Abstract

We recall the analytical expression that gives, for a rough surface illuminated at normal incidence, the light scattered in the half-space containing the specular reflection direction. Two cases are studied: the bare substrate and the substrate coated with one transparent layer. It is shown, for this specular direction, that the light scattered from a single layer can be equal to zero (perfect antiscattering) in certain conditions relative to the roughnesses of the two layer interfaces. Data calculation proves that this antiscattering effect occurs in all directions of the half-space of the reflected light. The experimental results are in good agreement with this theoretical analysis for five different dielectric materials. This study brings out most information about the grain of the material, which is responsible for the residual roughness.

© 1986 Optical Society of America

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References

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  1. P. Roche, P. Bousquet, F. Flory, J. Garcin, E. Pelletier, G. Albrand, “Determination of Interface Roughness Cross-Correlation Properties of an Optical Coating from Measurements of the Angular Scattering,” J. Opt. Soc. Am. A 1, 1028 (1984).
    [CrossRef]
  2. P. Roche, E. Pelletier, G. Albrand, “Antiscattering Transparent Monolayers: Theory and Experiment,” J. Opt. Soc. Am. A 1, 1032 (1984).
    [CrossRef]
  3. P. Bousquet, F. Flory, P. Roche, “Scattering from Multilayer Thin Films: Theory and Experiment,” J. Opt. Soc. Am 71, 1115 (1981).
    [CrossRef]
  4. J. M. Elson, J. M. Bennett, “Relation Between the Angular Dependence of Scattering and the Statistical Properties of Optical Surfaces,” J. Opt. Soc. Am. 69, 31 (1979).
    [CrossRef]
  5. E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353 (1976).
    [CrossRef]
  6. J. Garcin, “Diffusion de la lumière: étude expérimentale et représentation théorique des rugosités des interfaces de filtres multidiélectriques,” Doctoral Dissertation, Université d’Aix-Marseille III, Marseille, France (1982), unpublished.
    [PubMed]
  7. L. Névot, “Contribution à l’étude de la réflexion et de la diffusion rasante des rayons X par les surfaces et les couches minces. Influence des interfaces rugueux et diffus,” Doctoral Dissertation 1954, Université de Paris-Sud, Orsay, France (1978).
  8. H. K. Pulker, K. H. Guenther, “Electron Optical Investigation of the Cross-Sectional Structure of Vacuum Deposited Multilayer Systems,” Vak. Technol. 21, 201 (1972).
  9. K. H. Guenther, H. K. Pulker, “Electron Microscopic Investigations of Cross Sections of Optical Thin Films,” Appl. Opt. 15, 2992 (1976).
    [CrossRef] [PubMed]
  10. L. Névot, P. Croce, “Sur l’étude des couches superficielles monoatomiques par réflexion rasante (spéculaire ou diffuse) de rayons X, par la méthod de l’empilement Sandwich,” J, Appl. Cryst. Part 2 8, 304 (1975).
    [CrossRef]

1984 (2)

1981 (1)

P. Bousquet, F. Flory, P. Roche, “Scattering from Multilayer Thin Films: Theory and Experiment,” J. Opt. Soc. Am 71, 1115 (1981).
[CrossRef]

1979 (1)

1976 (2)

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

K. H. Guenther, H. K. Pulker, “Electron Microscopic Investigations of Cross Sections of Optical Thin Films,” Appl. Opt. 15, 2992 (1976).
[CrossRef] [PubMed]

1975 (1)

L. Névot, P. Croce, “Sur l’étude des couches superficielles monoatomiques par réflexion rasante (spéculaire ou diffuse) de rayons X, par la méthod de l’empilement Sandwich,” J, Appl. Cryst. Part 2 8, 304 (1975).
[CrossRef]

1972 (1)

H. K. Pulker, K. H. Guenther, “Electron Optical Investigation of the Cross-Sectional Structure of Vacuum Deposited Multilayer Systems,” Vak. Technol. 21, 201 (1972).

