Abstract

For grating periods larger than the excitation wavelength, multiple-grating orders couple incident optical radiation to the surface plasma waves (SPW’s) characteristic of the metal–air interface. For a grating period that is an integral multiple of the wave vector of these surface modes, two resonances become degenerate in coupling angle. There are also permitted diffraction orders at this coupling angle. The vicinity of this multiple-mode coupling resonance, where several free-space electromagnetic modes, as well as two surface modes, are coupled by different orders of a grating, is known as a minigap region. Not surprisingly, the response surface displays complex dependences on frequency, angle, and grating profile. A detailed experimental and theoretical study is presented of the optical response at 633 nm in the (+1, −2) minigap region for Ag films deposited on photolithographically defined 870-nm-period gratings. Measurements of both the 0-order reflectance and the −1-order diffraction are presented for a wide progression of grating depths. The SPW resonances depend on the grating depth, and this variation is used to tune through the minigap region for a fixed wavelength and period. Similar measurements are presented for a single grating as a function of wavelength through the minigap region. In both measurements the 0-order response shows only a single broad minimum as the resonances approach degeneracy, while the −1-order diffraction shows clearly defined momentum gaps. A simple theoretical model based on the Rayleigh hypothesis is presented that gives a good qualitative picture of the response. The response surfaces are sensitive to the grating profile, and detailed modeling requires inclusion of higher-order grating components.

© 1991 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. M. C. Hutley, Opt. Acta 20, 607 (1973).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. M. G. Weber and D. L. Mills, Phys. Rev. B 34, 2893 (1986).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. K. Utagawa, J. Opt. Soc. Am. 69, 333 (1979).
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    [CrossRef]
  27. H. Numata, J. Phys. Soc. Jpn. 51, 2575 (1982).
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  30. P. M. Van den Berg and J. J. Fokkema, J. Opt. Soc. Am. 69, 27 (1979).
    [CrossRef]

1991 (1)

1988 (2)

1987 (1)

D. Heitman, N. Kroo, C. Schulz, and Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

1986 (2)

M. G. Weber and D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

M. G. Weber, Phys. Rev. B 33, 909 (1986).
[CrossRef]

1984 (1)

N. E. Glass, M. Weber, and D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

1983 (2)

M. Yamashita and M. Tsuji, J. Phys. Soc. Jpn. 52, 2462 (1983).
[CrossRef]

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

1982 (1)

H. Numata, J. Phys. Soc. Jpn. 51, 2575 (1982).
[CrossRef]

1981 (2)

B. Laks, D. L. Mills, and A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

N. Kroo, Z. S. Szentirmay, and J. Felszerfalvi, Phys. Lett. A 86, 445 (1981).
[CrossRef]

1979 (2)

1977 (3)

D. Heitman, Opt. Commun. 20, 292 (1977).
[CrossRef]

W. Rothballer, Opt. Commun. 20, 429 (1977).
[CrossRef]

E. H. Rosengart and I. Pockrand, Opt. Lett. 1, 194 (1977).
[CrossRef]

1974 (1)

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 6577 (1974).
[CrossRef]

1973 (2)

M. C. Hutley, Opt. Acta 20, 607 (1973).
[CrossRef]

M. C. Hutley and V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

1968 (1)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Lett. 21, 1530 (1968).
[CrossRef]

1965 (1)

J. L. Uretski, Ann. Phys. 33, 400 (1965).
[CrossRef]

1964 (2)

R. Petit and M. Cadilhac, C. R. Acad. Sci. 259, 2077 (1964).

A. Wirgin, Rev. Opt. 9, 449 (1964).

1962 (1)

1958 (1)

E. A. Stern, as quoted in R. A. Ferrel, Phys. Rev. 111, 1214 (1958).
[CrossRef]

1941 (1)

1935 (1)

R. M. Wood, Phys. Rev. 48, 928 (1935).
[CrossRef]

Arakawa, E. T.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Lett. 21, 1530 (1968).
[CrossRef]

Ballantyne, J. M.

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

Bird, V. M.

