Orbital angular momentum superposition states in transmission electron microscopy and bichromatic multiphoton ionization

For the generation of OAM electron beams in a TEM, we utilize a hologram-based approach [5, 59–61] (experimental sketch in figure 2(a)), in which a diffraction mask $\mathcal{M}$ is derived from the superposition of an apertured plane wave ${\text{circ}}_{R}\left(r\right){\text{e}}^{\text{i}{k}_{0}x}$ and a real-space target electron wave function

$$\begin{equation}{\Phi}\left(r,\phi \right)={\text{circ}}_{R}\left(r\right)\cdot \left({\text{e}}^{\text{i}m\phi }\cdot {\text{e}}^{\text{i}{\kappa }_{\text{SPM}}}+{\text{e}}^{-\text{i}n\phi }\right),\end{equation} \tag{ 3 }$$

yielding the real-valued mask function

$$\begin{equation}\mathcal{M}\left(r,\phi \right)={\text{circ}}_{R}\left(r\right)\cdot \vert \left({\text{e}}^{\text{i}m\phi }\cdot {\text{e}}^{\text{i}{\kappa }_{\text{SPM}}}+{\text{e}}^{-\text{i}n\phi }\right)+{\mathrm{e}}^{\mathrm{i}{k}_{0}x}{\vert }^{2}.\end{equation} \tag{ 4 }$$

Here, circ_R (r) is the circular aperture function with radius R and κ_SPM a relative phase. By illuminating this mask with an electron plane wave, the mask's diffraction pattern Γ(k_x , k_y ) (cf figure 1(a)) is formed in the far-field:

$$\begin{equation}{\Gamma}\left(k,\xi \right)=\vert \mathcal{F}\left\{\mathcal{M}\right\}{\vert }^{2}.\end{equation} \tag{ 5 }$$

As shown in appendix A, $\mathcal{F}\left\{\mathcal{M}\right\}$ consists of a central component and two side lobes at distances ±k₀ in k_x −direction. Following the general principle of holography, the Fourier transform of the real-space target state, $\mathcal{F}\left\{\vert {\Phi}\left(r,\phi \right){\vert }^{2}\right\}$, and its complex conjugate are forming the side lobes. For a given reciprocal distance k^∓ = k_eq around the center of the first diffraction order of the reference wave, the phase behavior of the Fourier-transformed target wave follows equation (1). Moreover, for k^∓ ≈ k_eq the probability density of the superposition state can be approximated by

$$\begin{equation}\vert {{\Psi}}_{\text{SPM}}\left({k}^{\mp },{\xi }^{\mp }\right){\vert }^{2}\approx 2\vert {G}_{\text{SPM}}\left({k}^{\mp }\right){\vert }^{2}\left[1+\mathrm{cos}\left(\left[n+m\right]{\xi }^{\mp }+\gamma \right)\right],\end{equation} \tag{ 6 }$$

in which G_SPM(k^∓) is a function governed by the choice of the circular aperture radius, k^∓ and ξ^∓ denote shifted momentum coordinates and $\gamma ={\kappa }_{\text{SPM}}{\pm}\left(m-n\right)\frac{\pi }{2}$ (cf appendix A). As an example, we show in figure 2(a) the amplitude and phase of the numerical 2D-Fourier transform of a mask structure for m = 4 and n = 3.

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Experimentally, we utilized a binarized version of the calculated mask $\mathcal{M}$ (binarization threshold at half intensity maximum), which we fabricated by focused ion beam milling of a 30 nm gold thin film on a silicon nitride membrane (15 nm thickness). In the cut-through sections of the mask, the electron wave is transmitted. Non-cut sections result in wave components scattered by large angles and are subsequently blocked by apertures. For the hologram, we chose k₀ = 15 μm⁻¹ and an aperture diameter of 3.7 μm. The mask is illuminated by an electron beam with large coherence length (coherence length: ${d}_{\text{c}}=\frac{\lambda }{2\alpha }=$1.74 μm, angular spread: 2α = 1.44 μrad) formed in a transmission electron microscope (JEOL JEM-2100F, 200 kV acceleration voltage). Using the post-specimen imaging lens system, the diffraction pattern of the mask is projected on the detector (GATAN Orius SC600 charge-coupled device (CCD) camera; effective camera length: 300 m), giving access to the spatial profile of the probability density of the electron state (cf figure 1).

