Abstract

The quasistatic field around a circular hole in a two-dimensional hyperbolic medium is studied. As the loss parameter goes to zero, it is found that the electric field diverges along four lines each tangent to the hole. In this limit, the power dissipated by the field in the vicinity of these lines, per unit length of the line, goes to zero but extends further and further out so that the net power dissipated remains finite. Additionally the interaction between polarizable dipoles in a hyperbolic medium is studied. It is shown that a dipole with small polarizability can dramatically influence the dipole moment of a distant polarizable dipole, if it is appropriately placed. We call this the searchlight effect, as the enhancement depends on the orientation of the line joining the polarizable dipoles and can be varied by changing the frequency. For some particular polarizabilities the enhancement can actually increase the further the polarizable dipoles are apart.

© 2013 OSA

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of εand μ,”Uspekhi Fizicheskikh Nauk92, 517–526 (1967). English translation in Sov. Phys. Uspekhi10:509–514 (1968).
    [CrossRef]
  2. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B49, 8479–8482 (1994).
    [CrossRef]
  3. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
    [CrossRef]
  4. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” P. R. Soc. A462, 3027–3059 (2006).
    [CrossRef]
  5. N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
    [CrossRef] [PubMed]
  6. O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys.102, 124502 (2007).
    [CrossRef]
  7. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
    [CrossRef]
  8. N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
    [CrossRef]
  9. G. Bouchitté and B. Schweizer, “Cloaking of small objects by anomalous localized resonance,” Quantum J. Mech. Appl. Math.63, 437–463 (2010).
    [CrossRef]
  10. N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex21, 248–277 (2011).
    [CrossRef]
  11. H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
    [CrossRef]
  12. R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].
  13. H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].
  14. M. Xiao, X. Huang, J. W. Dong, and C. T. Chan, “On the time evolution of the cloaking effect of a metamaterial slab,” Opt. Lett.37, 4594–4596 (2012).
    [CrossRef] [PubMed]
  15. H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].
  16. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000).
    [CrossRef] [PubMed]
  17. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E64, 056625 (2001).
    [CrossRef]
  18. F. D. M. Haldane, “Electromagnetic surface modes at interfaces with negative refractive index make a ’not-quite-perfect’ lens,” (2002). ArXiv:cond-mat/0206420 v3 (2002).
  19. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett.88, 207403 (2002).
    [CrossRef]
  20. A. L. Pokrovsky and A. L. Efros, “Diffraction in left-handed materials and theory of veselago lens,” (2002). ArXiv:cond-mat/0202078 v2 (2002).
  21. S. A. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett.82, 1503–1505 (2003).
    [CrossRef]
  22. A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B338, 333–337 (2003). See also arXiv:cond-mat/0202078 v2 (2002).
    [CrossRef]
  23. X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B68, 113103 (2003).
    [CrossRef]
  24. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B67, 035109 (2003).
    [CrossRef]
  25. R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett.84, 1290–1292 (2004).
    [CrossRef]
  26. S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett.30, 1204–1206 (2005).
    [CrossRef] [PubMed]
  27. V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett.30, 75–77 (2005).
    [CrossRef] [PubMed]
  28. G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B394, 171–175 (2007).
    [CrossRef]
  29. J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
    [CrossRef]
  30. J. B. Pendry and S. A. Ramakrishna, “Refining the perfect lens,” Physica B338, 329–332 (2003).
    [CrossRef]
  31. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
    [CrossRef] [PubMed]
  32. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8247–8256 (2006).
    [CrossRef] [PubMed]
  33. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006).
    [CrossRef]
  34. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
    [CrossRef] [PubMed]
  35. J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).
  36. S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (2012).
    [CrossRef]
  37. H. C. Yang and Y. T. Chou, “Antiplane strain problems of an elliptic inclusion in an anisotropic medium,” J. Appl. Mech.44, 437–441 (1977).
    [CrossRef]
  38. A. H. Sihvola, “On the dielectric problem of isotropic sphere in anisotropic medium,” Electromagnetics17, 69–74 (1997).
    [CrossRef]
  39. G. W. Milton, The Theory of Composites (Cambridge University Press, 2002).
    [CrossRef]
  40. A. Sihvola, “Metamaterials and depolarization factors,” Prog. Electromagn. Res.51, 65–82 (2005).
    [CrossRef]

2013

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

2012

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (2012).
[CrossRef]

M. Xiao, X. Huang, J. W. Dong, and C. T. Chan, “On the time evolution of the cloaking effect of a metamaterial slab,” Opt. Lett.37, 4594–4596 (2012).
[CrossRef] [PubMed]

