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Performance estimates for a multicube retroreflector design
Optics Communications 441 (2019) 26–32
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Performance estimates for a multicube retroreflector design J.W. Lewellen a , J.R. Harris b ,∗ a b
Los Alamos National Laboratory, Los Alamos, NM 87544, USA Air Force Research Laboratory, Albuquerque, NM 87117, USA
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Keywords: Laser Retroreflector Dispersed performance
ABSTRACT We describe a novel type of retroreflector, which we call a ‘‘multicube’’ retroreflector, optimized to provide improved optical power return when a large number of retroreflectors are dispersed through a volume or on a surface. While it does not match the performance of a ‘‘multiplane cross’’ type retroreflector on an individual level, it offers improved performance compared to a standard corner cube while retaining the corner cube’s advantages: fabrication via stamping, and the ability to densely pack the retroreflectors. When dispersed as part of a cloud of retroreflectors, the multicube design provides significant performance improvements over both simple corner cubes and ‘‘multiplane cross’’ retroreflectors.
1. Introduction Retroreflectors are optical elements which can direct a significant fraction of the light incident upon them back towards their source. Retroreflectors occur in nature [1,2] as well as being produced artificially in a wide variety of configurations [1,3,4], and are employed in many applications including traffic safety [1,2,5–7], space applications [6,8–10], distance measurements on construction sites [11], communications [3,4,12,13], and remote sensing [4,10]. Different types of retroreflectors have different advantages and limitations, such that the retroreflector design should be tailored to its intended application [6, 14]. For example, in the case of road signs, perfect retroreflection is not actually desired, as the light source, retroreflector, and observer are generally not collinear [2], while perfect retroreflection with minimal losses and distortions is generally desired in space applications [6,8–10]. For our present application, we sought a retroreflector design that enables a large number of low-cost, rugged retroreflectors to be packaged compactly before use and then dispersed into a larger volume through which a large beam of light is propagating, and in which our objective is to maximize the total power of retroreflected light. This means that a reduction in the performance of each individual retroreflector is acceptable provided that it is more than offset by increasing the number of retroreflectors that can be fit into a given initial volume. It also means that the orientation of any given retroreflector with respect to the light source will not be known. This regime of operation is quite different than is typically encountered. Frequently, the light to be retroreflected is in a beam whose size is considerably smaller than the retroreflector, and only a single retroreflector is used. In this regime, to first order, one can expect that either 100% of the incident light, or none, is reflected back towards the source, depending upon whether the incident beam is within the
retroreflector’s angular acceptance or not. An example of this regime would be the use of a corner cube as part of an interferometer or similar optical instrument. A second regime occurs when the incident optical beam has a transverse size equal to or larger than the retroreflector; many retroreflectors may be used, in either a specific or random orientation, as part of the optical system. In this case, the incident beam ‘‘fills’’ the retroreflector(s), and the fraction of incident light reflected back towards the source is a more-or-less smooth function of the orientation of the retroreflector within the light beam. Additionally, our requirement to maximize the number of retroreflectors that can fit within the initial, pre-deployment volume requires designs which are ‘‘nestable’’, while their unknown post-deployment orientation requires that we consider retroreflection in all directions. These considerations rule out certain types of retroreflectors, such as spherical retroreflectors, which have larger acceptance angles than other types but do not allow efficient use of pre-deployment volume and tend to retroreflect smaller amounts of incident light than other designs [11]. Perhaps the simplest retroreflector is the corner cube, consisting of three plane surfaces. Light rays entering the corner cube’s acceptance window are reflected back towards their source along parallel but offset trajectories. Fig. 1 shows a conventional corner cube’s geometry. Given at least approximate prior knowledge of an optical system’s configuration, corner cubes can be highly effective retroreflectors, and can in principle be made easily from sheet metal via a stamping process. For our purposes, however, the standard corner cube is not ideal. The unknown post-deployment orientation means that a retroreflector capable of returning light with a wider angular acceptance than a conventional corner cube is desirable. This could be accomplished by filling the corner cube with a dielectric material [1] or by using
∗ Corresponding author.
https://doi.org/10.1016/j.optcom.2019.02.013 Received 1 November 2018; Received in revised form 29 January 2019; Accepted 4 February 2019 Available online 14 February 2019 0030-4018/Published by Elsevier B.V.
J.W. Lewellen and J.R. Harris
Optics Communications 441 (2019) 26–32
Nomenclature 𝜃, 𝜙
𝛷I 𝛷m 𝛷o 𝛷r 𝛷s
rotation angles that specify the orientation of the retroreflector relative to a fixed point, e.g. an optical source flux emitted by the source in the numerical simulation optical power emitted by the source that does not impinge upon the retroreflector the flux incident on the retroreflector optical power directed back towards the source by the retroreflector optical power incident on the retroreflector but not directed back towards the source
Fig. 2. A retroreflector consisting of eight joined corner cubes. The darker planes highlight a single corner cube.
associated with fabrication which might degrade their performance in practice. The multicube retroreflector will be shown to have limitations in scenarios where a single retroreflector can be deliberately placed with respect to the light source, but significantly improved performance when used as intended, with multiple retroreflectors dispersed from a small volume into a larger volume, with random orientations with respect to the light source.
Fig. 1. Corner cube geometry.
