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随着光学技术的发展, 人们对光场与微粒相互作用的研究越来越深入. 通过研究驻波场中非均匀手性粒子的辐射力特性, 可以深入了解光场对微粒的影响机制, 为微纳米尺度下分层手征粒子的操控和应用提供新思路. 本文对双高斯波束照射下非均匀手征分层粒子的辐射力展开研究. 从广义洛伦兹-米氏理论(generalized Lorentz-Mie theory, GLMT)和球矢量波函数(spherical vector wave functions, SVWFs)出发, 推导了双高斯波束(double Gaussian beams, DGBS)的总入射场展开系数. 基于边界连续条件和电磁动量守恒定理, 得到双高斯波对粒子的辐射力表达式. 通过与现有文献进行比较, 证明了理论和程序的正确性. 详细分析了各种参数对辐射力的影响, 如束腰宽度、偏振形式、粒子半径、内外手征参数、折射率、最外层厚度等. 研究表明, 与单个高斯束相比, 反向传播的高斯驻波在捕获或限制非均匀手性分层粒子方面表现出显著优势, 提供了更强的粒子操控能力. 此外, 通过选择合适的偏振态入射, 可以在这些参数之间实现微妙的平衡, 从而有效地稳定俘获非均匀手性粒子. 这些研究对于分析和理解形状复杂多层生物细胞的光学特性具有重要意义, 并在多层生物结构微操控方面具有重要应用价值.
Objective With the development of optical technology, the investigation of light-field-particle interactions has gained significant momentum. Such studies find widespread applications in optical manipulation, precision laser ranging, laser gas spectroscopy, and related fields. In optical manipulation techniques, employing two or more laser beams proves more effective for capturing and manipulating particles than using a single beam alone. In addition, with the increasing demand for manipulating particles with complex structures, it is necessary to conduct in-depth research on the radiation force characteristics of double Gaussian beams on non-uniform chiral particles. This research aims to deepen our understanding of how optical fields influence particles, thereby offering fresh perspectives in manipulating and utilizing non-uniform chiral layered particles on both a microscale and a nanoscale. Method Based on the generalized Lorentz-Mie theory (GLMT) and spherical vector wave functions (SVWFs), the total incident field of a double Gaussian beam can be expanded by using the coordinate addition theorem. The incident field coefficient and scattering coefficient of each region of the multilayer chiral sphere are obtained by enforcing boundary continuity and employing multilayer sphere scattering theory. The radiation force acting on non-uniform chiral layered particles within a double Gaussian beam is then derived through application of the electromagnetic momentum conservation theorem. Results and Discussions The theory and programs in this paper is compared with those in existing literature. The influence of various parameters on the radiation force is analyzed in detail, such as the incident angle, polarization angle, beam waist width, beam center position, and internal and external chiral parameters. These results indicate that compared with a single Gaussian beam, counter-propagating Gaussian standing waves exhibit significant advantages in capturing or confining inhomogeneous chiral layered particles, offering enhanced particle manipulation capabilities. Additionally, by selecting an appropriate polarization state of the incident light, a delicate balance can be achieved among these parameters, effectively stabilizing the capture of inhomogeneous chiral particles. Conclusions This study employs the generalized Lorenz-Mie theory and the principle of electromagnetic momentum conservation to derive analytical expressions for the transverse and axial radiation forces exerted by dual Gaussian beams on multi-layered chiral particles propagating in arbitrary directions. The research provides an in-depth analysis of how standing