A new optical concept for compact digital image acquisition devices with large field of view is developed and proofed experimentally. Archetypes for the imaging system are compound eyes of small insects and the Gabor-Superlens. A paraxial 3×3 matrix formalism is used to describe the telescope arrangement of three microlens arrays with different pitch to find first order parameters of the imaging system. A 2mm thin imaging system with 21×3 channels, 70°×10° field of view and 4.5mm×0.5mm image size is optimized and analyzed using sequential and non-sequential raytracing and fabricated by microoptics technology. Anamorphic lenses, where the parameters are a function of the considered optical channel, are used to achieve a homogeneous optical performance over the whole field of view. Captured images are presented and compared to simulation results.
©2005 Optical Society of America
Compact digital cameras and optical sensors are important features of next generations of customer products. Key specifications are resolution, sensitivity, power consumption, manufacturing and packaging costs and, maybe most important, overall size. Today’s digital microcameras which are based on miniaturized “classical” lens designs are rarely smaller than 5×5×5mm3. Recent improvements of CMOS image sensors allow for a further miniaturization. However, due to diffraction effects, a miniaturization of the optics will drastically reduce resolution and image finesse . How to overcome these limitations of optics? A fascinating approach is to look how nature has successfully solved similar problems in the case of very small creatures [2, 3]. For small invertebrates having an external skeleton, eyes are very expensive in weight and metabolic energy consumption. If the budget is tight, nature prefers to distribute the image capturing to a matrix of some small eye sensors instead of using a single eye. The resolution of so called compound or fly’s eyes is usually poor compared to the single aperture eyes. In nature, this lack of resolution is counterbalanced by additional functionality like a very large field of view (FOV), polarization sensitivity or fast movement detection. The optical performance of the two different types of compound eyes, the so called apposition and the superposition compound eye was discussed exhaustively  and several technical realizations or concepts of imaging optical sensors based on this principle of image transfer through different channels were presented [5, 6, 7, 8, 9, 10, 11].
We report of and experimentally demonstrate for the first time an optical system, where the images of different viewing directions transferred by separated optical channels form an upright overall image of a large FOV, called cluster eye (Fig. 1). The arrangement of the refracting surfaces is similar to that of a Gabor-Superlens [14, 15] but with the implementation of a field lens and a field aperture array. The microlens arrays (MLA) with different pitch can be seen as a cluster of single pupil microcameras which have tilted optical axes to obtain the large overall FOV. Each channel images only a small angular section. The widths and positions of the field apertures determine the amount of overlap and spatial annexation of the elemental images. Anamorphic lenses with elliptical lens bases as shown in Fig. 2(b) are used for correction of astigmatism for oblique incidence as well as to keep a constant projected aperture size . Due to small lens sags the cluster eye is well suited for microoptical fabrication technologies. Figure 2(c) shows, that the proposed approach allows for wafer-level assembly and packaging.
Compared to conventional imaging systems, the approach of the cluster eye has a very large solution space because of the great number of parameters which are coupled to each other. A full description of such a system has to be based on an analytical model to find all reasonable parameter sets and the fundamental relationships. In Section 2 we report on the determination and validation of the first order parameters of a cluster eye in the one dimensional case. In Section 3 the obtained paraxial parameters are transferred to parameters of real lenses which are optimized and analyzed with respect to obtainable sensitivity and resolution by sequential raytracing. Additionally, a non-sequential raytracing analysis helps to evaluate the ghost images and stray light to be expected. In Sections 4 and 5 fabrication and experimental characterization of a telescope compound eye imaging system are shown. Finally, in Section 6 conclusions are presented.
2. Paraxial model of the system using matrix formalism
2.1. 3×3 matrices for representation of MLAs
Arrays of optical elements can be treated as a special case of misaligned optical elements. As a convenient way we see the 3×3 matrix formalism which explicitly contains the misalignments [18, 19] in the paraxial transfer matrix M. The output vector including the paraxial output ray height hout and the paraxial output ray angle αout is calculated from the known input vector including the paraxial input ray height hin and the paraxial input ray angle αin using
This formalism is used exemplarily to trace a paraxial ray through an off-axis thin lens with the focal length f. First the ray has to be transformed to the local coordinate system of the laterally shifted lens (lateral shift σ) then the ray is traced through the lens and finally the ray has to be transformed back to the global coordinate system. The complete matrix of the off-axis lens is given in Eq. (2).