2. Performance analysis
a combination of eight corner cubes, joined at their inner vertices as shown in Fig. 2; we refer to the latter as a ‘‘multiplane cross’’ retroreflector. However, neither of these approaches allows efficient nesting of the retroreflectors to make best use of the available predeployment volume. In comparison, conventional corner cubes made from thin sheets can be densely packed, allowing more retroreflectors to be fit into a given volume. To address these issues, we propose a retroreflector configuration combining the potential for ease of fabrication of conventional corner cubes and the ability to easily and densely package them, with a greater range of angles over which light will be reflected back towards the source. We call this new design a ‘‘multicube’’ retroreflector; it is shown in Fig. 3 [15]. The multicube retroreflector is a corner cube retroreflector, modified to provide nonzero reflection when oriented with its primary axis of retroreflection directed away from the observer, and intended to maximize the number of retroreflectors that can be fit into a given volume. This concept is similar to the multiple cornercube configuration discussed in Ref. [8], with the difference that the multicube retroreflector is intended to be produced from thin materials to maximize the number that can be packed into a given pre-deployment volume. In the following sections, we will provide an initial analysis of the multicube retroreflector and compare its performance to the standard corner cube and the multiplane cross designs. Performance comparisons between different retroreflector types will primarily be based on simulation results. For the multicube design, a prototype was built and its performance compared to the simulated results for that design. This serves both to generally verify the simulation code performance in order to provide confidence in it and therefore in the numerically-based comparisons between the different designs, and more importantly to identify any minor differences between the performance of an actual and an idealized multicube in order to identify potential difficulties
We compare the performance of a conventional corner cube, a multiplane cross, and a multicube retroreflector by determining the fraction of the light incident on the retroreflector from a given angle, that is returned towards the source. The orientation of an object in space can be set by three angles; however, in our specific case the third angle would provide rotation of the retroreflector about the axis between the source and the retroreflector, and this rotation will have no effect upon the fraction of light returned to the source so long as the source is larger than the retroreflector. In effect we use a spherical coordinate system, with the retroreflector at the origin and the source sitting at (r,𝜃, 𝜙), where r is large compared to the dimensions of the retroreflector. (In practice we fix the source location and retroreflector, and rotate the retroreflector about two axes.) To analyze the performance of a retroreflector, we constructed a simple model in the ray-tracing code TracePro [16]. Our model uses a source–receiver pair with the retroreflector placed between them, as shown in Fig. 4; the retroreflector surfaces are assumed to be perfect mirrors, the mirrors are made from 0.005’’ thick sheets, and diffraction effects are assumed to be negligible. Overall outer dimensions of all retroreflectors is taken to be 1 cm. As shown, the angle 𝜙 is a rotation about the 𝑦-axis and ranges from 0–180◦ , and 𝜃 is rotation about the 𝑧-axis and runs from 0–360◦ . 𝛷𝐼 is the flux emitted by the source (perpendicular to the disk and directed towards –x; blue arrows in Fig. 4). 𝛷𝑚 is the flux that misses the retroreflector and is incident upon the receiver disk. 𝛷𝑜 = 𝛷𝐼 – 𝛷𝑚 is the total flux incident on the retroreflector, and depending on the retroreflector design, will vary somewhat with the orientation of the retroreflector with respect to the source. Finally, 𝛷𝑟 is the flux directed back towards the source (red arrow), and 𝛷𝑠 (yellow arrow) is flux reflected away from the source by the retroreflector; 𝛷𝑜 =𝛷𝑟 +𝛷𝑠 . 27
J.W. Lewellen and J.R. Harris
Optics Communications 441 (2019) 26–32
Fig. 3. Multicube retroreflector, front (left) and back (right) side views.
incident on the other surfaces (shown in gray) will be reflected away from the source. The total projected area of the ‘‘retroreflective’’ region never exceeds 50% of the total projected area, thus, the efficiency is low. The multicube exhibits the same high efficiency – approaching 100% – as the conventional corner cube when the corner is ‘‘aimed’’ at the source. However, it also exhibits reasonably high efficiency when the ‘‘back’’ is aimed at the source, as anticipated from the geometry. Here again the reduction in efficiency is simply the projected area of the ‘‘inset’’ cube versus the total projected area when viewed from the back. The multicube reflector therefore provides the same ease of fabrication as conventional corner cubes, for instance by a stamping process, while improving the overall retroreflection efficiency. 2.2. Random angle settings Fig. 4. Simulation geometry (not to scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
For each retroreflector, we also generated 15,000 random number pairs for (𝜙, 𝜃), uniformly distributed between 0 – 180◦ and 0-360◦ , respectively.1 This was done as a check against the possibility of the uniform-step angular sweep incorporating an unforeseen systematic error. For each pair we performed ray-tracing analysis, and analyzed the probability of returning a given fraction of incident flux to the source. The results are both a probability distribution for the fraction of retroreflected flux, and a weighted-average fraction of incident power that is retroreflected back towards the source. Fig. 6 shows a histogram of the fraction of incident power retroreflected towards the source, versus the frequency of that fraction being returned, for the multiplane cross retroreflector. The most likely retroreflection will be a few percent of incident power returned to the source; and, as would be anticipated considering Fig. 5(b), in general no more than 40% of the incident power would be returned. The very small increase near 100% corresponds to the case where one of the planes is almost exactly normal to the direction to the source. However, at least some power is returned to the source for more than 90% of possible orientations. Fig. 7 compares the performances of the corner cube and the multicube designs. As expected from the results above, the majority of the time, both designs return zero reflected power back to the source, but the multicube is approximately twice as likely to return non-zero power
We define the efficiency of the retroreflector as a function of rotation angles as 𝛷 𝛷𝑟 𝜂 (𝜑, 𝜃) = 𝛷𝑟 = 𝛷 −𝛷 . The value of 𝛷𝐼 is set by the source 𝑜 𝐼 𝑚 parameters, and 𝛷𝑟 and 𝛷𝑚 are obtained using TracePro’s analysis functions. 2.1. Rotation angle sweeps A separate trace of 10,000 rays was performed for each pair of rotation angles, to generate contour maps of the retroreflector efficiency versus the rotation angles; the results are shown in Fig. 5. The efficiency map for the conventional corner cube is not surprising. For approximately 1/8 of the total range of available angles, the corner cube provides continuous retroreflection, and approaches 100% when the corner is ‘‘aimed’’ directly at the source. Otherwise, the corner cube presents an outer surface to the source, and the incident flux is mostly directed away from the source except in the particular instances when the outer surface is parallel to the source; at which point, retroreflection is 100% (Here we are assuming that all surfaces of the corner cube are mirrored). The centrally-peaked response is a known feature of corner cube retroreflectors [1]. The efficiency map for the multiplane cross may appear to be surprising, because (except when one of the planes is normal to the direction to the source) the retroreflection efficiency never exceeds approximately 50%. However, consider Fig. 2 above. Flux incident on the blue highlighted planes in Fig. 2 will be retroreflected; however, flux
1 To verify the earlier statement regarding rotation of the retroreflector about the axis between the source and retroreflector, we also performed a similar study setting all three possible rotation angles randomly for the multicube retroreflector. The weighted-average retroreflected power fraction agreed to 1.5% between the two calculations.
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J.W. Lewellen and J.R. Harris
Optics Communications 441 (2019) 26–32
Fig. 5. Retroreflector efficiency as a function of angle for (a) conventional corner cube, (b) a multiplane cross, and (c) the new multicube design. 𝜙 and 𝜃 were swept in 5-degree steps from 0 – 180◦ and 0 – 360◦ , respectively. Table 1 The average return power fraction for each retroreflector is given. Retroreflector design
Average fraction of incident power retroreflected
Multiplane cross Corner cube Multicube
11.5% 3.3% 4.6%
simple increase of projected area, the orientation of the fourth cube also increases the range of angles over which some light is returned towards the source. So, a multicube offers increased performance over a corner cube in two respects: the greater average returned power fraction, and the greater range of angles over which some light is returned to the source. For an ideal design, e.g. infinitely thin and perfectly reflective surfaces, the ‘‘front side’’ of a multicube behaves as a standard corner cube; in practice, due to material thickness and non-unity reflectivity, the multicube design will sacrifice some reflected power versus a corner cube of the same overall dimensions. The amount of decrease will depend upon the details of manufacture. In terms of average power return to the source, as a single retroreflector the multiplane cross is clearly the most effective. A corner cube will on average return 29% of the power, and a multicube 40%, of the power that a multiplane cross will return.
