Method for synchronous and independent control of axial intensity and trajectory of stokes vector skyrmion

Abstract

This invention relates to the field of vector light field manipulation technology, and provides a method for synchronously and independently controlling the axial intensity and trajectory of a Stokes vector skyrmion. The method includes: calculating the required zero-order and higher-order angular spectrum functions based on a preset target axial intensity distribution and a three-dimensional propagation trajectory; then, performing Fourier transforms on these two sets of angular spectra using two lenses with the same focal length, thereby generating corresponding zero-order and higher-order Bessel beams at their respective back focal planes; subsequently, assigning right-handed and left-handed circular polarization to the customized zero-order and higher-order Bessel beams, respectively, and coherently superimposing them using a polarization beam splitter; finally, a Stokes vector skyrmion with a specific intensity distribution and a preset trajectory is generated. This invention enables the synchronous and independent control of the axial intensity and trajectory of a Stokes vector skyrmion, allowing for customization of the intensity and trajectory of the Stokes vector skyrmion as needed.

Description

Technical Field

[0001] This invention relates to the field of vector light field manipulation technology, and in particular to a method for synchronous and independent control of the axial intensity and trajectory of a Stokes vector skyrmion. Background Technology

[0002] Skyrmions were first proposed in 1961 by British physicist Skyrme as topological solitons in classical nonlinear field theory to explain the interaction between mesons and baryons. Subsequently, they attracted widespread attention in various fields such as liquid crystal materials, Bose-Einstein condensates, and quantum Hall systems. In 2009, Mühlbauer, through the small-angle neutron scattering (SANS) experiment, first verified the existence of topologically stable magnetic skyrmions in cubic chiral magnets. The unique topological properties of magnetic skyrmions endow them with a series of significant advantages, such as small size, low operating energy consumption, high storage density, and topological stability. This makes magnetic skyrmions of great value in information storage, logic operations, neuromorphic computing, and other fields, and opens up new directions for future technological development.

[0003] In 2018, as counterparts to magnetic skyrmions, S. Tsesses's team and Yuan Xiaocong's team respectively realized Néel-type skyrmions in evanescent electric fields using surface plasmon interference. In recent years, various optical skyrmions based on three-dimensional vector fields (including electromagnetic field vectors, spin vectors, Stokes vectors, Poynting vectors, and pseudo-spin vectors) have been discovered. Notably, Stokes vector skyrmions generated by superimposing a pair of orthogonal polarization modes have received widespread attention in recent studies. However, most skyrmions generated in free space can only propagate in straight lines, and the trajectory of self-accelerated skyrmions generated by superimposing Airy beams is still limited to a parabolic path.

[0004] Controlling the axial intensity variation and arbitrary trajectory of Stokes vector skyrmions is crucial for particle manipulation and information storage. Achieving axial intensity control and propagation along arbitrary trajectories is essential in these fields. However, most optical skyrmions generated in current research are limited to linear or parabolic propagation. Even in parabolic propagation, this relies solely on the inherent self-acceleration properties of Airy beams, fundamentally limiting the possibility of achieving arbitrary propagation paths. Furthermore, while skyrmions exhibit inherent anti-interference capabilities in perturbed and complex environments due to their topologically protected stable configuration, their intensity tends to weaken in lossy media due to scattering or absorption. Therefore, modulating the intensity distribution of skyrmions along the propagation direction is critical, but this problem remains unsolved. Thus, how to synchronously and independently control the intensity distribution and propagation trajectory of Stokes vector skyrmions still faces numerous challenges. Summary of the Invention

[0005] This invention primarily addresses the key technical problems in the prior art where Stokes vector skyrmions cannot propagate along arbitrary curved trajectories and where the axial (propagation direction) intensity distribution is difficult to control. It proposes a method for synchronously and independently controlling the axial intensity and trajectory of Stokes vector skyrmions, thereby achieving synchronous and independent control of the axial intensity and trajectory of Stokes vector skyrmions. The intensity and trajectory of Stokes vector skyrmions can be customized as needed, and the type of Stokes vector skyrmions can be arbitrarily adjusted.

