## Abstract

We introduce a design methodology for nonimaging, single-reflection mirrors with polygonal inlet apertures that generate a uniform irradiance distribution on a polygonal outlet aperture, enabling a multitude of applications within the domain of concentrated photovoltaics. Notably, we present single-mirror concentrators of square and hexagonal perimeter that achieve very high irradiance uniformity on a square receiver at concentrations ranging from 100 to 1000 suns. These optical designs can be assembled in compound concentrators with maximized active area fraction by leveraging tessellation. More advanced multi-mirror concentrators, where each mirror individually illuminates the whole area of the receiver, allow for improved performance while permitting greater flexibility for the concentrator shape and robustness against partial shading of the inlet aperture.

© 2017 Optical Society of America

## 1. INTRODUCTION

Concentrating photovoltaic (CPV) systems leverage concentration to decrease the required photovoltaic (PV) cell area for a given inlet aperture. As a consequence, the overall system cost can be decreased by using comparatively inexpensive optics, while higher system efficiencies can be reached by allowing the use of multijunction solar cells. Since the first successful implementation [1], CPV technology has made considerable improvements [2,3]. On the cell level, efficiency has continuously increased, with a recently demonstrated record of 46.0% [4]. On the system level, the variety of applications is substantial [2,3,5], with significant advances in efficiency and cost reduction.

Aside from concentration, the second parameter of major importance in CPV is irradiance uniformity across the receiver. On the single cell, irradiance nonuniformities result in a gradient in the charge carrier density, which leads to a drop in the open-circuit voltage, and therefore a lower electrical cell efficiency [6–9]. Nonuniform irradiance can reduce the electrical efficiency even more severely in systems with dense-array CPV receivers because of the current mismatch created between the cells [6,10,11]. The individual cells are often interconnected in series, such that the current through the array is limited by the cell with the lowest average irradiance.

Unfortunately, both concentration and irradiance uniformity are subject to a trade-off. Direct solar radiation reaching the Earth has near-perfect spatial uniformity. When the concentration is increased, typically with a point-focus concentrator such as a parabolic dish or Fresnel lens, a uniform irradiance distribution is increasingly difficult to uphold. Various nonimaging secondary optics have been developed to partially counteract this phenomenon, i.e., improve the irradiance uniformity while matching the square aperture of CPV cells/arrays without major penalties in concentration. Examples include kaleidoscope flux homogenizers [12–15], Köhler integrators [16–19], and other dielectric total internal reflection-based optics [20]. Alternative approaches use concurrent tailoring, e.g., with the simultaneous multisurface (SMS) method [21], of the primary and secondary optics. The advantage of tailoring both surfaces is that certain optical aberrations inherent to point-focus concentrators can be eliminated, which leads to higher concentrations and thus to a somewhat relaxed trade-off between concentration and irradiance uniformity. In each of these approaches, however, a certain amount of optical efficiency with respect to the single-stage concentrator is sacrificed to improve the irradiance uniformity, e.g., by absorption from additional reflections on a secondary mirror or within a dielectric optic, decrease of the acceptance angle (ray rejection), reduction of the intercept factor, or a combination of the above. At the same time, the system complexity and cost are increased.

Alternatively, to avoid these disadvantages, a single reflecting surface can be tailored such that it produces a uniform irradiance distribution with only one attenuation. The absence of dielectric optics (lenses) mitigates issues with cooling as well as chromatic aberrations. The single reflection ensures a high optical efficiency. However, one-reflection optics tailored for uniformity are not common for solar energy applications, due in part to the inherently lower achievable concentrations [22–24]. A notable exception is a design where a controlled combination of surface deformations, described using Zernike polynomials, is employed for the shaping of the mirror [25], but it is limited to circular mirrors. Other concepts relying on an array of carefully arranged small, flat mirror segments to produce uniform distributions have been successfully demonstrated [26,27].

Here, we present an alternative design method for single-reflection (primary) mirrors producing a uniform irradiance distribution that, in contrast to previous designs, uses freeform continuous mirrors with polygonal inlet apertures, which, as will be shown in the analysis that follows, has several intriguing advantages. Section 2 introduces the proposed method, Section 3 discusses various applications of the method for medium- and high-concentration PV, and Section 4 presents results with exemplary single- and multi-mirror concentrators.

## 2. METHODS

Achieving a prescribed irradiance distribution with a single reflection amounts to redirecting incident rays to different locations in the focal plane, which are determined as a function of the rays’ point of intersection with the mirror. The crucial task is to find a mapping technique that defines the relationship between a point on the mirror and a point on the receiver such that a continuous mirror and the desired irradiance distribution are produced. The proposed method allows one to achieve a uniform irradiance distribution on a polygonal receiver by single reflection on a primary mirror having a polygonal perimeter. It comprises the following steps: (i) 2D mapping is performed for a regular nodal grid between the two polygons while preserving the fractional area between points; (ii) the mapped points are subsequently arranged in 3D space via linear transformations; (iii) the final mirror shape is numerically optimized; (iv) final evaluation is carried out using Monte Carlo ray-tracing techniques.

#### A. Area-Conserving Mapping Between Two Polygons

The presented mapping method ${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{P}2}$ between two polygons ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ is based on a technique for low-distortion mapping between a disk $\mathcal{D}$ and a square $\mathcal{S}$ (${\mathrm{\Gamma}}_{\mathcal{D}\to \mathcal{S}}$), first published in [28]. The mapping method is bicontinuous, i.e., mapping can be performed from $\mathcal{S}$ to $\mathcal{D}$ and vice versa, and conserves the fractional area between points. Both these properties are essential for the mapping method presented here, as a disk serves as intermediate geometry to map between two polygons. For this purpose, the original method is extended to the more general case of mapping between a disk and a regular polygon with ${N}_{\mathcal{P}}$ sides. Importantly, the method is further adapted such as to conserve the total area between the mapped geometries (i.e., the areas of the polygon and the disk are equal; ${A}_{\mathcal{P}}={A}_{\mathcal{D}}$) as this simplifies its usage within the context of geometric optics.

