2.1.

Signatures of the Driving Structured Fields

The full Poincaré beam is a typical vector field, whose polarization state covers the full surface of the Poincaré sphere. It is constructed by an arbitrary-order Laguerre–Gaussian beam and a fundamental Gaussian mode with an orthogonal polarization state. Its electric field can be given by: ${\mathbf{E}}_{i,\omega }({\mathbf{r}}_i,t)=e^{-i\omega t}[u_{\omega }{e}^{i\ell \omega \phi i}({\mathbf{e}}_x+i{\mathbf{e}}_y)+u_{\omega }^{\prime }{e}^{i{\ell }^{\prime }\omega \phi i}({\mathbf{e}}_x-i{\mathbf{e}}_y)]+c.c.$, where $t$ is the time, $\omega$ is the angular frequency, ${\mathbf{r}}_i=({\rho }_i,{\phi }_i)$ denotes the spatial coordinate for the incident driving fields [the notation “$i$” indicates the variables in the incident plane, as shown in the inset of Fig. 1(a)], $u_{\omega }$ ($u_{\omega }^{\prime }$) is the spatial complex amplitude, and ${\ell }_{\omega }$ (${\ell }_{\omega }^{\prime }$) is the topological charge (${\ell }_{\omega }$ or ${\ell }_{\omega }^{\prime }$ equals zero). Without loss of generality, $u_{\omega }$ ($u_{\omega }^{\prime }$) can be approximated as a complex constant for the incident laser. For conciseness, we introduce a geometrical parameter defined as ${\eta }_{\omega }=({\ell }_{\omega },{\ell }_{\omega }^{\prime },|u_{\omega }^{\prime }|/|u_{\omega }|$, $\arg (u_{\omega }^{\prime })-\arg (u_{\omega }))$. As shown later, ${\eta }_{\omega }$ directly determines the topology of EUV fields.

Fig. 1

(a) Schematic of generating EUV skyrmions with HHG. A synthesized two-color ($800\mathrm{nm}+400\mathrm{nm}$) CVVB field is employed to generate HHG with non-trivial spatial spin distribution. The red and blue arrows represent the instantaneous electric field vectors of 800- and 400-nm incident fields, respectively, and the accompanied pseudocolor maps show their angular phase structures. We control the beam width of the incident 800-nm field that is twice that of the 400-nm field, so the focal distributions of intensities of 800- and 400-nm CVVBs are the same. We note here that if they are not the same, one cannot generate the skyrmions presented in this work. Here, we show the EUV skyrmionic fields can be generated, and the 13th harmonic spins down at the beam center and spins up at the surrounding. The inset illustrates the coordinates in the incident plane and focal plane, respectively. (b) The distributions of intensity (upper panel) and normalized $z$-direction SAM density (bottom panel) of 800- and 400-nm CVVBs in the focal plane individually. (c) The focal electric field structure of the two-color synthesized CVVB. The pseudocolor map indicates the intensity distribution of the driving beam. The overlapped Lissajous figures illustrate the local polarization states, in which the red profiles show the sixfold spatial symmetry of the two-color synthesized field.

To demonstrate our scheme, we start by considering one typical kind of full Poincaré beams, namely, cylindrical vortex vector beams (CVVBs). We illustrate the physical scenario in Fig. 1(a), where the incident field is spatiotemporally superposed by two CVVBs with different wavelengths ($800\mathrm{nm}+400\mathrm{nm}$). The geometrical parameters of the 800- and 400-nm CVVBs are given by ${\eta }_{\omega 1}=(-\mathrm{2,0},\mathrm{1,0})$ and ${\eta }_{\omega 2}=(\mathrm{0,2},\mathrm{1,0})$. The notations ${\omega }_1$ and ${\omega }_2$ represent the angular frequencies of 800 and 400 nm, respectively. As shown in Fig. 1(a), for each CVVB, it carries both radial polarization state (i.e., optical polarization singularity) and spiral phase (optical phase singularity).

