Abstract

A method for simulating latent image formation in a photoresist illuminated by partially coherent light is described. The latent image is calculated by repeatedly solving a boundary-value problem for the Helmholtz equation on a two-dimensional domain, using the finite-element method. The dependence of the developed image on defocus is investigated for the case of high numerical aperture. The obtained results are compared with those predicted by the vertical propagation model and the first-order model.

© 1989 Optical Society of America

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  1. M. Kameyama, K. Ushida, “The way to one-half micrometer lithography,” Opt. Eng. 26, 304–310 (1987); “Excimer laser stepper for submicron lithography,” in Symposium on Lasers in Lithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 147–154 (1987).
    [CrossRef]
  2. D. A. Bernard, “Simulation of focus effects in photolithography,”IEEE Trans. Semiconductor Manuf. SM-1, 85–97 (1988).
    [CrossRef]
  3. F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
    [CrossRef]
  4. H. P. Urbach, “Analysis of a model for imaging in photolithography,” J. Math. Anal. Appl. (to be published).
  5. M. S. C. Yeung, “Modeling high numerical aperture optical lithography,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 149–167 (1988).
  6. T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
    [CrossRef]
  7. D. A. Bernard, “Optical lithography simulation: introduction to SPESA,” Microelectron. Eng. 3, 379–386 (1985); in Microcircuit Engineering 1985, K. D. van der Mast, J. Radelaar, eds. (North-Holland, Amsterdam, 1985).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  10. M. Yeung, “Modeling aerial images in two and three dimensions,” in Proceedings of Kodak Microelectronics Seminar: Interface ’85, Kodak Publ. G-154 (Eastman Kodak, Rochester, N.Y., 1986), pp. 115–126.
  11. D. M. Eidus, “The principle of limiting absorption,” Mat. Sb. 57, 13–44 (1961); AMS Transl. 47, 157–191 (1965).
  12. J. L. Lions, E. Magenes, Nonhomogeneous Boundary Value Problems and Applications (Springer-Verlag, New York, 1972).
  13. H. P. Urbach, “On the convergence of the finite element method for some electromagnetic problems,” submitted to SIAM J. Numer. Anal.
  14. F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
    [CrossRef]
  15. D. A. Bernard, “Simulation of post exposure bake effects on photolithographic performance of a resist film,” Philips J. Res. 42, 566–582 (1987).
  16. O. C. Zienkiewicz, The Finite Element Method (McGraw-Hill, London, 1977).
  17. P. Hammond, Energy Methods in Electromagnetism (Clarendon, Oxford, 1986).
  18. G. H. Golub, C. F. Van Loan, Matrix Computations (North Oxford Academic, London, 1986).

1988 (1)

D. A. Bernard, “Simulation of focus effects in photolithography,”IEEE Trans. Semiconductor Manuf. SM-1, 85–97 (1988).
[CrossRef]

1987 (3)

T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
[CrossRef]

M. Kameyama, K. Ushida, “The way to one-half micrometer lithography,” Opt. Eng. 26, 304–310 (1987); “Excimer laser stepper for submicron lithography,” in Symposium on Lasers in Lithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 147–154 (1987).
[CrossRef]

D. A. Bernard, “Simulation of post exposure bake effects on photolithographic performance of a resist film,” Philips J. Res. 42, 566–582 (1987).

1985 (1)

D. A. Bernard, “Optical lithography simulation: introduction to SPESA,” Microelectron. Eng. 3, 379–386 (1985); in Microcircuit Engineering 1985, K. D. van der Mast, J. Radelaar, eds. (North-Holland, Amsterdam, 1985).
[CrossRef]

1975 (2)

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
[CrossRef]

F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
[CrossRef]

1961 (1)

D. M. Eidus, “The principle of limiting absorption,” Mat. Sb. 57, 13–44 (1961); AMS Transl. 47, 157–191 (1965).

Bernard, D. A.

D. A. Bernard, “Simulation of focus effects in photolithography,”IEEE Trans. Semiconductor Manuf. SM-1, 85–97 (1988).
[CrossRef]

D. A. Bernard, “Simulation of post exposure bake effects on photolithographic performance of a resist film,” Philips J. Res. 42, 566–582 (1987).

