Abstract

Previously published vector equations describing angle-resolved scattering from single-layer- and multilayer-coated optics have been integrated numerically and analytically over all angles in the reflecting hemisphere to obtain numerical results and analytical expressions for total integrated scattering (TIS). The effects of correlation length, polarization, angle of incidence, roughness height distribution, scattered light missed by the collecting hemisphere, and roughness cross-correlation properties of the multilayer stack on the TIS expression are considered. Background material on TIS from optics coated with single opaque reflecting layers is given for completeness and comparison to corresponding multilayer TIS results. It is shown that errors can occur in calculating the true rms surface roughness from actual TIS measurements; ways to correct these errors are discussed.

© 1983 Optical Society of America

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References

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  1. H. Davies, Proc. IEE London 101, 209 (1954).
  2. H. E. Bennett, J. O. Porteus, J. Opt. Soc. Am. 51, 123 (1961).
    [CrossRef]
  3. J. O. Porteus, J. Opt. Soc. Am. 53, 1394 (1963).
    [CrossRef]
  4. J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” Dissertation, U. Rochester, Rochester, N.Y. (1974).
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 379.
  6. J. M. Elson, Phys. Rev. B 12, 2541 (1975);J. M. Elson, Proc. Soc. Photo-Opt. Instrum. Eng. 240, 296 (1981).
    [CrossRef]
  7. The angle-resolved scattering equations pertaining to surfaces coated with single opaque reflecting films are given in the present notation in Ref. 6. These equations have been obtained previously byD. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9 and subsequently by numerous other workers using various methods.
  8. J. M. Elson, J. M. Bennett, J. Opt. Soc. Am. 69, 31 (1979).
    [CrossRef]
  9. E. L. Church, H. A. Jenkinson, J. M. Zavada, Opt. Eng. 18, 125 (1979).
    [CrossRef]
  10. H. E. Bennett, Opt. Eng. 17, 480 (1978).
    [CrossRef]
  11. For properties of G(τ) and g(k), see, e.g., J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), p. 18.
  12. J. M. Bennett, J. H. Dancy, Appl. Opt. 20, 1785 (1981).
    [CrossRef] [PubMed]
  13. H. Raether, “Surface Plasmons and Roughness,” in Surface Polaritons, V. M. Agranovich, D. L. Mills, Eds. (North-Holland, Amsterdam, 1982), Chap. 9.
  14. J. M. Bennett, J. P. Rahn, P. C. Archibald, D. L. Decker, “Specifying the Surface Finish of Diamond-Turned Optics—A Study of the Relation Between Surface Profiles and Scattering,” in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981).
  15. A two-Gaussian autocorrelation function was used in earlier work by J. M. Elson, J. P. Rahn, J. M. Bennett, Appl. Opt. 19, 669 (1980).
    [CrossRef] [PubMed]
  16. J. M. Elson, J. Opt. Soc. Am. 69, 48 (1979).
    [CrossRef]

1981 (1)

1980 (1)

1979 (3)

1978 (1)

H. E. Bennett, Opt. Eng. 17, 480 (1978).
[CrossRef]

1975 (1)

J. M. Elson, Phys. Rev. B 12, 2541 (1975);J. M. Elson, Proc. Soc. Photo-Opt. Instrum. Eng. 240, 296 (1981).
[CrossRef]

1963 (1)

1961 (1)

1954 (1)

H. Davies, Proc. IEE London 101, 209 (1954).

Archibald, P. C.

J. M. Bennett, J. P. Rahn, P. C. Archibald, D. L. Decker, “Specifying the Surface Finish of Diamond-Turned Optics—A Study of the Relation Between Surface Profiles and Scattering,” in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981).

Barrick, D. E.

The angle-resolved scattering equations pertaining to surfaces coated with single opaque reflecting films are given in the present notation in Ref. 6. These equations have been obtained previously byD. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9 and subsequently by numerous other workers using various methods.

Bendat, J. S.

For properties of G(τ) and g(k), see, e.g., J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), p. 18.

Bennett, H. E.

Bennett, J. M.

J. M. Bennett, J. H. Dancy, Appl. Opt. 20, 1785 (1981).
[CrossRef] [PubMed]

A two-Gaussian autocorrelation function was used in earlier work by J. M. Elson, J. P. Rahn, J. M. Bennett, Appl. Opt. 19, 669 (1980).
[CrossRef] [PubMed]

J. M. Elson, J. M. Bennett, J. Opt. Soc. Am. 69, 31 (1979).
[CrossRef]

J. M. Bennett, J. P. Rahn, P. C. Archibald, D. L. Decker, “Specifying the Surface Finish of Diamond-Turned Optics—A Study of the Relation Between Surface Profiles and Scattering,” in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 379.

