Abstract

We determine the range of thicknesses and refractive indices for which omnidirectional reflection from quasiperiodic dielectric multilayers occurs. By resorting to the notion of area under the transmittance curve, we assess in a systematic way the performance of the different Fibonacci multilayers.

© 2005 Optical Society of America

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References

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  1. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-62 (1987).
    [CrossRef] [PubMed]
  2. S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-9 (1987).
    [CrossRef] [PubMed]
  3. A complete and up-to-date bibliography on the subject can be found at <a href="http://home.earthlink.net/~jpdowling/pbgbib.html">http://home.earthlink.net/~jpdowling/pbgbib.html</a>.
  4. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  5. J. Lekner, Theory of Reflection (Dordrecht, The Netherlands, 1987).
  6. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, �??A dielectric omnidirectional reflector,�?? Science 282, 1679-82 (1998).
    [CrossRef] [PubMed]
  7. J. P. Dowling, �??Mirror on the wall: you�??re omnidirectional after all?,�?? Science 282, 1841-2 (1998).
    [CrossRef]
  8. E. Yablonovitch, �??Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,�?? Opt. Lett. 23, 1648-9 (1998).
    [CrossRef]
  9. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, �??Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,�?? Appl. Phys. A 68, 25-8 (1999).
    [CrossRef]
  10. W. H. Southwell, �??Omnidirectional mirror design with quarter-wave dielectric stacks,�?? Appl. Opt. 38, 5464-7 (1999).
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  11. J. Lekner �??Omnidirectional reflection by multilayer dielectric mirrors,�?? J. Opt. A 2, 349-53 (2000).
    [CrossRef]
  12. M. Kohmoto, B. Sutherland, and K. Iguchi, �??Localization of optics: Quasiperiodic media,�?? Phys. Rev. Lett. 58, 2436-8 (1987).
    [CrossRef] [PubMed]
  13. C. Schwartz, �??Reflection properties of pseudorandom multilayers,�?? Appl. Opt. 27, 1232-4 (1988).
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  14. M. Dulea, M. Severin, and R. Riklund, �??Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,�?? Phys. Rev. B 42, 3680-9 (1990).
    [CrossRef]
  15. A. Latgé and F. Claro, �??Optical propagation in multilayered systems,�?? Opt. Commun. 94, 389-96 (1992).
    [CrossRef]
  16. N. H. Liu, �??Propagation of light waves in Thue-Morse dielectric multilayers,�?? Phys. Rev. B 55, 3543-7 (1997.
    [CrossRef]
  17. M. S. Vasconcelos and E. L. Albuquerque, �??Transmission fingerprints in quasiperiodic dielectric multilayers,�?? Phys. Rev. B 59, 11128-31 (1999).
    [CrossRef]
  18. E. Maciá, �??Exploiting quasiperiodic order in the design of optical devices,�?? Phys. Rev. B 63, 205421 (2001).
    [CrossRef]
  19. E. Maciá, �??Optical engineering with Fibonacci dielectric multilayers,�?? Appl. Phys. Lett. 73, 3330-2 (1998).
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  20. E. Cojocaru, �??Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence,�?? Appl. Opt. 40, 6319-26 (2001).
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  21. D. Lusk, I. Abdulhalim and F. Placido, �??Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,�?? Opt. Commun. 198, 273-9 (2001).
    [CrossRef]
  22. R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, �??Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,�?? Appl. Phys. Lett. 80, 3063-5 (2002).
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  23. J. W. Dong, P. Han, and H. Z. Wang, �??Broad omnidirectional reflection band forming using the combination of Fibonacci quasi-periodic and periodic one-dimensional photonic crystals,�?? Chin. Phys. Lett. 20, 1963-5 (2003).
    [CrossRef]
  24. T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, �??Optimizing omnidirectional reflection by multilayer mirrors,�?? J. Opt. A 6, 127-31 (2004).
    [CrossRef]
  25. M. Kohmoto, L. P. Kadanoff, and C. Tang, �??Localization problem in one dimension: Mapping and escape,�?? Phys. Rev. Lett. 50, 1870-2 (1983).
    [CrossRef]
  26. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, �??All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,�?? J. Lightw. Technol. 17, 2018-24 (1999).
    [CrossRef]

Appl. Opt.

Appl. Phys. A

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, �??Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,�?? Appl. Phys. A 68, 25-8 (1999).
[CrossRef]

Appl. Phys. Lett.

E. Maciá, �??Optical engineering with Fibonacci dielectric multilayers,�?? Appl. Phys. Lett. 73, 3330-2 (1998).
[CrossRef]

R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, �??Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,�?? Appl. Phys. Lett. 80, 3063-5 (2002).
[CrossRef]

Chin. Phys. Lett.