Fig. 6. Histogram of returned power fraction vs frequency for multiplane cross.
(left panel). The right panel in Fig. 7 suppresses the zero-return case for a more clear comparison between the two designs. The performance improvement of the multicube over a simple corner cube is clearly evident. The two designs are approximately equal in performance when more than approximately 25% of power is returned to the source. However, for return powers between 0 and 25%, the multicube is much more likely to return at least some power to the source. Again, this is what would be expected given the 𝜙−𝜃 plots shown above, and from consideration of the basic device geometry. Finally, the expected fraction of incident power return can be calculated as the simple average power returned over the set of random orientations; the results are summarized in Table 1. The results for the corner cube and multicube are surprisingly high, given the histograms shown above. In more than 90% of possible orientations, the multiplane cross will return at least some incident power to the source. While this is not true of either the corner cube or the multicube, when they do retroreflect, these designs are more likely to reflect a higher fraction of the incident power back to the source. The increased performance of the multicube compared to the conventional corner cube is clear; the approximately 40% gain in average reflected power is in line with what would be expected from consideration of the above results, as well as the multicube geometry (see Fig. 3). One can conceptualize the multicube retroreflector as three identical corner cubes facing ‘‘front’’, and a fourth facing ‘‘back’’. (Note that this is not the actual geometry, merely an aid to interpreting the results.) The addition of the ‘‘back’’ corner cube would increase the returned light by 33%, averaged across all angles, all else being equal. In addition to the
2.3. Packing density Given a priori ability to set the orientation and distribution of retroreflectors with respect to a source, such as on a highway traffic sign, corner cubes are effective and economical choices. Given only a single retroreflector to be deployed with unknown orientation relative to a source, a multiplane cross is the clear choice. If the intent is to be able to deploy multiple retroreflectors, however, the ability to store retroreflectors prior to deployment can be an important consideration. Multiplane crosses, due to their geometry, do not pack tightly. Multiplane crosses with a transverse dimension L and made from infinitely thin material, can be packed at a maximum density of 2/L3 . Assuming infinitely thin material for corner or multicube retroreflectors would result in an infinite storage density, so we make several assumptions to calculate a comparable packing density. First, assume that the material thickness is 10−2 of the transverse dimension of the retroreflector. Further assume that the retroreflectors must be separated by 21 their material thickness. Adding an additional corner or multicube 29
J.W. Lewellen and J.R. Harris
Optics Communications 441 (2019) 26–32
Fig. 7. Histograms for corner cube (black) and multicube (red). Left: full scale; right: zero reflected power suppressed.
to a stack increases the total volume occupied by the stack by 1.5⋅ 10−2 L3 . The maximum possible packing density for either a corner cube or a multicube is thus 67/L3 , given these assumptions for sheet thickness and separation relative to the basic dimensions. Thinner sheets and smaller separations yield correspondingly higher packing fractions. For a practical example, a container of dimensions 20.5 cm × 2.5 cm × 2.5 cm could hold approximately 17,000 crosses that measure 0.5 cm per side, again assuming zero thickness for the planes and perfect packing. In contrast, consider either corner cubes or multicubes formed from 0.002" sheet stock, and with a transverse dimension of 0.5 cm. The same container could, given a separation of 0.001" between stacked retroreflectors, hold approximately 65,000 multicubes or corner cubes, or approximately 4 times as many retroreflectors compared to packing with multiplane crosses. (Note that the larger the size of the retroreflector, the larger the ratio becomes for a given container. If the retroreflectors are 2.5 cm on a side, the container could hold ∼18 multiplane crosses, or ∼2700 corner cubes or multiplane crosses, for a ratio of 150:1.) If dispersed into a cloud of equal volume, an incident optical beam would therefore be approximately 3.8 times as likely to illuminate a corner cube or multicube than a multiplane cross, all else being equal. A cloud composed of corner cubes would be expected to return about the same amount of light as one composed of multiplane crosses, assuming random orientation of the retroreflectors following dispersal. A cloud composed of multicubes, however, would be expected to return approximately 50% greater light to the source than one composed of multiplane crosses.
Fig. 8. Multicube retroreflector assembly jig; one 10 mm × 10 mm mirror has been placed in the jig for scale.
Note that this estimate assumes that any light incident upon a retroreflector but not returned to the source is simply lost; it does not account for the possibility of light returned to the source after encountering multiple retroreflectors.
3. Experimental validation To provide a preliminary validation of the multicube retroreflector concept, we constructed a test article of this design and measured its performance in an arrangement conceptually similar to that shown in Fig. 4. Fig. 9. Assembled multicube retroreflector, mounted on rotation axis support.
3.1. Construction The retroreflector was constructed using 12, 1-cm-square aluminized mirrors which can be used as either front- or back-surface reflectors [17] bonded together using Norland Optical Adhesive #81 [18]. A custom jig, shown in Fig. 8, was used to provide initial alignment of the mirrors prior to bonding. A photo of the assembled retroreflector is shown in Fig. 9.
3.2. Measurement setup A photo of the experimental apparatus is shown in Fig. 10. A ThorLabs M450LP1 mounted LED was used as the illumination source for the experiment, combined with a 2’’ aspheric lens to provide a quasi-parallel beam. A 2’’ pellicle beamsplitter (approximately 50% 30
J.W. Lewellen and J.R. Harris
Optics Communications 441 (2019) 26–32
a ‘‘light pipe’’ to transport some light through the retroreflector rather than returning it to the source. While Norland 81 is optically clear, it has an index of refraction of 1.56 when cured, and small droplets of cured adhesive on the mirror surfaces may be perturbing the retroreflection optical paths. 3.4. Potential improvements While we believe the results demonstrate the basic validity of the multicube retroreflector concept, we have identified several areas where the experiment can be improved. The largest single improvements would result from improved retroreflector construction. While conceptually simple, the physical assembly of the multicube retroreflector from individual panels proved problematic for several reasons. Planarity and orthogonality were difficult to maintain over the course of construction, and minimal adhesive use was required to help reduce the possibility of the mirrors bonding to the jig as well as to each other. A total of three multiplane retroreflectors were constructed; the adhesive joints of one retroreflector failed in several locations during removal from the jig; a second retroreflector’s joints failed as it was being placed into the measurement apparatus. For future one-off tests, we believe a better approach may be to use a polished glass cube bonded to three appropriately sized optical flats to form the basic geometry, and then to mirror-coat the ‘‘inside’’ of the multicube via vapor deposition; this would reduce the possibility of ‘‘light-guiding’’ effects influencing the results. Alternately, a series of improved assembly jigs can be designed to provide improved coplanarity and orthogonality when bonding individual mirrors into subassemblies, and the subassembly into the cube, and to reduce the probability of adhesive bonding the jig to the pieces being assembled. If many retroreflectors are desired for field testing, the best approach would probably be to fabricate molds appropriate for optical glass or plastic. Additionally, any future work should include an analysis of the impact of assembly errors on system performance to identify the required assembly tolerances. Second, the uniformity of the illumination source could be improved. Third, the experiment can be repeated at longer distances. This would reduce the amount of non-retroreflected light impinging upon the sensor, e.g. from planar surfaces nearly but not exactly perpendicular to the source. Finally, the retroreflector positioner could be improved by using computer-controlled rotation stages, so as to provide faster measurements with finer angular resolution, and by mechanically stiffening and improving the orthogonality of the axes.
