## Abstract

We study theoretically the electron quantum dynamics in atoms driven by intense IR laser pulses that are phase and/or polarization structured. The extremely non-linear electron dynamics causes high harmonic emission, which we calculate, analyze, and characterize. Results are presented for three different types of structured lasers: radially polarized and azimuthally polarized beams and optical skyrmions. We identify a topological index that is inherent to the driving pulse topology and is taken over by the high harmonics. All harmonics are found to have the same topological index. For vector IR pulses as driving fields, the far-field emitted beam tightens with a higher topological order and remains unchanged when the atom is driven by an optical skyrmion.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, employing phase and polarization structured light fields for extremely non-linear charge dynamics as manifested in high harmonic generation (HHG) [1] has received increasing research activities [2–4]. A substantial body of theoretical and experimental works [5–10] laid the foundations of the physics behind transferring the spatio-temporal characteristics of a structured infrared (IR) light field, for example its phase singularities, to the emitted HHG in the (X)UV regime. Related applications to attosecond metrology is one of the prominent example for the use of such harmonics. To appreciate the new features that structured laser pulses bring in, let us consider the coherent superposition of emitted light waves of dipole radiators located in the focal plane of an optical vortex with topological numbers $m$. Such an oscillating charge and current distribution generate higher harmonics with individual orbital angular momentum (OAM) $m$ times $n$ where $n$ is the harmonic order. Hence, the phase information of the incident optical vortex is imprinted on the radiated harmonic burst by the non-linear conversion processes of the HHG. Important for possible applications is the OAM-independent divergence of the harmonic spectrum. In this context, we refer to the finding that all harmonics diffract similarly [7,11].

Another example is the generation of higher harmonics upon irradiation with spatially varying polarization ranging from radial to azimuthal, such beams are commonly known as vector beams. Experimentally and theoretically, it was demonstrated that the polarization state of the fundamental IR beam is taken over by the higher frequency radiation [10]. Radial vector beams exhibit the capability for an extraordinary sharp focus [12] in comparison to conventional linearly polarized optical fields. Moreover, spot sizes on the sub-nm scale can be achieved in the (X)UV regime using state-of-the-art focusing techniques [13]. Combining these aspects results in fascinating applications ranging from areas such as ultra-fast diffraction imaging [14] or lithography [15,16] to ultrafast nanomagnetism [17,18].

It is well known that vector beams [19] are characterized by a longitudinal on-optical axis component. For a radial vector beam (RVB) an electric component is present along the optical axis. The origin of this component becomes clear when viewing the radially-polarized beam as a superposition of linearly polarized plane waves incident with a fixed angle with respect to the propagation direction of the vector beam. The finite projections of the electric fields of these plane waves subsum to the longitudinal component of vector beam. From this picture also follows that this component increases with tighter focusing (larger incident angle of the contributing plane waves). For the case of an azimuthally polarized type (AVB), a magnetic pendant can be found. To capture all the facets of the driving fields including the longitudinal components, a proper mathematical description of the propagating vector beams that obey Maxwell’s equations is mandatory. The corresponding spatially inhomogeneous electric (for the RVB) and magnetic (for the AVB) field amplitudes can be quantified in the topological sense, as will be discussed below.

In addition to the vector beams that have spatially dependent polarization states, optical fields resembling magnetic skyrmions are attracting substantial research [20]. Their spatial phase structure is intertwined with their polarization landscape. These new properties are expected to result in new optical phenomena. Here, we focus on the quantum dynamics of atoms strongly driven by intense IR skyrmionic pulses and trace their signature in the dynamics via the properties of the high harmonic signal. We find that the particular profile of the optical skyrmion allows for the emission of harmonics with all possible polarizations and imprints on the high harmonics specific phase information.

We calculate the HHG driven by three different topological optical fields and perform a characterization of the higher harmonics. We aim at a systematic study of the HHG and contrast HHG with conventional, vector beams, and optical skyrmions. Atomic units are used throughout the text.

