A new type of very compact optical element for a near-eye display (NED) that uses a pair of microlens arrays (MLAs) is presented. The MLA pair works in conjunction to form a magnifier (collimator). The purpose of this is to aid in the accommodation of the eye on a head-up display that is positioned within several centimeters from the eye; the MLA pair collimates the light rays departing from the display thereby generating a virtual image of the display at optical infinity. By using the MLA pair, we are able to make a collimator that retains a thin profile of about 2 mm in thickness with a system focal length of about 7 mm.
© 2015 Optical Society of America
A near-eye-display (NED) is a type of head-up display (HUD) that is positioned close to the eye, and it typically takes the form of a pair of glasses, helmet, or a headgear to secure the display near the eye within several centimeters. However, an average adult human eye cannot accommodate on objects closer than on average around 10 cm . Therefore all NEDs require an optical element (collimator), in one form or another , that generates a virtual image further away from the eye so that the image can be comfortably viewed.
The weight and size are the obvious points of consideration when designing an NED, because the head is the platform on which the NEDs are anchored. Excessive weight can make them uncomfortable to wear, or even lead to neck and head injuries . Excessively large NEDs can impede the wearer’s sight, and have a negative form-factor; it may even come across as having a disproportionate/unattractive appearance in the eye of the beholder. Therefore, the weight of the system needs to be reduced below the loading limits, above which will induce tissue and/or mechanical damages to the neck and the head (i.e. the neck muscles and the cervical vertebrae), but preferably as low as possible. In the same context, the size of the NED needs to be minimized, given the desired performance criteria such as the field of view (FOV) are not compromised.
Refractive singlet magnifiers (simple lenses) are typically used as the optical element of the NED to generate virtual images, thanks in part to their well-studied optics. However, the profile of the optical element can be quite thick when simple lenses are used, because the thickness of the lenses tends to grow as an inverse function of the focal length. When a large FOV is desired, the diameter of the optical element needs to be large enough as well to make room for the visual field of the eye. Oculus Rift DK2 (Oculus VR, Menlo Park, CA) for example, uses a pair of lenses with a diameter of ~40 mm and a thickness of almost 15 mm.
Some NEDs also incorporate reflecting surfaces which allow the optical path to be folded. Recon Snow goggles (Recon Instruments Inc., Vancouver, BC) and Vuzix M100 (Vuzix Corp., Rochester, NY) are examples of NEDs that use derivatives of singlet magnifiers with reflecting surfaces. Google Glass (Google Inc., Mountain View, CA) uses a Birdbath-type optical element that incorporates a curved mirror and beam splitters. Despite the folding strategies, NEDs can still have a considerable profile, i.e. about 13 mm in Google Glass and 20 mm in Recon Snow goggles, which can still impinge on the form-factor. In general, the weight of an NED is roughly proportional to the size of the optical element thus reducing the size of the optical element will decrease the system weight.
Several approaches are available in reducing the overall size of the refractive magnifier. For example, the frontal area of the lens can be reduced by using a smaller section of the lens. However, this diminishes the FOV as less of the object will be seen through the smaller aperture. Also, in order to compress the side profile (system thickness) the focal length can be decreased such that the object can be brought closer. However, decreasing the focal length requires a smaller radius of curvature (ROC) of the magnifier, which introduces more severe lens aberrations and increases the sag (or the lens thickness) of the magnifier. Suffice to say, refractive lenses, with their own limitations, have clear drawbacks in making a compact NED.
Diffractive waveguides are used in some NEDs to make the optical element very thin (as in Lumus DK-32, Lumus Ltd., Rehovot, Israel). However, the light being coupled into the waveguide still needs to be collimated, which can lead to the input collimator being bulky. Also, the waveguide needs to be long enough such that the exit-coupling region (the viewing area) is aligned with the line-of-sight of the wearer while the collimating optics and the display are positioned away from the view. This can lead to an increase of the overall volume of the waveguide and subsequently the system weight.