Fig. 10. Experimental apparatus. Blue lines represent the optical path.
transmission/reflection) was positioned between the light source and the retroreflector, oriented at approximately 20 degrees off normal to the axis between the light source and retroreflector. This allowed a reasonable physical separation between the retroreflector and detector, while maintaining a sufficiently large aperture to completely illuminate the retroreflector. A second aspheric lens focused the light from the retroreflector down onto a ThorLabs S121C silicon detector. Two rotation stages were used to orient the multicube retroreflector with respect to the light source. The power of the reflected light was measured as a function of the rotation angles; both axes were varied in 5-degree increments. 3.3. Results and discussion Contour plots of the reflected power, versus the two rotation angles, are plotted in Fig. 11(a) and (b) for the front and back sides of the multicube retroreflector prototype, respectively. The angles have been normalized to reference angles where the corner cube faces are perpendicular to the light source. The angular range of 𝜃 for the back plot does not extend past 70 degrees; this is because the horizontal axis support for the multicube retroreflector impinges on the light path and cuts off the reflected signal. The basic form of the plots are similar to those shown in Fig. 5, with some notable discrepancies. The front of the ideal multicube retroreflector shows a maximum return at angles of 𝜃=35◦ , 𝜙=45◦ off normal incidence. In Fig. 11, however, the maximum return occurs at 𝜃=35◦ , 𝜙=65◦ off normal incidence. We also note that instead of relatively ‘‘clean’’ normal-incidence reflections as shown in Fig. 5, nearnormal incidence angles in Fig. 11 show significant smearing out. The ratio of maximum non-normal-incidence returned power from the front and back sides of the multicube is expected to be approximately 2:1, based both on area-projection arguments and on the results from Fig. 5, and that the maximum non-normal-incidence returned power from the ‘‘front’’ side should be approximately that from the normalincidence return. The ratio of maximum non-normal-incidence returned power is as expected, however, the ratio of maximum non-normalincidence to normal-incidence returned power from the ‘‘front’’ side is 2:1, instead of 1:1. Taking together the shift in angular position of the maximum nonnormal-incidence returned power, the smeared near-normal-incidence reflected power, and the discrepancy in front-side power return ratios, we conclude that the most likely explanation is angular misalignment between the individual mirror segments used to make up the multicube retroreflector prototype. This exhibits as both non-parallelism between segments making up a single projected face, and non-orthogonality at corners. Other factors contributing to the discrepancies may be finite mirror thickness, and effects from the optical adhesive. With dimensions of 10 mm × 10 mm × 1 mm thick, the mirror thickness is non-negligible, in comparison to the simulated model, and a significant fraction of the projected surface area, depending upon orientation, can serve as
4. Conclusions We have evaluated the expected performance of a new retroreflector design, the multicube, compared to two conventional retroreflector designs, the corner cube and the multiplane cross. Numerical results were used to perform this comparison, while a prototype multicube retroreflector was built and tested to identify potential issues which might degrade its performance in practice. The multicube retains the advantages of corner cube retroreflectors – ease of fabrication and high packing density – while significantly increasing the average fraction of light returned to the source compared to a corner cube. A preliminary comparison of the performance of multicubes compared to multiplane retroreflectors, taking into account estimates for achievable packing fraction, suggests that the multicube could significantly outperform multiplane retroreflectors when randomly distributed from a fixed-size container. Acknowledgment This work was funded by the Office of Naval Research, USA. 31
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Optics Communications 441 (2019) 26–32
Fig. 11. Measured retroreflected light power as a function of angles for (a) the front and (b) the back side of the multicube retroreflector. In this figure, theta represents rotation about the vertical axis, and phi represents rotation about a horizontal axis. The direction of phi is opposite that in Fig. 5. Also note the difference in intensity scales between (a) and (b).
References
[10] A. Minato, S. Ozawa, N. Sugimoto, Appl. Opt. 40 (2001) 1459. [11] J.P. Oakley, Appl. Opt. 46 (2007) 1026. [12] L. Zhou, K.S.J. Pister, J.M. Kahn, Assembled corner-cube retroreflector quadruplet, in: Technical Digest, Fifteenth IEEE International Conference on Micro Electro Mechanical Systems, 2002. [13] W.S. Rabinovich, R. Mahon, P.G. Goetz, L. Swingen, J. Murphy, M. Ferraro, R. Burris, M. Suite, C.I. Moore, G.C. Gilbreath, S. Binari, Proc. SPIE 6304 (2006) 63040Q. [14] R. Beer, D. Marjaniemi, Appl. Opt. 5 (1966) 1191. [15] J.W. Lewellen, J.R. Harris, U.S. Provisional Patent Application 62/425097, 22 NOV 2016. [16] Tracepro, 2018, http://www.lambdares.com/tracepro, (Accessed 29 October 2018). [17] Edmund Optics part number 45-517. [18] Norland products, 2018, https://www.norlandprod.com/, (Accessed 29 October 2018).
[1] R.B. Nilsen, X.J. Lu, Proc. SPIE 5616 (2004) 47. [2] N.R. Greene, B.J. Filko, Ophthalmic Physiol. Opt. 30 (2010) 76–84. [3] P.G. Goetz, W.S. Rabinovich, R. Mahon, L. Swingen, G.C. Gilbreath, J.L. Murphy, H.R. Burris, M.F. Stell, Practical considerations of retroreflector choice in modulating retroreflector systems, in: Digest of the LEOS Summer Topical Meetings, 2005. [4] A. Arbabi, E. Arbabi, Y. Horie, S.M. Kamali, A. Faraon, Nat. Photon. 11 (2017) 415. [5] J.M. Wood, R.A. Tyrrell, P. Lacherez, A.A. Black, Ophthalmic Physiol. Opt. 37 (2017) 184–190. [6] P.O. Minott, Design of Retrodirector Arrays for Laser Ranging of Satellites, NASA Report NASA-TM-X-70657, 1974. [7] R.A. Tyrrell, J.M. Wood, A. Chaparro, T.P. Carberry, B.-S. Chu, R.P. Marszalek, Accid. Anal. Prev. 41 (2009) 506. [8] E.G. Schmidtlin, S.B. Shaklan, A.E. Carlson, Proc. SPIE 3350 (1998) 81–88. [9] A. Minato, N. Sugimoto, Appl. Opt. 37 (1998) 438.
