Abstract

We present an exact solution to the problem of electromagnetic scattering by nanosphere clusters embedded in a liquid crystal cell, based on the Mie theory. The dependence of the scattering property on the structure parameters is investigated in detail. It is shown that strong transverse light currents at the optical frequency can be obtained from these complex structures. Furthermore, we find that sign reversal of the transverse light current can be realized by changing frequency and voltage. The physical origins of these phenomena have been analyzed. The transverse light current for subwavelength nanoscale dimensions is of practical significance. Thus, the application of these phenomena to optical devices is anticipated.

© 2018 Chinese Laser Press

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    [Crossref]
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  7. B. A. van Tiggelen, R. Maynard, and T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
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2017 (1)

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the Hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).
[Crossref]

2014 (3)

M. Zhang and X. Zhang, “Electric field tunable photonic Hall effect with liquid crystals,” Phys. Lett. A 378, 1571–1577 (2014).
[Crossref]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

2013 (1)

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

2012 (3)

J. W. Taylor, L. K. Kurihara, and L. J. Martinez-Miranda, “Interaction of a bi-molecular liquid crystal film with functionalized nanoparticles,” Appl. Phys. Lett. 100, 173115 (2012).
[Crossref]

J. A. Scholl, A. l. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref]

J. Xu and X. Zhang, “Second harmonic generation in three-dimensional structures based on homogeneous centrosymmetric metallic spheres,” Opt. Express 20, 1668–1684 (2012).
[Crossref]

2011 (1)

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115, 19470–19475 (2011).
[Crossref]

2009 (1)

M. Briane and G. W. Milton, “Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient,” Arch. Ration. Mech. Anal. 193, 715–736 (2009).
[Crossref]

2006 (1)

Q. Zhao, L. Kang, B. Li, and J. Zhou, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89, 221918 (2006).
[Crossref]

2005 (1)

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

2004 (1)

Z. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[Crossref]

2003 (1)

H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E 67, 056607 (2003).
[Crossref]

2002 (1)

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

2001 (1)

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

2000 (1)

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

1999 (1)

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[Crossref]

1998 (2)

G. L. J. A. Rikken, A. Sparenberg, and B. A. van Tiggelen, “Photonic magneto-transport,” Physica B 246–247, 188–194 (1998).
[Crossref]

D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rikken, and A. Sparenberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
[Crossref]

1997 (1)

A. Sparenberg, G. L. J. A. Rikken, and B. A. van Tiggelen, “Observation of photonic magnetoresistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[Crossref]

1996 (2)

G. L. J. A. Rikken and B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature 381, 54–55 (1996).
[Crossref]

B. A. van Tiggelen, R. Maynard, and T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[Crossref]

1995 (3)

B. A. van Tiggelen, “Transverse diffusion of light in Faraday-active media,” Phys. Rev. Lett. 75, 422–424 (1995).
[Crossref]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[Crossref]

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
[Crossref]

1993 (1)

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[Crossref]

1988 (1)

F. C. MacKintosh and S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[Crossref]

1985 (1)

1973 (1)

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
[Crossref]

1961 (1)

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
[Crossref]

1954 (1)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
[Crossref]

Alexander, R. W.

Bell, R. J.

Berreman, D. W.

Bohren, C. F.

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

Bozhevolnyi, S. I.

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

Briane, M.

M. Briane and G. W. Milton, “Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient,” Arch. Ration. Mech. Anal. 193, 715–736 (2009).
[Crossref]

Chan, C. T.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Christensen, T.

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

Chui, S. T.

Z. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[Crossref]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
[Crossref]

David, C.

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115, 19470–19475 (2011).
[Crossref]

Dintinger, J.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Dionne, J. A.

J. A. Scholl, A. l. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref]

Feldmann, J.

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

Friedman, B.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
[Crossref]

García de Abajo, F. J.

C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115, 19470–19475 (2011).
[Crossref]

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[Crossref]

Ha, Y.-K.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Harmon, B. N.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[Crossref]

Huffman, D. R.

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

Jauho, A.-P.

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

John, S.

F. C. MacKintosh and S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[Crossref]

Kadic, M.

