Abstract

We investigate the transfer function of the discretized perfect lens in finite-difference time-domain (FDTD) and transfer matrix method (TMM) simulations; the latter allow to eliminate the problems associated with the explicit time dependence in FDTD simulations. We also find that the finite discretization mesh acts like imaginary deviations from μ=ε=1 and leads to a crossover in the transfer function from constance to exponential decay around k,max limiting the attainable super-resolution. We propose a simple qualitative model to describe the impact of the discretization. k,max is found to depend logarithmically on the mesh constant in qualitative agreement with the TMM simulations.

© 2006 Optical Society of America

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  1. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  2. S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
    [CrossRef]
  3. D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
    [CrossRef]
  4. R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  5. N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
    [CrossRef]
  6. S. A. Cummer, "Simulated causal subwavelength focusing by a negative refractive index slab," Appl. Phys. Lett. 82, 1503-1505 (2003).
    [CrossRef]
  7. X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003).
    [CrossRef]
  8. X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
    [CrossRef]
  9. R. Ruppin, "Surface polaritons of a left-handed material slab," J. Phys. Condens. Matter 13, 1811-1818 (2001).
    [CrossRef]
  10. G. Gómez-Santos, "Universal features of the time evolution of evanescent modes in a left-handed perfect lens," Phys. Rev. Lett. 90, 077401 (2003).
    [CrossRef] [PubMed]
  11. F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," eprint, cond-mat/0206420 (2002).
  12. P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
    [CrossRef]
  13. A. Taflove and S. C. Hagness, Computational Electrodynamics—The Finite Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

2003 (6)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

S. A. Cummer, "Simulated causal subwavelength focusing by a negative refractive index slab," Appl. Phys. Lett. 82, 1503-1505 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
[CrossRef]

G. Gómez-Santos, "Universal features of the time evolution of evanescent modes in a left-handed perfect lens," Phys. Rev. Lett. 90, 077401 (2003).
[CrossRef] [PubMed]

2002 (2)

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

2001 (2)

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

R. Ruppin, "Surface polaritons of a left-handed material slab," J. Phys. Condens. Matter 13, 1811-1818 (2001).
[CrossRef]

2000 (1)

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

Cummer, S. A.

S. A. Cummer, "Simulated causal subwavelength focusing by a negative refractive index slab," Appl. Phys. Lett. 82, 1503-1505 (2003).
[CrossRef]

Fang, N.

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

Gómez-Santos, G.

G. Gómez-Santos, "Universal features of the time evolution of evanescent modes in a left-handed perfect lens," Phys. Rev. Lett. 90, 077401 (2003).
[CrossRef] [PubMed]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics—The Finite Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Haldane, F. D.

F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," eprint, cond-mat/0206420 (2002).

Heyman, E.

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Markos, P.

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

Ong, C. K.

X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
[CrossRef]

Pendry, J. B.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

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

Ramakrishna, S. A.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

Rao, X. S.

X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003).
[CrossRef]

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," 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, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

Ruppin, R.

R. Ruppin, "Surface polaritons of a left-handed material slab," J. Phys. Condens. Matter 13, 1811-1818 (2001).
[CrossRef]

Schultz, S.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

Smith, D. R.

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

Soukoulis, C. M.

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics—The Finite Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Zhang, X.

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Appl. Phys. Lett. (3)

D. R. Smith, D. Schurig, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, "Limitations on subdiffraction imaging with a negative refractive index slab," Appl. Phys. Lett. 82, 1506-1508 (2003).
[CrossRef]

N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003).
[CrossRef]

S. A. Cummer, "Simulated causal subwavelength focusing by a negative refractive index slab," Appl. Phys. Lett. 82, 1503-1505 (2003).
[CrossRef]

J. Mod. Opt. (1)

S. A. Ramakrishna, J. B. PendryD. Schurig, D. R. Smith, and S. Schultz, "The asymmetric lossy near-perfect lens," J. Mod. Opt. 49, 1747-1762 (2002).
[CrossRef]

