Abstract

Transmission scattering from medium to air was used to measure the slope distribution of the rough plane surface of a transparent glass hemisphere. A facet model successfully explained the measured results of refraction, scattering, and polarization: Transmission scattering existed for incident angles greater than the critical angle, all measured curves for the normalized scattered intensity versus the facet slope angle for different detection directions overlapped, and the measured polarization of scattering was approximately constant for >99% of the facets. The slope distribution obtained by transmission scattering agrees with those of the surface profiles in the valid range of the profiler and can represent the slope distribution of the rough surface.

© 2000 Optical Society of America

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References

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  1. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Part I.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. M. Nieto-Vesperinas, J. A. Sánchez-Gil, A. J. Sant, J. C. Dainty, “Light transmission from a randomly rough dielectric diffuser: theoretical and experimental results,” Opt. Lett. 15, 1261–1263 (1990).
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    [CrossRef]
  16. M. Sidran, “Broadband reflectance and emissivity of specular and rough water surface,” Appl. Opt. 20, 3176–3183 (1981).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. B. van Ginneken, M. Stavrid, J. J. Koenderink, “Diffuse and specular reflectance from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
    [CrossRef]
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    [CrossRef]
  25. W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass ed. (McGraw-Hill, New York, 1995), p. 33.67, Table 23.
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    [CrossRef]
  27. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).
  28. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 174.
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    [CrossRef]

1999

1998

1997

1996

1995

R. E. Luna, E. R. Mendez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

1993

1992

1991

1990

1987

1984

1981

1978

H. E. Bennett, “Scattering characteristics of optical materials,” Opt. Eng. 17, 480–488 (1978).
[CrossRef]

P. C. Archibald, H. E. Bennett, “Scattering from infrared missile domes,” Opt. Eng. 17, 647–649 (1978).
[CrossRef]

1961

1954

Archibald, P. C.

P. C. Archibald, H. E. Bennett, “Scattering from infrared missile domes,” Opt. Eng. 17, 647–649 (1978).
[CrossRef]

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering, and polarization from a very rough black surface,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 174.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 174.

Bass, M.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass ed. (McGraw-Hill, New York, 1995), p. 33.67, Table 23.

Beckmann, P.

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

Bennett, H. E.

P. C. Archibald, H. E. Bennett, “Scattering from infrared missile domes,” Opt. Eng. 17, 647–649 (1978).
[CrossRef]

H. E. Bennett, “Scattering characteristics of optical materials,” Opt. Eng. 17, 480–488 (1978).
[CrossRef]

H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
[CrossRef]

S. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical Systems, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

Bennett, J. M.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering, and polarization from a very rough black surface,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

Cao, L.

Chapman, R. D.

Churnside, J. H.

Cox, C.

Dainty, J. C.

Gu, Z. H.

R. E. Luna, E. R. Mendez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Harris, T. J.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass ed. (McGraw-Hill, New York, 1995), p. 33.67, Table 23.

Harvey, J. E.

J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z. Gu, A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
[CrossRef]

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).

Irani, G. B.

Jakeman, E.

Jordan, D. L.

Koenderink, J. J.

Leridon, B.

Lettieri, T. R.

Lewis, G. D.

Lieberman, A. G.

Lu, J. Q.

R. E. Luna, E. R. Mendez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Luna, R. E.

R. E. Luna, E. R. Mendez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

Manor, Y.

Marx, E.

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

Mendez, E. R.

R. E. Luna, E. R. Mendez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

Munk, W.

Nee, S. F.

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering, and polarization from a very rough black surface,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

S. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical Systems, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. Gu, A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

Nee, S.-M. F.

Nee, T. W.

S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. Gu, A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

Nieto-Vesperinas, M.

O’Donnell, K. A.

Porteus, J. O.

Sánchez-Gil, J. A.

Sant, A. J.

Shaw, J. A.

Sidran, M.

Song, J. F.

Spizzichino, A.

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

Stavrid, M.

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Optical Engineering Press, Bellingham, Wash., 1995).
[CrossRef]

Thomas, M. E.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass ed. (McGraw-Hill, New York, 1995), p. 33.67, Table 23.

Tropf, W. J.

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass ed. (McGraw-Hill, New York, 1995), p. 33.67, Table 23.

van Ginneken, B.

Vernold, C. L.

J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z. Gu, A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
[CrossRef]

Vorburger, T. V.

Wilf, I.

Appl. Opt.

M. Sidran, “Broadband reflectance and emissivity of specular and rough water surface,” Appl. Opt. 20, 3176–3183 (1981).
[CrossRef] [PubMed]

R. D. Chapman, G. B. Irani, “Errors in estimating slope spectra from wave images,” Appl. Opt. 20, 3645–3652 (1981).
[CrossRef] [PubMed]

I. Wilf, Y. Manor, “Simulation of sea surface images in the infrared,” Appl. Opt. 23, 3174–3180 (1984).
[CrossRef] [PubMed]

E. Marx, T. V. Vorburger, “Direct and inverse problems for light scattered by rough surfaces,” Appl. Opt. 29, 3613–3626 (1990).
[CrossRef] [PubMed]

L. Cao, T. V. Vorburger, A. G. Lieberman, T. R. Lettieri, “Light scattering measurement of the rms slopes of rough surfaces,” Appl. Opt. 30, 3221–3227 (1991).
[CrossRef] [PubMed]

S.-M. F. Nee, “Ellipsometric view on reflection and scattering from optical blacks,” Appl. Opt. 31, 1549–1556 (1992).
[CrossRef] [PubMed]

E. Marx, B. Leridon, T. R. Lettieri, J. F. Song, T. V. Vorburger, “Autocorrelation functions from optical scattering for one-dimensionally rough surfaces,” Appl. Opt. 32, 67–76 (1993).
[CrossRef] [PubMed]

