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.

[Optical Society of America ]

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References

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  1. J. D. Joannopoulos , P. R. Villeneuve , and S. Fan , Nature (London) NATUAS 386 , 143 149 ( 1997
    [CrossRef]
  2. E. Yablonovich , Phys. Rev. Lett. PRLTAO 58 , 2059 2062 ( 1987
    [CrossRef]
  3. S. John , Phys. Rev. Lett. PRLTAO 58 , 2486 2489 ( 1987
    [CrossRef] [PubMed]
  4. J. He and M. Cada , IEEE J. Quantum Electron. IEJQA7 27 , 1182 1189 ( 1991
    [CrossRef]
  5. E. Yablonovich and T. J. Gmitter , Phys. Rev. Lett. PRLTAO 67 , 2295 2298 ( 1991
    [CrossRef]
  6. J. P. Dowling and C. M. Bowden , Phys. Rev. A PLRAAN 46 , 612 615 ( 1992
    [CrossRef] [PubMed]
  7. J. M. Bendickson , J. P. Dowling , and M. Scalora , Phys. Rev. E PLEEE8 53 , 4107 4121 ( 1996
    [CrossRef]
  8. M. Scalora , J. P. Dowling , A. S. Manka , and C. M. Bowden , Phys. Rev. A PLRAAN 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 PLEEE8 54 , R1078 R1081 ( 1996
    [CrossRef]
  10. M. Bertolotti , P. Masciulli , and C. Sibilia , Opt. Lett. OPLEDP 19 , 777 779 ( 1994
    [CrossRef] [PubMed]
  11. M. Scalora , M. J. Bloemer , A. S. Manka , J. P. Dowling , C. M. Bowden , R. Viswanathan , and J. W. Haus , Phys. Rev. A PLRAAN 56 , 3166 3172 ( 1997
    [CrossRef]
  12. D. J. Lovering , G. Fino , C. Simonneau , R. Kuszelewicz , R. Azoulay , and J. A. Levenson , Electron. Lett. ELLEAK 32 , 1782 1783 ( 1997
    [CrossRef]
  13. A. M. Steinberg and R. Y. Chiao , Phys. Rev. A PLRAAN 51 , 3525 3531 ( 1995
    [CrossRef] [PubMed]

Fino, G

D. J. Lovering , G. Fino , C. Simonneau , R. Kuszelewicz , R. Azoulay , and J. A. Levenson , Electron. Lett. ELLEAK 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 PLEEE8 54 , R1078 R1081 ( 1996
[CrossRef]

Simonneau, C

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

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 PLEEE8 54 , R1078 R1081 ( 1996
[CrossRef]

Other (13)

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

E. Yablonovich , Phys. Rev. Lett. PRLTAO 58 , 2059 2062 ( 1987
[CrossRef]

S. John , Phys. Rev. Lett. PRLTAO 58 , 2486 2489 ( 1987
[CrossRef] [PubMed]

J. He and M. Cada , IEEE J. Quantum Electron. IEJQA7 27 , 1182 1189 ( 1991
[CrossRef]

E. Yablonovich and T. J. Gmitter , Phys. Rev. Lett. PRLTAO 67 , 2295 2298 ( 1991
[CrossRef]

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

J. M. Bendickson , J. P. Dowling , and M. Scalora , Phys. Rev. E PLEEE8 53 , 4107 4121 ( 1996
[CrossRef]

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

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 PLEEE8 54 , R1078 R1081 ( 1996
[CrossRef]

M. Bertolotti , P. Masciulli , and C. Sibilia , Opt. Lett. OPLEDP 19 , 777 779 ( 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 PLRAAN 56 , 3166 3172 ( 1997
[CrossRef]

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

A. M. Steinberg and R. Y. Chiao , Phys. Rev. A PLRAAN 51 , 3525 3531 ( 1995
[CrossRef] [PubMed]

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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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