Compound eye-inspired imaging devices can find vast applications due to their remarkable imaging characteristics, such as extremely large field of view angle, low aberrations, high acuity to motion, and infinite depth of field. Recently, researchers have successfully developed a digital camera that resembles the structure and functions of apposition compound eyes of arthropod, by combining an elastic array of microlenses with a stretchable array of photodetectors in their planar form and then transforming into a hemispherical shape. Designing an elastomeric microlens array that can be mechanically stretched to very large extent without deteriorating the optical performance is critical to this development. In this study, mechanics and optics of the stretchable microlens array, in which each hemispherical microlens sits on top of a supporting post connected to a base membrane, are studied. The results show that proper designs of the microlenses, supporting posts and base membrane are critically important to meet both mechanical and optical requirements simultaneously. This study can have important implications in not only the design of artificial compound eye cameras, but also other developments that require stretchable optical elements.

## I. INTRODUCTION

The in-depth understanding of light-sensing organs in biology^{1–4} provides opportunities for the development of digital cameras with operational features beyond the existing technologies.^{5–10} For example, hemispherical photodetector arrays that mimic the vertebrate eyes have shown superior imaging characteristics with simple optics.^{11–17} Different fabrication strategies have been developed to realize such devices. Another type of imaging system, i.e., compound eyes, commonly seen in insects has drawn even more attention, due to their unique imaging characteristics that are very different from vertebrate eyes and existing commercial cameras, such as extremely large field of view angle, low aberrations, acute sensitivity to motion, and infinite depth of field.^{18–20} These features can have huge potential in medical, industrial, and military applications. Therefore, a lot of efforts have been invested to study and fabricate imaging devices inspired by compound eyes.^{21–27}

Recently, Song *et al.*^{28} have successfully developed a digital camera that resembles the structure and functions of apposition compound eyes of arthropods. Figure 1(a) schematically illustrates the fabrication processes.^{28} The optical element enables optical focusing and defines the overall mechanics. It is a molded piece of elastomeric polydimethylsiloxane (PDMS, Sylgard 184) microlens array that consists of densely packed microlenses. Each microlens is a hemisphere, sitting on a matching cylindrical supporting post, which is connected to a continuous elastomeric base membrane. The optoelectronic element provides photodetection and electrical readout. It consists of a stretchable array of thin silicon photodiodes and blocking diodes. The optical and optoelectronic elements are fabricated separately and then matched and bonded together by using transfer printing at their planar geometries. The combination of microlens, supporting post and photodetector resembles an ommatidium in natural compound eyes. Afterwards, the system is transformed into an almost full hemispherical shape to realize the artificial compound eye camera. An optical picture of the artificial compound eye camera is shown in Fig. 1(c).^{28} During the geometrical transformation, very large strains are introduced into both the optical and optoelectronic elements. Therefore, it is critically important to design both systems such that the strains do not deteriorating their optical and electronic performances, respectively.

Figure 1(b) shows cross-sectional view of the compound eye camera before and after deformation, with key parameters illustrated.^{28} Before deformation, the distance between neighbor microlenses is L_{0} = 0.92 mm. After deformation, the planar geometry is transformed into a full hemispherical shape with the radius of curvature *R* = 6.96 mm. The three key parameters of the optical element, radius of the hemispherical microlens *r* = 0.4 mm, height of the supporting post *h*, and thickness of the base membrane *t*, are critical for proper optical performance. Since the photodetector (the red bricks in Fig. 1(b)) should be placed at the focal point of the microlens to gain best imaging performance, the summation of these three parameters, i.e., *r** **+** **h** **+** **t*, should equal the effective focal length of the microlens $f=rn/(n\u22121)=1.33\u2009mm$, where n = 1.43 is the refractive index of PDMS.^{28} On the other hand, the key mechanics design in the stretchable microlens array is by utilizing the strain isolation effect provided by the supporting post. As the height of supporting post increases, the strain in the microlens decreases. Therefore, it is important to properly design the stretchable microlens array such that both optics and mechanics requirements are met simultaneously. In this paper, we analyze the strain and deformation of the microlens under both uniaxial and equibiaxial strains by using finite element simulations. Effects of different ratios of supporting post height versus the base membrane thickness (i.e., *h/t*) are studied. The influence of microlens deformation on the optical performance is also investigated by optical ray tracing simulations. This study can have important implications for not only the design of artificial compound eye cameras, but also other developments that require stretchable optical elements.