D. A. Bernard, “Optical lithography simulation: introduction to SPESA,” Microelectron. Eng. 3, 379–386 (1985); in Microcircuit Engineering 1985, K. D. van der Mast, J. Radelaar, eds. (North-Holland, Amsterdam, 1985).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Dill, F. H.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
[CrossRef]

F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
[CrossRef]

Eidus, D. M.

D. M. Eidus, “The principle of limiting absorption,” Mat. Sb. 57, 13–44 (1961); AMS Transl. 47, 157–191 (1965).

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (North Oxford Academic, London, 1986).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hammond, P.

P. Hammond, Energy Methods in Electromagnetism (Clarendon, Oxford, 1986).

Hasegawa, N.

T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
[CrossRef]

Hauge, P. S.

F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
[CrossRef]

Hornberger, W. P.

F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
[CrossRef]

Kameyama, M.

M. Kameyama, K. Ushida, “The way to one-half micrometer lithography,” Opt. Eng. 26, 304–310 (1987); “Excimer laser stepper for submicron lithography,” in Symposium on Lasers in Lithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 147–154 (1987).
[CrossRef]

Lions, J. L.

J. L. Lions, E. Magenes, Nonhomogeneous Boundary Value Problems and Applications (Springer-Verlag, New York, 1972).

Magenes, E.

J. L. Lions, E. Magenes, Nonhomogeneous Boundary Value Problems and Applications (Springer-Verlag, New York, 1972).

Matsuzawa, T.

T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
[CrossRef]

Moniwa, A.

T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
[CrossRef]

Neureuther, A. R.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
[CrossRef]

Shaw, J. M.

F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
[CrossRef]

Sunami, H.

T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
[CrossRef]

Tuttle, J. A.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
[CrossRef]

Urbach, H. P.

H. P. Urbach, “Analysis of a model for imaging in photolithography,” J. Math. Anal. Appl. (to be published).

H. P. Urbach, “On the convergence of the finite element method for some electromagnetic problems,” submitted to SIAM J. Numer. Anal.

Ushida, K.

M. Kameyama, K. Ushida, “The way to one-half micrometer lithography,” Opt. Eng. 26, 304–310 (1987); “Excimer laser stepper for submicron lithography,” in Symposium on Lasers in Lithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 147–154 (1987).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (North Oxford Academic, London, 1986).

Walker, E. J.

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Yeung, M.

M. Yeung, “Modeling aerial images in two and three dimensions,” in Proceedings of Kodak Microelectronics Seminar: Interface ’85, Kodak Publ. G-154 (Eastman Kodak, Rochester, N.Y., 1986), pp. 115–126.

Yeung, M. S. C.

M. S. C. Yeung, “Modeling high numerical aperture optical lithography,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 149–167 (1988).

Zienkiewicz, O. C.

O. C. Zienkiewicz, The Finite Element Method (McGraw-Hill, London, 1977).

IEEE Trans. Comput. Aided Design (1)

T. Matsuzawa, A. Moniwa, N. Hasegawa, H. Sunami, “Two-dimensional simulation of photolithography on a reflective stepped substrate,”IEEE Trans. Comput. Aided Design CAD-6, 446–451 (1987).
[CrossRef]

IEEE Trans. Electron Devices (2)

F. H. Dill, A. R. Neureuther, J. A. Tuttle, E. J. Walker, “Modeling projection printing of positive photoresist,”IEEE Trans. Electron Devices ED-22, 456–463 (1975).
[CrossRef]

F. H. Dill, W. P. Hornberger, P. S. Hauge, J. M. Shaw, “Characterization of positive photoresist,”IEEE Trans. Electron Devices ED-22, 445–452 (1975).
[CrossRef]

IEEE Trans. Semiconductor Manuf. (1)

D. A. Bernard, “Simulation of focus effects in photolithography,”IEEE Trans. Semiconductor Manuf. SM-1, 85–97 (1988).
[CrossRef]

Mat. Sb. (1)

D. M. Eidus, “The principle of limiting absorption,” Mat. Sb. 57, 13–44 (1961); AMS Transl. 47, 157–191 (1965).

Microelectron. Eng. (1)