Albrand, G.

Bennett, J. M.

Bousquet, P.

Croce, P.

L. Névot, P. Croce, “Sur l’étude des couches superficielles monoatomiques par réflexion rasante (spéculaire ou diffuse) de rayons X, par la méthod de l’empilement Sandwich,” J, Appl. Cryst. Part 2 8, 304 (1975).
[CrossRef]

Elson, J. M.

Flory, F.

Garcin, J.

P. Roche, P. Bousquet, F. Flory, J. Garcin, E. Pelletier, G. Albrand, “Determination of Interface Roughness Cross-Correlation Properties of an Optical Coating from Measurements of the Angular Scattering,” J. Opt. Soc. Am. A 1, 1028 (1984).
[CrossRef]

J. Garcin, “Diffusion de la lumière: étude expérimentale et représentation théorique des rugosités des interfaces de filtres multidiélectriques,” Doctoral Dissertation, Université d’Aix-Marseille III, Marseille, France (1982), unpublished.
[PubMed]

Guenther, K. H.

K. H. Guenther, H. K. Pulker, “Electron Microscopic Investigations of Cross Sections of Optical Thin Films,” Appl. Opt. 15, 2992 (1976).
[CrossRef] [PubMed]

H. K. Pulker, K. H. Guenther, “Electron Optical Investigation of the Cross-Sectional Structure of Vacuum Deposited Multilayer Systems,” Vak. Technol. 21, 201 (1972).

Névot, L.

L. Névot, P. Croce, “Sur l’étude des couches superficielles monoatomiques par réflexion rasante (spéculaire ou diffuse) de rayons X, par la méthod de l’empilement Sandwich,” J, Appl. Cryst. Part 2 8, 304 (1975).
[CrossRef]

L. Névot, “Contribution à l’étude de la réflexion et de la diffusion rasante des rayons X par les surfaces et les couches minces. Influence des interfaces rugueux et diffus,” Doctoral Dissertation 1954, Université de Paris-Sud, Orsay, France (1978).

Pelletier, E.

Pulker, H. K.

K. H. Guenther, H. K. Pulker, “Electron Microscopic Investigations of Cross Sections of Optical Thin Films,” Appl. Opt. 15, 2992 (1976).
[CrossRef] [PubMed]

H. K. Pulker, K. H. Guenther, “Electron Optical Investigation of the Cross-Sectional Structure of Vacuum Deposited Multilayer Systems,” Vak. Technol. 21, 201 (1972).

Roche, P.

P. Roche, P. Bousquet, F. Flory, J. Garcin, E. Pelletier, G. Albrand, “Determination of Interface Roughness Cross-Correlation Properties of an Optical Coating from Measurements of the Angular Scattering,” J. Opt. Soc. Am. A 1, 1028 (1984).
[CrossRef]

P. Roche, E. Pelletier, G. Albrand, “Antiscattering Transparent Monolayers: Theory and Experiment,” J. Opt. Soc. Am. A 1, 1032 (1984).
[CrossRef]

P. Bousquet, F. Flory, P. Roche, “Scattering from Multilayer Thin Films: Theory and Experiment,” J. Opt. Soc. Am 71, 1115 (1981).
[CrossRef]

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

Vidal, B.

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

Appl. Opt. (1)

J, Appl. Cryst. Part 2 (1)

L. Névot, P. Croce, “Sur l’étude des couches superficielles monoatomiques par réflexion rasante (spéculaire ou diffuse) de rayons X, par la méthod de l’empilement Sandwich,” J, Appl. Cryst. Part 2 8, 304 (1975).
[CrossRef]

J. Opt. Soc. Am (1)

P. Bousquet, F. Flory, P. Roche, “Scattering from Multilayer Thin Films: Theory and Experiment,” J. Opt. Soc. Am 71, 1115 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nouv. Rev. Opt. (1)

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Opt. 7, 353 (1976).
[CrossRef]

Vak. Technol. (1)

H. K. Pulker, K. H. Guenther, “Electron Optical Investigation of the Cross-Sectional Structure of Vacuum Deposited Multilayer Systems,” Vak. Technol. 21, 201 (1972).

