Abstract

We present an analytical study in the structure-modulated plasmonic angular momentum, which is trapped in the core region of a sectorial indefinite metamaterial. This metamaterial consists of periodically arranged metal-dielectric nano-wedges along the azimuthal direction. Employing a transfer-matrix calculation and a conformal-mapping technique, our theory can deal with an arbitrary number of wedges with realistically rounded tips. We demonstrate that in the deep-subwavelength regime, strong electric fields that carry large azimuthal variations can exist only within ten-nanometer length scale around the structural center. They are naturally bounded by a characteristic radius on the order of a hundred nanometers from the center. These extreme confining properties suggest that the structure under investigation can be superior to the conventional metal-dielectric cavities in terms of nanoscale photonic manipulation.

© 2013 Optical Society of America

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  1. 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]
  2. I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402 (2010).
    [Crossref] [PubMed]
  3. J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
    [Crossref] [PubMed]
  4. X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
    [Crossref]
  5. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
    [Crossref] [PubMed]
  6. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
    [Crossref] [PubMed]
  7. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
    [Crossref] [PubMed]
  8. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
    [Crossref] [PubMed]
  9. J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
    [Crossref]
  10. J. Li, L. Thylen, A. Bratkovsky, S.-Y. Wang, and R. S. Williams, “Optical magnetic plasma in artificial flowers,” Opt. Express 17, 10800–10805 (2009).
    [Crossref] [PubMed]
  11. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).
    [Crossref]
  12. A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981).
    [Crossref]
  13. R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B 31, 121–126 (1985).
    [Crossref]
  14. E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
    [Crossref] [PubMed]
  15. A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E 72, 036612 (2005).
    [Crossref]
  16. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [Crossref] [PubMed]
  17. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
    [Crossref] [PubMed]
  18. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [Crossref] [PubMed]
  19. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
    [Crossref] [PubMed]
  20. Z. Shen, Z. J. Hu, G. H. Yuan, C. J. Min, H. Fang, and X.-C. Yuan, “Visualizing orbital angular momentum of plasmonic vortices,” Opt. Lett. 37, 4627–4629 (2012).
    [Crossref] [PubMed]
  21. C. Yeh and F. Shimabukuro, The Essence of Dielectric Waveguides (Springer, 2008).
    [Crossref]
  22. Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
    [Crossref]
  23. Q. Li and M. Qiu, “Plasmonic wave propagation in silver nanowires: guiding modes or not?” Opt. Express 21, 8587–8595 (2013).
    [Crossref] [PubMed]
  24. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
    [Crossref] [PubMed]
  25. V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004).
    [Crossref]
  26. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
    [Crossref]
  27. J. D. Jackson, Classical Electrodynamics (John Wiley, 1998).
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  29. A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond. A 455, 2655–2686 (1999).
    [Crossref]
  30. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Brooks/Cole, 1976).
  31. P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B 69, 045418 (2004).
    [Crossref]
  32. Y. Ma, X. Li, H. Yu, L. Tong, Y. Gu, and Q. Gong, “Direct measurement of propagation losses in silver nanowires,” Opt. Lett. 35, 1160–1162 (2010).
    [Crossref] [PubMed]
  33. L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).
    [Crossref]
  34. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University, 1951).
  35. H. Benisty, “Dark modes, slow modes, and coupling in multimode systems,” J. Opt. Soc. Am. B 26, 718–724 (2009).
    [Crossref]
  36. F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82, 209–275 (2010).
    [Crossref]

2013 (1)

2012 (4)

Z. Shen, Z. J. Hu, G. H. Yuan, C. J. Min, H. Fang, and X.-C. Yuan, “Visualizing orbital angular momentum of plasmonic vortices,” Opt. Lett. 37, 4627–4629 (2012).
[Crossref] [PubMed]

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

2011 (1)

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

2010 (4)

