Abstract

Based on full-wave electromagnetic theory, we derive the zero-forward and zero-backward scattering conditions for radially anisotropic spheres within the quasi-static limit. We find that the near-field intensity can be tuned dramatically through the adjustment of the radial anisotropy, while the far-field light scattering diagrams are similar under the zero-forward or zero-backward scattering conditions. Generalized “Brewster’s angle” for anisotropic spheres is also derived, at which the scattering light is totally polarized. In addition, the high-quality polarized scattering wave and the tunable polarization conversion can be achieved for the radially anisotropic spheres.

© 2013 OSA

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    [CrossRef] [PubMed]
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2013 (1)

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, 1527 (2013).
[CrossRef] [PubMed]

2012 (5)

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Y. X. Ni, L. Gao, A. E. Miroshnichenko, and C. W. Qiu, “Non-Rayleigh scattering behavior for anisotropic Rayleigh particles,” Opt. Lett.37, 3390–3392 (2012).
[CrossRef]

H. L. Chen and L. Gao, “Anomalous electromagnetic scattering from radially anisotropic nanowires,” Phys. Rev. A86, 033825 (2012).
[CrossRef]

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

P. Bhatia and B. D. Gupta, “Fabrication and characterization of a surface plasmon resonance based fiber optic urea sensor for biomedical applications,” Sens. Actuators B161, 434–438 (2012).
[CrossRef]

2011 (3)

2010 (5)

2009 (2)

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of spin angular momentum of a light field,” Phys. Rev. Lett.102, 113602 (2009).
[CrossRef] [PubMed]

A. E. Miroshnichenko, “Non-Rayleigh limit of the Lorenz-Mie solution and supression of scattering by spheres of negative refractive index,” Phys. Rev. A80, 013808 (2009).
[CrossRef]

2008 (6)

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonance,” Phys. Rev. Lett.100, 043903 (2008).
[CrossRef] [PubMed]

B. García-Cámara, F. González, F. Moreno, and J. M. Saiz, “Exception for the zero-forward-scattering theory,” J. Opt. Soc. Am. A25, 2875–2878 (2008).
[CrossRef]

B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A25, 327–334 (2008).
[CrossRef]

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rew. E78, 046609 (2008).
[CrossRef]

C. W. Qiu and B. S. Luk’yanchuk, “Peculiarities in light scattering by spherical particles with radial anisotropy,” J. Opt. Soc. Am. A25, 1623–1628 (2008).
[CrossRef]

B. S. Luk’yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A92, 773–776 (2008).
[CrossRef]

2007 (4)

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

L. Gao and X. P. Xu, “Second- and third-harmonic generations for a nondilute suspension of coated particles with radial dielectric anisotropy,” Eur. Phys. J. B55, 403–409 (2007).
[CrossRef]

J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express15, 2622–2653 (2007).
[CrossRef] [PubMed]

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

2006 (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

M. I. Tribelsky and B. S. Luk’yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett.97, 263902 (2006).
[CrossRef]

T. Ambjornsson, G. Mukhopadhyay, S. P. Apell, and M. Kall, “Resonant coupling between localized plasmons and anisotropic molecular coatings in ellipsoidal metal nanoparticles,” Phys. Rew. B73, 085412 (2006).
[CrossRef]

2005 (1)

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

2004 (1)

Z. Liu, Z. Lin, and S. T. Chui, “Electromagnetic scattering by spherically negative-refractive-index particles: low frequency resonance and localization parameters,” Phys. Rev. E69, 016609 (2004).
[CrossRef]

2001 (1)

V. L. Sukhorukov, G. Meedt, M. Kurschner, and U. Zimmermann, “A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane,” J. Electrost.50, 191–204 (2001).
[CrossRef]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

1991 (1)

1983 (1)

1973 (1)

1908 (1)

G. Mie, “Contributions to the optics of turbid media, particularly colloidal metal solutions,” Ann. Phys.25, 377–445 (1908).
[CrossRef]

1871 (1)

L. Rayleigh, “On the light from the sky, its polarization and color appendix,” Philos. Mag.41, 107–120 (1871).

Alaverdyan, Y.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Albaladejo, S.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of spin angular momentum of a light field,” Phys. Rev. Lett.102, 113602 (2009).
[CrossRef] [PubMed]

Albella, P.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Alcaraz de la Osa, R.

Alu, A.

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

Alù, A.

