Abstract

The analysis of gratings of arbitrary depth, profile, and permittivity is conducted by cutting the modulated region into different slices for which the differential theory of gratings is able to compute the diffracted field for both TE and TM polarization without numerical instabilities. The use of a suitable transition matrix (R matrix) then allows one to analyze the entire stack without encountering the numerical instabilities that generally occur with use of the T-transmission matrix, which is well known in stratified media theory. The use of the R-matrix propagation algorithm provides a breakthrough for grating theoreticians in the sense that it not only permits the study of grating of arbitrary depth but also eliminates the numerical instabilities that have plagued the differential theory in TM polarization during the past 20 years.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. T. Gale, K. Knop, R. Morf, “Zero order diffractive microstructure for security applications,” presented at the Optical Security and Anticounterfeiting Systems Conference, Los Angeles, Calif., January 15–16, 1990.
  2. E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
    [CrossRef] [PubMed]
  3. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  4. B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
    [CrossRef]
  5. M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
    [CrossRef]
  6. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 and 706–782 (1950).
  7. 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]
  8. D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
    [CrossRef]
  9. 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]
  10. 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]
  11. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2593.
  12. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction gratings,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  13. 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]
  14. L. Li, “A modal analysis of lamellar diffraction gratings in conical mounting,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  15. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  16. M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. B 5, 1522–1524 (1988).
    [CrossRef]
  17. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
    [CrossRef]
  18. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).
  19. H. Iwaoka, K. Akiyama, “A high-resolution laser scale interferometer,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 135–139 (1984).
    [CrossRef]
  20. A. Teimel, “Technology and application of grating interferometers in high-precision measurements,” in Progress in Precision Engineering, P. Seysried, H. Kunzmann, T. McKeown, eds. (Springer-Verlag, Berlin, 1991), pp. 131–147.

1994 (1)

1993 (1)

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

1991 (2)

1988 (1)

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. B 5, 1522–1524 (1988).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction gratings,” 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]

1977 (1)

1976 (2)

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]

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 and 706–782 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 and 706–782 (1950).

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 gratings,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Akiyama, K.

H. Iwaoka, K. Akiyama, “A high-resolution laser scale interferometer,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 135–139 (1984).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction gratings,” 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.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction gratings,” 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]

Craig, M. S.

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 gratings,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

DeSandre, L. F.

Dhez, P.

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

Elson, J. M.

Gale, M. T.

M. T. Gale, K. Knop, R. Morf, “Zero order diffractive microstructure for security applications,” presented at the Optical Security and Anticounterfeiting Systems Conference, Los Angeles, Calif., January 15–16, 1990.

Henrici, P.

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).

Iwaoka, H.

H. Iwaoka, K. Akiyama, “A high-resolution laser scale interferometer,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 135–139 (1984).
[CrossRef]

Knop, K.

M. T. Gale, K. Knop, R. Morf, “Zero order diffractive microstructure for security applications,” presented at the Optical Security and Anticounterfeiting Systems Conference, Los Angeles, Calif., January 15–16, 1990.

Li, L.

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

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2593.

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]

Loewen, E. G.

Maystre, D.

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 gratings,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Morf, R.

M. T. Gale, K. Knop, R. Morf, “Zero order diffractive microstructure for security applications,” presented at the Optical Security and Anticounterfeiting Systems Conference, Los Angeles, Calif., January 15–16, 1990.

Nevière, M.

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

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. B 5, 1522–1524 (1988).
[CrossRef]

E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
[CrossRef] [PubMed]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

Pai, D. M.

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Teimel, A.

A. Teimel, “Technology and application of grating interferometers in high-precision measurements,” in Progress in Precision Engineering, P. Seysried, H. Kunzmann, T. McKeown, eds. (Springer-Verlag, Berlin, 1991), pp. 131–147.

Vidal, B.

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. B 5, 1522–1524 (1988).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[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]

Ann. Phys. (Paris) (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 and 706–782 (1950).

