## Manufacturability estimates for optical aspheres |

Optics Express, Vol. 19, Issue 10, pp. 9923-9941 (2011)

http://dx.doi.org/10.1364/OE.19.009923

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### Abstract

Some simple measures of the difficulty of a variety of steps in asphere fabrication are defined by reference to fundamental geometric considerations. It is shown that effective approximations can then be exploited when an asphere’s shape is characterized by using a particular orthogonal basis. The efficiency of the results allows them to be used not only as quick manufacturability estimates at the production end, but more importantly as part of an efficient design process that can boost the resulting optical systems’ cost-effectiveness.

© 2011 OSA

## 1. Introduction

7. C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE **5494**, 113–121 (2004). [CrossRef]

8. G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics,” SPIE Optifab (2009), paper TD06–25 (1), http://www.qedmrf.com.

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

## 2. A tailored characterization of shape

*c*and conic constant

*κ*, this takes the form

*m*. Notice that the added deviation from the conic vanishes at both the aperture’s center and its edge (where

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

*c*is then the curvature of the best-fit sphere.

*z*to displacement along the normal. The particular family of polynomials treated in [9

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

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

*λ*on an

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

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

## 3. Benefits of the option for a non-zero conic constant

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

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

**15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

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

3. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. **39**(13), 2198–2209 (2000). [CrossRef]

**15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

**15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

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

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

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

## 4. Significance of the principal curvatures

*j*th coefficient from the analogous expansion for

### 4.1 Pad polishing

7. C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE **5494**, 113–121 (2004). [CrossRef]

*T*and a spherical face of curvature

*T*in order to obtain the size of the largest tool that respects some maximal acceptable level of

### 4.2 Subaperture stitching

*χ*gets larger, the test becomes slower and the end result’s measurement uncertainty grows.

*x*axis is taken to lie in the asphere’s associated plane of symmetry. If the interferometer is focused at the part and the center of curvature of the TS is displaced by

*j*axis, the phase change between adjacent pixels therefore reaches a prescribed fraction, say

*γ*, of the Nyquist sampling rate when

*j*:

*k*axis. Notice that, while the size of the illuminated region on the part grows linearly with

*η*and the value given by Eq. (4.13) that is desired. Furthermore, since only a handful of TS’s are typically available in practice, the preferred one most closely matches this resulting value while also having a transmission surface of sufficiently large radius of curvature to accommodate the part. The key is that Eq. (4.13) is a core component of stitchability estimation. As remarked at Eq. (4.5), the chief subject of what follows is the determination of an efficient process for evaluating averages like those that appear inside these square roots. The end result cannot be as simple as the final one in Eq. (2.8), but that expression sets the standard for what is sought next.

## 5. First-order approximations for the principal curvatures

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

*Eqs. (5.8)*

*are exact equalities on axis*, i.e. at

## 6. Stitchability matrices

*χ*and use the result for any specific asphere to evaluate its associated extension factor, i.e. the ratio of the NA of the surface to the NA of the optimal TS. This not only determines the number of subapertures required for total cover of the surface but gives some idea of the uncertainty in the final metrology map for any particular stitching platform. Alternatively, a maximal acceptable value, say

*χ*in Eq. (6.2). A constraint is then applied during design to keep the expression on the right-hand side of Eq. (6.2) bounded accordingly.

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

*η*; for faster parts, there is then an additional factor of up to two or three. Notice also how the orientation of these “ellipses of stitchability” depends on the NA of the part. That is, the strongest aspheres that can be stitched have two coefficients of the same sign for slow parts, but their signs are different on fast parts (say

*M*, eigen-reduction of the

*χ*(now less than

*η*, each of Eqs. (6.7) leads trivially to a minimal value for the part size, i.e.

## 7. Concluding Remarks

3. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. **39**(13), 2198–2209 (2000). [CrossRef]

## Appendix A

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

*m*that satisfies

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

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

*θ*, the first piece inside the braces of Eq. (6.1) is found to be

*m*and

*n*. By changing variables to

**a**of Eq. (A.1) rather than

**b**, the change-of-basis matrix of [9

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

8. G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics,” SPIE Optifab (2009), paper TD06–25 (1), http://www.qedmrf.com.

## Acknowledgments

## References and links

1. | A. Epple and H. Wang, “‘Design to manufacture’ from the perspective of optical design and fabrication,” in |

2. | J. P. McGuire Jr., “Manufacturable mobile phone optics: higher order aspheres are not always better,” Proc. SPIE |

3. | R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. |

4. | J. W. Foreman Jr., “Simple numerical measure of the manufacturability of aspheric optical surfaces,” Appl. Opt. |

5. | J. W. Foreman Jr., “Mercier’s aspheric manufacturability index,” Appl. Opt. |

6. | J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE |

7. | C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE |

8. | G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics,” SPIE Optifab (2009), paper TD06–25 (1), http://www.qedmrf.com. |

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

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

11. | P. Murphy Jon Fleig, “Greg Forbes, Dragisha Miladinovic, Gary DeVries, and Stephen O'Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE |

**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: March 16, 2011

Revised Manuscript: April 27, 2011

Manuscript Accepted: April 30, 2011

Published: May 5, 2011

**Citation**

G.W. Forbes, "Manufacturability estimates for optical aspheres," Opt. Express **19**, 9923-9941 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9923

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### References

- A. Epple and H. Wang, “‘Design to manufacture’ from the perspective of optical design and fabrication,” in Optical Fabrication and Testing, OSA Technical Digest (Optical Society of America, 2008), OFB1.
- J. P. McGuire., “Manufacturable mobile phone optics: higher order aspheres are not always better,” Proc. SPIE 7652, 76521O, 76521O-8 (2010). [CrossRef]
- R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency surface errors on image quality,” Appl. Opt. 39(13), 2198–2209 (2000). [CrossRef]
- J. W. Foreman., “Simple numerical measure of the manufacturability of aspheric optical surfaces,” Appl. Opt. 25(6), 826–827 (1986). [CrossRef] [PubMed]
- J. W. Foreman., “Mercier’s aspheric manufacturability index,” Appl. Opt. 26(22), 4711–4712 (1987). [CrossRef] [PubMed]
- J. Kumler, “Designing and specifying aspheres for manufacturability,” Proc. SPIE 5874, 121 (2005), doi:.
- C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004). [CrossRef]
- G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics,” SPIE Optifab (2009), paper TD06–25 (1), http://www.qedmrf.com .
- G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010). [CrossRef] [PubMed]
- G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007). [CrossRef] [PubMed]
- P. Murphy Jon Fleig, “Greg Forbes, Dragisha Miladinovic, Gary DeVries, and Stephen O'Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006).

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