## II. MECHANICS OF STRETCHABLE MICROLENS ARRAY

When the microlens array is deformed from flat geometry to hemispherical shape, microlenses at the periphery are subject to uniaxial strains due to the constraint from the circular fixture, microlenses at the center are subject to equibiaxial strains, and the other microlenses are subject to biaxial strains. Therefore, in this study, uniaxial and equibiaxial strains are applied to microlenses to investigate the mechanics of stretchable microlens array. Finite element analysis (FEA) is adopted to simulate microlens deformation. A unit cell (an artificial ommatidium) that contains a microlens, supporting post and base membrane, as shown in the bottom frame of Fig. 1(b), is chosen for FEA simulation, and periodic boundary conditions are applied to the side walls of the base membrane. Eight-node, hexahedral brick elements (C3D8) are used to discretize the geometry. Yeoh hyperelastic model is adopted to characterize the material behavior of PDMS, with the strain energy density given by

where $I1=\lambda 12+\lambda 22+\lambda 32$, $\lambda 1$, $\lambda 2$, and $\lambda 3$ are the principal stretches, and the material constants *C _{1}*

*=*0.285 MPa,

*C*

_{2}

*=*0.015 MPa, and

*C*

_{3}

_{ }

_{=}0.019 MPa.

^{13,16}External tensile strains are kept at 10% in all the FEA simulations.

The key dimensions of the microlens unit cell include the radius of the hemispherical microlens *r* = 0.4 mm, height of the supporting post *h*, thickness of the base membrane *t*, and the size of the unit cell L_{0} = 0.92 mm, as shown in the bottom frame of Fig. 1(b). Since the photodiode should be placed at the focal point of the microlens, the summation of *r*, *h*, and *t* should be equal to the effective focal length of the microlens *f*, i.e., $r+h+t=f=1.33\u2009mm$. In this design, the dimension of the microlens is fixed; therefore, $h+t=f\u2212r=0.93\u2009mm$. This provides the design space to engineer the mechanics of the stretchable microlens array, by choosing different ratios of the supporting post height versus the base membrane thickness (i.e., *h/t*).

Figure 2 shows the maximum principal strain contours of the microlens unit cell, when the height ratio h/t = 0 (i.e., no supporting post) and 1. Results for microlens unit cell subject to uniaxial strains are shown in Figs. 2(a) (h/t = 0, i.e., no supporting post) and 2(b) (h/t = 1). When h/t = 0, the microlens is directly connected to the base membrane, the tensile strain can be effectively transferred into the microlens. The maximum principal strain in the microlens reaches 23%, which is 74% of the maximum strain in the unit cell (31%) or 2.3 times of the applied strain (10%). When a supporting post is introduced between the microlens and the base membrane with h/t = 1, the maximum strain in the microlens drastically reduces to only 0.61%, only 2.2% of the maximum strain in the unit cell (28%), or 6.1% of the applied strain. The introduction of a supporting post of height ratio h/t = 1 reduces the maximum strain in the microlens by 38 times, which clearly demonstrates the strain isolation effect provided by the supporting post. Results for microlens unit cell subject to equibiaxial strains are also shown in Figs. 2(c) (h/t = 0, i.e., no supporting post) and 2(d) (h/t = 1). The introduction of a supporting post of height ratio h/t = 1 reduces the maximum principal strain in the microlens by 24 times.

The relationship between the maximum tensile strain in the microlens (normalized by the applied strain) versus the height ratio h/t for both uniaxial and equibiaxial strain conditions are shown in Figs. 3(a) and 3(b), respectively. For uniaxial strain, the normalized maximum tensile strain in the microlens drastically reduces from 1.28 when h/t = 0 to 0.035 when h/t = 1. Further increasing the height ratio h/t does not significantly reduce the strain in the microlens, as the normalized maximum strain only slightly reduces to 0.014 when h/t = 2. For equibiaxial strain, similar phenomenon is demonstrated in Fig. 3(b). When h/t = 0, the normalized maximum tensile strain in the microlens is 1.13, which drastically reduces to 0.039 when h/t = 1, and slightly reduces to 0.011 when h/t = 2. In addition to strain, the effect of supporting post on the shape change of the microlens subject to stretching is also studied. When subject to uniaxial strain, the bottom shape of the microlens changes from circle to ellipse. Figure 3(c) shows the major and minor diameters of the elliptical bottom normalized by the original circle diameter versus the height ratio h/t. As the height ratio h/t increases, the normalized major diameter decreases rapidly from 1.08 when h/t = 0 to 1.00 when h/t = 1, and minor diameter increases rapidly from 0.97 when h/t = 0 to 1.00 when h/t = 1. Further increasing the height ratio h/t induces no change to both the normalized major and minor diameters. When subject to equibiaxial strain, the bottom shape of the microlens keeps circular, but the diameter increases due to the stretching effect. Figure 3(d) shows the bottom diameter normalized by the original bottom diameter versus the height ratio h/t. Obviously, as height ratio h/t increases from 0 to 1, the normalized bottom diameter decreases from 1.07 to 1. Further increasing the height ratio h/t to 2, the normalized bottom diameter keeps constant 1. As the microlens unit cell is stretched, the strain causes the bottom diameter of the microlens to increase and the height to decrease. To characterize this effect, the ratio of microlens height over the bottom diameter is also shown in Fig. 3(d), when the unit cell is under equibiaxial strain. As the height ratio h/t increases from 0 to 1, the ratio of microlens height/diameter rapidly increases from 0.45 to 0.5. This ratio keeps constant 0.5 as the height ratio h/t further increases to 2. All these results clearly indicate that increasing the height ratio of the supporting post over the base membrane thickness can rapidly reduce the strain and deformation in the microlens. A good threshold for this height ratio is h/t = 1. Above h/t = 1, further increasing the height ratio h/t does not significantly improve any more.

