Abstract

Surface finish measurements are usually fitted to models of the finish correlation function which are parametrized in terms of root-mean-square roughnesses, σ, and correlation lengths, l. Highly finished optical surfaces, however, are frequently better described by fractal models, which involve inverse power-law spectra and are parametrized by spectral strengths, Kn, and spectral indices, n. Analyzing measurements of fractal surfaces in terms of σ and l gives results which are not intrinsic surface parameters but which depend on the bandwidth parameters of the measurement process used. This paper derives expressions for these pseudoparameters and discusses the errors involved in using them for the characterization and specification of surface finish.

© 1988 Optical Society of America

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References

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  1. 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 (1976); J. M. Elson, J. M. Bennett, “Relation Between the Angular Dependence of Scattering and the Statistical Properties of Optical Surfaces,” J. Opt. Soc. Am. 69, 31 (1979); J. M. Elson, J. P. Rahn, J. M. Bennett, “Light Scattering from Multilayer Optics: Comparison of Theory and Experiment,” Appl. Opt. 19, 669 (1980); J. M. Bennett, J. H. Dancy, “Stylus Profiling Instrument for Measuring Statistical Properties of Smooth Optical Surfaces,” Appl. Opt. 20, 1785 (1981); J. M. Elson, H. E. Bennett, “Image Degradation Caused by Direct Scatter from Optical Components into the Image Plane,” Proc. Soc. Photo-Opt. Instrum. Eng. 511, 7 (1984); J. M. Bennett, J. J. Shaffer, Y. Shibano, Y. Namba, “Float Polishing of Optical Materials,” Appl. Opt. 26, 696 (1987).
    [CrossRef] [PubMed]
  2. R. J. Noll, P. Glenn, “Mirror Surface Autocovariance Functions and Their Associated Visible Scattering,” Appl. Opt. 21, 1824 (1982).
    [CrossRef] [PubMed]
  3. E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986). This paper discusses a number of definitions of the correlation length in addition to those in this paper.
  4. E. L. Church, “The Precision Measurement and Characterization of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 86 (1983).
  5. E. L. Church, G. M. Sanger, P. Z. Takacs, “The Comparison Between WYKO and TIS Measurements of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 749, 65 (1987).
  6. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979). (It is the n = 3 rather than the n = 2 fractal which is invariant under isotropic magnification.)
    [CrossRef]
  7. E. L. Church, “The Role of Surface Topography in X-Ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 196 (1979).
  8. E. L. Church, “Interpretation of High-Resolution X-Ray Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 257, 254 (1980). (Numerical results for an 1/f spectrum contain an error.)
  9. P. Z. Takacs, E. L. Church, “Comparison of Profiler Measurements Using Different Magnification Objectives,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 132 (1986).
  10. E. L. Church, “Statistical Fluctuations of Total Integrated Scatter Measurements,” J. Opt. Soc. Am. 71, 1602 (1981).
  11. E. L. Church, “Statistical Effects in the Measurement and Characterization of Smooth Scattering Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 511, 18 (1984).
  12. M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979); E. Jakeman, “Fresnel Scattering by a Corrugated Random Surface with Fractal Slope,” J. Opt. Soc. Am. 72, 1034 (1982). Much elegant work has been done on fractal scattering by British workers. These papers and the numerous references therein provide an introduction to that large body of literature.
    [CrossRef]
  13. E. L. Church, P. Z. Takacs, “The Interpretation of Glancing-Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 126 (1986); E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 107 (1986); H. Kunieda et al., “Roughness Measurement of X-Ray Mirror Surfaces,” Jpn. J. Appl. Phys. 25, 1292 (1986).
    [CrossRef]
  14. R. S. Sayles, T. R. Thomas, “Surface Topography as a Nonstationary Random Process,” Nature London 271, 431 (1978); Nature London 273, 573 (1978); M. V. Berry, J. H. Hannay, “Topography of Rough Surfaces,” Nature London 273, 573 (1978).
    [CrossRef]
  15. E. L. Church, “Small-Angle Scattering from Smooth Surfaces,” J. Opt. Soc. Am. 70, 1592 (1980).
  16. E. L. Church, “Direct Comparison of Mechanical and Optical Measurements of Precision Machined Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 105 (1983).
  17. E. L. O’Neill, A. Walther, “Problem in the Determination of Correlation Functions,” J. Opt. Soc. Am. 67, 1125 (1977); E. Freniere, E. L. O’Neill, A. Walther, “Problem in the Determination of Correlation Functions. II,” J. Opt. Soc. Am. 69, 634 (1979).
    [CrossRef]
  18. E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision Machined Optical Surfaces,” Opt. Eng. 24, 388 (1985).
    [CrossRef]
  19. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  20. H. Hogrefe, C. Kunz, “Soft X-Ray Scattering from Rough Surfaces: Experimental and Theoretical Analysis,” Appl. Opt. 26, 2851 (1987).
    [CrossRef] [PubMed]
  21. Detrending effects enter into scattering measurements by various means. The structure function is independent of piston displacements, but tilt detrending is accomplished by adjusting the optic axis to the specular direction for each measurement, and curvature can be removed by focusing the incident beam. In the smooth-surface limit these procedures are equivalent to ordinary least-squares detrending as described in the text for stylus measurements.
  22. To a first approximation a detrended fractal surface can be viewed as a conventional surface with a structure function corresponding to the sharp-cutoff spectrum given in Eq. (15). This suggests quantitative if not qualitative changes in the scattering pattern. A related but different question is the validity of the phase-screen model for fractal surfaces since that model presumes small surface slopes.

