We present the microoptical adaption of the natural superposition compound eye, which is termed “Gabor superlens”. Enabled by state-of-the-art microoptics technology, this well known principle has been adapted for ultra-compact imaging systems for the first time. By numerical ray tracing optimization, and by adding diaphragm layers and a field lens array, the optical performance of the Gabor superlens is potentially comparable to miniaturized conventional lens modules, such as currently integrated in mobile phones. However, in contrast to those, the Gabor superlens is fabricated using a standard microlens array technology with low sag heights and small diameter microlenses. Hence, there is no need for complex diamond turning for the generation of the master structures. This results in a simple and well controllable lens manufacturing process with the potential to high yield.
© 2009 OSA
Miniaturized reflowable lens modules, with a total track length of about 2mm, recently found their way in the consumer market, namely in VGA-CMOS-cameras for mobile phones. However, problems are especially the mastering of the lenses in an array and the effort of stacking of the objective components. The lenses of such a single-chamber camera usually have aspherical shape and are comparatively large for the world of microoptics. In order to fabricate these lenses in wafer-scale, a complex diamond turning of a single pin with a following step-and-repeat process or even a diamond turning of the full wafer is needed. Furthermore, in the replication process, the molded lenses suffer from shrinkage, due to their “large” sag height. UV-molding is commonly used for wafer-scale replication technology, at which the shrinkage through polymerization is rather high (e.g. compared to injection molding). This has to be compensated for by adjusting the master dimensions in case of lens sags exceeding approximately 100µm. Since shrinkage in UV-molding depends generally on the shape rather than on height only, it might be a problem to compensate the dimensions of the master for aspheres. Furthermore, local shrinkage and surface adhesion could cause local defects at the lens surface . This limits the yield and with more elaborate fabrication steps, the costs of a single-chamber camera lens increase.
An alternative approach for ultra-compact imaging systems is the adaption of the superposition compound eye, the Gabor superlens. This multi-channel imaging approach is well suited for microoptics. Hence, it benefits from a well established technology for the fabrication of microlens masters and their replication . The mastering includes photolithography, reflow processes and, for the fabrication of aspheres, additional reactive ion etching. Also the shrinkage of the UV-molded shallow lenses is well controllable. This leads to a better yield and a reduction of fabrication costs.
In this article we present the optical design, fabrication and experimental characterization of the first “microoptical” Gabor superlens. Using a modern optical simulation tool for numerical ray tracing, adding diaphragm layers and a field lens array this yields an ultra-compact imaging system with an optical performance that is comparable to wafer-level lens modules.
In section 2 the two main eye principles which are found in nature are outlined and their artificial counterparts are described. The adaption of the superposition compound eye, the Gabor superlens, is introduced. In section 3 we present the optical design and simulation of a microoptical Gabor superlens with an enhanced performance due to ray tracing optimization, adding diaphragm layers and a field lens array. The fabrication, that applies state-of-the-art microoptics fabrication technology, is described in section 4. Finally, a demonstration system is presented in section 5 and the performance predictions found by numerical simulations are experimentally verified.
2. Compound eyes and the Gabor superlens
When looking for a solution to miniaturized optics, it is worthwhile to examine some of the smallest creatures found in nature, the insects. Their compound eyes are divided in two main types, the apposition and the superposition type. In the former (Fig. 1(a) ), each ommatidium images a small section of a large field of view (FOV) in a single point and the final image is composed of the responses of the independent ommatidia. The apposition compound eye is found in diurnal insects, whereas nocturnal ones and some mid-water crustaceans have a superposition eye. This eye (Fig. 1(b)) forms an image point by superposing light from one direction through adjacent channels. To achieve this, a clear zone is needed, which leads to longer systems. On the other hand, these eyes are much more light-sensitive due to an effective pupil size that is larger than the diameter of a single channel [3–5]. The superposition compound eye is also found in some day-active species, e.g. the hawk-moth (Macroglossum stellatarum, Fig. 1(c)). This eye type does not necessarily have a low resolution. Especially for the theoretical case of aberration-free ommatidia, which are perfectly aligned, the resolution could be about ten times higher than that of the apposition counterpart .