2011

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex21, 248–277 (2011).
[CrossRef]

2010

G. Bouchitté and B. Schweizer, “Cloaking of small objects by anomalous localized resonance,” Quantum J. Mech. Appl. Math.63, 437–463 (2010).
[CrossRef]

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

2008

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
[CrossRef]

2007

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys.102, 124502 (2007).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B394, 171–175 (2007).
[CrossRef]

N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
[CrossRef] [PubMed]

2006

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8247–8256 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006).
[CrossRef]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” P. R. Soc. A462, 3027–3059 (2006).
[CrossRef]

2005

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
[CrossRef]

A. Sihvola, “Metamaterials and depolarization factors,” Prog. Electromagn. Res.51, 65–82 (2005).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett.30, 75–77 (2005).
[CrossRef] [PubMed]

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

2004

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett.84, 1290–1292 (2004).
[CrossRef]

2003

J. B. Pendry and S. A. Ramakrishna, “Refining the perfect lens,” Physica B338, 329–332 (2003).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
[CrossRef] [PubMed]

S. A. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett.82, 1503–1505 (2003).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B338, 333–337 (2003). See also arXiv:cond-mat/0202078 v2 (2002).
[CrossRef]

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B68, 113103 (2003).
[CrossRef]

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B67, 035109 (2003).
[CrossRef]

2002

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett.88, 207403 (2002).
[CrossRef]

2001

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E64, 056625 (2001).
[CrossRef]

2000

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

1997

A. H. Sihvola, “On the dielectric problem of isotropic sphere in anisotropic medium,” Electromagnetics17, 69–74 (1997).
[CrossRef]

1994

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B49, 8479–8482 (1994).
[CrossRef]

1977

H. C. Yang and Y. T. Chou, “Antiplane strain problems of an elliptic inclusion in an anisotropic medium,” J. Appl. Mech.44, 437–441 (1977).
[CrossRef]

1967

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of εand μ,”Uspekhi Fizicheskikh Nauk92, 517–526 (1967). English translation in Sov. Phys. Uspekhi10:509–514 (1968).
[CrossRef]

Alekseyev, L. V.

Ammari, H.

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

Bartal, G.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Botten, L. C.

N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex21, 248–277 (2011).
[CrossRef]

N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
[CrossRef] [PubMed]

Bouchitté, G.

G. Bouchitté and B. Schweizer, “Cloaking of small objects by anomalous localized resonance,” Quantum J. Mech. Appl. Math.63, 437–463 (2010).
[CrossRef]

Bruno, O. P.

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys.102, 124502 (2007).
[CrossRef]

Chan, C. T.

M. Xiao, X. Huang, J. W. Dong, and C. T. Chan, “On the time evolution of the cloaking effect of a metamaterial slab,” Opt. Lett.37, 4594–4596 (2012).
[CrossRef] [PubMed]

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

Cherednichenko, K.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

Choi, H.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Chou, Y. T.

H. C. Yang and Y. T. Chou, “Antiplane strain problems of an elliptic inclusion in an anisotropic medium,” J. Appl. Mech.44, 437–441 (1977).
[CrossRef]

Ciraolo, G.

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

Cummer, S. A.

S. A. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett.82, 1503–1505 (2003).
[CrossRef]

Dong, J. W.

M. Xiao, X. Huang, J. W. Dong, and C. T. Chan, “On the time evolution of the cloaking effect of a metamaterial slab,” Opt. Lett.37, 4594–4596 (2012).
[CrossRef] [PubMed]

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B338, 333–337 (2003). See also arXiv:cond-mat/0202078 v2 (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Diffraction in left-handed materials and theory of veselago lens,” (2002). ArXiv:cond-mat/0202078 v2 (2002).

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006).
[CrossRef]

Enoch, S.

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
[CrossRef]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett.88, 207403 (2002).
[CrossRef]

Gralak, B.

Guenneau, S.

Haldane, F. D. M.

F. D. M. Haldane, “Electromagnetic surface modes at interfaces with negative refractive index make a ’not-quite-perfect’ lens,” (2002). ArXiv:cond-mat/0206420 v3 (2002).

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E64, 056625 (2001).
[CrossRef]

Huang, X.

Jacob, Z.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8247–8256 (2006).
[CrossRef] [PubMed]

Kang, H.

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

Kivshar, Y. S.

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (2012).
[CrossRef]

Kohn, R. V.

R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].

Kruk, S. S.

