## Design of a spherical focal surface using close-packed relay optics |

Optics Express, Vol. 19, Issue 17, pp. 16132-16138 (2011)

http://dx.doi.org/10.1364/OE.19.016132

Acrobat PDF (886 KB)

### Abstract

This paper presents a design strategy for close-packing circular finite-conjugate optics to create a spherical focal surface. Efficient packing of circles on a sphere is commonly referred to as the Tammes problem and various methods for packing optimization have been investigated, such as iterative point-repulsion simulations. The method for generating the circle distributions proposed here is based on a distorted icosahedral geodesic. This has the advantages of high degrees of symmetry, minimized variations in circle separations, and computationally inexpensive generation of configurations with *N* circles, where *N* is the number of vertices on the geodesic. These properties are especially beneficial for making a continuous focal surface and results show that circle packing densities near steady-state maximum values found with other methods can be achieved.

© 2011 OSA

## 1. Introduction

1. S.-B. Rim, P. B. Catrysse, R. Dinyari, K. Huang, and P. Peumans, “The optical advantages of curved focal plane arrays,” Opt. Express **16**(7), 4965–4971 (2008). [CrossRef] [PubMed]

2. D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express **17**(13), 10659–10674 (2009). [CrossRef] [PubMed]

3. R. Dinyari, S.-B. Rim, K. Huang, P. B. Catrysse, and P. Peumans, “Curving monolithic silicon for nonplanar focal plane array applications,” Appl. Phys. Lett. **92**(9), 091114 (2008). [CrossRef]

4. I. Jung, J. Xiao, V. Malyarchuk, C. Lu, M. Li, Z. Liu, J. Yoon, Y. Huang, and J. A. Rogers, “Dynamically tunable hemispherical electronic eye camera system with adjustable zoom capability,” Proc. Natl. Acad. Sci. U.S.A. **108**(5), 1788–1793 (2011). [CrossRef] [PubMed]

5. H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes 3rd, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature **454**(7205), 748–753 (2008). [CrossRef] [PubMed]

2. D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express **17**(13), 10659–10674 (2009). [CrossRef] [PubMed]

9. R. H. Anderson, “Close-up imaging of documents and displays with lens arrays,” Appl. Opt. **18**(4), 477–484 (1979). [CrossRef] [PubMed]

*N*is the number of circles,

*a*is the surface area of the sphere enclosed by a single circle, and

*A*is the total surface area of the sphere.

10. B. W. Clare and D. L. Kepert, “The Closest Packing of Equal Circles on a Sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. **405**(1829), 329–344 (1986). [CrossRef]

*N*= 75 circles [11

11. R. F. Bruinsma, W. M. Gelbart, D. Reguera, J. Rudnick, and R. Zandi, “Viral self-assembly as a thermodynamic process,” Phys. Rev. Lett. **90**(24), 248101 (2003). [CrossRef] [PubMed]

*N*(such as the trivial cases of 3 or 4 circles), but the densities seem to plateau around an experimentally determined upper limit of about 0.83 as

*N*increases. As

*N*approaches infinity, one might predict that the maximum packing density will start to approach that of hexagonal packing on a plane which is 0.91. However, further speculation on this topic is beyond the scope of this paper and for the purposes of this work, 0.83 packing density will be taken as the standard against which the packing performance will be compared.

*N*. As

*N*grows larger, so do the computational resources required to solve for the lowest energy. For example, in order to capture close to 2 gigapixels over a field of view (FOV) of ± 60° with 10 megapixel sensors, approximately 200 micro-optics are needed. This corresponds to an

*N*of about 800, an order of magnitude larger than the largest

*N*solved with the point-repulsion methods.

## 2. Geodesic packing

*N*values [13

13. D. L. D. Caspar and A. Klug, “Physical Principles in the Construction of Regular Viruses,” *Cold Spring Harbor Symposia on Quantitative Biology*, (Cold Spring Harbor Laboratory Press, 1962), pp. 1–24. [PubMed]

*ν*, where frequency is the number of times the sides of the base triangle are divided, as shown in Fig. 2(a) . The vertices of the resulting triangles are then projected onto the sphere that circumscribes the original icosahedron as shown in the last step of Fig. 2(a). These vertices can then be used as the centers of the circles. Using a geodesic as the basis for close-packing limits

*N*to

*N*= 10

*ν*

^{2}+ 2.

