Abstract

Scattering data taken at 0.6328, 3.39, and 10.6 μm are compared with theoretical calculations based on vector and scalar theories of scattering. Methods are described for obtaining the rms surface height using various degrees of dependence on the models of scattering.

© 1983 Optical Society of America

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References

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  1. H. Davies, "The reflection of electromagnetic waves from a rough surface," Proc. IEE 101, 209–214 (1954).
  2. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  3. For instance, J. E. Harvey, "Light-scattering characteristics of optical surfaces," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1976); P. J. Chandley and W. T. Welford, "A reformulation of some results of P. Beckmann for scattering from rough surfaces," Opt. Quantum Electron. 7, 393–397 (1975).
    [CrossRef]
  4. S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1947), p. 161.
  5. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  6. E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360–374 (1977); J. E. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. 18, 116–124 (1979); A. Marvin, F. Toigo, and V. Celli, "Light scattering from rough surfaces: general incidence angle and polarization," Phys. Rev. B 11, 2777–2782 (1975); A. A. Maradudin and D. L. Mills, "Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness," Phys. Rev. B 11, 1392–1415 (1975); D. E. Barrick and W. E. Peake, "Scattering from surfaces with different roughness scales: analysis and interpretation," Tech. Rep. AD 662751 (Battelle Memorial Institute, 1967); A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); E. Kroger and E. Kretschmann, "Scattering of light by slightly rough surfaces or thin film including plasma resonance emission," Ann. Phys. 237, 1–15 (1970).
    [CrossRef]
  7. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.
  8. D. Beaglehole and O. Hunderi, "Study of the interaction of light with rough metal surfaces, I. Experiment," Phys. Rev. B 2, 309–329 (1970); J. M. Elson and J. M. Bennett, "Relation between the angular dependence of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979); R. J. Noll and P. Glenn, "Mirror surface autocovariance functions and their associated visible scattering," Appl. Opt. 21,1824–1838 (1982); J. Eastman and P. W. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12,418–420 (1974); P. J. Chandley, "Determination of the autocorrelation function of height on a rough surface from coherent light scattering," Opt. Quantum Electron. 8, 329–333 (1976); J. M. Bennett, "Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer," Appl. Opt. 15, 2705–2721 (1976).
    [CrossRef] [PubMed]
  9. R. P. Young, "Metal-optics scatter measurement," Proc. Soc. Photo-Opt. Instrum. Eng. 65, 57–62 (1975).
  10. F. E. Nicodemus, "Reflectance nomenclature and directional reflectance and emissivity," Appl. Opt. 9,1474–1475 (1970).
    [CrossRef] [PubMed]
  11. J. M. Bennett and J. H. Dancy, "Stylus profiling instrument for measuring statistical properties of smooth optical surfaces," Appl. Opt. 20, 1785–1802 (1981).
    [CrossRef] [PubMed]
  12. L. D. Brooks, "Microprocessor-based instrumentation for BRDF measurement from visible to FIR," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1982).
  13. Measured by J. Wyant, Optical Sciences Center, University of Arizona, Tucson, Arizona. For reference see C. Koliopoulos, "Interferometric optical phase measurement techniques," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1981).

1981

1975

R. P. Young, "Metal-optics scatter measurement," Proc. Soc. Photo-Opt. Instrum. Eng. 65, 57–62 (1975).

1970

D. Beaglehole and O. Hunderi, "Study of the interaction of light with rough metal surfaces, I. Experiment," Phys. Rev. B 2, 309–329 (1970); J. M. Elson and J. M. Bennett, "Relation between the angular dependence of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979); R. J. Noll and P. Glenn, "Mirror surface autocovariance functions and their associated visible scattering," Appl. Opt. 21,1824–1838 (1982); J. Eastman and P. W. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12,418–420 (1974); P. J. Chandley, "Determination of the autocorrelation function of height on a rough surface from coherent light scattering," Opt. Quantum Electron. 8, 329–333 (1976); J. M. Bennett, "Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer," Appl. Opt. 15, 2705–2721 (1976).
[CrossRef] [PubMed]

F. E. Nicodemus, "Reflectance nomenclature and directional reflectance and emissivity," Appl. Opt. 9,1474–1475 (1970).
[CrossRef] [PubMed]

Beaglehole, D.

D. Beaglehole and O. Hunderi, "Study of the interaction of light with rough metal surfaces, I. Experiment," Phys. Rev. B 2, 309–329 (1970); J. M. Elson and J. M. Bennett, "Relation between the angular dependence of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979); R. J. Noll and P. Glenn, "Mirror surface autocovariance functions and their associated visible scattering," Appl. Opt. 21,1824–1838 (1982); J. Eastman and P. W. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12,418–420 (1974); P. J. Chandley, "Determination of the autocorrelation function of height on a rough surface from coherent light scattering," Opt. Quantum Electron. 8, 329–333 (1976); J. M. Bennett, "Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer," Appl. Opt. 15, 2705–2721 (1976).
[CrossRef] [PubMed]

Beckmann, P.