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the Hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).
[Crossref]

Kang, L.

Q. Zhao, L. Kang, B. Li, and J. Zhou, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89, 221918 (2006).
[Crossref]

Kee, C.-S.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Kern, C.

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the Hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).
[Crossref]

Kienzler, T.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Kim, J.-E.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Klar, T. A.

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

Koh, A. l.

J. A. Scholl, A. l. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref]

Kurihara, L. K.

J. W. Taylor, L. K. Kurihara, and L. J. Martinez-Miranda, “Interaction of a bi-molecular liquid crystal film with functionalized nanoparticles,” Appl. Phys. Lett. 100, 173115 (2012).
[Crossref]

Lacoste, D.

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rikken, and A. Sparenberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
[Crossref]

Lee, J.-C.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Li, B.

Q. Zhao, L. Kang, B. Li, and J. Zhou, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89, 221918 (2006).
[Crossref]

Lim, H.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Lin, Z.

Z. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[Crossref]

Lin, Z. F.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Liu, F.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Long, L. L.

MacKintosh, F. C.

F. C. MacKintosh and S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[Crossref]

Martinez-Miranda, L. J.

J. W. Taylor, L. K. Kurihara, and L. J. Martinez-Miranda, “Interaction of a bi-molecular liquid crystal film with functionalized nanoparticles,” Appl. Phys. Lett. 100, 173115 (2012).
[Crossref]

Maynard, R.

B. A. van Tiggelen, R. Maynard, and T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[Crossref]

Mehl, G. H.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Milton, G. W.

M. Briane and G. W. Milton, “Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient,” Arch. Ration. Mech. Anal. 193, 715–736 (2009).
[Crossref]

Moroz, A.

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[Crossref]

Mortensen, N. A.

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

Muller, J.

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

Ng, J.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Nieuwenhuizen, T. M.

B. A. van Tiggelen, R. Maynard, and T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[Crossref]

Ordal, M. A.

Park, H. Y.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Querry, M. R.

Raza, S.

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

Rikken, G. L. J. A.

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

G. L. J. A. Rikken, A. Sparenberg, and B. A. van Tiggelen, “Photonic magneto-transport,” Physica B 246–247, 188–194 (1998).
[Crossref]

D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rikken, and A. Sparenberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
[Crossref]

A. Sparenberg, G. L. J. A. Rikken, and B. A. van Tiggelen, “Observation of photonic magnetoresistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[Crossref]

G. L. J. A. Rikken and B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature 381, 54–55 (1996).
[Crossref]

Rockstuhl, C.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Russek, J.

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
[Crossref]

Scharf, T.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Scholl, J. A.

J. A. Scholl, A. l. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref]

Sheng, P.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

Søndergaard, T.

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

Sonnichsen, C.

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

Sparenberg, A.

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

G. L. J. A. Rikken, A. Sparenberg, and B. A. van Tiggelen, “Photonic magneto-transport,” Physica B 246–247, 188–194 (1998).
[Crossref]

D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rikken, and A. Sparenberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
[Crossref]

A. Sparenberg, G. L. J. A. Rikken, and B. A. van Tiggelen, “Observation of photonic magnetoresistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[Crossref]

Stein, A.

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
[Crossref]

Stratton, A.

A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Takeda, H.

H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E 67, 056607 (2003).
[Crossref]

Tang, B.-J.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Taylor, J. W.

J. W. Taylor, L. K. Kurihara, and L. J. Martinez-Miranda, “Interaction of a bi-molecular liquid crystal film with functionalized nanoparticles,” Appl. Phys. Lett. 100, 173115 (2012).
[Crossref]

Ungar, G.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

van Tiggelen, B. A.

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

G. L. J. A. Rikken, A. Sparenberg, and B. A. van Tiggelen, “Photonic magneto-transport,” Physica B 246–247, 188–194 (1998).
[Crossref]

D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rikken, and A. Sparenberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
[Crossref]

A. Sparenberg, G. L. J. A. Rikken, and B. A. van Tiggelen, “Observation of photonic magnetoresistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[Crossref]

B. A. van Tiggelen, R. Maynard, and T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[Crossref]

G. L. J. A. Rikken and B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature 381, 54–55 (1996).
[Crossref]

B. A. van Tiggelen, “Transverse diffusion of light in Faraday-active media,” Phys. Rev. Lett. 75, 422–424 (1995).
[Crossref]

von Plessen, G.