[0006] This invention provides a method for synchronously and independently controlling the axial strength and trajectory of a Stokes vector skyrmion, comprising the following steps:

[0007] Step S1: Set beam control parameters; the beam control parameters include the axial intensity distribution function I(z) and the propagation trajectory function s(z) of the main lobe of the beam;

[0008] Step S2: Calculate any l-order angular spectrum function that satisfies the preset axial intensity distribution and propagation trajectory based on the beam control parameters;

[0009] Step S3: Use two identical lenses to examine the zero-order angular spectral function. and higher-order angular spectrum functions By performing a Fourier transform, zero-order and higher-order Bessel beams with axial intensity and propagation trajectory consistent with the set beam control parameters can be obtained.

[0010] Step S4: The zero-order Bessel beam and the higher-order Bessel beam are respectively assigned right-hand circular polarization and left-hand circular polarization. The customized zero-order right-hand circularly polarized Bessel beam and the higher-order left-hand circularly polarized Bessel beam are coherently superimposed by a polarization beam splitter. The phase difference between the two beams and the order of the higher-order angular spectrum are controlled to generate Stokes vector skyrmions with specific intensity distribution and propagation trajectory.

[0011] Furthermore, in step S1, the axial intensity distribution function I(z) is the light intensity distribution function at position z on the propagation axis of the main beam, and s(z) = [f(z), g(z)] is the displacement of the energy center of the main beam in a plane perpendicular to the propagation direction;

[0012] By controlling the axial intensity distribution function I(z) and the propagation trajectory function s(z) of the main lobe of the beam, Stokes vector skyrmion beams with different axial intensities and propagation trajectories can be generated, thereby achieving synchronous and independent control of the intensity and trajectory of the Stokes vector skyrmion.

[0013] Furthermore, in step S2, the arbitrary l-order angular spectrum function satisfying the preset axial strength and propagation trajectory is calculated using formula (1):

[0014]

[0015] Among them, (k x ,k y ,k z Let be a three-dimensional Cartesian coordinate system in Fourier space (k-space), whose components satisfy... It is the wave number, λ is the wavelength of the incident light; e (.) It is a complex exponential function; the rect() function represents a rectangular function, whose constraint k z It belongs to the range (-k, k); r m Represents the Bessel function J l ( ) the radius of the vortex ring, and k zm For Bessel beams in r m The axial wavenumber component at the location; z represents the axial coordinate of the beam propagation direction.

[0016] Furthermore, in step S3, the zero-order Bessel beam is reflected by mirror M1 and irradiates the polarization beam splitter.

[0017] Furthermore, the focal length of the lens is 50-500mm, and the size is 1-100mm.

[0018] Furthermore, in step S4, the formula for generating the Stokes vector skyrmion with a specific intensity distribution and propagation trajectory is as follows:

[0019]

[0020] Where (r,φ,z) is a three-dimensional cylindrical coordinate system in real space. and These are the l1 and l2 order angular spectral functions, respectively; This represents the inverse Fourier transform process; and These are the l1 and l2 order Bessel beams customized by the angular spectrum function, respectively; θ0 is the phase difference between the two Bessel beams; |R> and |L> are right-handed circularly polarized and left-handed circularly polarized, respectively.

[0021] Furthermore, in step S4, the l1-order Bessel beam and the l2-order Bessel beam are given orthogonal polarization states.

[0022] Furthermore, the l1st-order Bessel beam is endowed with right-hand circular polarization, and the l2nd-order Bessel beam is endowed with left-hand circular polarization.

[0023] This invention provides a method for synchronously and independently controlling the axial intensity and trajectory of a Stokes vector skyrmion. The method calculates the required zero-order and higher-order angular spectrum functions based on the target axial intensity distribution and three-dimensional trajectory. Then, using two lenses with the same focal length, Fourier transforms are performed on these two sets of angular spectra, generating corresponding zero-order and higher-order Bessel beams on their respective back focal planes. Subsequently, the customized zero-order and higher-order Bessel beams are assigned right-handed and left-handed circular polarization, respectively, and coherently superimposed using a polarization beam splitter. Finally, a Stokes vector skyrmion with a specific intensity distribution and a preset trajectory is generated.