A. Alù and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem,” J. Nanophoton.4, 041590 (2010).
[CrossRef]

Ambjornsson, T.

T. Ambjornsson, G. Mukhopadhyay, S. P. Apell, and M. Kall, “Resonant coupling between localized plasmons and anisotropic molecular coatings in ellipsoidal metal nanoparticles,” Phys. Rew. B73, 085412 (2006).
[CrossRef]

Apell, S. P.

T. Ambjornsson, G. Mukhopadhyay, S. P. Apell, and M. Kall, “Resonant coupling between localized plasmons and anisotropic molecular coatings in ellipsoidal metal nanoparticles,” Phys. Rew. B73, 085412 (2006).
[CrossRef]

Argyropoulos, C.

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

Bhatia, P.

P. Bhatia and B. D. Gupta, “Fabrication and characterization of a surface plasmon resonance based fiber optic urea sensor for biomedical applications,” Sens. Actuators B161, 434–438 (2012).
[CrossRef]

Bohren, C. F.

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

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th (expanded) ed. (Cambridge, 1999).

Chan, C. T.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Chantada, L.

Chen, H. L.

H. L. Chen and L. Gao, “Anomalous electromagnetic scattering from radially anisotropic nanowires,” Phys. Rev. A86, 033825 (2012).
[CrossRef]

Chen, P. Y.

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

Chui, S. T.

Z. Liu, Z. Lin, and S. T. Chui, “Electromagnetic scattering by spherically negative-refractive-index particles: low frequency resonance and localization parameters,” Phys. Rev. E69, 016609 (2004).
[CrossRef]

D’Aguanno, G.

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

Dahlin, A.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Dean, C. E.

Digman, M. J.

Durant, S.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Engheta, N.

A. Alù and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem,” J. Nanophoton.4, 041590 (2010).
[CrossRef]

Eyraud, C.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Fan, X.

Fang, N.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Flach, S.

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonance,” Phys. Rev. Lett.100, 043903 (2008).
[CrossRef] [PubMed]

Froufe-Pérez, L. S.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Fu, Y. H.

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, 1527 (2013).
[CrossRef] [PubMed]

Fung, T. H.

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rew. E78, 046609 (2008).
[CrossRef]

Gao, L.

H. L. Chen and L. Gao, “Anomalous electromagnetic scattering from radially anisotropic nanowires,” Phys. Rev. A86, 033825 (2012).
[CrossRef]

Y. X. Ni, L. Gao, A. E. Miroshnichenko, and C. W. Qiu, “Non-Rayleigh scattering behavior for anisotropic Rayleigh particles,” Opt. Lett.37, 3390–3392 (2012).
[CrossRef]

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rew. E78, 046609 (2008).
[CrossRef]

L. Gao and X. P. Xu, “Second- and third-harmonic generations for a nondilute suspension of coated particles with radial dielectric anisotropy,” Eur. Phys. J. B55, 403–409 (2007).
[CrossRef]

Garcia-Camara, B.

B. Garcia-Camara, J. M. Saiz, F. Gonzalez, and F. Moreno, “Nanoparticles with unconventional scattering properties: size effects,” Opt. Commun.283, 490–496 (2010).
[CrossRef]

García-Cámara, B.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

B. García-Cámara, R. Alcaraz de la Osa, J. M. Saiz, F. González, and F. Moreno, “Directionality in scattering by nanoparticles: Kerker’s null-scattering conditions revisited,” Opt. Lett.36, 728–730 (2011).
[CrossRef] [PubMed]

B. García-Cámara, F. González, and F. Moreno, “Linear polarization degree for detecting magnetic properties of small particles,” Opt. Lett.35, 4084–4086 (2010).
[CrossRef] [PubMed]

B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A25, 327–334 (2008).
[CrossRef]

B. García-Cámara, F. González, F. Moreno, and J. M. Saiz, “Exception for the zero-forward-scattering theory,” J. Opt. Soc. Am. A25, 2875–2878 (2008).
[CrossRef]

Geffrin, J. M.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Giles, C. L.

Gómez-Medina, R.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

M. Nieto-Vesperinas, R. Gómez-Medina, and J. J. Sáenz, “Angle-supressed scattering and optical forces on submicrometer dieletric particles,” J. Opt. Soc. Am. A28, 54–60 (2011).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express18, 11428–11443 (2010).
[CrossRef] [PubMed]

Gonzalez, F.

