Abstract

We propose an approach to far-field optical imaging beyond the diffraction limit. The proposed system allows image magnification, is robust with respect to material losses and can be fabricated by adapting existing metamaterial technologies in a cylindrical geometry.

© 2006 Optical Society of America

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References

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  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]
  2. J. B. Pendry , “ Negative refraction makes a perfect lens ,” Phys. Rev. Lett.   85 , 3966 – 3969 ( 2000 ).
    [Crossref] [PubMed]
  3. V. G. Veselago , “ The electrodynamics of substances with simultaneously negative values of permittivity and permeability ,” Sov. Phys. Usp.   10 , 509 ( 1968 ).
    [Crossref]
  4. R. A. Shelby , D. R. Smith , and S. Schultz ,“ Experimental verification of a negative index of refraction ,” Science   292 , 77 – 79 ( 2001 ).
    [Crossref] [PubMed]
  5. 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 ), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-24-3356 .
    [Crossref]
  6. V. A. Podolskiy and E. E. Narimanov , “ Near-sighted superlens ,” Opt. Lett.   30 , 75 – 77 ( 2005 ), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-1-75 .
    [Crossref] [PubMed]
  7. R. Merlin , “ Analytical solution of the almost-perfect-lens problem ,” Appl. Phys. Lett.   84 , 1290 – 1292 ( 2004 ).
    [Crossref]
  8. K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 2004 ).
    [Crossref]
  9. C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
    [Crossref]
  10. V. A. Podolskiy and E. E. Narimanov , “ Strongly anisotropic waveguide as a nonmagnetic left-handed system ,” Phys. Rev. B   71 , 201101 ( 2005 ).
    [Crossref]
  11. V. A. Podolskiy , L. Alekseyev , and E. E. Narimanov , “ Strongly anisotropic media: the THz perspectives of left-handed materials ”, J. Mod. Opt.   52 (16) 2343 ( 2005 ).
    [Crossref]
  12. R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
    [Crossref]
  13. A. A. Govyadinov and V. A. Podolskiy , “ Meta-material photonic funnels for sub-diffraction light compression and propagation ,” Phys. Rev. B   73 (15), 155108 ( 2006 ).
    [Crossref]
  14. 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]
  15. T. C. T. Ting , “ New solutions to pressuring, shearing, torsion, and extension of a cylindrically anisotropic elastic circular tube or bar ,” Proc. Roy. Soc. London ,   A455 , 3527 – 3542 ( 1999 ).
  16. 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]

2006 (2)

R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
[Crossref]

A. A. Govyadinov and V. A. Podolskiy , “ Meta-material photonic funnels for sub-diffraction light compression and propagation ,” Phys. Rev. B   73 (15), 155108 ( 2006 ).
[Crossref]

2005 (5)

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

V. A. Podolskiy , L. Alekseyev , and E. E. Narimanov , “ Strongly anisotropic media: the THz perspectives of left-handed materials ”, J. Mod. Opt.   52 (16) 2343 ( 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 , “ Near-sighted superlens ,” Opt. Lett.   30 , 75 – 77 ( 2005 ), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-1-75 .
[Crossref] [PubMed]

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 ), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-24-3356 .
[Crossref]

2004 (2)

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

K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 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]

2002 (1)

C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
[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)

T. C. T. Ting , “ New solutions to pressuring, shearing, torsion, and extension of a cylindrically anisotropic elastic circular tube or bar ,” Proc. Roy. Soc. London ,   A455 , 3527 – 3542 ( 1999 ).

1968 (1)

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

Alekseyev, L.

V. A. Podolskiy , L. Alekseyev , and E. E. Narimanov , “ Strongly anisotropic media: the THz perspectives of left-handed materials ”, J. Mod. Opt.   52 (16) 2343 ( 2005 ).
[Crossref]

Cai, W.

Chettiar, U. K.

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.

Elser, J.

R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
[Crossref]

Fang, N.

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]

Govyadinov, A. A.

A. A. Govyadinov and V. A. Podolskiy , “ Meta-material photonic funnels for sub-diffraction light compression and propagation ,” Phys. Rev. B   73 (15), 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]

Joannopoulos, J. D.

C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
[Crossref]

Johnson, S. G.

C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
[Crossref]

Kildishev, A. V.

Lee, H.

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]

Luo, C.

C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
[Crossref]

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]

Merlin, R.

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

Narimanov, E. E.

R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
[Crossref]

V. A. Podolskiy , L. Alekseyev , and E. E. Narimanov , “ Strongly anisotropic media: the THz perspectives of left-handed materials ”, J. Mod. Opt.   52 (16) 2343 ( 2005 ).
[Crossref]

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

V. A. Podolskiy and E. E. Narimanov , “ Near-sighted superlens ,” Opt. Lett.   30 , 75 – 77 ( 2005 ), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-1-75 .
[Crossref] [PubMed]

Nelson, K. A.

K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 2004 ).
[Crossref]

Pendry, J. B.

C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
[Crossref]

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

Podolskiy, V. A.

R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
[Crossref]

A. A. Govyadinov and V. A. Podolskiy , “ Meta-material photonic funnels for sub-diffraction light compression and propagation ,” Phys. Rev. B   73 (15), 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]

V. A. Podolskiy , L. Alekseyev , and E. E. Narimanov , “ Strongly anisotropic media: the THz perspectives of left-handed materials ”, J. Mod. Opt.   52 (16) 2343 ( 2005 ).
[Crossref]

V. A. Podolskiy and E. E. Narimanov , “ Near-sighted superlens ,” Opt. Lett.   30 , 75 – 77 ( 2005 ), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-1-75 .
[Crossref] [PubMed]

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]

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]

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]

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]

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]

R. A. Shelby , D. R. Smith , and S. Schultz ,“ Experimental verification of a negative index of refraction ,” Science   292 , 77 – 79 ( 2001 ).
[Crossref] [PubMed]

Sun, C.

