Abstract

We present the photonic spin Hall effect on an ellipsoidal Rayleigh particle, which amounts to a polarization-dependent shift in scattering far-field. Based on the dipole model, we demonstrate that such shift is unavoidable when the light incidence is inclined with respect to the main axis of the ellipsoidal Rayleigh particle. The result has general validity and can be applied to metal and dielectric materials. In addition, the photonic spin Hall effect also manifests itself in the optical force and torque exerted on the particle, which is promising for precision metrology, spin-optics devices and optical driven micro-machines. Due to wide existence of the Rayleigh particles in nature, we believe that our findings might provide a useful toolset for investigating polarization-dependent scattering of particles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (6)

G. Araneda, S. Walser, Y. Colombe, D. B. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeute, “Wavelength-scale errors in optical localization due to spin-orbit coupling of light,” Nat. Phys. 15(1), 17–21 (2019).
[Crossref]

T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
[Crossref]

J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. U. S. A. 116(23), 11137–11140 (2019).
[Crossref]

M. Neugebauer, S. Nechayev, M. Vorndran, G. Leuchs, and P. Banzer, “Weak measurement enhanced spin hall effect of light for particle displacement sensing,” Nano Lett. 19(1), 422–425 (2019).
[Crossref]

R. Shi, D. L. Gao, H. Hu, Y. Q. Wang, and L. Gao, “Enhanced broadband spin hall effects by core-shell nanoparticles,” Opt. Express 27(4), 4808–4817 (2019).
[Crossref]

D. V. Zhirihin, S. V. Li, D. Y. Sokolov, A. P. Slobozhanyuk, M. A. Gorlach, and A. B. Khanikaev, “Photonic spin hall effect mediated by bianisotropy,” Opt. Lett. 44(7), 1694–1697 (2019).
[Crossref]

2018 (2)

D. Gao, R. Shi, A. E. Miroshnichenko, and L. Gao, “Enhanced spin hall effect of light in spheres with dual symmetry,” Laser Photonics Rev. 12(11), 1800130 (2018).
[Crossref]

X. Zhou, L. Sheng, and X. Ling, “Photonic spin hall effect enabled refractive index sensor using weak measurements,” Sci. Rep. 8(1), 1221 (2018).
[Crossref]

2017 (1)

X. Ling, X. Zhou, K. Huang, Y. Liu, C. W. Qiu, H. Luo, and S. Wen, “Recent advances in the spin hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017).
[Crossref]

2016 (1)

2015 (1)

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light: Sci. Appl. 4(5), e290 (2015).
[Crossref]

2014 (6)

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3(5), e167 (2014).
[Crossref]

M. Neugebauer, P. Banzer, T. Bauer, S. Orlov, N. Lindlein, A. Aiello, and G. Leuchs, “Geometric spin hall effect of light in tightly focused polarization-tailored light beams,” Phys. Rev. A 89(1), 013840 (2014).
[Crossref]

D. O’Connor, P. Ginzburg, F. J. Rodríguezfortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5(1), 5327 (2014).
[Crossref]

J. Korger, A. Aiello, V. Chille, P. Banzer, and G. Leuchs, “Observation of the geometric spin hall effect of light,” Phys. Rev. Lett. 112(11), 113902 (2014).
[Crossref]

A. Banerjee, J. Soni, N. Ghosh, S. D. Gupta, and S. Mansha, “Giant Goos–Hänchen shift in scattering: the role of interfering localized plasmon modes,” Opt. Lett. 39(14), 4100–4103 (2014).
[Crossref]

P. L. Truong, X. Ma, and S. J. Sim, “Resonant rayleigh light scattering of single au nanoparticles with different sizes and shapes,” Nanoscale 6(4), 2307–2315 (2014).
[Crossref]

2013 (6)

J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38(13), 2295–2297 (2013).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-hänchen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun. 4(1), 1527 (2013).
[Crossref]

Y. Liu and X. Zhang, “Metasurfaces for manipulating surface plasmons,” Appl. Phys. Lett. 103(14), 141101 (2013).
[Crossref]

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref]

G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13(9), 4148–4151 (2013).
[Crossref]

2012 (5)

L.-J. Kong, S.-X. Qian, Z.-C. Ren, X.-L. Wang, and H.-T. Wang, “Effects of orbital angular momentum on the geometric spin hall effect of light,” Phys. Rev. A 85(3), 035804 (2012).
[Crossref]

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012).
[Crossref]