In the ultrafast bichromatic MPI approach, sodium atoms are ionized with polarization-tailored bichromatic laser pulses [51] to induce interference of pre-selected electron wave functions [54, 62–64]. Here, we consider atomic MPI (with N_p photons) from the sodium 3s ground state (figure 2(b)) using counter-rotating circularly polarized (CRCP) bichromatic propeller-type pulses (shown in figures 2 and 3) to generate OAM superposition states, yielding the photoemitted electron wave function in spherical momentum coordinates (k, ξ, ϑ) as

$$\begin{equation}{{\Psi}}_{\text{MPI}}\left(k,\xi ,\vartheta \right)\propto {\psi }_{m,m}\left(k,\xi ,\vartheta \right)\cdot {\text{e}}^{\text{i}{\kappa }_{\text{MPI}}}+{\psi }_{n,-n}\left(k,\xi ,\vartheta \right),\end{equation} \tag{ 7 }$$

using

$$\begin{equation}{\psi }_{l,m}\left(k,\xi ,\vartheta \right)={\mathrm{i}}^{{N}_{\text{p}}}{\mathcal{R}}_{l}\left(k\right){\mathcal{P}}_{l,m}\left[\mathrm{cos}\;\left(\vartheta \right)\right]{\text{e}}^{\text{i}m\xi },\end{equation} \tag{ 8 }$$

with the radial part ${\mathcal{R}}_{l}\left(k\right)$ of the continuum wave function (determined by the N_pth order spectrum of the laser pulse), the associated Legendre polynomials ${\mathcal{P}}_{l,m}\left[\mathrm{cos}\left(\vartheta \right)\right]$ and the phases ${\text{i}}^{{N}_{\text{p}}}$ from N_pth order perturbation theory (for details see appendix B and [54]). The relative phase κ_MPI is adjusted by optical phases introduced by the pulse shaper [51, 65] and the polarization–rotation optics. The two wavefunction components in equation (7) originate from the 3- and 4-photon–ionization channels and interfere in the same energy window (about 0.5 eV) of the continuum states, as depicted in figure 2(b), resulting in the superposition of two OAM electron states ψ_n,−n (k, ξ, ϑ) and ψ_m,m (k, ξ, ϑ) with topological charges ℓ₁ = −n and ℓ₂ = m, respectively. The target states are addressed via σ^±-transitions due to the left and right circular polarized (LCP/RCP) laser electric fields, for which the respective quantum numbers are determined by selection rules (Δl = 1, Δm = ±1). As discussed in appendix B, around $\vartheta =\frac{\pi }{2}$, the momentum–space wave function in equation (7) factorizes and follows equation (1). Hence, the electron wave function exhibits a phase structure in the (k_x , k_y )-plane similar to the spatially phase modulated electron beam in the TEM. As a consequence, the resulting electron density around $\vartheta =\frac{\pi }{2}$ can be written as

$$\begin{equation}\vert {{\Psi}}_{\text{MPI}}\left(k,\xi ,\vartheta \right){\vert }^{2}\approx 2\vert {G}_{\text{MPI}}\left(k\right){\vert }^{2}\left[1+\mathrm{cos}\left(\left[n+m\right]\xi +\gamma \right)\right],\end{equation} \tag{ 9 }$$

in which ${G}_{\text{MPI}}\left(k\right)={\mathcal{R}}_{n}\left(k\right){\mathcal{P}}_{n,-n}\left[\mathrm{cos}\left(\vartheta \right)\right]$ and $\gamma ={\kappa }_{\text{MPI}}+\left(m-n\right)\frac{\pi }{2}+m\pi$ in full analogy to equation (6).

