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 [24]. A substantial body of theoretical and experimental works [510] 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 $\pmb {r}=\left \{\rho ,\varphi ,z\right \}$:

$$\begin{aligned}\pmb{A}_\textrm{OV}^{m,\sigma}=&\left[J_m(q_\perp\rho)e^{im\varphi}\hat{e}_\sigma-i\sigma\frac{q_\perp}{q_z}J_{m+\sigma}(q_\perp\rho)e^{i(m+\sigma)\varphi}\hat{e}_z\right] \\ &\times A_0\Omega(t)e^{i(q_zz-\omega t)} + \textrm{c.c.}, \end{aligned}$$
where we introduced the (circular) polarization vector $\hat {e}_\sigma =(\hat {e}_\rho +i\sigma \hat {e}_\varphi )e^{i\sigma \varphi }$ $(\sigma =\pm 1$) in terms of the cylindrical unit vectors $\hat {e}_\rho$, $\hat {e}_\varphi$. Furthermore, $A_0$ is the vector potential amplitude, $q_\perp =q\sin \alpha$ ($q_z=q\cos \alpha$) is the transversal (longitudinal) projection of the wave vector $q=\omega /c$ along the propagation direction, and $J_m(x)$ is the $m$th Bessel function of the first kind. The opening angle of the Bessel cone is $\alpha$, typically between $0^\circ$ and $90^\circ$, is and characterizes the focusing of the optical field: A larger $\alpha$ indicates a tighter beam spot. The temporal envelope is modelled by $\Omega (t)=\sin (\pi t/T_p)^2$ for $t\in [0,T_p]$ where the pulse length $T_p=2\pi n_p/\omega$ is measured by the number of optical cycles $n_p$. The vector potential in Eq. (1) states that in the $antiparallel$ case, meaning sign $(m)=-$sign($\sigma$) the near-optical axis behavior of the optical vortex is dominated by the longitudinal component since $J_{m-1}(x)>J_m(x)$ for $x\ll 1$.

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):

$$\pmb{A}_\textrm{AVB}(\pmb{r},t)=-A_0J_1(q_\perp\rho)\Omega(t)e^{i(q_zz-\omega t)}\hat{e}_\varphi + \textrm{c.c.},$$
while the coherent difference results in the radially polarized beam:
$$\begin{aligned}\pmb{A}_\textrm{RVB}(\pmb{r},t)&=A_0\left[\vphantom{\frac{q_\perp}{q_\parallel}}J_1(q_\perp\rho)\hat{e}_\rho + i\frac{q_\perp}{q_\parallel}J_0(q_\perp\rho)\hat{e}_z\right] \\ &\quad\times\Omega(t)e^{i(q_zz-\omega t)} + \textrm{c.c.} \\ &=A_0\pmb{F}(\rho)\Omega(t)e^{i(q_zz-\omega t)} + \textrm{c.c.} \end{aligned}$$
Interestingly, the region around the RVB optical axis is dominated by a strong electric longitudinal component, which is absent in the case of the AVB. The AVB is characterized, in contrast, by an intense longitudinal magnetic field which the RVB lacks. The longitudinal components are the key to a proper description of the topological properties: Let us define the two (unit) vectors:
$$\pmb{m}_\textrm{RVB}=\left(\pmb{F}_{\perp}(\rho),-iF_z(\rho)\right)^T/\left|\pmb{F}(\rho)\right|$$
and
$$\pmb{m}_\textrm{AVB}=\left(\pmb{G}_{\perp}(\rho),-iG_z(\rho)\right)^T/\left|\pmb{G}(\rho)\right|$$
where the vector $\pmb {G}(\rho )$ is stemming from the AVB magnetic field $\pmb {B}_\textrm {AVB}=\pmb {\nabla }\times \pmb {A}_\textrm {AVB}=A_0\pmb {G}(\rho )e^{i(q_zz-\omega t)} + \textrm {c.c.}$. In other words, $\pmb {m}_\textrm {RVB}$ and $\pmb {m}_\textrm {AVB}$ are constructed out of the 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