In a first step, points on ${\mathcal{P}}_{1}$ are mapped to $\mathcal{D}$ (${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{D}}$). Using the inverse mapping (${\mathrm{\Gamma}}_{\mathcal{D}\to \mathcal{P}2}$), the points are then mapped from $\mathcal{D}$ to ${\mathcal{P}}_{2}$. The property of the disk of having rotational symmetry around its center for every angle allows the transition between two polygons with different ${N}_{\mathcal{P}1}$ and ${N}_{\mathcal{P}2}$. The mapping ${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{D}}$ from a regular polygon to a disk is illustrated by means of a hexagon (${N}_{\mathcal{P}1}=6$) in the top of Fig. 1. For convenience, the polygon and the disk both have an area ${A}_{\mathcal{P}}={A}_{\mathcal{D}}=1$, unlike in the original method where the polygon apothem and the disk radius are equal. The polygon can be divided into ${N}_{\mathcal{P}1}$ rotationally symmetric triangular regions, delimited by the lines connecting the center and the corners, $y=x\text{\hspace{0.17em}}\mathrm{tan}[(2j-1)\pi /{N}_{\mathcal{P}1}]$ (or $\phi =(2j-1)\pi /{N}_{\mathcal{P}1}$ in polar coordinates; and where $j=1,\dots ,{N}_{\mathcal{P}1}/2$), and the perimeter of the polygon, defined by the apothem ${a}_{\mathcal{P}1}={[{N}_{\mathcal{P}1}\text{\hspace{0.17em}}\mathrm{tan}(\pi /{N}_{\mathcal{P}1})]}^{-1/2}$ and circumradius ${R}_{\mathcal{P}1}={a}_{\mathcal{P}1}/\mathrm{cos}(\pi /{N}_{\mathcal{P}1})$. These regions are mapped into the corresponding sectors within the disk with radius ${R}_{\mathcal{D}}={(\pi )}^{-1/2}$.

${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{D}}$ is expressed in the following for a point ${\mathbf{p}}_{\mathcal{P}1}$, belonging to the first region on the polygon, which is defined by $\phi \in [-\pi /{N}_{\mathcal{P}1},\pi /{N}_{\mathcal{P}1}]$ and is shaded gray in Fig. 1. The mapping of a point belonging to a different region can be easily obtained by rotational symmetry around the origin. For ${\mathbf{p}}_{\mathcal{P}1}=[{x}_{1};{y}_{1}]$, the coordinates of ${\mathbf{p}}_{\mathcal{D}}$ on the disk become

Since the mapping method is bicontinuous, its inverse ${\mathrm{\Gamma}}_{\mathcal{D}\to \mathcal{P}2}$ is straightforward. It is illustrated by means of a square (${N}_{\mathcal{P}2}=4$), representative of the most common receiver geometry, in the bottom of Fig. 1. By reversing the coordinate transformation from Eq. (1), the Cartesian coordinates of ${\mathbf{p}}_{\mathcal{P}2}$ can be determined from ${\mathbf{p}}_{\mathcal{D}}$:

The complete mapping ${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{P}2}$ between the two polygons then yields

This elegant expression, however, only holds true for points that are located in the first region in both polygons, i.e., $\phi \in [-\pi /\mathrm{max}\{{N}_{\mathcal{P}1},{N}_{\mathcal{P}2}\},\pi /\mathrm{max}\{{N}_{\mathcal{P}1},{N}_{\mathcal{P}2}\}]$. For points that lie outside of the first region in at least one of the polygons, intermediate rotations around the origin are necessary between the two steps of the mapping. If mirror and receiver have the same perimeter shape (${N}_{\mathcal{P}1}={N}_{\mathcal{P}2}$), then Eq. (3) reduces to the obvious result ${\mathbf{p}}_{\mathcal{P}1}={\mathbf{p}}_{\mathcal{P}2}$, i.e., the initial shape remains undistorted. While such concentrators have certain advantages, it is a crucial benefit of the presented method that it allows designs with ${N}_{\mathcal{P}1}\ne {N}_{\mathcal{P}2}$, which have several interesting applications, as outlined in Section 3. Throughout a major part of this work, designs with ${N}_{\mathcal{P}1}=6$ and ${N}_{\mathcal{P}2}=4$ are used as examples to illustrate a general case and showcase the flexibility of the method.

#### B. Grid Generation

For the implementation of the mapping technique, ${\mathbf{p}}_{\mathcal{P}1}$ needs to be discretized into a grid of $N$ nodes ${\mathbf{p}}_{\mathcal{P}1,n}$. For the subsequent least mean square optimization of the mirror surface, it is advantageous if all surface elements within the grid, i.e., the areas defined by connecting three neighboring nodes, have equal area, as all nodes can then be weighted the same. Appendix A describes a convenient way of generating a regular grid having these properties on ${\mathcal{P}}_{1}$, which was applied in this paper.

#### C. From 2D to 3D: Scaling and Translation

Having established the relationship between nodes on ${\mathcal{P}}_{1}$ and their image on ${\mathcal{P}}_{2}$, the nodes now have to be assigned their final location in 3D space such as to make up the desired solar concentrator. This procedure is illustrated by Fig. 2. In addition to ${N}_{\mathcal{P}1}$ and ${N}_{\mathcal{P}2}$, the required design parameters of the concentrator are (1) the geometric design concentration ${C}_{g,\text{design}}={A}_{1}/{A}_{2}$, which defines the size of the image relative to the inlet aperture (${A}_{1}$ and ${A}_{2}$ are the respective projected areas of the mirror and the image); and (2) the focal length $f$ of the concentrator. For the purpose of demonstration in this paper, concentrators are normalized to unity inlet area, i.e., ${A}_{1}={A}_{\mathcal{P}1}=1$. Each node on the primary mirror is defined as

The area of the image ${\mathbf{p}}_{\mathcal{P}2,n}=[{x}_{2,n};{y}_{2,n}]$ is reduced by the factor ${C}_{g,\text{design}}^{-1/2}$ (the apothem of the scaled image is ${a}_{2}={C}_{g,\text{design}}^{-1/2}{a}_{\mathcal{P}2}$), and the nodes are elevated to the focal plane, thus creating the receiver