Other (2)

J. Garcin, “Diffusion de la lumière: étude expérimentale et représentation théorique des rugosités des interfaces de filtres multidiélectriques,” Doctoral Dissertation, Université d’Aix-Marseille III, Marseille, France (1982), unpublished.
[PubMed]

L. Névot, “Contribution à l’étude de la réflexion et de la diffusion rasante des rayons X par les surfaces et les couches minces. Influence des interfaces rugueux et diffus,” Doctoral Dissertation 1954, Université de Paris-Sud, Orsay, France (1978).

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Figures (11)

Fig. 1
Fig. 1

Notations used for the calculation of the field scattered by a single layer.

Fig. 2
Fig. 2

Variation of the coefficient C(θ) when the energy scattered in the specular reflection direction is equal to zero (arbitrary units): for a λ/2 high-index layer, the ratio σ0/σ1 calculated from Eq. (1) is 0.69; for a λ/4 low-index layer, the ratio σ0/σ1 calculated from Eq. (2) is 0.66. In these calculations, the substrate is transparent and we calculate backscattering (0° < θ < 90°) and forward scattering (90° < θ < 180°).

Fig. 3
Fig. 3

For a λ/2 high-index layer, comparison between the angular scattering curve of a bare substrate and of the same substrate coated with one layer. Influence of the ratio σ0/σ1 on the amount of scattered light (BRDF · cosθ characterizes the scattered energy in a given directions3). In these calculations, the substrate is transparent.

Fig. 4
Fig. 4

For a λ/4 low-index layer, comparison between the angular scattering curve of a bare substrate and of the same substrate coated with one layer. Influence of the ratio σ0/σ1 on the amount of scattered light. In these calculations, the substrate is transparent.

Fig. 5
Fig. 5

Plane section of the angular scattering curve of an absorbent glass, before and after the deposition of a λ/4 layer of Na3(AlF6); D0 = 13.5 ppm. There is no forward scattering (90° < θ < 180°).

Fig. 6
Fig. 6

Plane section of the angular scattering curve of an absorbent glass, before and after the deposition of a λ/4 layer of MgF2; D0 = 29.5 ppm (no forward scattering).

Fig. 7
Fig. 7

Plane section of the angular scattering curve of an absorbent glass, before and after the deposition of a λ/4 layer of SiO2; D0 = 112 ppm (no forward scattering).

Fig. 8
Fig. 8

Variation with time of the angular scattering curve of a Na3(AlF6) layer after return to the atmospheric pressure (t = 0). Curves (1)–(8) correspond respectively, to t = 0 h; t = 1 h (one hour); 2 h; 3 h; 5 h; 6 h; 7 h; 24 h (no forward scattering).

Fig. 9
Fig. 9

Plane section of the angular scattering curve of an absorbent substrate before and after the deposition of a λ/2 layer of SiO2; D0 = 4.6 ppm, Dc = 4.3 ppm (no forward scattering).

Fig. 10
Fig. 10

Plane section of the angular scattering curve of an absorbent substrate before and after the deposition of a λ/2 layer of SiO2; D0 = 107 ppm; Dc = 101 ppm (no forward scattering).

Fig. 11
Fig. 11

Plane section of the angular scattering curve of an absorbent substrate before and after the deposition of a λ/2 layer of ZnS; D0 = 92.8 ppm; Dc = 103 ppm (no forward scattering).

