Abstract

We investigated the propagation of electromagnetic waves in a finite superlattice bounded by a substrate and vacuum and separated from them by a buffer layer and a cap layer. In this realistic geometry, we show the existence of different kinds of localized and resonant modes whose behaviors are strongly dependent on the nature and thickness of the buffer and cap layers as well as on the width of the superlattice. These modes can be confined inside the superlattice or at its boundaries and give rise to different possibilities for the guidance of optical waves. The localized and resonant modes are obtained from an analytic calculation of the local and total densities of states within a Green’s function formalism for electromagnetic waves of s-polarization (transverse electric). We also evaluate the reflection and transmission coefficients and compare their behaviors with the densities of states.

© 1999 Optical Society of America

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  1. See, for example, Highlights in Condensed Matter Physics and Future Prospects, L. Esaki, ed. (Plenum, New York1991).
  2. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219; P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  3. L. H. Qin, Y. D. Zheng, R. Zhang, “Study of GexSi1-x/Si superlattices by ellipsometry,” Appl. Phys. A: Solids Surf. 55, 297–300 (1992).
    [CrossRef]
  4. T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
    [CrossRef]
  5. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
    [CrossRef]
  6. J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
    [CrossRef]
  7. R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
    [CrossRef]
  8. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media: I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  9. P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
    [CrossRef]
  10. W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
    [CrossRef]
  11. A. A. Bulgakov, V. R. Kovtun, “Surface optical oscillations in a limited stratified-periodic medium,” Opt. Spectrosc. 56, 269–274 (1984).
  12. A. A. Bulgakov, V. R. Kovtun, “Study of surface optical oscillations in periodical multilayer media,” Solid State Commun. 56, 781–785 (1985).
    [CrossRef]
  13. X. I. Saldana, G. Gonzalez de la Cruz, “Electromagnetic surface waves in semi-infinite superlattices,” J. Opt. Soc. Am. A 8, 36–40 (1991).
    [CrossRef]
  14. M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
    [CrossRef]
  15. F. Ramos-Mendieta, P. Halevi, “Electromagnetic surface modes of a dielectric superlattice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
    [CrossRef]
  16. Y. F. Li, J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4, 671–677 (1987).
    [CrossRef]
  17. T. Hattori, N. Tsurumachi, H. Nakatsuka, “Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 14, 348–355 (1997).
    [CrossRef]
  18. R. Wang, J. Dong, D. Y. Xing, “Defect studies in a one-dimensional photonic band gap structure,” Phys. Status Solidi B 200, 529–534 (1997).
    [CrossRef]
  19. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
    [CrossRef] [PubMed]
  20. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
    [CrossRef]
  21. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
    [CrossRef] [PubMed]
  22. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  23. M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
    [CrossRef] [PubMed]
  24. Y. F. Li, K. Iizuka, J. W. Y. Lit, “Periodic stratified structure in a multilayer planar optical waveguide,” J. Opt. Soc. Am. A 9, 559–568 (1992).
    [CrossRef]
  25. L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119–157 (1986);“Interface response theory of continuous composite systems,” Surf. Sci. Rep. 11, 139–178 (1990).
    [CrossRef]
  26. M. G. Cottam, A. A. Maradudin, “Surface linear response functions” in Surface Excitations, V. M. Agranovich, R. Loudon, eds. (Mod. Probl. Condens. Matter Sci.9) (North-Holland, Amsterdam, 1986), pp. 5–20.
  27. M. L. Bah, A. Akjouj, L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95–132 (1992).
    [CrossRef]
  28. J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
    [CrossRef]
  29. E. N. Economou, Green’s Functions in Quantum Physics (Spring-Verlag, Heidelberg, 1990), pp. 3–18.
  30. R. D. Meade, K. D. Brommer, A. M. Rappe, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772–13774 (1991).
    [CrossRef]

1997 (3)

1996 (4)

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1994 (2)

T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
[CrossRef]

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

1993 (2)

D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B 10, 314–321 (1993).
[CrossRef]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

1992 (3)

