Topology is an intrinsic property of the orbital symmetry and elemental spin–orbit interaction, but also, intriguingly, designed vectorial optical fields can break existing symmetries, to impose (dress) topology through coherent interactions with trivial materials. Through photonic spin–orbit interaction, light can transiently turn on topological interactions, such as chiral chemistry, or induce non-Abelian physics in matter. Employing electromagnetic simulations and ultrafast, time-resolved photoemission electron microscopy, we describe the geometric transformation of a normally incident plane wave circularly polarized light carrying a defined spin into surface plasmon polariton field carrying orbital angular momentum which converges into an array of plasmonic vortices with defined spin textures. Numerical simulations show how within each vortex domain, the photonic spin–orbit interaction molds the plasmonic orbital angular momentum into quantum chiral spin angular momentum textures resembling those of a magnetic meron quasiparticles. We experimentally examine the dynamics of such meron plasmonic spin texture lattice by recording the ultrafast nanofemto plasmonic field evolution with deep subwavelength resolution and sub-optical cycle time accuracy from which we extract the linear polarization, L-line singularity distribution, that defines the periodic lattice boundaries. Our results reveal how vectorial optical fields can impress their topologically nontrivial spin textures by coherent dressing or chiral excitations of matter.

## I. INTRODUCTION

Topology is a singular topic in physics, which explores emergent properties of matter that are defined by the geometry of the internal and external interactions, and is impervious to continuous distortions of size or shape.^{1} Topological properties play defining roles in physical systems, ranging from high energy physics, where they are hardly accessible, to solid state physics, where they impart captivating emergent phenomena, and serve as a bridge between the mesoscopic and microscopic, atomic, and even subatomic scales.^{2} Quantum chiral vector field textures describe elementary particles that are postulated in high energy physics, but find more accessible counterparts in superfluid ^{3}He vortices,^{3} chiral nematic liquid crystals,^{4} Bose-Einstein condensates,^{5} quantum Hall states, chiral magnets, and photonic vortices.^{6} Merons, which are the subject of this research, are vector textures, such as those of spin angular momentum (SAM), that wrap a Poincaré polarization vector sphere by half;^{7} they were first derived to represent the spin of electrons in three dimensional space as solutions of a non-Abelian gauge SU(2) theory.^{8} Meron quasiparticles were first discovered in the solid state context in chiral ferromagnetic materials as stable vortex-like topological spin textures on the nanometer scale.^{9,10} Because their nontrivial spin topology is described by a quantized half-integer topological charge and is robust against spin relaxation and randomization,^{11} they have aroused interest in manipulating dissipationless spin currents and for information storage in nonvolatile quantum memory devices^{11} and spin-based electronics.^{1} The formation and evolution of spin texture (e.g., meron) arrays describe the universal physics of topological phase transitions spanning contexts from cosmology to high-T_{c} superconductivity.^{10,12}

Optics describes the physical structuring of vectorial electromagnetic fields^{13} and is, thus, a natural domain for designing topological singularities and exploring their interactions with matter on the nanofemto scale.^{6} Whereas meron spin textures in chiral magnets originate from attractive atomic spin-spin interactions,^{9} meron plasmonic spin textures have been generated by illumination of plasmonic coupling metastructures that design their phase and *k*-vector distributions to perform the engineering of two orthogonal polarization components that define their three-dimensional (3D) Stokes pseudospin vectors on a Poincaré sphere. Such spin textures are generated when an optical wave carrying SAM interacts with a plasmonic coupling structure of a designed geometrical charge to undergo photonic spin–orbit interaction (SOI)^{14,15} of surface plasmon polaritons (SPPs) through the quantum spin-Hall effect of light.^{16,17} Indeed, a variation of these defining parameters led to the discovery of plasmonic topological spin and field textures with an integer topological charge, resembling magnetic skyrmion quasiparticle and their lattice textures^{18–20} by near-field optical methods^{21} and ultrafast nonlinear photoemission electron microscopy.^{22–24} The field textures' sign, however, oscillates at PHz frequencies, whereas the photonic spin textures are stable over the duration of the generation optical field.^{6} The spin pseudovectors are created and controlled on the nanofemto scale by defining their matter dressing on a polarization Poincaré sphere,^{6,13,25} but their component fields propagate at the local speed of light, and thus meron textures exist only on the time scales of their generation fields.

