1. Introduction

Using artificial apposition compound eye objectives for imaging applications with low required spatial resolution lead to extremely short vision systems with a thickness smaller than 250μm [1, 2, 3, 4]. Natural archetypes are the compound eyes of many insects such as the house fly. They consist of arrays of microlenses each with associated photo receptors both arranged on a curved basis. Due to this curvature compound eyes possess a very large field of view (FOV) but each optical channel (also known as ”ommatidium”) focuses light only onto a photoreceptor if it is coming from object points laying on the channels’ optical axes. Since each channel is used under normal incidence no off-axis aberrations such as astigmatism, field curvature, coma or distortion occur which would decrease the angular resolution with increasing FOV.

Artificial apposition compound eye objectives are limited to planar substrates since today’s micro-electronics fabrication technology is bonded to planar artificial receptor arrays such as CMOS or CCD sensors. Consequently the optical channels cannot be arranged in on-axis configurations inherently connected with the appearance of off-axis aberrations when using spherical lenses. In classical macroscopic optical systems, where one optical channel transfers the overall FOV, many optical elements of different refractive indices have to be used in order to minimize off-axis aberrations leading to very complex, bulky and expensive optical systems while they only represent a compromise of aberration correction for all viewing directions.

In contrast, for the apposition compound eye objective each lenslet is assigned only to one angle of the overall FOV. Consequently an individual correction of the channels for aberrations is feasible [5]. Due to the small numerical aperture of the lenslets of the objective astigmatism and field curvature are by far dominant compared to coma which is of minor influence. Therefore efficient channel-wise focusing of the oblique angle to be transferred is possible by using different and differently oriented anamorphic lenses for each channel [6, 7, 8]. The radii of curvature of the lenses in the two orthogonal directions have to be different and chosen in order to compensate for astigmatism due to the oblique incidence. Furthermore they are both chosen in such a way that the focal plane of all cells with their different angles of incidence is fixed at the position of the paraxial image plane. This leads to a planarized moiré magnified image [9, 10, 11] in the detector surface. A torus segment having two radii of curvature in perpendicular directions is the most appropriate 3D surface type for such anamorphic lens.

The fabrication of microlens arrays by melting of photoresist is a well established technology yielding to very smooth and well determined spherical surfaces [12, 13, 14, 15, 16, 17] often used for imaging applications [18]. Here the 3D surface is the result of surface tension effects and depends on the volume of the resist cylinder and the shape of the rim of the lens. Consequently stringent limitations to viable geometries apply. A suitable approximation of the desired torus segment is an ellipsoidal lens which can be easily formed by melting a photoresist cylinder on an ellipsoidal basis [19, 20, 21, 22].

In Section 2 we present design considerations for anamorphic microlenses for correction of astigmatism and field curvature based on Gullstrand’s equations [23, 24]. Furthermore in Sections 3 and 4 the suitability of ellipsoidal lenses fabricated by the reflow process for this purpose as well as the definition of the lens shape by its rim are shown. In Section 5, finally the application of the described approach to an artificial apposition compound eye objective for resolution homogenization over the entire FOV is presented.

2. Correction of astigmatism and field curvature

When using single spherical lenses under oblique incidence 3rd order aberration of astigmatism and field curvature will occur (Fig. 1). Rays laying in the tangential plane of the lens will experience higher optical power compared to rays running in the sagittal plane. Consequently different focal planes for tangential and sagittal rays exist, where the spots are blurred to lines. The difference in optical powers and therefore in axial position of the focal planes for the tangential and sagittal rays increases with increasing chief ray angle. For calculating the tangential and sagittal back focal lengths (BFL) s′_t and s′_s of a spherical surface the Gullstrand’s equations (2) and (3) are employed [23, 24]. Herein Δ denotes Picht’s operator being the difference between object and image space variables. In this notation the Snell’s law is written in the form

with σ and σ′ being the chief ray angles in object and image space, respectively, measured with respect to the optical axis. n and n′ are the corresponding refractive indexes on the two sides of a spherical surface. The tangential and sagittal Gullstrand’s equations (2) and (3), respectively, describe astigmatism and field curvature as a function of the chief ray angle and apply to small ray bundles passing trough a spherical surface with a radius of the curvature R.

 

Fig. 1. Circular lens and ellipsoidal lens under perpendicular and oblique incidence and related spot diagrams (Arrows indicate the position of the corresponding spot.). A circular lens with radius of curvature R=339μm and diameter D=242.8μm in fused silica (n=1.46 at 550nm wavelength) under perpendicular incidence produces a diffraction limited focus (geometrical spot size smaller than Airy-disk diameter indicated by black circle). However, if illuminated under oblique incidence astigmatism and especially field curvature lead to very large spots in the Gaussian image plane. The tangential and sagittal image planes are separated from the Gaussian image plane (here -165μm and -262μm, respectively) and the foci are blurred to lines. Using an anamorphic lens with adapted tangential and sagittal radii of curvature (R_t =579μm, R_s=451μm) for this special angle of incidence a diffraction limited spot size is achieved.

