Abstract

Optical properties of a silver-island film are interpreted by taking account of retarded dipole–dipole interactions between island particles instead of the static-field interaction. The discrepancies between the values of the structural parameters evaluated from electron micrographs and those evaluated optically with the assumption of the static-field dipole–dipole interaction can be thus interpreted. The retarded dipole–dipole interactions as well as the size effect also broaden the width of the plasma-resonance absorption peaks.

© 1974 Optical Society of America

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References

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  1. T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 18, 63 (1973).
    [Crossref]
  2. T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
    [Crossref]
  3. T. Yamaguchi and A. Kinbara, Sixth International Vacuum Congress, Kyoto, March 1974.
  4. J. Vlieger, Physica 64, 63 (1973).
    [Crossref]
  5. D. C. Skillman, J. Opt. Soc. Am. 61, 1264 (1971).
    [Crossref]
  6. G. B. Irani, T. Huen, and F. Wooten, J. Opt. Soc. Am. 61, 128 (1971).
    [Crossref]
  7. Equations (17) and (18) shows that, when F+g= 0, the imaginary parts of the left-hand sides of the equations become maximum values. On the other hand, the approximate expressions of T0 and Tp of an anisotropic film include Im (∊||) and Im(∊||)+Im(1/∊⊥), respectively, in their denominators. Because the bulk dielectric constant of the particle can be used as ∊i in Eq. (21), g is known as a function of the wavelength of light λ. Therefore, if we measure T0 and Tp as functions of λ, we can determine the values of F from the observed dip wavelength of the transmittance spectra. Detailed descriptions are given in Refs. 1 and 2.
  8. Recently, Vlieger and his co-worker proposed an improved theory in Physica 73, 287 (1974).

1974 (2)

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

Recently, Vlieger and his co-worker proposed an improved theory in Physica 73, 287 (1974).

1973 (2)

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 18, 63 (1973).
[Crossref]

J. Vlieger, Physica 64, 63 (1973).
[Crossref]

1971 (2)

Huen, T.

Irani, G. B.

Kinbara, A.

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 18, 63 (1973).
[Crossref]

T. Yamaguchi and A. Kinbara, Sixth International Vacuum Congress, Kyoto, March 1974.

Skillman, D. C.

Vlieger, J.

J. Vlieger, Physica 64, 63 (1973).
[Crossref]

Wooten, F.

Yamaguchi, T.

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 18, 63 (1973).
[Crossref]

T. Yamaguchi and A. Kinbara, Sixth International Vacuum Congress, Kyoto, March 1974.

Yoshida, S.

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 18, 63 (1973).
[Crossref]

J. Opt. Soc. Am. (2)

Physica (2)

J. Vlieger, Physica 64, 63 (1973).
[Crossref]

Recently, Vlieger and his co-worker proposed an improved theory in Physica 73, 287 (1974).

Thin Solid Films (2)

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 18, 63 (1973).
[Crossref]

T. Yamaguchi, S. Yoshida, and A. Kinbara, Thin Solid Films 21, 173 (1974).
[Crossref]

Other (2)

T. Yamaguchi and A. Kinbara, Sixth International Vacuum Congress, Kyoto, March 1974.

Equations (17) and (18) shows that, when F+g= 0, the imaginary parts of the left-hand sides of the equations become maximum values. On the other hand, the approximate expressions of T0 and Tp of an anisotropic film include Im (∊||) and Im(∊||)+Im(1/∊⊥), respectively, in their denominators. Because the bulk dielectric constant of the particle can be used as ∊i in Eq. (21), g is known as a function of the wavelength of light λ. Therefore, if we measure T0 and Tp as functions of λ, we can determine the values of F from the observed dip wavelength of the transmittance spectra. Detailed descriptions are given in Refs. 1 and 2.

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Figures (5)

FIG. 1
FIG. 1

Computed values of C as functions of a/λ for two cases, (a) normal incidence and (b) oblique incidence at 60°.

FIG. 2
FIG. 2

Observed values of F|| (open circle) and F - ( 18 / s + 1 ) × ( d w a / λ 2 ) r - 1 2 (solid circle) as functions of r - 3 2 for five films of different thickness.

FIG. 3
FIG. 3

Values of a, D, γ, and η as functions of dw. Of each pair of values connected by a line, the left is obtained with the static-field approximation and the right with the retarded dipole–dipole interactions. Open circles and triangles show the values obtained by use of the bulk dielectric constant of silver given by Skillman (Ref. 5) and Irani et al. (Ref. 6), respectively. Filled circles show values determined from electron micrographs.

FIG. 4
FIG. 4

Variations of a, D, γ, η, and Δg with 14/R for three films of different thickness. Dotted lines show the relation D = 2R.

FIG. 5
FIG. 5

Width of the absorption peaks as functions of dw. Open circles show the observed width. Δgshape Δgsize, Δginteraction, and Δgbulk mean, respectively, the contributions from the variety of the particle shape, the size effect, the retarded dipole–dipole interactions, and the bulk dielectric constant.