L.-J. Kong, X.-L. Wang, S.-M. Li, Y. Li, J. Chen, B. Gu, and H.-T. Wang, “Spin hall effect of reflected light from an air-glass interface around the brewster’s angle,” Appl. Phys. Lett. 100(7), 071109 (2012).
[Crossref]

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
[Crossref]

2011 (4)

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19(27), 26132–26149 (2011).
[Crossref]

H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84(4), 043806 (2011).
[Crossref]

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric spin hall effect of light at polarizing interfaces,” Appl. Phys. B: Lasers Opt. 102(3), 427–432 (2011).
[Crossref]

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin hall effects in plasmonic chains,” Nano Lett. 11(5), 2038–2042 (2011).
[Crossref]

2010 (2)

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett. 104(25), 253601 (2010).
[Crossref]

2009 (2)

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref]

2008 (5)

K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Geometrodynamics of spinning light,” Nat. Photonics 2(12), 748–753 (2008).
[Crossref]

H. Onur and K. Paul, “Observation of the spin hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008).
[Crossref]

F. Nori, “Geometrical optics: the dynamics of spinning light,” Nat. Photonics 2(12), 717–718 (2008).
[Crossref]

K. Y. Bliokh, G. Yuri, K. Vladimir, and H. Erez, “Coriolis effect in optics: unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in goos-hanchen and imbert-fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008).
[Crossref]

2007 (1)

K. Y. Bliokh and Y. P. Bliokh, “Polarization, transverse shifts, and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75(6), 066609 (2007).
[Crossref]

2006 (2)

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96(7), 073903 (2006).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

2004 (1)

M. Onoda, S. Murakami, and A. N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref]

2003 (1)

P. A. Belov, S. I. Maslovski, K. R. Simovski, and S. A. Tretyakov, “A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer,” Tech. Phys. Lett. 29(9), 718–720 (2003).
[Crossref]

2002 (1)

2001 (1)

1974 (1)

J. E. Sipe and J. V. Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9(5), 1806–1822 (1974).
[Crossref]

Aiello, A.

M. Neugebauer, P. Banzer, T. Bauer, S. Orlov, N. Lindlein, A. Aiello, and G. Leuchs, “Geometric spin hall effect of light in tightly focused polarization-tailored light beams,” Phys. Rev. A 89(1), 013840 (2014).
[Crossref]

J. Korger, A. Aiello, V. Chille, P. Banzer, and G. Leuchs, “Observation of the geometric spin hall effect of light,” Phys. Rev. Lett. 112(11), 113902 (2014).
[Crossref]

K. Y. Bliokh and A. Aiello, “Goos-hänchen and imbert-fedorov beam shifts: an overview,” J. Opt. 15(1), 014001 (2013).
[Crossref]

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric spin hall effect of light at polarizing interfaces,” Appl. Phys. B: Lasers Opt. 102(3), 427–432 (2011).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in goos-hanchen and imbert-fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008).
[Crossref]

Alonso, M. A.

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19(27), 26132–26149 (2011).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 063825 (2010).
[Crossref]

Araneda, G.

G. Araneda, S. Walser, Y. Colombe, D. B. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeute, “Wavelength-scale errors in optical localization due to spin-orbit coupling of light,” Nat. Phys. 15(1), 17–21 (2019).
[Crossref]

Banerjee, A.

Banzer, P.

M. Neugebauer, S. Nechayev, M. Vorndran, G. Leuchs, and P. Banzer, “Weak measurement enhanced spin hall effect of light for particle displacement sensing,” Nano Lett. 19(1), 422–425 (2019).
[Crossref]

M. Neugebauer, S. Grosche, S. Rothau, G. Leuchs, and P. Banzer, “Lateral spin transport in paraxial beams of light,” Opt. Lett. 41(15), 3499–3502 (2016).
[Crossref]

J. Korger, A. Aiello, V. Chille, P. Banzer, and G. Leuchs, “Observation of the geometric spin hall effect of light,” Phys. Rev. Lett. 112(11), 113902 (2014).
[Crossref]