B. Garcia-Camara, J. M. Saiz, F. Gonzalez, and F. Moreno, “Nanoparticles with unconventional scattering properties: size effects,” Opt. Commun.283, 490–496 (2010).
[CrossRef]

González, F.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

B. García-Cámara, R. Alcaraz de la Osa, J. M. Saiz, F. González, and F. Moreno, “Directionality in scattering by nanoparticles: Kerker’s null-scattering conditions revisited,” Opt. Lett.36, 728–730 (2011).
[CrossRef] [PubMed]

B. García-Cámara, F. González, and F. Moreno, “Linear polarization degree for detecting magnetic properties of small particles,” Opt. Lett.35, 4084–4086 (2010).
[CrossRef] [PubMed]

B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A25, 327–334 (2008).
[CrossRef]

B. García-Cámara, F. González, F. Moreno, and J. M. Saiz, “Exception for the zero-forward-scattering theory,” J. Opt. Soc. Am. A25, 2875–2878 (2008).
[CrossRef]

Gorbach, A. V.

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonance,” Phys. Rev. Lett.100, 043903 (2008).
[CrossRef] [PubMed]

Gordon, J. A.

Gupta, B. D.

P. Bhatia and B. D. Gupta, “Fabrication and characterization of a surface plasmon resonance based fiber optic urea sensor for biomedical applications,” Sens. Actuators B161, 434–438 (2012).
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Hao, J. M.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Hook, F.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Huffman, D. R.

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

Jiang, T.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Kall, M.

T. Ambjornsson, G. Mukhopadhyay, S. P. Apell, and M. Kall, “Resonant coupling between localized plasmons and anisotropic molecular coatings in ellipsoidal metal nanoparticles,” Phys. Rew. B73, 085412 (2006).
[CrossRef]

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Kerker, M.

Kivshar, Y. S.

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonance,” Phys. Rev. Lett.100, 043903 (2008).
[CrossRef] [PubMed]

Kong, J. A.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Kurschner, M.

V. L. Sukhorukov, G. Meedt, M. Kurschner, and U. Zimmermann, “A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane,” J. Electrost.50, 191–204 (2001).
[CrossRef]

Kuznetsov, A. I.

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, 1527 (2013).
[CrossRef] [PubMed]

Laroche, M.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of spin angular momentum of a light field,” Phys. Rev. Lett.102, 113602 (2009).
[CrossRef] [PubMed]

Lee, H.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Lin, Z.

Z. Liu, Z. Lin, and S. T. Chui, “Electromagnetic scattering by spherically negative-refractive-index particles: low frequency resonance and localization parameters,” Phys. Rev. E69, 016609 (2004).
[CrossRef]

Litman, A.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Liu, Z.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Z. Liu, Z. Lin, and S. T. Chui, “Electromagnetic scattering by spherically negative-refractive-index particles: low frequency resonance and localization parameters,” Phys. Rev. E69, 016609 (2004).
[CrossRef]

Luk’yanchuk, B.

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, 1527 (2013).
[CrossRef] [PubMed]

Luk’yanchuk, B. S.

X. Fan, Z. Shen, and B. S. Luk’yanchuk, “Huge light scattering from anisotropic spherical particles,” Opt. Express18, 24868–24880 (2010).
[CrossRef] [PubMed]

C. W. Qiu and B. S. Luk’yanchuk, “Peculiarities in light scattering by spherical particles with radial anisotropy,” J. Opt. Soc. Am. A25, 1623–1628 (2008).
[CrossRef]

B. S. Luk’yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A92, 773–776 (2008).
[CrossRef]

M. I. Tribelsky and B. S. Luk’yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett.97, 263902 (2006).
[CrossRef]

Marques, M. I.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of spin angular momentum of a light field,” Phys. Rev. Lett.102, 113602 (2009).
[CrossRef] [PubMed]

Marston, P. L.

Meedt, G.

V. L. Sukhorukov, G. Meedt, M. Kurschner, and U. Zimmermann, “A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane,” J. Electrost.50, 191–204 (2001).
[CrossRef]

Mie, G.

G. Mie, “Contributions to the optics of turbid media, particularly colloidal metal solutions,” Ann. Phys.25, 377–445 (1908).
[CrossRef]

Miroshnichenko, A. E.

