## Comparative assessment of freeform polynomials as optical surface descriptions |

Optics Express, Vol. 20, Issue 20, pp. 22683-22691 (2012)

http://dx.doi.org/10.1364/OE.20.022683

Acrobat PDF (3089 KB)

### Abstract

Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enables the fabrication of freeform optics, or more specifically, optical surfaces for imaging applications that are not rotationally symmetric. Various forms of polynomials for describing freeform optical surfaces exist in optical design and to support fabrication. A popular method is to add orthogonal polynomials onto a conic section. In this paper, recently introduced gradient-orthogonal polynomials are investigated in a comparative manner with the widely known Zernike polynomials. In order to achieve numerical robustness when higher-order polynomials are required to describe freeform surfaces, recurrence relations are a key enabler. Results in this paper establish the equivalence of both polynomial sets in accurately describing freeform surfaces under stringent conditions. Quantifying the accuracy of these two freeform surface descriptions is a critical step in the future application of these tools in both advanced optical system design and optical fabrication.

© 2012 OSA

## 1. Introduction

2. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express **19**(22), 21919–21928 (2011). [CrossRef] [PubMed]

3. J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. **16**(2), 99–102 (2009). [CrossRef]

4. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica **1**(7-12), 689–704 (1934). [CrossRef]

5. 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]

7. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express **16**(3), 1583–1589 (2008). [CrossRef] [PubMed]

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

*gradient-orthogonal Q-polynomials*following from the Q-polynomial form developed earlier for rotationally symmetric surfaces [9

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

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

## 2. Gradient-orthogonal Q-polynomials and Zernike polynomials

*Q*and

^{con}*Q*polynomials [9

^{bfs}9. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express **15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

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

*c*is the curvature of the best fit sphere,

*u = ρ/ρ*with

_{max}*ρ*being a position in the aperture and

*ρ*the radius of the enclosing circular aperture,

_{max}**20**(3), 2483–2499 (2012). [CrossRef] [PubMed]

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

*m*[10].

## 3. Ray grids for data site sampling and test cases

12. I. Kaya, K. P. Thompson, and J. P. Rolland, “Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces,” Opt. Express **19**(27), 26962–26974 (2011). [CrossRef] [PubMed]

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

## 4. Numerical simulations

*9*k*, where

^{2}*k*is the highest order of the polynomial in the polynomial fit. Truncation of the sums is carried out based upon the condition

*k<T*, for some given integer T, and

*k*equals

*m + 2n*for the gradient-orthogonal Q-polynomials.

*f*test case. We found that both polynomials performed almost identically for this test case. We have made use of approximately 54845 samples and 3320 polynomials with either set of polynomials. We have seen that for the

_{1}*f*test case, which is the 80 mm diameter F/1 parabola with a Gaussian bump placed away from the edge of the aperture, using hexagonal uniform ray grids both polynomials have peak-to-valley (PV) fit residuals ~10 nm (see Fig. 3(a)). Edge clustered ray grids result in a remarkable improvement on the overall fit residual profile, as shown in Fig. 3(b). Both polynomials produced PV fit residuals on the order of sub-nanometers with edge clustered ray grids.

_{1}*f*, when the Gaussian bump is placed at the edge of the aperture. Also in this case, the fit residuals for gradient-orthogonal Q-polynomials and Zernike polynomials are quite indistinguishable. In Fig. 4(a), the hexagonal uniform ray grid is used to create data sites for the least squares fitting, and we see that the PV fit residuals are ~4 nm with the gradient-orthogonal Q-polynomials and Zernike polynomials. The outcome is more compelling with the edge clustered ray grid, which increases the density of data sites towards the edge of the aperture. As seen in earlier work, this ray grid strategy significantly reduces the PV fit residuals, as seen in Fig. 4(b). We observe that the Zernike polynomials and the gradient-orthogonal Q-polynomials produced a sub-nanometer fit residual with the edge clustered sampling as shown in Fig. 4(b).

_{2}*f*and

_{1}*f*with hexagonal uniform and edge clustered ray grids with the two polynomial sets. We have gradually increased the degree of the Zernike polynomials and the gradient-orthogonal Q-polynomials as the truncation parameter in the sum is moved from

_{2}*T = 5*to

*T = 80*in steps of 10. As the number of basis elements goes up from 19 to 3319, the number of data samples in the fit increases from 226 to 54845. As found previously, we have observed that the edge clustered sampling consistently produces better fits when compared to hexagonal uniform sampling as demonstrated by the solid black lines in Fig. 5(a) and Fig. 5(b). We have also shown that gradient-orthogonal Q-polynomials and Zernike polynomials produced effectively exact representations with edge clustered sampling for both the less stressing

*f*case, with the bump away from the edge and the more stressing

_{1}*f*case, with the bump at the edge as marked with solid black lines in Fig. 5(a) and Fig. 5(b).

