Abstract

We numerically investigate the electromagnetic field radiated by a line source within a perfect lens [Phys. Rev. Lett. 85, 3966 (2000) ] consisting of two orthogonal planes delimiting positive and negative index media. Use of a coordinate transformation [J. Phys. Condens Matter 15, 6345 (2003) ] together with a well-adapted transfer-matrix method permits rigorous calculation of the vector field. We find that two negative corners combine to make a cavity that traps light along closed trajectories. Finally, we numerically show that the field presents some spatial oscillations with a period that is proportional to absorption σ inside the negative materials as 1lnσ and that it is associated with an infinite density of states when σ tends toward 0.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  2. J. B. Pendry and S. A. Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
    [CrossRef]
  3. M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
    [CrossRef]
  4. R. Merlin, Appl. Phys. Lett. 84, 1290 (2004).
    [CrossRef]
  5. B. Gralak, S. Guenneau, and J. B. Pendry are preparing a manuscript to be called “Transfer matrix formalism for point sources radiating in metamaterials.”
  6. D. R. Smith, D. Schuring, M. Rosenbluth, S. Schultz, S. A. Ramakrishna, and J. B. Pendry, Appl. Phys. Lett. 82, 1506 (2003).
    [CrossRef]
  7. S. P. Apell, P. M. Echenique, and R. H. Ritchie, Ultramicroscopy 65, 53 (1996).
    [CrossRef]

2004

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

2003

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

J. B. Pendry and S. A. Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

2002

M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
[CrossRef]

2000

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

1996

S. P. Apell, P. M. Echenique, and R. H. Ritchie, Ultramicroscopy 65, 53 (1996).
[CrossRef]

Apell, S. P.

S. P. Apell, P. M. Echenique, and R. H. Ritchie, Ultramicroscopy 65, 53 (1996).
[CrossRef]

Echenique, P. M.

S. P. Apell, P. M. Echenique, and R. H. Ritchie, Ultramicroscopy 65, 53 (1996).
[CrossRef]

Gralak, B.

B. Gralak, S. Guenneau, and J. B. Pendry are preparing a manuscript to be called “Transfer matrix formalism for point sources radiating in metamaterials.”

Guenneau, S.

B. Gralak, S. Guenneau, and J. B. Pendry are preparing a manuscript to be called “Transfer matrix formalism for point sources radiating in metamaterials.”

Merlin, R.

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

Notomi, M.

M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
[CrossRef]

Pendry, J. B.

J. B. Pendry and S. A. Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

D. R. Smith, D. Schuring, 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]

B. Gralak, S. Guenneau, and J. B. Pendry are preparing a manuscript to be called “Transfer matrix formalism for point sources radiating in metamaterials.”

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

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

Ritchie, R. H.

S. P. Apell, P. M. Echenique, and R. H. Ritchie, Ultramicroscopy 65, 53 (1996).
[CrossRef]

Rosenbluth, M.

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

Schultz, S.

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

Schuring, D.

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

Smith, D. R.

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

Appl. Phys. Lett.

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

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

J. Phys. Condens. Matter

J. B. Pendry and S. A. Ramakrishna, J. Phys. Condens. Matter 15, 6345 (2003).
[CrossRef]

Opt. Quantum Electron.

M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
[CrossRef]

Phys. Rev. Lett.

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

Ultramicroscopy

S. P. Apell, P. M. Echenique, and R. H. Ritchie, Ultramicroscopy 65, 53 (1996).
[CrossRef]

Other

B. Gralak, S. Guenneau, and J. B. Pendry are preparing a manuscript to be called “Transfer matrix formalism for point sources radiating in metamaterials.”

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

(a) Geometry of the 1D problem. (b) Ray diagram for the double corner reflector.

Fig. 2
Fig. 2

Real part of the electric field radiated by a periodic set of line sources in a 1D photonic crystal: The number of layers in one period is M = 4 . The four layers have the same thickness, h 4 . The values of permittivity and permeability are ϵ m = μ m = 1 for m = 1 , 3 and ϵ m = μ m = 1 + i σ for m = 2 , 4 . The harmonic line sources are located at ( x 1 , x 3 ) = ( 0 , h 8 + n h ) , where n Z , and have frequency ω h c = 16 , where c is the speed of light in vacuum. The value of absorption σ is 10 2 in (a) and 10 4 in (b).

Fig. 3
Fig. 3

Inverse spatial period L of field oscillations as a function of absorption σ.

Fig. 4
Fig. 4

Modulus of the (a) electric and (b) magnetic fields radiated by a line source in the presence of the double corner reflector. The absorption in negative media is σ = 10 4 . The harmonic line source is located at ( x 1 , x 3 ) = ( r 0 2 , r 0 2 ) and has a frequency of ω r 0 c = 4 .

Fig. 5
Fig. 5

Constant frequency dispersion diagram at ω h c = 16 in a 1D crystal as described in the caption of Fig. 2, but for σ = 10 6 . (a) Imaginary part of k 3 gives attenuation exp [ 2 π Im ( k 3 ) ] of the field through one period. (b) Real part of k 3 provides its phase shift exp [ 2 i π Re ( k 3 ) ] . If k 1 = ( k 1 2 + k 2 2 ) 1 2 5 , both Im ( k 3 ) and Re ( k 3 ) are close to 0.

Equations (10)

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

F ν ( n h + 0 + ) = F ν ( n h 0 + ) + P ν , ν = ε , μ ,
T ν , m = [ cos ( β m h m ) β m 1 ν m sin ( β m h m ) β m ν m 1 sin ( β m h m ) cos ( β m h m ) ] ,
F ν ( x 3 , m + n h ) = T ν , m F ν ( x 3 , m h m + n h ) , n Z ,
F ν [ ( n + 1 ) h ] = T ν F ν ( n h ) + P ν = F ν ( n h ) ,
F ν ( n h ) = [ I T ν ] 1 P ν , ν = ε , μ .
l = l 0 2 ln ( x 1 2 + x 3 2 r 0 2 ) , Φ = arctan ( x 3 x 1 ) [ π ] ,
ε ̃ i = ε Q 1 Q 2 Q 3 ( Q i ) 2 , μ ̃ i = μ Q 1 Q 2 Q 3 ( Q i ) 2 , i = 1 , 2 , 3 ,
Q 1 = ( x 3 2 + l 0 2 x 1 2 ) 1 2 x 1 2 + x 3 2 , Q 2 = 1 , Q 3 = ( x 1 2 + l 0 2 x 3 2 ) 1 2 x 1 2 + x 3 2 .
ε ̃ 1 = μ ̃ 1 = ε ̃ 3 = μ ̃ 3 = a , ε ̃ 2 = μ ̃ 2 = a ( x 1 2 + x 3 2 ) ,
E ̃ i = Q i E i , H ̃ i = Q i H i , i { 1 , 2 , 3 } ,

Metrics