## A new family of optical systems employing φ-polynomial surfaces |

Optics Express, Vol. 19, Issue 22, pp. 21919-21928 (2011)

http://dx.doi.org/10.1364/OE.19.021919

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### Abstract

Unobscured optical systems have been in production since the 1960s. In each case, the unobscured system is an intrinsically rotationally symmetric optical system with an offset aperture stop, a biased input field, or both. This paper presents a new family of truly nonsymmetric optical systems that exploit a new fabrication degree of freedom enabled by the introduction of slow-servos to diamond machining; surfaces whose departure from a sphere varies both radially and azimuthally in the aperture. The benefit of this surface representation is demonstrated by designing a compact, long wave infrared (LWIR) reflective imager using nodal aberration theory. The resulting optical system operates at F/1.9 with a thirty millimeter pupil and a ten degree diagonal full field of view representing an order of magnitude increase in both speed and field area coverage when compared to the same design form with only conic mirror surfaces.

© 2011 OSA

## 1. Introduction

1. 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]

3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A **22**(7), 1389–1401 (2005). [CrossRef] [PubMed]

3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A **22**(7), 1389–1401 (2005). [CrossRef] [PubMed]

4. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express **18**(16), 17433–17447 (2010). [CrossRef] [PubMed]

## 2. Optical surface representation with φ-polynomials

*z*, is represented by a function that depends on the radial component,

*ρ*, and the azimuthal component, φ, within the aperture of the part. Until recently, methods of fabrication have constrained the shape of optical surfaces to depend on the radial component only, that is,

*z = F(ρ)*. This limitation has been a severe constraint in the optical design of unobscured optical systems. It is well known that when any powered optical surface in an optical system is tilted or decentered with respect to the optical axis, third order coma will appear on-axis. While there are some special configurations where axial coma is eliminated (1:1 systems and systems that use the coma free pivot design principle), in general, the ubiquitous presence of axial coma in tilted and decentered systems has prevented access to a substantial family of optical design forms.

*z*is the sag of the surface,

*c*is the vertex curvature,

*k*is the conic constant,

*ρ*is the radial component in the aperture,

*Z*is the

_{j}*j*FRINGE Zernike polynomial, and

^{th}*C*is the magnitude coefficient of

_{j}*Z*. Each

_{j}*Z*is a polynomial in polar coordinates (

_{j}*R*and

*φ*) where

*R*is a quantity normalized to a radius

*R*, that is,

_{norm}*R = ρ/R*. A table summarizing the FRINGE Zernike polynomials can be found in [5]. Figure 1 illustrates the sag of a powered Zernike polynomial surface with spherical aberration (Z9), coma (Z7, Z8), and astigmatism (Z5, Z6), (in order of increasing field dependence). As can be seen from the resulting sag, the surface is asymmetric due to the comatic contribution and anamorphic due to the astigmatic contribution. When the sag is evaluated with respect to the base conic, as illustrated in the lower series of Fig. 1, the dominance of the comatic Zernike term appears, as is typical for this new design family.

_{norm}6. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A **24**(3), 569–577 (2007). [CrossRef] [PubMed]

## 3. Optical design with tilted φ-polynomial surfaces: a new optical system family

### 3.1 The new optical design

8. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A **26**(6), 1503–1517 (2009). [CrossRef] [PubMed]

10. T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt. **44**(5), 776–783 (2005). [CrossRef] [PubMed]

### 3.2 Creating the starting form

### 3.3 Creating the unobscured form

### 3.4 Creating field constant aberration correction

### 3.5 Creating field dependent aberration correction

12. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express **16**(25), 20345–20353 (2008). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | 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 |

2. | R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE |

3. | K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A |

4. | T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express |

5. | Synopsys Inc, “Zernike polynomials,” in |

6. | L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A |

7. | Air Force Avionics Laboratory, “Three mirror objective,” in |

8. | K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A |

9. | J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” Optical Research Associates, US Patent 5,309,276 (1994). |

10. | T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt. |

11. | J. W. Figoski, “Aberration characteristics of nonsymmetric systems,” in |

12. | K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express |

**OCIS Codes**

(220.1920) Optical design and fabrication : Diamond machining

(220.2740) Optical design and fabrication : Geometric optical design

(080.4035) Geometric optics : Mirror system design

(080.4228) Geometric optics : Nonspherical mirror surfaces

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: August 3, 2011

Revised Manuscript: September 27, 2011

Manuscript Accepted: September 28, 2011

Published: October 21, 2011

**Citation**

Kyle Fuerschbach, Jannick P. Rolland, and Kevin P. Thompson, "A new family of optical systems employing φ-polynomial surfaces," Opt. Express **19**, 21919-21928 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21919

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### References

- 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]
- R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).
- K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A22(7), 1389–1401 (2005). [CrossRef] [PubMed]
- T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express18(16), 17433–17447 (2010). [CrossRef] [PubMed]
- Synopsys Inc, “Zernike polynomials,” in CODE V Reference Manual, (2011), Volume IV, Appendix C.
- L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A24(3), 569–577 (2007). [CrossRef] [PubMed]
- Air Force Avionics Laboratory, “Three mirror objective,” in Aerial Camera Lenses, Report 027000 from RECON Central, the Reconnaissance Division/Reconnaissance Applications Branch, (1967), pp. 2–109.
- K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A26(6), 1503–1517 (2009). [CrossRef] [PubMed]
- J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” Optical Research Associates, US Patent 5,309,276 (1994).
- T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt.44(5), 776–783 (2005). [CrossRef] [PubMed]
- J. W. Figoski, “Aberration characteristics of nonsymmetric systems,” in 1985 International Optical Design Conference, W.H. Taylor, and D.T. Moore, eds. (SPIE, 1985), pp. 104–111.
- K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express16(25), 20345–20353 (2008). [CrossRef] [PubMed]

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