#### D. Mirror Surface Optimization

The next task then consists in finding ${z}_{1,n}$ of each node on the primary mirror such that (1) an on-axis ray impinging on that node is reflected to the corresponding node on the image; and (2) the resulting mirror surface is continuous. The optimization is thus based on on-axis rays only. Figure 3(a) depicts the initial situation after scaling and translation of the nodes, where ${z}_{1,n}=0$. The (inverse) incident ray direction is ${\hat{\mathbf{r}}}_{i}=[0;0;1]$ for all nodes. At each node ${\mathbf{x}}_{1,n}$ on the mirror, the desired reflected ray direction is ${\hat{\mathbf{r}}}_{o,n}=({\mathbf{x}}_{2,n}-{\mathbf{x}}_{1,n})/\Vert {\mathbf{x}}_{2,n}-{\mathbf{x}}_{1,n}\Vert $. If the ray is specularly reflected, the required surface normals at ${\mathbf{x}}_{1,n}$ to fulfill condition (1) are

To simultaneously fulfill condition (2), ${z}_{1,n}$ needs to be determined such that the tangent vector of the surface ${\widehat{\mathbf{t}}}_{n}$ is perpendicular to ${\widehat{\mathbf{n}}}_{n}$, i.e.,

The required boundary condition is the fixed position of the central node, as mentioned in Section 2.C. The detailed optimization procedure applied to fulfill Eq. (7) is outlined in Appendix B. At the end of the optimization, on-axis rays impinging at each node on the primary mirror are reflected to the corresponding node on the image with minimized error, as indicated in Fig. 3(b).

#### E. Monte Carlo Ray Tracing

The optimized surfaces are evaluated using Monte Carlo ray tracing. An in-house code [29] has been extended with ${C}^{0}$-continuous Nagata patches that are ideal for use with a mesh where the node positions (${\mathbf{x}}_{1,n}$) and normal vectors (${\widehat{\mathbf{n}}}_{n}$) are known, and are commonly employed for ray-tracing problems [30,31]. For reasonably smooth surfaces, such as the ones encountered in this paper, a mesh made up of ${C}^{0}$-continuous Nagata patches has ${C}^{1}$ continuity. All ray-tracing simulations were performed using a diffuse (equicosine) source of cone-angle ${\theta}_{\text{sun}}=4.65\text{\hspace{0.17em}}\mathrm{mrad}$. A spatially uniform flux distribution over the inlet aperture was assumed throughout. Additionally, the following assumptions apply: the shape is ideal; the reflectivity is uniform, independent of incident and outgoing directions, and perfectly specular; and geometric optics applies. Unless otherwise stated, simulations were performed with ${10}^{8}$ rays. For the generation of smooth irradiance distributions, grids with approximately 2500 nodes are used. Otherwise a lower resolution is sufficient, as shown in the grid size study in Appendix C. Irradiance intensities are expressed for a direct normal irradiance (DNI) of $1000\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{m}}^{2}$.

## 3. APPLICATIONS

Several CPV applications can benefit from concentrators that on the one hand have mirrors with a polygonal perimeter and on the other hand produce a uniform irradiance distribution on a polygonal receiver. The most straightforward example is a dish-type concentrator consisting of a single on-axis mirror and a square PV receiver. Exemplary designs with square, hexagonal, and circular [32] perimeter are shown in Fig. 4. Due to their simplicity, these exemplary concentrators are ideally suited to analyze the fundamental behavior and performance of the design method (Section 4.B) before it is employed to construct more elaborate designs (Section 4.C). They are, however, also imaginable as standalone concentrators for decentralized, small-scale energy production as well as for large-scale applications in fields. Comparable concentrator form factors (square continuous mirror) have recently emerged in commercial CPV applications, albeit with a parabolic profile and secondary optics [17].

The facilitated imaging to a square PV cell array makes square concentrators particularly attractive. One of the benefits of the presented method is that it is not limited to square perimeters but instead enables simple imaging between arbitrary polygons with minimal distortion. Hexagonal concentrators have the advantage compared to square concentrators of having a more compact mirror (smaller height difference between mirror center and corner) for the same focal length, especially at high rim angles. Consequently, it produces a more uniform angular distribution of the irradiance on the receiver. Meanwhile, it still allows tessellation, which has several practical advantages. The circular design is equivalent to the limit case of a polygon with ${N}_{\mathcal{P}1}\to \infty $ and the most compact mirror design. Additionally, circular dishes of parabolic and close-to-parabolic profile are state of the art in the solar energy and antenna industries. The well-established fabrication processes could readily be adapted to the presented designs. Recently, shaping of square glass mirrors for solar dish concentrators by radiative heating has been successfully demonstrated with small slope errors ($\mathrm{rms}<1.5\text{\hspace{0.17em}}\mathrm{mrad}$) [33], representing another potential avenue for the fabrication of the presented mirrors, regardless of perimeter shape.

The most interesting advantage of polygonal mirrors is their potential for tessellation, which allows various compound designs with high active area fraction (AAF). This potential can optimally be leveraged with CPV concentrator modules, where several on-axis mirrors, each with its own receiver, are mounted together on the same tracker to reduce complexity, e.g., with square concentrators [26,34], as exemplarily shown in Fig. 5(a). An additional advantage of such systems is that they can be designed with rectangular outer perimeters, which has a positive impact on the field-shading efficiency [35]. The same concept can be applied to hexagonal concentrators, as depicted in Fig. 5(b), allowing different (e.g., circular) outer perimeters, while still maximizing the AAF and uniformly illuminating square receivers. Circular perimeters, notably, allow a rotationally symmetric mirror support structure, which has practical advantages.

When increasing the number of mirrors, smoother and more versatile perimeters can be obtained. Concentrator panels, where an array of small, densely packed mirrors each concentrate onto a single CPV cell held in place by a transparent cover, as suggested in Fig. 5(c), are an extension of this concept. To facilitate the installation of electrical and thermal sinks for the cells, such concentrators are often designed as Cassegrain systems, as shown in Fig. 5(d). By using a flat secondary mirror, the ray redirection introduced in Section 2 is conserved. Here, the more compact hexagonal mirrors have the additional advantage of allowing the fabrication of a more compact CPV module with less steep gradients between the individual mirrors.