J. W. Dong, P. Han, and H. Z. Wang, �??Broad omnidirectional reflection band forming using the combination of Fibonacci quasi-periodic and periodic one-dimensional photonic crystals,�?? Chin. Phys. Lett. 20, 1963-5 (2003).
[CrossRef]

J. Lightw. Technol.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, �??All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,�?? J. Lightw. Technol. 17, 2018-24 (1999).
[CrossRef]

J. Opt. A

T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, �??Optimizing omnidirectional reflection by multilayer mirrors,�?? J. Opt. A 6, 127-31 (2004).
[CrossRef]

J. Lekner �??Omnidirectional reflection by multilayer dielectric mirrors,�?? J. Opt. A 2, 349-53 (2000).
[CrossRef]

Opt. Commun.

A. Latgé and F. Claro, �??Optical propagation in multilayered systems,�?? Opt. Commun. 94, 389-96 (1992).
[CrossRef]

D. Lusk, I. Abdulhalim and F. Placido, �??Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,�?? Opt. Commun. 198, 273-9 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. B

N. H. Liu, �??Propagation of light waves in Thue-Morse dielectric multilayers,�?? Phys. Rev. B 55, 3543-7 (1997.
[CrossRef]

M. S. Vasconcelos and E. L. Albuquerque, �??Transmission fingerprints in quasiperiodic dielectric multilayers,�?? Phys. Rev. B 59, 11128-31 (1999).
[CrossRef]

E. Maciá, �??Exploiting quasiperiodic order in the design of optical devices,�?? Phys. Rev. B 63, 205421 (2001).
[CrossRef]

M. Dulea, M. Severin, and R. Riklund, �??Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,�?? Phys. Rev. B 42, 3680-9 (1990).
[CrossRef]

Phys. Rev. Lett.

M. Kohmoto, B. Sutherland, and K. Iguchi, �??Localization of optics: Quasiperiodic media,�?? Phys. Rev. Lett. 58, 2436-8 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-62 (1987).
[CrossRef] [PubMed]

S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-9 (1987).
[CrossRef] [PubMed]

M. Kohmoto, L. P. Kadanoff, and C. Tang, �??Localization problem in one dimension: Mapping and escape,�?? Phys. Rev. Lett. 50, 1870-2 (1983).
[CrossRef]

Science

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, �??A dielectric omnidirectional reflector,�?? Science 282, 1679-82 (1998).
[CrossRef] [PubMed]

J. P. Dowling, �??Mirror on the wall: you�??re omnidirectional after all?,�?? Science 282, 1841-2 (1998).
[CrossRef]

Other

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

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

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

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Figures (4)

Fig. 1.
Fig. 1.

Regions where ODR (for p polarization) occurs for the Fibonacci systems S 2={LH}, S 3={LHL}, S 4={LHLLH}, and S 5={LHLLHLHL}. We have taken nL =1.75 and nH =3.35 at λ=10 µm. The inset identifies the filled ellipses and shows the corresponding bandwidths B calculated according to Eq. (15). The marked points correspond to the minimum area for each one of the systems.

Fig. 2.
Fig. 2.

Area under the transmittance curve, defined in Eq. (12), as a function of nLdL /λ and nHdH /λ for the system S 2, with the same data as in Fig. 1.

Fig. 3.
Fig. 3.

Regions of ODR for the same Fibonacci multilayers as in Fig. 1 in the plane (nL,nH ) of refractive indices. The curves show the limit of ODR for each stack with the optimum thicknesses marked in Fig. 1.

Fig. 4.
Fig. 4.

Logarithm of the area computed for systems [Sj ] N (with j=2,3,4, 5) as a function of the number of layers.

Tables (1)

Tables Icon

Table 1. Optimum parameters nLdL /λ and nHdH /λ, and the corresponding area, for systems [Sj ] N containing up to 26 layers.

Equations (17)

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M 0 = M H , M 1 = M L ,
M j = M j 1 M j 2 , j 2 .
M H = ( cos β H q H sin β H 1 q H sin β H cos β H ) ,
q H ( p ) = n H cos θ n cos θ H , q H ( s ) = n cos θ n H cos θ H ,
M j ( N ) = ( M j ) N .
𝓣 j ( N ) = 4 M j ( N ) 2 + 2 ,
Tr ( M j ) 2 .
Tr ( M j + 1 ) = Tr ( M j ) Tr ( M j 1 ) Tr ( M j 2 ) .
cos β L cos β H Λ L H sin β L sin β H 1 ,
cos ( 2 β L ) cos β H Λ L H sin ( 2 β L ) sin β H 1 .
Λ L H = 1 2 ( qL qH + qH qL ) ,
Λ L H ( p ) Λ L H ( s ) 1 .
n L d L λ = n H d H λ = 1 4 ,
𝓐 j ( N ) = 0 π 2 𝓣 j ( N ) ( θ ) d θ ,
n L d L λ = 0.34305 , n H d H λ = 0.25416 .
n L d L λ = 1 8 , n H d H λ = 1 4 .
B = λ long λ short 1 2 ( λ long + λ short ) .

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