Abstract

The electromagnetic scattering by a radially inhomogeneous isotropic metamaterial sphere whose electric permittivity is described by the lossless graded Drude model is studied according to the generalized Mie theory in full-wave condition. The distribution of electromagnetic field is calculated by solving Maxwell’s equations, and the exact analytic solutions are obtained in terms of confluent Heun and hypergeometry functions. This allows us to achieve the full-wave scattering cross section (SCS) analytically. The corresponding numerical analysis indicates that the full-wave SCS can be extremely small over a broad frequency band, representing a broadband electromagnetic transparency. Moreover, the analytic expression of the full-wave SCS also reveals the conditions for achieving the broadband electromagnetic transparency and makes tunable electromagnetic transparency feasible.

© 2011 Optical Society of America

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2010 (2)

T. Tyc, H. Y. Chen, C. T. Chan, and U. Leonhardt, “Non-Euclidean cloaking for light waves,” IEEE J. Sel. Top. Quantum Electron. 16, 418–426 (2010).
[CrossRef]

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

2009 (5)

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. 103, 073901 (2009).
[CrossRef] [PubMed]

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Broadband exterior cloaking,” Opt. Express 17, 14800–14805 (2009).
[CrossRef] [PubMed]

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

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef] [PubMed]

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

2008 (4)

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

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

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

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008).
[CrossRef]

2007 (4)

A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318–3332 (2007).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15, 7578–7590 (2007).
[CrossRef] [PubMed]

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[CrossRef]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

2006 (6)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431, 87–172 (2006).
[CrossRef]

D. A. B. Miller, “On perfect cloaking,” Opt. Express 14, 12457–12466 (2006).
[CrossRef] [PubMed]

2005 (1)

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

2004 (1)

2003 (1)

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

1986 (1)

E. W. Leaver, “Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics,” J. Math. Phys. 27, 1238–1265 (1986).
[CrossRef]

1978 (2)

C. G. Gray, “Multipole expansions of electromagnetic fields using Debye potentials,” Am. J. Phys. 46, 169–179 (1978).
[CrossRef]

C. G. Gray, “Debye potential representation of vector fields,” Am. J. Phys. 46, 735–736 (1978).
[CrossRef]

1976 (1)

1975 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Alù, A.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef] [PubMed]

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

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

A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15, 7578–7590 (2007).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318–3332 (2007).
[CrossRef] [PubMed]

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

Bohren, C. F.

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

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge, 1999).
[PubMed]

Cai, W. S.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[CrossRef]

Chan, C. T.

T. Tyc, H. Y. Chen, C. T. Chan, and U. Leonhardt, “Non-Euclidean cloaking for light waves,” IEEE J. Sel. Top. Quantum Electron. 16, 418–426 (2010).
[CrossRef]

Chen, H. S.

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Chen, H. Y.

T. Tyc, H. Y. Chen, C. T. Chan, and U. Leonhardt, “Non-Euclidean cloaking for light waves,” IEEE J. Sel. Top. Quantum Electron. 16, 418–426 (2010).
[CrossRef]

Chettiar, U. K.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[CrossRef]

Chew, H.

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Ding, K.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Dong, L.

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

Du, H.

Edwards, B.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef] [PubMed]

Engheta, N.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef] [PubMed]

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

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

A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15, 7578–7590 (2007).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Plasmonic materials in transparency and cloaking problems: mechanism, robustness, and physical insights,” Opt. Express 15, 3318–3332 (2007).
[CrossRef] [PubMed]

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

Fung, T. H.

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008).
[CrossRef]

Gao, L.

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008).
[CrossRef]

Gao, Y.

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

Gray, C. G.

C. G. Gray, “Multipole expansions of electromagnetic fields using Debye potentials,” Am. J. Phys. 46, 169–179 (1978).
[CrossRef]

C. G. Gray, “Debye potential representation of vector fields,” Am. J. Phys. 46, 735–736 (1978).
[CrossRef]

Gu, G. Q.

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

Guo, D. R.

Z. X. Wang and D. R. Guo, Introduction to Special Functions (Peking University, 2004).

Huang, J. P.

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431, 87–172 (2006).
[CrossRef]

Huangfu, J. T.

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Huffman, D. R.