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (2012).
[CrossRef]

Lai, Y.

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

Lee, H.

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

Lintner, S.

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys.102, 124502 (2007).
[CrossRef]

Liu, Z.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Lu, J.

R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].

McPhedran, R. C.

N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex21, 248–277 (2011).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B394, 171–175 (2007).
[CrossRef]

N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
[CrossRef] [PubMed]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B49, 8479–8482 (1994).
[CrossRef]

Merlin, R.

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett.84, 1290–1292 (2004).
[CrossRef]

Milton, G. W.

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B394, 171–175 (2007).
[CrossRef]

N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
[CrossRef] [PubMed]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” P. R. Soc. A462, 3027–3059 (2006).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B49, 8479–8482 (1994).
[CrossRef]

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

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

Minovich, A.

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (2012).
[CrossRef]

Narimanov, E.

Narimanov, E. E.

Neshev, D. N.

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (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. B49, 8479–8482 (1994).
[CrossRef]

Nicorovici, N.-A. P.

N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex21, 248–277 (2011).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B394, 171–175 (2007).
[CrossRef]

N.-A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express15, 6314–6323 (2007).
[CrossRef] [PubMed]

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” P. R. Soc. A462, 3027–3059 (2006).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
[CrossRef]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett.88, 207403 (2002).
[CrossRef]

Ong, C. K.

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B68, 113103 (2003).
[CrossRef]

Pendry, J. B.

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, “Refining the perfect lens,” Physica B338, 329–332 (2003).
[CrossRef]

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

Podolskiy, V. A.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett.30, 75–77 (2005).
[CrossRef] [PubMed]

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B338, 333–337 (2003). See also arXiv:cond-mat/0202078 v2 (2002).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Diffraction in left-handed materials and theory of veselago lens,” (2002). ArXiv:cond-mat/0202078 v2 (2002).

Powell, D. A.

S. S. Kruk, D. A. Powell, A. Minovich, D. N. Neshev, and Y. S. Kivshar, “Spatial dispersion of multilayer fishnet metamaterials,” Opt. Express20, 15101–15105 (2012).
[CrossRef]

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, “Refining the perfect lens,” Physica B338, 329–332 (2003).
[CrossRef]

Rao, X. S.

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B68, 113103 (2003).
[CrossRef]

Rho, J.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006).
[CrossRef]

Schurig, D.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
[CrossRef] [PubMed]

Schweizer, B.

G. Bouchitté and B. Schweizer, “Cloaking of small objects by anomalous localized resonance,” Quantum J. Mech. Appl. Math.63, 437–463 (2010).
[CrossRef]

R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].

Shvets, G.

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B67, 035109 (2003).
[CrossRef]

Sihvola, A.

A. Sihvola, “Metamaterials and depolarization factors,” Prog. Electromagn. Res.51, 65–82 (2005).
[CrossRef]

Sihvola, A. H.

A. H. Sihvola, “On the dielectric problem of isotropic sphere in anisotropic medium,” Electromagnetics17, 69–74 (1997).
[CrossRef]

Smith, D. R.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Tayeb, G.

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of εand μ,”Uspekhi Fizicheskikh Nauk92, 517–526 (1967). English translation in Sov. Phys. Uspekhi10:509–514 (1968).
[CrossRef]

Wang, H. Z.

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

Weinstein, M. I.

R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].

Xiao, M.

Xiong, Y.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Yang, H. C.

H. C. Yang and Y. T. Chou, “Antiplane strain problems of an elliptic inclusion in an anisotropic medium,” J. Appl. Mech.44, 437–441 (1977).
[CrossRef]

Ye, Z.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Yin, X.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Zhang, X.

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Zheng, H. H.

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E64, 056625 (2001).
[CrossRef]

Appl. Phys. Lett.

S. A. Cummer, “Simulated causal subwavelength focusing by a negative refractive index slab,” Appl. Phys. Lett.82, 1503–1505 (2003).
[CrossRef]

R. Merlin, “Analytical solution of the almost-perfect-lens problem,” Appl. Phys. Lett.84, 1290–1292 (2004).
[CrossRef]

Arch. Ration. Mech. Anal.