$$n_\textrm{i}=\frac{1}{4\pi}\int_{\rho_n}^{\rho_{n+1}}\textrm{d}A\,\,\pmb{m}_\textrm{i}\cdot\left[\partial_x\pmb{m}_\textrm{i}\times \partial_y\pmb{m}_\textrm{i}\right]=\pm1$$
for i$=$RVB,AVB. Consequently, the electric (RVB) and magnetic fields (AVB) can be characterized by a topological index as long as an accurate description of the longitudinal field components is included. We proofed numerically that this is also the case for the representation via Laguerre-Gaussian modes when the electromagnetic scalar potential is included. We note that – in our interpretation – cylindrical vector beams cannot be counted as optical skyrmions despite their topological properties. This can be explained by the spin angular momentum density $\pmb {s}_\textrm {i}=\pmb {E}_\textrm {i}\times \pmb {A}_\textrm {i}$ (i$=$RVB,AVB): in both cases a $z$-component is absent, i.e., $\pmb {s}_\textrm {i}\cdot \left [\partial _x\pmb {s}_\textrm {i}\times \partial _y\pmb {s}_\textrm {i}\right ]\equiv 0$. Accordingly, the spin angular momentum of the vector beams reveals no topological properties.

 

Fig. 1. Characterization of three different types of topological light fields. Left column: radial vector beam, middle column: azimuthal vector beam, right column: optical skyrmion. The upper row shows the topological vector $\pmb {m}_\textrm {i}$ (i=RVB,AVB,OS), as presented in the main text. The color scale represents the local direction of $\pmb {m}_\textrm {i}$. The middle row presents the corresponding Poynting vectors ($z$-component) in the focal plane $z=0$. The lower row shows cross sections along a chosen axis presenting the evolution of the direction of $\pmb {m}_\textrm {i}$ for the three cases.

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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

$$n_\textrm{OS}=\frac{1}{4\pi}\int\textrm{d}A\,\,\pmb{m}_\textrm{OS}\cdot\left[\partial_x\pmb{m}_\textrm{OS}\times \partial_y\pmb{m}_\textrm{OS}\right]=\pm1$$
over an area with boundary $\rho _\textrm {sk}$ yields in a topological index $n_\textrm {OS}=\pm 1$. In contrast to the vector beams, this topological index refers to the photon spin in analogy to the magnetic skyrmion.

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

$$i\partial_t\Psi_\textrm{i}(\pmb{r},t)=\hat{H}_\textrm{i}(t)\Psi_\textrm{i}(\pmb{r},t),$$
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:
$$\hat{H}_\textrm{i}(t)=\left[\hat{\pmb{p}}-\pmb{A}(\pmb{r}-\pmb{r}_\textrm{i},t)\right]^2,$$
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 $\pmb {r}_\textrm {d}$:

$$\Phi_\textrm{i}(\pmb{r}_\textrm{d},t)=\frac{1}{4\pi\epsilon_0}\int\textrm{d}\pmb{r}\,\frac{\rho_\textrm{i}(\pmb{r},t_R)}{|r_\textrm{d}-r_\textrm{i}-\pmb{r}|}$$
and
$$\pmb{A}_\textrm{i}(\pmb{r}_\textrm{d},t)=\frac{\mu_0}{4\pi}\int\textrm{d}\pmb{r}\,\frac{\pmb{j}_\textrm{i}(\pmb{r},t_R)}{|r_\textrm{d}-r_\textrm{i}-\pmb{r}|},$$
where we introduce the retarded time $t_R=t-|r_\textrm {d}-r_\textrm {i}-\pmb {r}|/c$. The time-dependent charge and current densities can be directly extracted from the propagated wave function via $\rho _\textrm {i}(\pmb {r},t)=|\Psi _\textrm {i}(\pmb {r},t)|^2$ and $\pmb {j}_\textrm {i}(\pmb {r},t)=\textrm {im}\left \{\Psi ^*_\textrm {i}(\pmb {r},t)\pmb {\nabla }\Psi _\textrm {i}(\pmb {r},t)\right \}$. The total (detected) vector and scalar potentials of the whole atomic distribution are found by a coherent sum, i.e.
$$\begin{aligned}\pmb{A}(\pmb{r}_\textrm{d},t)&=\sum_{i}\pmb{A}_\textrm{i}(\pmb{r}_\textrm{d},t) \\ \Phi(\pmb{r}_\textrm{d},t)&=\sum_{i}\Phi_\textrm{i}(\pmb{r}_\textrm{d},t)\end{aligned}.$$
The corresponding electric and magnetic fields can be found by $\pmb {E}(\pmb {r},t)=-\pmb {\nabla }\Phi (\pmb {r},t)-\partial _t\pmb {A}(\pmb {r},t)$ and $\pmb {B}(\pmb {r},t)=\pmb {\nabla }\times \pmb {A}(\pmb {r},t)$.

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:

$$n(\rho,z)=n_0e^{-\frac{\rho^2}{2\rho_g^2}}e^{-\frac{z^2}{2z_g^2}},$$
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).

