Download Design of a spherical focal surface using close

Design of a spherical focal surface using closepacked relay optics
Hui S. Son, Daniel L. Marks, Joonku Hahn, Jungsang Kim, and David J. Brady*
Department of Electrical and Computer Engineering, Duke University
Durham, North Carolina 27708, USA
*
[email protected]
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 Optical Society of America
OCIS codes: (040.1240) Detectors, arrays; (080.3620) Geometric optics, Lens system design;
(120.4880) Instrumentation, measurement, and metrology, optomechanics.
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ZEMAX, Radiant ZEMAX LLC, 112th Avenue NE, Bellevue, WA 98004.
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Received 22 Jun 2011; revised 21 Jul 2011; accepted 22 Jul 2011; published 8 Aug 2011
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1. Introduction
Traditional isomorphic imaging systems typically correct geometric aberrations with proper
design of optical elements so that the image can be captured on a flat sensor. This constraint
of a flat focal surface and the resulting need to convert a spherically distributed field to a flat
field necessitates the need to correct for field curvature. As evidenced by the increase in
complexity of optical designs with larger field ranges, compensating for curvature by
manipulating the focal surface shape rather than the optical elements offers attractive benefits
[1,2].
Various curved sensors have been proposed to account for field curvature. Dinyari et al.
propose a system where the pixel elements are connected via flexible interconnects and
mounted on a flexible membrane [3]. The substrate is then deformed by pressing with a
shaped rod. Similar systems where a flat sensor pattern is overlaid onto a flexible substrate are
viable options for small-scale, low resolution imagers [4,5]. However, limitations in fine
control of surface profile, susceptibility to environmental changes, and variability in pixel
density along the sensor face prevent this type of sensor from being used for high resolution,
diffraction limited imagers where optical tolerances are tight and even coverage of the field is
required.
Recent investigations into monocentric imaging systems illustrate the possibilities for
wide-field, high pixel count imaging with the availability of a spherical focal surface [6,7].
Such a focal surface can be approximated by incorporation of a monocentric lens as the
primary optic in a multiscale design [2]. By utilizing an array of secondary optics with
overlapping fields, called micro-optics, to capture the spherical image produced by a
monocentric lens such as the Gigagon [8], a spherical focal surface with no gaps can be
realized from an array of flat sensors as illustrated in Fig. 1. The field covered by each set of
micro-optics is relatively small, and thus the amount of curvature correction needed to image
each subfield onto a flat focal plane is minimal. The process is similar to close-up imaging of
flat objects using a lens array [9]. For such a plane-to-plane imaging system, optimum
strategy for close-packing of identical round lenses to obtain maximum light collection is
trivially hexagonal packing. However, close-packing of round lenses on a spherical surface is
far from trivial and is commonly referred to as the Tammes problem. How tightly the lenses
can be packed determines how much of the field each lens needs to cover. Thus close-packing
minimizes the amount of correction each lens needs to perform, and therefore indirectly
improves image quality by reducing lens complexity. Given an optical design following the
multiscale design methodology, such as in Fig. 1(b), the challenge of making a curved focal
surface then lies in finding an adequate solution to the Tammes problem.
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].
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The Tammes problem originated from a study on the distribution of pores on pollen
grains. The goal is to pack a given number of points on the surface of a sphere such that the
minimum chord length between two points is maximized. An equivalent way to view the
problem is to pack a given number of equal circles on the sphere such that the diameter of the
circles is maximized without overlap. Since the minimum chord length determines the
diameter of the circles, the claim is that a solution to the Tammes problem will maximize the
packing density defined by:
ρ=
Na
,
A
(1)
where 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.
Various computational optimization methods have been proposed for solving the Tammes
Problem. A common technique is treating the points as equal point charges floating on the
surface of the sphere and iteratively allowing them to repel one another until a stable
minimum energy configuration is reached [10]. This method has been applied to obtain
distributions for up to N = 75 circles [11]. Few of these configurations have been proven to be
the maximum packing density possible for a given 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.
For practical applications, such as the current goal of packing circular optics on a sphere,
these experimentally derived solutions have fundamental weaknesses. The configurations
resulting from these solutions tend to lack symmetry. Having a high degree of symmetry is
desirable for taking cross sections of the sphere for use as a focal surface. Also, the circle
distributions are not optimized to reduce the variation in the chord lengths between
neighboring points. This can lead to “rattle” where a circle is significantly farther from its
neighbors than average, which in a lens array means either gaps in field coverage or
inefficient field overlap. These computer optimized distributions must also be generated for
each new value of 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
A packing strategy based on a distorted icosahedral geodesic addresses the above issues while
still maintaining a relatively high packing density. A detailed description of geodesic design is
given by Kenner [12]. For the purposes of developing a packing strategy, use of a regular
icosahedron as the base polyhedron gives the best initial conditions for high N values [13]. To
build a standard geodesic, the faces of the icosahedron are divided into equilateral triangles
with frequency 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 = 10ν2 + 2.
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Fig. 2. (a) Example generation of frequency 3 standard icosahedral geodesic. (b) Trilinear
coordinate system.
The vertices defined this way form a baseline for further optimization. For values of 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 + z =ν.
(2)
Maintaining this condition ensures that the vertices span the icosahedral faces from end to
end. As shown later, it can also be used to normalize the vertices after some arbitrary
distortion. Since the base icosahedron is composed of 20 identical equilateral triangle faces,
optimization can be reduced to maximizing packing density on just one of the faces. On each
face, any distortion applied in one of the directions (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.
A few constraints are imposed in deriving the distortion equation. The first is that at the
edges of the icosahedron, the vertices can be moved along the edge but never normal to the
edge. For example, when distorting along the 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.
