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In this work, we present the design and fabrication of a progressive rear-view mirror for motorcycles. In the context of physiological and physical background knowledge, we first analyze the geometric relationships among the profile of the mirror, the blind spot, the field of view, and the reflected image size. On the basis of Walker’s eye model, the binocular disparity is further calculated according to the image size on each retina. We present the polynomial expansion that specifies our progressive mirror’s profile, as well as the fused deposition modeling process for fabricating physical mirrors. Compared with a conventional aspheric or flat mirror, this progressive mirror can achieve a wider horizontal viewing angle and shows a more stable image, thus enhancing riding safety.
© 2016 Optical Society of America
1. Introduction
In transportation, the blind spot is an area within a vehicle that the driver or passenger cannot observe directly. The A-pillar, side rear-view mirror, and interior rear-view mirror can all limit a driver’s view and create blind spots. A straightforward way to eliminate the side blind spot is to enlarge the field of view (FOV) by adopting a convex or aspheric rear-view mirror. In the United States, however, only planar mirrors are permitted for use on the driver side of a vehicle, because a planar mirror provides a reflected image that appears to have the same size as an identical object at an equivalent distance being viewed without the mirror. A flat mirror’s “unit magnification” characteristic does not distort the relative distance or approaching speed of vehicles seen in the mirror [1].
To alleviate distortion and maintain unit magnification simultaneously, vehicle designers have offered aspheric mirrors and multiradius mirrors [2–4], most of which have a relatively flat section and a curved section. The flat part can be used for correct distance perception, and the convex section can be used as a “presence” detector. As shown in Fig. 1(a), the rearview mirror is composed of a convex portion and a flat portion. This combination, however, has a side-effect, image jump, which is produced by the sudden introduction of the prismatic power when the eye looks across from the flat portion into the convex portion. A similar problem can also occur as the driver looks through the borders between sections of the multi-radius mirror, as illustrated in Fig. 1(b). Consequently, most users have to learn to physiologically adapt to the jump. One way to eliminate image jump completely is to use a mirror with a progressive reflecting power, which can be realized by a freeform mirror. The freeform mirrors do not utilize borders at all, so reflected images transits smoothly between various reflecting powers, allowing the eye to adjust well as it moves. Moreover, freeform optics is becoming increasingly useful in the imaging, illumination, and simplification of optical systems [5–7], because freeform optical devices provide more degrees of design freedom. In traffic applications, freeform reflectors can be used to enlarge the viewing angle and minimize distortion. Lee [8] designed a freeform transition zone between an upper zone (for distance vision) and a lower zone (for viewing nearby objects) of a side rear-view mirror for cars. A conventional two-section aspheric mirror has an inflection that extends from the boundary of the aspheric outer zone and to the edge of the mirror. A progressive mirror with an added transition zone has inflection only within the progressive zone between the inner and outer zones; such a mirror can provide an FOV of 36.3°.
Fig. 1 Image jump can be observed when the eye sees across the borders between sections with different orientations and curvatures. The borders of (a) a mirror composed of a convex portion and a flat portion and (b) a multiradius mirror.
A recent report [9] indicated that in each year from 2011 to 2014, approximately 450,000 motorcycles were sold to customers in the United States. However, the design of side rear-view mirrors for motorcycles has been seldom mentioned. One reason is that regarding mirrors being required on motorcycles, as well as other motorcycle equipment, the United States has no national standard. Motorcycle riders often have limited views of the road, and motorcycle accidents often result in severe, life-threatening injuries for the rider. Therefore, it is crucial to design adequate side rear-view mirrors for motorcycles.
In this work, we show that the design considerations for motorcycle side rear-view mirrors are similar to those for car mirrors. The main difference is that a motorcycle rider’s eyes are higher and closer to the mirror compared with those of a car driver. Thus, our proposed method simultaneously considers the FOV, reflected image size, and the binocular image disparity for optimizing the profile of the mirror. The resultant mirror has a freeform horizontal profile and a constant vertical curvature; therefore, it provides favorable image quality.
2. Design and simulation
Among the competing priorities in the design of a side rear-view mirror, a horizontal FOV, reflected image size, and distortion are the first three to consider.
