Abstract

The plane-wave method is modified for tunneling evanescent solutions of Maxwell’s equations with complex k vectors (tunneling modes). In our formulation the imaginary part of the k vectors is not necessarily parallel to the real part of the k vectors. We present computed complex photonic band structures of the simple-cubic (sc) and face-centered-cubic (fcc) lattice structures at the symmetric points, such as X, M, and R for the sc and X, L, W, U, and K for the fcc with various directions of imaginary k vectors. In addition, relations between imaginary k vectors and tunneling modes are examined at points X, M, R, and K. Tunneling electromagnetic modes can mathematically exist in the photonic band gaps among propagating eigenmodes even in an infinite photonic crystal. With the concept of electromagnetic tunnelings in photonic crystals, we explain classically various kinds of important phenomena such as inhibition of the spontaneous emission and localized defect modes. The method developed here for solutions with complex eigenfrequencies and k vectors can be readily extended to media with loss or gain by adoption of complex dielectric constants.

© 1995 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  2. E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
    [Crossref] [PubMed]
  3. S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
    [Crossref] [PubMed]
  4. K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
    [Crossref]
  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 (1990).
    [Crossref] [PubMed]
  6. Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (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 (1990).
    [Crossref] [PubMed]
  8. H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
    [Crossref]
  9. H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” Opt. Soc. Am. B 10, 296 (1993).
    [Crossref]
  10. E. Yablonovitch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Physica B 175, 81 (1991);E. Yablonovitch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
    [Crossref] [PubMed]
  11. G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
    [Crossref]
  12. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
    [Crossref]
  13. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
    [Crossref]
  14. E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
    [Crossref] [PubMed]
  15. K. M. Leung, “Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions,” J. Opt. Soc. Am. B 10, 303 (1993).
    [Crossref]
  16. N. Stefanous, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” Phys. Condens. Matter 4, 7389 (1992).
    [Crossref]
  17. Y. Qiu and K. M. Leung, “Complex band structures and transmission spectra of two-dimensional periodic crystals,” in Modeling and Simulation of Laser Systems III, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2117, 32 (1994).
    [Crossref]
  18. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
    [Crossref]
  19. K. M. Leung, “Plane wave calculation of photonic band structure,” in Classical Waves in Random and Periodic Structures, C. M. Soukoulis, ed., NATO Advanced Research Studies on Localizations and Propagations (Plenum, New York, 1993), p. 269.
  20. J. B. Pendry and A. MacKinnon, “Calculations of photon dispersion relation,” Phys. Rev. Lett. 69, 2772 (1992).
    [Crossref] [PubMed]
  21. T. Suzuki and P. K. L. Yu, “Dispersion relation at point L in the photonic band structure of the face-centered-cubic lattice with active or conductive dielectric media,” J. Opt. Soc. Am. B 12, 538 (1995).
    [Crossref]
  22. S. L. McCall and P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899 (1985).
    [Crossref]
  23. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1983), Chap. 6.
  24. W. C. Chew, Wave and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), p. 295.
  25. For example, C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).
  26. M. Tsuji, H. Shigesawa, and A. A. Oliner, “Simultaneous propagation of both bound and leaky dominant modes on conductor-backed coplanar strips,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1295.
  27. M. Tsuji, H. Shigesawa, and A. A. Oliner, “Dominant mode power leakage from printed-circuit wave guide”Radio Sci. 26, 559–564 (1991).
    [Crossref]
  28. D. Ngheim, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Existence of a leaky dominant mode on microstrip line with an isotropic substrate: theory and measurement,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1291.