From Eq. (2) we can derive that a laterally shifted lens adds an additional angular shift σ/f to the paraxial ray.
2.2. Paraxial description of the cluster eye
For a basic understanding of the performance of the cluster eye a simplified arrangement as shown in Fig. 3 was examined. Three lens arrays with different pitches form telescopes with field lenses and tilted optical axes. Each telescope-channel transmits a certain part of the overall FOV which is determined by a field stop array in the intermediate image plane at the position of the field lens array. The individually transferred parts of the image superimpose in the image plane to reconstruct the overall image. The focussing lens array (pitch p 1, focal length f 1) focuses the light in the intermediate image plane (object assumed to be at infinity). The field lens array (pitch pf , focal length ff ) is located in the intermediate image plane and redirects all light of the intermediate images into the relay lens array. The relay lens array (pitch p2, focal length f 2) magnifies the intermediate images onto the detector. The total number of channels is Ntot . The half angle of the maximum FOV is while the half maximum angle of the individual channels FOV is given by . The paraxial transfer function of the cluster eye is found by applying the 3×3 matrix formalism to
N is the number of the considered telescope-channel. The ray offset — N p1 representing the array arrangement of the optical channels can be included at any arbitrary position of the matrix multiplication because all matrices are referred to the same global coordinate system. We decided to implement it after the last lens array. Thus it is sufficient to consider just one single channel up to this point.
A number of different conditions have to be fulfilled to guarantee the desired optical performance of the cluster eye. In the following we only present two examples for a principle understanding of the formalism, all other conditions have to be treated in a similar manner and are only mentioned below.
The image formation condition for the individual telescopes requires that all rays entering the entrance pupil of a telescope under a certain angle must focus at one position. This implies that hout must be independent of hin , thus M 11=0. This is accomplished by
As expected, F is independent of the parameters of the field lenses.
The Gabor or Superposition condition has to be met in order to form one overall image. This is achieved, if all rays entering different channels under a certain angle meet at the same point in the detector plane. This implies that hout must be independent of the channel number N of the considered channel, thus M 13=0. It follows that
In a similar way conditions on magnification, segmentation of detector and FOV, F/# of image formation as well as tilt of optical axes, system length and vignetting (by the use of field lenses) are established. For a full understanding of the optical behavior of the cluster eye it is essential to determine its geometrical parameters uncoupled as a function of the desired optical system parameters. This is achieved by solving the equation system consisting of the conditions given above for the geometrical parameters.
If the necessary input parameters D (detector extension), (half angle of FOV), γ (half angle of image forming light cone), L (system length) and F (distance from imaging lens array to detector surface) are given, we find the conditional equations:
For simplification we substitute the value of f 1 in the equations for pf and ff. From the resolvability of the equation system follows the free choice of the length F in the input parameters, which is not to be seen as an overall system parameter, but gives a degree of freedom for the design.
2.3. Paraxial example system
The paraxial geometrical parameter sets obtained by the formulas of the previous section were implemented as paraxial lenses into a raytracing program. These simulations show that the obtained equation system delivers paraxial optical systems which satisfy the desired optical specifications. Table 1 presents the major parameters of the paraxial example system. The corresponding paraxial optical system is given in Fig. 4(a). Tilted optical axes of the telescopes enable a large overall FOV. Each channel is assigned to a portion of the overall FOV. In the paraxial model this is given by the spectrum of input angles for each individual channel. In a “real” system, field stops in the plane of the intermediate image determine the individual FOV and additional baffles in different surfaces will block the spurious light. The maximum marginal field of one channel is the minimum marginal field of the adjacent one. A perfect annexation of the individual sub images to one overall image can be noticed because the same field angle transmitted by two adjacent channels delivers one image point.
Figure 4(b) demonstrates the behavior of this system as a Gabor-Superlens where all channels contribute to the image point for a given object point if no constraints for the individual FOVs are given and thus all field angles can be transmitted by all channels.
