Abstract

The optical response of graded index spherical particles is studied using an effective medium approach, where the homogenization of the graded particle is achieved by using a static effective dielectric function available in the literature. Full-wave calculation using the standard Mie theory for this “homogenized system” shows that for a plasmonic particle, such an approximation can lead to highly-accurate results compared to the exact ones, especially for slowly and smoothly varying index profiles. An illustration is provided via an application of this method to the design of an optical cloak using a graded plasmonic coating based on the scattering cancellation scheme. This approach thus surpasses the various long-wavelength approximations currently available in the literature and provides an efficient numerical treatment of light scattering from these inhomogeneous particles without having to solve directly the Maxwell’s equations with spatially varying dielectric functions.

© 2012 Optical Society of America

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References

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  1. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).
  2. A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
    [CrossRef]
  3. A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
    [CrossRef]
  4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef]
  5. A. Alu and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. Pure Appl. Opt. 10, 093002 (2008).
    [CrossRef]
  6. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
    [CrossRef]
  7. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
    [CrossRef]
  8. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623(2005).
    [CrossRef]
  9. A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
    [CrossRef]
  10. L. Sun and K. W. Yu, “Broadband electromagnetic transparency by graded metamaterials: scattering cancellation scheme,” J. Opt. Soc Am. B 28, 994–1001 (2011).
    [CrossRef]
  11. G. Mie, “Beitrige zur Optik trüber Medien, speziell kolloidaler Metallösungen (Contributions to the optics of turbid media, especially colloidal metal suspensions),” Ann. Phys. 25, 376–445(1908).
  12. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  13. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960–1967 (1985).
    [CrossRef]
  14. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
    [CrossRef]
  15. P. J. Wyatt, “Errata: Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 134, AB1 (1964).
  16. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), Sec. 5.6.
  17. L. Dong, G. Q. Gu, and K. W. Yu, “First-principles approach to dielectric response of graded sphere particles,” Phys. Rev. B 67, 224205 (2003).
    [CrossRef]
  18. K. W. Yu and G. Q. Gu, “Effective conductivity of composites of graded spherical particles,” Phys. Lett. Sect. A 345, 448–452 (2005).
    [CrossRef]
  19. H. Y. Chung, P. T. Leung, and D. P. Tsai, “Modified long wavelength approximation for the optical response of a graded index plasmonic nanoparticle,” Plasmonics 7, 13–18 (2012).
  20. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8, 581–583 (1983).
    [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 14.
  22. M. Kerker, L. H. Kauffman, and W. A. Farone, “Scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. Numerical results,” J. Opt. Soc. Am. 56, 1053–1556 (1966).
    [CrossRef]
  23. L. Kai and P. Massoli, “Scattering of electromagnetic-plane wave by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
    [CrossRef]
  24. Note that our graded Drude function is slightly different from that used in [10] and damping is included in our model.
  25. V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
    [CrossRef]
  26. T. C. Choy, Effective Medium Theory: Principles and Applications (Clarendon, 1999).
  27. J. Schilling, “The quest for zero refractive index,” Nature Photonics 5, 449–451 (2011).
    [CrossRef]
  28. H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131, 124122 (2009).
    [CrossRef]

2012 (1)

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Modified long wavelength approximation for the optical response of a graded index plasmonic nanoparticle,” Plasmonics 7, 13–18 (2012).

2011 (2)

L. Sun and K. W. Yu, “Broadband electromagnetic transparency by graded metamaterials: scattering cancellation scheme,” J. Opt. Soc Am. B 28, 994–1001 (2011).
[CrossRef]

J. Schilling, “The quest for zero refractive index,” Nature Photonics 5, 449–451 (2011).
[CrossRef]

2010 (2)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
[CrossRef]

2009 (2)

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131, 124122 (2009).
[CrossRef]

2008 (4)

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

A. Alu and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. Pure Appl. Opt. 10, 093002 (2008).
[CrossRef]

V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
[CrossRef]

A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
[CrossRef]

2005 (2)

K. W. Yu and G. Q. Gu, “Effective conductivity of composites of graded spherical particles,” Phys. Lett. Sect. A 345, 448–452 (2005).
[CrossRef]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623(2005).
[CrossRef]

2003 (1)

L. Dong, G. Q. Gu, and K. W. Yu, “First-principles approach to dielectric response of graded sphere particles,” Phys. Rev. B 67, 224205 (2003).
[CrossRef]

2000 (1)

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

1994 (1)

1985 (1)

1983 (1)

1966 (1)

1964 (1)

P. J. Wyatt, “Errata: Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 134, AB1 (1964).

1962 (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1908 (1)

G. Mie, “Beitrige zur Optik trüber Medien, speziell kolloidaler Metallösungen (Contributions to the optics of turbid media, especially colloidal metal suspensions),” Ann. Phys. 25, 376–445(1908).

