Abstract

We present a study of the reliability of the supercell method in calculations of surface electromagnetic modes. For a truncated superlattice constituted of nonabsorbing dielectric layers, we demonstrate that the numerical solutions obtained by this method for transverse-electric waves agree with those based on the Bloch theory for the semi-infinite superlattice. A slab of superlattice with at least nine unit cells yields satisfactory convergence to an analytic dispersion relation for the surface modes. In addition, we apply the supercell method to study in detail the dependence of transverse-electric and transverse-magnetic surface waves on the cut-off position in the cell next to the surface. As a specific case, we choose a TiO2/SiO2 superlattice—layers with relatively high dielectric contrast in the visible spectrum. We find the surface modes strongly dependent on the position of the surface. In fact, they appear only for certain terminations. By plotting the field amplitudes, we show that there exist different possibilities for the guidance of surface waves. The variation of the penetration depth of these modes is also discussed.

© 1997 Optical Society of America

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References

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  1. R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
    [CrossRef]
  2. R. E. Camley and D. L. Mills, “Collective excitations of semi-infinite superlattice structure: surface plasmon, bulk plasmons, and the electron-energy-loss spectrum,” Phys. Rev. B 29, 1695 (1984).
    [CrossRef]
  3. E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233(2), 67 (1993).
    [CrossRef]
  4. L. H. Qin, Y. D. Zheng, and R. Zhang, “Study of  Gex Si1-x/Si  superlattices by ellipsometry,” Appl. Phys. A 55, 297 (1992).
    [CrossRef]
  5. T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
    [CrossRef]
  6. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423 (1977).
    [CrossRef]
  7. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  8. M. L. Bah, A. Akjouj, and L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95 (1992).
    [CrossRef]
  9. J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612 (1992).
    [CrossRef] [PubMed]
  10. J. P. Dowling and C. M. Bowden, “Beat radiation from dipoles near a photonic band edge,” J. Opt. Soc. Am. B 10, 353 (1993).
    [CrossRef]
  11. 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]
  12. D. Kossel, “Analogies between thin-film optics and electron-band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).
  13. P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
    [CrossRef]
  14. W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
    [CrossRef]
  15. A. A. Bulgakov and V. R. Kovtun, “Surface optical oscillations in a bounded layered-periodic medium,” Opt. Spektrosk. 56, 769 (1984).
  16. R. Haupt and L. Wendler, “Dispersion and damping properties of plasmon polaritons in superlattice structures,” Phys. Status Solidi B 142, 423 (1987); M. S. Kushwaha, “Intrasubband plasmons in semi-infinite n-i-p-i semiconductor superlattice,” Phys. Rev. B 45, 6050 (1992); R. F. Wallis, R. Szenics, J. J. Quinn, and G. F. Giuliani, “Theory of surface magnetoplasmon polaritons in truncated superlattices,” Phys. Rev. B 36, 1218 (1987).
    [CrossRef]
  17. W. L. Bloss, “Surface states of a semi-infinite superlattice,” Phys. Rev. B 44, 8035 (1990).
    [CrossRef]
  18. X. I. Saldaña and G. González de la Cruz, “Electromagnetic surface waves in semi-infinite superlattices,” J. Opt. Soc. Am. A 8, 36 (1991).
    [CrossRef]
  19. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  20. A. V. Vinogradov and I. V. Kozhevnikov, “X-ray surface waves in a superlattice,” JETP Lett. 40, 1222 (1984).
  21. R. F. Wallis, “Surface phonons: theoretical developments,” Surf. Sci. 299/300, 612 (1994).
    [CrossRef]
  22. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528 (1993).
    [CrossRef] [PubMed]

1994 (2)

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
[CrossRef]

R. F. Wallis, “Surface phonons: theoretical developments,” Surf. Sci. 299/300, 612 (1994).
[CrossRef]

1993 (3)

1992 (3)

L. H. Qin, Y. D. Zheng, and R. Zhang, “Study of  Gex Si1-x/Si  superlattices by ellipsometry,” Appl. Phys. A 55, 297 (1992).
[CrossRef]

