Abstract

Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Li, J. Hirsh, “All-dielectric high-efficiency reflection gratings made with multilayer thin film coatings,” Opt. Lett. 20, 1349–1351 (1995).
    [CrossRef] [PubMed]
  2. C. Heine, R. H. Morf, “Submicrometer gratings for solar energy applications,” Appl. Opt. 34, 2476–2482 (1995).
    [CrossRef] [PubMed]
  3. M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
    [CrossRef]
  4. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  5. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  6. L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  7. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  8. N. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  9. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  10. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  11. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  12. L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
    [CrossRef]
  13. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  14. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  15. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  16. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  17. L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to coated gratings with different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
    [CrossRef]
  18. The statement “matrix Ais of order O(1)” means that every element of Ais of order O(1). In the context of this paper saying that a matrix is of order O(1) means that it contains no exponentially growing functions with respect to layer thickness and matrix truncation order.
  19. A. K. Cousins, S. C. Gottschalk, “Application of the impedance formalism to diffraction gratings with multiple coating layers,” Appl. Opt. 29, 4268–4271 (1990).
    [CrossRef] [PubMed]
  20. R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, E. F. Beckenbach, ed. (McGraw-Hill, New York, 1961), Chap. 12, pp. 282–337.
  21. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1983), Chap. 3, p. 30, and Chap. 4, p. 52.

1996 (1)

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to coated gratings with different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

1995 (6)

1994 (6)

1993 (1)

1991 (2)

1990 (1)

1982 (1)

Awada, K. A.

Chandezon, J.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to coated gratings with different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Chateau, N.

Cotter, N. P. K.

Cousins, A. K.

DeSandre, L. F.

Elson, J. M.

Gaylord, T. K.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1983), Chap. 3, p. 30, and Chap. 4, p. 52.

Gottschalk, S. C.

Granet, G.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to coated gratings with different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Grann, E. B.

Heine, C.

Hirsh, J.

Hugonin, J. P.

Li, L.

Maystre, D.

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Moharam, M. G.

Montiel, F.

Morf, R. H.

Nevière, M.

Pai, D. M.

Plumey, J. P.

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to coated gratings with different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Pommet, D. A.

Preist, T. W.

Redheffer, R.

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, E. F. Beckenbach, ed. (McGraw-Hill, New York, 1961), Chap. 12, pp. 282–337.

Sambles, J. R.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1983), Chap. 3, p. 30, and Chap. 4, p. 52.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

N. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
[CrossRef]

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
[CrossRef]

L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmission matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (3)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to coated gratings with different profiles,” Pure Appl. Opt. 5, 141–156 (1996).
[CrossRef]

Other (3)

The statement “matrix Ais of order O(1)” means that every element of Ais of order O(1). In the context of this paper saying that a matrix is of order O(1) means that it contains no exponentially growing functions with respect to layer thickness and matrix truncation order.

R. Redheffer, “Difference equations and functional equations in transmission-line theory,” in Modern Mathematics for the Engineer, E. F. Beckenbach, ed. (McGraw-Hill, New York, 1961), Chap. 12, pp. 282–337.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1983), Chap. 3, p. 30, and Chap. 4, p. 52.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

General layered grating. All periodic medium interfaces share a common period, but otherwise they are arbitrary.

Fig. 2
Fig. 2

Abstract layered grating structure, where the horizontal lines represent either actual material interfaces or numerical interfaces. The fields in each layer can be represented either (a) as a superposition of upward- and downward-propagating and decaying waves or (b) as a superposition of two sets of orthogonally polarized eigenmodes.

Fig. 3
Fig. 3

Schematic diagrams for alternative interpretations of (a) the S matrix and (b) the R matrix. In Fig. 3(a), I and O stand for inputs to and outputs from the system represented by the square box. In Fig. 3(b), i and v stand for currents and voltages at the terminals of the circuit represented by the square box.