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, Opt. Eng. 18, 125 (1979).
[CrossRef]

Dancy, J. H.

Davies, H.

H. Davies, Proc. IEE London 101, 209 (1954).

Decker, D. L.

J. M. Bennett, J. P. Rahn, P. C. Archibald, D. L. Decker, “Specifying the Surface Finish of Diamond-Turned Optics—A Study of the Relation Between Surface Profiles and Scattering,” in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981).

Eastman, J. M.

J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” Dissertation, U. Rochester, Rochester, N.Y. (1974).

Elson, J. M.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, Opt. Eng. 18, 125 (1979).
[CrossRef]

Piersol, A. G.

For properties of G(τ) and g(k), see, e.g., J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), p. 18.

Porteus, J. O.

Raether, H.

H. Raether, “Surface Plasmons and Roughness,” in Surface Polaritons, V. M. Agranovich, D. L. Mills, Eds. (North-Holland, Amsterdam, 1982), Chap. 9.

Rahn, J. P.

A two-Gaussian autocorrelation function was used in earlier work by J. M. Elson, J. P. Rahn, J. M. Bennett, Appl. Opt. 19, 669 (1980).
[CrossRef] [PubMed]

J. M. Bennett, J. P. Rahn, P. C. Archibald, D. L. Decker, “Specifying the Surface Finish of Diamond-Turned Optics—A Study of the Relation Between Surface Profiles and Scattering,” in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 379.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, Opt. Eng. 18, 125 (1979).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

Opt. Eng. (2)

E. L. Church, H. A. Jenkinson, J. M. Zavada, Opt. Eng. 18, 125 (1979).
[CrossRef]

H. E. Bennett, Opt. Eng. 17, 480 (1978).
[CrossRef]

Phys. Rev. B (1)

J. M. Elson, Phys. Rev. B 12, 2541 (1975);J. M. Elson, Proc. Soc. Photo-Opt. Instrum. Eng. 240, 296 (1981).
[CrossRef]

Proc. IEE London (1)

H. Davies, Proc. IEE London 101, 209 (1954).

Other (6)

The angle-resolved scattering equations pertaining to surfaces coated with single opaque reflecting films are given in the present notation in Ref. 6. These equations have been obtained previously byD. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9 and subsequently by numerous other workers using various methods.

H. Raether, “Surface Plasmons and Roughness,” in Surface Polaritons, V. M. Agranovich, D. L. Mills, Eds. (North-Holland, Amsterdam, 1982), Chap. 9.

J. M. Bennett, J. P. Rahn, P. C. Archibald, D. L. Decker, “Specifying the Surface Finish of Diamond-Turned Optics—A Study of the Relation Between Surface Profiles and Scattering,” in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981).

For properties of G(τ) and g(k), see, e.g., J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), p. 18.

J. M. Eastman, “Surface Scattering in Optical Interference Coatings,” Dissertation, U. Rochester, Rochester, N.Y. (1974).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 379.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of apparatus to measure TIS. Most of the normally incident laser light scattered by sample S is collected by the aluminized hemisphere (Coblentz sphere) C and focused onto detector D. Some scattered light (— · ·) is lost through the hole along with the specular beam and beyond the large angle limits.

Fig. 2
Fig. 2

Schematic diagram showing notation for the ARS formulas. Light is incident at angle θ0, the polar and azimuthal scattering angles are θ and ϕ, respectively.

Fig. 3
Fig. 3

Normalized TIS/TIS vs σ/λ for an opaque highly reflecting surface. The light is assumed to be normally incident on the surface, and the scattered light is collected for all angles between 0 and 90°. TIS is for the same surface when σ ≫ λ.

Fig. 4
Fig. 4

Same as Fig. 3 except that the angle of incidence θ0 is 30°. TIS is for normal incidence and σ ≫ λ. The solid (dashed) curve is for p-polarized (s-polarized) incident light. For σ/λ > 2, the TIS/TIS curve is nearly constant at cos230° = 0.75.

Fig. 5
Fig. 5

Same as Figs. 3 and 4, except that the angle of incidence θ0 is 45°. The solid (dashed) curve is for p-polarized (s-polarized) incident light. For σ/λ > 2, the TIS/TIS curve is nearly constant at cos245° = 0.50.