D. A. Bernard, “Optical lithography simulation: introduction to SPESA,” Microelectron. Eng. 3, 379–386 (1985); in Microcircuit Engineering 1985, K. D. van der Mast, J. Radelaar, eds. (North-Holland, Amsterdam, 1985).
[CrossRef]

Opt. Eng. (1)

M. Kameyama, K. Ushida, “The way to one-half micrometer lithography,” Opt. Eng. 26, 304–310 (1987); “Excimer laser stepper for submicron lithography,” in Symposium on Lasers in Lithography, D. J. Ehrlich, J. Y. Tsao, J. S. Batchelder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 147–154 (1987).
[CrossRef]

Philips J. Res. (1)

D. A. Bernard, “Simulation of post exposure bake effects on photolithographic performance of a resist film,” Philips J. Res. 42, 566–582 (1987).

Other (10)

O. C. Zienkiewicz, The Finite Element Method (McGraw-Hill, London, 1977).

P. Hammond, Energy Methods in Electromagnetism (Clarendon, Oxford, 1986).

G. H. Golub, C. F. Van Loan, Matrix Computations (North Oxford Academic, London, 1986).

J. L. Lions, E. Magenes, Nonhomogeneous Boundary Value Problems and Applications (Springer-Verlag, New York, 1972).

H. P. Urbach, “On the convergence of the finite element method for some electromagnetic problems,” submitted to SIAM J. Numer. Anal.

H. P. Urbach, “Analysis of a model for imaging in photolithography,” J. Math. Anal. Appl. (to be published).

M. S. C. Yeung, “Modeling high numerical aperture optical lithography,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 149–167 (1988).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

M. Yeung, “Modeling aerial images in two and three dimensions,” in Proceedings of Kodak Microelectronics Seminar: Interface ’85, Kodak Publ. G-154 (Eastman Kodak, Rochester, N.Y., 1986), pp. 115–126.

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Figures (13)

Fig. 1
Fig. 1

Standard optical system for projection lithography with Koehler illumination.

Fig. 2
Fig. 2

Menu for source discretization in the σs direction as a function of the normalized period b ˜ = b/(λ/NA)(see the text).

Fig. 3
Fig. 3

One period of configuration of a bumpy resist on a substrate consisting of two layers. The region Ω′ is hatched.

Fig. 4
Fig. 4

(a) Comparison of developed resist profiles obtained from the bulk latent image predicted by the Helmholtz (solid curve) and VP (dotted curve) models. The focus offset zF = zF* − 1 μm corresponds to an ambient focus above the resist surface. The substrate has cleared at x = 0, where the incident aerial image has its maximum. (b) As in (a), cross sections of developed resist profiles exposed on a fully reflective substrate (with a postexposure bake diffusion length of 0.5λ/4n′). The focus offset zF = zF* + 1 μm corresponds to an ambient focus below the base of the resist.

Fig. 5
Fig. 5

Effect of source discretization, illustrated by resist profiles developed from the latent image: dashed–dotted curve, Γ = 0 (3 point sources); solid curve, Γ = 2 (5 sources) or Γ = 4 (10 sources) [see Eq. (25)].

Fig. 6
Fig. 6

Focus–exposure plot for a resist layer on a matched substrate, illuminated by partially coherent light, σ = 0.5, at a numerical aperture of 0.6. The nominal linewidth is 0.6 pm in a period of 1.2 μm. The predicted base linewidth of the resist profile is shown for the FO (dashed curves), VPeff (dotted curves), and Helmholtz (solid curves) models. The three exposure doses are 55.2 (squares), 70.6 (circles), and 107 (triangles) mJ/cm2.

Fig. 7
Fig. 7

Focus–exposure plot for a resist layer on a totally reflective substrate, illuminated by partially coherent light, with σ = 0.5. Predicted base linewidth of the resist profile is shown for the FO (dashed curves), VPeff (dotted curves), and Helmholtz (solid curves) models. The three exposure doses are 43.4 (squares), 51.5 (circles), and 74.3 (triangles) mJ/cm2.

Fig. 8
Fig. 8

Developed resist profiles at 60 sec on a matched substrate for the Helmholtz (solid curves), FO (dashed curves), and VP (dotted curves) models, shown for a sequence of focus offsets zF at a constant dose of 70.6 mJ/cm2. Lithographic parameters are identical to those described for Fig. 6.

