Abstract

We study ray dynamics inside the hyperlens, a device capable of sub-diffraction-limited far-field imaging. An analytical result for the ray trajectories inside the hyperlens is obtained using Hamiltonian optics, which offers an alternative description of the device. It is also found that the ray trajectories can exhibit a unique spiraling nature inside the device. Numerical simulations of plane wave and Gaussian beam scattering from the hyperlens confirm the semiclassical description.

© 2007 Optical Society of America

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  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]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  3. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
    [CrossRef]
  4. N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
    [CrossRef] [PubMed]
  5. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  6. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  7. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a sliver superlens," Science 308, 534-537 (2005).
    [CrossRef] [PubMed]
  8. B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
    [CrossRef]
  9. 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]
  10. A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev. B 74, 075103 (2006).
    [CrossRef]
  11. V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101 (2005).
    [CrossRef]
  12. A. A. Govyadinov and V. A. Podolskiy, "Meta-material photonic funnels for subdiffraction light compression and propagation," Phys. Rev. B 73, 155108 (2006).
    [CrossRef]
  13. I. Smolyaninov, Y. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
    [CrossRef] [PubMed]
  14. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, 2nd ed. (Reed Ltd., 1984).
  15. P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105 (2005).
    [CrossRef]
  16. M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
    [CrossRef]
  17. R. K. Fisher and R. W. Gould, "Resonance cones in the field pattern of a short antenna in an anisotropic plasma," Phys. Rev. Lett. 22, 1093-1095 (1969).
    [CrossRef]
  18. E. Arbel and L. B. Felsen, "Theory of radiation from sources in anisotropic media," in Electromagnetic Theory and Antennas, E.C.Jordan, ed. (Pergamon, 1963), pp. 391-421.

2007 (2)

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]

I. Smolyaninov, Y. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

2006 (4)

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]

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

A. A. Govyadinov and V. A. Podolskiy, "Meta-material photonic funnels for subdiffraction light compression and propagation," Phys. Rev. B 73, 155108 (2006).
[CrossRef]

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

2005 (4)

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
[CrossRef]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a sliver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101 (2005).
[CrossRef]

P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105 (2005).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (2)

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

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

1999 (1)

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
[CrossRef]

1969 (1)

R. K. Fisher and R. W. Gould, "Resonance cones in the field pattern of a short antenna in an anisotropic plasma," Phys. Rev. Lett. 22, 1093-1095 (1969).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Abram, R. A.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
[CrossRef]

Alekseyev, L. V.

Ambati, M.

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

Arbel, E.

E. Arbel and L. B. Felsen, "Theory of radiation from sources in anisotropic media," in Electromagnetic Theory and Antennas, E.C.Jordan, ed. (Pergamon, 1963), pp. 391-421.

Belov, P. A.

P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105 (2005).
[CrossRef]

Cai, W.

Chettiar, U. K.

Davis, C.

I. Smolyaninov, Y. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Deckert, V.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Drachev, V. P.

Engheta, N.

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Fang, N.

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a sliver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Felsen, L. B.

E. Arbel and L. B. Felsen, "Theory of radiation from sources in anisotropic media," in Electromagnetic Theory and Antennas, E.C.Jordan, ed. (Pergamon, 1963), pp. 391-421.

Fisher, R. K.

R. K. Fisher and R. W. Gould, "Resonance cones in the field pattern of a short antenna in an anisotropic plasma," Phys. Rev. Lett. 22, 1093-1095 (1969).
[CrossRef]

Gould, R. W.

R. K. Fisher and R. W. Gould, "Resonance cones in the field pattern of a short antenna in an anisotropic plasma," Phys. Rev. Lett. 22, 1093-1095 (1969).
[CrossRef]

Govyadinov, A. A.

A. A. Govyadinov and V. A. Podolskiy, "Meta-material photonic funnels for subdiffraction light compression and propagation," Phys. Rev. B 73, 155108 (2006).
[CrossRef]

Hecht, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Hung, Y.

I. Smolyaninov, Y. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Ikonen, P.

P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105 (2005).
[CrossRef]

Jacob, Z.

Kaliteevski, M. A.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
[CrossRef]

Kildishev, A. V.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, 2nd ed. (Reed Ltd., 1984).

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 sliver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, 2nd ed. (Reed Ltd., 1984).

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]

Martin, O. J. F.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Narimanov, E.

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101 (2005).
[CrossRef]

Nikolaev, V. V.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
[CrossRef]

Pendry, J. B.

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

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, 2nd ed. (Reed Ltd., 1984).

Podolskiy, V. A.

A. A. Govyadinov and V. A. Podolskiy, "Meta-material photonic funnels for subdiffraction light compression and propagation," Phys. Rev. B 73, 155108 (2006).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101 (2005).
[CrossRef]

Pohl, D. W.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Salandrino, A.