Tables (2)

Tables Icon

Table I Antiscattering Range on the Air Side

Tables Icon

Table II Measurements on the Air Side of the Total Integrated Scattering of a Substrate Before (D0) and After (Dc) the Deposition of a Single Layera

Equations (16)

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E 0 ( d ) - = [ δ H ˜ 0 / ( Y 0 - Y 0 ) ] [ cos ( 2 π n e / λ ) - i ( n 0 / n ) sin ( 2 π n e / λ ) ] , E 1 ( d ) - = δ H ˜ 1 / ( Y 1 - Y 1 ) with i 2 = - 1.
E T ( d ) - = E 0 ( d ) - = E 1 ( d ) - { 1 + [ E 0 ( d ) - / E 1 ( d ) - ] } = E 1 ( d ) - ( 1 + X ) ,
X = ( σ 0 / ( σ 1 ) [ ( n 2 - n 0 2 ) / ( n s 2 - n 2 ) ] { [ cos 2 ( 2 π n e / λ ) - ( n s / n ) 2 sin 2 ( 2 π n e / λ ) ] - i ( 2 n s / n ) sin ( 2 π n e / λ ) cos ( 2 π n e / λ ) } ,
( 2 π n e ) / λ = K π / 2 where K is an integer , if K = 2 p n e = p λ / 2 , if K = 2 p + 1 n e = ( 2 p + 1 ) λ / 4.
X = ( σ 0 / σ 1 ) · ( n 2 - n 0 2 ) / ( n s 2 - n 2 ) = - 1 ( σ 0 / σ 1 ) = ( n 2 - n s 2 ) / ( n 2 - n 0 2 ) .
X = - ( σ 0 / σ 1 ) [ ( n 2 - n 0 2 ) / ( n s 2 - n 2 ) ] ( n s / n ) 2 = - 1 , σ 0 / σ 1 = ( n / n s ) 2 ( n s 2 - n 2 ) / ( n 2 - n 0 2 ) .
σ 0 / σ 1 = [ ( n / n s ) 2 - 1 ] / [ ( n 0 / n ) 2 - 1 ] , σ 0 / σ 1 1 n n 0 n s .
E T ( d ) - ( λ / 2 ) / E N ( d ) - = [ 1 / ( n s 2 - n 0 2 ) ] [ n s 2 - n 2 ) + ( σ 0 / σ 1 ) ( n 2 - n 0 2 ) ] = g ( σ 0 / σ 1 ) .
E T ( d ) - ( λ / 2 ) / E N ( d ) - 1 ( σ 0 / σ 1 ) [ ( n 2 + n 0 2 - 2 n s 2 ) / ( n 2 - n 0 2 ) , 1 ] .
E T ( d ) - ( λ / 4 ) / E N ( d ) - = α { ( n 2 - n s 2 ) + ( σ 0 / σ 1 ) [ n s 2 - ( n 0 n s / n ) 2 ] } ,
α = - n 2 ( n 0 + n s ) / [ ( n s - n 0 ) ( n 0 n s + n 2 ) 2 ] < 0. E T ( d ) - ( λ / 4 ) / E N ( d ) - 1 σ 0 / σ 1 [ a , b ] ,
a = ( n / n s ) 2 [ n s 2 - n 2 + ( 1 / α ) ] / ( n 2 - n 0 2 ) , b = ( n / n s ) 2 [ n s 2 - n 2 - ( 1 / α ) ] / ( n 2 - n 0 2 ) .
E T ( d ) - ( λ / 2 ) / E T ( d ) - ( λ / 4 ) 1 σ 0 / σ 1 [ 0.68 , 0.71 ] .
Na 3 ( AlF 6 ) D 0 = 13.5 ppm σ 1 m 1 nm , B = 2 thus g 2 nm ; MgF 2 D 0 = 14.7 ppm σ 1 m 1 nm , B = 1.5 thus g 1.5 nm ; TiO 2 The high - index layers imply :
D 0 = 61.6 ppm σ 1 m 1.1 nm , B = 1 thus g 1.1 nm
D 0 = 9.0 ppm σ 1 m 0.8 nm , A = 0.4 thus g 0.3 nm .

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