Abstract

We present to the best of our knowledge the first successful demonstration of a planar, self-tracking solar concentrator system capable of a 2-dimensional angular acceptance of over 40°. The light responsive mechanism allows for efficient waveguide coupling and light concentration independently of the angle of incidence within the angular range. A coupling feature is created at the focal spot of the optical system by locally melting a phase change material which acts as an actuator due to the large thermal expansion. A dichroic prism membrane reflects the visible light so that it is efficiently coupled into a waveguide at the point of the created coupling feature. We show simulation results for concentration and efficiency, validated by an experimental proof of concept demonstration of a self-tracking concentrator array element. Simulations show that a system based on this approach can achieve 150X effective concentration by scaling the system collecting area to reasonable dimensions (40 x 10 cm2).

© 2014 Optical Society of America

1. Introduction

Due to the high cost of solar cell material, concentration photovoltaic (CPV) aims to decrease the amount of photovoltaic cell material used, by using less expensive optical elements to concentrate sunlight onto a smaller solar cell. This has the potential to decrease the cost per watt. Any CPV module regardless of the technology used has a certain acceptance angle and a concentration factor. The acceptance angle describes the part of the sky the concentrator is looking at and the concentration factor describes the relation between an incoming aperture and the exiting aperture, in this case a solar cell. It is fundamentally not possible to achieve both a high acceptance angle and a high concentration factor as can be seen in Eq. (1) where theta-max-in is the half angle of acceptance [1]

CCmax=1sin2(θmax,in).

Due to this tradeoff, we can identify three classes of concentration systems: low, medium and high concentration systems. Low concentration systems (<10x) have a large acceptance angle operate in a fixed position (no tracking). On the other end, high concentration systems (100x – 1000x) operate with very narrow acceptance angles and require precise mechanical tracking. These sun trackers are active mechanical systems, which consume energy in order to function, thereby reducing the overall efficiency of the system. Commercial CPV systems are typically large area, high concentration which results in large form factors. Hence HCPV systems are mostly used in solar farms. The middle ground, medium concentration, (10x – 200x) is not as common. A few solutions exist that operate in this area, like Entech’s Solar Volt module [2], Whitfield Solar’s solar concentrator [3] or parabolic trough concentrators. The drawback of MCPV is that it still requires accurate tracking despite not reaching very high concentration factors. This can lead to the point where the high price of a multi-junction cell is no longer justifiable. Because of the low efficiency of silicon based solar cells (ca. 15%) compared to III-V multi-junction cells (ca. 40%), only passive concentration (2x – 6x) makes sense economically for MCPV, since the cost of a 2-dimensional tracker is prohibitive. However passive concentration has been shown to be in the range of 2 to 6x, too low for MCPV. The proposed approach in this paper is a MCPV system where the tracking complexity is significantly reduced by providing self-tracking over more than ± 20°.

Planar concentrators fall into the LCPV and MCPV range and feature a flat form factor, but can suffer from either a low efficiency or low concentration, like luminescent [4, 5] or holographic concentrators [6]. Karp et al. [7] proposed a planar micro-optic concentrator in 2010 that features a high concentration factor (73x-300x) with a planar slim form factor and a high optical efficiency (90% - 82%). Increasing the acceptance angle of lens based planar concentrators can be achieved by lateral shifting of either the lens array or the coupling feature in a waveguide [8, 9].

New concepts of self-tracking systems are being developed that use the infrared energy form the solar spectrum to perform tracking, e.g. a moving liquid, warmed by the sun rotates a flat panel automatically due a differential heating system [10, 11]. Self-tracking CPV systems aim at using one axis to self-track seasonal changes ( ± 20 degrees), while diurnal tracking is performed with a coarse one-axis mechanical tracker. The proposed self-tracking mechanism features an acceptance angle high enough to account for the seasonal change in angle. The second direction, the daily change, is covered by using a coarse 1-dimensional tracker because, even with an acceptance angle of 40° the complete angular range of a day is not covered. The proposed self-tracking mechanism allows for an inexpensive coarse 1-dimensional tracking system to be used, potentially making operation in the medium CPV concentration range economically viable.

Several mechanisms have been proposed for self-tracking planar concentrators, including a light induced refractive index change [12], the phase-change of a hydrogel [13] and vapor bubbles [14, 15] to couple light into a waveguide. Recently, we proposed and characterized a light induced thermal phase change actuator [16]. Based on this work, we report in this paper, an experimental demonstration of a planar, self-tracking solar concentrator capable of achieving a high concentration (150x) over 40 degrees acceptance angle without any external mechanical tracking. The working principle of the self-tracking mechanism is illustrated in Fig. 1.The self-tracking mechanism uses a light activated coupling feature, which makes a point contact with a planar waveguide. The coupling element is a dichroic line prism array embedded in a 100 µm PDMS membrane. Between the membrane and the waveguide is a 15-20 µm air gap. The dichroic material reflects the short wavelength light for coupling to the PV cell and transmits the long wavelength light for device actuation. In our current devices a dichroic “cold mirror” with 750 nm transition was chosen for convenience. The transition wavelength of the dichroic can be adjusted to suit the absorption characteristic of the PV cell, with larger transitions requiring larger lens elements to enable device actuation.