wave beams affect the radiation force behavior of non-uniform chiral particles. Numerical analysis reveals significant influences of beam waist, particle size, chiral parameters, polarization angle and mode, as well as particle refractive index on both transverse and axial radiation forces. This research is important in analyzing and understanding the optical properties of complex-shaped multilayer biological cells and realizing the applications in the micromanipulation of multilayer biological structures. [1] Ashkin A 1970 Phys. Rev. Lett. 24 156Google Scholar
[2] Ashkin A 1980 Science 210 1081Google Scholar
[3] Leach J, Howard D, Roberts S, Gibson G, Gothard D, Cooper J, Buttery L 2009 J. Mod. Optic. 56 448Google Scholar
[4] Molloy J E, Dholakia K, Padgett M J 2003 J. Mod. Optic. 50 1501Google Scholar
[5] Parlatan U, Başar G, Başar G 2019 J. Mod. Optic. 66 228Google Scholar
[6] Jordan P, Clare H, Flendrig L, Leach J, Cooper J, Padgett M 2004 J. Mod. Optic. 51 627Google Scholar
[7] Tang Q, Liu P Z, Tang S 2022 Chin. Phys. B 31 044301Google Scholar
[8] Barton J P, Alexander D R, Schaub S A 1989 J. Appl. Phys. 66 4594Google Scholar
[9] Yang A H, Moore S D, Schmidt B S, Klug M, Lipson M, Erickson D 2009 Nature 457 71Google Scholar
[10] Padgett M, Bowman R 2011 Nat. Photon. 5 343Google Scholar
[11] Wang Z L, Yin J P 2008 Chin. Phys. B 17 2466Google Scholar
[12] Kiselev A D, Plutenko D O 2016 Phys. Rev. A 94 013804Google Scholar
[13] Zang Y C, Lin W J, Su C, Wu P F 2021 Chin. Phys. B 30 044301Google Scholar
[14] Dong F B, Chang C H, Jun F H, Yi W 2009 Chin. Phys. B 18 2853Google Scholar
[15] Ng J, Lin Z F, Chan C T 2010 Phys. Rev. Lett. 104 103601Google Scholar
[16] Liu X Y, Sun C, Deng D M 2021 Chin. Phys. B 30 024202Google Scholar
[17] 王焱, 彭妙, 程伟, 彭政, 成浩, 臧圣寅, 刘浩, 任孝东, 帅雨贝, 黄承志, 吴加贵, 杨俊波 2023 72 027801Google Scholar
Wang Y, Peng M, Cheng W, Peng Z, Cheng H, Zang S Y, Liu H, Ren X D, Shuai Y B, Huang C Z, Wu J G, Yang J B 2023 Acta Phys. Sin. 72 027801Google Scholar
[18] 殷杰, 陶超, 刘晓峻 2015 64 098102Google Scholar
Yin J, Tao C, Liu X J 2015 Acta Phys. Sin. 64 098102Google Scholar
[19] Ashkin A, Dziedzic J M 1971 Appl. Phys. Lett. 19 283Google Scholar
[20] Zemánek P, Jonáš A, Šrámek L, Liška M 1998 Opt. Commun. 151 273Google Scholar
[21] Zemánek P, Jonáš A, Liška M 2002 J. Opt. Soc. Am. A 19 1025Google Scholar
[22] Gauthier R C, Frangioudakis A 2000 Appl. Opt. 39 26Google Scholar
[23] Ren K F, Greha G, Gouesbet G 1994 Opt. Commun. 108 343Google Scholar
[24] Gouesbet G, Lock J A 1994 J. Opt. Soc. Am. A 11 2516Google Scholar
[25] Zemánek P, Jonáš A, Jákl P, Šerý M, Liška M 2003 Opt. Commun. 220 401Google Scholar
[26] Cizmar T, Garces-Chavez V, Dholakia K, Zemanek P 2004 Opt. Trap. Micro. 5514 643Google Scholar
[27] Van der Horst A, van Oostrum P D J, Moroz A, van Blaaderen A, Dogterom M 2008 Appl. Opt. 47 3196Google Scholar
[28] Zhao L, Li Y, Qi J, Xu J, Sun Q 2010 Opt. Express 18 5724Google Scholar
[29] Zhang T, Mahdy M R C, Dewan S S, Hossain M N, Rivy H M, Masud N, Jony Z R 2018 arXiv: 1811.01874 [physics. optics]
[30] Li Z J, Li S, Li H Y, Qu T, Shang Q C 2021 J. Opt. Soc. Am. A 38 616Google Scholar
[31] Wang S L, Liu X, Mourdikoudis S, Chen J, Fu W W, Sofer Z, Zhang Y, Zhang S P, Zheng G C 2022 ACS Nano. 16 19789Google Scholar