Tables (1)

Tables Icon

Table 1 Operation Counts (in N3 Flops) per Grating Layer for Different Variants of the S-Matrix and R-Matrix Algorithms

Equations (63)

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

σ ( p ) ± = { λ m ( p ) ; Re λ m ( p ) + Im λ m ( p ) 0 , λ m ( p ) σ ( p ) } .
W ( p + 1 ) [ u ( p + 1 ) ( y p + 0 ) d ( p + 1 ) ( y p + 0 ) ] = W ( p ) [ u ( p ) ( y p - 0 ) d ( p ) ( y p - 0 ) ] ,
[ u ( p ) ( y p - 0 ) d ( p ) ( y p - 0 ) ] = ϕ ( p ) [ u ( p ) ( y p - 1 + 0 ) d ( p ) ( y p - 1 + 0 ) ] ,
ϕ ( p ) = [ exp ( i λ m ( p ) + h p ) 0 0 exp ( i λ m ( p ) - h p ) ] ,
[ u ( p + 1 ) ( y p + 0 ) d ( p + 1 ) ( y p + 0 ) ] = t ˜ ( p ) [ u ( p ) ( y p - 1 + 0 ) d ( p ) ( y p - 1 + 0 ) ] ,
t ˜ ( p ) = t ( p ) ϕ ( p ) ,
t ( p ) = W ( p + 1 ) - 1 W ( p ) .
W ( p + 1 ) [ U ( p + 1 ) ( y p + 0 ) V ( p + 1 ) ( y p + 0 ) ] = W ( p ) [ U ( p ) ( y p - 0 ) V ( p ) ( y p - 0 ) ] ,
[ U ( p ) ( y p - 0 ) V ( p ) ( y p - 0 ) ] = ϕ ( p ) [ U ( p ) ( y p - 1 + 0 ) V ( p ) ( y p - 1 + 0 ) ] ,
ϕ ( p ) = [ cos ( λ m ( p ) h p ) η m ( p ) sin ( λ m ( p ) h p ) - η m ( p ) - 1 sin ( λ m ( p ) h p ) cos ( λ m ( p ) h p ) ] ,
[ U ( p + 1 ) ( y p + 0 ) V ( p + 1 ) ( y p + 0 ) ] = t ˜ ( p ) [ U ( p ) ( y p - 1 + 0 ) V ( p ) ( y p - 1 + 0 ) ] .
[ u ( p + 1 ) d ( 0 ) ] = S ( p ) [ u ( 0 ) d ( p + 1 ) ] .
[ u ( p + 1 ) d ( 0 ) ] = [ T u u ( p ) R u d ( p ) R d u ( p ) T d d ( p ) ] [ u ( 0 ) d ( p + 1 ) ] .
[ u ( p + 1 ) ( y p + 0 ) d ( p ) ( y p - 0 ) ] = s ( p ) [ u ( p ) ( y p - 0 ) d ( p + 1 ) ( y p + 0 ) ] ,
[ u ( p + 1 ) d ( p ) ] = s ˜ ( p ) [ u ( p ) d ( p + 1 ) ] ,
[ u ( p + 1 ) d ( p ) ] = [ t ˜ u u ( p ) r ˜ u d ( p ) r ˜ d u ( p ) t ˜ d d ( p ) ] [ u ( p ) d ( p + 1 ) ] .
s ˜ ( p ) = [ 1 0 0 exp ( - i λ m ( p ) - h p ) ] s ( p ) [ exp ( i λ m ( p ) + h p ) 0 0 1 ] ,
s ( p ) = [ t 11 ( p ) - t 12 ( p ) t 22 ( p ) - 1 t 21 ( p ) t 12 ( p ) t 22 ( p ) - 1 - t 22 ( p ) - 1 t 21 ( p ) t 22 ( p ) - 1 ] .