The adaption of the apposition compound eye led to optical systems with an overall thickness of about 200µm [7,8]. Latest results in this scope have an overall thickness of 300µm, an increased resolution of 144×96 pixels and a larger FOV of 85°×52°. However, artificial apposition compound eyes suffer from a trade-off between resolution and sensitivity. Even when incorporating neural superposition to increase sensitivity, a large image sensor is needed due to the lateral dilatation of the device to achieve an acceptable spatial resolution . This increases the sensor costs. Another approach, closer to the example found in nature, is to arrange the microlenses on a curved surface . However, state-of-the-art microelectronics fabrication technology is restricted to planar artificial receptor arrays, e.g. CMOS sensors. For this reason, camera applications of artificial compound eyes are limited to planar setups as well.
One technical application of the superposition principle is found in scanner optics, e.g. in facsimile and copying machines. For these unity magnification imaging systems usually arrays of gradient-index lenses are used, which are packed in a two or three row arrangement. Each lens forms an erect image, which superposes with the images of adjacent channels. The whole image is generated by scanning this line perpendicular to the orientation of the array [11–14]. More examinations concerning systems with unity magnification yield that an arrangement of two microlens arrays (MLAs) with low numerical aperture forms a more homogeneous image at the costs of resolution and compactness .
The principle of superposing optics, that forms an erect image and could be seen as the direct adaption of the superposition compound eyes, was already described by Dennis Gabor in 1940 . Hence this invention is termed the “Gabor Superlens”. As seen in Fig. 2 , the basic GSL consists of two MLAs, which are separated by the sum of their focal lengths. Every lens pair forms a microtelescope with inclined axis. For the first MLA having a slightly larger pitch than the second, the images of adjacent channels superpose in a distance from the GSL that depends on the object distance. Thus, this arrangement forms a real and erect image of the object. The other way around, if the pitch of the first MLA is lower than the pitch of the second, the GSL forms a virtual image. As it can be seen, the GSL acts similar to a conventional lens of larger dimensions, with the difference that the image is erect, its distance is directly proportional to the object distance and the GSL cannot be simply reversed. A parallel arrangement of two periodic MLAs with different pitches results in a periodicity of the combined structure. The dimension of such a period is p 1 2/(p 1-p 2), with p 1 and p 2 being the pitch of the first and the second MLA, respectively. Hence, if the MLAs are sufficiently large they build more than one GSL pattern. Furthermore, a GSL pattern has no sharp edge but blends into the adjacent ones [17,18]. This can be seen in Fig. 2, where light from an on-axis field forms two images. In the following only one pattern is considered and termed GSL.
To our knowledge, the first GSL was realized by Hembd-Sölner et al. It had a total track length of approximately 65mm with a FOV of about 10° and a resolution in image space of 2Linepairs/mm (LP/mm) .
With state-of-the-art microoptics technology at hand, we intended to drastically reduce the size of the GSL and increase its FOV in order to achieve an ultra-compact lens for miniaturized camera applications.
3. Design and simulation
In the following section the imaging equation of the GSL, in analogy to the equation of a conventional thin lens, is presented and a short introduction to the 3×3-matrix formalism is given. This is used for the derivation of a set of paraxial equations (i.e. neglecting any aberrations) that describe the GSL. The starting parameters for the numerical optical design are calculated with the help of these equations. Finally, the GSL is optimized with numerical ray tracing and the simulated optical properties are presented.
3.1 Lens equation
Analogous to a conventional thin lens the GSL satisfies an equivalent imaging equation. For infinite object distance, Hembd-Sölner et al.  derived its back focal length FS, which is given byEquation (1) and (2) hold for p 1 > p 2, i.e. for the GSL forming a real image, which is the contemplated usage.