 

Fig. 1 The self-tracking planar concentrator device concept. a) and b) show the focused light for different incoming angles. The coupling feature shifts due to the paraffin actuator reacting to the shifted focal spot. Inset: Long wavelength light (red) is transmitted through a dichroic facet array to heat up the phase-change actuator below (black). The expanded actuator presses the membrane against the waveguide (grey) allowing short wavelength light (bright yellow) to be coupled into the waveguide. The lens array is shown with a largely reduced scale for this conceptual drawing.

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Light focused by an array of lenses onto the coupling features (one for each lens) is reflected and trapped by total internal reflection into the waveguide, which transports and homogenizes the coupled light. The membrane is positioned on top of the actuator, consisting of a steel hexagonal hole array with 500 µm holes on a 600 µm spacing. These holes are filled with a paraffin wax/carbon black mixture that serves as the actuating element. Paraffin wax is a phase-change-material (PCM) that undergoes a volume expansion of about 10% when transiting from a solid phase into a liquid phase. It is a commonly used material in microfluidic valves or MEMS devices [17, 18]. As the light is focused on the membrane, the transmitted long wavelength part of the solar spectrum is absorbed by the carbon black, heating up the paraffin wax. Upon undergoing phase change transition at 48 °C, the paraffin wax expands upwards, pushing the membrane against the waveguide, allowing the reflected, short wavelength light to be coupled into the waveguide.

2. System components

The proposed self-tracking concentrator is an assembly of several individual elements. In the current work, the single elements are dealt with as individual systems.

2.1 Optical system

The optical system that focuses light onto the actuator is primarily responsible for the acceptance angle of the self-tracking concentrator, due to the angle of incidence and shape of the focal spot on the coupling feature. In order to track seasonal changes, the optical system has to achieve a small focus spot over an angle of ± 23 degrees. Small aberrations and a flat field curvature keep the focus spot small over the entire spectrum of incoming angles, which is needed to achieve efficient coupling. The size of the lenses is finally also responsible for the amount of power that is usable for each coupling features actuation. This means that there is a direct lens pupil to power relation and power can be freely scaled by choosing the right lens size.

One solution yielding a good compromise between performance and complexity is the combination of two plano-aspheric lenses shown in Fig. 2(a).This lens combination has been shown to give good results over the desired angular range [8, 9]. A previous design was reported earlier this year [19]. The current design (Fig. 2(a)) is an improvement over the earlier design as the spot size is maintained over a larger field angle by using aspheric lens elements.

 

Fig. 2 a) The optical system using two off-the-shelf aspheric lenses to create a flat Petzval field curvature over ± 23° incoming angle. b) The spot size only changes marginally over the set of incoming angles for the two aspheric lens system compared to two plano-convex lenses or a single lens. c,d) Change of the spot size for a single plano-convex lens and the two aspheric lenses. The different colors represent different parts of the optical spectrum (blue = 400nm, green = 550nm, red = 800nm, yellow = 1200nm).

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For our current system we have limited ourselves to off-the-shelf lenses and have selected two plastic aspheric lenses (Edmund Optics 66-010 & Edmund Optics 66-008) in Zeonex E48R with 25 mm diameter yielding an effective focal length of 36.7 mm and a numerical aperture of 0.44. These lenses make practical candidates for the construction of a lens array by hexagonal aperture edging and bonding into a lens array.

Figure 2(b) shows a comparison of the focal spot for selected incidence angles for a single plano-convex lens, two plano-convex lenses and the current solution using two plano-aspheric lenses. The plano-aspheric pair is shown to have good performance over the targeted angular range of ± 23°. This is crucial to keep the actuation constant over the desired angular range. Figure 2(c) and Fig. 2(d) show the spot sizes over field for the single plano-convex and the plano-aspheric system.

2.2 Waveguide

The waveguide slab transports the light from its entry point at the coupling feature to the exit aperture at the edge of the waveguide. Due to the line prism array (section 2.3) the light is only guided in two principal directions, so that only two facets of the waveguide can serve as an exit facet. In addition to the transport, the purpose of the waveguide is a homogenization of the coupled light. The waveguide also acts as the element providing the concentration, since any lens in this design is only used to keep the coupling feature small to achieve a higher coupling efficiency. Geometrical concentration is the ratio of the entrance aperture (top surface) over the exit aperture (edge surface). Assuming that coupling lenses cover the entire waveguide, the geometrical concentration factor only depends on the length of the waveguide divided by its thickness [7]. This means that a long and thin waveguide achieves a higher concentration. In order to maximize the geometrical concentration factor, only one edge of the waveguide is used as the exit aperture.

The increase in concentration factor with a longer and thinner geometry is diminished by a decreasing coupling efficiency of the system. Light coupled into the waveguide is guided until it undergoes a secondary interaction with another coupling feature or reaches the solar cell (at the edge of the waveguide). A long and thin waveguide will therefore have a higher chance of secondary interactions, since a longer waveguide will increase the absolute amount of coupling features, while a thinner waveguide will increase the ratio of coupling feature size to waveguide cross section. The product of the geometrical concentration factor and the coupling efficiency results in the effective concentration, which is used to describe the concentration experienced at the waveguide edge.

A solar cell can be placed on one of the two exit facets of the waveguide. Its opposing edge can then be coated with a reflective material in order to minimize losses. Another possibility is to place a second solar cell on the opposing edge so as to improve the optical to electrical efficiency. The geometrical concentration factor however scales much faster than the coupling efficiency decreases. Even taking into account absorption of the reflective material, a solution with only one solar cell gives a much higher effective concentration factor. The two long edges are left as is, since all the light will be reflected at angles that meet the TIR conditions on this surface. The refractive index of the dichroic membrane affects the accepted NA of the system. From geometric analysis any light focused on the coupling feature from a lens with an NA < 0.35 will be completely coupled into the waveguide having a refractive index of 1.4 for the dichroic membrane. This introduces a negligible loss factor in the experimental results due to the optical system being slightly bigger. The infrared light is transmitted through the membrane and focused onto the actuator situated below.