Plasmonic vortices have been investigated by continuous wave (cw) near field microscopic imaging^{21,26} and ultrafast interferometric time-resolved photoemission electron microscopy (ITR-PEEM).^{27–30} It has been established by analytical theory that SPP vortices with defined orbital angular momentum (OAM) undergo photonic SOI to form *single* topological meron^{6} and skyrmion^{21,23} spin textures. With ultrafast pulse generation, these topological spin textures persist on the generating pulse ∼20 femtosecond (fs) timescale and locally break the time-reversal symmetry on the nanometer scale.^{6} Moreover, square plasmonic vortex arrays have also been studied in the cw limit at a gold/air interface to generate spatially alternating spin textures as a potential method to interact with spin polarized electrons at topological material surfaces.^{26} Here, we extend the research by ultrafast, coherent, nonlinear microscopy from the imaging of single localized nontrivial meron plasmonic spin textures to their collective arrays. By ultrafast ITR-PEEM macroscopic imaging, we record a lattice of topological plasmonic SAM textures resembling a lattice of magnetic meron quasiparticles,^{10} which is designed by launching SPP wave packets from a nano-patterned square coupling structure that defines their SOI and interference guiding the formation of a meron-like spin texture (hereafter meron) array. The robustness of the SAM texture over the excitation pulse duration is demonstrated by extracting deep-subwavelength resolution images of linear polarization singularity distributions that define the meron domain boundaries. The designed meron SAM lattice, which carries spatially and temporally distributed chirality, is primed to drive nanofemto interactions with broken space and time inversion symmetry in their near field^{31} and can potentially dress trivial materials to Poincaré engineer transient non-trivial topological states of matter.^{13}

## II. RESULTS

### A. SPP vortex lattice

Figure 1(a) shows a schematic of an ITR-PEEM experimental setup.^{30} Normally incident, circularly polarized light (CPL) pulses of ∼20 fs duration and *λ _{L}* = 550 nm center wavelength illuminate a square SPP coupling structure. The coupling structure supplies momentum that is necessary to couple the optical field into SPP wave packets and therefore, defines their phase and

*k*-vectors;

^{32,33}it is nanofabricated by depositing an ∼100 nm thick Ag film onto an n-type Si(111) substrate by thermal evaporation, followed by etching a square shaped coupling structure by focused ion beam lithography. The coupling structure consists of three concentric square slits separated by the SPP wavelength (

*λ*= 530 nm), such that the SPP fields generated at each slit add in-phase. Figure 1(b) shows a scanning electron micrograph of the fabricated structure, with the slit width and depth of nominal ∼100 nm dimensions, and the inner-most square structure having lateral dimensions of 30

_{SPP}*λ*= 15.9

_{SPP}*μ*m.

*λ*wavelength is chosen because two-photon interaction imparts sufficient energy to excite a nonlinear two-photon photoemission (2PP) process; meanwhile, the signal from the bulk plasmonic photoemission is minimal.

_{L}^{34}Moreover, at this wavelength, the SPP decay is sufficiently slow for their wave packets to propagate beyond the coupling structure diameter.

^{30}

The excitation by a left circularly polarized (LCP) single (pump) pulse train in Fig. 1(c) records a spatially modulated static image of the 2PP signal at the center of the coupling structure. The image shows interferences among fields of the optical excitation and the SPP wave packet it creates. The photoelectron count rates of typically ∼10^{2} electrons per pulse to have a minimal space-charge distortion effect.^{35} A single pump pulse excitation signal is integrated over multiple independent pulse excitations to record the image. The pump pulse excitation image records (1) self-interference with a spatial period of *λ _{SPP}*, where at the structure edges the optical field launches the SPP wave packet and interferes with it,

^{32}and (2) SPP interference

^{6}of the fields that are launched from the four sides with phases defined by the optical CPL field. A π-phase shift of the SPP fields propagating in the opposite directions causes them to interfere with a period of

*λ*/2 at the square structure center.

_{SPP}^{36}This interference of the oppositely propagating, π-phase shifted SPP fields defines the topological character of the vortex array, as will be explained further.

In ITR-PEEM measurements, a pair of identical pump-probe pulses with the delay *τ* varied in 0.1 fs steps excites the sample. The pump-probe excitation introduces a *τ* dependent signal, which captures the interference between the SPP wave packet generated by the pump pulse and the optical field of the probe pulse.^{13} Thereby, the ITR-PEEM method records the vectorial SPP field flow in the sample.^{13} Scanning *τ* with interferometric precision and imaging the generated 2PP signal produces sequences of PEEM images, such as in Fig. 1(d), that have a precise phase relation to the driving field oscillation with a period of 1.83 fs. These images starting from *τ* = 36.2 fs (*τ* = 0 fs is when the pump and probe optical pulses overlap in time; *τ*_{I} = 0 fs is when the pump excited SPP pulse and the optical probe overlap at the vortex core), show how advancing *τ* in 0.2 fs increments modifies the pump SPP and probe optical field interference. Such image sequences are extracted from the ITR-PEEM experimental data by Fourier time filtering of the raw images (Fig. S1) in a pixel-wise fashion and then extracting the *τ* dependent signal by performing inverse Fourier transform of the linear, 1*ω _{L}*, signal component (

*ω*is the oscillation frequency of the laser).

