Photonics Research, Volume. 11, Issue 12, 2042(2023)

Physical conversion and superposition of optical skyrmion topologies

Houan Teng1, Jinzhan Zhong1,2, Jian Chen1,2, Xinrui Lei1,2, and Qiwen Zhan1,2、*
Author Affiliations
  • 1School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technologyhttps://ror.org/00ay9v204, Shanghai 200093, China
  • 2Zhangjiang Laboratory, Shanghai 201204, China
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    Optical skyrmions are quasiparticles with nontrivial topological textures that have significant potential in optical information processing, transmission, and storage. Here, we theoretically and experimentally achieve the conversion of optical skyrmions among Néel, Bloch, intermediate skyrmions, and bimerons by polarization devices, where the fusion and annihilation of optical skyrmions are demonstrated accordingly. By analyzing the polarization pattern in Poincaré beams, we reveal the skyrmion topology dependence on the device, which provides a pathway for the study of skyrmion interactions. A vectorial optical field generator is implemented to realize the conversion and superposition experimentally, and the results are in good agreement with the theoretical predictions. These results enhance our comprehension of optical topological quasiparticles, which could have a significant impact on the transfer, storage, and communication of optical information.

    1. INTRODUCTION

    Skyrmions are topologically protected quasiparticles originating from high-energy and condensed matter physics [1], and they hold great promise for future applications in spintronics, information processing, and high-density data storage due to their ultracompact size, topology-protected stability, and low current requirements [25]. Recently, the photonic counterpart of magnetic skyrmions was observed in either guided waves or paraxial vector beams. On the one hand, the 2D confinement of light in guided waves provides a smooth domain for skyrmion generation, which can be constructed by electric/magnetic field and spin angular momentum [617], with the topology constrained by the symmetry of a field. On the other hand, skyrmions formed with Stokes parameters corresponding to a spatial variation of a polarization pattern was proposed in paraxial vector beams [1822] or microcavity [2326], where the topological feature is manipulated by the selection of Poincaré beams. The variety of structured vector beams inspires a significant effort to investigate more kinds of optical quasiparticles, such as multiple π-twist target skyrmion, antiskyrmions and bimerons [27,28]. In addition, the electromagnetic fields supertoroidal pulses also exhibit skyrmionic structure [29,30]. The ultrasmall and ultrafast characteristics of optical skyrmions have demonstrated advanced applications in super-resolution imaging, metrology, and light–matter interactions [12,2932].

    While various skyrmionic textures have been demonstrated in optics, the generation and manipulation of Stokes skyrmions rely on digital hologram method assisted by spatial light modulator [19,22,33]. The conversion between different topologies upon propagation and diffraction has not yet been achieved, which is important for skyrmion dynamics and interactions. The wave plate (WP) is a fundamental optical component extensively employed in polarization optics and facilitates the manipulation of polarization states without changing the amplitude of the optical field [34,35]. The Stokes vector provides a valuable representation of polarization states, making the WP an essential tool for skyrmion conversion without altering the skyrmion number.

    In this work, we proposed a method to realize the homogeneous conversion between skyrmions, including Néel, Bloch, intermediate skyrmion types, and bimerons (topological quasiparticle homeomorphic to skyrmions), and explored the superposition of skyrmionic textures. We have theoretically derived the Mueller matrix of devices to achieve the conversion of skyrmions topology, which we identify as a series of WPs. With the conversion between different skyrmionic textures, the skyrmion number remains invariant, manifesting the topological protection. More significantly, we have experimentally verified the conversion by implementing a vectorial optical field generator, which showed good agreement with theoretical predictions. Based on the conversion between skyrmions, the fusion and annihilation of optical skyrmions were demonstrated by their superpositions. This work lays the foundation for exploiting the potential of topological photonics in the fields of optical signal processing and information transmission.

    2. THEORY

    A. Optical Topological Quasiparticle

    Optical topology quasiparticle can be formed by the Stokes vectors of a Poincaré beam composed of a pair of Laguerre–Gaussian (LG) modes with orthogonal polarization states, which can be expressed as [18,36] Ψ=cosc0·LG0,0e0+sinc0·eiϕγLG0,le1,where e0,e1 represent orthogonal polarization states, and LGp,l is the Laguerre–Gaussian mode [37] characterized by radial and azimuthal indices (p,l). The global phase difference between the two orthogonal polarization components is denoted by ϕγ, which controls the helicity textures of topological quasiparticle. c0 is a constant that controls the amplitude ratio of the orthogonal modes.

    Selecting the eigenstates as circular polarization (e0,e1)=(eR,eL), the normalized Stokes vector can be expressed as s=(cosβ(ρ)cosα(ϕ)cosβ(ρ)sinα(ϕ)sinβ(ρ)),where cosβ(ρ)=2ρ1+ρ2, sinβ(ρ)=1ρ21+ρ2, ρ(r)=tanc0·(2rw0)|l|, and α(ϕ)=lϕ+ϕγ.