The waveguide used in the experiments has the dimensions 5 x 50 x 1 mm3, fabricated in fused silica glass and polished on all sides. These dimensions were chosen so that the exit facet of the waveguide was small enough to be completely coupled into a photodetector for measurement.

2.3 Dichroic prism membrane

The dichroic prism membrane separates the solar spectrum into the short wavelength part, which is coupled into the waveguide, and the long wavelength part which delivers the energy to drive the actuation. For this, an array of 30° line prisms, coated with a dichroic layer stack, is embedded in a 150 µm PDMS membrane. The membrane itself appears as white as can be seen in Fig. 3 in the center.

 

Fig. 3 Waveguide (transparent vertical rectangle) and dichroic prism membrane (white square layer in the center) fixed in the holder. The phase-change actuator is square and located below the membrane.

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The 30° design of the line prism array features is chosen in order to minimize shadowing by neighboring prisms. Normal incident light is then reflected at a 60° angle, directly parallel to the slope of the line prisms. If light hits the prism at any other angle, it might undergo a secondary reflection at a neighboring prism, and be ultimately reflected at an angle <60°.

The current membrane separates the spectrum at approx. 750 nm (Fig. 4). The coating is custom and consists of a stack of dichroic materials deposited on PDMS. This cutoff was chosen in order to have enough power available to provide actuation and build a proof of concept. Ultimately the separation could be shifted to 1100 nm to make complete use of the absorption spectrum of silicon solar cells.

 

Fig. 4 The solar spectrum (blue) is divided by the dichroic membrane into long (yellow) and short (green) wavelength parts at roughly 750nm. The long wavelength portion is then used to drive the actuator while the short wavelength part is coupled into the waveguide to the PV cell.

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2.4. Phase change material actuator

The actuator responds to the focus of the sunlight with a vertical thermal expansion, creating the coupling feature at the waveguide. It is made of a mixture of paraffin wax and carbon black (Fig. 5(a)), filled into a honeycomb hole array (Fig. 5(b)). Paraffin wax has a very high (10%) expansion coefficient when it changes its phase from solid to liquid. The paraffin wax chosen for this actuator has a melting point at 48 °C. The carbon black in the mixture absorbs the energy of the infrared light and transmits it to the paraffin wax as heat. Due to the focusing lens enough heat is generated to melt the paraffin. The honeycomb array is manufactured in steel by laser cutting and features 500 µm holes. The holes confine the expansion to the vertical direction. The paraffin wax/carbon black mixture is filled into the holes by heating it in vacuum in order to remove residual air from the actuation cylinders. A glass-slide seals the actuator on the back, while a flexible PDMS membrane keeps the paraffin wax inside the actuator on the front (Fig. 5(c)). At the same time this membrane provides enough flexibility so that actuation can occur while the steel frame keeps the different actuation cylinders separate (Fig. 5(d)).

 

Fig. 5 Paraffin wax is mixed with carbon black (a) and filled into the honeycomb hole array (b). PDMS is spincoated on top to create a flexible membrane and the actuator is sealed on the back with a glass slide(c). d) Top view onto the steel actuator (bright) filled with paraffin wax/carbon black.

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3. Simulations of a full concentrator system

The plano-aspheric lens pair described above was modeled with ray-tracing software (Zemax) to design a functional system with off-the-shelf lenses. The optical system was then imported into a second raytracing software (Lighttools) where the full concentrator system was simulated to optimize the parameters/dimensions. Three parameters are used to describe the system: the geometric concentration factor, the coupling efficiency and effective concentration factor.

Simulations were conducted for different lengths (y) and waveguide thicknesses (t), while the width (x) of the waveguide was kept constant since it has no influence on the result. The results of these simulations are shown in Fig. 6.By increasing the length of the concentrator, the concentration factor increases linearly, according to Eq. (2). It is assumed in this simulation that the waveguide has the same lateral dimensions as the lens array, which acts as the input aperture. In the case of using only one edge of the waveguide as an exit aperture the geometrical concentration factor is only described by the ratio of the length in y direction and the thickness t of the waveguide.

 

Fig. 6 The effective concentration factor (c) is the product of the geometric concentration factor (a) and the coupling efficiency (b). Reasonable dimensions (40 x 10 cm2, 2mm waveguide) allow an effective concentration of 150X.

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CF=AinAout=xarrayyarrayxwaveguidetwaveguide=yarraytwaveguide.

