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Abstract
The recent emerging appearance of optical analogs of magnetic quasiparticles, i.e., optical skyrmions constructed via spin, field, and Stokes vectors, has garnered substantial interest from deep-subwavelength imaging and quantum entanglement. Here, we investigate systematically the topological state transitions of skyrmionic beams constructed by the Stokes vectors in the focusing configuration. We theoretically demonstrated that in the weak focusing, the skyrmion topological number is protected. Whereas, in the tight focusing, a unique topological transformation with skyrmion number variation is exhibited for the optical skyrmion, anti-skyrmion, and 2nd-order skyrmion structures. The significant difference between the topological state transitions of these two cases originates from the transformation from the paraxial optical system to the nonparaxial optical system, and the approximate two-dimensional polarization structure to the three-dimensional polarization structure. The results provide new insights into the topological state transitions in topological structures, which promote applications in information processing, data storage, and free-space optical communications.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Skyrmions are topologically protected quasiparticles with a topologically non-trivial spin structure formed at the micro- and nanoscale [1–3]. As a broad and profound topological soliton configuration, it has been predicted and investigated in many areas of physics due to its intriguing physical properties. Therein, the magnetic skyrmions, which have the characteristics of topological stability, small size, can serve as the information storage carriers for the next-generation spintronics [4,5]. In recent years, the concept of topology has been introduced into the field of optics and the skyrmionic textures were constructed by the confined electromagnetic (EM) field vectors of surface plasmon polaritons [6–12], spin vectors of evanescent EM fields [13–18], Poynting vectors [19,20], and pseudospin vectors in nonlinear photonic crystals [21]. The optical skyrmions possess the topological geometry and deep-subwavelength properties, resulting in plentiful interesting applications [22,23], including the picometre metrology [24], magnetic domain detection [25], ultrafast vector imaging [7], optical encryption [26], and topological Hall devices [21], and the topological textures were also formed in other classical fields such as acoustics [27] and water waves [28].
Besides, a new concept to form optical skyrmions has been reported recently, which constructs vector topological structures based on Stokes vectors in paraxial optical systems [29–37]. The skyrmionic textures can be generated by the superposition of two Laguerre-Gaussian (LG) beams associated with two orthogonal circular polarization basises, also known as the skyrmionic beams. Moreover, vector beams with spatially non-uniform polarization distributions, especially under the focusing conditions, have garnered significant attention due to their utilization in research fields such as optical super-resolution imaging [38], optical tweezers [39,40], nanofabrication [41] and quantum entanglement [42]. Previously, the focusing properties of spatial-variant polarized vector beams and high-order Poincare beams and also modulation of optical skyrmions in free space have been investigated intensely [43–48]. However, the topological evolution of skyrmionic beams under a focusing configuration has been rarely investigated.
Here, we investigate systematically the transformation of topological properties of paraxial skyrmionic beams in both weak focusing and tight focusing systems. Diverse topological skyrmionic beams are theoretically simulated and analyzed using the vectorial diffraction theory. The results show that in the weakly focusing process, the topological state evolution of skyrmionic textures is topologically protected. In contrast, skyrmionic beams undergo a topological transformation with their skyrmion number variation under a tight focusing condition. These topological state transitions of paraxial skyrmionic beams originate from the transformation between the paraxial optical system and the nonparaxial optical system. Meanwhile, the canonical momentum proportional to the mean wavevectors contains the non-negligible azimuthal components, resulting in the transformation from the two-dimensional (2D) polarization structures to the three-dimensional (3D) polarization structures. Our research opens up a new avenue for understanding the topological transition in optical system with potential applications in optical information processing and data storage.
2. Theory
The topological invariants of paraxial skyrmionic beams are evaluated by the skyrmion number through stereographic projection. The skyrmion number is defined as [2,23]:
Here, S = (S1, S2, S3)/S0 is the normalized Stokes vector, and σ denotes the integration area of the (x, y) plane. The calculated result n represents the number of times a unit sphere is wrapped. For the ordinary Néel- and Bloch-type skyrmions, their skyrmion numbers are n = 1 [22,23].
Skyrmionic beams are 3D topological vector fields localized in 2D the plane, typically constructed by the superposition of a pair of orthogonally polarized LG modes $\textrm{LG}_p^l$ (l and p are azimuthal and radial indices, respectively) [49]. The skyrmionic beams in the circularly polarized basises can be mathematically described using the following formula [29,32,42]:
The schematic diagram of the focusing of the paraxial skyrmionic beams is shown in Fig. 1. When the numerical aperture (NA) of the focusing lens is relatively small, the focal field can be described by the fractional Fourier transform, and the ordinary Fourier transform of LG mode can be expressed as [50]:
From the expressions (2)-(4), a LG mode is maintained after the Fourier transform except for an additional phase factor, which will lead to the evolution of topological state in skyrmionic beams owing to Gouy phase [29].
