Abstract

A second-order theory is developed for the reflection and transmission of light by a slightly rough surface. The general results are discussed, especially for a rough interface between vacuum and a free-electron-like metal, where surface-plasma excitation can occur. The influence of the damping of the electron gas and of the shape of the roughness is discussed. Some different previous theories for reflection by a rough surface are shown to be limiting cases of the present theory.

© 1975 Optical Society of America

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References

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  1. H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
    [CrossRef]
  2. J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).
  3. A. Daude, A. Savary, and S. Robin, J. Opt. Soc. Am. 62, 1 (1972).
    [CrossRef]
  4. J. G. Endriz and W. E. Spicer, Phys. Rev. B 4, 4144 (1971).
    [CrossRef]
  5. E. Kretschmann, thesis (Universität Hamburg, 1972).
  6. J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1971); J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970); E. Kretschmann, Z. Phys. 237, 1 (1970).
    [CrossRef]
  7. J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
    [CrossRef]
  8. D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
    [CrossRef]
  9. H. E. Bennett, J. Opt. Soc. Am. 51, 123 (1961); H. Davies, Proc. IEEE 101, 209 (1954); P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1973).
    [CrossRef]
  10. E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).
  11. S. O. Rice, Comm. Pure Appl. Math. 4, 351 (1951).
    [CrossRef]
  12. H. J. Juranek, Z. Phys. 233, 324 (1970).
    [CrossRef]
  13. Reference 11, Eq. (7.29).
  14. Equation (22) is valid for complex ∊ also.

1974 (1)

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[CrossRef]

1972 (1)

1971 (2)

J. G. Endriz and W. E. Spicer, Phys. Rev. B 4, 4144 (1971).
[CrossRef]

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1971); J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970); E. Kretschmann, Z. Phys. 237, 1 (1970).
[CrossRef]

1970 (3)

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[CrossRef]

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).

H. J. Juranek, Z. Phys. 233, 324 (1970).
[CrossRef]

1968 (1)

J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).

1963 (1)

1961 (1)

1951 (1)

S. O. Rice, Comm. Pure Appl. Math. 4, 351 (1951).
[CrossRef]

Arakawa, E. T.

J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).

Ashley, E. J.

J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).

Beaglehole, D.

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[CrossRef]

Bennett, H. E.

Bennett, J. M.

J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).

Daude, A.

Elson, J. M.

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[CrossRef]

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1971); J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970); E. Kretschmann, Z. Phys. 237, 1 (1970).
[CrossRef]

Endriz, J. G.

J. G. Endriz and W. E. Spicer, Phys. Rev. B 4, 4144 (1971).
[CrossRef]

Hunderi, O.

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[CrossRef]

Juranek, H. J.

H. J. Juranek, Z. Phys. 233, 324 (1970).
[CrossRef]

Kretschmann, E.

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).

E. Kretschmann, thesis (Universität Hamburg, 1972).

Kröger, E.

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).

Rice, S. O.

S. O. Rice, Comm. Pure Appl. Math. 4, 351 (1951).
[CrossRef]

Ritchie, R. H.

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[CrossRef]

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1971); J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970); E. Kretschmann, Z. Phys. 237, 1 (1970).
[CrossRef]

Robin, S.

Savary, A.

Spicer, W. E.

J. G. Endriz and W. E. Spicer, Phys. Rev. B 4, 4144 (1971).
[CrossRef]

Stanford, J. L.

J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).

Bull. Am. Phys. Soc. (1)

J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968).

Comm. Pure Appl. Math. (1)

S. O. Rice, Comm. Pure Appl. Math. 4, 351 (1951).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Rev. B (3)

D. Beaglehole and O. Hunderi, Phys. Rev. B 2, 309 (1970).
[CrossRef]

J. M. Elson and R. H. Ritchie, Phys. Rev. B 4, 4129 (1971); J. Crowell and R. H. Ritchie, J. Opt. Soc. Am. 60, 794 (1970); E. Kretschmann, Z. Phys. 237, 1 (1970).
[CrossRef]

J. G. Endriz and W. E. Spicer, Phys. Rev. B 4, 4144 (1971).
[CrossRef]

Phys. Status Solidi B (1)

J. M. Elson and R. H. Ritchie, Phys. Status Solidi B 62, 461 (1974).
[CrossRef]

Z. Phys. (2)

E. Kröger and E. Kretschmann, Z. Phys. 237, 1 (1970).

H. J. Juranek, Z. Phys. 233, 324 (1970).
[CrossRef]

Other (3)

Reference 11, Eq. (7.29).

