Abstract

Two dimensional refractive index profiles of ion exchanged channel waveguides in glass have been measured using an interferometric method. In order to obtain depth data, a shallow bevel is produced in the glass by polishing. A regularization algorithm for the extraction of the phase data from the interferometer image is presented. The method is applied to waveguides formed by the electric field assisted diffusion of Cu+ ions into a borosilicate glass. The index change obtained from the interferometer is in good agreement with that obtained from measurements on planar waveguides.

© 2009 Optical Society of America

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  1. S. I. Najafi, ed., Introduction to Glass Integrated Optics (Artech, 1992).
  2. G. L. Yip, P. C. Noutsios, and L. Chen, “Improved propagation-mode near-field method for refractive index profiling of optical waveguides,” Appl. Opt. 35, 2060-2068 (1996).
    [CrossRef] [PubMed]
  3. J. Steffen, A. Neyer, E. Voges, and N. Hecking, “Refractive index profile measurement techniques by reflectivity profiling: vidicon imaging, beam scanning, and sample scanning,” Appl. Opt. 29, 4468-4472 (1990).
    [CrossRef] [PubMed]
  4. D. Jestel and E. Voges, “Refractive index profiling of ion-exchanged glass waveguides by RNF--measurements,” Proc. SPIE 1128, 87-89 (1989).
  5. R. A. Betts, F. Lui, and T. W. Whitbread, “Nondestructive two-dimensional refractive-index profiling of integrated-optical wave guides by an interferometric method,” Appl. Opt. 30, 4384-4389 (1991).
    [CrossRef] [PubMed]
  6. A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
    [CrossRef]
  7. X. P. Blanco, “Interferometric characterization and analysis of silver-exchanged glass waveguides buried by electromigration: slab, channel and slab-sided channel configurations,” J. Opt. A Pure Appl. Opt. 8, 123-133 (2006).
    [CrossRef]
  8. P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Determination of refractive index profiles of Ag+─Na+ ion-exchanged multimode strip waveguides by variable wavefront shear double-refracting interferometry microinterferometry,” Appl. Opt. 45, 756-763 (2006).
    [CrossRef] [PubMed]
  9. M. Sochacka, E. L. Lago, and Z. Jarosze, “Refractive-index profiling of planar gradient-index waveguides by phase-measuring microinterferometry,” Appl. Opt. 33, 3342-3344(1994).
    [CrossRef] [PubMed]
  10. A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133-138 (2008).
    [CrossRef]
  11. R. Oven and M. Yin, “Assessment of ion exchanged channel waveguides in glass using interference microscopy,” Opt. Commun. 260, 506-510 (2006).
    [CrossRef]
  12. R. Oven, “Surface expansion of channel waveguides formed by ion exchange in glass,” J. Appl. Phys. 100, 053513(2006).
    [CrossRef]
  13. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  14. J. Yanez-Mendiola, M. Servin, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-296(2000).
    [CrossRef]
  15. M. Servin and R. Rodriguez-Vera, “Two-dimensional phase locked loop demodulation of interferograms,” J. Mod. Opt. 40, 2087-2094 (1993).
    [CrossRef]
  16. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm,” Appl. Opt. 41, 5479-5487 (2002).
    [CrossRef] [PubMed]
  17. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 4540-4548(1997).
    [CrossRef] [PubMed]
  18. C. Galvan and M. Rivera, “Second-order robust regularization cost function for detecting and reconstructing phase discontinuities,” Appl. Opt. 45, 353-359 (2006).
    [CrossRef] [PubMed]
  19. J. C. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47, 5446-5453 (2008).
    [PubMed]
  20. H. Liu, N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850-857 (2004).
    [CrossRef] [PubMed]
  21. R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted, ion exchange in glass,” J. Phys. D 37, 2207-2215(2004).
    [CrossRef]
  22. M. Yin, R. Oven, and P. A. Davies, “Low-loss Cu+─Na+ ion exchanged optical channel waveguides in glass,” Electron. Lett. 40, 1265-1266 (2004).
    [CrossRef]

