Abstract

The backward-wave phenomenon in an isotropic medium is investigated from a purely wave propagation point of view. The functional form for the index of refraction necessary to produce such behavior is derived using the condition that the phase and group-velocity vectors are antiparallel. It is shown that, in the case considered, the backward-wave propagation can be attained only in a medium where the index is negative. A more general case is then considered where the angle between the phase velocity and group velocity is allowed to vary between 90° and 270°. It is shown that such behavior requires propagation through an anisotropic medium where at least one of the axes has a negative index and the general form of the index along each of the three principal axes is derived. The condition that the group velocity must be positive in the transmission passband is then used to obtain the required minimum dispersion for a medium with negative index for both the isotropic and the anisotropic cases.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
    [CrossRef]
  2. J. Woodley and M. Mojahedi, "Negative group velocity and group delay in left-handed media," Phys. Rev. E 70, 046603 (2004).
    [CrossRef]
  3. D. R. Smith, D. Schuring, and J. B. Pendry, "Negative refraction of modulated electromagnetic waves," Appl. Phys. Lett. 81, 2713-2715 (2002).
    [CrossRef]
  4. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
    [CrossRef]
  5. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  6. M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
    [CrossRef]
  7. E. L. Bolda and R. Y. Chiao, "Two theorems for the group velocity in dispersive media," Phys. Rev. A 48, 3890-3894 (1993).
    [CrossRef] [PubMed]
  8. S. Ramo, Fields and Waves in Communication Electronics (Wiley, 1994).
  9. A. K. Iyer and G. V. Eleftheriades, "Negative refractive index metamaterials supporting 2-D waves," IEEE MTT-S Int. Microwave Symp. Dig. 2, 1067-1070 (2002).
  10. C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip 'LH line'," in 2002 IEEE Antennas and Propagation Symposium (IEEE Press, 2002), pp. 412-415.

2004 (1)

J. Woodley and M. Mojahedi, "Negative group velocity and group delay in left-handed media," Phys. Rev. E 70, 046603 (2004).
[CrossRef]

2002 (3)

D. R. Smith, D. Schuring, and J. B. Pendry, "Negative refraction of modulated electromagnetic waves," Appl. Phys. Lett. 81, 2713-2715 (2002).
[CrossRef]

T. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

A. K. Iyer and G. V. Eleftheriades, "Negative refractive index metamaterials supporting 2-D waves," IEEE MTT-S Int. Microwave Symp. Dig. 2, 1067-1070 (2002).

2001 (1)

V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

2000 (2)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
[CrossRef]

1993 (1)

E. L. Bolda and R. Y. Chiao, "Two theorems for the group velocity in dispersive media," Phys. Rev. A 48, 3890-3894 (1993).
[CrossRef] [PubMed]

Agi, K.

M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
[CrossRef]

Bolda, E. L.

E. L. Bolda and R. Y. Chiao, "Two theorems for the group velocity in dispersive media," Phys. Rev. A 48, 3890-3894 (1993).
[CrossRef] [PubMed]

Caloz, C.

C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip 'LH line'," in 2002 IEEE Antennas and Propagation Symposium (IEEE Press, 2002), pp. 412-415.

Chiao, R. Y.

E. L. Bolda and R. Y. Chiao, "Two theorems for the group velocity in dispersive media," Phys. Rev. A 48, 3890-3894 (1993).
[CrossRef] [PubMed]

Eleftheriades, G. V.

A. K. Iyer and G. V. Eleftheriades, "Negative refractive index metamaterials supporting 2-D waves," IEEE MTT-S Int. Microwave Symp. Dig. 2, 1067-1070 (2002).

T. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Grbic, T.

T. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Ilvonen, S.

V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Itoh, T.

C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip 'LH line'," in 2002 IEEE Antennas and Propagation Symposium (IEEE Press, 2002), pp. 412-415.

Iwai, H.

C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip 'LH line'," in 2002 IEEE Antennas and Propagation Symposium (IEEE Press, 2002), pp. 412-415.

Iyer, A. K.

A. K. Iyer and G. V. Eleftheriades, "Negative refractive index metamaterials supporting 2-D waves," IEEE MTT-S Int. Microwave Symp. Dig. 2, 1067-1070 (2002).

Lindell, V.

V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Malloy, K. J.

M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
[CrossRef]

Mojahedi, M.

J. Woodley and M. Mojahedi, "Negative group velocity and group delay in left-handed media," Phys. Rev. E 70, 046603 (2004).
[CrossRef]

M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Nikoskinen, K. I.

V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Okabe, H.

C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip 'LH line'," in 2002 IEEE Antennas and Propagation Symposium (IEEE Press, 2002), pp. 412-415.

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, D. Schuring, and J. B. Pendry, "Negative refraction of modulated electromagnetic waves," Appl. Phys. Lett. 81, 2713-2715 (2002).
[CrossRef]

Ramo, S.

S. Ramo, Fields and Waves in Communication Electronics (Wiley, 1994).

Schamiloglu, E.

M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Schuring, D.