3.2 Matrix formalism
For a paraxial description of a complex optical system the 2×2-matrix formalism is well known [19,20]. This method is applicable if the elements are centered on the same optical axis. But in order to handle more complex systems, e.g. where the components have an offset and/or tilt, the common matrix formalism can be extended . The entire optical system is represented by a matrix Msys that is derived by a multiplication of all the individual element matrices of the offset microlenses and the corresponding propagations between them. The ray-height (hout) and the angle to the optical axis (αout) at the output of the optical system is calculated from the ray-height (hin) and angle (αin) of the incident ray by
For a representation of the elements of the GSL in the matrix formalism, first the optical properties and the arrangement of the individual components are considered. Figure 3 shows the notation of the variables used for the GSL. Between the first and the second MLA a field MLA is inserted in a distance f 1 from the first MLA. This field MLA redirects light into the aperture of the second MLA and has ideally no influence on the image formation. Instead it reduces vignetting, increases the usable FOV and the contrast of the image due to reducing stray light from adjacent channels [20,22]. Finally, the second MLA relays the light to the overall image plane. In contrast to other approaches, where each channel works for a (at least partially) separate field angle, different channels of the GSL transmit the light with the same angle of incidence and superpose it in the image plane .
According to the matrix formalism, in Eq. (3) Msys is replaced by the system matrix of the GSL. Hence, the parameters of the particular elements are not independent of each other, but connected according to the system matrix. With some theoretical considerations on the imaging principle of the GSL, a set of paraxial equations is derived from the matrix elements. An extraction of this set is presented in the following.
Condition 1 – “focusing condition”: A general concept of the GSL is that all rays from the same object point superpose in a common point in the image plane with height hout, regardless of the impact height (hin) at the front lens of a channel of the GSL. Regarding to Eq. (3) this requires M 11 to be zero. With this, the back focal length of the GSL is
Condition 2 – “Gabor condition”: For the same reason, the height of a ray in the image plane must be independent of the channel through which it passed. Since M 13 is a function of the specific channel, it has to be zero as well. The resulting equation is the same as Eq. (1).
Condition 3 – “magnification”: The GSL forms a conventional and erect image. Thus the FOV is corresponding to the size of the image. The diameter of the image circle is therefore
The full paraxial equations are solved for “internal” parameters (such as the focal lengths of the MLAs, pitches and spacings) for a given set of “external” variables (such as FOV, F-number and overall thickness) . These parameters are then transferred into the numerical ray tracing program ZEMAX. For verification of the matrix formalism, Fig. 4 shows a layout diagram of a paraxial (i.e. ideal) GSL where all rays from the three plotted field angles perfectly superimpose in their corresponding image points.
3.3 Optical design and numerical ray tracing
In the paraxial approach described above the GSL works as an ideal lens. This means that all rays from an arbitrary field angle come to a perfect image point. Beginning from this, all paraxial represented components were gradually substituted by real lenses in ZEMAX. Consequently, aberrations must be considered, which reduce the imaging quality drastically. The thicknesses of layers, of spacings and radii of curvature (ROC) of the microlenses were optimized using a special merit function, to achieve a small RMS-spotradius (the root mean square radius of the point image in the focal plane) while maintaining the superposition principle.
In order to suppress inter-channel cross-talk and ghost images, four aperture layers have been added to the system. These are shown in Fig. 5 . The first diaphragm layer works as the aperture stop of the system, since it limits the cone of light accepted from an axial point on the object. Cross-talk between adjacent channels is additionally controlled by the “vignetting-stop”, while the “field-stop” determines the amount of channels contributing to an image point and thus also the light-sensitivity of the superlens. The “second MLA stop” prevents light to pass through the gaps between the microlenses on the second array.
The aberrations like spherical aberrations, astigmatism and field curvature of the first and the second MLA are critical for the image quality of the GSL. To reduce the former, conic constants were added as further degrees of freedom to these microlenses that lead to a strong increase of the resolution.
