Abstract

We study the transient behaviors in focusing a point source with a metamaterial lens (ϵ=μ=1+δ). We find the time evolutions of image formation are dictated by two relaxation mechanisms, namely, surface wave transport and absorption, determined, respectively, by Re(δ) and Im(δ). We show that image oscillations are inevitable in this three-dimensional configuration when Re(δ)0 and establish relationships among the relaxation time, the resolution enhancement, and δ.

© 2005 Optical Society of America

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  1. V. C. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  2. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  3. R. W. Ziolkowski and E. Heyman, Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  4. X. S. Rao and C. K. Ong, Phys. Rev. B 68, 113103 (2003).
    [CrossRef]
  5. X. S. Rao and C. K. Ong, Phys. Rev. E 68, 067601 (2003).
    [CrossRef]
  6. S. A. Cummer, Appl. Phys. Lett. 82, 1503 (2003).
    [CrossRef]
  7. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
    [CrossRef]
  8. L. Chen, S. L. He, and L. F. Shen, Phys. Rev. Lett. 92, 107404 (2004).
    [CrossRef]
  9. N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, 207403 (2002).
    [CrossRef]
  10. J. B. Pendry, Phys. Rev. Lett. 91, 099701 (2003).
    [CrossRef]
  11. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
    [CrossRef]
  12. R. Merlin, Appl. Phys. Lett. 84, 1290 (2004).
    [CrossRef]
  13. Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
    [CrossRef]
  14. G. Gomez-Santos, Phys. Rev. Lett. 90, 077401 (2003).
    [CrossRef]
  15. L. Zhou and C. T. Chan, Appl. Phys. Lett. 86, 101104 (2005).
    [CrossRef]
  16. The peak heights are finite because ? and ? are finite.
  17. R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
    [CrossRef]
  18. L. Zhou and C. T. Chan, Appl. Phys. Lett. 84, 1444 (2004).
    [CrossRef]
  19. B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
    [CrossRef]
  20. We restrict our attention to the simple case ?=?. There could be other choices for solving the field divergence problems, such as ?=?1+?, ?=1?(?1+?), ?=?1, ?=?1+?, and ?=?1+?, ?=?1. The time evolutions may be even more complex in those situations.
  21. We have chosen other criteria to define tR and have found that the functional relations do not change although the absolute values of tR change.

2005 (1)

L. Zhou and C. T. Chan, Appl. Phys. Lett. 86, 101104 (2005).
[CrossRef]

2004 (3)

R. Merlin, Appl. Phys. Lett. 84, 1290 (2004).
[CrossRef]

L. Chen, S. L. He, and L. F. Shen, Phys. Rev. Lett. 92, 107404 (2004).
[CrossRef]

L. Zhou and C. T. Chan, Appl. Phys. Lett. 84, 1444 (2004).
[CrossRef]

2003 (8)

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, Phys. Rev. B 68, 113103 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, Phys. Rev. E 68, 067601 (2003).
[CrossRef]

S. A. Cummer, Appl. Phys. Lett. 82, 1503 (2003).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

J. B. Pendry, Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

G. Gomez-Santos, Phys. Rev. Lett. 90, 077401 (2003).
[CrossRef]

2002 (2)

Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef]

2001 (2)

R. W. Ziolkowski and E. Heyman, Phys. Rev. E 64, 056625 (2001).
[CrossRef]

R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
[CrossRef]

2000 (1)

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

1968 (1)

V. C. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Chan, C. T.

L. Zhou and C. T. Chan, Appl. Phys. Lett. 86, 101104 (2005).
[CrossRef]

L. Zhou and C. T. Chan, Appl. Phys. Lett. 84, 1444 (2004).
[CrossRef]

Chen, L.

L. Chen, S. L. He, and L. F. Shen, Phys. Rev. Lett. 92, 107404 (2004).
[CrossRef]

Cummer, S. A.

S. A. Cummer, Appl. Phys. Lett. 82, 1503 (2003).
[CrossRef]

Forester, D. W.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef]

Gomez-Santos, G.

G. Gomez-Santos, Phys. Rev. Lett. 90, 077401 (2003).
[CrossRef]

Grzegorczyk, T. M.

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

He, S. L.

L. Chen, S. L. He, and L. F. Shen, Phys. Rev. Lett. 92, 107404 (2004).
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Kong, J. A.

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

Loschialpo, P. F.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

Merlin, R.

R. Merlin, Appl. Phys. Lett. 84, 1290 (2004).
[CrossRef]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef]

Ong, C. K.

X. S. Rao and C. K. Ong, Phys. Rev. B 68, 113103 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, Phys. Rev. E 68, 067601 (2003).
[CrossRef]

Pendry, J. B.

J. B. Pendry, Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Rachford, F. J.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

Ramakrishna, S. A.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

Rao, X. S.

