Abstract

In this theoretical paper the formulas expressing the reflectance of randomly rough surfaces are derived within the framework of the scalar theory of diffraction. The Fresnel approximation has been used in the mathematical procedure since geometrical conditions of some real reflectometers do not correspond to the conditions of the Fraunhofer diffraction of light. The formulas found are generally different from the formulas derived within the framework of the Fraunhofer approximation. By means of the numerical analysis of these formulas the differences between both the Fresnel and the Fraunhofer approximations are shown. It is also found that the formulas corresponding to the Fresnel approximation give the same results as the formulas derived by means of the Fraunhofer approximation if certain geometrical conditions are valid. These geometrical conditions are determined in this paper. The formulas corresponding to thin films with randomly rough boundaries are also introduced.

© 1980 Optical Society of America

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References

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  1. H. E. Bennett, J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [CrossRef]
  2. H. E. Bennett, J. Opt. Soc. Am. 53, 1389 (1963).
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  3. J. O. Porteus, J. Opt. Soc. Am. 53, 1394 (1963).
    [CrossRef]
  4. I. Ohlidal, K. Navratil, F. Lukes, J. Opt. Soc. Am. 61, 1630 (1971).
    [CrossRef]
  5. D. H. Hensler, Appl. Opt. 11, 2522 (1972).
    [CrossRef] [PubMed]
  6. J. Bauer, Phys. Status Solidi A: 39, 411 (1977).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  8. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  9. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  10. I. Ohlidal, F. Lukes, K. Navratil, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. 2, 83 (1975).
  11. H. Cramer, Mathematical Methods of Statistics (Princeton U. P., Princeton, 1946).
  12. I. Ohlidal, F. Lukes, K. Navratil, J. Phys. Paris 38, C5 (1977).
    [CrossRef]
  13. A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960).
  14. I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
    [CrossRef]
  15. I. Ohlidal, K. Navratil, F. Lukes, Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. 16, 1 (1974).
  16. I. Ohlidal, Ph.D. Thesis, Brno (1977) (in Czech).
  17. H. R. Philipp, E. A. Taft, Phys. Rev. 120, 37 (1960).
    [CrossRef]

1977 (2)

I. Ohlidal, F. Lukes, K. Navratil, J. Phys. Paris 38, C5 (1977).
[CrossRef]

J. Bauer, Phys. Status Solidi A: 39, 411 (1977).
[CrossRef]

1975 (1)

I. Ohlidal, F. Lukes, K. Navratil, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. 2, 83 (1975).

1974 (2)

I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

I. Ohlidal, K. Navratil, F. Lukes, Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. 16, 1 (1974).

1972 (1)

1971 (1)

1963 (2)

1961 (1)

1960 (1)

H. R. Philipp, E. A. Taft, Phys. Rev. 120, 37 (1960).
[CrossRef]

Bauer, J.

J. Bauer, Phys. Status Solidi A: 39, 411 (1977).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Bennett, H. E.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton U. P., Princeton, 1946).

Hensler, D. H.

Lukes, F.

I. Ohlidal, F. Lukes, K. Navratil, J. Phys. Paris 38, C5 (1977).
[CrossRef]

I. Ohlidal, F. Lukes, K. Navratil, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. 2, 83 (1975).

I. Ohlidal, K. Navratil, F. Lukes, Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. 16, 1 (1974).

I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

I. Ohlidal, K. Navratil, F. Lukes, J. Opt. Soc. Am. 61, 1630 (1971).
[CrossRef]

Navratil, K.

I. Ohlidal, F. Lukes, K. Navratil, J. Phys. Paris 38, C5 (1977).
[CrossRef]

I. Ohlidal, F. Lukes, K. Navratil, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. 2, 83 (1975).

I. Ohlidal, K. Navratil, F. Lukes, Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. 16, 1 (1974).

I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

I. Ohlidal, K. Navratil, F. Lukes, J. Opt. Soc. Am. 61, 1630 (1971).
[CrossRef]

Ohlidal, I.