In addition to these compound designs made up of individual on-axis concentrators, the presented design method also allows more elaborate concentrators composed of multiple off-axis mirrors. Within these multi-mirror concentrators, each mirror individually distributes the incident radiation uniformly over the entire area of a single common receiver. Exemplary concentrators based on square mirrors are shown in Fig. 6. As with the compound designs, tessellation of the polygonal mirrors can be exploited to create a continuous reflective surface with high AAF and versatile perimeter shape.

The simplest multi-mirror designs can be created by subdividing the reflective surface of the square single-mirror concentrator from Fig. 4(a) into multiple mirrors of equal size. Exemplary designs with four and nine mirrors are shown in Figs. 6(a)–6(b). Despite their similarity to the single-mirror concentrator, the multi-mirror designs have several additional advantages, as will be discussed in Section 4.C. Figures 6(c)–6(d) show other examples of multi-mirror designs, having essentially the same properties as the simpler designs from Figs. 6(a)–6(b), but with more complex perimeter shapes. This, notably, allows the same advantages as for the designs from Fig. 5, e.g., rectangular perimeters to improve the field-shading efficiency or circular perimeters for a rotationally symmetric structure, but with the potential advantage of having only one single receiver.

Finally, by leveraging the tessellating nature of polygons, not only for the reflective surface but also for the receiver, a design is conceivable where multiple mirrors redirect solar radiation on multiple adjacent receivers, as suggested in Fig. 7. A potential advantage of such a design is that mirrors can be slightly offset radially such that shading by the receiver can be eliminated. On the downside, this reduces the AAF of the concentrator and produces slightly asymmetric irradiance distributions.

## 4. RESULTS AND DISCUSSION

#### A. Mapping Method

Figure 8 shows the results of the mapping of a uniform grid of equally-spaced nodes between two polygons by the method outlined in Section 2. The nodes needed for the design of a hexagonal concentrator with square receiver [Fig. 4(b); ${N}_{\mathcal{P}1}=6$, ${N}_{\mathcal{P}2}=4$] are shown as an example. All geometries have an equal projected area ${A}_{\mathcal{P}1}={A}_{\mathcal{D}}={A}_{\mathcal{P}2}=1$. Figure 8(a) shows the regular nodal grid ${\mathbf{p}}_{\mathcal{P}1,n}$ generated on the hexagon. The node colors indicate their initial location within the hexagon to track the transformations. All triangular elements spanned among three neighboring nodes are equilateral and have equal area. The result of the first part of the mapping (${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{D}}$) that transforms the initial grid into ${\mathbf{p}}_{\mathcal{D},n}$ with minimal distortion can be seen in Fig. 8(b). The nodes are mapped into the sector within the disk corresponding to their initial region on the polygon. The triangular elements are no longer equilateral. However, their area is conserved. Figure 8(c) shows the result of the complete mapping from the hexagon to the square after the second step, ${\mathrm{\Gamma}}_{\mathcal{D}\to \mathcal{P}2}$, which transforms the intermediate nodes on the disk into the final image ${\mathbf{p}}_{\mathcal{P}2,n}$. Because the intermediate grid on the disk does not respect the rotational symmetry of the square, small irregularities can be observed along the diagonals of the square. However, the areas of the grid elements are still conserved and the irregularities become less important with a larger number of nodes than used for illustration here.

#### B. Single-Mirror Concentrators

The hexagonal concentrator from Fig. 4(b) (${N}_{\mathcal{P}1}=6$, ${N}_{\mathcal{P}2}=4$, ${A}_{1}=1$, $f=1$) at ${C}_{g,\text{design}}=500\times $ is used as a benchmark throughout this section, as the effects of various design parameters on the concentrator shape and performance are examined.

### 1. Mirror Shape

Figure 9 shows the profile difference $\mathrm{\Delta}z={z}_{m}-{z}_{p}$ between hexagonal concentrators with ${C}_{g,\text{design}}=100\times $, $500\times $, and $1000\times $, representative of medium- to high-concentration PV applications, and a parabolic dish, normalized by the apothem of the primary mirror, ${a}_{1}={a}_{\mathcal{P}1}$.

Figure 9(a) displays $\mathrm{\Delta}z$ as a contour plot over the whole mirror aperture. The profile deviation increases with increasing radial distance from the mirror center (due to the positioning of the mirror center on a parabolic dish, the deviation in the center is 0). It is, however, not rotationally symmetric but instead exhibits a combination of the symmetries of both mirror and receiver shapes. This results in the lowest deviation appearing along $y=0$, while the highest deviation appears along the lines $y=\mathrm{tan}(\pm \pi /6)x$. As ${C}_{g,\text{design}}$ is increased, the mirror profile approaches that of the parabolic dish. Figure 9(b) shows the radial difference along a direction with minimal offset ($x>0$, $y=0$; solid lines) compared with a direction of maximal offset ($x>0$, $y=\mathrm{tan}(\pi /6)x$; dashed lines), which underlines the asymmetric concentrator profile and its convergence to a parabolic dish for increasing ${C}_{g,\text{design}}$. In fact, in the limit case ${C}_{g,\text{design}}\to \infty $, all nodes on the image would merge into the focal point [0; 0; $f$], and hence $\mathrm{\Delta}z\to 0$ for the whole mirror. Essentially, the mirror profile for all other designs (${C}_{g,\text{design}}<\infty $) is a parabolic dish that is nonuniformly defocused, i.e., the extent of the divergence from the parabolic shape varies both in radial and circumferential direction. Due to the deviation from the parabolic shape, the rim angle of the mirror, which can be of interest for the receiver design, is affected. Appendix D addresses the matter of the rim angle and shows that its variation with ${C}_{g,\text{design}}$ is marginal.

### 2. Irradiance Distribution at the Receiver

Figure 10 compares the irradiance distribution on a square receiver (${N}_{\mathcal{P}2}=4$) with square, hexagonal, and circular primary mirrors designed with ${C}_{g,\text{design}}=100\times $, $500\times $, and $1000\times $. The irradiance distribution is plotted versus the receiver coordinate, normalized by the apothem of the scaled image, ${a}_{2}$. The square concentrator (${N}_{\mathcal{P}1}=4={N}_{\mathcal{P}2}$), represents the simplest design case, where mirror and receiver have the same perimeter shape. As noted in Section 2.A, the mapping between the nodes on the primary mirror and their image introduces no distortions. The irradiance distribution on the receiver is very uniform for all ${C}_{g,\text{design}}$, with a large central part where the irradiance is equal to ${C}_{g,\text{design}}$, and a fringe region of gradually decreasing irradiance, which is centered at the border of the image area.

