangle$ centered at the kinetic energy ${\varepsilon }_{3}$. Their weights a_nj are given by the sum over all products of the transition amplitudes of all permissible pathways and result in an $| {f},{m}=0\rangle$-type wave packet—briefly termed 'f-wave' in the following—rotated by 90° about the x-axis and aligned along the polarization direction of the blue pulse, i.e. the y-axis [39, 41]. The red s-polarized pulse ionizes via the same pathways to create an electron wave packet at ${\varepsilon }_{0}$. However, the additional relative phase of π rotates the 'f-wave' by 90° about the y-axis. As a result, O-LP ionization creates an 'f- within an f-wave', one of which is rotated with respect to the other. To describe the contributions at ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ all permissible frequency mixing pathways (e.g. one blue p-polarized photon and two red s-polarized photons) are considered and added coherently taking into account the relative phases. The resulting coefficients a_nj for photoionization with O-LP pulses presented in table 2(a) show that all superposition states ${\psi }_{n}(\varepsilon ,\theta ,\phi )$ at the respective energies ${\varepsilon }_{n}$ have contributions from all angular momentum states $| f,{m}_{j}\rangle$.

Because the states $| {\psi }_{0}({\varepsilon }_{0})\rangle$ and $| {\psi }_{3}({\varepsilon }_{3})\rangle$ are rotated $| f,0\rangle$ states, the amplitudes are also obtained by decomposition into the angular momentum states $| f,{m}_{j}\rangle$ and thus proportional to the elements of the Wigner D-matrices ${D}_{{m}_{j},0}^{(3)}\left(0,\tfrac{\pi }{2},0\right)$ and ${D}_{{m}_{j},0}^{(3)}\left(\tfrac{\pi }{2},\tfrac{\pi }{2},0\right)$ [41]. The photoelectron distributions at ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ are mainly superpositions of the $| f,3\rangle$ and $| f,-3\rangle$ states with minor contributions from the $| f,\pm 1\rangle$ states. The alternation of the signs of the coefficients to describe the superposition states $| {\psi }_{1}\rangle$ and $| {\psi }_{2}\rangle$ at the energies ${\varepsilon }_{1}$ and ${\varepsilon }_{2}$ shown in table 2, i.e. ${a}_{2j}={(-1)}^{\tfrac{{m}_{j}+1}{2}}{a}_{1j}$, corresponds to the rotation of the $| {\psi }_{1}\rangle$ state by 90° about the z-axis in accordance with the drawings shown in the insets (${\varepsilon }_{1}$ and ${\varepsilon }_{2}$) to figure 1(c).

Recently, we introduced an optical common-path scheme based on a $4f$ 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 (s- and p-polarized) is mounted in the Fourier plane of the $4f$ setup to enable independent spectral amplitude and phase modulation of two orthogonally polarized spectral bands. Application of the phase functions

$$\begin{eqnarray}{\varphi }_{A}(\omega )=\left\{\begin{array}{cc}{\varphi }_{1}(\omega )-\mathrm{acos}\left(\tfrac{{{ \mathcal A }}_{1}(\omega )}{\tilde{E}(\omega )}\right), & \omega \lt {\omega }_{0}\\ {\varphi }_{2}(\omega )+\mathrm{acos}\left(\tfrac{{{ \mathcal A }}_{2}(\omega )}{\tilde{E}(\omega )}\right), & \omega \geqslant {\omega }_{0}\end{array}\right.\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}{\varphi }_{B}(\omega )=\left\{\begin{array}{cc}{\varphi }_{1}(\omega )-\mathrm{acos}\left(-\tfrac{{{ \mathcal A }}_{1}(\omega )}{\tilde{E}(\omega )}\right), & \omega \lt {\omega }_{0}\\ {\varphi }_{2}(\omega )-\mathrm{acos}\left(\,\,\tfrac{{{ \mathcal A }}_{2}(\omega )}{\tilde{E}(\omega )}\right), & \omega \geqslant {\omega }_{0}\end{array}\right.,\end{eqnarray} \tag{ 9 }$$

with ${\omega }_{0}$ denoting the center frequency of the input spectral amplitude $\tilde{E}(\omega )$, to the LC displays A and B of the LC-SLM yields O-LP bichromatic fields with individually adjustable amplitude profiles ${{ \mathcal A }}_{1/2}(\omega )$, phase modulation functions ${\varphi }_{1/2}(\omega )$, and polarization states (s- or 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 ${{ \mathcal A }}_{1/2}(\omega )$ of the same bandwidth ${\rm{\Delta }}\omega =35\,\mathrm{mrad}\,{\mathrm{fs}}^{-1}$ corresponding to a pulse duration of ${\rm{\Delta }}t=80\,\mathrm{fs}$. The center frequency of the blue pulse was tuned to the potassium resonance $4p\leftarrow 4s$ at ${\omega }_{b}=2.45\,\mathrm{rad}\,{\mathrm{fs}}^{-1}$ (${\lambda }_{b}=769\,\mathrm{nm}$). The red pulse was centered at ${\omega }_{r}=2.29\,\mathrm{rad}\,{\mathrm{fs}}^{-1}$ (${\lambda }_{r}=823\,\mathrm{nm}$). 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

