1. Introduction

Multi-junction solar cells are the industry standard for providing solar power on board spacecraft. They are typically implemented as coverglass-interconnected cells (CICs) that are optimized for high efficiency (${\sim }31{}\%$ at 1-sun irradiance), low mass, and reliability. However, as space becomes increasingly commercialized, the high cost of existing CICs has driven exploration into alternative technologies [1–7]. In this context, microcell concentrating photovoltaics (µCPV) have been proposed as a path to improve the economics of space power while preserving the high efficiency, low mass, and compact form factor of existing CICs [8,9]. In addition to shrinking the size of the concentrator optics, microscale cell dimensions <1 mm$^2$ facilitate passive heat dissipation [8,10], enable greater freedom for physical layout and electrical interconnection on the solar wing, and may allow grid fingers and their associated shadowing loss on each cell to be eliminated [11]. The emergence of µCPV therefore represents an opportunity for space photovoltaics to enhance performance and reduce cost.

The metrics for space CPV are different than for terrestrial CPV. High launch costs and limited stowage volume mean that space solar power systems must achieve high specific power (W kg^-1) and power density (W m^-3) in a compact form factor. Moreover, to ensure reliable power generation in the presence of small pointing errors [12,13], it is generally agreed that the angular acceptance of space CPV systems must be $\pm 5$° or more, which limits them to lower concentration ratio ($<300\times$) than their terrestrial counterparts. In our previous work [8], we found that a simple parabolic dish reflector provides the best balance of specific power, power conversion efficiency, and angular tolerance for space µCPV based on a single optical surface. Adding a second optical surface would naturally be expected to improve performance and enable operation closer to the sine limit [14]. However, while the field of nonimaging optics offers numerous sine-limiting solutions in this context [14], the low mass, low volume, intermediate (but still sine-limiting) concentration regime relevant for space is largely unexplored, raising the possibility that more compact, higher specific power solutions may exist.

In this work, we derive, model, and experimentally test a new total internal reflection (TIR)-based V-cone tailored edge-ray concentrator (V-TERC) for space µCPV. This novel design is motivated by the TERC formalism for nonimaging solar concentration and extends our previous work [15] by demonstrating that the V-TERC design approaches the sine limit with a high degree of compactness at large acceptance angle, significantly improving the specific power and concentration ratio compared to the simple parabolic reflector case. We demonstrate a 3.1 mm-thick prototype V-TERC optic with a geometric gain of 137× and test it with a 650 µm square triple junction (3J) microcell. In outdoor testing under the terrestrial spectrum, this prototype achieves $30.1{}\%$ power conversion efficiency with a specific power of 90 W kg^-1 and an acceptance angle of $\pm 4.5$°. This is a record combination for µCPV to date and represents an important step toward increasing efficiency and lowering the cost of solar power in space.

2. Results

2.1 Ultra-compact concentrators at the sine limit

We previously showed that there is a fundamental trade-off between concentration ratio and compactness for µCPV concentrators based on a single reflective surface [8]. In two dimensions (2D), concentrators of this type can attain half the sine limit ($C_\textrm {max}=n/\sin \theta$ for a given refractive index, $n$, and acceptance half-angle, $\theta$, with the corresponding 3D limit being $C_\textrm {max}=(n/\sin \theta )^2$) at an axial thickness-to-aperture width aspect ratio ($\xi$, which serves as a measure of compactness) of 0.5, with a smaller achievable fraction of $C_\textrm {max}$ as $\xi$ decreases.

This tradeoff can be improved with minimal added manufacturing complexity by sloping the top surface of the optic to form an inverted V-cone as illustrated in Fig. 1(a). Incident light rays 1 and 2 refract across the V-cone surface and then reflect off the bottom surface of the optic (which is taken to be an ideal Fresnel reflector for simplicity) to reach the downward-facing receiver. Crucially, the V-cone surface also serves to redirect ray 2 (following its reflection from the bottom surface) to the receiver via total internal reflection. Mirroring the receiver about the dashed line in Fig. 1(a), it is evident that rays 1 and 2 will both strike its left edge at $x=-u_\textrm {i}$. Since the edge ray principle guarantees that all rays within the angular wedge set by rays 1 and 2 will also reach the receiver [16], it is then straightforward to relate the concentrator acceptance angle ($\pm \theta$), geometric gain ($G=u_\textrm {i}/u_\textrm {o}$), and aspect ratio ($\xi =y_\textrm {i}/(2u_\textrm {i})$) via:

