1. Introduction

Concentrating photovoltaic (CPV) systems usually require precise tracking to achieve high concentration. Since most of the CPV optics are designed to take advantage of only direct incident sunlight, solar tracking becomes one of the most crucial parts in a CPV system. Conventional solar trackers use two-axis mechanical rotation to accurately track the Sun daily and seasonally, maintaining alignment with the Sun all the time throughout the year. Such systems, however, are usually complex and bulky, especially when the collection aperture is large; therefore single-axis tracking is more desired instead.

Recently, planar waveguide solar concentrators have received intense investigations since they are compact, capable of achieving high concentration with uniform output, and potentially cheap for fabrication [1–10]. A typical waveguide concentrator consists of a lens array for primary concentration, couplers to redirect light into the waveguides, and waveguides to output homogenized light directly to the attached solar cells or secondary optics. Previously we proposed a lens-to-channel waveguide solar concentrator using a tilted lens array and channel waveguides [5, 6], as shown in Fig. 1.One end of each channel waveguide is angled by $45°$, acting as a coupling feature. Light hits this surface is reflected into the waveguides via total internal reflections (TIRs). As the lens array is tilted at an angle with respect to the waveguides, the channel waveguides can be placed closely side by side. Therefore light already coupled into the waveguides would not hit subsequent couplers, which avoids the decoupling issue associated with Karp’s initial designs that leads to reduced efficiency [1]. It achieves 800x concentration (using a tapered waveguide as the 2D secondary concentrator) with 90% efficiency using conventional mechanical trackers with angular tolerance of $\pm 0.7°$ [6].

 

Fig. 1 (a) A side view and (b) a top view of the previously proposed lens-to-channel waveguide solar concentrator. A lens array concentrates incoming light onto the channel waveguides below. One end of each channel waveguide is angled by $45°$, acting as the coupler to redirect light into the waveguides via total internal reflections. PV cells or other secondary optics are attached at the other end of the channel waveguides.

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Several novel compatible tracking structures are also proposed for planar waveguide concentrators. Baker et al. use light induced nonlinear response from reactive particles inside the waveguide to actively change the coupler location [11]. It requires a minimal change in waveguide material refractive index of 0.3, which is not easy to realize. The reactive material also needs precise control and complicated configurations. Similarly, Zagolla et al. report using light generated bubbles as a coupling feature [12, 13]. They achieve around 40% optical efficiency experimentally using laser light. This structure, however, is not compatible with III-V solar cells because the infrared portion of the spectrum is absorbed to generate heat for the bubble. The same issue goes to phase change materials (PCMs) [14, 15]. Furthermore, all the above configurations are designed and optimized based on Karp’s initial structure [1], which has an inherent loss mechanism at higher concentration as stated before.

Alternatively in this paper, we explore the possibility of using a hybrid tracking scheme for our previously proposed lens-to-channel waveguide concentrator, which includes a single-axis mechanical tracker and a translation stage. The conventional single-axis tracker rotates the whole structure on a daily basis by following the sun path from east to west; while the translation stage is designed to accommodate seasonal sun movement by adjusting the relative position between the lens array and the waveguides. The merit of such a tracking method incorporated with the lens array-waveguide system is to break a large energy collection aperture into small individual components, so that the movement of the translation stage is associated with single lenses instead of the whole collection area, leading to minimum position change. Hence less energy consumption is expected comparing to the double-axis rotation tracking (covering full daily and seasonal angles). A similar structure is previously discussed by Halls et al. [16]. However, their structure is either complicated due to the use of double-layered lens array or inefficient by using a reflective coupler surface. In contrast, only one lens array will be used in our structure and higher efficiency is achieved. We first review the sun movement briefly and introduce the overall system in section 2. Then in section 3, optical designs and parameter tradeoffs are carefully studied using a math model. Simulation results and experimental data are shown in section 4. Conclusions are made in section 5.

