Abstract

We suggest a geometrical framework in which to discuss periodic layered structures in the unit disk. Bandgaps appear when the point representing the system approaches the unit circle. We show that the trace of the matrix describing the basic period allows for a classification in three families of orbits with quite different properties. The laws of convergence of the iterates to the unit circle can then be considered universal features of the reflection.

© 2003 Optical Society of America

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References

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  1. A complete and up-to-date bibliography on the subject can be found at http://home.earthlink.net/~jpdowling/pbgbib.html.
  2. E. Centeno and D. Felbacq, J. Opt. Soc. Am. A 17, 320 (2000).
    [CrossRef]
  3. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
    [CrossRef] [PubMed]
  4. J. P. Dowling, Science 282, 1841 (1998).
    [CrossRef]
  5. E. Yablonovitch, Opt. Lett. 23, 1648 (1998).
    [CrossRef]
  6. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
    [CrossRef]
  7. J. Lekner, J. Opt. A 2, 349 (2000).
    [CrossRef]
  8. B. Gralak, G. Tayeb, and S. Enoch, J. Opt. Soc. Am. A 17, 1012 (2000).
    [CrossRef]
  9. M. Notomi, Phys. Rev. B 62, 10696 (2000).
    [CrossRef]
  10. T. Yonte, J. J. Monzón, L. L. Sánchez-Soto, J. F. Cariñena, and C. López-Lacasta, J. Opt. Soc. Am. A 19, 603 (2002).
    [CrossRef]
  11. J. J. Monzón, T. Yonte, L. L. Sánchez-Soto, and J. F. Cariñena, J. Opt. Soc. Am. A 19, 985 (2002).
    [CrossRef]
  12. J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
    [CrossRef]
  13. J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2013 (1999).
    [CrossRef]
  14. L. L. Sánchez-Soto, J. J. Monzón, T. Yonte, and J. F. Cariñena, Opt. Lett. 26, 1400 (2001).
    [CrossRef]
  15. J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Lett. 26, 370 (2001).
    [CrossRef]
  16. D. Felbacq, J. Phys. A 33, 7137 (2000).
    [CrossRef]
  17. D. Felbacq, B. Guizal, and F. Zolla, “Do Fresnel coefficients exist?” April 23, 2001, arXiv.org e-Print archive, Preprint physics/0104074, http://xxx.lanl.gov.
  18. J. Kapeluszny, T. Kuczumow, and S. Reich, Adv. Math. 143, 111 (1999).
    [CrossRef]
  19. J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Commun. 218, 43 (2003).
    [CrossRef]
  20. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  21. J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).

2003 (1)

J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Commun. 218, 43 (2003).
[CrossRef]

2002 (2)

2001 (2)

2000 (5)

1999 (4)

J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, J. Opt. Soc. Am. A 16, 2013 (1999).
[CrossRef]

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
[CrossRef]

J. Kapeluszny, T. Kuczumow, and S. Reich, Adv. Math. 143, 111 (1999).
[CrossRef]

1998 (3)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

J. P. Dowling, Science 282, 1841 (1998).
[CrossRef]

E. Yablonovitch, Opt. Lett. 23, 1648 (1998).
[CrossRef]

Cariñena, J. F.

Centeno, E.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
[CrossRef]

Dowling, J. P.

J. P. Dowling, Science 282, 1841 (1998).
[CrossRef]

Enoch, S.

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Felbacq, D.

E. Centeno and D. Felbacq, J. Opt. Soc. Am. A 17, 320 (2000).
[CrossRef]

D. Felbacq, J. Phys. A 33, 7137 (2000).
[CrossRef]

D. Felbacq, B. Guizal, and F. Zolla, “Do Fresnel coefficients exist?” April 23, 2001, arXiv.org e-Print archive, Preprint physics/0104074, http://xxx.lanl.gov.

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Gaponenko, S. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
[CrossRef]

Gralak, B.