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N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin hall effects in plasmonic chains,” Nano Lett. 11(5), 2038–2042 (2011).
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X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light: Sci. Appl. 4(5), e290 (2015).
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M. Neugebauer, S. Nechayev, M. Vorndran, G. Leuchs, and P. Banzer, “Weak measurement enhanced spin hall effect of light for particle displacement sensing,” Nano Lett. 19(1), 422–425 (2019).
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G. Araneda, S. Walser, Y. Colombe, D. B. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeute, “Wavelength-scale errors in optical localization due to spin-orbit coupling of light,” Nat. Phys. 15(1), 17–21 (2019).
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X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
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X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
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G. Li, M. Kang, S. Chen, S. Zhang, E. Y.-B. Pun, K. W. Cheah, and J. Li, “Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light,” Nano Lett. 13(9), 4148–4151 (2013).
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Y. Liu and X. Zhang, “Metasurfaces for manipulating surface plasmons,” Appl. Phys. Lett. 103(14), 141101 (2013).
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J. Zhou, H. Qian, C.-F. Chen, J. Zhao, G. Li, Q. Wu, H. Luo, S. Wen, and Z. Liu, “Optical edge detection based on high-efficiency dielectric metasurface,” Proc. Natl. Acad. Sci. U. S. A. 116(23), 11137–11140 (2019).
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X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light: Sci. Appl. 4(5), e290 (2015).
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H. Luo, X. Zhou, W. Shu, S. Wen, and D. Fan, “Enhanced and switchable spin hall effect of light near the brewster angle on reflection,” Phys. Rev. A 84(4), 043806 (2011).
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T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
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T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
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T. Zhu, Y. Lou, Y. Zhou, J. Zhang, J. Huang, Y. Li, H. Luo, S. Wen, S. Zhu, Q. Gong, M. Qiu, and Z. Ruan, “Generalized spatial differentiation from the spin hall effect of light and its application in image processing of edge detection,” Phys. Rev. Appl. 11(3), 034043 (2019).
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Appl. Phys. B: Lasers Opt. (1)

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, “Geometric spin hall effect of light at polarizing interfaces,” Appl. Phys. B: Lasers Opt. 102(3), 427–432 (2011).
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Appl. Phys. Lett. (3)

X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012).
[Crossref]

L.-J. Kong, X.-L. Wang, S.-M. Li, Y. Li, J. Chen, B. Gu, and H.-T. Wang, “Spin hall effect of reflected light from an air-glass interface around the brewster’s angle,” Appl. Phys. Lett. 100(7), 071109 (2012).
[Crossref]

Y. Liu and X. Zhang, “Metasurfaces for manipulating surface plasmons,” Appl. Phys. Lett. 103(14), 141101 (2013).
[Crossref]

Chin. Phys. Lett. (1)

X.-H. Ling, H.-L. Luo, M. Tang, and S.-C. Wen, “Enhanced and tunable spin hall effect of light upon reflection of one-dimensional photonic crystal with a defect layer,” Chin. Phys. Lett. 29(7), 074209 (2012).
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J. Opt. (1)

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Laser Photonics Rev. (1)

D. Gao, R. Shi, A. E. Miroshnichenko, and L. Gao, “Enhanced spin hall effect of light in spheres with dual symmetry,” Laser Photonics Rev. 12(11), 1800130 (2018).
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Light: Sci. Appl. (2)

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3(5), e167 (2014).
[Crossref]

X. Ling, X. Zhou, X. Yi, W. Shu, Y. Liu, S. Chen, H. Luo, S. Wen, and D. Fan, “Giant photonic spin hall effect in momentum space in a structured metamaterial with spatially varying birefringence,” Light: Sci. Appl. 4(5), e290 (2015).
[Crossref]

Nano Lett. (3)

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Figures (5)

