Abstract

Circular slab waveguides are conformally transformed into straight inhomogeneous waveguides, whereupon electromagnetic fields in the core are expanded in terms of Legendre polynomial basis functions. Thereafter, different analytical expression of electromagnetic fields in the cladding region, viz. Wentzel–Kramers–Brillouin solution, modified Airy function expansion, and the exact field solution for circular waveguides, i.e., Hankel function of complex order, are each matched to the polynomial expansion of the transverse electric field within the guide. This field matching process renders different boundary conditions to be satisfied by the set of orthogonal Legendre polynomial basis functions. In this fashion, the governing wave equation is converted into an algebraic and easy to solve eigenvalue problem, which is associated with a matrix whose elements are analytically given. Various numerical examples are presented and the accuracy of each of the abovementioned different boundary conditions is assessed. Furthermore, the computational efficiency and the convergence rate of the proposed method with increasing number of basis functions are briefly discussed.

© 2007 Optical Society of America

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References

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  1. L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures, Vol. 2 of IEE Electromagnetic Waves Series (Peter Peregrinus, Ltd., 1977), p. 205.
  2. F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE Press, Peter Peregrinus Ltd., 1979).
  3. S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996).
    [CrossRef]
  4. K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, "Analytic approach to dielectric optical bent slab waveguides," Opt. Quantum Electron. 37, 37-61 (2005).
    [CrossRef]
  5. T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993).
    [CrossRef]
  6. W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, "Scalar BPM analyses of TE and TM polarized fields in bent waveguides," IEEE Trans. Antennas Propag. 51, 1185-1198 (2003).
    [CrossRef]
  7. R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightwave Technol. 14, 634-639 (1996).
    [CrossRef]
  8. D. Marcuse, "Bending loss of the asymmetric slab waveguide," Bell Syst. Tech. J. 50, 2551-2563 (1971).
  9. E. A. J. Marcatili, "Bends in optical dielectric guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).
  10. R. Jedidi and R. Pierre, "Efficient analytical and numerical methods for the computation of bent loss in planar waveguides," J. Lightwave Technol. 23, 2278-2284 (2005).
    [CrossRef]
  11. W. Kim and C. Kim, "Radiation losses of bent planar waveguides," Fiber Integr. Opt. 21, 219-232 (2002).
    [CrossRef]
  12. M. Heiblum and J. H. Harris, "Analysis of curved optical wave-guides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-85 (1975).
    [CrossRef]
  13. D. Marcuse, "Curvature loss formula for optical fibers," J. Opt. Soc. Am. 66, 216-220 (1976).
    [CrossRef]
  14. A. Melloni, F. Carniel, R. Costa, and M. Martinelli, "Determination of bend mode characteristics in dielectric waveguides," J. Lightwave Technol. 19, 571-577 (2001).
    [CrossRef]
  15. K. Thyagarajan, M. R. Shenoy, and A. K. Ghatak, "Accurate numerical method for the calculation of bending loss in optical waveguides using a matrix approach," Opt. Lett. 12, 296-298 (1987).
    [PubMed]
  16. W. Berglund and A. Gopinath, "WKB analysis of bend losses in optical waveguides," J. Lightwave Technol. 18, 1161-1166 (2000).
    [CrossRef]
  17. C. Kim, Y. Kim, and W. Kim, "Leaky modes of circular slab waveguides: modified Airy functions," IEEE J. Sel. Top. Quantum Electron. 8, 1239-1245 (2002).
    [CrossRef]
  18. S. G. Mikhlin and K. L. Smolitskii, Approximate Method for Solution of Differential and Integral Equations (Elsevier, 1976), pp. 250-252.
  19. C. A. J. Fletcher, Computational Galerkin methods (Springer-Verlag, 1984), p. 309.
  20. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).
  21. A. Weisshaar, "Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides," J. Lightwave Technol. 12, 1943-1951 (1994).
    [CrossRef]
  22. K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
    [CrossRef]
  23. M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-976 (2006).
    [CrossRef]
  24. A. K. Ghatak, R. L. Gallawa, and I. C. Goyal, "Modified Airy function and WKB solutions to the wave equation," National Institute of Standards and Technology, Monograph 176 (United States Department of Commerce, 1991).
  25. R. E. Langer, "On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order," Trans. Am. Math. Soc. 33, 23-64 (1931).
    [CrossRef]
  26. I. C. Goyal, R. Jindal, and A. K. Ghatak, "Planar optical waveguides with arbitrary index profile: an accurate method of analysis," J. Lightwave Technol. 15, 2179-2182 (1997).
    [CrossRef]
  27. D. Rownald, "Nonperturbative calculation of bend loss for a pulse in a bent planar waveguide," IEE Proc.: Optoelectron. 144, 91-96 (1997).
    [CrossRef]
  28. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Ninth printing (Dover, 1970), p. 376.
  29. T. Tamir and R. C. Alferness, Guided-Wave Optoelectronics 2nd ed. (Springer-Verlag, 1990).
    [CrossRef]