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For the experimental implementation of MPI-based electron state generation and characterization, we combine bichromatic polarization pulse shaping (more details of the experimental setup are given in [50]) with a VMI based photoelectron tomography [66–68] sketched in figure 1. Near-infrared femtosecond pulses from a multipass chirped pulse amplifier (FEMTOLASERS Rainbow 500, CEP4 module, Femtopower HR, 3 kHz repetition rate; λ₀ = 790 nm, 1.0 mJ pulse energy) with actively stabilized carrier–envelope phase (CEP) are employed to seed a neon-filled hollow-core fiber for the generation of an octave-spanning white light supercontinuum (WLS). The white light pulses are modulated in the frequency domain using a home-built 4f polarization pulse shaping setup [51, 69, 70], which consists of a dual-layer liquid crystal spatial light modulator (LC-SLM; Jenoptik SLM-640d) in combination with a custom polarizer. For the conversion from linear to counter-rotating circular polarization, we utilize a superachromatic λ/4-waveplate at the shaper output. The generated bichromatic (3ω: 4ω) CRCP field (cf inset figure 2(b)), consisting of a red (λ_r = 880 nm, Δt_r ≈ 25 fs, LCP) and blue (λ_b = 660 nm, Δt_b ≈ 25 fs, RCP) component, is CEP-stabilized using an external active stabilization loop [53]. To this end, an f–2f interferometer is implemented, fed by an additional (ω: 2ω)-field extracted from the spectral edges of the WLS. The bichromatic fields are focused into the interaction region of a VMI spectrometer (peak intensity I₀ ≈ 2 × 10¹² W cm⁻²) filled with sodium vapor. The photoelectron wave packets created by atomic MPI are projected onto a position-sensitive 2D detector consisting of a Chevron micro-channel-plate (MCP) and a phosphor screen and are recorded by a CCD camera. For tomographic reconstruction of the 3D photoelectron momentum distribution (PMD) the input pulse sequence is rotated around the laser propagation direction by using a superachromatic λ/2-waveplate [67, 68]. Each PMD was retrieved from 45 projections, measured with an angular step size of δϕ = 4°, employing the Fourier slice algorithm [71]. Note that the VMI detection scheme, results in a projection of the PMD which scales linearly in the radial direction with the electron energy. Therefore, the displayed MPI results show a non-linear radial dependence, as compared to the SPM results.

The experimental OAM superposition states generated by SPM and MPI are compared in figure 3 for m = 4 and n = 3. Both, the intensity distribution in the first-order sidebands in SPM (figure 3(a)) and the MPI results (figure 3(b)), show a 7-fold rotationally symmetric flower-petal-like structure, as expected from equation (2). Due to the phase singularity of OAM states at k^± = 0, the intensity of the OAM superposition vanishes at the center in both cases. We note that the radial behaviors of the SPM- and MPI-generated states, depending on the mask aperture via G_SPM(k) and the radial part of the atomic wave function in G_MPI(k) respectively, are different in the SPM and MPI cases. Furthermore, in the TEM approach the generated mixed-OAM states exhibit no k_z -dependence, whereas in MPI an additional ϑ-dependence is introduced via ${\mathcal{P}}_{n,-n}\left[\mathrm{cos}\left(\vartheta \right)\right]$. It is instructive to further consider the symmetries of the spatial mask $\mathcal{M}$ and the electric field distribution in the MPI light field. The mask $\mathcal{M}$ can be explicitly expressed as

$$\begin{align}\hfill \mathcal{M}& ={\text{circ}}_{R}\left(r\right)\left[3+2\enspace \mathrm{cos}\left(\left[n+m\right]\phi +{\kappa }_{\text{SPM}}\right)\right.\hfill \\ \hfill & \quad \left.+4\enspace \mathrm{cos}\left(\frac{\left(n-m\right)\phi -{\kappa }_{\text{SPM}}+2{k}_{0}x}{2}\right)\mathrm{cos}\left(\frac{\left(m+n\right)\phi +{\kappa }_{\text{SPM}}}{2}\right)\right]\hfill \end{align} \tag{ 10 }$$

using equation (4). Despite the low symmetry of the mask function, the overall lobular structure of the mask, depicted in figure 3(a), follows a 7-fold rotational symmetry due to the second term in equation (10). The high-frequency mask components (third term in equation (10)) are formed by a spatial beating between the target wave function and the reference plane wave, yielding the OAM sidebands in the diffraction pattern of the mask. In the MPI scenario, the time-dependent CRCP optical electric field E (t) with commensurable frequencies nω and mω and equal envelopes ${\mathcal{E}}_{0}\left(t\right)$ is given by

$$\begin{equation}\boldsymbol{E}\left(t\right)=\sqrt{2}{\mathcal{E}}_{0}\left(t\right)\enspace \mathrm{cos}\left(\frac{\left(n+m\right)}{2}\omega t\right)\left(\begin{matrix}\hfill \mathrm{cos}\left(\frac{\left(n-m\right)}{2}\omega t\right)\hfill \\ \hfill \mathrm{sin}\left(\frac{\left(n-m\right)}{2}\omega t\right)\hfill \end{matrix}\right),\end{equation} \tag{ 11 }$$

