Abstract

The differential (angle-resolved) light scattering characteristics of spheres deposited on an optically smooth polished nickel surface were studied. In the experimental work a He–Ne laser beam (632.8 nm) illuminated individual polystyrene spheres of diameters 0.50, 1.09, 2.02, and 4.10 μm. The laser beam was directed onto the surface at 45° angle of incidence and focused to 15-μm 1/e2 diameter. A ring/wedge photodiode detector array centered about the specularly reflected beam collected the light scattered into twenty one ring-shaped elements ranging from ~17° to 62° from the specular direction. For comparison with experiment a theoretical model which partially uncoupled the scattering by the surface and the particle was developed based on extensions of Lorenz-Mie theory. The scattering measurements showed reasonable agreement with the model and indicated that the formulation can be adapted for first-order predictions of light scattering by spherical particles on optically smooth surfaces.

© 1988 Optical Society of America

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References

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  1. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  4. R. P. Young, “Low Scatter Mirror Degradation by Particle Contamination,” Opt. Eng. 15, 516 (1976).
    [Crossref]
  5. K. B. Nahm, W. L. Wolfe, “Light-Scattering Models for Spheres on a Conducting Plane: Comparison with Experiment,” Appl. Opt. 26, 2995 (1987).
    [Crossref] [PubMed]
  6. D. A. Thomas, “Light Scattering From Reflecting Optical Surfaces,” Ph.D. Dissertation, U. Arizona, Tucson (1980).
  7. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  8. D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).
  9. H. E. Bennett, J. O. Porteus, “Relation Between Surface Roughness and Specular Reflectance at Normal Incidence,” J. Opt. Soc. Am. 51, 123 (1961).
    [Crossref]
  10. K. J. Allardyce, N. George, “Diffraction Analysis of Rough Reflective Surfaces,” Appl. Opt. 26, 2364 (1987).
    [Crossref] [PubMed]
  11. J. M. Elson, J. M. Bennett, “Vector Scattering Theory,” Opt. Eng. 18, 116 (1979).
    [Crossref]
  12. D. C. Weber, “Light Scattering by Micron and Submicron Spheres on Optically Smooth Surfaces,” M.S. Thesis, Mechanical & Aerospace Engineering Department, Arizona State U., Tempe (1986).
  13. E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response Characteristics of Laser Diffraction Particle Size Analyzers: Optical Sample Volume Extent and Lens Effects,” Opt. Eng. 23, 610 (1984).
    [Crossref]
  14. V. R. Weidner, J. J. Hsia, “Reflection Properties of Pressed Polytetrafluoroethylene Powder,” J. Opt. Soc. Am. 71, 856 (1981).
    [Crossref]

1987 (2)

1985 (1)

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

1984 (1)

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response Characteristics of Laser Diffraction Particle Size Analyzers: Optical Sample Volume Extent and Lens Effects,” Opt. Eng. 23, 610 (1984).
[Crossref]

1981 (1)

1979 (1)

J. M. Elson, J. M. Bennett, “Vector Scattering Theory,” Opt. Eng. 18, 116 (1979).
[Crossref]

1976 (1)

R. P. Young, “Low Scatter Mirror Degradation by Particle Contamination,” Opt. Eng. 15, 516 (1976).
[Crossref]

1961 (1)

Allardyce, K. J.

Beckmann, P.

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

Bennett, H. E.

Bennett, J. M.

J. M. Elson, J. M. Bennett, “Vector Scattering Theory,” Opt. Eng. 18, 116 (1979).
[Crossref]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Chigier, N. A.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response Characteristics of Laser Diffraction Particle Size Analyzers: Optical Sample Volume Extent and Lens Effects,” Opt. Eng. 23, 610 (1984).
[Crossref]

Elson, J. M.

J. M. Elson, J. M. Bennett, “Vector Scattering Theory,” Opt. Eng. 18, 116 (1979).
[Crossref]

George, N.

Giauque, C.

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Gilsinn, D. E.

