Abstract

High-permittivity dielectric particles with resonant magnetic properties are being explored as constitutive elements of new metamaterials and devices. Magnetic properties of low-loss dielectric nanoparticles in the visible or infrared are not expected due to intrinsic low refractive index of optical media in these regimes. Here we analyze the dipolar electric and magnetic response of lossless dielectric spheres made of moderate permittivity materials. For low material refractive index (≲ 3) there are no sharp resonances due to strong overlapping between different multipole contributions. However, we find that Silicon particles with index of refraction ∼ 3.5 and radius ∼ 200nm present strong electric and magnetic dipolar resonances in telecom and near-infrared frequencies, (i.e. at wavelengths ≈ 1.2 – 2μm) without spectral overlap with quadrupolar and higher order resonances. The light scattered by these Si particles can then be perfectly described by dipolar electric and magnetic fields.

© 2011 Optical Society of America

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2010 (2)

2009 (9)

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. Cabuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[Crossref] [PubMed]

Q. Zhao, J. Zhou1, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[Crossref]

R. Carminati and J. J. Sáenz, “Density of states and extinction mean free path of waves in random media: dispersion relations and sum rules,” Phys. Rev. Lett. 102, 093902 (2009).
[Crossref] [PubMed]

M. Ibisate, D. Golmayo, and C. López, “Photonic crystals: silicon direct opals,” Adv. Mater. 28, 2899–2902 (2009).
[Crossref]

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

P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009).
[Crossref] [PubMed]

A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009).
[Crossref] [PubMed]

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[Crossref]

2008 (4)

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[Crossref]

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modelling of all-dielectric metamaterials,” Phys. Rev. B 77, 045104 (2008).
[Crossref]

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

2007 (7)

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]

R. C. J. Hsu, A. Ayazi, B. Houshmand, and B. Jalali, “All-dielectric photonic-assisted radio front-end technology,” Nat. Photonics 1, 535–538 (2007).
[Crossref]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[Crossref]

M. M. Sigalas, D. A. Fattal, R. S. Williams, S.Y. Wang, and R. G. Beausoleil, “ Electric field enhancement between two Si microdisks,” Opt. Express 15, 14711–14716 (2007).
[Crossref] [PubMed]

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

2006 (2)

J. B. Pendry, “Beyond metamaterials,” Nature Mater. 5, 763–764 (2006).
[Crossref]

L. Jyhlä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102 (2006).
[Crossref]

2005 (4)

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]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).
[Crossref]

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

N. Engheta and R.W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microwave Theory Tech. 53, 1535–1556 (2005).
[Crossref]

2004 (1)

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543–545 (2004).
[Crossref]

2003 (2)

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

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

2000 (2)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2000).
[Crossref]

1994 (1)

R. K. Mongia and P. Bhartia, “Dielectric resonator antennas-A review and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter-Wave Comput.-Aided Eng. 4, 230–247 (1994).
[Crossref]

1992 (1)

G. Videen and W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[Crossref] [PubMed]

1988 (1)

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1978 (1)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978)
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Ahmadi, A.

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modelling of all-dielectric metamaterials,” Phys. Rev. B 77, 045104 (2008).
[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, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[Crossref]

Aizpurua, J.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Albaladejo, S.

Alú, A.

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

Alù, A.

Armelles, G.

Ayazi, A.

R. C. J. Hsu, A. Ayazi, B. Houshmand, and B. Jalali, “All-dielectric photonic-assisted radio front-end technology,” Nat. Photonics 1, 535–538 (2007).
[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 sphercal particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Beausoleil, R. G.

Bhartia, P.

R. K. Mongia and P. Bhartia, “Dielectric resonator antennas-A review and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter-Wave Comput.-Aided Eng. 4, 230–247 (1994).
[Crossref]

Bickel, W. S.

G. Videen and W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[Crossref] [PubMed]

Blanco, A.

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Sec. 2.3.

Brongersma, M. L.

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[Crossref]

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. Cabuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett. 102, 133901 (2009).
[Crossref] [PubMed]

Carminati, R.

Cassagne, D.

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

Chantada, L.

Chaumet, P. C.

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]

Chylek, P.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978)
[Crossref]

Cornelius, T. W.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Draine, B. T.

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Eiden, S.

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

Engheta, N.

A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009).
[Crossref] [PubMed]

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

N. Engheta and R.W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microwave Theory Tech. 53, 1535–1556 (2005).
[Crossref]

Fattal, D. A.