Equation (22) is valid for complex ∊ also.

E. Kretschmann, thesis (Universität Hamburg, 1972).

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Figures (6)

FIG. 1
FIG. 1

Geometry of the rough surface.

FIG. 2
FIG. 2

Reflectance R0 and transmittance T0 at the smooth interface vacuum–metal (free-electron-like) for different damping constants γ.

FIG. 3
FIG. 3

(a) Surface-roughness function k · g(k) for various correlation lengths. (b) Dispersion curve of surface-plasma waves at a free-electron-like metal; the curve denotes the maximum of Re[w(k)]. (c) Real and imaginary part of w(k) for λ / λ p = 1.6 and γ = 0.04.

FIG. 4
FIG. 4

Relative change of reflectance −ΔR/R0 = −(RR0)/R0 from a rough-surface vacuum–free-electron-like metal (γ = 0.04) for various correlation lengths.

FIG. 5
FIG. 5

Relative decrease of reflectance for various damping constants of the metal ( 2 π σ / λ p = 2 ). The curve for γ = 0 is equal to Δ from Eq. (24). Dashed line - - -: (I/I0)sp from Eq. (23); dashed line — —: scalar limit Eq. (26) (independent of ∊).

FIG. 6
FIG. 6

Relative change of transmittance. Same parameters as in Fig. 4. σ = ∞ denotes the scalar limit [Eq. (27)].

Equations (35)