L. H. Qin, Y. D. Zheng, R. Zhang, “Study of GexSi1-x/Si superlattices by ellipsometry,” Appl. Phys. A: Solids Surf. 55, 297–300 (1992).
[CrossRef]

M. L. Bah, A. Akjouj, L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95–132 (1992).
[CrossRef]

Y. F. Li, K. Iizuka, J. W. Y. Lit, “Periodic stratified structure in a multilayer planar optical waveguide,” J. Opt. Soc. Am. A 9, 559–568 (1992).
[CrossRef]

1991 (4)

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

R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

X. I. Saldana, G. Gonzalez de la Cruz, “Electromagnetic surface waves in semi-infinite superlattices,” J. Opt. Soc. Am. A 8, 36–40 (1991).
[CrossRef]

1987 (2)

Y. F. Li, J. W. Y. Lit, “General formulas for the guiding properties of a multilayer slab waveguide,” J. Opt. Soc. Am. A 4, 671–677 (1987).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1986 (1)

L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119–157 (1986);“Interface response theory of continuous composite systems,” Surf. Sci. Rep. 11, 139–178 (1990).
[CrossRef]

1985 (1)

A. A. Bulgakov, V. R. Kovtun, “Study of surface optical oscillations in periodical multilayer media,” Solid State Commun. 56, 781–785 (1985).
[CrossRef]

1984 (1)

A. A. Bulgakov, V. R. Kovtun, “Surface optical oscillations in a limited stratified-periodic medium,” Opt. Spectrosc. 56, 269–274 (1984).

1978 (2)

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[CrossRef]

1977 (1)

Akjouj, A.

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

M. L. Bah, A. Akjouj, L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95–132 (1992).
[CrossRef]

Bah, M. L.

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

M. L. Bah, A. Akjouj, L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95–132 (1992).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Bowden, C. M.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

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

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Bulgakov, A. A.

A. A. Bulgakov, V. R. Kovtun, “Study of surface optical oscillations in periodical multilayer media,” Solid State Commun. 56, 781–785 (1985).
[CrossRef]

A. A. Bulgakov, V. R. Kovtun, “Surface optical oscillations in a limited stratified-periodic medium,” Opt. Spectrosc. 56, 269–274 (1984).

Chan, C. T.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Chen, P. C.

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[CrossRef]

Cho, A. Y.

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

Cottam, M. G.

M. G. Cottam, A. A. Maradudin, “Surface linear response functions” in Surface Excitations, V. M. Agranovich, R. Loudon, eds. (Mod. Probl. Condens. Matter Sci.9) (North-Holland, Amsterdam, 1986), pp. 5–20.

Dalichaouch, R.

Djafari-Rouhani, B.

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

Dobrzynski, L.

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

M. L. Bah, A. Akjouj, L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95–132 (1992).
[CrossRef]

L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119–157 (1986);“Interface response theory of continuous composite systems,” Surf. Sci. Rep. 11, 139–178 (1990).
[CrossRef]

Dong, J.

R. Wang, J. Dong, D. Y. Xing, “Defect studies in a one-dimensional photonic band gap structure,” Phys. Status Solidi B 200, 529–534 (1997).
[CrossRef]

Dowling, J. P.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Economou, E. N.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

E. N. Economou, Green’s Functions in Quantum Physics (Spring-Verlag, Heidelberg, 1990), pp. 3–18.

El Boudouti, E. H.

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Gonzalez de la Cruz, G.

Halevi, P.

Hattori, T.

T. Hattori, N. Tsurumachi, H. Nakatsuka, “Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 14, 348–355 (1997).
[CrossRef]

T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
[CrossRef]

Ho, K. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Hong, C. S.

Iizuka, K.

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Kawato, S.

T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
[CrossRef]

Kovtun, V. R.

A. A. Bulgakov, V. R. Kovtun, “Study of surface optical oscillations in periodical multilayer media,” Solid State Commun. 56, 781–785 (1985).
[CrossRef]

A. A. Bulgakov, V. R. Kovtun, “Surface optical oscillations in a limited stratified-periodic medium,” Opt. Spectrosc. 56, 269–274 (1984).