    The Stokes vector in Eq. (2) describes a mapping to the unit sphere, which represents the swirling structure of a skyrmion [1,2]. The topological property of an optical skyrmion is determined by the skyrmion number, which represents the number of times the Stokes vector wraps around a unit sphere, expressed as Ns=14πs·(sx×sy)dxdy.

    Considering the radial symmetry, the skyrmion number can be calculated through Ns=p·m=12[sinβ(r)]r=0r=rσ·12π[α(ϕ)]ϕ=0ϕ=2π [1,2,38] (see Appendix A for details). The polarity p can be expressed as p=12[sinβ(ρ)]r=0r=rσ, describing the variation of out-of-plane Stokes vector orientation. The vorticity m is expressed in terms of the integer m=12π[α(ϕ)]ϕ=0ϕ=2π describing the in-plane Stokes vector orientation, which is equal to the topological charge l of LG0,l. The offset ϕγ of α(ϕ)=lϕ+ϕγ is the helicity representing the initial phase of Stokes vector orientation.

    The skyrmion topology is determined by the polarity, vorticity, and helicity. The helicity of optical quasiparticle controls the orientation of Stokes vector in the S1S2 plane (skyrmion) or S2S3 plane (bimeron) corresponding to the global phase difference between orthogonal polarization states. The topological quasiparticle is classified as Néel-type with ϕγ=0,π [39], and as Bloch-type with ϕγ=±π/2 [40,41]. The polarity of optical quasiparticle controls the out-of-plane Stokes vector orientation, which corresponds to Stokes vector s3. For p=1 (p=1), the orientation of Stokes vector is downward (upward) at the center and upward (downward) at the boundary. The vorticity determined by l controls the distribution of the transverse Stokes vector orientation. As l=1 antiskyrmions with saddle textures are formed [42]. According to the relationship of the Stokes vector, we can achieve the conversion between optical skyrmions with different helicity, opposite polarity (with simultaneous opposite vorticity), and even bimeron by employing optical WPs, which reveals different mapping from the finite plane (representing the normal skyrmion configuration) to the Poincaré sphere, as illustrated in Fig. 1.

    Schematic diagram of the conversion of optical topological quasiparticle with polarization devices.

    Figure 1.Schematic diagram of the conversion of optical topological quasiparticle with polarization devices.

    B. Conversion of Optical Skyrmion by Optical Devices

    In Section 2.A, we have demonstrated the skyrmion topology dependence on the property of structured vector beam, which enables the conversion of different topological quasiparticles using optical devices, as expressed by the equation sout=M·sin,where sin is the Stokes vector field configuration of the initial topological texture, M is Mueller matrices of optical devices, which can consist of one or more wave plates, and sout is the Stokes vector field configuration of output topological texture. The fundamental theory of the skyrmion conversion is given in Appendix B.

    First, we examine the conversion of skyrmions with different helicity. Skyrmions with different helicities, such as Néel, Bloch, and intermediate types, exhibit distinct orientations of the Stokes vector in the S1S2 plane, which are controlled by the parameter ϕγ, as described in Eq. (2). Without loss of generality, we consider the input type as a Néel skyrmion with ϕγ=0. It has been demonstrated that the Stokes vector of a structured light can be tuned by an optical polarization rotator (PR) [43]. The PR is composed of two crossed quarter-wave plates (QWPs) with fast axes at 0 and π/2 and a retarder with fast axis at π/4 (phase delay of ϕd) positioned between them. The Mueller matrices of the PR are given by MPR(ϕd/2)=MQWP(π/2)MWP(ϕd,π/4)MQWP(0)=[cosϕdsinϕd0sinϕdcosϕd0001].

    By substituting Eq. (5) into Eq. (4), the output Stokes vector field configuration can be written as sout=MPR(ϕd/2)·sin=(cosβ(ρ)cos(lϕ+ϕd)cosβ(ρ)sin(lϕ+ϕd)sinβ(ρ)),demonstrating that a PR with a rotation of ϕd/2 can induce a rotation of ϕd in the helicity of optical skyrmions while keeping the polarity and vorticity unchanged. Therefore, the skyrmion number remains constant. Figure 2(a) shows the input skyrmion of Néel type with l=1 and ϕγ=0, and the three components of Stokes vector are depicted in the inset of Fig. 2. By employing a PR with a rotation of π/4, the spatial configuration of Stokes parameters s1 and s2 performs a clockwise rotation of π/2, while the corresponding Stokes vector undergoes a counterclockwise rotation, resulting in the generation of a Bloch skyrmion with ϕγ=ϕd=π/2, as shown in Fig. 2(b). Thus, the helicity texture of optical skyrmions can be fully controlled by adjusting the phase delay of the retarder of the PR. Furthermore, the utilization of an electrically adjustable retardance as a retarder enables rapid conversion between skyrmion types, thereby presenting potential applications in areas like optical signal processing and information transfer.