M. L. Bah, A. Akjouj, and L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95 (1992).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[CrossRef] [PubMed]

1991 (2)

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]

X. I. Saldaña and G. González de la Cruz, “Electromagnetic surface waves in semi-infinite superlattices,” J. Opt. Soc. Am. A 8, 36 (1991).
[CrossRef]

1990 (1)

W. L. Bloss, “Surface states of a semi-infinite superlattice,” Phys. Rev. B 44, 8035 (1990).
[CrossRef]

1987 (1)

R. Haupt and L. Wendler, “Dispersion and damping properties of plasmon polaritons in superlattice structures,” Phys. Status Solidi B 142, 423 (1987); M. S. Kushwaha, “Intrasubband plasmons in semi-infinite n-i-p-i semiconductor superlattice,” Phys. Rev. B 45, 6050 (1992); R. F. Wallis, R. Szenics, J. J. Quinn, and G. F. Giuliani, “Theory of surface magnetoplasmon polaritons in truncated superlattices,” Phys. Rev. B 36, 1218 (1987).
[CrossRef]

1984 (3)

A. A. Bulgakov and V. R. Kovtun, “Surface optical oscillations in a bounded layered-periodic medium,” Opt. Spektrosk. 56, 769 (1984).

A. V. Vinogradov and I. V. Kozhevnikov, “X-ray surface waves in a superlattice,” JETP Lett. 40, 1222 (1984).

R. E. Camley and D. L. Mills, “Collective excitations of semi-infinite superlattice structure: surface plasmon, bulk plasmons, and the electron-energy-loss spectrum,” Phys. Rev. B 29, 1695 (1984).
[CrossRef]

1983 (1)

R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
[CrossRef]

1978 (2)

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
[CrossRef]

1977 (1)

1966 (1)

D. Kossel, “Analogies between thin-film optics and electron-band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).

Akjouj, A.

M. L. Bah, A. Akjouj, and L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95 (1992).
[CrossRef]

Albuquerque, E. L.

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233(2), 67 (1993).
[CrossRef]

Arjavalingam, G.

Bah, M. L.

M. L. Bah, A. Akjouj, and L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95 (1992).
[CrossRef]

Bloss, W. L.

W. L. Bloss, “Surface states of a semi-infinite superlattice,” Phys. Rev. B 44, 8035 (1990).
[CrossRef]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Beat radiation from dipoles near a photonic band edge,” J. Opt. Soc. Am. B 10, 353 (1993).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[CrossRef] [PubMed]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528 (1993).
[CrossRef] [PubMed]

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]

Bulgakov, A. A.

A. A. Bulgakov and V. R. Kovtun, “Surface optical oscillations in a bounded layered-periodic medium,” Opt. Spektrosk. 56, 769 (1984).

Camley, R. E.

R. E. Camley and D. L. Mills, “Collective excitations of semi-infinite superlattice structure: surface plasmon, bulk plasmons, and the electron-energy-loss spectrum,” Phys. Rev. B 29, 1695 (1984).
[CrossRef]

R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
[CrossRef]

Chen, P. C.

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
[CrossRef]

Cho, A. Y.

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

Cottam, M. G.

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233(2), 67 (1993).
[CrossRef]

Djafari-Rouhani, B.

R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
[CrossRef]

Dobrzynski, L.

M. L. Bah, A. Akjouj, and L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95 (1992).
[CrossRef]

R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
[CrossRef]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Beat radiation from dipoles near a photonic band edge,” J. Opt. Soc. Am. B 10, 353 (1993).
[CrossRef]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[CrossRef] [PubMed]

González de la Cruz, G.

Hattori, T.

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
[CrossRef]

Haupt, R.

R. Haupt and L. Wendler, “Dispersion and damping properties of plasmon polaritons in superlattice structures,” Phys. Status Solidi B 142, 423 (1987); M. S. Kushwaha, “Intrasubband plasmons in semi-infinite n-i-p-i semiconductor superlattice,” Phys. Rev. B 45, 6050 (1992); R. F. Wallis, R. Szenics, J. J. Quinn, and G. F. Giuliani, “Theory of surface magnetoplasmon polaritons in truncated superlattices,” Phys. Rev. B 36, 1218 (1987).
[CrossRef]

Hong, C.-S.