M. C. Hutley and V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

Brueck, S. R. J.

Cadilhac, M.

R. Petit and M. Cadilhac, C. R. Acad. Sci. 259, 2077 (1964).

Celli, V.

Chen, Y. J.

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

Cowan, J. J.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Lett. 21, 1530 (1968).
[CrossRef]

Fano, U.

Felszerfalvi, J.

N. Kroo, Z. S. Szentirmay, and J. Felszerfalvi, Phys. Lett. A 86, 445 (1981).
[CrossRef]

Fokkema, J. J.

Gallaway, W. S.

Glass, N. E.

N. E. Glass, M. Weber, and D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

Hamm, R. N.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Lett. 21, 1530 (1968).
[CrossRef]

Heitman, D.

D. Heitman, N. Kroo, C. Schulz, and Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

D. Heitman, Opt. Commun. 20, 292 (1977).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Opt. Acta 20, 607 (1973).
[CrossRef]

M. C. Hutley and V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

Koteles, E. S.

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

Kroo, N.

D. Heitman, N. Kroo, C. Schulz, and Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

N. Kroo, Z. S. Szentirmay, and J. Felszerfalvi, Phys. Lett. A 86, 445 (1981).
[CrossRef]

Laks, B.

B. Laks, D. L. Mills, and A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

Maradudin, A. A.

P. Tran, V. Celli, and A. A. Maradudin, Opt. Lett. 13, 530 (1988).
[CrossRef] [PubMed]

B. Laks, D. L. Mills, and A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

Mills, D. L.

M. G. Weber and D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

N. E. Glass, M. Weber, and D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

B. Laks, D. L. Mills, and A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

Nevière, M.

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 6577 (1974).
[CrossRef]

Numata, H.

H. Numata, J. Phys. Soc. Jpn. 51, 2575 (1982).
[CrossRef]

Petit, R.

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 6577 (1974).
[CrossRef]

R. Petit and M. Cadilhac, C. R. Acad. Sci. 259, 2077 (1964).

Pockrand, I.

Ritchie, R. H.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Lett. 21, 1530 (1968).
[CrossRef]

Rosengart, E. H.

Rothballer, W.

W. Rothballer, Opt. Commun. 20, 429 (1977).
[CrossRef]

Schulz, C.

D. Heitman, N. Kroo, C. Schulz, and Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

Seymour, R. J.

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

Sonek, G. J.

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

Stern, E. A.

E. A. Stern, as quoted in R. A. Ferrel, Phys. Rev. 111, 1214 (1958).
[CrossRef]

Stewart, J. E.

Szentirmay, Z. S.

N. Kroo, Z. S. Szentirmay, and J. Felszerfalvi, Phys. Lett. A 86, 445 (1981).
[CrossRef]

Szentirmay, Zs.

D. Heitman, N. Kroo, C. Schulz, and Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

Tran, P.

Tsuji, M.

M. Yamashita and M. Tsuji, J. Phys. Soc. Jpn. 52, 2462 (1983).
[CrossRef]

Uretski, J. L.

J. L. Uretski, Ann. Phys. 33, 400 (1965).
[CrossRef]

Utagawa, K.

Van den Berg, P. M.

Vincent, P.

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 6577 (1974).
[CrossRef]

Weber, M.

N. E. Glass, M. Weber, and D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

Weber, M. G.

M. G. Weber, Phys. Rev. B 33, 909 (1986).
[CrossRef]

M. G. Weber and D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

Wirgin, A.

A. Wirgin, Rev. Opt. 9, 449 (1964).

Wood, R. M.

R. M. Wood, Phys. Rev. 48, 928 (1935).
[CrossRef]

Yamashita, M.

M. Yamashita and M. Tsuji, J. Phys. Soc. Jpn. 52, 2462 (1983).
[CrossRef]

Yousaf, M.

Zaidi, S. H.