[0024] This invention enables the simultaneous and independent control of Stokes vector skyrmions in terms of axial intensity and trajectory. The intensity and trajectory of the Stokes vector skyrmion can be customized as needed, and the type of Stokes vector skyrmion can be arbitrarily adjusted. Through a novel method based on angular spectrum theory, it achieves for the first time the simultaneous, independent, and precise control of Stokes vector skyrmions in both axial intensity distribution and three-dimensional propagation trajectory. This technology directly solves the core limitation in existing research that Stokes vector skyrmions cannot propagate along arbitrary free paths, while filling the research gap in controlling the light field intensity along the propagation direction, enabling on-demand design of axial intensity distribution and spatial trajectory. Its core advantage lies in the comprehensive degrees of freedom of multi-dimensional collaborative control: it can control its propagation along arbitrarily complex curved trajectories and independently customize the axial intensity distribution, providing unprecedented control flexibility for optical Stokes vector skyrmions. This method based on angular spectrum theory not only solves the dual challenges of trajectory and intensity control but also establishes a new technical system for the dynamic manipulation of complex structured light fields. This achievement is revolutionary in the fields of precise particle manipulation (such as optical tweezers transport along irregular paths) and high-density optical information storage (multidimensional encoding of fused trajectories, intensities, and topological states), and has significantly promoted the in-depth development of optical skyrmion research towards practical application and high degree of freedom. Attached Figure Description

[0025] Figure 1 This is a schematic diagram of the method for synchronously and independently controlling the axial strength and trajectory of the Stokes vector skyrmion provided by the present invention.

[0026] Figure 2 shows the angular spectra of order l=0 and order l=1 when I(z)=1, f(z)=0.15cos[2π(z / f+1)] and g(z)=0 in this invention; the intensity distribution on the xz plane after coherent superposition by focusing through the lens (b); the one-dimensional normalized intensity distribution propagating along the z-axis (c); the transverse polarization distribution at different propagation distances (d1-d4); and the three-dimensional diagrams of the Stokes vector (e1-e4): z=-0.075m, 0m, 0.075m and 0.14m.

[0027] Figure 3 shows the invention. f(z) = 0.15 × 10 -3 (z / f) 2 -0.05×10 -3 When g(z) = 0, the angular spectra of order l = 0 and order l = 1 (a); the intensity distribution on the xz plane after coherent superposition by focusing through a lens (b); the one-dimensional normalized intensity distribution propagating along the z-axis (c); the transverse polarization distribution at different propagation distances (d1-d4); the three-dimensional diagram of the Stokes vector (e1-e4): z = 0m, 0.055m, 0.095m and 0.14m. Detailed Implementation

[0028] To make the technical problems solved by this invention, the technical solutions adopted, and the technical effects achieved clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings, not all of them.

[0029] like Figure 1 As shown in the figure, an embodiment of the present invention provides a method for synchronously and independently controlling the axial strength and trajectory of a Stokes vector skyrmion, comprising the following steps:

[0030] Step S1: Set the beam control parameters.

[0031] The beam control parameters include the axial intensity distribution function I(z) and the propagation trajectory function s(z) of the main lobe of the beam; where s(z) = [f(z), g(z)], f(z) represents the trajectory modulation function used to control the displacement of the beam in the x direction, and g(z) represents the trajectory modulation function used to control the displacement of the beam in the y direction.

[0032] In step S1, the axial intensity distribution function I(z) is the light intensity distribution function at position z on the propagation axis of the main beam, and s(z) = [f(z), g(z)] is the displacement of the energy center of the main beam in a plane perpendicular to the propagation direction. I(z) and s(z) = [f(z), g(z)] are the key parameters for controlling the axial intensity and three-dimensional trajectory of the skyrmion, respectively. By controlling the axial intensity distribution function I(z) and the propagation trajectory function s(z) of the main lobe of the beam, skyrmion beams with different axial intensities and trajectories can be generated, achieving synchronous and independent control of the intensity and trajectory of the skyrmion.

