Abstract

The waveguide model has been extended so that light scattering problems in semiconductor structures due to illumination of arbitrary polarization can be solved. A numerical algorithm has been developed so that once the characteristic matrix of the structure is set up, the reflected scattering matrix can be calculated efficiently. The validity of the model was examined both theoretically and experimentally, and good results have been obtained. With this modeling capability, we can successfully simulate accurate optical images of semiconductor structures obtained from various microscopes including polarization-dependent ones such as Nomarski microscopes.

© 1991 Optical Society of America

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  1. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grading,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [Crossref]
  2. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
    [Crossref]
  3. D. Nyyssonen, “Theory of optical edge detection and imaging of thick layers,”J. Opt. Soc. Am. 72, 1425–1436 (1982).
    [Crossref]
  4. C. P. Kirk, “Precision measurement of microscope images,” Ph.D. dissertation (University of Leeds, Leeds, UK, 1985).
  5. D. Nyyssonen, C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry: theory,” J. Opt. Soc. Am. A 5, 1270–1280 (1988).
    [Crossref]
  6. C. Yuan, “Modeling of optical alignment and metrology in VLSI manufacturing,” Ph.D. dissertation (Carnegie Mellon University, Pittsburgh, Pa., 1989).
  7. C. Yuan, A. J. Strojwas, “Modeling of optical alignment and metrology schemes used in IC manufacturing,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 203–218 (1990).
    [Crossref]
  8. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1940), pp. 412–415.
  9. D. Maystre, “Rigorous vector theory of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 15.
    [Crossref]
  10. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 465–467.
  11. H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
    [Crossref]
  12. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 101–107.
  13. J. P. H. Benschop, “Phase detection using scanning optical microscopy,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.921, 123–130 (1988).
    [Crossref]
  14. The microscopes and experiments were built and performed by Jerry Shaw. J. Shaw is with the IBM T. J. Watson Research Center.
  15. D. L. Lessor, J. S. Hartman, R. L. Gordon, “Quantitative surface topography determination by Nomarski reflection microscopy. I. Theory,”J. Opt. Soc. Am. 69, 357–366 (1979).
    [Crossref]
  16. M. J. Fairlie, J. G. Akkerman, R. S. Timsit, J. M. Zavislan, “Surface roughness evaluation by image analysis in Nomarski DIC microscopy,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 105–113 (1987).
    [Crossref]
  17. C. P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966), pp. 11–16.
  18. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 283.

1988 (1)

1982 (1)

1979 (1)

1973 (1)

1966 (1)

Akkerman, J. G.

M. J. Fairlie, J. G. Akkerman, R. S. Timsit, J. M. Zavislan, “Surface roughness evaluation by image analysis in Nomarski DIC microscopy,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 105–113 (1987).
[Crossref]

Benschop, J. P. H.

J. P. H. Benschop, “Phase detection using scanning optical microscopy,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.921, 123–130 (1988).
[Crossref]

Burckhardt, C. B.

Clemmow, C. P.

C. P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966), pp. 11–16.

Fairlie, M. J.

M. J. Fairlie, J. G. Akkerman, R. S. Timsit, J. M. Zavislan, “Surface roughness evaluation by image analysis in Nomarski DIC microscopy,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 105–113 (1987).
[Crossref]

Gordon, R. L.

Hartman, J. S.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 465–467.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 283.

Kameyama, M.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Kaspar, F. G.

Kawai, H.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Kirk, C. P.

D. Nyyssonen, C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry: theory,” J. Opt. Soc. Am. A 5, 1270–1280 (1988).
[Crossref]

C. P. Kirk, “Precision measurement of microscope images,” Ph.D. dissertation (University of Leeds, Leeds, UK, 1985).

Lessor, D. L.

Matsumoto, K.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Maystre, D.

D. Maystre, “Rigorous vector theory of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 15.
[Crossref]

Murakami, S.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Nakamura, S.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Nei, M.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Nyyssonen, D.

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 101–107.

Strojwas, A. J.

C. Yuan, A. J. Strojwas, “Modeling of optical alignment and metrology schemes used in IC manufacturing,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 203–218 (1990).
[Crossref]

Timsit, R. S.