1995 (1)

T. Suzuki and P. K. L. Yu, “Dispersion relation at point L in the photonic band structure of the face-centered-cubic lattice with active or conductive dielectric media,” J. Opt. Soc. Am. B 12, 538 (1995).
[Crossref]

1993 (2)

1992 (6)

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

N. Stefanous, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” Phys. Condens. Matter 4, 7389 (1992).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

J. B. Pendry and A. MacKinnon, “Calculations of photon dispersion relation,” Phys. Rev. Lett. 69, 2772 (1992).
[Crossref] [PubMed]

1991 (4)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
[Crossref]

E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Physica B 175, 81 (1991);E. Yablonovitch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref] [PubMed]

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Dominant mode power leakage from printed-circuit wave guide”Radio Sci. 26, 559–564 (1991).
[Crossref]

1990 (5)

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

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 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (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 (1990).
[Crossref] [PubMed]

1989 (1)

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

1987 (1)

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

1985 (1)

S. L. McCall and P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899 (1985).
[Crossref]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
[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 (1990).
[Crossref] [PubMed]

Chew, W. C.

W. C. Chew, Wave and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), p. 295.

Gmitter, T. J.

E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Physica B 175, 81 (1991);E. Yablonovitch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

Haus, J. W.

H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” Opt. Soc. Am. B 10, 296 (1993).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

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 (1990).
[Crossref] [PubMed]

Inguva, R.

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

Jackson, D. R.

D. Ngheim, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Existence of a leaky dominant mode on microstrip line with an isotropic substrate: theory and measurement,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1291.

Joannopoulos, J. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
[Crossref]

Karathanos, V.

N. Stefanous, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” Phys. Condens. Matter 4, 7389 (1992).
[Crossref]

Kittel, C.

For example, C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).

Leung, K. M.

K. M. Leung, “Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions,” J. Opt. Soc. Am. B 10, 303 (1993).
[Crossref]

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

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 (1990).
[Crossref] [PubMed]

Y. Qiu and K. M. Leung, “Complex band structures and transmission spectra of two-dimensional periodic crystals,” in Modeling and Simulation of Laser Systems III, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2117, 32 (1994).
[Crossref]

K. M. Leung, “Plane wave calculation of photonic band structure,” in Classical Waves in Random and Periodic Structures, C. M. Soukoulis, ed., NATO Advanced Research Studies on Localizations and Propagations (Plenum, New York, 1993), p. 269.

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 (1990).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

MacKinnon, A.

J. B. Pendry and A. MacKinnon, “Calculations of photon dispersion relation,” Phys. Rev. Lett. 69, 2772 (1992).
[Crossref] [PubMed]

McCall, S. L.

S. L. McCall and P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899 (1985).
[Crossref]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
[Crossref]

Modinos, A.

N. Stefanous, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” Phys. Condens. Matter 4, 7389 (1992).
[Crossref]

Ngheim, D.

D. Ngheim, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Existence of a leaky dominant mode on microstrip line with an isotropic substrate: theory and measurement,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1291.

Oliner, A. A.

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Dominant mode power leakage from printed-circuit wave guide”Radio Sci. 26, 559–564 (1991).
[Crossref]

D. Ngheim, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Existence of a leaky dominant mode on microstrip line with an isotropic substrate: theory and measurement,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1291.

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Simultaneous propagation of both bound and leaky dominant modes on conductor-backed coplanar strips,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1295.

Pendry, J. B.

J. B. Pendry and A. MacKinnon, “Calculations of photon dispersion relation,” Phys. Rev. Lett. 69, 2772 (1992).
[Crossref] [PubMed]

Platzman, P. M.

S. L. McCall and P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899 (1985).
[Crossref]

Qian, G.-X.

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

Qiu, Y.

Y. Qiu and K. M. Leung, “Complex band structures and transmission spectra of two-dimensional periodic crystals,” in Modeling and Simulation of Laser Systems III, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2117, 32 (1994).
[Crossref]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
[Crossref]

Salehpour, M. R.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Satpathy, S.

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Shigesawa, H.

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Dominant mode power leakage from printed-circuit wave guide”Radio Sci. 26, 559–564 (1991).
[Crossref]

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Simultaneous propagation of both bound and leaky dominant modes on conductor-backed coplanar strips,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1295.

