Abstract

We derive a general theory for imaging by a flat lens without optical axis. We show that the condition for imaging requires a material having elliptic dispersion relations with negative group refraction. This medium is characterized by two intrinsic parameters σ and κ. Imaging can be achieved with both negative and positive wave vector refraction if σ is a positive constant. The Veselago-Pendry lens is a special case with σ = 1 and κ = 0. A general law of refraction for anisotropic media is revealed. Realizations of the imaging conditions using anisotropic media and inhomogeneous media, particularly photonic crystals, are discussed. Numerical examples of imaging and requirements for sub-wavelength imaging are also presented.

© 2005 Optical Society of America

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References

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  1. B. Mahon, The man who changed everything: The life of James Clerk Maxwell, John Wiley & Sons 2004.
  2. M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press (2003).
  3. E. W. Marchland, Gradient Index Optics, Academic Press, New York (1978).
  4. P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, "Electromagnetic wave chaos in gradient refractive index optical cavities," Phys. Rev. Lett. 86, 5466 (2001).
    [CrossRef] [PubMed]
  5. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
    [CrossRef]
  6. S. A. Tretyakov, "Research on negative refraction and backward-wave media: A historical perspective," in Negative Refraction: Revisiting Electromagnetics from Microwave to Optics, EPFL Latsis Symposium, Lausanne, Switzerland, 2005, pp.30-35.
  7. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and µ ," Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  8. R. A. Silin, "Possibility of creating plane-parallel lenses," Opt. Spectrosc. 44, 109 (1978).
  9. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  10. R. A. Shelby, D. R. Smith, and S. Shultz, "Experimental verification of a negative index of refraction," Science 292, 77 (2001).
    [CrossRef] [PubMed]
  11. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature (London) 423, 604 (2003).
    [CrossRef] [PubMed]
  12. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
    [CrossRef] [PubMed]
  13. C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
    [CrossRef]
  14. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature (London) 426, 404 (2003).
    [CrossRef] [PubMed]
  15. J. T. Shen and P. M. Platzman, "Near field imaging with negative dielectric constant lenses," Appl. Phys. Lett. 80, 3286 (2002).
    [CrossRef]
  16. W. T. Lu and S. Sridhar, "Near-field imaging by negative permittivity media," Microwave Opt. Tech. Lett. 39, 282 (2003).
    [CrossRef]
  17. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-diffraction-dimited optical imaging with a silver superlens," Science 308, 534 (2005).
    [CrossRef] [PubMed]
  18. D. O. S. Melville and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127 (2005).
    [CrossRef] [PubMed]
  19. Note that the Veselago-Pendry lens also operates at a single frequency for which n = ε= µ= -1.
  20. L. Landau and I.M. Lifshitz, Electrodynamics of continuous media, 2nd Ed. (Elsevier 1984).
  21. 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]
  22. For a PhC with negative group refraction,κ is always nonzero. For example for the second band, κ2~ 2π/a instead of κ~ 0.
  23. Y. Zhang, B. Fluegel, and A. Mascarenhas, "Total negative refraction in real crystals for ballistic electrons and light," Phys. Rev. Lett. 91, 157404 (2003).
    [CrossRef] [PubMed]
  24. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
    [CrossRef]

Appl. Phys. Lett. (1)

J. T. Shen and P. M. Platzman, "Near field imaging with negative dielectric constant lenses," Appl. Phys. Lett. 80, 3286 (2002).
[CrossRef]

EPFL Latsis Symposium 2005 (1)

S. A. Tretyakov, "Research on negative refraction and backward-wave media: A historical perspective," in Negative Refraction: Revisiting Electromagnetics from Microwave to Optics, EPFL Latsis Symposium, Lausanne, Switzerland, 2005, pp.30-35.