M. J. Fairlie, J. G. Akkerman, R. S. Timsit, J. M. Zavislan, “Surface roughness evaluation by image analysis in Nomarski DIC microscopy,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 105–113 (1987).
[Crossref]

Ushida, K.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1940), pp. 412–415.

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1940), pp. 412–415.

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 101–107.

Yuan, C.

C. Yuan, “Modeling of optical alignment and metrology in VLSI manufacturing,” Ph.D. dissertation (Carnegie Mellon University, Pittsburgh, Pa., 1989).

C. Yuan, A. J. Strojwas, “Modeling of optical alignment and metrology schemes used in IC manufacturing,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 203–218 (1990).
[Crossref]

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 465–467.

Zavislan, J. M.

M. J. Fairlie, J. G. Akkerman, R. S. Timsit, J. M. Zavislan, “Surface roughness evaluation by image analysis in Nomarski DIC microscopy,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 105–113 (1987).
[Crossref]

J. Opt. Soc. Am. (4)

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

Other (13)

C. Yuan, “Modeling of optical alignment and metrology in VLSI manufacturing,” Ph.D. dissertation (Carnegie Mellon University, Pittsburgh, Pa., 1989).

C. Yuan, A. J. Strojwas, “Modeling of optical alignment and metrology schemes used in IC manufacturing,” in Optical/Laser Microlithography III, V. Pol, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1264, 203–218 (1990).
[Crossref]

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1940), pp. 412–415.

D. Maystre, “Rigorous vector theory of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, p. 15.
[Crossref]

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), pp. 465–467.

H. Kawai, M. Nei, S. Murakami, M. Kameyama, S. Nakamura, K. Ushida, K. Matsumoto, “New generation optical stepper with high n.a. g-line lens,” in Optical/Laser Microlithography II, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1088, 170–177 (1989).
[Crossref]

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984), pp. 101–107.

J. P. H. Benschop, “Phase detection using scanning optical microscopy,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.921, 123–130 (1988).
[Crossref]

The microscopes and experiments were built and performed by Jerry Shaw. J. Shaw is with the IBM T. J. Watson Research Center.

M. J. Fairlie, J. G. Akkerman, R. S. Timsit, J. M. Zavislan, “Surface roughness evaluation by image analysis in Nomarski DIC microscopy,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 105–113 (1987).
[Crossref]

C. P. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966), pp. 11–16.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 283.

C. P. Kirk, “Precision measurement of microscope images,” Ph.D. dissertation (University of Leeds, Leeds, UK, 1985).

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Figures (10)

Fig. 1
Fig. 1

Waveguide model.

Fig. 2
Fig. 2

(a) Traditional method of characterizing a junction. (b) Our approximation for a junction.

Fig. 3
Fig. 3

Results of conservation-of-energy examination.

Fig. 4
Fig. 4

Reciprocity theorem in grating theory.

Fig. 5
Fig. 5

Results of reciprocity-theorem examination.

Fig. 6
Fig. 6

Comparisons between simulation results from the waveguide model and the Abbe imaging theory.

Fig. 7
Fig. 7

Comparisons between simulation results from the waveguide and Fresnel models. WG, waveguide; A, angstrom.

Fig. 8
Fig. 8

Comparisons between the measured differential phase contrast images and simulation results.

Fig. 9
Fig. 9

Schematic of the scanning Nomarski microscope.

Fig. 10
Fig. 10

Comparisons between the measured differential Nomarski images and simulation results.

Tables (2)

Tables Icon

Table 1 Comparison between the Measured and the Simulated Linewidths from the Differential Phase Contrast Scheme

Tables Icon

Table 2 Comparison between the Measured and the Simulated Linewidths from the Differential Nomarski Scheme

Equations (61)