In HHG, atoms directly interact with the focal field of the incident laser. It is important to know the signatures of focal field distribution. We, therefore, simulate the focal field of CVVB using the Richards–Wolf vectorial diffraction method.²⁵ We use the 20th order Gauss–Legendre integral formula to numerically calculate the integral, and the grid size is chosen as $50\mu \mathrm{m}$ in the focal plane with a spacing of $0.05\mu \mathrm{m}$. In our simulation, the numerical aperture (NA) is selected to be $\mathrm{NA}=0.02$. It is a typical focusing condition of HHG experiments. In this way, we can obtain the focal intensity distributions of both 800- and 400-nm fields [upper panel in Fig. 1(b)]. One can notice that the focal intensity distributions manifest as Gaussian-like structures, which differ from the typical donut-shaped structure of individual phase or polarization singularity. Such exotic distribution largely enriches the highly nonlinear interaction in the spatial domain. Furthermore, we calculate the normalized spin angular momentum (SAM) density of focal fields.^26,27 As shown in the bottom panel of Fig. 1(b), spatial-varying polarization is generated. It is interesting to find the presence of circular polarizations in the focal plane despite the incident light being linearly polarized throughout. Here, we emphasize that both the peculiar intensity distribution and SAM distribution of the focal field originate from spin-orbit interaction of light.²⁸ The underlying origin of such spin-orbit interaction comes from the time-varying polarization distribution in the wavefront of incident CVVB.^28,29 When the 800- and 400-nm CVVBs are spatiotemporally synthesized at focus, three-leaf polarizations show up [Fig. 1(c)]. This polarization has played an important role in the studies of HHG, which allows us to emit circularly polarized EUV light in HHG.³⁰ Most of the relevant works only involve laser fields with a spatially homogeneous three-leaf polarization. In our case, this two-color synthesized field, known as a Lissajous beam,^31,32 carries a versatile polarization state, and the polarization varies rapidly in space.

2.2.

Generation of EUV Skyrmions with CVVBs

To explore the HHG driven by the synthesized fields, we perform numerical simulation by solving the Schrödinger equation for hydrogen within strong-field approximation.³³ The high harmonic field with angular frequency $\omega$ is calculated from the Fourier components of the dipole moment, $\mathbf{P}(\mathbf{r},\omega )={\omega }^2\int \mathbf{D}(\mathbf{r},t)e^{-i\omega t}\mathrm{d}t$. Here, the dipole moment depends on both space ($\mathbf{r}$) and time ($t$), given by $\mathbf{D}(\mathbf{r},t)=-i{\int }_{-\infty }^t{\mathrm{d}t}^{\prime }{[-2\pi i/(t-t^{\prime }-i\delta )]}^{3/2}{\mathbf{d}}^{*}[{\mathbf{p}}_s+\mathbf{A}(\mathbf{r},t)]·\mathbf{E}(\mathbf{r},t^{\prime })·\mathbf{d}[{\mathbf{p}}_s+\mathbf{A}(\mathbf{r},t^{\prime })]\exp [-iS(\mathbf{r},{\mathbf{p}}_s,t)]+c.c.$, in which $\delta$ is an arbitrary small positive constant, ${\mathbf{p}}_s$ is the saddle point momentum calculated by saddle point equation ${\nabla }_{\mathbf{p}}S(\mathbf{r},{\mathbf{p}}_s,t)=0$, $S(\mathbf{r},{\mathbf{p}}_s,t)={\int }_{t^{\prime }}^t\mathrm{d}{t}^{″}\{{[\mathbf{p}+\mathbf{A}(\mathbf{r},t^{″})]}^2/2+I_p\}$ is the quasi-classical action, $d^{*}[.]$ and $d[.]$ contribute to the photoelectron recombination and ionization process, $\mathbf{E}$ is the focal electric field obtained by the Richards–Wolf vectorial diffraction method, and $\mathbf{A}$ is the vector potential of light. We insert the focal electric field into the expression of the HHG dipole moment and obtain the spatially resolved HHG field of different orders by changing the harmonic frequency $\omega$.