2008 (2)

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133-138 (2008).
[CrossRef]

J. C. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47, 5446-5453 (2008).
[PubMed]

2006 (5)

R. Oven and M. Yin, “Assessment of ion exchanged channel waveguides in glass using interference microscopy,” Opt. Commun. 260, 506-510 (2006).
[CrossRef]

R. Oven, “Surface expansion of channel waveguides formed by ion exchange in glass,” J. Appl. Phys. 100, 053513(2006).
[CrossRef]

X. P. Blanco, “Interferometric characterization and analysis of silver-exchanged glass waveguides buried by electromigration: slab, channel and slab-sided channel configurations,” J. Opt. A Pure Appl. Opt. 8, 123-133 (2006).
[CrossRef]

C. Galvan and M. Rivera, “Second-order robust regularization cost function for detecting and reconstructing phase discontinuities,” Appl. Opt. 45, 353-359 (2006).
[CrossRef] [PubMed]

P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Determination of refractive index profiles of Ag+─Na+ ion-exchanged multimode strip waveguides by variable wavefront shear double-refracting interferometry microinterferometry,” Appl. Opt. 45, 756-763 (2006).
[CrossRef] [PubMed]

2004 (3)

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted, ion exchange in glass,” J. Phys. D 37, 2207-2215(2004).
[CrossRef]

M. Yin, R. Oven, and P. A. Davies, “Low-loss Cu+─Na+ ion exchanged optical channel waveguides in glass,” Electron. Lett. 40, 1265-1266 (2004).
[CrossRef]

H. Liu, N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850-857 (2004).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

J. Yanez-Mendiola, M. Servin, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-296(2000).
[CrossRef]

1997 (2)

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 4540-4548(1997).
[CrossRef] [PubMed]

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

1996 (1)

1994 (1)

1993 (1)

M. Servin and R. Rodriguez-Vera, “Two-dimensional phase locked loop demodulation of interferograms,” J. Mod. Opt. 40, 2087-2094 (1993).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

D. Jestel and E. Voges, “Refractive index profiling of ion-exchanged glass waveguides by RNF--measurements,” Proc. SPIE 1128, 87-89 (1989).

1982 (1)

Basaran, C.

Betts, R. A.

Blahut, M.

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Blanco, X. P.

X. P. Blanco, “Interferometric characterization and analysis of silver-exchanged glass waveguides buried by electromigration: slab, channel and slab-sided channel configurations,” J. Opt. A Pure Appl. Opt. 8, 123-133 (2006).
[CrossRef]

Burton, D. R.

Cartwright, N.

Chen, L.

Cuevas, F. J.

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 4540-4548(1997).
[CrossRef] [PubMed]

Darudi, A.

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133-138 (2008).
[CrossRef]

Davies, P. A.

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted, ion exchange in glass,” J. Phys. D 37, 2207-2215(2004).
[CrossRef]

M. Yin, R. Oven, and P. A. Davies, “Low-loss Cu+─Na+ ion exchanged optical channel waveguides in glass,” Electron. Lett. 40, 1265-1266 (2004).
[CrossRef]

Galvan, C.

Gdeisat, M. A.

Gut, K.

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Hecking, N.

Hosseini, S. M. R. S.

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133-138 (2008).
[CrossRef]

Ina, H.

Jarosze, Z.

Jestel, D.

D. Jestel and E. Voges, “Refractive index profiling of ion-exchanged glass waveguides by RNF--measurements,” Proc. SPIE 1128, 87-89 (1989).

Karasinski, P.

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Kobayashi, S.

Lago, E. L.

Lalor, M. J.

Liu, H.

Lui, F.

Lukaszewicz, T.

Malacara-Hernandez, D.