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, 1527 (2013).
[CrossRef] [PubMed]

Y. X. Ni, L. Gao, A. E. Miroshnichenko, and C. W. Qiu, “Non-Rayleigh scattering behavior for anisotropic Rayleigh particles,” Opt. Lett.37, 3390–3392 (2012).
[CrossRef]

A. E. Miroshnichenko, “Non-Rayleigh limit of the Lorenz-Mie solution and supression of scattering by spheres of negative refractive index,” Phys. Rev. A80, 013808 (2009).
[CrossRef]

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonance,” Phys. Rev. Lett.100, 043903 (2008).
[CrossRef] [PubMed]

Monticone, F.

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

Moreno, F.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

B. García-Cámara, R. Alcaraz de la Osa, J. M. Saiz, F. González, and F. Moreno, “Directionality in scattering by nanoparticles: Kerker’s null-scattering conditions revisited,” Opt. Lett.36, 728–730 (2011).
[CrossRef] [PubMed]

B. García-Cámara, F. González, and F. Moreno, “Linear polarization degree for detecting magnetic properties of small particles,” Opt. Lett.35, 4084–4086 (2010).
[CrossRef] [PubMed]

B. Garcia-Camara, J. M. Saiz, F. Gonzalez, and F. Moreno, “Nanoparticles with unconventional scattering properties: size effects,” Opt. Commun.283, 490–496 (2010).
[CrossRef]

B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A25, 327–334 (2008).
[CrossRef]

B. García-Cámara, F. González, F. Moreno, and J. M. Saiz, “Exception for the zero-forward-scattering theory,” J. Opt. Soc. Am. A25, 2875–2878 (2008).
[CrossRef]

Mukhopadhyay, G.

T. Ambjornsson, G. Mukhopadhyay, S. P. Apell, and M. Kall, “Resonant coupling between localized plasmons and anisotropic molecular coatings in ellipsoidal metal nanoparticles,” Phys. Rew. B73, 085412 (2006).
[CrossRef]

Ni, Y. X.

Nieto-Vesperinas, M.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

M. Nieto-Vesperinas, R. Gómez-Medina, and J. J. Sáenz, “Angle-supressed scattering and optical forces on submicrometer dieletric particles,” J. Opt. Soc. Am. A28, 54–60 (2011).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express18, 11428–11443 (2010).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

Picus, Y.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Qiu, C. W.

Y. X. Ni, L. Gao, A. E. Miroshnichenko, and C. W. Qiu, “Non-Rayleigh scattering behavior for anisotropic Rayleigh particles,” Opt. Lett.37, 3390–3392 (2012).
[CrossRef]

C. W. Qiu and B. S. Luk’yanchuk, “Peculiarities in light scattering by spherical particles with radial anisotropy,” J. Opt. Soc. Am. A25, 1623–1628 (2008).
[CrossRef]

B. S. Luk’yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A92, 773–776 (2008).
[CrossRef]

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rew. E78, 046609 (2008).
[CrossRef]

Ran, L. X.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Rayleigh, L.

L. Rayleigh, “On the light from the sky, its polarization and color appendix,” Philos. Mag.41, 107–120 (1871).

Rindzevicius, T.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Roth, J.

Sáenz, J. J.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

M. Nieto-Vesperinas, R. Gómez-Medina, and J. J. Sáenz, “Angle-supressed scattering and optical forces on submicrometer dieletric particles,” J. Opt. Soc. Am. A28, 54–60 (2011).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express18, 11428–11443 (2010).
[CrossRef] [PubMed]

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of spin angular momentum of a light field,” Phys. Rev. Lett.102, 113602 (2009).
[CrossRef] [PubMed]

Saiz, J. M.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Sctherland, D. S.

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Shen, Z.

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Suárez-Lacalle, I.

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

Sukhorukov, V. L.

V. L. Sukhorukov, G. Meedt, M. Kurschner, and U. Zimmermann, “A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane,” J. Electrost.50, 191–204 (2001).
[CrossRef]

Sun, C.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Tribelsky, M. I.

M. I. Tribelsky, S. Flach, A. E. Miroshnichenko, A. V. Gorbach, and Y. S. Kivshar, “Light scattering by a finite obstacle and Fano resonance,” Phys. Rev. Lett.100, 043903 (2008).
[CrossRef] [PubMed]

M. I. Tribelsky and B. S. Luk’yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett.97, 263902 (2006).
[CrossRef]

Vaillon, R.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[CrossRef]

Videen, G.

Wang, D. S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th (expanded) ed. (Cambridge, 1999).

Xiong, Y.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Xu, X. P.