Kroll, N.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Li, Y. F.

Lit, J. W. Y.

Maradudin, A. A.

M. G. Cottam, A. A. Maradudin, “Surface linear response functions” in Surface Excitations, V. M. Agranovich, R. Loudon, eds. (Mod. Probl. Condens. Matter Sci.9) (North-Holland, Amsterdam, 1986), pp. 5–20.

McCall, S. L.

Meade, R. D.

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

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Nakatsuka, H.

T. Hattori, N. Tsurumachi, H. Nakatsuka, “Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 14, 348–355 (1997).
[CrossRef]

T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
[CrossRef]

Ng, W.

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[CrossRef]

Platzman, P. M.

Qin, L. H.

L. H. Qin, Y. D. Zheng, R. Zhang, “Study of GexSi1-x/Si superlattices by ellipsometry,” Appl. Phys. A: Solids Surf. 55, 297–300 (1992).
[CrossRef]

Ramos-Mendieta, F.

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

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

Rapper, A. M.

R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Saldana, X. I.

Scalora, M.

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Schultz, S.

Sigalas, M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Smith, D. R.

Soukoulis, C. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Tocci, M. D.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Tsurumachi, N.

T. Hattori, N. Tsurumachi, H. Nakatsuka, “Analysis of optical nonlinearity by defect states in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 14, 348–355 (1997).
[CrossRef]

T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
[CrossRef]

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

Wang, R.

R. Wang, J. Dong, D. Y. Xing, “Defect studies in a one-dimensional photonic band gap structure,” Phys. Status Solidi B 200, 529–534 (1997).
[CrossRef]

Xing, D. Y.

R. Wang, J. Dong, D. Y. Xing, “Defect studies in a one-dimensional photonic band gap structure,” Phys. Status Solidi B 200, 529–534 (1997).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yariv, A.

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[CrossRef]

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media: I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219; P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Yeh, P.

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[CrossRef]

P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media: I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219; P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zhang, R.

L. H. Qin, Y. D. Zheng, R. Zhang, “Study of GexSi1-x/Si superlattices by ellipsometry,” Appl. Phys. A: Solids Surf. 55, 297–300 (1992).
[CrossRef]

Zheng, Y. D.

L. H. Qin, Y. D. Zheng, R. Zhang, “Study of GexSi1-x/Si superlattices by ellipsometry,” Appl. Phys. A: Solids Surf. 55, 297–300 (1992).
[CrossRef]

Appl. Phys. A: Solids Surf. (1)

L. H. Qin, Y. D. Zheng, R. Zhang, “Study of GexSi1-x/Si superlattices by ellipsometry,” Appl. Phys. A: Solids Surf. 55, 297–300 (1992).
[CrossRef]

Appl. Phys. Lett. (2)

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105 (1978).
[CrossRef]

W. Ng, P. Yeh, P. C. Chen, A. Yariv, “Optical surface waves in periodic layered medium grown by liquid phase epitaxy,” Appl. Phys. Lett. 32, 370–371 (1978).
[CrossRef]

J. Mod. Opt. (1)

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (3)

J. Phys. Condens. Matter (1)

M. L. Bah, A. Akjouj, E. H. El Boudouti, B. Djafari-Rouhani, L. Dobrzynski, “Surface and interface optical waves in superlattices: transverse electric localized and resonant modes,” J. Phys. Condens. Matter 8, 4171–4188 (1996).
[CrossRef]

Opt. Spectrosc. (1)

A. A. Bulgakov, V. R. Kovtun, “Surface optical oscillations in a limited stratified-periodic medium,” Opt. Spectrosc. 56, 269–274 (1984).

