Microcavity-based Generation of Full Poincar\'{e} Beams with Arbitrary Skyrmion Numbers

A full Poincar\'e (FP) beam possesses all possible optical spin states in its cross-section, which constitutes an optical analogue of a skyrmion. Until now, FP beams have been exclusively generated using bulk optics. Here, we propose a generation scheme of an FP beam based on an optical microring cavity. We position two different angular gratings along with chiral lines on a microring cavity and generate an FP beam as a superposition of two light beams with controlled spin and orbital angular momenta. We numerically show that FP beams with tailored skyrmion numbers can be generated from this device, opening a route for developing compact light sources with unique optical spin fields.

Spin angular momentum (SAM; s) and orbital angular momentum (OAM; l) are fundamental parameters characterizing photons [1,2] in terms of the polarization state and a helical wavefront, respectively. Recent progress in the control of SAM and OAM in optical beams has led to the generation of various vector beams with textured optical spin fields [3-6]. Among them, full Poincaré (FP) beams exhibit extremely distinctive photonic spin distributions: an FP beam possesses all possible photonic spin states in its beam cross-section [7]. This unique optical spin field is receiving significant attention with potential for various applications, such as novel polarization sensing [8] and optical tweezers [9,10]. It is known that FP beams can be generated by superposing two light beams with opposite SAM states and different OAM states. Until now, this method has been examined using bulk optics, resulting in the generation of Laguerre-Gaussian [7, 11-13] and Bessel-type [14, 15] FP beams.
A notable property of FP beams is that their optical spin textures constitute skyrmions in two dimensions [16,17]. A skyrmion is a topological elementary excitation characterized by a topological number or a skyrmion number, N sk . Skyrmions have been observed in various condensed-matter systems, such as chiral magnetic materials [18], Bose-Einstein condensates [19], and chiral liquid crystals [20]. Optical skyrmions and their derivatives have been observed in some optical fields, including confined modes in nanophotonic structures [21][22][23][24][25][26] as well as in FP beams [16,27]. In conjunction with the concept of optical skyrmion, FP beams offer an interesting area for exploring rich physics, such as the transfer of topological charges between light and matter [16] and the formation of photonic Möbius strips [28].
Until now, FP beams have been exclusively generated using bulk optics, which is not suitable for practical applications that require robustness and compactness of light sources. For unlocking the potential of FP beams in various applications and physics, it is highly desired to develop FP beam generators on chip.
Here, we propose a scheme for generating an FP beam based on an optical microring cavity. Our approach utilizes a tightly confined whispering-galley mode (WGM), in which the spin-orbit interaction of light provides polarization singularities [29][30][31], enabling SAM-controlled light diffraction transferring the OAM into free space by the tailored perturbation of the structure. We perturb the cavity by patterning two types of angular diffraction gratings [32][33][34]. Accordingly, an FP beam is synthesized as a superposition of two diffracted beams with controlled SAM and OAM in the far field. We analytically show that this method in principle can produce FP beams with arbitrary N sk , and numerically demonstrate the generation of FP beams with various N sk s. Our scheme could facilitate the integration of FP beam sources with different N sk s on a single chip, which may widen the application of FP beams.
An arbitrary polarization state or a spin state of a photon can be described using a normalized Stokes vector composed of three parameters: s = (S 1 , S 2 , S 3 ) /S 0 . Each parameter corresponds to an expectation value of a Pauli matrix for a photonic spin state [35]. s denotes the orientation of a photonic spin, which can be visualized by a vector arrow drawn in a unit Poincaré sphere, as shown in Fig. 1(a). Figure 1(b) schematically depicts a cross-sectional Stokes vector field of an example FP beam and its projection to the surface of a unit Poincaré sphere. An FP beam possesses any possible spin states in the cross-section, and hence, projecting them to the surface completely wraps the sphere. The topological property or the order of an FP beam is characterized by a skyrmion number that counts the number of times the photonic spins in a certain area (A) wrap a unit Poincaré  1. (a) Unit Poincaré sphere defined in a space spanned by the normalized Stokes parameters, which represent the degree of linear polarization (s1), diagonal polarization (s2), and circular polarization (s3), respectively. A vector from the origin to a point on the sphere surface defines a Stokes vector and indicates the orientation of the optical spin. (b) Schematic of a cross-sectional Stokes vector field of an example FP beam and its projection to a unit Poincaré sphere.