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The influence of the object distances on the tangential and sagittal back focal lengths is negligible for distances larger than 10 time the focal length. Since s′_t and s′_s are measured along the direction of the chief ray their projection onto the optical axis has to be taken into account to find the position of the corresponding image planes and their distances to the Gaussian image plane. This is determined by the paraxial focal length s′ ₀.

Astigmatism for one angle of incidence is now eliminated if the BFLs for tangential and sagittal rays coincide for this angle. By furthermore requiring

the tangential and sagittal image planes are fixed at the position of the Gaussian image plane. For the compound eye imaging system a planarized moiré image [9, 10, 11] results. The index i, j indicates the considered channel within the array.

Inserting the Gullstrand’s equations (2, 3) into Eq. (4) the tangential and sagittal radii of curvature of the lens for correction of astigmatism and field curvature are calculated as a function of the chief ray angle to

Here infinite distant objects are assumed. Figure 2 provides a visualization of the nomenclature used in the Gullstrand’s equations.

 

Fig. 2. Geometrical parameters of an ellipsoidal lens for explanation of the notation in Gullstrands equations. n, n′: index of refraction in object and image space, respectively; σ, σ′: chief ray angle in object and image space, respectively; s′ ₀: paraxial back focal length; R_t: radius of curvature in tangential plane; R_s: radius of curvature in sagittal plane; a_t: axis of ellipsoidal at rim of lens laying in the tangential plane; a_s: axis of ellipsoidal at rim of lens laying in the sagittal plane.

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In Fig. 3 a plot of the calculated optimum tangential and sagittal radii of curvature as a function of the chief ray angle is given for a spherical lens with a paraxial focal length of 145μm. For validation of these results additionally the same lens was implemented as ”Biconic” surface in the ray-tracing software ZEMAX and optimized for the same angles of incidence. The obtained radii are marked as crosses in Fig. 3. Only for large chief ray angles significant deviations between numerically optimized and analytic values can be observed.

 

Fig. 3. Optimum tangential and sagittal radii of curvature under oblique incidence calculated by Gullstrand’s equations (lines) and using ray-tracing optimization (crosses); parax-ial focal length is 145μm, NA is 0.23.

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3. Ellipsoidal microlenses by melting of photoresist

Because of its superior capability of producing high quality microlenses at moderate cost melting of photoresist is the most appropriate fabrication technology for the proposed anamorphic microlenses [12, 13, 20, 21, 22]. Ellipsoidal lenses are one attractive type of anamorphic lenses because of having two separate paraxial radii of curvature and their ability of being produced by melting technology if the rim of the resist cylinder has an ellipsoidal shape [19]. In Fig. 4 the surface deviation of an ellipsoidal lens and a torus segment having the same paraxial radii of curvature is plotted. The peak-to-valley (PV) deviation for the given values of radii of curvature which are typical for the considered microlenses are less than λ/14 and can be neglected. Therefore ellipsoidal lenses are an appropriate approach to the desired anamorphic lenses.

The resulting minimum surface after the melting process of a resist cylinder with an ellipsoidal rim cannot be modeled completely analytically. As a good approximation the surface can be assumed to be a portion of a rotational symmetric ellipsoid and a numerical parametric modeling can be derived [19]. To proof this assumption we compared a surface generated by numerical iterative surface minimization with the constraints of volume conservation and the conservation of the resist boundary [25] with the results of the numerical parametric model for the same starting parameters of the resist cylinder. Figure 5(c) shows the height difference between the expected ideal ellipsoidal lens and the computer iterated surface. A good correspondence can be observed. The deviations occur along the rim of the lens and are due to the effect of coarse sampling during simulation.

 

Fig. 4. Surface deviation between torus segment and ellipsoidal lens with identical paraxial radii of curvature R_t = 485.7μm and R_s = 609.6μm, lens height 20.2μm and conical constants for representation of the ellipsoidal lens k_t = -0.096 and k_s = 0.134. For the representation of the torus segment k_x = k_y = 0 is set.

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In reflow fabrication the resulting shape of the structure depends on the geometry of the rim and the height of the cylinder to be melted. Therefore for mask generation the dependence of the radii of curvature on the major and minor axes of the ellipsoidal base is requested. Simple geometrical considerations on a circle lead to:

with R being the radius of curvature, h_L the vertex height and r the half diameter of a circular lens. Further geometrical aspects for ellipsoidal lenses lead to [19]:

Herein a_t and a_s are the major and the minor half axes of the ellipsoidal base of the resist cylinder. h_L is assumed to be small with respect to a_t and a_s and can be neglected.

