Abstract

I use the Rayleigh–Fano theory to calculate the optical response of a stack of N thin films with rough interfaces, defined by arbitrary periodic functions of the same period, bounded by two semi-infinite media. Each medium is assumed to be homogeneous, isotropic, local, and linear and characterized by a frequency-dependent complex dielectric constant. I derive two matrix equations that relate the reflected and the transmitted fields to the incident wave. These matrices can be calculated in a recursive way. The solution involves Fourier coefficients of functions that are dependent on the roughness profiles. This treatment is valid for both TM(p) or TE(s) polarizations. I apply this formulation to some particular cases.

© 1994 Optical Society of America

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References

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  1. U. Fano, J. Opt. Soc. Am. 31, 213 (1941).
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  2. F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
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  3. I. Pockrand, Opt. Commun. 13, 311 (1975).
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  4. I. Pockrand and H. Raether, Appl. Opt. 16, 1784 (1977).
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  5. T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).
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    [CrossRef]
  7. Z. Chen and H. J. Simon, J. Opt. Soc. Am. B 5, 1396 (1988).
    [CrossRef]
  8. M. G. Weber and D. L. Mills, Phys. Rev. B 32, 6238 (1985).
    [CrossRef]
  9. O. Mata-Méndez and P. Halevi, Phys. Rev. B 36, 1007 (1987).
    [CrossRef]
  10. M. G. Cavalcante, G. A. Farias, and A. A. Maradudin, J. Opt. Soc. Am. B 4, 1372 (1987).
    [CrossRef]
  11. P. Halevi and O. Mata-Méndez, Phys. Rev. B 39, 5694 (1989).
    [CrossRef]
  12. S. Wang and P. Halevi, Phys. Rev. B 47, 10815 (1993).
    [CrossRef]
  13. G. S. Agarwal, Phys. Rev. B 31, 3534 (1985).
    [CrossRef]
  14. C. R. Greco and M. L. Rustgi, J. Opt. Soc. Am. B 7, 877 (1990).
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  15. D. T. Nguyen and M. L. Rustgi, J. Opt. Soc. Am. B 9, 1850 (1992).
    [CrossRef]
  16. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]

1993 (1)

S. Wang and P. Halevi, Phys. Rev. B 47, 10815 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (1)

P. Halevi and O. Mata-Méndez, Phys. Rev. B 39, 5694 (1989).
[CrossRef]

1988 (1)

1987 (2)

1985 (4)

G. S. Agarwal, Phys. Rev. B 31, 3534 (1985).
[CrossRef]

M. G. Weber and D. L. Mills, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

1977 (2)

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
[CrossRef]

I. Pockrand and H. Raether, Appl. Opt. 16, 1784 (1977).
[CrossRef] [PubMed]

1975 (1)

I. Pockrand, Opt. Commun. 13, 311 (1975).
[CrossRef]

1941 (1)

Agarwal, G. S.

G. S. Agarwal, Phys. Rev. B 31, 3534 (1985).
[CrossRef]

Arakawa, E. T.

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

Cavalcante, M. G.

Celli, V.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
[CrossRef]

Chen, Z.

Fano, U.

Farias, G. A.

Goudonnet, J. P.

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).

Greco, C. R.

Halevi, P.

S. Wang and P. Halevi, Phys. Rev. B 47, 10815 (1993).
[CrossRef]

P. Halevi and O. Mata-Méndez, Phys. Rev. B 39, 5694 (1989).
[CrossRef]

O. Mata-Méndez and P. Halevi, Phys. Rev. B 36, 1007 (1987).
[CrossRef]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
[CrossRef]

Inagaky, T.

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

Maradudin, A. A.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
[CrossRef]

Mata-Méndez, O.

P. Halevi and O. Mata-Méndez, Phys. Rev. B 39, 5694 (1989).
[CrossRef]

O. Mata-Méndez and P. Halevi, Phys. Rev. B 36, 1007 (1987).
[CrossRef]

Mills, D. L.

M. G. Weber and D. L. Mills, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

Motosuga, M.

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).

Nguyen, D. T.

Pockrand, I.

Raether, H.

Rustgi, M. L.

Simon, H. J.

Toigo, F.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
[CrossRef]

Wang, S.

S. Wang and P. Halevi, Phys. Rev. B 47, 10815 (1993).
[CrossRef]

Weber, M. G.