The layout of the resulting GSL after the optimization is shown in Fig. 5.
3.4 Optical properties of the Gabor Superlens
For the characterization of the GSL, its important optical properties were studied in ray tracing simulation experiments. The results that can be compared to laboratory experiments are presented in section 5.
The superposition property of the GSL is related to the diameter of the field-stop. The larger it is, the more channels contribute to one image point. Hence, for an increasing number of superposing channels, the sensitivity of the system increases as well. Contrarily, the geometric radius of the focus spot increases and consequently the resolution decreases, because rays contributing from outer channels are stronger aberrated. This is demonstrated in Fig. 6 (movie online).
The RMS-spotradius of the GSL, such as for a conventional lens, depends on the object distance as it is shown in Fig. 7(a) . If one readjusts the image distance according to Eq. (2), the RMS-spotradius can be kept constant, whereas by keeping the image distance fixed (at FS) the RMS-spotradius more than triples its size.
Figure 7(b) shows the effect of off-axis aberrations on the resolution of the considered superlens. The RMS-spotradius increases with increasing FOV. Part of the increasing blur with angle of incidence is due to (i) aberrations of the individual microlenses and (ii) cumulative effects between the channels caused by superposition.
4. Technology and implementation
Compared to conventional single-chamber optics, the microlenses in the arrays of the GSL are much smaller and lower. Hence, the GSL is well suitable for state-of-the-art microoptics technology. For the fabrication of the microlens masters some technologies are available such as laser beam writing, grayscale-lithography, droplet dispense, a photothermal process  and reflow of photoresist. The latter is the most accurate and simple for small spherical lenslets and so best suited for our MLAs. The fabrication steps using this technology are presented in the following.
The steps for fabricating the MLA masters and the replication tool are presented in Table 1 . The diaphragm layers, made of a highly absorbing polymer, were structured by photolithography on precise thin glass substrates. Two thin glass substrates (nd = 1.523) of component 1 (cp. Figure 5) were bonded together, subsequently the MLAs were UV-molded (p 1 = 191µm, pf = 179µm, refractive index of the polymer nd = 1.521) on the substrate-sandwich with included apertures. For this demonstrator both components of the GSL were fabricated in wafer-scale and diced in a further step. Thus, the assembly of component 1 and 2 (p 2 = 167µm) was made at die-level. Potentially the assembly could be carried out on wafer-scale as well with a subsequent wafer-dicing.
The molded first and field MLA showed a deviation of 1μm (less than 1%) of the radii of curvature (ROC) from the desired values, with a deviation of the conic constants of 0.001 (less than 0.5%), which is well within the tolerances of this technology. The ROC of the second MLA fits the specifications and the deviation of the conic constant is 0.003 (0.4%).
The assembly of the two components of the GSL was aided by alignment marks on each chip. Alternatively, for a better control of this process, the alignment was carried out in an active way, by mounting the two components separately in front of a CCD sensor (Fig. 8(a) ) and optimizing the image quality of a presented test target while adjusting. The lateral accuracy of the alignment marks of the passive assembly is about 5µm with respect to each other. The active alignment is thought to be more accurate than 1µm. The axial distance of 120µm between the two components has been fixed with glass spacers after the alignment. Figure 8(b) shows the final GSL in comparison to the size of a one-cent coin.
5. Experimental results
The F-number is a well suited measure of the light-sensitivity of the GSL. It is defined as the ratio of the effective focal length and the effective diameter of the entrance pupil. We measured the F-number with a setup according to the ISO 517:2008.
