Abstract

We investigate the effective parameters and quasi-static resonances for periodic arrays of dielectric spheres. The effective medium model based on the homogenization of normal modes for spherical particles is used to determine the effective permittivity ϵeff and permeability μeff in the quasi-static regime. Major features of ϵeff and μeff are characterized by the Lorentz-type anomalous dispersion around the frequencies pertaining to the leading-order electric and magnetic resonances, respectively. In particular, the anomalous dispersion is depicted by a resonance function associated with the spherical cavity. The underlying mechanism of quasi-static resonance is illustrated with the localized and dipole-like field patterns at the resonant frequencies. A comparison with the effective parameters for periodic arrays of dielectric circular cylinders is also discussed.

© 2010 Optical Society of America

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2009 (4)

R. L. Chern and D. Felbacq, “Artificial magnetism and anticrossing interaction in photonic crystals and split-ring structures,” Phys. Rev. B 79, 075118 (2009).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

R. L. Chern and Y. T. Chen, “Effective parameters for photonic crystals with large dielectric contrast,” Phys. Rev. B 80, 075118 (2009).
[CrossRef]

2008 (5)

X. Hu, K. M. Ho, C. T. Chan, and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid,” Phys. Rev. B 77, 172301 (2008).
[CrossRef]

W. Śmigaj and B. Gralak, “Validity of the effective-medium approximation of photonic crystals,” Phys. Rev. B 77, 235445 (2008).
[CrossRef]

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

S. T. Chui and Z. Lin, “Long wavelength behavior of two dimensional photonic crystals,” Phys. Rev. E 78, 065601(R) (2008).
[CrossRef]

R. L. Chern and S. D. Chao, “Optimal higher-lying band gaps for photonic crystals with large dielectric contrast,” Opt. Express 16, 16600-16608 (2008).
[CrossRef] [PubMed]

2007 (4)

V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys. A 87, 259-264 (2007).
[CrossRef]

Y. Wu, Y. Lai, and Z. Q. Zhang, “Effective medium theory for elastic metamaterials in two dimensions,” Phys. Rev. B 76, 205313 (2007).
[CrossRef]

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

2006 (5)

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391-403 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[CrossRef]

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

N. A. Mortensen, S. Xiao, and D. Felbacq, “Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening,” J. Eur. Opt. Soc. Rapid Publ. 1, 06019 (2006).
[CrossRef]

2005 (2)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

2003 (1)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

2002 (1)

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (2)

A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B 62, 8531-8539 (2000).
[CrossRef]

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246 (2000).
[CrossRef]

1999 (2)

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359-5365 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1997 (1)

1992 (1)

A. Lagarkov, A. Sarychev, Y. Smychkovich, and A. Vinogradov, “Effective medium theory for microwave dielectric constant and magnetic permeability of conducting stick composites,” J. Electromagn. Waves Appl. 6, 1159-1176 (1992).

1988 (1)

G. D. Mahan, “Long-wavelength absorption of cermets,” Phys. Rev. B 38, 9500-9502 (1988).
[CrossRef]

1986 (1)

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609-2618 (1986).
[CrossRef]

1983 (1)

P. Chýlek and V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098-5106 (1983).
[CrossRef]

1981 (1)

1980 (1)

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248-2266 (1980).
[CrossRef]

1978 (1)

D. Stroud and F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: Application to model granular metals,” Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

1977 (1)

G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39-L42 (1977).
[CrossRef]

1974 (1)

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

1963 (1)

J. J. Hopfield, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563-572 (1963).
[CrossRef]

1947 (1)

L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65-68 (1947).

1904 (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and metal films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Ashcroft, N. W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248-2266 (1980).
[CrossRef]

Baker-Jarvis, J.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

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, 7th ed. (Cambridge, 1999).

Bouchitté, G.

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

Brongersma, M. L.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

Cabuz, A. I.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

Cassagne, D.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

Centeno, E.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

Chan, C. T.