The size of the area encompassing the entire fringe region is prescribed by the edge rays reflected from the outer rim of the primary mirror. To collect all rays, the perimeter of ${A}_{2}$ has to be increased in either direction by the half-width of the fringe region (half-width of the focal image of the sun, produced by reflection form a point on the outer rim), ${w}_{\text{fringe}}={R}_{\mathcal{P}1}\text{\hspace{0.17em}}\mathrm{sin}(2{\theta}_{\text{sun}})/\mathrm{sin}(2{\mathrm{\varphi}}_{\mathcal{P}1}^{+})$ [23], where ${\mathrm{\varphi}}_{\mathcal{P}1}^{+}$ is the major rim angle of the primary mirror and a function of its circumradius and the focal length (Appendix D). As a consequence, the full-collection concentration ratio is lower than ${C}_{g,\text{design}}$. In general, it is limited to values below the fundamental limit for convex single-reflection concentrators with axial symmetry, ${C}_{g,1,\mathrm{max},3\mathrm{D}}={\mathrm{sin}}^{2}(2{\mathrm{\varphi}}_{\mathcal{P}1}^{+})/{\mathrm{sin}}^{2}(2{\theta}_{\text{sun}})$ [23,36,37]. In particular, for a square receiver (${N}_{\mathcal{P}2}=4$), such as the one used throughout this work, the full-collection geometrical concentration limit is given by ${C}_{g,1,\mathrm{max},\text{square}}={C}_{g,\text{design}}/{(2{w}_{\text{fringe}}{({C}_{g,\text{design}})}^{1/2}+1)}^{2}$ [38], with a maximum of $1/(4{w}_{\text{fringe}}^{2})={C}_{g,1,\mathrm{max},3\mathrm{D}}/2$ for ${C}_{g,\text{design}}\to \infty $, i.e., the smallest receiver collecting all radiation is a square circumscribing the ideal image of the sun in the focal plane. For ${N}_{\mathcal{P}2}\to \infty $, the maximum of the full-collection geometrical concentration limit converges to ${C}_{g,1,\mathrm{max},3\mathrm{D}}$. Since ${w}_{\text{fringe}}$ is independent of ${C}_{g,\text{design}}$, the fringe region occupies a larger portion of the image area with increasing ${C}_{g,\text{design}}$. These results demonstrate the high performance achievable with the presented design method over a wide range of concentrations.

The examples of the hexagonal and disk concentrators (${N}_{\mathcal{P}1}\ne {N}_{\mathcal{P}2}$) are more complex, because the mapping leads to distortions in the image of the nodal grid. These distortions are more severe the larger the difference between ${N}_{\mathcal{P}1}$ and ${N}_{\mathcal{P}2}$ becomes (a disk is equivalent to a polygon with ${N}_{\mathcal{P}1}\to \infty $) and subsequently cause increasingly important nonuniformities in the irradiance distributions. Nevertheless, these nonuniformities are within reasonable bounds and become less severe at high ${C}_{g,\text{design}}$. At least the hexagonal concentrator, which allows several interesting applications and produces good uniformity, remains of interest.

Table 1 summarizes the average irradiance within the image area, $\u27e8E\u27e9$, obtained with these concentrators. $\u27e8E\u27e9$ is in the vicinity of ${C}_{g,\text{design}}$, but is diminished by the increasing relative size of the fringe region for high ${C}_{g,\text{design}}$. If the receiver is chosen smaller than the image area, an average irradiance close to ${C}_{g,\text{design}}$ can be maintained. A more quantitative evaluation of the performance taking this into consideration is provided in the following section for the hexagonal concentrator.

### 3. Intercept Factor and Uniformity

Of major importance for the design of a solar concentrator is the total system efficiency ${\eta}_{\mathrm{tot}}$, mainly made up of the optical efficiency ${\eta}_{\mathrm{opt}}$ (fraction of incident rays that reach the receiver after traveling through the optical system) and receiver efficiency ${\eta}_{\mathrm{el}}$ (electrical efficiency of a PV array): ${\eta}_{\mathrm{tot}}\propto {\eta}_{\mathrm{opt}}{\eta}_{\mathrm{el}}$. In an ideal optical system (mirror reflectance $\rho =1$, receiver absorptance $\alpha =1$), ${\eta}_{\mathrm{opt}}$ is equal to the intercept factor, defined here as the ratio of the radiative power intercepted by a central, square receiver of area ${A}_{i}$ and the total radiative power reaching the focal plane

The electrical efficiency of a PV array, on the other hand, is closely linked to the irradiance uniformity on the receiver [10]. Commonly in CPV, ${\eta}_{\mathrm{el}}$ is related to the cell-to-cell uniformity, as it is physically proportional to the mismatch losses with series-connected PV cells. It is defined as

where ${E}_{\mathrm{av},i}$ are the cell-averaged irradiances of each cell within the array and $\u27e8{E}_{\mathrm{av}}\u27e9$ is the mean array irradiance. $U$ varies between $U=1$ (perfectly uniform) and $U=0$ (highly nonuniform) and is highly dependent on the number of cells in the array.In an optical system, high intercept factor and irradiance uniformity (high optical and receiver efficiencies) are subject to a fundamental trade-off. In the system presented here, the remaining tunable parameter for a chosen design ${N}_{\mathcal{P}1},{N}_{\mathcal{P}2},{C}_{g,\text{design}}$) that has an impact on both $\gamma $ and $U$ is the size of the receiver compared to the image area. To find a configuration that maximizes total efficiency, it is therefore important to know the relationship between intercept factor and uniformity, which is shown in Fig. 11 for hexagonal mirrors and the three ${C}_{g,\text{design}}$ with a $6\times 6$ cell array. The circles (•) indicate the intercept factor and uniformity for a receiver having the size of the image area. The corresponding values are also summarized in the left column of Table 2. With increasing ${C}_{g,\text{design}}$, the trade-off between intercept factor and uniformity becomes more severe due to the fringe region becoming larger relative to the image area.

