Abstract

The connection between the Bremmer (reflection) series method and the R-matrix propagation method for modeling diffraction gratings is revealed. Both methods have been previously demonstrated to be immune from the problem of loss of significant digits caused by the growing and decaying exponential functions. It is mathematically proved that under certain conditions these two methods are formally equivalent. Consequently, the validity of the Bremmer series as a solution of Maxwell’s equation to the grating problem is established. Comparisons are also made between the Bremmer series method and the R-matrix propagation method in terms of their physical interpretations, algorithmic structures, numerical stabilities, and ranges of applicability.

© 1994 Optical Society of America

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References

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  1. Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. 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]
  3. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  4. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  5. 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]
  6. M. Nevière, E. Popov, “Analysis of dielectric gratings of arbitrary profiles and thicknesses: comment,” J. Opt. Soc. Am. A 9, 2095–2096 (1992).
    [CrossRef]
  7. H. Bremmer, “The W.K.B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
    [CrossRef]
  8. R. Bellman, R. Kalaba, “Functional equations, wave propagation and invariant imbedding,”J. Math. Mech. 8, 683–704 (1959).
  9. F. V. Atkinson, “Wave propagation and the Bremmer series,”J. Math. Anal. Appl. 1, 255–276 (1960).
    [CrossRef]
  10. H. L. Berk, D. L. Book, D. Pfirsch, “Convergence of the Bremmer series for the spatially inhomogeneous Helmholtz equation,”J. Math. Phys. 8, 1611–1619 (1967).
    [CrossRef]
  11. L. J. F. Broer, “Note on approximate solutions of the wave equation,” Appl. Sci. Res. Sect. B 10, 111–118 (1963).
  12. I. Kay, “Some remarks concerning the Bremmer series,”J. Math. Anal. Appl. 3, 40–49 (1961).
    [CrossRef]
  13. H. B. Keller, J. B. Keller, “Exponential-like solutions of systems of linear ordinary differential equations,”J. Soc. Ind. Appl. Math. 10, 246–259 (1962).
    [CrossRef]
  14. F. W. Sluijter, “Generalizations of the Bremmer series based on physical concepts,”J. Math. Anal. Appl. 27, 282–302 (1969).
    [CrossRef]
  15. 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]
  16. J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,”J. Chem. Phys. 65, 4272–4282 (1976).
    [CrossRef]
  17. S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).
  18. H. G. Booker, “The elements of wave propagation using the impedance concept,” J. Inst. Elect. Eng. Part 394, 171–184 (1947).
  19. L. M. Bekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  20. K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, U.K., 1961), Chap. 22, pp. 482–501.
  21. C. Altman, H. Cory, “The generalized thin-film optical method in electromagnetic wave propagation,” Radio Sci. 4, 457–470 (1969).
  22. 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.
  23. B. L. N. Kennett, “Reflections, rays, and reverberations,” Bull. Seismol. Soc. Am. 64, 1685–1696 (1974).
  24. K. G. Budden, “Full wave solutions for radio waves in a stratified magnetoionic medium,” Alta Freq. 38, 167–179 (1968).
  25. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, U.K., 1985), Chap. 5, p. 301.
  26. K. Knopp, Theory and Application of Infinite Series (Dover, New York, 1990), Chap. 4, Sections 16 and 17, pp. 136–151.
  27. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), Chap. 4, pp. 63–66.
  28. M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
    [CrossRef]

1994 (3)

1993 (1)

1992 (1)

1991 (2)

1976 (1)

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,”J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

1974 (1)

B. L. N. Kennett, “Reflections, rays, and reverberations,” Bull. Seismol. Soc. Am. 64, 1685–1696 (1974).

1969 (2)

F. W. Sluijter, “Generalizations of the Bremmer series based on physical concepts,”J. Math. Anal. Appl. 27, 282–302 (1969).
[CrossRef]

C. Altman, H. Cory, “The generalized thin-film optical method in electromagnetic wave propagation,” Radio Sci. 4, 457–470 (1969).

