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Abstract
Manipulating polarization is of significance for the application of light. Spin-orbit coupling provides a prominent pathway for manipulating the polarization of light field but generally requires tight focusing conditions or anisotropic media. In this paper, we construct ring Airy beams with hybrid polarizations and reveal the controllable polarization transforms in their autofocusing dynamics by manipulating concomitant spin-orbit coupling in free space. The numerical and experimental results show that the polarization transform is dependent on the azimuthal orders of amplitude and vortex phases of two spin constituents of ring Airy beams, that the focal spots present pure linear polarization whose orientation is determined by the initial phase when the vortex phase topological charge is equal to the amplitude angular factor, otherwise, the focal fields present cylindrical vector polarizations whose orders depend on the difference of amplitude angular orders and topological charges. Our work provides new insights for studying spin-orbit interactions and the depolarization of complex polarization.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization is an intrinsic and fundamental nature of light, manipulating states of polarization (SoPs) plays an important role in controllable light-matter interactions [1–6]. Until now, many vector beams with unique polarization distributions have been proposed and experimentally realized, such as classic cylindrical vector beams [7], double-mode vector beams [8], hybrid vector beams [9–11] and so on. These vector beams with spatially inhomogeneous SoPs have encouraging application prospects in micro fabrication [12], optical tweezers [13], high-resolution imaging [14], optical data storage [15,16], encryption [17], etc. Recently, the hybrid vector beams [9] have attracted intensive interest, because their nonuniform polarization could induce the orbital angular momentum (OAM) [13]. Importantly, the spin angular momentum (SAM) related to the SoP distribution and OAM may interact and induce the spin-orbit coupling [18–25], exhibiting great potentials in applications of precision metrology [26], polarimetry of scattering media [27] and nano-optics under the subwavelength scales [28].
The ring Airy beam (RAB), which manifests itself as abruptly autofocusing dynamics and controllable propagation trajectories [29–37], its energy abruptly increases several orders of magnitude before the focus, which makes it have great application values in biomedical treatment optical micromachining [29], laser ablation [30], light-bullet [31,32], super-resolution imaging [33,34], and so on. The autofocusing property also provides a new strategy for abrupt polarization transition of vector RAB focal point by manipulating spin-orbit coupling [38].
In this paper, we numerically and experimentally investigate the polarization transforms in the autofocusing of RABs with hybrid polarizations. The influences on the SoPs and intensity distributions at the initial and focal planes are represented by changing the amplitude angular factor, the topological charges of vortex phase, and the initial phase of two spin constituents. Furthermore, the interaction between SAM and OAM is also discussed. The mechanism of polarization transformation induced by spin-orbit coupling we presented could be expected to unique effects and phenomena.
2. OAM in ring Airy beams with hybrid polarizations
Vector beams could be created by the coherent superposition of left- and right-handed circularly polarized beams with presupposed phases, so we select the orthogonal circularly polarized basis vectors {eL, eR} as a typical case, we add amplitude angular factors to these basis vectors for introducing hybrid SoPs, the RAB with hybrid polarizations in the cylindrical coordinate system (r, φ, z) thus can be written as
The constructed RABs with hybrid polarizations at the initial planes are shown in Fig. 1, where the angular factors (l, m) are (0.5, 0.5), (1, 1), (1, 2), (2, 1) and (3, 1) with φ0 = π/4, respectively. The figures from top to bottom give the total intensity (I0, the ellipses represent the local SoP), two spin constituents, OAM density, and mapping trajectories of hybrid SoPs on the Poincaré sphere, respectively. As shown in Figs. 1(a1)−1(e1), the change of l and m do not affect the total intensities distributions but affect the SoP distributions, namely, the mapping trajectories on the classical Poincaré sphere. According to the relationship of Stokes parameters and longitude and latitude angles (2ψ,2χ) of Poincaré sphere [39], we can get tan(2ψ) = -tan(2mφ) and sin(2χ) = cos(2lφ+φ0), where the ψ (0≤ψ≤π) is the orientation angle and χ (-π/4≤χ≤π/4) is the ellipticity angle of the polarization ellipse, their periods along the azimuthal orientation are determined by l and m. As shown in Figs. 1(a3) and 1(b3), the SoPs always map to 8-shaped trajectories and follow 2 l times along them when l = m. If l is kept constant as shown in Figs. 1(b1) and 1(c1), the ranges of the two spin constituents don’t change, but the orientation angle of local SoPs will change faster with the increasing of m. If m is kept constant as shown in Figs. 1(b1), 1(d1), and 1(e1), the period of the ellipticity angle along the azimuthal orientation will shorter with the increasing of l, so the light lobes of the single spin constituents are two and four times as much as l, as the second row of Fig. 1. Their mapping trajectories are no longer regular but always connect the north and south poles of the Poincaré sphere in Figs. 1(c3), 1(d3) and 1(e3).