To enlarge a horizontal FOV, aspheric mirrors have become increasingly common on light vehicles in Europe under European Directive 2003/97/EC. The inner two-thirds of a typical aspheric mirror of the type currently used in the European Union is spherical and convex; it may have a larger horizontal radius of curvature than that of a typical spherical convex mirror. The outer one-third is an aspheric portion intended to increase the overall FOV. In [10], Wiegand et al. statically tested various driver-side rear-view mirrors. As illustrated in Fig. 2(a), in the case of a flat mirror, the FOV was approximately 12°, whereas for a convex mirror with 2000-mm radii of curvature, the FOV was enhanced to approximately 21°, and for an aspheric convex mirror with 2000-mm radii of curvature, the FOV was as high as approximately 40°. However, these tests were based on the assumption that a nominal driver sees from a single point on the bridge of the nose between the eyes (the right and left monocular vision segments). The side rear-view mirror in a typical car is approximately 16 cm wide, and the distance from the middle eye centroid to the mirror is approximately 90 cm. Using ZEMAX [11], the calculated binocular FOV of this configuration is approximately 14°, 2° more than the measured value in [10]. The whole FOV, therefore, depends on the mirror profile, mirror width, interpupillary distance (between the centers of the pupils), and the rider’s direction and distance to the mirror. To completely cover the blind spot, the horizontal FOV from a rear-view mirror must be over 45°. As illustrated in Fig. 2(b), our goal for aspheric motorcycle mirrors is an FOV greater than 45° using two eyes, below which the blind spot is not sufficiently large to conceal another motorcycle.
Fig. 2 Diagram depicting (a) the FOV of various rear-view mirrors for cars, and (b) the FOV goal of the proposed mirror for motorcycles.
The reflected image size is determined by the “image minification” factor, which is the ratio of the image size seen in a convex mirror compared with that seen in a corresponding flat mirror. As depicted in Fig. 3, the motorcycle rider views an image at a distance, -l, from the mirror, whether it is flat or convex. For a flat mirror, the distance from the mirror to the virtual image is lf. For a convex mirror with a curvature radius R, the corresponding distance is the ratio of the angle ωc subtended by the convex mirror to the angle ωf subtended by the flat mirror, which is expressed in Eq. (1) [10]:
It is immediately apparent that the minification factor is less than 1. The convex mirror accepted by most car drivers generally reduces the apparent size of the image to approximately half. Our minification factor for the spherical convex portion of a motorcycle mirror is therefore set to 0.5. When the rider is 0.51 m ( = -l) from the spherical convex portion and a vehicle is located 30 m ( = lf) behind the rider, these values can be substituted in Eq. (1) to show that the curvature radius R is approximately 1000 mm.
A complex rear-view mirror has an outer aspheric portion that distorts images. This distortion narrows the horizontal dimension relative to the vertical dimension. It is caused by astigmatism, which is defined as the difference between the powers in the horizontal and vertical directions [8]. The power difference between the inner spherical and outer aspheric portions can cause binocular image disparity when viewed with two eyes. If the image disparity is above a certain threshold, the viewer sees a double image with two depths. The calculation of binocular image disparity is typically based on a polar plot [12] derived by measuring images in the mirror as angular displacements relative to the driver’s eye. To simplify, we modify the method by directly adopting the image length on the retina. Our binocular image disparity (ID) is defined as follows [13]:
where lR is the image length (in microns) of the target projected on the rider’s right eye, and lL is the length on the rider’s left eye. ID measures the percentage change between the image sizes on the retinas of both eyes.Our mirror is 160 mm wide and 90 mm high, as shown in Fig. 4(a). To enlarge the horizontal FOV, the curvature radius between the far and near views is horizontally progressive (from Ri for far viewing to Ro for near viewing) and vertically constant (Ri). The FOV is the angle between the near and far views, as depicted in Fig. 4(b). The mirror has an extended polynomial front surface, and its sag equation is Eq. (3), which is composed of two portions: the conic aspheric surface and the extended polynomial deviation:
where N is the number of polynomial terms, Ai is the ith polynomial coefficient, m and n are integers, and x and y are the normalized coordinates. It was discovered in this research that higher-order (≥ 6) polynomial terms only slightly improve the optical performance and can be disregarded. In addition, the freeform surface is symmetric with respect to the y–z plane. Therefore, only three terms are included in the extended polynomial to describe the freeform surface:Fig. 4 (a) Rear-view mirror dimension. Dotted line indicates the horizontal line. (b) Design layout.
Once the sag terms were selected, the optical performance of the mirror was simulated using ZEMAX software. As shown in Fig. 5, three viewed targets were modeled on the computer. Most cars or motorcycles travel above 60 kilometers per hour. For a rider preparing to make turns, if a car 30 m behind can be well viewed, the rider has enough reaction time (about 2 seconds) to avoid road crashes. The first target is therefore set to 30 m behind in the direction of 0°. On the other hand, some fatal crashes occur as two vehicles are side-by-side. The sensing of the presence of a near-by car or motorcycle is important. Consequently, the other two “nearby” targets are set to 4 m behind in the direction of 22.5° and 2 m behind in the direction of 45°. Under all conditions, the distance from the rider’s middle eye centroid to the mirror was held constant at 510 mm. The mirror was aimed vertically to keep the target image in the center of the mirror.
