Abstract

Electrostatic properties of two-dimensional nanosystems can be completely described by their non-trivial geometry modes. In this paper we prove that these modes as well as the corresponding eigenvalues are invariant under any conformal transformation. This invariance suggests a new way to study electrostatic conformal transformations, while also providing an in-depth interpretation of the behavior exhibited by singular plasmonic nanoparticles.

© 2011 OSA

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  5. M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18, 12027–12032 (2010).
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  6. J. Renger, M. Kadic, G. Dupont, S. S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18, 15757–15768 (2010).
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  9. 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]
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    [CrossRef] [PubMed]
  31. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
    [CrossRef] [PubMed]
  32. M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2011 (1)

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

2010 (12)

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

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010).
[CrossRef] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010).
[CrossRef]

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (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] [PubMed]

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and its application,” IEEE Antennas Propag. Mag. 52, 24–45 (2010).
[CrossRef]

Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010).
[CrossRef] [PubMed]

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18, 12027–12032 (2010).
[CrossRef] [PubMed]

J. Renger, M. Kadic, G. Dupont, S. S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18, 15757–15768 (2010).
[CrossRef] [PubMed]

2009 (4)

P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009).
[CrossRef]

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

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

2008 (3)

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16, 18731–18738 (2008).
[CrossRef]

2007 (1)

B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media 17, 581–614 (2007).
[CrossRef]

2006 (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] [PubMed]

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006).
[CrossRef] [PubMed]

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

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

2005 (3)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[CrossRef] [PubMed]

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys. 7, 164 (2005).
[CrossRef]

2004 (1)

M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004).
[CrossRef]

2003 (3)

D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

J. B. Pendry and S. A. Ramakrishna, “Focusing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

2002 (1)

J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

2001 (1)

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef] [PubMed]

1992 (1)

D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol.  46, pp. 148–270.
[CrossRef]

1979 (1)

D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979).
[CrossRef]

Acimovic, S. S.

Aubry, A.

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010).
[CrossRef]

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010).
[CrossRef] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010).
[CrossRef]

Bergman, D. J.

M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef] [PubMed]

D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol.  46, pp. 148–270.
[CrossRef]

D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979).
[CrossRef]

Boardman, A.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Bohren, C. F.

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

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] [PubMed]

Catrysse, P. B.

P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009).
[CrossRef]

Chan, C. T.

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

Chen, H.

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

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Cummer, S. A.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[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]

Dupont, G.

Enoch, S.

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] [PubMed]

Faleev, S. V.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef] [PubMed]

Fan, S.

P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009).
[CrossRef]

Favaro, A.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Fernández-Domínguez, A. I.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Fredkin, D. R.

D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

Gralak, B.

B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media 17, 581–614 (2007).
[CrossRef]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[CrossRef] [PubMed]

Guenneau, S.

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18, 12027–12032 (2010).
[CrossRef] [PubMed]

J. Renger, M. Kadic, G. Dupont, S. S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18, 15757–15768 (2010).
[CrossRef] [PubMed]

B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media 17, 581–614 (2007).
[CrossRef]

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys. 7, 164 (2005).
[CrossRef]

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[CrossRef] [PubMed]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

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]

Kadic, M.

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

Kinsler, P.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Kobayashi, T.

M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004).
[CrossRef]

Kwon, D.-H.

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and its application,” IEEE Antennas Propag. Mag. 52, 24–45 (2010).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16, 18731–18738 (2008).
[CrossRef]

Lei, D. Y.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010).
[CrossRef]

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010).
[CrossRef]

Leonhardt, U.

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, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

Luo, Y.

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010).
[CrossRef] [PubMed]

Maier, S. A.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010).
[CrossRef]

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Mayergoyz, I. D.

D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

McCall, M. W.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

Milton, G. W.

G. W. Milton, The Theory of Composites (Cambridge University Press, 2002).
[CrossRef]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[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]

Narimanov, E. E.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[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] [PubMed]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010).
[CrossRef]

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010).
[CrossRef] [PubMed]

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010).
[CrossRef]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

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. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005).
[CrossRef] [PubMed]

J. B. Pendry and S. A. Ramakrishna, “Focusing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).
[CrossRef]

J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

Philbin, T. G.

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

Quidant, R.

Rahm, M.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys. 7, 164 (2005).
[CrossRef]

J. B. Pendry and S. A. Ramakrishna, “Focusing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).
[CrossRef]

J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

Renger, J.

Schurig, D.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

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]

Shen, Y. R.

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006).
[CrossRef] [PubMed]

Sheng, P.