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Performance estimates for a multicube retroreflector design J.W. Lewellen a , J.R. Harris b ,∗ a b
Los Alamos National Laboratory, Los Alamos, NM 87544, USA Air Force Research Laboratory, Albuquerque, NM 87117, USA
ARTICLE
INFO
Keywords: Laser Retroreflector Dispersed performance
ABSTRACT We describe a novel type of retroreflector, which we call a ‘‘multicube’’ retroreflector, optimized to provide improved optical power return when a large number of retroreflectors are dispersed through a volume or on a surface. While it does not match the performance of a ‘‘multiplane cross’’ type retroreflector on an individual level, it offers improved performance compared to a standard corner cube while retaining the corner cube’s advantages: fabrication via stamping, and the ability to densely pack the retroreflectors. When dispersed as part of a cloud of retroreflectors, the multicube design provides significant performance improvements over both simple corner cubes and ‘‘multiplane cross’’ retroreflectors.
1. Introduction Retroreflectors are optical elements which can direct a significant fraction of the light incident upon them back towards their source. Retroreflectors occur in nature [1,2] as well as being produced artificially in a wide variety of configurations [1,3,4], and are employed in many applications including traffic safety [1,2,5–7], space applications [6,8–10], distance measurements on construction sites [11], communications [3,4,12,13], and remote sensing [4,10]. Different types of retroreflectors have different advantages and limitations, such that the retroreflector design should be tailored to its intended application [6, 14]. For example, in the case of road signs, perfect retroreflection is not actually desired, as the light source, retroreflector, and observer are generally not collinear [2], while perfect retroreflection with minimal losses and distortions is generally desired in space applications [6,8–10]. For our present application, we sought a retroreflector design that enables a large number of low-cost, rugged retroreflectors to be packaged compactly before use and then dispersed into a larger volume through which a large beam of light is propagating, and in which our objective is to maximize the total power of retroreflected light. This means that a reduction in the performance of each individual retroreflector is acceptable provided that it is more than offset by increasing the number of retroreflectors that can be fit into a given initial volume. It also means that the orientation of any given retroreflector with respect to the light source will not be known. This regime of operation is quite different than is typically encountered. Frequently, the light to be retroreflected is in a beam whose size is considerably smaller than the retroreflector, and only a single retroreflector is used. In this regime, to first order, one can expect that either 100% of the incident light, or none, is reflected back towards the source, depending upon whether the incident beam is within the
retroreflector’s angular acceptance or not. An example of this regime would be the use of a corner cube as part of an interferometer or similar optical instrument. A second regime occurs when the incident optical beam has a transverse size equal to or larger than the retroreflector; many retroreflectors may be used, in either a specific or random orientation, as part of the optical system. In this case, the incident beam ‘‘fills’’ the retroreflector(s), and the fraction of incident light reflected back towards the source is a more-or-less smooth function of the orientation of the retroreflector within the light beam. Additionally, our requirement to maximize the number of retroreflectors that can fit within the initial, pre-deployment volume requires designs which are ‘‘nestable’’, while their unknown post-deployment orientation requires that we consider retroreflection in all directions. These considerations rule out certain types of retroreflectors, such as spherical retroreflectors, which have larger acceptance angles than other types but do not allow efficient use of pre-deployment volume and tend to retroreflect smaller amounts of incident light than other designs [11]. Perhaps the simplest retroreflector is the corner cube, consisting of three plane surfaces. Light rays entering the corner cube’s acceptance window are reflected back towards their source along parallel but offset trajectories. Fig. 1 shows a conventional corner cube’s geometry. Given at least approximate prior knowledge of an optical system’s configuration, corner cubes can be highly effective retroreflectors, and can in principle be made easily from sheet metal via a stamping process. For our purposes, however, the standard corner cube is not ideal. The unknown post-deployment orientation means that a retroreflector capable of returning light with a wider angular acceptance than a conventional corner cube is desirable. This could be accomplished by filling the corner cube with a dielectric material [1] or by using
∗ Corresponding author.
https://doi.org/10.1016/j.optcom.2019.02.013 Received 1 November 2018; Received in revised form 29 January 2019; Accepted 4 February 2019 Available online 14 February 2019 0030-4018/Published by Elsevier B.V.
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Optics Communications 441 (2019) 26–32
Nomenclature 𝜃, 𝜙
𝛷I 𝛷m 𝛷o 𝛷r 𝛷s
rotation angles that specify the orientation of the retroreflector relative to a fixed point, e.g. an optical source flux emitted by the source in the numerical simulation optical power emitted by the source that does not impinge upon the retroreflector the flux incident on the retroreflector optical power directed back towards the source by the retroreflector optical power incident on the retroreflector but not directed back towards the source
Fig. 2. A retroreflector consisting of eight joined corner cubes. The darker planes highlight a single corner cube.
associated with fabrication which might degrade their performance in practice. The multicube retroreflector will be shown to have limitations in scenarios where a single retroreflector can be deliberately placed with respect to the light source, but significantly improved performance when used as intended, with multiple retroreflectors dispersed from a small volume into a larger volume, with random orientations with respect to the light source.
Fig. 1. Corner cube geometry.