The use of microlens arrays (MLAs) is another alternative to making compact collimators. MLAs are thin, and with the short focal length of the microlenses the display can be positioned nearly adjacent to the MLA, allowing a slim system profile. However, a single MLA layer does not form a cohesive image of an object. Rather, each microlens on the MLA forms an individual image of the different perspectives of the same object. There are workarounds to this such as the use of a lightfield display reported in a recent work , where a single-layer MLA in conjunction with a display showing multiple tiles of shifted duplicate images. Each of the tiles is then imaged by a single microlens on the MLA layer. Another approach is using a Gabor superlens-derived magnifier, as previously discussed by us in . A Gabor superlens (just superlens from here on) uses a pair of MLAs (sometimes more) with microlenses of different focal lengths and pitches, where fixed pairs of microlenses of both MLAs work in conjunction so that the two MLAs optically act like a singlet lens. The use of multi-layer MLAs was pioneered by Dennis Gabor , who coined the term superlens to describe his two-sided lenticular sheet with the lenticules on each side having a different pitch. Imaging systems that make use of the 2D MLA-based superlens have been studied some decades later in other literatures [7–12]. Contrary to the use of superlenses in previous works as a focusing lens to gather light onto an image plane (i.e. image sensors), we use the superlens concept to collimate light from a display in an NED. With the intention of making the superlens-based NED as compact as possible, we use two single-sided MLA layers: the minimum to form a superlens. Furthermore, we use a concave MLA as one of the two MLAs, which reduces the overall thickness of the superlens. To the best of our knowledge, this is the first report of the use of concave MLAs in superlenses and the application of a superlens in an NED. Unlike the single-layer lightfield display, the use of a superlens magnifier eliminates the need for tiling images, thereby substantially increasing the achievable display resolution, although the system gains an additional MLA layer. This paper lays out the optical principles of the superlens having a concave MLA, largely through the ray transfer analysis, and demonstrates the use of superlens through simulation and a fabricated prototype.
2. Design of microlens array magnifier
2.1 Theoretical analysis
The analysis of ray propagation through an array of lenses can be carried out by treating the lens array as multiple single lenses that are systematically decentered from the center optical axis of the MLA. The ray analysis can be performed in just one dimension of the MLA plane if we assume that the microlenses have the same periodicity in both dimensions of the MLA plane, and the MLA is thin. A simplified ray propagation model through the MLA magnifier is shown in Fig. 1. In Fig. 1, the propagation of a light ray (denoted as the input ray) through the MLA system is shown, where it departs from an arbitrary point on the object plane, taking an arbitrary path through the MLA system. The two MLA layers are represented as a single rectangle for simplicity. We will let the optical axis be in the direction of the x-axis in the xyz coordinate system. The light rays originate from a point on the object plane at a height of hin from the optical axis and at a distance u from the MLA system. The rays entering the system make an angle of θin with respect to the optical axis, and the exiting rays converge at a point an arbitrary distance v away. B represents the actual path taken by the ray through a pair of microlenses, one on each MLA layer, that are each decentered vertically by some distance. A represents the hypothetical path taken by the input ray, as if it were to go through the same pair of microlenses, had they not been decentered and were axially coincidental with the origin of the input ray on the object plane. Then, the difference in the ray height and angle at the image plane between the two optical paths can be represented by ∆h and ∆θ. In order to take into account the height and angle shifts of the ray paths, we use 3 × 3 transfer matrices in a similar manner presented in .
The input ray can be described in terms of the ray height and angle modified by the MLA system asFig. 2, for matrices numbered between 1 and 5 .
For an imaging condition, all of the rays departing from the same point on the object plane having different θin must converge at the same point on the image plane. This suggests that hout needs to be independent of the θin requiring M12 = 0. The rays that go through different microlenses on the array arrive at the image plane with different ∆h, since ∆h is dependent on the microlens location variables N1 and N2, which are integers that describe the relativeness of the microlens location from the center of the array as in Fig. 2. For the same reason, hout needs to be independent of ∆h, yielding ∆h = 0. Since we want to collimate the exit rays, the image plane should be at infinity. Accordingly, the exit angle θout needs to be identical for the rays departing from the same point on the object plane.