Phys. Rev. A (1)

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructure,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (4)

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

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

R. D. Meade, K. D. Brommer, A. M. Rapper, J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasi-crystal,” Phys. Rev. B 50, 4220–4223 (1994).
[CrossRef]

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

R. Wang, J. Dong, D. Y. Xing, “Defect studies in a one-dimensional photonic band gap structure,” Phys. Status Solidi B 200, 529–534 (1997).
[CrossRef]

Solid State Commun. (1)

A. A. Bulgakov, V. R. Kovtun, “Study of surface optical oscillations in periodical multilayer media,” Solid State Commun. 56, 781–785 (1985).
[CrossRef]

Surf. Sci. Rep. (2)

M. L. Bah, A. Akjouj, L. Dobrzynski, “Response functions in layered dielectric media,” Surf. Sci. Rep. 16, 95–132 (1992).
[CrossRef]

L. Dobrzynski, “Interface response theory of discrete composite systems,” Surf. Sci. Rep. 6, 119–157 (1986);“Interface response theory of continuous composite systems,” Surf. Sci. Rep. 11, 139–178 (1990).
[CrossRef]

Other (4)

M. G. Cottam, A. A. Maradudin, “Surface linear response functions” in Surface Excitations, V. M. Agranovich, R. Loudon, eds. (Mod. Probl. Condens. Matter Sci.9) (North-Holland, Amsterdam, 1986), pp. 5–20.

E. N. Economou, Green’s Functions in Quantum Physics (Spring-Verlag, Heidelberg, 1990), pp. 3–18.

See, for example, Highlights in Condensed Matter Physics and Future Prospects, L. Esaki, ed. (Plenum, New York1991).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), pp. 155–219; P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of a finite superlattice (i=1, 2) with a buffer layer (n=0, i=a) and a cap layer (n=N, i=b) and sandwiched between a semi-infinite substrate (S) and vacuum. da, db, d1 and d2 are, respectively, the thicknesses of the buffer layer a, the cap layer b, and the two different slabs out of which the finite superlattice is built. D is the period of the superlattice. The reflection and transmission of TE electromagnetic waves are also represented. θi, θr, θt are the incident, reflected, and transmitted angles, respectively.

Fig. 2
Fig. 2

Dispersion of localized and resonant modes (dotted lines) induced by a finite SL and buffer and cap layers. The SL contains N=5 layer pairs of 2–1 dielectrics. The heavy solid and the heavy dashed lines are the vacuum light line and the substrate light line, respectively. The dotted lines below the substrate light lines are the localized modes, and those above the substrate light line are the resonant modes. The thin solid lines below the substrate light line are the confined modes of the SL, and those above the substrate light line are resonant modes.

Fig. 3
Fig. 3

Frequency dependence of (a) the DOS, (b) the transmission coefficient, and (c) the reflection coefficient for kD=4. The geometrical parameters are the same as in Fig. 2. The bulk contribution of the substrate and vacuum to the DOS are subtracted. Bs and Bv are δ functions of weight (-1/4) appearing at the bottom of the substrate and the vacuum bulk band, respectively.

Fig. 4
Fig. 4

Spatial representation of the local DOS corresponding to the case depicted in Fig. 2 for (a) kD=7.5 and ωD/c=2.726, (b) 3.194, (c) 3.791, (d) 3.899, (e) 4.222, (f) 4.438, (g) 4.887, (h) 5.451, and (i) 5.607. The positions of the different interfaces are indicated by vertical lines.

Fig. 5
Fig. 5

(a) DOS versus ωD/c for kD=7.5. Curves (b), (c), (d), and (e) represent the local densities of states versus ωD/c and for kD=7.5 calculated at the interfaces: substrate–buffer-layer, buffer-layer–superlattice, superlattice–cap-layer, and cap-layer–vacuum, respectively.

Fig. 6
Fig. 6

Variation of the frequencies of the localized and the resonant modes as a function of the width da of the buffer layer, for the system depicted in Fig. 1, for kD=6. Symbols Ii, (i=1,2), Si, (i=1, 2, 3, 4), and Ri (i=1, 2) (open-dotted lines) correspond to the localized modes at the SL–cap layer interface or inside the cap layer, the localized modes in the buffer layer–SL interface or inside the buffer layer, and the resonant modes, respectively. The thin solid lines represent the modes of the SL. The solid rectangles are the modes whose local DOS are presented in Fig. 7. The upper heavy solid line represents the vacuum light line, and the heavy dashed line represents the substrate light line.