Next, the mirror was combined with the Walker schematic eye model [14] with a 4-mm pupil diameter. This eye model is one of the most commonly used models for designing visual instruments because it achieves a useful compromise between simplicity and accuracy.
The reason why a 4-mm pupil diameter is set has to do with the typical riding condition. The normal pupil size in adults varies from 2 to 4 mm in diameter in bright light to 4 to 8 mm in the dark. To ride safely in the dark, motorcycles headlights are commonly used to enhance the ambient light level and provide enough visibility. As a result, a pupil diameter of 4 mm is assumed in the analysis.
The ZEMAX implementation using Walker’s eye model to design the freeform mirror is reported in Fig. 6(a), as it appears in the ZEMAX “Sequential Component Editor” at the beginning of the optimization procedure. The surface of object (OBJ) with a thickness 2000 mm represents a target point that locates 2000 mm away from the freeform mirror. Surfaces from no. 1 to no. 3 represent a tilted freeform mirror. Surfaces from no. 4 to no. 11 represent the middle eye centroid, composed of the dummy, cornea, anterior chamber, pupil, crystal lens, vitreous humor and retina, respectively. Among the surfaces, surface no. 2 is described by an extended polynomial. The three terms, A1y2, A2y4 and A3y6 in Eq. (4), selected in the extended polynomial to describe the freeform surface are defined in the columns X0Y2, X0Y4 and X0Y6 of Surface no. 2 in the ZEMAX “Extra Data Editor”, as shown in Fig. 6(b). The corresponding Radius and Coefficients are labeled by a V, denoting variables for the optimization procedure.
Fig. 6 Screenshots of (a) the ZEMAX sequential component editor showing the initial layout and (b) the ZEMAX extra data editor.
Calibration of the freeform mirror in the ZEMAX environment has been performed using the custom merit function reported in Fig. 7, as it appears in the ZEMAX Merit Function Editor in the middle of optimization procedure. For example, the operand of edge thickness value (ETVA) is used to set a constraint on the maximal deviation between the edge and vertex of the mirror. Operand no. 6 is “default merit function start” (DMFS), which here represents all operands after DMFS are used to minimize the spot diagram on the retina.
Fig. 7 Screenshot of the ZEMAX Merit Function Editor showing the custom merit function implemented for determining the profile of freeform mirror.
All the simulations were performed for monochromatic light (λ = 560 nm). There were three points (“far”, “middle”, and “near” points at which the power for far, middle, and near vision were measured; see Fig. 8), for which the retinal image quality of the eyes (binocular disparity and modulation transfer function (MTF) were estimated. Thus, the eye model was initially located in the middle eye centroid and could rotate around the rotation point of the eye. We optimized the horizontal line (shown in Fig. 4a) to have the chosen points’ real image heights projected to the intersection of the optical axis and the retina under a maximal deviation of 20 mm between the edge and vertex of the mirror, because the overall angle between the near and far views was over 45°. The final profile can be described by the following expression:
After determining the profile, we simulated a pair of eyes with an interpupillary distance of 65 mm. To calculate the binocular disparity, we further set the target in the three riding scenarios to be horizontal linear bars, which are 100, 180 and 1220 mm for the near, middle and far scenarios, respectively, and measured the image lengths on the retina of each eye. The eyes were simulated to gaze at a point moving across the horizontal line; the maximum binocular disparities for the three scenarios are shown in Table 1. The maximum ID changed only slightly when the target distance was increased from 2 m (1.33%) to 4 m (2.1%) and then to 30 m (2.61%). The acceptable ID is dependent on the image contrast and driver’s state of adaptation to the mirror image. If the ID is too high, double vision occurs, which makes the rider perceive two images of a single object. According to Qin’s research about Penum’s fusional area [15], the acceptable horizontal disparity limit in fovea has to be below 32 arcmin or 155 micron on retina, which is much larger than the image length difference on both retinas for the three riding scenarios. Similarly, the maximum binocular disparities for the three scenarios of a commercial mirror are also calculated for comparison and shown in Table 2. The mirror is composed of two sections. The inner two-thirds spherical portion has a radius of 2000 mm and the outer one-third aspherical portion a radius of 200 mm. The maximum ID also changed only slightly when the target distance was increased from 2 m (2.22%) to 4 m (0.22%) and then to 30 m (1.14%). For a near-by object, the freeform mirror induces less ID and less double image. As a result, the disparity values of our design were sufficiently low and also comparable with those of a conventional rear-view mirror for cars.