On the other hand, when light passes through a high NA objective lens, it is focused in a nonparaxial manner. This means that the vectorial characteristics of the light beam have a significant impact on the distribution of the light field in the focal plane region. As a result, the polarization distribution of the outgoing light differs from that of the incident light. According to the Richards-Wolf vector diffraction theory, tightly focused fields can be expressed as [51]:
3. Results and discussion
In this section, besides the weak focusing system, we will investigate the tight focusing properties of paraxial skyrmionic beams to understanding the topological state transitions in optical system. Theoretically, the process of focusing with low NA is a paraxial approximation, and the longitudinal electric field component of the focal field can be negligible comparing to the transverse electric field component of the focal field. However, in the tight focusing, as elaborated in Section 2, skyrmionic beams are focused in a nonparaxial manner. To obtain the vector properties for the nonparaxial focal beam, the intensity, phase, and polarization properties of the skyrmionic beam must be taken into account. It is anticipated that the skyrmionic beams will maintain its topological geometry in weak focusing system, whereas the topological features will be different under a tight focusing.
In paraxial approximation, we first generate Néel- and Bloch-type skyrmionic beams with a superposition of orthogonally polarized $\textrm{LG}_0^0$ and $\textrm{LG}_0^1$ modes, and it can be treated as the incident field in the focusing system. In this paper, we set the wavelength of the incident beam to be λ = 633 nm, the beam waist radius w0 = 100 µm, and the normalized Stokes vectors ${S_{1,2}}N = \sqrt {S_1^2 + S_2^2} /{S_0}$ and S3N = S3/S0. Figure 2 shows a numerical simulation of Néel-type (ϑ = π/2 and Φ = 0) and Bloch-type (ϑ = π/2 and Φ = π/2) skyrmionic beams. The normalized intensity distributions of spatial modes $\textrm{LG}_0^0$ (top panel) and $\textrm{LG}_0^1$ (bottom panel) and the inset show the corresponding phases, respectively, as shown in Figs. 2(a1) and 2(a2). As for incident Néel- and Bloch-type skyrmionic beams given by Eq. (2), their distributions of normalized intensity, polarization, and skyrmionic textures of Stokes vector fields are exhibited in Figs. 2(b) and 2(c), respectively.
Fig. 2. Illustration of incident Néel- and Bloch-type skyrmionic beams, which can be decomposed as: (a) Normalized intensity distributions of (a1) $\textrm{LG}_0^0$ (top) and (a2) $\textrm{LG}_0^1$ (bottom) modes, where the insets display the corresponding phase. (b) Normalized intensity and polarization distributions overlapped together of incident (b1) Néel-type (ϑ = π/2 and Φ = 0) and (b2) Bloch-type (ϑ = π/2 and Φ = π/2) skyrmionic beams, where the corresponding skyrmionic textures in the Stokes vector field are simulated in (c1) and (c2), respectively. The wavelength of the incident beam is set to λ= 633 nm, with a waist radius of w0 = 100 µm.
In the case of weak focusing, a lower NA causes a smaller deflection of the wavevector during beam propagation. For weakly focused Néel-type skyrmionic beam, the focal field normalized intensity, polarization, and skyrmionic structure are plotted in Figs. 3(a1) and 3(b1). It can be found that the Bloch-type skyrmionic texture (Fig. 3(b1)) is constructed in the focusing field, which is different from the Néel-type skyrmionic texture (Fig. 2(c1)) of the incident field. This indicates that the topological state transition happens after focusing. However, the skyrmion number of the topological texture remains unchanged, as expected. It is noted that the alteration in topological state results from the Gouy phase evolution between two distinct spatial LG modes during transmission [29,49].
Fig. 3. The focusing of Néel- and Bloch-type Skyrmionic beams. Left: (a) Normalized intensity distribution and polarization distribution in the focal plane of incident Néel-type skyrmionic beam (Fig. 2(c1)) after (a1) weak focusing and (a2) tight focusing. Stokes vector textures of (b1) Bloch type and (b2) skyrmionium-like type are simulated in the focal plane, where the top of panel shows the Bloch-type skyrmionic texture from the center area of the skyrmioniumic texture. (e)The radial variations of the normalized Stokes vectors S1,2N and S3N in tightly focused field. Right: (c) Normalized intensity distribution and polarization distribution in the focal plane of incident Bloch-type skyrmionic beam (Fig. 2(c2)) after (c1) weak focusing and (c2) tight focusing. Stokes vector textures of (d1) Néel type and (d2) skyrmionium-like type are simulated in the focal plane, where the top of panel shows the Néel-type skyrmionic texture from the center area of the skyrmioniumic texture. (f) The radial variations of the normalized Stokes vectors S1,2N and S3N in tightly focused field.
