Abstract

This paper describes the use of the area structure function (SF) for the specification and characterization of optical surfaces. A two-quadrant area SF is introduced because the one-quadrant area SF does not completely describe surfaces with certain asymmetries. Area SF calculations of simulation data and of a diamond turned surface are shown and compared to area power spectral density (PSD) and area autocorrelation function (ACF) representations. The direct relationship between SF, PSD, and ACF for a stationary surface does not apply to non-stationary surfaces typical of optics with figure errors.

© 2012 OSA

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References

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  1. D. M. Aikens, C. R. Wolfe, and J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the national ignition facility,” Proc. SPIE 2576, 281–292 (1995).
    [Crossref]
  2. J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
    [Crossref]
  3. J. M. Tamkin and T. D. Milster, “Effects of structured mid-spatial frequency surface errors on image performance,” Appl. Opt. 49(33), 6522–6536 (2010).
    [Crossref] [PubMed]
  4. R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE 5869, 58690U, 58690U-11 (2005).
    [Crossref]
  5. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  6. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55(11), 1427–1435 (1965).
    [Crossref]
  7. R. E. Parks, “Specifications: figure and finish are not enough,” Proc. SPIE 7071, 70710B, 70710B-9 (2008).
    [Crossref]
  8. A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE 6671, 66710A, 66710A-10 (2007).
    [Crossref]
  9. D. J. Whitehouse, The Properties of Random Surfaces of Significance in their Contact (University of Leicester, 1971).
  10. R. S. Sayles and T. R. Thomas, “The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation,” Wear 42(2), 263–276 (1977).
    [Crossref]
  11. D. J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc.- Inst. Mech. Eng. 215(2), 207–210 (2001).
    [Crossref]
  12. T. R. Thomas and B. G. Rose’n, “Determination of the optimum sampling interval for rough contact mechanics,” Tribol. Int. 33(9), 601–610 (2000).
    [Crossref]
  13. T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear 232(1), 41–50 (1999).
    [Crossref]
  14. T. R. Thomas and B. G. Rose’n, “Surfaces generated by abrasive finishing processes as self-affine fractals,” Int. J. Surf. Sci. Eng. 3(4), 275–285 (2009).
    [Crossref]
  15. E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics. vol. I, M. Bass, ed. (McGraw-Hill, New York, 2009).
  16. E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32(19), 3344–3353 (1993).
    [Crossref] [PubMed]

2010 (1)

2009 (1)

T. R. Thomas and B. G. Rose’n, “Surfaces generated by abrasive finishing processes as self-affine fractals,” Int. J. Surf. Sci. Eng. 3(4), 275–285 (2009).
[Crossref]

2008 (1)

R. E. Parks, “Specifications: figure and finish are not enough,” Proc. SPIE 7071, 70710B, 70710B-9 (2008).
[Crossref]

2007 (1)

A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE 6671, 66710A, 66710A-10 (2007).
[Crossref]

2005 (1)

R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE 5869, 58690U, 58690U-11 (2005).
[Crossref]

2001 (1)

D. J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc.- Inst. Mech. Eng. 215(2), 207–210 (2001).
[Crossref]

2000 (1)

T. R. Thomas and B. G. Rose’n, “Determination of the optimum sampling interval for rough contact mechanics,” Tribol. Int. 33(9), 601–610 (2000).
[Crossref]

1999 (1)

T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear 232(1), 41–50 (1999).
[Crossref]

1995 (2)

D. M. Aikens, C. R. Wolfe, and J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the national ignition facility,” Proc. SPIE 2576, 281–292 (1995).
[Crossref]

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

1993 (1)

1977 (1)

R. S. Sayles and T. R. Thomas, “The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation,” Wear 42(2), 263–276 (1977).
[Crossref]

1965 (1)

Aikens, D. M.

D. M. Aikens, C. R. Wolfe, and J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the national ignition facility,” Proc. SPIE 2576, 281–292 (1995).
[Crossref]

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Amini, N.

T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear 232(1), 41–50 (1999).
[Crossref]

Burge, J. H.

A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE 6671, 66710A, 66710A-10 (2007).
[Crossref]

Church, E. L.

English, R. E.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Fried, D. L.

Gallagher, B. B.

R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE 5869, 58690U, 58690U-11 (2005).
[Crossref]

Hvisc, A. M.

A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE 6671, 66710A, 66710A-10 (2007).
[Crossref]

Lawson, J. K.

D. M. Aikens, C. R. Wolfe, and J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the national ignition facility,” Proc. SPIE 2576, 281–292 (1995).
[Crossref]

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Manes, K. R.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Milster, T. D.

Parks, R. E.

R. E. Parks, “Specifications: figure and finish are not enough,” Proc. SPIE 7071, 70710B, 70710B-9 (2008).
[Crossref]

Rose’n, B. G.

T. R. Thomas and B. G. Rose’n, “Surfaces generated by abrasive finishing processes as self-affine fractals,” Int. J. Surf. Sci. Eng. 3(4), 275–285 (2009).
[Crossref]

T. R. Thomas and B. G. Rose’n, “Determination of the optimum sampling interval for rough contact mechanics,” Tribol. Int. 33(9), 601–610 (2000).
[Crossref]

T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear 232(1), 41–50 (1999).
[Crossref]

Sayles, R. S.

R. S. Sayles and T. R. Thomas, “The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation,” Wear 42(2), 263–276 (1977).
[Crossref]

Stamper, B. L.