*ν*greater than 2, where edge chords vary, improved solutions can be found by distorting this baseline. It is easier to keep track of the coordinate system if the distortion scheme is applied to the vertices immediately before projection onto the sphere. The positions of the vertices on the icosahedral face can be described by a trilinear coordinate system as shown in Fig. 2(b). Mapping two-dimensional space using this trilinear coordinate system imposes the relation expressed by:

*x*,

*y*, or

*z*) must also be applied in the other two directions in order to maintain the 3-fold symmetry of the equilateral triangles. This further simplifies the task of optimization to an application of a one-dimensional distortion scheme along the three directions. All that is left is to derive the form of this distortion.

*x*direction, at

*x*= 0 the distortion should leave the coordinate unchanged. At the other edges of the face the coordinates of the distorted distribution can be normalized to conform to the trilinear condition of Eq. (2) to ensure that these end vertices also lie along the edges. The second constraint is that there is no chirality in the distorted configuration. For example, when distorting along the

*x*direction, any dependences on

*y*and

*z*should be equal such that the distorted configuration is symmetric about the centerline normal to the

*x*= 0 edge. The third is an assumption that the standard geodesic is close to the desired solution and that there exists some smoothly varying function

*f*that when multiplied by the base geodesic coordinates will perturb the baseline geodesic into the distorted configuration.

*f*can be approximated by its Taylor expansion. The distortion along the

*x*-axis is expressed to 3rd order in the normalized coordinates

*x’ = x/ν*,

*y’ = y/ν*, and

*z’ = z/ν*by:

*ν*and the distortion function has been normalized such that the 0th order constant is 1. Due to symmetry, distortions in the

*y*and

*z*directions can be expressed by identical functions where the coordinate axes are permuted. After application of Eq. (4), it is critical to impose Eq. (2) to the distorted coordinates to easily convert them to spherical coordinates using the methods described by Kenner [12].

*N*will show that only up to the 1

^{st}order terms produce significant effects and the cross terms can effectively be ignored.

## 3. Results

*N*= 25,002) shows that the absolute difference in density between optimizing all terms up to 3

^{rd}order (including cross terms) and only optimizing the

*A*coefficient stays below 10

_{1}^{−3}. This difference in density is 0 at frequency 1 and generally increases rapidly at lower

*N*values and slowly at higher

*N*. For the purposes of building an arrayed focal surface, a fit using only one coefficient seems to be more than sufficient. The packing densities as a function of

*N*for the baseline geodesic and the distorted geodesic with 1

^{st}order correction are compared in Fig. 3(a) . The theoretical maximum packing density for geodesic packing, shown as a black dashed line, was determined by assuming that the circles will be most closely packed along the edges of the icosahedral face. This is true for the geodesic since the vertices on the edges are the closest to the sphere and thus experience the least amount of expansion upon projection onto the sphere. The maximum packing density is then obtained by assuming a circle size that ensures adjacent circles along the projected edge are in contact.

^{st}order distortion coefficients for up to frequency 10. Results indicate that the coefficient values vary slightly with frequency but remain relatively stable around an average value of −0.191, and indeed analysis of up to frequency 50 confirms this stability.

*N*physical system such as a curved focal surface. Similarly, going to non-circular apertures could improve the coverage of the sphere, at the cost of manufacturing complexity.

## 4. Summary and conclusions

*N*make this scheme highly applicable to the formation of a close-packed array of optics. Although the packing densities of the distorted geodesic (approximately 0.76 at high

*N*values) are slightly lower than those produced by point-repulsion methods and only certain values of

*N*can be achieved by our model (

*N*= 10

*ν*

^{2}+ 2), the benefits mentioned above are valuable in designing practical, high

*N*systems.

## Acknowledgments

## References and links

1. | S.-B. Rim, P. B. Catrysse, R. Dinyari, K. Huang, and P. Peumans, “The optical advantages of curved focal plane arrays,” Opt. Express |

2. | D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express |

3. | R. Dinyari, S.-B. Rim, K. Huang, P. B. Catrysse, and P. Peumans, “Curving monolithic silicon for nonplanar focal plane array applications,” Appl. Phys. Lett. |

4. | I. Jung, J. Xiao, V. Malyarchuk, C. Lu, M. Li, Z. Liu, J. Yoon, Y. Huang, and J. A. Rogers, “Dynamically tunable hemispherical electronic eye camera system with adjustable zoom capability,” Proc. Natl. Acad. Sci. U.S.A. |

5. | H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes 3rd, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature |

6. | J. E. Ford and E. Tremblay, “Extreme Form Factor Imagers,” in |

7. | O. S. Cossairt, D. Miau, and S. K. Nayar, “Gigapixel Computational Imaging,” in |

8. | D. L. Marks and D. J. Brady, “Gigagon: A Monocentric Lens Design Imaging 40 Gigapixels,” in |