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402 (2010).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Y. Ma, X. Li, H. Yu, L. Tong, Y. Gu, and Q. Gong, “Direct measurement of propagation losses in silver nanowires,” Opt. Lett. 35, 1160–1162 (2010).
[Crossref] [PubMed]

F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82, 209–275 (2010).
[Crossref]

2009 (3)

2008 (1)

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[Crossref] [PubMed]

2007 (1)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

2006 (3)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[Crossref] [PubMed]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

2005 (2)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E 72, 036612 (2005).
[Crossref]

2004 (2)

V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004).
[Crossref]

P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B 69, 045418 (2004).
[Crossref]

2003 (1)

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]

2001 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

1999 (1)

A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond. A 455, 2655–2686 (1999).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

1985 (1)

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B 31, 121–126 (1985).
[Crossref]

1981 (1)

A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981).
[Crossref]

1976 (1)

L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).
[Crossref]

1972 (1)

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).
[Crossref]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Abramowitz, M.

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

Aers, G. C.

A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981).
[Crossref]

Alekseyev, L. V.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Brooks/Cole, 1976).

Atwater, H. A.

P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B 69, 045418 (2004).
[Crossref]

Bartal, G.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Benisty, H.

Boardman, A. D.

A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981).
[Crossref]

Bozhevolnyi, S. I.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[Crossref] [PubMed]

Bratkovsky, A.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

Chang, D. E.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

Cho, S.-W.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Davis, L. C.

L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).
[Crossref]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

Dobrzynski, L.

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).
[Crossref]

Ducloy, M.

V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004).
[Crossref]

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Fang, H.

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Ferrando, A.

A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E 72, 036612 (2005).
[Crossref]

Fok, L.

J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
[Crossref]

García de Abajo, F. J.

F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82, 209–275 (2010).
[Crossref]

Garcia-Molina, R.

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B 31, 121–126 (1985).
[Crossref]

García-Vidal, F. J.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[Crossref] [PubMed]

Gong, Q.

Gras-Marti, A.

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B 31, 121–126 (1985).
[Crossref]

Gu, Y.

Hemmer, P. R.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

Hu, Q.

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

Hu, Z. J.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley, 1998).

Jacob, Z.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[Crossref] [PubMed]

Kang, M.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Kik, P. G.

P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B 69, 045418 (2004).
[Crossref]

Kim, H.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Klimov, V. V.

V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004).
[Crossref]

Kretzschmar, I.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

Krishnamoorthy, H. N. S.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

Lee, B.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Lee, S.-Y.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Li, J.

J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
[Crossref]

J. Li, L. Thylen, A. Bratkovsky, S.-Y. Wang, and R. S. Williams, “Optical magnetic plasma in artificial flowers,” Opt. Express 17, 10800–10805 (2009).
[Crossref] [PubMed]

Li, Q.

Li, X.

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

Lukin, M. D.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

Ma, Y.

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

Maier, S. A.

P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B 69, 045418 (2004).
[Crossref]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Maradudin, A. A.

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).
[Crossref]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Martín-Moreno, L.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[Crossref] [PubMed]

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University, 1951).

Menon, V. M.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Brooks/Cole, 1976).

Min, C. J.

Moreno, E.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[Crossref] [PubMed]

Narimanov, E.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[Crossref] [PubMed]

Narimanov, E. E.

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402 (2010).
[Crossref] [PubMed]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Park, J.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

Peng, R.-W.

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

Qiu, M.

Rawlins, A. D.

A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond. A 455, 2655–2686 (1999).
[Crossref]

Rho, J.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

Ritchie, R. H.

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B 31, 121–126 (1985).
[Crossref]

Rodrigo, S. G.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[Crossref] [PubMed]

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]

Shen, Z.

Shimabukuro, F.

C. Yeh and F. Shimabukuro, The Essence of Dielectric Waveguides (Springer, 2008).
[Crossref]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

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]

Smolyaninov, I. I.

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402 (2010).
[Crossref] [PubMed]

Sørensen, A. S.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Stegun, I. A.

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

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Teshima, R.

