## Two-quadrant area structure function analysis for optical surface characterization |

Optics Express, Vol. 20, Issue 21, pp. 23275-23280 (2012)

http://dx.doi.org/10.1364/OE.20.023275

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### 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

## 1. Introduction

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]

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]

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]

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]

6. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. **55**(11), 1427–1435 (1965). [CrossRef]

*r'*,

*z(r')*is the phase at position

*r'*,

*z(r' + r)*is the phase a distance

*r*away from the point

*r'*, and

*r*is the correlation length that depends on the wavelength and scales to the 6/5 power (for a default wavelength of 0.5 μm) [7

_{0}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]

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]

*τ*are selected, their squared height differences calculated, and then averaged to obtain SF(

_{1}*τ*

_{1}). This process is repeated for all separations

*τ*within the measurement area, generating the SF (Fig. 1(b)). The structure function can be computed over any chosen dynamic range for any aperture shape, albeit the number of points contributing to the average decreases as

*τ*tends toward the aperture size. Separation into figure, roughness and mid-spatial frequencies can still be done but it is not necessary. The linear SF compresses 2D height information to 1D and consequently is insensitive to anisotropic structure. Figure 2 shows calculated linear SFs for two quite different surfaces generated with unit coefficients for two different Zernike terms. The two surfaces in Fig. 2(a) and 2(c) have markedly different azimuthal frequencies, but the two linear SFs are almost indistinguishable.

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]

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]

## 2. Calculation of the area structure function

*m*equally spaced points in the

*x*direction and

*n*in the

*y*direction;

*m*; and

*n*[14

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]

*x*- and

*y*-axes require the two-quadrant area SF for a complete description of the spatial content.

## 3. Physical interpretation of the area structure function

### 3.1 Simulation data

### 3.2 Diamond turned surface

*τ*and

_{x}*τ*for which

_{y}## 4. Area structure function compared to other area analyses

### 4.1 Stationary surface

*E{ }*indicates an expectation. The function

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]

*z*value (piston). Therefore,The ACF is defined as

*τ*0 and shows local maxima for

_{x}= τ_{y}=*τ*and

_{x}*τ*length scales for which the surface has periodicity. Letting

_{y}*R*(

*τ*,

_{x}*τ*) represent the ACF, the expression of the SF (Eq. (5)) can be written as

_{y}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]

### 4.2 Non-stationary surfaces

**42**(2), 263–276 (1977). [CrossRef]

## 5. Conclusions and future work

## Acknowledgments

## References and links

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 |

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 |

3. | J. M. Tamkin and T. D. Milster, “Effects of structured mid-spatial frequency surface errors on image performance,” Appl. Opt. |

4. | R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE |

5. | V. I. Tatarski, |

6. | D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. |

7. | R. E. Parks, “Specifications: figure and finish are not enough,” Proc. SPIE |

8. | A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE |

9. | D. J. Whitehouse, |

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 |

11. | D. J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc.- Inst. Mech. Eng. |

12. | T. R. Thomas and B. G. Rose’n, “Determination of the optimum sampling interval for rough contact mechanics,” Tribol. Int. |

13. | T. R. Thomas, B. G. Rose’n, and N. Amini, “Fractal characterisation of the anisotropy of rough surfaces,” Wear |

14. | T. R. Thomas and B. G. Rose’n, “Surfaces generated by abrasive finishing processes as self-affine fractals,” Int. J. Surf. Sci. Eng. |

15. | E. L. Church and P. Z. Takacs, “Surface scattering,” in |

16. | E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(120.6660) Instrumentation, measurement, and metrology : Surface measurements, roughness

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 26, 2012

Revised Manuscript: September 4, 2012

Manuscript Accepted: September 19, 2012

Published: September 25, 2012

**Citation**

Liangyu He, Chris J. Evans, and Angela Davies, "Two-quadrant area structure function analysis for optical surface characterization," Opt. Express **20**, 23275-23280 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23275

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### References

- 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. SPIE2576, 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. SPIE2536, 38–50 (1995). [CrossRef]
- 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]
- R. N. Youngworth, B. B. Gallagher, and B. L. Stamper, “An overview of power spectral density (PSD) calculations,” Proc. SPIE5869, 58690U, 58690U-11 (2005). [CrossRef]
- V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
- D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am.55(11), 1427–1435 (1965). [CrossRef]
- R. E. Parks, “Specifications: figure and finish are not enough,” Proc. SPIE7071, 70710B, 70710B-9 (2008). [CrossRef]
- A. M. Hvisc and J. H. Burge, “Structure function analysis of mirror fabrication and support errors,” Proc. SPIE6671, 66710A, 66710A-10 (2007). [CrossRef]
- D. J. Whitehouse, The Properties of Random Surfaces of Significance in their Contact (University of Leicester, 1971).
- R. S. Sayles and T. R. Thomas, “The spatial representation of surface roughness by means of the structure function: a practical alternative to correlation,” Wear42(2), 263–276 (1977). [CrossRef]
- D. J. Whitehouse, “Some theoretical aspects of structure functions, fractal parameters and related subjects,” Proc.- Inst. Mech. Eng.215(2), 207–210 (2001). [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,” Wear232(1), 41–50 (1999). [CrossRef]
- 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]
- E. L. Church and P. Z. Takacs, “Surface scattering,” in Handbook of Optics. vol. I, M. Bass, ed. (McGraw-Hill, New York, 2009).
- 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]

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