Abstract

We develop an electromagnetic analysis and its numerical implementation for gratings of arbitrary shape etched into a plane multilayer. The multilayer consists of a stack of identical bilayers, each bilayer including two materials (dielectrics and/or metals) with different thicknesses. The analysis is based on the differential theory of gratings, which is generalized to the case of many superimposed gratings by computation of the T matrix of each elementary grating linking the field above the modulated region to the field below it. The product of all the T matrices gives the T matrix of the stack, from which the outgoing wave condition permits derivation of the grating efficiencies. The method, developed for TE polarization, can be used throughout the entire spectrum, but it is particularly useful in the soft-x-ray region. Examples of performances of multilayer gratings in this spectral domain are computed.

© 1994 Optical Society of America

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  1. T. W. Barbee, “Combined microstructure x-ray optics,” Rev. Sci. Instrum. 60, 1588–1595 (1989).
    [CrossRef]
  2. V. V. Aristov, A. I. Erko, V. V. Martynov, “Principles of Bragg–Fresnel multilayers optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
    [CrossRef]
  3. A. I. Erko, “Synthesized multilayer Fresnel x-ray optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
    [CrossRef]
  4. A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
    [CrossRef]
  5. 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]
  6. M. Nevière, “Multilayer coated gratings for x-ray diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–473 (1991).
    [CrossRef]
  7. A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
    [CrossRef]
  8. M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).
  9. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  10. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  11. R. Petit, “Diffraction par un réseau,” in Ondes Électromagnétiques en Radioélectricité et en Optique (Masson, Paris, 1993), pp. 296–317.
  12. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
    [CrossRef]
  13. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).
  14. C. Rife, W. R. Hunter, T. W. Barbee, R. G. Cruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. 28, 2984–2986 (1989).
    [CrossRef] [PubMed]

1993 (1)

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

1991 (1)

1990 (1)

A. I. Erko, “Synthesized multilayer Fresnel x-ray optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
[CrossRef]

1989 (2)

1988 (1)

V. V. Aristov, A. I. Erko, V. V. Martynov, “Principles of Bragg–Fresnel multilayers optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

1978 (1)

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. 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, 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, 706–782 (1950).

Agafonov, Yu.

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

Agafonov, Yu. A.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

Aristov, V. V.

V. V. Aristov, A. I. Erko, V. V. Martynov, “Principles of Bragg–Fresnel multilayers optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Barbee, T. W.

Brinkman, A. C.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Brunel, M.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

Craig, W. W.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Cruddace, R. G.

de Korte, P. A. J.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Den Boggende, A. J. F.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

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]

Erko, A.

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

Erko, A. I.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

A. I. Erko, “Synthesized multilayer Fresnel x-ray optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
[CrossRef]

V. V. Aristov, A. I. Erko, V. V. Martynov, “Principles of Bragg–Fresnel multilayers optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Hailey, C. J.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Hunter, W. R.

Kahn, S. M.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Martynov, V. V.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

V. V. Aristov, A. I. Erko, V. V. Martynov, “Principles of Bragg–Fresnel multilayers optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

Maystre, D.

Nevière, M.

M. Nevière, “Multilayer coated gratings for x-ray diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–473 (1991).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. 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]

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Petit, R.

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

R. Petit, “Diffraction par un réseau,” in Ondes Électromagnétiques en Radioélectricité et en Optique (Masson, Paris, 1993), pp. 296–317.

Rife, C.

Roschupkin, D. V.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

Vidal, B.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

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]

Videler, P. H.

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[CrossRef]

Vincent, P.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

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

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

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

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]

Yuakshin, A.

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

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, 706–782 (1950).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

J. X-Ray Sci. Technol. (1)

A. I. Erko, “Synthesized multilayer Fresnel x-ray optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
[CrossRef]

Nouv. Rev. Opt. (1)

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

Nucl. Instrum. Methods Phys. Res. A (1)

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333, 599–606 (1993).
[CrossRef]

Rev. Phys. Appl. (1)

V. V. Aristov, A. I. Erko, V. V. Martynov, “Principles of Bragg–Fresnel multilayers optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Rev. Sci. Instrum. (1)

T. W. Barbee, “Combined microstructure x-ray optics,” Rev. Sci. Instrum. 60, 1588–1595 (1989).
[CrossRef]

Other (5)

A. J. F. Den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of x-ray reflection gratings,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Soc. Proc. Photo-Opt. Instrum. Eng.982, 283–298 (1988).
[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]

R. Petit, “Diffraction par un réseau,” in Ondes Électromagnétiques en Radioélectricité et en Optique (Masson, Paris, 1993), pp. 296–317.

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

M. Brunel, D. V. Roschupkin, B. Vidal, P. Vincent, Yu. Agafonov, A. Erko, V. V. Martynov, A. Yuakshin, “Comparison of modal and differential method for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. A (to be published).

