Abstract

The transmission properties of quasi-periodic Cantor-like and Fibonacci photonic bandgap structures that have the same optical paths are studied and compared with one-dimensional, finite, N-period stacks. The electromagnetic mode densities and group velocities are also discussed. We show that the density of modes at the edge of the bandgap is greater for Cantor-like multilayers, so at the band edge a sharp decrease in the group velocity is found.

© 1998 Optical Society of America

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  1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature (London) 386, 143–149 (1997).
    [Crossref]
  2. E. Yablonovich, Phys. Rev. Lett. 58, 2059–2062 (1987).
    [Crossref]
  3. S. John, Phys. Rev. Lett. 58, 2486–2489 (1987).
    [Crossref] [PubMed]
  4. J. He and M. Cada, IEEE J. Quantum Electron. 27, 1182–1189 (1991).
    [Crossref]
  5. E. Yablonovich and T. J. Gmitter, Phys. Rev. Lett. 67, 2295–2298 (1991).
    [Crossref]
  6. J. P. Dowling and C. M. Bowden, Phys. Rev. A 46, 612–615 (1992).
    [Crossref] [PubMed]
  7. J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107–4121 (1996).
    [Crossref]
  8. M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
    [Crossref] [PubMed]
  9. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
    [Crossref]
  10. W. Heitler, The Quantum Theory of Radiation, 3rd ed. (Dover, New York, 1984), Chap. IV, Sec. 14.
  11. P. Tran, J. Opt. Soc. Am. B 14, 2598–2595 (1997),M. Scalora, P. J. Dowling, C. M. Bowden, and M. J. Bloemer, Phys. Rev. Lett. 73, 1368–1371 (1994).
    [Crossref] [PubMed]
  12. M. Bertolotti, P. Masciulli, and C. Sibilia, Opt. Lett. 19, 777–779 (1994).
    [Crossref] [PubMed]
  13. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
    [Crossref]
  14. D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
    [Crossref]
  15. A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525–3531 (1995).
    [Crossref] [PubMed]

1997 (4)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature (London) 386, 143–149 (1997).
[Crossref]

P. Tran, J. Opt. Soc. Am. B 14, 2598–2595 (1997),M. Scalora, P. J. Dowling, C. M. Bowden, and M. J. Bloemer, Phys. Rev. Lett. 73, 1368–1371 (1994).
[Crossref] [PubMed]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

1996 (2)

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

1995 (2)

M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
[Crossref] [PubMed]

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525–3531 (1995).
[Crossref] [PubMed]

1994 (1)

1992 (1)

J. P. Dowling and C. M. Bowden, Phys. Rev. A 46, 612–615 (1992).
[Crossref] [PubMed]

1991 (2)

J. He and M. Cada, IEEE J. Quantum Electron. 27, 1182–1189 (1991).
[Crossref]

E. Yablonovich and T. J. Gmitter, Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref]

1987 (2)

E. Yablonovich, Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref]

S. John, Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Azoulay, R.

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

Bertolotti, M.

Bloemer, M. J.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
[Crossref] [PubMed]

J. P. Dowling and C. M. Bowden, Phys. Rev. A 46, 612–615 (1992).
[Crossref] [PubMed]

Cada, M.

J. He and M. Cada, IEEE J. Quantum Electron. 27, 1182–1189 (1991).
[Crossref]

Chiao, R. Y.

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525–3531 (1995).
[Crossref] [PubMed]

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
[Crossref] [PubMed]

J. P. Dowling and C. M. Bowden, Phys. Rev. A 46, 612–615 (1992).
[Crossref] [PubMed]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature (London) 386, 143–149 (1997).
[Crossref]

Fino, G.

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

Gmitter, T. J.

E. Yablonovich and T. J. Gmitter, Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref]

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

He, J.

J. He and M. Cada, IEEE J. Quantum Electron. 27, 1182–1189 (1991).
[Crossref]

Heitler, W.

W. Heitler, The Quantum Theory of Radiation, 3rd ed. (Dover, New York, 1984), Chap. IV, Sec. 14.

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature (London) 386, 143–149 (1997).
[Crossref]

John, S.

S. John, Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Kuszelewicz, R.

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

Leavit, R. P.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

Levenson, J. A.

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

Lovering, D. J.

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
[Crossref] [PubMed]

Masciulli, P.

Reinhardt, S. B.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

Scalora, M.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
[Crossref] [PubMed]

Sibilia, C.

Simonneau, C.

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525–3531 (1995).
[Crossref] [PubMed]

Tocci, N. D.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

Tran, P.

P. Tran, J. Opt. Soc. Am. B 14, 2598–2595 (1997),M. Scalora, P. J. Dowling, C. M. Bowden, and M. J. Bloemer, Phys. Rev. Lett. 73, 1368–1371 (1994).
[Crossref] [PubMed]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature (London) 386, 143–149 (1997).
[Crossref]

Viswanathan, R.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

Yablonovich, E.