M. G. Weber and D. L. Mills, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

I. Pockrand, Opt. Commun. 13, 311 (1975).
[CrossRef]

Phys. Rev. B (8)

M. G. Weber and D. L. Mills, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

O. Mata-Méndez and P. Halevi, Phys. Rev. B 36, 1007 (1987).
[CrossRef]

P. Halevi and O. Mata-Méndez, Phys. Rev. B 39, 5694 (1989).
[CrossRef]

S. Wang and P. Halevi, Phys. Rev. B 47, 10815 (1993).
[CrossRef]

G. S. Agarwal, Phys. Rev. B 31, 3534 (1985).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 15, 5618 (1977).
[CrossRef]

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 31, 248 (1985).

T. Inagaky, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Multilayered structure with roughness in all the interfaces as considered in this paper. The interfaces have different profiles with period a and amplitudes hj.

Fig. 2
Fig. 2

Reflection coefficient (solid curve) and the squared modulus of |r−1|2 (dashed curve) versus the angle of incidence for a two-film system with d2 = 50 nm, d3 = 50 nm, a = 400 nm, h1 = h2 = 0, and h3/a = 0.02. The profile function is sinusoidal.

Fig. 3
Fig. 3

Same as in Fig. 2 for h1/a = h2/a = h3/a = 0.02. A stronger coupling compared with that in Fig. 2 is observed.

Fig. 4
Fig. 4

Reflection coefficient (solid curve) and the squared modulus of |r+1|2 (short-dashed curve) and |r−1|2 (long-dashed curve) as a function of angle of incidence for a two-metallic-film system with d2 = 30 nm and d3 = 50 nm, a = 600 nm and h1/a = h2/a = h3/a = 0.03. The profile function is triangular.

Equations (60)