[32] 马晓亮, 李雄, 郭迎辉, 赵泽宇, 罗先刚 2017 66 147802Google Scholar
Ma X L, Li X, Guo Y H, Zhao Z Y, Luo X G 2017 Acta Phys. Sin. 66 147802Google Scholar
[33] Rohrbach A, Stelzer E H K 2001 J. Opt. Soc. Am. A 18 839Google Scholar
[34] 史书姝, 肖姗, 许秀来 2022 71 067801Google Scholar
Shi S S, Xiao S, Xu X L 2022 Acta Phys. Sin. 71 067801Google Scholar
[35] 王志全, 施卫 2022 71 188704Google Scholar
Wang Z Q, Shi W 2022 Acta Phys. Sin. 71 188704Google Scholar
[36] Habashi A, Ghobadi C, Nourinia J, R Naderali 2023 Opt. Commun. 547 129840Google Scholar
[37] 米利, 周宏伟, 孙祉伟, 刘丽霞, 徐升华 2013 62 134704Google Scholar
Mi L, Zhou H W, Sun Z W, Liu L X, Xu S H 2013 Acta Phys. Sin. 62 134704Google Scholar
[38] Worasawate D, Mautz J R, Arvas E 2003 IEEE Trans. Antennas Propag. 51 1077Google Scholar
[39] Yuceer M, Mautz J R, Arvas E 2005 IEEE Trans. Antennas Propag. 53 1163Google Scholar
[40] Demir V, Elsherbeni A Z, Arvas E 2005 IEEE Trans. Antennas Propag. 53 3374Google Scholar
[41] Kuzu L, Demir V, Elsherbeni A Z, Arvas E 2007 Prog. Electromagn. Res. 67 1Google Scholar
[42] Cooray M F R, Ciric I R 1993 J. Opt. Soc. Am. A 10 1197Google Scholar
[43] Ermutlu M E, Sihvola A H 1994 Prog. Electromagn. Res. 9 87Google Scholar
[44] Jaggard D L, Liu J C 1999 IEEE Trans. Antennas Propag. 47 1201Google Scholar
[45] Yan B, Liu C H, Zhang H Y, Shi Y 2015 Opt. Commun. 338 261Google Scholar
[46] Wang W J, Sun Y F, Zhang H Y 2017 Opt. Commun. 385 54Google Scholar
[47] Gao X, Zhang H 2017 Optik 129 43Google Scholar
[48] Zheng M, Zhang H Y, Sun Y F, Wang Z G 2015 J. Quant. Spectrosc. Ra. 151 192Google Scholar
[49] Li L W, Dan Y, Leong M S, Kong J A 1999 Prog. Electromagn. Res. 23 239Google Scholar
[50] Shang Q C, Wu Z S, Qu T, Li Z J, Bai L 2016 J. Quant. Spectrosc. Ra. 173 72Google Scholar
[51] Qu T, Wu Z S, Shang Q C, Wu J, Bai L 2018 J. Quant. Spectrosc. Ra. 217 363Google Scholar
[52] Bai J, Liu X, Ge C X, Li Z J, Xiao C, Wu Z S, Shang Q C 2024 Opt. Commun. 554 130136Google Scholar
[53] Edmonds A R, Mendlowitz H 1958 Phys. Today 11 34Google Scholar
[54] Gouesbet G, Gréhan G 1999 J Opt. A-Pure. Appl. Opt. 1 706Google Scholar
[55] Geng Y L, Wu X B, Li L W, Guan B R 2004 Phys. Rev. E 70 056609Google Scholar
[56] Lock J A, Gouesbet G 1994 J. Opt. Soc. Am. A 11 2503Google Scholar
[57] Gouesbet G, Gréhan G, Maheu B 1990 J. Opt. Soc. Am. A 7 998Google Scholar
[58] Doicu A, Wriedt T 1997 Appl. Opt. 36 2971Google Scholar
[59] Brown A J 2014 J. Opt. Soc. Am. A 31 2789Google Scholar
[60] Edmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton: Princeton University Press) pp24−37
[61] Lakhtakia A 1994 Beltrami Fields in Chiral Media (World Scientific Pub. Co. Inc. ) pp5−26
[62] Sarkar D, Halas N J 1997 Phys. Rev. E 56 1102Google Scholar
[63] Aden A L, Kerker M 1951 J. Appl. Phys. 22 1242Google Scholar
[64] Shang Q C, Wu Z S, Qu T, Li Z J, Bai L 2013 Opt. Express 21 8677Google Scholar
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图 2 具有相同偏振角的反向传播双高斯波束$zox$平面强度分布图 (a)${w_{01}}\ ( {w_{02}} ) = 1.7\lambda $; (b)${w_{01}} ({w_{02}}) = 1.8\lambda $; (c) ${w_{01}}({w_{02}}) = 1.9\lambda $
Fig. 2. Intensity distribution of counter propagating DGBs in $zox$ plane with different $ {w_1},\; {w_2} $: (a) ${w_{01}}({w_{02}}) = 1.7\lambda $; (b) ${w_{01}}({w_{02}}) = 1.8\lambda $; (c) ${w_{01}}({w_{02}}) = 1.9\lambda $.