L. Gao and X. P. Xu, “Second- and third-harmonic generations for a nondilute suspension of coated particles with radial dielectric anisotropy,” Eur. Phys. J. B55, 403–409 (2007).
[CrossRef]

Yu, K. W.

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rew. E78, 046609 (2008).
[CrossRef]

Yu, Y. F.

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, 1527 (2013).
[CrossRef] [PubMed]

Yuan, Y.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Zhang, X.

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

Zhou, L.

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

Zimmermann, U.

V. L. Sukhorukov, G. Meedt, M. Kurschner, and U. Zimmermann, “A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane,” J. Electrost.50, 191–204 (2001).
[CrossRef]

Ziolkowski, R. W.

Ann. Phys. (1)

G. Mie, “Contributions to the optics of turbid media, particularly colloidal metal solutions,” Ann. Phys.25, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A (1)

B. S. Luk’yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A92, 773–776 (2008).
[CrossRef]

Eur. Phys. J. B (1)

L. Gao and X. P. Xu, “Second- and third-harmonic generations for a nondilute suspension of coated particles with radial dielectric anisotropy,” Eur. Phys. J. B55, 403–409 (2007).
[CrossRef]

J. Electrost. (1)

V. L. Sukhorukov, G. Meedt, M. Kurschner, and U. Zimmermann, “A single-shell model for biological cells extended to account for the dielectric anisotropy of the plasma membrane,” J. Electrost.50, 191–204 (2001).
[CrossRef]

J. Nanophoton. (2)

A. Alù and N. Engheta, “How does zero forward-scattering in magnetodielectric nanoparticles comply with the optical theorem,” J. Nanophoton.4, 041590 (2010).
[CrossRef]

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton.5, 053512 (2011).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Nano Lett. (2)

Z. Liu, S. Durant, H. Lee, Y. Picus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett.7, 403–408 (2007).
[CrossRef] [PubMed]

T. Rindzevicius, Y. Alaverdyan, A. Dahlin, F. Hook, D. S. Sctherland, and M. Kall, “Plasmonics sensing characteristis of single nanometric holes,” Nano Lett.5, 2335–2339 (2005).
[CrossRef] [PubMed]

Nat. Commun. (2)

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3, 1171 (2012).
[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, 1527 (2013).
[CrossRef] [PubMed]

Opt. Commun. (1)

B. Garcia-Camara, J. M. Saiz, F. Gonzalez, and F. Moreno, “Nanoparticles with unconventional scattering properties: size effects,” Opt. Commun.283, 490–496 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Philos. Mag. (1)

L. Rayleigh, “On the light from the sky, its polarization and color appendix,” Philos. Mag.41, 107–120 (1871).

Phys. Rev. A (2)

H. L. Chen and L. Gao, “Anomalous electromagnetic scattering from radially anisotropic nanowires,” Phys. Rev. A86, 033825 (2012).
[CrossRef]

A. E. Miroshnichenko, “Non-Rayleigh limit of the Lorenz-Mie solution and supression of scattering by spheres of negative refractive index,” Phys. Rev. A80, 013808 (2009).
[CrossRef]

Phys. Rev. E (1)

Z. Liu, Z. Lin, and S. T. Chui, “Electromagnetic scattering by spherically negative-refractive-index particles: low frequency resonance and localization parameters,” Phys. Rev. E69, 016609 (2004).
[CrossRef]

Phys. Rev. Lett. (6)

J. M. Hao, Y. Yuan, L. X. Ran, T. Jiang, J. A. Kong, C. T. Chan, and L. Zhou, “Manipulating electromagnetic wave polarizations by anisotropic metamaterials,” Phys. Rev. Lett.99, 063908 (2007).
[CrossRef] [PubMed]

C. Argyropoulos, P. Y. Chen, F. Monticone, G. D’Aguanno, and A. Alu, “Nonlinear plasmonic cloaks to realize giant all-optical scattering switching,” Phys. Rev. Lett.108, 263905 (2012).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef] [PubMed]

Phys. Rew. B (1)

T. Ambjornsson, G. Mukhopadhyay, S. P. Apell, and M. Kall, “Resonant coupling between localized plasmons and anisotropic molecular coatings in ellipsoidal metal nanoparticles,” Phys. Rew. B73, 085412 (2006).
[CrossRef]

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[CrossRef]

Science (1)

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[CrossRef] [PubMed]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Scattering intensity (logarithmic scale) as a function of the scattering angle calculated by full-wave scattering theory for spherical particles with q = 0.1, μr = μt = 2 when they are illuminated from left by a TM-polarized incident wave. The physical parameters satisfy either zero-backward scattering condition (a) or zero-forward scattering condition (b).