1968 (1)

K. G. Budden, “Full wave solutions for radio waves in a stratified magnetoionic medium,” Alta Freq. 38, 167–179 (1968).

1967 (1)

H. L. Berk, D. L. Book, D. Pfirsch, “Convergence of the Bremmer series for the spatially inhomogeneous Helmholtz equation,”J. Math. Phys. 8, 1611–1619 (1967).
[CrossRef]

1963 (1)

L. J. F. Broer, “Note on approximate solutions of the wave equation,” Appl. Sci. Res. Sect. B 10, 111–118 (1963).

1962 (1)

H. B. Keller, J. B. Keller, “Exponential-like solutions of systems of linear ordinary differential equations,”J. Soc. Ind. Appl. Math. 10, 246–259 (1962).
[CrossRef]

1961 (1)

I. Kay, “Some remarks concerning the Bremmer series,”J. Math. Anal. Appl. 3, 40–49 (1961).
[CrossRef]

1960 (1)

F. V. Atkinson, “Wave propagation and the Bremmer series,”J. Math. Anal. Appl. 1, 255–276 (1960).
[CrossRef]

1959 (1)

R. Bellman, R. Kalaba, “Functional equations, wave propagation and invariant imbedding,”J. Math. Mech. 8, 683–704 (1959).

1951 (1)

H. Bremmer, “The W.K.B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

1947 (1)

H. G. Booker, “The elements of wave propagation using the impedance concept,” J. Inst. Elect. Eng. Part 394, 171–184 (1947).

1938 (1)

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).

Altman, C.

C. Altman, H. Cory, “The generalized thin-film optical method in electromagnetic wave propagation,” Radio Sci. 4, 457–470 (1969).

Atkinson, F. V.

F. V. Atkinson, “Wave propagation and the Bremmer series,”J. Math. Anal. Appl. 1, 255–276 (1960).
[CrossRef]

Awada, K. A.

Bekhovskikh, L. M.

L. M. Bekhovskikh, Waves in Layered Media (Academic, New York, 1960).

Bellman, R.

R. Bellman, R. Kalaba, “Functional equations, wave propagation and invariant imbedding,”J. Math. Mech. 8, 683–704 (1959).

Berk, H. L.

H. L. Berk, D. L. Book, D. Pfirsch, “Convergence of the Bremmer series for the spatially inhomogeneous Helmholtz equation,”J. Math. Phys. 8, 1611–1619 (1967).
[CrossRef]

Book, D. L.

H. L. Berk, D. L. Book, D. Pfirsch, “Convergence of the Bremmer series for the spatially inhomogeneous Helmholtz equation,”J. Math. Phys. 8, 1611–1619 (1967).
[CrossRef]

Booker, H. G.

H. G. Booker, “The elements of wave propagation using the impedance concept,” J. Inst. Elect. Eng. Part 394, 171–184 (1947).

Bremmer, H.

H. Bremmer, “The W.K.B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

Broer, L. J. F.

L. J. F. Broer, “Note on approximate solutions of the wave equation,” Appl. Sci. Res. Sect. B 10, 111–118 (1963).

Budden, K. G.

K. G. Budden, “Full wave solutions for radio waves in a stratified magnetoionic medium,” Alta Freq. 38, 167–179 (1968).

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, U.K., 1961), Chap. 22, pp. 482–501.

Chateau, N.

Cory, H.

C. Altman, H. Cory, “The generalized thin-film optical method in electromagnetic wave propagation,” Radio Sci. 4, 457–470 (1969).

Desandre, L. F.

Elson, J. M.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), Chap. 4, pp. 63–66.

Horn, R. A.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, U.K., 1985), Chap. 5, p. 301.

Hugonin, J.-P.

Johnson, C. R.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, U.K., 1985), Chap. 5, p. 301.

Kalaba, R.

R. Bellman, R. Kalaba, “Functional equations, wave propagation and invariant imbedding,”J. Math. Mech. 8, 683–704 (1959).

Kay, I.

I. Kay, “Some remarks concerning the Bremmer series,”J. Math. Anal. Appl. 3, 40–49 (1961).
[CrossRef]

Keller, H. B.