[0033] Step S2: Based on the beam control parameters, calculate any l-order angular spectrum function that satisfies the preset axial intensity distribution and propagation trajectory using formula (1):

[0034]

[0035] Among them, (k x ,k y ,k z Let be a three-dimensional Cartesian coordinate system in Fourier space (k-space), whose components satisfy... It is the wave number, λ is the wavelength of the incident light; e (.) It is a complex exponential function; the rect() function represents a rectangular function, whose constraint k z It belongs to the range (-k, k); r m Represents the Bessel function J l() the radius of the vortex ring, and k zm For Bessel beams in r m The axial wavenumber component at the location; z represents the axial coordinate of the beam propagation direction.

[0036] Any l-order angular spectral function contains a zero-order (l=0) angular spectral function. and higher-order (l≠0) angular spectrum functions

[0037] Step S3: Use two identical lenses to examine the zero-order angular spectral function. and higher-order angular spectrum functions By performing a Fourier transform, zero-order and higher-order Bessel beams with axial intensity and propagation trajectory consistent with the set beam control parameters can be obtained.

[0038] Specifically, a zero-order Bessel beam and a higher-order Bessel beam are generated at the back focal plane of the lenses (L1, L2). The zero-order Bessel beam is reflected by mirror M1 and illuminates the polarizing beam splitter (PBS).

[0039] The focal lengths of the two lenses are 50-500mm, and the dimensions are 1-100mm. For example, the focal length is f=150mm, and the dimension is 5mm.

[0040] Step S4: The zero-order Bessel beam and the higher-order Bessel beam are respectively assigned right-hand circular polarization and left-hand circular polarization. The customized zero-order (l=0) right-hand circularly polarized Bessel beam and the higher-order (l≠0) left-hand circularly polarized Bessel beam are coherently superimposed using a polarization beam splitter. The phase difference between the two beams and the order of the higher-order angular spectrum are controlled to generate a Stokes vector skyrmion beam with a specific intensity distribution and propagation trajectory.

[0041] The formula for generating Stokes vector skyminions with specific intensity distributions and propagation trajectories is as follows:

[0042]

[0043] Where (r,φ,z) is a three-dimensional cylindrical coordinate system in real space. and These are the l1 and l2 order angular spectral functions, respectively; This represents the inverse Fourier transform process; and These are the l1 and l2 order Bessel beams customized by the angular spectrum function, respectively; θ0 is the phase difference between the two Bessel beams; |R> and |L> are right-handed circularly polarized and left-handed circularly polarized, respectively.

[0044] Zero-order right-hand circularly polarized Bessel beams and higher-order left-hand circularly polarized Bessel beams are coherently superimposed using a polarization beam splitter (PBS).

[0045] Controlling the phase difference θ0 between two beams: By adjusting θ0 = 0, the Néel-type Stokes vector skyrmion can be obtained; Bloch-type Stokes vector skyrmions can be obtained.

[0046] Controlling the order of higher-order angular spectra: Coherent superposition of a zero-order (l1=0) right-hand circularly polarized Bessel beam and a higher-order (l2≠0) left-hand circularly polarized Bessel beam can produce higher-order Stokes vector skyrmions of orders 1, 2, 3, ... by keeping l1=0 constant and l2=1, 2, 3, ... constant; keeping l1=0 constant and l2=-1 can produce inverse Stokes vector skyrmions.

[0047] Assigning right-handed and left-handed circular polarization to the modulated zero-order and higher-order Bessel beams, respectively, is key to generating Stokes vector skyrmions with tunable intensity and trajectory. The intensity and trajectory of the Stokes vector skyrmions can be controlled by adjusting the functions I(z) and s(z) in step S1. Different types and orders of Stokes vector skyrmions can be generated by controlling the phase difference between the two Bessel beams and the topological charge of the orbital angular momentum carried by the higher-order Bessel beam.

