Abstract

A simple comparison with predictions from a scattering surface model shows discrepancies in treatments of thin island films that are based on the effective-medium model. A new approach is thus proposed, and the transmission of a square network of spherical particles on a dielectric substrate is recalculated in a dipolar approximation. Although a convenient multipolar treatment, which would permit retardation-related effects to be handled in a proper way, is still missing, the present partially retarded treatment shows a clear improvement over available predictions.

© 1992 Optical Society of America

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  1. H. G. Craighead, G. A. Niklasson, “Characterization and optical properties of arrays of small gold particles,” Appl. Phys. Lett. 44, 1134–1136 (1984).
    [CrossRef]
  2. G. A. Niklasson, H. G. Craighead, “Optical response and fabrication of regular arrays of ultrasmall gold particles,” Thin Solid Films 125, 165–170 (1985).
    [CrossRef]
  3. T. Yamaguchi, S. Yoshida, A. Kinbara, “Optical effect of the substrate on the anomalous absorption of aggregated silver films,” Thin Solid Films 21, 173–187 (1984).
    [CrossRef]
  4. G. Bosi, B. de Dormale, “Substrate related effects on the optical behavior of a granular surface: the Maxwell Garnett theory revisited,” J. Appl. Phys. 58, 513–517 (1985).
    [CrossRef]
  5. M. M. Wind, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate I,” Physica (Amsterdam) 141A, 33–57 (1987).
  6. M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).
  7. P. A. Bobbert, J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica (Amsterdam) 147A, 115–141 (1987).
  8. V.-V. Truong, G. Bosi, T. Yamaguchi, “Optical behavior of granular metal films: single-image versus multiple-image approaches in the treatment of substrate effects,” J. Opt. Soc. Am. A 5, 1379–1381 (1988).
    [CrossRef]
  9. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Amsterdam) 137A, 209–242 (1986).
  10. P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica (Amsterdam) 137A, 243–257 (1986).
  11. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed., A. Jeffrey, ed. (Academic, London, 1980).
  12. G. Bosi, “Retardation-related effects on the optical behavior of a granular surface. I. Dipolar treatment,” J. Appl. Phys. 62, 237–242 (1987).
    [CrossRef]
  13. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6, 4370–4379 (1972).
    [CrossRef]
  14. S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
    [CrossRef]
  15. R. Ruppin, “Surface modes and optical absorption of a small sphere above a substrate,” Surf. Sci. 127, 108–118 (1983).
    [CrossRef]

1988 (1)

1987 (4)

M. M. Wind, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate I,” Physica (Amsterdam) 141A, 33–57 (1987).

M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).

P. A. Bobbert, J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica (Amsterdam) 147A, 115–141 (1987).

G. Bosi, “Retardation-related effects on the optical behavior of a granular surface. I. Dipolar treatment,” J. Appl. Phys. 62, 237–242 (1987).
[CrossRef]

1986 (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Amsterdam) 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica (Amsterdam) 137A, 243–257 (1986).

1985 (2)

G. A. Niklasson, H. G. Craighead, “Optical response and fabrication of regular arrays of ultrasmall gold particles,” Thin Solid Films 125, 165–170 (1985).
[CrossRef]

G. Bosi, B. de Dormale, “Substrate related effects on the optical behavior of a granular surface: the Maxwell Garnett theory revisited,” J. Appl. Phys. 58, 513–517 (1985).
[CrossRef]

1984 (2)

H. G. Craighead, G. A. Niklasson, “Characterization and optical properties of arrays of small gold particles,” Appl. Phys. Lett. 44, 1134–1136 (1984).
[CrossRef]

T. Yamaguchi, S. Yoshida, A. Kinbara, “Optical effect of the substrate on the anomalous absorption of aggregated silver films,” Thin Solid Films 21, 173–187 (1984).
[CrossRef]

1983 (1)

R. Ruppin, “Surface modes and optical absorption of a small sphere above a substrate,” Surf. Sci. 127, 108–118 (1983).
[CrossRef]

1978 (1)

S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
[CrossRef]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6, 4370–4379 (1972).
[CrossRef]

Anderson, T.