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 (1990).
[Crossref] [PubMed]

Sozuer, H. S.

H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” Opt. Soc. Am. B 10, 296 (1993).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

Stefanous, N.

N. Stefanous, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” Phys. Condens. Matter 4, 7389 (1992).
[Crossref]

Suzuki, T.

T. Suzuki and P. K. L. Yu, “Dispersion relation at point L in the photonic band structure of the face-centered-cubic lattice with active or conductive dielectric media,” J. Opt. Soc. Am. B 12, 538 (1995).
[Crossref]

Tsuji, M.

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Dominant mode power leakage from printed-circuit wave guide”Radio Sci. 26, 559–564 (1991).
[Crossref]

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Simultaneous propagation of both bound and leaky dominant modes on conductor-backed coplanar strips,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1295.

Williams, J. T.

D. Ngheim, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Existence of a leaky dominant mode on microstrip line with an isotropic substrate: theory and measurement,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1291.

Yablonovitch, E.

E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Physica B 175, 81 (1991);E. Yablonovitch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

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

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1983), Chap. 6.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1983), Chap. 6.

Yu, P. K. L.

T. Suzuki and P. K. L. Yu, “Dispersion relation at point L in the photonic band structure of the face-centered-cubic lattice with active or conductive dielectric media,” J. Opt. Soc. Am. B 12, 538 (1995).
[Crossref]

Zhang, Z.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

S. L. McCall and P. M. Platzman, “An optimized π/2 distributed feedback laser,” IEEE J. Quantum Electron. QE-21, 1899 (1985).
[Crossref]

J. Opt. Soc. Am. B (2)

T. Suzuki and P. K. L. Yu, “Dispersion relation at point L in the photonic band structure of the face-centered-cubic lattice with active or conductive dielectric media,” J. Opt. Soc. Am. B 12, 538 (1995).
[Crossref]

K. M. Leung, “Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions,” J. Opt. Soc. Am. B 10, 303 (1993).
[Crossref]

Opt. Soc. Am. B (1)

H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” Opt. Soc. Am. B 10, 296 (1993).
[Crossref]

Phys. Condens. Matter (1)

N. Stefanous, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” Phys. Condens. Matter 4, 7389 (1992).
[Crossref]

Phys. Rev. B (6)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961 (1991).
[Crossref]

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

Phys. Rev. Lett. (8)

J. B. Pendry and A. MacKinnon, “Calculations of photon dispersion relation,” Phys. Rev. Lett. 69, 2772 (1992).
[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 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (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 (1990).
[Crossref] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
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E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
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S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
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E. Yablonovitch and T. J. Gmitter, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380 (1991).
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Physica B (1)

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Physica B 175, 81 (1991);E. Yablonovitch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref] [PubMed]

Radio Sci. (1)

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Dominant mode power leakage from printed-circuit wave guide”Radio Sci. 26, 559–564 (1991).
[Crossref]

Other (7)

D. Ngheim, J. T. Williams, D. R. Jackson, and A. A. Oliner, “Existence of a leaky dominant mode on microstrip line with an isotropic substrate: theory and measurement,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1291.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, New York, 1983), Chap. 6.

W. C. Chew, Wave and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), p. 295.

For example, C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).

M. Tsuji, H. Shigesawa, and A. A. Oliner, “Simultaneous propagation of both bound and leaky dominant modes on conductor-backed coplanar strips,” in Vol. 3 of 1993 IEEE MTT-S International Microwave Symposium Digest Series (Institute of Electrical and Electronics Engineers, New York, 1993), p. 1295.

K. M. Leung, “Plane wave calculation of photonic band structure,” in Classical Waves in Random and Periodic Structures, C. M. Soukoulis, ed., NATO Advanced Research Studies on Localizations and Propagations (Plenum, New York, 1993), p. 269.