sphere, which is expressed as For an FP beam generated by superposing two beams with opposite SAMs (s 2 = −s 1 ) and different absolute OAMs (|l 2 | = |l 1 |), Eq. (1) yields the OAM difference, i.e., N sk = (l 2 − l 1 ) = ∆l. This expression is obtained by integrating Eq. (1) for a domain A where the spin state of the superposed beam flips from s 1 to −s 1 (derivation can be found in Supplemental Material (SM)). Therefore, FP beams with any skyrmion numbers can be generated by controlling the OAMs of the beams under superposition [11,12]. The capability of generating skyrmions with N sk s much larger than 1 could be of interest, because the extensively investigated magnetic skyrmions frequently exhibit N sk values of only 1 or 2 [36, 37]. Now, we discuss our scheme of the microcavity-based generation of FP beams. We consider a microring cavity supporting a WGM rotating within the cavity. The tight spatial confinement for the propagating mode induces the spin-orbit interaction of light [29][30][31], and thus leads to the coupling between th e spin and orbit degrees of freedom within the mode. Consequently, the WGM can be described by a superposition of spin-up and spindown components ψ ± with angular momentum states of |s, l = |±1, m ∓ 1 , where m is the azimuthal order of the WGM and yields the total angular momentum on multiplication with . Figure 2(a) schematically presents the distribution of the net spin density (S 3 = |ψ + | 2 − |ψ − | 2 ) of a counterclockwise transverse electric (TE)-like WGM in a microring cavity. The spatial profiles of the spin-up and spin-down modes differ, resulting in the emergence of purely spin polarized lines, known as C-lines [38-40]. We utilize this phenomenon for achieving SAM-controlled light scattering from the WGM. We apply a small refractive index perturbation aligned to a C-line, which selectively scatters circularly polarized photons, with the handedness depending on the polarity of the C-line, as schematically shown in the right inset in Fig. 2(a).
By arranging a circular, periodic array of index perturbations, we can control the OAM of the scattered light according to the angular momentum conservation law expressed as (m−s)−l = ng [32, 33]. Here, m−s is the effective OAM of the WGM that the angular grating feels, l is the OAM of the diffracted light, n is the diffraction order, and g is the number of grating elements. Hereafter, we focus on the first-order diffraction of n = 1. Figures 2(b) and (c) show example behaviors of the microcavity with gratings. When an angular grating with g = m−1 is patterned on the C-line of s = +1, a light beam described by |s, l = |+1, 0 will be generated ( Fig. 2(b)). Meanwhile, positioning a grating of g = m on the C-line of s = −1 generates a beam with |s, l = |−1, +1 (Fig. 2(c)).