X. Hu, K. M. Ho, C. T. Chan, and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid,” Phys. Rev. B 77, 172301 (2008).
[CrossRef]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[CrossRef]

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Chao, S. D.

Chen, H.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

Chen, J. I. L.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

Chen, Y. T.

R. L. Chern and Y. T. Chen, “Effective parameters for photonic crystals with large dielectric contrast,” Phys. Rev. B 80, 075118 (2009).
[CrossRef]

Chern, R. L.

R. L. Chern and Y. T. Chen, “Effective parameters for photonic crystals with large dielectric contrast,” Phys. Rev. B 80, 075118 (2009).
[CrossRef]

R. L. Chern and D. Felbacq, “Artificial magnetism and anticrossing interaction in photonic crystals and split-ring structures,” Phys. Rev. B 79, 075118 (2009).
[CrossRef]

R. L. Chern and S. D. Chao, “Optimal higher-lying band gaps for photonic crystals with large dielectric contrast,” Opt. Express 16, 16600-16608 (2008).
[CrossRef] [PubMed]

Chui, S. T.

S. T. Chui and Z. Lin, “Long wavelength behavior of two dimensional photonic crystals,” Phys. Rev. E 78, 065601(R) (2008).
[CrossRef]

Chýlek, P.

P. Chýlek and V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098-5106 (1983).
[CrossRef]

Felbacq, D.

R. L. Chern and D. Felbacq, “Artificial magnetism and anticrossing interaction in photonic crystals and split-ring structures,” Phys. Rev. B 79, 075118 (2009).
[CrossRef]

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

N. A. Mortensen, S. Xiao, and D. Felbacq, “Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening,” J. Eur. Opt. Soc. Rapid Publ. 1, 06019 (2006).
[CrossRef]

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

Giessen, H.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Gralak, B.

W. Śmigaj and B. Gralak, “Validity of the effective-medium approximation of photonic crystals,” Phys. Rev. B 77, 235445 (2008).
[CrossRef]

Granqvist, C. G.

Grzegorczyk, T. M.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

Guillon, P.

D. Kajfez and P. Guillon, Dielectric Resonators, 2nd ed. (Noble, 1998).

Guizal, B.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

Ho, K. M.

X. Hu, K. M. Ho, C. T. Chan, and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid,” Phys. Rev. B 77, 172301 (2008).
[CrossRef]

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Holloway, C. L.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Hopfield, J. J.

J. J. Hopfield, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563-572 (1963).
[CrossRef]

Hu, X.

X. Hu, K. M. Ho, C. T. Chan, and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid,” Phys. Rev. B 77, 172301 (2008).
[CrossRef]

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Huffman, D. R.

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

Hunderi, O.

Kabos, P.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Kajfez, D.

D. Kajfez and P. Guillon, Dielectric Resonators, 2nd ed. (Noble, 1998).

Kong, J. A.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

J. A. Kong, Electromagnetic Wave Theory (EMW, 2005).

Kranendonk, J. V.

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

Kroll, N.

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246 (2000).
[CrossRef]

Kuester, E. F.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Lagarkov, A.

A. Lagarkov, A. Sarychev, Y. Smychkovich, and A. Vinogradov, “Effective medium theory for microwave dielectric constant and magnetic permeability of conducting stick composites,” J. Electromagn. Waves Appl. 6, 1159-1176 (1992).

Lai, Y.

Y. Wu, Y. Lai, and Z. Q. Zhang, “Effective medium theory for elastic metamaterials in two dimensions,” Phys. Rev. B 76, 205313 (2007).
[CrossRef]

Lamb, W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248-2266 (1980).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon, 1977).

Lederer, F.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Lewin, L.

L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65-68 (1947).

Li, J.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[CrossRef]

Li, M.

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon, 1977).

Lin, Z.

S. T. Chui and Z. Lin, “Long wavelength behavior of two dimensional photonic crystals,” Phys. Rev. E 78, 065601(R) (2008).
[CrossRef]

Luo, R.

Mahan, G. D.

