Abstract

A numerically stable method is presented for the analysis of diffraction gratings of arbitrary profile, depth, and in conical mountings. It is based on the classical modal method and uses a stack of lamellar gratpermittivity to approximate an arbitrary profile. A numerical procedure known as the R-matrix propagation aling layers gorithm is used to propagate the modal fields through the layers. This procedure renders the implementation of this new method completely immune to the numerical instability that is associated with the conventional algorithm. Numerical examples including diffraction efficiencies of both dielectric and metallic propagation gratings of depths that range from subwavelength to hundreds of wavelengths are presented. Information about the convergence and the computation time of the method is also included.

© 1993 Optical Society of America

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References

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  1. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  2. L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  3. L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  4. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  5. J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,”J. Opt. (Paris) 14, 273–288 (1983).
    [CrossRef]
  6. K-T. Lee, T. F. George, “Theoretical study of laser-induced surface excitations on a grating,” Phys. Rev. B 31, 5106–5112 (1985).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  8. 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]
  9. D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
    [CrossRef]
  10. 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]
  11. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  12. 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]
  13. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  14. M. Nevière, E. Popov, “Analysis of dielectric gratings of arbitrary profiles and thicknesses: comment,” J. Opt. Soc. Am. A 9, 2095–2096 (1992).
    [CrossRef]
  15. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1984), Vol. 21, pp. 1–67.
    [CrossRef]
  16. S. L. Chuang, J. A. Kong, “Wavescattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  17. L. Li, Q. Gong, G. N. Lawrence, J. J. Burke, “Polarization properties of planar dielectric waveguide gratings,” Appl. Opt. 31, 4190–4197 (1992).
    [CrossRef] [PubMed]
  18. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
    [CrossRef]

1993

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

1992

1991

1985

K-T. Lee, T. F. George, “Theoretical study of laser-induced surface excitations on a grating,” Phys. Rev. B 31, 5106–5112 (1985).
[CrossRef]

1983

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,”J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

1982

S. L. Chuang, J. A. Kong, “Wavescattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,”J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

1981

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

1976

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

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]

1975

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Awada, K. A.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Brekhovskikh, L. M.

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

Burke, J. J.

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,”J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Wavescattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Desandre, L. F.

Elson, J. M.

Gaylord, T. K.

George, T. F.

K-T. Lee, T. F. George, “Theoretical study of laser-induced surface excitations on a grating,” Phys. Rev. B 31, 5106–5112 (1985).
[CrossRef]

Gong, Q.

Haggans, C. W.

Kong, J. A.

S. L. Chuang, J. A. Kong, “Wavescattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Lawrence, G. N.

Lee, K-T.

K-T. Lee, T. F. George, “Theoretical study of laser-induced surface excitations on a grating,” Phys. Rev. B 31, 5106–5112 (1985).
[CrossRef]

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]

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1984), Vol. 21, pp. 1–67.
[CrossRef]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
[CrossRef]

Mcphedran, R. C.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Moharam, M. G.

Nevière, M.

Pai, D. M.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Petit, R.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,”J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Popov, E.

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,”J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[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]

Zvijac, D. J.

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Appl. Opt.

Chem. Phys.

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Chem. Phys.

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. Mod. Opt.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. Opt. (Paris)

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,”J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Phys. Rev. B

K-T. Lee, T. F. George, “Theoretical study of laser-induced surface excitations on a grating,” Phys. Rev. B 31, 5106–5112 (1985).
[CrossRef]

Radio Sci.

S. L. Chuang, J. A. Kong, “Wavescattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Other

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
[CrossRef]

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

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1984), Vol. 21, pp. 1–67.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Coordinate system for a diffraction grating in a conical mounting.

Fig. 2
Fig. 2

Approximation of a grating of arbitrary profile by a stack of lamellar gratings.

Fig. 3
Fig. 3

(a) Efficiencies and (b) and (c) polarization angles of the diffraction orders generated by a sinusoidal grating as functions of the incident azimuthal angle ϕ. The incident polar angle is 60°, and the incident polarization is p. Other values of parameter are d = 0.3 μm, h = 0.15 μm, (+) = 2.25, (−1) = 1.0, λ = 0.5 μm, M = 30, and N = 31.

Fig. 4
Fig. 4

Diffraction efficiencies of a silver four-level binary grating in (a) TE and (b) TM polarizations. The binary grating approximates a right-angle triangular grating whose period is 0.8233 μm and whose depth is 0.341 μm. Other values of parameters are (+) = (1.46)2, (−1) = (0.1 + i5.58)2, λ = 0.85 μm, and N = 51. Here θ < 0 is equivalent to θ > 0 and ϕ > 180° in the polar coordinate system that is defined in Fig. 1.