A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981).
[Crossref]

Thylen, L.

Tong, L.

Wang, M.

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

Wang, S.-Y.

Williams, R. S.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

Xu, D.-H.

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

Yang, Q.-L.

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

Yang, X.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

Yao, J.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

Yeh, C.

C. Yeh and F. Shimabukuro, The Essence of Dielectric Waveguides (Springer, 2008).
[Crossref]

Yin, X.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
[Crossref]

Yu, H.

Yuan, G. H.

Yuan, X.-C.

Zhang, X.

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
[Crossref]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

Zhou, Y.

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

Europhys. Lett. (1)

Q. Hu, D.-H. Xu, R.-W. Peng, Y. Zhou, Q.-L. Yang, and M. Wang, “Tune the “rainbow” trapped in a multilayered waveguide,” Europhys. Lett. 99, 57007 (2012).
[Crossref]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010).
[Crossref] [PubMed]

Nat. Mater. (1)

J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, “Experimental demonstration of an acoustic magnifying hyperlens,” Nat. Mater. 11, 931–934 (2009).
[Crossref]

Nat. Photonics (1)

X. Yang, J. Yao, J. Rho, X. Yin, and X. Zhang, “Experimental realization of three-dimensional indefinite cavities at the nanoscale with anomalous scaling laws,” Nat. Photonics 6, 450–454 (2012).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Phys. Rev. A (2)

V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69, 013812 (2004).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. B (5)

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).
[Crossref]

A. D. Boardman, G. C. Aers, and R. Teshima, “Retarded edge modes of a parabolic wedge,” Phys. Rev. B 24, 5703–5712 (1981).
[Crossref]

R. Garcia-Molina, A. Gras-Marti, and R. H. Ritchie, “Excitation of edge modes in the interaction of electron beams with dielectric wedges,” Phys. Rev. B 31, 121–126 (1985).
[Crossref]

P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B 69, 045418 (2004).
[Crossref]

L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).
[Crossref]

Phys. Rev. E (1)

A. Ferrando, “Discrete-symmetry vortices as angular Bloch modes,” Phys. Rev. E 72, 036612 (2005).
[Crossref]

Phys. Rev. Lett. (5)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and focusing of electromagnetic fields with wedge plasmon polaritons,” Phys. Rev. Lett. 100, 023901 (2008).
[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]

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105, 067402 (2010).
[Crossref] [PubMed]

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97, 053002 (2006).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U. S. A. (1)

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U. S. A. 108, 11327–11331 (2011).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

A. D. Rawlins, “Diffraction by, or diffusion into, a penetrable wedge,” Proc. R. Soc. Lond. A 455, 2655–2686 (1999).
[Crossref]

Rev. Mod. Phys. (1)

F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. 82, 209–275 (2010).
[Crossref]

Science (4)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534–537 (2005).
[Crossref] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[Crossref] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
[Crossref] [PubMed]

Other (5)

C. Yeh and F. Shimabukuro, The Essence of Dielectric Waveguides (Springer, 2008).
[Crossref]

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University, 1951).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Brooks/Cole, 1976).

J. D. Jackson, Classical Electrodynamics (John Wiley, 1998).

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

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Figures (9)

Fig. 1
Fig. 1 Schematics of a sectorial construction of indefinite metamaterial consisting of medium 1 and medium 2, which are periodically arranged in the azimuthal ϕ-direction with alternating angular span γ1 and γ2 respectively.

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Fig. 2
Fig. 2 Plots of the complex-order modified Bessel function of the second kind Kiς (κrr)/Kiς (1) with ς = 1, 2.5, 5.5, 10, when the abscissa is taken as (a) κrr and (b) ln(κrr). The denominator Kiς (1) is introduced to cancel some large prefactors and optimize the visualization.

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Fig. 3
Fig. 3 Effective permittivities ε̃r and ε̃ϕ versus frequency ω for the metal and dielectric filling ratios η 1 = 1 3 and η 2 = 2 3.