Fig. 1. RABs with hybrid polarizations. Top row: simulated total intensities and polarization distributions (the green and red ellipses represent left- and right-handed SoPs, respectively); Second row: simulated left- and right-handed circularly polarized constituents; third row: simulated OAM density; Bottom row: SoPs mapping trajectories on the Poincaré sphere.
Meanwhile, we consider the influences of the parameters (l, m, φ0) on the OAM, the z-constituent of OAM density can be expressed as [13,39,40]
Equations (2) and (3) describe the OAM density induced by the angular phase gradient and the variety of SoPs, respectively. Where, the amplitude u = E0Ai[(r0-r)/w]exp[a(r0-r)/w], ϕ = (ϕL + ϕR)/2 = (-mφ+mφ)/2 = 0, ϕL and ϕR represent the phases of two spin constituents, α = cos(lφ+φ0)exp(-imφ) and β = sin(lφ+φ0)exp(imφ) determine the distribution of the SoPs. To better understand the variation of the OAM, we substitute the parameters of Eq. (1) into Eqs. (2) and (3), the total OAM density jz is obtained as
Equation (4) indicates that the jz is determined by the (l, m, φ0), and u is a circularly symmetric function, if l ≠ 0 and m ≠ 0, then the m limits the maximum value of jz. As shown in the third row of Fig. 1, the maximum peak of jz appears in the case of m = 2. Considering the symmetry of cosine function, the total OAM and SAM are both zero for the RABs with hybrid SoPs. However, the inhomogeneous OAM densities associated with SAM densities could manipulate the spin dynamics and polarization transform during beam propagation.
3. Experiment and results
We generated these RABs with hybrid polarizations by using an experimental setup as shown in Fig. 2(a). A p-polarized Gaussian beam (from He-Ne laser with wavelength λ = 632.8 nm) is converted into 45° linearly polarized light (based on the horizontal orientation, the references are same in the following) after the first half-wave plate (HWP1 with its fast axis along 22.5°), then passes through the beam deflector (BD1), it is equally divided into two orthogonal polarized light (i.e., p- and s-polarized constituents), and they are separated for a certain distance along the horizontal orientation. Both of them are reflected to the left and right split screens of the SLM by one side of the right-angle prism mirror (RAPM). Because the SLM only responds to p-polarized light, an HWP2 (with its fast axis along 45°) is placed on the path of the s-polarized constituent to make it convert into p-polarized light, then it is reflected by the SLM and still the s-polarized after traversing the HWP2 again. Next, two constituents reflected by the other side of the RAPM orderly pass the second inverted BD2 and a quarter-wave plate (QWP), then they are coaxially combined and converted into right- and left-handed polarized constituents. Finally, the output vector beam was observed by the charge-coupled device (CCD). In practical operation, the SLM’s plane was clearly imaged at the rear focal plane of a 4f filter system, the intensity distribution was detected and recorded by the CCD.
Fig. 2. (a) Experimental setup for generating RABs with hybrid polarizations. HWP1,2, half-wave plates; BD1,2, beam deflector; RAPM, right-angle prism mirror; SLM, spatial light modulator; QWP, quarter-wave plates; 4f, 4f filter system; P, polarizer; CCD, charge-coupled device. Left insets, CGH on the SLM; Right insets, measured p- and s- polarized constituents of radially polarized RAB at initial planes, respectively; (b)-(f) Measured total intensity distributions at the initial planes of RABs with hybrid polarizations; Bottom row: measured left- and right-handed circularly polarized constituents.