    Simulation results for optical skyrmion conversion with polarization devices. (a) Input Stokes vector configuration and transverse component of Néel skyrmion. Output Stokes vector configuration and transverse component of (b) Bloch skyrmion converted by PR, (c) antiskyrmion with p=−1 converted by HWP, and (d) bimeron converted by two QWPs. The Stokes parameters for input skyrmion are shown inside the dashed circle.

    Figure 2.Simulation results for optical skyrmion conversion with polarization devices. (a) Input Stokes vector configuration and transverse component of Néel skyrmion. Output Stokes vector configuration and transverse component of (b) Bloch skyrmion converted by PR, (c) antiskyrmion with p=1 converted by HWP, and (d) bimeron converted by two QWPs. The Stokes parameters for input skyrmion are shown inside the dashed circle.

    The helicity of optical skyrmion can be adjusted with the employment of PR. We further explore the conversion of the polarity of optical skyrmion. The polarity describes the variation of out-of-plane Stokes vector orientation, which is calculated by p=12[sinβ(ρ)]r=0r=rσ=12[1ρ2(r)1+ρ2(r)]r=0r=rσ, where ρ=tanc0·(2rw0)|l| represents the amplitude ratio of left- and right-handed polarized components. Consequently, as the ratio changes from ρ to 1/ρ, the polarity p is converted from 1 to 1. This conversion can be achieved using only a single half-wave plate (HWP) as the initial type is set as a Néel skyrmion with ϕγ=0, resulting in the derived output skyrmion as sout=MHWP(0)·sin=(cos(β)cos(lϕ)cos(β)sin(lϕ)sin(β)).

    It is worth noting that the HWP not only reverses the orientation of the out-of-plane Stokes vector s3 but also changes the in-plane Stokes vector S2, resulting in modulation on the vorticity of skyrmion. The Stokes vector field configuration of the output skyrmion is shown in Fig. 2(c), exhibiting an antiskyrmion. In this case, the polarity and vorticity are reversed, while the skyrmion number remains invariant, manifesting the topological protection of skyrmions.

    Finally, we investigate the conversion between skyrmions and skyrmion-homeomorphic quasiparticles, referred to as “homeomorphic conversion.” By employing two WPs with fast axes aligned at π/4 and π/2, skyrmions can be converted into bimerons. The Mueller matrix for this conversion can be represented as MQQ=MQWP(π/2)MQWP(π/4)=[001100010].

    Thus, the output type of skyrmion can be written as sout=MQQ·sin=(sinβ(ρ)cosβ(ρ)cos(lϕ+ϕγ)cosβ(ρ)sin(lϕ+ϕγ)).

    Equation (9) represents quasiparticles known as bimerons, which are homeomorphic to skyrmions, by applying linear polarization (e0,e1)=(eX,eY) as eigenstates, as described in Eq. (1). According to Eq. (9), the three components of the Stokes vector change from (s1,s2,s3)T to (s3,s1,s2)T, but this does not affect its skyrmion number because it still covers the Poincaré sphere only once. As the initial type is Néel skyrmion with l=1 and ϕγ=0, the field configuration of output Stokes vector is as shown in Fig. 2(d).

    3. EXPERIMENT AND DISCUSSION

    In order to investigate the conversion of optical skyrmions, a vectorial optical field generator [44] was constructed, as depicted in Fig. 3, which is capable of generating optical skyrmions. The setup utilizes two spatial light modulators (SLMs) divided into four sections to precisely control the phase, amplitude, polarization ratio, and retardation of vectorial optical fields [44]. A He–Ne laser beam was employed as the input light source, and the intensity of the Gaussian beam was regulated using a polarizer and an HWP. The beam was subsequently directed through SLM Section 1, where its phase was directly modulated. The phase-modulated beam was then projected onto SLM Section 2, which served as a PR in conjunction with a QWP and polarizer to modulate the beam’s amplitude. The second 4-f imaging system projected the horizontally polarized beam with the desired amplitude and phase onto SLM Section 3, where the polarization ratio was further adjusted using another PR constructed from a QWP and SLM Section 3. Subsequently, this beam traversed SLM Section 4 to regulate retardation by manipulating only the phase of the horizontal polarization component. The optical skyrmion was constructed and then passed through the skyrmion convertor to achieve the conversion. Finally, the conversed optical skyrmion was imaged on the CCD via the last 4-f imaging system. It is worth noting that optical Stokes skyrmions can also be experimentally generated using alternative vectorial optical field generators, such as splitting a single SLM into two parts to independently modulate the complex amplitudes of two orthogonal polarizations [19].

    Schematic diagram of the vectorial optical field generator. SLM, spatial light modulator; SF, spatial filter; HWP, half-wave plate; QWP, quarter-wave plate; P, polarizer; M, mirror; BS, beam splitter; L, lens; C, optical skyrmion converter.

    Figure 3.Schematic diagram of the vectorial optical field generator. SLM, spatial light modulator; SF, spatial filter; HWP, half-wave plate; QWP, quarter-wave plate; P, polarizer; M, mirror; BS, beam splitter; L, lens; C, optical skyrmion converter.