[0048] Stokes vector skyrmions have Bloch, Néel, inverse, and higher-order forms. Several specific embodiments are given below to demonstrate the effectiveness of the method proposed in this invention.

[0049] Example 1: Constructing a Bloch-type Stokes vector skyrmion with uniform intensity distribution and propagating along a serpentine trajectory.

[0050] With the intensity distribution function set to I(z) = 1, and the trajectory control functions f(z) = 0.15cos[2π(z / f+1)] and g(z) = 0, the angular spectra modulated by the 0th order (l = 0) and 1st order (l = 1) are calculated using the angular spectrum generation formula. and As shown in Figure 2(a). Subsequently, the two angular spectra are subjected to Fourier transforms through lenses, and right-handed and left-handed circular polarizations are assigned respectively. The phase difference θ0 between the two beams is controlled as follows: A Bloch-type Stokes vector skyrmion with a uniform intensity distribution and propagating along a serpentine trajectory can be obtained. Figures 2(b)-(c) show the longitudinal propagation distribution of the Stokes vector skyrmion in the xz plane and the normalized intensity distribution along the z-axis, respectively. It can be seen that the Stokes vector skyrmion with a uniform intensity distribution exhibits a serpentine trajectory in the xz plane, which is consistent with the preset parameters.

[0051] To investigate the topological protection properties of skyrmions propagating along curved trajectories, four random spatial locations were selected to illustrate the polarization elliptic distribution and Stokes vector diagrams of the transverse profiles, as shown in Figures 2(d)-(e). These figures demonstrate the evolution of the transverse field distribution of the Stokes vector skyrmions along the propagation direction, specifically presenting transverse profiles at z = -0.075m, 0m, 0.075m, and 0.14m, arranged in order of their appearance during propagation. Notably, in the transverse plane of the "bent chain" Stokes vector skyrmions, the topological structure of the main lobe maintains a Bloch-type distribution during propagation.

[0052] Example 2: Constructing a Néel-type Stokes vector skyrmion with a cosine intensity distribution that propagates along a parabolic trajectory.

[0053] Set the intensity distribution function The trajectory control function f(z) = 0.15 × 10 -3 (z / f) 2 -0.05×10 -3 Given g(z) = 0, and based on the aforementioned parameters, the angular spectra modulated at order 0 (l = 0) and order 1 (l = 1) are calculated using the angular spectrum generation formula. and As shown in Figure 3(a), the two angular spectra were then subjected to Fourier transforms through lenses and assigned right-handed and left-handed circular polarization, respectively. By controlling the phase difference θ0 between the two beams to be 0, a Néel-type Stokes vector skyrmion with a cosine intensity distribution and propagating along a parabolic trajectory was obtained. Its longitudinal cross-section in the xz plane and the normalized intensity distribution along the z-axis are shown in Figures 3(b)-(c). Clearly, the electric field exhibits a parabolic trajectory and a periodic intensity distribution, similar to a "bent chain," and the simulation results are in excellent agreement with the theoretical function. Figures 3(d) and (e) show the polarization distribution and Stokes vector at three discrete cross-sectional planes: z = 0 m, 0.055 m, 0.095 m, and 0.14 m. The transverse profiles of the Stokes vector skyrmions are given in the order of their propagation. It is noteworthy that the position of the main lobe of the Stokes vector skyrmion changes in the transverse plane. However, the polarization ellipse distribution and the Stokes vector diagram show that the Stokes vector skyrmion still maintains the Néel-type topology.

[0054] Through the aforementioned Examples 1-2, it was found that, based on the angular spectrum manipulation theory proposed in this invention, Stokes vector skyrmions with synchronously and independently controllable intensity and trajectory can be constructed. The axial intensity distribution and trajectory of the Stokes vector skyrmion can be independently controlled by functions I(z) and s(z) = [f(z), g(z)]. Furthermore, the type and order of the Stokes vector skyrmion can be arbitrarily adjusted by controlling the phase difference between Bessel beams and the order of higher-order Bessel beams. The work of this invention provides new insights into the study of optical skyrmions and demonstrates their great potential in optical manipulation and particle classification.