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

von Poschinger, H.

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

Wang, X. D.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[Crossref]

Wegener, M.

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the Hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).
[Crossref]

Wiebel, S.

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

Wubs, M.

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

Xu, J.

Xu, Y. L.

Yan, W.

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

Yang, Y.-C.

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

Yoshino, K.

H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E 67, 056607 (2003).
[Crossref]

Yu, Q. L.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[Crossref]

Zeng, X.

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Zhang, M.

M. Zhang and X. Zhang, “Electric field tunable photonic Hall effect with liquid crystals,” Phys. Lett. A 378, 1571–1577 (2014).
[Crossref]

Zhang, X.

M. Zhang and X. Zhang, “Electric field tunable photonic Hall effect with liquid crystals,” Phys. Lett. A 378, 1571–1577 (2014).
[Crossref]

J. Xu and X. Zhang, “Second harmonic generation in three-dimensional structures based on homogeneous centrosymmetric metallic spheres,” Opt. Express 20, 1668–1684 (2012).
[Crossref]

Zhang, X.-G.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[Crossref]

Zhao, Q.

Q. Zhao, L. Kang, B. Li, and J. Zhou, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89, 221918 (2006).
[Crossref]

Zhou, J.

Q. Zhao, L. Kang, B. Li, and J. Zhou, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89, 221918 (2006).
[Crossref]

ACS Nano (1)

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8, 1745–1758 (2014).
[Crossref]

Adv. Mater. (1)

J. Dintinger, B.-J. Tang, X. Zeng, F. Liu, T. Kienzler, G. H. Mehl, G. Ungar, C. Rockstuhl, and T. Scharf, “A self-organized anisotropic liquid-crystal plasmonic metamaterial,” Adv. Mater. 25, 1999–2004 (2013).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

Q. Zhao, L. Kang, B. Li, and J. Zhou, “Tunable negative refraction in nematic liquid crystals,” Appl. Phys. Lett. 89, 221918 (2006).
[Crossref]

J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann, “Electrically controlled light scattering with single metal nanoparticles,” Appl. Phys. Lett. 81, 171–173 (2002).
[Crossref]

J. W. Taylor, L. K. Kurihara, and L. J. Martinez-Miranda, “Interaction of a bi-molecular liquid crystal film with functionalized nanoparticles,” Appl. Phys. Lett. 100, 173115 (2012).
[Crossref]

Y.-K. Ha, Y.-C. Yang, J.-E. Kim, H. Y. Park, C.-S. Kee, H. Lim, and J.-C. Lee, “Tunable omnidirectional reflection bands and defect modes of a one-dimensional photonic band gap structure with liquid crystals,” Appl. Phys. Lett. 79, 15–17 (2001).
[Crossref]

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M. Briane and G. W. Milton, “Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient,” Arch. Ration. Mech. Anal. 193, 715–736 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

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

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C. David and F. J. García de Abajo, “Spatial nonlocality in the optical response of metal nanoparticles,” J. Phys. Chem. C 115, 19470–19475 (2011).
[Crossref]

Nat. Commun. (1)

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nat. Commun. 5, 3809 (2014).
[Crossref]

Nature (2)

J. A. Scholl, A. l. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
[Crossref]

G. L. J. A. Rikken and B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature 381, 54–55 (1996).
[Crossref]

Opt. Express (1)

Phys. Lett. A (1)

M. Zhang and X. Zhang, “Electric field tunable photonic Hall effect with liquid crystals,” Phys. Lett. A 378, 1571–1577 (2014).
[Crossref]

Phys. Rev. B (5)

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[Crossref]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
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F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[Crossref]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: numerical simulations,” Phys. Rev. B 72, 085130 (2005).
[Crossref]

F. C. MacKintosh and S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[Crossref]

Phys. Rev. E (4)

