## Optimal local shape description for rotationally non-symmetric optical surface design and analysis

Optics Express, Vol. 16, Issue 3, pp. 1583-1589 (2008)

http://dx.doi.org/10.1364/OE.16.001583

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### Abstract

A local optical surface representation as a sum of basis functions is proposed and implemented. Specifically, we investigate the use of linear combination of Gaussians. The proposed approach is a local descriptor of shape and we show how such surfaces are optimized to represent rotationally non-symmetric surfaces as well as rotationally symmetric surfaces. As an optical design example, a single surface off-axis mirror with multiple fields is optimized, analyzed, and compared to existing shape descriptors. For the specific case of the single surface off-axis magnifier with a 3 mm pupil, >15mm eye relief, 24 degree diagonal full field of view, we found the linear combination of Gaussians surface to yield an 18.5% gain in the average MTF across 17 field points compared to a Zernike polynomial up to and including 10th order. The sum of local basis representation is not limited to circular apertures.

© 2008 Optical Society of America

## 1. Introduction

*c*represents the curvature,

*r*is the

*k*is the conic constant.

1. G.D. Wasserman and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B **62**, 2–8 (1949). [CrossRef]

6. H. Chase, “Optical design with rotationally symmetric NURBS,” *Proc. SPIE* **4832**, 10–24 (2002). [CrossRef]

## 2. Optical Surface Representation using Local Basis Functions

*ϕ*are the basis functions and

_{i}*w*are the coefficients. In eqn.(2) we have shown the surface to be a function of two variables for clarity. Alternatively, we will use the vector

_{i}**x**to represent these two dimensions. In this paper, we propose and explore the use of 2D Gaussians as basis functions to describe optical surfaces. A 2D Gaussian is described by

*μ*represents the mean vector and Σ represents the covariance matrix. Gaussians have several desirable properties in the context of optical design. First, Gaussians can be considered to be local functions since the value of a Gaussian outside of 3 sigma is small (<0.011 for a zero mean, unit variance Gaussian). Second, Gaussians are smooth (

*C*

^{∞}) having derivatives of all orders. Third, the Fourier transform of a Gaussian is a Gaussian which gives us an analytical description of the Power Spectral Density (PSD) function that is represented by a linear combination of Gaussians. Figure 1 illustrates the concept of a linear combination of Gaussian basis functions summing to approximate a sphere along with 1-dimensional slices through the original and the fit.

*mxn*Φ matrix contains the basis functions,

*w*is a vector of weights, and

*Z*is the resulting surface. Further theoretical background into this framework is given by Buhmann[8

8. M.D. Buhmann, *Radial Basis Functions: Theory and Implementations* (Cambridge University Press, 2003). [CrossRef]

## 3. Optimization Procedure

*c*is the curvature in the

*x*and

*y*directions and

*k*is the conic constant. The third step was the initialization of the starting point. During optimization we set the weights of the basis functions to zero, starting with the base conic. The fourth step was the construction of the Φ matrix. Each column of the Φ matrix contains a vectorized form of equation 3 and is written as

*x*pieces in the x-dimension and

_{num}*y*pieces in the y-dimension. The number of columns in the matrix was set by the product of

_{num}*x*and

_{num}*y*. The number of rows in the Φ matrix controlled the spatial resolution of the Gaussians and was set by the user. We make a rectangular aperture assumption in this case. However, the sum of basis representation accommodates any aperture shape since the Gaussians can be moved spatially using their means. We divided the aperture diameter into

_{num}*x*pieces in the x-dimension and placed each x-mean 1/

_{num}*x*apart from each other. Similarly, we divided the aperture into

_{num}*y*pieces in the y-dimension and placed each y-mean 1/

_{num}*y*apart from each other. The variances in the covariance matrix were set to 1. The weights of 400 Gaussians in the illustration given in Fig. 1 were found through least squares by

_{num}*w*in equation 4 with the goal of reaching a minimum of the merit function given a starting point. The fifth step is an optional step to represent a starting point only with the sum of basis, without the base conic. The intention with step 5 is to allow the exploration of alternative optimization techniques (for example using the MATLAB optimization toolbox) such as the trust region dogleg, Gauss-Newton or simplex. We remind that the addition of the base conic was only required for the paraxial calculations in the raytrace code. Alternative optimization environments could utilize step 5 as the preferred way to experiment with this surface representation. The sixth step is the construction or choice of the error function. We used the transverse error in the image plane, which is the sum of squares of the deviations of the rays from their respective reference wavelength chief rays, as our merit function. The seventh step was to choose an optimization technique and to optimize the error function. The results reported in this paper are based on the damped least squares algorithm.