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528 (1993).
[CrossRef] [PubMed]

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]

Kawato, S.

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
[CrossRef]

Kossel, D.

D. Kossel, “Analogies between thin-film optics and electron-band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).

Kovtun, V. R.

A. A. Bulgakov and V. R. Kovtun, “Surface optical oscillations in a bounded layered-periodic medium,” Opt. Spektrosk. 56, 769 (1984).

Kozhevnikov, I. V.

A. V. Vinogradov and I. V. Kozhevnikov, “X-ray surface waves in a superlattice,” JETP Lett. 40, 1222 (1984).

Maradudin, A. A.

R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
[CrossRef]

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528 (1993).
[CrossRef] [PubMed]

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]

Mills, D. L.

R. E. Camley and D. L. Mills, “Collective excitations of semi-infinite superlattice structure: surface plasmon, bulk plasmons, and the electron-energy-loss spectrum,” Phys. Rev. B 29, 1695 (1984).
[CrossRef]

Nakatsuka, H.

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
[CrossRef]

Ng, W.

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
[CrossRef]

Qin, L. H.

L. H. Qin, Y. D. Zheng, and R. Zhang, “Study of  Gex Si1-x/Si  superlattices by ellipsometry,” Appl. Phys. A 55, 297 (1992).
[CrossRef]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528 (1993).
[CrossRef] [PubMed]

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]

Robertson, W. M.

Saldaña, X. I.

Tsurumachi, N.

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
[CrossRef]

Vinogradov, A. V.

A. V. Vinogradov and I. V. Kozhevnikov, “X-ray surface waves in a superlattice,” JETP Lett. 40, 1222 (1984).

Wallis, R. F.

R. F. Wallis, “Surface phonons: theoretical developments,” Surf. Sci. 299/300, 612 (1994).
[CrossRef]

Wendler, L.

R. Haupt and L. Wendler, “Dispersion and damping properties of plasmon polaritons in superlattice structures,” Phys. Status Solidi B 142, 423 (1987); M. S. Kushwaha, “Intrasubband plasmons in semi-infinite n-i-p-i semiconductor superlattice,” Phys. Rev. B 45, 6050 (1992); R. F. Wallis, R. Szenics, J. J. Quinn, and G. F. Giuliani, “Theory of surface magnetoplasmon polaritons in truncated superlattices,” Phys. Rev. B 36, 1218 (1987).
[CrossRef]

Yariv, A.

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
[CrossRef]

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
[CrossRef]

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zhang, R.

L. H. Qin, Y. D. Zheng, and R. Zhang, “Study of  Gex Si1-x/Si  superlattices by ellipsometry,” Appl. Phys. A 55, 297 (1992).
[CrossRef]

Zheng, Y. D.

L. H. Qin, Y. D. Zheng, and R. Zhang, “Study of  Gex Si1-x/Si  superlattices by ellipsometry,” Appl. Phys. A 55, 297 (1992).
[CrossRef]

Appl. Phys. A (1)

L. H. Qin, Y. D. Zheng, and R. Zhang, “Study of  Gex Si1-x/Si  superlattices by ellipsometry,” Appl. Phys. A 55, 297 (1992).
[CrossRef]

Appl. Phys. Lett. (2)

P. Yeh, A. Yariv, and A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen, and A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

D. Kossel, “Analogies between thin-film optics and electron-band theory of solids,” J. Opt. Soc. Am. 56, 1434 (1966).

P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423 (1977).
[CrossRef]

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

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

JETP Lett. (1)

A. V. Vinogradov and I. V. Kozhevnikov, “X-ray surface waves in a superlattice,” JETP Lett. 40, 1222 (1984).

Opt. Lett. (1)

Opt. Spektrosk. (1)

A. A. Bulgakov and V. R. Kovtun, “Surface optical oscillations in a bounded layered-periodic medium,” Opt. Spektrosk. 56, 769 (1984).