J. Phys. Condens. Matter (1)

R. Ruppin, "Surface polaritons of a left-handed material slab," J. Phys. Condens. Matter 13, 1811-1818 (2001).
[CrossRef]

Phys. Rev. B (1)

X. S. Rao and C. K. Ong, "Amplification of evanescent waves in a lossy left-handed material slab," Phys. Rev. B 68, 113103 (2003).
[CrossRef]

Phys. Rev. E (3)

X. S. Rao and C. K. Ong, "Subwavelength imaging by a left-handed material superlens," Phys. Rev. E 68, 067601 (2003).
[CrossRef]

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
[CrossRef]

P. Markos and C. M. Soukoulis, "Numerical studies of left-handed materials and arrays of split ring resonators," Phys. Rev. E 65, 036622 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

G. Gómez-Santos, "Universal features of the time evolution of evanescent modes in a left-handed perfect lens," Phys. Rev. Lett. 90, 077401 (2003).
[CrossRef] [PubMed]

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

Other (2)

F. D. M. Haldane, "Electromagnetic surface modes at interfaces with negative refractive index make a 'not-quite-perfect' lens," eprint, cond-mat/0206420 (2002).

A. Taflove and S. C. Hagness, Computational Electrodynamics—The Finite Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

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Figures (3)

Fig. 1
Fig. 1

(Color online) The distribution of the electric field in TE mode is shown from source ( z = 0 ) to focus ( z = 12 ) across the lossless (a) and lossy (b) perfect lens for successively finer discretization. The left-hand slab between the interfaces at z = 3 and z = 9 has μ = ε = 1 for the lossless PL and μ = ε = 1 + γ with γ = 0.03 i for the lossy PL. We discretized using a uniform cubic mesh with a linear resolution ranging from 214 to 12857 mesh points per vacuum-wavelength. The curves for 12857 mesh points practically coincide with the expected analytical result.

Fig. 2
Fig. 2

(Color online) The transfer function from source to focus is shown for the lossy PL (symbols) with μ = ε = 1 + γ for two different small imaginary parts γ = 0.002 i (a) and γ = 0.005 i (b) and different discretizations. The dashed lines show the corresponding transfer function for the lossless PL. We discretized using a uniform cubic mesh with a linear resolution ranging from 643 to 51429 mesh points per vacuum-wavelength. The additional rightmost dashed line corresponds to 102858 λ .

Fig. 3
Fig. 3

(Color online) The logarithmic scaling of the crossover parallel momentum k , max with the number of linear mesh points per vacuum-wavelength is shown for the lossless ( γ = 0 ) and two lossy PLs with γ = 0.002 i and γ = 0.005 i . The dotted line is a fit k , max d = ( 24 27 ) log ( 4 λ N mesh ) for the lossless PL.

Equations (8)

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E focus ( k , t ) = d ω t ( k , ω ) E source ( k ) g ( ω , ω 0 ) exp ( i ω t ) ,
S TE = Im ( A B * ) i k ω μ + k ω μ { A 2 2 exp [ 2 Im ( k r ) ] + B 2 2 exp [ + 2 Im ( k r ) ] + Re ( A B * ) }
τ imaging = τ 0 ( b ) [ τ 3 ( δ ) τ 2 ( d ) τ 1 ( δ ) ] τ 0 ( a ) .
τ i ( d ) = ( α i ( d ) β i ( d ) β i ( d ) α i ( d ) )
α i ( d ) = cos ( k i d ) + i 2 ( ζ i + 1 ζ i ) sin ( k i d ) ,
β i ( d ) = i 2 ( ζ i 1 ζ i ) sin ( k i d ) .
t = [ 1 + o ( δ ) ] exp [ i k 0 ( a + b ) ] cos ( k 2 d ) + [ 1 + δ 2 ( 2 ω 4 ω 2 k 2 ) ] i sin ( k 2 d ) .
k max d = W ( ω 2 δ d ) log δ ,

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