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α 0 j + β 10 k = ω c [ 1 ( sin θ ) j + 1 ( cos θ ) k ] ,
U i ( y , z ) = u exp [ i ( α 0 y + β 10 z ) ] ,             z < F 1 ( y ) .
α n = α 0 + ( 2 π / a ) n ,             n = - 2 , - 1 , 0 , 1 , 2 , .
β j n = [ ( ω / c ) j - α n 2 ] 1 / 2 ,             j = 2 , 3 , , N + 2.
U r ( y , z ) = n = - r n exp [ i ( α n y - β 1 n z ) ] ,
U t ( y , z ) = n = - t n exp { i [ α n y + β ( N + 2 ) n z ] } ,
U j ( y , z ) = n = - { α j n exp [ i ( α n y + β j n z ) ] + b j n exp [ i ( α n y - β j n z ) } .
U j + 1 z = F j ( y ) = U j z = F j ( y ) ,
1 w j + 1 n U j + 1 z = F j ( y ) = 1 w j n U j z = F j ( y ) ,
U n n ^ · U .
n exp ( i { α n y ± β j n z [ F j ( y ) ] } ) = i [ α n v j ( y ) ± β j n ] [ 1 + v j ( y ) 2 ] 1 / 2 exp ( i { α n y ± β j n z [ F j ( y ) ] } ) , v j ( y ) = F j ( y ) y .
n = - ( a 2 n exp { i [ α n y + β 2 n F 1 ( y ) ] } + b 2 n exp { i [ α n y - β 2 n F 1 ( y ) ] } ) = n = - ( δ n , 0 u exp { i [ α n y + β 1 n F 1 ( y ) ] } + r n exp { i [ α n y - β 1 n F 1 ( y ) ] } ) ,
n = - ( i [ α n v 1 ( y ) + β 2 n ] w 2 [ 1 + v 1 2 ( y ) ] 1 / 2 a 2 n exp { i [ α n y + β 2 n F 1 ( y ) ] } + i [ α n v 1 ( y ) - β 2 n ] w 2 [ 1 + v 1 2 ( y ) ] 1 / 2 b 2 n exp { i [ α n y - β 2 n F 1 ( y ) ] } ) = n = - ( i [ α n v 1 ( y ) + β 1 n ] w 1 [ 1 + v 1 2 ( y ) ] 1 / 2 δ n , 0 u exp { i [ α n y + β 1 n F 1 ( y ) ] } + i [ α n v 1 ( y ) + β 1 n ] w 1 [ 1 + v 1 2 ( y ) ] 1 / 2 r n exp { i [ α n y - β 1 n F 1 ( y ) ] } ) ;
n = - ( a ( j + 1 ) n exp { i [ α n y + β ( j + 1 ) n F j ( y ) ] } + b ( j + 1 ) n exp { i [ α n y - β ( j + 1 ) n F j ( y ) ] } ) = n = - ( a j n exp { i [ α n y + β j n F j ( y ) ] } + b j n exp { i [ α n y - β j n F j ( y ) ] } ) ,
n = - ( i [ α n v j ( y ) + β ( j + 1 ) n ] w j + 1 [ 1 + v j 2 ( y ) ] 1 / 2 a ( j + 1 ) n × exp { i [ α n y + β ( j + 1 ) n F j ( y ) ] } + i [ α n v j ( y ) - β ( j + 1 ) n ] w j + 1 [ 1 + v j 2 ( y ) ] 1 / 2 b ( j + 1 ) n × exp { i [ α n y - β ( j + 1 ) n F j ( y ) ] } ) = n = - ( i [ α n v j ( y ) + β j n ] w j [ 1 + v j 2 ( y ) ] 1 / 2 a j n exp { i [ α n y + β j n F j ( y ) ] } + i [ α n v j ( y ) - β j n ] w j [ 1 + v j 2 ( y ) ] 1 / 2 b j n exp { i [ α n y - β j n F j ( y ) ] } ) ,
n = - t n exp { i [ α n y + β ( N + 2 ) n F N + 1 ( y ) ] } = n = - ( a ( N + 1 ) n exp { i [ α n y + β ( N + 1 ) n F N + 1 ( y ) ] } + b ( N + 1 ) n exp { i [ α n y - β ( N + 1 ) n F N + 1 ( y ) ] } ) ,
n = - 1 [ α n v N + 1 ( y ) + β ( N + 2 ) n ] w N + 2 [ 1 + v N + 1 2 ( y ) ] 1 / 2 × t n exp { i [ α n y + β ( N + 2 ) n F N + 1 ( y ) ] } = n = - ( i [ α n v N + 1 ( y ) + β ( N + 1 ) n ] w N + 1 [ 1 + v N + 1 2 ( y ) ] 1 / 2 × a ( N + 1 ) n exp { [ α n y + β ( N + 1 ) n F N + 1 ( y ) ] } + i [ α n v n + 1 ( y ) - β ( N + 1 ) n ] w N + 1 [ 1 + v N + 1 2 ( y ) ] 1 / 2 × b ( N + 1 ) exp { i [ α n y - β ( N + 1 ) n F N + 1 ( y ) ] } ) .