In order to elucidate the advantages of the two-color scenario in generating EUV skyrmions, we compare the HHG driven by three different fields, i.e., the 800-nm CVVB, 400-nm CVVB, and their synthesized two-color light field. In simulation, the peak intensities of each driving field are taken to be $I=5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and their temporal envelopes are selected to be a ${\sin }^2$ function of $\tau =21.3\text{-}\mathrm{fs}$ duration time. In Figs. 2(a)–2(c), we show the HHG spectra for different drivers. Due to the joint restriction taken by parity, energy, and momentum conservations in HHG, harmonic spectra reveal the orders of ($2n-1$)th, ($4n-2$)th, and $n$th for the three different drivers, respectively.

Fig. 2

Spatio–spectral signatures of HHG driven by the CVVBs. (a)–(c) High harmonic spectra. (d)–(f) Spatial intensity distributions of logarithmic scales. (g)–(i) Spatial distributions of normalized SAM densities along the $z$-direction. The HHG drivers are (a), (d), (g) the 800-nm CVVB, (b), (e), (h) 400-nm CVVB, and (c), (f), (i) their synthesized two-color CVVB. The harmonic orders marked by black arrows in (a)–(c) correspond to each column in (d)–(i).

In time domain, the generated field in HHG is spatially structured attosecond pulse trains. To identify the real-space topology of the harmonics, we confine one specific frequency of harmonics and extract its spatial intensity distributions [Figs. 2(d)–2(f)]. It can be noticed that when the driver is a single-color CVVB, the harmonics manifest as annulus structures. By contrast, the intensity of harmonics with the two-color driving field is different, which reveals a Gaussian-like structure for ($3n+1$)th and ($3n-1$)th harmonic or annulus structure for $3n$th harmonic. Due to the peculiar polarization texture of the driving two-color beam, the $3n$th harmonic has significantly lower yield than the ($3n\pm 1$)th harmonics. In Figs. 2(g)–2(i), we show the corresponding SAM distributions. For each harmonic, the SAM density is spatially varied. Upon closer inspection, in the case of the two-color CVVB driver, the SAM distribution of produced harmonics reveals a sixfold symmetrical distribution [Fig. 2(i)]. This unique distribution can be attributed to the inherent sixfold symmetry of the driving field, which is illustrated by the red profiles depicted in Fig. 1(c).

To form a skyrmionic field, local SAM should orient oppositely at the center of the field compared to the surrounding. Our simulation indicates that one cannot obtain an explicit EUV skyrmionic field with single-color CVVB drivers in HHG. This is because the recollision of electrons is rigidly forbidden in a single-color circularly polarized local field,¹⁵ resulting in an extremely low photon yield at the center of the harmonic beam [as shown in Figs. 2(d) and 2(e)]. Consequently, the spatial spin reversal is not complete. By contrast, such obstacles can be overcome in the ($3n\pm 1$)th harmonics with a two-color driver. In this case, the yield at the center is substantial, and the SAM is oriented oppositely at the center compared with the surrounding, leading to a non-trivial real-space topology.

To prove that the created ($3n\pm 1$)th harmonics from two-color CVVB fields are skyrmions, we then analyze their spatial vector texture. Conventionally, skyrmions are characterized by skyrmion number $Q$, defined as³⁴ $Q=(1/4\pi )·{\iint }_{\rho 0}\mathbf{S}·({\partial }_x\mathbf{S}\times {\partial }_y\mathbf{S})\mathrm{d}x\mathrm{d}y$, where $\mathbf{S}(x,y)$ and $\rho \le {\rho }_0$ denote the vector fields and the defined region for constructing skyrmions in the $(x,y)$ plane. The vector fields of a two-dimensional skyrmion can be written as $\mathbf{S}[\rho \cos (\phi ),\rho \sin (\phi )]=\{\cos [\alpha (\phi )]\sin [\beta (\rho )]$, $\sin [\alpha (\phi )]\sin [\beta (\rho )],\cos [\beta (\rho )]\}$, where $\rho$ and $\phi$ are polar coordinates in the focal plane, and $\alpha$ and $\beta$ are spherical coordinates in the Poincaré sphere, as shown in Fig. 3(a). In this way, $Q$ reads