Appl. Opt. (1)

Chem. Phys. (1)

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[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. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. B 5, 1522–1524 (1988).
[CrossRef]

Nou. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction gratings,” 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]

Other (7)

M. T. Gale, K. Knop, R. Morf, “Zero order diffractive microstructure for security applications,” presented at the Optical Security and Anticounterfeiting Systems Conference, Los Angeles, Calif., January 15–16, 1990.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).

H. Iwaoka, K. Akiyama, “A high-resolution laser scale interferometer,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 135–139 (1984).
[CrossRef]

A. Teimel, “Technology and application of grating interferometers in high-precision measurements,” in Progress in Precision Engineering, P. Seysried, H. Kunzmann, T. McKeown, eds. (Springer-Verlag, Berlin, 1991), pp. 131–147.

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 (6)

Fig. 1
Fig. 1

Decomposition of the modulated region into M different slices.

Fig. 2
Fig. 2

Illustration of a stack of superimposed dielectric gratings.

Fig. 3
Fig. 3

Evolution of the accuracy ΔM on 0, +1, and +2 reflected efficiencies and on the total diffracted energy as functions of number of slices, for grating configuration 3 and TE polarization.

Fig. 4
Fig. 4

Same as for Fig. 3 but for TM polarization.

Fig. 5
Fig. 5

Evolutions of the −1- and 0-order efficiencies as functions of the groove depth for a symmetric triangular-profile gold grating with Littrow mount, for TE polarization.

Fig. 6
Fig. 6

0-Order efficiency of a dielectric coated sinusoidal silver grating as a function of dielectric thickness (M = 10, N = 6, Q = 20), for TE polarization.

Tables (5)

Tables Icon

Table 1 Evolution of the Transmitted Efficiencies and Total Diffracted Energy as Functions of M for Grating Configuration 1 and TE and TM Polarizations

Tables Icon

Table 2 Same as Table 1 for Grating Configuration 2 Made by Rods of Chromium Embedded in a Dielectric Grating

Tables Icon

Table 3 Evolution of the Reflected Efficiencies and Total Diffracted Energy as Functions of M for Grating Configuration 3 and TE and TM Polarizations

Tables Icon

Table 4 Convergence of the 0- and −1-Order Efficiencies and Total Diffracted Energy of a Symmetrical Lamellar Chromium Grating as N Increases, with TM Polarization and Littrow Mounta

Tables Icon

Table 5 Evolution of the Transmitted and Reflected Efficiencies of a Grating Made with Rectangular Chromium Rods as Functions of Na

Equations (53)