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2 E j - μ 0 0 j ( x ) 2 E j t 2 + [ E j · j ( x ) j ( x ) ] = 0 ,
2 E j - μ 0 0 j ( x ) 2 E j t 2 = 0.
E y 0 = exp ( i k 0 z ) + l = - L + L E l 0 exp ( i 2 π l b x ) × exp { - i k 0 [ 1 - ( l b λ 1 ) 2 ] 1 / 2 z } , H x 0 = - i k 0 exp ( i k 0 z ) + l = - L + L i k 0 [ 1 - ( l b λ 0 ) 2 ] 1 / 2 E l 0 × exp ( i 2 π l b x ) exp { - i k 0 [ 1 - ( l b λ 0 ) 2 ] 1 / 2 z } ,
E y s = l = - L + L E l s exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } , H x s = - l = - L + L i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 E l s × exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } ,
2 X j x 2 + [ k 0 2 q = L + L q j exp ( i 2 π q b x ) + ( α j ) ] X j = 0 , 2 Z j z 2 - ( α j ) 2 Z j = 0 ,
j ( x ) = q = - L + L q j exp ( i 2 π q b x ) .
X j ( x ) = l = - L + L B l , m j exp ( i 2 π l b x ) ,
E y j = m = - L + L { [ A m j exp ( α m j z ) + A m j exp ( - α m j z ) ] × l = - L + L B l , m j exp ( i 2 π l b x ) } , H x j = - m = - L + L { [ A m j exp ( α m j z ) - A m j exp ( - α m j z ) ] × α m j l = - L + L B l , m j exp ( i 2 π l b x ) } .
E l 0 = m = - L + L ( A m 1 + A m 1 ) B l , m 1 - δ l , 0 , [ 1 - ( l b λ 0 ) 2 ] 1 / 2 E l 0 = - 1 i k 0 m = - L + L ( A m 1 - A m 1 ) α m 1 B l , m 1 + δ l , 0 ,
m = - L + L { [ 1 - ( l b λ 0 ) 2 ] 1 / 2 + α m 1 i k 0 } B l , m 1 A m 1 + m = - L + L { [ 1 - ( l b λ 1 ) 2 ] 1 / 2 - α m 1 i k 0 } B l , m 1 A m 1 = 2 δ l , 0
[ C 11 0 C 12 0 0 0 ] [ A 1 A 1 ] = [ R 0 ] ,
C 11 0 ( l , m ) = { [ 1 - ( l b λ 0 ) 2 ] 1 / 2 + α m 1 i k 0 } B l , m 1 , C 12 0 ( l , m ) = { [ 1 - ( l b λ 0 ) 2 ] 1 / 2 - α m 1 i k 0 } B l , m 1 , R l = 2 δ l , 0 .
[ C 11 j C 12 j C 21 j C 22 j ] [ A j A j ] = [ E 11 j + 1 E 12 j + 1 E 21 j + 1 E 22 j + 1 ] [ A j + 1 A j + 1 ] ,
C 11 j ( l , m ) = exp ( α m j z j ) B l , m j , C 12 j ( l , m ) = exp ( - α m j z j ) B l , m j , C 21 j ( l , m ) = α m j exp ( α m j z j ) B l , m j , C 22 j ( l , m ) = - α m j exp ( - α m j z j ) B l , m j , E 11 j + 1 ( l , m ) = exp ( α m j + 1 z j ) B l , m j + 1 , E 12 j + 1 ( l , m ) = exp ( - α m j + 1 z j ) B l , m j + 1 , E 21 j + 1 ( l , m ) = α m j + 1 exp ( α m j + 1 z j ) B l , m j + 1 , E 22 j + 1 ( l , m ) = - α m j + 1 exp ( - α m j + 1 z j ) B l , m j + 1 .
[ C 11 n C 12 n 0 0 ]             [ A n A n ] = [ 0 0 ] ,
C 11 n ( l , m ) = { [ s - ( l b λ 0 ) 2 ] 1 / 2 - α m n i k 0 } exp ( α m n T ) B l , m n , C 12 n ( l , m ) = { [ s - ( l b λ 0 ) 2 ] 1 / 2 + α m n i k 0 } exp ( - α m n T ) B l , m n .
[ 0 0 ] = [ C 11 n C 12 n 0 0 ] [ E 11 n E 12 n E 21 n E 22 n ] - 1 [ C 11 n - 1 C 12 n - 1 C 21 n - 1 C 22 n - 1 ] × [ E 11 n - 1 E 12 n - 1 E 21 n - 1 E 22 n - 1 ] - 1 [ C 11 n - 2 C 12 n - 2 C 21 n - 2 C 22 n - 2 ] [ E 11 2 E 12 2 E 21 2 E 22 2 ] - 1 × [ C 11 1 C 12 1 C 21 1 C 22 1 ] [ A 1 A 1 ] = [ C 21 0 C 22 0 0 0 ] [ A 1 A 1 ] .