If the receiver size is chosen smaller than the image area (towards the top left of Fig. 11), $U$ increases up to the point where the fringe region lies completely outside the receiver. If the receiver size is decreased further, the uniformity decreases slightly due to the larger impact of the irradiance peaks at the receiver edges before it increases again as these peaks come to lie outside the receiver. However, due to the quickly decreasing intercept factor at these receiver sizes, a design to the left of the uniformity peak (dotted lines) is irrational. If the receiver size is chosen larger than the image area (towards the bottom right of Fig. 11), the uniformity drops quickly, with only marginal increase in intercept factor. A receiver the size of the image area therefore offers a relatively good trade-off between high uniformity and high intercept factor. The optimum design point for a specific application might deviate slightly from this point.

As mentioned in Section 2.E, simulations have been performed with ideally shaped mirrors. If a realistic mirror shape error, i.e., 1.5 mrad, is introduced, the absolute size of the fringe region is slightly increased and the sharp corners of the irradiance distribution are rounded off, while the irradiance within a major part of the aiming area remains uniform. Compared to the values given in Table 2, the intercept factor then decreases to 0.941, 0.863, and 0.806 for ${C}_{g,\text{design}}=100\times $, $500\times $, and $1000\times $, respectively.

### 4. Systems with Secondary Optics

Up until this point, shading of incident rays by the receiver has not been accounted for to show where spatially uniform rays reflected by the mirror would impinge on a receiver under ideal conditions. Figure 12 (left) shows the effect on the local and cell-averaged irradiance distribution caused by receiver shading for the hexagonal concentrator with ${C}_{g,\text{design}}=500\times $ and a $6\times 6$ cell array that has the exact dimensions of the image area (${C}_{g}={C}_{g,\text{design}}$). The performance metrics for all hexagonal concentrators (${C}_{g,\text{design}}=100\times $, $500\times $, $1000\times $) are summarized in Table 2 (middle column) and marked by the downward pointing triangles (▾) in Fig. 11. The decrease in intercept factor and uniformity as a result of shading is negligible for the high-concentration designs ${C}_{g,\text{design}}=500\times $ and $1000\times $, while a small drop in performance can be observed for ${C}_{g,\text{design}}=100\times $ [39].

A much more important change can be observed if a secondary optic is added in front of the receiver. If designed correctly, it can serve the double purpose of (1) collecting the radiation that would otherwise impinge on the outside of the receiver, thereby increasing the optical efficiency of the concentrator (fraction of radiation intercepted within the receiver area); and (2) redirecting this radiation predominantly to the outer region of the receiver where irradiance would otherwise be low, thereby increasing the uniformity.

Perhaps the simplest nonimaging concentrator can be constructed by orthogonally intersecting two V-troughs [40–42], a design usually referred to as crossed V-trough or pyramid concentrator, e.g., [43]. If dimensioned such that its outlet aperture is the same size as the receiver (the geometric concentration of the system remains unchanged; ${C}_{g}={C}_{g,\text{design}}$), it can predominantly collect fringe radiation while not interfering with the irradiance directed towards the receiver center, and therefore has a low average number of reflections and high optical efficiency. When designing the crossed V-trough, mirror size and vertex angle have to be selected such that the best overall performance is achieved. If the vertex angle $\psi $ of the V-troughs is chosen such that $\psi >{\mathrm{\varphi}}^{+}+{\theta}_{\text{sun}}$ (where ${\mathrm{\varphi}}^{+}=34.47\xb0$ is the major rim angle of the hexagonal concentrator, as discussed in Appendix D), then the secondary optic will theoretically accept all incident rays, provided the mirror length $l$ is large enough to collect all the rays reflected to the outside of the image area and ${C}_{g,\text{design}}\ll 1/\mathrm{sin}\text{\hspace{0.17em}}{({\theta}_{\text{sun}})}^{2}$. A small fraction of the incident radiation is, however, lost due to rejection caused by the 3D design (for rays that are reflected on two perpendicular walls, the ideal properties of the V-trough no longer apply). On the other hand, long walls increase the shaded area in the center of the mirror. Thus, they result in a decrease of the irradiance in the center of the receiver, and, more importantly, as a consequence, a decrease of the irradiance uniformity. A design with $\psi =35\xb0$ and $l=0.3$ provides a good trade-off and overall performance.

Figure 12 (right) shows the resulting local and cell-averaged irradiance distributions for the hexagonal concentrator with ${C}_{g,\text{design}}=500\times $ with the secondary optic installed [44,45]. The corresponding performance metrics are plotted with the upward pointing triangles (▴) in Fig. 11 and summarized in the right column in Table 2. For the concentrator with ${C}_{g,\text{design}}=100\times $, the additional shading of the incident radiation caused by the secondary further decreases the intercept factor and, to a larger extent, the irradiance uniformity achieved on the receiver. A secondary optic is therefore not advantageous in this case. For the higher concentrations, however, the secondary optic is highly beneficial. While the additional shading of the incident rays by the secondary does result in a lower irradiance in the receiver center, the cell-averaged irradiance in the center is only marginally affected. The overall uniformity is considerably improved as an effect of the redirection of part of the fringe radiation to the edge of the receiver (e.g., from 77.6% to 92.1% for ${C}_{g,\text{design}}=500\times $). The intercept factor is also significantly increased (from 90.6% to 98.8%) such that almost all radiation is collected by the receiver. In conclusion, a small addition to the design can considerably improve the concentrator performance at high concentrations without doing a lot of work, by only attenuating the rays at the outer region of the image area.

#### C. Multi-mirror Concentrators

As discussed in Section 3, the presented method also allows the design of multi-mirror concentrators, where each mirror is individually imaged onto the whole area of the receiver, allowing greater flexibility for the concentrator shape. For assessment of the properties of such multi-mirror designs, we compare the three simplest concentrators based on square mirrors. Inspecting these simple designs will allow an extrapolation to complex concentrators having a higher number of mirrors. Square mirrors are the ideal choice, as they achieve the best performance with a square receiver, due to the lack of distortions introduced during the mapping. As the number of mirrors is increased, the flexibility for the perimeter shape of the multi-mirror concentrator is increased such that the shape of the individual mirrors becomes less important, provided they allow tessellation such that the AAF can be maximized.

