Generation and control of extreme ultraviolet free-space optical skyrmions with high harmonic generation

Abstract. Optical skyrmion serves as a crucial interface between optics and topology. Recently, it has attracted great interest in linear optics. Here, we theoretically introduce a framework for the all-optical generation and control of free-space optical skyrmions in extreme ultraviolet regions via high harmonic generation (HHG). We show that by employing full Poincaré beams, the created extreme ultraviolet fields manifest as skyrmionic structures in Stokes vector fields, whose skyrmion number is relevant to harmonic orders. We reveal that the generation of the skyrmionics structure is attributed to spatial-resolved spin constraint of HHG. Through qualifying the geometrical parameters of full Poincaré beams, the topological texture of extreme ultraviolet fields can be completely manipulated, generating the Bloch-type, Néel-type, anti-type, and higher-order skyrmions. We promote the investigation of topological optics in optical highly nonlinear processes, with potential applications toward ultrafast spintronics with structured light fields.


Introduction
Skyrmions 1 are topological defects in vector fields, that can be created, moved, and annihilated.
They are ubiquitous and important in condensed matter physics 2,3 . Recently, the idea of skyrmions was introduced into the communities of photonics and optics 4 . The optical analogies of skyrmions have found advanced applications in various fields, e.g., topological Hall devices 5 and deepsubwavelength microscopy 6 . Optical skyrmion can be realized when the values of electromagnetic vector fields 4,[6][7][8] or Stokes vector fields [9][10][11] are taken to fully cover a unit sphere (4π) within the skyrmionic boundaries. It was first discovered that one can generate optical skyrmions in the form of surface plasmon polaritons by interacting light with nanostructures 4 . Later, the generation of directly determines the topology of EUV fields.
In HHG, atoms directly interact with the focal field of incident laser. It is important to know the signatures of focal field distribution. We therefore simulate the focal field of CVVB by using Richards-Wolf vectorial diffraction method 25 . We use 20th order Gauss-Legendre integral formula to numerically calculate the integral, and the grid size was chosen as 50 μm in the focal plane with a spacing of 0.05 μm. In our simulation, the numerical aperture is selected to be 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-nm and 400-nm field [up panel in Fig. 1(b)]. One can notice that the focal intensity distributions manifest as Gaussian-like structures, which differ from the typical donutshaped structure of individual phase or polarization singularity. Such exotic distribution largely enriches the highly nonlinear interaction in spatial domain. Furthermore, we calculate the normalized spin angular momentum (SAM) density of focal fields 26,27 . As shown in the bottom Fig. 1 (a) Schematic of generating EUV skyrmions with HHG. A synthesized two-color (800-nm + 400-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 is twice that of 400-nm field, so that the focal distributions of intensities of 800-nm 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 incident plane and focal plane, 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 6-fold spatial symmetry of the two-color synthesized field. panel of Fig. 1(b), spatial-varying polarization is generated. It is interesting to find the presence of circular polarizations in focal plane despite the incident light being linearly polarized throughout.
Here, we emphasize that both the peculiar intensity distribution and SAM distribution of 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-nm 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 to emit circularly polarized EUV light in HHG 30 . Most of the relevant works only involve laser field 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.

Generation of EUV skyrmions with CVVBs
To explore the HHG driven by the synthesized fields, we perform numerical simulation by solving Schrödinger equation for Hydrogen within strong-field approximation (SFA) 33 . The high harmonic field with angular frequency ω was calculated from the Fourier components of dipole moment, P(r, ω) = ω 2 ∫D(r, t)e −iωt dt. Here, the dipole moment depends on both space(r) and time(t), given We insert the focal electric field into the expression of HHG dipole moment and obtain the spatially resolved HHG field of different orders by changing the harmonic frequency ω.
In order to elucidate the advantages of 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 fields are taken to be I = 5×10 14 W/cm 2 , and their temporal envelops are selected to be a sin 2 function of τ = 21.3-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 nth for the three different drivers, respectively. 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 are different, which reveals a Gaussian-like structure for (3n+1)th and (3n−1)th harmonic or annulus structure for 3nth harmonic. Due to the peculiar polarization texture of the driving two-color beam, the 3nth harmonic has significantly lower yield than the (3n±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 twocolor CVVB driver, the SAM distribution of produced harmonics reveals a 6-fold symmetrical distribution [ Fig. 2(i)]. This unique distribution can be attributed to the inherent 6-fold 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 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±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 to the surrounding, leading to a non-trivial real-space topology.
To prove that the created (3n±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 as 34 : Q = (1/4π)•∫∫ρ0S•(∂xS× ∂yS)dxdy, where S(x, y) and ρ  ρ0 denote the vector fields and the defined region for constructing skyrmions in (x, y) plane. The vector fields of a twodimensional skyrmion can be written as cos[β(ρ)]), where ρ and φ are polar coordinate in focal plane, and α and β are spherical coordinates in Poincaré sphere, as shown in Fig. 3(a). In this way, Q reads: 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 ρ = 0 with its confinement ρ = ρ0, and meanwhile it should also cover the unit Poincaré sphere as ρ increases from ρ = 0 to ρ = ρ0.
In our configuration, the focal light field can be approximated as a paraxial beam, and thus we explore the possibility of generating EUV field with a skyrmionic structure in Stokes vector fields, 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 (nω1, nω2), in which nω1 and nω2 denote for the number of photons absorbed from 800-nm and 400-nm CVVBs. Since the driving field is spatially structured, the allowed photon channel is spatially varied. Although, the allowed photon absorption channels cannot be solely determined at the positions in 0 < ρ < ρ0, but the channel is well known at ρ = 0 and ρ = ρ0. 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. Therefore, the local spin states at ρ = 0 and ρ = ρ0 are the same as that of 800-nm photon for 13th harmonic [ Fig. 3(e)]. Similarly, the corresponding photon channel of 14th harmonic is (4,5), and thus the local spin is the same as 400-nm photon at ρ = 0 and ρ = ρ0 [ Fig. 3(f)]. As indicated in Fig. 1(b), the local spin of 800-nm and 400-nm CVVBs are inverse, leading to the 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-nm and 400-nm CVVBs, which is independent of the harmonic order.
Hence, the vorticities are the same for 13th and 14th skyrmions. As a result, the skyrmion numbers of 13th and 14th skyrmions are opposite.