    Our vectorial optical field generator facilitates the generation of Stokes skyrmions with customizable polarity and helicity topology texture, as described by Eq. (1). In order to experimentally reconstruct an optical skyrmion, the Stokes parameters S0,S1,S2, and S3 were obtained by measuring the intensity of various polarization components. The relationship between Stokes parameters and individual polarization component intensities can be expressed as follows: S0=I(0,0)+I(π/2,π/2),S1=I(0,0)I(π/2,π/2),S2=I(π/4,π/4)I(π/4,π/4),S3=I(π/4,0)I(π/4,0),where I(θ1,θ2) represents the optical intensity after passing through a QWP at θ1 and a polarizer at θ2.

    A. Conversion of Optical Skyrmion

    While it is feasible to experimentally generate various types of optical skyrmions by manipulating their polarity, vorticity, and helicity, there is currently a lack of solutions to implement interconversion between different types of skyrmions. Based on the aforementioned investigation, we have theoretically demonstrated that this conversion can be accomplished through the utilization of fundamental WPs called an “optical skyrmion converter.” In our study, we have performed an experimental demonstration showcasing the capability to convert between different types of optical skyrmions. The input in our experiment was an optical field that exhibited a classical Néel skyrmion topology. For the conversion process, we employed a specifically designed optical skyrmion converter comprising QWPs and an HWP, which are elaborated on in Section 2. Subsequently, the output skyrmion was detected and analyzed using a polarization analyzer (PA) consisting of a QWP and a polarizer.

    We have designed an optical skyrmion converter to experimentally achieve the conversion between different types of skyrmions. First, we present the conversion of optical skyrmions with different helicity. By superposing a pair of orthogonally polarized LG modes, we can experimentally create a fundamental Néel skyrmion of p=1, l=1, and ϕγ=0, as shown in Fig. 4(a). To convert the helicity of optical skyrmion, we adjust the rotation angle of PR from 0 to π/4. As a result, the Néel skyrmion with ϕγ=0 evolves into a Bloch skyrmion with ϕγ=π/2 [Fig. 4(b)]. The Stokes vector performs a π/2 rotation counterclockwise around the S3 axis. The Stokes vector S3 remains constant [Fig. 4(b1)], preserving the polarity, while s1 and s2 take a π/2 rotation counterclockwise in S1S2 plane, which does not change the vorticity [Fig. 4(b2)], indicating the topological protection of skyrmionic textures. The invariance of the skyrmion number is further confirmed through numerical integration, as described in Eq. (3), using the experimental results shown in Figs. 4(a) and 4(b). The experimental errors in skyrmion number arise from the limited numerical aperture of the experimental system and system noise fluctuations, especially in low-intensity regions [45]. Note that, if the rotation angle of PR is π, the optical skyrmion will revert back to a Néel skyrmion. In this demonstration, we specifically show the conversion using a PR with a rotation angle of π/4, but the same principle can be applied to achieve arbitrary helicity conversion by using a PR with the corresponding rotation angle.

    Experimental results for optical skyrmion conversion. (a) Input Néel skyrmion and top view below. Output Stokes vector of (b) Bloch skyrmion converted by PR, (c) anti-skyrmion with p=−1 converted by HWP, and (d) bimeron converted by two QWPs with top view of the skyrmionic texture right. Numerically calculated skyrmion numbers are indicated in their respective panels.

    Figure 4.Experimental results for optical skyrmion conversion. (a) Input Néel skyrmion and top view below. Output Stokes vector of (b) Bloch skyrmion converted by PR, (c) anti-skyrmion with p=1 converted by HWP, and (d) bimeron converted by two QWPs with top view of the skyrmionic texture right. Numerically calculated skyrmion numbers are indicated in their respective panels.

    It is important to note that Néel, Bloch, and intermediate skyrmions can naturally evolve into each other due to the different Gouy phase shifts between two orthogonal spatial modes, but this requires a long propagation distance [18,28]. For example, to achieve this conversion between Néel and Bloch types, it requires at least one Rayleigh distance (for a laser with a wavelength of 633 nm and a beam waist radius of 1 mm, this corresponds to approximately 5 m). In our approach, the distance limitation depends on the thickness of the WP combination, which typically remains within 10 cm. Our methods offer the advantages of a short conversion distance, ease of control, and cost-effectiveness, making them relevant for future applications in optical communication and optical information storage.

    To convert the polarity of the optical skyrmion, we utilize an HWP to convert the Néel skyrmion to antiskyrmion with p=1, as depicted in Fig. 4(c). The Stokes vector performs a π clockwise rotation around the S1 axis. As a result, the Stokes vector s1 remains constant, while s2 is reversed [Fig. 4(c2)], leading to a change in vorticity from 1 to 1. Additionally, the Stokes vector s3 undergoes a reversal [Fig. 4(c1)], resulting in a reversal of the polarity. As previously discussed in the theoretical section, the HWP not only reverses the polarity but also the vorticity while keeping the skyrmion number invariant, as shown in Figs. 4(a) and 4(c). By adjusting the rotation angle of PR, we can effectively convert the helicity and obtain similar outcomes to those presented in Fig. 4(b) but with an opposite polarity. It is worth noting that, despite slight deviations in the optical field, the type of skyrmion remains unaffected.