2 β 2 m a w 2 a 2 m = n = - { C m n ( 1 ) I ( 1 ) [ + β ( 1 ) n ( 2 ) m - ] δ 0 , n u - D m n ( 1 ) I ( 1 ) [ - β ( 1 ) n ( 2 ) m + ] r n } ,
- 2 β 2 m a w 2 b 2 m = n = - { D m n ( 1 ) I ( 1 ) [ + β ( 1 ) n ( 2 ) m + ] δ 0 , n u - C m n ( 1 ) I ( 1 ) [ - β ( 1 ) n ( 2 ) m - ] r n } ,
2 β ( j + 1 ) m a w j + 1 a ( j + 1 ) m = n = - { C m n ( j ) I ( j ) [ + β ( j ) n ( j + 1 ) m - ] a ( j ) n - D m n ( j ) I ( j ) [ - β ( j ) n ( j + 1 ) m + ] b ( j ) n } ,
- 2 β ( j + 1 ) m a w j + 1 b ( j + 1 ) m = n = - { D m n ( j ) I ( j ) [ + β ( j ) n ( j + 1 ) m + ] a ( j ) n - C m n ( j ) I ( j ) [ - β ( j ) n ( j + 1 ) m - ] b ( j ) n } ,
0 = n = - { D m n ( N + 1 ) I ( N + 1 ) [ + β ( N + 1 ) n ( N + 2 ) m + ] a ( N + 1 ) n - C m n ( N + 1 ) I ( N + 1 ) [ - β ( N + 1 ) n ( N + 2 ) m - ] b ( N + 1 ) n } ,
D m n ( j ) = [ ( α n / w j ) + ( α m / w j + 1 ) ] ( α n - α m ) β ( j ) n + β ( j + 1 ) m + [ β ( j ) n w j - β ( j + 1 ) m w j + 1 ] ,
C m n ( j ) = [ ( α n / w j ) + ( α m / w j + 1 ) ] ( α n - α m ) β ( j ) n - β ( j + 1 ) m + [ β ( j ) n w j + β ( j + 1 ) m w j + 1 ] ,
I ( j ) [ ± β ( j ) n ( j + 1 ) m ± ] = 1 a 0 a exp [ i ( α n - α m ) y ] × exp [ ± i β ( j ) n ( j + 1 ) m ± F j ( y ) ] d y ,
± β ( j ) n ( j + 1 ) m ± = ± [ β ( j ) n ± β ( j + 1 ) m ] .
2 β 1 m a w 1 δ m , 0 u = n = - { A m n ( 1 ) I ( 1 ) [ + β ( 2 ) n ( 1 ) m - ] a 2 m - B m n ( 1 ) I ( 1 ) [ - β ( 2 ) n ( 1 ) m + ] b 2 m } ,
2 β ( j ) m a w j a ( j ) m = n = - { A m n ( j ) I ( j ) [ + β ( j + 1 ) n ( j ) m - ] a ( j + 1 ) n - B m n ( j ) I ( j ) [ - β ( j + 1 ) n ( j ) m + ] b ( j + 1 ) n } ,
- 2 β ( j ) m a w j b ( j ) m = n = - { B m n ( j ) I ( j ) [ + β ( j + 1 ) n ( j ) m + ] a ( j + 1 ) n - A m n ( j ) I ( j ) [ - β ( j + 1 ) n ( j ) m - ] b ( j + 1 ) n } ,
2 β ( N + 1 ) m a w N + 1 a ( N + 1 ) m = n = - A m n ( N + 1 ) I ( N + 1 ) [ + β ( N + 2 ) n ( N + 1 ) m - ] t n ,
- 2 β ( N + 1 ) m a w N + 1 b ( N + 1 ) m = n = - B m n ( N + 1 ) I ( N + 1 ) [ + β ( N + 2 ) n ( N + 1 ) m + ] t n
B m n ( j ) = [ ( α n / w j + 1 ) + ( α m / w j ) ] ( α n - α m ) β ( j + 1 ) n + β ( j ) m + [ β ( j + 1 ) n w j + 1 - β ( j ) m w j ] ,
A m n ( j ) = [ ( α n / w j + 1 ) + ( α m / w j ) ] ( α n - α m ) β ( j + 1 ) n - β ( j ) m + [ β ( j + 1 ) n w j + 1 + β ( j ) m w j ] ,
I ( j ) [ ± β ( j + 1 ) n ( j ) m ± ] = 1 a 0 a exp [ i ( α n - α m ) y ] × exp [ ± i β ( j + 1 ) n ( j ) m ± F j ( y ) ] d y ,
± β ( j + 1 ) n ( j ) m ± = ± [ β ( j + 1 ) n ± β ( j ) m ] .
n = - G m n ( N + 1 ) r n = H m o ( N + 1 ) u ,
n = - J m n ( N + 1 ) t n = K m o ( 1 ) u ,