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

Bennett, J. M.

Brooks, L. D.

L. D. Brooks, "Microprocessor-based instrumentation for BRDF measurement from visible to FIR," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1982).

Church, E. L.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360–374 (1977); J. E. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. 18, 116–124 (1979); A. Marvin, F. Toigo, and V. Celli, "Light scattering from rough surfaces: general incidence angle and polarization," Phys. Rev. B 11, 2777–2782 (1975); A. A. Maradudin and D. L. Mills, "Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness," Phys. Rev. B 11, 1392–1415 (1975); D. E. Barrick and W. E. Peake, "Scattering from surfaces with different roughness scales: analysis and interpretation," Tech. Rep. AD 662751 (Battelle Memorial Institute, 1967); A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); E. Kroger and E. Kretschmann, "Scattering of light by slightly rough surfaces or thin film including plasma resonance emission," Ann. Phys. 237, 1–15 (1970).
[CrossRef]

Dancy, J. H.

Davies, H.

H. Davies, "The reflection of electromagnetic waves from a rough surface," Proc. IEE 101, 209–214 (1954).

Harvey, J. E.

For instance, J. E. Harvey, "Light-scattering characteristics of optical surfaces," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1976); P. J. Chandley and W. T. Welford, "A reformulation of some results of P. Beckmann for scattering from rough surfaces," Opt. Quantum Electron. 7, 393–397 (1975).
[CrossRef]

Hunderi, O.

D. Beaglehole and O. Hunderi, "Study of the interaction of light with rough metal surfaces, I. Experiment," Phys. Rev. B 2, 309–329 (1970); J. M. Elson and J. M. Bennett, "Relation between the angular dependence of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979); R. J. Noll and P. Glenn, "Mirror surface autocovariance functions and their associated visible scattering," Appl. Opt. 21,1824–1838 (1982); J. Eastman and P. W. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12,418–420 (1974); P. J. Chandley, "Determination of the autocorrelation function of height on a rough surface from coherent light scattering," Opt. Quantum Electron. 8, 329–333 (1976); J. M. Bennett, "Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer," Appl. Opt. 15, 2705–2721 (1976).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360–374 (1977); J. E. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. 18, 116–124 (1979); A. Marvin, F. Toigo, and V. Celli, "Light scattering from rough surfaces: general incidence angle and polarization," Phys. Rev. B 11, 2777–2782 (1975); A. A. Maradudin and D. L. Mills, "Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness," Phys. Rev. B 11, 1392–1415 (1975); D. E. Barrick and W. E. Peake, "Scattering from surfaces with different roughness scales: analysis and interpretation," Tech. Rep. AD 662751 (Battelle Memorial Institute, 1967); A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); E. Kroger and E. Kretschmann, "Scattering of light by slightly rough surfaces or thin film including plasma resonance emission," Ann. Phys. 237, 1–15 (1970).
[CrossRef]

Nicodemus, F. E.

Rayleigh, Lord

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1947), p. 161.

Spizzichino, A.

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

Wyant, J.

Measured by J. Wyant, Optical Sciences Center, University of Arizona, Tucson, Arizona. For reference see C. Koliopoulos, "Interferometric optical phase measurement techniques," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1981).

Young, R. P.

R. P. Young, "Metal-optics scatter measurement," Proc. Soc. Photo-Opt. Instrum. Eng. 65, 57–62 (1975).

Zavada, J. M.

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360–374 (1977); J. E. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. 18, 116–124 (1979); A. Marvin, F. Toigo, and V. Celli, "Light scattering from rough surfaces: general incidence angle and polarization," Phys. Rev. B 11, 2777–2782 (1975); A. A. Maradudin and D. L. Mills, "Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness," Phys. Rev. B 11, 1392–1415 (1975); D. E. Barrick and W. E. Peake, "Scattering from surfaces with different roughness scales: analysis and interpretation," Tech. Rep. AD 662751 (Battelle Memorial Institute, 1967); A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); E. Kroger and E. Kretschmann, "Scattering of light by slightly rough surfaces or thin film including plasma resonance emission," Ann. Phys. 237, 1–15 (1970).
[CrossRef]

Appl. Opt.

Phys. Rev.