G. D. Mahan, “Long-wavelength absorption of cermets,” Phys. Rev. B 38, 9500-9502 (1988).
[CrossRef]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in metal glasses and metal films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

McPhedran, R. C.

A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B 62, 8531-8539 (2000).
[CrossRef]

Meyrath, T. P.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Modinos, A.

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359-5365 (1999).
[CrossRef]

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Mortensen, N. A.

N. A. Mortensen, S. Xiao, and D. Felbacq, “Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening,” J. Eur. Opt. Soc. Rapid Publ. 1, 06019 (2006).
[CrossRef]

Niklasson, G. A.

O'Brien, S.

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

Ozin, G. A.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

Pan, F. P.

D. Stroud and F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: Application to model granular metals,” Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

Paul, T.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Pedersen, N. E.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609-2618 (1986).
[CrossRef]

Pendry, J. B.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391-403 (2006).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Peng, L.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

Pertsch, T.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Pozar, D. M.

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).

Ran, L.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Rockstuhl, C.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Sarychev, A.

A. Lagarkov, A. Sarychev, Y. Smychkovich, and A. Vinogradov, “Effective medium theory for microwave dielectric constant and magnetic permeability of conducting stick composites,” J. Electromagn. Waves Appl. 6, 1159-1176 (1992).

Sarychev, A. K.

A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B 62, 8531-8539 (2000).
[CrossRef]

Schuller, J. A.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246 (2000).
[CrossRef]

Shalaev, V. M.

A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B 62, 8531-8539 (2000).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sipe, J. E.

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

Smigaj, W.

W. Śmigaj and B. Gralak, “Validity of the effective-medium approximation of photonic crystals,” Phys. Rev. B 77, 235445 (2008).
[CrossRef]

Smith, D. R.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391-403 (2006).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246 (2000).
[CrossRef]

Smith, G. B.

G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39-L42 (1977).
[CrossRef]

Smychkovich, Y.

A. Lagarkov, A. Sarychev, Y. Smychkovich, and A. Vinogradov, “Effective medium theory for microwave dielectric constant and magnetic permeability of conducting stick composites,” J. Electromagn. Waves Appl. 6, 1159-1176 (1992).

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics, 2nd ed. (Academic, 1949).

Srivastava, V.

P. Chýlek and V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098-5106 (1983).
[CrossRef]

Stefanou, N.

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359-5365 (1999).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Stratton, J. A.

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

Stroud, D.

D. Stroud and F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: Application to model granular metals,” Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

Taubner, T.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

Vier, D. C.

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246 (2000).
[CrossRef]

Vinogradov, A.

A. Lagarkov, A. Sarychev, Y. Smychkovich, and A. Vinogradov, “Effective medium theory for microwave dielectric constant and magnetic permeability of conducting stick composites,” J. Electromagn. Waves Appl. 6, 1159-1176 (1992).

Vynck, K.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

Waterman, P. C.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609-2618 (1986).
[CrossRef]

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

Wood, D. M.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248-2266 (1980).
[CrossRef]

Wu, Y.

Y. Wu, Y. Lai, and Z. Q. Zhang, “Effective medium theory for elastic metamaterials in two dimensions,” Phys. Rev. B 76, 205313 (2007).
[CrossRef]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[CrossRef]

Xiao, S.

N. A. Mortensen, S. Xiao, and D. Felbacq, “Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening,” J. Eur. Opt. Soc. Rapid Publ. 1, 06019 (2006).
[CrossRef]

Yannopapas, V.

V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys. A 87, 259-264 (2007).
[CrossRef]

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359-5365 (1999).
[CrossRef]

Zentgraf, T.

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

Zhang, H.

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

Zhang, Z. Q.

Y. Wu, Y. Lai, and Z. Q. Zhang, “Effective medium theory for elastic metamaterials in two dimensions,” Phys. Rev. B 76, 205313 (2007).
[CrossRef]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[CrossRef]

Zi, J.