R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE 5869, 58690U, 58690U-11 (2005).
[Crossref]

Takacs, P. Z.

Tamkin, J. M.

Thomas, T. R.

T. R. Thomas and B. G. Rose’n, “Surfaces generated by abrasive finishing processes as self-affine fractals,” Int. J. Surf. Sci. Eng. 3(4), 275–285 (2009).
[Crossref]

T. R. Thomas and B. G. Rose’n, “Determination of the optimum sampling interval for rough contact mechanics,” Tribol. Int. 33(9), 601–610 (2000).
[Crossref]

T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear 232(1), 41–50 (1999).
[Crossref]

R. S. Sayles and T. R. Thomas, “The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation,” Wear 42(2), 263–276 (1977).
[Crossref]

Trenholme, J. B.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Whitehouse, D. J.

D. J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc.- Inst. Mech. Eng. 215(2), 207–210 (2001).
[Crossref]

Wolfe, C. R.

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

D. M. Aikens, C. R. Wolfe, and J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the national ignition facility,” Proc. SPIE 2576, 281–292 (1995).
[Crossref]

Youngworth, R. N.

R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE 5869, 58690U, 58690U-11 (2005).
[Crossref]

Appl. Opt. (2)

Int. J. Surf. Sci. Eng. (1)

T. R. Thomas and B. G. Rose’n, “Surfaces generated by abrasive finishing processes as self-affine fractals,” Int. J. Surf. Sci. Eng. 3(4), 275–285 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. SPIE (5)

R. E. Parks, “Specifications: figure and finish are not enough,” Proc. SPIE 7071, 70710B, 70710B-9 (2008).
[Crossref]

A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE 6671, 66710A, 66710A-10 (2007).
[Crossref]

R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE 5869, 58690U, 58690U-11 (2005).
[Crossref]

D. M. Aikens, C. R. Wolfe, and J. K. Lawson, “The use of power spectral density (PSD) functions in specifying optics for the national ignition facility,” Proc. SPIE 2576, 281–292 (1995).
[Crossref]

J. K. Lawson, C. R. Wolfe, K. R. Manes, J. B. Trenholme, D. M. Aikens, and R. E. English, “Specification of optical components using the power spectral density function,” Proc. SPIE 2536, 38–50 (1995).
[Crossref]

Proc.- Inst. Mech. Eng. (1)

D. J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc.- Inst. Mech. Eng. 215(2), 207–210 (2001).
[Crossref]

Tribol. Int. (1)

T. R. Thomas and B. G. Rose’n, “Determination of the optimum sampling interval for rough contact mechanics,” Tribol. Int. 33(9), 601–610 (2000).
[Crossref]

Wear (2)

T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear 232(1), 41–50 (1999).
[Crossref]

R. S. Sayles and T. R. Thomas, “The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation,” Wear 42(2), 263–276 (1977).
[Crossref]

Other (3)

E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics. vol. I, M. Bass, ed. (McGraw-Hill, New York, 2009).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

D. J. Whitehouse, The Properties of Random Surfaces of Significance in their Contact (University of Leicester, 1971).

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Figures (6)

Fig. 1
Fig. 1

Calculation of the linear SF. (a) Input map. (b) Linear SF as a function of separation.

Fig. 2
Fig. 2

Comparison of linear SFs. (a) Map 1. (b) Linear SF of map 1. (c) Map 2. (d) Linear SF of map 2.

Fig. 3
Fig. 3

Calculation of the area SF. (a) Input map. (b) Move the duplicated map in 2 directions. (c and d) area SF in 2 quadrants (the value is between 0 and 8 μm2).

Fig. 4
Fig. 4

Interpretation of the 2-quadrant SF calculation for a diamond-turned aluminum flat. (a) Input map. (b) Residual error after removing the first 36 Zernike terms. (c) Area SF for part (a) of the input map in 2 quadrants. (d) Area SF of the residual error, part (c) in 2 quadrants.

Fig. 5
Fig. 5

Comparison between SF, ACF and PSD for a polished surface after removing 36 Zernikes. (a) Area SF of the residual error. (b) Area ACF of the residual error. (c) Deviation = SF-2σ2(1-ACF). (d) Area PSD of the residual error.

Fig. 6
Fig. 6

Analysis of a diamond-turned aluminum flat. (a) Residual error after removing the first 36 Zernike terms. (b) Area SF. (c) Area ACF. (d) Area PSD.

Equations (8)

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S(r)= [z(r')z(r'+r)] 2 = ( λ 2π ) 2 6.88 ( r r 0 ) 5 3 ,
S( τ x , τ y )= 1 (m τ x )(n τ y ) i=1 m τ x j=1 n τ y {z(i,j)z(i+τ , x j+ τ y )} 2 ,
S( τ x , τ y )=E{ [ z(x,y)z(x+τ , x y+ τ y ) ] 2 }=E{ z 2 (x,y) }+E{ z 2 (x+τ , x y+ τ y ) }2E{ z(x,y)z(x+τ , x y+ τ y ) },
E{ z 2 (x,y) }=E{ z 2 (x+τ , x y+ τ y ) }=ψ(0,0)= σ 2 .
S( τ x , τ y )=2{ σ 2 ψ( τ x , τ y ) }.
S( τ x , τ y )=2 σ 2 { 1R( τ x , τ y ) }.
P( ω x , ω y )=F[ψ( τ x , τ y )],
S( τ x , τ y )=2{ σ 2 F [P( ω x , ω y )] 1 }.

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