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

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Kerker, M.

Kildishev, A. V.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[CrossRef]

Kong, J. A.

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1990).

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Leaver, E. W.

E. W. Leaver, “Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics,” J. Math. Phys. 27, 1238–1265 (1986).
[CrossRef]

Leonhardt, U.

T. Tyc, H. Y. Chen, C. T. Chan, and U. Leonhardt, “Non-Euclidean cloaking for light waves,” IEEE J. Sel. Top. Quantum Electron. 16, 418–426 (2010).
[CrossRef]

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

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

Liu, Y. M.

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

Luo, Y.

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Miller, D. A. B.

Milton, G. W.

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. 103, 073901 (2009).
[CrossRef] [PubMed]

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Broadband exterior cloaking,” Opt. Express 17, 14800–14805 (2009).
[CrossRef] [PubMed]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Onofrei, D.

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Broadband exterior cloaking,” Opt. Express 17, 14800–14805 (2009).
[CrossRef] [PubMed]

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. 103, 073901 (2009).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

Popa, B. I.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Qiu, C. W.

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008).
[CrossRef]

Ronveaux, A.

A. Ronveaux, Heun’s Differential Equations (Oxford, 1995).

Schurig, D.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[CrossRef]

Silveirinha, M. G.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Tsang, L.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Tyc, T.

T. Tyc, H. Y. Chen, C. T. Chan, and U. Leonhardt, “Non-Euclidean cloaking for light waves,” IEEE J. Sel. Top. Quantum Electron. 16, 418–426 (2010).
[CrossRef]

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

van de Hulst, H. C.

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

Vasquez, F. G.

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Broadband exterior cloaking,” Opt. Express 17, 14800–14805 (2009).
[CrossRef] [PubMed]

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. 103, 073901 (2009).
[CrossRef] [PubMed]

Wang, Z. X.

Z. X. Wang and D. R. Guo, Introduction to Special Functions (Peking University, 2004).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge, 1999).
[PubMed]

Wu, B. I.

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Yu, K. W.

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008).
[CrossRef]

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431, 87–172 (2006).
[CrossRef]

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

Zhang, B.

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

Zhang, J. J.

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Zhang, X.

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

Am. J. Phys. (2)

C. G. Gray, “Multipole expansions of electromagnetic fields using Debye potentials,” Am. J. Phys. 46, 169–179 (1978).
[CrossRef]

C. G. Gray, “Debye potential representation of vector fields,” Am. J. Phys. 46, 735–736 (1978).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

T. Tyc, H. Y. Chen, C. T. Chan, and U. Leonhardt, “Non-Euclidean cloaking for light waves,” IEEE J. Sel. Top. Quantum Electron. 16, 418–426 (2010).
[CrossRef]

J. Math. Phys. (1)

E. W. Leaver, “Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics,” J. Math. Phys. 27, 1238–1265 (1986).
[CrossRef]

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

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

J. Opt. Soc. Am. (2)

Nat. Photon. (1)

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photon. 1, 224–227 (2007).
[CrossRef]

Opt. Express (4)

Phys. Rep. (1)

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431, 87–172 (2006).
[CrossRef]

Phys. Rev. B (2)

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

J. J. Zhang, J. T. Huangfu, Y. Luo, H. S. Chen, J. A. Kong, and B. I. Wu, “Cloak for multilayered and gradually changing media,” Phys. Rev. B 77, 035116 (2008).
[CrossRef]

Phys. Rev. E (3)

L. Gao, T. H. Fung, K. W. Yu, and C. W. Qiu, “Electromagnetic transparency by coated spheres with radial anisotropy,” Phys. Rev. E 78, 046609 (2008).
[CrossRef]

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

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

H. S. Chen, B. I. Wu, B. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” Phys. Rev. Lett. 99, 063903 (2007).
[CrossRef] [PubMed]

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

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef] [PubMed]

Y. Gao, J. P. Huang, Y. M. Liu, L. Gao, K. W. Yu, and X. Zhang, “Optical negative refraction in ferrofluids with magnetocontrollability,” Phys. Rev. Lett. 104, 034501 (2010).
[CrossRef] [PubMed]

F. G. Vasquez, G. W. Milton, and D. Onofrei, “Active exterior cloaking for the 2D Laplace and Helmholtz equations,” Phys. Rev. Lett. 103, 073901 (2009).
[CrossRef] [PubMed]

Prog. Opt. (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).
[CrossRef]

Science (4)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[CrossRef] [PubMed]

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

Other (8)

A. Ronveaux, Heun’s Differential Equations (Oxford, 1995).

Z. X. Wang and D. R. Guo, Introduction to Special Functions (Peking University, 2004).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

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 (Wiley, 1983).