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Fig. 4
Fig. 4 Calculated eigen-spectrum ω(ς, lz) versus lz for several fixed values of ς. (a) From the actual medium theory with N = 24, γ = π/12, and lz cutoff at the 1st Brillouin zone boundary ±π/γ = ±12. (b) From the effective medium theory with lz manually cutoff at ±12 for the sake of comparison. Note that a physical lz can only take discrete integers denoted by grey dots according to Eq. (12).

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Fig. 5
Fig. 5 Conformal mapping from the (x, y)-coordinates to the (u, v)-coordinates with different total number of units N = 1 to 6. The red lines show the branch cuts and the semi-focal length. The green areas are the intended areas for metal filling at the ratio η 1 = 1 3.

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Fig. 6
Fig. 6 Plots of the normalized Hermite function with m = 0, 1, 4, 15, 40.

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Fig. 7
Fig. 7 Calculated eigen-spectrum ω(kza, m, lz) versus kza in the case of N = 4. (a) lz = 1, m = 0, 1, 2, 100. (b) lz = 2, m = 0, 1, 2, 100. The dot-dashed lines are the light line of the dielectric taking kza as the abscissa when a = 10 nm. As a result of the non-retarded approximation, only the right portion of the calculated dispersion curves outside the light cone is physical. The dotted lines are the limiting dispersion curves as m → ∞, converging to the surface plasma frequency ωsp.

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Fig. 8
Fig. 8 Profiles of the electrostatic potential close to the wedge tips at a representative frequency ω = 3.54×1015 s−1 (532 nm free-space wavelength) for the unit number N = 4, the structure-modulated angular momentum lz = 1 and 2, and the radial oscillation order m = 0. The semi-focal length a = 10 nm. The black curves indicate the wedge shapes. The white lines indicate the branch cuts from the conformal mapping.

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Fig. 9
Fig. 9 Profiles of the electrostatic potential away from the wedge tips at a representative frequency ω = 3.54 × 1015 s−1 (532 nm free-space wavelength) for the number of units N = 6 and 24, and several structure-modulated angular momentum lz chosen between 1 and N/2. The coordinates are measured in kzx and kzy in view of the non-retarded assumption κrr ≃ kzr. The white lines indicate the wedge interfaces. The blank central areas are where the severe oscillations and field intensity occur.

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Equations (29)