9. | R. H. Anderson, “Close-up imaging of documents and displays with lens arrays,” Appl. Opt. |

10. | B. W. Clare and D. L. Kepert, “The Closest Packing of Equal Circles on a Sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

11. | R. F. Bruinsma, W. M. Gelbart, D. Reguera, J. Rudnick, and R. Zandi, “Viral self-assembly as a thermodynamic process,” Phys. Rev. Lett. |

12. | H. Kenner, |

13. | D. L. D. Caspar and A. Klug, “Physical Principles in the Construction of Regular Viruses,” |

14. | S. Aoyama, “Golf Ball Dimple Pattern,” U.S. Patent 6 358 161, March 19, 2002. |

15. | ZEMAX, Radiant ZEMAX LLC, 112th Avenue NE, Bellevue, WA 98004. |

**OCIS Codes**

(040.1240) Detectors : Arrays

(080.3620) Geometric optics : Lens system design

(120.4880) Instrumentation, measurement, and metrology : Optomechanics

**ToC Category:**

Geometric Optics

**History**

Original Manuscript: June 22, 2011

Revised Manuscript: July 21, 2011

Manuscript Accepted: July 22, 2011

Published: August 8, 2011

**Citation**

Hui S. Son, Daniel L. Marks, Joonku Hahn, Jungsang Kim, and David J. Brady, "Design of a spherical focal surface using close-packed relay optics," Opt. Express **19**, 16132-16138 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16132

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### References

- S.-B. Rim, P. B. Catrysse, R. Dinyari, K. Huang, and P. Peumans, “The optical advantages of curved focal plane arrays,” Opt. Express 16(7), 4965–4971 (2008). [CrossRef] [PubMed]
- D. J. Brady and N. Hagen, “Multiscale lens design,” Opt. Express 17(13), 10659–10674 (2009). [CrossRef] [PubMed]
- R. Dinyari, S.-B. Rim, K. Huang, P. B. Catrysse, and P. Peumans, “Curving monolithic silicon for nonplanar focal plane array applications,” Appl. Phys. Lett. 92(9), 091114 (2008). [CrossRef]
- I. Jung, J. Xiao, V. Malyarchuk, C. Lu, M. Li, Z. Liu, J. Yoon, Y. Huang, and J. A. Rogers, “Dynamically tunable hemispherical electronic eye camera system with adjustable zoom capability,” Proc. Natl. Acad. Sci. U.S.A. 108(5), 1788–1793 (2011). [CrossRef] [PubMed]
- H. C. Ko, M. P. Stoykovich, J. Song, V. Malyarchuk, W. M. Choi, C.-J. Yu, J. B. Geddes, J. Xiao, S. Wang, Y. Huang, and J. A. Rogers, “A hemispherical electronic eye camera based on compressible silicon optoelectronics,” Nature 454(7205), 748–753 (2008). [CrossRef] [PubMed]
- J. E. Ford and E. Tremblay, “Extreme Form Factor Imagers,” in Imaging Systems, OSA technical Digest (CD) (Optical Society of America, 2010), paper IMC2.
- O. S. Cossairt, D. Miau, and S. K. Nayar, “Gigapixel Computational Imaging,” in Proceedings of IEEE Conference on Computational Photography (IEEE, 2011), pp.1–8.
- D. L. Marks and D. J. Brady, “Gigagon: A Monocentric Lens Design Imaging 40 Gigapixels,” in Imaging Systems, OSA technical Digest (CD) (Optical Society of America, 2010), paper ITuC2.
- R. H. Anderson, “Close-up imaging of documents and displays with lens arrays,” Appl. Opt. 18(4), 477–484 (1979). [CrossRef] [PubMed]
- B. W. Clare and D. L. Kepert, “The Closest Packing of Equal Circles on a Sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. 405(1829), 329–344 (1986). [CrossRef]
- R. F. Bruinsma, W. M. Gelbart, D. Reguera, J. Rudnick, and R. Zandi, “Viral self-assembly as a thermodynamic process,” Phys. Rev. Lett. 90(24), 248101 (2003). [CrossRef] [PubMed]
- H. Kenner, Geodesic math and how to use it, 2nd ed. (University of California Press 2003).
- D. L. D. Caspar and A. Klug, “Physical Principles in the Construction of Regular Viruses,” Cold Spring Harbor Symposia on Quantitative Biology, (Cold Spring Harbor Laboratory Press, 1962), pp. 1–24. [PubMed]
- S. Aoyama, “Golf Ball Dimple Pattern,” U.S. Patent 6 358 161, March 19, 2002.
- ZEMAX, Radiant ZEMAX LLC, 112th Avenue NE, Bellevue, WA 98004.

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