Fig. 1.
Fig. 1. The three-dimensional schematic of the system and polarization dependence of light scattering. (a) The blue and red rotatory arrows represent the incident left and right circularly polarized lights, respectively. The incident directions are all in the $y$-$z$ plane. $\beta$ is the angle between the incident light and the $z$-axis. The blue and red lines represent the scattering directions of left and right circularly polarized lights, respectively. (b) The far-field scattering diagram of a metallic (Au) ellipsoidal Rayleigh particle in ${Oxy}$ plane simulated in COMSOL Multiphysics software for different incident polarized lights. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 3 nm and 30 nm, respectively. The wavelength of the incident light is 650 nm and the incident angle $\beta$ is $45^\circ$. $\sigma \equiv 0$, +1, and -1 indicate the linear, right-circular and left-circular polarizations, respectively.
Fig. 2.
Fig. 2. The simulated cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field. The material of the ellipsoidal Rayleigh particle is metallic (Au). The wavelength of the incident light is 650 nm. The incident angle is $45^\circ$. (a) and (b) The changes in the cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field with the radius of the particle in $y$ direction ($a_y$) through simulation. The radii of the particle in the $x$ and $z$ directions are 10 nm and 30 nm, respectively. (c) and (d) The changes in the cosine values of azimuthal angles and elevation angles for the centroids of scattering far-field with the radius of the particle in $x$ direction ($a_x$) through simulation. The radii of the particle in the $y$ and $z$ directions are 10 nm and 30 nm, respectively.
Fig. 3.
Fig. 3. The dependences of the cosine values of azimuthal angles for the centroids of scattering far-field on the incident angle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 20 nm and 30 nm, respectively.
Fig. 4.
Fig. 4. The simulated optical forces exerted on the metallic (Au) ellipsoidal Rayleigh particle. The wavelength of the incident light is 650 nm. (a) - (c) The changes in the $x$, $y$ and $z$ components of optical forces with the radius of the particle in $y$ direction ($a_y$). The radii of the particle in the $x$ and $z$ directions are 10 nm and 30 nm, respectively. The incident angle is $45^\circ$. (d) - (f) The changes in the $x$, $y$ and $z$ components of optical forces with the incident angle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 20 nm and 30 nm, respectively.
Fig. 5.
Fig. 5. The simulated optical torques exerted on the metallic (Au) ellipsoidal Rayleigh particle. The wavelength of the incident light is 650 nm. (a) - (c) The changes in the $x$, $y$ and $z$ components of optical torques with the radius of the particle in $y$ direction ($a_y$). The radii of the particle in the $x$ and $z$ directions are 10 nm and 30 nm, respectively. The incident angle is $45^\circ$. (d) - (f) The changes in the $x$, $y$ and $z$ components of optical torques with the incident angle. The radii of the particle in the $x$, $y$ and $z$ directions are 10 nm, 20 nm and 30 nm, respectively.

Equations (21)

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E inc = E 0 ( σ i x ^ + cos β y ^ + sin β z ^ ) exp [ i k 0 ( sin β y + cos β z ) ] ,
H inc = H 0 ( x ^ + σ i cos β y ^ + σ i sin β z ^ ) exp [ i k 0 ( sin β y + cos β z ) ] ,
E sc = k 0 2 4 π ϵ 0 e i k 0 r r [ ( n ^ × p ) × n ^ 1 c n ^ × m ] ,
H sc = 1 η 0 ( n ^ × E sc ) ,
p = α ¯ ee E inc ,
m = α ¯ mm H inc ,
α ¯ ee = ( α ee1 x x + i α ee2 x x 0 0 0 α ee1 y y + i α ee2 y y 0 0 0 α ee1 z z + i α ee2 z z ) ,
α ¯ mm = ( α mm1 x x + i α mm2 x x 0 0 0 α mm1 y y + i α mm2 y y 0 0 0 α mm1 z z + i α mm2 z z ) .
S = 1 2 Re ( E sc × H sc ) .
cos φ = cos φ S r r 2 sin θ d θ d φ S r r 2 sin θ d θ d φ ,
cos θ = cos θ S r r 2 sin θ d θ d φ S r r 2 sin θ d θ d φ ,
cos φ σ sin 2 β η 0 W 3 η 0 2 W 1 + ( 1 / c 2 ) W 2 ,
cos θ cos β η 0 W 4 η 0 2 W 1 + ( 1 / c 2 ) W 2 ,
W 1 = ( α ee1 x x 2 + α ee2 x x 2 ) + ( α ee1 y y 2 + α ee2 y y 2 ) cos 2 β + ( α ee1 z z 2 + α ee2 z z 2 ) sin 2 β ,
W 2 = ( α mm1 x x 2 + α mm2 x x 2 ) + ( α mm1 y y 2 + α mm2 y y 2 ) cos 2 β + ( α mm1 z z 2 + α mm2 z z 2 ) sin 2 β ,
W 3 = ( α mm2 z z α ee1 y y α mm1 z z α ee2 y y ) ( α mm2 y y α ee1 z z α mm1 y y α ee2 z z ) ,
W 4 = ( α mm1 y y α ee1 x x + α mm2 y y α ee2 x x ) + ( α mm1 x x α ee1 y y + α mm2 x x α ee2 y y ) .
cos φ = σ sin 2 β C 1 C 2 + C 3 cos 2 β ,
F = S n ^ T ¯ d S ,
T ¯ = 1 2 Re [ ϵ E E + μ H H 1 2 ( ϵ E E + μ H H ) I ¯ ] ,
Γ = S n ^ ( T ¯ × r ) d S ,

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