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Figures (10)

Fig. 1
Fig. 1

Schematic representation of a grating etched into a multilayer.

Fig. 2
Fig. 2

Notation for the Rayleigh coefficients in a stack of gratings.

Fig.3
Fig.3

Decomposition of the multilayer grating in terms of elementary ones; in this figure e1+e1′= e3.

Fig. 4
Fig. 4

Representation and notation for an arbitrary plane multilayer covered by a grating.

Fig. 5
Fig. 5

Representation of the first elementary grating derived from Figs. 1 and 3 (for this case q = 4).

Fig. 6
Fig. 6

Blazing of a Bragg–Fresnel triangular grating.

Fig. 7
Fig. 7

0- and −1-order efficiencies of a triangular profile multilayer grating as a function of grazing incidence at 1.33-nm wavelength. The corresponding flat multilayer reflectivity is shown as a reference.

Fig. 8
Fig. 8

Absolute efficiencies in the 0 and ± 1 orders of a lamellar multilayer grating as a function of incidence at 0.154-nm wavelength.

Fig. 9
Fig. 9

Same as Fig. 8 but for a sinusoidal profile.

Fig. 10
Fig. 10

Same as Fig. 9 but for a triangular profile.

Tables (1)

Tables Icon

Table 1 Absolute Efficiencies of a 10.92-μm, 0.52° Blaze-Angle Bragg–Fresnel Grating as Functions of Grazing Incidence Angle

Equations (58)