Fig. 7
Fig. 7

Spatial representation of the LDOS for the three modes (belonging to the branch S1) indicated by the solid rectangles in Fig. 6.

Fig. 8
Fig. 8

Same as in Fig. 6, but the variation of the frequencies is given versus db/D.

Fig. 9
Fig. 9

Evolution of the transmission coefficient as function of the number of cells N, for kD=1. The parameters (both material and geometrical) are the same as in Fig. 2.

Equations (80)

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n(ω2, k; n, i, x3)
=-iπc2 Im d+(ω2, k; n, i, x3; n, i, x3),
d+(ω2)=limΓ0[d(ω2+iΓ)],
n(ω2)=n1(ω2)+n2(ω2)+na(ω2)+nb(ω2)+Δsn(ω2)+Δvn(ω2),
n1(ω2)=-1πc2 n=1N Im -d1/2d1/2d(n, 1, x3; n, 1, x3)dx3,
n2(ω2)=-2πc2 n=0N-1 Im -d2/2d2/2d(n, 2, x3; n, 2, x3)dx3,
na(ω2)=-aπc2 Im -da/2da/2d(0, a, x3; 0, a, x3)dx3,
nb(ω2)=-bπc2 Im -db/2db/2d(N, b, x3; N, b, x3)dx3,
Δsn(ω2)=-sπc2 Im-0[d(x3, x3)-Gs(x3, x3)]dx3,
Δvn(ω2)=-vπc2 Imxb+[d(x3, x3)-Gv(x3, x3)]dx3,
xb=da+ND+db
αi2(k, ω)=k2-ω2c2 i,
Ci=cosh(αidi),
Si=sinh(αidi),
Fi=αi,
Rs=1+FaSa/FsCa1+FsSa/FaCa,
Rv=1+FbSb/FvCb1+FvSb/FbCb,
t=exp(ik3D),
cos(k3D)=C1C2+12 F1F2+F2F1S1S2.
n1(ω2)=-1πc2 Im t(t2-1)Δ- t 1-t2Nt2-1 (AB0t+A0B)S1α1F1 C2S1+12 C1S2F1F2+F2F1+S2d12F2 1-F22F12+NΔ+d1F1 C2S1+12 C1S2F1F2+F2F1+S1S22α1F2 1-F22F12,
n2(ω2)=-2πc2 Im t(t2-1)Δ- t 1-t2Nt2-1 (AB0+A0Bt)S2α2F2 C1S2+12 C2S1F1F2+F2F1+S1d22F1 1-F12F22+NΔ+d2F2 C1S2+12 C2S1F1F2+F2F1+S1S22α2F1 1-F12F22,
na(ω2)=a2πc2 Im 11+FsSaFaCaΔ- ySaαaCa+da+daSaFaCa (z+yFs)-zFsFa2 SaαaCa-daB0t-t2NySaαaCa+da+daSaFaCa (z0+yFs)-z0FsFa2 SaαaCa-daBt,
nb(ω2)=b2πc2 Im 11+FvSbFbCbΔ- xSbαbCb+db-dbSbFbCb (k-xFv)+kFvFb2 SbαbCb-dbA0-t2Nx0SbαbCb+db-dbSbFbCb (k0-x0Fv)+k0FvFb2 SbαbCb-dbA,
Δsn(ω2)=-sπc2 Im 12αs 12Fs-y+z SaFaCaB0t-t2Ny+z0 SaFaCaBt1+FsSaFaCaΔ-,
Δvn(ω2)=-vπc2 Im 12αv 12Fv+-x+k SbFbCbA0-t2N-x0+k0 SbFbCbA1+FvSbFbCbΔ-,
Δ±=ytA0B0x0±t2N ABx,
A0=FsRsx0+k0,
A=FsRsx+k,
B0=FvRvy-z0,
B=FvRvy-z,