By arranging the above two gratings in parallel, an FP beam can be synthesized in far field as a superposition of the light beams diffracted by the two grati ngs. The farfield amplitude profile, A(θ, φ), of an FP beam generated by our scheme can be analytically approximated by where J |l| is the |l|th-order Bessel function of the first kind, k is the wavenumber, d 0 is the radius of the circle where the angular grating is patterned, φ is the azimuth angle, and θ is the elevation angle (θ = 0 corresponds to the direction normal to the device plane). This analytical expression was derived with a toy model that assumes each grating element acts as a small electric dipole (see SM and Refs. [41,42]). As a specific design of an FP beam generator, we consider a silicon microring cavity with a radius of 3 µm. The waveguide of the cavity has a width of 450 nm and a height of 200 nm, confining a single TE mode at the telecommunication C-band. The refractive indices of silicon and the background environment were set to 3.4 and 1.0, respectively. The ring cavity supports a WGM of an azimuthal order of 24 around a wavelength of 1.57 µm with a large free spectral range of ≈ 30 nm. This mode has a high Q factor of ≈ 3×10 8 without any index perturbation. We focus on the counterclockwise WGM and analyze it using finite-difference time-domain (FDTD) sim- Schematic showing a photonic spin distribution for a counterclockwise WGM in a micoring cavity. The region with high positive (negative) spin density is indicated by red (blue). There are two singular lines with pure spin polarization of s = +1 or −1, called as C-lines. The inset shows a zoomed image of the microring, presenting light scattering due to the refractive index perturbations on the C-lines. The scattered light is circularly polarized, and its polarity corresponds to that of the perturbed C-line. ulations. A calculation domain of 3.5 × 3.5 × 1.0 µm 3 and a grid size of 10 × 10 × 10 nm 3 were employed in the simulation. We selectively excited the mode in the simulator and continued the computation until the stationary state was reached. Figure 3(a) shows a computed mode profile of the unperturbed micoring cavity. We found Clines of s = +1 and s = −1 at the positions respectively deviated by −0.14 µm and +0.125 µm from the waveguide center in the radial direction. Along each C-line, we arranged an array of air holes with a radius and a depth of both 40 nm. The far fields radiated from the structure were calculated from the obtained near fields using the near-to-far field conversion technique.
First, we designed an FP beam generator emitting a beam with N sk = +1. We patterned 23 holes on the Cline of s = +1 and 24 holes on the C-line of s = −1, so that the beams in the states of |s, l = |+1, 0 and |s, l = |−1, +1 will be respectively diffracted and superposed. Figure 3(b) presents the intensity profile of the calculated far field within an elevation angle of θ ≤ 30 • projected onto a flat plane after collimating the beam virtually. The red dashed line in Fig. 3 The values of θ =13 • and 30 • approximately correspond to the first and second minimums in the intensity profile of the light diffracted only by the angular grating of s = +1 (see SM). Thus, the directions of the spins are expected to be aligned downward (s ≈ −1) at these angles. Figure 3(c) shows the spatial distribution of the optical spin field within θ ≤ 13 • . Spins are directed upward near the center, rotate as changing azimuth angle, and gradually flip their direction to downward as the domain edge is approached. Figure 3(d) plots a projection of the spin texture to a unit Poincaré sphere. The full coverage of the surface demonstrates that the generated beam is indeed an FP beam. The projection map suggests that the observed optical spin field forms a Blochtype skyrmion. We note that other types of skyrmions, such as Néel-type one, can be generated by rotating one of the angular gratings with respect to the other (not shown).
Subsequently, we calculated N sk according to Eq. (1). The integration was performed from the center to a particular θ and for all φ. Figure 3(e) displays the θ dependence of N sk . At θ ≈ 13 • , N sk reaches a near unity value of 0.98, which compares well with the designed value of N sk = +1. The remaining error of 0.02 may be caused by the imperfect circular polarization of the scattered light, resulting from the finite size of the scatter and/or from the multiple reflection of the scattered light before exiting the structure. We also analytically deduced values of N sk using our toy model and overlaid the result on the plot in Fig. 3(e). The analytical curve explains well the FDTD results, in particular when θ is small, in which case the toy model can be regarded as a good approximation. For both the FDTD and analytical results, we observed oscillations in N sk for θ > 13 • . The maxima and minima of the oscillations correspond to the point where the intensity of one of the two scattered beams under superposition approaches zero. Analytically, N sk should oscillate between 0 and the designed N sk , leading to the generation of a skyrmion and an antiskyrmion alternately. This skyrmionic structure is known as a skyrmion multiplex [43], the simplest case of which is known as skyrmionium. Another skyrmion state with N sk = 1/2, called as half skyrmion or meron [44], could also be produced by spatially filtering the beam by a diaphragm passing only θ < 8 • .