## 2. Topological light fields – overview

We consider topological optical fields composed as a linear combination of optical vortices [21]. In addition to phase information, it is essential to account for the longitudinal field components. Depending on the direction of the optical orbital angular momentum (OAM) relative to the spin angular momentum (SAM, helicity), the longitudinal electric field component can be even stronger than the transversal field component [22]. Mathematically, a widely used description of optical vortices is given by Laguerre-Gaussian modes [23]. However, the associated vector potential does not fulfill the Coulomb gauge condition so that the electric scalar potential via $\pmb {\nabla }\cdot \pmb {A}(\pmb {r},t)=-(1/c^2)\partial _t\Phi (\pmb {r},t)$ has to be considered. For a transparent treatment and a clear interpretation of the results the Bessel modes are ideally suited. Theses beams are exact solutions of the Helmholtz equation and fulfill the Coulomb gauge conditions. Rhe electromagnetic fields read $\pmb {E}(\pmb {r},t)=-\partial _t\pmb {A}(\pmb {r},t)$ and $\pmb {B}(\pmb {r},t)=\pmb {\nabla }\times \pmb {A}(\pmb {r},t)$. Accordingly, the accurate description of the longitudinal fields is included.

The vector potential of an optical vortex with a topological charge $m$ and helicity $\sigma$ has the form in cylindrical coordinates $\pmb {r}=\left \{\rho ,\varphi ,z\right \}$:

We note that the winding number $m$ characterizes the (quantized) amount of the OAM carried by the optical vortex. The $z$-projection of the total angular momentum associated with optical field is then given by $(m+\sigma )\hbar$, which is presented by the $z$-component of $\pmb {A}_\textrm {OV}^{m,\sigma }$.

#### 2.1 Cylindrical vector beams

Cylindrical vector beams share some properties with optical vortices like the inhomogeneous intensity distribution. However, the vector potential exhibits space-dependent polarization states (but no orbital phase dependence). Mathematically, we can express such a beam as a linear combination of two optical vortices with $\left \{m=+1,\sigma =-1\right \}$ and $\left \{m=-1,\sigma =+1\right \}$. The coherent sum yields a vector beam with azimuthal polarization (AVB):

*spatial*part of the respective fields with a non-vanishing longitudinal component (RVB $\leftrightarrow$ electric field, AVB $\leftrightarrow$ magnetic field).

The spatial dependencies of $\pmb {m}_\textrm {RVB}$ and $\pmb {m}_\textrm {AVB}$ in Figs. 1(a) and (b): both vectors reflect the radial symmetry of the cylindrical vector beams. Furthermore, both $\pmb {m}_\textrm {RVB}$ and $\pmb {m}_\textrm {RVB}$ change their directions continuously from orthonormal to in-plane to orthonormal in a periodic manner, a behavior that is reminiscent of a magnetic hedgehog skyrmion. Let us examine the corresponding Poynting vector $\pmb {S}=(\pmb {E}\times \pmb {B})/\mu _0$, whose (only non-vanishing) $z$-component is shown in Figs. 1(d) and (e): the radial symmetric energy flux reveals several zeros $\rho _n$. An analytical calculation reveals that for both vector beams $|\pmb {S}|\sim J_1(q_\perp \rho )^2$ so that the radii $\rho _i=x_{n+1}/q_\perp$ where $x_{n}$ is the $n$th zero of $J_1(x)$. The zeros of the energy flux are crucial for the topological characterization: in analogy to the magnetic skyrmion we find that