Under this assumption, the function 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:
f x ( x′, y ′, z ′) ≈ A0 + A11 x′ + A12 y ′ + A13 z ′
+ A21 x′2 + A22 y ′2 + A23 z ′2 + a21 x′y ′ + a22 x ′z ′ + a23 y ′z ′
+ A31 x ′3 + A32 y ′3 + A33 z ′3 + a31 x′2 y ′ + a32 x′2 z ′
+ a33 y ′2 z ′ + a34 y ′2 x′ + a35 z ′2 x′ + a36 z ′2 y ′ + a37 x ′y ′z ′.
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(3)
Received 22 Jun 2011; revised 21 Jul 2011; accepted 22 Jul 2011; published 8 Aug 2011
15 August 2011 / Vol. 19, No. 17 / OPTICS EXPRESS 16135
Applying the trilinear condition of Eq. (2) and the constraint of non-chirality, Eq. (3) can
be simplified to:
f x ( x′, y ′, z ′,ν ) ≈ 1 + A1 (ν ) x′ + A2 (ν ) x′2 + A3 (ν ) x′3 + C2 (ν ) y ′z ′ + C3 (ν ) x′y ′z ′,
(4)
where the constants are now a function of 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].
Under the assumption that the distortion function is smoothly varying and produces small
perturbations from the baseline, it stands to reason that the higher order terms in the Taylor
series expansion, representing higher power derivatives, would be of lower significance. In
fact, results for higher values of N will show that only up to the 1st order terms produce
significant effects and the cross terms can effectively be ignored.
3. Results
The coefficients of Eq. (4) can be obtained numerically by optimizing to produce the largest
minimum chord length. Analysis of up to frequency 50 (corresponding to N = 25,002) shows
that the absolute difference in density between optimizing all terms up to 3rd order (including
cross terms) and only optimizing the A1 coefficient stays below 10−3. 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 1st 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.
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.
As a metric to quantify the variation in the separation of the vertices, we define the chord
ratio as:
Chord Ratio =
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Maximum Chord Length − Minimum Chord Length
,
Minimum Chord Length
(5)
Received 22 Jun 2011; revised 21 Jul 2011; accepted 22 Jul 2011; published 8 Aug 2011
15 August 2011 / Vol. 19, No. 17 / OPTICS EXPRESS 16136
where chord length indicates the distance between neighboring vertices.
Figure 3(b) compares the chord ratios for the distorted geodesic with that of the baseline
geodesic. Recalling from above, high variability can lead to “rattle” and gaps in the field. The
baseline geodesic arrangement is initially very ordered. Creating small perturbations from the
baseline to maximize the minimum chord length naturally reduces the chord ratio, as shown in
the above figure.
Table 1 lists the 1st 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.
Table 1. First order distortion coefficient
ν
3
4
5
6
7
8
9
10
A1
−0.19165
−0.19099
−0.19131
−0.19113
−0.19127
−0.19131
−0.19137
−0.19146
We apply this technique to design a spherical focal surface for a 2 gigapixel camera. In
this case a frequency 9 geodesic produces the appropriate number of micro-optics to image
120° FOV with 10 megapixel sensors. A comparison of the baseline geodesic packing with
the distorted packing is shown in Fig. (4). Only one triangle section of the icosahedron is
displayed for simplicity. It is immediately clear that the distorted geodesic in Fig. 4(b)
produces a significant improvement in coverage and consistency over the baseline solution in
Fig. 4(a). By adding distortion, packing density was improved from a baseline value of 0.5646
to 0.7615 and the chord ratio was decreased from 0.3832 to 0.1877. Comparison also shows
that in the baseline model vertices are more densely packed at the corners of the triangle and
distortion tends to spread these vertices out more evenly. Figure 4(c) shows vertex
distributions for the two models before projection and confirms that distortion does in fact
tend to push the vertices away from the corners.
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.
In principle, it is possible to increase the packing density further. We note that the circles
along the edges are essentially close-packed, but clearly there is room for improvement inside
the triangle. If we relax the constraint that the outer circles lie along the edge, a slight increase
in packing density could be achieved. For example, if alternating circles along the edges were
to be moved in and out of the triangle following a zig-zag pattern, the minimum chord length
of the system could be increased. In order to apply this type of distortion, a more involved
distortion function would be needed and is an interesting topic for future investigation.
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If the constraint of identically sized circles is relaxed and circle size variation is allowed (a
common practice in dimpling golf balls), packing densities as high as 0.85 can be achieved
[14]. However, this can greatly increase complexity and cost when applied to a high 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.
Application of our packing scheme for the production of a spherical focal surface is
currently underway. Figure 5(a) shows a machined mount for producing the 2 gigapixel focal
surface of Fig. 4(b). We have found that the ordered layout of the circles produced by the
distorted geodesic allows us to make the micro-optics in the shape of a regular hexagon
instead of a circle. The camera locations representing the vertices of the original icosahedron
require pentagonal optics, but with the use of these two shapes we can increase packing
density to near 0.85 which is on par with both point-repulsion optimization and using multiple
circle sizes.
Fig. 5. (a) Machined geodesic frame for creating a spherical 2 gigapixel focal surface. (b)
Hexagonal optic for increasing packing density.
4. Summary and conclusions
We have shown that a simple linear modification to an icosahedral geodesic produces
significant improvements when applied to the Tammes problem. The gain in packing density,
decrease in separation variability, retention of the inherent symmetry of the geodesic, and the
ease with which configurations can be generated for various 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 = 10ν2 + 2), the benefits mentioned above are valuable in designing practical, high
N systems.
Acknowledgments
We gratefully acknowledge the Defense Advanced Research Projects Agency (DARPA) for
financial support via award HR-0011-10-C-007.
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