_{L}^{6,13,37}This procedure follows that developed for extracting of coherent responses

^{38,39}from the incoherent ones (e.g., hot electron generation) in interferometric time-resolved two-photon photoemission spectroscopy.

^{40,41}The extracted 1

*ω*images in Fig. 1(d) represent the delay dependent first-order optical-SPP interaction consisting of interference between the probe field and the

_{L}*in-plane*component of the pump-induced SPP field.

^{6,37}Supplementary movie S1 presents thus extracted 1

*ω*SPP field evolution in 0.1 fs steps, and Fig. S2 presents a detailed explanation of the interference imaging.

_{L}The experimental Fourier filtered images in Fig. 1(d) show a sequence where τ is advanced by π optical cycle, starting from where the probe field points along one diagonal, and within ½ cycle points along the same diagonal, but with the opposite sign. The experimental measurement integrates the photoelectron signal over many optical cycles of the pump and probe pulses to obtain time-averaged fields shown in Figs. 1(d) and S2, and only τ is varied to obtain each image. The interference patterns form two-dimensional (2D) square lattice arrays of discrete spots of constructive interferences alternating with intermediate horizontal and vertical one-dimensional (1D) stripe patterns with *λ _{SPP}* periodicity (the details of the interference imaging of the SPP fields are explained in Fig. S2). The evolution of these images follows the gyration of SPP fields of the vortex array, such that their circularly polarized in-plane components interfere with the circulating probe field in a rotating frame. In other words, the probe field and the in-plane SPP field co-gyrate at the same angular frequency, thus maintaining their relative phase difference during the entire pulse illumination. The images evolve from a square lattice at τ of intense/weak photoemission at the vortex cores when their electric fields are parallel/antiparallel, causing the fields to be in-phase/out-of-phase. After τ evolves by π/2, the probe field points along the opposite diagonal, causing the adjacent sites with parallel/antiparallel SPP fields to become bright/dark. After τ evolves by π, as in Fig. 1(d), the lattice units that interfered constructively/destructively at τ now interfere destructively/constructively at τ + π. The 1D horizontal and vertical lines occur for τ + π/4 and τ + 3π/4 delays where the neighboring lines of lattice sites interfere with the vertical and horizontal components of the optical field, respectively.

^{23}

We employ the finite-difference time-domain (FDTD) method to show in Fig. S3 the calculated in-plane SPP field distributions excited from the square lattice by LCP light. The fields transition between the 2D lattice and 1D stripe patterns through gyration of SPP fields about the vortex cores. For the interference patterns to transition from the lattice to stripe patterns, the gyration of the neighboring horizontal and vertical vortex cells must alternate from clockwise (CW) to counterclockwise (CCW) with a spatial period of *λ _{SPP}*/2. Figure 1(e) shows the calculated pump excited SPP and the probe pulse optical field interference for LCP excitation that defines the ITR-PEEM imaging (see details in the supplementary material) as their mutual delay is advanced in steps of 0.2 fs or ∼π/4; this simulation confirms the origin of ITR-PEEM image contrast in Fig. 1(d).

The appearance of the vertical and horizontal lines in Figs. 1(d) and 1(e) is a graphical reminder that the CPL light is composed of linearly polarized fields with a ± π/2 phase shift. It suggests that the CPL excitation could also be performed with orthogonally linearly polarized light pulses with the appropriate phase control.^{42}

To further interpret how the LCP light carrying longitudinal SAM of *σ* = −1 generates the observed contrast, we consider generation of SPP wave packets carrying OAM, and how they undergo SOI and interference to produce a square array of vortices. To proceed, we simulate the SPP Poynting vector (*k _{SPP}*) distribution that describes the SPP energy flow under LCP excitation, which is shown by arrows in Fig. 2(a) for the central cluster of nine vortices. The out-of-plane fields of such a vortex array were imaged previously by near field microscopy for a square coupling structure.