The coupling efficiency (CE) is the ratio of the light entering the system through the lens array over the light exiting the system at the edge. Hence it describes the optical transmission efficiency of the whole system, including all losses in the different materials due to absorption, as well as Fresnel losses at all the interfaces. The effective concentration factor (EC) is the product of the geometric concentration and the coupling efficiency and describes the actual concentration experienced at the exit aperture considering optical losses. Figure 6(a) shows that high CFs (500x and higher) are achievable with the proposed system. From Fig. 6(b) we can also see that the highest possible CE that’s achievable with the current system is about 75% for a single lens concentrator system. Most of these losses are related to Fresnel reflections at the five air-glass interfaces due to passing through two lenses and the initial waveguide interface. This 75% will then decrease with an increase in length of the system as well as with a thinner waveguide, since both absorption increases and more secondary interactions of the coupled light with the dichroic prisms are occurring. More secondary interactions directly lead to a higher amount of light being lost due to secondary interactions. A straightforward increase of the coupling efficiency would be the application of an antireflective coating on the lenses for the visible part of the spectrum to increase the part of the spectrum that is concentrated. It’s not needed for the long wavelength part that is transmitted since experimental results have already shown that the amount of energy transmitted is high enough to provide actuation. Despite the coupling efficiency decreasing with the length of the system, the EC (Fig. 6(c)) will increase due to the strong linear increase of the concentration factor (Fig. 6(a)). The results here only show the behavior for systems with a maximal length of about 1 m, as the growth decreases significantly afterwards. However simulations were run for lengths up to 4 m and no saturation point was found.

4. Experiments and discussion of results

The experimental setup used to test the concentrator system measures the power output at one of the waveguide edges as a function of time for a given incidence angle of the light. The power is then measured by changing the angle of incidence of the light from the solar simulator. The dynamics can be studied over the full angular region of interest. Figure 7 shows a schematic of the experimental setup.

 

Fig. 7 Schematic of the experimental setup. A single lens-pair was used to couple light into a small waveguide (length y, thickness t).

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A single lens pair of the design described above was used to focus light from a 1-sun solar simulator on the actuator, which subsequently couples the short wavelength portion into the waveguide. Due to the geometry of this setup, the geometric concentration here is different from the one presented in Eq. (2) and Fig. 6. The results in Fig. 6(a) assume a waveguide of the same dimensions as the input aperture. Since the waveguide here is much smaller, a geometric concentration factor of about 100 is achieved under the assumption of a reflective second edge (Eq. (3)). However in the experimental setup, none of the waveguide edges are coated. The geometric concentration factor for this case is calculated in Eq. (3).

CF=AinAout=12.52mm2π25mm1mm50.

Simulations run on exactly this geometry predict a system efficiency of 67.3% (input aperture to exit aperture) with 32.7% being losses (Fresnel reflections, absorption and outcoupling) but not incorporating the spectral splitting by the dichroic membrane. This gives the performance of the system in the targeted wavelength range. Using the AM1.5 spectrum as an input to the simulations and using the dichroic membrane, 25.47% of the solar spectral light power is used to drive the actuator. This results in a simulated coupling efficiency of 41.83% with a detector directly attached to the exit facet of the waveguide and the opposing waveguide facet coated with a 90% reflective material.

Figure 8 shows a photo of the actual setup attached to a rotation stage. The resolution of the rotation stage was much higher than the actual degree step used in the experiments. The honeycomb actuator was fixed with the dichroic membrane on top and the waveguide was clamped on top of the actuator. The whole assembly was then adjusted and positioned at the focus point of the lens pair. A thermal power sensor is then brought as close as possible to the edge of the waveguide. Due to experimental, mechanical constraints, the distance between the edge of the waveguide and the sensor itself was of the order of a few millimeters. A solar simulator is used as the light source providing a simulated AM 1.5 spectrum with a high homogeneity.

 

Fig. 8 The assembled actuator and the waveguide were tested with a 1 sun solar simulator. A rotations stage (background) will set an angle and the thermal sensor (red) will measure the power output at the waveguide edge as a function of time.

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The output power of the waveguide’s exit facet was then measured over an angular range of ± 25°. The measurement provides: 1. the dynamics of the system and 2. Efficiency via the measured coupled optical power over the angular spectrum.

The results of the dynamics (optical coupled power as a function of time) are shown in Fig. 9(a) and Fig. 9(b). Figure 9(a) shows the dynamics in absolute value, which allows a comparison of the different angles and their coupling efficiency. Figure 9(b) uses normalized values between 0 and 1 to compare the speed of the actuation. At t = 0 sec. a ND filter, simulating low intensity conditions, was removed and the response measured by the light detector positioned close to the edge of the waveguide. For most angles, we can see a clear instant fast rise with a slower increase afterwards. The time for the actuator to reach 90% of the maximum value is less than 4s for smaller angles and between 6 and 10s for larger angles. Higher angles such as + 20° and −20° (not shown in the graphic) show a much slower dynamic and need a lot more time (approximately 15s) to achieve the 90% value. With the current optical system, the focal spot moves at a speed of 5 µm/s as a result of the sun’s rotation. An actuation time smaller than 15 s represents <75 µm of focal spot shift which is well within the diameter of one honeycomb hole (500 um).

 

Fig. 9 a) The dynamic curve shows the behavior of actuator as a function of time for sudden onset, high intensity conditions (0-20 s) and sudden onset, low intensity conditions (21-40 s) for different angles in absolute power values. b) The dynamics of different angles are compared in normalized values. Smaller angles reach 90% of the maximum faster (<4 s) than larger angles (6-10 s). This is a slightly faster than the 8-12 s the heat needs to dissipate and the coupling value reaches value lower than 10%.

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Actuation is obtained for all angles. At t = 20 s the ND filter blocks the light. The response is an instant drop in intensity with a slowly decaying curve until the initial value is reached. A small local minimum present in some curves directly after the setting of low intensity conditions can be regarded as a measurement error. As can be seen in Fig. 9(b), it takes about 8-12 s for the actuator to settle below a 10% value, as the heat is dissipated into the steel housing of the actuator.