Felbacq, D.

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

Foteinopoulou, S.

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2000).
[Crossref]

Fromm, D. P.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

Froufe-Pérez, L. S.

Garca-Etxarri, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

García, P. D.

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

García-Martín, A.

Golmayo, D.

M. Ibisate, D. Golmayo, and C. López, “Photonic crystals: silicon direct opals,” Adv. Mater. 28, 2899–2902 (2009).
[Crossref]

Gómez-Medina, R.

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]

Guizal, B.

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

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge Univ. Press, 2006).

Holloway, C. L.

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

Houshmand, B.

R. C. J. Hsu, A. Ayazi, B. Houshmand, and B. Jalali, “All-dielectric photonic-assisted radio front-end technology,” Nat. Photonics 1, 535–538 (2007).
[Crossref]

Hsu, R. C. J.

R. C. J. Hsu, A. Ayazi, B. Houshmand, and B. Jalali, “All-dielectric photonic-assisted radio front-end technology,” Nat. Photonics 1, 535–538 (2007).
[Crossref]

Huang, K. C.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543–545 (2004).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).
[Crossref]

Ibisate, M.

M. Ibisate, D. Golmayo, and C. López, “Photonic crystals: silicon direct opals,” Adv. Mater. 28, 2899–2902 (2009).
[Crossref]

Jalali, B.

R. C. J. Hsu, A. Ayazi, B. Houshmand, and B. Jalali, “All-dielectric photonic-assisted radio front-end technology,” Nat. Photonics 1, 535–538 (2007).
[Crossref]

Joannopoulos, J. D.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543–545 (2004).
[Crossref]

Jyhlä, L.

L. Jyhlä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102 (2006).
[Crossref]

Kabos, P.

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

Karim, S.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Kiehl, J. T.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978)
[Crossref]

Kino, G. S.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978)
[Crossref]

Kolmakov, I.

L. Jyhlä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102 (2006).
[Crossref]

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]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves-Advanced Topics (Wiley, 2001).

Kuester, E. F.

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

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Laroche, M.

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

Lippens, D.

Q. Zhao, J. Zhou1, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[Crossref]

López, C.

M. Ibisate, D. Golmayo, and C. López, “Photonic crystals: silicon direct opals,” Adv. Mater. 28, 2899–2902 (2009).
[Crossref]

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

Marinchio, H.

Marqués, M. I.

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

Maslovski, S.

L. Jyhlä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102 (2006).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Moerner, W. E.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

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, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[Crossref]

Mongia, R. K.

R. K. Mongia and P. Bhartia, “Dielectric resonator antennas-A review and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter-Wave Comput.-Aided Eng. 4, 230–247 (1994).
[Crossref]

Moroz, A.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).
[Crossref]

Mosallaei, H.

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modelling of all-dielectric metamaterials,” Phys. Rev. B 77, 045104 (2008).
[Crossref]

Neubrech, F.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge Univ. Press, 2006).

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]

Palik, E.D.

E.D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

Pendry, J. B.

J. B. Pendry, “Beyond metamaterials,” Nature Mater. 5, 763–764 (2006).
[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]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Povinelli, M. L.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543–545 (2004).
[Crossref]

Pucci, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[Crossref] [PubMed]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Rahmani, A.

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]

Reufer, M.

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

Rojas-Ochoa, L. F.

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

Sáenz, J. J.

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric magnetic particles,” Opt. Express 18, 11428–11443 (2010).
[Crossref] [PubMed]

S. Albaladejo, R. Gómez-Medina, L. S. Froufe-Pérez, H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18, 3556–3567 (2010).
[Crossref] [PubMed]

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

R. Carminati and J. J. Sáenz, “Density of states and extinction mean free path of waves in random media: dispersion relations and sum rules,” Phys. Rev. Lett. 102, 093902 (2009).
[Crossref] [PubMed]

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

Sapienza, R.

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

Scheffold, F.

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

Schuck, P. J.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

Schuller, J. A.

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[Crossref]

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]

Segerink, F. B.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[Crossref]

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[Crossref]

Sigalas, M. M.

Soukoulis, C. M.

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2000).
[Crossref]

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[Crossref]

Sundaramurthy, A.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

Taminiau, T. H.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[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]

Torrado, J. F.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Tretyakov, S.

L. Jyhlä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102 (2006).
[Crossref]

Tsang, L.