X. Hu, K. M. Ho, C. T. Chan, and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid,” Phys. Rev. B 77, 172301 (2008).
[CrossRef]

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

Zia, R.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. A (1)

V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys. A 87, 259-264 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

D. R. Smith, D. C. Vier, N. Kroll, and S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609-2618 (1986).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. Lagarkov, A. Sarychev, Y. Smychkovich, and A. Vinogradov, “Effective medium theory for microwave dielectric constant and magnetic permeability of conducting stick composites,” J. Electromagn. Waves Appl. 6, 1159-1176 (1992).

J. Eur. Opt. Soc. Rapid Publ. (1)

N. A. Mortensen, S. Xiao, and D. Felbacq, “Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening,” J. Eur. Opt. Soc. Rapid Publ. 1, 06019 (2006).
[CrossRef]

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

J. Phys. D (1)

G. B. Smith, “Dielectric constants for mixed media,” J. Phys. D 10, L39-L42 (1977).
[CrossRef]

J. Phys.: Condens. Matter (1)

S. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

Opt. Express (1)

Philos. Trans. R. Soc. London, Ser. A (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and metal films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

Phys. Rev. (1)

J. J. Hopfield, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563-572 (1963).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. B (16)

W. Śmigaj and B. Gralak, “Validity of the effective-medium approximation of photonic crystals,” Phys. Rev. B 77, 235445 (2008).
[CrossRef]

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008).
[CrossRef]

P. Chýlek and V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098-5106 (1983).
[CrossRef]

D. Stroud and F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: Application to model granular metals,” Phys. Rev. B 17, 1602-1610 (1978).
[CrossRef]

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248-2266 (1980).
[CrossRef]

G. D. Mahan, “Long-wavelength absorption of cermets,” Phys. Rev. B 38, 9500-9502 (1988).
[CrossRef]

R. L. Chern and D. Felbacq, “Artificial magnetism and anticrossing interaction in photonic crystals and split-ring structures,” Phys. Rev. B 79, 075118 (2009).
[CrossRef]

Y. Wu, Y. Lai, and Z. Q. Zhang, “Effective medium theory for elastic metamaterials in two dimensions,” Phys. Rev. B 76, 205313 (2007).
[CrossRef]

X. Hu, K. M. Ho, C. T. Chan, and J. Zi, “Homogenization of acoustic metamaterials of Helmholtz resonators in fluid,” Phys. Rev. B 77, 172301 (2008).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79, 073103 (2009).
[CrossRef]

R. L. Chern and Y. T. Chen, “Effective parameters for photonic crystals with large dielectric contrast,” Phys. Rev. B 80, 075118 (2009).
[CrossRef]

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60, 5359-5365 (1999).
[CrossRef]

A. K. Sarychev, R. C. McPhedran, and V. M. Shalaev, “Electrodynamics of metal-dielectric composites and electromagnetic crystals,” Phys. Rev. B 62, 8531-8539 (2000).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74, 085111 (2006).
[CrossRef]

Phys. Rev. E (1)

S. T. Chui and Z. Lin, “Long wavelength behavior of two dimensional photonic crystals,” Phys. Rev. E 78, 065601(R) (2008).
[CrossRef]

Phys. Rev. Lett. (5)

X. Hu, C. T. Chan, J. Zi, M. Li, and K. M. Ho, “Diamagnetic response of metallic photonic crystals at infrared and visible frequencies,” Phys. Rev. Lett. 96, 223901 (2006).
[CrossRef] [PubMed]

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[CrossRef] [PubMed]

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

L. Peng, L. Ran, H. Chen, H. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. 98, 157403 (2007).
[CrossRef] [PubMed]

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99, 107401 (2007).
[CrossRef] [PubMed]

Proc. Inst. Electr. Eng. (1)

L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65-68 (1947).

Science (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788-792 (2004).
[CrossRef] [PubMed]

Other (9)

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999).

D. Kajfez and P. Guillon, Dielectric Resonators, 2nd ed. (Noble, 1998).

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).