The left insets in Fig. 2(a) are two-dimensional computer-generated holograms (CGHs) [41] loaded on the SLM, and the right insets correspond to measured p- and s-polarized constituents of radially polarized RAB at the initial plane, respectively. In experiment, r0 = 461.50μm, w = 27.15µm, a = 0.05, and the focal length f = 6.23 cm. Figures 2(b)–2(f) show the measured Stokes parameters S3 (drawn with total intensity as background) of RABs with (l, m, φ0) = (0.5, 0.5, π/4), (1, 1, π/4), (1, 2, π/4), (2, 1, π/4), (3, 1, π/4), where the blue and red correspond to the left- and right-handed polarizations, respectively. The bottom row shows the intensity distributions of left- and right-handed circularly polarized constituents. The results are consistent with the simulated results in the front two rows of Fig. 1.
We first observed the autofocusing field of RABs with the same azimuthal orders of amplitude and vortex phases, i.e., l = m. Figure 3 shows the measured results of RABs with l = m = 0.5, 1, 2, 3, 10 and φ0 = π/4. From top to bottom, these figures give the measured total intensity, p- and s-polarized constituents. These insets show the corresponding simulated results. The focal fields present solid light spots of the same size when l and m are integers, as shown in Figs. 3(b1)−3(e1), which is significantly different from the highly bright hollow focal spot of radial vector RABs. Meanwhile, it is noteworthy that the local SoPs of these central spots are p-polarized, by comparing the results shown in Figs. 3(a2)−3(e2) and Figs. 3(a3)−3(d3). Consequently, the focal field forms a solid p-polarized spot with any case of the l = m, which is consistent with the SoPs of the trajectories’ intersection on the Poincaré sphere in Fig. 1(b3). The increasing of l only enhances the symmetry of the SoPs and the number of times around the 8-shaped trajectory. The solid circular spots mean that two spin constituents with l = m have uniform SAM (±ħ per photon) without vortex phases.
Fig. 3. Experimentally measured and simulated intensity distributions of the total (top row), p- (second row), and s- (bottom row) polarized constituents at the focal planes of RABs with hybrid polarizations and l = m = 0.5, 1, 2, 3, 10 and φ0=π/4, respectively (the white arrows are the polarization orientations). Insets, corresponding simulated results.
The cos(lφ+φ0) could be expanded into [exp(ilφ+iφ0) + exp(-ilφ−iφ0)]/2, meanwhile, sin(lφ) = -i[exp(ilφ+iφ0) - exp(-ilφ−iφ0)]/2, we could substitute these equations to Eq. (1) and get
Figure 4 shows the experiment results in the cases of φ0 = 0, π/2, 3π/4, and we select l = m = 10 to eliminate the interference of the sidelobes. From top to bottom, these figures give the mapping trajectories, measured total intensity distributions, and two orthogonal components after polarization filtering, respectively. These insets show the corresponding simulation results. The shapes of mapping trajectories are same as each other with different locations on the Poincaré sphere, as shown in Figs. 4(a1)−4(c1). The second to fourth rows show that the sizes of the central focal spots are same with 45°, 135°, and s-linear polarization orientations corresponding with the SoPs of eL + ieR, -eL + ieR, eL-eR, respectively. Meanwhile, most of the energy is concentrated in the 45°, 135°, and s-polarized constituents, and these SoPs are consistent with the intersection points as shown in Figs. 4(a1)−4(c1).
Fig. 4. Polarization mapping trajectories (top row) on the Poincaré sphere, experimentally measured intensity distributions of the total (second row), and orthogonal polarized constituents (bottom two rows) at the focal planes of RABs with l = m = 10 and different φ0 (the white arrows are the polarized orientations), respectively. Insets, corresponding simulated results.
Figure 5 shows the experiment results with l = 10, m = 7, 8, 9, and φ0 = π/4, where the amplitude angular factor is not equal to the vortex phase topological charges. From top to bottom, these figures give measured and simulated total intensity distributions, Stokes parameters, normalized OAM and SAM densities (σz) of left-handed circularly polarized constituent at focal planes, respectively. Figures. 5(a)–5(f) show that the dark cores at the focal center get smaller with the increasing of m, and they are typical 1, 2, 3 order cylindrical vector beams by observing the S1, S2 and S3 distributions. It means that the two circularly polarized constituents are hollow spots embedded with ±(l-m) order vortex phases, respectively. The OAM and SAM densities of the bottom row represent that the sizes of dark cores of left-handed circularly polarized constituents depend on the difference of amplitude angular orders and initial topological charges. The experimentally measured S3 distributions of central parts are not exactly equal to 0, because the phase singularities of two spin constituents have tiny shifting with each other for perturbation during the autofocusing, and the higher order vortex occurs the splitting phenomenon [42,43], so the center of the vortex phases can’t completely superimpose.
