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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 27284–27285
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Design of a spherical focal surface using close-packed relay optics: erratum

Hui S. Son, Daniel L. Marks, Joonku Hahn, Jungsang Kim, and David J. Brady  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 27284-27285 (2013)
http://dx.doi.org/10.1364/OE.21.027284


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Abstract

A coding error was found in calculating the optimal packing distribution of our geodesic array. The error was corrected and the new optimization results in slightly improved packing density. The overall approach and algorithm remain unchanged.

© 2013 Optical Society of America

In the article [1

1. H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011). [CrossRef] [PubMed]

] we iteratively optimized the packing distribution of circles on a sphere using an icosahedral geodesic as the base and distorting the circle coordinates with a first order polynomial. The code used to project the vertices from the icosahedron onto the surface of a unit sphere was based off the method described by Kenner [2

2. H. Kenner, Geodesic Math and How to Use It, 2nd ed. (University of California 2003).

] on page 75, Eqs. (12).4-12.6. These Eqs. are listed below for reference.

x1=xsin(72°)
(1)
y1=y+xcos(72°)
(2)
z1=ν/2+2z/(1+5)
(3)

Equations (1) through (3) are used to compute the Cartesian coordinates of the geodesic vertices (x1, y1, z1) given the trilinear coordinates (x, y, z) used to describe the vertex locations on the triangular face of the icosahedron and the frequency (ν) of the geodesic [1

1. H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011). [CrossRef] [PubMed]

].

These Eqs. were used in our Matlab code, but an error was found in the implementation of Eq. (2) where x1 was used instead of x. Since the Eqs. were applied in the order shown above, the x value used in Eq. (2) was scaled by the sine term for non-zero values. This resulted in slightly decreased packing density and a violation of the constraint that edge vertices do not move normal to the edge [1

1. H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011). [CrossRef] [PubMed]

]. We have corrected this error and rerun the optimization which produces an improvement in packing density and slightly different coefficients for the distortion polynomial from those shown in Fig. 3 and Table 1

Table 1. First order distortion coefficient

table-icon
View This Table
. The revised comparison of packing density and chord ratio for baseline geodesic and distorted geodesic distributions are shown below in Fig. 3 and the new coefficients are listed in Table 1. Figure 3 shows that packing density and chord ratio have improved to 0.7666 and 0.1734, respectively, for a frequency 9 geodesic.

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.

References and links

1.

H. S. Son, D. L. Marks, J. Hahn, J. Kim, and D. J. Brady, “Design of a spherical focal surface using close-packed relay optics,” Opt. Express 19(17), 16132–16138 (2011). [CrossRef] [PubMed]

2.

H. Kenner, Geodesic Math and How to Use It, 2nd ed. (University of California 2003).

OCIS Codes
(040.0040) Detectors : Detectors
(040.1240) Detectors : Arrays
(080.0080) Geometric optics : Geometric optics
(080.3620) Geometric optics : Lens system design
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.4880) Instrumentation, measurement, and metrology : Optomechanics

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 30, 2013
Manuscript Accepted: September 3, 2013
Published: November 1, 2013

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: erratum," Opt. Express 21, 27284-27285 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-27284


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