X. S. Rao and C. K. Ong, Phys. Rev. B 68, 113103 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, Phys. Rev. E 68, 067601 (2003).
[CrossRef]

Rosenbluth, M.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

Ruppin, R.

R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
[CrossRef]

Schelleng, J.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

Shen, L. F.

L. Chen, S. L. He, and L. F. Shen, Phys. Rev. Lett. 92, 107404 (2004).
[CrossRef]

Smith, D. L.

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

Smith, D. R.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

Veselago, V. C.

V. C. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Wu, B. I.

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

Zhang, Y.

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

Zhou, L.

L. Zhou and C. T. Chan, Appl. Phys. Lett. 86, 101104 (2005).
[CrossRef]

L. Zhou and C. T. Chan, Appl. Phys. Lett. 84, 1444 (2004).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Appl. Phys. Lett. (5)

S. A. Cummer, Appl. Phys. Lett. 82, 1503 (2003).
[CrossRef]

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
[CrossRef]

R. Merlin, Appl. Phys. Lett. 84, 1290 (2004).
[CrossRef]

L. Zhou and C. T. Chan, Appl. Phys. Lett. 86, 101104 (2005).
[CrossRef]

L. Zhou and C. T. Chan, Appl. Phys. Lett. 84, 1444 (2004).
[CrossRef]

Electromagn. Waves (1)

Y. Zhang, T. M. Grzegorczyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

J. Appl. Phys. (1)

B. I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, J. Appl. Phys. 93, 9386 (2003).
[CrossRef]

J. Phys. Condens. Matter (1)

R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
[CrossRef]

Phys. Rev. B (1)

X. S. Rao and C. K. Ong, Phys. Rev. B 68, 113103 (2003).
[CrossRef]

Phys. Rev. E (3)

X. S. Rao and C. K. Ong, Phys. Rev. E 68, 067601 (2003).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, and J. Schelleng, Phys. Rev. E 67, 025602(R) (2003).
[CrossRef]

R. W. Ziolkowski and E. Heyman, Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett. (5)

G. Gomez-Santos, Phys. Rev. Lett. 90, 077401 (2003).
[CrossRef]

L. Chen, S. L. He, and L. F. Shen, Phys. Rev. Lett. 92, 107404 (2004).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, 207403 (2002).
[CrossRef]

J. B. Pendry, Phys. Rev. Lett. 91, 099701 (2003).
[CrossRef]

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. C. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other (3)

The peak heights are finite because ? and ? are finite.

We restrict our attention to the simple case ?=?. There could be other choices for solving the field divergence problems, such as ?=?1+?, ?=1?(?1+?), ?=?1, ?=?1+?, and ?=?1+?, ?=?1. The time evolutions may be even more complex in those situations.

We have chosen other criteria to define tR and have found that the functional relations do not change although the absolute values of tR change.

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Figures (4)

Fig. 1
Fig. 1

Time evolution of E (in units of μ 0 P 0 ) at the image point for lenses with several values of Re ( δ ) at the working frequency. Here Im ( δ ) = 0.0004 and d = 10 mm are fixed.

Fig. 2
Fig. 2

Time evolution of E (in units of μ 0 P 0 ) measured at the image point for lenses with three values of γ (in units of gigahertz) and thicknesses d (in units of millimeters). Re ( δ ) is fixed as 0.004 . Open circles, results with only propagating components included for a lens with δ = 0.08 + 0.0002 i and d = 10 mm .

Fig. 3
Fig. 3

(a) Calculated values of E ( ω ) ω 0 ( ω ω 0 + i η ) as functions of frequency for several values of d. The SW spectra (TE and TM modes are degenerate) and the corresponding group velocity are shown in (b) and (c), respectively, for two values of d. In (c), only the branch with a lower group velocity (higher k ) is considered when f > 10 GHz .

Fig. 4
Fig. 4

(a) Calculated (symbols) resolution enhancement R as a function of γ, fitted by a logarithmic relation (solid line). (b) Relaxation time t R in units of 2 d c (symbols) as a function of R achieved by a lens; solid line, best fit with ln ( t R ) = a + b R , where a and b are constants.

Equations (4)

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E ( r , t ) = 1 2 π d ω exp ( i ω t ) E ( r , ω ) ω ω ω 0 + i η ,
E y 3 D ( x , z ; ω ) = i μ 0 P 0 8 π 0 1 k 0 z exp ( i k 0 z z ) { T TE ( k ) [ J 0 ( k x ) J 2 ( k x ) ] + k 0 z 2 k 2 T TM ( k ) [ J 0 ( k x ) + J 2 ( k x ) ] } k d k ,
E ( t ) E av ( t ) + E osc sin ( ω ¯ t + ϕ ) exp ( β t ) ,
E ( t ) = E 0 exp ( i ω 0 t ) + E 2 exp ( i ω 1 t ) E 0 + E osc sin ( ω ¯ t + ϕ ) ,

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