T u u ( p ) = t ˜ u u ( p ) [ 1 - R u d ( p - 1 ) r ˜ d u ( p ) ] - 1 T u u ( p - 1 ) , R u d ( p ) = r ˜ u d ( p ) + t ˜ u u ( p ) R u d ( p - 1 ) [ 1 - r ˜ d u ( p ) R u d ( p - 1 ) ] - 1 t ˜ d d ( p ) , R d u ( p ) = R d u ( p - 1 ) + T d d ( p - 1 ) r ˜ d u ( p ) [ 1 - R u d ( p - 1 ) r ˜ d u ( p ) ] - 1 T u u ( p - 1 ) , T d d ( p ) = T d d ( p - 1 ) [ 1 - r ˜ d u ( p ) R u d ( p - 1 ) ] - 1 t ˜ d d ( p ) .
S ( - 1 ) = [ 1 0 0 1 ] ,
[ u ( n + 1 ) d ( 0 ) ] = [ T u u ( n ) R u d ( n ) R d u ( n ) T d d ( n ) ] [ u ( 0 ) d ( n + 1 ) ] .
u ( n + 1 ) = R u d ( n ) d ( n + 1 ) , d ( 0 ) = T d d ( n ) d ( n + 1 ) .
[ U ( p + 1 ) U ( 0 ) ] = R ( p ) [ V ( p + 1 ) V ( 0 ) ] .
[ U ( p + 1 ) ( y p + 0 ) U ( p ) ( y p - 0 ) ] = r ( p ) [ V ( p + 1 ) ( y p + 0 ) V ( p ) ( y p - 0 ) ] ,
[ U ( p + 1 ) U ( p ) ] = r ˜ ( p ) [ V ( p + 1 ) V ( p ) ] .
r ˜ 11 ( p ) = r 11 ( p ) - r 12 ( p ) ζ ( p ) r 21 ( p ) , r ˜ 12 ( p ) = r 12 ( p ) ζ ( p ) η m ( p ) csc [ λ m ( p ) h p ] , r ˜ 21 ( p ) = η m ( p ) csc [ λ m ( p ) h p ] ζ ( p ) r 21 ( p ) , r ˜ 22 ( p ) = η m ( p ) cot [ λ m ( p ) h p ] - η m ( p ) csc [ λ m ( p ) h p ] × ζ ( p ) η m ( p ) csc [ λ m ( p ) h p ] ,
ζ ( p ) = [ r 22 ( p ) + η m ( p ) cot ( λ m ( p ) h p ) ] - 1 .
r ( p ) = [ t 11 ( p ) t 21 ( p ) - 1 t 12 ( p ) - t 11 ( p ) t 21 ( p ) - 1 t 22 ( p ) t 21 ( p ) - 1 - t 21 ( p ) - 1 t 22 ( p ) ] .
r ˜ ( p ) = [ t ˜ 11 ( p ) t ˜ 21 ( p ) - 1 t ˜ 12 ( p ) - t ˜ 11 ( p ) t ˜ 21 ( p ) - 1 t ˜ 22 ( p ) t ˜ 21 ( p ) - 1 - t ˜ 21 ( p ) - 1 t ˜ 22 ( p ) ] .
R 11 ( p ) = r ˜ 11 ( p ) - r ˜ 12 ( p ) Z ( p ) r ˜ 21 ( p ) , R 12 ( p ) = r ˜ 12 ( p ) Z ( p ) R 12 ( p - 1 ) , R 21 ( p ) = - R 21 ( p - 1 ) Z ( p ) r ˜ 21 ( p ) , R 22 ( p ) = R 22 ( p - 1 ) + R 21 ( p - 1 ) Z ( p ) R 12 ( p - 1 ) ,
Z ( p ) = ( r ˜ 22 ( p ) - R 11 ( p - 1 ) ) - 1 .
R ( 0 ) = r ˜ ( 0 ) .
[ U ( n + 1 ) U ( 0 ) ] = [ R 11 ( n ) R 12 ( n ) R 21 ( n ) R 22 ( n ) ] [ V ( n + 1 ) V ( 0 ) ] .
[ 1 - R 11 ( n ) R 12 ( n ) - R 21 ( n ) 1 + R 22 ( n ) ] [ u ( n + 1 ) d ( 0 ) ] = [ - 1 - R 11 ( n ) R 12 ( n ) - R 21 ( n ) - 1 + R 22 ( n ) ] [ d ( n + 1 ) u ( 0 ) ] .