^{26}The vortex cores correspond to points where the out-of-plane fields interfere destructively so that the SPP phase is undefined. In Fig. 2(b), we show the calculated color phase map of the out-of-plane SPP electric field ($Ez$) for the same region as in Fig. 2(a) at an interaction time τ

_{I}= 0 fs when the SPP field reaches the maximum at the center of the square, to define the phase and spatial dispositions of the vortex lattice for the square coupling structure. Evidently, the phase for vortices lying on-diagonal sites circulates in the CCW direction. At each lattice site going horizontally or vertically from the diagonal, the circulation direction alternates between the CCW and CW, as anticipated from the experimental images. Such alternating vortex circulations are consistent with the field gyration in Fig. S3. Within each vortex, the phase winds by 2π, which results in a phase singularity extending between two vertically disposed vortex cores, and after advancing by π/2 between the horizontally disposed ones; this explains the vertical and horizontal 1D contrast in Figs. 1(d) and 1(e), as was also discussed by Spektor

*et al.*

^{26}Because the OAM of plasmonic vortex is defined by the number of 2π phase winding around the vortex core, we can assert that the generated SPP wave packets form a lattice of plasmonic vortices each having a defined OAM of ±1 with a spatial period of

*λ*/2, where the sign depends on whether the field circulation is CW or CCW.

_{SPP}^{43}

Before leaving the discussion of the plasmonic fields, it is important to note that for CPL light there is always the same +π/2 or a −π/2 phase shift between the *x*- and *y*-field components driving the vortex field circulation at each corner of the square lattice in the same sense, such that all the diagonal elements of the lattice also must circulate in the direction of the applied field. This can only happen if the array consists of an odd number of elements in the *x*- and *y*-directions. Thus, the square lattice vortex array resembles a Go game board, where one tiles a square lattice with an odd number of columns and rows with binary (black and white, rather than CW and CCW) tiles, which is topologically distinct from a chess board with an even number of squares. The significance of this distinction will be elaborated in discussion of the topological spin textures.

### B. Lattice of plasmonic meron SAM textures

Evanescent waves such as SPPs^{44} propagate in direction of their *k _{SPP}* wave vectors, and their electric fields oscillate between the transverse and longitudinal polarizations in the meridional plane. This oscillation causes a transverse locking of the SAM pseudovector,

^{17,45}which is given by $S=14\omega Im(\epsilon E*\xd7E+\mu H*\xd7H)$, where $E$ and $H$ are the electric and magnetic field components of the SPP wave, and $\epsilon $ and $\mu $ are the permittivity and permeability of the supporting materials. This transverse spin-momentum locking is a chiral property of the evanescent waves that is referred to as the quantum spin-Hall effect of light.

^{16,46}

Figure 2(c) shows a colormap of the out-of-plane SAM component normalized to its magnitude at each pixel by the local SAM magnitude, along with their normalized in-plane components (arrows) for the same instantaneous time and region as in Fig. 2(b). Note, while the field oscillates sinusoidally at the driving frequency, the spin texture remains constant for the excitation pulse duration. In addition to the out-of-plane plasmonic spin components, the chiral property of SPP also allows a 3D topologically nontrivial spin texture with the magnitude of the normalized in-plane SAM shown in Fig. 2(d), demonstrating a sharp square distribution that forms a boundary between the converging/diverging in-plane SAM pseudovectors in Fig. 2(c). For right circularly polarized (RCP) incident light, the incident photon spin is reversed, and consequently, the calculated SAM components in each square domain are simply reversed (and therefore are not shown). Moreover, Fig. 2(e) plots a 3D representation of the normalized SAM textures within the same central lattice regions in Fig. 2(b), which depicts more graphically the vectorial spin textures associated with the lattice of SPP vortices generated by LCP excitation. Such SAM texture originates from the SOI of the SPP field, which forms the alternating *k _{SPP}* circulation of the vortices. The out-of- and in-plane SAM directions are a consequence of spin-momentum locking and sign of the field circulation.

^{16,17,45}Furthermore, the SAM textures are most symmetrically defined in the central domains, but as amplitudes of the interfering fields with the opposite

*k*become unbalanced, the symmetry lessens further from the central domains.