S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
[CrossRef]

Bedeaux, D.

M. M. Wind, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate I,” Physica (Amsterdam) 141A, 33–57 (1987).

M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica (Amsterdam) 147A, 115–141 (1987).

M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Amsterdam) 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica (Amsterdam) 137A, 243–257 (1986).

Bosi, G.

V.-V. Truong, G. Bosi, T. Yamaguchi, “Optical behavior of granular metal films: single-image versus multiple-image approaches in the treatment of substrate effects,” J. Opt. Soc. Am. A 5, 1379–1381 (1988).
[CrossRef]

G. Bosi, “Retardation-related effects on the optical behavior of a granular surface. I. Dipolar treatment,” J. Appl. Phys. 62, 237–242 (1987).
[CrossRef]

G. Bosi, B. de Dormale, “Substrate related effects on the optical behavior of a granular surface: the Maxwell Garnett theory revisited,” J. Appl. Phys. 58, 513–517 (1985).
[CrossRef]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6, 4370–4379 (1972).
[CrossRef]

Craighead, H. G.

G. A. Niklasson, H. G. Craighead, “Optical response and fabrication of regular arrays of ultrasmall gold particles,” Thin Solid Films 125, 165–170 (1985).
[CrossRef]

H. G. Craighead, G. A. Niklasson, “Characterization and optical properties of arrays of small gold particles,” Appl. Phys. Lett. 44, 1134–1136 (1984).
[CrossRef]

de Dormale, B.

G. Bosi, B. de Dormale, “Substrate related effects on the optical behavior of a granular surface: the Maxwell Garnett theory revisited,” J. Appl. Phys. 58, 513–517 (1985).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed., A. Jeffrey, ed. (Academic, London, 1980).

Granqvist, C. G.

S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
[CrossRef]

Greef, R.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica (Amsterdam) 137A, 243–257 (1986).

Hunderi, O.

S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6, 4370–4379 (1972).
[CrossRef]

Kinbara, A.

T. Yamaguchi, S. Yoshida, A. Kinbara, “Optical effect of the substrate on the anomalous absorption of aggregated silver films,” Thin Solid Films 21, 173–187 (1984).
[CrossRef]

Niklasson, G. A.

G. A. Niklasson, H. G. Craighead, “Optical response and fabrication of regular arrays of ultrasmall gold particles,” Thin Solid Films 125, 165–170 (1985).
[CrossRef]

H. G. Craighead, G. A. Niklasson, “Characterization and optical properties of arrays of small gold particles,” Appl. Phys. Lett. 44, 1134–1136 (1984).
[CrossRef]

Norrman, S.

S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
[CrossRef]

Ruppin, R.

R. Ruppin, “Surface modes and optical absorption of a small sphere above a substrate,” Surf. Sci. 127, 108–118 (1983).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed., A. Jeffrey, ed. (Academic, London, 1980).

Truong, V.-V.

Vlieger, J.

M. M. Wind, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate I,” Physica (Amsterdam) 141A, 33–57 (1987).

P. A. Bobbert, J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica (Amsterdam) 147A, 115–141 (1987).

M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica (Amsterdam) 137A, 243–257 (1986).

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Amsterdam) 137A, 209–242 (1986).

Wind, M. M.

M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).

M. M. Wind, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate I,” Physica (Amsterdam) 141A, 33–57 (1987).

Yamaguchi, T.

V.-V. Truong, G. Bosi, T. Yamaguchi, “Optical behavior of granular metal films: single-image versus multiple-image approaches in the treatment of substrate effects,” J. Opt. Soc. Am. A 5, 1379–1381 (1988).
[CrossRef]

T. Yamaguchi, S. Yoshida, A. Kinbara, “Optical effect of the substrate on the anomalous absorption of aggregated silver films,” Thin Solid Films 21, 173–187 (1984).
[CrossRef]

Yoshida, S.