This suggests that θout needs to be independent of θin and also ∆θ, the angle shift introduced by microlens decentration, since it is dependent on N1 and N2, corresponding to M22 = ∆θ = 0. This leaves us with the conditions imposed on the entries of the system matrix
Then, the exit ray height,Eqs. (5) and (7), we see that the conditions are met only when N1 = N2 such that they can be factored out of the equations. This means that the two MLAs have the same number of microlenses, and the light from the N-th microlens of the first MLA shall pass through the N-th microlens of the second MLA for correct refraction. Now if we let v → ∞ for the collimating condition and rearranging the equations, we get
The MLA layer clearance d is equal to the sum of the image distance of the first MLA and the focal length of the second MLA, assuming the MLAs are infinitesimally thin. Alternatively from Eq. (10), d can be expressed in terms of f1, F, and f2 as
Equation (13) reveals an interesting aspect of the two-layer MLA system; it shows that the object distance F is a simple function of the ratio between microlens pitches p1 and p2. When the image distance v is at infinity, F essentially becomes the focal length of the MLA magnifier, which is achieved by having different pitches for each of the two MLAs. Three combinations of the MLA pair with convex and/or concave microlenses satisfy Eqs. (10)–(15) such that F becomes positive, which fulfills the condition for image magnification. Of the three possible combinations of the concave and convex MLAs, we note that using a concave MLA as the first layer and a convex MLA as the second layer results in θout < 0, and has the most number of microlenses that contribute to the light propagation from the object plane to the image plane. Both characteristics contribute to the formation of the largest eyebox, which is the volume in front of an NED within which the complete virtual image can be viewed without being cropped.
We aim at having a 20° angular FOV and a 20 mm eye-relief (center of the eyebox at 20 mm from the exit surface of the MLA) to benchmark against the Recon Snow Goggles. From Eqs. (10)–(15), we see that the image system of the two-layer MLA is spanned by the design variables F, f1, f2, p1, and p2. We treat f2 and p2 as constants, since an off-the-shelf convex MLA is used as the second MLA, purchased from Suss MicroOptics (part # 18-00047). Also, the second MLA needs to take the finite/infinite conjugate form such that the exit rays from the second convex are collimated (image formed at infinity), while the object is a focal length away from the second MLA. Therefore, the front focal plane of the second convex MLA must coincide with the image plane of the first concave MLA. This fixed relative position of the two MLA layers allows us to fix d, for a fixed f2 and the variables F and f1 from which we can calculate the image distance of the first MLA. Note that we are still under the assumption that the microlenses are thin and also using the small angle approximation. Under these conditions we are able to describe the MLA system by only F and f1, since p1 is also in the space spanned by those two variables, according to Eq. (11). Figure 3(a) is a design tradespace spanned by F and f1, which shows in color the appropriate pitch of the concave microlens p1 for each combination of F, f1, and a fixed p2 of 0.250 mm. The superposed double dot-dashed green line represents the combinations of F and f1 that yield the required eye-relief of 20 mm, and the dot-dashed red line represents the combinations that yield an angular FOV of 20°. Figure 3(b) shows the required volume of a single photoresist microlens on the mold, from which the concave microlenses will be cast. The shortest focal length of the microlens is achieved when its ROC is equal to its base radius (assuming a spherical profile). Then, the combinations of F and f1 with the ROC smaller than the base radius are unobtainable, represented as the shaded area in Fig. 3(b). Due to limitations in fabrication and the photoresist used, the maximum photoresist deposition thickness is restricted to 30 μm. This restriction increases the minimum ROC for the given base radius of the microlens, because the ROC depends on the volume of the mold microlens. This is indicated by the grey contours that identify the boundary between the obtainable design space (above the contour) and the unobtainable design space (below and left of the contour). The number above the contour denotes the ratio between the microlens base diameter and p1. The lines that correspond to the FOV and the eye-relief requirements are also superposed in Fig. 3(b). When the microlens diameter is equal to p1 (i.e. the ratio between them is 1), the minimum F that would satisfy both the FOV and the eye-relief requirements is around 8.5 mm, as indicated by the point A in Fig. 3(b), where the requirement lines and the grey contour intersect. In order to make the MLA system more compact, the point of intersection needs to be moved down and to the left, such that F becomes smaller. This can be achieved by reducing the microlens diameter, which in turn would allow a smaller ROC and a shorter focal length to be realized. For microlens diameters less than 80% of the pitch, the fill-factor of the microlens falls under 50% thus we want to limit the minimum microlens diameter to at least 80% of the pitch. The fill-factor is the ratio between the area occupied by just the microlenses, and the total area of the array. Then on the 0.8 contour, we choose a design point midway between both FOV and the eye-relief lines as a compromise between the two requirements as well as for maximum compactness and largest eyebox [point B in Fig. 3(b)]. This design option yields the MLA parameters listed in Table 1, with an eye-relief of 19.3 mm and an angular FOV of 18.9°.