[ C 11 0 C 12 0 C 21 0 C 22 0 ] [ A 1 A 1 ] = [ R 0 ] .
2 H j - μ 0 0 j ( x ) 2 H j t 2 + j ( x ) j ( x ) × ( × H j ) = 0.
2 H j x 2 + 2 H j z 2 + k 0 2 q = - L L q j exp ( i 2 π q b x ) H j - q = - L L ( i 2 π q b ) q j × exp ( i 2 π q b x ) q = - L L q j exp ( i 2 π q b x ) H j x = 0 ,
j ( x ) = q = - L + L q j exp ( i 2 π q b x ) , j ( x ) = 1 j ( x ) = q = - L + L q j exp ( i 2 π q b x )
2 X j x 2 - q = - L + L C q j exp ( i 2 π q b x ) X j x + [ k 0 2 q = - L + L q j exp ( i 2 π q b x ) + ( α j ) 2 ] X j = 0 , 2 Z j z 2 - ( α j ) 2 Z j = 0 ,
q = - L + L C q j exp ( i 2 π q b x ) = q = - L + L ( i 2 π q b x ) q j × exp ( i 2 π q b x ) q = - L + L q j exp ( i 2 π q b x ) .
l = - L + L ( 2 π l b ) 2 B l j exp ( i 2 π l b x ) + q = - L + L C q j exp ( i 2 π q x ) l = - L + L ( i 2 π l b ) B l j × exp ( i 2 π l b x ) - [ k 0 2 q = - L + L q j exp ( i 2 π q b x ) + ( α j ) 2 ] × l = - L + L B l j exp ( i 2 π l b x ) = 0.
D p q = { ( 2 π p b ) 2 - k 0 2 0 j + C 0 j ( i 2 π p b ) p = q - k 0 2 p - q j + C p - q j ( i 2 π q b ) p q .
H y 0 = exp ( i k 0 z ) + l = - L + L H l 0 exp ( i 2 π l b x ) × exp { - i k 0 [ 1 - ( l b λ 0 ) 2 ] 1 / 2 z } , E x 0 = i k 0 exp ( i k 0 z ) - l = - L + L i k 0 [ 1 - ( l b λ 0 ) 2 ] 1 / 2 H l 0 × exp ( i 2 π l b x ) exp { - i k 0 [ 1 - ( l b λ 0 ) 2 ] 1 / 2 z } .
H y j = m = - L + L { [ A m j exp ( α m j z ) + A m j exp ( - α m j z ) ] × l = - L + L B l , m j exp ( i 2 π l b x ) } , E x j = m = - L + L { [ A m j exp ( α m j z ) - A m j exp ( - α m j z ) ] α m j × l = - L + L B l , m j exp ( i 2 π l b x ) } ,
H y s = l = - L + L H l s exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } , E x s = 1 s l = - L + L i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 H l s × exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } .
H l 0 = m = - L + L ( A m 1 + A m 1 ) B l , m 1 - δ l , 0 [ 1 - ( l b λ 0 ) 2 ] H l 0 = - 1 i k 0 m = - L + L ( A m 1 - A m 1 ) α m 1 B l , m 1 + δ l , 0
m = - L + L { [ 1 - ( l b λ 0 ) 2 ] 1 / 2 B l , m 1 + α m 1 i k 0 B l , m 1 } A m 1 + m = - L + L { [ 1 - ( l b λ 0 ) 2 ] 1 / 2 B l , m 1 - α m 1 i k 0 B l , m 1 } A m 1 = 2 δ l , 0
[ C 11 0 C 12 0 0 0 ] [ A 1 A 1 ] = [ R 0 ] ,
C 11 0 ( l , m ) = [ 1 - ( l b λ 0 ) 2 ] 1 / 2 B l , m 1 + α m 1 i k 0 B l , m 1 , C 12 0 ( l , m ) = [ 1 - ( l b λ 0 ) 2 ] 1 / 2 B l , m 1 - α m 1 i k 0 B l , m 1 , R l = 2 δ l , 0 .
[ C 11 j C 12 j C 21 j C 22 j ] [ A j A j ] = [ E 11 j + 1 E 12 j + 1 E 21 j + 1 E 22 j + 1 ] [ A j + 1 A j + 1 ] ,