Fig. 6
Fig. 6

TIS/TIS vs σ/λ for a surface having a two-Gaussian autocorrelation function and measured using an apparatus similar to that shown in Fig. 1. The short-range correlation length σS is held constant at 0.35 μm, while the long-range correlation length varies from 0 to 16.0 μm. The short- and long-range rms roughness values are 33.6 and 47.5 Å, respectively. For this type of surface (similar to an actual diamond-turned surface), significant light is lost through the specular beam exit hole.

Fig. 7
Fig. 7

TIS/TIS vs σ/λ for light normally incident on a 23-layer dielectric stack, with the scattered light collected from all angles between 0 and 90°. The thin films are quarterwave optical thickness at normal incidence. The solid (dashed) curve is for a correlated (uncorrelated) multilayer stack with Gaussian autocorrelation functions assumed for the film interfaces. TIS is for an opaque highly reflecting surface with σ ≫ λ. For σ/λ > 2, the solid curve remains nearly constant at unity.

Fig. 8
Fig. 8

TIS/TIS vs σ/λ for p-polarized light incident at θ0 = 30° on the same dielectric stack as in Fig. 7, except that the thin films have a quarterwave optical thickness at 30° incidence. The solid (dashed) curve is for a correlated (uncorrelated) multilayer stack with Gaussian autocorrelation functions assumed for the film interfaces. TIS is for an opaque highly reflecting surface at θ0 = 0° and σ ≫ λ. For σ/λ > 2, the TIS/TIS curve for the correlated case is nearly constant at cos230° = 0.75.

Fig. 9
Fig. 9

Same as Fig. 8, except that the incident light is s-polarized.

Fig. 10
Fig. 10

Same as Fig. 8 (p-polarized incident light), except that the angle of incidence is 45°, and the films are quarterwave optical thickness at 45° incidence. For σ/λ > 4.5, the TIS/TIS curve for the correlated case is nearly constant at cos2 45° = 0.50.

Fig. 11
Fig. 11

Same as Fig. 10, except that the incident light is s-polarized.

Fig. 12
Fig. 12

Plot of the electric field intensity for p-polarized (solid curves) and s-polarized (dashed curves) light incident on a 23-layer dielectric stack. Angles of incidence are (a) 0, (b) 30, and (c) 45°. The optical thicknesses of the layers are quarterwave at the given angle of incidence. At normal incidence there is no difference between s- and p-polarized incident light.

Equations (51)