2006 (1)

2005 (2)

R. Jedidi and R. Pierre, "Efficient analytical and numerical methods for the computation of bent loss in planar waveguides," J. Lightwave Technol. 23, 2278-2284 (2005).
[CrossRef]

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, "Analytic approach to dielectric optical bent slab waveguides," Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

2003 (2)

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, "Scalar BPM analyses of TE and TM polarized fields in bent waveguides," IEEE Trans. Antennas Propag. 51, 1185-1198 (2003).
[CrossRef]

K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
[CrossRef]

2002 (2)

W. Kim and C. Kim, "Radiation losses of bent planar waveguides," Fiber Integr. Opt. 21, 219-232 (2002).
[CrossRef]

C. Kim, Y. Kim, and W. Kim, "Leaky modes of circular slab waveguides: modified Airy functions," IEEE J. Sel. Top. Quantum Electron. 8, 1239-1245 (2002).
[CrossRef]

2001 (1)

2000 (1)

1997 (2)

I. C. Goyal, R. Jindal, and A. K. Ghatak, "Planar optical waveguides with arbitrary index profile: an accurate method of analysis," J. Lightwave Technol. 15, 2179-2182 (1997).
[CrossRef]

D. Rownald, "Nonperturbative calculation of bend loss for a pulse in a bent planar waveguide," IEE Proc.: Optoelectron. 144, 91-96 (1997).
[CrossRef]

1996 (2)

R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightwave Technol. 14, 634-639 (1996).
[CrossRef]

S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

1994 (1)

A. Weisshaar, "Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides," J. Lightwave Technol. 12, 1943-1951 (1994).
[CrossRef]

1993 (1)

T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993).
[CrossRef]

1987 (1)

1976 (1)

1975 (1)

M. Heiblum and J. H. Harris, "Analysis of curved optical wave-guides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-85 (1975).
[CrossRef]

1971 (1)

D. Marcuse, "Bending loss of the asymmetric slab waveguide," Bell Syst. Tech. J. 50, 2551-2563 (1971).

1969 (1)

E. A. J. Marcatili, "Bends in optical dielectric guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).

1931 (1)

R. E. Langer, "On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order," Trans. Am. Math. Soc. 33, 23-64 (1931).
[CrossRef]

Bell Syst. Tech. J. (2)

D. Marcuse, "Bending loss of the asymmetric slab waveguide," Bell Syst. Tech. J. 50, 2551-2563 (1971).

E. A. J. Marcatili, "Bends in optical dielectric guides," Bell Syst. Tech. J. 48, 2103-2132 (1969).

Fiber Integr. Opt. (1)

W. Kim and C. Kim, "Radiation losses of bent planar waveguides," Fiber Integr. Opt. 21, 219-232 (2002).
[CrossRef]

IEE Proc.: Optoelectron. (1)

D. Rownald, "Nonperturbative calculation of bend loss for a pulse in a bent planar waveguide," IEE Proc.: Optoelectron. 144, 91-96 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Heiblum and J. H. Harris, "Analysis of curved optical wave-guides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-85 (1975).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

C. Kim, Y. Kim, and W. Kim, "Leaky modes of circular slab waveguides: modified Airy functions," IEEE J. Sel. Top. Quantum Electron. 8, 1239-1245 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

W. J. Song, G. H. Song, B. H. Ahn, and M. Kang, "Scalar BPM analyses of TE and TM polarized fields in bent waveguides," IEEE Trans. Antennas Propag. 51, 1185-1198 (2003).
[CrossRef]

J. Lightwave Technol. (8)

R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightwave Technol. 14, 634-639 (1996).
[CrossRef]

A. Weisshaar, "Impedance boundary method of moments for accurate and efficient analysis of planar graded-index optical waveguides," J. Lightwave Technol. 12, 1943-1951 (1994).
[CrossRef]

S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993).
[CrossRef]

I. C. Goyal, R. Jindal, and A. K. Ghatak, "Planar optical waveguides with arbitrary index profile: an accurate method of analysis," J. Lightwave Technol. 15, 2179-2182 (1997).
[CrossRef]