J. Yanez-Mendiola, M. Servin, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-296(2000).
[CrossRef]

Marroquin, J. L.

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 4540-4548(1997).
[CrossRef] [PubMed]

Mrozek, E.

Mrozek, P.

Neyer, A.

Noutsios, P. C.

Opilski, A.

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Opilski, Z.

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Oven, R.

R. Oven, “Surface expansion of channel waveguides formed by ion exchange in glass,” J. Appl. Phys. 100, 053513(2006).
[CrossRef]

R. Oven and M. Yin, “Assessment of ion exchanged channel waveguides in glass using interference microscopy,” Opt. Commun. 260, 506-510 (2006).
[CrossRef]

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted, ion exchange in glass,” J. Phys. D 37, 2207-2215(2004).
[CrossRef]

M. Yin, R. Oven, and P. A. Davies, “Low-loss Cu+─Na+ ion exchanged optical channel waveguides in glass,” Electron. Lett. 40, 1265-1266 (2004).
[CrossRef]

Peng, H.

Rivera, M.

Rodriguez-Vera, R.

M. Servin and R. Rodriguez-Vera, “Two-dimensional phase locked loop demodulation of interferograms,” J. Mod. Opt. 40, 2087-2094 (1993).
[CrossRef]

Rogozinski, R.

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Servin, M.

J. Yanez-Mendiola, M. Servin, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-296(2000).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 4540-4548(1997).
[CrossRef] [PubMed]

M. Servin and R. Rodriguez-Vera, “Two-dimensional phase locked loop demodulation of interferograms,” J. Mod. Opt. 40, 2087-2094 (1993).
[CrossRef]

Sochacka, M.

Steffen, J.

Takeda, M.

Voges, E.

Whitbread, T. W.

Xu, J. C.

Xu, Q.

Yanez-Mendiola, J.

J. Yanez-Mendiola, M. Servin, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-296(2000).
[CrossRef]

Yin, M.

R. Oven and M. Yin, “Assessment of ion exchanged channel waveguides in glass using interference microscopy,” Opt. Commun. 260, 506-510 (2006).
[CrossRef]

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted, ion exchange in glass,” J. Phys. D 37, 2207-2215(2004).
[CrossRef]

M. Yin, R. Oven, and P. A. Davies, “Low-loss Cu+─Na+ ion exchanged optical channel waveguides in glass,” Electron. Lett. 40, 1265-1266 (2004).
[CrossRef]

Yip, G. L.

Appl. Opt. (10)

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 4540-4548(1997).
[CrossRef] [PubMed]

R. A. Betts, F. Lui, and T. W. Whitbread, “Nondestructive two-dimensional refractive-index profiling of integrated-optical wave guides by an interferometric method,” Appl. Opt. 30, 4384-4389 (1991).
[CrossRef] [PubMed]

M. Sochacka, E. L. Lago, and Z. Jarosze, “Refractive-index profiling of planar gradient-index waveguides by phase-measuring microinterferometry,” Appl. Opt. 33, 3342-3344(1994).
[CrossRef] [PubMed]

G. L. Yip, P. C. Noutsios, and L. Chen, “Improved propagation-mode near-field method for refractive index profiling of optical waveguides,” Appl. Opt. 35, 2060-2068 (1996).
[CrossRef] [PubMed]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm,” Appl. Opt. 41, 5479-5487 (2002).
[CrossRef] [PubMed]

H. Liu, N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850-857 (2004).
[CrossRef] [PubMed]

C. Galvan and M. Rivera, “Second-order robust regularization cost function for detecting and reconstructing phase discontinuities,” Appl. Opt. 45, 353-359 (2006).
[CrossRef] [PubMed]

P. Mrozek, E. Mrozek, and T. Lukaszewicz, “Determination of refractive index profiles of Ag+─Na+ ion-exchanged multimode strip waveguides by variable wavefront shear double-refracting interferometry microinterferometry,” Appl. Opt. 45, 756-763 (2006).
[CrossRef] [PubMed]