T u u ( p ) = [ t 11 ( p ) - R u d ( p ) t 21 ( p ) ] ϕ + ( p ) T u u ( p - 1 ) , R u d ( p ) = [ t 12 ( p ) + t 11 ( p ) Ω ( p ) ] [ t 22 ( p ) + t 21 ( p ) Ω ( p ) ] - 1 , R d u ( p ) = R d u ( p - 1 ) - T d d ( p ) t 21 ( p ) ϕ + ( p ) T u u ( p - 1 ) , T d d ( p ) = T d d ( p - 1 ) ϕ - ( p ) - 1 [ t 22 ( p ) + t 21 ( p ) Ω ( p ) ] - 1 ,
Ω ( p ) = ϕ + ( p ) R u d ( p - 1 ) ϕ - ( p ) - 1 ,
T u u ( p ) = [ t ˜ 11 ( p ) - R u d ( p ) t ˜ 21 ( p ) ] T u u ( p - 1 ) , R u d ( p ) = [ t ˜ 12 ( p ) + t ˜ 11 ( p ) R u d ( p - 1 ) ] [ t ˜ 22 ( p ) + t ˜ 21 ( p ) R u d ( p - 1 ) ] - 1 , R d u ( p ) = R d u ( p - 1 ) - T d d ( p ) t ˜ 21 ( p ) T u u ( p - 1 ) , T d d ( p ) = T d d ( p - 1 ) [ t ˜ 22 ( p ) + t ˜ 21 ( p ) R u d ( p - 1 ) ] - 1 .
R 11 ( p ) = [ t ˜ 12 ( p ) + t ˜ 11 ( p ) R 11 ( p - 1 ) ] [ t ˜ 22 ( p ) + t ˜ 21 ( p ) R 11 ( p - 1 ) ] - 1 , R 12 ( p ) = [ t ˜ 11 ( p ) - R 11 ( p ) t ˜ 21 ( p ) ] R 12 ( p - 1 ) , R 21 ( p ) = R 21 ( p - 1 ) [ t ˜ 22 ( p ) + t ˜ 21 ( p ) R 11 ( p - 1 ) ] - 1 , R 22 ( p ) = R 22 ( p - 1 ) - R 21 ( p ) t ˜ 21 ( p ) R 12 ( p - 1 ) .
[ W 11 ( p + 1 ) - W 12 ( p ) W 21 ( p + 1 ) - W 22 ( p ) ] [ u ( p + 1 ) ( y p + 0 ) d ( p ) ( y p - 0 ) ] = [ W 11 ( p ) - W 12 ( p + 1 ) W 21 ( p ) - W 22 ( p + 1 ) ] [ u ( p ) ( y p - 0 ) d ( p + 1 ) ( y p + 0 ) ] .
s ( p ) = [ W 11 ( p + 1 ) - W 12 ( p ) W 21 ( p + 1 ) - W 22 ( p ) ] - 1 [ W 11 ( p ) - W 12 ( p + 1 ) W 21 ( p ) - W 22 ( p + 1 ) ] .
r ( p ) = [ W 11 ( p + 1 ) - W 11 ( p ) W 21 ( p + 1 ) - W 21 ( p ) ] - 1 [ - W 12 ( p + 1 ) W 12 ( p ) - W 22 ( p + 1 ) W 22 ( p ) ] .
[ a u u a u d a d u a d d ] * [ b u u b u d b d u b d d ] = [ b u u ( 1 - a u d b d u ) - 1 a u u b u d + b u u a u d ( 1 - b d u a u d ) - 1 b d d a d u + a d d b d u ( 1 - a u d b d u ) - 1 a u u a d d ( 1 - b d u a u d ) - 1 b d d ] ,
[ a 11 a 12 a 21 a 22 ] [ b 11 b 12 b 21 b 22 ] = [ b 11 - b 12 ( b 22 - a 11 ) - 1 b 21 b 12 ( b 22 - a 11 ) - 1 a 12 - a 21 ( b 22 - a 11 ) - 1 b 21 a 22 + a 21 ( b 22 - a 11 ) - 1 a 12 ] .
a * ( b * c ) = ( a * b ) * c ,
a ( b c ) = ( a b ) c .
S ( n ) = { [ ( s ˜ ( 0 ) * s ˜ ( 1 ) ) * s ˜ ( 2 ) ] * } * s ˜ ( n ) .
S ( n ) = s ˜ ( 0 ) * { * [ s ˜ ( n - 2 ) * ( s ˜ ( n - 1 ) * s ˜ ( n ) ) ] } .