1987 (2)

E. L. Church, G. M. Sanger, P. Z. Takacs, “The Comparison Between WYKO and TIS Measurements of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 749, 65 (1987).

H. Hogrefe, C. Kunz, “Soft X-Ray Scattering from Rough Surfaces: Experimental and Theoretical Analysis,” Appl. Opt. 26, 2851 (1987).
[CrossRef] [PubMed]

1986 (3)

E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986). This paper discusses a number of definitions of the correlation length in addition to those in this paper.

P. Z. Takacs, E. L. Church, “Comparison of Profiler Measurements Using Different Magnification Objectives,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 132 (1986).

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing-Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 126 (1986); E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 107 (1986); H. Kunieda et al., “Roughness Measurement of X-Ray Mirror Surfaces,” Jpn. J. Appl. Phys. 25, 1292 (1986).
[CrossRef]

1985 (1)

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision Machined Optical Surfaces,” Opt. Eng. 24, 388 (1985).
[CrossRef]

1984 (1)

E. L. Church, “Statistical Effects in the Measurement and Characterization of Smooth Scattering Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 511, 18 (1984).

1983 (2)

E. L. Church, “The Precision Measurement and Characterization of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 86 (1983).

E. L. Church, “Direct Comparison of Mechanical and Optical Measurements of Precision Machined Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 105 (1983).

1982 (1)

1981 (1)

E. L. Church, “Statistical Fluctuations of Total Integrated Scatter Measurements,” J. Opt. Soc. Am. 71, 1602 (1981).

1980 (2)

E. L. Church, “Interpretation of High-Resolution X-Ray Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 257, 254 (1980). (Numerical results for an 1/f spectrum contain an error.)

E. L. Church, “Small-Angle Scattering from Smooth Surfaces,” J. Opt. Soc. Am. 70, 1592 (1980).

1979 (3)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979); E. Jakeman, “Fresnel Scattering by a Corrugated Random Surface with Fractal Slope,” J. Opt. Soc. Am. 72, 1034 (1982). Much elegant work has been done on fractal scattering by British workers. These papers and the numerous references therein provide an introduction to that large body of literature.
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979). (It is the n = 3 rather than the n = 2 fractal which is invariant under isotropic magnification.)
[CrossRef]

E. L. Church, “The Role of Surface Topography in X-Ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 196 (1979).

1978 (1)

R. S. Sayles, T. R. Thomas, “Surface Topography as a Nonstationary Random Process,” Nature London 271, 431 (1978); Nature London 273, 573 (1978); M. V. Berry, J. H. Hannay, “Topography of Rough Surfaces,” Nature London 273, 573 (1978).
[CrossRef]

1977 (1)

1976 (1)

Beckmann, P.

