Abstract

We present an in-depth analysis of the resonant intermixing between optical orbital and spin angular momentum of Laguerre-Gaussian (LG) beams, mediated by chiral clusters made of silicon nanospheres. In particular, we establish a relationship between the spin and orbital quantum numbers characterizing the LG beam and the order q of the rotation symmetry group 𝒞q of the cluster of nanospheres for which resonantly enhanced coupling between the two components of the optical angular momentum is observed. Thus, similar to the case of diffraction grating-mediated transfer of linear momentum between optical beams, we demonstrate that clusters of nanospheres that are invariant to specific rotation transformations can efficiently transfer optical angular momentum between LG beams with different quantum numbers. We also discuss the conditions in which the resonant interaction between LG beams and a chiral cluster of nanospheres leads to the generation of superchiral light.

© 2016 Optical Society of America

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References

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2014 (1)

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref] [PubMed]

2013 (4)

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating Far-Field Orbital Angular Momenta from Near-Field Optical Chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref] [PubMed]

E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological Shaping of Light by Closed-Path Nanoslits,” Phys. Rev. Lett. 111, 193901 (2013).
[Crossref] [PubMed]

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

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, 724–726 (2013).
[Crossref] [PubMed]

2012 (4)

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

A. S. Rury and R. Freeling, “Mie scattering of purely azimuthal Laguerre-Gauss beams: Angular-momentum-induced transparency,” Phys. Rev. A 86, 053830 (2012).
[Crossref]

M. Schaferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Pys. Rev. X 2, 031010 (2012).

2011 (2)

2010 (3)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[Crossref]

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

2009 (4)

2008 (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

2007 (2)

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75, 063409 (2007).
[Crossref]

2006 (2)

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

E. Centeno and D. Felbacq, “Second-harmonic emission in two-dimensional photonic crystals,” J. Opt. Soc. Am. B 23, 2257–2264 (2006).
[Crossref]

2005 (1)

2004 (1)

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

2002 (2)

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[Crossref]

1995 (1)

1994 (1)

1993 (1)

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

1992 (3)

N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation Of Optical-Phase Singularities By Computer-Generated Holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

V. S. Liberman and B. Y. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

1986 (2)

R. Y. Chiao and Y. S. Wu, “Manifestations of Berrys topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[Crossref] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berrys topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[Crossref] [PubMed]

1984 (1)

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Adachi, H.

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75, 063409 (2007).
[Crossref]

Aiello, A.

K. Y. Bliokh, A. Aiello, and M. A. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), p. 174.
[Crossref]

Aizpurua, J.

Akahoshi, S.

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75, 063409 (2007).
[Crossref]

Allen, L.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Almazov, A.

Alonso, M. A.

K. Y. Bliokh, A. Aiello, and M. A. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), p. 174.
[Crossref]

Andrews, D. L.

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press-Elsevier, Burlington, 2008).

Auger, J. C.

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Biener, G.

Biris, C. G.

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[Crossref]

Bliokh, K. Y.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

K. Y. Bliokh, “Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium,” J. Opt. A 11, 094009 (2009).
[Crossref]

K. Y. Bliokh, A. Aiello, and M. A. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), p. 174.
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Brasselet, E.

E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological Shaping of Light by Closed-Path Nanoslits,” Phys. Rev. Lett. 111, 193901 (2013).
[Crossref] [PubMed]

Brongersma, M. L.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref] [PubMed]

Cai, X.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Centeno, E.

Chantada, L.

Cheah, K. W.

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

Chen, J.

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Chen, S.

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

Chiao, R. Y.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berrys topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[Crossref] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berrys topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[Crossref] [PubMed]

Chiu, D. T.

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17, 23316–23322 (2009).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

Cohen, A. E.

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
[Crossref] [PubMed]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Ding, J.

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Dogariu, A.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref] [PubMed]

Dregely, D.

M. Schaferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Pys. Rev. X 2, 031010 (2012).

Drezet, A.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating Far-Field Orbital Angular Momenta from Near-Field Optical Chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref] [PubMed]

Ebbesen, T. W.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating Far-Field Orbital Angular Momenta from Near-Field Optical Chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

Elfstrom, H.

Fan, P.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref] [PubMed]

Felbacq, D.

Freeling, R.

A. S. Rury and R. Freeling, “Mie scattering of purely azimuthal Laguerre-Gauss beams: Angular-momentum-induced transparency,” Phys. Rev. A 86, 053830 (2012).
[Crossref]

Froufe-Perez, L. S.

