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. 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).
2. 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]
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,” Physica1(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]
6. G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, 2007).
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. Express16(3), 1583–1589 (2008). [CrossRef] [PubMed]
8. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012). [CrossRef] [PubMed]
9. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express15(8), 5218–5226 (2007). [CrossRef] [PubMed]
10. M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
11. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010). [CrossRef] [PubMed]
12. 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]
13. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010). [CrossRef] [PubMed]

Benitez, P.
Cakmakci, O.
Forbes, G. W.
Foroosh, H.
Fuerschbach, K.
Jester, P.
Kaya, I.
Menke, C.
Miñano, J. C.
Moore, B.
Rolland, J. P.
Santamaria, A.
Thompson, K. P.
Urban, K.
Vo, S.
Zernike, F.

Appl. Opt.

• 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]

Inf. Display - J. Soc. I

• 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).

Opt. Express

• 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]
• 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]
• 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]

Opt. Rev.

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

Physica

• F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica1(7-12), 689–704 (1934). [CrossRef]

Other

• M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
• G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific Publishing, 2007).

2012, Forbes, Opt. Express
2011, Fuerschbach, Opt. Express
2011, Jester, Appl. Opt.
2011, Kaya, Opt. Express
2010, Forbes, Opt. Express
2010, Forbes, Opt. Express
2009, Miñano, Opt. Rev.
2008, Cakmakci, Opt. Express
2008, Cakmakci, Inf. Display - J. Soc. I
2007, Forbes, Opt. Express
1934, Zernike, Physica

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