T. Yamaguchi, S. Yoshida, A. Kinbara, “Optical effect of the substrate on the anomalous absorption of aggregated silver films,” Thin Solid Films 21, 173–187 (1984).
[CrossRef]

Appl. Phys. Lett. (1)

H. G. Craighead, G. A. Niklasson, “Characterization and optical properties of arrays of small gold particles,” Appl. Phys. Lett. 44, 1134–1136 (1984).
[CrossRef]

J. Appl. Phys. (2)

G. Bosi, “Retardation-related effects on the optical behavior of a granular surface. I. Dipolar treatment,” J. Appl. Phys. 62, 237–242 (1987).
[CrossRef]

G. Bosi, B. de Dormale, “Substrate related effects on the optical behavior of a granular surface: the Maxwell Garnett theory revisited,” J. Appl. Phys. 58, 513–517 (1985).
[CrossRef]

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

Phys. Rev. (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. B (1)

S. Norrman, T. Anderson, C. G. Granqvist, O. Hunderi, “Optical properties of discontinuous gold films,” Phys. Rev. B 18, 674–695 (1978).
[CrossRef]

Physica (Amsterdam) (5)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Amsterdam) 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres—comparison with an ellipsometric experiment,” Physica (Amsterdam) 137A, 243–257 (1986).

M. M. Wind, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate I,” Physica (Amsterdam) 141A, 33–57 (1987).

M. M. Wind, P. A. Bobbert, J. Vlieger, D. Bedeaux, “The polarizability of a truncated sphere on a substrate II,” Physica (Amsterdam) 143A, 164–182 (1987).

P. A. Bobbert, J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica (Amsterdam) 147A, 115–141 (1987).

Surf. Sci. (1)

R. Ruppin, “Surface modes and optical absorption of a small sphere above a substrate,” Surf. Sci. 127, 108–118 (1983).
[CrossRef]

Thin Solid Films (2)

G. A. Niklasson, H. G. Craighead, “Optical response and fabrication of regular arrays of ultrasmall gold particles,” Thin Solid Films 125, 165–170 (1985).
[CrossRef]

T. Yamaguchi, S. Yoshida, A. Kinbara, “Optical effect of the substrate on the anomalous absorption of aggregated silver films,” Thin Solid Films 21, 173–187 (1984).
[CrossRef]

Other (1)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed., A. Jeffrey, ed. (Academic, London, 1980).

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Figures (3)

Fig. 1
Fig. 1

Spherical grains on a flat substrate and related images. Also shown are the incident, reflected, and refracted wave vectors.

Fig. 2
Fig. 2

Square lattice oriented at an angle ψ relative to the parallel component (k||) of the incident wave vector.

Fig. 3
Fig. 3

Transmittance (at normal incidence) of a square lattice of gold particles on a sapphire substrate: curve a, experimental1,2; curve b, spherical particles (diameter 29.5 nm, lattice constant 50 nm), dipolar approximation, scattering-surface approach (filled circles represent calculated points); curve c, spherical particles (diameter 32 nm, lattice constant 50 nm), dipolar approximation, effective-medium approach.8

Equations (50)

