Abstract

Propagation of light in a negative dispersion regime (antiparallel phase and group velocities) may be attributed to either fast light or a backward wave. We show that by applying causality, only one of these is valid for each scenario. A nanoplasmonic structure is shown to support both types of solution depending on the parameters.

©2009 Optical Society of America

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References

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  1. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48(1), R34–R37 (1993).
    [Crossref] [PubMed]
  2. G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
    [Crossref] [PubMed]
  3. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
    [Crossref] [PubMed]
  4. R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” Prog. Opt. 43, 497–530 (2002).
    [Crossref]
  5. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [Crossref]
  6. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
    [Crossref] [PubMed]
  7. M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
    [Crossref]
  8. V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).
  9. V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Physics Usp. 49(10), 1029–1044 (2006).
    [Crossref]
  10. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003).
    [Crossref]
  11. B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44(24), 13556–13572 (1991).
    [Crossref]
  12. E. Feigenbaum and M. Orenstein, “Modeling of complementary (void) plasmon waveguiding,” J. Lightwave Technol. 25(9), 2547–2562 (2007).
    [Crossref]
  13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
    [Crossref] [PubMed]
  14. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
    [Crossref] [PubMed]
  15. A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007).
    [Crossref]
  16. B. E. Sernelius, Surface modes in physics, 1st ed. (Wiley, 2001), Chap. 4.
  17. A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33(1), 231–233 (1962).
    [Crossref]
  18. A. Alú and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23(3), 571–583 (2006).
    [Crossref]
  19. E. D. Palik, Handbook of optical constants of solids, 2nd ed. (Academic, 1998).

2007 (4)

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[Crossref]

E. Feigenbaum and M. Orenstein, “Modeling of complementary (void) plasmon waveguiding,” J. Lightwave Technol. 25(9), 2547–2562 (2007).
[Crossref]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007).
[Crossref]

2006 (6)

A. Alú and N. Engheta, “Optical nanotransmission lines: synthesis of planar left-handed metamaterials in the infrared and visible regimes,” J. Opt. Soc. Am. B 23(3), 571–583 (2006).
[Crossref]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[Crossref] [PubMed]

V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Physics Usp. 49(10), 1029–1044 (2006).
[Crossref]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

2003 (1)

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003).
[Crossref]

2002 (1)

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

1993 (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48(1), R34–R37 (1993).
[Crossref] [PubMed]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44(24), 13556–13572 (1991).
[Crossref]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

1962 (1)

A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33(1), 231–233 (1962).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Physics Usp. 49(10), 1029–1044 (2006).
[Crossref]

Alú, A.

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Barsi, C.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

Boyd, R. W.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

Braginsky, L.

V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Chiao, R. Y.

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48(1), R34–R37 (1993).
[Crossref] [PubMed]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Dolling, G.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

Engheta, N.

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[Crossref] [PubMed]

Feigenbaum, E.

Gartstein, Y. N.

V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Physics Usp. 49(10), 1029–1044 (2006).
[Crossref]

Gauthier, D. J.

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

Gehring, G. M.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

Govyadinov, A. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007).
[Crossref]

Hafner, Ch.

V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Kostinski, N.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44(24), 13556–13572 (1991).
[Crossref]

Noginov, M. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007).
[Crossref]

Oliner, A. A.

A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33(1), 231–233 (1962).
[Crossref]

Orenstein, M.

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Podolskiy, V. A.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007).
[Crossref]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44(24), 13556–13572 (1991).
[Crossref]

Schweinsberg, A.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[Crossref] [PubMed]

Shklover, V.

V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Shvets, G.

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003).
[Crossref]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[Crossref]

Tamir, T.

A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33(1), 231–233 (1962).
[Crossref]

Veselago, V.

V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44(24), 13556–13572 (1991).
[Crossref]

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91(19), 191103 (2007).
[Crossref]

J. Appl. Phys. (1)

A. A. Oliner and T. Tamir, “Backward waves on isotropic plasma slabs,” J. Appl. Phys. 33(1), 231–233 (1962).
[Crossref]

J. Comput. Theor. Nanosci. (1)

V. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Phys. Rev. A (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48(1), R34–R37 (1993).
[Crossref] [PubMed]

Phys. Rev. B (2)

G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B 67(3), 035109 (2003).
[Crossref]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44(24), 13556–13572 (1991).
[Crossref]

Phys. Rev. Lett. (2)

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[Crossref] [PubMed]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98(17), 177404 (2007).
[Crossref]

Physics Usp. (1)

V. M. Agranovich and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Physics Usp. 49(10), 1029–1044 (2006).
[Crossref]

Prog. Opt. (1)

R. W. Boyd and D. J. Gauthier, “‘Slow’ and ‘Fast’ Light,” Prog. Opt. 43, 497–530 (2002).
[Crossref]

Science (4)

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[Crossref] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[Crossref] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

Other (2)

E. D. Palik, Handbook of optical constants of solids, 2nd ed. (Academic, 1998).

B. E. Sernelius, Surface modes in physics, 1st ed. (Wiley, 2001), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic presentation of a negative dispersion curve. Block blue arrows, group direction; block red arrows, phase direction. (b) Schematic roots in the complex plane of modal propagation constant for negative dispersion. Green points on the real axis indicate the solution pair for the lossless case, and the red/blue arrows indicate the revolution of the roots into the causal fast-light/backward-wave quadrants.
Fig. 2
Fig. 2 Plane wave dispersion for a bulk material having equal dielectric constant and magnetic permeability, each with a Lorentian dispersion. (a) Stationary solutions of the real effective index (nr) presented in the positive quadrant (energy propagation direction is not determined). (b) Causal solution for a source located at -∞ [imaginary part of the effective index (ni) overlay by dotted line; the resonance frequency is at the peak of ni(ω)].
Fig. 3
Fig. 3 (a) A plasmonic-gap waveguide scheme with magnetic-field amplitude of TM0 and TM1 modes. (b) Non-causal dispersion curves of (lossless) gold–silicon–gold plasmonic gap for TM0 and TM1 at different gap widths. Textbook dispersion is curve shown in the first quadrant (positive n eff) marked by the bold blue frame. (c) Poynting vector in the propagation direction (z) for the negative index TM1. (d) Causal normalized dispersion curve for energy propagating to + z. Stars (yellow, green) for the (30nm, 20nm) gaps indicating the cutoff transition (zero group velocity) between forward and backward waves.
Fig. 4
Fig. 4 Causal complex normalized dispersion curves for actual gold–silicon–gold plasmonic gap with gap thickness d = 30nm. Red, green ellipses indicating regions of interest, as discussed in the text. (a) TM1, (b) TM0.
Fig. 5
Fig. 5 Energy velocity and power flow ratios for TM1 and TM0 modes of the plasmonic gap waveguide in Fig. 4. Red ellipses are related to those of Fig. 4; (a) TM1 and (c) TM0 ratio of total power (z direction) in the dielectric to that in the metallic cladding, (b) TM1 and (d) TM0 ratio of energy velocity to group velocity.

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