B. A. van Tiggelen, R. Maynard, and T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[Crossref]

Z. Lin and S. T. Chui, “Electromagnetic scattering by optically anisotropic magnetic particle,” Phys. Rev. E 69, 056614 (2004).
[Crossref]

S. Wiebel, A. Sparenberg, G. L. J. A. Rikken, D. Lacoste, and B. A. van Tiggelen, “Photonic Hall effect in absorbing media,” Phys. Rev. E 62, 8636–8639 (2000).
[Crossref]

H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E 67, 056607 (2003).
[Crossref]

Phys. Rev. Lett. (3)

B. A. van Tiggelen, “Transverse diffusion of light in Faraday-active media,” Phys. Rev. Lett. 75, 422–424 (1995).
[Crossref]

A. Sparenberg, G. L. J. A. Rikken, and B. A. van Tiggelen, “Observation of photonic magnetoresistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[Crossref]

C. Kern, M. Kadic, and M. Wegener, “Experimental evidence for sign reversal of the Hall coefficient in three-dimensional metamaterials,” Phys. Rev. Lett. 118, 016601 (2017).
[Crossref]

Physica B (1)

G. L. J. A. Rikken, A. Sparenberg, and B. A. van Tiggelen, “Photonic magneto-transport,” Physica B 246–247, 188–194 (1998).
[Crossref]

Q. Appl. Math. (3)

B. Friedman and J. Russek, “Addition theorems for spherical waves,” Q. Appl. Math. 12, 13–23 (1954).
[Crossref]

A. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
[Crossref]

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
[Crossref]

Other (3)

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

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

A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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Figures (7)

Fig. 1.
Fig. 1. Geometry of the scattering problem for an ensemble of N spheres embedded in an LC cell. Here, θ and φ denote the polar and azimuthal angles of the wave vector k, respectively. The α represents tilt angle between the principal axis of the nematic LC molecule and the x axis.
Fig. 2.
Fig. 2. (a) Transverse asymmetry parameter gy and (b) scattering efficiency Qs as a function of incident frequency at external voltage V/Vc=1.0239 in the non-centrosymmetric case. (c) and (d) show the results in the centrosymmetric case. The geometrical parameters of two structures as shown in the inset of (b) and (d) are taken as R=50  nm, r=13  nm, and dy=20  nm. Red, blue, and green lines represent left circularly polarized, right circularly polarized, and linearly polarized incident waves, denoted by LCP, RCP, and LP10 with (pθ,pφ)=1/2(1,i), (pθ,pφ)=1/2(1,i), and (pθ,pφ)=(1,0), respectively. The black lines denote the results for the pure LC cell, with r=0  nm illuminated by the left circularly polarized wave. The A, B, and C points in (b) indicate the values of gy and Qs in Figs. 3(a)3(c), respectively.
Fig. 3.
Fig. 3. In the xy plane, the distributions of the electric field intensity (a), (b), and (c) correspond to the points A, B, and C in Fig. 2(b), respectively. (d), (e), and (f) show the time-average scattering Poynting vector distributions of points A, B, and C separately.
Fig. 4.
Fig. 4. (a) Transverse asymmetry parameter gy as a function of incident frequency. Black solid and red dashed lines represent two linearly polarized incident waves, denoted by LP10 and LP01 with (pθ,pφ)=(1,0) and (pθ,pφ)=(0,1), respectively. The blue square line represents the LP10 result calculated using COMSOL Multiphysics. (b), (c) Electric field intensity for the points A and B, respectively, in (a). (d), (e) The corresponding time-average scattering Poynting vector. The asymmetric chain structure is shown in the inset of (a). The value of external voltage is V/Vc=1.0239. The geometrical parameters are described in the text.
Fig. 5.
Fig. 5. Asymmetric chain structure with R=50  nm is illuminated by the linearly polarized wave LP10. (a) Transverse asymmetry parameter gy as a function of the ratio r/R at external voltages V/Vc=1.0239. Black, red, and blue lines represent the incident frequencies ω=3.85  eV, ω=4.15  eV, and ω=4.85  eV, respectively. (b) gy as a function of the external voltage V/Vc when the incident wave frequency is ω=4.0  eV. Black square, red circle, blue triangle, and green triangle lines represent Ag sphere radius r=11, 12, 13, and 14 nm, respectively.
Fig. 6.
Fig. 6. (a) Transverse asymmetry parameter gy as a function of incident frequency. The red and black lines indicate the centrosymmetry and non-centrosymmetry cases, respectively. (b), (c) and (d) Electric field intensity for the points A, B, and C, respectively, in (a). (e), (f), and (g) Corresponding time-average scattering Poynting vector. The triangle structure is shown in the inset of (a). The value of external voltage is V/Vc=1.0239. The radii of the Ag spheres are r=20  nm; other parameters are described in the text.
Fig. 7.
Fig. 7. Transverse asymmetry parameter gy as a function of (a) the ratio r/R and (b) external voltages for the triangle system as shown in the inset of (b). The linearly polarized wave LP10 propagates along the x axis. Three Ag spheres are tangent to the LC cell. The other parameters are described in the text. (a) Solid lines and dashed lines represent the external voltages V/Vc=1.0239 and 1.9529, respectively. Black, red, and blue curves denote the incident frequencies ω=3.6  eV, ω=3.63  eV, and ω=3.66  eV, respectively. (b) Green square, orange circle, and pink triangle lines represent the incident frequencies ω=3.61  eV, ω=3.63  eV, and ω=3.65  eV, respectively. The other parameters in (b) are consistent with the vertical light green line in (a).