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1990).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge, 1999).
[PubMed]

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Electric permittivity and magnetic permeability of the graded isotropic sphere.

Fig. 2
Fig. 2

Contour plots and the cross section diagrams of the SCS for (a)  r 0 = 0.1 , (b) the full-wave condition, and (c), (d) quasi-static approximation versus the frequency ω of the incident electromagnetic wave and the gradient parameter c 1 . Darker regions in the plots correspond to lower SCS.

Fig. 3
Fig. 3

Similar to Fig. 2, but for (a)  r 0 = 1.0 , (b) the full-wave condition, and (c), (d) quasi-static approximation.

Fig. 4
Fig. 4

Full-wave SCS of a homogenous isotropic sphere.

Equations (54)

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ϵ ( r ) = ϵ ϵ s ( r ) I ,
μ ( r ) = μ μ s ( r ) I ,
E = 1 i ω ϵ ( r ) × × ( r u ) + × ( r v ) ,
H = 1 i ω μ ( r ) × × ( r v ) × ( r u ) .
ϵ ( r ) r [ 1 ϵ ( r ) ( r u ) r ] + 1 r 2 sin θ θ [ sin θ ( r u ) θ ] + 1 r 2 sin 2 θ 2 ( r u ) ϕ 2 + ω 2 ϵ ( r ) μ ( r ) r u = 0 ,
μ ( r ) r [ 1 μ ( r ) ( r v ) r ] + 1 r 2 sin θ θ [ sin θ ( r v ) θ ] + 1 r 2 sin 2 θ 2 ( r v ) ϕ 2 + ω 2 ϵ ( r ) μ ( r ) r v = 0 ,
ϵ ( r ) d d r [ 1 ϵ ( r ) d ( r f u ( r ) ) d r ] + [ ω 2 ϵ ( r ) μ ( r ) l ( l + 1 ) r 2 ] r f u ( r ) = 0 ,
μ ( r ) d d r [ 1 μ ( r ) d ( r f v ( r ) ) d r ] + [ ω 2 ϵ ( r ) μ ( r ) l ( l + 1 ) r 2 ] r f v ( r ) = 0 ,
u ( 1 ) ( r 0 ) = u ( 2 ) ( r 0 ) ,
v ( 1 ) ( r 0 ) = v ( 2 ) ( r 0 ) ,
1 ϵ ( 1 ) ( r u ( 1 ) ) r | r = r 0 = 1 ϵ ( 2 ) ( r u ( 2 ) ) r | r = r 0 ,
1 μ ( 1 ) ( r v ( 1 ) ) r | r = r 0 = 1 μ ( 2 ) ( r v ( 2 ) ) r | r = r 0 ,
E in = e x e i k r cos θ ,
H in = ( e y / Z ) e i k r cos θ ,
ϵ s ( r ) = 1 ω p 2 ( 0 ) ω 2 ( c 0 c 1 ( r / r 0 ) n ) ,
u in ( r , θ , ϕ ) = ω ϵ k l = 1 i l 2 l + 1 l ( l + 1 ) j l ( k r ) P l ( 1 ) ( cos θ ) cos ϕ ,
v in ( r , θ , ϕ ) = ω μ k Z l = 1 i l 2 l + 1 l ( l + 1 ) j l ( k r ) P l ( 1 ) ( cos θ ) sin ϕ ,
u sc ( r , θ , ϕ ) = ω ϵ k l = 1 i l 2 l + 1 l ( l + 1 ) A u ( l ) h l ( 1 ) ( k r ) P l ( 1 ) ( cos θ ) cos ϕ ,