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

u 1 = n { A n ( 1 ) exp [ - i β n ( 1 ) y ] + B n ( 1 ) exp [ i β n ( 1 ) y ] } exp ( i α n x ) ,
u 2 = n { A n ( 2 ) exp [ - i β n ( 2 ) y ] + B n ( 2 ) exp [ i β n ( 1 ) y ] } exp ( i α n x ) ,
α n = α + n 2 π d ,             α = k 0 ν 1 sin θ , k 0 = 2 π λ ,             β m ( i ) = k 0 2 ν i 2 - α n 2 , i [ 1 , 2 ] ,             Re β n ( i ) + Im β n ( i ) > 0 ,
u = n U n ( y ) exp ( i α n x ) ,
[ U n ( y j ) V n ( y j ) ] = t ( j ) [ U n ( y j - 1 ) V n ( y j - 1 ) ] .
[ U n ( h ) V n ( h ) ] = t ( M ) t ( M - 1 ) t ( j ) t ( 2 ) t ( 1 ) [ U n ( 0 ) V n ( 0 ) ] .
[ U n ( y j - 1 ) U n ( y j ) ] = r ( j ) [ V n ( y j - 1 ) V n ( y j ) ]
r ( j ) = [ r 11 ( j ) r 12 ( j ) r 21 ( j ) r 22 ( j ) ] .
r 11 ( j ) = - [ t 21 ( j ) ] - 1 t 22 ( j ) , r 12 ( j ) = [ t 21 ( j ) ] - 1 , r 21 ( j ) = t 12 ( j ) - t 11 ( j ) [ t 21 ( j ) ] - 1 t 22 ( j ) , r 22 ( j ) = t 11 ( j ) [ t 21 ( j ) ] - 1 .
[ U n ( 0 ) U n ( y j ) ] = R ( j ) [ V n ( 0 ) V n ( y j ) ]
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 .
[ U n ( 0 ) U n ( y M ) ] = R ( M ) [ V n ( 0 ) V n ( y M ) ] .
U n ( h ) = A n ( 1 ) exp ( - i β 1 , n h ) + B n ( 1 ) exp ( i β 1 , n h ) , V n ( h ) = q 1 ( - i β 1 , n ) [ A n ( 1 ) exp ( - i β 1 , n h ) - B n ( 1 ) exp ( i β 1 , n h ) ] , U n ( 0 ) = A n ( 2 ) + B n ( 2 ) , V n ( 0 ) = q 2 ( - i β 2 , n ) [ A n ( 2 ) - B n ( 2 ) ] ,
q i = { 1 i for TE polarization 1 k 0 2 ν i 2 for TM polarization , i = 1 , 2 .
[ A n ( 2 ) + B n ( 2 ) A n ( 1 ) exp ( - i β 1 , n h ) + B n ( 1 ) exp ( i β 1 , n h ) ] = [ R 11 ( M ) R 12 ( M ) R 21 ( M ) R 22 ( M ) ] × [ q 2 ( - i β 2 , n ) [ A n ( 2 ) - B n ( 2 ) ] q 1 ( - i β 1 , n ) [ A n ( 1 ) exp ( - i β 1 , n h ) - B n ( 1 ) exp ( i β 1 , n h ) ] ] .
A ˜ n ( 1 ) = A n ( 1 ) exp ( - i β 1 , n h ) , B ˜ n ( 1 ) = B n ( 1 ) exp ( i β 1 , n h ) .
{ A n ( 2 ) } + { B n ( 2 ) } = - i q 2 R 11 ( M ) [ β 2 , n ] ( { A n ( 2 ) } - { B n ( 2 ) } ) - i q 1 R 12 ( M ) [ β 1 , n ] ( { A ˜ n ( 1 ) } - { B ˜ n ( 1 ) } ) , { A ˜ n ( 1 ) } + { B ˜ n ( 1 ) } = - i q 2 R 21 ( M ) [ β 2 , n ] ( { A n ( 2 ) } - { B n ( 2 ) } ) - i q 1 R 22 ( M ) [ β 1 , n ] ( { A ˜ n ( 1 ) } - { B ˜ n ( 1 ) } ) ,
P 11 = i q 2 R 11 ( M ) [ β 2 , n ] , P 12 = i q 1 R 12 ( M ) [ β 1 , n ] , P 21 = i q 2 R 21 ( M ) [ β 2 , n ] , P 22 = i q 1 R 22 ( M ) [ β 1 , n ] ,
