Abstract

Surface-plasmon excitation on a corrugated Ag surface in conical (nonplanar) geometry is studied. Both p and s polarizations of the incident wave are considered. The Rayleigh method is applied to obtain the infinite set of simultaneous linear equations with respect to the reflected and transmitted amplitudes of the spatial harmonics. The truncated system is numerically solved to explain the experimental data of Inagaki et al. [ J. Opt. Soc. Am. B 3, 992 ( 1986)]. Mode conversions, p to s and s to p, are fully taken into account. The optimal behavior of the minimal total reflectivity with respect to the corrugation amplitudes is studied for two azimuthal angles.

© 1987 Optical Society of America

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  1. The literature on the theory of the scattering of electromagnetic radiation from a rough surface is exhaustive. See, for example, A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975); A. A. Maradudin and W. Zierau, Phys. Rev. B 14, 484 (1976); A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 405; G. S. Agarwal, Phys. Rev. B 15, 2371 (1977).
    [Crossref]
  2. For experiments on electromagnetic scattering with rough surfaces, see, for example, H. Raether, in Surface Polaritons, A. A. Maradudin and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 331.
  3. For a review on surface plasmons, see, for example, G. J. Kovacs, in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), p. 143.
  4. See, for example, R. Petit, ed., Electromagnetic Theory of Grating (Springer-Verlag, Berlin, 1980).
    [Crossref]
  5. For a description of conical diffraction geometry, see, for example, D. Maystre, in Electromagnetic Theory of Grating, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 85; R. Petit, in Electromagnetic Theory of Grating, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 31.
  6. D. L. Mills and M. Weber, Phys. Rev. B 26, 1075 (1982).
    [Crossref]
  7. G. S. Agarwal, Phys. Rev. B 31, 3534 (1985).
    [Crossref]
  8. G. S. Agarwal and S. S. Jha, Phys. Rev. B 26, 482 (1982).
    [Crossref]
  9. G. S. Agarwal and S. Dutta Gupta, Phys. Rev. B 31, 5239 (1986).
    [Crossref]
  10. M. T. Wlodarczyk and S. R. Seshadri, J. Opt. Soc. Am. A 2, 171 (1985).
    [Crossref]
  11. T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
    [Crossref]
  12. T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, J. Opt. Soc. Am. B 3, 992 (1986).
    [Crossref]
  13. See, for example, H. Raether, Opt. Commun. 42, 217 (1982); W. Rothballer, Opt. Commun. 20, 429 (1977).
    [Crossref]
  14. F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
    [Crossref]
  15. R. Reinisch and M. Neviere, Phys. Rev. B 26, 1870 (1983).
    [Crossref]
  16. D. Agassi and T. F. George, Phys. Rev. B 33, 2393 (1986).
    [Crossref]
  17. A. A. Maradudin, J. Opt. Soc. Am. 73, 759 (1983).
    [Crossref]
  18. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [Crossref]
  19. We use the same normalization as in Ref. 12.
  20. We could not go beyond seven harmonics because of our computer limitations. The reliability of our results for large θg and +2 resonance is less.
  21. T. Inagaki, M. Motosuga, E. T. Arakawa, and J. P. Goudonnet, Phys. Rev. B 32, 6238 (1985).
    [Crossref]
  22. See, for example, D. Sarid, Phys. Rev. Lett. 47, 1927 (1981); D. Sarid, R. T. Deck, and J. J. Fasano, J. Opt. Soc. Am. 72, 1345 (1982).
    [Crossref]

1986 (3)

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. B 31, 5239 (1986).
[Crossref]

T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, J. Opt. Soc. Am. B 3, 992 (1986).
[Crossref]

D. Agassi and T. F. George, Phys. Rev. B 33, 2393 (1986).
[Crossref]

1985 (3)

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

M. T. Wlodarczyk and S. R. Seshadri, J. Opt. Soc. Am. A 2, 171 (1985).
[Crossref]

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

1983 (3)

T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
[Crossref]

A. A. Maradudin, J. Opt. Soc. Am. 73, 759 (1983).
[Crossref]

R. Reinisch and M. Neviere, Phys. Rev. B 26, 1870 (1983).
[Crossref]

1982 (3)

See, for example, H. Raether, Opt. Commun. 42, 217 (1982); W. Rothballer, Opt. Commun. 20, 429 (1977).
[Crossref]

G. S. Agarwal and S. S. Jha, Phys. Rev. B 26, 482 (1982).
[Crossref]

D. L. Mills and M. Weber, Phys. Rev. B 26, 1075 (1982).
[Crossref]

1981 (1)