A. Melloni, F. Carniel, R. Costa, and M. Martinelli, "Determination of bend mode characteristics in dielectric waveguides," J. Lightwave Technol. 19, 571-577 (2001).
[CrossRef]

W. Berglund and A. Gopinath, "WKB analysis of bend losses in optical waveguides," J. Lightwave Technol. 18, 1161-1166 (2000).
[CrossRef]

R. Jedidi and R. Pierre, "Efficient analytical and numerical methods for the computation of bent loss in planar waveguides," J. Lightwave Technol. 23, 2278-2284 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

K. R. Hiremath, M. Hammer, R. Stoffer, L. Prkna, and J. Ctyroky, "Analytic approach to dielectric optical bent slab waveguides," Opt. Quantum Electron. 37, 37-61 (2005).
[CrossRef]

Trans. Am. Math. Soc. (1)

R. E. Langer, "On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order," Trans. Am. Math. Soc. 33, 23-64 (1931).
[CrossRef]

Other (8)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Ninth printing (Dover, 1970), p. 376.

T. Tamir and R. C. Alferness, Guided-Wave Optoelectronics 2nd ed. (Springer-Verlag, 1990).
[CrossRef]

L. Lewin, D. C. Chang, and E. F. Kuester, Electromagnetic Waves and Curved Structures, Vol. 2 of IEE Electromagnetic Waves Series (Peter Peregrinus, Ltd., 1977), p. 205.

F. Sporleder and H. G. Unger, Waveguides Tapers Transitions and Couplers (IEE Press, Peter Peregrinus Ltd., 1979).

A. K. Ghatak, R. L. Gallawa, and I. C. Goyal, "Modified Airy function and WKB solutions to the wave equation," National Institute of Standards and Technology, Monograph 176 (United States Department of Commerce, 1991).

S. G. Mikhlin and K. L. Smolitskii, Approximate Method for Solution of Differential and Integral Equations (Elsevier, 1976), pp. 250-252.

C. A. J. Fletcher, Computational Galerkin methods (Springer-Verlag, 1984), p. 309.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, 2001).

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Figures (9)

Fig. 1
Fig. 1

Index profile of a typical circular slab waveguide with arbitrarily inhomogeneous refractive index: (a) primary index profile in a cylindrical coordinate system ( r , ϕ , z ) , and (b) conformally mapped index profile in a Cartesian coordinate system ( u , v , z ) .

Fig. 2
Fig. 2

Normalized bending loss of a slab waveguide with parabolic permittivity profile, n max / n 1 = 1.003 , and V = k 0 d n max 2 n 1 2 , versus normalized curvature radius r 0 / d . Square: MAF-based boundary condition; circle, WKB-based boundary condition; solid curve, Hankel-function-based boundary condition; dots, matrix approach.

Fig. 3
Fig. 3

Relative error in calculation of the bending loss by using our proposed approach with different ( u d u t ) 's in the same slab waveguide with parabolic permittivity profile, r 0 / d = 1000 , and V = 3 .

Fig. 4
Fig. 4

Electric field profile calculated by using Legendre polynomial expansion together with (a) WKB- and (b) MAF-based boundary conditions at λ = 1.3   μm , d = 4   μm , n 1 = 2.00 , and n max = 2.006 .

Fig. 5
Fig. 5

Typical test case to study the convergence, accuracy, and computation time of the proposed method. (a) Straight inhomogeneous slab waveguide with parabolic permittivity profile. (b) Conformally mapped circularly bent inhomogeneous slab waveguide with the same parabolic permittivity profile and normalized curvature radius r 0 / d = 1000 .

Fig. 6
Fig. 6

Relative error in extraction of the propagation constant β of the typical straight inhomogeneous slab waveguide as shown in Fig. 5(a) versus M, i.e., the number of retained Legendre polynomial basis functions in the proposed method (solid curve), the number of homogeneous sublayers in applying the transfer matrix method based on the staircase approximation (dashed line).

Fig. 7
Fig. 7

Computation time in extraction of the propagation constant β versus the achieved relative error in the same typical straight inhomogeneous slab waveguide: the proposed method (solid line), the transfer matrix method based on the staircase approximation (dashed curve).

Fig. 8
Fig. 8

Relative error in extraction of the propagation constant β of the typical circularly bent inhomogeneous slab waveguide as shown in Fig. 5(b) versus M, i.e., the number of retained Legendre polynomial basis functions in the proposed method (solid curve) and the number of homogeneous sublayers in applying the transfer matrix method based on the staircase approximation (dashed curve).