J. Steffen, A. Neyer, E. Voges, and N. Hecking, “Refractive index profile measurement techniques by reflectivity profiling: vidicon imaging, beam scanning, and sample scanning,” Appl. Opt. 29, 4468-4472 (1990).
[CrossRef] [PubMed]

J. C. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47, 5446-5453 (2008).
[PubMed]

Electron. Lett. (1)

M. Yin, R. Oven, and P. A. Davies, “Low-loss Cu+─Na+ ion exchanged optical channel waveguides in glass,” Electron. Lett. 40, 1265-1266 (2004).
[CrossRef]

J. Appl. Phys. (1)

R. Oven, “Surface expansion of channel waveguides formed by ion exchange in glass,” J. Appl. Phys. 100, 053513(2006).
[CrossRef]

J. Mod. Opt. (1)

M. Servin and R. Rodriguez-Vera, “Two-dimensional phase locked loop demodulation of interferograms,” J. Mod. Opt. 40, 2087-2094 (1993).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

X. P. Blanco, “Interferometric characterization and analysis of silver-exchanged glass waveguides buried by electromigration: slab, channel and slab-sided channel configurations,” J. Opt. A Pure Appl. Opt. 8, 123-133 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. D (1)

R. Oven, M. Yin, and P. A. Davies, “Characterization of planar optical waveguides formed by copper-sodium electric field assisted, ion exchange in glass,” J. Phys. D 37, 2207-2215(2004).
[CrossRef]

Opt. Commun. (2)

J. Yanez-Mendiola, M. Servin, and D. Malacara-Hernandez, “Iterative method to obtain the wrapped phase in an interferogram with a linear carrier,” Opt. Commun. 178, 291-296(2000).
[CrossRef]

R. Oven and M. Yin, “Assessment of ion exchanged channel waveguides in glass using interference microscopy,” Opt. Commun. 260, 506-510 (2006).
[CrossRef]

Opt. Eng. (1)

A. Opilski, R. Rogozinski, M. Blahut, P. Karasinski, K. Gut, and Z. Opilski, “Technology of ion exchange in glass and its application in waveguide planar sensors,” Opt. Eng. 36, 1625-1638 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

A. Darudi and S. M. R. S. Hosseini, “An interferometric method for refractive index profiling of planar gradient index waveguides,” Opt. Lasers Eng. 47, 133-138 (2008).
[CrossRef]

Proc. SPIE (1)

D. Jestel and E. Voges, “Refractive index profiling of ion-exchanged glass waveguides by RNF--measurements,” Proc. SPIE 1128, 87-89 (1989).

Other (1)

S. I. Najafi, ed., Introduction to Glass Integrated Optics (Artech, 1992).

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Figures (9)

Fig. 1
Fig. 1

Coordinate system with respect to the beveled sample and channel waveguide.

Fig. 2
Fig. 2

Schematic diagram of the interference microscope and sample.

Fig. 3
Fig. 3

Typical interferogram of a beveled channel waveguide.

Fig. 4
Fig. 4

Results using model refractive index profile (a) phase function and (b) integrated index function Δ N ( x , y ) . (c) Refractive index profile Δ n ( z , y ) determined from algorithm, (d) model refractive index profile Δ n ( z , y ) ..

Fig. 5
Fig. 5

Profile of bevel.

Fig. 6
Fig. 6

(a) Experimental phase function using no regularization. (b) Experimental phase function with regularization ( Γ = 0.2 ).

Fig. 7
Fig. 7

Experimental integrated index function Δ N ( x , y ) corresponding to the phase function of Fig. 6b.

Fig. 8
Fig. 8

Experimental refractive index profile Δ n ( z , y ) corresponding to the phase function of Fig. 6.

Fig. 9
Fig. 9

Convergence of algorithm.