$$\begin{eqnarray}&&{\varphi }_{1/2}(\omega )=\pm \displaystyle \frac{\tau }{2}\cdot (\omega -{\omega }_{r/b}),\end{eqnarray} \tag{ 10 }$$

in equations (8) and (9). By applying the linear phases with respect to the center frequencies ${\omega }_{r}$ and ${\omega }_{b}$, 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].

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 $5\times {10}^{-7}\,\mathrm{mbar}$) provided by a dispenser source. The laser intensity used in the experiments was ${I}_{0}\approx {10}^{11}{{\rm{W}}\mathrm{cm}}^{-2}$. 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 $\lambda /2$ 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 Fourier-based 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.

Figure 3(a) shows sections through the tomographically reconstructed 3D-EDs from CNR-CP ionization. The data shown in different columns corresponds to three different time delays τ. Different rows represent the three cartesian principal planes. For $\tau =0$ (middle column) both colors coincide in time. The respective sections exhibit four contributions centered around momenta ${p}_{n}=\sqrt{2{m}_{{\rm{e}}}{\varepsilon }_{n}}$, with m_e denoting the electron mass and ${\varepsilon }_{n}$ being the central energy of the nth energy window as introduced in section 2. For best visibility, the weaker p₁- and p₂-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 p₀ (red arrow) and p₃ (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 $| {\psi }_{0}\rangle \propto | f,3\rangle$ and $| {\psi }_{3}\rangle \propto | f,-3\rangle$ predicted 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 p₁ (purple arrow) and p₂ (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 ${Y}_{3,\pm 1}(\theta ,\phi )$ associated with states $| {\psi }_{1}\rangle \propto | f,1\rangle$ and $| {\psi }_{2}\rangle \propto | f,-1\rangle$.

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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 p₀ and p₃ (in [–30°..30°]) as well as p₁ and p₂ (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 $\tau =-300\,\mathrm{fs}$ (left column of figure 3(a)) the red component precedes the blue. In this case, only the p₀- and p₃-contribution are observed in the photoelectron spectrum, confirming the one-color character of these signals. In contrast, for reversed pulse ordering at $\tau =300\,\mathrm{fs}$ (right column), the term around p₁ persists. Since the blue band is resonant with the potassium $4p\leftarrow 4s$ transition, excitation by the blue prepulse generates population in the $4p$ state. After time τ the excited system is probed by the red postpulse in a non-resonant two-photon ionization step. Similar to the $\tau =0$ case, the corresponding wave packet detected around p₁ 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 $4{p}_{1/2}$ and $4{p}_{3/2}$. 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 p₁-contribution. Both one-color ionization pathways are insensitive to the time delay and the p₂-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 pump-probe 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 p₀ to p₃ (insets to figure 4)—much like peeling an onion—permits selective inspection of all four angular momentum states $| f,{m}_{j}\rangle$.

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The reconstructed 3D-ED created by ionization with O-LP pulses at $\tau =0\,\mathrm{fs}$ is shown in figure 6. Sections through the 3D distributions at $\tau =0\,\mathrm{fs}$ and $\tau =\pm 300\,\mathrm{fs}$ are depicted in figure 5(a). Weak contributions were enhanced for better visibility using enhancement factors up to 3.9. At $\tau =0\,\mathrm{fs}$ (middle column) we observe contributions from all four frequency mixing pathways. The outermost contribution at p₃, resulting from one-color ionization by the blue pulse, exhibits an $| f,0\rangle$-type shape in the x–y-plane (upper row). Due to the p-polarization 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 p₀ 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 p₁ 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) (${\varepsilon }_{1}$-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 $| f,\pm 3\rangle$. A similar structure is observed for the p₂-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 p₂-wave packet relative to the p₁-wave packet about the laser propagation direction (z-axis), as discussed in section 2.2.

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By introducing a time delay of $\tau =-300\,\mathrm{fs}$ (left column) only the one-color contributions around p₀ and p₃ 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 p₂ 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 p₁-wave packet evolves in time due to the spin–orbit interaction in the excited $4p$ state. As a result, the contributions in the x–y- and the x–z-plane exchange their roles between $\tau =0$ and 300 fs, indicating a 90° rotation 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 'f- within 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 p₀ to p₃, we compare simulated and measured contributions individually (see insets to figure 6) to find excellent agreement, although the equatorial node in the reconstructed p₂-wave packet is not fully resolved due to the energetic overlap with the neighboring lobe of the p₃-wave packet.