Y. Qiu and K. M. Leung, “Complex band structures and transmission spectra of two-dimensional periodic crystals,” in Modeling and Simulation of Laser Systems III, A. D. Schnurr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2117, 32 (1994).
[Crossref]

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Figures (11)

Fig. 1
Fig. 1

(a) Real eigenfrequency versus the imaginary k vector at point X for the sc lattice structure. The horizontal axis ki indicates the y component of the imaginary k vector [i.e. ki = (0, ki, 0)]. The tunneling modes exist up to point Xc because the corresponding imaginary eigenfrequencies remain 0 as shown in (b). The gap at ki = 0 indicates the original mode gap at point X for propagating modes. At point Xc the lower and the upper (branch 1 and branch 2) modes degenerate almost at the midpoint of the mode gap at point X. All k vectors are in units of (2π/a). (b) Imaginary eigenfrequency versus the imaginary k vector at point X for the sc lattice structure. The imaginary eigenfrequencies are 0 up to point Xc in spite of the increase in the magnitude of the imaginary k vector. (c) Real and imaginary k vectors in the first Brillouin zone (in units of 2π/a). k vector k = kr + iki; real k vector kr = (0, 0.5, 0); imaginary k vector ki = (0, ki, 0) = (0, |ki|, 0); magnitude of the imaginary k vector ki = |ki|.

Fig. 2
Fig. 2

(a) Photonic band structure for the sc lattice. NP = 125, a = 1.0, b = 12.25, βυ = 0.81. The shaded area shows a photonic band gap. The photonic band gap appears in the fifth and sixth modes. (b) Tunneling modes at point X. NP = 150. The imaginary k vector ki is in the (0, 1, 0) direction at point X [kr = (0, 0.5, 0)]. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. (c) Tunneling modes at point M. NP = 152. The imaginary k vector ki is in the (0, 1, 0) direction at point M [kr = (0.5, 0.5, 0)]. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. (d) Tunneling modes at point R. NP = 160. The imaginary k vector ki is in the (1, 1, 1) direction at point R [kr = (0.5, 0.5, 0.5)]. The horizontal axis ki indicates the x, y, z component of the imaginary k vector [i.e., ki = (ki, ki, ki)]. (e) Tunneling modes at point R. NP = 160. The imaginary k vector ki is in the (0, 1, 0) direction at point X [kr = (0.5, 0.5, 0.5)]. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. All k vectors are in units of (2π/a).

Fig. 3
Fig. 3

Convergence of the real eigenfrequencies for the sc lattice structure. (a) At point X. The real eigenfrequencies, at point X, are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (0, −0.1, 0). (b) At point M. The real eigenfrequencies at point M are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (0, −0.1, 0). (c) At point R. The real eigenfrequencies at point R are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (−0.1, −0.1, −0.1). All k vectors are in units of (2π/a).

Fig. 4
Fig. 4

(a) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.2) at point X. The imaginary k vector ki is rotated by π rad from (1, 0, 0) to (−1, 0, 0) through the (0, 1, 0) direction with the real k vector fixed at point X, kr = (0, 0.5, 0). The imaginary eigenfrequency is plotted as the imaginary k vector (|ki| = 0.2) is rotated. The points or projected points that intercept the horizontal axis indicate possible directions of the imaginary k vector at that magnitude. All k vectors are in units of (2π/a). (b) Rotation of the imaginary k vector in the first Brillouin zone for the sc lattice structure from ki ‖ (1, 0, 0) to ki ‖ (−1, 0, 0) at point X by rotated angle θ.

Fig. 5
Fig. 5

(a) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.2) at point M. The imaginary k vector ki is rotated by π rad from (1, 1, 0) to (−1, −1, 0) through the (0, 0, 1) direction with the real k vector fixed at point M, kr = (0.5, 0.5, 0). The imaginary eigenfrequency is plotted as the imaginary k vector (|ki| = 0.2) is rotated. The points or projected points that intercept the horizontal axis indicate possible directions of the imaginary k vector at that magnitude. All k vectors are in units of (2π/a). (b) Rotation of the imaginary k vector in the first Brillouin zone for the sc lattice structure from ki ‖ (1, 1, 0) to ki ‖ (−1, −1, 0) at point M by rotated angle θ.