Ann. Phys. (1)

J. L. Uretski, Ann. Phys. 33, 400 (1965).
[CrossRef]

Appl. Opt. (2)

C. R. Acad. Sci. (1)

R. Petit and M. Cadilhac, C. R. Acad. Sci. 259, 2077 (1964).

J. Opt. Soc. Am. (3)

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

J. Phys. Soc. Jpn. (2)

H. Numata, J. Phys. Soc. Jpn. 51, 2575 (1982).
[CrossRef]

M. Yamashita and M. Tsuji, J. Phys. Soc. Jpn. 52, 2462 (1983).
[CrossRef]

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 6577 (1974).
[CrossRef]

Opt. Acta (2)

M. C. Hutley, Opt. Acta 20, 607 (1973).
[CrossRef]

M. C. Hutley and V. M. Bird, Opt. Acta 20, 771 (1973).
[CrossRef]

Opt. Commun. (2)

D. Heitman, Opt. Commun. 20, 292 (1977).
[CrossRef]

W. Rothballer, Opt. Commun. 20, 429 (1977).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

N. Kroo, Z. S. Szentirmay, and J. Felszerfalvi, Phys. Lett. A 86, 445 (1981).
[CrossRef]

Phys. Rev. (2)

R. M. Wood, Phys. Rev. 48, 928 (1935).
[CrossRef]

E. A. Stern, as quoted in R. A. Ferrel, Phys. Rev. 111, 1214 (1958).
[CrossRef]

Phys. Rev. B (5)

N. E. Glass, M. Weber, and D. L. Mills, Phys. Rev. B 29, 6548 (1984).
[CrossRef]

B. Laks, D. L. Mills, and A. A. Maradudin, Phys. Rev. B 23, 4965 (1981).
[CrossRef]

M. G. Weber and D. L. Mills, Phys. Rev. B 34, 2893 (1986).
[CrossRef]

D. Heitman, N. Kroo, C. Schulz, and Zs. Szentirmay, Phys. Rev. B 35, 2660 (1987).
[CrossRef]

M. G. Weber, Phys. Rev. B 33, 909 (1986).
[CrossRef]

Phys. Rev. Lett. (1)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Lett. 21, 1530 (1968).
[CrossRef]

Rev. Opt. (1)

A. Wirgin, Rev. Opt. 9, 449 (1964).

Solid State Commun. (1)

Y. J. Chen, E. S. Koteles, R. J. Seymour, G. J. Sonek, and J. M. Ballantyne, Solid State Commun. 46, 95 (1983).
[CrossRef]

Other (2)

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

V. M. Agranovich and D. L. Mills, eds., Surface Polaritons (North-Holland, Amsterdam, 1982).

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Figures (17)

Fig. 1
Fig. 1

Dispersion relation of SPW’s for a lossless, free-electron metal ( = 1 − ωp2/ω2). The axes are normalized to ωp and kp = ωp/c. The grating vectors corresponding to the n = ±1 and n = ±2 orders of a grating of period d are shown as vertical dashed lines. The range of wave vectors accessible by varying the input angle from normal to grazing incidence for n +1 and n = −2 is shown as horizontal lines. Note that for this choice of parameters (λ/d ~ 0.73), there are two SPW resonances and that the −1 diffraction order propagates throughout the angular range.

Fig. 2
Fig. 2

Experimental arrangement for reflection and diffraction-order measurements.

Fig. 3
Fig. 3

0-order reflectances at 633 nm for 870-nm-period gratings with varying grating depths h. The left-hand column presents experimental results; the right-hand column presents theoretical modeling. See the text for details.

Fig. 4
Fig. 4

Same as Fig. 3 but for deeper gratings.

Fig. 5
Fig. 5

Cross-section SEM photographs of the gratings used for the experiments. The measured depths are (a) 95 nm, (b) 110 nm, (c) 122 nm, and (d) 150 nm.

Fig. 6
Fig. 6

0-order reflectances at 633 nm for gratings with (a) residual SPW effects and (b) absorptive behavior.

Fig. 7
Fig. 7

Cross-section SEM photographs of the gratings used for the measurements in Fig. 6.

Fig. 8
Fig. 8

−1-diffraction-order scans for the gratings used in Fig. 3. The left-hand column presents experimental results; the right-hand column presents theoretical modeling. See the text for details.