2. Solar path and system overview

Figure 2 briefly reviews the solar path seen by a solar system at a certain location. For a fixed solar system, it is usually set at the angle of its latitude towards south by default (northern hemisphere) so that the system normal is aligned with the Equator. Hence the system receives symmetrical sunlight angle while the Sun travels between the Cancer and the Capricorn throughout the year, which is $\pm 23.5°$ with respect to the Equator. Figure 2(b) illustrates the sky dome seen by the solar system. At Equinox, the Sun travels right overhead from east to west during the day. In the proposed tracking scheme, the single-axis mechanical tracker would follow this trajectory for each single day to track the daily sun path; the translation stage adjusts the relative position between the lens array and the waveguides to accommodate any variance brought by the seasonal movement of the Sun.

 

Fig. 2 (a) A solar system is usually tilted at the angle of its latitude towards south (northern hemisphere) and therefore it sees symmetrical movement of the Sun throughout the year. (b) The sky dome seen by a solar system tilted as in (a). A single-axis tracker is used to rotate the whole structure daily from east to west (green line). As the Sun moves between the Cancer and the Capricorn (blue lines), the translation stage would adjust the relative position between the lens array and the waveguides to accommodate the $\pm 23.5°$ seasonal angle variation.

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Around the Equinox, incoming sunlight shines at a small angle onto the lens array. In this case, chromatic aberration from the lens array is the main factor in determining the real spot size, which in turns determines the primary concentration. However, when the Sun moves towards either the Cancer or the Capricorn, incoming light angle becomes so large that field curvature and astigmatism at the image plane instead becomes the major problem [Fig. 3(a)].In other words, the focal points from the lens array do not lie in a same plane over the year. If the distance between the lens array and the waveguides is kept constant by moving waveguides only in x direction, couplers will see a large spot size at oblique angles, which dramatically decreases coupling efficiency. Therefore we propose to adjust the air gap between the lens array and the waveguides as well to compensate the field curvature aberration. The translation stage thus moves in an x-y manner.

 

Fig. 3 (a) Illustration of x-y and x-only movement of the translation stage using a lens-waveguide pair in XY plane (looking into the waveguide). When incident angle is large, field curvature and astigmatism become significant; focal points are not located in the same plane. If the relative movement is x-only, large spot size at oblique angles would lead to efficiency decrease. Therefore x-y movement is important to compensate for the increased spot size. (b) A brief 3D view of the proposed tracking scheme. Only chief rays are shown. At the Equinox, the lens array and the waveguides are aligned in all three dimensions. The system is placed so that the Sun travels in YZ plane at this time of the year. Departure from the Equinox, the single-axis mechanical tracker (x-axis in this figure) rotates the whole structure daily to ensure that the lens array and the channel waveguides are always aligned in z-direction. Relative movement by the translation stage happens only in x-axis and y-axis.

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To sum up, the single-axis mechanical tracker ensures that light does not deviate from the designed focal point in z direction, while the translation stage moves in an x-y fashion to accommodate the seasonal shift of the focal points. Such vertical and lateral translation movement is illustrated in Fig. 3(b). We stress again that the movements in all three dimensions are associated with single lenses only and therefore less position change and less energy consumption are expected comparing to conventional tracking methods.

3. Optical design

As a start, we study the angular space to explore conditions that have to be met in order to remain high efficiency at large incident angles ($\pm 23.5°$). Assuming paraxial approximation still applies to the lens array and the focused light rays are symmetrical with respect to the chief ray (which may be inaccurate at large incident angles, though), $\theta$ is defined as the angle between the ray after lens and y-axis. The range for $\theta$ is [Fig. 4(a)]

 

Fig. 4 (a) The aberration-free lens focuses incoming light onto its paraxial image plane, from which we can estimate the maximum angle after the lens array. (b) A comparison between normal incidence and oblique incidence onto the coupler surface.