The angular response of the actuator is shown in Fig. 10(a).The blue bars are the initial value of the measurement and relates to an “off” actuated state. The yellow bars show the value for the same angle after 20 s. The minimum and maximum values presented in Fig. 9(a) can also be found here at the according angles. It is clear that the actuator works over an angular range of at least ± 20°, even though there is still space for optimization at larger angles. At higher angles, the amount of outcoupling from the exit facet is less than at smaller angles. The reason here lies partly in less light and therefore less energy available due to the cosine-theta obliquity factor and in the aberrations for the visible light at higher angles. The spot size for the visible wavelengths increases at higher angles (Fig. 2(d)) while the coupling feature size decreases due to a lower amount of energy. There are also two dips (around 14° and 20°) in the right side of the graph. These non-optimal actuation values are caused by an imperfectly filled actuator, a consistency problem during fabrication that will be addressed in future work.

 

Fig. 10 a) The actuator allows for actuation from −20° to + 20°. The blue curve shows the minimum amount of sunlight at for every angle whereas the yellow curve shows the maximum amount of coupling. b) Comparison of simulated optical efficiencies and the measured experimental result.

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Figure 10(b) shows the optical efficiencies obtained for different simulation cases and the experimental result. When the air gap between the exit facet and the detector is eliminated and a reflective material added on the opposing facet of the waveguide, simulation results in an optical to optical efficiency of 41.83% as described above. In the experimental setup, one side of the waveguide was left without a reflective coating which causes an expected drop in simulated efficiency from 41.83% to 28.14%. Finally, since the detector is not directly attached to the waveguide in the experimental setup, further additional losses are expected and can be included in the simulation. Considering a 1 cm2 detector area in close proximity to the waveguide edge (100 µm distance), the simulation shows that the measured efficiency drops from 28.14% (no air gap, no coated waveguide facet) to 14.54% (100 µm air gap between detector and waveguide edge). As the distance d (Fig. 7) increases, the measured efficiency decreases as shown in the black curve in Fig. 10(b).

The current measurements (Fig. 10(a)) show a maximum of 4 mW coupling efficiency, which corresponds to a coupling efficiency of approximately 1% (green line in Fig. 10(b)) in agreement with our simulation as shown in Fig. 10(b). These losses were caused largely by the distance d to the detector and losses from an opposing uncoated waveguide facet. Due to the bulkiness of the sensor and its thermal insulation it was experimentally not possible to decrease the distance between the waveguide edge and the sensor surface further than a few millimeters. Despite this limitation, experimental results match our predicted simulations suggesting that good efficiencies can be straightforwardly achieved in such system.

Figure 11(a) shows a screenshot from a movie featuring the phase change actuator system. The movie is set to 4x speed. The waveguide is visible in the center and the dichroic membrane (white) can be seen below. As the actuator is turned “on”, the waveguide edge continues to show outcoupling. The angular speed is set to exceed the actuation speed. As a result the exit facet goes dark after turning and then lights up a few moments later when actuation is occurring again. The video in Fig. 11(b) shows the same device. Here the rotating speed is now set to match the actuation speed. As a result the exit facet is continuously lit up, regardless of the angle.

 

Fig. 11 a) Screenshot from a video made in the lab showing the actuation for different angles. The angular speed is set to exceed the actuation speed. The exit facet therefore darkens after the system being turned until the actuation occurs again. See Media 1. b) Screenshot from a video made in the lab showing the actuation for different angles. The angular speed is set to match the actuation speed. The exit facet is continuously lit throughout the angular range. See Media 2.

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5. Conclusion

In this paper we have successfully demonstrated a proof-of concept self-tracking solar concentrator. The concentrator itself consists of four elements: the focusing optical system, the dichroic prism membrane, the waveguide and the phase change actuator. The focusing lens system has been optimized with off-the-shelf components. Each building block has been independently characterized before assembly to make the concentrator. To the best of our knowledge this is the first successful demonstration of a planar self-tracking solar concentrator system. Simulations show that for a reasonably sized concentrator, effective concentrations of 150x are possible with angular range of ± 20 degrees, large enough to cover seasonal variations in conjunction with a relatively inexpensive 1-dimensional coarse tracking system.

An experimental demonstration of the self-tracking concentrator element showed effective self-tracking over a ± 20 degree range. Although measured optical efficiencies were low and found to be roughly 1%, these values were attributed to difficulties and constraints in the measurement rather than the actual system efficiency. Simulations including these experimental constraints support the experimental measurements and suggest that these losses can be eliminated in future work bringing optical efficiencies closer to the predicted 41% for a single lens system. A detector directly attached to the exit facet of the waveguide will give the highest improvement and allow for much more exact measurements. Further a better prism membrane master and higher quality waveguides would add to improving the results.

Practical and realistic future use of the self-tracking concentrator system we have described is largely a question of system economics and outside of the scope of this paper. Also of high importance in the future of this technology is the choice of PV cell to make efficient use of the self-tracking mechanism and spectral utilization. Due to the spectral cut-off of the long wavelength portion of the solar spectrum, III-V multi-junction cells that make use of the complete spectrum are not suitable with the proposed self-tracking system. For use with crystalline silicon solar cells the cut-off of the dichroic membrane should be shifted towards 1100 nm. In case the power is not sufficient for actuation, the collecting aperture of each lenslet can be scaled until it meets the power required for actuation. With the proposed system multi-junction cells based on µc-silicon or amorphous silicon could be used [20]. Also under consideration are novel, thin GaAs solar cells produced as produced by Alta-Devices [21]. The efficiency from their commercial product is higher than 24% and GaAs would benefit from a concentrated input [22]. It is also more expensive than silicon, justifying the use of a solar concentrator for a higher amount of light.

