Abstract

Here we demonstrate by theoretical analysis a novel way to enhance the omnidirectional total-reflection wavelength range in one-dimensional photonic bandgap material by using a ternary periodic structure (i.e., three material layers constituting a period of the lattice). The omnidirectional total reflection range using a binary (BaF2PbS) periodic structure was enhanced by 108nm when the structure was modified by sandwiching a thin layer of ZrO2 between every two layers, constituting a period of the lattice. A shift of the omnidirectional range toward higher wavelengths was also observed. When the sandwiched layer was CeF3, the enhancement in the range was 120nm. As the sandwiched layer is very thin, there will not be any significant increase in the size of the reflector, contrary to the case of heterostructured photonic crystals (PCs) where two or more PCs are clubbed together to achieve enhancement.

© 2006 Optical Society of America

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  1. D. N. Chigrin, 'Omnidirectional Bragg mirror,' in Electromagnetic Waves Propagation in Photonic Crystals with Incomplete Photonic Band Gap, Thesis (Univ. Wuppertal, 2003), pp. 62-75.
  2. M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, 'Dielectric omnidirectional visible reflector,' Opt. Lett. 26, 1197-1199 (2001).
    [CrossRef]
  3. 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]
  4. 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]
  5. S. K. Singh, K. B. Thapa, J. P. Pandey, and S. P. Ojha, 'Si/SiO2 one-dimensional omnidirectional photonic crystals,' in Proceedings of International Conference on Optics and Optoelectronics (IRDE Dehradun, 2005), PP-PBS-6.
  6. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
    [CrossRef]
  7. E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
    [CrossRef]
  8. J. A. E. Wasey and W. L. Barnes, 'Efficiency of spontaneous emission from planar microcavities,' J. Mod. Opt. 47, 725-741 (2000).
    [CrossRef]
  9. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, 'Guiding optical light in air using an all-dielectric structure,' J. Lightwave Technol. 17, 2039-2041 (1999).
    [CrossRef]
  10. 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]
  11. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (2002).
    [CrossRef] [PubMed]
  12. I. Abdulhalim, 'Reflective polarization conversion Fabry-Perot resonator using omnidirectional mirror of periodic anisotropic stack,' Opt. Commun. 215, 225-230 (2003).
    [CrossRef]
  13. S.-S. Lo, M.-S. Wang, and C.-C. Chen, 'Semiconductor hollow optical waveguides formed by omnidirectional reflectors,' Opt. Express 12, 6589-6593 (2004).
    [CrossRef] [PubMed]
  14. M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
    [CrossRef]
  15. M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), pp. 1-70.
  16. M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), p. 60.

2005 (1)

E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
[CrossRef]

2004 (2)

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
[CrossRef]

S.-S. Lo, M.-S. Wang, and C.-C. Chen, 'Semiconductor hollow optical waveguides formed by omnidirectional reflectors,' Opt. Express 12, 6589-6593 (2004).
[CrossRef] [PubMed]

2003 (1)

I. Abdulhalim, 'Reflective polarization conversion Fabry-Perot resonator using omnidirectional mirror of periodic anisotropic stack,' Opt. Commun. 215, 225-230 (2003).
[CrossRef]

2002 (2)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

2001 (1)

2000 (2)

J. A. E. Wasey and W. L. Barnes, 'Efficiency of spontaneous emission from planar microcavities,' J. Mod. Opt. 47, 725-741 (2000).
[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]

1999 (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]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, 'Guiding optical light in air using an all-dielectric structure,' J. Lightwave Technol. 17, 2039-2041 (1999).
[CrossRef]

1998 (1)

Abdulhalim, I.

I. Abdulhalim, 'Reflective polarization conversion Fabry-Perot resonator using omnidirectional mirror of periodic anisotropic stack,' Opt. Commun. 215, 225-230 (2003).
[CrossRef]

Barnes, W. L.

J. A. E. Wasey and W. L. Barnes, 'Efficiency of spontaneous emission from planar microcavities,' J. Mod. Opt. 47, 725-741 (2000).
[CrossRef]

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), pp. 1-70.

M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), p. 60.

Chen, C.

Chen, C.-C.

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, 'Omnidirectional Bragg mirror,' in Electromagnetic Waves Propagation in Photonic Crystals with Incomplete Photonic Band Gap, Thesis (Univ. Wuppertal, 2003), pp. 62-75.

Deopura, M.

Fan, S.

Ferre-Borrull, J.

E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
[CrossRef]

Fink, Y.

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]

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Hu, X.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Ibanescu, M.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
[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]

Jia, W.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Joannopoulos, J. D.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
[CrossRef]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (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]

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, 'Guiding optical light in air using an all-dielectric structure,' J. Lightwave Technol. 17, 2039-2041 (1999).
[CrossRef]

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]

Johnson, S. G.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
[CrossRef]

Lavrinenko, A. 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]

Li, Y.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Liu, X.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Lo, S.-S.