D. R. Smith, D. Schuring, and J. B. Pendry, "Negative refraction of modulated electromagnetic waves," Appl. Phys. Lett. 81, 2713-2715 (2002).
[CrossRef]

Smith, D. R.

D. R. Smith, D. Schuring, and J. B. Pendry, "Negative refraction of modulated electromagnetic waves," Appl. Phys. Lett. 81, 2713-2715 (2002).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Tretyakov, S. A.

V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Woodley, J.

J. Woodley and M. Mojahedi, "Negative group velocity and group delay in left-handed media," Phys. Rev. E 70, 046603 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

D. R. Smith, D. Schuring, and J. B. Pendry, "Negative refraction of modulated electromagnetic waves," Appl. Phys. Lett. 81, 2713-2715 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Mojahedi, E. Schamiloglu, K. Agi, and K. J. Malloy, "Frequency-domain detection of superluminal group velocity in a distributed Bragg reflector," IEEE J. Quantum Electron. 36, 418-424 (2000).
[CrossRef]

IEEE MTT-S Int. Microwave Symp. Dig. (1)

A. K. Iyer and G. V. Eleftheriades, "Negative refractive index metamaterials supporting 2-D waves," IEEE MTT-S Int. Microwave Symp. Dig. 2, 1067-1070 (2002).

J. Appl. Phys. (1)

T. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, "BW media with negative parameters, capable of supporting backward waves," Microwave Opt. Technol. Lett. 31, 129-133 (2001).
[CrossRef]

Phys. Rev. A (1)

E. L. Bolda and R. Y. Chiao, "Two theorems for the group velocity in dispersive media," Phys. Rev. A 48, 3890-3894 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Woodley and M. Mojahedi, "Negative group velocity and group delay in left-handed media," Phys. Rev. E 70, 046603 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Other (2)

S. Ramo, Fields and Waves in Communication Electronics (Wiley, 1994).

C. Caloz, H. Okabe, H. Iwai, and T. Itoh, "Application of the transmission line theory of left-handed (LH) metamaterials to the realization of a microstrip 'LH line'," in 2002 IEEE Antennas and Propagation Symposium (IEEE Press, 2002), pp. 412-415.

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 (4)

Fig. 1
Fig. 1

Dispersion relation for the index given in Eq. (7).

Fig. 2
Fig. 2

(a) 3D and (b) two-dimensional (2D) k-space diagrams for RHM and LHM two-sheeted hyperboloids. In both cases the media are uniaxial with ϵ x = ϵ y = ϵ s . In the RHM case the parameters are ϵ s = 1 , ϵ z = 2 , and μ r = 1 . The parameters in the LHM case are ϵ s = 1 , ϵ z = 2 , and μ r = 1 . Because the parameters in the LHM case are simply the negatives of those in the RHM case, the k surfaces are identical. The angle between the phase velocity and group velocity for the RHM and LHM cases are shown on the 2D plots.

Fig. 3
Fig. 3

(a) 3D and (b) two-dimensional (2D) k-space diagrams for RHM and LHM one-sheeted hyperboloids. In both cases the media are uniaxial with ϵ x = ϵ y = ϵ s . In the RHM case the parameters are ϵ s = 1 , ϵ z = 2 , and μ r = 1 . The parameters in the LHM case are ϵ s = 1 , ϵ z = 2 , and μ r = 1 . Because the parameters in the LHM case are simply the negatives of those in the RHM case, the k surfaces are identical. The angle between the phase velocity and group velocity for the RHM and LHM cases are shown on the 2D plots.

Fig. 4
Fig. 4

Calculated angle between the phase and group-velocity vectors for the RHM and LHM hyperboloidal k surfaces considered. (a) Two-sheeted hyperboloid (Fig. 2). (b) One-sheeted hyperboloid (Fig. 3).

Tables (1)

Tables Icon

Table 1 Index Seen by Different Polarizations in the Two- and One-Sheeted Hyperboloids

Equations (18)

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

v p = c 0 n ( k ) k ̂ ,
v g = k c 0 k n ( k ) .
( v g x , v g y , v g z ) = ( d x v p x , d y v p y , d z v p z ) ,
v g = d v p .
k c 0 k n ( k ) = d c 0 n ( k ) k ̂ .
n ( k ) = b k 2 ,
ω ( k ) = c 0 k n ( k ) = c 0 b k .
v g i = d i v p i ,
c 0 ( 1 n ( k ) k n ( k ) 2 n ( k ) k i ) k i k = d i c 0 n ( k ) k i k .
( 1 d i ) n ( k ) = k n ( k ) 2 n ( k ) k i .
n ( k ) = α i ( k 2 ) ( 1 d i ) 2 ,
v g = [ c 0 n ( k ) c 0 k n ( k ) 2 n ( k ) k ] k ̂ .
n ( k ) = γ k p ,
p > 1 for γ < 0 ,
p < 1 for γ > 0 .
v g = c 0 b k 2 > 0 .
ϵ ̿ = ϵ 0 [ ϵ s 0 0 0 ϵ s 0 0 0 ϵ z ] ,
μ = μ 0 μ r ,

Metrics