 

Fig. 1. (a) Geometric construction of the V-TERC with an ideal Fresnel back reflector (e.g., with infinitely small facets). The blue shaded region is a transparent dielectric (e.g., glass) with refractive index $n$ located in an ambient background where the refractive index $n$′ is unity. (b) Relationship between aspect ratio, optical efficiency, and geometric gain for different acceptance angles as given by Eq. (1) in the text. (c) Fraction of the sine concentration limit achieved by the V-TERC (solid lines) and the ideal single surface reflective concentrator (dashed lines) from Ref. [8] at different aspect ratios.

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The flux concentration ratio ($C$) of this V-TERC is less than $G$ because a fraction of the incident light ($1/G$) is blocked by the back of the receiver. Additionally, when $\xi$ and $G$ are large, some of the incident light close to the receiver edges gets refracted across the middle of the optic and rejected (’crossover’ loss) since the bottom reflector is only designed to work for rays entering on the same side of the optic. The onset of of this crossover loss occurs when $G>(2\xi \sin {\beta })^{-1}$, where $\beta =\delta -\sin ^{-1}{[\tfrac {1}{n}\sin {(\delta -\theta })]}$. Together, these losses establish the ideal geometric optical efficiency, $\eta _\textrm {opt} = C/G$ of the V-TERC, which is further reduced in practice by material-specific reflection, absorption, and scattering losses.

Figure 1(b) shows how $\xi$ and $\eta _\textrm {opt}$ vary as $G$ increases for different acceptance angles. At small $G$, the aspect ratio can be low and the receiver shading loss determines the optical efficiency for all acceptance angles. As $G$ increases, the aspect ratio must also increase and the crossover loss noted above sets in, most significantly when the acceptance angle is large. Figure 1(c) casts the results of Eq. (1) in the context of the sine limit, demonstrating that the V-TERC achieves $>95\%$ of $C_\textrm {max}$ when $\xi =0.5$ and approximately $87\%$ of $C_\textrm {max}$ when $\xi =0.3$. Compared to the 2D single surface microconcentrator limit established in Ref. [8], the V-TERC design roughly doubles the achievable $C$ for a given acceptance angle and aspect ratio.

While the flat, Fresnel-style back reflector in Fig. 1(a) could in principle be realized, it is challenging to do so in practice (because the facets must be extremely small at the scale relevant for µCPV) and thus a curved geometric reflector represents a more realistic near-term solution. Figure 2(a) shows that the solution is a compound parabolic reflector whose right-hand surface is described by a parabola focused at $(-u_\textrm {o},0)$ and rotated by angle $\beta$ to reflect ray $1$ from the right edge of the input aperture to the left edge of the receiver (the focal point) [16]. The vertical position of this parabola (and thus the concentrator’s $\xi$) is then fixed by the need for the top surface slope (set by $\delta$) to mirror the receiver so that it intercepts the nominal reflection of ray 2 (thus ensuring that it will arrive at the focal point of the rotated parabola after total internal reflection from the V-TERC top surface).

 

Fig. 2. (a) Construction of the dielectric V-TERC with a compound parabolic primary reflector. The right-hand side of the back reflector consists of the parabola with focal length, $f$, rotated by angle, $\beta$, to focus on the left-hand edge of the receiver. The angular extent of incident light ($2\theta$) is transformed to $2\theta '$ upon refraction into the optic. (b) Comparison of the V-TERC to a simple parabolic reflective concentrator for different acceptance angles. The inset overlays a V-TERC and simple parabolic concentrator that both have $C = 10$ and $\theta =4$°. The aspect ratio of the former is $\xi = 0.31$ versus $\xi = 0.6$ for the latter. (c) Optical efficiency as a function of incidence angle for two-dimensional $C = 10$ V-TERCs with three different aspect ratios; the inset displays the cross-section of each design. The analytical model assumes ideal conditions with no dispersion, reflection, or absorption losses.