### 1. Mirror Shape

Figure 13 compares the normalized mirror profile difference to a parabolic dish, $\mathrm{\Delta}z/{a}_{1}$, of the three concentrators as a contour plot: (a) a square, single-mirror, on-axis concentrator [cf. Fig. 4(a)]; (b) a square, four-mirror concentrator [cf. Fig. 6(a)]; and (c) a square, nine-mirror concentrator [cf. Fig. 6(b)], all with ${C}_{g,\text{design}}=500\times $. As expected, the shape of the single-mirror concentrator, shown in Fig. 13(a), is similar to the hexagonal design shown in Fig. 9, with radially increasing divergence from the parabola shape. However, as the circumradius of the square is larger than that of the hexagon, $\mathrm{\Delta}z/{a}_{1}$ for the same ${C}_{g,\text{design}}$ is larger in the corners. Another variation is that the profile difference is rotationally symmetric, because the mirror and the receiver have the same perimeter shape (${N}_{\mathcal{P}2}={N}_{\mathcal{P}1}$). The dissimilar mirror profiles of the multi-mirror designs, plotted in Figs. 13(b)–13(c), become immediately apparent. By design, the centers of each individual mirror are placed on an on-axis paraboloid of revolution. The divergence from this paraboloid increases radially with respect to each mirror center. However, due to the off-axis positioning of the mirrors, $\mathrm{\Delta}z$ is not rotationally symmetric. Additionally, the maximum divergence becomes considerably smaller with an increasing number of mirrors. Naturally, the mirror slope is not continuous between mirrors, i.e., the concentrator does not have ${C}^{1}$-continuity. In the four-mirror design, where the individual mirrors have line symmetry about the $x$- and $y$-axis, the concentrator has ${C}^{0}$-continuity. However, this is not entirely the case for the nine-mirror design, although differences are almost negligible.

In theory, the number of mirrors can be increased to infinity. In this limit case, each point on the concentrator images to the entire receiver area, and hence the individual infinitesimal mirrors are convex. In between, there has to exist a design where the mirrors change curvature from concave to convex, i.e., the mirrors become approximately flat [46], and which has an interesting potential for applications. The number of mirrors that results in such a design varies with focal length and concentration. This special case bears resemblance to similar designs, where a large number of flat mirrors is distributed over a concave surface with the goal of producing a uniform square irradiance distribution [26,27]. However, these designs have subtle differences, such as a different arrangement and outer shape of the individual mirrors.

### 2. Irradiance Distribution at the Receiver

Figure 14 compares the irradiance distribution on the receiver produced by the four- and nine-mirror concentrators to that produced by the single-mirror design (cf. Fig. 10). At the first glance, it appears as though there is no difference among the distributions produced by the different concentrators. However, if the local irradiance difference $\mathrm{\Delta}E={E}_{\text{single}}-{E}_{\text{multi}}$ is plotted, dissimilarities in the irradiance distribution become apparent. The single-mirror concentrator reflects a larger amount of radiation to the outside and the central region of the image area, whereas the multi-mirror concentrators tend to focus more to the inner edge of the image. The difference between both multi-mirror designs is relatively small.

Despite the seemingly important difference in radiation at the outside of the image, the intercept factor within the image area is only slightly improved from 0.89 with the single-mirror design to 0.90 with the four and nine-mirror designs. Considering that, in addition, a large part of the radiation outside of the image area can be collected by a secondary optic such as the one described in Section 4.B, the intercept factor improvement alone might not justify the added complexity of using multiple mirrors with square designs. However, with other concentrator perimeters, multi-mirror designs based on square mirrors remain of significant interest, even if only taking into account the above results. When contemplating the comparatively poor irradiance uniformity produced by a circular, single-mirror concentrator (cf. Section 4.B.2), especially at low ${C}_{g,\text{design}}$, it is suggested that using a multi-mirror concentrator with a near-circular overall perimeter but composed of square individual mirrors, such as exemplarily shown in Fig. 6(d), might be the better solution to achieve a uniform irradiance with a circular perimeter.

Importantly, an additional advantage of multi-mirror concentrators is their robustness to partial shading of the inlet aperture, e.g., in a field of solar dishes at low solar elevation angles. Figure 15 illustrates the reason for this robustness using the simple designs from Fig. 13. It compares the irradiance distributions produced by the three concentrators if a square area in the bottom right corner corresponding to, respectively, one-ninth (top row) and one-fourth (bottom row) of the inlet aperture is obstructed. Table 3 provides an overview of the corresponding cell-to-cell uniformities on a $6\times 6$ cell array. The conclusions that can be drawn here extend to designs with a higher number of mirrors. With the single-mirror concentrator [Fig. 15(a)], the obstructed portion of the inlet aperture is directly imaged to the same location at the receiver. This results in zero irradiance in the affected location, while the remaining part of the receiver sees full irradiance ($500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$), as indicated by the fractions overlaid on the plot. The result is a very poor irradiance uniformity under shaded conditions ($U=0\%$). With the multi-mirror designs, however, the total irradiance distribution is the sum of the irradiances received by the individual mirrors. The shading therefore only affects the irradiance distributions produced by the individual shaded mirrors, while the unshaded mirrors still illuminate the entire image. In the ideal case, the shaded inlet area exactly corresponds to the surface of one or multiple individual mirrors. This is the case for the four-mirror design when one-fourth of the inlet is shaded [Fig. 15(b), bottom row] and the nine-mirror design with one-ninth shading [Fig. 15(c), top row]. The unshaded mirrors then each distribute the irradiance over the whole receiver, while the shaded mirror is entirely inactive. The resulting irradiance distribution therefore has the same uniformity as in the unshaded case ($U=77.2\%$ resp. 77.6%) with an average intensity proportional to the fraction of the unshaded mirrors ($3/4\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=375\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$ for the four-mirror design with one-fourth shading, and $8/9\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=444\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$ for the nine-mirror design with the one-ninth shading, respectively).

