1. Introduction

Optical concentration is a powerful strategy for improving the economics of photovoltaic solar energy conversion because it dramatically reduces the cell area needed to generate a given amount of power. The advent of nonimaging optics by Winston and associates over 30 years ago pushed geometric optical concentrators (e.g. those based on lenses or mirrors) to the so-called sine limit [1–5], which describes a fundamental tradeoff between acceptance angle (θ_acc) and concentration ratio (CR) given by CR ≤ (1/sin θ_acc)². This relationship requires that all passive concentrators track the Sun in order to achieve high concentration (CR > 100), which is a disadvantage due to the added expense of the tracking system and the high sensitivity of concentrator throughput on tracking error [4].

Luminescent solar concentrators (LSCs) operate differently by absorbing sunlight and re-emitting it at longer wavelength within the confines of a transparent slab, where the majority (~75%) is trapped by total internal reflection and absorbed by solar cells located at the edges [6–8]. Thermodynamically, LSCs can surpass the sine limit because entropy is produced in the Stoke’s shift, consideration of which leads to a theoretical maximum CR > 100 for typical LSC dye molecules [9–11]. Despite their extraordinary potential, experimental LSC concentration ratios to date are about an order of magnitude lower [7,12,13], though recent developments have demonstrated paths for improvement [14–16]. In the following, we explore the opportunity to improve LSC performance by incorporating elements from the field of nonimaging optics [1–5] into their design.

2. Nonimaging optics in luminescent concentration

The potential for geometric improvement of LSC concentration ratio was noted early on [17] and follows directly from the “brightness theorem,” which states that the radiance of light (W/sr/m²) cannot be increased by a passive optical system [4]. Loosely, this means that the product of geometric and angular optical extent must remain fixed (or decrease), and that an ideal concentrator transforms light with limited angular extent at its input to fill the full 2π steradian half-space at its output [2,3].

Since the angular extent of luminescence reaching the edges of an LSC is naturally restricted by total internal reflection to be outside the critical angle, θ_crit, its output intensity (W/m²) can in principle be increased by a factor β = 1/sin(90 − θ_crit) with no loss of throughput assuming concentration in one transverse dimension, by appropriately tapering its edges. Most LSCs are made of fluorescent acrylic plastic with a refractive index n ~1.5 and thus there exists potential for ~34% increase in concentration ratio that is essentially free to implement since it only involves changing the mold in which the plastic is cast.

A modest additional intensity increase can also be realized by increasing the refractive index of the tapered LSC edges [17], however, the added complexity and expense of such an effort arguably outweighs its benefit and therefore is not considered here.

Previous work explored the use of a straight, “wedge”-tapered edge to increase LSC output intensity, however, this approach falls short of the sine-limiting enhancement above by a factor of two [17]. Here, we approach the geometric enhancement of LSC concentration from the rigorous standpoint of nonimaging optics, which provides the optimum framework and tools needed to reach the sine limit. In addition, we show that intensity can be increased beyond the 1D sine limit (i.e. light is effectively concentrated in the second dimension as well) through a ray-recycling effect that takes place between the different LSC edges.

3. CPC-enhanced luminescent concentrators

The trough compound parabolic concentrator (CPC) [1–5] is a well-known nonimaging optical element that achieves the 1D sine limit for rays propagating within a plane, and thus is well suited for use as the edge-profile of an LSC. We have investigated the performance of LSCs with CPC-tapered edges through ray-tracing simulation performed using Zemax optical modeling software. Figure 1(a) shows the physical layout of a typical simulation for the case of a 100 x 100 x 2 mm LSC composed of dye-doped poly(methyl methacrylate), or acrylic plastic. Light incident normally on the LSC (blue rays) is absorbed and then re-emitted isotropically (green rays) with unity photoluminescent quantum yield and random polarization, with all Fresnel reflections properly accounted for. Those rays trapped within the slab are guided toward perfectly absorbing solar cells located at the edges.