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Figure 2(b) compares the aspect ratio of this V-TERC with that of a simple dielectric-filled parabolic reflector as a function of $C$ for different acceptance angles. Although the V-TERC in this case exhibits slightly higher $\xi$ and crossover loss than the idealized flat reflector version from Fig. 1(b), it is still significantly more compact than the simple parabolic optic (Fig. 2(b), inset), enabling roughly twice the concentration at the same $\xi$. Figure 2(c) plots the optical efficiency of several $C = 10$ V-TERCs with varying aspect ratio (illustrated in the inset) as a function of incidence angle, neglecting all absorption, reflection, and dispersion-related losses. This plot highlights the trade-off between optical efficiency and angular acceptance that occurs as the V-TERC becomes more compact (i.e. the crossover loss decreases with $\xi$ at the expense of reduced acceptance angle).

2.2 Three dimensional V-TERC

The three dimensional (3D) extension of the V-TERC is achieved by rotating the darker solid portion of Fig. 2(a) symmetrically about the y-axis. Here we assume a circular microcell that matches the output aperture of the concentrator for simplicity, noting that in situations where this is not the case, there is either a loss of light or the geometric gain is reduced (e.g. as in the experimental prototype below, where a square microcell overfills the circular output aperture). As with most other concentrators, the axisymmetric 3D V-TERC is less ideal than its 2D counterpart due to some skew ray rejection [14]. Figure 3(a) illustrates this point with 3D ray tracing simulations of $C = 100$ V-TERCs derived from the 2D designs in Fig. 2(c). The on-axis optical efficiency is higher in 3D since the relative impact of shading and crossover losses is reduced. However, ray rejection at higher incidence angle increases because skew rays outside the x-y plane are less likely to reach the receiver following total internal reflection from the top surface. This loss is most obvious for high $\xi$ designs, as the sine-limiting angular acceptance of the $\xi = 0.54$ case in 2D (Fig. 2(c)) falls below that of lower $\xi$ designs in 3D (Fig. 3(a)). We note that a 2D ’trough-style’ concentrator with an elongated receiver might also be considered; however, this greatly decreases the attainable concentration ratio, which in turn reduces the cost leverage of concentration, the power conversion efficiency, and the specific power of the system (because the mass/area of a 2D design is higher than the 3D counterpart).

 

Fig. 3. (a) Optical efficiency calculated for 3D axisymmetric $C = 100$ V-TERCs derived from the 2D designs in Fig. 2(c). The calculations are based on ray tracing and assume the same ideal conditions as in Fig. 2(c). The vertical dashed line indicates the sine-limiting acceptance angle for $C=100$. The ticks on the top axis denote the nominal acceptance angle $\theta _\textrm {a}$ (defined at $90\%$ of the on-axis optical efficiency) of each V-TERC design and the inset shows a rendering of the $\xi = 0.39$ V-TERC. (b) Specific power computed for the ideal V-TERC and simple parabolic concentrators optically bonded to a 170 µm-diameter microcell receiver.

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To assess the potential of the V-TERC for space µCPV, Fig. 3(b) shows its calculated specific power when it is optically bonded to a $30\%$-efficient, 170 µm diameter 3J microcell [17]. We assume a glass optic ($n=1.51$ with density $\rho = 2.5$ g cm^-3) and illumination by the AM0 solar spectrum (1366 W m^-2). For ease of comparison, only the mass of the optic is considered. Also, microcell efficiency is assumed to be independent of light intensity for the range of $G$ values considered, while recognizing that cell efficiency can increase linearly with $\log ({C})$ when the series resistance is sufficiently low. In all cases, the specific power decreases with $C$ because the size (and thus mass) of the optic increases in proportion to $G$. Nevertheless, this calculation shows that the V-TERC has the potential to achieve specific power levels that are competitive with existing multi-junction CIC technology ($\sim 200$ W kg^-1), and well in excess of what simple parabolic µCPV can deliver. For example, a $C = 100$ V-TERC (relatively high concentration for space) with an acceptance half-angle of $\theta = 6$° has the potential to deliver up to 670 W kg^-1 with a total thickness of just 570 µm. By contrast, the concentration ratio of a simple parabolic µCPV system with the same specific power and acceptance angle is just $C = 25$.