4. Experimental verification of aberration correction under oblique incidence

For implementation into an artificial cluster compound eye microoptical sensor [26] ellipsoidal lenses have been fabricated, optimized for an angle of incidence of 35°. A spherical micro lens with the same paraxial focal length of 750.8μm was fabricated for comparison. Measurements of the spots when illuminating the lenses under different angles of incidence were realized using a setup as drawn in Fig. 6. Figure 7(a) and (b) illustrate the influence of astigmatism and field curvature due to oblique incidence on a spherical surface leading to blurred spots connected with poor optical performance. Good aberration correction with an adapted ellipsoidal lens is demonstrated in Fig. 7(c) and (d) leading to diffraction limited spot sizes. For comparison we additionally present the experimentally obtained foci in the tangential and sagittal plane and in the plane of least confusion of the circular lens under oblique incidence and of the used ellipsoidal lens under perpendicular incidence in Fig. 8.

 

Fig. 5. (a) Resist cylinder on an ellipsoidal basis to be melted, height: 10.18μm, lens bases: a_t = 138.9μm, a_s = 155.6μm; (b) Computer simulated surface by iterative melting of the elliptical resist cylinder, lens height: 20.3μm; (c) Surface deviation (PV) of ideal ellipsoidal lens and iterated surface is maximum λ/7 and thus diffraction limited.

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Fig. 6. Experimental setup for focus evaluation under oblique incidence. Rotation axis is aligned with vertex/center of microlens under test. Resolution of the measurement was determined imaging Ronchi rulings to 0.18μm/Pixel on CCD.

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5. Ultra-thin camera objective with chirped array of ellipsoidal micro-lenses

The proposed ultra-thin camera objective is based on artificial apposition compound eye sensors and consists of a micro-lens array combined with an array of pinholes and/or photodetectors in their focal plane [3, 4]. Due to a decenter of the lenses with respect to the corresponding pinholes, which differs for each channel, a comparatively large FOV results. Since each optical channel is assigned to one chief ray angle only, a channel-wise correction of astigmatism and field curvature can be accomplished. This leads in consequence to an array of individually shaped ellipsoidal micro-lenses which we like to call a chirped array. For the definition of an ellipsoidal lens five parameters are needed (decenter in X and Y and orientation angle with respect to the global coordinate system, radii of curvature in tangential and sagittal plane). The functions for calculating the parameters of each cell, which is identified by its index (i,j), are derived completely analytically [27]. Hereby the decenter and orientation angle are determined by geometrical aspects. The radii of curvature are calculated using Eqs. (6) and (5). Eq. (8) leads to the major and minor axes of the ellipsoidal lens bases necessary for mask generation.

 

Fig. 7. Experimentally obtained spots. (a) Circular lens under 0° (design angle), window width is 18μm, 1/e ²-width is 4.3μm, Strehl ratio is 0.98. (b) Circular lens under 35°, window width is 92μm, image plane is the same as in (a). (c) Ellipsoidal lens under 35° (design angle), window width is 18μm, Strehl ratio is 0.52, image plane is the same as in (a). (d) Ellipsoidal lens under 32° (best angle), window width is 18μm, Strehl ratio is 0.94, image plane is moved 80μm away from lens compared to (a). Elliptical lens was optimized for 35° angle of incidence. Design radii of curvature: R_t = 485.7μm, R_s = 609.6μm; resulting minor and major axes: a_t = 138.9μm, a_s = 155.6μm; measured radii of curvature of resist lenses transferred into quartz: R_t = 451μm, R_s = 579μm; circular lens parameters: design radius of curvature: R = 375.2μm; diameter: d = 242.8μm; measured radius of curvature: R = 339μm.

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Fig. 8. Experimentally obtained spots. Window width is 92μm for all images: (a) Circular lens under 35°, 170μm axial distance from paraxial focus towards the lens = tangential image plane. (b) Circular lens under 35°, 240μm axial distance from paraxial focus towards the lens = circle of least confusion. (c) Circular lens under 35°, 300μm axial distance from paraxial focus towards the lens = sagittal image plane. (d) Ellipsoidal lens under 0°, 380μm axial distance from paraxial focus off the lens = circle of least confusion.

