Abstract

The spectral transmission properties of a self-similar optical Fabry–Perot resonator are theoretically studied. The structure considered can be realized by alternation of two dielectric layers of different refractive indices such that the highest refractive-index layers belong to a triadic Cantor set. The transmission spectrum exhibits localized peaks inside forbidden frequency band gaps. This feature could be useful in the design of novel filters. Moreover the self-similarity of the structure is reflected in a weak self-similarity in the transmission spectrum. A comparison with a regular periodic structure is also presented.

© 1994 Optical Society of America

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  1. R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
    [CrossRef] [PubMed]
  2. C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
    [CrossRef]
  3. L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
    [CrossRef]
  4. F. Laruelle, B. Etienne, Phys. Rev. B 37, 4816 (1988).
    [CrossRef]
  5. Rammal, G. Toulouse, J. Phys. Lett. 44, L13 (1983).
    [CrossRef]
  6. F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
    [CrossRef] [PubMed]
  7. Y. Sakurada, J. Uosumi, T. Asakura, Pure Appl. Opt. 1, 29 (1992).
    [CrossRef]
  8. K. Uno, J. Uozumi, T. Asakura, Appl. Opt. 32, 2722 (1993).
    [CrossRef] [PubMed]
  9. M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Freeman, New York, 1991), Chaps. 1–10.
  10. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 51–70.

1993

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

K. Uno, J. Uozumi, T. Asakura, Appl. Opt. 32, 2722 (1993).
[CrossRef] [PubMed]

1992

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Y. Sakurada, J. Uosumi, T. Asakura, Pure Appl. Opt. 1, 29 (1992).
[CrossRef]

1988

F. Laruelle, B. Etienne, Phys. Rev. B 37, 4816 (1988).
[CrossRef]

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

1985

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

1983

Rammal, G. Toulouse, J. Phys. Lett. 44, L13 (1983).
[CrossRef]

Alippi, A.

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Asakura, T.

K. Uno, J. Uozumi, T. Asakura, Appl. Opt. 32, 2722 (1993).
[CrossRef] [PubMed]

Y. Sakurada, J. Uosumi, T. Asakura, Pure Appl. Opt. 1, 29 (1992).
[CrossRef]

Bajema, K.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

Bettucci, A.

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Bhattacharya, P. K.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 51–70.

Brennan, T. M.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

Clarke, R.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

Craciun, F.

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Etienne, B.

F. Laruelle, B. Etienne, Phys. Rev. B 37, 4816 (1988).
[CrossRef]

Gourley, L.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

Hammons, B. E.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

Hurley, C.

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

Juang, F. Y.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

Laruelle, F.

F. Laruelle, B. Etienne, Phys. Rev. B 37, 4816 (1988).
[CrossRef]

McDonald, A. E.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

Merlin, R.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

Molinari, E.

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Nagle, J.

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

Petri, A.

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Ploog, K.

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

Rammal,

Rammal, G. Toulouse, J. Phys. Lett. 44, L13 (1983).
[CrossRef]

Sakurada, Y.

Y. Sakurada, J. Uosumi, T. Asakura, Pure Appl. Opt. 1, 29 (1992).
[CrossRef]

Schneider, R. P.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

Schroeder, M.

M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Freeman, New York, 1991), Chaps. 1–10.

Tamura, S.

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

Tiggs, C. P.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

Toulouse, G.

Rammal, G. Toulouse, J. Phys. Lett. 44, L13 (1983).
[CrossRef]

Uno, K.

Uosumi, J.

Y. Sakurada, J. Uosumi, T. Asakura, Pure Appl. Opt. 1, 29 (1992).
[CrossRef]

Uozumi, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 51–70.

Wolfe, J. P.

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

L. Gourley, C. P. Tiggs, R. P. Schneider, T. M. Brennan, B. E. Hammons, A. E. McDonald, Appl. Phys. Lett. 62, 1736 (1993).
[CrossRef]

J. Phys. Lett.

Rammal, G. Toulouse, J. Phys. Lett. 44, L13 (1983).
[CrossRef]

Phys. Rev. B

F. Laruelle, B. Etienne, Phys. Rev. B 37, 4816 (1988).
[CrossRef]

C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37, 8829 (1988).
[CrossRef]

Phys. Rev. Lett.

R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
[CrossRef] [PubMed]

F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi, Phys. Rev. Lett. 68, 1555 (1992).
[CrossRef] [PubMed]

Pure Appl. Opt.

Y. Sakurada, J. Uosumi, T. Asakura, Pure Appl. Opt. 1, 29 (1992).
[CrossRef]

Other

M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (Freeman, New York, 1991), Chaps. 1–10.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 51–70.

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Figures (5)

Fig. 1
Fig. 1

Cantor set for the levels N = 0, 1, 2, and 3. The structure for N = 0 is the initiator, and the one corresponding to N = 1 is the generator. The black regions correspond to the layers with n2 =3, and the white regions correspond to the layers with n1 = 1.524. The refractive index of the two embedding layers is n = 1.515.

Fig. 2
Fig. 2

(a) Refractive-index distribution for the level N = 3, (b) the resonator configuration.

Fig. 3
Fig. 3

Transmission spectrum magnitude versus ϕ/π for the first four Cantor levels: (a) N = 1, (b) N = 2, (c) N = 3, (d) N = 4.

Fig. 4
Fig. 4

Transmission spectrum magnitude versus ϕ/π for a periodic resonator with the same total optical path as the fractal one shown in Fig. 3(d).

Fig. 5
Fig. 5

Transmission spectrum magnitude versus ϕ/π around ϕ = 3π for N = 4: (a) one-period bandwidth, (b) magnified central third of (a).

Equations (4)

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t ( k 0 , L ) = 2 T 22 ( N ) ( k 0 , L ) - T 21 ( N ) ( k 0 , L ) i k 0 n - i k 0 n T 12 ( N ) ( k 0 , L ) + T 11 ( N ) ( k 0 , L ) ,
T ( k ) ( ϕ ) = T ( k - 1 ) ( ϕ ) T 1 ( 3 k - 1 ϕ ) T ( k - 1 ) ( ϕ ) ,             k = 1 , 2 , , N ,
T ( 0 ) ( ϕ ) = T 2 ( ϕ ) ,
T h ( ϕ ) = [ cos ( ϕ 3 ) 1 k 0 n h sin ( ϕ 3 ) - k 0 n h sin ( ϕ 3 ) cos ( ϕ 3 ) ] ,             h = 1 , 2 ,

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