I. Ohlidal, F. Lukes, K. Navratil, J. Phys. Paris 38, C5 (1977).
[CrossRef]

I. Ohlidal, F. Lukes, K. Navratil, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. 2, 83 (1975).

I. Ohlidal, K. Navratil, F. Lukes, Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. 16, 1 (1974).

I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

I. Ohlidal, K. Navratil, F. Lukes, J. Opt. Soc. Am. 61, 1630 (1971).
[CrossRef]

I. Ohlidal, Ph.D. Thesis, Brno (1977) (in Czech).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Philipp, H. R.

H. R. Philipp, E. A. Taft, Phys. Rev. 120, 37 (1960).
[CrossRef]

Porteus, J. O.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Taft, E. A.

H. R. Philipp, E. A. Taft, Phys. Rev. 120, 37 (1960).
[CrossRef]

Vasicek, A.

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Appl. Opt. (1)

Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. (1)

I. Ohlidal, K. Navratil, F. Lukes, Folia Fac. Sci. Nat. Univ. Purkynianae Brun. XV, Phys. 16, 1 (1974).

J. Opt. Soc. Am. (4)

J. Phys. Paris (1)

I. Ohlidal, F. Lukes, K. Navratil, J. Phys. Paris 38, C5 (1977).
[CrossRef]

Phys. Rev. (1)

H. R. Philipp, E. A. Taft, Phys. Rev. 120, 37 (1960).
[CrossRef]

Phys. Status Solidi A: (1)

J. Bauer, Phys. Status Solidi A: 39, 411 (1977).
[CrossRef]

Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. (1)

I. Ohlidal, F. Lukes, K. Navratil, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun. Phys. 2, 83 (1975).

Surf. Sci. (1)

I. Ohlidal, F. Lukes, K. Navratil, Surf. Sci. 45, 91 (1974).
[CrossRef]

Other (6)

I. Ohlidal, Ph.D. Thesis, Brno (1977) (in Czech).

H. Cramer, Mathematical Methods of Statistics (Princeton U. P., Princeton, 1946).

A. Vasicek, Optics of Thin Films (North-Holland, Amsterdam, 1960).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (8)

Fig. 1
Fig. 1

Geometrical representation of the experimental conditions: k1 and/or k2 is the wave vector of incident and/or scattered wave, Θ2 and Θ3 are the angles characterizing the geometry of light scattering, D is the area of the detector (a circle), M is the mean plane of the randomly rough surface, α0 is the acceptance angle of the detector, R0 is the distance between M and D, r0 is the distance between a point lying in M and a point lying in D, and 2X and/or 2Y is the linear dimensions of the irradiated area.

Fig. 2
Fig. 2

Geometrical representation of a randomly rough surface: ζ(x,y) is a random function describing the surface mathematically, n is the local normal of the surface, r is the distance between a point lying in D and a point lying in the surface, n0 and/or n is the refractive index of the ambient and/or the substrate.

Fig. 3
Fig. 3

Relative incoherent reflectance as a function of σ/λ. Curve 1, Fraunhofer approximation; curves 2,3,4, Fresnel approximation (2: L/R0 = 0.01; 3: L/R0 = 0.02; 4: L/R0 = 0.033). In all cases it is valid that α0 = 0.04 rad; 1 T / λ 20. For example, σ = 0.03 μm and T = 3 μm and 0.15 μm ≤ λ ≲ 3 μm.

Fig. 4
Fig. 4

Relative total reflectance ρT as a function of σ/λ. Curves 14 correspond to curves 14 in Fig. 3. Curve 5 denotes the relative coherent reflectance ρC corresponding to curves 14.