For the determination of the effective focal length of the GSL a setup as shown in Fig. 9 was used. The principle is a measurement of the magnification of the GSL. A test target (e.g. a glass plate with linepairs of known spatial frequency) was put in the focal plane of a well corrected achromatic lens (with known focal length). The illuminated test target was relayed to infinity by the achromatic lens and imaged by the GSL to an image sensor with a given pixel pitch. Hence, measuring the image height himg, the effective focal length of the GSL results from
The diameter of the effective entrance pupil (index: EP) of the GSL was measured from its image as shown in Fig. 10 . Therefore an iris diaphragm was put between a CCD sensor and a well corrected achromatic lens in the focal distance of the latter. The GSL was put on the same optical axis as the former components with the front towards the achromatic lens. Illuminated with a diffuse backlight, the GSL was moved on this optical axis until a sharp image of the entrance pupil was formed on the sensor. The size of the image (Aimg) was measured, by counting illuminated pixels. To calibrate this assembly, the GSL was replaced by an object (e.g. pinhole, index: PH) of well known diameter (). The diameter of the effective entrance pupil results from
The experimentally determined F-number of the GSL is 2.8, which is in good agreement with the simulated F-number of 2.7. The good conformity is attributed to the possibility to adjust the two components of the GSL individually.
However, care must be taken in measuring with the CCD sensor. Due to the short effective focal length and the small entrance pupil of the GSL, small errors caused by inaccurate distance or area measurements based on the not-negligible pixel size of the CCD can lead to severe (in our case about 10%) deviance in the calculation of the F-number.
For visualization of the resolution performance of the GSL, the contrast transfer function is plotted in Fig. 11 . Linepair targets of different spatial frequencies were imaged by the GSL and the contrast of the on-axis part of the image was measured (Fig. 11(a), the object contrast is 1 per definition). The experimental data of the on-axis configuration is plotted as the green solid line. This was also done for two configurations of the simulation – first for a configuration where the resolution is optimized to be a compromise for the whole FOV (black solid line) and second for a configuration, which was optimized for best on-axis resolution (blue dashed line). The maximum resolution in image space was measured to be 49LP/mm, for an arbitrary defined minimal image contrast of 0.2.
The dependency of the contrast on the field angle is shown in Fig. 11(b) for a spatial frequency of 30LP/mm. The derivation between measured and simulated curves is within the tolerances of this method. The image contrast also strongly depends on the distance to the image plane, due to field curvature. Furthermore, it is limited by noise and stray light.
In order to determine the field dependent vignetting or relative illumination, a homogenous white test target was imaged by the GSL. A line-scan through this image, normalized to its maximum value, is shown in Fig. 12 . The relative illumination reduces drastically down to zero at the edge of the FOV, which is a result of the decreasing number of channels contributing at the larger field angles. The experimental data deviates only slightly from the simulated one. This vignetting effect can also be observed in the grayscale images below (Fig. 13 ).
A set of different test targets were presented to the GSL (Fig. 11). This provides a good impression on the image quality of the microoptical implementation of a GSL.
Table 2 summarizes the parameters of the investigated GSL in a comparison between simulated and experimentally obtained values.
6. Conclusion and Outlook
We presented the optical design, technological realization and experimental characterization of the first “microoptical” Gabor superlens. Enabled by state-of-the-art microoptical technology, this well known principle was adapted to an ultra-compact imaging system. In previous work it has been shown that this technology is well suited for artificial compound eyes due to its high precision and reproducibility for the fabrication of microlens array masters. Contrarily to the mastering of conventional single-chamber optics, no complex diamond turning is needed that also has to be adapted to wafer-scale as an elaborate, serial and time consuming step-and-repeat process. Regarding shrinkage as well, the diamond turned masters have to be iteratively adjusted to reach the desired shape for the replica. Lens mastering by lithography in contrast is done in parallel on wafer scale and is well adapted to the required feature size and to the UV replication step. This makes the production process less complex and hence more cost effective.
Furthermore, additional diaphragm layers and a field lens array have been implemented, which results in an improved imaging performance of this Gabor superlens in comparison to former realizations. Due to its increased complexity and higher number of components, the Gabor superlens will be thicker than the artificial apposition compound eye. On the other hand, it is more light-sensitive and yields a higher angular resolution in object space, due to the larger effective focal length. Furthermore, a very small, conventional image sensor could be used, that is the main cost-factor of a compact camera system. Compared to conventional single-chamber optics, the Gabor superlens reaches a smaller total track length.
