Abstract

We investigate electromagnetic (EM) scattering and plasmonic cloaking in a system composed of a dielectric cylinder coated with a magneto-optical shell. In the long-wavelength limit we demonstrate that the application of an external magnetic field can not only switch on and off the cloaking mechanism but also mitigate losses, as the absorption cross section is shown to drop sharply precisely at the cloaking operation frequency band. We also show that the angular distribution of the scattered radiation can be effectively controlled by applying an external magnetic field, allowing for a swift change in the scattering pattern. By demonstrating that these results are feasible with realistic, existing magneto-optical materials, such as graphene epitaxially grown on SiC, we suggest that magnetic fields could be used as effective, versatile external agents to tune plasmonic cloaks and to dynamically control EM scattering in an unprecedented way. We hope that these results may find use in disruptive photonic technologies.

© 2014 Optical Society of America

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  1. N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
    [CrossRef]
  2. J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef]
  3. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef]
  4. 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]
  5. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005).
    [CrossRef]
  6. A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A 10, 093002 (2008).
    [CrossRef]
  7. 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]
  8. D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
    [CrossRef]
  9. P. Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP284 (2012).
  10. D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
    [CrossRef]
  11. W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
    [CrossRef]
  12. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
    [CrossRef]
  13. T. H. Anderson, T. G. Mackay, and A. Lakhtakia, “Toward a cylindrical cloak via inverse homogenization,” J. Opt. Soc. Am. A 29, 239–243 (2012).
    [CrossRef]
  14. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010).
    [CrossRef]
  15. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
    [CrossRef]
  16. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
    [CrossRef]
  17. A. Alú, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
    [CrossRef]
  18. J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
    [CrossRef]
  19. S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
    [CrossRef]
  20. P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
    [CrossRef]
  21. P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
    [CrossRef]
  22. N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Nonlinear control of invisibility cloaking,” Opt. Express 20, 14954–14959 (2012).
    [CrossRef]
  23. 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]
  24. F. G. Vasquez, G. W. Milton, and D. Onofrei, “Broadband exterior cloaking,” Opt. Express 17, 14800–14805 (2009).
    [CrossRef]
  25. P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
    [CrossRef]
  26. M. Farhat, C. Rockstuhl, and H. Bagci, “A 3D tunable and multi-frequency graphene plasmonic cloak,” Opt. Express 21, 12592–12603 (2013).
    [CrossRef]
  27. F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
    [CrossRef]
  28. W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
    [CrossRef]
  29. T. K. Xia, P. M. Hui, and D. Stroud, “Theory of Faraday rotation in granular magnetic materials,” J. Appl. Phys. 67, 2736–2741 (1990).
    [CrossRef]
  30. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  31. J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. AP-34, 1243–1249 (1986).
    [CrossRef]
  32. [30] Note that Eqs. (5)–(8) simplify considerably in the isotropic regions 1 and 3.
  33. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 9th ed. (Dover, 1965).
  34. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
    [CrossRef]
  35. D. B. Hough and L. R. White, “The calculation of hamaker constants from liftshitz theory with applications to wetting phenomena,” Adv. Colloid Interface Sci. 14, 3–41 (1980).
    [CrossRef]

2013

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
[CrossRef]

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[CrossRef]

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
[CrossRef]

M. Farhat, C. Rockstuhl, and H. Bagci, “A 3D tunable and multi-frequency graphene plasmonic cloak,” Opt. Express 21, 12592–12603 (2013).
[CrossRef]

2012

T. H. Anderson, T. G. Mackay, and A. Lakhtakia, “Toward a cylindrical cloak via inverse homogenization,” J. Opt. Soc. Am. A 29, 239–243 (2012).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Nonlinear control of invisibility cloaking,” Opt. Express 20, 14954–14959 (2012).
[CrossRef]

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[CrossRef]

D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
[CrossRef]

P. Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP284 (2012).

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

2011

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[CrossRef]

2010

P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
[CrossRef]

P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
[CrossRef]

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

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

2009

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

A. Alú, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[CrossRef]

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (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]

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]

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

2008

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

2006

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

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (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]

2005

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

1994

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

1990

T. K. Xia, P. M. Hui, and D. Stroud, “Theory of Faraday rotation in granular magnetic materials,” J. Appl. Phys. 67, 2736–2741 (1990).
[CrossRef]

1986

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. AP-34, 1243–1249 (1986).
[CrossRef]

1980

D. B. Hough and L. R. White, “The calculation of hamaker constants from liftshitz theory with applications to wetting phenomena,” Adv. Colloid Interface Sci. 14, 3–41 (1980).
[CrossRef]

Alitalo, P.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[CrossRef]

Alú, A.