图 3 与图2相比具有不同偏振角的反向传播双高斯波束$zox$平面强度分布图 (a)${w_{01}}({w_{02}}) = 1.7\lambda $; (b)${w_{01}}({w_{02}}) = 1.8\lambda $; (c) ${w_{01}}({w_{02}}) = 1.9\lambda $
Fig. 3. Intensity distribution of counter propagating DGBs in $zox$ plane with different ${\beta _1}$, ${\beta _2}$ compared with Fig. 2: (a) ${w_{01}}({w_{02}}) = 1.7\lambda $; (b) ${w_{01}}({w_{02}}) = 1.8\lambda $; (c) ${w_{01}}({w_{02}}) = 1.9\lambda $.
图 4 与图2相比具有不同入射角度的双高斯波束$zox$平面强度分布图 (a) ${w_{01}}({w_{02}}) = 1.7\lambda $; (b)${w_{01}}({w_{02}}) = 1.8\lambda $; (c)${w_{01}}({w_{02}}) = 1.9\lambda $
Fig. 4. Intensity distribution of counter propagating DGBs in $zox$ plane with different ${\alpha _1},\; {\alpha _2}$ compared with Fig. 2: (a) ${w_{01}}({w_{02}}) = 1.7\lambda $; (b) ${w_{01}}({w_{02}}) = 1.8\lambda $; (c) ${w_{01}}({w_{02}}) = 1.9\lambda $.
图 8 不同偏振角下, 双高斯波束入射双层手性球的辐射力随轴(横)向位置的变化曲线 (a)沿z轴辐射力${F_z}$;(b)沿x轴辐射力${F_x}$; (c)沿y轴辐射力${F_y}$
Fig. 8. Radiative force of DGBS in different polarization angles on double-layer chiral sphere; (a) Radiative force along the z-axis ${F_z}$; (b) radiative force along the x-axis ${F_x}$; (c) radiative force along the y-axis ${F_y}$.
图 9 不同内层手性下, 双高斯波束入射双层手性球的辐射力随轴(横)向位置的变化曲线 (a)沿z轴辐射力${F_z}$; (b)沿x轴辐射力${F_x}$; (c)沿y轴辐射力${F_y}$; (d)双高斯波束对双层手性球远场散射RCS
Fig. 9. Radiative force of DGBS in different inner layer chiral on double-layer chiral sphere: (a) z-axis radiative force ${F_z}$; (b) x-axis radiative force ${F_x}$; (c) y-axis radiative force ${F_y}$; (d) the RCS of dual Gaussian beams scattering double-layer chiral sphere.
图 10 不同外层手性下, 双高斯波束入射双层手性球的辐射力随轴(横)向位置的变化曲线 (a)沿z轴辐射力${F_z}$; (b)沿x轴辐射力${F_x}$; (c)沿y轴辐射力${F_y}$
Fig. 10. Radiative force of DGBS in different outer layer chiral on double-layer chiral sphere: (a) z-axis radiative force ${F_z}$; (b) x-axis radiative force ${F_x}$; (c) y-axis radiative force ${F_y}$.
图 11 不同球内层半径下, 双高斯波束入射双层手性球的辐射力随轴(横)向位置的变化曲线 (a)沿z轴辐射力${F_z}$; (b)沿x轴辐射力${F_x}$; (c)沿y轴辐射力${F_y}$
Fig. 11. Radiative force of DGBS in different radii of the inner layers on double-layer chiral sphere: (a) z-axis radiative force ${F_z}$; (b) x-axis radiative force ${F_x}$; (c) y-axis radiative force ${F_y}$.
图 12 不同球外层半径下, 双高斯波束入射双层手性球的辐射力随轴(横)向位置的变化曲线 (a)沿z轴辐射力${F_z}$; (b)沿x轴辐射力${F_x}$; (c)沿y轴辐射力${F_y}$
Fig. 12. Radiative force of DGBS in different radii of the outer layers on double-layer chiral sphere: (a) z-axis radiative force ${F_z}$; (b) x-axis radiative force ${F_x}$; (c) y-axis radiative force ${F_y}$.
图 13 不同偏振状态下, 双高斯波束入射双层手性球的辐射力随轴(横)向位置的变化曲线 (a)沿z轴辐射力${F_z}$; (b)沿x轴辐射力${F_x}$; (c)沿y轴辐射力${F_y}$
Fig. 13. Radiative force of DGBS in different polarization states on double-layer chiral sphere: (a) z-axis radiative force ${F_z}$; (b) x-axis radiative force ${F_x}$; (c) y-axis radiative force ${F_y}$.