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance,” Arch. Ration. Mech. Anal.208, 667–692 (2013).
[CrossRef]

Electromagnetics

A. H. Sihvola, “On the dielectric problem of isotropic sphere in anisotropic medium,” Electromagnetics17, 69–74 (1997).
[CrossRef]

J. Appl. Mech.

H. C. Yang and Y. T. Chou, “Antiplane strain problems of an elliptic inclusion in an anisotropic medium,” J. Appl. Mech.44, 437–441 (1977).
[CrossRef]

J. Appl. Phys.

O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” J. Appl. Phys.102, 124502 (2007).
[CrossRef]

Nature

J. Rho, Z. Ye, Y. Xiong, X. Yin, Z. Liu, H. Choi, G. Bartal, and X. Zhang, “Spherical hyperlens for two-dimensional sub-diffractional imaging at visible frequencies,” Nature1, 143 (2010).

New J. Phys.

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko, and Z. Jacob, “Solutions in folded geometries, and associated cloaking due to anomalous resonance,” New J. Phys.10, 115021 (2008).
[CrossRef]

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch, and G. Tayeb, “Finite wavelength cloaking by plasmonic resonance,” New J. Phys.10, 115020 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

P. R. Soc. A

G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” P. R. Soc. A462, 3027–3059 (2006).
[CrossRef]

Phys. Rev. B

J. W. Dong, H. H. Zheng, Y. Lai, H. Z. Wang, and C. T. Chan, “Metamaterial slab as a lens, a cloak, or an intermediate,” Phys. Rev. B83, 115124 (2011).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B49, 8479–8482 (1994).
[CrossRef]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006).
[CrossRef]

X. S. Rao and C. K. Ong, “Amplification of evanescent waves in a lossy left-handed material slab,” Phys. Rev. B68, 113103 (2003).
[CrossRef]

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B67, 035109 (2003).
[CrossRef]

Phys. Rev. E

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett.

N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett.88, 207403 (2002).
[CrossRef]

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

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
[CrossRef] [PubMed]

Physica B

J. B. Pendry and S. A. Ramakrishna, “Refining the perfect lens,” Physica B338, 329–332 (2003).
[CrossRef]

G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect lenses,” Physica B394, 171–175 (2007).
[CrossRef]

A. L. Pokrovsky and A. L. Efros, “Diffraction theory and focusing of light by a slab of left-handed material,” Physica B338, 333–337 (2003). See also arXiv:cond-mat/0202078 v2 (2002).
[CrossRef]

Proc. Roy. Soc. A

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance,” Proc. Roy. Soc. A461, 3999–4034 (2005).
[CrossRef]

Prog. Electromagn. Res.

A. Sihvola, “Metamaterials and depolarization factors,” Prog. Electromagn. Res.51, 65–82 (2005).
[CrossRef]

Quantum J. Mech. Appl. Math.

G. Bouchitté and B. Schweizer, “Cloaking of small objects by anomalous localized resonance,” Quantum J. Mech. Appl. Math.63, 437–463 (2010).
[CrossRef]

Science

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science315, 1686 (2007).
[CrossRef] [PubMed]

Uspekhi Fizicheskikh Nauk

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of εand μ,”Uspekhi Fizicheskikh Nauk92, 517–526 (1967). English translation in Sov. Phys. Uspekhi10:509–514 (1968).
[CrossRef]

Wave. Random Complex

N.-A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states and cloaking in finite clusters of coated cylinders,” Wave. Random Complex21, 248–277 (2011).
[CrossRef]

Other

R. V. Kohn, J. Lu, B. Schweizer, and M. I. Weinstein, “A variational perspective on cloaking by anomalous localized resonance,” (2012). ArXiv:1210.4823 [math.AP].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Spectral theory of a neumann-poincaré-type operator and analysis of cloaking due to anomalous localized resonance ii,” (2013). ArXiv:1212.5066 [math.AP].

H. Ammari, G. Ciraolo, H. Kang, H. Lee, and G. W. Milton, “Anomalous localized resonance using a folded geometry in three dimensions,” (2013). ArXiv:1301.5712 [math-ph].

A. L. Pokrovsky and A. L. Efros, “Diffraction in left-handed materials and theory of veselago lens,” (2002). ArXiv:cond-mat/0202078 v2 (2002).

F. D. M. Haldane, “Electromagnetic surface modes at interfaces with negative refractive index make a ’not-quite-perfect’ lens,” (2002). ArXiv:cond-mat/0206420 v3 (2002).

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

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Figures (4)

Fig. 1
Fig. 1

Plot of the absolute value of the potential V(x,y) around the disk given by (2.26) with an applied field directed along the x-axis and with parameters γ′ = 1, εx = 3 and c = 0.01 − i. The potential has kinks at the four characteristic lines given by (3.41) which are also drawn. Near these lines the electric field is huge.