 

Fig. 2. Characterization of the azimuthally polarized higher harmonics in the far-field: a) Harmonic spectrum with classical cut off energy (belonging to the intensity maximum). b) transversal magnetic field in the dependence on the axial distance (in form of the divergence angle). c) $z$-component of the radiated Poynting vector. d) The longitudinal magnetic field distribution. Panels in bottom right part: unit vector $\pmb {m}_\textrm {AVB}$ for the different harmonic orders. The results for the individual harmonics (11th, 13th and 15th) correspond to integration along the bandwidth of the respective harmonic peak in the spectrum.

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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.

 

Fig. 3. Characterization of the radially polarized higher harmonics in the far-field: a) Harmonic spectrum with classical cut off energy (belonging to the intensity maximum). b) transversal electric field in the dependence on the axial distance. c) $z$-component of the radiated Poynting vector. d) longitudinal electric field distribution. Panels in bottom right part: unit vector $\pmb {m}_\textrm {RVB}$ for the different harmonic orders. The results for the individual harmonics (11th, 13th and 15th) correspond to integration along the bandwidth of the respective harmonic peak in the spectrum.

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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 $\pmb {m}_\textrm {RVB}$.

 

Fig. 4. Role of the focusing in the composition of the radiated higher harmonics (here harmonic order 13): a) longitudinal components for two different angles $\alpha$. The dashed curves indicate the corresponding transversal fields. b) The (normalized) Stokes parameter for the two different angles $\alpha$. The dashed line indicates the sum $S_1+S_3$. The blue arrows represent $\pmb {m}_\textrm {RVB}$ as defined in the main text.

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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].

 

Fig. 5. Characterization of the higher harmonics in the far-field: a) Azimuthal Fourier transformation of the emitted $E_z$ revealing the winding number $\ell _n$. b) Diffraction properties of the radiated harmonics.

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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

$$n_\textrm{OS}=\frac{1}{4\pi}\int\textrm{d}A\,\,\pmb{m}_\textrm{OS}\cdot\left[\partial_x\pmb{m}_\textrm{OS}\times \partial_y\pmb{m}_\textrm{OS}\right]=-1$$
for all harmonics if the integration boundary is given by the location of the maximum of the corresponding Poynting vector component $S_z$, as indicated by the black paths in Figs. 6(a) and (b). Hence, similar to the HHG driven by vector beams, the topological index is a conserved quantity, which is transferred into the higher frequency regime.

 

Fig. 6. Far-field Poynting vectors (upper row) and SAM density (lower row) for two selected higher harmonics corresponding to HHG driven by an optical skyrmion.

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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.

 

Fig. 7. Normalized Stokes parameters $S_i/S_0$ in the transversal plane for the 5th harmonic.

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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.

References

1. P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004). [CrossRef]  

2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]  

3. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010). [CrossRef]  

4. M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]  

5. Y. Toda, S. Honda, and R. Morita, “Dynamics of a paired optical vortex generated by second-harmonic generation,” Opt. Express 18(17), 17796–17804 (2010). [CrossRef]  

6. M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012). [CrossRef]  

7. C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013). [CrossRef]  

8. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014). [CrossRef]  

9. R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016). [CrossRef]  

10. C. Hernández-García, A. Turpin, J. San Román, A. Picón, R. Drevinskas, A. Cerkauskaite, P. G. Kazansky, C. G. Durfee, and Í. J. Sola, “Extreme ultraviolet vector beams driven by infrared lasers,” Optica 4(5), 520–526 (2017). [CrossRef]  

11. W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019). [CrossRef]  

12. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]  

13. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005). [CrossRef]  

14. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015). [CrossRef]  

15. C. Wagner and N. Harned, “Euv lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010). [CrossRef]  

16. G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010). [CrossRef]  

17. T. Bose and J. Berakdar, “Nonlinear magneto-optical response to light carrying orbital angular momentum,” J. Opt. 16(12), 125201 (2014). [CrossRef]  

18. J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016). [CrossRef]  

19. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009). [CrossRef]  

20. S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018). [CrossRef]  

21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]  

22. G. F. Quinteiro, D. Reiter, and T. Kuhn, “Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge,” Phys. Rev. A 91(3), 033808 (2015). [CrossRef]  

23. M. Padgett and L. Allen, “The poynting vector in laguerre-gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]  

24. M. Nurhuda and F. H. Faisal, “Numerical solution of time-dependent schrödinger equation for multiphoton processes: A matrix iterative method,” Phys. Rev. A 60(4), 3125–3133 (1999). [CrossRef]  

25. I. Ivanov, “Time delay in strong-field photoionization of a hydrogen atom,” Phys. Rev. A 83(2), 023421 (2011). [CrossRef]  

26. A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010). [CrossRef]  

27. C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010). [CrossRef]  

28. C. Hernández-García, “Coherent attosecond light sources based on high-order harmonic generation: influence of the propagation effects,” Ph.D. thesis, Salamanca (2013).

