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

Full Article  |  PDF Article

Errata

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: erratum," Opt. Express 21, 27284-27285 (2013)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-22-27284

References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  15. ZEMAX, Radiant ZEMAX LLC, 112th Avenue NE, Bellevue, WA 98004.

2011 (1)

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]

2009 (1)

2008 (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]

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]

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]

2003 (1)

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]

1986 (1)

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]

1979 (1)

Anderson, R. H.

Brady, D. J.

Bruinsma, R. F.

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]

Catrysse, P. B.

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]

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]

Choi, W. M.

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]

Clare, B. W.

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]

Dinyari, R.

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]

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]

Geddes, J. B.

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]

Gelbart, W. M.

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]

Hagen, N.

Huang, K.

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]

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]

Huang, Y.

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]

Jung, I.

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]

Kepert, D. L.

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]

Ko, H. C.

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]

Li, M.

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]

Liu, Z.

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]

Lu, C.

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]

Malyarchuk, V.

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]

Peumans, P.

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]

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]

Reguera, D.

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]

Rim, S.-B.

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]

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]

Rogers, J. A.

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]

Rudnick, J.

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]

Song, J.

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]

Stoykovich, M. P.

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]

Wang, S.

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]

Xiao, J.

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]

Yoon, J.

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]

Yu, C.-J.

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]

Zandi, R.

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]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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]

Nature (1)

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]

Opt. Express (2)

Phys. Rev. Lett. (1)

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]

Proc. Natl. Acad. Sci. U.S.A. (1)

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]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

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]

Other (7)

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.

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.

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Figures (5)

Fig. 1
Fig. 1

Scheme for a spherical focal surface. (a) Illustration of how a spherically curved image projected by the main lens can be captured by an array of micro-opitcs and flat sensors. (b) Cross section of optical model [15].

Fig. 2
Fig. 2

(a) Example generation of frequency 3 standard icosahedral geodesic. (b) Trilinear coordinate system.

Fig. 3
Fig. 3

(a) Packing densities as a function of N. Blue line is baseline geodesic, red line is the distorted geodesic with 1st order correction, and black dashed line is the theoretical maximum. (b) Chord ratios as a function of N. Blue line is baseline geodesic and red line is the distorted geodesic with 1st order correction.

Fig. 4
Fig. 4

Comparison of circle packing in baseline (a) and distorted (b) frequency 9 geodesic. (c) Comparison of vertex distributions in baseline and distorted geodesics before projection.

Fig. 5
Fig. 5

(a) Machined geodesic frame for creating a spherical 2 gigapixel focal surface. (b) Hexagonal optic for increasing packing density.

Tables (1)

Tables Icon

Table 1 First order distortion coefficient

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ρ = N a A ,
x + y + z = ν .
f x ( x , y , z ) A 0 + A 11 x + A 12 y + A 13 z + A 21 x 2 + A 22 y 2 + A 23 z 2 + a 21 x y + a 22 x z + a 23 y z + A 31 x 3 + A 32 y 3 + A 33 z 3 + a 31 x 2 y + a 32 x 2 z + a 33 y 2 z + a 34 y 2 x + a 35 z 2 x + a 36 z 2 y + a 37 x y z .
f x ( x , y , z , ν ) 1 + A 1 ( ν ) x + A 2 ( ν ) x 2 + A 3 ( ν ) x 3 + C 2 ( ν ) y z + C 3 ( ν ) x y z ,
Chord Ratio = Maximum Chord Length Minimum Chord Length Minimum Chord Length ,

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