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

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[CrossRef] [PubMed]

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]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Sonnefraud, Y.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[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]

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] [PubMed]

Stockman, M. I.

M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef] [PubMed]

Stroud, D.

D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol.  46, pp. 148–270.
[CrossRef]

Vutha, A. C.

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys. 7, 164 (2005).
[CrossRef]

Wang, F.

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006).
[CrossRef] [PubMed]

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] [PubMed]

Werner, D. H.

Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010).
[CrossRef] [PubMed]

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and its application,” IEEE Antennas Propag. Mag. 52, 24–45 (2010).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16, 18731–18738 (2008).
[CrossRef]

Wu, Q.

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Zeng, Y.

Appl. Phys. Lett. (2)

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and its application,” IEEE Antennas Propag. Mag. 52, 24–45 (2010).
[CrossRef]

J. Opt. (1)

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011).
[CrossRef]

J. Phys. C: Solid State Phys. (1)

D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979).
[CrossRef]

J. Phys.: Condens. Matter (2)

J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

J. B. Pendry and S. A. Ramakrishna, “Focusing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345–6364 (2003).
[CrossRef]

N. J. Phys. (3)

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys. 7, 164 (2005).
[CrossRef]

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008).
[CrossRef]

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010).
[CrossRef]

Nano Lett. (2)

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010).
[CrossRef] [PubMed]

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010).
[CrossRef] [PubMed]

Nat. Mater. (1)

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

Opt. Express (3)

Opt. Lett. (2)

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[CrossRef]

Phys. Rev. B (3)

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010).
[CrossRef]

M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004).
[CrossRef]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010).
[CrossRef]

Phys. Rev. Lett. (6)

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001).
[CrossRef] [PubMed]

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006).
[CrossRef] [PubMed]

D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. 91, 253902 (2003).
[CrossRef]

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008).
[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 (5)

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

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

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009).
[CrossRef] [PubMed]

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] [PubMed]

Solid State Physics (1)

D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol.  46, pp. 148–270.
[CrossRef]

Waves Random Complex Media (1)

B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media 17, 581–614 (2007).
[CrossRef]

Other (3)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

G. W. Milton, The Theory of Composites (Cambridge University Press, 2002).
[CrossRef]

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

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Figures (2)

Fig. 1
Fig. 1

A one-dimensional finite-thickness slab in xy coordinates (a) is transformed to an annulus (b) with a conformal transformation of w = ez, a crescent (c) and two kissing cylinders (d) with a conformal transformation of w = 1/z, in the new uv coordinates.

Fig. 2
Fig. 2

The dependence of the expansion coefficient on the mode index k, when it is assumed that pu = pv = 1. Notice that two eigenmodes with identical k while different symmetry have identical amplitudes of the expansion coefficients, mode symmetry therefore is not specified. Here d is an arbitrary real quantity.

Equations (64)