[0055] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications to the technical solutions described in the foregoing embodiments, or equivalent substitutions for some or all of the technical features, do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for independently controlling axial intensity and trajectory of a stokes vector skyrmion, characterized in that, Includes the following steps: Step S1: Set beam control parameters; the beam control parameters include the axial intensity distribution function. I ( z The propagation trajectory function of the main lobe of the beam. s ( z ); Step S2: Based on the beam control parameters, calculate any beam that satisfies the preset axial intensity distribution and propagation trajectory. Angular spectral function; In step S2, any condition satisfying the preset axial strength and propagation trajectory is calculated using formula (1). Angular spectral function: ； in, Let be a three-dimensional Cartesian coordinate system in Fourier space (k-space), whose components satisfy... ; It is the wave number. It is the wavelength of the incident light; It is a complex exponential function; the rect() function represents a rectangular function, whose constraints... It falls within the range (-k, k); Representing the Bessel function () the radius of the vortex ring, and For Bessel beams in The axial wavenumber component at the location; z represents the axial coordinate of the beam propagation direction; Step S3: Use two identical lenses to examine the zero-order angular spectral function. and higher-order angular spectrum functions By performing a Fourier transform, zero-order and higher-order Bessel beams with axial intensity and propagation trajectory consistent with the set beam control parameters can be obtained. Step S4: The zero-order Bessel beam and the higher-order Bessel beam are respectively assigned right-hand circular polarization and left-hand circular polarization. The customized zero-order right-hand circularly polarized Bessel beam and the higher-order left-hand circularly polarized Bessel beam are coherently superimposed by a polarization beam splitter. The phase difference between the two beams and the order of the higher-order angular spectrum are controlled to generate a Stokes vector skyrmion beam with a specific intensity distribution and propagation trajectory. In step S4, the formula for generating the Stokes vector skyrmion with a specific intensity distribution and propagation trajectory is as follows: ； in,( () is a three-dimensional cylindrical coordinate system in real space. and They are respectively as well as Angular spectral function; () indicates the inverse Fourier transform process; and Each is customized by the angular spectrum function. and Bessel beam; It is the phase difference between two Bessel beams; and These are right-handed circular polarization and left-handed circular polarization, respectively.

2. The method for synchronously and independently controlling the axial strength and trajectory of the Stokes vector skyrmion according to claim 1, characterized in that, In step S1, the axial strength distribution function I ( z ) is the light intensity distribution function at position z on the propagation axis of the main beam. s ( z ) = [ f ( z ) , g ( z The displacement of the main beam energy center in a plane perpendicular to the propagation direction; by controlling the axial intensity distribution function. I ( z and the propagation trajectory function of the main lobe of the beam. s ( z It can generate Stokes vector skyrmion beams with different axial intensities and propagation trajectories, thereby achieving synchronous and independent control of the intensity and trajectory of Stokes vector skyrmions.

3. The method for synchronously and independently controlling the axial strength and trajectory of the Stokes vector skyrmion according to claim 1, characterized in that, In step S3, the zero-order Bessel beam is reflected by mirror M1 and illuminates the polarization beam splitter.

4. The method for synchronously and independently controlling the axial strength and trajectory of the Stokes vector skyrmion according to claim 1, characterized in that, The lens has a focal length of 50-500mm and a size of 1-100mm.

5. The method for synchronously and independently controlling the axial strength and trajectory of the Stokes vector skyrmion according to claim 1, characterized in that, In step S4, Bessel beams and The Bessel beam is given an orthogonal polarization state.

6. The method for synchronously and independently controlling the axial strength and trajectory of the Stokes vector skyrmion according to claim 5, characterized in that, The Bessel beam is endowed with right-hand circular polarization. The Bessel beam is endowed with left-handed circular polarization.

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