Q = { exp ( - i β 1 , n h ) δ n , 0 } ,
[ I + P 11 - P 12 - P 21 - I + P 22 ] [ { T n } { B ˜ n ( 1 ) } ] = [ - P 12 Q ( I + P 22 ) Q ] ,
[ U n ( y j - 1 ) V n ( y j ) ] = r ( j ) [ V n ( y j - 1 ) U n ( y j ) ]
r ( j ) = [ r 11 ( j ) r 12 ( j ) r 21 ( j ) r 22 ( j ) ] .
r 11 ( j ) = - [ t 11 ( j ) ] - 1 t 12 ( j ) , r 12 ( j ) = [ t 11 ( j ) ] - 1 , r 21 ( j ) = t 22 ( j ) - t 21 ( j ) [ t 11 ( j ) ] - 1 t 12 ( j ) , r 22 ( j ) = t 21 ( j ) [ t 11 ( j ) ] - 1 .
[ U n ( 0 ) V n ( y j ) ] = R ( j ) [ V n ( 0 ) U n ( y j ) ] .
R 11 ( j ) = R 11 ( j - 1 ) + R 12 ( j - 1 ) Z ( j ) r 11 ( j ) R 21 ( j - 1 ) , R 12 ( j ) = R 12 ( j - 1 ) Z ( j ) r 12 ( j ) , R 21 ( j ) = r 21 ( j ) Y ( j ) R 21 ( j - 1 ) , R 22 ( j ) = r 22 ( j ) + r 21 ( j ) Y ( j ) R 22 ( j - 1 ) r 12 ( j ) ,
Z ( j ) = [ I - r 11 ( j ) R 22 ( j - 1 ) ] - 1 , Y ( j ) = [ I - R 22 ( j - 1 ) r 11 ( j ) ] - 1 ,
R ( 0 ) = [ 0 I I 0 ]
H ˜ x y = - 2 E z x 2 - k 2 ( x , y ) E z , E z y = H ˜ x ,
H z y = k 2 E ˜ x , E ˜ x y = - x ( 1 k 2 ( x , y ) H z x ) - H z ,
d H ˜ x , n d y = α n 2 E z , n = m = - + ( k 2 ) n - m E z , m , d E z , n d y = H ˜ x , n
d H z , n d y = m = - + ( k 2 ) n - m E ˜ x , m , d E ˜ x , n d y = α n m = - + α m ( 1 k 2 ) n - m H z , m - H z , n
Δ M = log 10 | e p ( M + M 0 , N ) - e p ( M , N ) e p ( M , N ) | , Δ N = log 10 | e p ( M , N + N 0 ) - e p ( M , N ) e p ( M , N ) | ,
[ U m ( y j ) V m ( y j ) ] = [ t 11 ( j ) t 12 ( j ) t 21 ( j ) t 22 ( j ) ] [ U m ( y j - 1 ) V m ( y j - 1 ) ] ,
[ U m ( y j - 1 ) V m ( y j ) ] = [ r 11 ( j ) r 12 ( j ) r 21 ( j ) r 22 ( j ) ] [ V m ( y j - 1 ) U m ( y j ) ] .
U m ( y j ) = t 11 ( j ) U m ( y j - 1 ) + t 12 ( j ) V m ( y j - 1 ) , V m ( y j ) = t 21 ( j ) U m ( y j - 1 ) + t 22 ( j ) V m ( y j - 1 ) ,
U m ( y j - 1 ) = [ t 11 ( j ) ] - 1 U m ( y j ) - [ t 11 ( j ) ] - 1 t 12 ( j ) V m ( y j - 1 ) , V m ( y j ) = t 21 ( j ) [ t 11 ( j ) ] - 1 U m ( y j ) + { t 22 ( j ) - t 21 ( j ) [ t 11 ( j ) ] - 1 t 12 ( j ) } V m ( y j - 1 ) .
r 11 ( j ) = - [ t 11 ( j ) ] - 1 t 12 ( j ) , r 12 ( j ) = [ t 11 ( j ) ] - 1 , r 21 ( j ) = t 22 ( j ) - t 21 ( j ) [ t 11 ( j ) ] - 1 t 12 ( j ) , r 22 ( j ) = t 21 ( j ) [ t 11 ( j ) ] - 1 ,
[ U m ( 0 ) V m ( y j - 1 ) ] = [ R 11 ( j - 1 ) R 12 ( j - 1 ) R 21 ( j - 1 ) R 22 ( j ) ] [ V m ( 0 ) U m ( y j - 1 ) ] ,
[ U m ( 0 ) V m ( y j ) ] = [ R 11 ( j ) R 12 ( j ) R 21 ( j ) R 22 ( j ) ] [ V m ( 0 ) U m ( y j ) ] .