Fig. 2
Fig. 2

Plot of the absolute value of the dipole amplitude β1x near the line where e ≈ 0 with polarizabilities α1x = 2, and α1y = 1, α2x = 0.2, and α2y = 0.1 and parameter c = 0.01 − i. We use rotated coordinates ξ = ( x 0 + y 0 ) / 2 and τ = ( x 0 y 0 ) / 2. Note the long-range interaction.

Fig. 3
Fig. 3

Same as Fig. 2 but with polarizabilities α1x = 2, and α1y = 1, α2x = 0.1, and α2y = 0.2 and parameter c = 0.01 − i.

Fig. 4
Fig. 4

Plot of the absolute value of the dipole amplitude β1x near the line where e ≈ 0 with polarizabilities α1x = 2, and α1y = 1, α2x = α2y = 0.1 and parameter c = 0.01 − i. We use rotated coordinates ξ = ( x 0 + y 0 ) / 2 and τ = ( x 0 y 0 ) / 2. Note that the interaction along the line τ = 0 first becomes stronger as ξ increases, then weakens.

Equations (83)

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

V ˜ e ( z ) = 1 z z 0 ( 1 ε c 1 + ε c ) a 2 z 0 2 ( z a 2 / z 0 ) ,
ε V = 0 , ε = [ ε x 0 0 ε y ] .
2 V y 2 = μ 2 2 V x 2 ,
z + r 2 z = 2 w ,
u = ± ( r 0 + r 2 / r 0 ) / 2 , v = ± ( r 0 r 2 / r 0 ) / 2 .
z = w + w 2 r 2 , 1 z = w w 2 r 2 r 2 .
w 2 r 2 = ( w + r ) ( w r ) / ( w + r ) ,
γ ( z + r 2 / z ) / 2 = γ ( r 0 e i θ + r 2 e i θ / r 0 ) / 2 ,
γ ¯ 2 ( r 0 2 r 0 e i θ + r 2 r 0 e i θ r 0 2 ) = γ ¯ t ,
t = 1 2 ( r 0 2 z + r 2 z r 0 2 ) = w 2 ( r 2 r 0 2 + r 0 2 r 2 ) + w 2 r 2 2 ( r 2 r 0 2 r 0 2 r 2 ) .
x = u , y = v / c ,
x = ± ( r 0 + r 2 / r 0 ) / 2 , y = ± ( r 0 r 2 / r 0 ) / ( 2 c ) .
r 0 = 1 + c , r = 1 c 2 ,
( r 0 + r 2 / r 0 ) = 2 , ( r 0 r 2 / r 0 ) = 2 c .
ε = [ ε x 0 0 ε y ] ,
ε y = ε x / c 2 .
V ( x , y ) = Re ( β w + γ ¯ t ) for x 2 + y 2 1 , = δ x x + δ y y for x 2 + y 2 < 1 ,
t = ( 1 + c 2 ) ( x + i c y ) 1 c 2 + 2 c ( x + i c y ) 2 + c 2 1 c 2 1 .
Re [ ( β + γ ) w ] = δ x x + δ y y .
n x ε x x Re ( β w + γ ¯ t ) + n y ε y y Re ( β w + γ ¯ t ) = n x δ x + n y δ y ,
n x ε x x Re ( γ ¯ t ) + n y ε y y Re ( γ ¯ t ) = n x ε x x Re ( γ w ) + n y ε y y Re ( γ w ) ,
n x ε x x Re [ ( β γ ) w ] + n y ε y y Re [ ( β γ ) w ] = n x δ x + n y δ y .
β + γ = δ x , ε x ( β γ ) = δ x .
δ x = 2 γ ε x ε x 1 , β = γ ( ε x + 1 ) ε x 1 ,
V ( x , y ) = β x + γ Re ( t ) ,
V ( x , y ) = γ ( ε x + 1 ) x ε x 1 + γ ( 1 + c 2 ) x 1 c 2 + γ c [ ( x + i c y ) 2 + c 2 1 + ( x i c y ) 2 + c 2 1 ] c 2 1 ,