29. M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015). [CrossRef]  

30. J. Wätzel, C. Granados-Castro, and J. Berakdar, “Magnetoelectric response of quantum structures driven by optical vector beams,” Phys. Rev. B 99(8), 085425 (2019). [CrossRef]  

31. W. H. McMaster, “Polarization and the stokes parameters,” Am. J. Phys. 22(6), 351–362 (1954). [CrossRef]  

32. J. Wätzel and J. Berakdar, “Multipolar, polarization-shaped high-order harmonic generation by intense vector beams,” Phys. Rev. A 101(4), 043409 (2020). [CrossRef]  

33. W. Paufler, B. Böning, and S. Fritzsche, “Tailored orbital angular momentum in high-order harmonic generation with bicircular laguerre-gaussian beams,” Phys. Rev. A 98(1), 011401 (2018). [CrossRef]  

34. C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015). [CrossRef]  

References

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  • |
  • |

  1. P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004).
    [Crossref]
  2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
    [Crossref]
  3. T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
    [Crossref]
  4. M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [Crossref]
  5. Y. Toda, S. Honda, and R. Morita, “Dynamics of a paired optical vortex generated by second-harmonic generation,” Opt. Express 18(17), 17796–17804 (2010).
    [Crossref]
  6. M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
    [Crossref]
  7. C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
    [Crossref]
  8. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
    [Crossref]
  9. R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
    [Crossref]
  10. C. Hernández-García, A. Turpin, J. San Román, A. Picón, R. Drevinskas, A. Cerkauskaite, P. G. Kazansky, C. G. Durfee, and Í. J. Sola, “Extreme ultraviolet vector beams driven by infrared lasers,” Optica 4(5), 520–526 (2017).
    [Crossref]
  11. W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019).
    [Crossref]
  12. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref]
  13. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
    [Crossref]
  14. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
    [Crossref]
  15. C. Wagner and N. Harned, “Euv lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
    [Crossref]
  16. G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010).
    [Crossref]
  17. T. Bose and J. Berakdar, “Nonlinear magneto-optical response to light carrying orbital angular momentum,” J. Opt. 16(12), 125201 (2014).
    [Crossref]
  18. J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
    [Crossref]
  19. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
    [Crossref]
  20. S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
    [Crossref]
  21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  22. G. F. Quinteiro, D. Reiter, and T. Kuhn, “Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge,” Phys. Rev. A 91(3), 033808 (2015).
    [Crossref]
  23. M. Padgett and L. Allen, “The poynting vector in laguerre-gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
    [Crossref]
  24. M. Nurhuda and F. H. Faisal, “Numerical solution of time-dependent schrödinger equation for multiphoton processes: A matrix iterative method,” Phys. Rev. A 60(4), 3125–3133 (1999).
    [Crossref]
  25. I. Ivanov, “Time delay in strong-field photoionization of a hydrogen atom,” Phys. Rev. A 83(2), 023421 (2011).
    [Crossref]
  26. A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
    [Crossref]
  27. C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
    [Crossref]
  28. C. Hernández-García, “Coherent attosecond light sources based on high-order harmonic generation: influence of the propagation effects,” Ph.D. thesis, Salamanca (2013).
  29. M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
    [Crossref]
  30. J. Wätzel, C. Granados-Castro, and J. Berakdar, “Magnetoelectric response of quantum structures driven by optical vector beams,” Phys. Rev. B 99(8), 085425 (2019).
    [Crossref]
  31. W. H. McMaster, “Polarization and the stokes parameters,” Am. J. Phys. 22(6), 351–362 (1954).
    [Crossref]
  32. J. Wätzel and J. Berakdar, “Multipolar, polarization-shaped high-order harmonic generation by intense vector beams,” Phys. Rev. A 101(4), 043409 (2020).
    [Crossref]
  33. W. Paufler, B. Böning, and S. Fritzsche, “Tailored orbital angular momentum in high-order harmonic generation with bicircular laguerre-gaussian beams,” Phys. Rev. A 98(1), 011401 (2018).
    [Crossref]
  34. C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
    [Crossref]