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. Pure Appl. Opt. 10, 093002 (2008).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
[CrossRef]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623(2005).
[CrossRef]

Bhandari, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 14.

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Chen, Y.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Choy, T. C.

T. C. Choy, Effective Medium Theory: Principles and Applications (Clarendon, 1999).

Chung, H. Y.

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Modified long wavelength approximation for the optical response of a graded index plasmonic nanoparticle,” Plasmonics 7, 13–18 (2012).

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131, 124122 (2009).
[CrossRef]

Dong, L.

L. Dong, G. Q. Gu, and K. W. Yu, “First-principles approach to dielectric response of graded sphere particles,” Phys. Rev. B 67, 224205 (2003).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
[CrossRef]

A. Alu and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. Pure Appl. Opt. 10, 093002 (2008).
[CrossRef]

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623(2005).
[CrossRef]

Farone, W. A.

Fedotov, V. A.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Gu, G. Q.

K. W. Yu and G. Q. Gu, “Effective conductivity of composites of graded spherical particles,” Phys. Lett. Sect. A 345, 448–452 (2005).
[CrossRef]

L. Dong, G. Q. Gu, and K. W. Yu, “First-principles approach to dielectric response of graded sphere particles,” Phys. Rev. B 67, 224205 (2003).
[CrossRef]

Janicki, V.

V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
[CrossRef]

Kai, L.

Kauffman, L. H.

Kerker, M.

M. Kerker, L. H. Kauffman, and W. A. Farone, “Scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. Numerical results,” J. Opt. Soc. Am. 56, 1053–1556 (1966).
[CrossRef]

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), Sec. 5.6.

Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
[CrossRef]

Khardikov, V. V.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Kivshar, Y. S.

A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
[CrossRef]

Leonhardt, U.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

Leung, P. T.

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Modified long wavelength approximation for the optical response of a graded index plasmonic nanoparticle,” Plasmonics 7, 13–18 (2012).

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131, 124122 (2009).
[CrossRef]

Massoli, P.

Meier, M.

Mie, G.

G. Mie, “Beitrige zur Optik trüber Medien, speziell kolloidaler Metallösungen (Contributions to the optics of turbid media, especially colloidal metal suspensions),” Ann. Phys. 25, 376–445(1908).

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
[CrossRef]

Pendry, J. B.

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

Prosvirnin, S. L.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Sancho-Parramon, J.

V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
[CrossRef]

Schilling, J.

J. Schilling, “The quest for zero refractive index,” Nature Photonics 5, 449–451 (2011).
[CrossRef]

Schwanecke, A. S.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Shalaev, V.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Shevts, G.

A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
[CrossRef]

Sun, L.

L. Sun and K. W. Yu, “Broadband electromagnetic transparency by graded metamaterials: scattering cancellation scheme,” J. Opt. Soc Am. B 28, 994–1001 (2011).
[CrossRef]

Tsai, D. P.

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Modified long wavelength approximation for the optical response of a graded index plasmonic nanoparticle,” Plasmonics 7, 13–18 (2012).

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131, 124122 (2009).
[CrossRef]

Tyc, T.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

Wokaun, A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 14.

Wyatt, P. J.

P. J. Wyatt, “Errata: Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 134, AB1 (1964).

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Yu, K. W.

L. Sun and K. W. Yu, “Broadband electromagnetic transparency by graded metamaterials: scattering cancellation scheme,” J. Opt. Soc Am. B 28, 994–1001 (2011).
[CrossRef]

K. W. Yu and G. Q. Gu, “Effective conductivity of composites of graded spherical particles,” Phys. Lett. Sect. A 345, 448–452 (2005).
[CrossRef]

L. Dong, G. Q. Gu, and K. W. Yu, “First-principles approach to dielectric response of graded sphere particles,” Phys. Rev. B 67, 224205 (2003).
[CrossRef]

Zheludev, N. I.

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Zorc, H.