Fig. 5
Fig. 5

Relative incoherent reflectance as a function of σ/λ. Curves 1a, 2a, 3b, and/or 1b, 2b, 3b correspond to the Fresnel and/or Fraunhofer approximation (1 − α0 = 0.01 rad; 2 − α0 = 0.04 rad; 3 − α0 = 0.06 rad). In all cases it is valid that L/R0 = 0.02; 1 ≲ T/λ ≤ 20. For example, σ = 0.03 μm and T = 3 μm and 0.15 μm ≤ λ ≲ 3 μm.

Fig. 6
Fig. 6

Relative incoherent reflectance as a function of σ/λ. Curves 1a, 2a, 3a and/or 1b, 2b, 3b correspond to the Fresnel and/or Fraunhofer approximation (1 − 1 ≲ T/λ ≤ 6,7,2 − 1 ≲ T/λ ≤ 20,3 − 1 ≲ T/λ ≤ 40). In all cases it is valid that α0 = 0.04 rad; L/R0 = 0.02. For example, σ = 0.03 μm; 1 − T = 1 μm; 2 − T = 3 μm; 3 − T = 6 μm; and 0.15 μm ≤ λ ≲ λmax, where 1 − λmax = 1 μm, 2 − λmax = 3 μm, 3 − λmax = 6 μm.

Fig. 7
Fig. 7

Relative incoherent reflectance as a function of λ. Curves 1a, 2a and/or 1b, 2b correspond to the Fresnel and/or Fraunhofer approximation. 1 − σ = 0.03 μm, 2 − σ = 0.05. In all cases it is valid that T = 3 μm; α0 = 0.04 rad; L/R0 = 0.02.

Fig. 8
Fig. 8

Spectral dependences of absolute total reflectance RT and absolute incoherent reflectance Ri for the randomly rough surface of silicon, σ = 0.03 μm; T = 3 μm; α0 = 0.04 rad; L/R0 = 0.02.

Equations (49)