E. Yablonovich and T. J. Gmitter, Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref]

E. Yablonovich, Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref]

Electron. Lett. (1)

D. J. Lovering, G. Fino, C. Simonneau, R. Kuszelewicz, R. Azoulay, and J. A. Levenson, Electron. Lett. 32, 1782–1783 (1997).
[Crossref]

IEEE J. Quantum Electron. (1)

J. He and M. Cada, IEEE J. Quantum Electron. 27, 1182–1189 (1991).
[Crossref]

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

P. Tran, J. Opt. Soc. Am. B 14, 2598–2595 (1997),M. Scalora, P. J. Dowling, C. M. Bowden, and M. J. Bloemer, Phys. Rev. Lett. 73, 1368–1371 (1994).
[Crossref] [PubMed]

Nature (London) (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature (London) 386, 143–149 (1997).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (4)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166–3172 (1997).
[Crossref]

A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 51, 3525–3531 (1995).
[Crossref] [PubMed]

J. P. Dowling and C. M. Bowden, Phys. Rev. A 46, 612–615 (1992).
[Crossref] [PubMed]

M. Scalora, J. P. Dowling, A. S. Manka, and C. M. Bowden, Phys. Rev. A 52, 726–734 (1995).
[Crossref] [PubMed]

Phys. Rev. E (2)

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, N. D. Tocci, J. P. Dowling, C. M. Bowden, and R. P. Leavit, Phys. Rev. E 54, R1078–R1081 (1996).
[Crossref]

J. M. Bendickson, J. P. Dowling, and M. Scalora, Phys. Rev. E 53, 4107–4121 (1996).
[Crossref]

Phys. Rev. Lett. (3)

E. Yablonovich, Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref]

S. John, Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

E. Yablonovich and T. J. Gmitter, Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref]

Other (1)

W. Heitler, The Quantum Theory of Radiation, 3rd ed. (Dover, New York, 1984), Chap. IV, Sec. 14.

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Figures (6)

Fig. 1
Fig. 1

Layer configurations under study: (a) An N-period stack composed of two-layer unit cells of refractive indices n1=2.5 and n2=1.5 with quarter-wave thicknesses a=λ0/(4n1) and b=λ0/(4n2) (λ0=1.558 μm), respectively, (b) A three-stage Cantor-like multilayer, (c) Fibonacci multilayer S8 generated from the same materials with quarter-wave thickness.

Fig. 2
Fig. 2

Plots of the dimensionless DOM, ρ˜N=vbulkρN, and dimensionless transmittance TN=|t|2 for (a) a 14-period stack (b) a 3-stage Cantor-like multilayer, and (c) Fibonacci multilayer S8 versus normalized k/k0=λ0/λ with normal wave incidence.

Fig. 3
Fig. 3

Plots of the dimensionless DOM, ρ˜N=vbulkρN, and dimensionless transmittance TN=|t|2 for (a) a Fabry–Perot resonator with a half-wavelength central layer of refractive index n2 (the mirrors are dielectric periodic quarter-wavelength stacks of refractive indices n1 and n2) and (b) a central layer of 3/4(λ0) optical path. The total optical path is of the same order of magnitude as the structures in Fig. 2.

Fig. 4
Fig. 4

Plot of the dimensionless group velocity vN/c=(cρN)-1 for a 14-period stack and 3-stage Cantor-like and Fibonacci multilayers versus normalized k/k0=λ0/λ with normal wave incidence.

Fig. 5
Fig. 5

Plots of the dimensionless DOM, ρ˜N=vbulkρN, and dimensionless transmittance TN=|t|2 for the case of TM polarization for (a) a 14-period stack, (b) a 3-stage Cantor-like multilayer, and (c) Fibonacci multilayer S8 versus normalized k/k0=λ0/λ.

Fig. 6
Fig. 6

Transmittance TN=|t|2 (a) for a 41-period stack and (b) for 4-stage Cantor-like multilayer with band-edge group velocity. The dimensionless group velocity vN/c=(cρN)-1 for both structures is shown in (c).

Equations (6)

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

ρ(ω)dkdω=1D yx-xyx2+y2,
t(k0, L)=2T22(N)(k0, 3NL)-T21(N)(k0, 3NL)ik0n-ik0nT12(N)(k0, 3NL)+T11(N)(k0, 3NL),
T(k)(3kφ)=T(k-1)(3k-1φ)T1(3k-1φ)T(k-1)(3k-1φ),
k=1, 2, , N,
T(0)(φ)=T2(φ),
Th(φ)=cosφ31k0nh sinφ3-k0nh sinφ3cosφ3,h=1, 2

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