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

A. Alú, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[CrossRef]

Alù, A.

F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[CrossRef]

P. Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP284 (2012).

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[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]

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

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

Anderson, T. H.

Argyropoulos, C.

F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[CrossRef]

Bagci, H.

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Belov, P. A.

D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
[CrossRef]

Bohren, C. F.

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

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

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, J.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Chen, P. Y.

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

P. Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP284 (2012).

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[CrossRef]

Crassee, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

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]

Damaskos, N. J.

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. AP-34, 1243–1249 (1986).
[CrossRef]

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]

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]

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

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

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

Farhat, M.

Farina, C.

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
[CrossRef]

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
[CrossRef]

Filonov, D. S.

D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
[CrossRef]

Gaponenko, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Hough, D. B.

D. B. Hough and L. R. White, “The calculation of hamaker constants from liftshitz theory with applications to wetting phenomena,” Adv. Colloid Interface Sci. 14, 3–41 (1980).
[CrossRef]

Huffman, D. R.

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

Hui, P. M.

T. K. Xia, P. M. Hui, and D. Stroud, “Theory of Faraday rotation in granular magnetic materials,” J. Appl. Phys. 67, 2736–2741 (1990).
[CrossRef]

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]

Kerkhoff, A.

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

Kivshar, Y. S.

D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
[CrossRef]

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Nonlinear control of invisibility cloaking,” Opt. Express 20, 14954–14959 (2012).
[CrossRef]

Kort-Kamp, W. J. M.

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
[CrossRef]

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
[CrossRef]

Kuzmenko, A. B.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Lakhtakia, A.

Leonhardt, U.

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

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Li, P.

P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
[CrossRef]

P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
[CrossRef]

Liu, Y.

P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
[CrossRef]

P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
[CrossRef]

Luukkonen, O.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[CrossRef]

Mackay, T. G.

McPhedran, R. C.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Melin, K.

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

Meng, Y.

P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
[CrossRef]

P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
[CrossRef]

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]

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

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

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]

Monticone, F.

F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[CrossRef]

Monzon, J. C.

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. AP-34, 1243–1249 (1986).
[CrossRef]

Moreno, G.

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

Onofrei, D.

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

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]

Orlita, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Ostler, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (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]

Pinheiro, F. A.

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
[CrossRef]

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
[CrossRef]

Potemski, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Rainwater, D.

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

Rockstuhl, C.

Rosa, F. S. S.

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
[CrossRef]

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
[CrossRef]

Schurig, D.

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]

Seyller, T.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Shadrivov, I. V.

Sheng, P.

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

Shurig, D.

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

Simovski, C.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[CrossRef]

Slobozhanyuk, A. P.

D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (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]

Soric, J.

P. Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP284 (2012).

Soric, J. C.

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

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]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

Stroud, D.

T. K. Xia, P. M. Hui, and D. Stroud, “Theory of Faraday rotation in granular magnetic materials,” J. Appl. Phys. 67, 2736–2741 (1990).
[CrossRef]

Tretyakov, S.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (2009).
[CrossRef]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Vasquez, F. G.

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]

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

Walter, A. L.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

White, L. R.

D. B. Hough and L. R. White, “The calculation of hamaker constants from liftshitz theory with applications to wetting phenomena,” Adv. Colloid Interface Sci. 14, 3–41 (1980).
[CrossRef]

Xia, T. K.

T. K. Xia, P. M. Hui, and D. Stroud, “Theory of Faraday rotation in granular magnetic materials,” J. Appl. Phys. 67, 2736–2741 (1990).
[CrossRef]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

Zharov, A. A.

Zharova, N. A.

Zheludev, N. I.

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[CrossRef]

Zhu, M.

P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
[CrossRef]

P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
[CrossRef]

ACS Nano

P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[CrossRef]

Adv. Colloid Interface Sci.

D. B. Hough and L. R. White, “The calculation of hamaker constants from liftshitz theory with applications to wetting phenomena,” Adv. Colloid Interface Sci. 14, 3–41 (1980).
[CrossRef]

Adv. Mater.

P. Y. Chen, J. Soric, and A. Alù, “Invisibility and cloaking based on scattering cancellation,” Adv. Mater. 24, OP281–OP284 (2012).

IEEE Trans. Antennas Propag.