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TIS = diffuse reflectance specular + diffuse reflectance ,
( 4 π δ λ ) 2 = TIS ,
G ( τ ) = ( 1 2 π ) 2 d 2 kg ( k ) exp ( i k τ )
g ( k ) = d 2 τ G ( τ ) exp ( i k τ )
0 dkkg ( k ) = 2 π δ 2 .
G ( τ ) = δ 2 exp ( τ 2 / σ 2 ) ,
g ( k ) = π δ 2 σ 2 exp ( k 2 σ 2 / 4 ) .
1 P 0 d P d Ω = ( ω / c ) 4 π 2 cos θ 0 cos 2 θ | 1 | 2 g ( k k 0 ) × ( | χ θ | 2 | q + q | 2 + | χ ϕ | 2 | q + q 0 | 2 ) ,
χ θ = ( q q 0 cos ϕ k k 0 ) cos ϕ q 0 + q 0 + ( ω / c ) q sin ϕ sin ϕ q 0 + q 0 ,
χ ϕ = ( ω c ) [ q 0 sin ϕ cos ϕ q 0 + q 0 ( ω / c ) cos ϕ sin ϕ q 0 + q 0 ] .
1 P 0 d P d Ω = ( ω / c ) 4 π 2 cos 2 θ | 1 | 2 | 1 + | 2 g ( k ) × [ | q | 2 cos 2 ϕ | q + q | 2 + ( ω / c ) 2 sin 2 ϕ | q + q | 2 ] .
TIS = P / P 0 R 0 + P / P 0 P R 0 P 0 ,
P = SH d P d Ω d Ω
1 P 0 d P d Ω = ( ω / c ) 4 π 2 ( cos 2 ϕ + sin 2 ϕ cos 2 θ ) g [ ( ω / c ) sin θ ] ,
TIS = P P 0 = 1 P 0 SH d P d Ω d Ω ,
TIS = ( ω / c ) 4 π 0 π / 2 d θ sin θ ( 1 + cos 2 θ ) g [ ( ω / c ) sin θ ] .
TIS = P P 0 = ( ω / c ) 2 π 0 2 π / λ d k k [ 2 ( k λ / 2 π ) 2 ] [ 1 ( k λ / 2 π ) 2 ] 1 / 2 g ( k ) ,
g ( k ) 2 π δ 2 [ δ ( k ) / k ] ,
TIS = ( 16 π 2 δ 2 ) / λ 2 .
1 P 0 d P d Ω = ( ω / c ) 4 π 2 | 1 N 1 + N | 2 cos 2 θ g ( k ) .
1 P 0 d P d Ω d Ω = P P 0 = R 0 16 π 2 δ 2 λ 2 ,
TIS = P R 0 P 0 = 16 π 2 δ 2 λ 2 ,
g ( k ) π δ 2 σ 2 .
TIS 0 = 64 3 π 4 δ 2 σ 2 λ 4 ,
( δ 0 δ ) 2 = 3 4 π 2 ( σ / λ ) 2 ,
TIS = ( 4 π δ cos θ 0 λ ) 2 ,
E ( r ) = cos θ 0 2 λ i A d a E ( x , y ) r exp { 2 π i λ [ r 0 + 2 ζ ( x , y ) cos θ 0 ] } ,
D ( ζ ) d ζ = 1 ,
D ( ζ ) ζ d ζ = ζ ¯ ,
D ( ζ ) ζ 2 d ζ = δ 2 ,
D ( ζ ) f ( ζ ) d ζ = f ( ζ ) ,
E ( r ) = cos θ 0 2 λ i A d A E ( x , y ) r 0 d ζ D ( ζ ) × exp { 2 π i λ [ r 0 + 2 ζ ( x , y ) cos θ 0 ] } ,
E ( r ) = E 1 ( r ) d ζ D ( ζ ) exp ( 4 π i λ ζ cos θ 0 ) ,
E 1 ( r ) = cos θ 0 2 λ i A d A E ( x , y ) r 0 exp ( 2 π i r 0 λ ) .
E ( r ) = E 1 = E S .
E R = E ( r ) E 1 d ζ D ( ζ ) × [ 1 + 4 π i λ ζ cos θ 0 + 1 2 ( 4 π i cos θ 0 λ ) 2 ζ 2 ] .
TIS = 1 [ 1 8 π 2 δ 2 cos 2 θ 0 / λ 2 ] 2 16 π 2 δ 2 cos 2 θ 0 λ 2 .
δ e 2 = 1 2 π α β dk k g ( k ) ,
δ e 2 = α β dk k 0 d τ τ G ( τ ) J 0 ( k τ ) ,
( δ e δ ) 2 = exp ( α 2 σ 2 4 ) exp ( β 2 σ 2 4 )
( δ e δ ) 2 = 1 1 + ( α σ ) 2 1 1 + ( β σ ) 2 ,
( e δ δ ) 2 { σ 2 ( β 2 α 2 ) / 4 σ λ , exp ( α 2 σ 2 / 4 ) σ λ ,
( e δ δ ) 2 { σ 2 ( β 2 α 2 ) / 2 σ λ , 1 / α σ σ λ ,
( e δ δ ) 2 { π 2 σ 2 / λ 2 σ λ , 1 σ λ ,
( e δ δ ) 2 { 2 π 2 σ 2 / λ 2 σ λ , 1 σ λ ,
TIS = ( 16 π 2 δ e 2 ) / λ 2 ,
TIS = 16 π 2 δ 2 λ 2 [ 1 exp ( π 2 σ 2 / λ 2 ) ] ,
TIS = { 16 π 2 δ 2 / λ 2 σ λ , 16 π 4 δ 2 σ 2 / λ 4 σ λ .
G ( τ ) = δ S 2 exp ( τ 2 / σ S 2 ) + δ L 2 exp ( τ 2 / σ L 2 ) .
g ( k ) = π [ δ S 2 σ S 2 exp ( k 2 σ S 2 / 4 ) + δ L 2 σ L 2 exp ( k 2 σ L 2 / 4 ) ] .
( δ e δ ) 2 = ( δ L δ ) 2 [ exp ( α 2 σ L 2 / 4 ) exp ( β 2 σ L 2 / 4 ) ] + ( δ S δ ) 2 [ exp ( α 2 σ S 2 / 4 ) exp ( β 2 σ S 2 / 4 ) ] ,

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