Fig. 6
Fig. 6

(a) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.2) at point R. The imaginary k vector ki is rotated by π rad from (1, 0, 1) to (−1, 0, −1) through the (0, 1, 0) direction with the real k vector fixed at point R, kr = (0.5, 0.5, 0.5). The imaginary eigenfrequency is plotted as the imaginary k vector (|ki| = 0.2) is rotated. The points or projected points that intercept the horizontal axis indicate possible directions of the imaginary k vector at that magnitude. All k vectors are in units of (2π/a). (b) Rotation of the imaginary k vector in the first Brillouin zone for the sc lattice structure from ki ‖ (1, 0, 1) to ki ‖ (−1, 0, −1) at point R by rotated angle θ.

Fig. 7
Fig. 7

(a) Real eigenfrequency versus the imaginary k vector at point X for the fcc lattice structure. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. The lowest four tunneling modes (branch 1) exist up to point Xc1 because the corresponding imaginary eigenfrequencies remain 0, as shown in (b). The seventh and eighth tunneling modes couple to the eleventh and twelfth modes at points Xc2 and Xc3 (branch 3). Between the two points these tunneling modes become unphysical because of their nonzero imaginary eigenfrequencies. The fifth and sixth (branch 2) and ninth and tenth (branch 4) modes decrease monotonically. All k vectors are in units of (2π/a). (b) Imaginary eigenfrequency versus the imaginary k vector at point X for the fcc lattice structure. The imaginary eigenfrequencies for the lowest four tunneling modes are 0 up to point Xc1 in spite of the increase in the magnitude of the imaginary k vectors. Between points Xc1 and Xc3 the imaginary eigenfrequencies for the seventh, eighth, eleventh, and twelfth tunneling modes show a bulge, a temporary nonzero region of the imaginary eigenfrequencies. (c) Real and imaginary k vector in the first Brillouin zone for the fcc lattice structure (in units of 2π/a). k vector, k = kr + iki; real k vector, kr = (0, 1, 0); imaginary k vector, ki = (0, |ki|, 0) = (0, ki, 0); magnitude of the imaginary k vector, |ki| = ki.

Fig. 8
Fig. 8

(a) Photonic band structure for the fcc lattice. NP = 343, a = 1.0, b = 12.25, βυ = 0.74. The shaded area shows a photonic band gap. For the fcc structure with these parameters the photonic band gap appears in the eighth and ninth modes. (b) Tunneling modes at point X. NP = 150. The imaginary k vector ki is in the (0, 1, 0) direction at point X [kr = (0, 1, 0)]. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. (c) Tunneling modes at point L. NP = 120. The imaginary k vector ki is in the (1, 1, 1) direction at point L [kr = (0.5, 0.5, 0.5)]. The horizontal axis ki indicates the x, y, z component of the imaginary k vector [i.e., ki = (ki, ki, ki)]. (d) Tunneling modes at point W. NP = 144. The imaginary k vector ki is in the (0, 1, 0) direction at point W [kr = (1, 0.5, 0)]. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. (e) Tunneling modes at point K. NP = 160. The imaginary k vector ki is in the (0, 1, 0) direction at point K [kr = (0.5, 0.5, 0.5)]. The horizontal axis ki indicates the y component of the imaginary k vector [i.e., ki = (0, ki, 0)]. Tunneling modes at point U behave in the same way as at point K. All k vectors are in units of (2π/a).

Fig. 9
Fig. 9

Convergence of the real eigenfrequency for the fcc lattice structure. (a) At point X. The real eigenfrequencies at point X are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (0, −0.6, 0). (b) At point L. The real eigenfrequencies at point L are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (−0.3, −0.3, −0.3). (c) At point W. The real eigenfrequencies at point W are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (0, −0.3, 0). (d) At point K. The real eigenfrequencies at point K are plotted as the diffraction order corresponding to the plane-wave cutoff (see Table 3) is increased. The imaginary k vector is set to (0, −0.3, 0). All k vectors are in units of (2π/a).