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]

Ting, T. C. T.

T. C. T. Ting , “ New solutions to pressuring, shearing, torsion, and extension of a cylindrically anisotropic elastic circular tube or bar ,” Proc. Roy. Soc. London ,   A455 , 3527 – 3542 ( 1999 ).

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]

Wangberg, R.

R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
[Crossref]

Ward, D.W.

K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 2004 ).
[Crossref]

Webb, K. J.

K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 2004 ).
[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]

Yang, M.

K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 2004 ).
[Crossref]

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.

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]

Appl. Phys. Lett. (1)

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

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)

V. A. Podolskiy , L. Alekseyev , and E. E. Narimanov , “ Strongly anisotropic media: the THz perspectives of left-handed materials ”, J. Mod. Opt.   52 (16) 2343 ( 2005 ).
[Crossref]

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

R. Wangberg , J. Elser , E. E. Narimanov , and V. A. Podolskiy , “ Non-magnetic nano-composites for optical andinfrared negative refraction index media ,” J. Opt. Soc. Am. B.   23 , 498 ( 2006 ).
[Crossref]

Opt. Lett. (2)

Phys. Rev. B (3)

C. Luo , S. G. Johnson , J. D. Joannopoulos , and J. B. Pendry , “ All-angle negative refraction without negativeeffective index ,” Phys. Rev. B   65 , 201104-1 – 201104-4 ( 2002 ).
[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 sub-diffraction light compression and propagation ,” Phys. Rev. B   73 (15), 155108 ( 2006 ).
[Crossref]

Phys. Rev. E (1)

K. J. Webb , M. Yang , D.W. Ward , and K. A. Nelson , “ Metrics for negative-refractive-index material ,” Phys. Rev. E   70 , 035602(R) ( 2004 ).
[Crossref]

Phys. Rev. Lett. (2)

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]

Proc. Roy. Soc. London (1)

T. C. T. Ting , “ New solutions to pressuring, shearing, torsion, and extension of a cylindrically anisotropic elastic circular tube or bar ,” Proc. Roy. Soc. London ,   A455 , 3527 – 3542 ( 1999 ).

Science (2)

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]

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]

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Figures (8)

Fig. 1.
Fig. 1.

(a) A conventional imaging system transforms propagating waves, but does not operate on the decaying evanescent waves; these waves can only be detected in the near field. (b) “Superlens” amplifies the evanescent waves but does not change their decaying character. (c) An ideal device would convert evanescent waves to propagating waves for ease of detection and processing; these waves should not mix with the propagating waves emanating from the object.

Fig. 2.
Fig. 2.

(a) Scattering of an incident plane wave by a target (yellow object) can be represented as scattering of various angular momentum modes (the regions of high intensity are shown in black and low intensity in white). Higher order modes are exponentially small at the center (b). This results from an upper bound on values of kθ and the formation of the caustic shown in red in (c).

Fig. 3.
Fig. 3.

Dispersion relation for isotropic medium (a) and for a material with εr < 0, εθ > 0 (b). Note that for a fixed frequency, the wave vector k can take on arbitrarily large values (within the effective medium approximation).

Fig. 4.
Fig. 4.

(a) High angular momentum states in an isotropic dielectric cylinder (b) High angular momentum states in a cylinder made of εθ > 0, εr < 0 metamaterial (in the effective medium approximation); note that the field penetrates to the center.

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

(a) Top view of the hyperlens made of 50 alternating layers of metal(dark regions) with εm = -2 and dielectric (grey regions) with εd = 5. The outer radius is 2.2μm and the inner radius is 250nm. (b) Calculated light intensity for m=20 angular momentum state in false color representation where red denotes high intensity and blue corresponds to low intensity. Note the penetrating nature due to the achieved cylindrical anisotropy. (c) A hollow core cylinder of the same geometry made from a uniform dielectric ε uniform = 1.5 (average of εm and εd ) (d) Corresponding intensity for m=20 mode

Fig. 7.
Fig. 7.

(a) Schematics of imaging by the hyperlens. Two point sources separated by λ/3 are placed within the hollow core of the hyperlens consisting of 160 alternating layers of metal (ε = -1+0.01i) and dielectric (ε = 1.1) each 10 nm thick (the inner layer of the device is dielectric). The radius of the hollow core is R inner=250 nm, the outer radius R outer=1840 nm, the operating wavelength is 300 nm and the distance between the sources is 100 nm. (b) False color plot of intensity in the region bounded by the rectangle showing the highly directional nature of the beams from the two point sources. The boundary is shown in black where the separation between the beams is much greater than λ due to magnification.

Fig. 8.
Fig. 8.

Demonstration of subwavelength resolution in the composite hyperlens containing two sources placed a distance λ/4.5 apart inside the core. (a): Field at the source. (b): Field outside the hyperlens.

Equations (9)

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

exp ( i k x ) = m = m = i m J m ( k r ) exp ( i m ϕ ) ,
m = k θ r ,
k r 2 + k θ 2 = ε ω 2 c 2 .
k 2 ε + k 2 ε = ω 2 c 2 ,
k r 2 ε θ + k θ 2 ε r = ω 2 c 2 ,
B z J m ε r ε θ ( ω c ε θ ) exp ( i m ϕ ) .
Δ R inner R outer λ .
ε θ = ε m + ε d 2
ε r = 2 ε m ε d ε m + ε d ,

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