Equations (22)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos ( ϕ ( x , y ) + ω x ) ,
ϕ ( x , y ) = Δ ϕ ( x , y ) + ϕ 0 ( x , y ) = [ 4 π λ x tan θ W Δ n ( z , y ) d z + 4 π λ ( n s n o ) x tan θ ] + ϕ 0 ( x , y ) ,
Δ ϕ ( x , Y ) = 4 π λ ( n s n o ) x tan θ .
λ 4 π [ Δ ϕ ( x , y ) Δ ϕ ( x , Y ) ] = Δ N ( x , y ) = x tan θ W Δ n ( z , y ) d z ,
Δ n ( z = x tan θ , y ) = 1 tan θ d d x Δ N ( x , y ) .
ϕ ( x , y ) = Δ ϕ ( x , y ) + ϕ 0 ( x , y ) = 4 π λ 0 W Δ n ( z , y ) d z + 4 π λ ( n s n o ) Δ h ( y ) + ϕ 0 ( x , y ) ,
λ 4 π [ Δ ϕ ( x , y ) Δ ϕ ( x , Y ) ] = Δ N ( 0 , y ) = 0 W Δ n ( z , y ) d z ,
ϕ = ϕ m , N + ϕ m , N + 1 ϕ m , N x N + 1 x N ( x x N ) for     x N < x < x N + 1 ,
S = n = 1 N D [ a m , N + b m , N cos ( ϕ m , N + ϕ m , N + 1 ϕ m , N x m , N + 1 x m , N ( x n x N ) + ω x n ) I m , n ] 2 + Γ N D [ r N ( ϕ m , N + 1 2 ϕ m , N + ϕ m , N 1 ) 2 + ( ϕ m + 1 , N 2 ϕ m , N + ϕ m 1 , N ) 2 + r N ( ϕ m 1 , N + 1 2 ϕ m , N + ϕ m + 1 , N 1 ) 2 + r N ( ϕ m + 1 , N + 1 2 ϕ m , N + ϕ m 1 , N 1 ) 2 ] ,
Δ n ( z , y ) = Δ n o 1 + exp ( z d α ) 1 2 [ erf ( y + W m / 2 d L ) erf ( y W m / 2 d L ) ] ,
Δ z 2 π Δ p f tan θ ω ,
c 1 , m , N = a m , N , c 2 , m , N = b m , N cos ϕ m , N , c 3 , m , N = b m , N sin ϕ m , N .
S c 1 , m , N = 0 , S c 2 , m , N = 0 , S c 3 , m , N = 0 ,
[ N D n = 1 N D cos ω x n n = 1 N D sin ω x n n = 1 N D cos ω x n n = 1 N D cos 2 ω x n n = 1 N D sin ω x n cos ω x n n = 1 N D sin ω x n n = 1 N D cos ω x n sin ω x n n = 1 N D sin 2 ω x n ] [ c 1 , m , N c 2 , m , N c 3 , m , N ] = [ n N D I m , n n N D I m , n cos ω x n n N D I m , n sin ω x n ] ,
φ m , N = tan 1 ( c 3 , m , N c 2 , m , N ) .
R = Γ N D [ r N { c 3 , m , N b m , N ( c 2 , m , N 1 b m , N 1 + c 2 , m , N + 1 b m , N + 1 ) c 2 , m , N b m , N ( c 3 , m , N 1 b , m , N 1 + c 3 , m , N + 1 b m , N + 1 ) } 2 + { c 3 , m , N b m , N ( c 2 , m 1 , N b m 1 , N + c 2 , m + 1 , N b m + 1 , N ) c 2 , m , N b m , N ( c 3 , m 1 , N b m 1 , N + c 3 , m + 1 , N b m + 1 , N ) } 2 + r N { c 3 , m , N b m , N ( c 2 , m 1 , N 1 b m 1 , N 1 + c 2 , m + 1 , N + 1 b m + 1 , N + 1 ) c 2 , m , N b m , N ( c 3 , m 1 , N 1 b m 1 , N 1 + c 3 , m + 1 , N + 1 b m + 1 , N + 1 ) } 2 + r N { c 3 , m , N b m , N ( c 2 , m + 1 , N 1 b m + 1 , N 1 + c 2 , m 1 , N + 1 b m 1 , N + 1 ) c 2 , m , N b m , N ( c 3 , m + 1 , N 1 b m + 1 , N 1 + c 3 , m 1 , N + 1 b m 1 , N + 1 ) } 2 ] .
