Abstract

The application of angle-resolved light scattering (ARLS) to the measurement of the surface roughness of glossy coatings on paper was investigated. To this end, ARLS patterns were measured for laser light scattered from several glossy paper samples, and these patterns were compared with those calculated using a theoretical model based on plane-wave scattering from an isotropic rough surface. Mechanical stylus profilometry data for the rms roughnesses and the autocorrelation functions of the coatings were used as input to calculate the patterns. For all the paper samples measured, as well as for all the incidence angles used, there was good agreement between the experimental and the calculated patterns when all the rms roughnesses measured by profilometry were reduced by 30%. The indication from these experiments is that ARLS may be used to determine the roughness parameters of the coatings. As a check on these results, measurements were also performed with a commercial optical surface probe; these data agreed well with both the ARLS and the stylus profilometry results.

© 1991 Optical Society of America

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References

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  1. P. Richter, E. Lorincz, I. Peczeli, F. Engard, “Coherent optical measurement of surface roughness and its application in the paper industry,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 258–265 (1983).
  2. E. Lorincz, P. Richter, F. Engard, “Interferometric statistical measurement of surface roughness,” Appl. Opt. 25, 2778– 2784 (1986).
    [CrossRef] [PubMed]
  3. J. M. Lucas, “An on-line paper machine, sensor for roughness and other measurements,” presented at the International Symposium on Process Control, Vancouver, Canada, (1977).
  4. J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).
  5. E. Marx, J.-F. Song, T. V. Vorburger, T. R. Lettieri, “Light scattered by coated paper,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 826–834 (1990).
  6. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989).
  7. Certain commercial equipment, instruments, or materials are identified in this paper to specify adequately the experimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
  8. J.-F. Song, T. Vorburger, “Stylus profiling at high resolution and low force,” Appl. Opt. 30, 42–51 (1991).
    [CrossRef]
  9. ANSI/ASME B46.1-1985, Surface Texture (American Society of Mechanical Engineers, New York, 1985).
  10. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
    [CrossRef]
  11. E. Marx, T. Vorburger, “Direct and inverse problems for light scattered by rough surfaces,” Appl. Opt. 29, 3613–3626 (1990).
    [CrossRef] [PubMed]
  12. T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
    [CrossRef]
  13. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  14. E. L. Church, P. Z. Takacs, T. L. Leonard, “The prediction of BRDFs from surface profile measurements,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 136–150 (1989).
  15. E. Church, U.S. Army Research, Development, and Engineering Command, Picatinny Arsenal, Dover, N.J. 07801 (personal communication).
  16. R. Brodmann, O. Gerstorfer, G. Thurn, “Optical roughness measuring instrument for fine-machined surfaces,” Opt. Eng. 24, 408–413 (1985).
    [CrossRef]
  17. 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).
    [CrossRef]
  18. B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).
  19. J. Rakels, “Recognized surface finish parameters obtained from diffraction patterns of rough surfaces,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 119–125 (1988).
  20. P. Z. Takacs, R. C. Hewitt, E. L. Church, “Correlation between the performance and metrology of glancing-incidence synchrotron-radiation mirrors containing millimeter-wavelength shape errors,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 119–124 (1987).
  21. R. Brodmann, M. Allgauer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 111–118 (1988).
  22. J. Zimmerman, T. Vorburger, H. Moncarz, “Automated optical roughness inspection,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 252–264 (1988).

1991 (1)

1990 (1)

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

1986 (1)

1985 (1)

R. Brodmann, O. Gerstorfer, G. Thurn, “Optical roughness measuring instrument for fine-machined surfaces,” Opt. Eng. 24, 408–413 (1985).
[CrossRef]

1984 (1)

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

1979 (1)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

1976 (1)

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).
[CrossRef]

Allgauer, M.

R. Brodmann, M. Allgauer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 111–118 (1988).

Beckmann, P.

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

Bennett, J. M.

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

Brodmann, R.

R. Brodmann, O. Gerstorfer, G. Thurn, “Optical roughness measuring instrument for fine-machined surfaces,” Opt. Eng. 24, 408–413 (1985).
[CrossRef]

R. Brodmann, M. Allgauer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 111–118 (1988).

Chandley, P. J.

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).
[CrossRef]

Church, E.

E. Church, U.S. Army Research, Development, and Engineering Command, Picatinny Arsenal, Dover, N.J. 07801 (personal communication).

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

E. L. Church, P. Z. Takacs, T. L. Leonard, “The prediction of BRDFs from surface profile measurements,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 136–150 (1989).

P. Z. Takacs, R. C. Hewitt, E. L. Church, “Correlation between the performance and metrology of glancing-incidence synchrotron-radiation mirrors containing millimeter-wavelength shape errors,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 119–124 (1987).

Engard, F.

E. Lorincz, P. Richter, F. Engard, “Interferometric statistical measurement of surface roughness,” Appl. Opt. 25, 2778– 2784 (1986).
[CrossRef] [PubMed]

P. Richter, E. Lorincz, I. Peczeli, F. Engard, “Coherent optical measurement of surface roughness and its application in the paper industry,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 258–265 (1983).