 

Fig. 1 (a) Physical layout of a typical ray-tracing simulation for a conventional luminescent concentrator. Incident light is indicated by blue rays and luminescence by green rays. (b) Radiant intensity distribution of light reaching the edge of a conventional LSC as indicated by the side-view schematic above. Sagittal (S) and transverse (T) angles are defined according the inset of (a) for the cell highlighted in red. (c) Tapering the edge into a compound parabolic concentrator (CPC) geometry as shown in the wireframe side-view above transforms the radiant intensity distribution in (b) to fill the full 2π steradian half-space.

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As coherence is justifiably neglected on the LSC length scales treated here, the only influence of wavelength in these simulations is through the LSC absorption spectrum and refractive index dispersion. For the sake of clarity in presenting our results, the simulations here are thus restricted to monochromatic incident (λ_i = 550 nm) and re-emitted (λ_em = 650 nm) wavelengths, which are typical for LSC dyes [16,18,19]. We choose an absorption length, L_abs = 2 mm for incident light and parameterize reabsorption loss at λ_em via the self-absorption (SA) ratio [16], defined as the self-absorption length, L_sa, at the emission wavelength divided by L_abs. Each simulation is carried out using 5x10⁵ rays, with variation of less than 0.2% between nominally identical simulation runs.

Figure 1(b) shows the radiant intensity distribution (W/sr), relative to the cell normal for a conventional LSC as a function of its sagittal and transverse angles (see Fig. 1(a)). As expected, the distribution does not include angles that correspond to less than the LSC critical angle (θ_crit ~42°), evidenced by the “missing” portions at the top and bottom of the circle. However, as shown in Fig. 1(c), tapering the same edge to form a trough CPC with an acceptance angle, θ_acc = 48°, reduces the output aperture and expands the distribution to fill the full 2π steradian half-space admitted by the solar cell. Note that the CPC faces must be coated with a reflective metal layer (aluminum here, 94% reflectance at normal incidence) to redirect rays impinging below the local critical angle; angle-dependent Fresnel reflection losses are accounted for.

Figure 2(a) explores the intensity (or equivalently concentration ratio) enhancement, β, relative to the original LSC that results from this angular transformation as a function of the CPC acceptance angle and its length, L_CPC, defined in Fig. 1(c). We begin by treating the special case of a quasi-1D CPC LSC, which is long and narrow with perfectly absorbing transverse sides as depicted in the inset of Fig. 2(b). This arrangement filters out rays propagating in the transverse plane to approximate the case of a CPC LSC that operates predominantly in one dimension for rays incident in the sagittal plane.

 

Fig. 2 (a) Intensity increase realized for a 2 mm thick, quasi-1 dimensional CPC LSC relative to its conventional LSC counterpart calculated as a function of the acceptance angle and CPC length. As noted by the dashed green line, there is a limiting “natural” CPC length dependent upon on acceptance angle that is enforced to prevent the CPC edges from closing back in on one another at the input aperture; shorter lengths reflect a truncated CPC. The CPC input aperture is locked to the LSC edge thickness and thus the output aperture varies with CPC length. (b) Relative intensity (left-hand axis) and optical efficiency (right-hand axis) obtained for a “natural” length CPC LSC [e.g. following the green dashed line in (a)] as a function of acceptance angle. The inset illustrates the quasi-1 dimensional approximation used in these calculations, where the LSC is long and narrow with absorbing side faces to eliminate rays propagating significantly outside the sagittal plane.

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In Fig. 2(a), beginning at θ_acc = 90°, which is equivalent to no CPC at all, the intensity increases with decreasing acceptance angle and reaches an initial plateau with β = 1.23 at θ_acc = 48°. Stated more directly, at this point, the CPC LSC requires ~20% less solar cell area than the original LSC to absorb the same amount of light.