    Skyrmions and bimerons are two types of optical quasiparticles characterized by homeomorphic topological textures. To achieve the homeomorphic conversion, we apply two QWPs with a fast axis at π/4 and π/2 to modulate the Néel skyrmion into a Néel bimeron, as illustrated in Fig. 4(d). In this case, the rotation of Stokes vector is more complicated. The Stokes vector performs a π/2 counterclockwise rotation around the S2 axis, followed by a π/2 counterclockwise rotation around the S1 axis. This rotation operation does not change the fact that the Stokes vector covers the Poincaré sphere once, so the skyrmion number remains unchanged, as confirmed by the experiment’s calculated skyrmion numbers [Figs. 4(a) and 4(d)]. Helicity control can also be applied to the bimeron by replacing the PR with a WP. This adjustment is necessary because the orthogonal state of polarization comprises horizontally and vertically linearly polarized components. The experimental results align well with the corresponding theoretical predictions (see Appendix C for more experimental results).

    B. Superposition of Optical Skyrmion with Different Types

    In this section, we explore the superposition of optical skyrmions with different helicity or polarity, which can be achieved through the conversion of Néel skyrmions. A classical Néel skyrmion [Fig. 5(a)] was constructed by the above experimental setup and then split into two skyrmions (SK1 and SK2) of the same helicity by BS. We also modulate the helicity or polarity of the SK2, as shown in Fig. 5(b). The modulated skyrmions then pass through the next BS to synthesize a co-axis skyrmion (SK3) and superposition occurs.

    Experimental results for the superposition of two optical skyrmions SK1 and SK2. (a) The same input SK1: Néel skyrmion and top view below. (b) Schematic diagram of the superposition of two optical skyrmions. Superposition with same helicity of (c) input SK2 Néel skyrmion and (d) output SK3 Néel skyrmion. Superposition with different helicity of (e) input SK2 Bloch skyrmion and (f) output SK3 intermediate skyrmion. Superposition with different polarity of (g) input SK2 antiskyrmion and (h) output SK3 nonskyrmion.

    Figure 5.Experimental results for the superposition of two optical skyrmions SK1 and SK2. (a) The same input SK1: Néel skyrmion and top view below. (b) Schematic diagram of the superposition of two optical skyrmions. Superposition with same helicity of (c) input SK2 Néel skyrmion and (d) output SK3 Néel skyrmion. Superposition with different helicity of (e) input SK2 Bloch skyrmion and (f) output SK3 intermediate skyrmion. Superposition with different polarity of (g) input SK2 antiskyrmion and (h) output SK3 nonskyrmion.

    First, we investigate two skyrmions with different helicity. The SK1 has a helicity of ϕγ=ϕ0, while the SK2 has a helicity of ϕγ=ϕ0+Δϕ, which was converted by PR according to Eq. (6). It is easy to show that the superposition between two optical skyrmions with arbitrary helicity results in the intermediate skyrmion with ϕγ=ϕ0+Δϕ/2, and the orientation of the outplane changes to s3=2ρ2(1+cosΔϕ)2+ρ2(1+cosΔϕ), which does not affect the type of skyrmion. The superposition of skyrmions with different helicity changes only the helicity, not the polarity and the vorticity. Thus, the skyrmion number remains invariant. More details on the derivation are given in Appendix D. In our experimental investigation, we initially studied the superposition of two skyrmions of the Néel type with ϕγ=0, as shown in Fig. 5(c). The experimental result confirmed that the output was still a Néel skyrmion with ϕγ=0 [see Fig. 5(d)], indicating the successful functioning of our system. Next, we proceeded to superpose Néel and Bloch (ϕγ=π/2) skyrmions [Fig. 5(e)] in our experiment. The result of this superposition was an intermediate state with ϕγ=π/4, as shown in Fig. 5(f). The change in the distribution of s3 is not significant, indicating that the superposition of skyrmions with different helicity does not have a significant effect on the s3 component of the Stokes vector.

    In addition, we also investigated the superposition of two skyrmions with different polarity. Without loss of generality, we set SK1 to be of Néel type, and we convert SK2 to an antiskyrmion using an HWP according to Eq. (7). The Stokes vectors s1 of SK1 and SK2 have the same orientation, while s2 and s3 have opposite orientations, as shown in Fig. 5(g). The superposition of SK1 and SK2 leads to the cancellation of s2 and s3, resulting in the annihilation of the skyrmion, and the formation of a nonskyrmion state with skyrmion number equal to zero, as depicted in Fig. 5(h). More experimental results are presented in detail in Appendix E.

    The superposition of optical skyrmions with different types demonstrates fusion and annihilation, enabling the creation of complex and versatile structures in optical systems. This capability allows for the generation of customized light patterns with unique properties by combining skyrmions with different helicity or polarity, including controlled spin textures or the formation of nonskyrmion states. This opens possibilities for applications in high-dimensional encoding, optical information processing, and advanced imaging techniques.