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

Bennett, J. M.

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979); E. Jakeman, “Fresnel Scattering by a Corrugated Random Surface with Fractal Slope,” J. Opt. Soc. Am. 72, 1034 (1982). Much elegant work has been done on fractal scattering by British workers. These papers and the numerous references therein provide an introduction to that large body of literature.
[CrossRef]

Church, E. L.

E. L. Church, G. M. Sanger, P. Z. Takacs, “The Comparison Between WYKO and TIS Measurements of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 749, 65 (1987).

E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986). This paper discusses a number of definitions of the correlation length in addition to those in this paper.

P. Z. Takacs, E. L. Church, “Comparison of Profiler Measurements Using Different Magnification Objectives,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 132 (1986).

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing-Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 126 (1986); E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 107 (1986); H. Kunieda et al., “Roughness Measurement of X-Ray Mirror Surfaces,” Jpn. J. Appl. Phys. 25, 1292 (1986).
[CrossRef]

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision Machined Optical Surfaces,” Opt. Eng. 24, 388 (1985).
[CrossRef]

E. L. Church, “Statistical Effects in the Measurement and Characterization of Smooth Scattering Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 511, 18 (1984).

E. L. Church, “The Precision Measurement and Characterization of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 86 (1983).

E. L. Church, “Direct Comparison of Mechanical and Optical Measurements of Precision Machined Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 105 (1983).

E. L. Church, “Statistical Fluctuations of Total Integrated Scatter Measurements,” J. Opt. Soc. Am. 71, 1602 (1981).

E. L. Church, “Interpretation of High-Resolution X-Ray Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 257, 254 (1980). (Numerical results for an 1/f spectrum contain an error.)

E. L. Church, “Small-Angle Scattering from Smooth Surfaces,” J. Opt. Soc. Am. 70, 1592 (1980).

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979). (It is the n = 3 rather than the n = 2 fractal which is invariant under isotropic magnification.)
[CrossRef]

E. L. Church, “The Role of Surface Topography in X-Ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 196 (1979).

Glenn, P.

Hogrefe, H.

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979). (It is the n = 3 rather than the n = 2 fractal which is invariant under isotropic magnification.)
[CrossRef]

Kunz, C.

Noll, R. J.

O’Neill, E. L.

Sanger, G. M.

E. L. Church, G. M. Sanger, P. Z. Takacs, “The Comparison Between WYKO and TIS Measurements of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 749, 65 (1987).

Sayles, R. S.

R. S. Sayles, T. R. Thomas, “Surface Topography as a Nonstationary Random Process,” Nature London 271, 431 (1978); Nature London 273, 573 (1978); M. V. Berry, J. H. Hannay, “Topography of Rough Surfaces,” Nature London 273, 573 (1978).
[CrossRef]

Spizzichino, A.

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

Takacs, P. Z.

E. L. Church, G. M. Sanger, P. Z. Takacs, “The Comparison Between WYKO and TIS Measurements of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 749, 65 (1987).

P. Z. Takacs, E. L. Church, “Comparison of Profiler Measurements Using Different Magnification Objectives,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 132 (1986).

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing-Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 126 (1986); E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 107 (1986); H. Kunieda et al., “Roughness Measurement of X-Ray Mirror Surfaces,” Jpn. J. Appl. Phys. 25, 1292 (1986).
[CrossRef]

Thomas, T. R.

R. S. Sayles, T. R. Thomas, “Surface Topography as a Nonstationary Random Process,” Nature London 271, 431 (1978); Nature London 273, 573 (1978); M. V. Berry, J. H. Hannay, “Topography of Rough Surfaces,” Nature London 273, 573 (1978).
[CrossRef]

Vorburger, T. V.