Garcia-Etxarri, A.

Genet, C.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating Far-Field Orbital Angular Momenta from Near-Field Optical Chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Gervinskas, G.

E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological Shaping of Light by Closed-Path Nanoslits,” Phys. Rev. Lett. 111, 193901 (2013).
[Crossref] [PubMed]

Giessen, H.

M. Schaferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Pys. Rev. X 2, 031010 (2012).

Gomez-Medina, R.

Gorodetski, Y.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating Far-Field Orbital Angular Momenta from Near-Field Optical Chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Gouesbet, G.

Grehan, G.

Guo, C.

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Haefner, D.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref] [PubMed]

Hanna, S.

Hasman, E.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref] [PubMed]

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, 724–726 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

Heckenberg, N.

Hentschel, M.

M. Schaferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Pys. Rev. X 2, 031010 (2012).

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, 1998).

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Juodkazis, S.

E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological Shaping of Light by Closed-Path Nanoslits,” Phys. Rev. Lett. 111, 193901 (2013).
[Crossref] [PubMed]

Kang, M.

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

Khersonskii, V. K.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[Crossref]

Khonina, S.

Kleiner, V.

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, 724–726 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002).
[Crossref]

Kotlyar, V.

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Laci, A.

M. Mishchenko, L. Travis, and A. Laci, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University, Cambridge, 2002).

Lafait, J.

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[Crossref]

Li, G.

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

Li, J.

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

Li, Y.

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Liberman, V. S.

V. S. Liberman and B. Y. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Lin, D.

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref] [PubMed]

Lock, J. A.

Lopez, C.

Maguid, E.

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, 724–726 (2013).
[Crossref] [PubMed]

Manzo, C.

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

Marchesini, S.

Marrucci, L.

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

Maystre, D.

McDuff, R.

McGloin, D.

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17, 23316–23322 (2009).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

Mishchenko, M.

M. Mishchenko, L. Travis, and A. Laci, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University, Cambridge, 2002).

Miyakawa, K.

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75, 063409 (2007).
[Crossref]

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[Crossref]

Murakami, S.

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

Nagaosa, N.

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

Nienhuis, G.

G. Nienhuis, “Angular Momentum and Vortices in Optics,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D. L. Andrews, ed. (Elsevier, New York, 2008), p. 19.
[Crossref]

Nieto-Vesperinas, M.

Niv, A.

OBrien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Onoda, M.

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

Ozeri, D.

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, 724–726 (2013).
[Crossref] [PubMed]

Panoiu, N. C.

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[Crossref]

Paparo, D.

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

Pun, E. Y.

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

Rury, A. S.

A. S. Rury and R. Freeling, “Mie scattering of purely azimuthal Laguerre-Gauss beams: Angular-momentum-induced transparency,” Phys. Rev. A 86, 053830 (2012).
[Crossref]

Saenz, J. J.

Schaferling, M.

M. Schaferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Pys. Rev. X 2, 031010 (2012).

Scheffold, F.

Seniutinas, G.

E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological Shaping of Light by Closed-Path Nanoslits,” Phys. Rev. Lett. 111, 193901 (2013).
[Crossref] [PubMed]

Shapiro, D.

Shitrit, N.

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, 724–726 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Simpson, S. H.

Smith, C.

Soifer, V.

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Stein, B.

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

Stout, B.

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[Crossref]

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Sukhov, S.

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref] [PubMed]

Tang, Y.

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
[Crossref] [PubMed]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Tatian, B.

Tayeb, G.

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Tomita, A.

A. Tomita and R. Y. Chiao, “Observation of Berrys topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[Crossref] [PubMed]

Travis, L.

M. Mishchenko, L. Travis, and A. Laci, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University, Cambridge, 2002).

Turunen, J.

Vanderveen, H.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[Crossref]

Veksler, D.

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, 724–726 (2013).
[Crossref] [PubMed]

Wang, H.

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

Wang, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Wang, X.

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

White, A.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Wu, Y. S.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berrys topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[Crossref] [PubMed]

Yu, S.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Yulevich, I.

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, 724–726 (2013).
[Crossref] [PubMed]

Zeldovich, B. Y.

V. S. Liberman and B. Y. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Zhang, S.

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

Zhao, Y.