3. Simulation of imaging system with real lenses
The calculated paraxial parameters count only as starting values for the layout of a system consisting of real MLA telescopes. When transferring the paraxial parameters to the parameters of real lenses, the focal widths and apertures are a function of the main field angles (tilt of the optical axis) of the considered channel in order to minimize off-axis aberrations, especially astigmatism and field curvature and to keep the light flux through the projected aperture constant, respectively. Anamorphic lenses with different radii of curvature and lens sizes as a function of the channel number are established. From raytracing simulations and third order aberration theory  it is derived, that the focal widths of the lenses must change with 1/cos2(2 ) in the direction where the field angle is increased and with 1/cos(2 ) in the perpendicular direction in order to keep the image surface of each channel at the position of the detector surface. In order to minimize the dependence of the light flux for field angle, the lens aperture has to change with 1/cos(2 ) in the direction of increasing field angle.
For the validation of the paraxial design strategy and to establish a recipe for the channel dependent transfer of the paraxial lens parameters to real lenses, we developed an optical system to be fabricated by microoptics technology whose parameters can be found in Table 1. For this set of parameters we obtain reasonable numerical apertures (NA) of the microlenses which result in good field performance.
The results of optimization in sequential raytracing for a wavelength of 550nm and lenses with spherical curvatures are presented in Fig. 5 and Fig. 6. We optimized the lens curvatures in x- and y-direction as well as lens positions after the parameter transfer to obtain minimum spot size and good image annexation. Using anamorphic lenses for the focussing lens array we can minimize astigmatism and field curvature to a large extent but coma is still present. Only a small increase of the geometrical spot size is observed over the entire FOV. The magnification is almost kept constant for all channels and we observe nearly perfect alignment of the sub-images to one overall image. However, due to the large individual channel F/#, the airy disk diameter is in the magnitude of 10 to 15µm and the system sensitivity determined by 1/(2F/#)2 is comparatively low.
Our sequential raytracing treatment of the cluster eye is sufficient to optimize each channels parameter in order to achieve the desired imaging properties. However, it is not possible to observe any effect of interaction between channels, because all channels are treated completely separately in different configurations. A non-sequential raytracing analysis of this imaging system is necessary in order to determine the influence of light which passes from one channel to another and/or is reflected at the metal apertures on all lenses. The resulting effects are ghost images and an increased background light level due to stray light. The cluster eye was implemented in non-sequential raytracing software with a detector-surface placed in the image plane. By filtering the ray database of the non-sequential raytrace with respect to the path a ray took through the different channels, we can calculate the ratio of maximum signal intensity to maximum spurious intensity and also the corresponding amount of signal power and spurious power distributed over the image plane reducing the overall contrast (Fig. 7(a)). In Fig. 7(b) and (c) movies of the image of a 2°×2° extended source moving through the FOV of the cluster eye in discrete steps of 1.8° are presented in linear and logarithmic scale. Spurious light appears not to be dominant. Bar targets were imaged in the non-sequential model, the results are given in Fig. 7(d) and (e). The observable oscillation of the signal to noise ratio in Fig. 7(a) is due to the overlap of marginal fields transmitted by adjacent channels, thus delivering two times the power compared to the center field. We can conclude, that, even if the effect of spurious light reduces the image contrast, images of a spatial frequency of at least 1 LP/° can be resolved.
A cluster eye with 21×3 optical channels was fabricated by microoptics technology. The lens shapes are defined by reflow of photoresist cylinders on variable (ellipsoidal) bases . The lenses are subsequently transferred into fused silica by reactive ion etching . Arrays of apertures with ellipsoidal (focussing array), rectangular (field lens array) and circular shapes (relay lens array) are applied onto the corresponding lens arrays by chromium etching or lift off. Finally the three lens array wafers are stacked in a modified SUSS mask aligner MA6 with active control of axial distances, wedge error compensation and lateral alignment using appropriate marks. The principle arrangement of the realized cluster eye is given in Fig. 8.