2.3 Experimental prototype

To validate the V-TERC design, we constructed a single cell prototype shown in Fig. 4(a), which consists of a $650\times 650$ µm² square 3J InGaP/GaAs/InGaAsNSb microcell (designed for the AM1.5D terrestrial spectrum [18]) and a diamond-turned glass (Schott B270) optic. The axial thickness of the optic is 3.1 mm and the diameter of its input aperture is 8.6 mm, which yields $\xi = 0.36$ and a geometric gain of $G=137$ (here $G$ is the ratio of the input aperture area to the square cell area). The back side of the V-TERC is mirrored with a 150 nm-thick layer of evaporated Ag and the front V-cone surface is coated with a 190 nm-thick layer of evaporated Teflon AF ($n=1.31$) to reduce reflection loss. Averaged over the solar spectrum, the reflectance of the silver mirror is 96% and the transmittance of the antireflection-coated front surface is 97%. The microcell and its contact traces are supported on a 1 mm-thick, $2\times 1$ mm² glass substrate (diced from a larger array of transfer-printed microcells) that is aligned with the exit aperture of the V-TERC (the circular output aperture is inscribed within the square microcell) and bonded with index-matched optical adhesive (NOA 63, Norland) as shown in the inset of Fig. 4(a).

 

Fig. 4. (a) Current-voltage characteristic for the V-TERC µCPV prototype ($G=137$) measured outdoors pointing directly at the sun on a clear day. The current density is expressed relative to the aperture area of the optic. The inset photographs show the prototype optic and the testing apparatus mounted on a rotation stage. (b) Measured short-circuit current density (black line; left-hand axis) and simulated optical efficiency (dashed red line; right-hand axis) as a function of incidence angle. (c) Summary of the losses that account for the difference between the measured µCPV efficiency and that which would result for the same cell in an ideal, lossless concentrator with perfect heat sinking.

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The performance of this µCPV system was measured outdoors on a clear sunny day at the National Oceanic and Atmospheric Administration Surface Radiation (NOAA SURFRAD) monitoring site located near State College, PA as described previously [8,10]. The prototype is mounted on a motorized rotation stage equipped with a sundial to orient it toward the Sun and controllably vary the incidence angle. Figure 4(a) shows the current-voltage characteristic recorded at normal incidence which, based on the direct normal irradiance of 101 mW cm^-2, yields a power conversion efficiency of $(30.1\pm 0.1){}\%$. Figure 4(b) plots the short-circuit current density measured as a function of incidence angle, demonstrating an angular response that agrees well with our Zemax ray tracing model and the nominal $\pm 4.5$° acceptance angle of the V-TERC design.

The power conversion efficiency of this µCPV system is slightly lower than the bare cell under similar irradiance conditions ($\eta _\textrm {PV}=(31.5\pm 0.2){}\%$ based on the global horizontal irradiance), and noticeably lower than a bare cell measured under concentrated light from a solar simulator ($\eta _\textrm {PV}=40.7{}\%$ at $C = 140$) [18]. The difference is due mostly to optical losses and, to a lesser extent, cell heating. The net optical efficiency of the V-TERC is estimated to be 78% based on the ratio of the µCPV short-circuit current to that of the bare cell, though this may be a slight underestimate because the bare cell measurement ($J_{\textrm {sc}}=14.5$ mA/cm², $V_{\textrm {oc}}=2.58$ V, and $FF=0.84$ [10]) includes the contribution from diffuse light while the µCPV does not (i.e. the concentrator can only collect $1/G$ of the diffuse light, which is negligible). Figure 4(c) breaks down the optical loss, which is dominated by form error in the V-TERC optic, followed by cell and contact shading, Fresnel reflections, and material absorption. If the form error can be corrected and shading limited only to the microcell, the power conversion efficiency could realistically exceed 36%.

To understand the impact of cell heating, we carried out a combination of transient photovoltage measurements and finite element simulations. Figure 5 plots the drop in open-circuit voltage that occurs as the microcell heats up following sudden exposure to direct sunlight. Using the known temperature coefficient of the cell’s open-circuit voltage (-4.7 mV °C^-1) then enables an estimation of the microcell temperature, which reaches $\sim 22$ °C above the ambient temperature after 50 s. The inset shows the temperature profile of an axisymmetric finite element heat transfer simulation of this operating condition, which predicts a transient temperature rise in agreement with the data. Altogether, we conclude that cell heating accounts for roughly a 1% absolute loss in power conversion efficiency. Note that this heating loss decreases for smaller cells (owing to more efficient heat dissipation [10]), and also when the cell is operating at its maximum power point rather than at open-circuit (since some of the input energy is converted to work).