Guizal, B.

D. Felbacq, B. Guizal, and F. Zolla, “Do Fresnel coefficients exist?” April 23, 2001, arXiv.org e-Print archive, Preprint physics/0104074, http://xxx.lanl.gov.

Joannopoulos, J. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Kapeluszny, J.

J. Kapeluszny, T. Kuczumow, and S. Reich, Adv. Math. 143, 111 (1999).
[CrossRef]

Kuczumow, T.

J. Kapeluszny, T. Kuczumow, and S. Reich, Adv. Math. 143, 111 (1999).
[CrossRef]

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
[CrossRef]

Lekner, J.

J. Lekner, J. Opt. A 2, 349 (2000).
[CrossRef]

J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).

López-Lacasta, C.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Monzón, J. J.

Notomi, M.

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Reich, S.

J. Kapeluszny, T. Kuczumow, and S. Reich, Adv. Math. 143, 111 (1999).
[CrossRef]

Sánchez-Soto, L. L.

Tayeb, G.

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Yablonovitch, E.

Yarotsky, D. A.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
[CrossRef]

Yeh, P.

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

Yonte, T.

Zolla, F.

D. Felbacq, B. Guizal, and F. Zolla, “Do Fresnel coefficients exist?” April 23, 2001, arXiv.org e-Print archive, Preprint physics/0104074, http://xxx.lanl.gov.

Adv. Math. (1)

J. Kapeluszny, T. Kuczumow, and S. Reich, Adv. Math. 143, 111 (1999).
[CrossRef]

Appl. Phys. A (1)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, Appl. Phys. A 68, 25 (1999).
[CrossRef]

J. Opt. A (1)

J. Lekner, J. Opt. A 2, 349 (2000).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Phys. A (1)

D. Felbacq, J. Phys. A 33, 7137 (2000).
[CrossRef]

Opt. Commun. (2)

J. J. Monzón and L. L. Sánchez-Soto, Opt. Commun. 162, 1 (1999).
[CrossRef]

J. J. Monzón, T. Yonte, and L. L. Sánchez-Soto, Opt. Commun. 218, 43 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. B (1)

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Science (2)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

J. P. Dowling, Science 282, 1841 (1998).
[CrossRef]

Other (4)

A complete and up-to-date bibliography on the subject can be found at http://home.earthlink.net/~jpdowling/pbgbib.html.

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

J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 1987).

D. Felbacq, B. Guizal, and F. Zolla, “Do Fresnel coefficients exist?” April 23, 2001, arXiv.org e-Print archive, Preprint physics/0104074, http://xxx.lanl.gov.

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Figures (2)

Fig. 1
Fig. 1

Plots of typical orbits in the unit disk for (a) canonical transfer matrices as given in Eqs. (7) and (b) arbitrary transfer matrices.

Fig. 2
Fig. 2

Plots of the successive iterates (N=1,,5) for elliptic, hyperbolic, and parabolic actions starting from the origin as the initial point. The physical system is described in the text. Only hyperbolic and parabolic actions tend to the unit circle.

Equations (12)

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

Ea+Ea(-)=MasEs+Es(-),
Mas=1/TasRas*/Tas*Ras/Tas1/Tas*αββ*α*.
z=E(-)/E+
za=ΦMas,zs=β*+α*zsα+βzs,
zf±=12β-2i Imα±TrMas2-41/2.
Mˆas=CMasC-1,
Kˆφ=expiφ/200exp-iφ/2, Aˆχ=coshχ/2i sinhχ/2-i sinhχ/2coshχ/2, Nˆη=1-iη/2η/2η/21+iη/2,
zN=ΦMas,zN-1=ΦMasN,z0,
zN=1-ξN1-ξNzf+/zf-zf+,
zN=Nβzf2Nβzf-1,
zN2=β2β2+sinhχ/sinhNχ2,
zN2=β2β2+1/N2,

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