V ( x , y ) = δ x x = 2 γ ε x x ε x 1 .
c ( β + γ ) = δ y , ε y c ( β γ ) = δ y .
δ y = 2 c γ ε y 1 ε y , β = γ ( ε y + 1 ) ε y 1 ,
V ( x , y ) = β c y + γ Im ( t ) ,
V ( x , y ) = γ ( ε y + 1 ) c y 1 ε y + γ ( 1 + c 2 ) c y 1 c 2 + γ c [ ( x + i c y ) 2 + c 2 1 ( x i c y ) 2 + c 2 1 ] i ( c 2 1 ) ,
V ( x , y ) = δ y y = 2 c γ ε y y 1 ε y .
( x + i c y ) 2 + c 2 1 = ( x + i c y + 1 c 2 ) x + i c y 1 c 2 x + i c y + 1 c 2 ,
( x i c y ) 2 + c 2 1 = ( x i c y + 1 c 2 ) x i c y 1 c 2 x i c y + 1 c 2 ,
x + i c y 1 c 2 x + i c y + 1 c 2 = x + i c ¯ y 1 c 2 ¯ x i c ¯ y + 1 c 2 ¯ ,
y Re [ c ¯ 1 c 2 ] = x Im [ 1 c 2 ] .
( x , y ) = ± ( Re [ c ¯ 1 c 2 ] Re [ c ] , Im [ 1 c 2 ] Re [ c ] ) ,
( Re [ c ] ) 2 ( Re [ c ¯ 1 c 2 ] ) 2 ( Im [ 1 c 2 ] ) 2 > 0 .
ε y = ε x / ( i μ + η ) 2 ε x / μ 2 + 2 i ε x η / μ 3
( x + i c y ) 2 + c 2 1 = ( x + μ y + i η y + 1 c 2 ) ( x + μ y + i η y 1 c 2 ) , ( x i c y ) 2 + c 2 1 = ( x μ y i η y + 1 c 2 ) ( x μ y i η y 1 c 2 ) ,
x + μ y + 1 + μ 2 = 0 , x + μ y 1 + μ 2 = 0 , x μ y + 1 + μ 2 = 0 , x μ y 1 + μ 2 = 0 ,
( x , y ) = ( ± 1 / 1 + μ 2 , ± μ / 1 + μ 2 ) .
V ( x , y ) = H ( x , y ) + G ( x , y ) x + μ y + i η y + r ,
r = 1 c 2 1 + μ 2 + i μ η 1 + μ 2 , H ( x , y ) = γ ( ε x + 1 ) x ε x 1 + γ ( 1 + c 2 ) x 1 c 2 + γ c ( x i c y ) 2 r 2 c 2 1 , G ( x , y ) = γ c x + i c y r c 2 1 .
V y μ G 0 x + μ y + i η y + r μ G 0 h + i η s ,
G 0 = γ i μ 2 1 + μ 2 1 + μ 2 = γ μ 2 ( 1 + μ 2 ) 3 / 4
h = x + μ y + 1 + μ 2 , s = y + μ 1 + μ 2 .
Im ( ε y ) | V y | 2 | G 0 | 2 μ 2 Im ( ε y ) | h + i η s | 2 η ε x | G 0 | 2 μ h 2 + η 2 s 2 ,
0 h 0 d h h 2 + η 2 s 2 = ln ( h 0 + h 2 + η 2 s 2 ) ln ( η | s | ) ,
Im ( ε y ) | V y | 2 d h 4 ε x | G 0 | 2 η ln ( η | s | ) .
2 0 1 / η 4 ε x | G 0 | 2 η ln ( η | s | ) d s = 8 ε x | G 0 | 2 .
( x + i c y ) 2 + c 2 1 x + i c y + c 2 1 2 ( x + i c y ) , ( x i c y ) 2 + c 2 1 x i c y + c 2 1 2 ( x i c y ) ,
V ( x , y ) γ ( ε x + 1 ) x ε x 1 + γ ( 1 + c 2 ) x 1 c 2 2 γ c x 1 c 2 + γ c 2 ( x + i c y ) + γ c 2 ( x i c y ) γ x ( ε x + 1 ε x 1 + 1 c 1 + c ) + γ c x x 2 + c 2 y 2 γ ( ε x + 1 ε x 1 + 1 c 1 + c ) ( x x α x x 2 + c 2 y 2 ) ,
α x = c ( ε x + 1 ε x 1 + 1 c 1 + c ) = c ( 1 + c ) ( 1 ε x ) 2 ( ε x + c ) ,