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision Machined Optical Surfaces,” Opt. Eng. 24, 388 (1985).
[CrossRef]

Walther, A.

Wyant, J. C.

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision Machined Optical Surfaces,” Opt. Eng. 24, 388 (1985).
[CrossRef]

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979). (It is the n = 3 rather than the n = 2 fractal which is invariant under isotropic magnification.)
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

E. L. O’Neill, A. Walther, “Problem in the Determination of Correlation Functions,” J. Opt. Soc. Am. 67, 1125 (1977); E. Freniere, E. L. O’Neill, A. Walther, “Problem in the Determination of Correlation Functions. II,” J. Opt. Soc. Am. 69, 634 (1979).
[CrossRef]

E. L. Church, “Statistical Fluctuations of Total Integrated Scatter Measurements,” J. Opt. Soc. Am. 71, 1602 (1981).

E. L. Church, “Small-Angle Scattering from Smooth Surfaces,” J. Opt. Soc. Am. 70, 1592 (1980).

J. Phys. A (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781 (1979); E. Jakeman, “Fresnel Scattering by a Corrugated Random Surface with Fractal Slope,” J. Opt. Soc. Am. 72, 1034 (1982). Much elegant work has been done on fractal scattering by British workers. These papers and the numerous references therein provide an introduction to that large body of literature.
[CrossRef]

Nature London (1)

R. S. Sayles, T. R. Thomas, “Surface Topography as a Nonstationary Random Process,” Nature London 271, 431 (1978); Nature London 273, 573 (1978); M. V. Berry, J. H. Hannay, “Topography of Rough Surfaces,” Nature London 273, 573 (1978).
[CrossRef]

Opt. Eng. (2)

E. L. Church, T. V. Vorburger, J. C. Wyant, “Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision Machined Optical Surfaces,” Opt. Eng. 24, 388 (1985).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979). (It is the n = 3 rather than the n = 2 fractal which is invariant under isotropic magnification.)
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (9)

E. L. Church, “The Role of Surface Topography in X-Ray Scattering,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 196 (1979).

E. L. Church, “Interpretation of High-Resolution X-Ray Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 257, 254 (1980). (Numerical results for an 1/f spectrum contain an error.)

P. Z. Takacs, E. L. Church, “Comparison of Profiler Measurements Using Different Magnification Objectives,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 132 (1986).

E. L. Church, “Statistical Effects in the Measurement and Characterization of Smooth Scattering Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 511, 18 (1984).

E. L. Church, “Comments on the Correlation Length,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 102 (1986). This paper discusses a number of definitions of the correlation length in addition to those in this paper.

E. L. Church, “The Precision Measurement and Characterization of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 86 (1983).

E. L. Church, G. M. Sanger, P. Z. Takacs, “The Comparison Between WYKO and TIS Measurements of Surface Finish,” Proc. Soc. Photo-Opt. Instrum. Eng. 749, 65 (1987).

E. L. Church, P. Z. Takacs, “The Interpretation of Glancing-Incidence Scattering Measurements,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 126 (1986); E. L. Church, P. Z. Takacs, “Statistical and Signal Processing Concepts in Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 645, 107 (1986); H. Kunieda et al., “Roughness Measurement of X-Ray Mirror Surfaces,” Jpn. J. Appl. Phys. 25, 1292 (1986).
[CrossRef]

E. L. Church, “Direct Comparison of Mechanical and Optical Measurements of Precision Machined Surfaces,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 105 (1983).

Other (3)

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

Detrending effects enter into scattering measurements by various means. The structure function is independent of piston displacements, but tilt detrending is accomplished by adjusting the optic axis to the specular direction for each measurement, and curvature can be removed by focusing the incident beam. In the smooth-surface limit these procedures are equivalent to ordinary least-squares detrending as described in the text for stylus measurements.