Next, we synthesized an FP beam with N sk = −1 or an antiskyrmionic beam. For this, g of the grating on the Cline of s = −1 was changed from 24 to 26, so that a beam in a state of |s, l = |−1, −1 will be radiated from the grating.  an antivortex behavior, as expected. The calculated N sk reaches −1.07 at θ = 13 • , as presented in Fig. 4(c). These results demonstrate that an antiskyrmionic FP beam can also be generated within our scheme. Finally, we present an example device generating an FP beam with a large N sk of −5. For this demonstration, g of the grating on the C-line of s = −1 was increased to 30, which radiatesa beam in the state of |s, l = |−1, −5 . Figures 4(d) and (e) show the computed far-field intensity profile and the corresponding spin field, respectively. From the latter plot, it can be observed that the optical spins wind five times around the field center. max. diffracted beam in the state of |s, l = |−1, −5 close to θ = 13 • . The weak signal is obsecured by the background noise in the simulator, and thus, yields an unpredictable result. Here, the large l of the beam induces the weak signal at small θ values. More robust generation of FP beams with high N sk s could be possible by engineering the size and/or shape of the microring and the pattern of the far fields.
In summary, we demonstrated FP beam generation from a microring cavity. We augmented the microring by angular gratings patterned on the C-lines, which diffracted SAM-controlled beams with different OAMs. We showed that the superposition of the beams from the microcavity results in an FP beam with an arbitrary controllable N sk . We examined concrete designs of siliconbased microring cavities and verified the generation of FP beams with various N sk s. We note that the clarified Bessel-like beam patterns of the generated FP beams may lead to their self-healing propagation [45][46][47]. We believe that the compact FP beam generators could find broad applications in optical communication and optical sensing and play a significant role in exploring condensed matter physics.

S-I. A TOY MODEL FOR DESCRIBING DIFFRACTION PATTERNS
We develop a toy model consisting of a circular array of radiative dipoles for analytically deriving diffraction patterns of angular diffraction gratings interacting with a whisperinggalley mode (WGM). We consider infinitesimal dipoles, the respective polarization of which emulates the local optical polarization of the WGM. We assumed that other scattering mechanisms occurring in the microring cavity are negligible and solely considered the radiation from the dipoles located equidistantly. The radiation from one of the dipoles can be written using a well-known approximated form given by where r 0 is the position of the dipole and k is the wavenumber. The source dipole moment is replaced by a local electric field vector of the WGM, E WGM (r 0 ), so as to express the scattered radiation of the mode by a grating element at r 0 . In far-field region (kr 1 and r r 0 ), Eq. (S1) can be further approximated to a spherical wave: In the last equal of this equation, we decomposed the far field into three components. The first factor, P far,single (φ, θ) = r × [r × E WGM ] /r 2 , represents the vectorial nature of the field.
The set of the cross products, r × r×, projects the polarization of the source dipole field (which equals E WGM ) to the plane normal to the wavevector (or to the vector r). The second factor, R(r) = e ikr /r, expresses the propagation of a scalar spherical wave. The third factor,Ẽ far,single (φ, θ) = exp [−ikr · r 0 /r], is also a scalar part of the far field. The last two factors merely describe the field intensity distribution in the far field and do not scramble the light polarization. Therefore, the polarization distribution can be solely represented by P far,single (φ, θ). We evaluate the vector part or the light polarization on a coordinate described by  S-2 E X and E Y components correspond to the light polarization in the collimated far field.
Then, P far,single can be expressed as The second term in the right hand side becomes negligible as far as both E z and θ are small. These conditions match with our case, where light scattering from a TE mode (E z ≈ 0) is considered mainly for θ of no more than 30 • . We note that, even at θ = 30 • , radiation from a circular dipole exhibit a high degree of circular polarization of |s 3 | = Now, we discuss far field created by an array of the scattering elements. Considering the phase difference of radiation among the grating elements, the superposed radiation field can be expressed as Here g is the number of grating elements.
This equation takes a form of inverse discrete Fourier transform. We consider the limit of infinite g and obtain a continuous Fourier transform given bỹ By replacing t → e iξ , we get a Fourier series which yields 1 2π into Eq. (S7), we finally obtaiñ (S11) This equation gives a good approximation to Eq. (S6) when the "spatial sampling rate" or g is large and when the "spatial frequency" or θ is small. These conditions also meet with the situation discussed in the main text, where the first order diffraction is treated and the dominant radiation is concentrated to the region with small θ as far as m ≈ g.