Fig. 9
Fig. 9

Computation time in extraction of the propagation constant β versus the achieved relative error in the same typical circularly bent inhomogeneous slab waveguide: the proposed method (solid line), the transfer matrix method based on the staircase approximation (dashed curve).

Tables (2)

Tables Icon

Table 1 Real and Imaginary Parts of the Propagation Constants β Computed by Applying Various Methods and Boundary Conditions

Tables Icon

Table 2 Real and Imaginary Parts of the Propagation Constants β Computed by an Analytical Method and Various Boundary Conditions

Equations (50)

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1 r r ( r Ψ r ) β 2 r 0 2 r 2 Ψ + k 0 2 n 2 ( r ) Ψ = 0 ,
E z = Ψ ( r ) exp ( j β r 0 ϕ ) .
u = r 1 ln ( r r 1 ) ,
v = r 1 ϕ .
d 2 Ψ d u 2 + k 0 2 ( n 2 ( u ) e 2 u / r 1 N 2 ) Ψ = 0 ,
N = β k 0 r 0 r 1 .
Ψ ( u ) = { m = 0 + M q m 1 P m ( ξ 1 ) u a = u 0 u u 1 m = 0 + M q m 2 P m ( ξ 2 ) u 1 u u 2 m = 0 + M q m j P m ( ξ j ) u j 1 u u j m = 0 + M q m N L P m ( ξ N L ) u N L 1 u u N L = u d  ,
ξ j = ( 2 u ( u j + u j 1 ) ) / ( u j u j 1 ) .
m = 2 + M q m j d 2 d x 2 P m ( ξ j ) + k 0 2 [ n 2 ( ξ j d j + Δ u j 2 ) × exp ( ξ j d j + Δ u j r 1 ) N 2 ] m = 0 + M q m j P m ( ξ j ) = 0 ,
d j = u j u j 1 ,
Δ u j = u j + u j 1 .
n 2 ( u ) exp ( 2 u r 1 ) = a 0 j + a 1 j ξ j + a 2 j ξ j 2 + a 3 j ξ j 3 + + a h j ξ j h ,
j = 1 , 2 , 3 , , N L ,
[ A ( M 1 ) × ( M + 1 ) j ] [ q ¯ ( M + 1 ) × 1 j ] = 0 ,
[ S ( u j 1 ) U ( u j 1 ) ] = B 2 × ( M + 1 ) j [ q ¯ ( M + 1 ) × 1 j ] .
B 2 × ( M + 1 ) j = [ P 0 ( 1 ) P 1 ( 1 ) P M ( 1 ) λ π d j P 0 ( 1 ) λ π d j P 1 ( 1 ) λ π d j P M ( 1 ) ] ,
[ S ( u j ) U ( u j ) ] = C 2 × ( M + 1 ) j [ q ¯ ( M + 1 ) × 1 j ] ,
C 2 × ( M + 1 ) j = [ P 0 ( 1 ) P 1 ( 1 ) P M ( 1 ) λ π d j P 0 ( 1 ) λ π d j P 1 ( 1 ) λ π d j P M ( 1 ) ] .
[ S ( u j ) U ( u j ) ] = [ Q 2 × 2 j ] [ S ( u j 1 ) U ( u j 1 ) ] ,
[ Q 2 × 2 j ] = [ C 2 × ( M + 1 ) j ] [ A ( M 1 ) × ( M + 1 ) j B 2 × ( M + 1 ) j ] 1 [ 0 ( M 1 ) × 2 I 2 × 2 ] ,
[ S ( u d ) U ( u d ) ] = Q 2 × 2 [ S ( u a ) U ( u a ) ] ,
[ Q 2 × 2 ] = [ A B C D ] = j = 1 N L [ Q 2 × 2 j ] .
g 2 A + g 2 g 1 B + C + g 1 D = 0.
1 k ( u ) exp ( ± j u k ( ξ ) d ξ ) .
Ψ ( u ) = { A κ ( u ) exp ( u u a κ ( ξ ) d ξ )     u < u a B k ( u ) exp ( j u d u k ( ξ ) d ξ ) u d < u ,