H. B. Keller, J. B. Keller, “Exponential-like solutions of systems of linear ordinary differential equations,”J. Soc. Ind. Appl. Math. 10, 246–259 (1962).
[CrossRef]

Keller, J. B.

H. B. Keller, J. B. Keller, “Exponential-like solutions of systems of linear ordinary differential equations,”J. Soc. Ind. Appl. Math. 10, 246–259 (1962).
[CrossRef]

Kennett, B. L. N.

B. L. N. Kennett, “Reflections, rays, and reverberations,” Bull. Seismol. Soc. Am. 64, 1685–1696 (1974).

Knopp, K.

K. Knopp, Theory and Application of Infinite Series (Dover, New York, 1990), Chap. 4, Sections 16 and 17, pp. 136–151.

Li, L.

Light, J. C.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,”J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Montiel, F.

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Nevière, M.

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

M. Nevière, E. Popov, “Analysis of dielectric gratings of arbitrary profiles and thicknesses: comment,” J. Opt. Soc. Am. A 9, 2095–2096 (1992).
[CrossRef]

Pai, D. M.

Pfirsch, D.

H. L. Berk, D. L. Book, D. Pfirsch, “Convergence of the Bremmer series for the spatially inhomogeneous Helmholtz equation,”J. Math. Phys. 8, 1611–1619 (1967).
[CrossRef]

Popov, E.

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.

Schelkunoff, S. A.

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).

Sluijter, F. W.

F. W. Sluijter, “Generalizations of the Bremmer series based on physical concepts,”J. Math. Anal. Appl. 27, 282–302 (1969).
[CrossRef]

Walker, R. B.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,”J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Alta Freq. (1)

K. G. Budden, “Full wave solutions for radio waves in a stratified magnetoionic medium,” Alta Freq. 38, 167–179 (1968).

Appl. Sci. Res. Sect. B (1)

L. J. F. Broer, “Note on approximate solutions of the wave equation,” Appl. Sci. Res. Sect. B 10, 111–118 (1963).

Bell Syst. Tech. J. (1)

S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bell Syst. Tech. J. 17, 17–48 (1938).

Bull. Seismol. Soc. Am. (1)

B. L. N. Kennett, “Reflections, rays, and reverberations,” Bull. Seismol. Soc. Am. 64, 1685–1696 (1974).

Commun. Pure Appl. Math. (1)

H. Bremmer, “The W.K.B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

J. Chem. Phys. (1)

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,”J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

J. Inst. Elect. Eng. Part (1)

H. G. Booker, “The elements of wave propagation using the impedance concept,” J. Inst. Elect. Eng. Part 394, 171–184 (1947).

J. Math. Anal. Appl. (3)

I. Kay, “Some remarks concerning the Bremmer series,”J. Math. Anal. Appl. 3, 40–49 (1961).
[CrossRef]

F. W. Sluijter, “Generalizations of the Bremmer series based on physical concepts,”J. Math. Anal. Appl. 27, 282–302 (1969).
[CrossRef]

F. V. Atkinson, “Wave propagation and the Bremmer series,”J. Math. Anal. Appl. 1, 255–276 (1960).
[CrossRef]

J. Math. Mech. (1)

R. Bellman, R. Kalaba, “Functional equations, wave propagation and invariant imbedding,”J. Math. Mech. 8, 683–704 (1959).

J. Math. Phys. (1)

H. L. Berk, D. L. Book, D. Pfirsch, “Convergence of the Bremmer series for the spatially inhomogeneous Helmholtz equation,”J. Math. Phys. 8, 1611–1619 (1967).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Soc. Ind. Appl. Math. (1)

H. B. Keller, J. B. Keller, “Exponential-like solutions of systems of linear ordinary differential equations,”J. Soc. Ind. Appl. Math. 10, 246–259 (1962).
[CrossRef]

Opt. Commun. (1)

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Radio Sci. (1)

C. Altman, H. Cory, “The generalized thin-film optical method in electromagnetic wave propagation,” Radio Sci. 4, 457–470 (1969).

Other (7)

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.

Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
[CrossRef]

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, U.K., 1985), Chap. 5, p. 301.