_{SPP}To evaluate the SAM textures quantitatively, the SPP fields are subdivided into regions where at the center the SAM points into- or out-of- the surface plane, corresponding to the north or south poles of a Poincaré polarization sphere, and at their boundaries between them, the SAM lies in the surface plane, corresponding to points on a Poincaré sphere equator.^{47–49} In addition, we observe repeating square shaped domains that tile Fig. 2(d) with edges defined by yellow lines of the maximum value of the in-plane SAM repeating with a period of *λ _{SPP}*/2. This implies that the square domains are bounded by deep subwavelength scale edges where the SAM lies purely in the

*x-y*plane, and gradually angles orthogonal to them to point in the ±

*z*directions at the vortex cores. Thus, the SAM is not extinguished, but merely rotates into- and out-of- the surface plane. These SAM textures are reminiscent of the ones previously discovered in the magnetic

^{9}and plasmonic merons,

^{6}and therefore we refer to them as meron spin textures forming a lattice where the SAM is modulated with a

*λ*periodicity. These SAM textures form stable domains as long as the optical pulse is applied, as we will show. According to Faraday's law, these circulating SPP vortex fields generate a surface normal magnetic field at cores, which breaks the time-reversal symmetry; the CW/CCW circulation causes the optical magnetization to oscillate spatially with

_{SPP}*λ*periodicity.

_{SPP}^{50}The influence of magnetic polarization leading to Lorentz reciprocity violating optoelectronic processes such as the inverse Faraday effect

^{51,52}has been explained by theory,

^{53,54}invoked in optical magnetization by circularly polarized light,

^{55}and detected experimentally in, for example, gold plasmonic colloids.

^{56–58}

Considering the field and spin textures of the vortex array in more detail, we focus on the S_{z} = 0 lines, which appear as white contrast in Fig. 2(c). These are periodic boundaries of the meron spin textures, which are naturally linked to polarization singularities of the SPP field. Figure 3(a) shows a map of the in-plane SPP electric field [$ESPP(x,y)]$ polarization ellipses for the central square domain in Fig. 2(c). The map is obtained by drawing orbits that the electric field vector traces at each point in the 2D space during one cycle of the vortex gyration (i.e., the carrier frequency). Two characteristics stand out: (1) all points where the polarization becomes linear lie on boundaries of each domain (red dashed rectangle); (2) the isolated points of circular polarization exist at the center of each domain (green dashed circle), as well as at each corner (full green circle; with the opposite phase on each diagonal such that their contributions to the topological charge cancel) of each domain where the vertical and horizontal linearly polarized edges meet. In vectorial optics, these limiting linear polarizations at the vortex edges, and circular polarizations at its center and corners, respectively, are called the L-line and C-point singularities.^{59,60} Figure 3(b) shows the polarization ellipticity distribution of the in-plane SPP field, where large values immediately reveal the L-lines that define the domain boundaries. Thus, the singular point mapping identifies the SAM topology^{6} by their association with geographic points on a Poincaré polarization sphere; specifically, the L-lines fall on the sphere equator, where S_{z} = 0, and C-points, on its north or south poles. Above, we surmised that the square domains are defined by boundaries where the SAM vectors lie in-plane. We confirm this by identifying L-lines as boundaries of the topological spin textures as defined by the Poincaré sphere. Thus, the wrapping of Poincaré spheres by the surface normal S_{z} vector at the poles and S_{x,y} vectors at the equator identifies the vector field as that of a meron.

To quantitatively establish the meron topology within each square shaped domain, we evaluate the quasiparticle topological density based on their SAM vectorial distribution. The topological quasiparticle density is defined as $D=n\xb7(\u2202n\u2202x\xd7\u2202n\u2202y)$, where $n$ is the unit vector that defines the direction of the order parameter and $D$ quantifies its curvature.^{20} Whereas for magnetic meron and skyrmion topological quasiparticles the vector $n=MM$ is defined by the normalized local magnetization ** M** vector, by analogy, for plasmonic spin textures, we take the

*normalized*SAM as the order parameter, $n=S\u2032=SS$.

^{6,23}Figure 3(c) shows the calculated distribution of D at τ

_{I}= 0 fs. Supplementary movie S2 presents the simulated femtosecond time evolution of D. Whereas D evolves spatiotemporally as the SPP lattice is established and eventually wanes, it is a stable texture over the ∼20 fs duration of the excitation pulse. By performing areal integration $N=14\pi \u222cDdxdy$ over the central square region with an effective area of four domains (see details in supplementary material and Fig. S4), we obtain the topological charge for the central square of

*N*= +0.5 for CW and

*N*= −0.5 for CCW CPL excitation. In fact, each square domain forming the lattice has a half-integer value of N confirming that the plasmonic SAM textures correspond to meron quasiparticles, but the sign is opposite in the neighboring vertical and horizontal domains depending on the vortex field circulation. Finally, in Fig. 3(d) we plot

*N*as a function of interaction time relative to τ

_{I}= 0 fs at the square center; this simulation shows that

*N*remains constant over the duration of the SPP wave packet, confirming that the topological spin texture is a robust property that breaks the time-reversal symmetry over the optical illumination time.