    4. CONCLUSION

    In conclusion, we have presented a theoretical approach to accomplish the conversion among different types of skyrmionic topological textures by polarization devices. Additionally, the conversion of optical skyrmions among Néel, Bloch, antiskyrmions, and bimerons was experimentally verified by using a series of WPs. Furthermore, we have experimentally demonstrated the superposition of skyrmions with different helicity or polarity based on the proposed approach. The experimental results align with our theoretical predictions, validating the efficacy of our approach. The superposition of optical skyrmions enables the creation of complex and versatile structures in optical systems. Additionally, our scheme can be further extended to incorporate a broader range of photonic quasiparticles by leveraging the diverse designs available for WP cascades. Although the Stokes pseudovector does not adhere to the conventional rules of vector summation, the Stokes skyrmion can be effectively manipulated and transformed between different types using polarization devices. This unique property of the Stokes skyrmion has significant potential for applications in optical information processing, optical computing, and optical storage, where precise control and transformation of its properties are of utmost importance.

    APPENDIX A: DETAIL OF OPTICAL SKYRMION

    In this appendix, we show how to construct the skyrmion by Stokes vector and derive the skyrmion number. Optical skyrmions can be generated by superposing a pair of LG modes with orthogonal polarization states, as described in Eq. (1) of the main text. For the convenience of our analysis, we will temporarily disregard the propagation of beams. By extracting the common factorization, the Eq. (1) can be rewritten as Ψ=C[eR+ρeiα(ϕ)eL],where ρ=tanc0·(2rw0)|l|;α(ϕ)=lϕ+ϕγ;C=cosc0exp(r2w02). It is easy to see from Eq. (A1) that the dimensionless parameter ρ=tanc0·(2rw0)|l| regulates the ratio of the two polarizations, so that the polarization approaches eR, as it gets closer to the axis, and approaches eL, as it moves away from it. To ensure that the significant intensity region of the beam, the width of the beam w, encompasses most polarizations, it is desirable to select a value for c0 such that tanc0 is of the order of unity.

    The skyrmion vector field is constructed by Stokes vector, so we need to calculate the field configuration of Stokes parameters of beam, which can be expressed as S=(|ER|2+|EL|22Re(EREL*)2Im(EREL*)|ER|2|EL|2)=C2(1+ρ22ρcosα(ϕ)2ρsinα(ϕ)1ρ2)=C2(1+ρ2)(12ρ1+ρ2cosα(ϕ)2ρ1+ρ2sinα(ϕ)1ρ21+ρ2).

    According to Pythagorean theorem cosβ(ρ)=2ρ1+ρ2 and sinβ(ρ)=1ρ21+ρ2, the normalized reduced Stokes vector can be given by s=(cosβ(ρ)cosα(ϕ)cosβ(ρ)sinα(ϕ)sinβ(ρ)),where α(ϕ)=lϕ+ϕγ. Equation (A3) describes the Stokes vector configuration of Eq. (1), which corresponds to the skyrmion texture. The topological property of an optical skyrmion is determined by the skyrmion number, as described in Eq. (3) of the main text. Exploiting the radial symmetry, the skyrmion number can be derived as s=14πσn·(nx×ny)dxdy=14π0rσdr02πdϕdβ(r)drdα(ϕ)dϕsinβ(r)=12[sinβ(r)]r=0r=rσ·12π[α(ϕ)]ϕ=0ϕ=2π=p·m,where p=12[sinβ(r)]r=0r=rσ and m=12π[α(ϕ)]ϕ=0ϕ=2π, which represent the vertical and horizontal wrapping times.

    APPENDIX B: FUNDAMENTAL THEORY OF THE SKYRMION CONVERSION

    In this appendix, we will delve into the effects of the WP on skyrmions, providing a detailed analysis of its influence on these optical quasiparticles. The Stokes vector configuration of skyrmion can be represented as (s1,s2,s3)T, allowing us to describe the effect of the WP on the skyrmion using Mueller matrices.

    First, we consider the action of the WP with fast axis horizontal with a phase delay ϕ, and the Mueller matrix can be expressed as MWP(ϕ)=[1000cosϕsinϕ0sinϕcosϕ].

    We observe that the matrix corresponds to a rotation matrix with the S1 direction as the rotation axis, and the rotation angle is determined by the phase delay. However, it is important to note that, if the fast axis is chosen to be the S2 direction instead, the rotation direction will be reversed.

    Then, we consider the action of the WP with fast axis π/4 with a phase delay ϕ, and the Mueller matrix can be written as MWP(ϕ)=[cosϕ0sinϕ010sinϕ0cosϕ].

    We found that this matrix also corresponds to the rotation matrix, except that the axis of rotation has changed from the S1 direction to the S2 direction.

