Abstract

The transmission-line matrix (TLM) method is shown to be an effective technique for calculating the propagation of electromagnetic waves in dielectric arrays exhibiting photonic band gaps. The simulator is particularly valuable for modeling processes whose parameters of interest are intrinsically in the time domain. To illustrate the capabilities of the technique for such temporal problems, the tunneling of electromagnetic pulses through the forbidden gap of a two-dimensional dielectric array is simulated. As researchers have recently measured in one-dimensional systems, pulses tunnel with a group velocity that exceeds the speed of light in vacuum. The TLM simulations show directly that for this superluminal tunneling process the group velocity continues to be a valid descriptor of the peak of the pulse. Furthermore, the results show that once sufficient periodicity exists to create a forbidden gap the tunneling time is constant independent of the thickness of the photonic crystal through which tunneling occurs.

© 1997 Optical Society of America

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    [CrossRef] [PubMed]
  2. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
    [CrossRef]
  3. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
    [CrossRef] [PubMed]
  4. G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
    [CrossRef] [PubMed]
  5. K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [CrossRef] [PubMed]
  6. Ze Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
    [CrossRef] [PubMed]
  7. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  8. J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  9. A. M. Steinberg, P. G. Kwait, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
    [CrossRef] [PubMed]
  10. Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
    [CrossRef] [PubMed]
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  15. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
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  24. S. A. Boothroyd, L. Chan, and W. M. Robertson, “Visualizing coherent light with an electromagnetic wave simulator,” IEEE Trans. Educ. 39, 29–39 (1996).
  25. W. M. Robertson, S. A. Boothroyd, and L. Chan, “Photonic band structure calculations using a two-dimensional electromagnetic simulator,” J. Mod. Opt. 41, 285–293 (1994).
    [CrossRef]
  26. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
    [CrossRef] [PubMed]
  27. M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
    [CrossRef] [PubMed]
  28. G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
    [CrossRef]
  29. E. R. Brown and O. B. McMahon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
    [CrossRef]

1995 (3)

S. M. Moniri-Ardakani and E. N. Glytsis, “Application of the transmission line matrix method to the analysis of slab and channel optical waveguides,” Appl. Opt. 34, 2704–2711 (1995).
[CrossRef] [PubMed]

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

E. R. Brown and O. B. McMahon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

1994 (2)

W. M. Robertson, S. A. Boothroyd, and L. Chan, “Photonic band structure calculations using a two-dimensional electromagnetic simulator,” J. Mod. Opt. 41, 285–293 (1994).
[CrossRef]

Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

1993 (2)

A. M. Steinberg, P. G. Kwait, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[CrossRef] [PubMed]

1992 (2)

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

1990 (5)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Ze Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
[CrossRef]

1988 (1)

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

1971 (1)

M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A 4, 2104–2108 (1971).
[CrossRef]

1970 (1)

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
[CrossRef]

Bolda, E. L.

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[CrossRef] [PubMed]

Boothroyd, S. A.

W. M. Robertson, S. A. Boothroyd, and L. Chan, “Photonic band structure calculations using a two-dimensional electromagnetic simulator,” J. Mod. Opt. 41, 285–293 (1994).
[CrossRef]

Bowden, C. M.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Brown, E. R.

E. R. Brown and O. B. McMahon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Chan, L.

W. M. Robertson, S. A. Boothroyd, and L. Chan, “Photonic band structure calculations using a two-dimensional electromagnetic simulator,” J. Mod. Opt. 41, 285–293 (1994).
[CrossRef]

Chiao, R. Y.

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwait, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Crisp, M. D.

M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A 4, 2104–2108 (1971).
[CrossRef]

Dowling, J. P.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

Garrett, C. G. B.

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

Garrison, J. C.

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[CrossRef] [PubMed]

Genack, A. Z.

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

Glytsis, E. N.

Halbout, J.-M.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
[CrossRef]

Haus, J. W.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

John, S.

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

Kopcsay, G. V.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
[CrossRef]

Krausz, F.

Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Kurizki, G.

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

Kwait, P. G.

A. M. Steinberg, P. G. Kwait, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Leung, K. M.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

MacKinnon, A.

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Manka, A. S.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

McCumber, D. E.

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

McMahon, O. B.

E. R. Brown and O. B. McMahon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Moniri-Ardakani, S. M.

Pastol, Y.

G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
[CrossRef]

Pendry, J. B.

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Robertson, W. M.

W. M. Robertson, S. A. Boothroyd, and L. Chan, “Photonic band structure calculations using a two-dimensional electromagnetic simulator,” J. Mod. Opt. 41, 285–293 (1994).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Satpathy, S.

Ze Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Scalora, M.

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Spielmann, Ch.

Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Steinberg, A. M.

A. M. Steinberg, P. G. Kwait, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Stingl, A.

Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Szipocs, R.

Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

Wang, J.

