Abstract

We propose and investigate the use of slanted surface-relief gratings with nonbinary profiles as high-efficiency broadband couplers for light guides. First, a Chandezon-method-based rigorous numerical formulation is presented for modeling the slanted gratings with overhanging profiles. Then, two typical types of slanted grating couplers—a sinusoidal one and a trapezoidal one—are studied and optimized numerically, both exhibiting a high coupling efficiency of over 50% over the full band of white LED under the normal illumination of unpolarized light. Reasonable structural parameters with nice tolerance have been obtained for the optimized designs. It is found that the performance of the couplers depends little on the grating profile shape, but primarily on the grating period and the slant angle of the ridge. The underlying mechanism is analyzed by the equivalence rules of gratings, which provide useful guidelines for the design and fabrication of the couplers. Preliminary investigation has been performed on the fabrication and replication of the slanted overhanging grating couplers, which shows the feasibility of fabrication with mature microfabrication techniques and the perspective for mass production.

© 2010 Optical Society of America

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References

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  1. K. Chien and H. D. Shieh, “Time-multiplexed three-dimensional displays based on directional backlights with fast-switching liquid-crystal displays,” Appl. Opt. 45, 3106–3110 (2006).
    [CrossRef] [PubMed]
  2. T. Levola, “Diffractive optics for virtual reality displays,” J. Soc. Inf. Disp. 14, 467–475 (2006).
    [CrossRef]
  3. S. Siitonen, P. Laakkonen, P. Vahimaa, M. Kuittinen, and N. Tossavainen, “White LED light coupling into light guides with diffraction gratings,” Appl. Opt. 45, 2623–2630 (2006).
    [CrossRef] [PubMed]
  4. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [CrossRef]
  5. M. Li and S. J. Sheard, “Waveguide couplers using parallelogramic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
    [CrossRef]
  6. M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, and E. Cambril, “Design and fabrication of slanted binary surface relief gratings for a planar optical interconnection,” Appl. Opt. 36, 5717–5727 (1997).
    [CrossRef] [PubMed]
  7. B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating couplers,” Opt. Express 12, 3313–3326 (2004).
    [CrossRef] [PubMed]
  8. S. Siitonen, P. Laakkonen, P. Vahimaa, K. Jefimovs, M. Kuittinen, M. Parikka, M. Mönkkönen, and A. Orpana, “Coupling of light from an LED into a thin light guide by diffractive gratings,” Appl. Opt. 43, 5631–5636 (2004).
    [CrossRef] [PubMed]
  9. S. Wu, E. N. Glytsis, and T. K. Gaylord, “Optimization of finite-length input volume holographic grating couplers illuminated by finite-width incident beams,” Appl. Opt. 44, 4435–4446 (2005).
    [CrossRef] [PubMed]
  10. J. Liu, R. T. Chen, B. M. Davies, and L. Li, “Modeling and design of planar slanted volume holographic gratings for wavelength-division-multiplexing applications,” Appl. Opt. 38, 6981–6986 (1999).
    [CrossRef]
  11. B. Chernov, M. Nevière, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
    [CrossRef]
  12. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  13. J. P. Plumey and G. Granet, “Generalization of the coordinate transformation method with application to surface-relief gratings,” J. Opt. Soc. Am. A 16, 508–516 (1999).
    [CrossRef]
  14. J. P. Plumey, B. Guizal, and J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
    [CrossRef]
  15. T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).
  16. L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am A 16, 2521–2531 (1999).
    [CrossRef]
  17. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  18. N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
    [CrossRef]
  19. C. Elsner, J. Dienelt, and D. Hirch, “3D-microstructure replication processes using UV-curable acrylates,” Microelectron. Eng. 65, 163–170 (2003).
    [CrossRef]
  20. T. Levola and P. Laakkonen, “Replicated slanted gratings with a high refractive index material for in and outcoupling of light,” Opt. Express 15, 2067–2074 (2007).
    [CrossRef] [PubMed]
  21. M. Gale, “Replication techniques for diffractive optical elements,” Microelectron. Eng. 34, 321–339 (1997).
    [CrossRef]
  22. E.D.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).
  23. M. Breidne and D. Maystre, “Equivalence of ruled, holographic, and lamellar gratings in constant deviation mountings,” Appl. Opt. 19, 1812–1821 (1980).
    [CrossRef] [PubMed]
  24. The trapezoidal profile a(x˜1) should be an odd function so that it could be expanded into a sine series. For this purpose, we have made proper translation of the origin of coordinate system Ox˜1x˜2x˜3 and assumed w=Δ+d/2 to fulfill this requirement. This change to the studied trapezoidal profile is negligibly small.