Future work will expand the current concentrator to an array of focusing lenses (instead of a single lens pair) and integrate the actuator and reflective prism membrane as one element and provide a techno-economic analysis.

Acknowledgment

The authors would like to acknowledge the support from the Swiss National Science Foundation: Nano-Tera project 20NA21_145936 “Solar integrated Nano Electrolyzer: SHINE”

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17. E. T. Carlen and C. H. Mastrangelo, “Electrothermally activated paraffin microactuators,” J. Microelectromech. Syst. 11(3), 165–174 (2002). [CrossRef]  

18. H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010). [CrossRef]  

19. E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013). [CrossRef]  

20. A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004). [CrossRef]  

21. http://www.altadevices.com/pdfs/single_cell.pdf, last access 13.01.2014.

22. S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012). [CrossRef]  

References

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  1. R. Swanson, “Photovoltaic Concentrators,” in Handbook of Photovoltaic Science A. Luque, and S. Hegedus (John Wiley & Sons, Ltd, 2005), pp. 449–503.
  2. http://www.entechsolar.com/products/solarvolt.htm , last access 13.01.2014.
  3. http://fn-solar.com/#2 , last access 13.01.2014.
  4. R. Reisfeld and S. Neuman, “Planar solar energy converter and concentrator based on uranyl-doped glass,” Nature 274(5667), 144–145 (1978).
    [Crossref]
  5. R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32(9), 850–856 (2010).
    [Crossref]
  6. J. M. Castro, D. Zhang, B. Myer, and R. K. Kostuk, “Energy collection efficiency of holographic planar solar concentrators,” Appl. Opt. 49(5), 858–870 (2010).
    [Crossref] [PubMed]
  7. J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18(2), 1122–1133 (2010).
    [Crossref] [PubMed]
  8. F. Duerr, Y. Meuret, and H. Thienpont, “Tracking integration in concentrating photovoltaics using laterally moving optics,” Opt. Express 19(S3Suppl 3), A207–A218 (2011).
    [Crossref] [PubMed]
  9. J. M. Hallas, K. A. Baker, J. H. Karp, E. J. Tremblay, and J. E. Ford, “Two-axis solar tracking accomplished through small lateral translations,” Appl. Opt. 51(25), 6117–6124 (2012).
    [Crossref] [PubMed]
  10. M. J. Clifford and D. Eastwood, “Design of a novel passive tracker,” Sol. Energy 77(3), 269–280 (2004).
    [Crossref]
  11. http://www.zomeworks.com/photovoltaic-tracking-racks/ , last access 14.11.2013.
  12. K. A. Baker, J. H. Karp, E. J. Tremblay, J. M. Hallas, and J. E. Ford, “Reactive self-tracking solar concentrators: concept, design, and initial materials characterization,” Appl. Opt. 51(8), 1086–1094 (2012).
    [Crossref] [PubMed]
  13. P. Schmaelzle and G. Whiting, “Lower critical solution temperature (LCST) polymers as a self-adaptive alternative to mechanical tracking for solar energy harvesting devices,” MRS Fall Meeting & Exhibit (2010).
  14. V. Zagolla, E. Tremblay, and C. Moser, “Light induced fluidic waveguide coupling,” Opt. Express 20(S6), A924–A931 (2012).
    [Crossref]
  15. V. Zagolla, E. Tremblay, and C. Moser, “Efficiency of a micro-bubble reflector based, self-adaptive waveguide solar concentrator,” Proc. SPIE 8620, 862010 (2013).
    [Crossref]
  16. E. J. Tremblay, D. Loterie, and C. Moser, “Thermal phase change actuator for self-tracking solar concentration,” Opt. Express 20(S6), A964–A976 (2012).
    [Crossref]
  17. E. T. Carlen and C. H. Mastrangelo, “Electrothermally activated paraffin microactuators,” J. Microelectromech. Syst. 11(3), 165–174 (2002).
    [Crossref]
  18. H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010).
    [Crossref]
  19. E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
    [Crossref]
  20. A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
    [Crossref]
  21. http://www.altadevices.com/pdfs/single_cell.pdf , last access 13.01.2014.
  22. S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012).
    [Crossref]

2013 (2)

V. Zagolla, E. Tremblay, and C. Moser, “Efficiency of a micro-bubble reflector based, self-adaptive waveguide solar concentrator,” Proc. SPIE 8620, 862010 (2013).
[Crossref]

E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
[Crossref]

2012 (5)

2011 (1)

2010 (4)

R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32(9), 850–856 (2010).
[Crossref]

J. M. Castro, D. Zhang, B. Myer, and R. K. Kostuk, “Energy collection efficiency of holographic planar solar concentrators,” Appl. Opt. 49(5), 858–870 (2010).
[Crossref] [PubMed]

J. H. Karp, E. J. Tremblay, and J. E. Ford, “Planar micro-optic solar concentrator,” Opt. Express 18(2), 1122–1133 (2010).
[Crossref] [PubMed]

H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010).
[Crossref]

2004 (2)

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

M. J. Clifford and D. Eastwood, “Design of a novel passive tracker,” Sol. Energy 77(3), 269–280 (2004).
[Crossref]

2002 (1)

E. T. Carlen and C. H. Mastrangelo, “Electrothermally activated paraffin microactuators,” J. Microelectromech. Syst. 11(3), 165–174 (2002).
[Crossref]

1978 (1)

R. Reisfeld and S. Neuman, “Planar solar energy converter and concentrator based on uranyl-doped glass,” Nature 274(5667), 144–145 (1978).
[Crossref]

Bailat, J.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Baker, K. A.