2. Performance analysis
a combination of eight corner cubes, joined at their inner vertices as shown in Fig. 2; we refer to the latter as a ‘‘multiplane cross’’ retroreflector. However, neither of these approaches allows efficient nesting of the retroreflectors to make best use of the available predeployment volume. In comparison, conventional corner cubes made from thin sheets can be densely packed, allowing more retroreflectors to be fit into a given volume. To address these issues, we propose a retroreflector configuration combining the potential for ease of fabrication of conventional corner cubes and the ability to easily and densely package them, with a greater range of angles over which light will be reflected back towards the source. We call this new design a ‘‘multicube’’ retroreflector; it is shown in Fig. 3 [15]. The multicube retroreflector is a corner cube retroreflector, modified to provide nonzero reflection when oriented with its primary axis of retroreflection directed away from the observer, and intended to maximize the number of retroreflectors that can be fit into a given volume. This concept is similar to the multiple cornercube configuration discussed in Ref. [8], with the difference that the multicube retroreflector is intended to be produced from thin materials to maximize the number that can be packed into a given pre-deployment volume. In the following sections, we will provide an initial analysis of the multicube retroreflector and compare its performance to the standard corner cube and the multiplane cross designs. Performance comparisons between different retroreflector types will primarily be based on simulation results. For the multicube design, a prototype was built and its performance compared to the simulated results for that design. This serves both to generally verify the simulation code performance in order to provide confidence in it and therefore in the numerically-based comparisons between the different designs, and more importantly to identify any minor differences between the performance of an actual and an idealized multicube in order to identify potential difficulties
We compare the performance of a conventional corner cube, a multiplane cross, and a multicube retroreflector by determining the fraction of the light incident on the retroreflector from a given angle, that is returned towards the source. The orientation of an object in space can be set by three angles; however, in our specific case the third angle would provide rotation of the retroreflector about the axis between the source and the retroreflector, and this rotation will have no effect upon the fraction of light returned to the source so long as the source is larger than the retroreflector. In effect we use a spherical coordinate system, with the retroreflector at the origin and the source sitting at (r,𝜃, 𝜙), where r is large compared to the dimensions of the retroreflector. (In practice we fix the source location and retroreflector, and rotate the retroreflector about two axes.) To analyze the performance of a retroreflector, we constructed a simple model in the ray-tracing code TracePro [16]. Our model uses a source–receiver pair with the retroreflector placed between them, as shown in Fig. 4; the retroreflector surfaces are assumed to be perfect mirrors, the mirrors are made from 0.005’’ thick sheets, and diffraction effects are assumed to be negligible. Overall outer dimensions of all retroreflectors is taken to be 1 cm. As shown, the angle 𝜙 is a rotation about the 𝑦-axis and ranges from 0–180◦ , and 𝜃 is rotation about the 𝑧-axis and runs from 0–360◦ . 𝛷𝐼 is the flux emitted by the source (perpendicular to the disk and directed towards –x; blue arrows in Fig. 4). 𝛷𝑚 is the flux that misses the retroreflector and is incident upon the receiver disk. 𝛷𝑜 = 𝛷𝐼 – 𝛷𝑚 is the total flux incident on the retroreflector, and depending on the retroreflector design, will vary somewhat with the orientation of the retroreflector with respect to the source. Finally, 𝛷𝑟 is the flux directed back towards the source (red arrow), and 𝛷𝑠 (yellow arrow) is flux reflected away from the source by the retroreflector; 𝛷𝑜 =𝛷𝑟 +𝛷𝑠 . 27
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Optics Communications 441 (2019) 26–32
Fig. 3. Multicube retroreflector, front (left) and back (right) side views.
incident on the other surfaces (shown in gray) will be reflected away from the source. The total projected area of the ‘‘retroreflective’’ region never exceeds 50% of the total projected area, thus, the efficiency is low. The multicube exhibits the same high efficiency – approaching 100% – as the conventional corner cube when the corner is ‘‘aimed’’ at the source. However, it also exhibits reasonably high efficiency when the ‘‘back’’ is aimed at the source, as anticipated from the geometry. Here again the reduction in efficiency is simply the projected area of the ‘‘inset’’ cube versus the total projected area when viewed from the back. The multicube reflector therefore provides the same ease of fabrication as conventional corner cubes, for instance by a stamping process, while improving the overall retroreflection efficiency. 2.2. Random angle settings Fig. 4. Simulation geometry (not to scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
For each retroreflector, we also generated 15,000 random number pairs for (𝜙, 𝜃), uniformly distributed between 0 – 180◦ and 0-360◦ , respectively.1 This was done as a check against the possibility of the uniform-step angular sweep incorporating an unforeseen systematic error. For each pair we performed ray-tracing analysis, and analyzed the probability of returning a given fraction of incident flux to the source. The results are both a probability distribution for the fraction of retroreflected flux, and a weighted-average fraction of incident power that is retroreflected back towards the source. Fig. 6 shows a histogram of the fraction of incident power retroreflected towards the source, versus the frequency of that fraction being returned, for the multiplane cross retroreflector. The most likely retroreflection will be a few percent of incident power returned to the source; and, as would be anticipated considering Fig. 5(b), in general no more than 40% of the incident power would be returned. The very small increase near 100% corresponds to the case where one of the planes is almost exactly normal to the direction to the source. However, at least some power is returned to the source for more than 90% of possible orientations. Fig. 7 compares the performances of the corner cube and the multicube designs. As expected from the results above, the majority of the time, both designs return zero reflected power back to the source, but the multicube is approximately twice as likely to return non-zero power
We define the efficiency of the retroreflector as a function of rotation angles as 𝛷 𝛷𝑟 𝜂 (𝜑, 𝜃) = 𝛷𝑟 = 𝛷 −𝛷 . The value of 𝛷𝐼 is set by the source 𝑜 𝐼 𝑚 parameters, and 𝛷𝑟 and 𝛷𝑚 are obtained using TracePro’s analysis functions. 2.1. Rotation angle sweeps A separate trace of 10,000 rays was performed for each pair of rotation angles, to generate contour maps of the retroreflector efficiency versus the rotation angles; the results are shown in Fig. 5. The efficiency map for the conventional corner cube is not surprising. For approximately 1/8 of the total range of available angles, the corner cube provides continuous retroreflection, and approaches 100% when the corner is ‘‘aimed’’ directly at the source. Otherwise, the corner cube presents an outer surface to the source, and the incident flux is mostly directed away from the source except in the particular instances when the outer surface is parallel to the source; at which point, retroreflection is 100% (Here we are assuming that all surfaces of the corner cube are mirrored). The centrally-peaked response is a known feature of corner cube retroreflectors [1]. The efficiency map for the multiplane cross may appear to be surprising, because (except when one of the planes is normal to the direction to the source) the retroreflection efficiency never exceeds approximately 50%. However, consider Fig. 2 above. Flux incident on the blue highlighted planes in Fig. 2 will be retroreflected; however, flux
1 To verify the earlier statement regarding rotation of the retroreflector about the axis between the source and retroreflector, we also performed a similar study setting all three possible rotation angles randomly for the multicube retroreflector. The weighted-average retroreflected power fraction agreed to 1.5% between the two calculations.
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Optics Communications 441 (2019) 26–32
Fig. 5. Retroreflector efficiency as a function of angle for (a) conventional corner cube, (b) a multiplane cross, and (c) the new multicube design. 𝜙 and 𝜃 were swept in 5-degree steps from 0 – 180◦ and 0 – 360◦ , respectively. Table 1 The average return power fraction for each retroreflector is given. Retroreflector design
Average fraction of incident power retroreflected
Multiplane cross Corner cube Multicube
11.5% 3.3% 4.6%
simple increase of projected area, the orientation of the fourth cube also increases the range of angles over which some light is returned towards the source. So, a multicube offers increased performance over a corner cube in two respects: the greater average returned power fraction, and the greater range of angles over which some light is returned to the source. For an ideal design, e.g. infinitely thin and perfectly reflective surfaces, the ‘‘front side’’ of a multicube behaves as a standard corner cube; in practice, due to material thickness and non-unity reflectivity, the multicube design will sacrifice some reflected power versus a corner cube of the same overall dimensions. The amount of decrease will depend upon the details of manufacture. In terms of average power return to the source, as a single retroreflector the multiplane cross is clearly the most effective. A corner cube will on average return 29% of the power, and a multicube 40%, of the power that a multiplane cross will return.