Phys. Rep. (1)

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233(2), 67 (1993).
[CrossRef]

Phys. Rev. A (1)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (5)

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]

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220 (1994).
[CrossRef]

R. E. Camley, B. Djafari-Rouhani, L. Dobrzynski, and A. A. Maradudin, “Transverse elastic waves in periodically layered infinite and semi-infinite media,” Phys. Rev. B 27, 7318 (1983).
[CrossRef]

R. E. Camley and D. L. Mills, “Collective excitations of semi-infinite superlattice structure: surface plasmon, bulk plasmons, and the electron-energy-loss spectrum,” Phys. Rev. B 29, 1695 (1984).
[CrossRef]

W. L. Bloss, “Surface states of a semi-infinite superlattice,” Phys. Rev. B 44, 8035 (1990).
[CrossRef]

Phys. Status Solidi B (1)

R. Haupt and L. Wendler, “Dispersion and damping properties of plasmon polaritons in superlattice structures,” Phys. Status Solidi B 142, 423 (1987); M. S. Kushwaha, “Intrasubband plasmons in semi-infinite n-i-p-i semiconductor superlattice,” Phys. Rev. B 45, 6050 (1992); R. F. Wallis, R. Szenics, J. J. Quinn, and G. F. Giuliani, “Theory of surface magnetoplasmon polaritons in truncated superlattices,” Phys. Rev. B 36, 1218 (1987).
[CrossRef]

Surf. Sci. (1)

R. F. Wallis, “Surface phonons: theoretical developments,” Surf. Sci. 299/300, 612 (1994).
[CrossRef]

Surf. Sci. Rep. (1)

M. L. Bah, A. Akjouj, and L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95 (1992).
[CrossRef]

Other (2)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (14)

Fig. 1
Fig. 1

(a) Geometry of the unit cell of width d=a+b. The repetition of this cell along the x axis generates an infinite superlattice constituted of alternating layers of dielectric constants a and b with thicknesses a and b, respectively. (b) Geometry of the semi-infinite superlattice bounded by air.

Fig. 2
Fig. 2

Geometry of a supercell of width L. This symmetric structure contains one slab of superlattice (width l) bounded by two air layers of thickness l0 each. The inner cells (here n =5) and the two surface cells (incomplete unit cells) give the slab width l=(n+2τ)d. Thus L=l+2l0.

Fig. 3
Fig. 3

Schematic representation for the termination of the superlattice. The dotted line is the position of the surface. Each figure shows both the surface cell of thickness τd and the first inner cell. In the uppermost figure, τ=1, the surface cell is a complete unit cell. (a) 0τ<b/2d; the surface cuts the superlattice through the inner b layer of the original complete surface cell. (b) b/2dτ<b/2d+a/d; the termination is realized through the a layer. (c) b/2d+a/dτ<1; the surface cuts the superlattice through the outermost layer of dielectric constant b. In each of the last three figures the surface cell is a portion of the unitary cell.

Fig. 4
Fig. 4

TE surface modes calculated by the supercell method (continuous curves) and by Eq. (11.5-6) of Ref. 17 (dotted curve). See text for material parameters. The shaded zone represents the first bulk band for β>ωb1/2/c at ωa/c=π. Dashed curves are the band edges. The cut parameter is τd=b/2 +a. (a) Solutions obtained with one inner cell. Only for large separations (b>a) do the surface modes in the slab converge to that of the semi-infinite superlattice. (b) Solutions with seven inner cells. The equivalence of both methods is much better. Only for small separations (b/a0.35) do the two surface modes in the slab have appreciable interaction.

Fig. 5
Fig. 5

Electromagnetic modes of a TiO2/SiO2 superlattice. The shaded zones are the bulk bands. Dashed curves are the surface modes. The larger-slope continuous line is the 0 light line (β=ω01/2/c). The lower-slope continuous line is the b light line (β=ωb1/2/c). With b=0.5a the cut parameter τ =0.75 represents a termination with a surface layer of dielectric constant a of thickness 0.875a [see Fig. 3(b)]. The figures show the different locations of the dispersion curves of the surface modes in the four lowest band gaps: (a) TE modes and (b) TM modes. The line satisfying the Brewster condition is also plotted.