See, for example, D. Sarid, Phys. Rev. Lett. 47, 1927 (1981); D. Sarid, R. T. Deck, and J. J. Fasano, J. Opt. Soc. Am. 72, 1345 (1982).
[Crossref]

1977 (1)

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
[Crossref]

1975 (1)

The literature on the theory of the scattering of electromagnetic radiation from a rough surface is exhaustive. See, for example, A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975); A. A. Maradudin and W. Zierau, Phys. Rev. B 14, 484 (1976); A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 405; G. S. Agarwal, Phys. Rev. B 15, 2371 (1977).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. B 31, 5239 (1986).
[Crossref]

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

G. S. Agarwal and S. S. Jha, Phys. Rev. B 26, 482 (1982).
[Crossref]

Agassi, D.

D. Agassi and T. F. George, Phys. Rev. B 33, 2393 (1986).
[Crossref]

Arakawa, E. T.

T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, J. Opt. Soc. Am. B 3, 992 (1986).
[Crossref]

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

T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
[Crossref]

Celli, V.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Dutta Gupta, S.

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. B 31, 5239 (1986).
[Crossref]

George, T. F.

D. Agassi and T. F. George, Phys. Rev. B 33, 2393 (1986).
[Crossref]

Goudonnet, J. P.

T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, J. Opt. Soc. Am. B 3, 992 (1986).
[Crossref]

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

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
[Crossref]

Inagaki, T.

T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, J. Opt. Soc. Am. B 3, 992 (1986).
[Crossref]

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

T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
[Crossref]

Jha, S. S.

G. S. Agarwal and S. S. Jha, Phys. Rev. B 26, 482 (1982).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Kovacs, G. J.

For a review on surface plasmons, see, for example, G. J. Kovacs, in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), p. 143.

Maradudin, A. A.

A. A. Maradudin, J. Opt. Soc. Am. 73, 759 (1983).
[Crossref]

The literature on the theory of the scattering of electromagnetic radiation from a rough surface is exhaustive. See, for example, A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975); A. A. Maradudin and W. Zierau, Phys. Rev. B 14, 484 (1976); A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 405; G. S. Agarwal, Phys. Rev. B 15, 2371 (1977).
[Crossref]

Marvin, A.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
[Crossref]

Maystre, D.

For a description of conical diffraction geometry, see, for example, D. Maystre, in Electromagnetic Theory of Grating, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 85; R. Petit, in Electromagnetic Theory of Grating, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 31.

Mills, D. L.

D. L. Mills and M. Weber, Phys. Rev. B 26, 1075 (1982).
[Crossref]

The literature on the theory of the scattering of electromagnetic radiation from a rough surface is exhaustive. See, for example, A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975); A. A. Maradudin and W. Zierau, Phys. Rev. B 14, 484 (1976); A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 405; G. S. Agarwal, Phys. Rev. B 15, 2371 (1977).
[Crossref]

Motosuga, M.

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

T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
[Crossref]

Neviere, M.

R. Reinisch and M. Neviere, Phys. Rev. B 26, 1870 (1983).
[Crossref]

Raether, H.

See, for example, H. Raether, Opt. Commun. 42, 217 (1982); W. Rothballer, Opt. Commun. 20, 429 (1977).
[Crossref]

For experiments on electromagnetic scattering with rough surfaces, see, for example, H. Raether, in Surface Polaritons, A. A. Maradudin and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 331.

Reinisch, R.

R. Reinisch and M. Neviere, Phys. Rev. B 26, 1870 (1983).
[Crossref]

Sarid, D.

See, for example, D. Sarid, Phys. Rev. Lett. 47, 1927 (1981); D. Sarid, R. T. Deck, and J. J. Fasano, J. Opt. Soc. Am. 72, 1345 (1982).
[Crossref]

Seshadri, S. R.

Toigo, F.

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
[Crossref]

Weber, M.

D. L. Mills and M. Weber, Phys. Rev. B 26, 1075 (1982).
[Crossref]

Wlodarczyk, M. T.

Yamamori, K.