Fig. 5. Experimentally measured and simulated total intensity distributions (top row), Stokes parameters (second to fourth rows), and normalized OAM and SAM densities of left-handed circularly polarized constituent (bottom row) at the focal planes of RABs with the different m and l = 10, φ0 = π/4.
From the above results, these RABs with hybrid polarizations finally form l-m order cylindrical vector beams at focal planes when l ≠ m. The single left-handed circularly polarized constituents carry SAM (ħ per photon) and OAM (-mħ per photon) at the initial planes, while SAM (ħ per photon) and OAM (lħ-mħ per photon) on the central focal fields. It means that the polarization transforms induced by spin-orbit coupling could be manipulated by the difference of amplitude angular orders and topological charges.
4. Conclusion
In conclusion, the amplitude angular factor, topological charges of the vortex phase, and the initial phase of two spin constituents could significantly influence the spin-orbit coupling in the autofocusing behaviors of RABs with hybrid polarizations. The controllable polarization transforms of autofocusing spots are represented, and the results show that the parameters (l, m, φ0) can make the RABs transform into l-m order cylindrical vector beams or pure linear polarized beams, and the polarized orientations determined by the initial phase are consistent with the orientations represented by intersection points of 8-shaped trajectories. We believe these results about the polarization transforms in the RABs with hybrid polarizations could contribute to the spin-orbit interactions and manufacturing of depolarization devices.
Funding
National Key Research and Development Program of China (2022YFA1404800); National Natural Science Foundation of China (12074312, 12074313, 12174309, 12304372); Fundamental Research Funds for the Central Universities (3102019JC008, 5000230111, xzy012022023); Shaanxi Fundamental Science Research Project for Mathematics and Physics (22JSY013); China Postdoctoral Science Foundation (2022M712553).
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.
References
1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]
2. Y. Zhao, Q. Zhan, Y. Zhang, et al., “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30(8), 848–850 (2005). [CrossRef]
3. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31(6), 820–822 (2006). [CrossRef]
4. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]
5. X. Yi, Y. Liu, X. Ling, et al., “Hybrid-order Poincaré sphere,” Phys. Rev. A 91(2), 023801 (2015). [CrossRef]
6. F. Yue, D. Wen, J. Xin, et al., “Vector Vortex Beam Generation with a Single Plasmonic Metasurface,” ACS Photonics 3(9), 1558–1563 (2016). [CrossRef]
7. Z. Liu, Y. Liu, Y. Ke, et al., “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017). [CrossRef]
8. X. L. Wang, J. Ding, W. J. Ni, et al., “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef]
9. H. T. Wang, X. L. Wang, Y. Li, et al., “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010). [CrossRef]
10. G. M. Lerman, L. Stern, and U. Levy, “Generation and tight focusing of hybridly polarized vector beams,” Opt. Express 18(26), 27650–27657 (2010). [CrossRef]
11. L. Han, S. Qi, S. Liu, et al., “Hybrid vector beams with non-uniform orbital angular momentum density induced by designed azimuthal polarization gradient,” Chin. Phys. B 29(9), 094203 (2020). [CrossRef]
12. T. Zhao, S. Zhang, Y. Guo, et al., “Dynamic plasmonic nano-traps for single molecule surface-enhanced Raman scattering,” Nanoscale 8(1), 233–242 (2016). [CrossRef]
13. X. L. Wang, J. Chen, Y. Li, et al., “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105(25), 253602 (2010). [CrossRef]
14. X. Xie, Y. Chen, K. Yang, et al., “Harnessing the Point-Spread Function for High-Resolution Far-Field Optical Microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014). [CrossRef]
15. V. Parigi, V. D’Ambrosio, C. Arnold, et al., “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” Nat. Commun. 6(1), 7706 (2015). [CrossRef]