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[1] Ashkin A 1970 Phys. Rev. Lett. 24 156Google Scholar
[2] Ashkin A 1980 Science 210 1081Google Scholar
[3] Leach J, Howard D, Roberts S, Gibson G, Gothard D, Cooper J, Buttery L 2009 J. Mod. Optic. 56 448Google Scholar
[4] Molloy J E, Dholakia K, Padgett M J 2003 J. Mod. Optic. 50 1501Google Scholar
[5] Parlatan U, Başar G, Başar G 2019 J. Mod. Optic. 66 228Google Scholar
[6] Jordan P, Clare H, Flendrig L, Leach J, Cooper J, Padgett M 2004 J. Mod. Optic. 51 627Google Scholar
[7] Tang Q, Liu P Z, Tang S 2022 Chin. Phys. B 31 044301Google Scholar
[8] Barton J P, Alexander D R, Schaub S A 1989 J. Appl. Phys. 66 4594Google Scholar
[9] Yang A H, Moore S D, Schmidt B S, Klug M, Lipson M, Erickson D 2009 Nature 457 71Google Scholar
[10] Padgett M, Bowman R 2011 Nat. Photon. 5 343Google Scholar
[11] Wang Z L, Yin J P 2008 Chin. Phys. B 17 2466Google Scholar
[12] Kiselev A D, Plutenko D O 2016 Phys. Rev. A 94 013804Google Scholar
[13] Zang Y C, Lin W J, Su C, Wu P F 2021 Chin. Phys. B 30 044301Google Scholar
[14] Dong F B, Chang C H, Jun F H, Yi W 2009 Chin. Phys. B 18 2853Google Scholar
[15] Ng J, Lin Z F, Chan C T 2010 Phys. Rev. Lett. 104 103601Google Scholar
[16] Liu X Y, Sun C, Deng D M 2021 Chin. Phys. B 30 024202Google Scholar
[17] 王焱, 彭妙, 程伟, 彭政, 成浩, 臧圣寅, 刘浩, 任孝东, 帅雨贝, 黄承志, 吴加贵, 杨俊波 2023 72 027801Google Scholar
Wang Y, Peng M, Cheng W, Peng Z, Cheng H, Zang S Y, Liu H, Ren X D, Shuai Y B, Huang C Z, Wu J G, Yang J B 2023 Acta Phys. Sin. 72 027801Google Scholar
[18] 殷杰, 陶超, 刘晓峻 2015 64 098102Google Scholar
Yin J, Tao C, Liu X J 2015 Acta Phys. Sin. 64 098102Google Scholar
[19] Ashkin A, Dziedzic J M 1971 Appl. Phys. Lett. 19 283Google Scholar
[20] Zemánek P, Jonáš A, Šrámek L, Liška M 1998 Opt. Commun. 151 273Google Scholar
[21] Zemánek P, Jonáš A, Liška M 2002 J. Opt. Soc. Am. A 19 1025Google Scholar
[22] Gauthier R C, Frangioudakis A 2000 Appl. Opt. 39 26Google Scholar
[23] Ren K F, Greha G, Gouesbet G 1994 Opt. Commun. 108 343Google Scholar
[24] Gouesbet G, Lock J A 1994 J. Opt. Soc. Am. A 11 2516Google Scholar
[25] Zemánek P, Jonáš A, Jákl P, Šerý M, Liška M 2003 Opt. Commun. 220 401Google Scholar
[26] Cizmar T, Garces-Chavez V, Dholakia K, Zemanek P 2004 Opt. Trap. Micro. 5514 643Google Scholar
[27] Van der Horst A, van Oostrum P D J, Moroz A, van Blaaderen A, Dogterom M 2008 Appl. Opt. 47 3196Google Scholar
[28] Zhao L, Li Y, Qi J, Xu J, Sun Q 2010 Opt. Express 18 5724Google Scholar
[29] Zhang T, Mahdy M R C, Dewan S S, Hossain M N, Rivy H M, Masud N, Jony Z R 2018 arXiv: 1811.01874 [physics. optics]
[30] Li Z J, Li S, Li H Y, Qu T, Shang Q C 2021 J. Opt. Soc. Am. A 38 616Google Scholar
[31] Wang S L, Liu X, Mourdikoudis S, Chen J, Fu W W, Sofer Z, Zhang Y, Zhang S P, Zheng G C 2022 ACS Nano. 16 19789Google Scholar
[32] 马晓亮, 李雄, 郭迎辉, 赵泽宇, 罗先刚 2017 66 147802Google Scholar