V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beitrige zur Optik trüber Medien, speziell kolloidaler Metallösungen (Contributions to the optics of turbid media, especially colloidal metal suspensions),” Ann. Phys. 25, 376–445(1908).

Appl. Opt. (2)

J. Appl. Phys. (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Chem. Phys. (1)

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131, 124122 (2009).
[CrossRef]

J. Opt. Pure Appl. Opt. (1)

A. Alu and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. Pure Appl. Opt. 10, 093002 (2008).
[CrossRef]

J. Opt. Soc Am. B (1)

L. Sun and K. W. Yu, “Broadband electromagnetic transparency by graded metamaterials: scattering cancellation scheme,” J. Opt. Soc Am. B 28, 994–1001 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

Nano Lett. (1)

A. S. Schwanecke, V. A. Fedotov, V. V. Khardikov, S. L. Prosvirnin, Y. Chen, and N. I. Zheludev, “Nanostructured metal film with asymmetric optical transmission,” Nano Lett. 8, 2940–2943 (2008).
[CrossRef]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
[CrossRef]

Nature Photonics (1)

J. Schilling, “The quest for zero refractive index,” Nature Photonics 5, 449–451 (2011).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. Sect. A (1)

K. W. Yu and G. Q. Gu, “Effective conductivity of composites of graded spherical particles,” Phys. Lett. Sect. A 345, 448–452 (2005).
[CrossRef]

Phys. Rev. (2)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

P. J. Wyatt, “Errata: Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 134, AB1 (1964).

Phys. Rev. B (1)

L. Dong, G. Q. Gu, and K. W. Yu, “First-principles approach to dielectric response of graded sphere particles,” Phys. Rev. B 67, 224205 (2003).
[CrossRef]

Phys. Rev. E (1)

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623(2005).
[CrossRef]

Phys. Rev. Lett. (3)

A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008).
[CrossRef]

A. B. Khanikaev, S. H. Mousavi, G. Shevts, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010).
[CrossRef]

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

Plasmonics (1)

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Modified long wavelength approximation for the optical response of a graded index plasmonic nanoparticle,” Plasmonics 7, 13–18 (2012).

Science (1)

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

Thin Solid Films (1)

V. Janicki, J. Sancho-Parramon, and H. Zorc, “Refractive index profile modelling of dielectric inhomogeneous coatings using effective medium theories,” Thin Solid Films 516, 3368–3373 (2008).
[CrossRef]

Other (5)

T. C. Choy, Effective Medium Theory: Principles and Applications (Clarendon, 1999).

Note that our graded Drude function is slightly different from that used in [10] and damping is included in our model.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), Sec. 5.6.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 14.

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of the accuracy and limitation of our proposed effective medium approach: plots of the extinction efficiencies of a graded sphere with dielectric function ε ( r ) = c r / a , where (a)  c = 2.5 , (b)  c = 2.5 + i , (c)  c = - 2.5 , and (d)  c = - 2.5 + i . The radius of the sphere is fixed at a = 1000 nm .

Fig. 2.
Fig. 2.

Extinction efficiency for a graded Drude sphere with ε = ε Ag r m / a m with m = 1 and a radius fixed at a = 100 nm . Results from our EMT are compared with those from the exact graded Mie theory as well as those from the static (Rayleigh) and MLWA approximations.

Fig. 3.
Fig. 3.

Extinction efficiency for a graded Drude sphere with ε = ε Ag r m / a m of different power indices: (a)  m = 1 , (b)  m = 2 , and (c)  m = 3 . The radius is fixed at a = 100 nm .

Fig. 4.
Fig. 4.

Extinction efficiency for a dielectric sphere of radius a coated by a graded index shell. The dielectric constant of the sphere is ε 1 = 4 and that of the shell has a linear profile ε 2 = - 3 r / b , where the outer radius of the shell is fixed to b = 0.01 λ . The effective dielectric function for the dipole resonance mode ε s , 1 is shown by the dashed line.

Fig. 5.
Fig. 5.

Same as in Fig. 4, except that loss is introduced for the shell with ε 2 = ( - 3 + 0.1 i ) r / b .