J. C. Monzon and N. J. Damaskos, “Two-dimensional scattering by a homogeneous anisotropic rod,” IEEE Trans. Antennas Propag. AP-34, 1243–1249 (1986).
[CrossRef]

J. Appl. Phys.

T. K. Xia, P. M. Hui, and D. Stroud, “Theory of Faraday rotation in granular magnetic materials,” J. Appl. Phys. 67, 2736–2741 (1990).
[CrossRef]

J. Opt. A

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

J. Opt. Soc. Am. A

J. Phys. D

P. Li, Y. Liu, Y. Meng, and M. Zhu, “A frequency-tunable cloak with semiconducting constituents,” J. Phys. D 43, 175404 (2010).
[CrossRef]

P. Li, Y. Liu, Y. Meng, and M. Zhu, “Frequency-tunable superconducting cloaking,” J. Phys. D 43, 485401 (2010).
[CrossRef]

Nano Lett.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett. 12, 2470–2474 (2012).
[CrossRef]

Nat. Mater.

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

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009).
[CrossRef]

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[CrossRef]

New J. Phys.

D. Rainwater, A. Kerkhoff, K. Melin, J. C. Soric, G. Moreno, and A. Alú, “Experimental verification of three-dimensional plasmonic cloaking in free-space,” New J. Phys. 14, 013054 (2012).
[CrossRef]

J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alú, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013).
[CrossRef]

Opt. Express

Phys. Rev. A

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Spontaneous emission in the presence of a spherical plasmonic metamaterial,” Phys. Rev. A 87, 023837 (2013).
[CrossRef]

Phys. Rev. B

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B 49, 8479–8482 (1994).
[CrossRef]

A. Alú, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009).
[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

S. Tretyakov, P. Alitalo, O. Luukkonen, and C. Simovski, “Broadband electromagnetic cloaking of long cylindrical objects,” Phys. Rev. Lett. 103, 103905 (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]

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]

F. Monticone, C. Argyropoulos, and A. Alù, “Multilayered plasmonic covers for comblike scattering response and optical tagging,” Phys. Rev. Lett. 110, 113901 (2013).
[CrossRef]

W. J. M. Kort-Kamp, F. S. S. Rosa, F. A. Pinheiro, and C. Farina, “Tuning plasmonic cloaks with an external magnetic field,” Phys. Rev. Lett. 111, 215504 (2013).
[CrossRef]

Phys. Status Solidi

D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Y. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi 6, 46–48 (2012).
[CrossRef]

Science

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[CrossRef]

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

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (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]

Other

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

[30] Note that Eqs. (5)–(8) simplify considerably in the isotropic regions 1 and 3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 9th ed. (Dover, 1965).

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Figures (8)

Fig. 1.
Fig. 1.

Scattering system: an isotropic cylinder with dielectric constant εc and radius a coated by a magneto-optical shell with permittivity tensor εs and outer radius b>a under the influence of a static magnetic field B and a TM-polarized monochromatic plane wave.

Fig. 2.
Fig. 2.

Scattering efficiency Qsc (normalized by its value in the absence of B, Qsc(0)) as a function of the frequency of the impinging wave for B=5T (solid line), 10 T (dashed line), and 20 T (dotted–dashed line) for (a) same losses as in graphene epitaxial growth on SiC (Γ=18.3×1012rad/s) and (b) losses 10 times smaller (Γ=1.83×1012rad/s). Contour plots of Qsc/Qsc(0) as a function of both frequency f and magnetic field B are shown in (c) and (d) for the same set of parameters chosen in (a) and (b), respectively.

Fig. 3.
Fig. 3.

Spatial distribution of the scattered field Hz in the xy plane for (a) B=0T and (b) B=20T. The frequency of the incident wave is f=4THz. In panel (a) the vertical arrow indicates the total electric dipole pt(0) of the whole system for a vanishing magnetic field, whereas in panel (b) the arrows represent the electric dipole induced on the cylinder, pc, on the shell, ps, and on the total electric dipole, pt(B), for B0.

Fig. 4.
Fig. 4.

Differential scattering cross section for B=0T (solid line) and B=20T (dashed line) for an incident wave of frequency 14 THz and two different values of loss parameters: (a) Γ=18.3×1012rad/s and (b) Γ=1.83×1012rad/s.

Fig. 5.
Fig. 5.

(a) Rotation angle φr as a function of frequency for B=5T (solid line), B=10T (dashed line), and B=20T (dotted–dashed line). The inset shows the definition of φr. (b) Angle φr as a function of the external magnetic field strength B for three different frequencies of the impinging wave, f=1THz (solid line), f=5THz (dashed line), and f=10THz (dotted–dashed line).

Fig. 6.
Fig. 6.

Rotation angle given by Eq. (30) as a function of both frequency f and external magnetic field B strength.