Whereas in the case of tight focusing, we obtain a skyrmionium-like texture with skyrmion number n = 0, as depicted in Fig. 3(b2). The top of the panel shows that the central region of this topological texture exhibits a Bloch-type skyrmionic texture. Notably, this skyrmioniumic texture owns a unique radially infinite π-step phase structure theoretically, similar to a Bessel function [52]. It can be illustrated by the radial variations of the normalized S3N in tightly focused field, as shown in Fig. 3(e).
Then, we discuss the evolution of topological state of skyrmionic beam with a Bloch-type texture under focusing configurations. In comparsion with Figs. 2(c2), 3(d1), and 3(d2), the weakly focused skyrmionic beam is also topologically protected, except for evolving from Bloch-type into Néel-type state. However, in tight focusing, there is a skyrmionium-like texture constructed in the Stokes vector field of the focal plane. Therefore, the skyrmion number changes from 1 to zero in the tight focusing configuration, which means that a topological transition occurs.
To understand this topological state transitions, we investigate the momentum properties of skyrmionic beams. NA = 0.9 is used here. It was known that light possesses momentum and angular momentum, which play a key role in understanding light-matter interactions [53,54]. The kinetic momentum density Π proportional to the Poynting vector can be decomposed into the canonical momentum density po and the Belinfante spin momentum density ps as Π = po + ps, where ${{\bf {p}}_o} = {\mathop{\rm Im}\nolimits} [\varepsilon {{\bf{E}}^ \ast } \cdot (\nabla){\bf{E}} + \mu {{\bf{H}}^ \ast} \cdot (\nabla){\bf{H}}]/4\omega = \hbar \bar{{\bf{k}}}$ is associated with the mean wavevector of structured light [52,55]. The canonical momentum density of the tightly focused Néel-type skyrmionic beam is calculated by the vectorial diffraction theory, as shown in Fig. 4. We can see that the axial component of the canonical momentum density dominates in the focal space and the radial component pρ vanishes. However, although the pϕ is small but it is non-negligible. In addition, for the focusing fields calculated by the vectorial diffraction theory, they satisfy the non-paraxial Maxwell’s equations automatically instead of the paraxial approximation.
Fig. 4. The components of canonical momentum density po of incident Néel-type skyrmionic beam after tight focusing: (a) radial pρ, (b) azimuthal pϕ, and (c) z-component pz.
Previous work has shown that an anti-type skyrmion (n = –1) can be constructed using orbital angular momentum Poincaré sphere beam, with a saddle-like texture [56]. Here, we also generate an anti-type skyrmion in the Stokes vector field by modulating the azimuthal index l of LG modes. The first and second columns of Fig. 5 show the normalized intensity and phase distributions of spatial modes $\textrm{LG}_0^0$ and $\textrm{LG}_0^{ - 1}$, respectively. An anti-type Stokes vector texture (ϑ = π/2 and Φ = 0) in the incident field is simulated, as plotted in Fig. 5(f).
Fig. 5. Illustration of incident anti-type (ϑ = π/2 and Φ = 0) skyrmionic beam, which can be decomposed as: Normalized intensity distributions of (a) $\textrm{LG}_0^0$ and (b) $\textrm{LG}_0^{ - 1}$ modes, where the corresponding phase are plotted in (d) and (e), respectively. (c) Normalized intensity distribution and polarization distribution of anti skyrmionic beam. (f) Topological texture of anti-type in a Stokes vector field.
Subsequently, the topological evolution of the incident anti-type skyrmionic beam is also investigated. In both the weak and focusing systems, the normalized intensity and polarization distributions in the focal plane are plotted in Figs. 6(a) and 6(c), respectively. Compared with Figs. 5(f) and 6(b), it can be found that the topological state transitions of skyrmionic texture occurs, while the skyrmion number maintains the same value. In Fig. 6(d), the topological state of skyrmionic texture evolves from the anti-type into the skyrmionium-like type. Moreover, the radial variations of the normalized Stokes vectors S1,2N and S3N in tightly focused field are shown in Fig. 6(e). Clearly, the variation in skyrmion number of anti-type skyrmionic beam implies that it undergoes topological transformations in the tightly focused system.