One might suspect that the alternating sign of the topological charge between neighboring square domains causes its cancelation for the total array making it trivial. As we already noted, however, the array generated by CPL illumination of a square lattice must have an odd number of domains and therefore cannot be tiled by unit cells of the opposite topological charge that sum to zero. The total topological charge, therefore, must be ±1/2, depending on the photon spin of the optical illumination. Thus, the illumination with CPL light guarantees that the meron lattice possesses a nontrivial topology.

### C. ITR-PEEM imaging of plasmonic meron lattice boundaries

The ITR-PEEM method introduced above directly records the electric field distributions, but not the spin dynamics. Moreover, whereas the spatial distribution of the SPP fields is defined by the wave diffraction,^{61} their spin can be textured on a deep sub-diffraction scale.^{21} Therefore, it is challenging to extract the spin information from the field imaging. A characterization of the ultrafast plasmonic meron spin texture, however, can be performed by analysis of the spatiotemporal evolution of the SPP fields with nanofemto precision. Specifically, we define the spatial evolution of a scalar field, i.e., the optical flow, from the ultrafast 3D (2D space and time dimension) ITR-PEEM movie S1 through the numerical Horn–Schunk (HS) artificial intelligence algorithm.^{62} The HS algorithm evaluates the spatial and temporal pixel wise variation of an image intensity, *I*, which it refers to as the optical flow, and defines it by the chain rule, $\u2202I\u2202rdrdt$, where *r* is a spatial coordinate, and its time derivative is a velocity vector. The HS algorithm thus performs an image motion velocity analysis by minimizing a global energy functional $FHS(\tau )=\u222cIx(\tau )vx(\tau )+\u2009Iy(\tau )vy(\tau )+It(\tau )dxdy$, where $Ix,y,t$ are the pixel-wise spatial and temporal derivatives of the image intensity, and $vx,y$ are the optical flow velocity vectors. ITR-PEEM movies obtained by the Fourier analysis record the time evolution of the in-plane fields, which can be described by $E||\u223ccosxx\u0302+i\sigma \u2009cos(y)y\u0302$,^{26} where $x\u0302$ and $y\u0302$ are unit vectors along *x* and *y* directions. The goal is to obtain $vx,y$, which are parallel to $Ex,y$. Because the SPP field oscillates from the transverse *E _{z}*, to the longitudinal $E||$, where the longitudinal component points in the direction of $vx,y$, it is convenient to perform the HS method on the out-of-plane E

_{z}component, and note that its field is shifted by π/2 with respect to the $E||$ phase while maintaining the same spatiotemporal dynamics as the latter, requiring the obtained $vx,y$, distribution to be shifted by ¼ λ

_{SPP}. The HS analysis thus obtains a sequence of flow velocity distributions ($vx$, $vy$) that align with the local in-plane SPP field directions, from which we extract the ellipticity of the velocity circulation as the degree of the linear polarization. The absolute values of the ellipticity of velocities range from +1 to +∞, representing circular and linear polarizations, respectively. Such analysis captures the linear polarization boundaries that border the meron domains, where the SPP propagation Poynting vectors

^{62–64}[see Fig. 2(a)] are the largest. The regions of high ellipticity as identified by the optical flow thus locate the regions of the linear polarization of SPP fields. Hence, the HS method obtains experimental L-line maps for each advance of τ by one optical cycle, as shown in Fig. 4; it thus takes advantage of the dynamical nanofemto imaging of the SPP fields to locate where S

_{z}∼ 0 and S

_{x}

_{(}

_{y}

_{)}is maximum, with a sub-diffraction limited resolution, based on a single optical cycle field evolution analysis. The HS analysis of the 3D nanofemto interference imaging, thus, validates formation of the meron SAM texture within individual cells of the 2D array. We expect that such imaging of linearly polarized optical or plasmonic fields by the optical flow analysis of interferometric data is broadly applicable to the characterization of vectorial spin textures of topological fields.

Figure 4 illustrates the time-stability of thus defined L-line distributions obtained from experimental ITR-PEEM movies over a span of 16 fs. It confirms the formation of a meron lattice consisting of square shaped domains, each having a dimension of ∼*λ _{SPP}*/2, or 265 nm. Although this lattice and its associated SAM texture are stable over >16 fs timescale, its extraction from the experimental data requires sub-optical cycle (sub-femtosecond) pump-probe delay scanning. The excellent agreement between the experimentally simulated and extracted L-lines in Fig. 3(b) and 4 confirms the formation of a lattice of plasmonic meron spin textures.

