Abstract

We investigate omnidirectional reflection from higher-order gaps in one-dimensional photonic crystals. Moreover, we present a designing criterion for omnidirectional reflection from several distinct gaps simultaneously, using only a single photonic crystal with a constant period. We show that for practical values of photonic crystals parameters, several relatively large omnidirectional gaps may be obtained. As an example, we demonstrate an omnidirectional reflector that exhibits two distinct wide omnidirectional ranges at near-infrared wavelengths. This omnidirectional reflector that operates in several ranges of wavelengths may have various potential applications.

© 2007 Optical Society of America

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References

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  1. J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573-1575 (1998).
    [CrossRef]
  2. 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, 25-28 (1999).
    [CrossRef]
  3. 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. Lightwave Technol. 17, 2018-2024 (1999).
    [CrossRef]
  4. Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
    [CrossRef]
  5. Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
    [CrossRef]
  6. S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
    [CrossRef] [PubMed]
  7. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
    [CrossRef] [PubMed]
  8. B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, "Low-loss infrared dielectric material system for dual-range omnidirectional reflectivity," Opt. Lett. 26, 1370-1372 (2001).
    [CrossRef]
  9. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  10. I. Nusinsky and A. A. Hardy, "Band gap analysis of one-dimensional photonic crystals and conditions for gap closing," Phys. Rev. B 73, 125104 (2006).
    [CrossRef]
  11. J. A. Gaspar-Armenta and F. Villa, "Band-structure properties of one-dimensional photonic crystal under the formalism of equivalent systems," J. Opt. Soc. Am. B 21, 405-412 (2004).
    [CrossRef]
  12. J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994).
    [CrossRef]
  13. M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
    [CrossRef]
  14. Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
    [CrossRef]

2007

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

2006

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

I. Nusinsky and A. A. Hardy, "Band gap analysis of one-dimensional photonic crystals and conditions for gap closing," Phys. Rev. B 73, 125104 (2006).
[CrossRef]

2004

2003

Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
[CrossRef]

2002

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

2001

2000

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

1999

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, 25-28 (1999).
[CrossRef]

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. Lightwave Technol. 17, 2018-2024 (1999).
[CrossRef]

1998

1994

J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892-2899 (1994).
[CrossRef]

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

Chen, Ming

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

Chigrin, D. N.

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, 25-28 (1999).
[CrossRef]

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. Lightwave Technol. 17, 2018-2024 (1999).
[CrossRef]

Cho, Chi-O

Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
[CrossRef]

Dapkus, P. D.

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

Fan, S.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573-1575 (1998).
[CrossRef]

Fink, Y.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, "Low-loss infrared dielectric material system for dual-range omnidirectional reflectivity," Opt. Lett. 26, 1370-1372 (2001).
[CrossRef]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573-1575 (1998).
[CrossRef]

Gaponenko, S. V.

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, 25-28 (1999).
[CrossRef]

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. Lightwave Technol. 17, 2018-2024 (1999).
[CrossRef]

Gaspar-Armenta, J. A.

Hardy, A. A.

I. Nusinsky and A. A. Hardy, "Band gap analysis of one-dimensional photonic crystals and conditions for gap closing," Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Hart, S. D.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

Ibanescu, M.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Jeon, Heonsu

Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
[CrossRef]

Joannopoulos, J. D.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, "Low-loss infrared dielectric material system for dual-range omnidirectional reflectivity," Opt. Lett. 26, 1370-1372 (2001).
[CrossRef]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, "Omnidirectional reflection from a one-dimensional photonic crystal," Opt. Lett. 23, 1573-1575 (1998).
[CrossRef]

Kao, Chih-Chiang

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Kuo, Hao-Chung

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Lavrinenko, A. V.

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. Lightwave Technol. 17, 2018-2024 (1999).
[CrossRef]

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, 25-28 (1999).
[CrossRef]

Lekner, J.

Leung, Kok-Ming

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Li, Chunfei

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

Lin, Chung-Hsiang

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Lo, Jun-Ren

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Ma, Shaojie

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

MacDougal, M. H.

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

Maskaly, G. R.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

Nusinsky, I.

I. Nusinsky and A. A. Hardy, "Band gap analysis of one-dimensional photonic crystals and conditions for gap closing," Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Park, Yeonsang

Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
[CrossRef]

Prideaux, P. H.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

Roh, Young-Geun

Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
[CrossRef]

Steier, W. H.

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

Temelkuran, B.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, "Low-loss infrared dielectric material system for dual-range omnidirectional reflectivity," Opt. Lett. 26, 1370-1372 (2001).
[CrossRef]

Thomas, E. L.

Tsai, Jui-Yen

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Villa, F.

Wang, Weibiao

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

Winn, J. N.

Xia, Yuxue

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

Xu, Mai

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

Yarotsky, D. 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, 25-28 (1999).
[CrossRef]

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. Lightwave Technol. 17, 2018-2024 (1999).
[CrossRef]

Yeh, P.

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

Yu, Chang-Chin

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Zhao, H.

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

Ziari, M.

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

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, 25-28 (1999).
[CrossRef]

Appl. Phys. Lett.

Yeonsang Park, Young-Geun Roh, Chi-O Cho, and Heonsu Jeon, "GaAs-based near-infrared omnidirectional reflector," Appl. Phys. Lett. 82, 2770-2772 (2003).
[CrossRef]

Electron. Lett.