2007 (1)

2006 (3)

2005 (1)

2004 (2)

2003 (1)

C. Elsner, J. Dienelt, and D. Hirch, “3D-microstructure replication processes using UV-curable acrylates,” Microelectron. Eng. 65, 163–170 (2003).
[CrossRef]

2002 (1)

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

2001 (1)

B. Chernov, M. Nevière, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

1999 (3)

1997 (4)

J. P. Plumey, B. Guizal, and J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
[CrossRef]

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).

M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, and E. Cambril, “Design and fabrication of slanted binary surface relief gratings for a planar optical interconnection,” Appl. Opt. 36, 5717–5727 (1997).
[CrossRef] [PubMed]

M. Gale, “Replication techniques for diffractive optical elements,” Microelectron. Eng. 34, 321–339 (1997).
[CrossRef]

1996 (1)

1994 (1)

M. Li and S. J. Sheard, “Waveguide couplers using parallelogramic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

1982 (1)

1980 (1)

1977 (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Bendfeldt, L.

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

Breidne, M.

Cambril, E.

Chandezon, J.

Chavel, P.

Chen, R. T.

Chernov, B.

B. Chernov, M. Nevière, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

Chien, K.

Cornet, G.

Davies, B. M.

de Beaucoudrey, N.

Dienelt, J.

C. Elsner, J. Dienelt, and D. Hirch, “3D-microstructure replication processes using UV-curable acrylates,” Microelectron. Eng. 65, 163–170 (2003).
[CrossRef]

Dupuis, M. T.

Elsner, C.

C. Elsner, J. Dienelt, and D. Hirch, “3D-microstructure replication processes using UV-curable acrylates,” Microelectron. Eng. 65, 163–170 (2003).
[CrossRef]

Fink, M.

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

Gale, M.

M. Gale, “Replication techniques for diffractive optical elements,” Microelectron. Eng. 34, 321–339 (1997).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Granet, G.

Guizal, B.

Harris, J. B.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).

Hirch, D.

C. Elsner, J. Dienelt, and D. Hirch, “3D-microstructure replication processes using UV-curable acrylates,” Microelectron. Eng. 65, 163–170 (2003).
[CrossRef]

Jefimovs, K.

Jiang, J.

Kuittinen, M.

Laakkonen, P.

Levola, T.

Li, L.

Li, M.

M. Li and S. J. Sheard, “Waveguide couplers using parallelogramic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

Liu, J.

Maystre, D.

Miller, M.

Mönkkönen, M.

Nevière, M.

B. Chernov, M. Nevière, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

Nordin, G. P.

Orpana, A.

Parikka, M.

Peng, S. T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Pfeiffer, K.

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

Plumey, J. P.

Popov, E.

B. Chernov, M. Nevière, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

Preist, T. W.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).

Roos, N.

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

Sambles, J. R.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).

Scheer, H. C.

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

Schulz, H.

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

Sheard, S. J.

M. Li and S. J. Sheard, “Waveguide couplers using parallelogramic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

Shieh, H. D.

Siitonen, S.

Tamir, T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Tossavainen, N.

Turunen, J.

Vahimaa, P.

Wang, B.

Wanstall, N. P.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).

Wu, S.

Appl. Opt. (7)

K. Chien and H. D. Shieh, “Time-multiplexed three-dimensional displays based on directional backlights with fast-switching liquid-crystal displays,” Appl. Opt. 45, 3106–3110 (2006).
[CrossRef] [PubMed]

S. Siitonen, P. Laakkonen, P. Vahimaa, M. Kuittinen, and N. Tossavainen, “White LED light coupling into light guides with diffraction gratings,” Appl. Opt. 45, 2623–2630 (2006).
[CrossRef] [PubMed]

S. Siitonen, P. Laakkonen, P. Vahimaa, K. Jefimovs, M. Kuittinen, M. Parikka, M. Mönkkönen, and A. Orpana, “Coupling of light from an LED into a thin light guide by diffractive gratings,” Appl. Opt. 43, 5631–5636 (2004).
[CrossRef] [PubMed]

S. Wu, E. N. Glytsis, and T. K. Gaylord, “Optimization of finite-length input volume holographic grating couplers illuminated by finite-width incident beams,” Appl. Opt. 44, 4435–4446 (2005).
[CrossRef] [PubMed]

J. Liu, R. T. Chen, B. M. Davies, and L. Li, “Modeling and design of planar slanted volume holographic gratings for wavelength-division-multiplexing applications,” Appl. Opt. 38, 6981–6986 (1999).
[CrossRef]

M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, and E. Cambril, “Design and fabrication of slanted binary surface relief gratings for a planar optical interconnection,” Appl. Opt. 36, 5717–5727 (1997).
[CrossRef] [PubMed]

M. Breidne and D. Maystre, “Equivalence of ruled, holographic, and lamellar gratings in constant deviation mountings,” Appl. Opt. 19, 1812–1821 (1980).
[CrossRef] [PubMed]

Appl. Phys. (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

J. Mod. Opt. (1)

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073–1080(1997).