Fig. 3

Characterization of the EUV skyrmions in HHG. (a) Unit Poincaré sphere defined by the normalized Stokes parameters, $\mathbf{S}=(S_1,S_2,S_3)$. At the north and south poles of the Poincaré sphere, polarizations are spin-up and spin-down, respectively. (b) The projection of Stokes parameters of 13th and 14th harmonics in the $S_1-S_3$ plane as a function of radial distance (upper panel) and the projection in the $S_1-S_2$ plane as a function of azimuthal angle (lower panel). (c), (d) Spatial distributions of Stokes vectors for the 13th and 14th harmonics in the focal plane, in which the color scheme of the Stokes vector is encoded by the value of $S_3$. (e), (f) Physical picture of generating skyrmions in HHG.

The skyrmion number is divided into two integers, $q$ and $m$, in which $q$ describes the polarity of skyrmions, and $m$ indicates the vorticity of skyrmions, respectively. In order to create a skyrmionic field, $q$ and $m$ should both be non-zero. That is, the field vector has to be reversed, comparing the center $\rho =0$ with its confinement $\rho ={\rho }_0$, and, meanwhile, it should also cover the unit Poincaré sphere as $\rho$ increases from $\rho =0$ to $\rho ={\rho }_0$.

In our configuration, the focal light field can be approximated as a paraxial beam, and thus we explore the possibility of generating EUV fields with a skyrmionic structure in Stokes vector fields, $\mathbf{S}=(S_1,S_2,S_3)$. Each parameter of $\mathbf{S}$ corresponds to an expectation value of a Pauli matrix for a photonic spin state. The skyrmionic boundary is selected to be ${\rho }_0=24\mu \mathrm{m}$, in which the focal 800- and 400-nm driving CVVBs are circularly polarized. We then inspect the topological character of the skyrmions by the stereographic projection of Stokes vectors [Figs. 3(b)–3(d)]. One can notice that the Stokes vectors point to the opposite direction at $\rho =0$ and $\rho ={\rho }_0$, and their orientations change in a vortex type [Figs. 3(c) and 3(d)]. To obtain the specific topological charge, $Q$, we inspect the azimuthal and radial variation of the Stokes vector distribution [Fig. 3(b)]. It can be seen that the direction of the Stokes vector is changed from downward (upward) to upward (downward) for the 13th (14th) harmonic. This indicates that the polarities of the 13th and 14th harmonics are given by $q_{13}=-1$ and $q_{14}=1$, respectively. Besides, their transverse components rotate twice counterclockwise from $\phi =0$ to $\phi =2\pi$, meaning the vorticity of the skyrmion is given by $m_{13}=m_{14}=2$. Taking both the polarity and vorticity of skyrmions into account, the skyrmion numbers can be determined to be $Q_{13}=q_{13}·m_{13}=-2$ and $Q_{14}=q_{14}·m_{14}=2$ for these two high harmonics, respectively. Such EUV skyrmionic fields can be categorized as two-order skyrmions.³

It is interesting to note that the skyrmion number is dependent on harmonic order. In the picture of absorbing photons, the allowed photon absorption channel for specific harmonics can be written by ($n_{\omega 1},n_{\omega 2}$), in which $n_{\omega 1}$ and $n_{\omega 2}$ denote the number of photons absorbed from 800- and 400-nm CVVBs. Since the driving field is spatially structured, the allowed photon channel is spatially varied. However, the allowed photon absorption channels cannot be solely determined at the positions in $0<\rho <{\rho }_0$, but the channel is well known at $\rho =0$ and $\rho ={\rho }_0$. There, the channel is given by (5, 4) for the 13th harmonic. That is, the number of absorbed 800-nm photons is one more than the number of 400-nm photons. Therefore, the local spin states at $\rho =0$ and $\rho ={\rho }_0$ are the same as that of the 800-nm photons for the 13th harmonic [Fig. 3(e)]. Similarly, the corresponding photon channel of the 14th harmonic is (4, 5), and thus the local spin is the same as the 400-nm photon at $\rho =0$ and $\rho ={\rho }_0$ [Fig. 3(f)]. As shown in Fig. 1(b), the local spin of 800- and 400-nm CVVBs is inverse, leading to opposite polarities of the 13th and 14th EUV skyrmions. On the other hand, the vorticity of EUV skyrmions is influenced by the focal phase distribution of the 800- and 400-nm CVVBs, which is independent of the harmonic order. Hence, the vorticities are the same for the 13th and 14th skyrmions. As a result, the skyrmion numbers of the 13th and 14th skyrmions are opposite.