Equations (79)

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ϵ=ϵL0ϵ^=ϵL0(no2+Δϵcos2α0Δϵcosαsinα0no20Δϵcosαsinα0no2+Δϵsin2α),
Es(j0)=n=1m=nniE¯mn[amnNmn(3)(k0,rj0)+bmnMmn(3)(k0,rj0)],
Hs(j0)=k0wμ0n=1m=nnE¯mn[bmnNmn(3)(k0,rj0)+amnMmn(3)(k0,rj0)],
Cmn=[2n+1n(n+1)(nm)!(n+m)!]1/2,
Ein(j0)=n=1m=nniE¯mn[pmnNmn(1)(k0,rj0)+qmnMmn(1)(k0,rj0)],
Hin(j0)=k0ωμ0n=1m=nnE¯mn[qmnNmn(1)(k0,rj0)+pmnMmn(1)(k0,rj0)].
pmn=[pθτ˜mn(cosθk)ipϕπ˜mn(cosθk)]eimϕk,
qmn=[pθπ˜mn(cosθk)ipϕτ˜mn(cosθk)]eimϕk,
EI0(j0)=n=1m=nniE¯mnlγlj0[cmn,lMmn(1)(kl,rj0)+dmn,lNmn(1)(kl,rj0)+w¯mn,lλlLmn(1)(kl,rj0)]ilγlj0kl2ω2μL0w¯00,lL00(1)(kl,rj0),
HI0(j0)=n=1m=nn1ωμL0E¯mnlklγlj0×[dmn,lMmn(1)(kl,rj0)+cmn,lNmn(1)(kl,rj0)],
EI(j0)=Et(j0)+j(1,N)Essj(j,j0),
HI(j0)=Ht(j0)+j(1,N)Hssj(j,j0),
Et(j0)=n,miE¯mnlαlj0[cmn,lMmn(1)(kl,rj0)+dmn,lNmn(1)(kl,rj0)+w¯mn,lλlLmn(1)(kl,rj0)]ilαlj0kl2ω2μL0w¯00,lL00(1)(kl,rj0),
Ht(j0)=n,m1ωμL0E¯mnlklαlj0×[dmn,lMmn(1)(kl,rj0)+cmn,lNmn(1)(kl,rj0)].
Essj(j)=n,miE¯mnlβlj[cmn,lMmn(3)(kl,rj)+dmn,lNmn(3)(kl,rj)+w¯mn,lλlLmn(3)(kl,rj)]ilβljkl2ω2μL0w¯00,lL00(3)(kl,rj),
Hssj(j)=n,m1ωμL0E¯mnlklβlj×[dmn,lMmn(3)(kl,rj)+cmn,lNmn(3)(kl,rj)].
EsIj(j)=n,miE¯mn×[emnjNmn(1)(kj,rj)+fmnjMmn(1)(kj,rj)],
HsIj(j)=kjωμjn,mE¯mn×[fmnjNmn(1)(kj,rj)+emnjMmn(1)(kj,rj)],
Esinj(j)=Et(j0,j)+ij(1,N)Essi(i,j),
Hsinj(j)=Ht(j0,j)+ij(1,N)Hssi(i,j).
Mmn=ν=0μ=νν(A0μνmnMμν+B0μνmnNμν),
Nmn=ν=0μ=νν(B0μνmnMμν+A0μνmnNμν),
Lmn=ν=0μ=ννΓ0μνmnLμν,