In addition to the strain and bottom shape change of the microlens when subject to stretching, the surface profile of the microlens is another important aspect that worth studying as it directly affects the optics of the microlens. Under uniaxial tension, the shape of the microlens surface changes from a hemisphere to ellipsoid. Figures 4(a) and 4(b) show the fitting of the cross sections along the major and minor axes for height ratio h/t = 0 and 1, respectively. The method of least squares was adopted to conduct the profile fitting. Due to the severe distortion around the edge as shown in Figs. 3(a) and 3(c), the edge region whose height is 1/8 of the total height of the microlens is excluded from the fitting. This exclusion does not significantly affect the usefulness of the fitting results on the optical study of the microlens, as the area of this region is only 1/8 of the entire hemispherical surface and light beams at the central region of the microlens are usually more important than those at the edge for optical imaging in artificial compound eye. Spherical curves were used to fit both cross sections along the major and minor axes, which show excellent agreement with surface profiles given by FEA simulations. As shown in Fig. 4(a), when there is no supporting post, i.e., height ratio h/t = 0, the radii of the cross sections along major and minor axes are *R*_{x} = 0.42 mm and *R*_{y} = 0.388 mm, respectively. When supporting post is introduced, the change of the microlens profile due to stretching is greatly reduced. As shown in Fig. 4(b), when the height ratio h/t reaches 1, the radii of the cross sections along major and minor axes are equal, *R*_{x} = *R*_{y} = 0.40 mm, which is the same to that of the undeformed microlens. Fig. 4(c) shows the radii of curvature of both cross sections normalized by the radius of the undeformed microlens versus the height ratio h/t. Clearly, as height ratio h/t increase, *R*_{x} decreases and *R*_{y} increases rapidly. When the height ratio reaches 1, the two curves collapse into one horizontal line, with both radii equal to the radius of the undeformed microlens.

Under equibiaxial tension, the shape of the microlens surface keeps hemispherical. Figures 4(d) and 4(e) show the fitting of FEA results with spherical curves for height ratio h/t = 0 and 1, respectively. Both fitting show excellent agreement. Figure 4(d) gives the radius of curvature of the deformed microlens surface to be 0.42 mm for height ratio h/t = 0, i.e., no supporting post. For height ratio h/t = 1, Fig. 4(e) gives the radius of curvature of the deformed microlens surface to be 0.4 mm, which is equal to that of the undeformed microlens. The relationship between the normalized radius of curvature of the deformed microlens surface and the height ratio h/t under equibiaxial tension is demonstrated in Fig. 4(f). The radius of curvature rapidly decreases with increasing height ratio h/t. When the height ratio reaches 1, the radius of curvature of the deformed microlens equals that of the undeformed microlens. Both uniaxial and equibiaxial cases clearly demonstrate the efficient strain isolation effect provided by the supporting post. As supporting post is introduced, the influence of the stretching in the base membrane on the surface profile of the microlens is greatly reduced. When the height ratio h/t reaches 1, this influence is negligible.

## III. OPTICS OF STRETCHABLE MICROLENS

The optics of the microlens unit cell subject to strain is also studied by ray-tracing simulation using FRED (Photon Engineering LLC). Figure 5(a) shows schematic illustration of the ray-tracing model of the microlens. For paraxial light rays, the focal length of the microlens can be given as^{29}

where *r* is the radius of the microlens, n is the refractive index of the PDMS microlens. In the experiment, the radius of curvature of the microlens is 0.4 mm, and PDMS has the refractive index of $n\u22481.43$,^{28} so the focal length is obtained as $f$ = 1.33 mm. When the size of light beam becomes large, the paraxial approximation does not hold, and Eq. (2) cannot give accurate prediction for the focal length any more. Figure 5(b) presents the relationship between the focal length and the radius of the light beam calculated from ray-tracing simulation. Here, the light beam is kept parallel to the optical axis. When the light beam size is very small, the focal length agrees well with the result given by Eq. (2). When the radius of light beam increases, the focal length decreases from 1.33 mm to 1.19 mm when light beam radius = 0.3 mm, and to 1.05 mm when light beam radius = 0.4 mm. In the following, we use collimated light beam of radius 0.3 mm for optical simulations, which corresponds to the focal length of the undeformed microlens *f* = 1.19 mm.