V ( x , y ) γ ( ε y + 1 ) c y 1 ε y + γ ( 1 + c 2 ) c y 1 c 2 2 γ c 2 y 1 c 2 + γ c 2 i ( x + i c y ) γ c 2 i ( x i c y ) γ c y ( ε y + 1 1 ε y + 1 c 1 + c ) γ c 2 y x 2 + c 2 y 2 γ c y ( ε y + 1 1 ε y + 1 c 1 + c ) ( y y α y x 2 + c 2 y 2 ) ,
α y = c ( ε y + 1 1 ε y + 1 c 1 + c ) = c ( 1 + c ) ( 1 ε y ) 2 ( 1 + c ε y ) .
u π c u ( 1 ε x ) 2 π ( u 2 + v 2 ) [ ε x + ( 1 ε x ) c 1 + c ] ,
x x α x x 2 + c 2 y 2 ,
V ( x , y ) x + a 1 α x / 2 x + i c y α x / 2 x i c y = x + a 1 x α x x 2 + c 2 y 2 ,
V ( x , y ) y + a 2 + α y / ( 2 i c ) x + i c y α y / ( 2 i c ) x i c y = y + a 2 y α y x 2 + c 2 y 2 .
V ( x , y ) γ x x + γ y y + a x γ x α x + y γ y α y x 2 + c 2 y 2 .
Im ( ε y ) | V y | 2 μ 2 | α x | 2 Im ( ε y ) 4 | g + i η y | 4 η ε x | α x | 2 2 μ ( g 2 + η 2 y 2 ) 2 ,
g 0 g 0 Im ( ε y ) | V y | 2 d g ε x | α x | 2 2 μ y 3 η 2 d ν ( ν 2 + 1 ) 2 ,
V ( x , y ) = x + x β 1 x + y β 1 y x 2 + c 2 y 2 + ( x x 0 ) β 2 x + ( y y 0 ) β 2 y ( x x 0 ) 2 + c 2 ( y y 0 ) 2 .
V ( x , y ) x β 1 x + y β 1 y x 2 + c 2 y 2 ( x 0 β 2 x + y 0 β 2 y ) x 0 2 + c 2 y 0 2 + x + x β 2 x + y β 2 y x 0 2 + c 2 y 0 2 ( 2 x x 0 + 2 y y 0 ) ( x 0 β 2 x + y 0 β 2 y ) ( x 0 2 + c 2 y 0 2 ) 2 ,
β 1 x = [ 1 β 2 x x 0 2 + c 2 y 0 2 + 2 x 0 ( x 0 β 2 x + y 0 β 2 y ) ( x 0 2 + c 2 y 0 2 ) 2 ] α 1 x , β 1 y = [ β 2 y x 0 2 + c 2 y 0 2 + 2 c 2 y 0 ( x 0 β 2 x + y 0 β 2 y ) ( x 0 2 + c 2 y 0 2 ) 2 ] α 1 y .
V ( x , y ) = x 0 + ( x x 0 ) + [ x 0 + ( x x 0 ) ] β 1 x + [ y 0 + ( y y 0 ) ] β 1 y [ x 0 + ( x x 0 ) ] 2 + c 2 [ y 0 + ( y y 0 ) ] 2 + ( x x 0 ) β 2 x + ( y y 0 ) β 2 y ( x x 0 ) 2 + c 2 ( y y 0 ) 2 ,
V ( x , y ) ( x x 0 ) β 2 x + ( y y 0 ) β 2 y ( x x 0 ) 2 + c 2 ( y y 0 ) 2 + x 0 + ( x 0 β 1 x + y 0 β 1 y ) x 0 2 + c 2 y 0 2 + ( x x 0 ) + ( x x 0 ) β 1 x + ( y y 0 ) β 1 y x 0 2 + c 2 y 0 2 [ 2 ( x x 0 ) x 0 + 2 ( y y 0 ) y 0 ] ( x 0 β 1 x + y 0 β 1 y ) ( x 0 2 + c 2 y 0 2 ) 2 .
β 2 x = [ 1 β 1 x x 0 2 + c 2 y 0 2 + 2 x 0 ( x 0 β 1 x + y 0 β 1 y ) ( x 0 2 + c 2 y 0 2 ) 2 ] α 2 x , β 2 y = [ β 1 y x 0 2 + c 2 y 0 2 + 2 c 2 y 0 ( x 0 β 1 x + y 0 β 1 y ) ( x 0 2 + c 2 y 0 2 ) 2 ] α 2 y .
e = x 0 2 + c 2 y 0 2 x 0 2 μ 2 y 0 2 2 i μ η y 0 2 ,