Fig. 10
Fig. 10

(a) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.2) at point X. The imaginary k vector ki is rotated from (1, 0, 0) to (−1, 0, 0) through the (0, 1, 0) direction with the real k vector fixed at point X, kr = (0, 1, 0). The imaginary eigenfrequency is plotted as the imaginary k vector (|ki| = 0.2) is rotated. The points or projected points that intercept the horizontal axis indicate possible directions of the imaginary k vector at that magnitude. (b) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.6) at point X. The imaginary k vector ki is rotated from (1, 0, 0) to (−1, 0, 0) through the (0, 0, 1) direction with the real k vector fixed at point X, kr = (0, 1, 0). The magnitude of the imaginary k vector |ki| is set to 0.6. All k vectors are in units of (2π/a). (c) Rotation of the imaginary k vector in the first Brillouin zone from ki = (1, 0, 0) to ki = (−1, 0, 0) at point X by rotated angle θ.

Fig. 11
Fig. 11

(a) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.1) at point K. The imaginary k vector ki is rotated from (1, 0, 0) to (−1, 0, 0) through the (0, 0, 1) direction with the real k vector fixed at point K (0.75, 0, 0.75). The imaginary eigenfrequency is plotted as the magnitude of the imaginary k vector (|ki| = 0.1) is rotated. (b) Imaginary eigenfrequency versus the rotated angle of the imaginary k vector (|ki| = 0.4) at point K. The imaginary k vector ki is rotated from (−1, 0, 0) to (1, 0, 0) through the (0, 0, 1) direction with the real k vector fixed at point K (0.75, 0.75). The imaginary eigenfrequency is plotted as the imaginary k vector (|ki| = 0.4) is rotated. All k vectors are in units of (2π/a). (c) Rotation of the imaginary k vector in the first Brillouin zone from ki = (1, 0, 0) to ki = (−1, 0, 0) at point K by rotated angle θ.

Tables (3)

Tables Icon

Table 1 Parameters of Computations (sc Lattice Structure)

Tables Icon

Table 2 Parameters of Computations (fcc Lattice Structure)

Tables Icon

Table 3 Diffraction Order and Plane-Wave Cutoff

Equations (45)