J. Opt. Soc. Am A (1)

L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am A 16, 2521–2531 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Soc. Inf. Disp. (1)

T. Levola, “Diffractive optics for virtual reality displays,” J. Soc. Inf. Disp. 14, 467–475 (2006).
[CrossRef]

Microelectron. Eng. (3)

N. Roos, H. Schulz, L. Bendfeldt, M. Fink, K. Pfeiffer, and H. C. Scheer, “First and second generation purely thermoset stamps for hot embossing,” Microelectron. Eng. 61–62, 399–405 (2002).
[CrossRef]

C. Elsner, J. Dienelt, and D. Hirch, “3D-microstructure replication processes using UV-curable acrylates,” Microelectron. Eng. 65, 163–170 (2003).
[CrossRef]

M. Gale, “Replication techniques for diffractive optical elements,” Microelectron. Eng. 34, 321–339 (1997).
[CrossRef]

Opt. Commun. (2)

M. Li and S. J. Sheard, “Waveguide couplers using parallelogramic-shaped blazed gratings,” Opt. Commun. 109, 239–245 (1994).
[CrossRef]

B. Chernov, M. Nevière, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

Opt. Express (2)

Other (2)

E.D.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).

The trapezoidal profile a(x˜1) should be an odd function so that it could be expanded into a sine series. For this purpose, we have made proper translation of the origin of coordinate system Ox˜1x˜2x˜3 and assumed w=Δ+d/2 to fulfill this requirement. This change to the studied trapezoidal profile is negligibly small.

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Figures (10)

Fig. 1
Fig. 1

Geometries of several grating couplers: (a) binary grating, (b) binary slanted grating, (c) asymmetric-profile triangular grating, (d) slanted overhanging sinusoidal grating, and (e) slanted overhanging trapezoidal grating. In (d) and (e), the grating parameters are indicated; the dashed and solid arrows mean that most of the incident energy is coupled to the 1 st transmitted order.

Fig. 2
Fig. 2

Dependence of η 1 of the slanted sinusoidal grating on d and b at three RGB wavelengths under the normal incidence of unpolarized light.

Fig. 3
Fig. 3

Dependence of η 1 of the slanted sinusoidal grating on d and ξ at three RGB wavelengths under the normal incidence of unpolarized light.

Fig. 4
Fig. 4

Coupling efficiency η 1 of the optimized slanted sinusoidal grating with b = 450 nm , ξ = 24 ° , and d = 580 nm under TE, TM, and unpolarized illumination.

Fig. 5
Fig. 5

Dependence of η 1 of the slanted sinusoidal grating on the incident angle θ for TE, TM, and unpolarized light in the radiance spectrum of a white LED.

Fig. 6
Fig. 6

Dependence of η 1 of the slanted trapezoidal grating on d and h at three RGB wavelengths under the normal incidence of unpolarized light.

Fig. 7
Fig. 7

Same as Fig. 3, but for the slanted trapezoidal grating.

Fig. 8
Fig. 8

Same as Fig. 4, but for the slanted trapezoidal grating with d = 570 nm , h = 600 nm , ξ = 25 ° , w = d / 2 , and Δ = 30 nm .

Fig. 9
Fig. 9

Schematic of the fabrication process of the slanted overhanging grating coupler.

Fig. 10
Fig. 10

Side view SEM image of a replicated slanted over hanging grating sample.

Tables (1)

Tables Icon

Table 1 Comparison of the Performance of Three Slanted Gratings

Equations (26)