Fig. 9
Fig. 9

Analog of Fig. 8 for a totally reflective substrate. The constant dose is 51.5 mJ/cm2, with lithographic parameters identical to those used for Fig. 7. The most accurate model is the Helmholtz model (solid curves), particularly at zFzF*.

Fig. 10
Fig. 10

Focus–exposure plot on a matched substrate for a fully coherent source, σ = 0. Lithographic conditions and symbols are otherwise identical to those used for Fig. 6.

Fig. 11
Fig. 11

Latent-image contours for the case of a resist atop a totally reflective flat substrate. The dose is 74.3 mJ/cm2, zF = 0.357 μm, and the other lithographic parameters have the values used for Fig. 7. The equiconcentration curves shown correspond to the values 0.15, 0.40, 0.65, and 0.90.

Fig. 12
Fig. 12

Latent-image contours for the case of a resist atop a bumpy totally reflective substrate. Lithographic parameters have the same values as in Fig. 9, and the m values of the curves are, again, 0.15, 0.40, 0.65, and 0.90.

Fig. 13
Fig. 13

Three-dimensional plot of the intensity at the end of bleaching for the case of a totally reflective bumpy substrate. The dose is 74.3 mJ/cm2; all other parameters have the values used for Fig. 7.

Tables (2)

Tables Icon

Table 1 Convergence Results for Matched Substrate

Tables Icon

Table 2 Convergence Results for Totally Reflective Substrate

Equations (68)