Et(j0,j)=n,miE¯mnlαlj0[cmn,lj0,jMmn(1)(kl,rj)+dmn,lj0,jNmn(1)(kl,rj)+w¯mn,lj0,jλlLmn(1)(kl,rj)]ilαlj0kl2ω2μL0w¯00,lj0,jL00(1)(kl,rj),
Ht(j0,j)=n,m1ωμL0E¯mnlklαlj0×[dmn,lj0,jMmn(1)(kl,rj)+cmn,lj0,jNmn(1)(kl,rj)],
dmn,lj0,j=μνE¯μνE¯mn×[dμν,lA0mnμν(j0,j)+cμν,lB0mnμν(j0,j)],
cmn,lj0,j=μνE¯μνE¯mn×[dμν,lB0mnμν(j0,j)+cμν,lA0mnμν(j0,j)],
w¯mn,lj0,j=μνE¯μνE¯mnw¯μν,lΓ0mnμν(j0,j).
Ess(i,j)=n,miE¯mnlβli[cmn,li,jMmn(1)(kl,rj)+dmn,li,jNmn(1)(kl,rj)+w¯mn,li,jλlLmn(1)(kl,rj)]ilβlikl2ω2μL0w¯00,li,jL00(1)(kl,rj),
Hss(i,j)=n,m1ωμL0E¯mnlklβli×[dmn,li,jMmn(1)(kl,rj)+cmn,li,jNmn(1)(kl,rj)],
dmn,li,j=μνE¯μνE¯mn×[dμν,lA0mnμν(i,j)+cμν,lB0mnμν(i,j)],
cmn,li,j=μνE¯μνE¯mn×[dμν,lB0mnμν(i,j)+E¯μνE¯mncμν,lA0mnμν(i,j)],
w¯mn,li,j=μνE¯μνE¯mnw¯μν,lΓ0mnμν(i,j).
[Esinj(j)+Essj(j)]×e^r=EsIj(j)×e^r,
[Hsinj(j)+Hssj(j)]×e^r=HsIj(j)×e^r.
(emnjfmnj)=(IVJ¯IVJ)α˜lj0+ij(1,N)(SVJ¯SVJ)β˜li+(SVH¯SVH)β˜lj,
(emnjfmnj)=(IUJIUJ¯)α˜lj0+ij(1,N)(SUJSUJ¯)β˜li+(SUHSUH¯)β˜lj,
IVJ¯mn,l=dmn,lj0,jkjklψn(klaj)ψn(kjaj)+w¯mn,lj0,jλlkjaj(klaj)2ψn(klaj)ψn(kjaj),
IVJmn,l=cmn,lj0,jkjklψn(klaj)ψn(kjaj),
SVJ¯mn,l=dmn,li,jkjklψn(klaj)ψn(kjaj)+w¯mn,li,jλlkjaj(klaj)2ψn(klaj)ψn(kjaj),
SVJmn,l=cmn,li,jkjklψn(klaj)ψn(kjaj),
SVH¯mn,l=dmn,lkjklξn(klaj)ψn(kjaj)+w¯mn,lλlkjaj(klaj)2ξn(klaj)ψn(kjaj),
SVHmn,l=cmn,lkjklξn(klaj)ψn(kjaj),
IUJmn,l=dmn,lj0,jμjμL0ψn(klaj)ψn(kjaj),
IUJ¯mn,l=cmn,lj0,jμjμL0ψn(klaj)ψn(kjaj),
SUJmn,l=dmn,li,jμjμL0ψn(klaj)ψn(kjaj),
SUJ¯mn,l=cmn,li,jμjμL0ψn(klaj)ψn(kjaj),