and contains an expression describing the beating, similar to the corresponding expression in equation (10). Projecting the trace of the temporally evolving electric field vector onto the transverse polarization plane, results in a propeller-shaped curve with ${\mathcal{S}}_{\text{opt}}$-fold rotational symmetry, given by ${\mathcal{S}}_{\text{opt}}=\left(n+m\right)/\mathrm{gcd}\left(n,m\right)$, where gcd(n, m) denotes the greatest common divisor of n and m [54]. A measured polarization profile of the generated phase-stable (3ω: 4ω) CRCP field is depicted in the inset of figure 3(b) (for more experimental details see [50]), highlighting the similarity to the corresponding mask structure shown in the inset of figure 3(a). The close similarity suggests to interpret the diffraction mask in figure 2(a) as an advanced double slit analogously to the spectral double slit [41] in the MPI framework (cf figure 2(b)).

For a more detailed discussion of the topological properties of the OAM superposition state from equation (1), we consider the probability current j of the evolved real-space electron wave function [72], yielding [3, 55–57, 73]

$$\begin{equation}\boldsymbol{j}=\frac{\hslash }{m}\enspace \mathfrak{Im}\left[{{\Psi}}^{{\ast}}\nabla {\Psi}\right]\propto \vert {\Psi}{\vert }^{2}\frac{m-n}{2}{\boldsymbol{e}}_{\xi }+\mathcal{O}\left({\beta }_{0}-1\right)\approx \rho \frac{m-n}{2}{\boldsymbol{e}}_{\xi },\end{equation} \tag{ 12 }$$

at a certain radius k, with the probability density ρ and $\mathcal{O}\left({\beta }_{0}-1\right)$ denoting terms proportional to (β₀ − 1). Although the resulting electron density is a static structure, i.e. a standing wave, j does not vanish for n ≠ m. The probability current shown in figure 2(b), top panel is curling around the center of the structure in azimuthal direction e _ξ with an angular spatial frequency ${\omega }_{\xi }\propto \frac{m-n}{2}$ and an amplitude determined by ρ.

For MPI, this observation is rationalized by the fact, that during photoionization, the probability current is driven by the electric field. The angular frequency of the laser electric field of a propeller-pulse with equal field amplitudes is given by ${\omega }_{{\Phi}}=\frac{m-n}{2}\omega$ which does not change within the pulse [74, 75] (cf appendix B) and therefore ω_Φ(m, n) ∝ ω_ξ (m, n). In the single color case both angular frequencies vanish, since m = n [54, 58, 76]. In general, the topological charge of the superposition state, ℓ(n, m; β₀) [3] is a function discontinuously depending on the parameter β₀ (cf appendix C). In the specific case of β₀ = 1, the topological charge takes a fractional value $\ell =\frac{m-n}{2}$, leading to a discontinuous topological charge in the experiment, depending on the respective amplitudes of the different OAM states in the superposition. For β₀ ≠ 1, the topological charge of the superposition state is given by the topological charge of the OAM state with the larger weight in equation (1). However, small variations around β₀ = 1 do not affect the wave packet structure since Ψ and j are continuous functions of β₀.

Finally, we note that for single OAM states e^imξ the topological charge m can be experimentally determined by applying an astigmatic defocus and counting the resulting number of intensity minima [77, 78]. A similar approach for the OAM superposition states only leads to a complex interference pattern with the number of intensity minima not directly connected to the topological charge.

The orientation of the flower-petal-like electron density is determined by the phase γ (cf equation (2)). In the experimental SPM approach, γ is set by the factor κ_SPM in the mask design. In figure 4(a), we show the simulated electron density profile in the diffraction pattern for κ_SPM = 0, π/2 and π, corresponding to γ = π/2, π and 3π/2. A rotation of the first-order sideband Γ⁽⁻¹⁾ around the respective center by an angle γ/(n + m) is visible (cf equation (6)). According to Friedel's law applicable for real-valued masks, the diffraction pattern remains inversion symmetric independent of κ_SPM [79, 80] (cf figure 5(a)). Notably, the approach does not correspond to a simple mask rotation, which would cause the whole diffraction pattern to rotate around its center at k = 0.