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× E ( r ) = i μ ω H ( r ) ,
× H ( r ) = i 0 ( r ) ω E ( r ) ,
· [ ( r ) E ( r ) ] = 0 ,
· H ( r ) = 0 .
E ( r ) = K [ x ̂ E x ( K ) + y ̂ E y ( K ) + z ̂ E z ( K ) ] exp [ i ( k + K ) · r ] ,
H ( r ) = K [ x ̂ H x ( K ) + y ̂ H y ( K ) + z ̂ H z ( K ) ] × exp [ i ( k + K ) · r ] .
( r ) K V ( K ) exp [ i ( k + K ) · r ] ,
( r ) a ( for scatterers ) b ( for the background ) .
H x [ H x ( K 1 ) H x ( K 2 ) H x ( K 3 ) H x ( K NP ) ] ,
H y [ H y ( K 1 ) H y ( K 2 ) H y ( K 3 ) H y ( K NP ) ] ,
H z [ H z ( K 1 ) H z ( K 2 ) H z ( K 3 ) H z ( K NP ) ] ,
E x [ E x ( K 1 ) E x ( K 2 ) E x ( K 3 ) E x ( K NP ) ] ,
E y [ E y ( K 1 ) E y ( K 2 ) E y ( K 3 ) E y ( K NP ) ] ,
E z [ E z ( K 1 ) E z ( K 2 ) E z ( K 3 ) E z ( K NP ) ] ,
K x [ ( k + K 1 ) x 0 0 0 0 ( k + K 2 ) x 0 0 0 0 ( k + K 3 ) x 0 · 0 0 0 ( k + K NP ) x ] ,
K y [ ( k + K 1 ) y 0 0 0 0 ( k + K 2 ) y 0 0 0 0 ( k + K 3 ) y 0 · 0 0 0 ( k + K NP ) y ] ,
K z [ ( k + K 1 ) z 0 0 0 0 ( k + K 2 ) z 0 0 0 0 ( k + K 3 ) z 0 · 0 0 0 ( k + K NP ) z ] ,
V [ V 11 V 12 V 13 V 1 NP V 21 V 22 V 23 V 2 NP V 31 V 32 V 33 V 3 NP · V NP 1 V NP2 V NP3 V NP NP ] ,
V m n = V ( K m K n ) .
K y E z K z E y = μ 0 ω H x , K z E x K x E z = μ 0 ω H y , K x E y K y E x = μ 0 ω H z ,
K y H z K z H y = 0 ω VE x , K z H x K x H z = 0 ω VE y , K x H y K y H x = 0 ω VE z ,
K x H x + K y H y + K z H z = 0 ,
K x VE x + K y VE y + K z VE z = 0 .
E = [ E x E y E z ] ,
H = [ H x H y H z ] ,
T = [ 0 K z K y K z 0 K x K y K x 0 ] ,
V = [ V 0 0 0 V 0 0 0 V ] ,
K = [ K x K y K z ] .
TE = μ ω H ,
TH = 0 ω VE ,
KH = 0 ,
KVE = 0 .
[ TV 1 T ] H = ω 2 c 2 H ,
[ ( K y V 1 K y + K z V V 1 K z ( K y V 1 K x ) ( K z V 1 K x ) ( K x V 1 K y ) ( K K x V 1 K x + K z V 1 K z ) ( K z V 1 K y ) ( K x V 1 K z ) ( K y V 1 K z ) ( K x V 1 K x + K y V 1 K y ) ] [ H x H y H z ] = ω 2 c 2 [ H x H y H c ] .
[ V 1 T 2 ] E = ω 2 c 2 E ,
[ ( K y 2 + K z 2 ) ( K y K x ) ( K z K x ) ( K x K y ) + ( K x 2 + K z 2 ) ( K z K y ) ( K x K z ) ( K y K z ) + ( K x 2 + K z 2 ) ] [ E x E y E z ] = ω 2 c 2 [ V 0 0 0 V 0 0 0 V ] [ E x E y E z ] ,
[ ( K y V 1 K y + K z V 1 K z + K y V 1 K x K y 1 K x ) ( K y V 1 K x K y 1 K z K z V 1 K x ) ( K y V 1 K z K y 1 K x K x V 1 K z ) ( K x V 1 K x + K y V 1 K y + K y V 1 K z K y 1 K z ) ] [ H x H z ] = ω 2 c 2 [ H x H x ] .
[ ( K y 2 + K z 2 + K y K x V 1 K y 1 K x V ) ( K y K x V 1 K y 1 K z V K z K x ) ( K y K z V 1 K y 1 K x V K x K z ) ( K x 2 + K y 2 + K y K z V 1 K y 1 K x V ) ] [ E x E z ] = ω 2 c 2 [ V 0 0 V ] [ E x E z ] .
K p * = K p , K p T = K p ( p = x , y , z ) , T T = T , T * = T .
ω 2 c 2 real and positive .
ω 2 c 2 complex ; ω = ω real + i ω imag { ω imag 0 for unphysical modes ω imag = 0 for tunneling modes .
k = k r + i k i complex .
| H ( K n ) | = [ H x 2 ( K n ) + H y 2 ( K n ) + H z 2 ( K n ) ] 1 / 2 .
S = ( NP ) K | H ( K ) | 2 log | H ( K ) | 2 ] ,
D | | k + K | 2 k 2 | .

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