M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and W. H. Steier, "Wide-bandwidth distributed Bragg reflectors using oxide/GaAs multilayers," Electron. Lett. 30, 1147-1149 (1994).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

Chung-Hsiang Lin, Jui-Yen Tsai, Chih-Chiang Kao, Hao-Chung Kuo, Chang-Chin Yu, Jun-Ren Lo, and Kok-Ming Leung, "Enhanced light output in InGaN-based light emitting diodes with omnidirectional one-dimensional photonic crystals," Jpn. J. Appl. Phys. 45, 1591-1593 (2006).
[CrossRef]

Opt. Laser Technol.

Ming Chen, Chunfei Li, Mai Xu, Weibiao Wang, Shaojie Ma, and Yuxue Xia, "Eye-protection glasses against YAG laser injury based on the band gap reflection of one-dimensional photonic crystal," Opt. Laser Technol. 39, 214-218 (2007).
[CrossRef]

Opt. Lett.

Phys. Rev. B

I. Nusinsky and A. A. Hardy, "Band gap analysis of one-dimensional photonic crystals and conditions for gap closing," Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Science

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, "External reflection from omnidirectional dielectric mirror fibers," Science 296, 510 (2002).
[CrossRef] [PubMed]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Other

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

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Figures (7)

Fig. 1
Fig. 1

Electromagnetic wave propagation in one-dimensional photonic crystal for TE and TM polarizations.

Fig. 2
Fig. 2

Photonic band structure of one-dimensional photonic crystal with n L = 1.45 , n H = 3.5 and d L / d H = 8 / 3 , for TE (right) and TM (left) polarizations. The gray and white areas are for the allowed and forbidden regions, respectively. The white circles represent the gap closing points. The black area represents the region of omnidirectional reflection. The dashed lines are the light lines of air.

Fig. 3
Fig. 3

Schematic location of the closing points as function of the ratio q / p .

Fig. 4
Fig. 4

Range to midrange ratio Δ ω / ω 0 in percents for the first three omnidirectional gaps as function of n H and d L / Λ , for n L = 1.45 .

Fig. 5
Fig. 5

Boundaries of omnidirectional reflection within the mth gap (for m = 1 , 2 , 3 , 4 ), as a function of n H and d L / Λ , for n L = 1.45 (solid and dashed curves). Solid curves are for omnidirectional gaps that have no closing points, whereas dashed curves represent omnidirectional gaps that have at least one closing point outside the line cone. Curves with circles show the necessary conditions for M omnidirectional gaps, defined by condition (9).

Fig. 6
Fig. 6

Photonic band structure of one-dimensional photonic crystal with n H = 3.5 , n L = 1.45 , and d L / d H = 1 / 1 . The notations are as in Fig. 2.

Fig. 7
Fig. 7

Reflection coefficient for TE (solid curves) and TM (dotted curves) polarizations, at several angles of incidence. The gray regions indicate the ranges of omnidirectional reflection. The parameters of photonic crystal are the same as in Fig. 6 and d L = d H = 182.5   nm .

Equations (23)

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

d L p n L 2 n 0 2 sin 2 θ 0 = d H q n H 2 n 0 2 sin 2 θ 0 ,
sin θ 0 = 1 n 0 q 2 n H 2 d H 2 p 2 n L 2 d L 2 q 2 d H 2 p 2 d L 2 .
d L n L 2 n 0 2 d H n H 2 n 0 2 < q p < d L n L d H n H .
0 < q p < d L n L 2 n 0 2 d H n H 2 n 0 2 .
q p > d L n L d H n H ,
d L n L d H n H < 1.
d L n L 2 n 0 2 d H n H 2 n 0 2 > 1.
d L n L d H n H < 1 / 2 ,
1 / 2 < d L n L 2 n 0 2 d H n H 2 n 0 2 < d L n L d H n H < 2 ,
d L n L 2 n 0 2 d H n H 2 n 0 2 > 2.
d L n L d H n H < 1 M 1 .
ω TM , m ( 90 ° ) < ω m + ( 0 ° ) ,
ω m ± ( θ 0 ) = c δ [ m π ± 2 r m ± ( θ 0 ) ] , m = 1 , 2 , 3 ,
r m + ( 0 ° ) δ ( 0 ° ) + r TM , m ( 90 ° ) δ ( 90 ° ) > m π 2 [ 1 δ ( 90 ° ) 1 δ ( 0 ° ) ] ,
d L n L d H n H < 1 M 1 ,
i + 1 M ( i + 1 ) < d L n L 2 n 0 2 d H n H 2 n 0 2 < d L n L d H n H < i + 2 M ( i + 2 ) , i = 0 , 1 ,   .   .   .   , M 3 ,
d L n L 2 n 0 2 d H n H 2 n 0 2 > M 1.
sin r 2 s ± = | sin γ δ ( s π ± r 2 s ± ) | B 2 B   for   s = 1 , 2 ,   .   .   .   ,
sin r 2 s + 1 ± = | cos γ δ ( s π + π 2 ± r 2 s + 1 ± ) | B 2 B   for   s = 0 , 1 , 2 ,   .   .   .   ,
B = { 1 + 1 2 ( n L cos θ L n H cos θ H + n H cos θ H n L cos θ L ) TE 1 + 1 2 ( n H cos θ L n L cos θ H + n L cos θ H n H cos θ L ) TM ,
δ = n L d L cos θ L + n H d H cos θ H ,
γ = n L d L cos θ L n H d H cos θ H ,
n 0 sin θ 0 = n L sin θ L = n H sin θ H .

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