The intensity continues to increase with acceptance angle decreasing below 48° however, the drawback to small acceptance angles is that the overall optical efficiency of the concentrator drops significantly due to ray rejection by the CPC. The optical efficiency, η_opt, is the ratio of the total power reaching the cells to that incident on the concentrator and it is related to the concentration ratio via CR = Gη_opt, where the geometric gain, G, is the ratio of LSC input aperture to total cell area [12,16,20]. For an ideal 1D CPC LSC, there would be no reduction in optical efficiency relative to the original LSC (i.e. no light would be lost) until θ_acc decreases below the threshold of 48°, below which point the CPC would begin rejecting light [2–4].

The trends in Fig. 2(b) attest to this qualitative assessment, demonstrating that the optical efficiency remains >90% that of the original LSC for θ_acc > 48°, falling steeply for smaller θ_acc. The intensity enhancement, β = 1.23, reached at this threshold is less than the sine limit of β = 1.34 discussed above due to reflection loss from the metal-coated CPC faces and the small fraction of “unfiltered” out-of-plane rays originating near the CPC that have significant transverse momentum components. It is also noteworthy that the intensity enhancement and relative optical efficiency trends shown in Fig. 2(b) have little dependence on the self-absorption ratio of the LSC itself and hence can be realized for a wide variety of LSC emitters.

Figure 3(a) shows simulation data obtained for the more practically relevant case of a 100 x 100 x 2 mm LSC in which all four edges are CPC-tapered. The intensity enhancement and relative optical efficiency trends observed here are qualitatively similar to that for the quasi-1D situation in Fig. 2, but in this case, both η_opt and β are significantly greater at small CPC acceptance angles below the θ_acc = 48° threshold. Additionally, there is also a strong dependence on self-absorption ratio in this region. As demonstrated in Fig. 4 below, these results are due to ray recycling between CPCs on different edges—that is, light rejected by one CPC is collected by another. This recycling effect is manifest most strongly for θ_acc < 48° because a larger fraction of light is rejected by each CPC to begin with, providing more opportunities for recycling. Similarly, the dependence on self-absorption grows stronger for θ_acc < 48° when ray recycling is significant since these rays must travel a greater distance within the LSC before being collected. A qualitatively similar trend toward greater intensity enhancement at small θ_acc is also observed upon decreasing the concentrator geometric gain for a fixed SA ratio since this reduces the path length of recycled rays and hence their probability of being reabsorbed.

 

Fig. 3 (a) Output intensity and optical efficiency of a 100 x 100 x 2 mm LSC with natural length (dependent on θ_acc) CPC-tapered edges relative to its conventional LSC counterpart. Data is included for several different self-absorption ratios, SA = ∞, SA = 243, SA = 118, and SA = 56, in the order indicated by the black arrow. (b) Similar data obtained for a 100 x 100 x 5 mm LSC with CPC edges truncated to a length of 1.5 mm, showing a significant increase in both intensity and efficiency at small acceptance angle due to improved ray-recycling that results from truncation.

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Fig. 4 (a) Schematic showing how light rejected at the right-hand edge of a CPC LSC (green rays) is recollected at the top edge. Rays are incident from a vertically oriented (i.e. normal to the LSC faces) line source 5 mm from the midpoint of the right edge within the nominal acceptance angle of its CPC. (b) Fraction of rays collected at the right and top cells as illustrated in (a) for increasing emission azimuth in a 2 mm thick LSC with natural length CPC edges. The ϕ > 85° yellow shaded region indicates the point at which rays are incident directly on the top edge. (c) Similar data obtained for the case of a 5 mm thick LSC with truncated CPC edges.