U m ( y j - 1 ) = r 11 ( j ) V m ( y j - 1 ) + r 12 ( j ) U m ( y j ) , V m ( y j - 1 ) = R 21 ( j - 1 ) V m ( 0 ) + R 22 ( j - 1 ) U m ( y j - 1 ) ,
[ I - r 11 ( j ) R 22 ( j - 1 ) ] U m ( y j - 1 ) = r 12 ( j ) U m ( y j ) + r 11 ( j ) R 21 ( j - 1 ) V m ( 0 ) , [ I - R 22 ( j - 1 ) r 11 ( j ) ] V m ( y j - 1 ) = R 21 ( j - 1 ) V m ( 0 ) + R 22 ( j - 1 ) r 12 ( j ) U m ( y j ) .
Z ( j ) = [ I - r 11 ( j ) R 22 ( j - 1 ) ] - 1 , Y ( j ) = [ I - R 22 ( j - 1 ) r 11 ( j ) ] - 1 .
U m ( 0 ) = R 11 ( j - 1 ) V m ( 0 ) + R 12 ( j - 1 ) U m ( y j - 1 ) , V m ( y j ) = r 21 ( j ) V m ( y j - 1 ) + r 22 ( j ) U m ( y j ) ,
U m ( 0 ) = R 11 ( j - 1 ) V m ( 0 ) + R 12 ( j - 1 ) Z ( j ) r 12 ( j ) U m ( y j ) + R 12 ( j - 1 ) Z ( j ) r 11 ( j ) R 21 ( j - 1 ) V m ( 0 ) , V m ( y j ) = r 21 ( j ) Y ( j ) R 21 ( j - 1 ) V m ( 0 ) + r 21 ( j ) Y ( j ) R 22 ( j - 1 ) r 12 ( j ) U m ( y j ) + r 22 ( j ) U m ( y j ) ,
R 11 ( j ) = R 11 ( j - 1 ) + R 12 ( j - 1 ) Z ( j ) r 11 ( j ) R 21 ( j - 1 ) , R 12 ( j ) = R 12 ( j - 1 ) Z ( j ) r 12 ( j ) , R 21 ( j ) = r 21 ( j ) Y ( j ) R 21 ( j - 1 ) , R 22 ( j ) = r 22 ( j ) + r 21 ( j ) Y ( j ) R 22 ( j - 1 ) r 12 ( j ) ,
U ( x , y ) = m [ A m exp ( - i β m y ) + B m exp ( i β m y ) ] exp ( i α m x ) ,
U m ( y j ) = A m exp ( - i β m y j ) + B m exp ( i β m y j ) ,
V m ( y j ) = - i χ β m [ A m exp ( - i β m y j ) - B m exp ( i β m y j ) ] ,
A m = 1 i χ β m 1 2 i sin [ β m ( y j - y j - 1 ) ] [ V m ( y j ) exp ( i β m y j - 1 ) - V m ( y j - 1 ) exp ( i β m y j ) ] , B m = 1 i χ β m 1 2 i sin [ β m ( y j - y j - 1 ) ] [ V m ( y j ) exp ( - i β m y j - 1 ) - V m ( y j - 1 ) exp ( - i β m y j ) ] .
U m ( y j ) = - 1 2 χ β m { 1 sin [ β m ( y j - y j - 1 ) ] } ( - 2 V m ( y j - 1 ) + { exp [ i β m ( y j - 1 - y j ) ] + exp [ i β m ( y j - y j - 1 ) ] } V m ( y j ) ) , U m ( y j - 1 ) = - 1 2 χ β m { 1 sin [ β m ( y j - y j - 1 ) ] } × ( { - exp [ i β m ( y j - y j - 1 ) ] - exp [ - i β m ( y j - y j - 1 ) ] } V m ( y j - 1 ) + 2 V m ( y j ) ) .
r 11 ( j ) = [ 1 χ β m 1 tan ( β m h j ) ] , r 12 ( j ) = [ - 1 χ β m 1 sin ( β m h j ) ] , r 21 ( j ) = [ 1 χ β m 1 sin ( β m h j ) ] , r 22 ( j ) = [ - 1 χ β m 1 tan ( β m h j ) ] ,
r 11 ( j ) = [ i χ β m tan ( β m h j ) ] , r 12 ( j ) = [ 1 cos ( β m h j ) ] , r 21 ( j ) = [ 1 cos ( β m h j ) ] , r 22 ( j ) = [ i χ β m tan ( β m h j ) ] .

Metrics