To a first approximation a detrended fractal surface can be viewed as a conventional surface with a structure function corresponding to the sharp-cutoff spectrum given in Eq. (15). This suggests quantitative if not qualitative changes in the scattering pattern. A related but different question is the validity of the phase-screen model for fractal surfaces since that model presumes small surface slopes.

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Figures (5)

Fig. 1
Fig. 1

Periodogram estimate of the power spectrum of a fused-silica surface taken with a Wyko profiling microscope with a 2.5× objective. Its inverse power-law form may be characterized by the fractal parameters n = 1.25 and Kn = 4(−9) (μm)3−n or, equivalently, by D = 1.875 and T = 7.3(−5) μm. (Figure from Ref. 9.)

Fig. 2
Fig. 2

Calculated power spectra of a Brownian fractal surface subjected to various types of detrending of the individual profile traces. A denotes piston only; B, piston and tilt; and C, piston, tilt, and quadratic curvature. The vertical axis is S(fx)/K2L2 and the horizontal axis is fxL.

Fig. 3
Fig. 3

Power spectrum of a Brownian fractal surface according to the sharp-cutoff model discussed in the text. The scales are the same as in Fig. 2.

Fig. 4
Fig. 4

Calculated correlation functions of a Brownian fractal surface obtained by taking the Fourier transformation of the spectra shown in Fig. 2. The vertical axis is C(τ)/K2L and the horizontal axis is τ/L. A, B, and C denote the different types of detrending explained in the caption to Fig. 2.

Fig. 5
Fig. 5

Correlation function of a Brownian fractal surface according to the sharp-cutoff model. This result is the Fourier transformation of the spectrum shown in Fig. 3. The scales are the same as in Fig. 4.

Tables (1)

Tables Icon

Table I Calculated Values of the Measured Finish Parameters and Coefficients of Variation for a Brownian Fractal Surface Subjected to Different Types of Detrending, A, B, and Ca

Equations (53)