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

[ θ ( r ) φ n ( r ) ] = s n 2 φ n ( r ) ,
( ϕ | ψ ) = θ ϕ * ψ d v ,
i e i ( θ j φ n e j ) = g i j i ( θ j φ n ) = g i ( θ i φ n ) ,
s n i e i ( j φ n e j ) = s n g i j i ( j φ n ) = s n g i i φ n .
n s ( ω ) ( φ n | φ 0 ) φ n s ( ω ) s n n β n ( φ n | φ 0 ) φ n ,
P a = ω ɛ 0 Im ( ɛ 1 ) I e / 2 ,
P ex = ω ɛ 0 2 n | ( φ n | φ 0 ) | 2 Im [ ( ɛ 1 1 ) β n ] .
s k , ± = [ 1 ± e | k | ( d 2 d 1 ) ] / 2 ,
{ [ e 2 | k | d 1 e | k | ( d 1 + d 2 ) ] e | k | x , x d 1 [ e | k | x e | k | ( d 1 + d 2 ) ] e | k | x , d 1 < x < d 2 ( 1 e | k | ( d 2 d 1 ) ) e | k | x , x d 2
( φ k , ± | φ 0 ) = 2 k α k [ ρ 2 p F k ρ 1 p * F k * ] ,
F k = d 1 d 2 d x d y g e k z * , p 1 2 ( p u i p v ) , g d w d z .
u x = v y = g + g * 2 , u y = v x = g * g 2 i ,
φ 0 x = g p + g * p * , φ 0 y = i ( g p g * p * ) .
( φ k , ± | φ 0 ) = α k p u e d 1 + d 2 δ ( | k | 1 ) / 4 .
{ ( r 1 2 r 2 2 ) ( 1 ɛ 1 ) , r r 1 r 1 2 ( 1 ɛ 1 ) + r 2 2 ( 1 + ɛ 1 ) 2 r 1 2 r 2 2 / r 2 , r 1 < r < r 2 r 2 2 ( r 2 2 r 1 2 ) ( 1 + ɛ 1 ) / r 2 , r r 2
( φ k , ± | φ 0 ) = α k [ p u + i sgn ( k ) p v ] / 4 ,
( φ k , ± | φ 0 ) = α k 4 [ p u e | k | ( d 2 + d 1 ) ± 1 e | k | ( d 2 d 1 ) ± 1 + i sgn ( k ) p v e | k | ( d 2 + d 1 ) 1 e | k | ( d 2 d 1 ) 1 ] .
s n φ n ( r ) = G ^ φ n ( r ) d v θ ( r ) G ( r , r ) φ n ( r ) ,
2 G ( r , r ) = δ ( r r ) .
( ϕ | ψ ) = θ ( r ) ϕ * ψ d v ,
( ϕ | G ^ ψ ) = θ ( r ) ϕ * ( G ^ ψ ) dv = dv θ ( r ) ϕ * [ d v θ ( r ) G ( r , r ) ψ ] = dv d v θ ( r ) θ ( r ) ϕ * [ G ( r , r ) ψ ] ,
( G ^ ϕ | ψ ) = θ ( r ) ( G ^ ϕ * ) ψ dv = dv θ ( r ) ψ [ d v θ ( r ) G ( r , r ) ϕ * ] = d v d v θ ( r ) θ ( r ) ψ [ G ( r , r ) ϕ * ] = dv d v θ ( r ) θ ( r ) ψ [ G ( r , r ) ϕ * ] = dv d v θ ( r ) θ ( r ) ψ [ G ( r , r ) ϕ * ] ,
ϕ * [ G ( r , r ) ψ ] = ψ [ G ( r , r ) ϕ * ] ,
| φ m | 2 dv = 1 s m θ ( r ) | φ m | 2 dv ,
( φ m | φ n ) = δ mn .
φ 0 = n ( φ n | φ 0 ) φ n .
( θ φ 0 ) = s ( ω ) 2 φ i ( θ φ i ) ,
φ i = n s n φ n s ( ω ) s n ( φ n | φ 0 ) .
φ t = φ i + φ 0 = n s ( ω ) φ n s ( ω ) s n ( φ n | φ 0 ) n β n ( φ n | φ 0 ) φ n .
P ex = ω ɛ 0 2 Im ( ɛ 1 1 ) E 0 * E t dv .
P ex = ω ɛ 0 2 n | ( φ n | φ 0 ) | 2 Im [ β n ( ɛ 1 1 ) ] .
I e = v θ E t * E t dv = n | β n ( φ n | φ 0 ) | 2 .
p a = ω ɛ 0 2 Im ( ɛ 1 ) θ E t * E t dv = ω ɛ 0 2 Im ( ɛ 1 ) I e .
( a b 1 ) m 2 = b 2 , a m 2 = ( s 1 ) s ( b 1 m 2 b 2 ) c b 2 = b 1 n 2 , c = ( 1 s ) s ( b 1 n 2 b 2 ) ,
s k , + = 1 + e | k | ( d 2 d 1 ) 2 ,
α k φ k , + e iky = { [ e 2 | k | d 1 e | k | ( d 1 + d 2 ) ] e | k | x , x d 1 e | k | ( d 1 + d 2 ) e | k | x + e | k | x , d 1 < x < d 2 ( 1 e | k | ( d 2 d 1 ) ) e | k | x , x d 2 .
s k , = 1 e | k | ( d 2 d 1 ) 2 ,
α k φ k , e iky = { [ e 2 | k | d 1 + e | k | ( d 1 + d 2 ) ] e | k | x , x d 1 e | k | ( d 1 + d 2 ) e | k | x + e | k | x , d 1 < x < d 2 ( 1 + e | k | ( d 2 d 1 ) ) e | k | x , x d 2 .