_{2}*f*and the

_{1}*f*test cases, as shown by the dash-dot blue curves in Fig. 5(a) and Fig. 5(b). Zernike polynomials and gradient-orthogonal Q-polynomials combined with edge clustered sampling consistently produced significantly better fits as the maximum degree of the polynomial is increased from

_{2}*T = 5*to

*T = 80*(see the black solid curves in Fig. 5(a) and Fig. 5(b)). For both test cases

*f*and

_{1}*f*, fits with Zernike polynomials and gradient-orthogonal Q-polynomials reached the required subnanometer levels (see point B in Fig. 5(a) and Fig. 5(b)). The gradient-orthogonal Q-polynomials performed as well as Zernike polynomials in achieving the accuracy levels in describing the optical surfaces, as given here as test cases

_{2}*f*and

_{1}*f*.

_{2}*T = 80*, with hexagonal uniform and edge clustered sampling with both polynomial sets for the height of the bump set at 12.5µm, 25µm, 50µm, and 100µm as shown in Fig. 6 . Dash-dot lines show the RMS fit residuals in the least-squares approximations with gradient-orthogonal Q-polynomials. Solid lines are used when the Zernike polynomials are used. Results show that there is a

*linear*relationship between the minimum RMS fit residual and the height of the bump. Specifically, in Fig. 6(a) that addressed a bump away from the edge of the aperture (i.e. case

*f*), Point A shows the RMS fit residual when the height of the bump is 12.5 µm using Zernike polynomials with edge clustered sampling. Point B shows the RMS fit residual when the height of the bump is 100 µm. The RMS fit residual increased from 4.5x10

_{1}^{−12}m to 3.6x10

^{−11}m that is 8 times. An equivalent relation is also found for the Points C and D. Moreover, we observe that with edge clustered sampling both polynomials produced two orders of magnitude better RMS fit residuals when compared with the performance of either polynomial with hexagonal uniform sampling (see blue and black curves in Fig. 6). The Point A records a RMS fit residual 4.5x10

^{−12}m, and Point C shows 1.8x10

^{−10}m residual fit departure.

*f*. Similarly for Fig. 6(b), the Zernike polynomials performed slightly better, while not significantly, with hexagonal uniform sampling for the test case

_{1}*f*(see blue lines in Fig. 6(b)). Similarly the black curves demonstrate the improved performance with edge clustered sampling. We can see for both Fig. 6(a) and Fig. 6(b) that the black dash-dot line and the black solid line coincide, which suggest that with edge clustered sampling Zernike polynomial and gradient-orthogonal Q-polynomials provide fits with identical fidelity for the test cases

_{2}*f*and

_{1}*f*

_{2}.## 5. Conclusion and future work

7. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express **16**(3), 1583–1589 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” Inf. Display - J. Soc. I |

2. | K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express |

3. | J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev. |

4. | F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica |

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

6. | G. E. Fasshauer, |

7. | O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express |

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

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

10. | M. Born and E. Wolf, |

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

12. | I. Kaya, K. P. Thompson, and J. P. Rolland, “Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces,” Opt. Express |

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

**OCIS Codes**

(220.0220) Optical design and fabrication : Optical design and fabrication

(220.4610) Optical design and fabrication : Optical fabrication

(220.4830) Optical design and fabrication : Systems design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: June 20, 2012

Manuscript Accepted: August 28, 2012

Published: September 19, 2012

**Citation**

Ilhan Kaya, Kevin P. Thompson, and Jannick P. Rolland, "Comparative assessment of freeform polynomials as optical surface descriptions," Opt. Express **20**, 22683-22691 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22683

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### References

- O. Cakmakci, S. Vo, K. P. Thompson, and J. P. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis eyewear display: Field-of-view limit,” Inf. Display - J. Soc. I16, 1089–1098 (2008).
- K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express19(22), 21919–21928 (2011). [CrossRef] [PubMed]
- J. C. Miñano, P. Benitez, and A. Santamaria, “Freeform optics for illumination,” Opt. Rev.16(2), 99–102 (2009). [CrossRef]
- F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica1(7-12), 689–704 (1934). [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. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, 2007).
- O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express16(3), 1583–1589 (2008). [CrossRef] [PubMed]
- G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012). [CrossRef] [PubMed]
- G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express15(8), 5218–5226 (2007). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
- G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010). [CrossRef] [PubMed]
- I. Kaya, K. P. Thompson, and J. P. Rolland, “Edge clustered fitting grids for φ-polynomial characterization of freeform optical surfaces,” Opt. Express19(27), 26962–26974 (2011). [CrossRef] [PubMed]
- G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010). [CrossRef] [PubMed]

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