2020 (1)

J. Wätzel and J. Berakdar, “Multipolar, polarization-shaped high-order harmonic generation by intense vector beams,” Phys. Rev. A 101(4), 043409 (2020).
[Crossref]

2019 (2)

J. Wätzel, C. Granados-Castro, and J. Berakdar, “Magnetoelectric response of quantum structures driven by optical vector beams,” Phys. Rev. B 99(8), 085425 (2019).
[Crossref]

W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019).
[Crossref]

2018 (2)

W. Paufler, B. Böning, and S. Fritzsche, “Tailored orbital angular momentum in high-order harmonic generation with bicircular laguerre-gaussian beams,” Phys. Rev. A 98(1), 011401 (2018).
[Crossref]

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

2017 (1)

2016 (2)

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
[Crossref]

2015 (4)

G. F. Quinteiro, D. Reiter, and T. Kuhn, “Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge,” Phys. Rev. A 91(3), 033808 (2015).
[Crossref]

C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
[Crossref]

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
[Crossref]

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
[Crossref]

2014 (2)

T. Bose and J. Berakdar, “Nonlinear magneto-optical response to light carrying orbital angular momentum,” J. Opt. 16(12), 125201 (2014).
[Crossref]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

2013 (1)

C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
[Crossref]

2012 (1)

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

2011 (1)

I. Ivanov, “Time delay in strong-field photoionization of a hydrogen atom,” Phys. Rev. A 83(2), 023421 (2011).
[Crossref]

2010 (6)

A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
[Crossref]

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

Y. Toda, S. Honda, and R. Morita, “Dynamics of a paired optical vortex generated by second-harmonic generation,” Opt. Express 18(17), 17796–17804 (2010).
[Crossref]

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

C. Wagner and N. Harned, “Euv lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010).
[Crossref]

2009 (2)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
[Crossref]

2005 (1)

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref]

2004 (1)

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004).
[Crossref]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

2001 (1)

M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

1999 (1)

M. Nurhuda and F. H. Faisal, “Numerical solution of time-dependent schrödinger equation for multiphoton processes: A matrix iterative method,” Phys. Rev. A 60(4), 3125–3133 (1999).
[Crossref]

1995 (1)

M. Padgett and L. Allen, “The poynting vector in laguerre-gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

1954 (1)

W. H. McMaster, “Polarization and the stokes parameters,” Am. J. Phys. 22(6), 351–362 (1954).
[Crossref]

Abeln, B.

A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
[Crossref]

Agostini, P.

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004).
[Crossref]

Allen, L.

M. Padgett and L. Allen, “The poynting vector in laguerre-gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Anderson, E. H.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref]

Arpin, P.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Attwood, D. T.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref]

Auguste, T.

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

Bartal, G.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Bartschat, K.

A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berakdar, J.

J. Wätzel and J. Berakdar, “Multipolar, polarization-shaped high-order harmonic generation by intense vector beams,” Phys. Rev. A 101(4), 043409 (2020).
[Crossref]

J. Wätzel, C. Granados-Castro, and J. Berakdar, “Magnetoelectric response of quantum structures driven by optical vector beams,” Phys. Rev. B 99(8), 085425 (2019).
[Crossref]

J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
[Crossref]

T. Bose and J. Berakdar, “Nonlinear magneto-optical response to light carrying orbital angular momentum,” J. Opt. 16(12), 125201 (2014).
[Crossref]

Böning, B.

W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019).
[Crossref]

W. Paufler, B. Böning, and S. Fritzsche, “Tailored orbital angular momentum in high-order harmonic generation with bicircular laguerre-gaussian beams,” Phys. Rev. A 98(1), 011401 (2018).
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Bose, T.

T. Bose and J. Berakdar, “Nonlinear magneto-optical response to light carrying orbital angular momentum,” J. Opt. 16(12), 125201 (2014).
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Boyd, R. W.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

Caillat, J.

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
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R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

Cerkauskaite, A.

Chao, W.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
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Chen, M.-C.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Cohen, K.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Corkum, P. B.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
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DiMauro, L. F.

P. Agostini and L. F. DiMauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67(6), 813–855 (2004).
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Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Dreischuh, A.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

Drevinskas, R.

Durfee, C. G.

Faisal, F. H.

M. Nurhuda and F. H. Faisal, “Numerical solution of time-dependent schrödinger equation for multiphoton processes: A matrix iterative method,” Phys. Rev. A 60(4), 3125–3133 (1999).
[Crossref]

Fritzsche, S.