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Figure 3(b) explores the results obtained for a 100 x 100 x 5 mm LSC with truncated CPC edges. Compound parabolic concentrators are often shortened from their natural length to reduce their size and expense at the cost of a slight decrease in concentration ratio [2–4]. A side effect of such truncation is that the CPC begins to collect light incident outside its nominal range of acceptance angles. This has important implications for the efficiency of ray-recycling in LSCs with low reabsorption according to Fig. 3(b), which shows that η_opt remains above 90% down to a new, secondary threshold θ_acc ~25°. The corresponding concentration ratio enhancement at this point reaches β = 1.35 for an LSC with no reabsorption, and remains above 30% for SA = 243 (blue dash-dotted line), which is readily achieved through energy transfer among different chromophores [16]. To be explicit, in this case, the CPC LSC is operating at 90% of the original LSC efficiency with a 1.5x reduction in solar cell area, leading to an overall 1.35x intensity increase on the cells. Alternatively, one could instead arrange to maintain the same overall power output as the original LSC by increasing the CPC LSC facial collection area (i.e. the area over which sunlight is incident) by approximately 11%.

4. Ray-recycling effects

Ray-recycling among CPCs at different edges of an LSC effectively enables 2D geometric concentration because light arriving with transverse angles outside the acceptance of one CPC is rejected and transformed through reflection to be within the acceptance of another, thereby enabling the use of smaller θ_acc than would be possible for the purely 1D situation (i.e. Fig. 2). Figure 4(a) shows this process qualitatively, where rays are emitted uniformly from a line source standing near the right-hand edge of a θ_acc = 40° CPC LSC at constant polar angle, θ_em = 55° (relative to the normal of the LSC faces) and varying azimuth, ϕ. Rays incident toward the right-hand CPC edge at small azimuth (red and blue rays) are within its nominal acceptance angle and are directly collected, whereas those incident at wide azimuth (green rays) are rejected by the right-hand CPC and collected by the top CPC.

Figure 4(b) shows this behavior from a more quantitative standpoint for ray bundles at different polar emission angles in a natural length CPC LSC with θ_acc = 40°. Starting at θ_acc = 50°, which corresponds to the nominal CPC acceptance threshold, rays are rejected by the right CPC for ϕ > 10° and only begin being recycled by the top CPC when ϕ exceeds ~75°. The “gap” between CPC rejection on the right and recollection on the top indicates emission angle combinations (i.e. θ_em, ϕ) at which light is not transformed through reflection from one CPC into the acceptance of another. This gap narrows as θ_em increases to 55° and closes completely at θ_em = 60°, at which point the majority of rays rejected at the right edge are recycled at the top edge. Note that the left and bottom edges also make a small recycling contribution, though these have been omitted for clarity.

As inferred from Fig. 3(b) and demonstrated in Fig. 4(c), truncated edge-CPCs have a greater capacity for ray-recycling due to their relaxed acceptance criteria. Here, even rays at θ_em = 45°—outside the nominal 45° CPC acceptance angle—are accepted at the right and top edges with reasonable efficiency. As in Fig. 2(b), the azimuth gap between rejection at the right edge and recollection at the top edge narrows with increasing θ_em, however in this case, a significant portion of the recycling (up to 40%) also takes place at the left and bottom edges (not shown). Thus, by increasing the likelihood of acceptance beyond θ_acc at any one edge, truncation significantly improves ray-recycling efficiency among all of the edges together, resulting in the high η_opt observed for small θ_acc in Fig. 3(b). Based on the interplay between θ_acc and rejection/recollection azimuth, it is likely that further improvement is possible through an optimal combination of LSC shape (e.g. square, triangle, etc.), CPC acceptance angle, and truncation length.

5. Conclusion

In conclusion, we have shown that incorporating nonimaging optics into LSC design can increase concentration ratio by 30% or more at little to no expense. We have identified a ray-recycling effect among nominally one-dimensional edge-CPCs that effectively enables two-dimensional concentration, thereby surpassing the 1D sine limit. More broadly, new LSC designs based on surface-mounted solar cell arrays [14] are likely to benefit from the integration of nonimaging optical elements, and in particular, LSCs with highly directional luminescence [15,21] stand to benefit enormously from nonimaging designs that leverage their low angular extent emission into high output intensity.