The MLA magnifier is simulated using Zemax (version 12) in order to estimate the spatial resolution of the MLA magnifier. The model is constructed based on the optimal microlens parameter values in Table 1. In constructing the simulation, we use full 3D models of the MLAs as we take into account the thickness of the MLAs (both the substrate thickness and the microlens sag). The dimensions of the first concave MLA is 12 mm × 12 mm × 1 mm, and the second convex MLA is 12 mm × 12 mm × 0.9 mm. Figure 4 shows a planar cut of the 3D model of the MLA magnifier with three point sources on the object plane at different heights (3.2 mm, 0 mm, and −3.2 mm from the optical axis) being collimated with different exit angles.
The spot diagram analysis in Zemax traces and plots the rays intersecting a plane. Using this, the dimensions of the eyebox is approximated by “slicing” the collimated beam with an intermediate plane placed a certain distance away and measuring the geometrical radius of the spot generated on the plane. The maximum size of the eyebox (maximum cross section in the yz-plane) according to the simulation is 5 mm in diameter, 20 mm from the last surface of the convex (second) MLA in the x-axis direction.
We use a simplified model of an eye provided in Zemax (the Eye Retinal Image model) that forms an image on the retina, to test collimation of the light rays as in Fig. 5. The model of the eye used in the simulation images a distant object at infinity, thus if the exit rays from the MLA magnifier are indeed collimated, the rays will form a focal spot on the retina of the eye model. We use a customized version of the MLA lens surface supplied in Zemax (us_array.dll), in order to allow microlens diameter to be different from the pitch. The simulation parameters are given in Table 2. Other parameters not listed in Table 2 are kept with default values.
The degree of collimation of the light rays (i.e. the spatial resolution of the magnifier) can be assessed from the modulation transfer function (MTF) plots. An MTF plot shows the contrast of the image generated by the optical system as a function of the spatial frequency of the features of the object. The spatial frequency is measured in cycles/mm and describes the size of features that can be resolved. The contrast is measured as a normalized difference between the maximum and minimum luminance of the image features, and generally decreases as the spatial frequency of the object is increased.
Figure 6 shows the two MTF plots which represent the worst and best case scenarios for the spatial resolution. The worst case represents the stray light rays propagating through the gaps between the microlenses that do not form an image on the retina. The best case scenario corresponds to the light rays that are refracted by the microlenses as intended.
Several techniques exist for producing microlens arrays, which include the embossing [14, 15], diffractive printing , ablation-etching , direct printing , and photolithography-reflow techniques [19–22]. We chose the photolithography-reflow technique as the resources were readily available in our lab. Figure 7 shows the process steps and the cross-sections of the substrate after each process step.
The concave MLA is fabricated using photolithography and reflow processes followed by PDMS casting. In our fabrication, we use a 4-inch silicon wafer as the substrate, and SPR-220 7.0 (Dow Chemical Company, Marlborough, MA) positive-tone photoresist as it is capable of being deposited in relatively thick layers. The MF-26A positive tone developer (Dow Chemical Company) is used to develop the photoresist. After the photolithography step, islands of photoresist cylinders are defined on the wafer, patterned from an array of circles printed on a bright-field mask.