Gerstorfer, O.

R. Brodmann, O. Gerstorfer, G. Thurn, “Optical roughness measuring instrument for fine-machined surfaces,” Opt. Eng. 24, 408–413 (1985).
[CrossRef]

Gilsinn, D.

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Hewitt, R. C.

P. Z. Takacs, R. C. Hewitt, E. L. Church, “Correlation between the performance and metrology of glancing-incidence synchrotron-radiation mirrors containing millimeter-wavelength shape errors,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 119–124 (1987).

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

Leonard, T. L.

E. L. Church, P. Z. Takacs, T. L. Leonard, “The prediction of BRDFs from surface profile measurements,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 136–150 (1989).

Leridon, B.

B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).

Lettieri, T.

B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).

Lettieri, T. R.

E. Marx, J.-F. Song, T. V. Vorburger, T. R. Lettieri, “Light scattered by coated paper,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 826–834 (1990).

Lorincz, E.

E. Lorincz, P. Richter, F. Engard, “Interferometric statistical measurement of surface roughness,” Appl. Opt. 25, 2778– 2784 (1986).
[CrossRef] [PubMed]

P. Richter, E. Lorincz, I. Peczeli, F. Engard, “Coherent optical measurement of surface roughness and its application in the paper industry,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 258–265 (1983).

Lucas, J. M.

J. M. Lucas, “An on-line paper machine, sensor for roughness and other measurements,” presented at the International Symposium on Process Control, Vancouver, Canada, (1977).

Marx, E.

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

B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).

E. Marx, J.-F. Song, T. V. Vorburger, T. R. Lettieri, “Light scattered by coated paper,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 826–834 (1990).

Mattsson, L.

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

McLay, M.

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Moncarz, H.

J. Zimmerman, T. Vorburger, H. Moncarz, “Automated optical roughness inspection,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 252–264 (1988).

Peczeli, I.

P. Richter, E. Lorincz, I. Peczeli, F. Engard, “Coherent optical measurement of surface roughness and its application in the paper industry,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 258–265 (1983).

Rakels, J.

J. Rakels, “Recognized surface finish parameters obtained from diffraction patterns of rough surfaces,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 119–125 (1988).

Richter, P.

E. Lorincz, P. Richter, F. Engard, “Interferometric statistical measurement of surface roughness,” Appl. Opt. 25, 2778– 2784 (1986).
[CrossRef] [PubMed]

P. Richter, E. Lorincz, I. Peczeli, F. Engard, “Coherent optical measurement of surface roughness and its application in the paper industry,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 258–265 (1983).

Scire, F.

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Song, J.-F.

J.-F. Song, T. Vorburger, “Stylus profiling at high resolution and low force,” Appl. Opt. 30, 42–51 (1991).
[CrossRef]

E. Marx, J.-F. Song, T. V. Vorburger, T. R. Lettieri, “Light scattered by coated paper,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 826–834 (1990).

B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).

Spizzichino, A.

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

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

Takacs, P. Z.

E. L. Church, P. Z. Takacs, T. L. Leonard, “The prediction of BRDFs from surface profile measurements,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 136–150 (1989).

P. Z. Takacs, R. C. Hewitt, E. L. Church, “Correlation between the performance and metrology of glancing-incidence synchrotron-radiation mirrors containing millimeter-wavelength shape errors,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 119–124 (1987).

Teague, E.

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Thurn, G.

R. Brodmann, O. Gerstorfer, G. Thurn, “Optical roughness measuring instrument for fine-machined surfaces,” Opt. Eng. 24, 408–413 (1985).
[CrossRef]

Vorburger, T.

J.-F. Song, T. Vorburger, “Stylus profiling at high resolution and low force,” Appl. Opt. 30, 42–51 (1991).
[CrossRef]

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

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

J. Zimmerman, T. Vorburger, H. Moncarz, “Automated optical roughness inspection,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 252–264 (1988).

Vorburger, T. V.

B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).

E. Marx, J.-F. Song, T. V. Vorburger, T. R. Lettieri, “Light scattered by coated paper,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 826–834 (1990).

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

Zimmerman, J.

J. Zimmerman, T. Vorburger, H. Moncarz, “Automated optical roughness inspection,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 252–264 (1988).

Appl. Opt. (1)

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

Appl. Opt. (2)

J. Res. Natl. Bur. Stand. (1)

T. Vorburger, E. Teague, F. Scire, M. McLay, D. Gilsinn, “Surface roughness studies with DALLAS—detector array for laser light angular scattering,”J. Res. Natl. Bur. Stand. 89, 3–16 (1984).
[CrossRef]

Opt. Eng. (2)

R. Brodmann, O. Gerstorfer, G. Thurn, “Optical roughness measuring instrument for fine-machined surfaces,” Opt. Eng. 24, 408–413 (1985).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
[CrossRef]

Opt. Quantum Electron. (1)

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).
[CrossRef]

Other (15)

B. Leridon, E. Marx, T. Lettieri, J.-F. Song, T. V. Vorburger, “Autocorrelation function from optical scattering for one-dimensionally rough surfaces” (preprint).