W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019).
[Crossref]

W. Paufler, B. Böning, and S. Fritzsche, “Tailored orbital angular momentum in high-order harmonic generation with bicircular laguerre-gaussian beams,” Phys. Rev. A 98(1), 011401 (2018).
[Crossref]

Frumker, E.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

Gariepy, G.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

Géneaux, R.

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

Gjonaj, B.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Gobert, O.

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

Granados-Castro, C.

J. Wätzel, C. Granados-Castro, and J. Berakdar, “Magnetoelectric response of quantum structures driven by optical vector beams,” Phys. Rev. B 99(8), 085425 (2019).
[Crossref]

Grum-Grzhimailo, A. N.

A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
[Crossref]

Hammond, T. J.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

Hansinger, P.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

Harned, N.

C. Wagner and N. Harned, “Euv lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref]

Hernández-García, C.

C. Hernández-García, A. Turpin, J. San Román, A. Picón, R. Drevinskas, A. Cerkauskaite, P. G. Kazansky, C. G. Durfee, and Í. J. Sola, “Extreme ultraviolet vector beams driven by infrared lasers,” Optica 4(5), 520–526 (2017).
[Crossref]

C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
[Crossref]

C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
[Crossref]

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

C. Hernández-García, “Coherent attosecond light sources based on high-order harmonic generation: influence of the propagation effects,” Ph.D. thesis, Salamanca (2013).

Honda, S.

Ishikawa, T.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
[Crossref]

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I. Ivanov, “Time delay in strong-field photoionization of a hydrogen atom,” Phys. Rev. A 83(2), 023421 (2011).
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Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
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Jarque, E. C.

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

Kapteyn, H. C.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Kazansky, P. G.

Kern, C.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

Kim, K. T.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

Krausz, F.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[Crossref]

Kuhn, T.

G. F. Quinteiro, D. Reiter, and T. Kuhn, “Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge,” Phys. Rev. A 91(3), 033808 (2015).
[Crossref]

Leach, J.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref]

Lindner, N.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

McMaster, W. H.

W. H. McMaster, “Polarization and the stokes parameters,” Am. J. Phys. 22(6), 351–362 (1954).
[Crossref]

Miao, J.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
[Crossref]

Morita, R.

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
[Crossref]

Y. Toda, S. Honda, and R. Morita, “Dynamics of a paired optical vortex generated by second-harmonic generation,” Opt. Express 18(17), 17796–17804 (2010).
[Crossref]

Murnane, M. M.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
[Crossref]

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Nurhuda, M.

M. Nurhuda and F. H. Faisal, “Numerical solution of time-dependent schrödinger equation for multiphoton processes: A matrix iterative method,” Phys. Rev. A 60(4), 3125–3133 (1999).
[Crossref]

Oka, K.

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
[Crossref]

Ostrovsky, E.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Padgett, M.

M. Padgett and L. Allen, “The poynting vector in laguerre-gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995).
[Crossref]

Paufler, W.

W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019).
[Crossref]

W. Paufler, B. Böning, and S. Fritzsche, “Tailored orbital angular momentum in high-order harmonic generation with bicircular laguerre-gaussian beams,” Phys. Rev. A 98(1), 011401 (2018).
[Crossref]

Pavlyukh, Y.

J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
[Crossref]

Pérez-Hernández, J.

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

Pert, G.

G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010).
[Crossref]

Picón, A.

C. Hernández-García, A. Turpin, J. San Román, A. Picón, R. Drevinskas, A. Cerkauskaite, P. G. Kazansky, C. G. Durfee, and Í. J. Sola, “Extreme ultraviolet vector beams driven by infrared lasers,” Optica 4(5), 520–526 (2017).
[Crossref]

C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
[Crossref]

C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
[Crossref]

Plaja, L.

C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
[Crossref]

C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
[Crossref]

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

Popmintchev, T.

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Quinteiro, G. F.

G. F. Quinteiro, D. Reiter, and T. Kuhn, “Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge,” Phys. Rev. A 91(3), 033808 (2015).
[Crossref]

Ramos, J.

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

Reiter, D.

G. F. Quinteiro, D. Reiter, and T. Kuhn, “Formulation of the twisted-light–matter interaction at the phase singularity: The twisted-light gauge,” Phys. Rev. A 91(3), 033808 (2015).
[Crossref]

Robinson, I. K.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
[Crossref]

Roso, L.