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves-Advanced Topics (Wiley, 2001).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Van Hulst, N. F.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[Crossref]

Videen, G.

G. Videen and W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[Crossref] [PubMed]

Vynck, K.

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

Wang, S.Y.

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, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72, 193103 (2005).
[Crossref]

Williams, R. S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Sec. 2.3.

Yannopapas, V.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).
[Crossref]

Zhang, F.

Q. Zhao, J. Zhou1, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[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]

Zhao, Q.

Q. Zhao, J. Zhou1, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[Crossref]

Zhou1, J.

Q. Zhao, J. Zhou1, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[Crossref]

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]

Ziolkowski, R.W.

N. Engheta and R.W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microwave Theory Tech. 53, 1535–1556 (2005).
[Crossref]

Adv. Mater. (2)

P. D. García, R. Sapienza, A. Blanco, and C. López, “Photonic glass: a novel random material for light,” Adv. Mater. 19, 2597–2602 (2007).
[Crossref]

M. Ibisate, D. Golmayo, and C. López, “Photonic crystals: silicon direct opals,” Adv. Mater. 28, 2899–2902 (2009).
[Crossref]

Appl. Phys. Lett. (2)

M. Reufer, L. F. Rojas-Ochoa, S. Eiden, J. J. Sáenz, and F. Scheffold, “Transport of light in amorphous photonic materials,” Appl. Phys. Lett. 91, 171904 (2007).
[Crossref]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85, 543–545 (2004).
[Crossref]

Astrophys. J. (2)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[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 sphercal particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596–2603 (2003).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

N. Engheta and R.W. Ziolkowski, “A positive future for double-negative metamaterials,” IEEE Trans. Microwave Theory Tech. 53, 1535–1556 (2005).
[Crossref]

Int. J. Microwave Millimeter-Wave Comput.-Aided Eng. (1)

R. K. Mongia and P. Bhartia, “Dielectric resonator antennas-A review and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter-Wave Comput.-Aided Eng. 4, 230–247 (1994).
[Crossref]

J. Appl. Phys. (1)

L. Jyhlä, I. Kolmakov, S. Maslovski, and S. Tretyakov, “Modeling of isotropic backward-wave materials composed of resonant spheres,” J. Appl. Phys. 99, 043102 (2006).
[Crossref]

J. Phys.: Condens. Matter (1)

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).
[Crossref]

Mater. Today (1)

Q. Zhao, J. Zhou1, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12, 60–69 (2009).
[Crossref]

Nat. Photonics (3)

R. C. J. Hsu, A. Ayazi, B. Houshmand, and B. Jalali, “All-dielectric photonic-assisted radio front-end technology,” Nat. Photonics 1, 535–538 (2007).
[Crossref]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[Crossref]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[Crossref]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Nature Mater. (1)

J. B. Pendry, “Beyond metamaterials,” Nature Mater. 5, 763–764 (2006).
[Crossref]

Opt. Express (6)

Phys. Rev. A (2)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978)
[Crossref]

G. Videen and W. S. Bickel, “Light-scattering resonances in small spheres,” Phys. Rev. A 45, 6008–6012 (1992).
[Crossref] [PubMed]

Phys. Rev. B (6)

A. Alú and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78, 085112 (2008).
[Crossref]

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modelling of all-dielectric metamaterials,” Phys. Rev. B 77, 045104 (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]

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. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2000).
[Crossref]

Phys. Rev. Lett. (7)

R. Carminati and J. J. Sáenz, “Density of states and extinction mean free path of waves in random media: dispersion relations and sum rules,” Phys. Rev. Lett. 102, 093902 (2009).
[Crossref] [PubMed]

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

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94, 017402 (2005).
[Crossref] [PubMed]

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garca-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett. 101, 157403 (2008).
[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]

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

Other (8)

E.D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999), Sec. 2.3.

O. N. Singh and A. Lakhtakia, eds. Electromagnetic Fields in Unconventional Materials and Structures (Wiley, 2000).

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves-Advanced Topics (Wiley, 2001).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).
[Crossref]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge Univ. Press, 2006).

Supplementary Material (2)

» Media 1: PDF (254 KB)     
» Media 2: PDF (278 KB)     

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Figures (6)

Fig. 1
Fig. 1 Incident field vector.