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U m , s ( x m , t ) A s ( t ) exp ( - i k x s x m / F c ) ,
A s ( t ) A s * ( t ) = A s 2 δ s , s ,
U m ( x m , t ) U m * ( x m , t ) sinc ( x m - x m λ / 2 NA c ) ,
U m , s ( x m ) = exp ( - i k x s x m / F c ) ,
t ( x m ) = n = - 1 ( - a / 2 , a / 2 ) ( x m + n b ) ,
U ^ o , s ( ξ o ) = n = - C ( n ) δ ( ξ o + x s λ F c + n b ) ,
C ( n ) = sin ( π n a / b ) π n .
P ( x l ) = 1 ( - D / 2 , D / 2 ) ( x l ) .
U ^ i , s ( ξ ) P ( - λ d i ξ ) U ^ o , s ( - M ξ ) ,
U ^ i , s ( ξ ) = P ( - λ d i ξ ) n = - C ( n ) δ ( M ξ - x s λ F c - n b ) .
x ¯ = λ / ( D / 2 d i ) λ / NA .
U i , s = 1 M n = - C ( n ) 1 ( - 1 , 1 ) ( σ s + n M b ˜ ) × exp [ i 2 π ( σ s + n M b ˜ ) x ˜ ] .
σ s = x s / F c D / 2 d o sin θ s N A o .
σ = X s / F c D / 2 d o NA c NA o .
1 = I i ( x ) = s A s 2 1 ( - 1 , 1 ) ( σ s ) .
U i , s ( x , z ) = n = - c s i ( n ) exp [ - i k γ s ( n ) z ] exp [ i 2 π ( σ s + n b ˜ ) x ˜ ] ,
c s i ( n ) = C ( n ) exp [ i k γ s ( n ) z i ] 1 ( - 1 , 1 ) ( σ s + n / b ˜ ) , γ s ( n ) = { 1 - [ NA ( σ s + n / b ˜ ) ] 2 } 1 / 2 .
α s = k NA σ s .
I i , s ( x , z ) = n = - m = - c s ( n ) c s ( m ) exp { - i k [ γ s ( n ) - γ s ( m ) ] z F } × exp [ i 2 π ( m - n x b ) ] ,
c s ( n ) = C ( n ) 1 ( - 1 , 1 ) ( σ s + n b ˜ ) .
z F = z - z i .
N σ s = { n Z 1 ( - 1 , 1 ) ( σ s + n / b ˜ ) 0 }
I n 1 , n 2 = { σ s ( - σ , σ ) σ s ( - 1 - n 1 b ˜ , - 1 - n 1 - 1 b ˜ ) ( 1 - n 2 + 1 b ˜ , 1 - n 2 b ˜ ) } .
k [ γ s ( n ) - γ s ( m ) ] z F π z F R [ ( m - n ) σ s b ˜ + ( m 2 - n 2 ) 1 2 b ˜ 2 ] ,
Δ σ s , max = 1 Γ z F / R .
u s ( x , z , t ) exp ( - i ω t )
k 2 n ( x , z , t ) 2 u s ( x , z , t ) + Δ u s ( x , z , t ) = 0 ,             ( x , z ) R 2 ,
u s ( x , z , t ) - u s i ( x , z ) satisfies the outgoing radiation condition ( o . r . c . ) for z , lim z - u s ( x , z , t ) = 0 or u s satisfies the o . r . c . for z - , depending on whether the lowest layer of the substrate is absorptive or not absorptive .
u s = 0             in the substrate .
u s i ( x + b , z ) = u s i ( x , z ) exp ( i α s b ) ,             ( x , z ) R 2 ,
v s i ( x , z ) u s i ( x , z ) exp ( - i α s x ) , v s ( x , z , t ) u s ( x , z , t ) exp ( - i α s x ) ,
[ k 2 n ( x , z , t ) 2 - α s 2 ] v s ( x , z , t ) + 2 i α s v s x ( x , z , t ) + Δ v s ( x , z , t ) = 0             on Ω , v s ( - b 2 , z , t ) = v s ( b 2 , z , t ) , v s x ( - b 2 , z , t ) = v s x ( b 2 , z , t ) , v s ( x , z , t ) - v s i ( x , z ) satisfies the o . r . c . for z , lim z - v s ( x , z , t ) = 0 or v s ( x , z , t ) satisfies the o . r . c . for z - , depending on whether the lower layer of the sub - strate is absorptive or nonabsorptive .
ω 2 π t - ( π / ω ) t + ( π / ω ) - · S ( x , z , t ˜ ) d t ˜ ω o n ( x , z , t ) n ( x , z , t ) u s ( x , z , t ) 2 = ω o n ( x , z , t ) n ( x , z , t ) v s ( x , z , t ) 2
ω o n ( x , z , t ) n ( x , z , t ) source v s ( x , z , t ) 2 d s .
n ( x , z , t ) = n r , n ( x , z , t ) = A m ( x , z , t ) + B
m t ( x , z , t ) = - C [ source v s ( x , z , t ) 2 d s ] m ( x , z , t ) ,             ( x , z ) Ω r
m t ( t ) = F [ m ( t ) ] ,
m ( x , z , 0 ) = m o ( x , z ) ,             ( x , z ) Ω r .