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

A. Sommerfeld, Partial Differential Equations in Physics, 2nd ed. (Academic, 1949).

J. A. Kong, Electromagnetic Wave Theory (EMW, 2005).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd ed. (Pergamon, 1977).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

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Figures (4)

Fig. 1
Fig. 1

Schematics of (a) the periodic array of dielectric spheres and (b) the equivalent unit cell used in the effective medium model.

Fig. 2
Fig. 2

(a) Resonance function α = α + i α and (b) effective permittivity ϵ eff = ϵ eff + i ϵ eff and permeability μ eff = μ eff + i μ eff for the periodic array of dielectric spheres with r 1 a = 0.3 , ϵ 1 = 200 + 3 i , and μ 1 = 1 embedded in the background with ϵ = 1 and μ = 1 .

Fig. 3
Fig. 3

Field contours of H y ( 1 ) at (a) a λ 0.116 and (b) 0.235, and field vectors of ( E x ( 1 ) , E z ( 1 ) ) at (c) a λ 0.116 (d) 0.235 in the symmetric ( x z ) plane for the periodic array of dielectric spheres in Fig. 2. The fields are normalized to have unity maximum magnitude. Red (dark gray) and green (light gray) colors correspond to positive and negative field values, respectively. In (c) and (d), the color represents the value of E x ( 1 ) . The solid and dotted lines denote the sphere surface and the equivalent unit cell boundary, respectively.

Fig. 4
Fig. 4

Field contours of E x ( 1 ) at (a) a λ 0.168 (b) 0.288, and field vectors of ( H y ( 1 ) , H z ( 1 ) ) at (c) a λ 0.168 (d) 0.288 in the symmetric ( y z ) plane for the periodic array of dielectric spheres in Fig. 2. The fields are normalized to have unity maximum magnitude. Red (dark gray) and green (light gray) colors correspond to positive and negative field values, respectively. In (c) and (d), the color represents the value of H y ( 1 ) . The solid and dotted lines denote the sphere surface and the equivalent unit cell boundary, respectively.

Equations (47)