C. Hernández-García, J. Pérez-Hernández, J. Ramos, E. C. Jarque, L. Roso, and L. Plaja, “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A 82(3), 033432 (2010).
[Crossref]

Ruchon, T.

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

San Román, J.

C. Hernández-García, A. Turpin, J. San Román, A. Picón, R. Drevinskas, A. Cerkauskaite, P. G. Kazansky, C. G. Durfee, and Í. J. Sola, “Extreme ultraviolet vector beams driven by infrared lasers,” Optica 4(5), 520–526 (2017).
[Crossref]

C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
[Crossref]

C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
[Crossref]

Schäffer, A.

J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
[Crossref]

Sola, Í. J.

Soskin, M.

M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Spielmann, C.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Suzuki, M.

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
[Crossref]

Taïeb, R.

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

Tallents, G.

G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010).
[Crossref]

Toda, Y.

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
[Crossref]

Y. Toda, S. Honda, and R. Morita, “Dynamics of a paired optical vortex generated by second-harmonic generation,” Opt. Express 18(17), 17796–17804 (2010).
[Crossref]

Tsesses, S.

S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361(6406), 993–996 (2018).
[Crossref]

Turpin, A.

Urness, T.

A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
[Crossref]

Vasnetsov, M.

M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[Crossref]

Wagenaars, E.

G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010).
[Crossref]

Wagner, C.

C. Wagner and N. Harned, “Euv lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

Wätzel, J.

J. Wätzel and J. Berakdar, “Multipolar, polarization-shaped high-order harmonic generation by intense vector beams,” Phys. Rev. A 101(4), 043409 (2020).
[Crossref]

J. Wätzel, C. Granados-Castro, and J. Berakdar, “Magnetoelectric response of quantum structures driven by optical vector beams,” Phys. Rev. B 99(8), 085425 (2019).
[Crossref]

J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
[Crossref]

Weflen, D.

A. N. Grum-Grzhimailo, B. Abeln, K. Bartschat, D. Weflen, and T. Urness, “Ionization of atomic hydrogen in strong infrared laser fields,” Phys. Rev. A 81(4), 043408 (2010).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Yamane, K.

M. Suzuki, K. Yamane, K. Oka, Y. Toda, and R. Morita, “Extended stokes parameters for cylindrically polarized beams,” Opt. Rev. 22(1), 179–183 (2015).
[Crossref]

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
[Crossref]

Zürch, M.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

Adv. Opt. Photonics (1)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1 (2009).
[Crossref]

Am. J. Phys. (1)

W. H. McMaster, “Polarization and the stokes parameters,” Am. J. Phys. 22(6), 351–362 (1954).
[Crossref]

Carbon (1)

J. Wätzel, Y. Pavlyukh, A. Schäffer, and J. Berakdar, “Optical vortex driven charge current loop and optomagnetism in fullerenes,” Carbon 99, 439–443 (2016).
[Crossref]

J. Opt. (2)

W. Paufler, B. Böning, and S. Fritzsche, “High harmonic generation with laguerre–gaussian beams,” J. Opt. 21(9), 094001 (2019).
[Crossref]

T. Bose and J. Berakdar, “Nonlinear magneto-optical response to light carrying orbital angular momentum,” J. Opt. 16(12), 125201 (2014).
[Crossref]

Nat. Commun. (1)

R. Géneaux, A. Camper, T. Auguste, O. Gobert, J. Caillat, R. Taïeb, and T. Ruchon, “Synthesis and characterization of attosecond light vortices in the extreme ultraviolet,” Nat. Commun. 7(1), 12583 (2016).
[Crossref]

Nat. Photonics (3)

T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent x-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

C. Wagner and N. Harned, “Euv lithography: Lithography gets extreme,” Nat. Photonics 4(1), 24–26 (2010).
[Crossref]

G. Tallents, E. Wagenaars, and G. Pert, “Optical lithography: Lithography at euv wavelengths,” Nat. Photonics 4(12), 809–811 (2010).
[Crossref]

Nat. Phys. (1)

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and C. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8(10), 743–746 (2012).
[Crossref]

Nature (1)

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[Crossref]

New J. Phys. (1)

C. Hernández-García, J. San Román, L. Plaja, and A. Picón, “Quantum-path signatures in attosecond helical beams driven by optical vortices,” New J. Phys. 17(9), 093029 (2015).
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Figures (7)