Marsal, L. F.

E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
[CrossRef]

Ojha, S. P.

S. K. Singh, K. B. Thapa, J. P. Pandey, and S. P. Ojha, 'Si/SiO2 one-dimensional omnidirectional photonic crystals,' in Proceedings of International Conference on Optics and Optoelectronics (IRDE Dehradun, 2005), PP-PBS-6.

Pallares, J.

E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
[CrossRef]

Pandey, J. P.

S. K. Singh, K. B. Thapa, J. P. Pandey, and S. P. Ojha, 'Si/SiO2 one-dimensional omnidirectional photonic crystals,' in Proceedings of International Conference on Optics and Optoelectronics (IRDE Dehradun, 2005), PP-PBS-6.

Povinelli, M. L.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
[CrossRef]

Ripin, D. J.

Singh, S. K.

S. K. Singh, K. B. Thapa, J. P. Pandey, and S. P. Ojha, 'Si/SiO2 one-dimensional omnidirectional photonic crystals,' in Proceedings of International Conference on Optics and Optoelectronics (IRDE Dehradun, 2005), PP-PBS-6.

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, 'Dielectric omnidirectional visible reflector,' Opt. Lett. 26, 1197-1199 (2001).
[CrossRef]

Thapa, K. B.

S. K. Singh, K. B. Thapa, J. P. Pandey, and S. P. Ojha, 'Si/SiO2 one-dimensional omnidirectional photonic crystals,' in Proceedings of International Conference on Optics and Optoelectronics (IRDE Dehradun, 2005), PP-PBS-6.

Thomas, E. L.

Ullal, C. K.

Wang, M.-S.

Wang, X.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Wasey, J. A. E.

J. A. E. Wasey and W. L. Barnes, 'Efficiency of spontaneous emission from planar microcavities,' J. Mod. Opt. 47, 725-741 (2000).
[CrossRef]

Winn, J. N.

Wolf, E.

M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), p. 60.

M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), pp. 1-70.

Xifre-Perez, E.

E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
[CrossRef]

Xu, C.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[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]

Zi, J.

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

Appl. Phys. A (1)

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. (2)

X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, 'Enlargement of omni-directional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures,' Appl. Phys. Lett. 80, 4291-4293 (2002).
[CrossRef]

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, 'Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,' Appl. Phys. Lett. 85, 1466-1468 (2004).
[CrossRef]

J. Appl. Phys. (1)

E. Xifre-Perez, L. F. Marsal, J. Pallares, and J. Ferre-Borrull, 'Porous silicon mirrors with enlarged omnidirectional band gap,' J. Appl. Phys. 97, 064503-1-064503-5 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

J. A. E. Wasey and W. L. Barnes, 'Efficiency of spontaneous emission from planar microcavities,' J. Mod. Opt. 47, 725-741 (2000).
[CrossRef]

Nature (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, 'Wavelength-scalable hollow optical fibers with large photonic band gaps for Co2 laser transmission,' Nature 420, 650-653 (2002).
[CrossRef] [PubMed]

Opt. Commun. (1)

I. Abdulhalim, 'Reflective polarization conversion Fabry-Perot resonator using omnidirectional mirror of periodic anisotropic stack,' Opt. Commun. 215, 225-230 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Science (1)

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 (4)

D. N. Chigrin, 'Omnidirectional Bragg mirror,' in Electromagnetic Waves Propagation in Photonic Crystals with Incomplete Photonic Band Gap, Thesis (Univ. Wuppertal, 2003), pp. 62-75.

S. K. Singh, K. B. Thapa, J. P. Pandey, and S. P. Ojha, 'Si/SiO2 one-dimensional omnidirectional photonic crystals,' in Proceedings of International Conference on Optics and Optoelectronics (IRDE Dehradun, 2005), PP-PBS-6.

M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), pp. 1-70.

M. Born and E. Wolf, 'Basic properties of the electromagnetic field,' in Principles of Optics (Cambridge U. Press, 1980), p. 60.

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Figures (8)

Fig. 1
Fig. 1

Depiction of a one-dimensional binary periodic lattice.

Fig. 2
Fig. 2

Reflectivity spectra of PC1 at various angles of incidence. The solid (dashed) curves are for TE (TM) polarization. The gray area is the total omnidirectional bandgap ( n L = 1.46 , n H = 4.35 , n A = 1.0 , n S = 1.52 , d L = 250 nm , η opt 1 = 0.56 , N = 25 ).

Fig. 3
Fig. 3

Photonic bandgap structure of PC1 in terms of wavelength and incident angle. The solid (dotted) curves are for TE (TM) polarization bands. The gray area is the total omnidirectional band gap ( n L = 1.46 , n H = 4.35 , n A = 1.0 , n S = 1.52 , d L = 250 nm , η opt 1 = 0.56 , N = 25 ).