Fig. 6. Histogram of returned power fraction vs frequency for multiplane cross.
(left panel). The right panel in Fig. 7 suppresses the zero-return case for a more clear comparison between the two designs. The performance improvement of the multicube over a simple corner cube is clearly evident. The two designs are approximately equal in performance when more than approximately 25% of power is returned to the source. However, for return powers between 0 and 25%, the multicube is much more likely to return at least some power to the source. Again, this is what would be expected given the 𝜙−𝜃 plots shown above, and from consideration of the basic device geometry. Finally, the expected fraction of incident power return can be calculated as the simple average power returned over the set of random orientations; the results are summarized in Table 1. The results for the corner cube and multicube are surprisingly high, given the histograms shown above. In more than 90% of possible orientations, the multiplane cross will return at least some incident power to the source. While this is not true of either the corner cube or the multicube, when they do retroreflect, these designs are more likely to reflect a higher fraction of the incident power back to the source. The increased performance of the multicube compared to the conventional corner cube is clear; the approximately 40% gain in average reflected power is in line with what would be expected from consideration of the above results, as well as the multicube geometry (see Fig. 3). One can conceptualize the multicube retroreflector as three identical corner cubes facing ‘‘front’’, and a fourth facing ‘‘back’’. (Note that this is not the actual geometry, merely an aid to interpreting the results.) The addition of the ‘‘back’’ corner cube would increase the returned light by 33%, averaged across all angles, all else being equal. In addition to the
2.3. Packing density Given a priori ability to set the orientation and distribution of retroreflectors with respect to a source, such as on a highway traffic sign, corner cubes are effective and economical choices. Given only a single retroreflector to be deployed with unknown orientation relative to a source, a multiplane cross is the clear choice. If the intent is to be able to deploy multiple retroreflectors, however, the ability to store retroreflectors prior to deployment can be an important consideration. Multiplane crosses, due to their geometry, do not pack tightly. Multiplane crosses with a transverse dimension L and made from infinitely thin material, can be packed at a maximum density of 2/L3 . Assuming infinitely thin material for corner or multicube retroreflectors would result in an infinite storage density, so we make several assumptions to calculate a comparable packing density. First, assume that the material thickness is 10−2 of the transverse dimension of the retroreflector. Further assume that the retroreflectors must be separated by 21 their material thickness. Adding an additional corner or multicube 29
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Optics Communications 441 (2019) 26–32
Fig. 7. Histograms for corner cube (black) and multicube (red). Left: full scale; right: zero reflected power suppressed.
to a stack increases the total volume occupied by the stack by 1.5⋅ 10−2 L3 . The maximum possible packing density for either a corner cube or a multicube is thus 67/L3 , given these assumptions for sheet thickness and separation relative to the basic dimensions. Thinner sheets and smaller separations yield correspondingly higher packing fractions. For a practical example, a container of dimensions 20.5 cm × 2.5 cm × 2.5 cm could hold approximately 17,000 crosses that measure 0.5 cm per side, again assuming zero thickness for the planes and perfect packing. In contrast, consider either corner cubes or multicubes formed from 0.002" sheet stock, and with a transverse dimension of 0.5 cm. The same container could, given a separation of 0.001" between stacked retroreflectors, hold approximately 65,000 multicubes or corner cubes, or approximately 4 times as many retroreflectors compared to packing with multiplane crosses. (Note that the larger the size of the retroreflector, the larger the ratio becomes for a given container. If the retroreflectors are 2.5 cm on a side, the container could hold ∼18 multiplane crosses, or ∼2700 corner cubes or multiplane crosses, for a ratio of 150:1.) If dispersed into a cloud of equal volume, an incident optical beam would therefore be approximately 3.8 times as likely to illuminate a corner cube or multicube than a multiplane cross, all else being equal. A cloud composed of corner cubes would be expected to return about the same amount of light as one composed of multiplane crosses, assuming random orientation of the retroreflectors following dispersal. A cloud composed of multicubes, however, would be expected to return approximately 50% greater light to the source than one composed of multiplane crosses.
Fig. 8. Multicube retroreflector assembly jig; one 10 mm × 10 mm mirror has been placed in the jig for scale.
Note that this estimate assumes that any light incident upon a retroreflector but not returned to the source is simply lost; it does not account for the possibility of light returned to the source after encountering multiple retroreflectors.
3. Experimental validation To provide a preliminary validation of the multicube retroreflector concept, we constructed a test article of this design and measured its performance in an arrangement conceptually similar to that shown in Fig. 4. Fig. 9. Assembled multicube retroreflector, mounted on rotation axis support.
3.1. Construction The retroreflector was constructed using 12, 1-cm-square aluminized mirrors which can be used as either front- or back-surface reflectors [17] bonded together using Norland Optical Adhesive #81 [18]. A custom jig, shown in Fig. 8, was used to provide initial alignment of the mirrors prior to bonding. A photo of the assembled retroreflector is shown in Fig. 9.
3.2. Measurement setup A photo of the experimental apparatus is shown in Fig. 10. A ThorLabs M450LP1 mounted LED was used as the illumination source for the experiment, combined with a 2’’ aspheric lens to provide a quasi-parallel beam. A 2’’ pellicle beamsplitter (approximately 50% 30
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a ‘‘light pipe’’ to transport some light through the retroreflector rather than returning it to the source. While Norland 81 is optically clear, it has an index of refraction of 1.56 when cured, and small droplets of cured adhesive on the mirror surfaces may be perturbing the retroreflection optical paths. 3.4. Potential improvements While we believe the results demonstrate the basic validity of the multicube retroreflector concept, we have identified several areas where the experiment can be improved. The largest single improvements would result from improved retroreflector construction. While conceptually simple, the physical assembly of the multicube retroreflector from individual panels proved problematic for several reasons. Planarity and orthogonality were difficult to maintain over the course of construction, and minimal adhesive use was required to help reduce the possibility of the mirrors bonding to the jig as well as to each other. A total of three multiplane retroreflectors were constructed; the adhesive joints of one retroreflector failed in several locations during removal from the jig; a second retroreflector’s joints failed as it was being placed into the measurement apparatus. For future one-off tests, we believe a better approach may be to use a polished glass cube bonded to three appropriately sized optical flats to form the basic geometry, and then to mirror-coat the ‘‘inside’’ of the multicube via vapor deposition; this would reduce the possibility of ‘‘light-guiding’’ effects influencing the results. Alternately, a series of improved assembly jigs can be designed to provide improved coplanarity and orthogonality when bonding individual mirrors into subassemblies, and the subassembly into the cube, and to reduce the probability of adhesive bonding the jig to the pieces being assembled. If many retroreflectors are desired for field testing, the best approach would probably be to fabricate molds appropriate for optical glass or plastic. Additionally, any future work should include an analysis of the impact of assembly errors on system performance to identify the required assembly tolerances. Second, the uniformity of the illumination source could be improved. Third, the experiment can be repeated at longer distances. This would reduce the amount of non-retroreflected light impinging upon the sensor, e.g. from planar surfaces nearly but not exactly perpendicular to the source. Finally, the retroreflector positioner could be improved by using computer-controlled rotation stages, so as to provide faster measurements with finer angular resolution, and by mechanically stiffening and improving the orthogonality of the axes.