    Finally, we consider the action of the WP with fast axis θ with a phase delay ϕ, and the Mueller matrix is found to be MWP(ϕ)=[cos22θ+cosϕsin22θ(1cosϕ)sin2θcos2θsinϕsin2θ(1cosϕ)sin2θcos2θsin22θ+cosϕcos22θsinϕcos2θsinϕsin2θsinϕcos2θcosϕ].

    This matrix also corresponds to the rotation matrix, except that the axis of rotation has changed from the S1 direction to the 2θ direction. It should be noted that the rotation axis of this rotation matrix is limited to the S1S2 plane. As a result, a single waveplate is unable to achieve rotation along the S3 direction.

    In order to achieve rotation around the S3 direction, we need to combine a series of WPs, e.g., by combining two HWPs [34] or by combining two QWPs and a retarder with a phase shift ϕ [43]. In the following, we demonstrate the latter approach, and the corresponding Mueller matrix can be expressed as M=QWP(0)WP(ϕ,π/4)QWP(π/2)=[100001010][cosϕ0sinϕ010sinϕ0cosϕ][100001010]=[cosϕsinϕ0sinϕcosϕ0001].

    Thus far, we have successfully implemented rotations around the S1,S2, and S3 directions. A WP will induce a uniform rotation in all Stokes vectors of skyrmions. For instance, a rotation along the S3 direction alters the helicity of the skyrmion, a π rotation along the S1 direction results in an antiskyrmion with p=1, and a π/2 rotation along the S1 direction followed by a π/2 rotation along the S2 direction leads to a bimeron.

    APPENDIX C: MORE EXPERIMENTAL RESULTS OF SKYRMION CONVERSION

    In this appendix, we present more detailed experimental results of skyrmion conversion to complement the main text. While the main text focuses on demonstrating three specific transformations of the skyrmion in terms of the Stokes vector, here we provide additional information of skyrmion conversion, including the intensity field distribution, polarization state distribution, and the three components of the Stokes vector, as shown in Fig. 6.

    More experimental results for optical skyrmion conversion. (a) The intensity, state of polarization distribution of input Néel skyrmion, and corresponding Stokes components below. Output intensity, state of polarization distribution of (b) Bloch skyrmion converted by PR, (c) antiskyrmion with p=−1 converted by HWP, and (d) bimeron converted by two QWPs with corresponding Stokes components below.

    Figure 6.More experimental results for optical skyrmion conversion. (a) The intensity, state of polarization distribution of input Néel skyrmion, and corresponding Stokes components below. Output intensity, state of polarization distribution of (b) Bloch skyrmion converted by PR, (c) antiskyrmion with p=1 converted by HWP, and (d) bimeron converted by two QWPs with corresponding Stokes components below.

    APPENDIX D: THEORY OF SUPERPOSITION OF DIFFERENT SKYRMIONS

    In this appendix, we present the detail theory of the superposition of a skyrmion with different types. First, we investigate the superposition between two optical skyrmions (SK1 with ϕγ=ϕ0 and SK2 with ϕγ=ϕ0+Δϕ) with arbitrary helicity, which can be written as SK1=cosc0·LG0,0·eR+sinc0·eiϕ0LG0,1·eL,SK2=cosc0·LG0,0·eR+sinc0·ei(ϕ0+Δϕ)LG0,1·eL.

    These normalized Stokes vectors can be derived as follows: s=(cosβcos(ϕ+ϕ0+Δϕ/2)cosβsin(ϕ+ϕ0+Δϕ/2)sinβ),where cosβ=2ρ(1+cosΔϕ)2+ρ2(1+cosΔϕ) and sinβ=2ρ2(1+cosΔϕ)2+ρ2(1+cosΔϕ).

    Equation (D2) shows that the superposition between two optical skyrmions with arbitrary helicity can also result in the intermediate skyrmion type with ϕγ=ϕ0+Δϕ/2 and slightly changes of the distribution of s3. Figure 7 shows the superposition between Néel- and Bloch-type. The first row shows the superposition between Néel-type-1 with ϕγ=0 and Bloch-type-1 with ϕγ=π/2 and results in the intermediate-type-1 with ϕγ=π/4, as shown in Figs. 7(a)–7(c). The second row shows the superposition between Néel-type-1 with ϕγ=0 and Bloch-type-2 with ϕγ=π/2 and results in the intermediate-type-2 with ϕγ=π/4, as shown in Figs. 7(d)–7(f).

    First row is the Stokes vector distribution, intensity with polarization distributions and transverse component of Stokes vector of the input SK1 Néel-type-1 with ϕγ=0 (a) and SK2 Bloch-type with ϕγ=π/2 (b), and output SK3 intermediate-type with ϕγ=π/4 (c). Second row corresponds to SK1 Néel-type-1 with ϕγ=0 (d) and SK2 Bloch-type with ϕγ=−π/2 (e), and output SK3 intermediate-type with ϕγ=−π/4 (f).