Figure 5(c) shows the influence of the height ratio h/t on the focal length of the microlens unit cell when subject to 10% uniaxial strain. Since the deformation changes the shape of the microlens from hemispherical to ellipsoidal, two focal lengths, *f*_{x} along the major axis and *f*_{y} along the minor axis, are used to characterize the deformed microlens. Both focal lengths are normalized by the focal length of the undeformed microlens in Fig. 5(c). Because the deformation makes the cross section along the major axis to be oblate, the focal length *f*_{x} is larger than that of the undeformed microlens *f*. On the other hand, the focal length *f*_{y} is smaller than *f*. However, as the height ratio h/t increases, the focal length along the major axis *f*_{x} decreases and the focal length along the minor axis *f*_{y} increases rapidly. When the height ratio h/t reaches 1, the two curves collapse into one, and both focal lengths equal the focal length of the undeformed hemispherical microlens. Figure 5(d) shows the normalized focal length of the microlens versus the height ratio h/t when the unit cell is subject to 10% equibiaxial tensile strain. Due to stretching, the focal length of the deformed microlens is larger than that of the undeformed microlens. When the supporting post is introduced, the normalized focal length decreases rapidly with increasing height ratio h/t. Once the height ratio h/t reaches 1, the focal length matches with that of the undeformed microlens.

In addition to focal length, the image spot size of the light beam focused by the microlens is also studied, which can be important for designing the photodetector. Here, we keep the imaging plane at the focal length of the undeformed microlens. The inset of Fig. 5(e) shows the shapes of image spots focused by the microlens unit cell with height ratio h/t = 0 and 1 under uniaxial 10% tensile strain. With height ratio h/t = 0, i.e., no supporting post, the microlens is deformed to an ellipsoidal shape. This shape change is also reflected in the image spot shape. The majority of the light rays are focused into an ellipse, with major axis a = 20.4 *μ*m and minor axis b = 4.3 *μ*m. When the height ratio reaches 1, the spot shape changes to a circle, with radius a = b = 11.3 *μ*m, which is equal to the spot size of the undeformed lens. Figure 5(e) shows the relationship between the major radius of spot normalized by the spot radius of the undeformed microlens and the height ratio h/t. The ratio of minor axis b to major axis a is also presented. As the height ratio h/t increases, the normalized major axis decreases rapidly from 1.8 to 1, and the ratio b/a increases rapidly from 0.2 to 1. Fig. 5(f) shows the results for the microlens unit cell under equibiaxial tensile strain. Since equibiaxial tensile strain stretches the microlens uniformly in horizontal plane, the shape of microlens keeps hemispherical; therefore, the shape of the image spot also keeps circular. With height ratio h/t = 0, the radius of the image spot is 17.7 *μ*m. As the height ratio h/t increases to 1, the radius of the image spot decreases to 11.3 *μ*m, i.e., the spot size of the undeformed microlens. The relationship between the normalized spot radius and the height ratio h/t is presented in Fig. 5(f). The curve rapidly decreases from 1.57 when h/t = 0 to 1 when h/t = 1, and keeps constant for h/t > 1.

## IV. CONCLUSIONS

The mechanics and optics of the stretchable elastomeric microlens array under uniaxial and equibiaxial tension are studied. The results show that the strain and deformation of the microlens can be dramatically reduced when the supporting post is introduced between the microlens and the base membrane. As the ratio of the supporting post height to the base membrane thickness h/t increases, the strain and deformation in the microlens reduces rapidly. Once the height ratio h/t reaches 1, the effect of strain in the microlens is negligible. Ray tracing simulations illustrate how the deformation in the microlens affect the optical properties of the microlens, such as focal length, spot shape, and size. This study provides critical information for designing artificial compound eye cameras or other systems that require microlens arrays to be mechanically stretchable without affecting the optical properties. The results could also have important implications on applications that require tuning of optics through mechanical means.

## ACKNOWLEDGMENTS

The authors gratefully acknowledge support from ACS Petroleum Research Fund (Grant No. 53780-DN17) and NSF (Grant No. CMMI-1405355).