Carlen, E. T.

E. T. Carlen and C. H. Mastrangelo, “Electrothermally activated paraffin microactuators,” J. Microelectromech. Syst. 11(3), 165–174 (2002).
[Crossref]

Castro, J. M.

Clifford, M. J.

M. J. Clifford and D. Eastwood, “Design of a novel passive tracker,” Sol. Energy 77(3), 269–280 (2004).
[Crossref]

Ding, D.

S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012).
[Crossref]

Droz, C.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Duerr, F.

Eastwood, D.

M. J. Clifford and D. Eastwood, “Design of a novel passive tracker,” Sol. Energy 77(3), 269–280 (2004).
[Crossref]

Ford, J. E.

Gale, B. K.

H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010).
[Crossref]

Hallas, J. M.

Ho, T.

H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010).
[Crossref]

Johnson, S. R.

S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012).
[Crossref]

Karp, J. H.

Kostuk, R. K.

Kroll, U.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Liu, S.

S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012).
[Crossref]

Loterie, D.

E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
[Crossref]

E. J. Tremblay, D. Loterie, and C. Moser, “Thermal phase change actuator for self-tracking solar concentration,” Opt. Express 20(S6), A964–A976 (2012).
[Crossref]

Mastrangelo, C. H.

E. T. Carlen and C. H. Mastrangelo, “Electrothermally activated paraffin microactuators,” J. Microelectromech. Syst. 11(3), 165–174 (2002).
[Crossref]

Meier, J.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Meuret, Y.

Moser, C.

V. Zagolla, E. Tremblay, and C. Moser, “Efficiency of a micro-bubble reflector based, self-adaptive waveguide solar concentrator,” Proc. SPIE 8620, 862010 (2013).
[Crossref]

E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
[Crossref]

V. Zagolla, E. Tremblay, and C. Moser, “Light induced fluidic waveguide coupling,” Opt. Express 20(S6), A924–A931 (2012).
[Crossref]

E. J. Tremblay, D. Loterie, and C. Moser, “Thermal phase change actuator for self-tracking solar concentration,” Opt. Express 20(S6), A964–A976 (2012).
[Crossref]

Myer, B.

Neuman, S.

R. Reisfeld and S. Neuman, “Planar solar energy converter and concentrator based on uranyl-doped glass,” Nature 274(5667), 144–145 (1978).
[Crossref]

Reisfeld, R.

R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32(9), 850–856 (2010).
[Crossref]

R. Reisfeld and S. Neuman, “Planar solar energy converter and concentrator based on uranyl-doped glass,” Nature 274(5667), 144–145 (1978).
[Crossref]

Sant, H. J.

H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010).
[Crossref]

Schade, H.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Shah, A.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Thienpont, H.

Tremblay, E.

V. Zagolla, E. Tremblay, and C. Moser, “Efficiency of a micro-bubble reflector based, self-adaptive waveguide solar concentrator,” Proc. SPIE 8620, 862010 (2013).
[Crossref]

E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
[Crossref]

V. Zagolla, E. Tremblay, and C. Moser, “Light induced fluidic waveguide coupling,” Opt. Express 20(S6), A924–A931 (2012).
[Crossref]

Tremblay, E. J.

Vallat-Sauvain, E.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Vanecek, M.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Wyrsch, N.

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Zagolla, V.

E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
[Crossref]

V. Zagolla, E. Tremblay, and C. Moser, “Efficiency of a micro-bubble reflector based, self-adaptive waveguide solar concentrator,” Proc. SPIE 8620, 862010 (2013).
[Crossref]

V. Zagolla, E. Tremblay, and C. Moser, “Light induced fluidic waveguide coupling,” Opt. Express 20(S6), A924–A931 (2012).
[Crossref]

Zhang, D.

Zhang, Y.-H.

S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012).
[Crossref]

Appl. Opt. (3)

J. Microelectromech. Syst. (1)

E. T. Carlen and C. H. Mastrangelo, “Electrothermally activated paraffin microactuators,” J. Microelectromech. Syst. 11(3), 165–174 (2002).
[Crossref]

J. Micromech. Microeng. (1)

H. J. Sant, T. Ho, and B. K. Gale, “An in situ heater for a phase-change-material-based actuation system,” J. Micromech. Microeng. 20(8), 085039 (2010).
[Crossref]

Nature (1)

R. Reisfeld and S. Neuman, “Planar solar energy converter and concentrator based on uranyl-doped glass,” Nature 274(5667), 144–145 (1978).
[Crossref]

Opt. Express (4)

Opt. Mater. (1)

R. Reisfeld, “New developments in luminescence for solar energy utilization,” Opt. Mater. 32(9), 850–856 (2010).
[Crossref]

Proc. SPIE (3)

V. Zagolla, E. Tremblay, and C. Moser, “Efficiency of a micro-bubble reflector based, self-adaptive waveguide solar concentrator,” Proc. SPIE 8620, 862010 (2013).
[Crossref]

E. Tremblay, V. Zagolla, D. Loterie, and C. Moser, “Self-tracking planar concentrator using a solar actuated phase-change mechanism,” Proc. SPIE 8620, 862011 (2013).
[Crossref]