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Light $\theta$ is refracted once by the air/cladding/waveguide ($n_w$) interface before hitting the coupler. The refracted angle $\gamma$ is easily calculated using $\sin \theta =n_w\sin \gamma$. Figure 4(b) shows the difference between light incidence onto the coupler surface at the Equinox (normal incidence) and that at other time (oblique incidence). At the Equinox, light is symmetric ($-{\gamma }_M<\gamma <{\gamma }_M$); in contrast, when departure from the Equinox, such symmetry is broken and ${\gamma }_1<\gamma <{\gamma }_2$.

Consider a beam of light $\theta$ after the lens (${\theta }_1\le \theta \le {\theta }_2$). Light incidence onto the coupler surface can be expressed as

An exemplar plot is shown in Fig. 5, using $n_w=1.62$,$n_s=1.43$ and $F/4$. The incident angles at the coupler surface are in fact larger for oblique angles. In other words, the TIR condition in Eq. (3) may be met by oblique angles while violated by normal incidence; hence an increase in coupling efficiency for oblique angles might be expected. It is also important to note that oblique incidence does not have any impact on y-axis reflections and it increases only the reflection angles in the x-axis. This is the critical part of such translation stage designs in that the large incident angles are all converted to the reflection angles on sidewalls which are waveguide/air interfaces, comparing to y-axis waveguide/substrate interfaces.

 

Fig. 5 Plots of (a) incident angles on the coupler surface ${\psi }_i$; (b) reflection angles at sidewalls ${\psi }_{sw}$; and (c) reflection angles at the waveguide-substrate interface${\psi }_{sub}$ . Data points from different incident angles collapse to one curve in (c), indicating the oblique incidence has no impact on the y-axis reflection angle. Blue lines mark the acceptable angular range for TIRs.

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Besides the effectiveness of the angular space discussed above, we further quantitatively examine the importance of x-y tuning instead of x-axis only, which allows compensation for aberrations from the lens array. As stated before, astigmatism and field curvature are more important factors than chromatic aberration when incident angle is large. Fortunately, if the waveguide plane moves in an x-y fashion, aberrations can be significantly minimized and spot sizes are acceptable for a relatively high concentration. In Fig. 6, we compare the spot radius for x-only and x-y tuning scenarios, using Edmund #48-172 aspherical lens ($D=23.9mm$, $F/4$) with incident half angle up to $25°$. The simulation results are found by optimizing the full field ($0~25°$) to give minimum spot size for each field angle. In the x-only case, the detector plane is intentionally moved towards the lens to increase the spot size at smaller angles, balancing aberrations at larger field angles. Therefore a V curve is observed with minimum spot size located at around $18.5°$. The spot diagrams in Fig. 6(b) clearly shows astigmatism in the x-only scenario. The result in Table 1 indicates that x-y tuning achieves over 6.5x more concentration under the same condition.

 

Fig. 6 (a) The simulation setup. The x-only detector plane is intentionally moved towards the lens to balance aberrations at large obliquities. (b) Spot radius plot using x-only and x-y tuning. They show huge difference (see Table 1) when incident angle is large. Four spot diagrams for $0°$, $15.1°$, $18.5°$, and $21.3°$ fields are also shown. While x-y tuning suffers only comatic aberration, astigmatism is easily observed in the x-only scenario.

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Table 1. Field angles and their corresponding spot sizes.

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We point out that, due to the conservation of etendue [17], the translation stage is used in trade of lower concentration, because the secondary concentration is compromised by the large reflection angle $\mathrm{arc}\tan \left(k_x/k_z\right)$ [Eq. (4)]. Furthermore, the lens array (primary concentration $C_1$) is not an ideal concentrator. As a result, the proposed tracking scheme is mainly designed for systems with concentration <250x when 2D secondary concentration ($C_2$) is used as described in [6] ($C_1~100\text{x}$,$C_2<1/\sin 23.5°=2.5\text{x}$).