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sick, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Simovski, C. R.

P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105 (2005).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smolyaninov, I.

I. Smolyaninov, Y. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[CrossRef] [PubMed]

Sokolovski, G. S.

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
[CrossRef]

Srituravanich, W.

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

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, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a sliver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Wild, U. P.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Xi, D.

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[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, J.

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

Yuan, H.-K.

Zenobi, R.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

Zhang, X.

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, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a sliver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, "Scanning near-field optical microscopy with aperture probes: fundamentals and applications," J. Chem. Phys. 112, 7761 (2000).
[CrossRef]

J. Mod. Opt. (1)

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46(5), 875-890 (1999).
[CrossRef]

Nat. Mater. (1)

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, "Ultrasonic metamaterials with negative modulus," Nat. Mater. 5, 452-456 (2006).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (4)

P. A. Belov, C. R. Simovski, and P. Ikonen, "Canalization of subwavelength images by electromagnetic crystals," Phys. Rev. B 71, 193105 (2005).
[CrossRef]

A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev. B 74, 075103 (2006).
[CrossRef]

V. A. Podolskiy and E. E. Narimanov, "Strongly anisotropic waveguide as a nonmagnetic left-handed system," Phys. Rev. B 71, 201101 (2005).
[CrossRef]

A. A. Govyadinov and V. A. Podolskiy, "Meta-material photonic funnels for subdiffraction light compression and propagation," Phys. Rev. B 73, 155108 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

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

R. K. Fisher and R. W. Gould, "Resonance cones in the field pattern of a short antenna in an anisotropic plasma," Phys. Rev. Lett. 22, 1093-1095 (1969).
[CrossRef]

Science (4)

N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-limited optical imaging with a sliver superlens," Science 308, 534-537 (2005).
[CrossRef] [PubMed]

I. Smolyaninov, Y. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-1701 (2007).
[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]

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other (2)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics, 2nd ed. (Reed Ltd., 1984).

E. Arbel and L. B. Felsen, "Theory of radiation from sources in anisotropic media," in Electromagnetic Theory and Antennas, E.C.Jordan, ed. (Pergamon, 1963), pp. 391-421.

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Figures (9)

Fig. 1
Fig. 1

Trajectory of two rays incident on the hyperlens with different impact parameters calculated using the analytical expression (a) η = 0.5 and (b) η = 0.1 . Note the strong spiraling behavior (c) “channeling” regime for large η = 100 , where rays travel in straight lines radially. Note that all rays travel toward the center.

Fig. 2
Fig. 2

Possible realizations of metacylinders. (a) Concentric alternate metallic layers and dielectric layers or (b) radially symmetric “slices” of metal and dielectric materials produce ( ϵ θ > 0 , ϵ r < 0 ) anisotropy. This results in a hyperbolic dispersion relation necessary for penetration of the field close to the center.

Fig. 3
Fig. 3

(a) Schematic of a Gaussian beam with impact parameter p impinging on the layered hyperlens (top view), consisting of alternate layers of metal and dielectric materials. The inner hollow region and region outside the hyperlens are vacuum. (b) Ray trajectories representing the Gaussian beam calculated for the effective medium parameters (Section 3) of the hyperlens using Eq. (19). Note the narrowing of the Gaussian beam, predicted by the semiclassical theory, toward the core of the hyperlens. We consider specular reflection at the inner core.

Fig. 4
Fig. 4

(a) Absolute value of the field shown in false color for a Gaussian beam scattering from the layered hyperlens with parameters p 4 λ , r m i n λ , r m a x 7 λ , h λ 100 , ϵ m 0.4 , ϵ d 2.4 . The inner and outer boundaries of the hyperlens are shown in red (online). The ray trajectory shown in black is calculated using Eq. (19) and specular reflection at the inner boundary. Note the narrowing of the Gaussian beam and also that the center of the beam moves along the calculated ray trajectory. (b) Real part of the field due to Gaussian beam scattering. The negative refraction of the Poynting vector, which is along the direction of the beam, is clearly seen. The wave vector ( k ) refracts positively, as can be seen by drawing the normal to the phase fronts.

Fig. 5
Fig. 5

Strong spiraling behavior for a Gaussian beam incident ( p 1.4 λ ) on the layered hyperlens with r m i n λ , r m a x 7 λ , and h λ 70 and metal and dielectric materials chosen to achieve ϵ θ 1 and ϵ r = 0.01 . The magnitude of the field is shown in false color (online), while the trajectory predicted by the semiclassical theory is plotted in the inset.

Fig. 6
Fig. 6

Deviations from effective medium theory for thick layers. Near the core, the trajectory obtained by Eq. (19) (shown in black) does not match the numerical field calculations obtained by Gaussian beam scattering.