S-II. DETAILED FDTD RESULTS
In this section, we discuss more details of the results computed by the finite-difference time-domain (FDTD) method. Figures S1(a) and (b) respectively show the calculated intensity distribution and photonic spin density distribution ( of a WGM without gratings. Here, we computed a TE-like counterclockwise-rotating WGM with the azimuthal order of 24. The intensity distribution is nearly constant in the azimuthal direction but with slight modulation. The wavy pattern is induced by the intrusion of the counter-rotating WGM, which was weakly excited in the simulator due to the imperfect mode selection. The similar distribution can be found in the spin density distribution, which shows two wavy lines with mutually opposite polarity. The spin-up (-down) region is found near the inner (outer) edge of the microring, which is much clearly displayed in the inset of We computed far fields based on the near-to-far field conversion method, using near fields recorded at a plane just above the microring cavity. Figures S2(a-d) show computed farfield patters for a perturbed ring cavity with a 23-element grating on the C-line of s = +1,  which is designed to diffract a beam in a state of |s, l = |+1, 0 . The intensity profile plotted in Fig. S2(a) exhibits a circularly-symmetric beam pattern. Figure S2(b) shows a phase profile of the diffracted beam. In the azimuth direction, the phase is almost constant, which corresponds to an OAM state of l = 0. Figure S2 where the spin state is not pure spin-up anymore, occur in the places where the far-field intensity itself is very weak. We consider that, at these regions in the far field, unspecified background noise in the simulator causes the unexpected spin flip. Figure S2

S-III. ANALYTICAL SKYRMION NUMBER FOR AN FP BEAM
Here, we analytically derive an expression for a skyrmion number of an FP beam generated by superposing two vortex beams with different OAM and opposite SAM values. We use a Jones vector |s 1 for expressing a polarization state. |s 1 is defined in a x-y plane and can be fully described with two parameters: an amplitude ratio (tan α) and an phase S-7 difference (δ) between the x and y electric field components. Then, |s 1 is expressed as The opposite (or orthogonal) polarization state |s 2 is written as which satisfies s 2 | s 1 = 0. The polarization state |s 1 can also be described using a threecomponent Stokes vector and the Stokes vector for the orthogonal state is s 2 = −s 1 . Meanwhile, the other angular momentum, orbital angular momentum (OAM), can be characterized by a spiral wavefront.
It is known that the mode cross section of an optical beam carrying a well-defined OAM is circularly symmetric in general, and the spiral wavefront can be written as where r is the radial distance from the beam center, φ is the azimuth angle, A is the amplitude, l is the order of OAM, and γ is the phase offset.

S-IV. DETAILED FAR-FIELD PROFILE OF AN FP BEAM
In this section, we take a closer look to the field pattern of an FP beam. We consider an FP beam with N sk = +1, which is synthesized from two beams in the angular momentum states of |s, l = |+1, 0 and |−1, +1 . Each beam in the different angular momentum state can be independently generated by solely placing a single angular grating on the microring.
The far-field pattern of such a beam generated by the single grating can be described by a Bessel function, as derived in a previous section. Figure S3 antiskyrmions appear alternately is knows as skyrmion multiplex [S3]. In the skyrmionic structure, one can find a variety of skyrmionic structures, such as skyrmionium, skyrmion S-10 and meron. These states may be obtained by spatially filtering the FP beam carrying the skyrmion multiplex. Figure S3(b) shows the same set of results calculated using the FDTD method. Overall, the FDTD simulations reproduce the results obtained by the toy model in particular when θ is small. One noticeable deviation can find in the behavior of N sk , which does not return to zero when varying θ. This is due to the imperfection in the spin flips at the zero field intensity points. In the case of FDTD simulations, unspecified background radiation scrambles the spin state where the diffracted signal is weak. The behavior of N sk also significantly deviates when θ is large, where the approximation used in the toy model does not stand. The use of a microring with a larger radius would be helpful for focusing the beam power to the region with small θ and thereby for generating more ideal FP beams.