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Ê ( P ) = 1 4 π S ( Ê 1 f ̂ n f ̂ Ê 1 n ) dS ,
f ̂ = exp ( i k 2 r ) / r ,
Ê 1 = ( 1 + R ̂ ) A 0 exp ( ik ζ ) Ê 1 n = i ( 1 R ̂ ) k n z A 0 exp ( i k ζ ) ,
r = r 0 R 0 ζ / r 0 ,
X 2 + Y 2 2 λ R 0 1 .
r 0 = R 0 + 1 2 R 0 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] .
X 2 λ R 0 , Y 2 λ R 0 1 ,
[ ( x x 0 ) 2 + ( y y 0 ) 2 ] max / R 0 2 1 ,
Ê ( P ) = ik 2 π R 0 R ̂ A 0 exp ( ik R 0 ) X X Y Y exp ( i 2 k ζ ) × exp { i k 2 R 0 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } dy dx .
D [ Ê ( P ) ] = | Ê ( P ) Ê ( P ) | 2 ,
Ê ( P ) Ê * ( P ) = Ê ( P ) Ê * ( P ) + D [ Ê ( P ) ] .
Ê ( P ) = R ̂ A 0 exp ( ik R 0 ) X ̂ ( k ) for P A , Ê ( P ) = 0 for P A ,
X ̂ ( k ) = exp ( i 2 kz ) w ( z ) dz ,
F c = k 0 4 XY A 0 2 | R ̂ | 2 | X ̂ ( k ) | 2 ,
R c = 0 | X ̂ ( k ) | 2 ,
ρ c = R c / 0 = | X ̂ ( k ) | 2 .
w ( z ) = 1 ( 2 π ) 1 / 2 σ exp ( z 2 / 2 σ 2 ) ,
X ̂ ( k ) = exp ( 8 π 2 σ 2 / λ 2 ) ,
D [ Ê ( P ) ] = x X X X X Y Y Y Y [ X ̂ ( k 1 k ) | X ̂ ( k ) | 2 ] × exp [ ik ( r 0 r 01 ) ] dyd y 1 dxd x 1 ,
x = ( 0 A 0 2 k 2 ) / ( 4 π 2 R 0 2 ) , ζ 1 ζ ( x 1 , y 1 ) , r 01 = r 0 ( x 1 , y 1 ) ,
X ̂ ( k 1 k ) = exp [ i 2 k ( z z 1 ) ] w ( z , z 1 ; τ ) dzd z 1 ,
x x 1 = τ cos φ , y y 1 = τ sin φ .
r 0 r 01 = [ ( x x 0 ) τ cos φ + ( y y 0 ) τ sin φ ] / R 0 .
w ( z , z 1 ; τ ) = w ( z ) δ ( z z 1 ) C ( τ ) + w ( z ) w ( z 1 ) [ 1 C ( τ ) ] ,
C ( τ ) = exp ( τ 2 / T 2 ) ,
D [ Ê ( P ) ] = π x T 2 [ 1 | X ̂ ( k ) | 2 ] X X Y Y × exp { k 2 T 2 4 R 0 2 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } dydx ,
lim T / λ F i = k 0 4 XY 0 A 0 2 ( 1 | X ̂ ( k ) | 2 ) .
a = kT 2 R 0 , b 1 = x 0 k 2 T 2 2 R 0 2 , b 2 = y 0 k 2 T 2 2 R 0 2 .
D [ Ê ( P ) ] = π x T 2 ( 1 | X ̂ ( k ) | 2 ) exp [ k 2 T 2 ( x 0 2 + y 0 2 ) / 4 R 0 2 ] 1 a 2 × aX aX aY aY exp [ ( x 2 + b 1 a x ) ] exp [ ( y 2 + b 2 a y ) ] dydx .
f ( x , y ) = 1 + n = 1 k = 0 n q = 0 n k p = 0 k K ( n , k , p , q ) × ( b 1 a ) n k q ( b 2 a ) k p x n k y k ,
K ( n , k , p , q ) = ( 1 ) n ( q + p ) / 2 2 ( q + 2 ) / 2 ( n k q ) ! ( k p ) ! q ! ! p ! ! .
D [ Ê ( P ) ] = π x T 2 ( 1 | X ̂ ( k ) | 2 ) exp [ k 2 T 2 ( x 0 2 + y 0 2 ) / 4 R 0 2 ] × 4 XY [ 1 + n = 1 k = 0 n q = 0 n k p = 0 k K ̅ ( n , k , p , q ) × a q + p b 1 n k q b 2 k p X n k Y k ] ,
K ̅ ( n , k , p , q ) = [ ( 1 ) ( q + p ) / 2 2 ( q + p ) / 2 ] / [ ( n k q ) ! ( k p ) ! q ! ! × p ! ! ( n k + 1 ) ( k + 1 ) ] .
