One-lens camera using a biologically based artificial compound eye with multiple focal lengths: supplementary material

This document provides supplementary information to “One-lens camera using a biologically based artificial compound eye with multiple focal lengths,” Provided below is information regarding the main benefits of using a hemispherical lens for a camera system, a description of the fabrication of a transparent planar hexagonal microlens array with convex microlens shapes, microscope images of the planar array, and measurements of the radius of curvature of the curved array using the S-4800 control program with a scanning electron microscope. We also provide measurements of the camera field of view, and the relationship between a captured image and the rectangular image sensor. Finally, additional images captured and stitched images by the camera system are presented.


BENEFITS OF USING A HEMISPHERICAL LENS IN
A CAMERA SYSTEM In discussing the advantages provided by a hemispherical lens, we consider the path of light from an object through a hemispherical lens onto an image plane. We assume a beam of light is incident onto the planar surface of a hemispherical lens; when the observation plane is a tangential plane, the ray is deflected resulting in a smaller angle with the optical axis. Because the angle of refraction is smaller than the angle of incidence, the aperture diameter will be enlarged from Dt1 to Dt2 [see Fig. S1(a)] . However, if the observation plane is a sagittal plane [see Fig. S1(b)], there will be no change of the aperture diameter (Ds). Although the aperture diameters in the tangential plane (Dt2) and the sagittal plane (Ds) are different when the light is incident onto the curved surface of the hemispherical lens, the difference decreases when the angle between the ray and the optical axis is reduced, as shown in Fig. S1(c). This causes astigmatic aberration to be dramatically reduced, and the result is shown in Fig. S1(d), which is an enlargement of the region outlined by the dotted line in Fig. S1(c). Without the refraction, lens will exhibit strong astigmatic aberration, especially at large incidence angles.
After the light rays pass through the planar surface of the hemispherical lens, the light rays in the beam will still be parallel to each other, as shown in Fig. S2. Because the aperture stop is at the center of the hemispherical lens, the marginal rays will be axisymmetric respect to the chief ray. Therefore, the parallel light rays focus at a single point, and there is no coma aberration.

TRANSPARENT PLANAR HEXAGONAL MLA WITH CONVEX MICROLENS SHAPES
To confirm that the microlenses in each hexagonal ring have the same focal length and that microlenses in different rings have different focal lengths, the planar hexagonal microlens array (MLA) with convex microlens shapes is transformed into a transparent one, starting from the opaque silicon wafer.
After the MLA membrane with concave microlens shapes is fabricated, we pour UV-curing adhesive onto the PDMS membrane as shown in Fig. S3(a) (see step ①), and then cover it with a glass substrate (see step ②). Because the PDMS membrane does not need to be pulled into a spherical shape by the vacuum pump, the PDMS membrane can be made into a thicker section. Norland Optical Adhesive 65 (NOA65) is used in this research. Next, the adhesive is cured using UV light, and the MLA membrane is peeled off, as shown in Fig. S3(b). A microscope is used to inspect the images projected by this transparent planar hexagonal MLA. Because the microlenses of each hexagonal ring have different focal lengths, when the focusing plane of the microlenses of one ring locates on the image sensor of the microscope, the other rings will produce blurry images. S3. Replication process for a planar hexagonal MLA: (a) MLA membrane with concave microlens shapes is covered with UV-curing adhesive (step ①), and a glass substrate is placed over it (step ②). (b) After adhesive is exposed to UV light, MLA membrane can be peeled off.

MICROSCOPE IMAGES OF THE PLANAR HEXAGONAL MLA
The microlenses of each hexagonal ring of the MLA have similar focal lengths because they have the same surface profile. However, different rings correspond to different focal lengths, as shown in Fig. S4.
As previously stated in the manuscript, a microscope is used to inspect the images projected by the transparent MLA. When rays of light which pass through one of the hexagonal microlens rings are focused onto the image sensor, the other rings will be out of focus. Therefore, the microscope can only observe clear images from one ring at a time. Figure S5 shows the first to the fourth hexagonal microlens rings in focus.   Figure S7 depicts a system for measuring the field of view (FOV) of a camera module that is composed of a one-piece lens with a curved hexagonal MLA and an image sensor. Three angular measurements of the FOV are of interest: diagonal, horizontal, and vertical.

MEASURING CAMERA FIELD OF VIEW
To measure the range of angles from which incident radiation can be collected by the image sensor, we need to identify those parts of an object which can imaged by the corner and peripheral microlenses of the MLA onto the image sensor. In order to find out precisely which rays proceed from the object via the aperture stop up to the edge of the image field, consecutive letters and numbers are displayed on the screen object. The outermost sub-images that are detected and conveyed by the image sensor correspond to the extent of the observable screen object.
An image formed by the optical system design is shown in Fig. S8. The number 5 and letters B, S, and X are captured by the corner microlenses of the MLA. The images of the letters B and S are formed by microlenses at two opposing vertices of the MLA. The distance L between the letters B and S on the screen object is 6.7 cm; the distance d between the screen object and the planar surface of the camera lens is 3.2 cm (see Fig. S7). Therefore, the diagonal FOV value can be calculated to be 92.6° based on 2 ( 2 ). In the same way, the distance between the letters H and N on the screen object is 5.8 cm, so the horizontal FOV value can be calculated to be 84.4°; the distance between the number 8 and letter V on the screen object is 3.8 cm, so the vertical FOV value can be calculated to be 61.4°.

RELATIONSHIP BETWEEN CAPTURED IMAGE AND
RECTANGULAR IMAGE SENSOR The optical axis lies along the centers of the camera lens and rectangular image sensor, so the captured image will be centered on the optical axis. Therefore, the center sub-image is located at the center of the rectangular image sensor. In addition, the four subimages of the microlenses of the MLA at the vertices need to be matched to the four vertices of the rectangular image sensor, as shown in Fig. S9.

CAPTURED AND STITCHED IMAGE
More photos captured by our camera system are shown in Figs. S10(a), S10(c), and S10(e). As described in the manuscript, the captured photos are trimmed, resulting in multiple sub-images from the hexagonal MLA, which are then stitched into combined images. The combined images are image-processed into final digital images. The completely stitched and processed photographs show good quality in Figs. S10(b), S10(d), and S10(f).