## Fitting freeform shapes with orthogonal bases |

Optics Express, Vol. 21, Issue 16, pp. 19061-19081 (2013)

http://dx.doi.org/10.1364/OE.21.019061

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### Abstract

Orthogonality is exploited for fitting analytically-specified freeform shapes in terms of orthogonal polynomials. The end result is expressed in terms of FFTs coupled to a simple explicit form of Gaussian quadrature. Its efficiency opens the possibilities for proceeding to arbitrary numbers of polynomial terms. This is shown to create promising options for quantifying and filtering the mid-spatial frequency structure within circular domains from measurements of as-built parts.

© 2013 OSA

## 1. Introduction

1. A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt. **51**(15), 3054–3058 (2012), doi:. [CrossRef] [PubMed]

4. R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE **8169**, 81690X, 81690X-9 (2011), doi:. [CrossRef]

5. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express **20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

6. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. **2**(1), 97–109 (2012), doi:. [CrossRef]

5. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express **20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

5. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express **20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

6. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. **2**(1), 97–109 (2012), doi:. [CrossRef]

*normal departure*.

8. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express **18**(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]

## 2. Fourier methods for the azimuthal fit

10. See Section 12.3 of [7]. Or see Sec. 12.4 in http://apps.nrbook.com/empanel/index.html#

## 3. Rotationally symmetric radial fit (i.e. m = 0)

8. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express **18**(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]

8. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express **18**(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]

**18**(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]

**18**(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]

**18**(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]

## 4. Radial fit for m > 0

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

## 5. Demonstration

6. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. **2**(1), 97–109 (2012), doi:. [CrossRef]

**2**(1), 97–109 (2012), doi:. [CrossRef]

**2**(1), 97–109 (2012), doi:. [CrossRef]

**2**(1), 97–109 (2012), doi:. [CrossRef]

**2**(1), 97–109 (2012), doi:. [CrossRef]

## 6. Fitting mid-spatial frequencies

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

**2**(1), 97–109 (2012), doi:. [CrossRef]

### 6.1 Projections of pure sinusoids

^{−6}times the peak-to-valley of the sag. The PSS is evidently almost all at third order for the odd case, and at second and fourth order for the even case.

### 6.2 Frequency filtering the Q-spectra for a sinusoid

### 6.3 Demonstration with synthetic data

^{6}. In this case, it is evident in the plot at left in Fig. 8 that only about a half of the 23,000 fitted coefficients are large enough to be significant for these purposes, viz. those within

## 7. Concluding remarks

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

12. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express **18**(13), 13851–13862 (2010), doi:. [CrossRef] [PubMed]

## Appendix: Derivation of the radial fit

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

## References and links

1. | A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt. |

2. | I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol. |

3. | P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math. |

4. | R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE |

5. | G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express |

6. | C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. |

7. | W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, |

8. | G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express |

9. | M. Abramowitz and I. Stegun, |

10. | See Section 12.3 of [7]. Or see Sec. 12.4 in http://apps.nrbook.com/empanel/index.html# |

11. | J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen. |

12. | G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express |

13. | J. K. Lawson, J. M. Auerbach, R. E. English Jr, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup III, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(220.1250) Optical design and fabrication : Aspherics

(220.4610) Optical design and fabrication : Optical fabrication

(220.4830) Optical design and fabrication : Systems design

(220.4840) Optical design and fabrication : Testing

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: June 11, 2013

Manuscript Accepted: July 19, 2013

Published: August 2, 2013

**Citation**

G. W. Forbes, "Fitting freeform shapes with orthogonal bases," Opt. Express **21**, 19061-19081 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-19061

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### References

- A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt.51(15), 3054–3058 (2012), doi:. [CrossRef] [PubMed]
- I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:. [CrossRef]
- P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).
- R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:. [CrossRef]
- G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012). [CrossRef] [PubMed]
- C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:. [CrossRef]
- W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 15.4.
- G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010), doi:. [CrossRef] [PubMed]
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978). See 25.4.38.
- See Section 12.3 of [7]. Or see Sec. 12.4 in http://apps.nrbook.com/empanel/index.html#
- J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:. [CrossRef]
- G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010), doi:. [CrossRef] [PubMed]
- J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:. [CrossRef]

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