k ( u ) = k 0 n 2 ( u ) exp ( 2 u r 1 ) N 2 ,
κ ( u ) = k 0 N 2 n 2 ( u ) exp ( 2 u r 1 ) .
Ψ ( u ) = { A κ ( u ) exp ( u u a κ ( ξ ) d ξ )    u < u a B ξ ( u )   Ai ( ξ ( u ) exp ( j 2 3 π ) ) u d < u ,
ξ ( u ) = ( 3 2 u s u k ( u ) d u ) 2 / 3 ,
Ψ ( u ) = { A J υ ( k 0 n ( u b ) exp ( u r 1 ) r 1 )    u < u b B H υ ( 2 ) ( k 0 n ( u c ) exp ( u r 1 ) r 1 ) u c < u ,
n 2 ( r ) = { n 1 2 r 2 < r   or   r < r 1 n 1 2 + ( n max 2 n 1 2 ) [ 1 ( r r 0 d ) 2 ] r 1 < r < r 2 ,
n 2 ( r ) = { n 1 2 r 2 < r   or   r < r 1 n max 2 r 1 < r < r 2 ,
m = 0 M q m j d 2 d x 2 P m ( ξ j ) + ( ( i = 0 h a i j ξ j i ) β 2 ) m = 0 M q m j P m ( ξ j ) = 0 ,
ξ j P m ( ξ j ) = m + 1 2 m + 1 P m + 1 ( ξ j ) + m 2 m + 1 P m 1 ( ξ j ) .
m = 0 M ξ j q m P m ( ξ j ) = m = 0 M χ m P m ( ξ j ) ,
χ m = m 2 m 1 q m 1 + m + 1 2 m + 3 q m + 1 .
χ = [ 0 1 3 0 0 0 1 0 2 5 0 0 0 2 3 0 0 0 0 0 0 0 0 0 M 2 M + 1 0 0 0 M 2 M 1 0 ] ( M + 1 ) × ( M + 1 ) ,
[ χ s , t ] ( M + 1 ) × ( M + 1 ) = { s 2 × s + 1     t s = 1 s 1 2 × s 3     s t = 1 0 | t s | 1 ,
[ χ ¯ m ] = [ χ ] [ q ¯ m ] .
d d ξ Ψ = m = 0 M q m d d ξ P m ( ξ ) = m = 0 M q m j = 1 ( n + 1 ) / 2 ( 2 m 4 j + 3 ) P m 2 j + 1 ( ξ ) = m = 0 M ζ m P m ( ξ ) .
R = [ 0 1 0 1 1 × ( 1 + ( 1 ) M 2 ) 1 × ( 1 + ( 1 ) M + 1 2 ) 0 0 3 0 3 × ( 1 + ( 1 ) M 1 2 ) 3 × ( 1 + ( 1 ) M 2 ) 0 0 0 5 5 × ( 1 + ( 1 ) M 2 ) 5 × ( 1 + ( 1 ) M + 1 2 ) 0 0 0 0 0 0 ( 2 × M 1 ) 0 0 0 0 0 0 0 ] ( M + 1 ) × ( M + 1 ) ,
[ ζ ¯ m ] = [ R ] [ q ¯ m ] .
[ ( 2 d j ) 2 R ( M + 1 ) × ( M + 1 ) 2 + i = 0 h a i j [ χ ( M + 1 ) × ( M + 1 ) ] i N 2 I ] × [ q ¯ ( M + 1 ) × 1 j ] = 0.
R 2 = [ 0 0 3 0 10 0 0 0 15 0 ( 2 s 1 ) × ( t + s 1 ) × ( t s ) × ( 1 + ( 1 ) s + t ) / 4 0 0 0 0 35 ( 2 M 3 ) × ( 2 M 1 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] ( M + 1 ) × ( M + 1 ) ,
R 2 = [ D ( M 1 ) × ( M + 1 ) 0 2 × ( M + 1 ) ] ,
[ D s , t ] ( M 1 ) × ( M + 1 ) = { ( 2 s 1 ) × ( t + s 1 ) × ( t s ) × ( 1 + ( 1 ) s + t ) / 4 t s > 1 0 t s 1 ,
[ 0 s , t ] ( 2 ) × ( M + 1 ) = 0.
[ ( 2 d j ) 2 D ( M 1 ) × ( M + 1 ) + F j ( M 1 ) × ( M + 1 ) N 2 I ( M 1 ) × ( M + 1 ) ] [ q ¯ ( M + 1 ) × 1 j ] = [ A ( M 1 ) × ( M + 1 ) ] [ q ¯ ( M + 1 ) × 1 j ] = 0 ,
[ F ( M 1 ) × ( M + 1 ) j G 2 × ( M + 1 ) j ] = i = 0 h a i j [ χ ( M + 1 ) × ( M + 1 ) ] i ,
[ I s , t ] ( M 1 ) × ( M + 1 ) = δ s , t .

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