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

Zhang, Ze

Ze Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. R. Brown and O. B. McMahon, “Large electromagnetic stop bands in metallodielectric photonic crystals,” Appl. Phys. Lett. 67, 2138–2140 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

G. Arjavalingam, Y. Pastol, J.-M. Halbout, and G. V. Kopcsay, “Broad-band microwave measurements with transient radiation from optoelectronically pulsed antennas,” IEEE Trans. Microwave Theory Tech. 38, 615–621 (1990).
[CrossRef]

J. Mod. Opt. (1)

W. M. Robertson, S. A. Boothroyd, and L. Chan, “Photonic band structure calculations using a two-dimensional electromagnetic simulator,” J. Mod. Opt. 41, 285–293 (1994).
[CrossRef]

Phys. Rev. A (4)

M. Scalora, J. P. Dowling, A. S. Manka, C. M. Bowden, and J. W. Haus, “Pulse propagation near highly reflective surfaces: application to photonic band-gap structures and the question of superluminal tunneling times,” Phys. Rev. A 52, 726–734 (1995).
[CrossRef] [PubMed]

C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A 1, 305–313 (1970).
[CrossRef]

M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A 4, 2104–2108 (1971).
[CrossRef]

E. L. Bolda, R. Y. Chiao, and J. C. Garrison, “Two theorems for the group velocity in dispersive media,” Phys. Rev. A 48, 3890–3894 (1993).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Photonics bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
[CrossRef]

Phys. Rev. Lett. (10)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Ze Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

A. M. Steinberg, P. G. Kwait, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993).
[CrossRef] [PubMed]

Ch. Spielmann, R. Szipocs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308–2311 (1994).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in two-dimensional dielectric arrays,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Other (10)

A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in Dispergierenden Medien,” Ann. Physik 44, 177–201 (1914).

A. Sommerfeld, “Ein einwand gegen die Relativtheorie der Elektrodynamik und seine Beseitigung,” Physik Z. 8, 841–860 (1907).

L. Brillouin, “Uber die Fortpflanzung des Lichtes in Dispergierenden Medien,” Ann. Physik 44, 203–240 (1914).

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960), pp. 17–42.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 302.

J. Brown, “Faster than the speed of light,” New Scientist, April 1995, pp. 26–30.

W. J. R. Hoefer and P. P. M. So, The Electromagnetic Wave Simulator (Wiley, New York, 1991). This book comes with a PC version of the TLM simulator suitable for exploring many of the features detailed in this paper.

W. J. R. Hoefer, “The Transmission Line Matrix (TLM) Method,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 496–591.

S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett. 48, 738–741 (1982). See also A. Katz and R. R. Alfano, “Comment, pulse propagation in an absorbing medium,” Phys. Rev. Lett. 49, 1292 (1982); and “Reply,” S. Chu and S. Wong, Phys. Rev. Lett. PRLTAO 49, 1293 (1982).
[CrossRef]

S. A. Boothroyd, L. Chan, and W. M. Robertson, “Visualizing coherent light with an electromagnetic wave simulator,” IEEE Trans. Educ. 39, 29–39 (1996).

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of a single node of the stub-loaded shunt network used to create a TLM mesh.

Fig. 2
Fig. 2

Schematic representation of the TLM mesh. For visual clarity, the figure shows a much shorter length of TLM mesh than was used in the actual simulations. The figure is not to scale; see text for the actual simulation parameters used. The magnetic walls make the array (of only two rods in this case) appear to be of infinite extent perpendicular to the direction of propagation of the electromagnetic pulse.

Fig. 3
Fig. 3

Transmission characteristic of a finite slab of photonic crystal eight rows of rods thick (solid curve). The fundamental photonic band gap can be seen to cover the range 38–56 GHz, and a second band gap appears centered at 82 GHz. Overlaid on the figure (dotted curve) is the frequency spectrum of the Gaussian pulse used for the tunneling simulations.

Fig. 4
Fig. 4

(a) Received signal for tunneling of the Gaussian pulse through vacuum (upper curve) and through a photonic crystal four rows thick (lower curve). (b) Received signal for tunneling of the Gaussian pulse through vacuum (upper curve) and through a photonic crystal eight rows thick (lower curve).

Fig. 5
Fig. 5

Effective group velocity versus the thickness of the photonic crystal. The points are the values extracted from the simulation results; the curve is a guide for the eye.

Fig. 6
Fig. 6

Tunneling time versus the thickness of the photonic crystal. The points are the values extracted from the simulation results; the curve is a guide for the eye.

Fig. 7
Fig. 7

Plot of the phase of the received pulse as a function of frequency for tunneling through slabs of photonic crystal between one and eight rows thick. The phase values are extracted from the simulation results as described in the text. For the cases of three to eight rows of rods, the dispersion is anomalous as indicated by the positive slope of the phase-versus-frequency curve. The Gaussian frequency components that make up the tunneling pulse are indicated by the solid curve referenced to the left-hand axis.

Tables (1)

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Table 1 Table of Numerical Values Used To Confirm the Continued Validity of the Group Velocitya

Equations (7)

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

vg=dωdk=cn+ωdndω,
V1V2V3V4V5n+1r
=1y2-y222222-y222222-y222222-y22y02y02y02y02y0-y×V1V2V3V4V5ni,
2Vyx2+2Vyz2=2LC1+y042Vyt2,
s(t)=A exp(iω0t)exp-tt02.
vg=cn+fdndf.
n(f)=cϕ(f)2πfT+1,

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