x 0 = R 0 sin Θ 2 cos Θ 3 , y 0 = R 0 sin Θ 2 cos Θ 3 .
F i = k 0 R 0 2 0 α 0 0 2 π D [ Ê ( P ) ] Θ 2 d Θ 3 d Θ 2 .
R i = 0 { 1 exp [ ( 4 π σ / λ ) 2 ] } { 1 exp [ ( π T λ α 0 ) 2 } + n = 2 k = 0 n q = 0 n k p = 0 k K ̅ 1 ( n , k , p , q ) ( π T R 0 λ ) n X n k Y k × { 1 exp [ ( π T λ α 0 ) 2 ] μ = 0 ( n p q ) / 2 1 μ ! ( π T λ α 0 ) 2 μ } ,
K ̅ 1 ( n , k , p , q ) = ( 1 ) ( p + q ) / 2 2 n / 2 [ ( n k q ) ! ! ( k p ) ! ! p ! ! q ! ! ( n k + 1 ) ( k + 1 ) ] .
R i = R iF + Δ R ,
R iF = 0 { 1 exp [ ( 4 π σ / λ ) 2 ] } { 1 exp [ ( π T λ α 0 ) 2 ] }
R T = R c + R i .
Δ R = 0 π 4 3 ( T λ ) 4 X 2 + Y 2 R 0 2 α 0 2 { 1 exp [ ( 4 π σ / λ ) 2 ] } × exp [ ( π T λ α 0 ) 2 ] .
| Δ R | 0 0.01 ,
( T λ ) 4 X 2 + Y 2 R 0 2 α 0 2 3 × 10 4 .
R i = R iF + 0 exp [ ( 4 π σ / λ ) 2 ] m = 1 n = 2 k = 0 n q = 0 n k p = 0 k × K ̅ 1 ( n , k , p , q ) ( 4 π σ / λ ) 2 m m ! × ( π T R 0 λ m ) n X n k Y k { 1 exp [ ( π T λ m α 0 ) 2 ] × μ = 0 ( n p q ) / 2 1 μ ! ( π T λ m α 0 ) 2 μ } ,
R iF = 0 exp [ ( 4 π σ / λ ) 2 ] m = 1 ( 4 π σ / λ ) 2 m m ! × { 1 exp [ ( π T α 0 λ m ) 2 ] } .
Ê 1 = { 1 + r 1 + ( 1 r 1 2 ) m = 0 ( 1 ) m r 1 m r 2 m + 1 × exp [ i ( m + 1 ) ( 4 π / λ ) n 1 d L ] } exp ( ik ζ 1 ) ,
R i = r 1 2 ρ i ( σ 1 , T 1 ) + m = 0 M r 1 2 m r 2 2 ( m + 1 ) ( 1 r 1 2 ) 2 { | X ̂ 1 ( B m ) | 2 [ 1 | X ̂ 2 ( M m ) | 2 ] × ρ i ( T 2 ) + | X ̂ 2 ( M m ) | 2 [ 1 | X ̂ 1 ( B m ) | 2 ] ρ i ( T 1 ) + [ 1 | X ̂ 2 ( M m ) | 2 ] × [ 1 | X ̂ 1 ( B m ) | 2 ] ρ i ( T ) } + 2 m = 0 M l = m + 1 M ( 1 ) m + l r 1 m + l r 2 m + l + 2 × ( 1 r 1 2 ) 2 cos Δ ml { | X ̂ 1 ( B m ) | | X ̂ 1 ( B l ) | [ 1 | X ̂ 2 ( M m ) | | X ̂ 2 ( M l ) | ] × ρ i ( T 2 ) + | X ̂ 2 ( M m ) | | X ̂ 2 ( M l ) | [ 1 | X ̂ 1 ( B m ) | | X ̂ 1 ( B l ) | ] ρ i ( T 1 ) + [ 1 | X ̂ 2 ( M m ) | | X ̂ 2 ( M l ) | ] [ 1 | X ̂ 1 ( B m ) | | X ̂ 1 ( B l ) | ] ρ i ( T ) } + 2 m = 0 M ( 1 ) m × r 1 m + 1 r 2 m + 1 ( 1 r 1 2 ) cos δ m { | X ̂ 2 ( M m ) | [ 1 | X ̂ 1 ( B m ) | | X ̂ 1 ( k ) | ] ρ i ( T 1 ) } ,
Δ ml = ( m l ) 4 π λ n 1 d ̅ , δ m = ( m + 1 ) 4 π λ n 1 d ̅ , B q = 4 π λ [ ( q + 1 ) n 1 1 ] , M q = ( q + 1 ) 4 π λ , q = m , l | X ̂ j ( s ) | = exp ( s 2 σ j 2 / 2 ) , J = 1 , 2 , s = B q , M q ,
ρ i ( T 1 ) = ρ i ( σ 1 , T 1 ) / Ω , ρ i ( T 2 ) = ρ i ( σ 1 , T 2 ) / Ω , ρ i ( T ) = ρ i ( σ 1 , T ) / Ω , Ω = 1 exp [ ( 4 π σ 1 / λ ) 2 ] , T = T 1 T 2 / ( T 1 2 + T 2 2 ) 1 / 2 ,

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