Fig. 9
Fig. 9

−1-diffraction-order scans for the gratings in Fig. 4. The left-hand column presents experimental results; the right-hand column presents theoretical modeling. See the text for details.

Fig. 10
Fig. 10

−1-diffraction-order scans for the gratings used in Fig. 6.

Fig. 11
Fig. 11

Energy-sum (0-order reflectance plus −1-diffraction-order) scans for the gratings used in Figs. 3, 4, 8, and 9.

Fig. 12
Fig. 12

0-order reflectance and −1-diffraction-order scans for the 95-nm-deep gratings (Figs. 4a, 9a, and 11e) as the wavelength is increased from 586 to 607 nm. The left-hand column presents the 0-order measurements; the right-hand column presents the −1-diffraction-order measurements.

Fig. 13
Fig. 13

Same as Fig. 12 but for an increase in wavelength from 610 to 617 nm.

Fig. 14
Fig. 14

Same as Fig. 12 but for an increase in wavelength from 619 to 633 nm.

Fig. 15
Fig. 15

Plot of the resonance behavior observed in Figs. 1214. The 0-order reflectivity (a) shows a single dominant resonance becoming a single broad minimum in the minigap region. In contrast, the −1-diffraction-order scans (b) show a well-defined momentum gap corresponding to the (+1, −2) minigap region.

Fig. 16
Fig. 16

Comparison between theory and experiment for a grating with a nearly sinusoidal profile. The left-hand column presents experimental results; the right-hand column presents theoretical modeling.

Fig. 17
Fig. 17

Comparison between theory and experiment for a profile showing a slight deviation from a sinusoidal shape. The left-hand column presents experimental results; the right-hand column presents theoretical modeling.

Equations (16)