Y. Zhao, D. Shapiro, D. McGloin, D. T. Chiu, and S. Marchesini, “Direct observation of the transfer of orbital angular momentum to metal particles from a focused circularly polarized Gaussian beam,” Opt. Express 17, 23316–23322 (2009).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

Zhu, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Appl. Opt. (2)

J. Mod. Opt. (1)

B. Stout, J. C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[Crossref]

J. Opt. A (1)

K. Y. Bliokh, “Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium,” J. Opt. A 11, 094009 (2009).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

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

Opt. Commun. (1)

M. W. Beijersbergen, L. Allen, H. Vanderveen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (5)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

A. S. Rury and R. Freeling, “Mie scattering of purely azimuthal Laguerre-Gauss beams: Angular-momentum-induced transparency,” Phys. Rev. A 86, 053830 (2012).
[Crossref]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

H. Adachi, S. Akahoshi, and K. Miyakawa, “Orbital motion of spherical microparticles trapped in diffraction patterns of circularly polarized light,” Phys. Rev. A 75, 063409 (2007).
[Crossref]

V. S. Liberman and B. Y. Zeldovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199–5207 (1992).
[Crossref] [PubMed]

Phys. Rev. B (1)

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81, 195102 (2010).
[Crossref]

Phys. Rev. Lett. (11)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref] [PubMed]

X. Wang, J. Chen, Y. Li, J. Ding, C. Guo, and H. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[Crossref]

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

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating Far-Field Orbital Angular Momenta from Near-Field Optical Chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref] [PubMed]

E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological Shaping of Light by Closed-Path Nanoslits,” Phys. Rev. Lett. 111, 193901 (2013).
[Crossref] [PubMed]

Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109, 013901 (2012).
[Crossref] [PubMed]

D. Haefner, S. Sukhov, and A. Dogariu, “Spin Hall effect of light in spherical geometry,” Phys. Rev. Lett. 102, 123903 (2009).
[Crossref] [PubMed]

R. Y. Chiao and Y. S. Wu, “Manifestations of Berrys topological phase for the photon,” Phys. Rev. Lett. 57, 933–936 (1986).
[Crossref] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berrys topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937–940 (1986).
[Crossref] [PubMed]

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

Pys. Rev. X (1)

M. Schaferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Pys. Rev. X 2, 031010 (2012).

Science (5)

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
[Crossref] [PubMed]

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref] [PubMed]

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, 724–726 (2013).
[Crossref] [PubMed]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Other (7)

G. Nienhuis, “Angular Momentum and Vortices in Optics,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D. L. Andrews, ed. (Elsevier, New York, 2008), p. 19.
[Crossref]

K. Y. Bliokh, A. Aiello, and M. A. Alonso, “Spin-orbit interactions of light in isotropic media,” in The Angular Momentum of Light, D. L. Andrews and M. Babiker, eds. (Cambridge University, 2012), p. 174.
[Crossref]

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley, 1998).

M. Mishchenko, L. Travis, and A. Laci, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University, Cambridge, 2002).

D. L. Andrews, Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Academic Press-Elsevier, Burlington, 2008).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Schematic of the system geometry and incident LG0l beam. The optical beam propagates along the z-axis, the cluster of spherical nanoparticles is located in the (x, y)-plane, which coincide with the plane where the beam waist reaches its minimum value, and the center of the cluster is on the z-axis.

Fig. 2
Fig. 2

Spatial distribution of the total electric field and its ρ-, φ-, and z-components, determined in the focal plane of a LG04 beam. Top, middle, and bottom panels correspond to σ = 0, σ = 1, and σ = −1, respectively, whereas the wavelength is λ = 2515nm.

Fig. 3
Fig. 3

Top panel shows the the spectra Cext(λ) calculated for a 500 nm silicon nanosphere, normalized to the cross-section area of the sphere. The excitation is a linearly polarized PW (σ = 0). The index in parentheses in the legend indicates the multipole order. In the bottom panel, the left (right) set of three plots represents, from left to right, the spatial distribution of the total electric (magnetic) field, the longitudinal component of the electric (magnetic) field of the first-order TE mode, TE(1), and the azimuthal component of the electric (magnetic) field of the first-order TM mode, TM(1).

Fig. 4
Fig. 4

Spectra of Cext vs. wavelength, calculated for a silicon nanosphere with radius a = 500nm. The incoming excitations are PWs and LG beams, LG0l, l = 1, 2, 3, 4. From top to bottom, the panels correspond to the spin number σ = 0, σ = 1, and σ = −1. The vertical line corresponds to the wavelength of the TE(2) resonance, λ = 2487nm.