S. Liu, D. Ding, S. R. Johnson, and Y.-H. Zhang, “Optimal optical designs for planar GaAs single-junction solar cells with textured and reflective surfaces,” Proc. SPIE 8256, 82560M (2012).
[Crossref]

Prog. Photovolt. Res. Appl. (1)

A. Shah, H. Schade, M. Vanecek, J. Meier, E. Vallat-Sauvain, N. Wyrsch, U. Kroll, C. Droz, and J. Bailat, “Thin-film silicon solar cell technology,” Prog. Photovolt. Res. Appl. 12(23), 113–142 (2004).
[Crossref]

Sol. Energy (1)

M. J. Clifford and D. Eastwood, “Design of a novel passive tracker,” Sol. Energy 77(3), 269–280 (2004).
[Crossref]

Other (6)

http://www.zomeworks.com/photovoltaic-tracking-racks/ , last access 14.11.2013.

R. Swanson, “Photovoltaic Concentrators,” in Handbook of Photovoltaic Science A. Luque, and S. Hegedus (John Wiley & Sons, Ltd, 2005), pp. 449–503.

http://www.entechsolar.com/products/solarvolt.htm , last access 13.01.2014.

http://fn-solar.com/#2 , last access 13.01.2014.

http://www.altadevices.com/pdfs/single_cell.pdf , last access 13.01.2014.

P. Schmaelzle and G. Whiting, “Lower critical solution temperature (LCST) polymers as a self-adaptive alternative to mechanical tracking for solar energy harvesting devices,” MRS Fall Meeting & Exhibit (2010).

Supplementary Material (2)

» Media 1: AVI (3468 KB)     
» Media 2: AVI (3865 KB)     

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Figures (11)

Fig. 1
Fig. 1 The self-tracking planar concentrator device concept. a) and b) show the focused light for different incoming angles. The coupling feature shifts due to the paraffin actuator reacting to the shifted focal spot. Inset: Long wavelength light (red) is transmitted through a dichroic facet array to heat up the phase-change actuator below (black). The expanded actuator presses the membrane against the waveguide (grey) allowing short wavelength light (bright yellow) to be coupled into the waveguide. The lens array is shown with a largely reduced scale for this conceptual drawing.
Fig. 2
Fig. 2 a) The optical system using two off-the-shelf aspheric lenses to create a flat Petzval field curvature over ± 23° incoming angle. b) The spot size only changes marginally over the set of incoming angles for the two aspheric lens system compared to two plano-convex lenses or a single lens. c,d) Change of the spot size for a single plano-convex lens and the two aspheric lenses. The different colors represent different parts of the optical spectrum (blue = 400nm, green = 550nm, red = 800nm, yellow = 1200nm).
Fig. 3
Fig. 3 Waveguide (transparent vertical rectangle) and dichroic prism membrane (white square layer in the center) fixed in the holder. The phase-change actuator is square and located below the membrane.
Fig. 4
Fig. 4 The solar spectrum (blue) is divided by the dichroic membrane into long (yellow) and short (green) wavelength parts at roughly 750nm. The long wavelength portion is then used to drive the actuator while the short wavelength part is coupled into the waveguide to the PV cell.
Fig. 5
Fig. 5 Paraffin wax is mixed with carbon black (a) and filled into the honeycomb hole array (b). PDMS is spincoated on top to create a flexible membrane and the actuator is sealed on the back with a glass slide(c). d) Top view onto the steel actuator (bright) filled with paraffin wax/carbon black.
Fig. 6
Fig. 6 The effective concentration factor (c) is the product of the geometric concentration factor (a) and the coupling efficiency (b). Reasonable dimensions (40 x 10 cm2, 2mm waveguide) allow an effective concentration of 150X.
Fig. 7
Fig. 7 Schematic of the experimental setup. A single lens-pair was used to couple light into a small waveguide (length y, thickness t).
Fig. 8
Fig. 8 The assembled actuator and the waveguide were tested with a 1 sun solar simulator. A rotations stage (background) will set an angle and the thermal sensor (red) will measure the power output at the waveguide edge as a function of time.
Fig. 9
Fig. 9 a) The dynamic curve shows the behavior of actuator as a function of time for sudden onset, high intensity conditions (0-20 s) and sudden onset, low intensity conditions (21-40 s) for different angles in absolute power values. b) The dynamics of different angles are compared in normalized values. Smaller angles reach 90% of the maximum faster (<4 s) than larger angles (6-10 s). This is a slightly faster than the 8-12 s the heat needs to dissipate and the coupling value reaches value lower than 10%.
Fig. 10
Fig. 10 a) The actuator allows for actuation from −20° to + 20°. The blue curve shows the minimum amount of sunlight at for every angle whereas the yellow curve shows the maximum amount of coupling. b) Comparison of simulated optical efficiencies and the measured experimental result.
Fig. 11
Fig. 11 a) Screenshot from a video made in the lab showing the actuation for different angles. The angular speed is set to exceed the actuation speed. The exit facet therefore darkens after the system being turned until the actuation occurs again. See Media 1. b) Screenshot from a video made in the lab showing the actuation for different angles. The angular speed is set to match the actuation speed. The exit facet is continuously lit throughout the angular range. See Media 2.

Equations (3)

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C C max = 1 sin 2 ( θ max,in ) .
CF= A in A out = x array y array x waveguide t waveguide = y array t waveguide .
CF= A in A out = 12.5 2 m m 2 π 25mm1mm 50.

Metrics