Fig. 5
Fig. 5

Convergence of the TE diffraction efficiencies with respect to the stratification order M for a shallow metallic grating [the first grating in Table 4; h/d = 0.1]. M* = 170.

Fig. 6
Fig. 6

Convergence of the TE diffraction efficiencies with respect to the stratification order M for a very deep metallic grating [the third grating in Table 4; h/d = 10].

Fig. 7
Fig. 7

(a) Convergence of the transmitted negative first-order TE diffraction efficiencies for the four dielectric gratings referenced in Table 3. N* = 105. (b) Convergence of the transmitted negative first-order TM diffraction efficiencies for the four dielectric gratings referenced in Table 3. N* = 105.

Fig. 8
Fig. 8

(a) Convergence of the negative first-order TE diffraction efficiencies for the four metallic gratings referenced in Table 4. N* = 105. (b) Convergence of the negative first-order TM diffraction efficiencies for the four metallic gratings referenced in Table 4. N* = 195.

Fig. 9
Fig. 9

Computation times for computing diffraction efficiencies of the gratings referenced in Tables 3 and 4. The CPU times were measured with a SPARCstation 2 made by SPARC International, Inc. The computer program runs in double precision. The stratification order is fixed at M = 10. The curves are the third-order polynomial fits.

Tables (4)

Tables Icon

Table 1 Illustration of the Numerical Instability of the T-Matrix Propagation Algorithma

Tables Icon

Table 2 Comparison of the Numerical Results of the Present Study and the Results of Chuang and Kong16 for a Sinusoidal Dielectric Grating in a Conical Mountinga

Tables Icon

Table 3 Diffraction Efficiencies of Sinusoidal Dielectric Gratings of Four Groove Depth-to-Period Ratiosa

Tables Icon

Table 4 Diffraction Efficiencies of Sinusoidal Metallic Gratings of Four Groove Depth-to-Period Ratiosa

Equations (54)