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Hirleman, E. D.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response Characteristics of Laser Diffraction Particle Size Analyzers: Optical Sample Volume Extent and Lens Effects,” Opt. Eng. 23, 610 (1984).
[Crossref]

Hsia, J. J.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

McLay, M. J.

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Nahm, K. B.

Oechsle, V.

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response Characteristics of Laser Diffraction Particle Size Analyzers: Optical Sample Volume Extent and Lens Effects,” Opt. Eng. 23, 610 (1984).
[Crossref]

Porteus, J. O.

Scire, F. E.

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Spizzichino, A.

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

Teague, E. C.

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Thomas, D. A.

D. A. Thomas, “Light Scattering From Reflecting Optical Surfaces,” Ph.D. Dissertation, U. Arizona, Tucson (1980).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Vorburger, T. V.

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Weber, D. C.

D. C. Weber, “Light Scattering by Micron and Submicron Spheres on Optically Smooth Surfaces,” M.S. Thesis, Mechanical & Aerospace Engineering Department, Arizona State U., Tempe (1986).

Weidner, V. R.

Wolfe, W. L.

Young, R. P.

R. P. Young, “Low Scatter Mirror Degradation by Particle Contamination,” Opt. Eng. 15, 516 (1976).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Opt. Eng. (3)

R. P. Young, “Low Scatter Mirror Degradation by Particle Contamination,” Opt. Eng. 15, 516 (1976).
[Crossref]

J. M. Elson, J. M. Bennett, “Vector Scattering Theory,” Opt. Eng. 18, 116 (1979).
[Crossref]

E. D. Hirleman, V. Oechsle, N. A. Chigier, “Response Characteristics of Laser Diffraction Particle Size Analyzers: Optical Sample Volume Extent and Lens Effects,” Opt. Eng. 23, 610 (1984).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. E. Gilsinn, T. V. Vorburger, E. C. Teague, M. J. McLay, C. Giauque, F. E. Scire, “Surface Roughness Metrology by Angular Distributions of Scattered Light,” Proc. Soc. Photo-Opt. Instrum. Eng. 525, 2 (1985).

Other (6)

D. C. Weber, “Light Scattering by Micron and Submicron Spheres on Optically Smooth Surfaces,” M.S. Thesis, Mechanical & Aerospace Engineering Department, Arizona State U., Tempe (1986).

D. A. Thomas, “Light Scattering From Reflecting Optical Surfaces,” Ph.D. Dissertation, U. Arizona, Tucson (1980).

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

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

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Figures (12)

Fig. 1
Fig. 1

Coordinate system for the study of light scattering by particles. The direction of the scattered light is specified by the scattering angles θs, measured from the z axis, and ϕs, measured from the x axis in the x-y plane. The shaded region is part of the scattering plane which contains the scattered ray and the optical axis (z axis).

Fig. 2
Fig. 2

Idealized schematic of light scattering by a particle on a surface, where scattering by the surface and the particle are assumed to be uncoupled. (The particle is shown separated from the surface to facilitate the explanation.) If it is also assumed that light scattered by the particle and then subsequently reflected by the surface proceeds unobstructed to the detector, a sum of Lorenz-Mie theory backscatter (Csc,b) and forward scatter (Csc,f) contributions can be used to model the scattering. R is the specular reflectance of the surface.

Fig. 3
Fig. 3

Plot of differential light scattering cross section dCsc/dΩ (integrated over ϕs from 0 to 2π) as a function of scattering angle θs calculated using the modified Lorenz-Mie theory of Eq. (4) for scattering by particles on surfaces. The three curves represent the three terms of Eq. (4), and the data were calculated for 1.09-μm diam polystyrene spheres with refractive index n = 1.586 − 0.0i for λ = 632.8 nm (unpolarized) on a polished nickel surface at 45° angle of incidence.

Fig. 4
Fig. 4

Schematic of the experimental apparatus used in the surface-scattering studies. The apparatus was situated on an optical table within a clean enclosure which maintained class 1 conditions at the sample surface.

Fig. 5
Fig. 5

Schematic of the surface and ring detector system with the relevant coordinate systems. θs is the scattering angle in the particle scattering coordinate system (measured from the specularly reflected beam), θi and θr are the incident and reflected angles relative to the surface normal, and θsur is the angle of the scattered light measured from the surface normal.