Fig. 6
Fig. 6

Electric-field profiles Re(E) of the lower three TE surface modes of a TiO2/SiO2 superlattice. They correspond to the modes in the lower three band gaps of Fig. 5(a) at β=1.2. At the bottom of the figure the separation between the arrows represents the width of the unit cell. The modes are mainly guided by the incomplete a layer of thickness 0.875a. Only the uppermost curve corresponds to the low-localization zone, between the 0 and the b light lines.

Fig. 7
Fig. 7

Electric-field profiles of two surface modes for the third dispersion curve of Fig. 5(a). The dashed line is the superlattice–air interface. The figure shows the variation of the field confinement of modes lying in the low-localization zone (β =0.9) and the high-localization zone (β=1.8).

Fig. 8
Fig. 8

TE surface-modes behavior as a function of the cut distance in the surface cell (TiO2/SiO2 superlattice). The interval 0τ1 corresponds to the unit cell width. The two arrows at the bottom of the figure (which delimit three regions in the cell) are the limits of the layer of dielectric constant a. By increasing τ, modes are lowered from the bulk bands and become surface modes. There are no surface modes for τ>b/2d+a/d = 0.833.

Fig. 9
Fig. 9

Electric-field profile Re(E) of a surface wave with the greatest extremum in the surface b layer.

Fig. 10
Fig. 10

As in Fig. 8 but for TM polarization, β=0.9, and n =15.

Fig. 11
Fig. 11

Electric-field profile Re(E) (continuous curve) and magnetic-field profile Re(H) (dotted curve) of surface waves with their local maximum in the second layer from the surface (the first a layer).

Fig. 12
Fig. 12

Representation of the τ regions in which surface modes appear for β=0.9 in the lower three band gaps of the band structure in Figs. 5(a) and 5(b).

Fig. 13
Fig. 13

Same as Fig. 12 but for β=1.5.

Fig. 14
Fig. 14

Detail of Fig. 5(b) showing parts of the two lower bands. Surface modes (dashed curves) appear in the band gap assuming that the cell next to the surface is a complete bulk cell (τ=1). The graph is obtained with a slab of n=13 complete cells. For n the two dashed curves coalesce and terminate at the intersection point of the bulk bands.

Tables (1)

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Table 1 Attenuation Coefficient α and Penetration Depth δ of Three Modes (β, ω) on the Third Dispersion Curve of Fig. 5(a)

Equations (17)

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×E=i ωcB,×H=-i ωc(x)E,
1(x)2x2+2y2E=-ω2c2E,
x1(x)x+1(x)2y2H=-ω2c2H.
Aκ(x, y)=GAκ(G)exp[i(κ+G)x]exp(iβy),
1(x)=GμG exp(iGx).
GμG-G[(κ+G)2+β2]Eκ(G)=ω2c2Eκ(G)(TE),
GμG-G[(κ+G)(κ+G)+β2]Hκ(G)
=ω2c2Hκ(G)(TM).
(x)=b+(a-b)Θa2-|x|,
μG=1d-d/2d/2 1(x)exp(-iGx)dx=1bδG,0+1a-1bf sin(Ga/2)Ga/2,
1(x)=10+1(x)-10Θl2-|x|,
1(x)=1b+1a-1bF(x)+xjΘa2-|x-xj|.
F(x)=00<τdb/2(a)xjΘτd-b/22-x-xjb/2<τdb/2+a(b)xjΘ(a/2-|x-xj|)b/2+a<τdd(c),
μG=1L-L/2L/2 1(x)exp(-iGx)dx.
μG=10+1b-10 lL+1a-1bnaL+F1δG,0+1b-10 lLsin(Gl/2)Gl/2(1-δG,0)+1a-1baLsin(Ga/2)Ga/2xj exp(-iGxj)+F2×(1-δG,0),
F1(x)=00τd<b/22(τd-b/2)Lb/2τd<b/2+a2aLb/2+aτd<d,
F2(x)=00τd<b/22(τd-b/2)Lsin[G(τd-b/2)/2]G(τd-b/2)/2cos(Gxj)b/2τd<b/2+a2aLsin(Ga/2)Ga/2cos(Gxj)b/2+aτd<d.

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