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ϵ eff = D ϵ 0 E , μ eff = B μ 0 H ,
2 E + k 2 E = 0 , 2 H + k 2 H = 0 ,
E 1 ( n ) = a n M o n ( 1 ) + b n N e n ( 1 ) , H 1 ( n ) = i k 1 ω μ 1 ( a n N o n ( 1 ) + b n M e n ( 1 ) ) ,
E i ( n ) = c n M o n ( 1 ) + d n N e n ( 1 ) , H i ( n ) = i k ω μ ( c n N o n ( 1 ) + d n M e n ( 1 ) ) ,
E s ( n ) = e n M o n ( 3 ) + f n N e n ( 3 ) , H s ( n ) = i k ω μ ( e n N o n ( 3 ) + f n M e n ( 3 ) ) ,
c n a n = i [ μ μ 1 ψ n ( x 1 ) ξ n ( x ) k k 1 ψ n ( x 1 ) ξ n ( x ) ] ,
e n a n = i [ k k 1 ψ n ( x 1 ) ψ n ( x ) μ μ 1 ψ n ( x 1 ) ψ n ( x ) ] ,
d n b n = i [ k k 1 ψ n ( x 1 ) ξ n ( x ) μ μ 1 ψ n ( x 1 ) ξ n ( x ) ] ,
f n b n = i [ μ μ 1 ψ n ( x 1 ) ψ n ( x ) k k 1 ψ n ( x 1 ) ψ n ( x ) ] ,
D 2 ϵ 0 { b 1 ϵ 1 f j 1 ( x 1 ) x 1 + d 1 ϵ [ j 1 ( x e ) x e f j 1 ( x ) x ] + f 1 ϵ [ h 1 ( x e ) x e f h 1 ( x ) x ] } ,
B 2 i μ 0 ω { a 1 k 1 f j 1 ( x 1 ) x 1 + c 1 k [ j 1 ( x e ) x e f j 1 ( x ) x ] + e 1 k [ h 1 ( x e ) x e f h 1 ( x ) x ] } ,
E d 1 ψ 1 ( x e ) x e + f 1 ξ 1 ( x ) x ,
H i k ω μ [ c 1 ψ 1 ( x e ) x e + e 1 ξ 1 ( x ) x ] ,
c 1 a 1 ψ 1 ( x 1 ) 2 x μ ̃ 1 + 2 μ μ 1 , e 1 a 1 x 2 ψ 1 ( x 1 ) 3 i μ μ ̃ 1 μ 1 ,
d 1 b 1 ψ 1 ( x 1 ) 2 x 1 ϵ ̃ 1 + 2 ϵ ϵ , f 1 b 1 k x 2 ψ 1 ( x 1 ) 3 i k 1 ϵ ϵ ̃ 1 ϵ ,
ϵ ̃ 1 = ϵ 1 α ( x 1 ) , μ ̃ 1 = μ 1 α ( x 1 ) , α ( x 1 ) = 2 j 1 ( x 1 ) ψ 1 ( x 1 ) .
D ϵ 0 b 1 ψ 1 ( x 1 ) 3 x 1 [ ( 1 + 2 f ) ϵ ̃ 1 + 2 ( 1 f ) ϵ ] ,
E b 1 ψ 1 ( x 1 ) 3 x 1 ϵ [ ( 1 f ) ϵ ̃ 1 + ( 2 + f ) ϵ ] ,
B μ 0 a 1 i k ω μ 1 ψ 1 ( x 1 ) 3 x [ ( 1 + 2 f ) μ ̃ 1 + 2 ( 1 f ) μ ] ,
H a 1 i k ω μ 1 ψ 1 ( x 1 ) 3 x μ [ ( 1 f ) μ ̃ 1 + ( 2 + f ) μ ] .
ϵ eff ϵ ( 1 + 2 f ) ϵ ̃ 1 + 2 ( 1 f ) ϵ ( 1 f ) ϵ ̃ 1 + ( 2 + f ) ϵ ,
μ eff μ ( 1 + 2 f ) μ ̃ 1 + 2 ( 1 f ) μ ( 1 f ) μ ̃ 1 + ( 2 + f ) μ .
α ( x 1 ) = 2 ( sin x 1 x 1 cos x 1 ) ( x 1 2 1 ) sin x 1 + x 1 cos x 1 ,
ϵ eff ϵ ( 1 + 2 f ) ϵ 1 + 2 ( 1 f ) ϵ ( 1 f ) ϵ 1 + ( 2 + f ) ϵ ,
μ eff μ ( 1 + 2 f ) μ 1 + 2 ( 1 f ) μ ( 1 f ) μ 1 + ( 2 + f ) μ .
M o n ( l ) = cos ϕ π n ( cos θ ) z n ( ρ ) θ ̂ sin ϕ τ n ( cos θ ) z n ( ρ ) ϕ ̂ ,
M e n ( l ) = sin ϕ π n ( cos θ ) z n ( ρ ) θ ̂ cos ϕ τ n ( cos θ ) z n ( ρ ) ϕ ̂ ,