    Figure 7.First row is the Stokes vector distribution, intensity with polarization distributions and transverse component of Stokes vector of the input SK1 Néel-type-1 with ϕγ=0 (a) and SK2 Bloch-type with ϕγ=π/2 (b), and output SK3 intermediate-type with ϕγ=π/4 (c). Second row corresponds to SK1 Néel-type-1 with ϕγ=0 (d) and SK2 Bloch-type with ϕγ=π/2 (e), and output SK3 intermediate-type with ϕγ=π/4 (f).

    First row is the Stokes vector distribution, intensity with polarization distributions and transverse component of Stokes vector of the input SK1 Néel-type-1 with ϕγ=0 (a) and SK2 antitype with p=−1,ϕγ=0 (b), and output SK3 nonskyrmion (c). Second row corresponds to SK1 Néel -type-1 with ϕγ=0 (d) and SK2 antitype-2 with p=−1,ϕγ=π (e), and output SK3 bimeron (f).

    Figure 8.First row is the Stokes vector distribution, intensity with polarization distributions and transverse component of Stokes vector of the input SK1 Néel-type-1 with ϕγ=0 (a) and SK2 antitype with p=1,ϕγ=0 (b), and output SK3 nonskyrmion (c). Second row corresponds to SK1 Néel -type-1 with ϕγ=0 (d) and SK2 antitype-2 with p=1,ϕγ=π (e), and output SK3 bimeron (f).

    As Δϕ=π, we can obtain the topology of a bimeron [see Figs. 8(d)–8(f)], whose normalized Stokes vector distribution can be expressed as s(Δϕ=π)=(sinβcosβcos(ϕ+ϕ0+π/2)cosβsin(ϕ+ϕ0+π/2)),where cosβ=2ρ1+ρ2 and sinβ=1ρ21+ρ2. From the above equations, we can observe that the distribution of the Stokes vector is similar to Eq. (9) in the main text, with the only difference being that s2 and s3 have a negative sign, corresponding to a phase change of π in helicity. Furthermore, using a similar approach, we can deduce that the bimeron has a polarity of 1, vorticity of 1, and helicity of π. The skyrmion number is also equal to 1.

    APPENDIX E: MORE EXPERIMENTAL RESULTS OF SUPERPOSITION OF DIFFERENT SKYRMIONS

    In this appendix, we present more detailed experimental results of Fig. 5 in the main text. Figure 9 shows the intensity field distribution, polarization state distribution, and the three components of the Stokes vector for the superposition between Néel-type-1 and Néel-type-1. Figures 10 and 11 show the superposition between Néel-type-1 and Bloch-type-1 and between Néel-type-1 and antitype-1, respectively.

    More experimental results for superposition of Néel-type-1 and Néel-type-1. (a) Intensity with state of polarization distribution and Stokes components of input SK1 Néel-type-1. (b) corresponds to the input SK2 Néel-type-1. (c) corresponds to the output SK3 Néel-type-1.

    Figure 9.More experimental results for superposition of Néel-type-1 and Néel-type-1. (a) Intensity with state of polarization distribution and Stokes components of input SK1 Néel-type-1. (b) corresponds to the input SK2 Néel-type-1. (c) corresponds to the output SK3 Néel-type-1.

    More experimental results for superposition of Néel-type-1 and Bloch-type-1. (a) Intensity with state of polarization distribution and Stokes components of input SK1 Néel-type-1. (b) corresponds to the input SK2 Bloch-type-1. (c) corresponds to the output SK3 intermediate-type-1.

    Figure 10.More experimental results for superposition of Néel-type-1 and Bloch-type-1. (a) Intensity with state of polarization distribution and Stokes components of input SK1 Néel-type-1. (b) corresponds to the input SK2 Bloch-type-1. (c) corresponds to the output SK3 intermediate-type-1.

    More experimental results for superposition of Néel-type-1 and antitype-1. (a) Intensity with state of polarization distribution and Stokes components of input SK1 Néel-type-1. (b) corresponds to the input SK2 antitype-1. (c) corresponds to the output SK3 nonskyrmion.

    Figure 11.More experimental results for superposition of Néel-type-1 and antitype-1. (a) Intensity with state of polarization distribution and Stokes components of input SK1 Néel-type-1. (b) corresponds to the input SK2 antitype-1. (c) corresponds to the output SK3 nonskyrmion.

    [20] Y. Shen, C. He, Z. Song, B. Chen, H. He, Y. Ma, J. A. J. Fells, S. J. Elston, S. M. Morris, M. J. Booth, A. Forbes. Topologically controlled multiskyrmions in photonic gradient-index lenses. arXiv(2023).

    [37] A. E. Siegman. Lasers(1986).

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    Houan Teng, Jinzhan Zhong, Jian Chen, Xinrui Lei, Qiwen Zhan. Physical conversion and superposition of optical skyrmion topologies[J]. Photonics Research, 2023, 11(12): 2042

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    Paper Information

    Category: Physical Optics

    Received: Jul. 4, 2023

    Accepted: Oct. 2, 2023

    Published Online: Nov. 20, 2023

    The Author Email: Qiwen Zhan (qwzhan@usst.edu.cn)

    DOI:10.1364/PRJ.499485

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