C 11 j ( l , m ) = exp ( α m j z j ) B l , m j , C 12 j ( l , m ) = exp ( - α m j z j ) B l , m j , C 21 j ( l , m ) = α m j exp ( α m j z j ) B l , m j , C 22 j ( l , m ) = - α m j exp ( - α m j z j ) B l , m j , E 11 j + 1 ( l , m ) = exp ( α m j + 1 z j ) B l , m j + 1 , E 12 j + 1 ( l , m ) = exp ( - α m j + 1 z j ) B l , m j + 1 , E 21 j + 1 ( l , m ) = α m j + 1 exp ( α m j + 1 z j ) B l , m j + 1 , E 22 j + 1 ( l , m ) = - α m j + 1 exp ( - α m j + 1 z j ) B l , m j + 1 .
[ C 11 n C 12 n 0 0 ] [ A n A n ] = [ 0 0 ] ,
C 11 n ( j , m ) = { [ s - ( l b λ 0 ) 2 ] 1 / 2 B l , m n - α m n i k 0 s B l , m n } exp ( α m n T ) , C 12 n ( l , m ) = { [ s - ( l b λ 0 ) 2 ] 1 / 2 B l , m n + α m n i k 0 s B l , m n } × exp ( - α m n T ) .
[ C 11 0 C 12 0 C 21 0 C 22 0 ] [ A 1 A 1 ] = [ R 0 ] ,
R l = 2 [ 1 - ( l 0 b λ 0 ) 2 ] 1 / 2 δ l , l 0 ,
E ( x ) = l E l exp ( i 2 π l b x ) ,
R l = 2 [ 1 - ( l b λ 0 ) 2 ] 1 / 2 E l ,
E l 0 = m = - L + L ( A m 1 + A m 1 ) B l , m 1 - E l .
E y j , H y j ~ m = - L + L { [ A m j exp ( α m j z ) + A m j exp ( - α m j z ) ] + l = - L + L B l , m i exp ( i 2 π l b x ) } .
2 H j - μ 0 0 j ( x ) 2 H j t 2 + j ( x ) j ( x ) × ( × H j ) = 0.
( x ) = 2 - 1 π tan - 1 γ x + 2 + 1 2 , ( x ) = x ^ 2 - 1 π γ 1 + γ 2 x 2 ,
γ = 20 λ n tan ( 0.4 π ) .
cos θ l 0 = l = - K + K E l 0 + l = - L + L E l s .
cos θ l 0 = l = - K + K E l 0 2 cos θ l + l = - L + L E l s 2 Re { [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 } .
cos θ l 0 = l = - K + K H l 0 2 cos θ l 0 + l = - L + L H l s 2 × Re { [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 n s 2 } .
β 1 2 = E 2 2 cos θ 2 E 1 2 cos θ 1 .
E 2 2 cos θ 2 E 1 2 cos θ 1 = E 1 2 cos θ 1 E 2 2 cos θ 2 .
E l ( x , z ) = y ^ E l exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } ,
E l ( x , z ) = y ^ E l exp ( i k 0 n s n · r ) ,
n · n = 1 ,             n · E l = 0 ,             n · H l = 0 ,
r = x ^ x + y ^ y + z ^ z ,
n = x ^ l b λ 0 n s + z ^ [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 n s .
H l ( x , z ) = { - x ^ [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 + z ^ l b λ 0 } E l × exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } .
E l = Re [ z ^ · ( E l × H l * ) ] = E l 2 Re [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 .
H l ( x , z ) = y ^ H l exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z }
H l ( x , z ) = y ^ H l exp ( i k 0 n s n · r ) ,
E l = { x ^ [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 n s 2 - z ^ l b λ 0 n s 2 } H l × exp ( i 2 π l b x ) exp { i k 0 [ s - ( l b λ 0 ) 2 ] 1 / 2 z } .
E l = Re [ z ^ · ( E l × H l * ) ] = H l 2 Re { [ n s 2 - ( l b λ 0 ) 2 ] 1 / 2 n s 2 } .

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