[ N D n = 1 N D cos Θ n n = 1 N D sin Θ n n = 1 N D cos Θ n n = 1 N D cos 2 Θ n + R 22 n = 1 N D sin Θ n cos Θ n R 23 n = 1 N D sin Θ n n = 1 N D cos Θ n sin Θ n R 32 n = 1 N D sin 2 Θ n + R 33 ] [ c 1 , m , N c 2 , m , N c 3 , m , N ] = [ n = 1 N D I m , n n = 1 N D I m , n cos Θ n n = 1 N D I m , n sin Θ n ] ,
Θ n = ϕ * m , N + 1 ϕ * m , N x N + 1 x N ( x n x N ) + ω x n ,
R 22 = Γ N D b * m , N 2 [ r N ( c * 3 , m , N 1 b * m , N 1 + c * 3 , m , N + 1 b * m , N + 1 ) 2 + ( c * 3 , m + 1 , N b * m + 1 , N + c * 3 , m 1 , N b * m 1 , N ) 2 + r N ( c * 3 , m 1 , N 1 b * m 1 , N 1 + c * 3 , m + 1 , N + 1 b * m + 1 , N + 1 ) 2 + r N ( c * 3 , m 1 , N + 1 b * m 1 , N + 1 + c * 3 , m + 1 , N 1 b * m + 1 , N_ 1 ) 2 ] ,
R 33 = Γ N D b * m , N 2 [ r N ( c * 2 , m , N 1 b * m , N 1 + c * 2 , m , N + 1 b * m , N + 1 ) 2 + ( c * 2 , m + 1 , N b * m + 1 , N + c * 2 , m 1 , N b * m 1 , N ) 2 + r N ( c * 2 , m 1 , N 1 b * m 1 , N 1 + c * 2 , m + 1 , N + 1 b * m + 1 , N + 1 ) 2 + r N ( c * 2 , m 1 , N + 1 b * m 1 , N + 1 + c * 2 , m + 1 , N 1 b * m + 1 , N 1 ) 2 ] ,
R 23 = R 32 = Γ N D b * m , N 2 [ r N ( c * 3 , m , N 1 b * m , N 1 + c * 3 , m , N + 1 b * m , N + 1 ) ( c * 2 , m , N 1 b * m , N 1 + c * 2 , m , N + 1 b * m , N + 1 ) + ( c * 3 , m + 1 , N b * m + 1 , N + c * 3 , m 1 , N b * m 1 , N ) ( c * 2 , m + 1 , N b * m + 1 , N + c * 2 , m 1 , N b * m 1 , N ) + r N ( c * 3 , m + 1 , N 1 b * m + 1 , N 1 + c * 3 , m 1 , N + 1 b * m 1 , N + 1 ) ( c * 2 , m + 1 , N 1 b * m + 1 , N 1 + c * 2 , m 1 , N + 1 b * m 1 , N + 1 ) + r N ( c * 3 , m + 1 , N + 1 b * m + 1 , N + 1 + c * 3 , m 1 , N 1 b * m 1 , N 1 ) ( c * 2 , m + 1 , N + 1 b * m + 1 , N + 1 + c * 2 , m 1 , N 1 b * m 1 , N 1 ) ] .
E = N = 1 N M A X m = 1 m max ( ϕ m , N ϕ * m , N ) 2 N max m max .

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