Microwave Opt. Tech. Lett. (1)

W. T. Lu and S. Sridhar, "Near-field imaging by negative permittivity media," Microwave Opt. Tech. Lett. 39, 282 (2003).
[CrossRef]

Nature (2)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature (London) 423, 604 (2003).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Imaging by flat lens using negative refraction," Nature (London) 426, 404 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Spectrosc. (1)

R. A. Silin, "Possibility of creating plane-parallel lenses," Opt. Spectrosc. 44, 109 (1978).

Phys. Rev. B (1)

C. Luo, S. G. Johnson, and J. D. Joannopoulos, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

Phys. Rev. B (1)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "Subwavelength imaging in photonic crystals," Phys. Rev. B 68, 045115 (2003).
[CrossRef]

Phys. Rev. E (1)

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, "Gradient index metamaterials," Phys. Rev. E 71, 036609 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, "Electromagnetic wave chaos in gradient refractive index optical cavities," Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

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

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, "Negative refraction and left-handed electromagnetism in microwave photonic crystals," Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef] [PubMed]

Y. Zhang, B. Fluegel, and A. Mascarenhas, "Total negative refraction in real crystals for ballistic electrons and light," Phys. Rev. Lett. 91, 157404 (2003).
[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]

Science (2)

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

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

Sov. Phys. Usp. (1)

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

Other (6)

B. Mahon, The man who changed everything: The life of James Clerk Maxwell, John Wiley & Sons 2004.

M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press (2003).

E. W. Marchland, Gradient Index Optics, Academic Press, New York (1978).

Note that the Veselago-Pendry lens also operates at a single frequency for which n = ε= µ= -1.

L. Landau and I.M. Lifshitz, Electrodynamics of continuous media, 2nd Ed. (Elsevier 1984).

For a PhC with negative group refraction,κ is always nonzero. For example for the second band, κ2~ 2π/a instead of κ~ 0.

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Figures (4)

Fig. 1.
Fig. 1.

Wave vector refraction for (a) a single interface between the vacuum and an ENGRM and (b) an ENGRM slab.

Fig. 2.
Fig. 2.

Group velocity for an ENGRM with elliptic EFS and (kr - κ ) ∙ ∇kr ω < 0 and (a) negative or (b) positive wave vector refraction. The dashed arrow is kr - κ . (c) Ray diagram of the group velocity for imaging by an ENGRM flat lens with u + v = σd.

Fig. 3.
Fig. 3.

Field intensity of two sources (J 0(k 0 r)) imaged by an ENGRM flat lens with σ = 0.5, κ = 0, and thickness d = 10. The two white lines indicate the surfaces of the lens. We set μx = -0.5 for the ENGRM for perfect transmission.

Fig. 4.
Fig. 4.

(a) EFS of the TE modes of a square lattice PhC calculated using plane wave expansion. (b) Far-field Hz of a point source across an ENGRM slab with dispersion relation krz = κ - κkz with κ = 3.2465 and σ = 0.08 which approximates the actual EFS of the PhC (dashed line in the left panel). The permittivity εx = 2.2 is used. Note the modulated field inside the homogeneous ENGRM. Due to impedance mismatch, the images are not “perfect” unlike Fig. 3.

Equations (13)

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u = σv
σ = d k rz d k z .
k rz = κ σ k z .
σ 2 ( κ k rz ) 2 + k rx 2 = ω 0 2 c 2 .
( k r κ z ̂ ) · k r ω < 0
tan β = σ tan θ .
u + v = σd ,
μ x = κ k 0 cos θ σ .
E far t ( r ) = i exp ( iκd ) 0 π 2 d θ 2 π exp { i k z [ z + u ( 1 + σ ) d ] } cos ( k x x ) .
E far in ( r ) = i exp ( iκz ) 0 π 2 d θ 2 π exp [ i k z ( σz u ) ] cos ( k x x )
k rz = κ + iσq
μ x = μ x 0 σ + q .
μ x = μ x ± ( σ + q ) tanh 1 ( σq ) d 2 .

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