#### 2.2 Optical skyrmion

In contrast to cylindrical vector beams, an optical skyrmion has topological features in the spin angular momentum density. In the following, we consider $\pmb {A}_\textrm {OS}=\pmb {A}_\textrm {OV}^{m_1=2,+1}+\pmb {A}_\textrm {OV}^{m_2=0,-1}$, which is the coherent superposition of a (left) circularly polarized vortex with winding number of two with a rotationally symmetric, (right) circularly polarized zero-order Bessel beam. As a result, the electromagnetic potentials fields are not cylindrical symmetric, while a phase is imprinted. As a consequence, the vector of the SAM density is three-dimensional. In Fig. 1(c), the unit vector $\pmb {m}_\textrm {OS}=\pmb {\sigma }(\rho ,\varphi )/|\pmb {\sigma }(\rho ,\varphi )|$, where $\pmb {\sigma }=\epsilon (\pmb {E}_\textrm {OS}\times \pmb {A}_\textrm {OS})$ (SAM density), is shown in the focal plane $z=0$. It reveals a similar topological character as $\pmb {m}_\textrm {RVB}$ and $\pmb {m}_\textrm {AVB}$, meaning the vector changes continuously from out-of-plane to in-plane to out-of-plane in a periodic manner when increasing the axial distance $\rho$. The corresponding Poynting vector is displayed in Fig. 1(f)). Interestingly, evaluating

## 3. Quantum mechanical description of the HHG process

As an example of the highly non-linear interaction of topological light fields with matter we performed quantum mechanical calculations for atoms driven by such fields and evaluated the far-field HHG which signifies the nonlinear dipole oscillations. Technically, we use a matrix iteration scheme [24] which has been well-tested in atomic strong-field physics phenomena (even HHG) [25,26]. In short, the 3D-TDSE for the hydrogen-type atom in the presence of an external vector beam field is given by

The vector $\pmb {r}$ refers to the local coordinate frame of the atomic emitter positioned at $\pmb {r}_\textrm {i}$ (cf. Fig. 1(b)). The light-matter interaction is provided by the minimal coupling Hamiltonian: where $\hat {\pmb {p}}$ is the momentum operator and $\pmb {A}(\pmb {r}-\pmb {b},t)$ is the electromagnetic vector potential of the incident external topological light field. Technically, the wave function was expanded in spherical harmonics, i.e., $\Psi _\textrm {i}(\pmb {r},t)=\sum _{\ell m}^{L_\textrm {max}}R_{\ell }(r)Y_{\ell ,m}(\Omega _{\pmb {r}})$, resulting in the numerical propagation of $(1+L_\textrm {max})^2$ 1D-TDSE corresponding to the angular channel. The Hamiltonian was discretized on an equidistant radial grid with $R_\textrm {max}=10000$ a.u. reaching so the far-field. $\Delta r=0.05$ a.u., while $L_\textrm {max}=15$ was chosen as to ensure convergence. As the simplest example for a single effective electron atomic target we use a hydrogen atom.The major contribution to the strong-field phenomena and HHG stems from the atoms in the vicinity of the intensity maximum (donut). Here, the local gradient of the vector potential is very weak so that those atoms are exposed to homogeneous and linearly polarized light waves (on the atomic scale). As a consequence, the HHG signal is mainly composed out of the constructive interference of the individual dipole radiation patterns. From the wave function, we can find the local time-dependent vector and scalar potentials radiated from the i-th source distribution around at a (detecting) observer located at $\pmb {r}_\textrm {d}$:

Finally, we assume a low-density target [27] and an optimal phase matching containing contributions from intensity, as well as dispersion effects from neutral gas and plasma [28].

## 4. Results

The photon energy of the incident AVB is $\hbar \omega =1.55$ eV ($\lambda =800$ nm) and has a pulse length of $n_p=4$ optical cycles. The considered peak intensity amounts to $1.82\times 10^{14}$ W/cm$^{2}$. We consider here a focused light spot with a diameter $\approx 1.2\lambda$, i.e., an opening angle $\alpha \approx 30^\circ$ is used in the calculations. The atomic gas phase target in our simulation is modelled by a cylindrically shaped gaussian distribution:

where $n_0=75$ atoms/$\mu$m$^3$ and $\rho _g=2$ $\mu$m, so that atoms are mainly distributed in the high laser intensity area of the Bessel beam. Furthermore, to model the thickness of the sample $z_g=1$ $\mu$m was chosen. Simulations evidence however that this thickness has only a minor effect on the spatial characteristics of the emitted harmonics, which we attribute to the absent Gouy phase in the driving Bessel beam (cf. Eq. (1)).Due to their localization compared to incident light field, the atoms are exposed to a linearly polarized light giving rise to dipole radiation in the far-field. Due to the coherence of the vector beam two emitters at the same distance to the optical axis are subject to the *same* time-varying vector potential, although the local polarization direction may be different. Constructive and destructive interference result in a unique spatial dependence of the total radiated electromagnetic fields.