Figure 8 shows the shape transformation of the cylindrical islands into spherical caps in the reflow process. In essence, the substrate is heated above the melting temperature of the photoresist and the surface tension turns the profile of the molten photoresist into spherical caps. Assuming the volume of the cylindrical island remains constant during the reflow yields the correlation between the dimension parameters
Thus in order to control the radius of curvature Rs, we can adjust the cylindrical island height (by changing the deposition thickness) tp and the cylinder base diameter r (by changing the diameter of the circles on the mask), given that the cylinder height is sufficiently large in comparison to the cylinder diameter such that the surface contour of the reflowed island can be assumed spherical.
However, upon inspection of the developed cylinders we notice that the cylinder sidewall is chamfered quite a bit towards the center of the mask, which makes the volume of the photoresist island less than the ideal cylinder. This could be due to the diffraction of light at the abrupt edges of the circular openings on the mask which creates a non-uniform exposure field with a tapered intensity profile towards the edges. In , the pre-reflow microlens structures are seen with similarly chamfered sides. We estimate the degree of inclination of the sidewall from the top and the base diameters of the chamfered cylinder as shown in Fig. 9.
The slope of the sidewall of the photoresist in Fig. 9 is approximately ~45°, and the slope is almost consistent for photoresist heights of 25.5, 28, and 29.5 µm. Therefore we assume that the sidewall slope to be fixed for similar photoresist thicknesses, and represent the volume of the chamfered cylinder as a function of its height tp and the base diameter r such that
The maximum thickness of the photoresist that can be reliably reproduced is about 30 μm for a spin speed of 455 rpm, as determined from the measured spin speed vs. deposition thickness curve as shown in Fig. 10. With a 30 μm thick layer, we find that the actual fabricated microlenses have a slightly larger diameter of 0.216 mm compared to the target diameter of 0.21 mm. This increases the actual ROC slightly to 0.16 mm.
The silicon wafer with the islands of spherical caps is used as a mold for casting. PDMS is used as the casting material for it has good mechanical and optical characteristics (e.g. rigidity and transparency). The wafer is prepared for casting by first cleaning the surface with an air gun and the MLA area is enclosed by a 3D-printed “fence” to contain the PDMS resin. The casting of PDMS is performed following the standard casting procedure consisting of mixing, degassing, and curing the PDMS. The PDMS resin mixed with the curing agent at a 5:1 ratio is used to yield ~1 mm thin MLAs that are mechanically robust enough to be handled. After curing for 2 hours at 60°C, the hardened PDMS is cut and peeled off from the wafer. The cast PDMS concave microlenses are shown in Fig. 11.
4. Testing of the MLA magnifier
The fabricated concave MLA is paired with the commercial convex MLA, and held together by a 3D-printed frame. The translational and rotational alignment of the MLAs relative to each other is performed manually using an XYZ-stage (DT12XYZ from Thorlabs). Figure 12 shows the virtual replica of our test setup modeled in Solidworks.
To quantify the spatial resolution of the MLA magnifier, we use test patterns consisting of alternating black and white lines with different spatial frequencies. The magnified images of the test patterns produced by the MLA magnifier are captured with a Canon S120 camera. Figure 13 shows the test patterns used in the measurement and the resulting images.