4. Tracking using a translation stage for a lens-to-channel waveguide solar concentrator

In order to validate the practical performance of the proposed tracking scheme, we construct a 50x system building block in ZEMAX. An ideal blackbody source from 400nm to 1000nm at 5777K with $\pm 0.5°$ incidence angle (to simulate tracking tolerance) is set as the light source to simulate the incoming sunlight. Edmund #48-172 aspherical lenses are used for the lens array. The input surface area is $\frac{\pi }{4}\times {\left(23.9mm\right)}^2$, while the output area is $3mm\times 3mm$ ($C_1=50\text{x}$,$C_2=1\text{x}$). We choose PMMA as the waveguide material ($n_w=1.49$ at 550nm). The refractive index of PMMA, however, is so small that the coupler cannot couple all the light into the waveguide at normal incidence. Therefore, when incident angle becomes larger, the coupling efficiency is supposed to be higher as predicted in section 3. Figure 7 shows the ZEMAX simulation results by different incident angles. Although the TIR coupling efficiency at the coupler surface increases up to 100% at higher obliquities, the overall efficiency reaches at its maximum at a certain angle ($~17.5°$ in this setup) and then begins to fall due to the projection factor $\cos \theta$. Still, this configuration functions at over 75% efficiency throughout the year. It is also to be noted that anti-reflection coatings and a larger waveguide refractive index is desired (e.g. F2 glass with $n_w=1.64$at 550nm) in practical applications for higher efficiency. The PMMA here is used only to illustrate the increasing coupling efficiency for oblique angles.

 

Fig. 7 Plot of the projection factor $\cos \theta$, the estimated Fresnel reflection loss, the TIR coupling efficiency at the coupler ${\eta }_{coupling}$ and the overall coupling efficiency ${\eta }_{overall}$.

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Experimentally, we fabricate the above $3mm\times 3mm$ building block using PMMA cut by a CO₂ laser cutter. The waveguide is then mounted at the focal point of the Edmund #48-172 aspherical lens. A Xeon white light source is collimated and used to simulate the incoming sunlight. The angle of the last mirror in this measurement setup is carefully controlled by a goniometer to simulate different incident angles [Fig. 8(a)]. The measurement results are plotted in Fig. 8(b) along with simulation results for both x-y and x-only adjustments. Note the x-only detector plane is not moved towards the lens as that in Fig. 6 in this setup but remains at the focal plane of normal incidence. Both measurements match their simulation counterparts very well except for large incident angles of x-y tuning, mainly resulting from the imperfect coupler surface. When the incident angle is small, the spot size covers only small portion of the coupler and therefore defects have little impact on the efficiency; in contrast, however, the spot size becomes as large as the coupler surface area at higher obliquity that any defect decreases the efficiency. The detailed fabrication process is beyond the scope of this work and will be presented in a separate article. Still, the big difference between x-y and x-only can be easily seen; the x-y efficiency remains at over 65% for all angles. It is worth mentioning that the required adjustment range for x and y directions are only $34mm$ and $24mm$, respectively, given the parameters used in this setup.

 

Fig. 8 (a) A schematic of the measurement setup. Incident angle is precisely controlled by the reflection mirror using a goniometer stage. (b) Comparison between simulation and experimental results for both x-y and x-only waveguide movement.

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5. Conclusion

We propose a two-axis tracking scheme designed for a lens-to-waveguide system with concentration <250x, which is realized by a single-axis mechanical tracker and a translation stage. The translation stage is used for adjusting relative positions between lens arrays and waveguides. A math model is built to validate the effectiveness of this idea in both angular and physical space. We show that the x-y movement is crucial to the waveguide in order to minimize aberrations from the lens array. A prototype building block with 50x concentration is fabricated and tested. It shows >65% optical efficiency for various incident angles, matching well with the simulation results from ZEMAX.