Equations (36)

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E l , j = E l , o exp ( - i k y j sin θ 0 ) ,
P j = α V E l , o exp ( - i k y j sin θ 0 ) ,
E l , o = E o + E + j E j ,
E x , y = α x , y V 4 π o l o 3 s - 1 s + 1 E l , o , x , y , E z = 2 α z V 4 π o l o 3 s - 1 s + 1 E l , o , z ,
P j , x , y = 2 s + 1 P j , x , y , P j , z = 2 s s + 1 P j , z .
E j , r = P j 4 π o 2 cos θ j r j 3 ( 1 + i k r j ) exp ( - i k r j ) , E j , θ = P j 4 π o sin θ j r j 3 [ 1 + i k r j - ( k r j ) 2 ] exp ( - i k r j ) ,
E j , x = E l , o , x o 2 s + 1 α x V A j [ ( 1 + i k r j ) ( 2 sin 2 θ j - cos 2 θ j ) + ( k r j ) 2 cos 2 θ j ] , E j , y = E l , o , y o 2 s + 1 α y V A j [ ( 1 + i k r j ) ( 2 cos 2 θ j - sin 2 θ j ) + ( k r j ) 2 sin 2 θ j ] , E j , z = E l , o , z o 2 s s + 1 α z V A j [ - ( 1 + i k r j ) + ( k r j ) 2 ] ,
A j = 1 4 π r j 3 exp [ - i k ( r j + y j sin θ 0 ) ] .
E l , o , x , y , z = E o , x , y , z 1 + α x , y , z β x , y , z ,
o β x , y = - γ 2 24 η 3 s - 1 s + 1 + C x , y 2 s + 1 d w a , o β z = - 2 γ 2 24 η 3 s - 1 s + 1 + C z 2 s s + 1 d w a .
C x = j A j [ ( 1 + i k r j ) ( 2 - 3 cos 2 θ j ) + ( k r j ) 2 cos 2 θ j ] , C y = j A j [ ( 1 + i k r j ) ( 3 cos 2 θ j - 1 ) + ( k r j ) 2 sin 2 θ j ] , C z = j A j [ - ( 1 + i k r j ) + ( k r j ) 2 ] .
ϕ = k ( r + y sin θ 0 ) 2 ,
x 2 x o 2 + ( y + y o sin θ 0 ) 2 y o 2 = 1 ,
x o = ϕ k cos θ 0 ,             y o = ϕ k cos 2 θ 0 .
C x , y , z = 0 c x , y , z ( ϕ ) d ϕ .
C x = C y C .
Re ( C , x , y , z ) = C 1 , , x , y , z + C 2 , , x , y , z ( a / λ ) 2 , Im ( C , x , y , z ) = C 3 , , x , y , z ( a / λ ) ,
C 1 , = C 1 , x = C 1 , y = - 0.358 ,             C 1 , z = 0.716 , C 2 , = 9.0 ,             C 2 , x = 7.5 ,             C 2 , y = 6.5 ,             C 2 , z = 7.0 , C 3 , = 3.0 ,             C 3 , x = 6.1 ,             C 3 , y = 1.4 ,             C 3 , z = 4.6.
( - 1 ) E = N P = q α E l ,
α , = ( i - 1 ) 0 1 + f , ( i - 1 ) ,
E x , y = E o , x , y ,             E z = E o , z / z ,
, x , y - 1 = q 1 F , x , y + g + i Δ g , x , y ,
1 - 1 z = q 1 F z + g + i Δ g z ,
F , x , y = f - γ 2 24 η 3 s - 1 s + 1 + C 1 , , x , y 2 s + 1 d w a + C 2 , , x , y 2 s + 1 d w a λ 2 ,
F z = f - 2 γ 2 24 η 3 s - 1 s + 1 + C 1 , z 2 s s + 1 d w a + C 2 , z 2 s s + 1 d w a λ 2 ,
g = Re ( 1 i - 1 ) ,
Δ g , x , y , z = Im ( 1 i - 1 ) + C 3 , , x , y , z 2 s + 1 d w λ .
F - 18 s + 1 d w a λ 2 r - 1 2 = f - γ 2 24 η 3 s - 1 s + 1 - 0.716 s + 1 d w a r - 1 2 .
U = f - γ 2 24 η 3 s - 1 s + 1 ,
V = - 0.716 s + 1 d w a .
W = 1 - ( F x + F y + F z ) = ( γ 2 6 η 3 - 1.432 d w a ) s - 1 s + 1 - 14 s + 2 s + 1 d w a λ 2 .
f = U + 1 4 [ W - 2 V ( s - 1 ) + 14 s + 2 s + 1 d w a λ 2 ] .
η = [ γ 2 24 ( f - U ) s - 1 s + 1 ] 1 3 ,
D = [ 6 π a 2 γ d w ] 1 3 ,
( F - 18 s + 1 d w a λ 2 r - 1 2 ) vs r - 3 2
Δ g = Δ g bulk + Δ g size + Δ g interaction + Δ g shape ,