Fig. 1.
Fig. 1. Characterization of three different types of topological light fields. Left column: radial vector beam, middle column: azimuthal vector beam, right column: optical skyrmion. The upper row shows the topological vector $\pmb {m}_\textrm {i}$ (i=RVB,AVB,OS), as presented in the main text. The color scale represents the local direction of $\pmb {m}_\textrm {i}$. The middle row presents the corresponding Poynting vectors ($z$-component) in the focal plane $z=0$. The lower row shows cross sections along a chosen axis presenting the evolution of the direction of $\pmb {m}_\textrm {i}$ for the three cases.
Fig. 2.
Fig. 2. Characterization of the azimuthally polarized higher harmonics in the far-field: a) Harmonic spectrum with classical cut off energy (belonging to the intensity maximum). b) transversal magnetic field in the dependence on the axial distance (in form of the divergence angle). c) $z$-component of the radiated Poynting vector. d) The longitudinal magnetic field distribution. Panels in bottom right part: unit vector $\pmb {m}_\textrm {AVB}$ for the different harmonic orders. The results for the individual harmonics (11th, 13th and 15th) correspond to integration along the bandwidth of the respective harmonic peak in the spectrum.
Fig. 3.
Fig. 3. Characterization of the radially polarized higher harmonics in the far-field: a) Harmonic spectrum with classical cut off energy (belonging to the intensity maximum). b) transversal electric field in the dependence on the axial distance. c) $z$-component of the radiated Poynting vector. d) longitudinal electric field distribution. Panels in bottom right part: unit vector $\pmb {m}_\textrm {RVB}$ for the different harmonic orders. The results for the individual harmonics (11th, 13th and 15th) correspond to integration along the bandwidth of the respective harmonic peak in the spectrum.
Fig. 4.
Fig. 4. Role of the focusing in the composition of the radiated higher harmonics (here harmonic order 13): a) longitudinal components for two different angles $\alpha$. The dashed curves indicate the corresponding transversal fields. b) The (normalized) Stokes parameter for the two different angles $\alpha$. The dashed line indicates the sum $S_1+S_3$. The blue arrows represent $\pmb {m}_\textrm {RVB}$ as defined in the main text.
Fig. 5.
Fig. 5. Characterization of the higher harmonics in the far-field: a) Azimuthal Fourier transformation of the emitted $E_z$ revealing the winding number $\ell _n$. b) Diffraction properties of the radiated harmonics.
Fig. 6.
Fig. 6. Far-field Poynting vectors (upper row) and SAM density (lower row) for two selected higher harmonics corresponding to HHG driven by an optical skyrmion.
Fig. 7.
Fig. 7. Normalized Stokes parameters $S_i/S_0$ in the transversal plane for the 5th harmonic.

Equations (14)

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A A OV m , σ = [ J m ( q ρ ) e i m φ e ^ σ i σ q q z J m + σ ( q ρ ) e i ( m + σ ) φ e ^ z ] × A 0 Ω ( t ) e i ( q z z ω t ) + c.c. ,
A A AVB ( r r , t ) = A 0 J 1 ( q ρ ) Ω ( t ) e i ( q z z ω t ) e ^ φ + c.c. ,
A A RVB ( r r , t ) = A 0 [ q q J 1 ( q ρ ) e ^ ρ + i q q J 0 ( q ρ ) e ^ z ] × Ω ( t ) e i ( q z z ω t ) + c.c. = A 0 F F ( ρ ) Ω ( t ) e i ( q z z ω t ) + c.c.
m m RVB = ( F F ( ρ ) , i F z ( ρ ) ) T / | F F ( ρ ) |
m m AVB = ( G G ( ρ ) , i G z ( ρ ) ) T / | G G ( ρ ) |
n i = 1 4 π ρ n ρ n + 1 d A m m i [ x m m i × y m m i ] = ± 1
n OS = 1 4 π d A m m OS [ x m m OS × y m m OS ] = ± 1
i t Ψ i ( r r , t ) = H ^ i ( t ) Ψ i ( r r , t ) ,
H ^ i ( t ) = [ p p ^ A A ( r r r r i , t ) ] 2 ,
Φ i ( r r d , t ) = 1 4 π ϵ 0 d r r ρ i ( r r , t R ) | r d r i r r |
A A i ( r r d , t ) = μ 0 4 π d r r j j i ( r r , t R ) | r d r i r r | ,
A A ( r r d , t ) = i A A i ( r r d , t ) Φ ( r r d , t ) = i Φ i ( r r d , t ) .
n ( ρ , z ) = n 0 e ρ 2 2 ρ g 2 e z 2 2 z g 2 ,
n OS = 1 4 π d A m m OS [ x m m OS × y m m OS ] = 1

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