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C ( τ ) = lim L 1 L 0 L τ d x Z ( x ) Z ( x + τ ) ,
S ( f x ) = lim L 2 L | L / 2 + L / 2 d x Z ( x ) exp ( i 2 π f x x ) | 2 .
C ( τ ) = j σ j 2 exp ( | τ | / l j ) + k σ k 2 exp [ ( τ / l k ) 2 ] .
σ 2 = C ( 0 ) = 0 d f x S ( f x ) ,
l = 2 σ 2 0 d τ C 2 ( τ ) = 1 2 σ 4 0 d f x S 2 ( f x ) .
S ( f x ) = K n / f x n with 1 < n < 3 ,
S ( f ) = Γ [ ( n + 1 ) / 2 ] 2 Γ ( 1 2 ) Γ ( n / 2 ) · K n f x n + 1 .
D = ( 5 n ) / 2 ,
T 3 n = 1 2 ( 2 π ) n Γ ( n ) cos ( n π / 2 ) K n .
Z r ( x ) = Z ( x ) + [ A + B x + C x 2 ] ,
Z m ( x ) = Z ( x ) [ a + b x + c x 2 ] ,
S m ( f x ) = L + d f x S ( f x ) K 2 ( f x , f x ) ,
K ( f x , f x ) = j 0 ( π [ f x f x ] L ) υ = 0 μ ( 2 υ + 1 ) j υ ( π f x L ) j υ ( π f x L ) ,
S m ( f x ) = S ( f x ) · T ( f x ) .
S m ( f x ) { 0 0 f x < 1 / L , k / f x n 1 / L < f x < ,
σ m = [ K n L n 1 n 1 ] 1 / 2 ,
l m = ( n 1 ) 2 L 2 ( 2 n 1 ) .
C ( τ ) = σ 0 2 exp ( | τ | / l 0 )
S ( f x ) = 4 σ 0 2 l 0 / [ 1 + ( 2 π l 0 f x ) 2 ] ,
K 2 = σ 0 2 / π 2 l 0 .
γ ( Q ) = rms value of Q average value of Q .
γ ( σ m 2 ) = ( 2 l f / L ) 1 / 2 ,
l f = 2 σ m 2 0 L d τ ( 1 τ L ) C 2 ( τ ) .
γ ( σ m 2 ) ( n 1 ) / ( 2 n 1 ) 1 / 2 .
1 I 0 R d I d θ s = F · P * + d τ exp ( i α τ ) exp [ 1 2 β 2 D ( τ ) ] ,
α = 2 π λ ( sin θ s sin θ i ) ,
β = 2 π λ ( cos θ s cos θ i ) ,
D ( τ ) = lim L 1 L 0 L τ d x [ Z ( x + τ ) Z ( x ) ] 2 = 4 0 d f x S ( f x ) sin 2 ( π f x τ ) .
D ( τ ) = 2 [ C ( 0 ) C ( τ ) ] ,
1 I 0 R d I d θ s = P * exp ( g ) · + d τ exp ( i α τ ) exp [ β 2 C ( τ ) ] .
g = ( 4 π σ λ cos θ i ) 2 1 .
1 I 0 R d I d θ s P * [ δ ( α 2 π ) + 1 2 β 2 S ( α 2 π ) ] .
D ( τ ) = T 2 | τ / T | n 1 ,
1 I 0 R d I d θ s = P * + d τ exp ( i α τ ) exp ( 1 2 β 2 T 2 | τ T | n 1 ) ,
1 I 0 R d I d θ s = P * 4 β 2 T ( β 2 T ) 2 + ( 2 α ) 2 ,
σ m = 1 / β and l m = 2 / β 2 T .
σ = [ 1 2 T 3 n L n 1 ] 1 / 2 ,
1 I 0 R d I d θ s = P * [ δ ( α 2 π ) + 1 2 β 2 K n ( 2 π α ) n ] ,
Strehl factor = [ Eq . ( 25 ) ] α = 0 / [ Eq . ( 25 ) ] α = β = 0 .
Strehl factor = exp ( g )
Strehl factor = 2 g 2 [ exp ( g ) 1 + g ] ,
σ = [ 1 2 T L ] 1 / 2 .
C ( τ ) = 0 d f x S ( f x ) cos ( 2 π f x τ ) ,
S ( f x ) = 4 0 d τ C ( τ ) cos ( 2 π f x τ ) .
C ( τ ) A = π 2 3 K 2 L ( 1 5 X + 6 X 2 2 X 3 ) ,
C ( τ ) B = π 2 15 K 2 L ( 2 19 X + 45 X 2 34 X 3 + 6 X 5 ) ,
C ( τ ) C = π 2 35 K 2 L ( 3 41 X + 140 X 2 156 X 3 + 45 X 5 20 X 7 ) ,
S ( f x ) A = 4 π 2 3 K 2 L 2 a 4 [ ( 12 + a 2 ) cos a + ( 12 + 5 a 2 ) ] ,
S ( f x ) B = 8 π 2 15 K 2 L 2 a 6 [ ( 360 a 3 a 3 ) sin a + ( 360 78 a 2 1 2 a 4 ) cos a + ( 360 102 a 2 + 19 2 a 4 ) ] ,
S ( f x ) C = 12 π 2 35 K 2 L 2 a 8 [ ( 33600 a 2640 a 3 + 16 3 a 5 ) sin a + ( 33600 13840 a 2 + 232 a 4 + 1 3 a 6 ) cos a + ( 33600 2960 a 2 312 a 4 + 41 3 a 6 ) ] ,
S ( f m ) A = K 2 L 2 2 m 2 ,
S ( f m ) B = K 2 L 2 6 / 5 m 2 [ 1 5 ( m π ) 2 ] ,
S ( f m ) C = K 2 L 2 6 / 5 m 2 [ 1 10 / 7 ( m π ) 2 75 ( m π ) 4 ] .

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