D. Beaglehole and O. Hunderi, "Study of the interaction of light with rough metal surfaces, I. Experiment," Phys. Rev. B 2, 309–329 (1970); J. M. Elson and J. M. Bennett, "Relation between the angular dependence of scattering and the statistical properties of optical surfaces," J. Opt. Soc. Am. 69, 31–47 (1979); R. J. Noll and P. Glenn, "Mirror surface autocovariance functions and their associated visible scattering," Appl. Opt. 21,1824–1838 (1982); J. Eastman and P. W. Baumeister, "The microstructure of polished optical surfaces," Opt. Commun. 12,418–420 (1974); P. J. Chandley, "Determination of the autocorrelation function of height on a rough surface from coherent light scattering," Opt. Quantum Electron. 8, 329–333 (1976); J. M. Bennett, "Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a FECO scanning interferometer," Appl. Opt. 15, 2705–2721 (1976).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng.

R. P. Young, "Metal-optics scatter measurement," Proc. Soc. Photo-Opt. Instrum. Eng. 65, 57–62 (1975).

Other

L. D. Brooks, "Microprocessor-based instrumentation for BRDF measurement from visible to FIR," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1982).

Measured by J. Wyant, Optical Sciences Center, University of Arizona, Tucson, Arizona. For reference see C. Koliopoulos, "Interferometric optical phase measurement techniques," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1981).

H. Davies, "The reflection of electromagnetic waves from a rough surface," Proc. IEE 101, 209–214 (1954).

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

For instance, J. E. Harvey, "Light-scattering characteristics of optical surfaces," Ph.D. Thesis (University of Arizona, Tucson, Ariz., 1976); P. J. Chandley and W. T. Welford, "A reformulation of some results of P. Beckmann for scattering from rough surfaces," Opt. Quantum Electron. 7, 393–397 (1975).
[CrossRef]

S. Silver, Microwave Antenna Theory and Design (McGraw-Hill, New York, 1947), p. 161.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

E. L. Church, H. A. Jenkinson, and J. M. Zavada, "Measurement of the finish of diamond-turned metal surfaces by differential light scattering," Opt. Eng. 16, 360–374 (1977); J. E. Elson and J. M. Bennett, "Vector scattering theory," Opt. Eng. 18, 116–124 (1979); A. Marvin, F. Toigo, and V. Celli, "Light scattering from rough surfaces: general incidence angle and polarization," Phys. Rev. B 11, 2777–2782 (1975); A. A. Maradudin and D. L. Mills, "Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness," Phys. Rev. B 11, 1392–1415 (1975); D. E. Barrick and W. E. Peake, "Scattering from surfaces with different roughness scales: analysis and interpretation," Tech. Rep. AD 662751 (Battelle Memorial Institute, 1967); A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978); E. Kroger and E. Kretschmann, "Scattering of light by slightly rough surfaces or thin film including plasma resonance emission," Ann. Phys. 237, 1–15 (1970).
[CrossRef]

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

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Figures (21)

Fig. 1
Fig. 1

BRDF data for 0.6328 μm (*), 3.39 μm (0), and 10.6 μm (+) for 10° incidence and the corresponding one-dimensional vector theory (ss) (solid lines).

Fig. 2
Fig. 2

BRDF data for 0.6328 μm (*), 3.39 μm (0), and 10.6 μm (+) for 30° incidence and corresponding one-dimensional vector theory (ss) (solid lines).

Fig. 3
Fig. 3

BRDF data for 0.6328 μm (*), 3.39 μm (0), and 10.6 μm (+) for 50° incidence and corresponding one-dimensional vector theory (ss) (solid lines).

Fig. 4
Fig. 4

BRDF data for 0.6328 μm (+), 0.5145 μm (×), and 0.4880 μm (0), and the corresponding one-dimensional vector theory (pp) (solid lines) for 30° incidence.

Fig. 5
Fig. 5

BRDF for 0.6328-μm unpolarized theory exponential model for 50° incidence.

Fig. 6
Fig. 6

Instrument function for positive (+) and negative (*) ββ0 at 0.6328 μm and noise level for positive (0) and negative (1) ββ0.

Fig. 7
Fig. 7

Instrument function for positive (+) and negative (*) ββ0 and noise level for positive (θ) and negative (1) ββ0 at 3.39 μm.

Fig. 8
Fig. 8

Instrument function for positive (+) and negative (*) ββ0 and noise level for negative ββ0 at 10.6 μm.

Fig. 9
Fig. 9

Power spectrum for 0.6328, 3.39, and 10.6 μm at 30° incidence with two-dimensional Beckmann model.

Fig. 10
Fig. 10

Power spectrum for 0.6328, 3.39, and 10.6 μm at 30° incidence with one-dimensional Beckmann model.