In future work the rectangular arrangement of the microlenses in the arrays of the Gabor superlens will be transformed into a hexagonal structure to increase the fill factor and therewith the sensitivity of the Gabor superlens. In addition, chirped microlenses  may be implemented to achieve a more homogenous resolution across the field of view, and also vignetting should be reduced. The Gabor superlens has the potential for further decreasing of the total track length while keeping or increasing the sensitivity. Therefore it could be implemented as a secondary camera in mobile phones. The resolution is currently too low to realistically achieve 1-2Megapixel or more. However, secondary cameras currently require CIF to VGA resolution at maximum. The Gabor superlens may also be a cost-efficient and compact approach for machine vision and medical imaging as well as an imaging system for automotive applications.
The work presented here was funded by the German Ministry of Education and Research (BMBF) within the initiative BIONA: “Nachwuchswissenschaftlergruppe: Insekten-inspirierte abbildende optische Systeme” (FKZ: 01RB0705A). Furthermore the authors like to express their gratitude to Sylke Kleinle, Antje Oelschläger, Andre Matthes and Bernd Höfer for their contributions to the fabrication and the assembly of the Gabor superlens.
References and links
1. J. Lim, M. Choi, H. Kim, and S. Kang, “Fabrication of Hybrid Microoptics Using UV Imprinting Process with Shrinkage Compensation Method,” Jpn. J. Appl. Phys. 47(8), 6719–6722 (2008). [CrossRef]
3. G. A. Horridge, “The compound eye of insects,” Sci. Am. 237, 108–120 (1977). [CrossRef]
4. J. S. Sanders and C. E. Halford, “Design and analysis of apposition compound eye optical sensors,” Opt. Eng. 34(1), 222–235 (1995). [CrossRef]
6. R. Navarro and N. Franceschini, “On image quality of microlens arrays in diurnal superposition eyes,” J. Opt. A, Pure Appl. Opt. 7, L69–L78 (1998).
7. J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, “Artificial apposition compound eye fabricated by micro-optics technology,” Appl. Opt. 43(22), 4303–4310 (2004). [CrossRef] [PubMed]
16. D. Gabor, UK Patent 541 753, (1940).
17. M. C. Hutley, and R. F. Stevens, “The formation of Integral Images by Afocal Pairs of Lens Arrays (“Superlens”),” IOP Short Meeting Series 30, 147–154 (Bristol: IOP Publishing, 1991).
18. C. Hembd-Sölner, R. F. Stevens, and M. C. Hutley, “Imaging properties of the Gabor Superlens,” J. Opt. A, Pure Appl. Opt. 1(1), 94–102 (1999). [CrossRef]
19. E. Hecht, Optics, 3rd edition (Addison-Wesley, 1994). [PubMed]
20. W. J. Smith, Modern Optical Engineering, third edition (McGraw-Hill, 2000).
21. N. Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A, Pure Appl. Opt. 4(4), S1–S9 (2002). [CrossRef]
23. J. Duparré, P. Schreiber, A. Matthes, E. Pshenay-Severin, A. Bräuer, and A. Tünnermann, “Microoptical telescope compound eye,” Opt. Exp. 13(3), 889–903 (2005). [CrossRef]
25. P. Dannberg, G. Mann, L. Wagner, and A. Bräuer, “Polymer UV-moulding for micro-optical systems and O/E-integration,” Proc. SPIE 4179(16), 137–145 (2000). [CrossRef]
26. J. Duparré, F. Wippermann, P. Dannberg, and A. Reimann, “Chirped arrays of refractive ellipsoidal microlenses for aberration correction under oblique incidence,” Opt. Exp. 13(26), 10539–10551 (2005). [CrossRef]