T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
[Crossref]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (1)

See, for example, H. Raether, Opt. Commun. 42, 217 (1982); W. Rothballer, Opt. Commun. 20, 429 (1977).
[Crossref]

Phys. Rev. B (11)

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, Phys. Rev. B 18, 5618 (1977).
[Crossref]

R. Reinisch and M. Neviere, Phys. Rev. B 26, 1870 (1983).
[Crossref]

D. Agassi and T. F. George, Phys. Rev. B 33, 2393 (1986).
[Crossref]

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

The literature on the theory of the scattering of electromagnetic radiation from a rough surface is exhaustive. See, for example, A. A. Maradudin and D. L. Mills, Phys. Rev. B 11, 1392 (1975); A. A. Maradudin and W. Zierau, Phys. Rev. B 14, 484 (1976); A. A. Maradudin, in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 405; G. S. Agarwal, Phys. Rev. B 15, 2371 (1977).
[Crossref]

D. L. Mills and M. Weber, Phys. Rev. B 26, 1075 (1982).
[Crossref]

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

G. S. Agarwal and S. S. Jha, Phys. Rev. B 26, 482 (1982).
[Crossref]

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. B 31, 5239 (1986).
[Crossref]

T. Inagaki, M. Motosuga, K. Yamamori, and E. T. Arakawa, Phys. Rev. B 28, 1740 (1983).
[Crossref]

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

Phys. Rev. Lett. (1)

See, for example, D. Sarid, Phys. Rev. Lett. 47, 1927 (1981); D. Sarid, R. T. Deck, and J. J. Fasano, J. Opt. Soc. Am. 72, 1345 (1982).
[Crossref]

Other (6)

For experiments on electromagnetic scattering with rough surfaces, see, for example, H. Raether, in Surface Polaritons, A. A. Maradudin and D. L. Mills, eds. (North-Holland, Amsterdam, 1982), p. 331.

For a review on surface plasmons, see, for example, G. J. Kovacs, in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), p. 143.

See, for example, R. Petit, ed., Electromagnetic Theory of Grating (Springer-Verlag, Berlin, 1980).
[Crossref]

For a description of conical diffraction geometry, see, for example, D. Maystre, in Electromagnetic Theory of Grating, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 85; R. Petit, in Electromagnetic Theory of Grating, R. Petit, ed. (Springer-Verlag, Berlin, 1980), p. 31.

We use the same normalization as in Ref. 12.

We could not go beyond seven harmonics because of our computer limitations. The reliability of our results for large θg and +2 resonance is less.

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Figures (4)

Fig. 1
Fig. 1

Top view of the corrugated structure with corrugation amplitude a. The plane of incidence is along OO′ and perpendicular to the XZ plane.

Fig. 2
Fig. 2

(a) Peak absorptance A of p-polarized photons for the plasma resonances +1 and +2 in an Ag grating with corrugation amplitude a = 25 nm as a function of the azimuthal angle θg. Dashed curves show the results of 1 − ∑ Rp − ∑ Rps calculations; dotted curve shows the joule heating calculations. Experimental results12 are shown by circled dots. Optical constants of Ag are taken from the work of Johnson and Christy.18 (b) Peak absorptance A with s-polarized incident photons for the plasma resonances +1 and +2. Dashed curves show the results of 1 − ∑ Rs − ∑ Rsp calculations; the dotted curve is the result of joule heating calculations.

Fig. 3
Fig. 3

(a) Same as in Fig. 2(a) but with corrugation amplitude a = 41 m. (b) Same as in Fig. 2(b) but with corrugation amplitude a = 41 nm.

Fig. 4
Fig. 4

Minimal total reflectivity ∑ R as a function of the corrugation amplitude a for incident p-polarization and +1 resonance. Different curves are labeled by different values of the azimuthal angle.

Equations (38)