K. Knopp, Theory and Application of Infinite Series (Dover, New York, 1990), Chap. 4, Sections 16 and 17, pp. 136–151.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955), Chap. 4, pp. 63–66.

L. M. Bekhovskikh, Waves in Layered Media (Academic, New York, 1960).

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, Cambridge, U.K., 1961), Chap. 22, pp. 482–501.

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Figures (5)

Fig. 1
Fig. 1

M-layer stratified grating. The periodic variations of the media in the horizontal direction are not shown.

Fig. 2
Fig. 2

Schematic for the physical interpretation of Eq. (12d).

Fig. 3
Fig. 3

Schematic for operator K(i, j, k).

Fig. 4
Fig. 4

Schematic for a typical term in operator Q.

Fig. 5
Fig. 5

Schematic for a typical term in operator J.

Equations (52)

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λ ( u ) : Re [ λ ( u ) ] + Im [ λ ( u ) ] < 0 , λ ( d ) : Im [ λ ( d ) ] + Im [ λ ( d ) ] > 0 ,
( u ( z j + 0 ) d ( z j + 0 ) ) = [ c u u ( j ) c u d ( j ) c d u ( j ) c d d ( j ) ] ( u ( z j - 0 ) d ( z j - 0 ) ) ,
( u ( z j - 0 ) d ( z j + 0 ) ) = [ t u u ( j ) t u d ( j ) t d u ( j ) t d d ( j ) ] ( u ( z j + 0 ) d ( z j - 0 ) ) ,
[ t u u ( j ) r u d ( j ) r d u ( j ) t d d ( j ) ] = [ c u u ( j ) - 1 - c u u ( j ) - 1 c u d ( j ) c d u ( j ) c u u ( j ) - 1 c d d ( j ) - c d u ( j ) c u u ( j ) - 1 c u d ( j ) ] .
u ( z j + 0 ) = L u ( z j , z j + 1 ) u ( z j + 1 - 0 ) , d ( z j + 1 - 0 ) = L d ( z j + 1 , z j ) d ( z j + 0 ) ,
L u ( z j , z j + 1 ) = exp [ i λ j + 1 ( u ) ( z j - z j + 1 ) ] , L d ( z j + 1 , z j ) = exp [ i λ j + 1 ( d ) ( z j + 1 - z j ) ]
( u ( z j - 0 ) d ( z j + 1 - 0 ) ) = [ t ˜ u u ( j ) r ˜ u d ( j ) r ˜ d u ( j ) t ˜ d d ( j ) ] ( u ( z j + 1 - 0 ) d ( z j - 0 ) ) ,
[ t ˜ u u ( j ) r ˜ u d ( j ) r ˜ d u ( j ) t ˜ d d ( j ) ] = [ t u u ( j ) L u ( z j , z j + 1 ) r u d ( j ) L d ( z j + 1 , z j ) r d u ( j ) L u ( z j , z j + 1 ) L d ( z j + 1 , z j ) t d d ( j ) ] .
( u ( z j + 1 ) d ( z j + 1 ) ) = [ c ˜ u u ( j ) c ˜ u d ( j ) c ˜ d u ( j ) c ˜ d d ( j ) ] ( u ( z j ) d ( z j ) ) ,
[ c ˜ u u ( j ) c ˜ u d ( j ) c ˜ d u ( j ) c ˜ d d ( j ) ] = [ L u ( z j + 1 , z j ) c u u ( j ) L u ( z j + 1 , z j ) c u d ( j ) L d ( z j + 1 , z j ) c d u ( j ) L d ( z j + 1 , z j ) c d d ( j ) ] ,
