Abstract

We propose an approach to optical imaging beyond the diffraction limit, based on transformation optics in concentric circular cylinder domains. The resulting systems allow image magnification and minimize reflection losses due to the impedance matching at the input or output boundaries. While perfect impedance matching at both surfaces can be obtained only in a system with radius-dependent magnetic permeability, we demonstrate that comparable performance can be achieved in an optimized nonmagnetic design.

© 2007 Optical Society of America

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References

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
    [CrossRef] [PubMed]
  2. V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356 (2005).
    [CrossRef]
  3. N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, Nat. Mater. 5, 452 (2006).
    [CrossRef] [PubMed]
  4. N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
    [CrossRef] [PubMed]
  5. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
    [CrossRef] [PubMed]
  6. A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas 24, 413 (2003).
    [CrossRef] [PubMed]
  7. J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
    [CrossRef] [PubMed]
  8. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
    [CrossRef] [PubMed]
  9. Z. Jacob, L. V. Alekseyev, and E. Narimanov, Opt. Express 14, 8247 (2006).
    [CrossRef] [PubMed]
  10. A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
    [CrossRef]
  11. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
    [CrossRef] [PubMed]
  12. I. Smolyaninov, Y. Hung, and C. Davis, Science 315, 1699 (2007).
    [CrossRef] [PubMed]
  13. S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, Phys. Rev. E 74, 036621 (2006).
    [CrossRef]
  14. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).
    [CrossRef]
  15. W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, Appl. Phys. Lett. 91, 111105 (2007).
    [CrossRef]
  16. A. Kildishev and E. Danielsson, Proceedings of the COMSOL Multiphysics User's Conference (COMSOL Multiphysics, 2005).
  17. The validation procedure will be discussed elsewhere.
  18. Note that forcing μz=1 in the set of Eq. without any impedance matching results in about one order of magnitude decrease of the output field intensity.

2007

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. Smolyaninov, Y. Hung, and C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

2006

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, Phys. Rev. E 74, 036621 (2006).
[CrossRef]

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, Nat. Mater. 5, 452 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, Opt. Express 14, 8247 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
[CrossRef]

2005

2003

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas 24, 413 (2003).
[CrossRef] [PubMed]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

Appl. Phys. Lett.

W. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, Appl. Phys. Lett. 91, 111105 (2007).
[CrossRef]

Nat. Mater.

N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, Nat. Mater. 5, 452 (2006).
[CrossRef] [PubMed]

Nat. Photonics

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. Photonics 1, 224 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006).
[CrossRef]

Phys. Rev. E

S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Physiol. Meas

A. Greenleaf, M. Lassas, and G. Uhlmann, Physiol. Meas 24, 413 (2003).
[CrossRef] [PubMed]

Science

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 (2006).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, Science 315, 1686 (2007).
[CrossRef] [PubMed]

I. Smolyaninov, Y. Hung, and C. Davis, Science 315, 1699 (2007).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, Science 308, 534 (2005).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef] [PubMed]

Other

A. Kildishev and E. Danielsson, Proceedings of the COMSOL Multiphysics User's Conference (COMSOL Multiphysics, 2005).

The validation procedure will be discussed elsewhere.

Note that forcing μz=1 in the set of Eq. without any impedance matching results in about one order of magnitude decrease of the output field intensity.

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Figures (4)

Fig. 1
Fig. 1

Transformations of concentric circular domains. (a) Virtual domain (circle A) is mapped onto physical domain (ring B) in a cloaking device. Both domains share the same boundary at ρ = r = b . (b) Mapping of the virtual domain (thin ring A) onto physical domain (thicker ring B) in a hyperlens with shared boundary at ρ = r = a .

Fig. 2
Fig. 2

Numerical simulations of the ideal lens. (a) H-field magnitude generated by test sources in free space. Inset, five test sources at a virtual circumference, ρ = a 30 nm . (b) Imaging of the same sources using an ideal hyperlens with the design rules of Eq. [5] and matched impedance at the internal boundary, ρ = r = a .

Fig. 3
Fig. 3

(a) Near-field imaging transformation of the lens. Dark gray curve, magnitude of the H field illuminating the internal boundary, ρ = 0.6 μ m , in front of the sources; light gray curve, field at the external boundary, ρ = 3 μ m . (b) Dark gray curve (internal), impedance matched at the internal boundary; medium gray curve (external), impedance matched at the external boundary; light gray curve (ideal), same as the light gray curve in (a).

Fig. 4
Fig. 4

Magnitude of the H field calculated with nonmagnetic design rules of Eqs. (6, 7). Light gray curves (orange online) indicate input and output arcs for Fig. 3b. Both designs use a = 600 nm , b = 610 nm , and l = 3 μ m , with the test sources of Fig. 2. (a) Impedance is matched at the internal interface, ρ = a , using the design rules of Eq. (6). (b) Impedance is matched at the external interface, ρ = l , using the rules of Eq. (7).

Equations (7)

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ε ρ ρ 1 ( ρ ε ϕ 1 h m ) + [ k 0 2 ε ρ μ z ( m ρ ) 2 ] h m = 0 .
( ρ r ε ρ 1 r ) r 1 [ ( ρ r ε ϕ r ) r h m ( r ) ] ( r ) + [ k 0 2 [ ε ρ μ z ( ρ r ) 2 ] ( m r ) 2 ] h m = 0 .
r 1 [ r h m ( r ) ] ( r ) + [ k 0 2 ( m r ) 2 ] h m = 0 ,
ε ϕ = ρ r r , ε ρ = 1 ε ϕ = r ( r ρ ) , μ z = r r ρ .
ε ρ = 1 + ( τ b l ) ρ , ε ϕ = 1 ε ρ , μ z = ε ρ τ 2 .
ε ϕ = ρ r , ε ρ = r ( r ρ ) , μ z = 1 ,
ε ϕ = ( ρ r ) ( b l ) , ε ρ = r ( r ρ ) , μ z = 1 .

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