Ω [ ( k 2 n 2 - α s 2 ) v s 2 + 2 i α s v s x v ¯ s - v s 2 ] d x d z = Ω [ ( k 2 n 2 - α s 2 ) v s + 2 i α s v s x + Δ v s ] v ¯ s d x d z = 0.
Ω k 2 n n v s 2 d x d z = 0 ,
v s ( x , z , t ) = m = - + { c s i ( m ) exp [ - i k s ( m ) z ] + c s r ( m , t ) exp [ i k s ( m ) z ] } exp ( 2 π i m x b ) ,
k s ( m ) = [ k 2 - ( 2 π m b + α s ) 2 ] 1 / 2 .
v s ( x , z , t ) = m = - + v ^ s ( m , z , t ) exp ( 2 π i m x b ) .
v ^ s z ( m , z 2 , t ) - i k s ( m ) v ^ s ( m , z 2 , t ) = - 2 i k s ( m ) c s i ( m ) ,             m Z .
v ^ s z ( m , z 1 , t ) - i k s j ( m ) χ s ( m ) v ^ s ( m , z 1 , t ) = 0 ,             m Z ,
k s j ( m ) = [ k 2 n j 2 - ( 2 π m b + α s ) 2 ] 1 / 2 ,
χ s ( m ) = - 1 + O [ exp ( - a m ) ]             for m ± .
B s 1 ( w ) ( x ) = m = - + i k s j ( m ) χ s ( m ) w ^ ( m , z 1 ) exp ( 2 π i m x b ) , B s 2 ( w ) ( x ) = m = - + i k s ( m ) w ^ ( m , z 2 ) exp ( 2 π i m x b ) .
[ k 2 n ( x , z , t ) 2 - α s 2 ] v s ( x , z , t ) + 2 i α s v s x ( x , z , t ) + Δ v s ( x , z , t ) = 0             on Ω , v s ( - b 2 , z , t ) = v s ( b 2 , z , t ) , v s x ( - b 2 , z , t ) = v s x ( b 2 , z , t )             for z 1 < z < z 2 , v s z ( x , z 1 , t ) - B s 1 [ v s ( · , z 1 , t ) ] ( x ) = 0             for - b / 2 < x < b / 2 , v s z ( x , z 2 , t ) - B s 2 [ v s ( · , z 2 , t ) ] ( x ) = f s i ( x )             for - b 2 < x < b 2 ,
f s i ( x ) = m = - + - 2 i k s ( m ) c s i ( m ) exp ( 2 π i m x b ) .
ω V [ Ω ( w 2 + | w x | 2 + | w z | 2 ) d x d z ] 1 / 2 .
a s ( v , w ) Ω { - [ k 2 n ( x , z ) 2 - α s 2 ] v w ¯ - 2 i α s v x w ¯ + v · w ¯ } d x d z , b s ( v , w ) - b / 2 b / 2 B s 1 ( v ) ( x ) w ¯ ( x , z 1 ) d x - - b / 2 b / 2 B s 2 ( v ) ( x ) w ¯ ( x , z 2 ) d x .
{ v s and v s are continuous , Δ v s is piecewise continuous and bounded on [ - b 2 , b 2 ] × [ z 1 , z 2 ] , and v s satisfies the coditions of problem ( 46 ) pointwise almost everywhere , } { v s V satisfies a s ( v s , w ) + b s ( v s , w ) = - b / 2 b / 2 f s i ( x ) w ¯ ( x , z 2 ) d x w V } .
v s ( x , z , t ) = 0             for ( x , z ) Γ 1 .
Ω ( - k 2 n 2 E y w ¯ + E y · w ¯ ) = Ω E y n w ¯ .
δ J = δ J y exp ( - i ω t ) i ^ y ,
k 2 n 2 δ E y + Δ δ E y = - i ω μ o δ J y .
Ω E y δ J y ¯ + Ω k 2 ( n 2 - n ¯ 2 ) i ω μ o E y δ E y ¯ = δ Ω 1 i ω μ o ( E y δ E y ¯ n - δ E y ¯ E y n ) .
Re [ A exp ( - i ω t ) ] Re [ B exp ( - i ω t ) ] = ½ Re ( A B ¯ ) + ½ Re [ A B exp ( - 2 i ω t ) ] ,
m ( x , z , t ) = m o ( x , z ) exp [ - C 0 t source v s ( x , z , t ˜ ) 2 d s d t ˜ ] .
m ( x , z , t j + 1 ) = m ( x , z , t j ) exp [ - C s i v s i ( x , z , t j ) 2 Δ t j ] ,             j = 0 , 1 , ,
V N V .
a s ( v s N , w ) + b s ( v s N , w ) = - b / 2 b / 2 f s i ( x ) w ¯ ( x , z 2 ) d x w V N .
[ a s ( φ ν , φ μ ) + b s ( φ ν , φ μ ) ] ν = 1 , μ = 1 N , N
- b / 2 b / 2 B s 1 ( φ ν ) ( x , z 1 ) φ μ ( x , z 1 ) d x = b m = - i k s j ( m ) χ s ( m ) φ ^ ν ( m , z 1 ) φ ^ ¯ μ ( m , z 1 ) , - b / 2 b / 2 B s 2 ( φ ν ) ( x , z 2 ) φ μ ( x , z 2 ) d x = b m = - i k s ( m ) φ ^ ν ( m , z 2 ) φ ^ ¯ μ ( m , z 2 ) ,
k s j ( m ) χ s ( m ) = O ( m ) , φ ^ ν ( m , z k ) = O ( m - 2 ) .
c s i ( - 1 ) c s i ( 0 ) = c s i ( 1 ) c s i ( 0 ) = 0.637.
D e f f ( z F ) 2 = ( z F - z F * ) 2 + ( z F * ) 2 ,

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