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p = η E loc ,
e ( r ) = i ( ω 3 / v 3 ) S ( ω r / v ) · p ,
S ( ω r / v ) = I h 0 ( ω r / v ) + ( rr / r 2 - I ) h 2 ( ω r / v ) ω 0 - i ( v 3 / ω 3 ) ( 3 rr / r 5 - I / r 3 ) ,
p = η [ 1 - i η ( ω 3 / v 3 ) j S ( ω r j / v ) ] - 1 E 0 ,
eff = [ 1 + 4 π η N v 1 - i η ( ω 3 / v 3 ) j S ( ω r j / v ) ] ,
T = | 1 - 2 π i ( ω / v ) η N s 1 - i η ( ω 3 / v 3 ) j S ( ω r j / v ) | - 2 ,
E ( z ) = E 0 exp ( - i ω z / v ) + N S e ( ρ ) d 2 ρ = [ E 0 + 2 π i ( ω / v ) N S / p ] exp ( - i ω z / v )             ( z < 0 ) .
T = 1 + 2 π i ( ω / v ) η N S / [ 1 - i η ( ω 3 / v 3 ) j S ( ω r j / v ) ] 2 ,
η = 3 2 i v 3 ω 3 { v v 0 j 1 ( ω b v 0 ) [ 2 j 0 ( ω b v ) - j 2 ( ω b v ) ] - j 1 ( ω b v ) [ 2 j 0 ( ω b v 0 ) - j 2 ( ω b v 0 ) ] } / { v v 0 j 1 ( ω b v 0 ) [ 2 h 0 ( ω b v ) - h 2 ( ω b v ) ] - h 1 ( ω b v 0 ) [ 2 j 0 ( ω b v 0 ) - j 2 ( ω b v 0 ) ] } ,
p m = η m E loc ( r m ) .
E loc ( r m ) = [ E in 0 exp ( i k in · r m ) + E re 0 exp ( i k re · r m ) ] + j m S [ ω ( r m - r j ) / v ] · p j + j S [ ω ( r m - r j + 2 b j z ^ ) / v ] · P j ,
k in = k - k z ^ ,             k re = k + k z ^ ,
p j = p exp ( i k · r j ) ,             P j = P exp ( i k · r j ) ,
[ I - η j 0 exp ( i k · r j ) S ( ω r j / v ) ] · p - η j × exp ( i k · r j ) S [ ω ( r j - 2 b z ^ ) / v ] · P = η ( E in 0 + E re 0 ) .
r j = a ( m cos ψ - n sin ψ , m sin ψ + n cos ψ , 0 ) ,
H = j exp ( i k · r j ) S [ ω ( r j - 2 b z ^ ) / v ] ,
H 11 - H 22 = - m n ( m 2 + n 2 ) ( m 2 + n 2 + 4 b 2 / a 2 ) - 1 × J 2 [ k a ( m 2 + n 2 ) 1 / 2 ] h 2 [ ( ω a / v ) × ( m 2 + n 2 + 4 b 2 / a 2 ) 1 / 2 ] ,
H 11 + H 22 + H 33 = 2 m n J 0 [ k a ( m 2 + n 2 ) 1 / 2 ] × h 0 [ ( ω a / v ) ( m 2 + n 2 + 4 b 2 / a 2 ) 1 / 2 ] ,
H 11 + H 22 - 2 H 33 = m n ( m 2 + n 2 - 8 b 2 / a 2 ) × ( m 2 + n 2 + 4 b 2 / a 2 ) - 1 × J 0 [ k a ( m 2 + n 2 ) 1 / 2 ] h 2 [ ( ω a / v ) × ( m 2 + n 2 + 4 b 2 / a 2 ) 1 / 2 ] ,
H 13 = - 2 i ( b / a ) m n ( m 2 + n 2 ) 1 / 2 × ( m 2 + n 2 + 4 b 2 / a 2 ) - 1 × J 1 [ k a ( m 2 + n 2 ) 1 / 2 ] h 2 [ ( ω a / v ) × ( m 2 + n 2 + 4 b 2 / a 2 ) 1 / 2 ] ,
H 12 = H 23 = 0 ,
J m ( x ) = i - m π 0 π exp ( i x cos φ ) cos ( m φ ) d φ