#### 4.1 HHG by an azimuthal vector beam

The polarization properties in HHG by azimuthal vector beams was already studied extensively by Hernández-García *et al.* [10]. The azimuthal polarization state will be transferred into the higher frequency-regime. We can confirm this study as the numerical calculation of the extended Stokes parameters for cylindrically polarized beams [29] revealed for all considered harmonics $S^\textrm {ext.}_{\ell =1}=(1,-1,0,0)^T$.

The study of the far-field properties gives some indication about the topological and diffraction properties of the higher harmonics, which are inferred from the Fourier transform of $\pmb {E}(\pmb {r}_\textrm {d},t)$ at an observe far away from the location of the atomic emitters. In Fig. 2, we present spatial dependencies of the electromagnetic field distributions for higher harmonics of orders $11$, $13$ and $15$. As usual, all fields are shown as a function of the diffraction angle $\beta =\arctan (\rho /z)$ where $\rho$ is the distance to the optical axis and $z$ is the on-axis distance to the focal plane (where the HHG takes place).

As a consequence of Maxwell’s equations, the corresponding magnetic field of the respective harmonic is given by $\pmb {B}_\textrm {HH}=B_\rho \hat {e}_\rho + B_z\hat {e}_z$, whereas the transversal field, shown in panel a), vanishes at the optical axis. In the vicinity of the optical axis, it reveals a linear slope with the axial distance (the dependence on the far-field divergence angle $\beta$ is shown) which is characteristic for (cylindrical) vector beams. In this context, the $z$-component of the Poynting vector in Fig. 2(b)) is cylindrically symmetric, vanishes at $\beta =0$ ($\rho =0$), and increases quadratically with the axial distance in the vicinity of the optical axis. Furthermore, it reveals a diffraction trend for the HHG process with vector beams: with increasing harmonic order, the beams spot tends to tighten, which is due to missing phase variation in the driving vector beams – in contrast to HHG with optical vortices, where all harmonics diverge similarly [11]. In panel c), we present the magnetic longitudinal component for the different harmonic orders. Characteristic for all harmonics is the strong on-axis magnetic field, which is in line with the prediction by Maxwell’s equations [30]. Moreover, the longitudinal and transversal field components exhibit a phase difference of $\pi /2$.

Similar to Eq. (5), we can define a unit vector $\pmb {m}_\textrm {AVB}= (B_\rho ,iB_z)^T /\sqrt {|B_\rho |^2+|B_z|^2}$ which is cylindrically symmetric and contains information about the topology of the (higher) harmonics. The evolution of $\pmb {m}_\textrm {AVB}$ for all three harmonic orders is presented in the three panels in the bottom right part of Fig. 2. Similar to the driving field, the direction of the vector changes from out-of-plane to in-plane to out-of-plane with axial distance, whereas the length for a full turnaround depends on the divergence angle/harmonic order. However, in all three cases, the evaluation of Eq. (6) yields in a topological index $n_\textrm {AVB}=1$ when limiting the integration region to the first zero (excluding $\beta =0$) of the Poynting component $S_z$ (Fig. 2(b)). As a consequence and important message, we can conclude that the topological index is a conserved quantity in the HHG process driven by an azimuthal vector beam.

#### 4.2 HHG by an radial vector beam

Similar to AVB, in the case of a radial beam all emitted harmonics are perfectly radially polarized as confirmed by numerical evaluation of the extended Stokes parameter $S_{\ell =1}^\textrm {ext.}=(1,+1,0,0)^T$.