v sc ( r , θ , ϕ ) = ω μ k Z l = 1 i l 2 l + 1 l ( l + 1 ) A v ( l ) h l ( 1 ) ( k r ) P l ( 1 ) ( cos θ ) sin ϕ ,
ϵ s ( r ) d d r [ 1 ϵ s ( r ) d ( r f u ( r ) ) d r ] + [ k 2 ϵ s ( r ) l ( l + 1 ) r 2 ] r f u ( r ) = 0 ,
d 2 ( r f v ( r ) ) d r 2 + [ k 2 ϵ s ( r ) l ( l + 1 ) r 2 ] r f v ( r ) = 0 .
x ( x x 0 ) g ( x ) + ( B 1 + B 2 x ) g ( x ) + [ w 2 x ( x x 0 ) 2 w η ( x x 0 ) + B 3 ] g ( x ) = 0 ,
x = r 2 ,
x 0 = a / b ,
w = k b / 2 ,
η = k a / ( 4 b ) ,
B 1 = a ( 3 + 2 l ) / ( 2 b ) ,
B 2 = ( 1 + 2 l ) / 2 ,
B 3 = ( 1 + l ) / 2 .
g ( x ) = e i w x H c ( a ) ( p , α , γ , δ , σ ; x / x 0 ) = e i w x j = 0 c j ( a ) ( x / x 0 ) j ,
p = i a k / ( 4 b ) ,
α = [ i a k + b ( 1 + 2 l ) ] / ( 4 b ) ,
γ = ( 3 + 2 l ) / 2 ,
δ = 1 ,
σ = [ ( b + i a k ) ( 2 b ( 1 + l ) + i a k ) ] / ( 4 b ) .
c 1 ( a ) = 0 ,
c 0 ( a ) = 1 ,
f j ( a ) c j + 1 ( a ) + g j ( a ) c j ( a ) + h j ( a ) c j 1 ( a ) = 0 ,
g j ( a ) = j ( j 4 p + γ + δ 1 ) σ ,
f j ( a ) = ( j + 1 ) ( j + γ ) ,
h j ( a ) = 4 p ( j + α 1 ) .
f u ( r ) = r l e 1 2 i k b r 2 H c ( a ) ( p , α , γ , δ , σ ; b r 2 / a ) .
4 x z ( x ) + ( 6 + 4 l ) z ( x ) + k 2 ( a + b x ) z ( x ) = 0 ,
z ( x ) = e 1 2 i k b x F 11 ( κ ; ν ; i k b x ) ,
κ = [ ( 3 + 2 l ) b + i k a ] / ( 4 b ) ,
ν = 3 / 2 + l .
f v ( r ) = r l e 1 2 i k b r 2 F 11 ( κ ; ν ; i k b r 2 ) .
u sp ( r , θ , ϕ ) = ω ϵ k l = 1 i l 2 l + 1 l ( l + 1 ) B u ( l ) f u ( r ) P l ( 1 ) ( cos θ ) cos ϕ ,
v sp ( r , θ , ϕ ) = ω μ k Z l = 1 i l 2 l + 1 l ( l + 1 ) B v ( l ) f v ( r ) P l ( 1 ) ( cos θ ) sin ϕ ,
A u ( l ) = ψ ( k r 0 ) F u ( r 0 ) ϵ s ( r 0 ) ψ ( k r 0 ) F u ( r 0 ) ϵ s ( r 0 ) ξ ( k r 0 ) F u ( r 0 ) ξ ( k r 0 ) F u ( r 0 ) ,
A v ( l ) = ψ ( k r 0 ) F v ( r 0 ) μ s ( r 0 ) ψ ( k r 0 ) F v ( r 0 ) μ s ( r 0 ) ξ ( k r 0 ) F v ( r 0 ) ξ ( k r 0 ) F v ( r 0 ) ,
C sca ( M ) = 2 π k 2 l = 1 ( 2 l + 1 ) ( | A u ( l ) | 2 + | A v ( l ) | 2 ) .
C sca ( M ) C sca ( R ) = 128 π 5 r 0 6 3 λ 4 ( | ϵ eff ϵ ϵ eff + 2 ϵ | 2 + | μ eff μ μ eff + 2 μ | 2 ) = 128 π 5 r 0 6 3 λ 4 | b | 2 ,
ε g ( ω ) = 1 c 0 · ω p 2 ( 0 ) / ω 2 ,

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