β 1 x α 1 x = 1 β 2 x e + 2 x 0 ( x 0 β 2 x + y 0 β 2 y ) e 2 , β 1 y α 1 y = β 2 y e + 2 ( e x 0 2 ) ( x 0 β 2 x + y 0 β 2 y ) y 0 e 2 , β 2 x α 2 x = 1 β 1 x e + 2 x 0 ( x 0 β 1 x + y 0 β 1 y ) e 2 , β 2 y α 2 y = β 1 y e + 2 ( e x 0 2 ) ( x 0 β 1 x + y 0 β 1 y ) y 0 e 2 .
β [ β 1 x β 1 y β 2 x β 2 y ] = A 1 [ 1 0 1 0 ] ,
A = [ 1 α 1 x 0 1 e 2 x 0 2 e 2 2 x 0 y 0 e 2 0 1 α 1 y 2 x 0 ( e x 0 2 ) e 2 y 0 1 e + 2 x 0 2 e 2 1 e 2 x 0 2 e 2 2 x 0 y 0 e 2 1 α 2 x 0 2 x 0 ( e x 0 2 ) e 2 y 0 1 e + 2 x 0 2 e 2 0 1 α 2 y ] ,
det ( A ) = [ 4 x 0 2 ( e x 0 2 ) ( α 1 x α 1 y ) ( α 2 x α 2 y ) + ( e 2 α 1 x α 2 x ) ( e 2 α 1 y α 2 y ) ] e 4 α 1 x α 1 y α 2 x α 2 y ,
4 x 0 2 ( e x 0 2 ) ( α 1 x α 1 y ) ( α 2 x α 2 y ) + ( e 2 α 1 x α 2 x ) ( e 2 α 1 y α 2 y ) = 0 .
β [ 2 x 0 2 α 1 x α 1 y [ 2 x 0 2 ( α 2 y α 2 x ) + α 2 y α 2 x ] α 1 y α 2 y α 1 x α 2 x 4 x 0 4 ( α 1 x α 1 y ) ( α 2 x α 2 y ) 2 x 0 3 α 1 x α 1 y [ 2 x 0 2 ( α 2 y α 2 x ) + α 2 y α 2 x ] y 0 [ α 1 y α 2 y α 1 x α 2 x 4 x 0 4 ( α 1 x α 1 y ) ( α 2 x α 2 y ) ] 2 x 0 2 α 2 x α 2 y [ 2 x 0 2 ( α 1 y α 1 x ) + α 1 x α 1 y ] α 1 y α 2 y α 1 x α 2 x 4 x 0 4 ( α 1 x α 1 y ) ( α 2 x α 2 y ) 2 x 0 3 α 2 y α 2 x [ 2 x 0 2 ( α 1 y α 1 x ) + α 1 x α 1 y ] y 0 [ α 1 y α 2 y α 1 x α 2 x 4 x 0 4 ( α 1 x α 1 y ) ( α 2 x α 2 y ) ] ] .
x 0 4 = α 1 x α 1 y α 2 x α 2 y 4 ( α 1 x α 1 y ) ( α 2 x α 2 y ) ,
β [ α 1 x α 1 y α 1 x α 1 y α 1 x α 1 y x 0 ( α 1 x α 1 y ) y 0 0 0 ] .
β [ α 1 x 0 0 0 ] .
β [ 2 x 0 2 2 x 0 3 y 0 2 x 0 2 ( α 1 x α 1 y 2 x 0 2 α 1 x + 2 x 0 2 α 1 y ) α 1 x α 1 y 2 x 0 3 ( α 1 x α 1 y 2 x 0 2 α 1 x + 2 x 0 2 α 1 y ) y 0 α 1 y α 1 x ] ,
β = [ ( e 2 + e α 2 x 2 x 0 2 α 2 x ) α 1 x e 2 α 1 x α 2 x 2 α 1 y x 0 ( e x 0 2 ) α 2 x ( e 2 α 2 x α 1 y ) y 0 α 2 x f ( e 2 α 2 x α 1 y ) ( e 2 α 1 x α 2 x ) 2 x 0 ( e x 0 2 ) α 2 x g y 0 ( e 2 α 2 x α 1 y ) ( e 2 α 1 x α 2 x ) ] ,
f = e 3 α 1 x 2 x 0 2 α 1 x e 2 e α 1 x α 2 x α 1 y + 2 x 0 2 α 1 x α 2 x α 1 y e 4 + 4 α 1 x α 2 x e x 0 2 4 α 1 x α 2 x x 0 4 + α 2 x α 1 y e 4 4 x 0 2 α 1 y α 2 x e + 4 x 0 4 α 1 y α 2 x , g = α 1 x e 2 α 1 y α 1 x α 2 x α 1 x α 2 x e + α 1 y α 2 x e + 2 α 1 x α 2 x x 0 2 2 x 0 2 α 1 y α 2 x .
β 1 x x 0 2 α 2 x α 1 x 2 μ 2 η 2 y 0 4 = α 2 x α 1 x 2 η 2 y 0 2 ,

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