Ma X L, Li X, Guo Y H, Zhao Z Y, Luo X G 2017 Acta Phys. Sin. 66 147802Google Scholar
[33] Rohrbach A, Stelzer E H K 2001 J. Opt. Soc. Am. A 18 839Google Scholar
[34] 史书姝, 肖姗, 许秀来 2022 71 067801Google Scholar
Shi S S, Xiao S, Xu X L 2022 Acta Phys. Sin. 71 067801Google Scholar
[35] 王志全, 施卫 2022 71 188704Google Scholar
Wang Z Q, Shi W 2022 Acta Phys. Sin. 71 188704Google Scholar
[36] Habashi A, Ghobadi C, Nourinia J, R Naderali 2023 Opt. Commun. 547 129840Google Scholar
[37] 米利, 周宏伟, 孙祉伟, 刘丽霞, 徐升华 2013 62 134704Google Scholar
Mi L, Zhou H W, Sun Z W, Liu L X, Xu S H 2013 Acta Phys. Sin. 62 134704Google Scholar
[38] Worasawate D, Mautz J R, Arvas E 2003 IEEE Trans. Antennas Propag. 51 1077Google Scholar
[39] Yuceer M, Mautz J R, Arvas E 2005 IEEE Trans. Antennas Propag. 53 1163Google Scholar
[40] Demir V, Elsherbeni A Z, Arvas E 2005 IEEE Trans. Antennas Propag. 53 3374Google Scholar
[41] Kuzu L, Demir V, Elsherbeni A Z, Arvas E 2007 Prog. Electromagn. Res. 67 1Google Scholar
[42] Cooray M F R, Ciric I R 1993 J. Opt. Soc. Am. A 10 1197Google Scholar
[43] Ermutlu M E, Sihvola A H 1994 Prog. Electromagn. Res. 9 87Google Scholar
[44] Jaggard D L, Liu J C 1999 IEEE Trans. Antennas Propag. 47 1201Google Scholar
[45] Yan B, Liu C H, Zhang H Y, Shi Y 2015 Opt. Commun. 338 261Google Scholar
[46] Wang W J, Sun Y F, Zhang H Y 2017 Opt. Commun. 385 54Google Scholar
[47] Gao X, Zhang H 2017 Optik 129 43Google Scholar
[48] Zheng M, Zhang H Y, Sun Y F, Wang Z G 2015 J. Quant. Spectrosc. Ra. 151 192Google Scholar
[49] Li L W, Dan Y, Leong M S, Kong J A 1999 Prog. Electromagn. Res. 23 239Google Scholar
[50] Shang Q C, Wu Z S, Qu T, Li Z J, Bai L 2016 J. Quant. Spectrosc. Ra. 173 72Google Scholar
[51] Qu T, Wu Z S, Shang Q C, Wu J, Bai L 2018 J. Quant. Spectrosc. Ra. 217 363Google Scholar
[52] Bai J, Liu X, Ge C X, Li Z J, Xiao C, Wu Z S, Shang Q C 2024 Opt. Commun. 554 130136Google Scholar
[53] Edmonds A R, Mendlowitz H 1958 Phys. Today 11 34Google Scholar
[54] Gouesbet G, Gréhan G 1999 J Opt. A-Pure. Appl. Opt. 1 706Google Scholar
[55] Geng Y L, Wu X B, Li L W, Guan B R 2004 Phys. Rev. E 70 056609Google Scholar
[56] Lock J A, Gouesbet G 1994 J. Opt. Soc. Am. A 11 2503Google Scholar
[57] Gouesbet G, Gréhan G, Maheu B 1990 J. Opt. Soc. Am. A 7 998Google Scholar
[58] Doicu A, Wriedt T 1997 Appl. Opt. 36 2971Google Scholar
[59] Brown A J 2014 J. Opt. Soc. Am. A 31 2789Google Scholar
[60] Edmonds A R 1957 Angular Momentum in Quantum Mechanics (Princeton: Princeton University Press) pp24−37
[61] Lakhtakia A 1994 Beltrami Fields in Chiral Media (World Scientific Pub. Co. Inc. ) pp5−26
[62] Sarkar D, Halas N J 1997 Phys. Rev. E 56 1102Google Scholar
[63] Aden A L, Kerker M 1951 J. Appl. Phys. 22 1242Google Scholar
[64] Shang Q C, Wu Z S, Qu T, Li Z J, Bai L 2013 Opt. Express 21 8677Google Scholar
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