N o n ( l ) = sin ϕ n ( n + 1 ) sin θ π n ( cos θ ) z n ( ρ ) ρ r ̂ + sin ϕ τ n ( cos θ ) [ ρ z n ( ρ ) ] ρ θ ̂ + cos ϕ π n ( cos θ ) [ ρ z n ( ρ ) ] ρ ϕ ̂ ,
N e n ( l ) = cos ϕ n ( n + 1 ) sin θ π n ( cos θ ) z n ( ρ ) ρ r ̂ + cos ϕ τ n ( cos θ ) [ ρ z n ( ρ ) ] ρ θ ̂ sin ϕ π n ( cos θ ) [ ρ z n ( ρ ) ] ρ ϕ ̂ ,
c n ψ n ( x ) + e n ξ n ( x ) = a n k k 1 ψ n ( x 1 ) ,
c n ψ n ( x ) + e n ξ n ( x ) = a n μ μ 1 ψ n ( x 1 ) ,
d n ψ n ( x ) + f n ξ n ( x ) = b n μ μ 1 ψ n ( x 1 ) ,
d n ψ n ( x ) + f n ξ n ( x ) = b n k k 1 ψ n ( x 1 ) ,
c n a n = i [ μ μ 1 ψ n ( x 1 ) ξ n ( x ) k k 1 ψ n ( x 1 ) ξ n ( x ) ] ,
e n a n = i [ k k 1 ψ n ( x 1 ) ψ n ( x ) μ μ 1 ψ n ( x 1 ) ψ n ( x ) ] ,
d n b n = i [ k k 1 ψ n ( x 1 ) ξ n ( x ) μ μ 1 ψ n ( x 1 ) ξ n ( x ) ] ,
f n b n = i [ μ μ 1 ψ n ( x 1 ) ψ n ( x ) k k 1 ψ n ( x 1 ) ψ n ( x ) ] ,
M o 1 ( l ) x ̂ = cos θ z 1 ( ρ ) ,
N e 1 ( l ) x ̂ = 2 sin 2 θ cos 2 ϕ z 1 ( ρ ) ρ + cos 2 θ cos 2 ϕ [ ρ z 1 ( ρ ) ] ρ + sin 2 ϕ [ ρ z 1 ( ρ ) ] ρ ,
M e 1 ( l ) y ̂ = cos θ z 1 ( ρ ) ,
N o 1 ( l ) y ̂ = 2 sin 2 θ sin 2 ϕ z 1 ( ρ ) ρ + cos 2 θ sin 2 ϕ [ ρ z 1 ( ρ ) ] ρ + cos 2 ϕ [ ρ z 1 ( ρ ) ] ρ .
I 1 ( l ) 0 r 1 N e 1 ( l ) x ̂ d v = 0 r 1 N o 1 ( l ) y ̂ d v = 2 V e f z 1 ( x 1 ) x 1 ,
I 2 ( l ) r 1 r e N e 1 ( l ) x ̂ d v = r 1 r e N o 1 ( l ) y ̂ d v = 2 V e [ z 1 ( x e ) x e f z 1 ( x ) x ] ,
D ϵ 0 V e ( b 1 ϵ 1 I 1 ( 1 ) + d 1 ϵ I 2 ( 1 ) + f 1 ϵ I 2 ( 3 ) ) = 2 ϵ 0 { b 1 ϵ 1 f j 1 ( x 1 ) x 1 + d 1 ϵ [ j 1 ( x e ) x e f j 1 ( x ) x ] + f 1 ϵ [ h 1 ( x e ) x e f h 1 ( x ) x ] } ,
B i μ 0 ω V e ( a 1 k 1 I 1 ( 1 ) + c 1 k I 2 ( 1 ) + e 1 k I 2 ( 3 ) ) = 2 i μ 0 ω { a 1 k 1 f j 1 ( x 1 ) x 1 + c 1 k [ j 1 ( x e ) x e f j 1 ( x ) x ] + e 1 k [ h 1 ( x e ) x e f h 1 ( x ) x ] } .
E 2 l e 0 π | ( c 1 M o 1 ( 1 ) + d 1 N e 1 ( 1 ) + e 1 M o 1 ( 3 ) + f 1 N e 1 ( 3 ) ) x ̂ r d θ | r = r e , ϕ = π 2 = d 1 ψ 1 ( x e ) x e + f 1 ξ 1 ( x ) x ,
H i k ω μ 2 l e 0 π | ( c 1 N o 1 ( 1 ) + d 1 M e 1 ( 1 ) + e 1 N o 1 ( 3 ) + f 1 M e 1 ( 1 ) ) y ̂ r d θ | r = r e , ϕ = 0 = i k ω μ [ c 1 ψ 1 ( x e ) x e + e 1 ξ 1 ( x ) x ] ,

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