Fig. 10. Experimental apparatus. Blue lines represent the optical path.
transmission/reflection) was positioned between the light source and the retroreflector, oriented at approximately 20 degrees off normal to the axis between the light source and retroreflector. This allowed a reasonable physical separation between the retroreflector and detector, while maintaining a sufficiently large aperture to completely illuminate the retroreflector. A second aspheric lens focused the light from the retroreflector down onto a ThorLabs S121C silicon detector. Two rotation stages were used to orient the multicube retroreflector with respect to the light source. The power of the reflected light was measured as a function of the rotation angles; both axes were varied in 5-degree increments. 3.3. Results and discussion Contour plots of the reflected power, versus the two rotation angles, are plotted in Fig. 11(a) and (b) for the front and back sides of the multicube retroreflector prototype, respectively. The angles have been normalized to reference angles where the corner cube faces are perpendicular to the light source. The angular range of 𝜃 for the back plot does not extend past 70 degrees; this is because the horizontal axis support for the multicube retroreflector impinges on the light path and cuts off the reflected signal. The basic form of the plots are similar to those shown in Fig. 5, with some notable discrepancies. The front of the ideal multicube retroreflector shows a maximum return at angles of 𝜃=35◦ , 𝜙=45◦ off normal incidence. In Fig. 11, however, the maximum return occurs at 𝜃=35◦ , 𝜙=65◦ off normal incidence. We also note that instead of relatively ‘‘clean’’ normal-incidence reflections as shown in Fig. 5, nearnormal incidence angles in Fig. 11 show significant smearing out. The ratio of maximum non-normal-incidence returned power from the front and back sides of the multicube is expected to be approximately 2:1, based both on area-projection arguments and on the results from Fig. 5, and that the maximum non-normal-incidence returned power from the ‘‘front’’ side should be approximately that from the normalincidence return. The ratio of maximum non-normal-incidence returned power is as expected, however, the ratio of maximum non-normalincidence to normal-incidence returned power from the ‘‘front’’ side is 2:1, instead of 1:1. Taking together the shift in angular position of the maximum nonnormal-incidence returned power, the smeared near-normal-incidence reflected power, and the discrepancy in front-side power return ratios, we conclude that the most likely explanation is angular misalignment between the individual mirror segments used to make up the multicube retroreflector prototype. This exhibits as both non-parallelism between segments making up a single projected face, and non-orthogonality at corners. Other factors contributing to the discrepancies may be finite mirror thickness, and effects from the optical adhesive. With dimensions of 10 mm × 10 mm × 1 mm thick, the mirror thickness is non-negligible, in comparison to the simulated model, and a significant fraction of the projected surface area, depending upon orientation, can serve as
4. Conclusions We have evaluated the expected performance of a new retroreflector design, the multicube, compared to two conventional retroreflector designs, the corner cube and the multiplane cross. Numerical results were used to perform this comparison, while a prototype multicube retroreflector was built and tested to identify potential issues which might degrade its performance in practice. The multicube retains the advantages of corner cube retroreflectors – ease of fabrication and high packing density – while significantly increasing the average fraction of light returned to the source compared to a corner cube. A preliminary comparison of the performance of multicubes compared to multiplane retroreflectors, taking into account estimates for achievable packing fraction, suggests that the multicube could significantly outperform multiplane retroreflectors when randomly distributed from a fixed-size container. Acknowledgment This work was funded by the Office of Naval Research, USA. 31
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Fig. 11. Measured retroreflected light power as a function of angles for (a) the front and (b) the back side of the multicube retroreflector. In this figure, theta represents rotation about the vertical axis, and phi represents rotation about a horizontal axis. The direction of phi is opposite that in Fig. 5. Also note the difference in intensity scales between (a) and (b).
References
[10] A. Minato, S. Ozawa, N. Sugimoto, Appl. Opt. 40 (2001) 1459. [11] J.P. Oakley, Appl. Opt. 46 (2007) 1026. [12] L. Zhou, K.S.J. Pister, J.M. Kahn, Assembled corner-cube retroreflector quadruplet, in: Technical Digest, Fifteenth IEEE International Conference on Micro Electro Mechanical Systems, 2002. [13] W.S. Rabinovich, R. Mahon, P.G. Goetz, L. Swingen, J. Murphy, M. Ferraro, R. Burris, M. Suite, C.I. Moore, G.C. Gilbreath, S. Binari, Proc. SPIE 6304 (2006) 63040Q. [14] R. Beer, D. Marjaniemi, Appl. Opt. 5 (1966) 1191. [15] J.W. Lewellen, J.R. Harris, U.S. Provisional Patent Application 62/425097, 22 NOV 2016. [16] Tracepro, 2018, http://www.lambdares.com/tracepro, (Accessed 29 October 2018). [17] Edmund Optics part number 45-517. [18] Norland products, 2018, https://www.norlandprod.com/, (Accessed 29 October 2018).
[1] R.B. Nilsen, X.J. Lu, Proc. SPIE 5616 (2004) 47. [2] N.R. Greene, B.J. Filko, Ophthalmic Physiol. Opt. 30 (2010) 76–84. [3] P.G. Goetz, W.S. Rabinovich, R. Mahon, L. Swingen, G.C. Gilbreath, J.L. Murphy, H.R. Burris, M.F. Stell, Practical considerations of retroreflector choice in modulating retroreflector systems, in: Digest of the LEOS Summer Topical Meetings, 2005. [4] A. Arbabi, E. Arbabi, Y. Horie, S.M. Kamali, A. Faraon, Nat. Photon. 11 (2017) 415. [5] J.M. Wood, R.A. Tyrrell, P. Lacherez, A.A. Black, Ophthalmic Physiol. Opt. 37 (2017) 184–190. [6] P.O. Minott, Design of Retrodirector Arrays for Laser Ranging of Satellites, NASA Report NASA-TM-X-70657, 1974. [7] R.A. Tyrrell, J.M. Wood, A. Chaparro, T.P. Carberry, B.-S. Chu, R.P. Marszalek, Accid. Anal. Prev. 41 (2009) 506. [8] E.G. Schmidtlin, S.B. Shaklan, A.E. Carlson, Proc. SPIE 3350 (1998) 81–88. [9] A. Minato, N. Sugimoto, Appl. Opt. 37 (1998) 438.
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