Fig. 5
Fig. 5

Spectrum of normalized Cext corresponding to chiral structures with q-fold symmetry made of silicon nanospheres with radius, a = 500nm. Left panels correspond to q = 3, incident PW with σ = 0, and LG04 beams with σ = ±1, whereas right panels correspond to q = 4, incident PW with σ = 0, and LG05 beams with σ = ±1. The vertical line corresponds to the wavelength of the TE(2) resonance, λ = 2515nm.

Fig. 6
Fig. 6

Spatial distribution of the magnitude of the electric field calculated at λ = 2515nm. From left to right column, the panels correspond to a PW with σ = 1, a PW with σ = −1, a LG04 with σ = 1, and a LG04 with σ = −1. The top and botom panels correspond to the total field and the field corresponding to the TE(2) quadrupole mode, respectively.

Fig. 7
Fig. 7

Maps of the spatial distribution of the enhancement of the optical chirality, ��̂(r), calculated at λ = 2515nm for PWs and LG04 beams with σ = ±1, incident on a chiral cluster of silicon nanoparticles with radius, a = 500nm, with 3-fold rotational symmetry. In the left (logarithmic plots) and middle panels the optical chirality is normalized to the optical chirality of a circularly-polarized PW, whereas in the right panel (logarithmic plots) the optical chirality is normalized to the local optical chirality corresponding to an incident circularly-polarized PW with the same value of σ.

Equations (21)

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E inc ( r ) = m n [ a m n M m n ( 1 ) ( k r ) + b m n N m n ( 1 ) ( k r ) ] ,
H inc ( r ) = i ζ m n [ b m n M m n ( 1 ) ( k r ) + a m n N m n ( 1 ) ( k r ) ] .
E inc ( r ) = E 0 inc e i k inc r ,
a m n PW = 4 π i n X m n * ( k ^ inc ) E 0 inc ,
b m n PW = 4 π i n 1 Z m n * ( k ^ inc ) E 0 inc ,
X m n ( r ^ ) = ( θ i m sin θ d 0 m n ( θ ) φ ^ d d θ d 0 m n ( θ ) ) e i m φ ,
Z m n ( r ^ ) = ( θ ^ d d θ d 0 m n ( θ ) + φ ^ i m sin θ d 0 m n ( θ ) ) e i m φ .
g 1 , n = e α n s 2 ,
g 3 , n = g 1 , n + α n s 4 ( 3 α n s 2 ) e α n s 2 ,
g 5 , n = g 3 , n + α n 2 s 8 ( 10 5 α n s 2 + 1 2 α n 2 s 4 ) e α n s 2 ,
C m n LG ( l , 1 ) = { G m n PW , m = l + 1 0 , otherwise , C m n LG ( l , 1 ) = { G m n PW , m = l + 1 0 , otherwise ,
E j sca ( r j P ) = m n [ p m n j M m n ( 3 ) ( k r j P ) + q m n j N m n ( 3 ) ( k r j P ) ] , r j P > a ,
H j sca ( r j P ) = i ζ m n [ q m n j M m n ( 3 ) ( k r j P ) + p m n j N m n ( 3 ) ( k r j P ) ] , r j P > a .
F tot ( r P ) = F inc ( r P ) + j = 1 N F sca ( r j P ) .
E j int ( r j P ) = m n [ c m n j M m n ( 1 ) ( k r j P ) + d m n j N m n ( 1 ) ( k r j P ) ] , r j P < a ,
H j int ( r j P ) = i ζ m n [ d m n j M m n ( 1 ) ( k r j P ) + c m n j N m n ( 1 ) ( k r j P ) ] , r j P < a .
[ p j q j ] = T sca j { β ( j , 0 ) [ a b ] + l = 1 l j N α ( j , l ) [ p j q j ] } , j = 1 , , N ,
S V = U ,
S = ( I T 1 α ( 1 , 2 ) T 1 α ( 1 , 3 ) T 1 α ( 1 , N ) T 2 α ( 2 , 1 ) I T 2 α ( 2 , 3 ) T 2 α ( 2 , N ) T 3 α ( 3 , 1 ) T 3 α ( 3 , 2 ) I T 3 α ( 3 , N ) T N α ( N , 1 ) T N α ( N , 2 ) T N α ( N , 3 ) I ) .
𝒞 ( r ) = ε ω 2 Im [ E * ( r ) B ( r ) ] .
𝒞 ^ ( r ) = 𝒞 { PW ± , LG ± } tot ( r ) | 𝒞 PW ± { tot , inc } ( r ) | .

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