J. Rakels, “Recognized surface finish parameters obtained from diffraction patterns of rough surfaces,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 119–125 (1988).

P. Z. Takacs, R. C. Hewitt, E. L. Church, “Correlation between the performance and metrology of glancing-incidence synchrotron-radiation mirrors containing millimeter-wavelength shape errors,” in Metrology: Figure and Finish, B. Truax, ed., Proc. Soc. Photo-Opt. Instrum. Eng.749, 119–124 (1987).

R. Brodmann, M. Allgauer, “Comparison of light scattering from rough surfaces with optical and mechanical profilometry,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 111–118 (1988).

J. Zimmerman, T. Vorburger, H. Moncarz, “Automated optical roughness inspection,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 252–264 (1988).

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

E. L. Church, P. Z. Takacs, T. L. Leonard, “The prediction of BRDFs from surface profile measurements,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 136–150 (1989).

E. Church, U.S. Army Research, Development, and Engineering Command, Picatinny Arsenal, Dover, N.J. 07801 (personal communication).

P. Richter, E. Lorincz, I. Peczeli, F. Engard, “Coherent optical measurement of surface roughness and its application in the paper industry,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 258–265 (1983).

ANSI/ASME B46.1-1985, Surface Texture (American Society of Mechanical Engineers, New York, 1985).

J. M. Lucas, “An on-line paper machine, sensor for roughness and other measurements,” presented at the International Symposium on Process Control, Vancouver, Canada, (1977).

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

E. Marx, J.-F. Song, T. V. Vorburger, T. R. Lettieri, “Light scattered by coated paper,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 826–834 (1990).

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

Certain commercial equipment, instruments, or materials are identified in this paper to specify adequately the experimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

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Figures (10)

Fig. 1
Fig. 1

Portion of the Talystep trace of sample 4.

Fig. 2
Fig. 2

Surface height distribution of the coating of sample 2.

Fig. 3
Fig. 3

Geometry for light scattering from glossy coated paper.

Fig. 4
Fig. 4

Closeup schematic diagram of the DALLAS apparatus.

Fig. 5
Fig. 5

Light-scattering patterns for uncoated paper: θi = −16°, −30°, and −46°.

Fig. 6
Fig. 6

Light-scattering patterns for four different locations on coated paper sample 2. The gap in the pattern at −30° is in the direction of the incident laser beam.

Fig. 7
Fig. 7

(a) Experimental light-scattering patterns for coated paper sample 4 for various angles of incidence θi; (b) calculated patterns.

Fig. 8
Fig. 8

Light-scattering patterns for all three coated paper samples: (a) θi= −16°; (b) θi= −60°.

Fig. 9
Fig. 9

High-resolution light-scattering patterns for all three coated paper samples: (a) measured; (b) computed using corrected stylus data; (c) computed using uncorrected stylus data; (d) computed using stylus data band limited in the spatial frequency.

Fig. 10
Fig. 10

Optical probe curves: (a) uncoated paper; (b) coated paper sample 3; and (c) blanchard ground metal surface.

Tables (3)

Tables Icon

Table I Reference Conditions for Mechanical Stylus Measurements

Tables Icon

Table II Surface Parameters from Mechanical Stylus Measurementa

Tables Icon

Table III Results of Optical Probe Measurements

Equations (12)

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

σ = [ 1 N k = 1 N z k 2 ] 1 / 2
C j = 1 σ 2 ( N j ) k = 1 N j z k z j + k , j = 0 , 1 , ,
C ( τ ) = exp ( | τ / T | α ) ,
ρ ( θ , ϕ ) = F ( θ , ϕ ) X X d x Y Y d y exp [ i v ( θ , ϕ ) r ( x , y ) ] ,
F ( θ , ϕ ) = 1 + cos θ i cos θ sin θ i sin θ cos ϕ cos θ i ( cos θ i + cos θ ) .
D ( θ , ϕ ) = 2 π κ [ F ( θ , ϕ ) ] 2 A 0 J 0 [ υ x y ( θ , ϕ ) τ ] exp { g ( θ ) [ 1 C ( τ ) ] } exp [ g ( θ ) ] τ d τ ,
υ x y ( θ , ϕ ) = 2 π λ ( sin 2 θ i + 2 sin θ i sin θ cos ϕ + sin 2 θ ) 1 / 2 ,
g ( θ ) = [ 2 π σ λ ( cos θ i + cos θ ) ] 2 ,
I ( θ ) = [ F ( θ , 0 ) ] 2 0 J 0 ( υ T τ ) × exp ( g ) { exp [ g exp ( τ α ) ] 1 } τ d τ ,
υ ( θ ) = ( 2 π / λ ) | sin θ i + sin θ | .
S N = K i ( θ i θ ̅ i ) 2 ( I i / I o ) ,
S N = K i θ i 2 ( I i I b I o N I b )

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