Finally, we emphasize that because the L-line singularity boundaries identify where S_{z} changes sign, they can be considered superpositions of the LCP and RCP SPP fields that exist on an atomic or even smaller spatial scale.^{65} In other words, they define “cat” states where the SPP fields could perform quantum computation operations.^{56} In Fig. 5, we show averaged horizontal line profiles of the experimental and calculated L-lines. While the L-lines occur at singular points, their measure, i.e., the polarization ellipticity, has a finite width defined by the image resolutions in space and time. The calculated L-line resolution, therefore, is reported as the full width at half-maximum (FWHM) of the ellipticity peak, is ∼25 nm. For the experimentally extracted L-lines, the average FWHM of the ellipticity is ∼40 nm for L-lines averaged over 7 × 7 domains, close to those obtained from the simulation. This demonstrates that our nanofemto resolved ITR-PEEM imaging enables location of polarization singularities and characterization their spin textures on a deep subwavelength and optical cycle timescale.

## III. DISCUSSION AND CONCLUSIONS

By electromagnetic simulations and ultrafast microscopy, we demonstrate the design of a lattice of meron plasmonic topological spin textures, based on the optical field polarization and SPP coupling structure geometry. Previously employed circular or Archimedean spiral coupling structures with designed geometrical charge for generation of plasmonic vortices can define properties of isolated plasmonic topological skyrmion and meron textures through the SOI of SPPs. We employ a polygonal SPP coupling structure combined with CPL optical excitation, such as has been used previously to generate lattices of SPP vortices,^{26} to create and image a transient lattice of meron topological spin textures. In agreement with the previous study,^{26} we confirm the generation of an array of SPP vortices and characterize them by experiment and theory as belonging to a vectorial SAM array with a meron texture within each element. Specifically, SPPs launched from square slit structures by CPL form square domains consisting of an odd number of meron units in the vertical and horizontal directions with spins alternating in pointing up and down at each vortex core, having topological charges of +1/2 and −1/2, respectively.^{9} The geometrical charge of the coupling structure and the circular polarization of the optical field ensure that the total number of meron units is odd so that the total topological charge of the array is ±1/2, the sign depending on the spin σ of the CPL light. In addition, unlike topological textures composed of plasmonic *fields*, which average to zero over one optical cycle,^{22,24} the reported plasmonic meron *spin* textures are robust on timescale of the applied optical field, which we confirm in Fig. 4 to be over ten optical cycles for our ultrafast pulses, but in principle, could be continuous. The interaction of CPL pulses with the plasmonic coupling structures defines the SOI of plasmonic fields, creating entanglement between the spin and orbital angular momenta of photons and photoelectrons participating in the excitation and imaging process.^{6,66} Such entanglement in meron arrays could find applications in quantum computation applications.^{66}

The nanofemto ITR-PEEM imaging of the SPP dynamics demonstrates that it is possible to image and extract from 3D data the complex spatial polarization distributions on a deep subwavelength scale by recording the delay dependent photoelectron distributions in ITR-PEEM movies.^{13,24} Besides showing the advantage of using massive photoelectrons to image massless photon fields, it also suggests that it might also be possible to achieve deep sub-wavelength resolution in lithographic processes with visible light, by taking advantage of polarization sensitivity of light-matter interactions^{67} as well as the sub-diffraction limited texturing of spin.

The generation of plasmonic vortices by vectorial optical fields and geometric coupling structures offers a rich platform for the design of plasmonic vortex arrays with topological SAM textures and their interactions on the nanofemto scale. Such fields transiently break the time-reversal symmetry in the near field and can be applied to drive charge and spin currents inducing chiral and non-trivial topological phenomena as envisaged in Ref. 26. They open the possibility to entangle spin and orbital angular momenta,^{6,23,66} to generate magnetic field lattices,^{50} to affect electron spin scattering by Aharonov-Bohm effect,^{68,69} to perform dynamical studies of topological phase transitions,^{12,70,71} to excite Majorana modes in chiral superconductors,^{72} to act as nodes for quantum information processing,^{73} and to interact with molecular materials through dipolar coupling.^{33} Finally, in the investigated excitation scheme, we employ effectively plane wave excitation, but one can envisage how the optical field for the SPP excitation could be spatially and temporally modulated to design polarization of the SPP generation fields and thereby influence the spin and orbital entanglement of SPPs for quantum computation applications.^{66} For example, considering CPL light as a linear superposition of vertically and horizontally linearly polarized optical fields interacting with the vertical and horizontal edges of the square coupling structure,^{32,33} it should be possible to switch between the LCP and RCP excitation by varying the phase between the two linear components^{42} and thereby switch the topological charge at each meron domain on a sub optical cycle timescale.^{74} Such excitation could individually address domain-specific topological spin textures for the purpose of multiple quantum-bit operations. Our research, thus, discloses crucial features of plasmonic meron topology, which can unveil new realizations of surface plasmon field and spin properties and their dynamics.