 

Fig. 5. Transient decrease in $V_\textrm {oc}$ following sudden exposure to direct sunlight. The right-hand axis rescales the voltage data to show the associated microcell temperature increase based on the known temperature coefficient (-4.7 mV °C^-1) of the cell. The dashed red line shows the microcell temperature rise predicted by finite element modeling of the µCPV system held under open-circuit conditions. The inset shows a cross-sectional view of the temperature profile at steady-state, where the cell reaches 27 °C above the ambient.

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3. Discussion

Based on the active area, mass, and efficiency of the V-TERC prototype, it achieves a specific power of 90 W kg^-1, which is lower than existing multi-junction CICs ($\sim 200$ W kg^-1). However, as Fig. 3(b) demonstrates, reducing the cell size and concentration ratio could significantly improve this metric. Moving from the standalone V-TERC considered here to a tiled lenslet array version that can scale to large area (e.g. the solar wing of a spacecraft) will inevitably sacrifice some of the specific power improvement because truncating the perimeter of a round V-TERC to form, e.g. a hexagon, trims material from the outermost and thinnest region of the optic (thus making the associated tradeoff between lower mass and lower geometric gain less favorable). As an example, truncating the $C = 100$, $\theta = 6$° V-TERC highlighted from Fig. 3(b) above, into a hexagonal tile reduces the concentration ratio to $C = 82$ and its specific power from 670 to 550 W kg^-1, roughly a 20% decrease in both cases. In practice, there is an additional cusp loss between lenslets; however, this is manufacturer-specific and depends strongly on the size of the lenslets and the technology used to produce them [8].

Like other microcell concentrators, the V-TERC design introduced here is compact, monolithic, and can be made from radiation-resistant glass. Compared to simple parabolic µCPV, the V-TERC opens up a higher concentration ratio regime that could further reduce cost and boost cell efficiency without sacrificing angular acceptance. While other sine-limiting concentrators with $\xi < 0.4$ exist [14,19–21], the V-TERC stands out for its relative simplicity [19,22] and because it does not require the cell to be embedded in the middle of the optic [20]. There may also be scope for further performance improvement by transitioning from a V-cone to a freeform top surface. Nevertheless, the clearest drawback of V-TERC µCPV remains the challenge of manufacturing an arrayed, two-surface lenslet array with µm-scale tolerances. We have previously demonstrated simple parabolic lenslet arrays for 650 µm [8] and 170 µm (unpublished) square microcells; however, the non-planar front surface of a V-TERC array requires additional precision during the glass molding process and also makes it more difficult to interconnect the microcells.

A potential solution to the latter issue would be to transfer print and lithographically interconnect the microcells on a planar sheet of cover glass, then align and bond this sheet to the V-TERC optic as done in Ref. [8]. This would, of course, leave an open volume between the cover glass and V-cone surface that could either be filled with a low-index dielectric adhesive (resulting in a mass penalty), or left open, which would require high-performance antireflection coatings [23] to suppress the additional Fresnel reflections. As with any space system, reliability is paramount and would need to be demonstrated for either implementation through extensive temperature cycling, environmental, and radiation hardness testing.

If the manufacturing and reliability challenges can be addressed, the high concentration of V-TERC µCPV could dramatically reduce the amount of multijunction epitaxial material needed for a given solar wing and thereby alter the cost structure of solar power in space. In particular, the V-TERC could be enabling for low intensity, low temperature (LILT) conditions in deep space, where fill-factor and shunt current losses significantly degrade solar cell performance [24,25]. A V-TERC µCPV array with $C=100$ could, for example, restore the illumination intensity on the cells to around one sun for missions as distant as Saturn, where the solar flux is ${\sim }100\times$ lower than in Earth orbit.

4. Conclusion

In summary, we have introduced a compact V-TERC concentrator for space µCPV that enables operation near the sine limit with high specific power. A proof-of-concept system consisting of a single 3J microcell bonded to a diamond-turned glass V-TERC optic was tested outdoors and achieved a power conversion efficiency of 30.1% at a geometric gain of $137\times$ with $\pm 4.5$° acceptance. These results push the boundary of what is possible for space µCPV and are the first to demonstrate power conversion efficiency on par with existing CICs, which should motivate further effort to pilot a large scale µCPV array that can be rigorously vetted for space operation.