F k = d 1 d 2 d x d y e z e k z * = π ( e 2 d 2 e 2 d 1 ) δ ( k 1 ) ,
F k * = d 1 d 2 d x d y e z * e k z = π ( e 2 d 2 e 2 d 1 ) δ ( k + 1 ) ,
e iky dy = 2 π δ ( k ) .
( φ k , ± | φ 0 ) = α k 4 p u e d 1 + d 2 δ ( | k | 1 ) .
φ i ( u , v ) = p u r 1 r 2 4 [ ( β 1 , 1 ) ( α 1 φ 1 , + α 1 φ 1 , ) ( β 1 , + 1 ) ( α 1 φ 1 , + + α 1 φ 1 , + ] ,
β 1 , ± 1 = ( 1 ɛ 1 ) ( r 2 ± r 1 ) ( 1 + ɛ 1 ) r 2 ( 1 ɛ 1 ) r 1 .
σ φ i φ 0 = { ( r 1 2 r 2 2 ) ( 1 ɛ 1 ) 2 , r r 1 r 1 2 ( 1 ɛ 1 ) 2 + r 2 2 ( 1 ɛ 1 2 ) 2 ( 1 ɛ 1 ) r 1 2 r 2 2 / r 2 , r 1 < r < r 2 r 2 2 ( r 2 2 r 1 2 ) ( 1 ɛ 1 2 ) / r 2 , r r 2
F k = d 1 d 2 d x dy 1 ( x + i y ) 2 e k ( x iy ) = d 1 d 2 e kx dx dy 1 ( x + i y ) 2 e iky .
F k < 0 = 2 π k d 1 d 2 e 2 kx dx = π ( e 2 k d 2 e 2 k d 1 ) .
F k * = d 1 d 2 dx dy 1 ( x iy ) 2 e k ( x + iy ) = π ( e 2 k d 2 e 2 k d 1 ) ,
dy 1 ( x iy ) 2 e iky = 2 π k e k x .
( φ k , ± | φ 0 ) = α k 4 [ p u + i sgn ( k ) p v ] .
F k = d 1 d 2 dx dy 1 ( x + i y ) 2 e k ( x iy ) = d 1 d 2 e kx dx dy 1 ( x + i y ) 2 e iky .
F k = π ( e 2 k d 2 1 ) , F k * = π ( e 2 k d 1 1 ) ,
F k = π ( e 2 k d 1 1 ) , F k * = π ( e 2 k d 2 1 ) ,
( φ k , ± | φ 0 ) = α k 4 [ p u e | k | ( d 2 + d 1 ) ± 1 e | k | ( d 2 d 1 ) ± 1 + i sgn ( k ) p v e | k | ( d 2 + d 1 ) 1 e | k | ( d 2 d 1 ) 1 ] .
E = w φ t = n β n ( φ n | φ 0 ) w φ n ,
E u = n β n ( φ n | φ 0 ) [ φ n x x u + φ n y y u ] , E v = n β n ( φ n | φ 0 ) [ φ n x x v + φ n y y v ] ,
x u = y v = z w + z w * 2 , x v = y u = i z w z w * 2 ,
φ k , ± ( x , y ) = 1 α k e iky ( a k , ± e | k | x + b k , ± e | k | x ) ,
E u = k , ± | k | e iky α k β k , ± ( φ k , ± | φ 0 ) [ ( a k , ± e | k | x z w b k , ± e | k | x z w * ) ϑ ( k ) + ( a k , ± e | k | x z w * b k , ± e | k | x z w ) ϑ ( k ) ] , E v = i k , ± k e iky α k β k , ± ( φ k , ± | φ 0 ) [ ( a k , ± e | k | x z w + b k , ± e | k | x z w * ) ϑ ( k ) + ( a k , ± e | k | x z w * + b k , ± e | k | x z w ) ϑ ( k ) ] ,
β k , ± = s ( ω ) s ( ω ) s k , ± = 2 ɛ 1 + 1 1 1 ± e γ | k | t ,
E u = p u + i p v 2 ( ɛ 1 + 1 ) 0 dkk e iky ( a k , + e | k | x z w b k , + e | k | x z w * 1 + e γ kt + a k , e | k | x z w b k , e | k | x z w * 1 e γ k t ) p u i p v 2 ( ɛ 1 + 1 ) 0 dkk e iky ( a k , + e | k | x z w * b k , + e | k | x z w 1 + e γ + kt + a k , e | k | x z w * b k , e | k | x z w 1 e γ + kt ) .
E u = i π γ ( p u + i p v ) ( ɛ 1 + 1 ) t 2 ( a e γ x / t e i γ y / t z w b e γ x / t e i γ y / t z w * )
E u = i π γ ( p u i p v ) ( ɛ 1 + 1 ) t 2 ( a e γ x / t e i γ y / t z w * b e γ x / t e i γ y / t z w )
E u = π γ ( i p u sgn ( y ) p v ) ( ɛ 1 + 1 ) t 2 ( 1 + e γ ) ( x + i | y | ) 2 e γ ( x + i | y | 2 d 1 ) / t .

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