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-nm and 400-nm) beams. To this end, we start by giving an analytical expression for the focal field of full Poincaré beams as where Jℓ stands for the ℓ-order Bessel function of the first kind, and θ is polar angle in the output pupil whose maximum value, θm, is determined by numerical aperture (NA) of lens. Here, it should be noted that for a full Poincaré beam, its topology is largely determined by the phase difference between its two orthogonal components, ex + iey and ex -iey. According to Eq. (3), the sign of Uℓ endows additional phase difference between these two components beyond the initial one, arg(u) −arg(uʹ). It implies that the focusing process itself modulates the topology of the laser field, exerting a significant influence on the topology of EUV skyrmions in HHG.
With Eq. (2), we then propose two empirical restriction rules for the geometrical parameters plane. It requires that the third components of their Stokes vectors (S3,ω1 and S3,ω2) exhibit opposite signs, leading to (ii), the transverse components of Stokes vectors (i.e., S1 and S2) determine the direction of longaxis of the focal electric field ellipse. To create regular spatial polarization distributions for harmonics, S1,ω1 and S2,ω1 need to be the same as S1,ω2 and S2,ω2, respectively. According to Eq.
(2), S1 and S2 are determined by the relative phase between right-and left-handed components, i.e., arg(uU ℓ ) −arg(uʹU ℓʹ ). Thus, the two-color full Poincaré beams are further confined by in which κ = [sign(ℓ)] |ℓ| (sign(ℓ) = 1 for ℓ ≥ 0 and sign(ℓ) = −1 for ℓ < 0) is related to the additional phase introduced by U ℓ , and arg(uʹ) − arg(u) is related with the initial phase of incident field. can be controlled by 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.

Discussion and Conclusion
This work demonstrates the generation and control of EUV skyrmions. On the one hand, the generation of EUV skyrmions roots in the transfer of photon angular momentum between driving fields and high harmonics. From the view of electron motion, the transfer of photon orbital angular momentum is realized by the spatially resolved recombination time of electrons in the interaction region. As for the transfer of photon SAM, it is first recorded by the ionized electron trajectory (or electron orbital angular momentum) with respect to nucleus and then taken by high harmonics at the instant of recombination. It should be noted that the spin-orbit interaction of light is critical in modifying the distribution of photon angular momentum in focal plane, giving rise to the generation of EUV skyrmions in HHG. On the other hand, the control of EUV skyrmions is realized by adjusting the spatial mode of two-color driving fields. In principle, the selection of two-color laser wavelength affects the spectrum of HHG, but it has little influence on the formation and control of EUV skyrmions. Compared with single-color scenario, the two-color scenario has significant advantages. It introduces more degrees of freedom that allows one to finely control the spatial structure of EUV skyrmionic fields. Moreover, two-color fields enable the creation of bright circularly polarized HHG, where the spin states of driving fields can be imprinted onto different orders of harmonics. Hence, one can sculpt the spatial mode of 3-dimensional stokes parameters for emitted EUV fields (Fig. 4c-h). Here, the employment of HyOPSP presents a clear picture for the configuration of two-color full Poincaré, and it can be easily expanded to many fields such as four-wave mixing.
To summarize, we have shown theoretically that HHG makes it possible to generate and manipulate the EUV free-space optical skyrmions. The robust control methodology shows its uniqueness in EUV optical skyrmions, which has never been unveiled for the optical skyrmions in surface plasmon polaritons or guided modes. This work presents an important interface between strong-field physics and topology 36,37 . The generation and control of EUV skyrmions open a door to access the skyrmionic dynamics in the broad ultraviolet region. Magnetic skyrmion Hall effect has been experimentally discovered recently 38 , however, to our knowledge, its optical analog has been only studied in theory 6 . Since the EUV optical skyrmions reveals a regularly spatial distribution of photon spin-orbital state, its highly nonlinear process and ability in ultrahigh spatial resolution may enable the observation of optical skyrmion Hall effect. Looking broadly, the EUV skyrmions endow one laser beam with totally opposite chiral responses when interacting with chiral matters, and thus it could offer new opportunities in the studies of laser-based spatial separation of enantiomers via photoionization. Furthermore, due to the elaborate spatial polarization structure of EUV skyrmion, it also has implications in HHG-induced EUV nanoscale imaging 39 .

Code, Data, and Materials Availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.