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n = ( S / x , S / y , 1 ) .
S av = 1 F F d 2 x S ( x ) = 0 ,
s ( k ) = 1 ( 2 π ) 2 d 2 x S ( x ) e i k x , k = ( k x , k y ) .
S 2 av = 1 F · F d 2 x · [ S ( x ) ] 2 ,
G ( x ) = 1 S 2 F · F d 2 x S ( x ) · S ( x + x ) ,
g ( k ) = ( 2 π ) 2 1 S 2 F | s ( k ) | 2 .
f ( k ) av = d 2 k · d 2 k 0 · s ( k k ) · s ( k k 0 ) · a ( k ) · a ( k 0 ) av
f ( k ) av = S 2 av · b ( k ) · d 2 k g ( k k ) · a ( k ) .
z = S ( x ) , Δ E × n = 0 , Δ H × n = 0 .
Δ E x ( S ) = Δ E z ( S ) · S x , Δ E y ( S ) = Δ E z ( S ) · S y , Δ H x ( S ) = Δ H z ( S ) · S x , Δ H y ( S ) = Δ H z ( S ) · S y .
Δ E x ( 0 ) = x Δ 0 S E z ( z ) d z i ω c Δ 0 S H y ( z ) d z , Δ E y ( 0 ) = y Δ 0 S E z ( z ) d z i ω c Δ 0 S H x ( z ) d z , Δ H x ( 0 ) = x Δ 0 S H z ( z ) d z i ω c Δ 0 S D y ( z ) d z , Δ H y ( 0 ) = y Δ 0 S H z ( z ) d z i ω c Δ 0 S D x ( z ) d z .
Δ E x ( 1 ) ( 0 ) = Δ E y ( 1 ) ( 0 ) = Δ H y ( 1 ) ( 0 ) = 0 , Δ H x ( 1 ) ( 0 ) = i · ω c · S D y ( 0 ) ( 0 ) .
Δ e x ( 1 ) ( k , 0 ) = Δ e y ( 1 ) ( k , 0 ) = Δ h y ( 1 ) ( k , 0 ) = 0 , Δ h x ( 1 ) ( k , 0 ) = i · ω c d 2 k 0 · s ( k k 0 ) · d y ( 0 ) ( k 0 , 0 ) .
h 2 p ( 1 ) ( k ) = 2 k 1 1 k 2 + 2 k 1 Δ h x ( 1 ) ( k ) sin ϕ , h 1 p ( 1 ) ( k ) = 1 k 2 2 k 1 h 2 p ( 1 ) ( k ) , e 2 s ( 1 ) ( k ) = ( ω / c ) k 1 + k 2 Δ h x ( 1 ) ( k ) cos ϕ , e 1 s ( 1 ) ( k ) = e 2 s ( 1 ) ( k ) ,
k n = [ n ( ω / c ) 2 k 2 ] 1 / 2 ; Im ( k n ) > 0 ; k = | k | ; ϕ = arc cos ( k x / k ) .
k = k sp ( ω ) = ( ω c ) ( 1 2 1 + 2 ) 1 / 2 .
I sc = c 8 π Re d 2 x ( ( E 2 ( 1 ) × H 2 ( 1 ) * ) z + ( E 1 ( 1 ) × H 1 ( 1 ) * ) z ) = c 8 π ( 2 π ) 2 Re d 2 k ( k 2 2 | h 2 p ( 1 ) | 2 + k 2 | e 2 s ( 1 ) | 2 + k 1 1 | h 1 p ( 1 ) | 2 + k 1 | e 1 s ( 1 ) | 2 ) .
I sc I 0 = 4 2 | 2 1 | 2 ( ω c ) 2 S 2 av Re ( Q )
Q = d 2 k g ( k ) w ( k ) , w ( k ) = w p ( k ) sin 2 ϕ + w s ( k ) cos 2 ϕ , w p ( k ) = k 1 k 2 ( c / ω ) 1 k 2 + 2 k 1 , w s ( k ) = ( w / c ) k 2 + k 1 .
Q = d k k g ( k ) π w ( k ) , w ( k ) = w p ( k ) + w s ( k ) .
Δ E x ( 2 ) = 0 , Δ E y ( 2 ) = i ω c ( S Δ H x ( 1 ) + 1 2 S 2 Δ z H x ( 0 ) ) = 1 2 ( ω c ) 2 S 2 Δ D y ( 0 ) , Δ H x ( 2 ) = i ω c ( S Δ D y ( 1 ) + 1 2 S 2 Δ z D y ( 0 ) ) , Δ H y ( 2 ) = i ω c S Δ D x ( 1 ) .
Δ d y ( 1 ) ( k ) = ( 2 1 ) w ( k ) c ω Δ h x ( 1 ) ( k ) ,
Δ e x ( 2 ) av = Δ h y ( 2 ) av = 0 , Δ e y ( 2 ) av = 1 2 ( ω / c ) 2 S 2 Δ d y ( 0 ) , Δ h x ( 2 ) av = 1 2 ( ω / c ) 2 S 2 { 1 + 2 ( 2 1 ) · Q } Δ d y ( 0 ) ,
r 0 = 2 1 2 + 1 , t 0 = 2 2 2 + 1
r = r 0 [ 1 2 ( ω / c ) 2 S 2 2 { 1 + ( 2 1 ) Q } ] , t = t 0 [ 1 + ( ω / c ) 2 S 2 ( 2 1 ) 2 { 1 2 ( 1 + 2 ) Q } ] ,
Δ R R 0 = ( | r | 2 | r 0 | 2 ) | r 0 | 2 , Δ T T 0 = ( | t | 2 | t 0 | 2 ) | t 0 | 2 , Δ R R 0 = 4 ( ω c ) 2 S 2 Re [ 2 { 1 + ( 2 1 ) Q } ] , Δ T T 0 = ( ω c ) 2 S 2 Re [ ( 2 1 ) 2 · { 1 2 ( 1 + 2 ) Q } ] .
d 2 k = k · d k · d ϕ = ( ω / c ) 2 sin θ cos θ d θ d ϕ = ( ω / c ) 2 cos θ d Ω ,
( d I I 0 d Ω ) light = 4 ( ω c ) 4 g ( ω c sin θ ) | 1 | 2 cos 2 θ S 2 × { | sin 2 θ | sin 2 ϕ | cos θ + ( sin 2 θ ) 1 / 2 | 2 + cos 2 ϕ | cos θ + ( sin 2 θ ) 1 / 2 | 2 } .
( I I 0 ) sp = ( 2 π ) 2 ( ω c ) 4 S 2 2 ( 1 ) 5 / 2 g ( k sp ) .
Δ R R 0 = Δ R = d Ω ( d I I 0 d Ω ) light ( I I 0 ) sp .
Q = w ( k = 0 ) = 1 1 + 2 .
Δ R R 0 = 4 ( ω c ) 2 S 2 ,
Δ T T 0 = ( ω c ) 2 S 2 Re ( 1 ) 2 .
1 = 1 ( λ λ p ) 2 1 1 + i γ ( λ / λ p ) ,
g ( k ) = g ( k ) = 1 4 π σ 2 · exp ( σ 2 k 2 / 4 ) .