Let us turn our interest to the higher harmonics in the far-field region, where the spatial characteristics of the electromagnetic components were inspected. In Fig. 3(a), we present the transversal electric field, which is radially polarized, for three different harmonic orders. Similar to the AVB driven counterpart, $E_\rho$ vanishes at $\rho =0$ and increases linearly in the vicinity of the optical axis. The field distribution is reminiscent of a Bessel function of order one (similar to the driving field). Furthermore, depending on the harmonic order, the location of the maximum of the transversal electric field amplitude changes: the consideration of a higher harmonic yields in a shrunk beam spot. The corresponding Poynting vector is presented in panel b). We find a similar shape as in the case of HHG driven by an AVB. Namely, $S_z$ is characterized by a dark spot at the optical axis resulting in a donut-shaped intensity distribution. A radially polarized electric field is always accompanied by a longitudinal field component, as clearly shown in Fig. 3(c). Striking features of $E_z$ are the pronounced on-axis amplitude and a phase shift of $\pi /2$ relative to $E_\rho$. In addition, the spatial profile of the longitudinal components is reminiscent of a Bessel function of zero order (again, similar to the driving field). Similar to the transversal field, the considered harmonic order influences the diffraction determinedly.

The characteristic profiles of the transversal and longitudinal electric field help us to characterize the topological state of the higher harmonics. Applying Eq. (4), we can define the unit vector $\pmb {m}_\textrm {RVB}=(E_\rho ,iE_z)^T/\sqrt {|E_\rho |^2+|E_z|^2}$ for every harmonic order. The evolution of the vector direction in dependence on the axial distance for the three examples is shown in the panels in the right bottom part of Fig. 3. They reveal the characteristic change from out-of-plane to in-plane to out-of-plane, while the length of distance depends on the divergence angle $\beta$ and is given by the first (non-zero) maximum of the Poynting component $S_z$. Performing now an integration similar to Eq. (6), where the integration area is limited by the zero of $S_z$, we find for all harmonics a topological index $n_\textrm {RVB}=1$. Consequently and similar to the HHG process initiated by an AVB, the topology of the driving field remains a conserved quantity.

Another interesting aspect is the role of focusing of the driving RVB in the composition of the radiated higher harmonics. As presented in Fig. 4(a), the focusing angle $\alpha$ (which is an experimentally accessible quantity) determines the ratio between the longitudinal and transversal fields of the higher harmonics. While an angle of $28^\circ$ results in higher harmonics where the transversal and longitudinal field components have similar amplitudes (cf. Fig. 3), a stronger focusing (or larger $\alpha$) yields a dominating on-axis ($z$-polarized) field. In this connection, a discussion of the Stokes parameters is intriguing: If we define the conventional Stokes parameters [31] with respect to the radiated $E_\rho$ and $E_z$ of the higher harmonics, we find a unique and cylindrically symmetric polarization landscape [32]. As shown in Fig. 4(b), the polarization of the emission is characterized by the two Stokes parameter $S_1$ and $S_3$ ($S_1+S_3=1$ in the whole beam profile), meaning the polarization state changes continuously between linear and circular in the dependence on the axial distance. The occurrence of $S_3$ is attributed to the phase difference of $\pi /2$ between $E_\rho$ and $E_z$. Further, we find a remarkable connection with the vector $\pmb {m}_\textrm {RVB}$, which characterizes the topology and is represented by the chain of blue arrows: The direction of $\pmb {m}_\textrm {RVB}$ represents the local polarization state of the emission. Pointing out-of-plane means the radiation is linearly polarized (in $z$-direction), while in-plane pointing corresponds to radial polarization. However, pointing in direction of $\pm 45^\circ$ characterizes a local circular polarization state with respect to the observed $\hat {e}_\rho -z$ plane. Hence, measuring the space-dependent Stokes parameters allows the characterization of the topology of the higher harmonics since the Stokes vector can be mapped into $\pmb {m}_\textrm {RVB}$.