G m n ( K + 1 ) = p w K + 1 2 β ( K + 1 ) p { D m p ( K + 1 ) I ( K ) [ + β ( K + 1 ) p ( K + 2 ) m + ] G p n ( K ) + C m p ( K + 1 ) I ( K ) [ - β ( K + 1 ) p ( K + 2 ) m - ] G p n ( K ) } ,
H m n ( K + 1 ) = p w K + 1 2 β ( K + 1 ) p { D m p ( K + 1 ) I ( K ) [ + β ( K + 1 ) p ( K + 2 ) m + ] H p n ( K ) + C m p ( K + 1 ) I ( K ) [ - β ( K + 1 ) p ( K + 2 ) m - ] H p n ( K ) } ,
G m n ( K + 1 ) = p w K + 1 2 β ( K + 1 ) p { D m p ( K + 1 ) I ( K ) [ - β ( K + 1 ) p ( K + 2 ) m + ] G p n ( K ) + C m p ( K + 1 ) I ( K ) [ + β ( K + 1 ) p ( K + 2 ) m - ] G p n ( K ) } ,
H m n ( K + 1 ) = p w K + 1 2 β ( K + 1 ) p { D m p ( K + 1 ) I ( K ) [ - β ( K + 1 ) p ( K + 2 ) m + ] H p n ( K ) + C m p ( K + 1 ) I ( K ) [ + β ( K + 1 ) p ( K + 2 ) m - ] H p n ( K ) } ,
G m n ( 1 ) = C m n ( 1 ) I ( 1 ) [ - β ( 1 ) n ( 2 ) m - ] ,
H m n ( 1 ) = D m n ( 1 ) I ( 1 ) [ + β ( 1 ) n ( 2 ) m + ] ,
G m n ( 1 ) = D m n ( 1 ) I ( 1 ) [ - β ( 1 ) n ( 2 ) m + ] ,
H m n ( 1 ) = C m n ( 1 ) I ( 1 ) [ + β ( 1 ) n ( 2 ) m - ] .
J m n ( K + 1 ) = p w K + 1 2 β ( K + 1 ) p { A m p ( K + 1 ) I ( K ) [ + β ( K + 2 ) p ( K + 1 ) m - ] J p n ( K ) + B m p ( K + 1 ) I ( K ) [ - β ( K + 2 ) p ( K + 1 ) m + ] J p n ( K ) } ,
J m n ( K + 1 ) = p w K + 1 2 β ( K + 1 ) p { A m p ( K + 1 ) I ( K ) [ - β ( K + 2 ) p ( K + 1 ) m - ] J p n ( K ) + B m p ( K + 1 ) I ( K ) [ + β ( K + 2 ) p ( K + 1 ) m + ] J p n ( K ) } ,
J m n ( 1 ) = A m n ( 1 ) I ( 1 ) [ + β ( N + 2 ) n ( N + 1 ) m - ] ,
J m n ( 1 ) = B m n ( 1 ) I ( 1 ) [ + β ( N + 2 ) n ( N + 1 ) m + ] ,
K m 0 ( 1 ) = 2 β 1 m a w 1 δ m , 0 .
G m n ( 2 ) = p w 2 2 β 2 p [ D m p ( 2 ) D p n ( 1 ) I ( 2 ) ( + β 2 p 3 m + ) I ( 1 ) ( - β 1 n 2 p + ) + C m p ( 2 ) C p n ( 1 ) I ( 2 ) ( - β 2 p 3 m - ) I ( 1 ) ( - β ln 2 p - ) ] ,
H m n ( 2 ) = p w 2 2 β 2 p [ D m p ( 2 ) C p n ( 1 ) I ( 2 ) ( + β 2 p 3 m + ) I ( 1 ) ( + β ln 2 p - ) + C m p ( 2 ) C p n ( 1 ) I ( 2 ) ( - β 2 p 3 m - ) I ( 1 ) ( + β ln 2 p + ) ] ,
J m n ( 2 ) = p w 2 2 β 2 p [ A m p ( 1 ) A m p ( 2 ) I ( 2 ) ( + β 3 n 2 p - ) I ( 1 ) ( + β 2 p 1 m - ) + B m p ( 1 ) B p n ( 2 ) I ( 2 ) ( + β 3 n 2 p + ) I ( 1 ) ( - β 2 p 1 m + ) ] .
I ( j ) [ ± β ( j + 1 ) n ( j ) m ± ] = exp [ ± i β ( j + 1 ) n ( j ) m ± F j ] δ m , n .
G m n ( 2 ) = w 2 2 β 2 n [ D m n ( 2 ) D n n ( 1 ) I ( 2 ) ( + β 2 n 3 m + ) + C m n ( 2 ) C n n ( 1 ) I ( 2 ) ( - β 2 n 3 m - ) ] ,
H m n ( 2 ) = w 2 2 β 2 n [ D m p ( 2 ) C n n ( 1 ) I ( 2 ) ( + β 2 n 3 m + ) + C m p ( 2 ) D p n ( 2 ) I ( 2 ) ( - β 2 n 3 m - ) ] ,
J m n ( 2 ) = w 2 2 β 2 n [ A m n ( 1 ) A n n ( 1 ) I ( 2 ) ( + β 3 n 2 n - ) + B m n ( 1 ) B n n ( 1 ) I ( 2 ) ( - β 3 n 2 n + ) ] .
R 0 = r 0 2 .
f j ( y ) = h j 2 sin ( 2 π y a ) .
f j ( y ) = - h j 2 + h j a y , 0 < y < b , f j ( y ) = h j ( a + b ) 2 ( a - b ) + h j a - b y , b < y < a ,

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