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

E ( i ) = E 0 e i ( k z - ω t ) e x ,
E r ( s ) = E 0 cos φ ( k r ) 2 = 1 ( + 1 ) B e ζ ( 1 ) ( k r ) P 1 ( cos θ ) ,
E θ ( s ) = - E 0 cos φ k r = 1 { B e ζ ( 1 ) ( k r ) P 1 ( cos θ ) sin θ - i B m ζ ( 1 ) ( k r ) P 1 ( cos θ ) 1 sin θ } ,
E φ ( s ) = - E 0 sin φ k r = 1 { B e ζ ( 1 ) ( k r ) P 1 ( cos θ ) 1 sin θ - i B m ζ ( 1 ) ( k r ) P 1 ( cos θ ) sin θ } ,
B e = i + 1 2 + 1 ( + 1 ) ε ( x ) ψ ( x ) W ( x ) - W ( x ) ψ ( x ) ε ( x ) ζ ( 1 ) ( x ) W ( x ) - W ( x ) ζ ( 1 ) ( x ) ,
B m = i + 1 2 + 1 ( + 1 ) ε ( x ) ψ ( x ) G ( x ) - G ( x ) ψ ( x ) ε ( x ) ζ ( 1 ) ( x ) G ( x ) - G ( x ) ζ ( 1 ) ( x ) ,
d 2 W d ρ 2 - d ln ε d ρ d W d ρ + [ ε - ( + 1 ) ρ 2 ] W = 0 ,
d 2 G d ρ 2 + [ ε - ( + 1 ) ρ 2 ] G = 0 .
d ε s d a = - ( ε s - ε ) 3 a ε [ ( 3 - x 2 ) ( ε s - ε ) + 9 ε ] ,
ε s ( a ) = z a m ,
z = η 2 [ - ( 1 + m ) + m 2 + 2 m + 9 ] .
B e = i + 1 2 + 1 ( + 1 ) β ψ ( x ) ψ ( β x ) - ψ ( β x ) ψ ( x ) β ζ ( 1 ) ( x ) ψ ( β x ) - ψ ( β x ) ζ ( 1 ) ( x ) ,
d 2 ψ ( x ) d x 2 + [ 1 - ( + 1 ) x 2 ] ψ ( x ) = 0 .
ε ( x ) ψ ( x ) - w ( x ) ψ ( x ) ε ( x ) ζ ( 1 ) ( x ) - w ( x ) ζ ( 1 ) ( x ) = β ( x ) ψ ( x ) - ψ ( x ) f ( x ) β ( x ) ζ ( 1 ) ( x ) - ζ ( 1 ) ( x ) f ( x ) ,
w ( x ) = ε ( x ) β ( x ) f ( x ) .
w ( x ) + w 2 ( x ) - ε ( x ) ε ( x ) w ( x ) + [ ε ( x ) - ( + 1 ) x 2 ] = 0 .
w ( x ) = ( ε ( x ) β ( x ) - ε ( x ) β ( x ) 2 β 3 / 2 ( x ) ) f ( x ) + ε ( x ) β ( x ) f ( x ) ,
β = 2 β 3 / 2 ε f [ ε 2 β f 2 + ε β f + ε - ( + 1 ) x 2 ] .
ψ ( x ) = x + 1 ( 2 + 1 ) !! [ 1 - x 2 2 ( 2 + 3 ) + ] .
β ( x ) = - β - ε x ε [ β + ( + 1 ) ε ] + β 2 x + 1 .
β ( x ) = - β - ε x ε [ β + ( + 1 ) ε ] ,
β = [ ε a + ( + 1 ) ε b ] b 2 + 1 + ( + 1 ) ( ε a - ε b ) a 2 + 1 [ ε a + ( + 1 ) ε b ] b 2 + 1 - ( ε a - ε b ) a 2 + 1 ε b .
( ε 1 - ε 2 ) [ ( + 1 ) ε 2 + ε h ] a 2 + 1 = ( ε 2 - ε h ) [ ε 1 + ( + 1 ) ε 2 ] b 2 + 1 .
β a = ε Ag 2 [ - ( 1 + m ) + ( m + 1 ) 2 + 4 ( + 1 ) ] a m L m ,
β b = ε Ag 2 [ - ( 1 + m ) + ( m + 1 ) 2 + 4 ( + 1 ) ] b m L m .
ε a β a , ε b β S , β β b ,
β S = - ξ ± ξ 2 + η ,
ξ = 1 2 ( b 2 + 1 - a 2 + 1 ) [ ( + 1 β a - β b ) b 2 + 1 - ( + 1 β b - β a ) a 2 + 1 ] ,
η = + 1 β a β b .

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