Fig. 6
Fig. 6

SEM micrograph of a 0.50-μm polystyrene sphere deposited on the nickel surface used in this study. The deep scratch above and to the left of the particle is somewhat anomalous, and the surface in the lower right of the photograph is more typical of the majority of the surface. The focused laser beam (15-μm diameter) covered an area approximately equal to that shown in the photo. This specific particle was not included in the data set due to the uncharacteristically large scratch near the particle.

Fig. 7
Fig. 7

Plot of Csc/Ω averaged over several polystyrene calibration spheres of each size deposited on an optically smooth nickel surface vs mean scattering angle θs of ring detectors. Ω was determined by integrating dΩ = sinθsss over the ring detector apertures. Measurements were made at 45° angle of incidence and λ = 632.8 nm (unpolarized). Also shown is the mean background scattering signal from the nominal surface microroughness.

Fig. 8
Fig. 8

Plot of Csc/Ω for different (1.09-μm) polystyrene calibration spheres deposited on a nickel surface vs mean scattering angle θs of ring detectors. Ω was determined by integrating dΩ = sinθssδϕs over the ring detector apertures. Measurements were made at 45° angle of incidence and λ = 632.8 nm (unpolarized). The data indicate the particle-to-particle variations in the mean data plotted in Fig. 7. The repeatability (1σ) of any one particle signature was of the order of the symbol width.

Fig. 9
Fig. 9

Plot of Csc/Ω vs mean scattering angle θs of ring detectors for 0.50-μm diam polystyrene spheres. Theoretical values for Csc were determined from Eq. (4) by integration of Eq. (3) over the finite apertures of the ring detector elements. Ω was determined by integrating dΩ = sinθsss over the apertures. The data apply to polystyrene spheres with refractive index n = 1.586 − 0.0i on an optically smooth nickel surface for λ = 632.8 nm (unpolarized) at 45° angle of incidence.

Fig. 10
Fig. 10

Plot of Csc/Ω vs mean scattering angle θs of ring detectors for 1.09-μm diam polystyrene spheres. Theoretical values for Csc were determined from Eq. (4) by integration of Eq. (3) over the finite apertures of the ring detector elements. Ω was determined by integrating dΩ = sinθsss over the apertures. The data apply to polystyrene spheres with refractive index n = 1.586 − 0.0i on an optically smooth nickel surface for λ = 632.8 nm (unpolarized) at 45° angle of incidence.

Fig. 11
Fig. 11

Plot of Ccs/Ω vs mean scattering angle θs of ring detectors for 2.02-μm diam polystyrene spheres. Theoretical values for Csc were determined from Eq. (4) by integration of Eq. (3) over the finite apertures of the ring detector elements. Ω was determined by integrating dΩ = sinθsss over the apertures. The data apply to polystyrene spheres with refractive index n = 1.586 − 0.0i on an optically smooth nickel surface for λ = 632.8 nm (unpolarized) at 45° angle of incidence.

Fig. 12
Fig. 12

Plot of Csc/Ω vs mean scattering angle θs of ring detectors for 4.1-μm diam polystyrene spheres. Theoretical values for Csc were determined from Eq. (4) by integration of Eq. (3) over the finite apertures of the ring detector elements. Ω was determined by integrating dΩ = sinθsss over the apertures. The data apply to polystyrene spheres with refractive index n = 1.586 − 0.0i on an optically smooth nickel surface for λ = 632.8 nm (unpolarized) at 45° angle of incidence.

Equations (4)

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I ( θ s , ϕ s ) = I 0 F ( θ s , ϕ s , α ) / k 2 r 2 ,
F ( θ s , ϕ s , α ) = i 1 ( α , θ s ) sin 2 ϕ s + i 2 ( α , θ s ) cos 2 ϕ s ,
C s c ( α , Ω det ) = ( 1 / k 2 ) F ( θ s , ϕ s , α ) sin θ s d θ s d ϕ s ,
C s c = C s c , b + R C s c , f ,

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