Fig. 7
Fig. 7

Dispersion relations (curves), wave vectors (solid arrows), Poynting vectors, and ray directions (dashed arrows). (a) Circular dispersion relation for an isotropic medium. The Poynting vectors are parallel to the respective wave vectors drawn from the origin to various points on the curve; hence a point source radiates isotropically. (b) Hyperbolic dispersion relation for a strongly anisotropic medium. The Poynting vector is not parallel to the wave vector, and they lie within a cone whose half-angle is related to the angle between the asymptotes of the hyperbola (dashed lines), leading to a beamlike point source radiation pattern.

Fig. 8
Fig. 8

Subdifraction imaging in the hyperlens. (a) Beamlike radiation obtained from Eq. (19) for two point sources kept near the inner boundary of the hyperlens for large η (“channeling” regime). The rays are negatively refracted at the inner surface and proceed radially outward, leading to magnification at the outer surface. The point source is represented as a source of rays in all directions (inset). (b) Numerical confirmation of the beamlike radiation using a layered metamaterial hyperlens made of alternating layers of metal ( ϵ m 1 ) and dielectric ( ϵ d 1.1 ) materials and two point sources near the inner boundary. The magnitude of the field is shown in false color.

Fig. 9
Fig. 9

(a) Parallel rays incident on the hyperlens in the spiraling regime intersect inside the hyperlens, bounce off the inner region, and proceed in a beamlike nature. This implies plane wave scattering from the hyperlens should show a focusing effect. (b) Numerical verification of the focusing effect for a layered hyperlens ( r m i n λ , r m a x 5 λ , h λ 60 , ϵ m 0.4 , ϵ d 2.4 ) realization. The boundaries of the hyperlens are shown by dark circles.

Equations (32)

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D = 0 , B = 0 , × B = 1 c D t , × E = 1 c B t .
× [ ϵ 1 ( × B ) ] = ω 2 c 2 B .
1 r r { r ϵ θ B z r } + 1 r 2 θ { 1 ϵ r B z θ } + ω 2 c 2 B z = 0 .
λ r ϵ i j θ 1
λ ϵ i j r 1 .
B z ( r , θ ) = B 0 exp ( i S ( r , θ ) ) ,
1 r r { r ϵ θ B z r } 1 ϵ θ 2 S r 2 B ,
1 r 2 θ { 1 ϵ r B z θ } 1 r 2 ϵ θ 2 S θ 2 B ,
1 ϵ θ 2 S r 2 + 1 r 2 ϵ r 2 S θ 2 = ω 2 c 2 .
S t ( r , θ ) = S ( r , θ ) ω t .
H = c p r 2 ϵ θ + p θ 2 r 2 ϵ r ,
r ̇ = H p r = c 2 p r H ϵ θ ,
θ ̇ = H p θ = c 2 p θ H ϵ r r 2 ,
p r ̇ = H r = c 2 p θ 2 H ϵ r r 3 ,
p θ ̇ = 0 ,
d r d θ = ϵ r r 2 ϵ θ p θ ϵ r ξ 2 ϵ θ p θ 2 ϵ r r 2 ,
r ( θ ) = p θ ξ ϵ r sinh ( η ( θ θ 0 ) ) .
η = ϵ r ϵ θ
r ( θ ) = ρ ϵ r sinh ( η ( θ θ 0 ) ) ,
θ 0 = sin 1 ( ρ r m a x ) 1 η sinh 1 ( ρ r m a x ϵ r ) .
ϵ θ = ϵ m + ϵ d 2 ,
ϵ r = 2 ϵ m ϵ d ϵ m + ϵ d ,
B z = R 1 ( r ) exp ( i m θ ) ,
r 2 d 2 R 1 d r 2 + r d R 1 d r + [ ϵ i k 2 r 2 m 2 ] R 1 = 0 ,
R 1 ( r ) c 1 f 1 ( ϵ i k 0 r ) + c 2 f 2 ( ϵ i k 0 r ) ,
B y i n c ( x , z ) = m = m = i m J m ( k r ) exp ( i m ϕ ) A m ,
A m = e k x 2 ( 2 σ k ) 2 + i x 0 k x + i z 0 k z + i m α d k x ,
B y s c a t t ( x , z ) = m = m = i m ( r m 1 ) H m ( 1 ) ( k r ) exp ( i m ϕ ) A m ,
B y o u t e r ( x , z ) = B y i n c ( x , z ) + B y s c a t t ( x , z ) .
B y j ( x , z ) = m = m = [ γ m H m ( 1 ) ( ϵ j k r ) exp ( i m ϕ ) + δ m H m ( 2 ) ( ϵ j k r ) exp ( i m ϕ ) ] ,
B y c o r e ( x , z ) = m = m = [ μ m J m ( k r ) exp ( i m ϕ ) ] .
tan ( θ c ) = ϵ θ ϵ r = 1 η ,

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