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ν l 1 , ν h 1 , a d 1 , λ d 1 .
curl E = j ω μ 0 H , curl H = j ω 0 r ( x , y ) E ,
Δ E ( x , y ) + α ( x , y ) E ( x , y ) = 0 ,
E ( x + d , y ) = E ( x , y ) exp ( j γ d ) ,
γ = 2 π λ ν 1 sin θ .
E ( x , y ) = n = + E n ( y ) exp ( j γ n x ) ,
E ( x , y ) = n = + { A n ( i ) exp [ j χ n ( i ) y ] + B n ( i ) × exp [ j χ n ( i ) y ] } exp ( j γ n x ) ,
V q = [ A n q B n q ] } 2 N + 1 } 2 N + 1 .
V q = T q V q + 1 .
[ A n ( 1 ) B n ( 1 ) ] = T 1 T 2 T 3 T 4 T M [ A n ( M ) B n ( M ) ] ,
T = T 1 T 2 T 3 T 4 T M .
[ A n ( 1 ) B n ( 1 ) ] = T [ A n ( M ) 0 ] } 2 N + 1 } 2 N + 1 .
A n ( 1 ) = n = 1 2 N + 1 T n m A m ( M ) = n = 1 2 N + 1 M A n m A m ( M ) .
A n ( M ) = m T n m A m ( 1 ) .
A n ( M ) = T n , 0 .
B n ( 1 ) = m = 1 2 N + 1 T n + 2 N + 1 , m A m ( M ) = n = 1 2 N + 1 M B n , m A m ( M ) .
B n ( 1 ) = n = 1 2 N + 1 M B n , m p = 1 2 N + 1 T m , p A p ( 1 ) .
B n ( 1 ) = m = 1 2 N + 1 R n m A m ( 1 ) .
B n ( 1 ) = R n , 0 = m = 1 2 N + 1 M B n , m T m , 0 .
2 N + 1 { 2 N + 1 { [ C n ( 1 ) D n ( 1 ) ] = T M [ C n ( P + 1 ) D n ( P + 1 ) ] .
E n , k ( y ) = C n ( k ) exp [ i χ n ( k + 1 ) y ] , E n , k + ( y ) = D n ( k ) exp [ i χ n ( k + 1 ) y ] , k [ 1 , P + 1 ] ,
χ n ( k ) = ( k 0 2 n k 2 γ n 2 ) 1 / 2 , Re [ χ n ( k ) ] + Im [ χ n ( k ) ] > 0 ,
E ( x , y ) = n = N + N [ E n , k ( y ) + E n , k + ( y ) ] exp ( i γ n x ) .
E n , k ( y 0 + δ ) = E n , k exp [ i χ n ( k + 1 ) δ ] .
E n , k ( b k ) + E n , k + ( b k ) = E n , k + 1 ( b k ) + E n , k + 1 + ( b k ) ,
χ n ( k + 1 ) E n , k ( b k ) + χ n ( k + 1 ) E n , k + ( b k ) = χ n ( k + 2 ) E n , k + 1 ( b k ) + χ n ( k + 2 ) E n , k + 1 + ( b k ) .
E n , k = [ E n , k E n , k + ] ,
E n , k ( y 0 + δ ) = C n , k , δ E n , k ( y 0 ) ,
C n , k , δ = [ exp [ i χ n ( k + 1 ) δ ] 0 0 exp [ i χ n ( k + 1 ) δ ] ] .
E n , k ( b k ) = T n , k E n , k + 1 ( b k ) ,
T n , k = [ χ n ( k + 1 ) + χ n ( k + 2 ) 2 χ n ( k + 1 ) χ n ( k + 1 ) χ n ( k + 2 ) 2 χ n ( k + 1 ) χ n ( k + 1 ) χ n ( k + 2 ) 2 χ n ( k + 1 ) χ n ( k + 1 ) + χ n ( k + 2 ) 2 χ n ( k + 1 ) ] .
N n , k = C n , k , b k 1 b k 2 T n , k ,
E n , k 1 ( b k 2 ) = N n , k E n , k ( b k 1 ) .
E n , 1 ( 0 ) = N n , 2 N n , 3 N n , k N n , P N n , P + 1 E n , P + 1 ( b N ) .
E n , 1 ( 0 ) = n E n , P + 1 ( b N ) .
[ C n ( 1 ) D n ( 1 ) ] = [ 1 , 1 , n 1 , 2 , n 2 , 1 , n 2 , 2 , n ] [ C n ( P + 1 ) exp [ + i χ n ( P + 2 ) b P ] D n ( P + 1 ) exp [ i χ n ( P + 2 ) b P ] ] ,
C n ( 1 ) = 1 , 1 , n exp [ + i χ n ( P + 2 ) b P ] C n ( P + 1 ) + 1 , 2 , n exp [ i χ n ( P + 2 ) b P ] D n ( P + 1 ) ,
D n ( 1 ) = 2 , 1 , n exp [ + i χ n ( P + 2 ) b P ] C n ( P + 1 ) + 2 , 2 , n exp [ i χ n ( P + 2 ) b P ] D n ( P + 1 ) .
d 2 E n ( y ) d y 2 = m = N + N υ n m ( y ) E m ( y ) ,
E n p ( 0 ) = δ n p if p [ 1 , 2 N + 1 ] , E n p ( 0 ) = δ n , p ( 2 N + 1 ) if p [ 2 N + 2 , 4 N + 2 ] .
d E n p ( y ) d y y = 0 = j χ n ( 1 ) δ n p if p [ 1 , 2 N + 1 ] , d E n p ( y ) d y y = 0 = j χ n ( 1 ) δ n , p ( 2 N + 1 ) if p [ 2 N + 2 , 4 N + 2 ] .
f 2 = f 1 + h f 1 + 1.5 h 2 12 υ 1 f 1 + 25 6 h 2 12 υ 2 g 2 + 1 3 h 2 12 υ 3 g 3 ,
g 2 = f 1 + 0.4 h f 1 + 24 25 h 2 12 υ 1 f 1 , g 3 = f 1 + h f 1 3 h 2 12 υ 1 f 1 + 9 h 2 12 υ 2 g 2 .
ξ ( y ) = f ( y ) h 2 12 f ( y ) = [ 1 h 2 12 υ ( y ) ] f ( y )
f ( y ) = [ 1 + h 2 12 υ ( y ) + h 4 144 υ 2 ( y ) ] ξ ( y ) + O ( h 6 ) ;
f ( y ) = [ 10 ξ ( y 7 h ) + 28 ξ ( y 6 h ) 485 ξ ( y 5 h ) + 1778 ξ ( y 4 h ) 3325 ξ ( y 3 h ) + 3740 ξ ( y 2 h ) 3150 ξ ( y h ) 360 ξ ( y ) ] / 720 h + 147 f ( y ) / 60 h + O ( h 6 ) .
A n , p q 1 = 1 2 [ E n , p ( e 1 ) 1 i χ n ( 1 ) d E n , p d y y = e 1 ] exp [ i χ n ( 1 ) e 1 ] ,
B n , p q 1 = 1 2 [ E n , p ( e 1 ) 1 i χ n ( 1 ) d E n , p d y y = e 1 ] exp [ i χ n ( 1 ) e 1 ] .
E ( x , y 1 ) = n = + { A n ( q ) exp [ j χ n ( 1 ) c ] exp [ j χ n ( 1 ) y ] + B n ( q ) exp [ j χ n ( 1 ) c ] exp [ j χ n ( 1 ) y ] } .
V q = [ A n ( q ) B n ( q ) ] = [ A n ( q ) exp [ j χ n ( 1 ) c ] B n ( q ) exp [ j χ n ( 1 ) c ] ] .
V q = D c V q ,
V q = T q V ( q + 1 ) ,
D c V q = T q D c V q + 1 V q = D c 1 T q D c V q + 1 .
T q = D c 1 T q D c .
λ = 2 D n sin θ ,
λ = ( 2 d sin φ sin α ) / m = [ 2 d sin φ sin ( θ + φ ) ] / m .
( D sin θ ) / n = [ d sin φ sin ( θ + φ ) ] / m .
D / n ( d sin φ ) / m .

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