#### 4.3 HHG by an optical skyrmion

The HHG process driven by the optical skyrmion, as discussed in section 2.2, exhibits interesting emission properties and conserved quantities. While in the case of both vector beams, the emitted harmonics reveal a spatial structure similar to the fundamental field, the high-frequency radiation associated with the optical skyrmion is markedly different. Analyzing the Poynting vector in Fig. 1(f), a quadrangle structure is revealed, which points to an internal phase changing in four cycles (or four multiplies of $4\pi$). In Fig. 5(a), we present the azimuthal Fourier transformation of the electric field’s $z$-component for different harmonic orders. Two striking features of the HHG process can be retrieved: First, we find two peaks in the spectrum while the difference is always four. In the following, we call it the *winding number* $\ell _n$. We note that $\ell _n$ corresponds to the z-projection of OAM + SAM of the $n$th optical skyrmion harmonic. Second, the winding number increases with every harmonic order $n$. Such a behavior is in agreement with the general knowledge about the HHG process driven by pure linearly polarized optical vortices, where the OAM (equal to winding number) equals $n\times m$. At the same time, $m$ is the OAM of the driving field [7,8,11]. In the case of an optical skyrmion, the general rule is altered: we found that for the $(p+q)$th harmonic $\ell _{p+q} = pm_1 + qm_2 + (p-q)$. For instance, at the $5th$ harmonic, the dominating contributions stem from $(p,q)=(3,2)$ and $(p,q)=(2,3)$ yielding $\ell _5=7$ or $\ell _5=3$. We note, that these rules are similar as tabulated in Ref. [33].

In Fig. 5(b), the diffraction properties in the far-field along the $x$-axis are shown. Here, we find another difference to the vector beam-driven HHG process, where the beam spot of the radiated harmonic shrinks with the harmonic order $n$. In the case of the optical skyrmion, the intensity peak of the location of the Poynting vector maximum converges against a fixed position. Hence, the diffraction appears to be independent of the harmonic order. Again, this behavior nicely reflects the diffraction properties of the OAM-HHG process, where all harmonics are emitted with similar divergence [34]. However, in our case, the diameter of the intensity donut tends to shrink with increasing order. To emphasize this behavior, we present the full profile of $S_z$ for two selected harmonics in panels a) and b) of Fig. 6. As already explained, a quadrangle structure as a result of the internal phase difference is visible. Interestingly, the on-axis intensity, which is dominating in the fundamental field (cf. Fig. 1(f)), decreases drastically in the high-frequency regime. Hence, we find a donut-shaped intensity distribution for the higher harmonics. The SAM density whose associated unit vector $\pmb {m}_\textrm {OS}$ is shown in panel c) (for HH5) and panel d) (for HH7), respectively, reveals a similar behavior as shown in Fig. 1(c). As a consequence, the topological aspects of the electromagnetic field is transported into the HHG process. Numerically, we proved that

As already mentioned in the introduction, an optical skyrmion shows the possibility to imprint all possible polarization states phase-locked and in a coherent manner onto the higher harmonics. Figure 7 shows the regular normalized Stokes parameter (referring to $E_x$ and $E_y$ of the emitted harmonic) spatially resolved. As expected, the polarization landscape is highly involved meaning that all polarization states can be found. Furthermore, we find numerically that $\sqrt {S_1^2+S_2^2+S_3^2}=+1$ so that the radiation is fully polarized. As a consequence, moving slightly the position of the observer can change the polarization state drastically. An important message is the possibility to produce circularly polarized higher harmonics via the HHG process driven by optical skyrmion which is not possible in the case of unstructured incident light mode.

## 5. Conclusions

We presented a summary of our theoretical results on extremely non-linear quantum dynamics of electrons in atoms driven by structured light and investigated the manifestation of this dynamics in the high harmonic generation signal. HHG are systematically analyzed with respect to their topological properties. In all three cases (radial vector beam, azimuthal vector beam, and optical propagating skyrmion) as driving fields, the topological index $n_\textrm {i}$ of the driving field is a conserved quantity, meaning that all harmonics have the same index evidencing that the topology of the fundamental light pulse is taken over by the emitted higher frequency radiation. Furthermore, in the case of vector beams, the far-field light spot shrinks with a higher topological order while it is unchanged in divergence for an incident optical skyrmion.

## Funding

Deutsche Forschungsgemeinschaft (SPP1840, TRR227, WA 4352/2-1).

## Disclosures

The authors declare no conflicts of interest.

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