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h j = y j - y j - 1 ,             j = 0 , 1 , , M .
j ( x ) = { 2 , j , x 1 , j x x 2 , j 1 , j otherwise ,             j = 1 , 2 , , M .
k j 2 ( x ) = j ( x ) μ j ( x ) k 2 ,             j = 0 , 1 , , M + 1 ,
k = k M + 1 ( x ^ sin θ cos ϕ - y ^ cos θ + z ^ sin θ sin ϕ ) .
k ˜ j 2 ( x ) = k j 2 ( x ) - k z 2 ,
( E z ( x , y ) H z ( x , y ) ) = ( I z ( e ) I z ( h ) ) exp [ - i β 0 ( + 1 ) y ] e 0 ( x ) + n = - + ( R n ( e ) R n ( h ) ) exp [ i β n ( + 1 ) y ] e n ( x ) ,             y y M + 1 ,
( E z ( x , y ) H z ( x , y ) ) = n = - + ( T n ( e ) T n ( h ) ) exp [ - i β n ( - 1 ) y ] e n ( x ) ,             y y 0 ,
e n ( x ) = exp ( i α n x ) ,
α n = α 0 + 2 n π / d ,             α 0 = k ( + 1 ) sin θ cos ϕ ,
β n ( j ) 2 = k ˜ ( j ) 2 - α n 2 ,             Re [ β n ( j ) ] + Im [ β n ( j ) ] > 0 ,             j = ± 1.
( E z ( x , y ) H z ( x , y ) ) = m = 0 ( ω m , j ( e ) ( y ) u m , j ( e ) ( x ) χ m , j ( e ) ( y ) w m , j ( e ) ( x ) ) + m = 0 ( χ m , j ( h ) ( y ) w m , j ( h ) ( x ) ω m , j ( h ) ( y ) u m , j ( h ) ( x ) ) ,             j = 1 , , M .
ω m , j ( s ) ( y ) = a m , j ( s ) cos [ λ m , j ( s ) y ] + b m , j ( s ) sin [ λ m , j ( s ) y ] ,
χ m , j ( s ) ( y ) = Λ m , j ( s ) { - b m , j ( s ) cos [ λ m , j ( s ) y ] + a m , j ( s ) sin [ λ m , j ( s ) y ] } ,
Λ m , j ( s ) = [ k z 2 + λ m , j ( s ) 2 ] / λ m , j ( s ) ,             j = 1 , 2 , , M ,
λ m , 0 ( s ) = β m ( - 1 ) ,             λ m , M + 1 ( s ) = β m ( + 1 ) ,
a m , 0 ( s ) = T m ( s ) ,             b m , 0 ( s ) = - i T m ( s ) ,
a m , M + 1 ( s ) = R m ( s ) + I z ( s ) δ m 0 ,             b m , M + 1 ( s ) = i [ R m ( s ) - I z ( s ) δ m 0 ] ,
u m , j ( s ) ( x ) = e m ( x ) ,             w m , j ( s ) ( x ) = 0 ,             j = 0 , M + 1.
Λ m , j ( s ) = i ,             j = 0 , M + 1.
( Ω j + 1 ( y i + 0 ) X j + 1 ( y j + 0 ) ) = [ A 11 ( j ) A 12 ( j ) A 21 ( j ) A 22 ( j ) ] ( Ω j ( y j - 0 ) X j ( y j - 0 ) ) ,
Ω j ( y ) = ( ω j ( e ) ( y ) ω j ( h ) ( y ) ) ,             X j ( y ) = ( χ j ( e ) ( y ) χ j ( h ) ( y ) ) .
( Ω j ( y j - 0 ) X j ( y j - 0 ) ) = [ C 11 ( j ) - S 12 ( j ) S 21 ( j ) C 22 ( j ) ] ( Ω j ( y j - 1 + 0 ) X j ( y j - 1 + 0 ) ) ,
C 11 ( j ) = C 22 ( j ) = [ cos [ λ m , j ( e ) h j ] 0 0 cos [ λ m , j ( h ) h j ] ] ,
S 12 ( j ) = [ [ 1 / Λ m , j ( e ) ] sin [ λ m , j ( e ) h j ] 0 0 [ 1 / Λ m , j ( h ) ] sin [ λ m , j ( h ) h j ] ] ,
S 21 ( j ) = [ Λ m , j ( e ) sin [ λ m , j ( e ) h j ] 0 0 Λ m , j ( h ) sin [ λ m , j ( h ) h j ] ] .
( Ω j + 1 ( y i + 0 ) X j + 1 ( y j + 0 ) ) = [ t 11 ( j ) t 12 ( j ) t 21 ( j ) t 22 ( j ) ] ( Ω j ( y j - 1 + 0 ) X j ( y j - 1 + 0 ) ) ,
[ t 11 ( j ) t 12 ( j ) t 21 ( j ) t 22 ( j ) ] = [ A 11 ( j ) A 12 ( j ) A 21 ( j ) A 22 ( j ) ] [ C 11 ( j ) - S 12 ( j ) S 21 ( j ) C 22 ( j ) ] .
T ( j ) = t ( j ) t ( j - 1 ) t ( 0 ) .
( Ω j ( y j - 1 + 0 ) Ω j + 1 ( y j + 0 ) ) = [ r 11 ( j ) r 12 ( j ) r 21 ( j ) r 22 ( j ) ] ( X j ( y j - 1 + 0 ) X j + 1 ( y j + 0 ) ) ,
[ r 11 ( j ) r 12 ( j ) r 21 ( j ) r 22 ( j ) ] = [ - t 21 ( j ) - 1 t 22 ( j ) t 21 ( j ) - 1 t 12 ( j ) - t 11 ( j ) t 21 ( j ) - 1 t 22 ( j ) t 11 ( j ) t 21 ( j ) - 1 ] .