## 4. Optical System Design Examples

*x*curvature of the base sphere, the second coefficient represented the conic constant in

*x*, and the third coefficient represented the conic constant in

*y*. The fourth coefficient represented the aperture size. The next 36 coefficients were the weights of the Gaussians (

*w*). The last 72 coefficients contained the

_{i}*x*and

*y*means of the Gaussians. During the optimization runs presented in this paper, the means were frozen (not variable) during optimization runs. Means were included in the user coefficient array solely for diagnostics purposes to make sure that the gridding functions were working properly. The variables in this elementary test case included the x and y curvatures, image plane defocus, and the 36 weights of the 6×6 Gaussians across the aperture. The conic constant was set to zero and not varied during optimization. The optimization converged in a few cycles (<10) to a parabola with a Strehl ratio of 1.

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

10. O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Tech. (**2**), 199–216 (2006). [CrossRef]

*λ*=550nm (no dispersion), for the optimized linear combination of Gaussians surface is shown in Fig. 2 (center). Interferogram of the surface with the base sphere subtracted is shown in Fig. 2 (right). Distortion characteristics exhibit similar behavior with each design having a maximum of about 3 to 4%. Table 1 shows a comparison of the surface representation proposed and implemented in this paper against an anamorphic asphere, Zernike polynomial up to and including 10th order, an x-y polynomial up to and including 10th order (good balancing achieved with order 5) with the maximum distortion and the average MTF across 17 field points as the comparison metrics. The sum of local basis representation proposed in this paper achieves the highest MTF performance averaged across 17 field points by 18.5% in the field with an acceptable level of maximum distortion, among the functions that were compared. Zernike optimization has been confirmed independently in Zemax and Code-V and the results agree to within 1.5% of the reported MTF value in Table 1.

## 5. Conclusion and Future Work

## Acknowledgements

## References and links

1. | G.D. Wasserman and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B |

2. | J. M. Rodgers, “Nonstandard representations of aspheric surfaces in optical design,” Ph.D. Thesis, University of Arizona (1984). |

3. | D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002). |

4. | J.E. Stacy, “Design of Unobscured Reflective Optical Systems with General Surfaces,” Ph.D. Thesis, University of Arizona (1983). |

5. | T. Davenport, “Creation of a uniform circular illuminance distribution using faceted reflective NURBS”, Ph.D. Thesis, University of Arizona (2002). |

6. | H. Chase, “Optical design with rotationally symmetric NURBS,” |

7. | S. Lerner, “Optical Design Using Novel Aspheric Surfaces,” Ph.D. Thesis, University of Arizona (2003). |

8. | M.D. Buhmann, |

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

10. | O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Tech. ( |

11. | O. Cakmakci and J. Rolland, “Dual-Element Off-Axis Near-Eye Optical Magnifier,” Opt. Lett. ( |

12. | J. M. Rodgers and K. P. Thompson, “Benefits of Freeform Mirror Surfaces in Optical Design,” |

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(220.1250) Optical design and fabrication : Aspherics

(220.4830) Optical design and fabrication : Systems design

(230.4040) Optical devices : Mirrors

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: December 6, 2007

Revised Manuscript: January 19, 2008

Manuscript Accepted: January 21, 2008

Published: January 23, 2008

**Virtual Issues**

Vol. 3, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Ozan Cakmakci, Brendan Moore, Hassan Foroosh, and Jannick P. Rolland, "Optimal local shape description for rotationally non-symmetric optical surface design and analysis," Opt. Express **16**, 1583-1589 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1583

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### References

- G. D. Wasserman and E. Wolf, "On the theory of aplanatic aspheric systems," Proc. Phys. Soc. B 62, 2-8 (1949). [CrossRef]
- J. M. Rodgers, "Nonstandard representations of aspheric surfaces in optical design," Ph.D. Thesis, University of Arizona (1984).
- D. Knapp, "Conformal Optical Design," Ph.D. Thesis, University of Arizona (2002).
- J. E. Stacy, "Design of unobscured reflective optical systems with general surfaces," Ph.D. Thesis, University of Arizona (1983).
- T. Davenport, "Creation of a uniform circular illuminance distribution using faceted reflective NURBS", Ph.D. Thesis, University of Arizona (2002).
- H. Chase, "Optical design with rotationally symmetric NURBS," Proc. SPIE 4832, 10-24 (2002). [CrossRef]
- S. Lerner, "Optical design using novel aspheric surfaces," Ph.D. Thesis, University of Arizona (2003).
- M. D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University Press, 2003). [CrossRef]
- G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express (15), 5218-5226 (2007). [CrossRef] [PubMed]
- O. Cakmakci and J. Rolland, "Head-worn displays: A review," J. Disp. Tech. 2, 199-216 (2006). [CrossRef]
- O. Cakmakci and J. Rolland, "Dual-element off-axis near-eye optical magnifier," Opt. Lett. 32, 1363-1365 (2007). [CrossRef] [PubMed]
- J. M. Rodgers and K. P. Thompson, "Benefits of freeform mirror surfaces in optical design," Proceedings of the American Society of Precision Engineering 2004 winter topical meeting on freeform optics, 31, 73-78 (2004).

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