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Similarly, in the MPI approach, γ is controlled by optical phases via the parameter κ_MPI (cf equation (9)), given by κ_MPI = −mφ_r + nφ_b − (m − n)φ_ce + (m + n)ζ (for details see appendix B), with the respective relative phases φ_r/b of the red and blue component of the bichromatic laser field, the carrier–envelope phase φ_ce and the relative angle ζ/2 of the λ/2-waveplate [54]. Similar to the SPM approach, the resulting photoelectron density is rotated by an angle of κ_MPI/(n + m). To highlight the analogy, three experimental examples of rotational phase control are depicted in figure 4(b) for the optical phases φ_r = φ_b = φ_ce = 0 (left), φ_ce = π (middle) and ${\varphi }_{\text{b}}=\frac{\pi }{3}$ (right). Furthermore, the control of the spatial rotation is not only accessible via relative spectral phases of the field but also by the adjustment of a λ/2-waveplate resulting in a rotation of the whole bichromatic laser field in the polarization plane (see e.g. equation (6) in [67]). In fact, this feature is crucial for the tomographical reconstruction of the full 3D PMD [56, 63, 81, 82]. Importantly, not only optical but also quantum phases accumulated during the photoemission process result in a rotation of the detected electron density [64, 73]. Hence, the MPI approach further enables to study imprinted time-dependent dynamics of quantum systems, e.g., due to spin–orbit coupling [63] or Rydberg states [64, 83]. Moreover, it has been demonstrated that the photoelectron wave packet's symmetry is also controlled via the laser intensity as the interaction evolves from the perturbative to the strong-field regime [76, 84]. Recently, spiral shaped electron wave packets (electron vortices) in MPI [54–56, 58, 73, 76, 88] have attracted significant attention. These vortices have a k-dependent phase due to the time-evolution of the wave packet [85].

In electron wave optics, spatial control of electron beams by magnetic lenses is conveniently described by spatial phase masks applied in the back focal plane of the imaging lens [86, 87]. The corresponding phase masks typically exhibit k-dependent phase functions. For example, the effect of a non-aberrated circular symmetric magnetic lens can be described by a phase mask ${\text{e}}^{\text{i}C{k}^{2}}$ with $C\in \mathbb{R}$, resulting in a converging parabolic wavefront [86, 87]. In the holographic TEM approach utilized here, an equivalent k-dependent phase function γ(k) can be imprinted onto the diffracted electron wave by choosing a more sophisticated mask design. In particular, the holographic mask is calculated by applying the (inverse) Hankel transform of the respective targeted k-dependence in position space. For this reason we choose the mth order normalized Hankel transform of ${\mathrm{e}}^{\mathrm{i}C{k}^{2}}\enspace {\mathrm{e}}^{-\frac{1}{2}{C}_{2}{k}^{2}}$ and the nth order normalized Hankel transform of ${\mathrm{e}}^{-\frac{1}{2}{C}_{2}{k}^{2}}$ as complex amplitudes for the respective partial states e^imϕ and e^−inϕ , respectively (for more details see appendix A). Experimental and calculated results for the far-field electron diffraction pattern and phase-distribution are shown in figure 5(a), exhibiting a spiral-shaped electron distribution.

In the MPI approach, a similar k-dependent phase appears during the time-evolution of the wave packets when an additional time-delay τ (applied to the blue pulse) is introduced between the two spectral components in the bichromatic field [54, 58, 76]. This delay yields $\gamma \left(k\right)={\gamma }_{0}+\frac{\hslash \tau }{2m}{k}^{2}$, where γ of equation (1) is denoted as γ₀. The resulting photoelectron density is displayed in figure 5(b), showing a pronounced tilt of the petal lobes relative to the radial direction. The time-delay leads to a τ-dependent radial component of the probability current j along with a radial dependence of the interference term, giving rise to the spiral shape [88]. The tilt angle of the lobes increases for larger time delays. In contrast to the PMD, for sufficiently large time-delays, i.e. when the delay exceeds the pulse duration, the spectral components in the bichromatic fields are temporally separated, such that the polarization profile is circular and does not show the propeller-type structure in the polarization plane. Hence, in the multiphoton regime the symmetry of the wave packet is completely described by the quantum interference of states with different angular momenta and not fully determined by the optical field structure. For a vanishing time-delay the photoelectron wave packet symmetry generally maps the field symmetry when the difference between both photonicities (n and m) equals 1, i.e., when the wave packet exhibits an odd (n + m)-fold rotational symmetry, as discussed in references [54–56]. While the MPI technique enables radial control via the N_pth order laser electric field's spectrum, the holographic TEM approach can be extended to other radial phase dependencies, allowing advanced transverse control of the resulting electron distribution.