( Ω 0 ( y - 1 + 0 ) Ω j + 1 ( y j + 0 ) ) = [ R 11 ( j ) R 12 ( j ) R 21 ( j ) R 22 ( j ) ] ( X 0 ( y - 1 + 0 ) X j + 1 ( y j + 0 ) ) .
R 11 ( j ) = R 11 ( j - 1 ) + R 12 ( j - 1 ) Z ( j ) R 21 ( j - 1 ) , R 12 ( j ) = - R 12 ( j - 1 ) Z ( j ) r 12 ( j ) , R 21 ( j ) = r 21 ( j ) Z ( j ) R 21 ( j - 1 ) , R 22 ( j ) = r 22 ( j ) - r 21 ( j ) Z ( j ) r 12 ( j ) ,
Z ( j ) = [ r 11 ( j ) - R 22 ( j - 1 ) ] - 1 .
[ R 11 ( 0 ) R 12 ( 0 ) R 21 ( 0 ) R 22 ( 0 ) ] = [ r 11 ( 0 ) r 12 ( 0 ) r 21 ( 0 ) r 22 ( 0 ) ] .
[ 1 + R 11 ( M ) - R 12 ( M ) R 21 ( M ) 1 - R 22 ( M ) ] ( T ˜ R ˜ ) = [ - 1 + R 11 ( M ) - R 12 ( M ) R 21 ( M ) - 1 - R 22 ( M ) ] ( 0 I ˜ ) ,
I ˜ = ( I z ( e ) I z ( h ) ) δ m 0 exp [ - β m ( + 1 ) y M ] , R ˜ = ( R m ( e ) R m ( h ) ) exp [ i β m ( + 1 ) y M ] , T ˜ = ( T m ( e ) T m ( h ) ) exp [ - i β m ( - 1 ) y - 1 ] .
[ R 11 ( - 1 ) R 12 ( - 1 ) R 21 ( - 1 ) R 22 ( - 1 ) ] = [ - 1 0 0 - 1 ] ,
[ 1 - R 22 ( M ) ] R ˜ = - [ 1 + R 22 ( M ) ] I ˜ .
[ β 0 ( + 1 ) / k ˜ ( + 1 ) 2 ] [ ( + 1 ) I z ( e ) 2 + μ ( + 1 ) I z ( h ) 2 ] = 1.
η n ( + 1 ) = [ β n ( + 1 ) / k ˜ ( + 1 ) 2 ] [ ( + 1 ) R n ( e ) 2 + μ ( + 1 ) R n ( h ) 2 ] ,
η n ( - 1 ) = [ β n ( - 1 ) / k ˜ ( - 1 ) 2 ] [ ( - 1 ) T n ( e ) 2 + μ ( - 1 ) T n ( h ) 2 ] ,
k n ( j ) = x ^ α n + y ^ sgn ( j ) β n ( j ) + z ^ k z ,             j = ± 1 ,
s ^ n ( j ) = [ k n ( j ) × y ^ ] / k n ( j ) × y ^ ,             p ^ n ( j ) = s ^ n ( j ) × k n ( j ) / k ( j ) .
α n ( j ) = arctan [ E n s ( j ) / E n p ( j ) ] , 0 α n ( j ) π / 2 , δ n ( j ) = - arg [ E n s ( j ) / E n p ( j ) ] , - π < δ n ( j ) π ,
E n s ( + 1 ) E n p ( + 1 ) = [ μ ( + 1 ) ( + 1 ) ] 1 / 2 α n ( + 1 ) k 0 R n ( e ) + β n ( + 1 ) k z R n ( k ) α n μ ( + 1 ) k 0 R n ( h ) - β n ( + 1 ) k z R n ( e ) .
Δ q = log 10 ( f q - f q * ) / f q * ,
T = a + M P 3 ( N ) ,
f q g = 0 d f ( x ) ¯ q ( x ) g ( x ) d x ,
[ e m + 1 / 1 u n , 1 ( h ) ] - 1 = [ u m , 1 + ( h ) e n ] ,
[ u m , j + ( h ) 1 / e j + 1 u n , j + 1 ( h ) ] - 1 = [ u m , j + 1 + ( h ) 1 / j u n , j ( h ) ] ,
r 11 ( 0 ) = i δ m n cot [ β n ( - 1 ) h 0 ] , r 12 ( 0 ) = ( - 1 ) β m ( - 1 ) csc [ β m ( - 1 ) h 0 ] e m + | 1 1 | u n , 1 ( h ) r 21 ( 0 ) = i u m , 1 + ( h ) | 1 1 | e n csc [ β n ( - 1 ) h 0 ] , r 22 ( 0 ) = l = 0 u m , 1 + ( h ) | 1 1 | e l ( - 1 ) β l ( - 1 ) cot [ β l ( - 1 ) h 0 ] e l + | 1 1 | u n , 1 ( h ) ;
r 11 ( j ) = - δ m n cot [ λ n , j ( h ) h j ] λ n , j ( h ) , r 12 ( j ) = csc [ λ m , j ( h ) h j ] λ m , j ( h ) u m , j + ( h ) | 1 j + 1 | u n , j + 1 ( h ) , r 21 ( j ) = - u m , j + 1 + ( h ) | 1 j + 1 | u n , j ( h ) csc [ λ n , j ( h ) h j ] λ n , j ( h ) , r 22 ( j ) = l = 0 u m , j + 1 + ( h ) | 1 j + 1 | u n , j ( h ) cot [ λ l , j ( h ) h j ] λ l , j ( h ) u l , j + ( h ) | 1 j + 1 | u n , j + 1 ( h ) ;
r 11 ( M ) = - δ m n cot [ λ n , M ( h ) h M ] λ n , M ( h ) , r 12 ( M ) = - i β m ( + 1 ) ( + 1 ) csc [ λ m , M ( h ) h M ] λ m , M ( h ) u m , M + ( h ) e n , r 21 ( M ) = - e m + u n , M ( h ) csc [ λ n , M ( h ) h M ] λ n , M ( h ) , r 22 ( M ) = - i l = 0 e m + u l , M ( h ) cot [ λ l , M ( h ) h M ] λ l , M ( h ) u l , M + ( h ) e n β n ( + 1 ) ( + 1 ) .
h M = h M + h M ,             exp ( i λ * h M ) < 10 8 .

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