16. X. Ouyang, Y. Xu, M. Xian, et al., “Synthetic helical dichroism for six-dimensional optical orbital angular momentum multiplexing,” Nat. Photonics 15(12), 901–907 (2021). [CrossRef]
17. X. Li, T. H. Lan, C. H. Tien, et al., “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012). [CrossRef]
18. O. Hosten and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319(5864), 787–790 (2008). [CrossRef]
19. H. Luo, S. Wen, W. Shu, et al., “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80(4), 043810 (2009). [CrossRef]
20. J. L. Ren, B. Wang, M. M. Pan, et al., “Spin separations in the spin Hall effect of light,” Phys. Rev. A 92(1), 013839 (2015). [CrossRef]
21. Y. Zhang, P. Li, S. Liu, et al., “Unveiling the photonic spin Hall effect of freely propagating fan-shaped cylindrical vector vortex beams,” Opt. Lett. 40(19), 4444–4447 (2015). [CrossRef]
22. H. Li, M. Tang, J. Wang, et al., “Spin Hall effect of Airy beam in inhomogeneous medium,” Appl. Phys. B 125(3), 51 (2019). [CrossRef]
23. S. Fu, C. Guo, G. Liu, et al., “Spin-Orbit Optical Hall Effect,” Phys. Rev. Lett. 123(24), 243904 (2019). [CrossRef]
24. R. Liu, G. Wang, D. Deng, et al., “Spin Hall effect of Laguerre-Gaussian beams in PT symmetric metamaterials,” Opt. Express 29(14), 22192–22201 (2021). [CrossRef]
25. S. Liu, S. Qi, Y. Li, et al., “Controllable oscillated spin Hall effect of Bessel beam realized by liquid crystal Pancharatnam-Berry phase elements,” Light: Sci. Appl. 11(1), 219 (2022). [CrossRef]
26. X. Zhou, Z. Xiao, H. Luo, et al., “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012). [CrossRef]
27. C. Schwartz and A. Dogariu, “Backscattered polarization patterns, optical vortices, and the angular momentum of light,” Opt. Lett. 31(8), 1121–1123 (2006). [CrossRef]
28. L. T. Vuong, A. J. L. Adam, J. M. Brok, et al., “Electromagnetic Spin-Orbit Interactions via Scattering of Subwavelength Apertures,” Phys. Rev. Lett. 104(8), 083903 (2010). [CrossRef]
29. P. Zhang, J. Prakash, Z. Zhang, et al., “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef]
30. A. Mathis, F. Courvoisier, L. Froehly, et al., “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012). [CrossRef]
31. Y. Jiang, K. Huang, and X. Lu, “Radiation force of abruptly autofocusing Airy beams on a Rayleigh particle,” Opt. Express 21(20), 24413 (2013). [CrossRef]
32. W. Lu, X. Sun, H. Chen, et al., “Abruptly autofocusing property and optical manipulation of circular Airy beams,” Phys. Rev. A 99(1), 013817 (2019). [CrossRef]
33. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014). [CrossRef]
34. T. Vettenburg, H. I. C. Dalgarno, J. Nylk, et al., “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014). [CrossRef]
35. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, et al., “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef]
36. Y. Jiang, Z. Cao, H. Shao, et al., “Trapping two types of particles by modified circular Airy beams,” Opt. Express 24(16), 18072–18081 (2016). [CrossRef]
37. B. Chen, C. Chen, X. Peng, et al., “Propagation of sharply autofocused ring Airy Gaussian vortex beams,” Opt. Express 23(15), 19288–19298 (2015). [CrossRef]
38. S. Liu, M. Wang, P. Li, et al., “Abrupt polarization transition of vector autofocusing Airy beams,” Opt. Lett. 38(14), 2416–2418 (2013). [CrossRef]
39. D. Zhang, X. Feng, K. Cui, et al., “Identifying Orbital Angular Momentum of Vectorial Vortices with Pancharatnam Phase and Stokes Parameters,” Sci. Rep. 5(1), 11982 (2015). [CrossRef]
40. Y. Pan, X. Z. Gao, Z. C. Ren, et al., “Arbitrarily tunable orbital angular momentum of photons,” Sci. Rep. 6(1), 29212 (2016). [CrossRef]
41. Y. Zhang, P. Li, C. Ma, et al., “Efficient generation of vector beams by calibrating the phase response of a spatial light modulator,” Appl. Opt. 56(17), 4956–4960 (2017). [CrossRef]
42. Y. Jiang, K. Huang, and X. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef]
43. X. Zhang, P. Li, S. Liu, et al., “Autofocusing of ring Airy beams embedded with off-axial vortex singularities,” Opt. Express 28(6), 7953–7960 (2020). [CrossRef]