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( Δ + k 0 2 ) ( E y H y ) = 0 ,
( 2 y 2 + k 0 2 ) E x = 2 E y x y - i k 0 H y z ,
( 2 y 2 + k 0 2 ) E z = 2 E y y z + i k 0 H y x ,
( 2 y 2 + k 0 2 ) H x = 2 H y x y + i k 0 E y z ,
( 2 y 2 + k 0 2 ) H z = H y z y - i k 0 E y x .
E y 1 = exp ( - i α 0 z ) [ A 0 exp ( - i β 0 y ) exp ( i γ 0 x ) + n B n exp ( i β n y ) exp ( i γ n x ) ] ;
H y 1 = exp ( i α 0 z ) [ A ˜ 0 exp ( - i β 0 y ) exp ( i γ 0 x ) + B ˜ n exp ( i β n y ) exp ( i γ n x ) ] .
α 0 = k 0 0 sin θ sin θ g ,
γ 0 = k 0 0 sin θ cos θ g ,
β 0 2 = k 0 2 0 - γ 0 2 - α 0 2 ,
γ n = γ 0 + n K ,             n = 0 , ± 1 , ± 2 , . ,
β n 2 = k 0 2 0 - γ n 2 - α n 2 .
E y 2 = exp ( i α 0 z ) [ n C n exp ( - i β ˜ n y ) exp ( i γ n x ) ] ,
H y 2 = exp ( i α 0 z ) [ n C ˜ n exp ( - i β ˜ n y ) exp ( i γ n x ) ] ,
β ˜ n 2 = k 0 2 - γ n 2 - α 0 2 ,
E z 1 y = a sin K x = E z 2 y = a sin K x ,
H z 1 y = a sin K x = H z 2 y = a sin K x ,
( E x 1 t x + E y 1 t y ) y = a sin K x = ( E x 2 t x + E y 2 t y ) y = a sin K x ,
( H x 1 t x + H y 1 t y ) y = a sin K x = ( H x 2 t x + H y 2 t y ) y = a sin K x ,
t ^ = { [ 1 + ( a K cos K x ) 2 ] - 1 / 2 , a K cos K x [ 1 + ( a K cos K x ) 2 ] - 1 / 2 , 0 } .
exp ( i β n a sin K x ) = - exp ( i m K x ) J m ( β n a )
n = - + α 0 β n B ˜ n - k 0 0 γ n B n k 0 2 0 - β n 2 J m - n ( β n a ) + α 0 β ˜ n C ˜ n + k 0 γ n C n k 0 2 - β ˜ n 2 × J n - m ( β ˜ n a ) = ( - 1 ) m α 0 β 0 A ˜ 0 + k 0 0 γ 0 A 0 k 0 2 0 - β 0 2 J m ( β 0 a ) .
n = - + α 0 β n B n + k 0 γ n B ˜ n k 0 2 0 - β n 2 J m - n ( β n a ) + α 0 β ˜ n C n + k 0 γ n C ˜ n k 0 2 - β ˜ n 2 × J n - m ( β n a ) = ( - 1 ) m α 0 β 0 A 0 - k 0 γ 0 A ˜ 0 k 0 2 0 - β 0 2 J m ( β 0 a ) ,
n = - + J m - n ( β n a ) [ β n γ n k 0 2 0 - β n 2 - ( m - n ) K β n ] × B n - k 0 α 0 k 0 2 0 - β n 2 J m - n ( β n a ) B ˜ n + J n - m ( β ˜ n a ) [ β ˜ n γ n k 0 2 - β ˜ n 2 - ( m - n ) K β ˜ n ] × C n + k 0 α 0 k 0 2 - β ˜ n 2 J n - m ( β ˜ n a ) C ˜ n = ( - 1 ) m J m ( β 0 a ) × { [ β 0 γ 0 k 0 2 0 - β 0 2 - m K β 0 ] A 0 + k 0 α 0 k 0 2 0 - β 0 2 A ˜ 0 } ,
n = - J m - n ( β n a ) [ β n γ n k 0 2 0 - β n 2 - ( m - n ) K β n ] × B ˜ n + k 0 α 0 0 k 0 2 0 - β n 2 J m - n ( β n a ) B n + J n - m ( β ˜ n a ) [ β ˜ n γ n k 0 2 - β ˜ n 2 - ( m - n ) K β ˜ n ] × C ˜ n - k 0 α 0 k 0 2 - β ˜ n 2 J n - m ( β ˜ n a ) C n = ( - 1 ) m J m ( β 0 a ) × { [ β 0 γ 0 k 0 2 0 - β 0 2 - m K β 0 ] A ˜ 0 - k 0 α 0 0 k 0 2 0 - β 0 2 A 0 } .
A 0 = ( sin θ 0 ) H .
A ˜ 0 = ( 0 sin θ ) E .
B n = k 0 2 0 - β n 2 k 0 0 α 2 + γ 2 H n r ,
C n = k 0 2 - β ˜ n 2 k 0 α 0 2 + γ n 2 H n t ,
B ˜ n = k 0 2 0 - β n 2 k 0 α 0 2 + γ n 2 E n r ,
C ˜ n = k 0 2 - β ˜ n 2 k 0 α 0 2 + γ n 2 E n t .
A X = F ,
x n = H n r ,             x n + l a = H n t ;             x n + 21 a = E n r ;             x n + 31 a = E n t ,
1 - R n p - R n p s
1 - R n s - R n s p
R n p = H n r 2 ,             R n p s = E n r 2             for incident p polarization ;
R n s = E n r 2 ,             R n s p = H n r 2             for incident s polarization .
θ 1 / 2 ( a ) = 2 θ 1 / 2 ( a = 0 ) ,

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