( u ( z 0 ) d ( z j + 1 ) ) = [ T u u ( j ) R u d ( j ) R d u ( j ) T d d ( j ) ] ( u ( z j + 1 ) d ( z 0 ) ) .
T u u ( j ) = T u u ( j - 1 ) [ 1 - r ˜ u d ( j ) R d u ( j - 1 ) ] - 1 t ˜ u u ( j ) ,
R u d ( j ) = R u d ( j - 1 ) + T u u ( j - 1 ) [ 1 - r ˜ u d ( j ) R d u ( j - 1 ) ] - 1 r ˜ u d ( j ) T d d ( j - 1 ) ,
R d u ( j ) = r ˜ d u ( j ) + t ˜ d d ( j ) [ 1 - R d u ( j - 1 ) r ˜ u d ( j ) ] - 1 R d u ( j - 1 ) t ˜ u u ( j ) ,
T d d ( j ) = t ˜ d d ( j ) [ 1 - R d u ( j - 1 ) r ˜ u d ( j ) ] - 1 T d d ( j - 1 ) .
[ T u u ( 0 ) R u d ( 0 ) R d u ( 0 ) T d d ( 0 ) ] = [ t ˜ u u ( 0 ) r ˜ u d ( 0 ) r ˜ d u ( 0 ) t ˜ d d ( 0 ) ] .
( 1 - X ) - 1 = k = 0 X k ,
( u ( z j + 1 ) d ( z j + 1 ) ) = c ( j ) c ( j - 1 ) c ( 0 ) ( u ( z 0 ) d ( z 0 ) ) ,
( u ( z 0 ) d ( z j + 1 ) ) = s ( 0 ) * s ( 1 ) * * s ( j ) ( u ( z j + 1 ) d ( z 0 ) ) ,
[ a u u a u d a d u a d d ] * [ b u u b u d b d u b d d ] = [ a u u ( 1 - b u d a d u ) - 1 b u u a u d + a u u ( 1 - b u d a d u ) - 1 b u d a d d b d u + b d d ( 1 - a d u b u d ) - 1 a d u b u u b d d ( 1 - a d u b u d ) - 1 a d d ] .
[ t ˜ u u ( j ) r ˜ u d ( j ) r ˜ d u ( j ) t ˜ d d ( j ) ] [ c ˜ u u ( j ) c ˜ u d ( j ) c ˜ d u ( j ) c ˜ d d ( j ) ] T ,
d ( z M + 1 ) = k = 0 d 2 k ( z M + 1 ) ,
u ( z 0 ) = k = 0 u 2 k + 1 ( z 0 ) .
d 0 ( z j ) = P d ( z j , z 0 ) d 0 ( z 0 ) ,
u 2 k + 1 ( z j ) = i j P u ( z j , z i ) r u d ( i ) d 2 k ( z i ) ,
d 2 k ( z j ) = i j P d ( z j , z i ) r d u ( i - 1 ) u 2 k - 1 ( z i ) ,
P u ( z j , z i ) = j q i - 1 t ˜ u u ( q ) ,             P d ( z j , z i ) = j - 1 p i t ˜ d d ( p ) .
i j k a j = { a i a i + 1 a k i k 1 i > k , i j k a j = { a i a i + 1 a k i k 1 i < k .
K ( i , j , k ) = [ i - 1 q j + 1 t ˜ d d ( q ) ] r ˜ d u ( j ) [ j + 1 p k - 1 t ˜ u u ( p ) ] .
Q 0 ( M ) = 1 ,
Q 2 p ( M ) = { i 1 , , i 2 p - 1 } I p ( M ) K ( M ) , i 2 p - 1 , i 2 p - 2 ) × r ˜ u d ( i 2 p - 2 ) K ( i 2 , i 1 , M ) r ˜ u d ( M ) ,             p 1 ,
i 0 = M , i 2 n - 3 + 1 i 2 n - 2 M , 0 i 2 n - 1 i 2 n - 2 - 1 , 1 n p .
Q 2 p ( 0 ) = 0 ,             p 1 ,
Q 2 p ( 1 ) = [ r ˜ d u ( 0 ) r ˜ u d ( 1 ) ] p .