J. A. Shaw, J. H. Churnside, “Scanning-laser glint measurements of sea-surface slope statistics,” Appl. Opt. 36, 4202–4213 (1997).
[CrossRef] [PubMed]

S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
[CrossRef]

D. L. Jordan, G. D. Lewis, E. Jakeman, “Emission polarization of roughened glass and aluminum surfaces,” Appl. Opt. 35, 3583–3590 (1996).
[CrossRef] [PubMed]

B. van Ginneken, M. Stavrid, J. J. Koenderink, “Diffuse and specular reflectance from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
[CrossRef]

J. Mod. Opt.

R. E. Luna, E. R. Mendez, J. Q. Lu, Z. H. Gu, “Enhanced backscattering due to total internal reflection at a dielectric–air interface,” J. Mod. Opt. 42, 257–269 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

H. E. Bennett, “Scattering characteristics of optical materials,” Opt. Eng. 17, 480–488 (1978).
[CrossRef]

P. C. Archibald, H. E. Bennett, “Scattering from infrared missile domes,” Opt. Eng. 17, 647–649 (1978).
[CrossRef]

Opt. Lett.

Other

S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. Gu, A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
[CrossRef]

W. J. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass ed. (McGraw-Hill, New York, 1995), p. 33.67, Table 23.

J. E. Harvey, C. L. Vernold, “Transfer function characterization of scattering surfaces: revisited,” in Scattering and Surface Roughness, Z. Gu, A. Maradudin, eds., Proc. SPIE3141, 113–127 (1997).
[CrossRef]

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 174.

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

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Optical Engineering Press, Bellingham, Wash., 1995).
[CrossRef]

S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering, and polarization from a very rough black surface,” in Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).

S. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical Systems, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Geometry of the hemispherical sample. (a) Side view of the sample; the shaded slice indicates the rough surface edge. (b) View as the sample is tilted slightly upward.

Fig. 2
Fig. 2

Measured surface profiles for the rough surface of the BK7 glass hemispherical sample for (a) a 1-mm scan length with a 10-µm/s speed and (b) a 254-µm scan length with a 2-µm/s speed.

Fig. 3
Fig. 3

Schematics for the transmission scattering experiment: (a) the experimental setup for transmission scattering from a sample, (b) geometry of transmission scattering by the hemispherical sample. A is the angle of incidence on the rough surface, and B is the angle of detection with respect to the incident beam.

Fig. 4
Fig. 4

Specular refraction by the BK7 glass hemispherical sample at incident angle A 0 for a detection direction at B. The wavelength is at 632.8 nm. Diamonds, A 0 measured for a setting of B; filled circles, settings of B for the scattering measurements.

Fig. 5
Fig. 5

Measured intensity of transmission scattering for the BK7 glass hemispherical sample versus variable incident angle A for the detector at a fixed direction B.

Fig. 6
Fig. 6

Light passes straight through only facets that are normal to the incident beam. (a) The intersections of the arrows with the rough surface mark the facets that permit straight-through passage. (b) Light passes straight through the facets with a slope angle β as the sample is rotated by an angle -β.

Fig. 7
Fig. 7

Normalized scattered intensity (=I/ I max) versus facet slope angle (β) for the curves of Fig. 5.

Fig. 8
Fig. 8

Normalized BTDF versus direction cosine η defined by Eqs. (4) for the curves of Fig. 5. The curve for A 0 = 30 deg has been broadened significantly.

Fig. 9
Fig. 9

Measured ψ versus angle of incidence A. ψ(A) is mostly flat in the middle for the straight-through case (A 0 = 0) and is still flat for values of A of as much as 40 deg for oblique refraction with A 0 = 20 deg.

Fig. 10
Fig. 10

Measured ψ versus facet slope β for A 0 = 0, 20 deg. ψ(β) for A 0 = 20 deg is similar and parallel to that for A 0 = 0 deg.

Fig. 11
Fig. 11

Measured Δ versus facet slope β for A 0 = 0, 20 deg. The changes in Δ from the value at β = 0 are small for both cases.

Fig. 12
Fig. 12

Slope distribution (%) reduced from the surface profiles with 1-mm scan length such as in Fig. 2(a). The thick curve is the average of the four curves obtained from scans 1–4 of Table 1.

Fig. 13
Fig. 13

Slope distribution obtained from the curves of Fig. 7. All the thin curves of Fig. 7 are averaged and plotted here as the thick curve.

Fig. 14
Fig. 14

Slope distributions obtained by transmission scattering and surface profiles. The two methods agree for 2 deg < |β| < 20 deg. The slope distribution by scattering can represent the slope distribution of the rough surface.

Tables (3)

Tables Icon

Table 1 Parameters Used and Surface Characteristics Obtained for Several Profile Scans Measured with a Tencor Stylus Profilometer

Tables Icon

Table 2 Facet Slope Angles β (deg) at Which the Normalized Scattered Intensity is at 0.1, 0.01, and 0.001 for Several Settings of B

Tables Icon

Table 3 Probability (%) for Normalized Scattered Intensity (In) in the Intervals Between 1, 0.1, 0.01, 0.001, and 0 for Several Settings of B

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

n=sinA0+B/sinA0.
β=A-A0.
BTDF A = power scattered in ΔΩ/incident power cosB+A.Normalized BTDF  BTDFA/BTDFA0.
kin-kout=2π/λxˆn sinA-sinB+A+zˆn cosA- cosB+A=2π/λ(xˆ sinβ+zˆ cosβ)×n cosA0-cosB+A0, η=-sinβn2+1-2n cosB1/2.
tp/ts=tan ψ expiΔ.

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