Fig. 7.
Fig. 7.

Differential scattering cross section for a incident wave of frequency 5 THz and (a) Γ=18.3×1012rad/s and (b) Γ=1.83×1012. As the direction of the magnetic field is reversed while keeping its strength constant, the scattered pattern rotates by (a) 60° and (b) 90°.

Fig. 8.
Fig. 8.

Absorption efficiency Qabs (normalized by its value in the absence of B, Qabs(0)) as a function of the frequency of the impinging wave for B=5T (solid line), 10 T (dashed line), and 20 T (dotted–dashed line) for (a) same losses as in graphene epitaxial growth on SiC (Γ=18.3×1012rad/s) and (b) 10 times smaller losses (Γ=1.83×1012rad/s). Panels (c) and (d) show the behavior of the imaginary part of κs(εs2γs2)/εs as a function of frequency for B=0T (dotted line), B=5T (solid line), 10 T (dashed line), and 20 T (dotted–dashed line) for Γ=18.3×1012rad/s and Γ=1.83×1012rad/s, respectively.

Equations (32)

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

Hi=Hi0eiω(tx/c)z^,Ei=(μ0c)x^×Hi=Ei0eiω(tx/c)y^,
εs=[εxxεxy0εyxεyy000εzz]=[εsiγs0iγsεs000εzz],
×E=iωμ0H,
×H=iωε·E,
(εi)xx2Hz(i)x2+(εi)yy2Hz(i)y2+[(εi)xy+(εi)yx]2Hz(i)xy+ω2αiμ0Hz(i)=0
Eρ(i)(ρ,φ)=iωαi{(εi)xyHz(i)ρ+(εi)xxρHz(i)φ}
Eφ(i)(ρ,φ)=iωαi{(εi)xxHz(i)ρ(εi)xyρHz(i)φ},
αi(εi)xx(εi)yy(εi)xy(εi)yx.
(2+ki2)Hz(p)=0
ki=ωμ0αi(εi)xx.
Hz(1)(ρ,φ)=m=+AmJm(kcρ)eimφ(ρa),
Hz(2)(ρ,φ)=m=+im{BmJm(ksρ)+CmNm(ksρ)}eimφ(a<ρ<b),
Hz(3)(ρ,φ)=m=+im{Jm(k0ρ)+DmHm(1)(k0ρ)}eimφ(bρ),
Hz(1)(a,φ)=Hz(2)(a,φ),Hz(2)(b,φ)=Hz(3)(b,φ),Eφ(1)(a,φ)=Eφ(2)(a,φ),Eφ(2)(b,φ)=Eφ(3)(b,φ).
dQscdφ=2k0π|imDmeiπ/4eim(φπ/2)|2,
Qsc=2k0b+|Dm|2,
Qext=2k0b+Re(Dm),
Qabs=QextQsc.
Dm=UmVm,
Um=|Jm(kca)Jm(ksa)Nm(ksa)0kcεcJ(kca)Jm(ksa)Nm(ksa)00Jm(ksb)Nm(ksb)Jm(k0b)0Jm(ksb)Nm(ksa)k0ε0Jm(k0b)|
Vm=|Jm(kca)Jm(ksa)Nm(ksa)0kcεcJ(kca)Jm(ksa)Nm(ksa)00Jm(ksb)Nm(ksb)Hm(1)(k0b)0Jm(ksb)Nm(ksa)k0ε0Hm(1)(k0b)|
Jm(x)=1εs2γs2[εsksJm(x)+mγsksxJm(x)]
Nm(x)=1εs2γs2[εsksNm(x)+mγsksxNm(x)],
εs(ω,B)=ε0+iσxx2D(ω,B)ω(ba)
γs(ω,B)=σxy2D(ω,B)ω(ba),
σxx2D(ω,B)=3.5σ0+σ+2D(ω,B)+σ2D(ω,B)2
σxy2D(ω,B)=σ+2D(ω,B)σ2D(ω,B)2i
σ±2D(ω,B)=2dπiωωcω02/ω+iΓ.
d/σ0=0.52eV,ω0=9.9×1012rad/s,Γ=18.3×1012rad/s,ωc=3.2×1012B(T)rad/s.
εc(ω)ε0=1+ωp12ωr12ω2iΓ1ω+ωp22ωr22ω2iΓ2ω,
ωp1=1.11×1014rad/s,ωr1=5.54×1014rad/s,ωp2=1.96×1016rad/s,ωr2=1.35×1016rad/s,Γ1=Γ2=0.1×1012rad/s.
φr=12tan1[Im(D1D1*)Re(D1D1*)].

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