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k SPW = k 0 [ / ( 1 + ) ] 1 / 2 ,
k 0 sin θ + 2 π n / d = ± k SPW ,
J p ( k z u ) + n B n v ( - 1 ) p - n J p - n ( α n u ) - n B n m J p - n ( β n u ) = 0 ,
( K z - p g k 0 k z ) J p ( k z u ) - n [ α n - ( p - n ) g k n α n ] B n v ( - 1 ) p - n J p - n ( α n u ) - 1 n [ β n - ( p - n ) g k n β n ] B n m J p - n ( β n u ) = 0 ,
z = u 1 sin ( g y ) + u 2 sin ( 2 g y ) .
J p ( k z u 1 ) J 0 ( k z u 2 ) + n ( - 1 ) p - n J p - n ( α n u 1 ) J 0 ( α 2 u 2 ) B n v - n B n m J p - n ( β n u 1 ) J 0 ( β n u 2 ) + s = 1 [ A ( p , s ) + n B ( p , n , s ) + n C ( p , n , s ) ] = 0
D ( p ) + n E ( p , n ) - 1 n F ( p , n ) + s = 1 [ G ( p , s ) + n I ( p , n , s ) - 1 n L ( p , n , s ) ] = 0 ,
B v ( y , z ) = exp [ i ( k y y + k z z ) ] + n B n v exp [ i ( K n y - α n z ) ] ,             z < 0 , B m ( y , z ) = n B n m exp [ i ( K n y + β n z ) ] ,             z > 0 ,
α n = i ( k n 2 - k 0 2 ) 1 / 2 ,             β n = i ( k n 2 - k 0 2 ) 1 / 2 .
f ( y ) = u sin ( g y ) ,
exp [ i γ f ( y ) ] = exp [ i γ u sin ( g y ) ] = p = - exp ( i p g y ) J p ( γ u ) .
B v ( y , z ) = B m ( y , z ) z = f ( y ) , n B v ( y , z ) = 1 n B m ( y , z ) z = f ( y ) ,
n = { 1 + [ f ( y ) y ] 2 } - 1 / 2 [ z - f ( y ) y y ] ,
f ( y ) = u 1 sin ( g y ) + u 2 sin ( 2 g y ) ,
exp [ i γ f ( y ) ] = p , q exp [ i ( p + 2 q ) g y ] J p ( γ u 1 ) J q ( γ u 2 ) .
A ( p , s ) = [ J p - 2 s ( k z u 1 ) + ( - 1 ) s J p + 2 s ( k z u 1 ) ] J s ( k z u 2 ) , B ( p , n , s ) = ( - 1 ) p - n B n v [ J p - n - 2 s ( α n u 1 ) + ( - 1 ) s J p - n + 2 s ( α n u 1 ) ] J s ( α n u 2 ) , C ( p , n , s ) = B n m [ J p - n - 2 s ( β n u 1 ) ] + ( - 1 ) s J p - n + 2 s ( β n u 1 ) ] J s ( β n u 2 ) , D ( p ) = [ K z - p g ( k y / k z ) ] J p ( k z u 1 ) J 0 ( k z u 2 ) - g k y u 2 × [ J p - 2 ( k z u 1 ) + J p + 2 ( k z u 1 ) ] J 0 ( k z u 2 ) , E ( p , n ) = - ( - 1 ) p - n B n v [ α n - ( p - n ) g ( k n / α n ) ] × J p - n ( α n u 1 ) J 0 ( α n u 2 ) - g k n u 2 B n v × [ J p - n - 2 ( α n u 1 ) + J p - n + 2 ( α n u 1 ) ] J 0 ( α n u 2 ) , F ( p , n ) = B n m [ β n - ( p - n ) g ( k n / β n ) ] × J p - n ( β n u 1 ) J 0 ( β n u 2 ) - g k n u 2 B n m × [ J p - n - 2 ( β n u 1 ) + J p - n + 2 ( β n u 1 ) ] J 0 ( β n u 2 ) , G ( p , s ) = [ k z - ( p - 2 s ) g ( k y / k z ) ] J p - 2 s ( k z u 1 ) J s ( k z u 2 ) + ( - 1 ) s [ k z - ( p + 2 s ) g ( k y / k z ) ] J p + 2 s ( k z u 1 ) × J s ( k z u 2 ) - g k y u 2 [ J p - 2 s - 2 ( k z u 1 ) + ( - 1 ) s J p + 2 s - 2 ( k z u 1 ) ] J s ( k z u 2 ) - g k y u 2 × [ J p - 2 s + 2 ( k z u 1 ) + ( - 1 ) s J p + 2 s + 2 ( k z u 1 ) ] J s ( k z u 2 ) , I ( p , n , s ) = - ( - 1 ) p - n B n v [ α n - ( p - n - 2 s ) g ( k n / α n ) ] × J p - n - 2 s ( α n u 1 ) J s ( α n u 2 ) + ( - 1 ) s B n v × [ α n - ( p - n + 2 s ) g ( k n / α n ) ] × J p - n + 2 s ( α n u 1 ) J s ( α n u 2 ) - g k n u 2 B n v × [ J p - n - 2 s - 2 ( α n u 1 ) + ( - 1 ) s J p - n + 2 s - 2 ( α n u 1 ) ] × J s ( α n u 2 ) - g k n u 2 B n v [ J p - n - 2 s + 2 ( α n u 1 ) + ( + 1 ) s J p - n + 2 s + 2 ( α n u 1 ) ] J s ( α n u 2 ) , L ( p , n , s ) = B n m [ β n - ( p - n - 2 s ) g ( k n / β n ) ] × J p - n - 2 s ( β n u 1 ) J s ( β n u 2 ) + ( - 1 ) s B n m × [ β n - ( p - n + 2 s ) g ( k n / β n ) ] × J p - n + 2 s ( β n u 1 ) J s ( β n u 2 ) - g k n u 2 B n m × [ J p - n - 2 s - 2 ( β n u 1 ) + ( - 1 ) s J p - n + 2 s - 2 ( β n u 1 ) ] × J s ( β n u 2 ) - g k n u 2 B n m [ J p - n - 2 s + 2 ( β n u 1 ) + ( - 1 ) s J p - n + 2 s + 2 ( β n u 1 ) ] J s ( β n u 2 ) .

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