The test patterns are displayed on a Sony microdisplay that has 1044 × 768 pixels and a pixel pitch of 7.8 μm. The images are taken with the same exposure time of 1/13 s, aperture of f/6.3, and the focal length of 5 mm (lowest zoom setting), on the lowest image sensor sensitivity setting (ISO 100). The parameters are chosen primarily to avoid saturating the brightness or making the recorded image too dim, while achieving a maximum contrast across the image. Note that the images are not taken under controlled lighting, and different camera parameters would have affected the measurement. The line width of each test pattern is defined in number of pixels, from which the spatial frequency in cycles/mm is calculated. In order to find the contrast of the test pattern images, the RGB values of the pixels in the recorded images need to be converted into relative luminance values. The first step to the relative luminance conversion is decoding or linearizing the gamma from the recorded RGB values, as the camera images are saved in the JPEG format. The detailed calculations needed for the gamma decoding and the relative luminance conversion are covered in . Once the relative luminance values are obtained from the RGB values, the contrast is averaged over the 1 mm2 area (roughly consisting of 16 microlenses for both the concave and the convex array) near the center of the MLA. The resulting contrast of the test pattern images at each spatial frequency is compared with the MTF response from the simulation (using an axial point source and the parameters in Table 1) as shown in Fig. 14. The contrast drops to nearly zero at ~12 cycles/mm for the horizontal test patterns (representing the sagittal resolution in the xz-plane), at which the corresponding spatial frequency represents the maximum resolution of the MLA magnifier. We note that the contrast for the vertical test patterns (representing the tangential resolution in the xy-plane) drops to zero at lower spatial frequency than the horizontal test patterns, even though the pitch of the microlenses for both MLAs is identical in both y and z-axis directions. This could have resulted from a higher degree of translational misalignment between the test components in the y-axis direction than the z-axis direction.
We also observe that the measured contrast over the entire range of spatial frequency is somewhere between the MTF response of the light coming only from the inter-lens gaps and the MTF response of the light propagating only through the microlenses. This is expected as the actual MTF response of the MLA magnifier should be a combination of these two extreme cases. The light rays passing through the regions between the microlenses arrive at the image plane as stray light (the worst case), thereby contributing to the image as the background luminance. This reduces the contrast of the image generated from the light rays correctly refracted by the microlenses (the best case). A set of test images are also displayed and the images seen with and without the MLA magnifier are captured using the same camera, as shown in Fig. 15. Since the angular FOV of the camera is known from its focal length and the sensor size, the location of the aperture of the camera can be calculated from a tangential ratio between the actual size of a feature used in the captured image (i.e. the square opening of the MLA magnifier holder) and the angular FOV subtended from it in the image. The location of the aperture is estimated to be about 18 mm from the MLA magnifier holder. To estimate the size of the eyebox, the MLA magnifier and the microdisplay, each mounted on an XYZ-stage, are moved relative to the camera in the y-axis and z-axis directions (parallel to the plane of the MLAs), until a 6.4 mm × 6.4 mm large test object starts to disappear from the view of the camera and appears cropped. Then, the displacement of the camera between the opposing sides of the test image corresponds to the eyebox size. The eyebox size is measured to be about 4 mm in both directions.
By capitalizing on the optical properties of the multi-layer MLA system, we have demonstrated the use of the multi-layer MLA system as a compact magnifier in a non-folded NED. The MLA magnifier has been designed with target angular FOV and the eye-relief of 20° and 20 mm respectively, benchmarked from Recon’s Snow HUD goggles. In making the prototype we settled with compromises for the MLA parameters due to fabrication limitations, and the resulting angular FOV and the eye-relief are slightly lower than the target values at 17.2° angular FOV at 18 mm from the MLA magnifier. The size of the eyebox is found to be similar to the estimated size from the ray-tracing simulation. In the end, we demonstrate that by using the MLA magnifier, objects very close to the eye can be brought into focus while maintaining a compact profile of ~10 mm, measured from the display to the exit surface of the convex MLA, including the thickness of the microdisplay used. This shows that the MLA magnifier can indeed be used as a collimator in NEDs, although the current spatial resolution of the prototype MLA magnifier leaves room for improvement.
This work is supported in part by Recon Instruments Inc. (Vancouver BC, Canada) and the Mitacs Accelerate program of Canada (awards IT02390 and IT03173). This work has also been partially supported by the Canada Research Chairs Program. We express our gratitude to Prof. Jonathan Holzman in the School of Engineering at UBC Okanagan for providing access to optical test equipment.
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