Table 1. Binocular Disparity of the Freeform Mirror for Each Scenario
Table 2. Binocular Disparity of a Commercial Mirror for Each Scenario
The layout of the system is shown in Fig. 9. It provided an overall FOV of approximately 52.4∘ (the combination of right and left vision) for the three scenarios, which is larger than the design specification, 45∘. The mirror had a rectangular size of 160 (W) × 90 (H) mm2. To evaluate the mirror, this research calculated the modulation transfer function (MTF) of images on the retinas of two eyes for the three riding scenarios. For a normal eye, the standard definition of normal visual acuity (20/20 vision) is the ability to resolve a spatial pattern separated by a visual angle of 1 minute of arc, which corresponds to 83 lps/mm in the scale of MTF. If the mirror yields serious aberration, the MTF at 83 lps/mm will perform far below the diffraction-limit requirement. For the final profile, the MTF plots of right and left eyes seeing an object at the three distances (30, 4, and 2 m backward from the mirror) are all higher than 0.3 at 83 lp/mm. Figure 10 shows the MTF plots for both eyes seeing an object at 30 m through the mirror. For the freeform mirror with a polished surface, the incident rays from a distant target passing through the rider’s pupil can be considered nearly paraxial and converges well to a small spot, which yields high image quality on the retina. On the other hand, we found the real image quality is mainly degraded by the irregularities of the freeform mirror’s surface. It can be seen that the MTF relies more on the surface condition and a suitable fabrication method is significant.
Fig. 9 Overall FOV in the horizontal direction, which is the combination of right and left vision, approaches 52.4°.
Fig. 10 MTF plots of the far view. MTF plots as (a) the left eye sees through the inner zone (Y ~0 mm, Z = 0 mm) and (b) the right eye sees through the inner zone (Y ~0 mm, Z = 0 mm).
3. Fabrication
Current designed freeform mirror is non-rotational-symmetrical, which is composed of different curvatures in different orientations and hard to precisely make in traditional turning machining. Usually a 5-axis ultra-precision computer numerical control (CNC) machine is the first choice to manufacture this optical surface with micrometer form accuracy, because it has additional degrees of freedom to reduce the geometric errors. But its fabrication fee was beyond our budget too much, we used three dimensional (3D) printing instead. The prototype of a motorcycle rear-view mirror was realized using fused deposition modeling (FDM) [16], which is a kind of 3D printing. Because our mirror is an asymmetric object, the required fabrication process was markedly complex. To simplify the process, a larger symmetric substrate, which was four times larger than the intended product, was fabricated and then partitioned into the design size after FDM. FDM comprised four steps. The first step involved exporting a stereolithography file format (STL file) to the machine and mathematically slicing the model for the printing process. In the second step, the substrate was produced by extruding small strings of molten acrylonitrile butadiene styrene (ABS) to form a coarse substrate layer-by-layer as the material hardened immediately after extrusion. Afterward, a computer-numeric-control machine was used to fine polish the surface of the ABS substrate. In the fourth step, the polished side of the ABS substrate was coated with aluminum to enhance reflectivity. After FDM, the object was scanned to confirm its profile. Figure 11 shows four views of the fabricated object, illustrating the geometry before the partitioning into four pieces. The right view shows that the vertical curvature radius approaches the designed value of 1000 mm and the height is 90 mm after the partitioning. The front view shows that the gap between the edge and center is close to 20 mm, which matches the maximal deviation value of 20 mm, as illustrated in Fig. 8.
Fig. 11 Four views of the fabricated object. (a) Top view; (b) perspective view; (c) right view; (d) front view.
To measure the performance of the partitioned mirror, two lasers were used to represent the viewing positions of the right and left eyes. As shown in Fig. 12, the beams from the lasers were traced to obtain an overall FOV of 52.7° in the horizontal direction, which was slightly larger than the design specification of 52.4°. The image in Fig. 13(a) shows a view of a scene with a traditional flat mirror, whereas the image in Fig. 13(b) shows a view of the same scene using our mirror. The red dash outlines the scene reflected by the flat mirror. The mirror gives an overall FOV of 52.7°, which is sufficiently large to eliminate the blind spot for riders. Our mirror can reflect a scene four times larger than the scene reflected by a flat mirror.
Fig. 12 Optical bench for measuring the overall FOV of the partitioned mirror by the ray-trace method.
4. Conclusion
We demonstrated the design and fabrication procedures of a progressive rear-view mirror for motorcycle riders. We presented the evaluation equations of the achievable FOV, the reflected image size, and the binocular disparity. For a 160 × 90 mm progressive mirror, the overall FOV can reach 52.4° in the horizontal direction, whereas the maximum binocular disparity is less than 5.5%. The proposed mirror can reflect a scene four times larger than that reflected by a flat mirror.
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