x0=S1F1 t+S2F2,z0=C1C2+F2F1 S1S2-1t,
k0=C2-C1t,
x=S1F1+S2F2 t,z=C1C2+F2F1 S1S2-t,
k=C2t-C1,
y=C1S2F2+C2S1F1.
Δ-=0.
R=1-FsSa/FaCa1+FsSa/FaCa ytA0-B0x0-t2N A-BxΔ-2,
T=FvFs 2yFstN(t2-1)CaCb(1+FsSa/FaCa)(1+FvSb/FbCb)Δ-2,
A0-=-FsRs-x0+k0,
A-=-FsRs-x+k,
Rs-=1-FaSa/FsCa1-FsSa/FaCa.
Gi(x3, x3)=-12Fi exp(-αi|x3-x3|)
d0,a, -da2; 0, a,-da2=-y+z SaFaCaB0t-t2Ny+z0 SaFaCaBt(1+FsSa/FaCa)Δ-,
d0,a, -da2; 0, a, +da2=d0, a, +da2; 0,a, -da2
=-yt B0-t2NBCa(1+FsSa/FaCa)Δ-,
d0,a, da2; 0, a, da2=-yt B0-t2NBΔ-.
dN, b, -db2; N, b, -db2=-xA0-t2Nx0AΔ-,
dN, b, db2; N, b, -db2=dN, b, -db2; N, b, db2
=-xA0-t2Nx0ACb(1+FvSb/FbCb)Δ-,
dN, b, db2; N, b, db2
=-x+k SbFbCbA0-t2N-x0+k0 SbFbCbA(1+FvSb/FbCb)Δ-.
dn, 1, -d12; n, 1, -d12
=tt2-1 {yt|n-n|-1Δ- [xAB0tn+n-1-x0ABt2N(tn-n+tn-n)+x0tA0Bt2N-n-n]},
dn, 1, -d12; n, 1, d12
=tt2-1 S2F2 t|n-n|+S1F1 t|n-n-1|-1Δ- yAB0tn+n-ytx ABt2N(xtn-n-1+x0tn-n)+ytA0Bt2N-n-n,
dn, 1, d12; n, 1, -d12=dn, 1, -d12; n, 1, d12,
dn, 1, d12; n, 1, d12
=tt2-1 {yt|n-n|-1Δ- [x0AB0tn+n-x0ABt2N(tn-n+tn-n)+xA0Bt2N-n-n]}.
d(x3, x3)
=-12Fs exp[-αs|x3-x3|]+12Fs-y+z SaFaCaB0t-t2Ny+z0 SaFaCaBt(1+FsSa/FaCa)Δ-
×exp[αs(x3+x3)].
d(0, a, x3; 0, a, x3)
=Ua(x3, x3)+1Sa2 sinhαada2-x3;sinhαada2+x3×d(Ma, Ma)sinhαada2-x3sinhαada2+x3,
Ua(x3, x3)
=-12Fa exp(-αa|x3-x3|)
+12FaSa sinhαada2-x3
×exp-αada2+x3+sinhαada2+x3
×exp-αada2-x3,
d(n, i, x3; n, i, x3)
=δnnδilUi(x3, x3)
+1SiSisinhαidi2-x3; sinhαidi2+x3
×d(Mm, Mm)sinhαidi2-x3sinhαidi2+x3,
Ui(x3, x3)
=-12Fi exp(-αi|x3-x3|)+12FiSi×sinhαidi2-x3exp-αidi2+x3+sinhαidi2+x3exp-αidi2-x3.
d(N, b, x3; N, b, x3)
=Ub(x3, x3)+1Sb2 sinhαbdb2-x3;sinhαbdb2+x3×d(Mb, Mb)sinhαbdb2-x3sinhαbdb2+x3,
Ub(x3, x3)
=-12Fb exp(-αb|x3-x3|)+12FbSb×sinhαbdb2-x3exp-αbdb2+x3+sinhαbdb2+x3exp-αbdb2-x3
d(x3, x3)
=-12Fv exp[-αv|x3-x3|]+12Fv+-x+k SbFbCbA0-t2N-x0+k0 SbFbCbA(1+FvSb/FbCb)Δ-×exp[-αv(x3+x3-2xb)].

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