d 2 k ( z M + 1 ) = t ˜ d d ( M ) p = 0 k Q 2 p ( M ) d 2 k - 2 p ( z M ) .
Q 2 p ( M ) = l = 1 p m = 1 l i m = 2 p - l m = 1 l [ J i m ( M - 1 ) r ˜ u d ( M ) ] ,             p 1 , M 1 ,
J 2 n - 1 ( 0 ) = 0 ,             n > 1 ,
J 2 n - 1 ( M - 1 ) = { i 1 , i 2 , , i 2 n - 1 } I n ( M ) K ( M , i 2 n - 1 , i 2 n - 2 ) × r ˜ u d ( i 2 n - 2 ) K ( i 2 , i 1 , M ) .
i 0 = M , i 2 l - 3 + 1 i 2 l - 2 M - 1 , 0 i 2 l - 1 i 2 l - 2 - 1 , 1 l n .
J 1 ( M - 1 ) = i = 0 M - 1 K ( M , i , M ) .
J 2 p - 1 ( M - 1 ) = t ˜ d d ( M - 1 ) × { m = 0 p - 1 l = 1 m q = 1 l i q = 2 m - l q = 1 l [ J i q ( M - 2 ) r ˜ u d ( M - 1 ) ] J 2 p - 2 m - 1 ( M - 2 ) } t ˜ u u ( M - 1 ) ,
J 2 p - 1 ( M - 1 ) = t ˜ d d ( M - 1 ) [ m = 0 p - 1 Q 2 m ( M - 1 ) J 2 p - 2 m - 1 ( M - 2 ) ] t ˜ u u ( M - 1 ) .
D ( z M + 1 ) = d ( z M + 1 )
k = 0 d 2 k ( z M + 1 ) = t ˜ d d ( M ) [ 1 - R d u ( M - 1 ) r ˜ u d ( M ) ] - 1 k = 0 d 2 k ( z M ) .
d 2 k ( z M + 1 ) = t ˜ d d ( M ) p = 0 k [ [ 1 - R d u ( M - 1 ) r ˜ u d ( M ) ] - 1 ] 2 p d 2 k - 2 p ( z M ) ,
[ [ 1 - R d u ( M - 1 ) r ˜ u d ( M ) ] - 1 ] 2 p = Q 2 p ( M ) .
[ [ 1 - R d u ( M - 1 ) r ˜ u d ( M ) ] - 1 ] 2 p = l = 1 p m = 1 l i m = 2 p - l m = 1 l { [ R d u ( M - 1 ) ] i m r ˜ u d ( M ) } .
[ R d u ( M - 1 ) ] 2 n - 1 = J 2 n - 1 ( M - 1 )
[ R d u ( M - 1 ) ] 1 = r ˜ d u ( M - 1 ) + t ˜ d d ( M - 1 ) [ R d u ( M - 2 ) ] 1 t ˜ u u ( M - 1 ) = = l = 0 M - 1 K ( M , l , M ) .
[ R d u ( 1 ) ] 2 n - 1 = t ˜ d d ( 1 ) [ [ 1 - r ˜ d u ( 0 ) r ˜ u d ( 1 ) ] - 1 ] 2 n - 1 r ˜ d u ( 0 ) t ˜ u u ( 1 ) = t ˜ d d ( 1 ) [ r ˜ d u ( 0 ) r ˜ u d ( 1 ) ] 2 n - 2 r ˜ d u ( 0 ) t ˜ u u ( 1 ) .
J 2 n - 1 ( 1 ) = t ˜ d d ( 1 ) [ m = 0 n - 1 Q 2 m ( 1 ) J 2 n - 2 m - 1 ( 0 ) ] t ˜ u u ( 1 ) = t ˜ d d ( 1 ) [ r ˜ d u ( 0 ) r ˜ u d ( 1 ) ] 2 n - 2 r ˜ d u ( 0 ) t ˜ u u ( 1 ) .
[ R d u ( M - 1 ) ] 2 n - 1 = t ˜ d d ( M - 1 ) ( m = 0 n - 1 [ [ 1 - R d u ( M - 2 ) r ˜ u d ( M - 1 ) ] - 1 ] 2 m × [ R d u ( M - 2 ) ] 2 n - 2 m - 1 ) t ˜ u u ( M - 1 ) = t ˜ d d ( M - 1 ) ( m = 0 n - 1 l = 1 m ( q = 1 l i q ) = 2 m - l q = 1 l { [ R d u ( M - 2 ) ] i q r ˜ u d ( M - 1 ) } × [ R d u ( M - 2 ) ] 2 n - 2 m - 1 ) t ˜ u u ( M - 1 )

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