2.3.

Complete Control Over the Texture of EUV Skyrmions

The behaviors and stabilization of skyrmions highly depend on their topological texture, and thus we then demonstrate how to control the topology of EUV skyrmions. We extend the driving fields from CVVBs to general full Poincaré beams, and we also resort to the synthetization of two-color (800 and 400 nm) beams. To this end, we start by giving an analytical expression for the focal field of full Poincaré beams as

where $C=i^{\ell }kf/2$ ($k$ is the wavenumber, $f$ is the focal distance) and $U_{\ell }$ are introduced by optical focusing. $U_{\ell }$ modulates the complex amplitude of focal field, and it is expressed as

With Eq. (2), we then propose two empirical restriction rules for the geometrical parameters $\eta$ of 800- and 400-nm incident full Poincaré beams, which ensure the generated EUV fields in HHG exhibit skyrmionic characteristics. (i) Considering the spin constraints in HHG, the local SAMs of 800- and 400-nm full Poincaré beams should have opposite orientations in the focal plane. It requires that the third components of their Stokes vectors ($S_{3,\omega 1}$ and $S_{3,\omega 2}$) exhibit opposite signs, leading to

(ii) The transverse components of Stokes vectors (i.e., $S_1$ and $S_2$) determine the direction of the long-axis of the focal electric field ellipse. To create regular spatial polarization distributions for harmonics, $S_{1,\omega 1}$ and $S_{2,\omega 1}$ need to be the same as $S_{1,\omega 2}$ and $S_{2,\omega 2}$, respectively. According to Eq. (2), $S_1$ and $S_2$ are determined by the relative phase between right- and left-handed components, i.e., $\arg (u{U}_{\ell })-\arg (u^{\prime }{U}_{{\ell }^{\prime }})$. Thus, the two-color full Poincaré beams are further confined by

Confined by the above rules, for each 800-nm full Poincaré beam, there is only one qualified 400-nm full Poincaré beam. Taking an 800-nm full Poincaré beam with ${\ell }_{\omega 1}=-1$ and ${\ell }_{\omega 1}^{\prime }=0$ as an example, the topological charges of a qualified 400-nm beam should be ${\ell }_{\omega 2}=0$ and ${\ell }_{\omega 2}^{\prime }=1$. Then, to visualize our controlling strategy of EUV skyrmions, we employ the concept of HyOPSP, which is comprised of two hybrid-order Poincaré spheres³⁵ at different wavelengths. Each point on the sphere represents one specific full Poincaré beam. In Figs. 4(a) and 4(b), we display the corresponding HyOPSP, in which $S_1^h$, $S_2^h$, and $S_3^h$ are the redefined Stokes parameters serving as the sphere’s Cartesian coordinates. $\mathrm{\Phi }={\tan }^{-1}(S_2^h/S_1^h)=\arg (u)-\arg (u^{\prime })$ and $\mathrm{\Theta }={\cos }^{-1}(S_3^h)$ are the azimuthal angle and pitching angle, respectively. We colorize some points on these two spheres ($\mathrm{\Phi }=0$, $\mathrm{\Theta }\in [0,\pi ]$ for 800 nm and $\mathrm{\Phi }=\pi$, $\mathrm{\Theta }\in [0,\pi ]$ for 400 nm). The points in the same colors construct the qualified two-color full Poincaré beams which can generate EUV skyrmions in HHG. For each point on the 800-nm hybrid-order Poincaré sphere, one can find a unique point on the 400-nm sphere that meets the above restriction rules [Eqs. (4) and (5)].