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For prototyping we chose the parameters of the lens array in order to meet the specifications of an available large-pitch CMOS imager [28] (Tab. 1). The master structures were originated by reflow of photo-resist. The actual objectives were subsequently fabricated by UV-embossing [29] of the lens arrays on the front side of a thin glass substrate. On the back side of this substrate the pinholes were structured into a metal layer located in the focal plane of the micro-lenses [4]. For comparison reasons we built two systems with same number of channels, paraxial focal length and FOV. The first objective consists of identical spherical lenses therefore using a regular micro-lens array (rMLA). The second one uses a chirped micro-lens array (cMLA) with ellipsoidal lenses for channel-wise correction of astigmatism and field curvature. Both prototypes are capturing a quadrant of a symmetrical FOV, since the other three quadrants are simply mirrored images of the considered area (Fig. 9).

Table 1. Parameters of fabricated artificial apposition eye objectives.

View Table

 

Fig. 9. Schematic drawing of a camera chip capturing a quadrant of the full FOV. The left and bottom margins serve as docking areas for the gripping tool during assembly. The channel in the lower left corner has a perpendicular viewing direction with respect to the objective-plane and consequently applies a circular lens. With increasing viewing angle of the channel the ellipticity of the corresponding lens is increased up to an angle of σ_max=32° on the diagonal.

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We displayed different representative test patterns to a vision system composed of a chirped lens array and for comparison of the regular lens array and investigated the captured images with respect to resolution homogeneity over the FOV. For simplicity only one quadrant of the entire symmetrical FOV is tested.

Figure 10 and the three left columns of Fig. 11 show original circular and bar test targets, respectively and the corresponding images taken by compound eye objectives applying chirped or regular lens arrays. It can be clearly observed that - as to be expected - the resolution in the center of the FOV is independent of using regular or chirped lens arrays (signal frequency responses: 0.36 for 16LP, 0.23 for 20LP and 0.14 for 24LP). However, with increasing view-ing angle the resolution is decreased down to zero when simply using the regular lens array while the resolution stays constant when applying the chirped lens array where each channel is individually optimized for its viewing direction.

 

Fig. 10. Circular symmetric test patterns of different spatial frequency (LP stands for line-pairs over the FOV) centered on the center of FOV and captured images of those by using a chirped lens array for channelwise aberration correction for the oblique incidence and by using a regular lens array for comparison.

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A radial star pattern is well suited to determine the optical cut-off frequency of an imaging system. Here the object frequency is a function of the radial coordinate with respect to the center of the star. The right column of Fig. 11 shows the test pattern, the captured image using a regular lens array and the image taken by using the chirped lens array. As expected the resolution in the lower left corner is the same for both systems since here we have perpendicular incidence. The fundamental difference in resolution as a function of angle of incidence can be observed in the upper left and lower right corners (σ_maxx=σ_maxy =23°) already but even stronger in the upper right corner (σ_max=32°) of the captured images. Using the chirped lens array the star pattern can be resolved deep in the center even for the maximum angle of FOV while using the regular lens array the resolution is reduced to approximately half of that in the center of FOV.

6. Conclusions and outlook

The use of chirped micro-lens arrays provides a new degree of freedom in the optical design of artificial compound eye imaging systems. cMLA enable for superior optical performance compared to regular arrays because each single lens can be designed individually. The description of the array parameters can either be derived from analytical functions or from numerical simulation. In case of the artificial apposition compound eye objective a correction of off-axis aberrations such as astigmatism and field curvature was achieved by employing a chirped array of ellipsoidal micro-lenses. This lead to a drastically improved resolution homogeneity over the objectives entire FOV since each channel is individually corrected for its direction of view. Ellipsoidal lenses were chosen because they are anamorphic and offer the opportunity of reflow fabrication leading to smooth surfaces and therefore excellent optical performance. In future we will also correct distortion by the adequate channel-wise positioning of the pinholes. In order to achieve an additional beam deflection by the microlenses for increasing the FOV and even the correction of spherical aberration and coma, more sophisticated surface profiles such as off-axis segments of aspherical lenses are necessary. Due to their non-symmetrical profile an origination by reflow is not possible. A promising candidate for the fabrication of such profiles is laser lithography giving the ability to directly writing arbitrary surface shapes. However, problems of this technology to be solved in the future are the required large sag heights in the range of several tenths of microns, steep slopes, high fill factor and surface roughness.

 

Fig. 11. Bar targets of different spatial frequency and captured images of those by using a chirped lens array for channel-wise aberration correction for the oblique incidence and by using a regular lens array for comparison. Additionally, a specially adopted 4×1/4 radial star test pattern demonstrates the obtainable resolution in the four image corners as a function of the angle of incidence by the different radii of vanishing contrast of the radial star patterns.

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Finally summarizing, besides the advantages of artificial compound eye imaging systems we already presented in previous papers [3, 4], such as compactness and large telephoto ratio, this paper demonstrates for the first time one of their major benefits: Because of the segmented image transfer each channel can be specially optimized for its individual viewing direction while classical single-channel-imaging-systems always have to be a compromise for all the angles of incidence represented in the FOV.