## IV. MATERIALS AND METHODS

### A. Experimental methods

ITR-PEEM is a highly parallel and non-perturbative ultrafast microscopy method for capturing sub-diffraction limited vectorial space-time SPP fields.^{13} It is based on recording spatiotemporal distributions of photoelectrons excited by the impressed optical field at the surface via the second-order nonlinear interaction^{75} that generates photoelectron emission. The 2PP signal from the sample is proportional to the integral $\u222bET4(x,y,z,t)dt$ of the total field where $ETx,y,z,t=ELx,y,z,t+ESPPx,y,z,t$, and $EL$ denotes the incident optical field and $ESPP$ the SPP field it creates. When the excitation is performed with identical pump and probe pulses with interferometrically defined time (phase) delay *τ*, the photoemission signal has contributions where 2PP is excited by the two pulses acting independent of the delay *τ*, and cross terms capturing the interference between the pump excited SPP and the *τ* delayed optical fields.^{30}

Specifically, we image the SPP field dynamics with a *SPECS* aberration corrected low energy electron microscopy/photoemission electron microscopy (AC-LEEM/PEEM) instrument with a specified spatial resolution ∼8 nm in the PEEM mode. The 2PP signal is excited by illumination of the sample with <20 fs laser pulses generated by a non-collinear optical parametric amplifier (NOPA) that is pumped by the third harmonic of an IMPULSE Clark-MXR Yb-doped fiber oscillator/amplifier system operating at a 1 MHz repetition rate. The NOPA produces excitation pulses at *λ _{L}* = 550 nm (2.25 eV), which, together with the SPP field they generate, excite photoelectrons by non-linear 2PP process. The excitation light is transmitted through a Mach–Zehnder interferometer (MZI), to generate identical pump-probe pulse pairs with the delay advanced in 0.1 fs steps under interferometric control.

^{76,77}The excitation field polarization is established by passing the linearly polarized NOPA output through a

*λ*/4 retardation plate. The optical pulse chirp is minimized by setting the number of light reflections from negative dispersion mirrors.

### B. Simulation framework

We simulate the nanofemto evolution of the vectorial SPP fields by the full-vectorial finite-difference time-domain (FDTD) method,^{78,79} which uses the constitutive relations to evolve electromagnetic fields by solving the Maxwell's equations at each point in space. We perform the FDTD simulations for the vacuum/Ag interface with a square coupling structure having 30*λ _{SPP}* long linear dimensions, which is illuminated by normally incident CPL. For

*λ*= 550 nm, the dielectric function of silver is described by a Drude relative permittivity, as implemented in the Lumerical FDTD software based on fitting of experimental dielectric functions from Johnson and Christy.

_{L}^{80}It gives the corresponding SPP wavelength of

*λ*= 530 nm. The HS optical flow analysis is performed with an algorithm from Matlab.

_{SPP}## SUPPLEMENTARY MATERIAL

See the supplementary material for the details of experimental and simulation data analysis. Figures S1–S4, Movies S1 and S2.

## ACKNOWLEDGMENTS

We acknowledge valuable research resources from the University of Pittsburgh Center for Research Computing and Nanoscale Fabrication and Characterization Facility as well as discussions with A. Kubo. This research was supported in part by ONR-MURI-N00014-20-S-F003 grant on Molecular Qubits for Synthetic Electronics, and Ministry of Science and Technology, Taiwan, Grant No. 109-2112-M-007-031-MY3. A.G. thanks the University of Pittsburgh Dietrich School of Arts and Sciences Graduate Fellowship.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no competing interest. Three patent disclosures have been submitted on related research. This information is currently confidential.

### Author Contributions

Conceptualization: C.B.H., Y.D., and H.P.; methodology: H.P., C.B.H., and Y.D.; investigation: A.G., Z.Z., S.Y., and T.W.; simulations: A.G., S.Y., and Y.D.; visualization: A.G., S.Y., and Y.D.; supervision: Y.D., C.B.H., and H.P.; writing—original draft: A.G.; writing—review and editing: Y.D., A.G., S.Y., C.B.H., and H.P.

## DATA AVAILABILITY

All raw data are available upon request. The simulations were performed with a commercial Lumerical code. Further data are available in the supplementary material.