SUHmn,l=dmn,lμjμL0ξn(klaj)ψn(kjaj),
SUH¯mn,l=cmn,lμjμL0ξn(klaj)ψn(kjaj).
β˜lj+ij(1,N)GS¯i,jβ˜li=GI¯j0,jα˜lj0,
GS¯i,j=(SVH¯SUHSVHSUH¯)1(SVJ¯SUJSVJSUJ¯),
GI¯j0,j=(SVH¯SUHSVHSUH¯)1(IUJIVJ¯IUJ¯IVJ).
β¯Nl=Tαβα˜lj0,
β¯Nl=(β˜l1β˜ljβ˜lN),
Tαβ=(I+(GS¯1,1GS¯N,1GS¯1,NGS¯N,N))1(GI¯j0,1GI¯j0,N).
Essj(j,j0)=n,miE¯mnlβlj[cmn,lj,j0Mmn(3)(kl,rj0)+dmn,lj,j0Nmn(3)(kl,rj0)+w¯mn,lj,j0λlLmn(3)(kl,rj0)]ilβljkl2ω2μL0w¯00,lj,j0L00(3)(kl,rj0),
Hssj(j,j0)=n,m1ωμL0E¯mnlklβlj×[dmn,lj,j0Mmn(3)(kl,rj0)+cmn,lj,j0Nmn(3)(kl,rj0)],
dmn,lj,j0=μνE¯μνE¯mn×[dμν,lA0mnμν(j,j0)+E¯μνE¯mncμν,lB0mnμν(j,j0)],
cmn,lj,j0=μνE¯μνE¯mn×[dμν,lB0mnμν(j,j0)+E¯μνE¯mncμν,lA0mnμν(j,j0)],
w¯mn,lj,j0=μνE¯μνE¯mnw¯μν,lΓ0mnμν(j,j0).
[Ein(j)+Es(j)]×e^r=EI(j)×e^r,
[Hin(j)+Hs(j)]×e^r=HI(j)×e^r.
(pq)=(Λ¯00Λ)(ab)+(VJ¯VJ)α˜lj0+j=1N(VH¯VH)β˜lj,
(pq)=(Λ00Λ¯)(ab)+(UJUJ¯)α˜lj0+j=1N(UHUH¯)β˜lj,
Λ¯mn,uv=ξn(k0R)ψn(k0R)δn,vδm,u,Λ=ξn(k0R)ψn(k0R)δn,vδm,u,
VJ¯=dmn,lk0klψn(klR)ψn(k0R)+w¯mn,lλlk0R(klR)2ψn(klR)ψn(k0R),
VJ=cmn,lk0klψn(klR)ψn(k0R),
VH¯=dmn,lk0klξn(klR)ψn(k0R)+w¯mn,lλlk0R(klR)2ξn(klR)ψn(k0R),
VH=cmn,lk0klξn(klR)ψn(k0R),
UJ¯=μ0μL0cmn,lψn(klR)ψn(k0R),UJ=μ0μL0dmn,lψn(klR)ψn(k0R),
UH¯=μ0μL0cmn,lξn(klR)ψn(k0R),UH=μ0μL0dmn,lξn(klR)ψn(k0R).
(ab)=S(pq),
Cs=4πk02n=1m=nn(|amn|2+|bmn|2).
Qs=4(k0R)2n=1m=nn(|amn|2+|bmn|2).
gy=4πF(θ,φ)πR2QssinθsinφdΩ,
F(θ,φ)=dσ(θ,φ)dΩ.
dσ(θ,φ)dΩ=|f(θ,φ)|2,
Es=E0f(θ,φ)eik0rr,r=|r|.
ϵj(ω)=1ωp2ω(ω+iγ),

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