H = j 0 exp ( i k · r j ) S ( ω r j / v ) ,
H 11 - H 22 = - m n J 2 [ k a ( m 2 + n 2 ) 1 / 2 ] h 2 [ ( ω a / v ) × ( m 2 + n 2 ) 1 / 2 ] ,
H 11 + H 22 + H 33 = 2 m n J 0 [ k a ( m 2 + n 2 ) 1 / 2 ] h 0 [ ( ω a / v ) × ( m 2 + n 2 ) 1 / 2 ] ,
H 11 + H 22 - 2 H 33 = m n J 0 [ k a ( m 2 + n 2 ) 1 / 2 ] h 2 [ ( ω a / v ) × ( m 2 + n 2 ) 1 / 2 ] ,
H 12 = H 13 = H 23 = 0.
[ I - i ( ω 3 / v 3 ) η H ] · p - i ( ω 3 / v 3 ) η H · P = η ( E in 0 + E re 0 ) ,
H 11 = H 22 = - ½ H 33 = - ½ ( i / a 3 ) ( v 3 / ω 3 ) × [ F ( 2 b / a ) - ¼ ( a 3 / b 3 ) ] ,
H 11 = H 22 = - ½ H 33 = - ½ ( i / a 3 ) ( v 3 / ω 3 ) F ( 0 ) ,
F ( x ) = m n [ ( m 2 + n 2 + x 2 ) - 3 / 2 - 3 x 2 ( m 2 + n 2 + x 2 ) - 5 / 2 ]
e ( r ) = i ( ω 3 / v 3 ) S ( ω r / v ) · p ,
e ( r ) = i ( ω 3 / v 3 ) { S ( ω r / v ) · p + S [ ω ( r + 2 b z ^ ) / v ] · P }
h m ( ω r / v ) = ( v / v ) m k = 0 2 - k ( 1 / k ! ) ( 1 - v 2 / v 2 ) k × ( ω r / v ) k h m + k ( ω r / v ) ,
P = [ ( - ) / ( + ) ] ( - p + p z z ^ ) ,
p = [ 2 / ( + ) ] p ,
{ I - i ( ω 3 / v 3 ) η [ H - H · ( x x ^ + y ^ y ^ - z ^ z ^ ) ( - ) / ( + ) ] } · p = η ( E in 0 + E re 0 ) ,
E ( r ) = i N S ( ω 3 / v 3 ) exp ( i k · r ) × { S [ ω ( - ρ + z z ^ ) / v ] exp ( i k · ρ ) d 2 ρ } · p ,
0 x ν + 1 ( x 2 + y 2 ) - n / 2 J ν ( u x ) h n [ v ( x 2 + y 2 ) 1 / 2 ] d x = u ν v - ( n + 1 ) ( v 2 - u 2 ) ( n - v ) / 2 y ν - n + 1 h n - ν - 1 [ y ( v 2 - u 2 ) 1 / 2 ] .
E ( r ) = 2 π i N ( ω / v ) M ( θ t ) · p exp [ i ( k - k z ^ ) · r ] ,
M ( θ ) = [ cos θ 0 sin θ 0 1 / cos θ 0 sin θ 0 sin θ tan θ ] ,
E tr = E tr 0 exp [ i ( k - k z ^ ) · r ] ,
E tr 0 exp ( i k b ) = t · E in 0 exp ( i k b ) ,
t = [ 2 cos θ t / ( cos θ i + cos θ t ) ] x ^ x ^ + [ 2 cos θ i / ( cos θ i + cos θ t ) ] y ^ y ^ + [ 2 ( / ) cos θ i / ( cos θ i + cos θ t ) ] z ^ z ^
E re 0 exp ( - i k b ) = r · E in 0 exp ( i k b ) ,
r = [ ( cos θ i - cos θ t ) / ( cos θ i + cos θ t ) ] ( - x ^ x ^ + z ^ z ^ ) + [ cos θ i - cos θ t ) / cos θ i + cos θ t ) ] y ^ y ^ .
T = E tr 0 + E 0 2 / E tr 0 2 ,
E 0 = 2 π i ( N / ) ( ω / v ) M ( θ t ) · p
( ω ) = B ( ω ) + ω P 2 ω 2 + i ω / τ B - ω P 2 ω 2 + i ω / τ ,
τ - 1 = τ B - 1 + v F / b ,

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