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κ r 2 = − k r 2 = k z 2 − μ ε ω 2 c 2 .
[ ∂ r 2 + 1 r 2 ∂ ϕ 2 + ω 2 c 2 μ ε − k z 2 ] Φ E = 0 ,                   [ ∂ r 2 + 1 r 2 ∂ ϕ 2 + ω 2 c 2 μ ε − k z 2 ] Φ H = 0 .
E = − ∇ Φ E + i μ ε ω 2 c 2 k z Φ E e z − μ ω c k z ∇ × [ Φ H e z ] ,
H = − ∇ Φ H + i μ ε ω 2 c 2 k z Φ H e z + ε ω c k z ∇ × [ Φ E e z ] ,
Φ E ( r , ϕ , z ; κ r , ς , k z ) = 1 ε κ r K i ς ( κ r r ) [ A ς e − ς ϕ + B ς e + ς ϕ ] e i k z z ,
Φ H ( r , ϕ , z ; κ r , ς , k z ) = 1 μ κ r K i ς ( κ r r ) [ C ς e − ς ϕ + D ς e + ς ϕ ] e i k z z ,
r ≃ ς κ r ≡ b .
κ r 2 ≃ k z 2 .
[ ∂ r 2 + 1 r 2 ∂ ϕ 2 − k z 2 ] Φ E ≃ 0 ,                         E ≃ − ∇ Φ E ,                     H ≃ 0 ,
[ ∂ r 2 + 1 r 2 ∂ ϕ 2 − k z 2 ] Φ H ≃ 0 ,                   H ≃ − ∇ Φ H ,                       E ≃ 0 .
T = ( e − ς γ 1 [ cosh ( ς γ 2 ) − ε 1 2 + ε 2 2 2 ε 1 ε 2 sinh ( ς γ 2 ) ] ε 1 2 − ε 2 2 2 ε 1 ε 2 sinh ( ς γ 2 ) ε 2 2 − ε 1 2 2 ε 1 ε 2 sinh ( ς γ 2 ) e + ς γ 1 [ cosh ( ς γ 2 ) + ε 1 2 + ε 2 2 2 ε 1 ε 2 sinh ( ς γ 2 ) ] ) .
det | T − e i l z γ I | = 0 ,                         ( l z = 0 , ± 1 , ± 2 , … , ± N 2 ) .
cos ( l z γ ) = cosh ( ς γ 1 ) cosh ( ς γ 2 ) + 1 2 ( ε 1 ε 2 + ε 2 ε 1 ) sinh ( ς γ 1 ) sinh ( ς γ 2 ) .
ς 2 ε ˜ ϕ + l z 2 ε ˜ r = 0 .
ε ˜ r = ε 1 η 1 + ε 2 η 2 ,                         ε ˜ ϕ = ε 1 ε 2 ε 1 η 2 + ε 2 η 1 ,
ε 1 ( ω ) ≈ ε h − ( ε s − ε h ) ω p 2 ω 2 − i ω Γ .
K i ς ( κ r r ) ~ − π ς sinh ς sin [ ς ln ( 1 2 κ r r ) − arg Γ ( 1 + i ς ) ] ,                     ( κ r r → 0 ) ,
[ ∂ x 2 + ∂ y 2 − k z 2 ] Φ E ( x , y ) = 0 .
[ ∂ u 2 + ∂ v 2 − | d w d s | ( u , v ) 2 k z 2 a 2 ] Φ E ( u , v ) = 0 .
w N = [ cosh s ] 2 ,                     i . e . ,                       x + i y = a [ cosh ( u + i v ) ] 2 N e i 2 π N n , ( u ∈ [ 0 , + ∞ ) ,                     v ∈ [ − π 2 , + π 2 ] ,                 n = 0 , 1 , 2 , … , N − 1 ) .
x + i y = r e i ϕ ≃ a 4 N e 2 N u e i ( 2 N v + 2 π N n ) ,                 i . e . ,                 r ≃ a 4 N e 2 N u ,                   ϕ ≃ 2 N v + 2 π N n .
| d w d s | 2 = | 2 N [ cosh s ] 2 N − 1 sinh s | 2 = 4 N 2 [ sinh 2 u + cos 2 v ] 2 N − 1 [ sinh 2 u + sin 2 v ] .
− [ ∂ u 2 + ∂ v 2 ] Φ E + 4 N 2 k z 2 a 2 [ sinh 2 u + cos 2 v ] 2 N − 1 [ sinh 2 u + sin 2 v ] Φ E = 0 .
− [ ∂ u 2 + ∂ v 2 ] Φ E + ( 2 N k z a ) 2 ( u 2 + v 2 ) Φ E = 0 .
[ − d 2 d u 2 + ( 2 N k z a ) 2 u 2 ] U ( u ) = + ( 2 N ς ) 2 U ( u ) ,
[ − d 2 d v 2 + ( 2 N k z a ) 2 v 2 ] V ( v ) = − ( 2 N ς ) 2 V ( v ) .
U m ( u ) = 1 2 m m ! ( 2 k z a N π ) 1 4 exp [ − 1 2 ( 2 N k z a ) u 2 ] ℋ m [ ( 2 N k z a ) 1 2 u ] ,
ς m = N k z a 2 ( 2 m + 1 ) ,               ( m = 0 , 1 , 2 , 3 , … ) .
V m ( v ) = A m 𝒟 − m − 1 [ + ( 4 N k z a ) 1 2 v ] + B m 𝒟 − m − 1 [ − ( 4 N k z a ) 1 2 v ] .

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