S ( n ) = [ s ˜ ( 0 ) * * s ˜ ( j - 1 ) ] * s ˜ ( j ) * [ s ˜ ( j - 1 ) * * s ˜ ( n ) ] ,
{ R u d ( p ) } ,             { R u d ( p ) , T d d ( p ) } ,             { R u d ( p ) , T u u ( p ) } , { R u d ( p ) , T u u ( p ) , T d d ( p ) } .
{ R 11 ( p ) } ,             { R 11 ( p ) , R 12 ( p ) } ,             { R 11 ( p ) , R 21 ( p ) } , { R 11 ( p ) , R 12 ( p ) , R 21 ( p ) } .
R ( - 1 ) = [ 1 0 0 - 1 ] .
ψ l ( p + 1 ) ( x ) f p ( x ) ψ n ( p ) ( x ) d x ,
r ^ ( p ) = [ - η m ( p ) cot ( λ m ( p ) h p ) η m ( p ) csc ( λ m ( p ) h p ) - η m ( p ) csc ( λ m ( p ) h p ) η m ( p ) cot ( λ m ( p ) h p ) ] .
[ U ( p + 1 ) U ( p ) ] = r ˜ ( p ) [ V ( p + 1 ) V ( p ) ]
[ U ( p ) U ( 0 ) ] = R ( p - 1 ) [ V ( p ) V ( 0 ) ]
[ U ( p + 1 ) U ( 0 ) ] = R ( p ) [ V ( p + 1 ) V ( 0 ) ] .
[ U ( p + 1 ) ( y p + 0 ) U ( p ) ( y p - 0 ) ] = r ( p ) [ V ( p + 1 ) ( y p + 0 ) V ( p ) ( y p - 0 ) ] ,
[ U ( p ) ( y p - 0 ) U ( p ) ( y p - 1 + 0 ) ] = r ^ ( p ) [ V ( p ) ( y p - 0 ) V ( p ) ( y p - 1 + 0 ) ] ,
[ U ( p + 1 ) ( y p + 0 ) U ( p ) ( y p - 1 + 0 ) ] = r ˜ ( p ) [ V ( p + 1 ) ( y p + 0 ) V ( p ) ( y p - 1 + 0 ) ] .
r ˜ ( p ) = [ [ t 12 ( p ) - t 11 ( p ) η m ( p ) cot ( λ m ( p ) h p ) ] t 22 ( p ) - 1 t 11 ( p ) η m ( p ) csc ( λ m ( p ) h p ) - η m ( p ) csc ( λ m ( p ) h p ) t 22 ( p ) - 1 η m ( p ) cot ( λ m ( p ) h p ) ] .
R ^ 11 ( p ) = - η m ( p ) cot ( λ m ( p ) h p ) + η m ( p ) csc ( λ m ( p ) h p ) × ω ( p ) η m ( p ) csc ( λ m ( p ) h p ) , R ^ 12 ( p ) = η m ( p ) csc ( λ m ( p ) h p ) ω ( p ) R 12 ( p - 1 ) , R ^ 21 ( p ) = R 21 ( p - 1 ) ω ( p ) η m ( p )   csc ( λ m ( p ) h p ) , R ^ 22 ( p ) = R 22 ( p - 1 ) + R 21 ( p - 1 ) ω ( p ) R 12 ( p - 1 ) ,
ω ( p ) = [ η m ( p ) cot ( λ m ( p ) h p ) - R 11 ( p - 1 ) ] - 1 .
R 11 ( p ) = [ t 12 ( p ) + t 11 ( p ) R ^ 11 ( p ) ] [ t 22 ( p ) + t 21 ( p ) R ^ 11 ( p ) ] - 1 , R 12 ( p ) = [ t 11 ( p ) - R 11 ( p ) t 21 ( p ) ] R ^ 12 ( p ) , R 21 ( p ) = R ^ 21 ( p ) [ t 22 ( p ) + t 21 ( p ) R ^ 11 ( p ) ] - 1 , R 22 ( p ) = R ^ 22 ( p ) - R 21 ( p ) t 21 ( p ) R ^ 12 ( p ) .

Metrics