## Characterizing the shape of freeform optics |

Optics Express, Vol. 20, Issue 3, pp. 2483-2499 (2012)

http://dx.doi.org/10.1364/OE.20.002483

Acrobat PDF (3456 KB)

### Abstract

A recently introduced method for characterizing the shape of rotationally symmetric aspheres is generalized here for application to a wide class of freeform optics. New sets of orthogonal polynomials are introduced along with robust and efficient algorithms for computing the surface shape as well as its derivatives of any order. By construction, the associated characterization offers a rough interpretation of shape at a glance and it facilitates a range of estimates of manufacturability.

© 2012 OSA

## 1. Introduction

4. W. T. Plummer, J. G. Baker, and J. Van Tassell, “Photographic optical systems with nonrotational aspheric surfaces,” Appl. Opt. **38**(16), 3572–3592 (1999). [CrossRef] [PubMed]

5. L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE **8167**, 81670M (2011). [CrossRef]

6. F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE **7061**, 70610G, 70610G-9 (2008). [CrossRef]

7. K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE **7652**, 76520C, 76520C-7 (2010). [CrossRef]

9. A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE **8167**, 816703, 816703-10 (2011). [CrossRef]

10. 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). [CrossRef]

11. P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. **50**(6), 822–828 (2011). [CrossRef] [PubMed]

## 2. Coordinates, aperture shapes, and sag expressions

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

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

## 3. Orthogonalization in terms of the mean square gradient

12. G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express **19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

12. G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express **19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

12. G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express **19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

13. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express **15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

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

15. C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials,” Opt. Express **15**(26), 18014–18024 (2007). [CrossRef] [PubMed]

## 4. Simple surface for demonstration

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

*y*means that

## 5. The option for a non-zero conic constant

**19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

**19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

**19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

**19**(10), 9923–9941 (2011). [CrossRef] [PubMed]

## 6. Concluding remarks

13. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express **15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

## Appendix A: Auxiliary polynomials

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

*m*and

*n*. As in [14

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

*m*and then in

*n*, by using

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

*m*— these constants are determined by starting with

*m*and

*n*. For verification of any implementation of these processes, I note that the initial sub-blocks of all the entities appearing in Eq. (A.18) are given by

## Appendix B: Additional recurrence-based processes

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

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

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

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

## Acknowledgments

## References and links

1. | H. J. Birchall, “Lenses and their combination and arrangement in various instruments and apparatus,” U.S. patent 2,001,952 (21 May 1935). |

2. | H. J. Birchall, “Lens of variable focal power having surfaces of involute form,” U.S. patent 2,475,275 (7 March 1949). |

3. | C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. patent 2,878,721 (24 March 1959). |

4. | W. T. Plummer, J. G. Baker, and J. Van Tassell, “Photographic optical systems with nonrotational aspheric surfaces,” Appl. Opt. |

5. | L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE |

6. | F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE |

7. | K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE |

8. | J. R. Rogers, “A comparison of anamorphic, keystone, and Zernike surface types for aberration correction,” Proc. SPIE |

9. | A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE |

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

11. | P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt. |

12. | G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express |

13. | G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express |

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

15. | C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials,” Opt. Express |

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

**OCIS Codes**

(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: November 21, 2011

Revised Manuscript: January 4, 2012

Manuscript Accepted: January 6, 2012

Published: January 19, 2012

**Citation**

G. W. Forbes, "Characterizing the shape of freeform optics," Opt. Express **20**, 2483-2499 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2483

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### References

- H. J. Birchall, “Lenses and their combination and arrangement in various instruments and apparatus,” U.S. patent 2,001,952 (21 May 1935).
- H. J. Birchall, “Lens of variable focal power having surfaces of involute form,” U.S. patent 2,475,275 (7 March 1949).
- C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. patent 2,878,721 (24 March 1959).
- W. T. Plummer, J. G. Baker, and J. Van Tassell, “Photographic optical systems with nonrotational aspheric surfaces,” Appl. Opt.38(16), 3572–3592 (1999). [CrossRef] [PubMed]
- L. Wang, P. Benítez, J. C. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE8167, 81670M (2011). [CrossRef]
- F. Muñoz, P. Benítez, and J. C. Miñano, “High-order aspherics: the SMS nonimaging design method applied to imaging optics,” Proc. SPIE7061, 70610G, 70610G-9 (2008). [CrossRef]
- K. H. Fuerschbach, K. P. Thompson, and J. P. Rolland, “A new generation of optical systems with φ-polynomial surfaces,” Proc. SPIE7652, 76520C, 76520C-7 (2010). [CrossRef]
- J. R. Rogers, “A comparison of anamorphic, keystone, and Zernike surface types for aberration correction,” Proc. SPIE7652, 76520B, 76520B-8 (2010). [CrossRef]
- A. Yabe, “Method to allocate freeform surfaces in axially asymmetric optical systems,” Proc. SPIE8167, 816703, 816703-10 (2011). [CrossRef]
- 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). [CrossRef]
- P. Jester, C. Menke, and K. Urban, “B-spline representation of optical surfaces and its accuracy in a ray trace algorithm,” Appl. Opt.50(6), 822–828 (2011). [CrossRef] [PubMed]
- G. W. Forbes, “Manufacturability estimates for optical aspheres,” Opt. Express19(10), 9923–9941 (2011). [CrossRef] [PubMed]
- G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express15(8), 5218–5226 (2007). [CrossRef] [PubMed]
- G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010). [CrossRef] [PubMed]
- C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials,” Opt. Express15(26), 18014–18024 (2007). [CrossRef] [PubMed]
- G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010). [CrossRef] [PubMed]

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