Fig. 11
Fig. 11

Power spectrum for 0.6328, 3,39, and 10.6 μm at 30° incidence with two-dimensional Davies model.

Fig. 12
Fig. 12

Power spectrum for 0.6328, 3.39, and 10.6 μm at 30° incidence with one-dimensional Davies model.

Fig. 13
Fig. 13

Power spectrum for 0.6328, 3.39, and 10.6 μm for 30° incidence with two-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 14
Fig. 14

Power spectrum for 0.6328, 3.39, and 10.6 μm for −30° incidence with one-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 15
Fig. 15

Power spectrum for 0.6328, 3.39, and 10.6 μm for −10° incidence with one-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 16
Fig. 16

Power spectrum for 0.6328, 3.39, and 10.6 μm for −50° incidence with one-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 17
Fig. 17

Power spectrum for 0.6328, 3.39, and 10.6 μm for 10° incidence with one-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 18
Fig. 18

Power spectrum for 0.6328, 3.39, and 10.6 μm for 30° incidence with one-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 19
Fig. 19

Power spectrum for 0.6328, 3.39, and 10.6 μm for 50° incidence with one-dimensional vector theory and corresponding theoretical curve (solid line).

Fig. 20
Fig. 20

Power spectrum for 0.488, 0.5145, and 0.6328 μm at 30° incidence with one-dimensional vector model and corresponding theoretical curves.

Fig. 21
Fig. 21

Power spectrum for uu polarization at 30° incidence based on one-dimensional vector model.

Tables (3)

Tables Icon

Table 1 Values of rms Surface Height for Different Measurements Obtained with Vector Theory (in angstroms)

Tables Icon

Table 2 Values of rms Surface Height Obtained by Fitting Scalar Theory to BRDF Data

Tables Icon

Table 3 Values of rms Surface Heights Obtained from the Vector Theory by (A) Integration and (B) Curve Fitting to W(p)

Equations (22)

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f r s s = 1 P 0 d P s s d Ω = 1 π 2 k 4 F s s W ( p ) ,
f r p p = 1 P 0 d P p p d Ω = 1 π 2 k 4 F p p W ( p ) ,
F s s = cos θ i cos θ s ,
F p p = F s s ( 1 - sin θ i sin θ s cos θ i cos θ s ) 2 .
F = ( cos θ i + cos θ s ) 4 / cos θ s .
d P u u = 1 2 ( d P s s + d P s p + d P p s + d P p p ) .
d P s s = τ s s d P u u ,
d P p p = τ p p d P u u .
f r u u = 1 2 ( f r s s + f r p p ) .
W 2 ( p ) = f r / ( k 4 F 1 π 2 ) ,
W 1 ( p ) = f r / ( k 3 F 2 π ) .
f r s s = BRDF = 1 π 2 k 4 × F s s × W 2 ( p ) .
f r s s = 1 π 2 k 4 × F s s - - σ 2 exp [ - ( x , y ) / l ] × exp [ - j 2 π ( x f x + y f y ) ] d x d y ,
d Ω = d θ s d ϕ s sin θ s .
d q = 2 π d f y = 2 π λ d k y = 2 π λ sin θ s cos ϕ s d ϕ s .
d Ω = d θ sin θ s d q 2 π λ sin θ s cos θ s = d θ × d q 2 π λ cos ϕ s .
d P d θ = 16 π 2 λ 4 [ F 0 ] σ 2 2 π λ cos ϕ s - - - exp ( - x / l ) × exp ( - j 2 π x f x ) d x exp ( - y / l ) exp ( - j 2 π x f y ) d y .
- exp ( - x / l ) exp ( - j 2 π x f x ) d x = 2 l 1 + ( 2 π f x l ) 2 = 2 l 1 + ( p l ) 2 ,
exp ( y / ) exp ( - j 2 π y f y ) d y = 2 l 1 + ( 2 π f y l ) 2 = 2 l 1 + ( q l ) 2 .
d P d θ = 16 π 2 λ 4 [ F 0 ] σ 2 2 π λ cos θ s 2 l 1 + ( p l ) 2 - 2 l 1 + ( q l ) 2 d q .
- l 1 + ( q l ) 2 d q = - d x 1 + x 2 = π             ( let q l = x ) .
d P d θ = 16 π 2 λ 4 [ F 0 ] σ 2 2 π λ cos ϕ s × 2 l 1 + ( p l ) 2 2 π = 16 π 2 λ 3 [ F 0 ] 1 cos ϕ s × 2 σ 2 l 1 + ( p l ) 2 = 16 π 2 λ 3 [ F 0 ] 2 σ 2 l 1 + ( p l ) 2             ( cos ϕ s = 1 in the plane ) .