The major issue of the lens array fabrication by reflow process is the predetermination of the lens shape by the resist height and the shape of the lens basis  within the used parameter space. As experiments showed, the lens height is constant for a variation of the resist cylinder basis within a certain accuracy  and so the radius of curvature of the lens is just given by the size of the lens base because of
with R being the radius of curvature, hL the lens sag and r the radius of a circular lens. Further geometrical aspects of ellipsoidal lenses lead to 
Herein ax and ay are the major and the minor half axes of the ellipsoidal base of the resist cylinder and Rx and Ry are the major and minor radii of curvature of the resulting ellipsoidal lens. hL is assumed to be small with respect to ax and ay . By using Eq. 18 the desired change of radii of curvature from channel to channel is tuned. Table 2 summarizes the ideal parameters of the three lens array layers. Figure 9 demonstrates exemplarily the measured radii of curvature
of the focussing array (ellipsoidal lenses) in comparison to the ideal values. The deviation of the lens cross sections from a circle is always below 90nm (RMS). The radius error is below 2%.
5. Experimental characterization
We presented different test patterns to the realized cluster eye and relayed the overall image onto a conventional CCD-camera by a microscope objective with a magnification of x5 and NA of 0.18. Imaging a white surface the image annexation of all the individual microimages can be observed because one smooth white image should be generated. Figure 10 demonstrates that perfect image stitching could not be obtained. Either the microimages are of roughly rectangular shape but do not connect to each other (Fig. 10(a)) or they connect in some portions (Fig. 10(b)) but have strong overlap in others (Fig. 10(c)) which causes a considerable intensity modulation even for a smooth white object. This problem only partly can be drawn back to fabrication errors of lens curvatures or axial misalignments. Analyzing the quality of the field apertures, we observed that they are not exactly rectangular due to fabrication problems as can be seen in Fig. 11. Due the non-telecentric behavior of the cluster eye and the limited NA of the relaying microscope objective the transmitted field angles are restricted and thus only a limited number of channels can be observed. However, with the central 8×3 channels the following images could be captured. Figure 12 shows images of a radial star pattern (“Siemens star”), captured at different axial positions from the cluster eye. It can be observed, that the matching of the image planes of the individual telescopes with the position of the perfect annexation of the microimages is particularly critical. Tolerances of lens array fabrication and assembly are very tight and there are no compensation possibilities without reducing either contrast of the microimages or degrading image stitching. However, it is proven that one overall image is generated by the transfer of the different image section through different channels with a strong demagnification. Each channel has an individual FOV of 4.1 °×4.1°, the size of the individual microimages is 192µm×192µm. This results in a magnification with an equivalent focal length of 2.75mm at a system length of the realized cluster eye (distance first lens to image plane) of only 1.99mm. This large telephoto ratio is due to the magnification in the telescopes. In Fig. 13 images of bar targets of different spatial frequencies are presented. Over a FOV of 33 °x12° a resolution of 3.3LP/° is achieved. Finally, in Fig. 14 images of different test patterns visualizing the optical performance of the cluster eye are presented. Distant text and faces can be clearly resolved. Introducing a thin ground diffusing glass in the image plane of the cluster eye, the relayed image becomes more coarse but is still visible (Fig. 15(a)) and larger FOVs can be relayed. Using a C-mount objective, the whole image plane of the cluster eye can be relayed onto the CCD-camera. Operation of the cluster eye for objects having an extension of almost the designed FOV is thus proven in Fig. 15(b).
We demonstrated experimentally for the first time a compact imaging principle for a large field of view based on artificial compound eye vision and fabricated by microoptics technology. Here the magnification is not determined by the focal length as for the conventional single channel imaging principle, but by the ratio of focal lengths in the array of microtelescopes. The optical channels have tilted optical axes with respect to each other and each images only a small section of the object. The microimages of all the different channels combine to one overall image. An imaging system with 21×3 channels and 70 °×10° was designed, analyzed by simulation and fabricated. We were able to demonstrate the functionality of 16×3 optical channels having a horizontal FOV of 63° and a resolution of 3.3LP/° for the central channels. The complexity of the presented system compared to other artificial compound eye imaging approaches  is much higher. However, the telescope compound eye allows for the first time the use of microoptical fabrication technology for a compact imaging system which produces a regular macroscopic image of a distant object with large magnification. It has the potential to achieve a resolution similar to that of conventional imaging systems because of the transfer of different image segments by separated optical channels.
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