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x ˜ 1 = x y tan ξ , x ˜ 2 = y sec ξ , x ˜ 3 = z ;
x 1 = x ˜ 1 , x 2 = x ˜ 2 a ( x ˜ 1 ) , x 3 = x ˜ 3 .
x ˜ 2 = a ( x ˜ 1 ) = b sin ( K x ˜ 1 ) ,
x ˜ 2 = a ( x ˜ 1 ) = { h x ˜ 1 / Δ ( 0 x ˜ 1 < Δ ) h ( Δ x ˜ 1 < w Δ ) h ( w x ˜ 1 ) / Δ ( w Δ x ˜ 1 < w ) 0 ( w x ˜ 1 < d ) .
a n sin = { b ( n = 1 ) 0 ( otherwise )
a n trap = { 2 h d π 2 n 2 Δ sin ( n 2 K Δ ) ( if   n   is odd ) 0 ( if   n   is even )
a ˙ n sin = { b K / 2 ( n = ± 1 ) 0 ( otherwise ) ,
a ˙ n trap = { 0 ( n = 0 ) i h 2 π n Δ { exp ( i n K Δ ) 1 exp ( i n K w ) + exp [ i n K ( w Δ ) ] } ( otherwise ) ,
g ( x 1 ) = sec 2 ξ [ 1 ( a ˙ + sin ξ ) 0 ( a ˙ + sin ξ ) 1 + 2 a ˙ sin ξ + a ˙ 2 0 0 0 cos 2 ξ ] ,
G = sec 2 ξ [ I ( [ [ a ˙ ] ] + sin ξ · I ) O ( [ [ a ˙ ] ] + sin ξ · I ) cos 2 ξ · I + ( [ [ a ˙ ] ] + sin ξ · I ) 2 O O O cos 2 ξ · I ] ,
[ [ a ˙ ] ] m n = a ˙ m n = 1 d 0 d a ˙ ( x 1 ) exp [ i ( m n ) K x 1 ] d x 1 ,
[ O I ( G 22 ) 1 Γ ± ( G 22 ) 1 ( [ α ] G 12 + G 21 [ α ] ) ] ( F 3 ± F 3 ± ) = ρ ( F 3 ± F 3 ± ) ,
E 3 ( + ) = I 3 e exp [ i ( α 0 x ˜ 1 + β 0 ( + ) x ˜ 2 ) ] + n U + R n e exp [ i ( α n x ˜ 1 + β n ( + ) + x ˜ 2 ) ] + m exp ( i α m x 1 ) q V + F 3 m q + exp ( i ρ q + x 2 ) u q e ,
H 3 ( + ) = I 3 h exp [ i ( α 0 x ˜ 1 + β 0 ( + ) x ˜ 2 ) ] + n U + R n h exp [ i ( α n x ˜ 1 + β n ( + ) + x ˜ 2 ) ] + m exp ( i α m x 1 ) q V + F 3 m q + exp ( i ρ q + x 2 ) u q h ,
E 3 ( ) = n U T n e exp [ i ( α n x ˜ 1 + β n ( ) x ˜ 2 ) ] + m exp ( i α m x 1 ) q V F 3 m q exp ( i ρ q x 2 ) d q e ,
H 3 ( ) = n U T n h exp [ i ( α n x ˜ 1 + β n ( ) x ˜ 2 ) ] + m exp ( i α m x 1 ) q V F 3 m q exp ( i ρ q x 2 ) d q h ,
( E 1 ( ± ) H 1 ( ± ) ) = [ i τ 3 ± 1 τ 2 ± M τ 1 ± M i τ 3 ± 1 ] ( E 3 ( ± ) H 3 ( ± ) ) ,
E 3 ( + ) = m exp ( i α m x 1 ) { L m [ β 0 ( + ) ] exp [ i β 0 ( + ) x 2 ] I 3 e + n U + L m n [ β n ( + ) + ] exp [ i β n ( + ) + x 2 ] R n e + q V + F 3 m q + exp ( i ρ q + x 2 ) u q e } ,
E 3 ( ) = m exp ( i α m x 1 ) { n U L m n [ β n ( ) ] exp [ i β n ( ) x 2 ] T n e + q V F 3 m q exp ( i ρ q x 2 ) d q e } ,
L m ( ν ) = 1 d 0 d exp [ i ν a ( x 1 ) i m K x 1 ] d x 1 ,
L m ( ν ) = exp ( i m π 2 ) I m ( i ν b ) ,
( R n e u q e T n e d q e R n h u q h T n h d q h ) T = [ A + A O O O O A + A τ 3 + α A + τ 3 α A τ 2 + B + τ 2 B τ 1 + B + τ 1 B τ 3 + α A + τ 3 α A ] 1 [ I O O I τ 3 + α τ 2 + P τ 1 + P τ 3 + α ] ( I m e I m h ) ,
A ± = [ L m n ( β n ( ± ) ± ) F 3 m q ± ] , ϕ ± = diag [ β n ( ± ) ± ρ q ± ] , B ± = sec 4 ξ [ ( [ [ a ˙ ] ] + sin ξ ) α A ± ( 1 + 2 sin ξ [ [ a ˙ ] ] + [ [ a ˙ ] ] 2 ) A ± ϕ ± ] , P = sec 4 ξ [ ( [ [ a ˙ ] ] + sin ξ ) α ( 1 + 2 sin ξ [ [ a ˙ ] ] + [ [ a ˙ ] ] 2 ) β 0 ( + ) ] , I m e = I 3 e L m [ β 0 ( + ) ] , and I m h = I 3 h L m [ β 0 ( + ) ] .
η n ( + ) = ( β n ( + ) + α n sin ξ ) ( ε + | R n e | 2 + μ | R n h | 2 ) / k + 2 ,
η n ( ) = ( β n ( ) α n sin ξ ) ( ε | T n e | 2 + μ | T n h | 2 ) / k 2 ,
( β 0 ( + ) α 0 sin ξ ) ( ε + | I e | 2 + μ | I h | 2 ) = k + 2 .

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