Fig. 4
Fig. 4

Depiction of a one-dimensional ternary periodic lattice.

Fig. 5
Fig. 5

Reflectivity spectra of PC2 at various angles of incidence. The solid (dashed) curves are for TE (TM) polarization. The gray area is the total omnidirectional bandgap ( n L = 1.46 , n C = 2.07 , n H = 4.35 , n A = 1.0 , n S = 1.52 , d L = 250 nm , η opt 2 = 0.045 , N = 25 ).

Fig. 6
Fig. 6

Photonic bandgap structure of PC2 in terms of wavelength and incident angle. The solid (dotted) curves are for TE (TM) polarization bands. The gray area is the total omnidirectional bandgap ( n L = 1.46 , n C = 2.07 , n H = 4.35 , n A = 1.0 , n S = 1.52 , d L = 250 nm , η opt 2 = 0.045 , N = 25 ).

Fig. 7
Fig. 7

Reflectivity spectra of PC3 at various angles of incidence. The solid (dashed) curves are for TE (TM) polarization. The gray area is the total omnidirectional bandgap ( n L = 1.46 , n C = 1.60 , n H = 4.35 , n A = 1.0 , n S = 1.52 , d L = 250 nm , η opt 3 = 0.06 , N = 25 ).

Fig. 8
Fig. 8

Photonic bandgap structure of PC3 in terms of wavelength and incident angle. The solid (dotted) curves are for TE- (TM-) polarization bands. The gray area is the total omnidirectional bandgap ( n L = 1.46 , n C = 1.60 , n H = 4.35 , n A = 1.0 , n S = 1.52 , d L = 250 nm , η opt 3 = 0.06 , N = 25 ).

Equations (19)

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

M [ d ] = i = 1 k ( cos γ i i p i sin γ i i p i sin γ i cos γ i ) ( M 11 M 12 M 21 M 22 ) ,
cos θ i = ( 1 n A 2 sin 2 θ A n i 2 ) 1 2 .
[ M ( d ) ] N = [ M 11 U N 1 ( a ) U N 2 ( a ) M 12 U N 1 ( a ) M 21 U N 1 ( a ) M 22 U N 1 ( a ) U N 2 ( a ) ] ( m 11 m 12 m 21 m 22 ) ,
M 11 = ( cos γ 1 cos γ 2 p 2 p 1 sin γ 1 sin γ 2 ) ,
M 12 = i ( 1 p 2 cos γ 1 sin γ 2 + 1 p 1 sin γ 1 cos γ 2 ) ,
M 21 = i ( p 1 sin γ 1 cos γ 2 + p 2 cos γ 1 sin γ 2 ) ,
M 22 = ( cos γ 1 cos γ 2 p 1 p 2 sin γ 1 sin γ 2 ) .
U N ( a ) = sin [ ( N + 1 ) cos 1 a ] ( 1 a 2 ) 1 2 ,
a = 1 2 ( M 11 + M 22 ) .
r = R A = ( m 11 + m 12 p s ) p A ( m 21 + m 22 p s ) ( m 11 + m 12 p s ) p A + ( m 21 + m 22 p s ) ,
p A = n A cos θ A , p s = n s cos θ s = n s ( 1 n A 2 sin 2 θ A n s 2 ) 1 2 .
R = r 2 = ( r r * ) .
p i = cos θ i n i ,
p A = cos θ A n A ,
p s = cos θ s n s = ( 1 n A 2 sin 2 θ A n s 2 ) 1 2 n s .
M 11 = ( cos γ 1 cos γ 2 cos γ 3 p 2 p 1 sin γ 1 sin γ 2 cos γ 3 p 3 p 2 cos γ 1 sin γ 2 sin γ 3 p 3 p 1 sin γ 1 cos γ 2 sin γ 3 ) ,
M 12 = i ( 1 p 1 sin γ 1 cos γ 2 cos γ 3 + 1 p 2 cos γ 1 sin γ 2 cos γ 3 + 1 p 3 cos γ 1 cos γ 2 sin γ 3 p 2 p 1 p 3 sin γ 1 sin γ 2 sin γ 3 ) ,
M 21 = i ( p 1 sin γ 1 cos γ 2 cos γ 3 + p 2 cos γ 1 sin γ 2 cos γ 3 + p 3 cos γ 1 cos γ 2 sin γ 3 p 1 p 3 p 2 sin γ 1 sin γ 2 sin γ 3 ) ,
M 22 = ( cos γ 1 cos γ 2 cos γ 3 p 1 p 2 sin γ 1 sin γ 2 cos γ 3 p 2 p 3 cos γ 1 sin γ 2 sin γ 3 p 1 p 3 sin γ 1 cos γ 2 sin γ 3 ) .

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