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Fig. 2
Fig. 2 Scattering cross section map of a non-absorbing Mie sphere as a function of the refractive index m and the y parameter, y = mka = m(2πa/λ). Green areas correspond to parameter ranges where the magnetic dipole contribution dominates the total scattering cross section, while red areas represent regions where the electric dipole contribution is dominating. The remaining blue-saturated areas are dominated by higher order multipoles. Brightness in the color-map is proportional to the total cross section. White horizontal lines represent the y-range covered by figures 3 (m = 3.5) and 5 (m = 2.45).

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Fig. 3
Fig. 3 Scattering cross-section σS versus the wavelength λ for a 230nm Si sphere (the refraction index m = 3.5 is constant and real in this wavelength range). The contribution of each term in the Mie expansion is also shown. The green line corresponds to the magnetic dipole contribution.

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Fig. 4
Fig. 4 Maps for the modulus of the total electric an magnetic vectors normalized to the incoming electric and magnetic field respectively (Etot/Einc and Htot/Hinc), for a Si nanoparticle of radius a = 230nm under plane wave illumination, (cf. Fig. 1). XZ planes crossing y = 0 are displayed. The left and central panels correspond to λ = 1250nm and λ = 1680nm of the electric and magnetic resonance peaks of Fig. 3, respectively. The right panel corresponds to the situation where the electric and magnetic resonances contribute equally to the scattering cross section (λ = 1525nm). The corresponding far field scattering radiation patterns for the three wavelengths are shown in the bottom row. The corresponding maps for YZ and XZ planes can be seen in the suplementary figure files ( Media 1) and ( Media 2), respectively.

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Fig. 5
Fig. 5 Scattering cross-section σS versus the wavelength λ for a sphere of radius a = 330nm (its relative refractive index is m = 6 ≈ 2.45, constant and real in this wavelength range). The contribution of each coefficient in the Mie expansion is also shown. The green peak at longer wavelengths corresponds to the magnetic dipole contribution.

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Fig. 6
Fig. 6 Effective real and imaginary permittivities and permeabilities for an arbitrary arrangement of Si spheres in an otherwise homogeneous medium with ɛh = μh = 1 for two different filling factors f = 0.25 (a) and f = 0.5 (b).

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Equations (15)

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E = E 0 u Z e i k X e − i ω t ,                       B = B 0 u Y e i k X e − i ω t
a n = 1 2 ( 1 − e − 2 i α n ) = i sin α n e − i α n
b n = 1 2 ( 1 − e − 2 i β n ) = i sin β n e − i β n
tan α n = m 2 j n ( y ) [ x j n ( x ) ] ′ − j n ( x ) [ y j n ( y ) ] ′ m 2 j n ( y ) [ x y n ( x ) ] ′ − y n ( x ) [ y j n ( y ) ] ′ ,
tan β n = j n ( y ) [ x j n ( x ) ] ′ − j n ( x ) [ y j n ( y ) ] ′ j n ( y ) [ x y n ( x ) ] ′ − y n ( x ) [ y j n ( y ) ] ′ ,
σ S = σ ext = 2 π k 2 ∑ n = 1 ∞ ( 2 n + 1 ) { sin 2 α n + sin 2 β n } = ∑ n = 1 ∞ { σ E , n + σ M , n }
σ S res = σ ext res | { x ≪ 1 ; m ≫ 1 } ≈ 2 π k 2 ( 2 n + 1 )
p = ɛ 0 ɛ h α E E ,                     m = 1 μ 0 μ h α M B ;
α E = i ( k 3 6 π ) − 1 a 1 ,                       α M = i ( k 3 6 π ) − 1 b 1 .
α E = α E ( 0 ) 1 − i k 3 6 π α E ( 0 ) ,               α M = α M ( 0 ) 1 − i k 3 6 π α M ( 0 ) ,
α E ( 0 ) = − 6 π k 3 tan α 1 ,                         α M ( 0 ) = − 6 π k 3 tan β 1 .
σ ext = k Im { α E + α M }
σ S = k 4 6 π { | α E | 2 + | α M | 2 }
α E ( 0 ) | y ≪ 1 ≈ 4 π a 3 m 2 − 1 m 2 + 2 ,                 α M ( 0 ) | y ≪ 1 ≈ 4 π a 3 ( m 2 − 1 ) k 2 a 3 30
ɛ eff − 1 ɛ eff + 2 = f α E 4 π a 3 ;                       μ eff − 1 μ eff + 2 = f α M 4 π a 3

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