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
ErrataHui 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)
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 . 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 . 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 , 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 . 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.
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:
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 . This method has been applied to obtain distributions for up to N = 75 circles . 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 . 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 . 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 = 10ν 2 + 2.
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:
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: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 .
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.
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.
As a metric to quantify the variation in the separation of the vertices, we define the chord ratio as:
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.
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.
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.
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 . 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.
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 = 10ν 2 + 2), the benefits mentioned above are valuable in designing practical, high N systems.
We gratefully acknowledge the Defense Advanced Research Projects Agency (DARPA) for financial support via award HR-0011-10-C-007.
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