Fig. 4

Control over the topological texture of EUV skyrmions. (a), (b) HyOPSP formed by an 800-nm Poincaré beam (${\ell }_{\omega 1}=-1$ and ${\ell }_{\omega 1}^{\prime }=0$) and a 400-nm Poincaré beam (${\ell }_{\omega 2}=0$ and ${\ell }_{\omega 2}^{\prime }=1$). The varying colors on the spheres are used to show the qualified combination of two-color Poincaré beams. Points $A$ and $B$ on the HyOPSP indicate the positions of two-color Poincaré beams, which can generate (c), (d) a Bloch-type EUV skyrmion and (e) a Néel-type EUV skyrmion in HHG. (f) If selecting the point $A$ in the 800-nm sphere and $A^{\prime }$ in the 400-nm sphere, the generated EUV field reveals a non-skyrmion structure. (g), (h) Formation of anti-skyrmion and third-order skyrmion, respectively. In each diagram, the lower panel shows the projection of Stokes vectors in the $S_1\text{-}{S}_2$ plane as a function of azimuthal angle. The harmonic orders are 13th for (c) and 14th for (d)–(f).

Then, we select some typical points on the HyOPSP to demonstrate control over topological textures of skyrmions. At point $A$, the geometrical parameters of 800- and 400-nm full Poincaré beams are ${\eta }_{\omega 1}=(-\mathrm{1,0},\mathrm{1,0})$ and ${\eta }_{\omega 2}=(\mathrm{0,1},1,\pi )$, respectively. As shown in Figs. 4(c) and 4(d), the HHG simulations demonstrate that the generated EUV fields manifest as Bloch-type skyrmions, whose skyrmion numbers are $Q=1$ ($q=1$, $m=1$) for the 13th harmonic and $Q=-1$ ($q=-1$, $m=1$) for the 14th harmonic. Through changing the positions on the HyOPSP, the topological texture can be adjusted. At the points $B$, the geometrical parameters are given by ${\eta }_{\omega 1}=(-\mathrm{1,0},1,\pi /2)$ and ${\eta }_{\omega 2}=(\mathrm{0,1},\mathrm{1,3}\pi /2)$, and the resulted harmonic field is revealed as the Néel-type skyrmions of $Q=-1$ ($q=-1$, $m=1$) [Fig. 4(e)]. Besides the Bloch-type and Neel-type skyrmions, one can also shape the EUV skyrmions with arbitrary topological textures, such as anti-skyrmion of $Q=1$ ($q=-1$, $m=-1$) [${\eta }_{\omega 1}=(\mathrm{1,0},\mathrm{1,0})$ and ${\eta }_{\omega 2}=(0,-\mathrm{1,1},\pi )$] [Fig. 4(g)] and third-order skyrmion of $Q=-3$ ($q=-1$, $m=3$) [${\eta }_{\omega 1}=(-\mathrm{3,0},\mathrm{1,0})$ and ${\eta }_{\omega 2}=(\mathrm{0,3},1,\pi )$] [Fig. 4(h)]. One can notice that the topology of EUV skyrmions in HHG is closely related with the geometrical parameters of the incident laser. The first two parameters of $\eta$ control the polarity and vorticity of EUV skyrmions, while the last two parameters of $\eta$ influence the skyrmionic texture (such as Néel-type and Bloch-type).

For comparison, if we select two-color full Poincaré beams that do not satisfy the restriction rules of $\eta$, e.g., point $A$ on the 800-nm sphere [${\eta }_{\omega 1}=(-\mathrm{1,0},\mathrm{1,0})$] and point $A^{\prime }$ on the 400-nm sphere [${\eta }_{\omega 2}=(\mathrm{0,1},\mathrm{1,0})$], one can see that the formed Stokes vectors of the EUV field distribute desultorily, and the HHG field is not a skyrmion [Fig. 4(f)]. Overall, the texture of EUV skyrmions can be controlled using the driving two-color full Poincaré beams with qualified geometrical parameters. Such a robust methodology shows its uniqueness in free-space optical skyrmions, which has never been unveiled for the optical skyrmions in guided modes or surface plasmon.