Fig. 2
Fig. 2

Distribution of the total near-field intensity (log10 |E|) calculated by full-wave scattering theory for the sphere with q = 0.6, μr = μt = 2 under the zero-backward scattering condition: εr = εt = 2 for (a) and (b), εr = 5t = 7/5 for (c) and (d), εr = 1, εt = 3 for (e) and (f). Note that (a), (c), (e): show the fields both inside and outside the spheres, whereas (b), (d), (f) show the fields inside the spheres only.

Fig. 3
Fig. 3

Distribution of the total near-field intensity (log10 |E|) calculated by full-wave scattering theory for the spheres with q = 0.6, μr = μt = 2 under the zero-forward scattering condition: εr = εt = 0.4 for (a), εr = 2, εt = 6/25 for (b), and εr = 1/4, εt = 13/25 for (c).

Fig. 4
Fig. 4

The polar diagram of absolute value of polarization (a) and the degree of polarization versus θ (b) for q = 0.5, εr = −1, μr = −5, and for different radial anisotropy.

Fig. 5
Fig. 5

Same as in Fig. 4 but with q = 1.

Equations (15)

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a n = ε t ψ n ( q ) ψ ν n ( m q ) μ t ψ n ( q ) ψ ν n ( m q ) ε t ξ n ( q ) ψ ν n ( m q ) μ t ξ n ( q ) ψ ν n ( m q ) ,
b n = ε t ψ n ( q ) ψ γ n ( m q ) μ t ψ n ( q ) ψ γ n ( m q ) ε t ξ n ( q ) ψ γ n ( m q ) μ t ξ n ( q ) ψ γ n ( m q ) ,
ν n = n ( n + 1 ) Ae + 1 4 1 2 and γ n = n ( n + 1 ) Am + 1 4 1 2 ,
I = I 1 + I 2 = λ 2 4 π 2 r 2 { | S | 2 sin 2 ϕ + | S | | | 2 cos 2 ϕ } ,
S = n = 1 2 n + 1 n ( n + 1 ) [ a n π n ( cos θ ) + b n τ n ( cos θ ) ] ,
S | | = n = 1 2 n + 1 n ( n + 1 ) [ a n τ n ( cos θ ) + b n π n ( cos θ ) ] ,
R = S , | | ( 180 ° ) = n = 1 2 n + 1 n ( n + 1 ) ( a n b n ) ,
T = S , | | ( 0 ° ) = n = 1 2 n + 1 n ( n + 1 ) ( a n + b n ) .
ψ 1 ( x ) ~ x 2 3 x 4 30 , ξ 1 ( x ) ~ i x i x 2 + x 2 3 , ψ ν ( x ) ~ π Γ ( ν + 3 2 ) ( x 2 ) ν + 1 π Γ ( ν + 5 2 ) ( x 2 ) ν + 3 .
a 1 2 ε t ( ν 1 + 1 ) 3 i ( ε t + ν 1 + 1 ) q 3 and b 1 2 μ t ( γ 1 + 1 ) 3 i ( μ t + γ 1 + 1 ) q 3 .
ε t ν 1 + 1 = μ t γ 1 + 1 and ε t ν 1 + 1 = 2 ( γ 1 + 1 ) μ t 4 μ t + ( γ 1 + 1 ) .
a 1 1 1 + 3 i ( ε t + ν 1 + 1 ) [ 2 ε t ( ν 1 + 1 ) ] q 3 and b 1 1 1 + 3 i ( μ t + γ 1 + 1 ) [ 2 μ t ( γ 1 + 1 ) ] q 3 .
ε t ν 1 + 1 3 [ 2 ( γ 1 + 1 ) μ t ] 2 i q 3 [ 2 μ t ( γ 1 + 1 ) ] 3 [ 4 μ t + ( γ 1 + 1 ) ] 4 i q 3 [ 2 μ t ( γ 1 + 1 ) ] .
cos θ = ( 2 ε t ( ν 1 + 1 ) ) ( μ t + ( γ 1 + 1 ) ) ( 2 μ t ( γ 1 + 1 ) ) ( ε t + ( ν 1 + 1 ) ) , cos θ | | = 1 cos θ ,
P = | S | 2 | S | | | 2 | S | 2 + | S | | | 2 .

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