Fig. 6. The focusing of anti-type skyrmionic beam. The distributions of normalized intensity and polarization in the focal plane of incident anti-type skyrmionic beam (Fig. 5(f)) after (a) weak focusing and (c) tight focusing. Stokes vector textures of (b) anti type and (d) skyrmionium-like type are plotted in the focal plane with the corresponding transverse component distribution inset. (e) Normalized Stokes vectors S1,2N and S3N vary along the radial direction in tightly focused field.
Finally, we further explore the topological properties of more sophisticated skyrmionic beams. For simplicity and without any loss of generality, we consider a higher-order skyrmion with the radial index l = 2 as the incident beam case, which can be decomposed into spatial modes $\textrm{LG}_0^0$ and $\textrm{LG}_0^2$. The first and second columns of Fig. 7 display the normalized intensity and phase information of these modes, respectively. We construct a 2nd-order skyrmionic texture (ϑ = π/2 and Φ = 0) in the incident field, as plotted in Fig. 7(f).
Fig. 7. Illustration of incident higher-order skyrmionic beam (ϑ = π/2 and Φ = 0) with azimuthal index l = 2, which can be decomposed as: Normalized intensity distributions of (a) $\textrm{LG}_0^0$ and (b) $\textrm{LG}_0^2$ modes, where the corresponding phase are plotted in (d) and (e), respectively. (c) The normalized intensity and polarization distributions, and (f) 2nd-order skyrmionic texture in a Stokes vector field with the corresponding transverse component distribution inset.
To further analyze the topological evolution of skyrmioinc beams situation, the distributions of normalized intensity, polarization, and Stokes vector textures were theoretically simulated for both the weakly and tightly focused fields, as illustrated in Figs.8(a)-8(d). The higher-order skyrmionium serves as a generalized extension of skyrmion, whose texture is constructed in the Stokes vector field after tight focusing, as plotted in Fig. 8(d). Our simulations show that a unique topological transformation of skyrmionic texture occurs after focusing through a high NA lens. We should point out that the Stokes vector can exhibit a 2nd-order skyrmioniumic texture in a certain region. However, beyond this region, the Stokes vector no longer exhibits the regular change from a positive to a negative state. Therefore, it cannot show a radially infinite π-step phase structure for a skyrmionium.
Fig. 8. The focusing of 2nd-order skyrmionic beam. Normalized intensity distribution and polarization distribution at the focus plane of incident 2nd-order skyrmionic beam (Fig. 7(f)) after (a) weak focusing and (c) tight focusing. Stokes vector textures of (b) 2nd-order skyrmion (the inset shows the corresponding transverse component distribution) and (d) higher-order skyrmionium-like type (bottom) with a 2nd-order skyrmionic texture (top) from the center area are plotted in the focal plane. (e) Normalized Stokes vectors S1,2N and S3N vary along the radial direction in tightly focused field.
4. Conclusions
In conclusion, we have studied the topological properties of skyrmionic beams in the focusing system, including the topological state transitions of Néel- and Bloch-type skyrmions, anti-skyrmion, and 2nd-order skyrmion in the weak and tight focusing configurations. We observed that the topological states of skyrmionic beams evolve in the weakly focusing process, while the skyrmion number maintains unchanged. Nevertheless, in tight focusing, a unique transformation of topological skyrmion number is shown. We reveal that the significant difference between these two cases is due to the transformation from the paraxial optical system to the nonparaxial optical system, where the azimuthal component of canonical momentum cannot be ignored. This makes the 2D polarization structure is transformed to the 3D polarization structure, and the axial EM field components are non-negligible. Our findings provide an avenue for understanding the topological transitions of optical topological structures in focusing, scattering, and imaging and offer applications in optical communications [57], imaging, and data storage.
Funding
National Natural Science Foundation of China (12174266, 92250304, 61935013, 62075139, 61427819, 61622504); Guangdong Major Project of Basic Research (2020B0301030009); Leadership of Guangdong Province Program (00201505); Science, Technology and Innovation Commission of Shenzhen Municipality (RCJC20200714114435063).
Acknowledgments
This work was supported, in part, by the National Natural Science Foundation of China grants 12174266, 92250304, 61935013, 62075139, 61427819, and 61622504, the Guangdong Major Project of Basic Research grant 2020B0301030009, the Leadership of Guangdong province program grant 00201505, and the Science and Technology Innovation Commission of Shenzhen grants RCJC20200714114435063.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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