In the general case, when the perimeter of the shaded area does not exactly overlap with perimeters of the individual mirrors, some mirrors are partly shaded. As a result, the corresponding fractional area on the receiver does not receive radiation from that mirror. Accordingly, with the four-mirror concentrator where one-ninth of the inlet is shaded [Fig. 15(b), top row], three mirrors are unshaded and illuminate the full receiver. An area corresponding to four-ninths of the bottom right mirror, however, is shaded. Consequently, five-ninths of the receiver sees full power ($4/4\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$), while four-ninths of the receiver receives only power from three mirrors ($3/4\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=375\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$). It can easily be verified that these numbers add up to a total of eight-ninths of radiation collected by the receiver, corresponding to the unshaded area fraction as expected ($5/9\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}+4/9\times 375\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=444\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=8/9\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$). Due to the fragmentation of the reflector into several mirrors, the uniformity is still acceptable despite the considerable shading ($U=66.6\%$).

Although the situation is slightly more complex, the distribution with the nine-mirror concentrator where one-fourth of the inlet is shaded [Fig. 15(c), bottom row] can be explained in the exact same way: five mirrors remain unshaded, on one mirror the bottom right quarter is shaded, two mirrors are half shaded (bottom half and right half, respectively), and one mirror is entirely shaded. By summing up the individual distributions of the mirrors on the receiver, the specific pattern from Fig. 15(c) emerges with average distributions of $5/9\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=278\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$, $7/9\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=389\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$, and $8/9\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=444\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$ in the different sectors of the receiver. Overall, the receiver collects $1/4\times 278\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}+2/4\times 389\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}+1/4\times 444\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=375\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}=3/4\times 500\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{m}}^{2}$, accurately corresponding to the unshaded area with comparatively good uniformity for the large amount of shading ($U=59.0\%$).

From these findings, it becomes apparent that, as a general rule, with increasing number of mirrors, an increasingly larger fraction of mirrors is either fully shaded or fully unshaded and hence has no negative impact on the uniformity. In theory, the best performance would therefore be achieved by a concentrator with an infinite number of mirrors, i.e., where each point on the concentrator images to the entire receiver area. However, there is a practical limit to the number of mirrors because of the difficulties associated with accurately reproducing the slopes, the penalty from small gaps in-between individual mirrors, and mirror alignment.

## 5. CONCLUSIONS

We have presented a design method for nonimaging mirrors with polygonal inlet apertures that produce a uniform irradiance distribution on a polygonal receiver with a single reflection. The method uses low-distortion, fractional-area-conserving mapping between two polygons, followed by numerical optimization of the 3D mirror shape. Potential CPV applications for the presented method are manifold, ranging from simple on-axis mirror designs, via compound concentrators that leverage tessellation to maximize the active area fraction (AAF), to advanced multi-mirror concentrators with additional advantages.

We evaluated the performance of single-mirror concentrators with square, hexagonal, and circular perimeter and concentrations of $100\times $, $500\times $, and $1000\times $, representative of medium- to high-concentration PV applications. For concentrators with a square inlet aperture, the achieved irradiance distribution on the receiver is highly uniform for all concentrations because no distortions are introduced in the mapping procedure. It is a significant strength of our method that it also allows the design of mirrors with other polygonal apertures. A good performance is noticeably achieved with hexagonal concentrators. Shading of the mirror by the receiver has almost no effect on the intercept factor and uniformity, except for the low-concentration designs. The addition to the design of simple secondary optics, however, can considerably improve the concentrator performance (optical efficiency and irradiance uniformity) at high concentrations by only attenuating the rays in the fringe region at the outer edge of the receiver.

We also assessed the performance of multi-mirror concentrators, where each mirror is individually imaged onto the whole area of the receiver, allowing greater flexibility for the concentrator shape and increased robustness against shading of large parts of the inlet aperture.

## APPENDIX A

For the application of the concentrator design method outlined in Section 2, the object area (polygon or disk) needs to be discretized into a set of nodes before mapping. It is beneficial for the successful optimization of the concentrator that these nodes be regularly spaced within the grid, i.e., that the area of all triangular surface elements spanned by three neighboring nodes have equal (or as close as possible to equal) area.

Figure 16(a) describes the procedure of creating $N$ regularly spaced nodes ${\mathbf{p}}_{\mathcal{P}1,n}$ on a polygon $\mathcal{P}$ with ${N}_{\mathcal{P}1}$ sides, using the example of a hexagon (${N}_{\mathcal{P}1}=6$). In the first region (shaded in gray), a basis $\{\mathbf{u}-\mathbf{v}\}$ with $\mathbf{u}={a}_{\mathcal{P}1}[1;\mathrm{tan}(\pi /{N}_{\mathcal{P}1})]$ and $\mathbf{v}={a}_{\mathcal{P}1}[1;-\mathrm{tan}(\pi /{N}_{\mathcal{P}1})]$ is created. Nodes are then generated such that ${\mathbf{p}}_{\mathcal{P}1}(i,j)=(i\mathbf{u}+j\mathbf{v})/(k-1)$, where $k$ is the desired number of nodes along one basis vector, $i=0,\dots ,k$, and $j=0,\dots ,k-i-1$. The nodes in the other regions $q=2,\dots ,{N}_{\mathcal{P}1}$ are generated simply by rotation of $\{\mathbf{u}-\mathbf{v}\}$ by $(q-1)2\pi /{N}_{\mathcal{P}1}$ around the origin. The total number of nodes generated this way is $N=1+{N}_{\mathcal{P}1}k(k-1)/2$. Delaunay triangulation [47] is subsequently performed on ${\mathbf{p}}_{\mathcal{P}1}(i,j,k)$ to create the triangular surface elements needed for the surface optimization and ray tracing. For ${N}_{\mathcal{P}1}=3$ and ${N}_{\mathcal{P}1}=6$ the elements are equilateral, while for ${N}_{\mathcal{P}1}=4$ and ${N}_{\mathcal{P}1}>6$, they are isosceles.

Figure 16(b) describes the procedure of creating $N$ uniformly spaced nodes on a disk, as required for the design of circular mirrors. ${k}^{\prime}$ is the desired number of nodes evenly spaced along the positive $x$ axis within the disk ($y=0$, $0<x<{R}_{\mathcal{D}}$; nodes marked in red). The radial distance between these nodes then is $\mathrm{\Delta}r=R/{k}^{\prime}$. The remaining nodes are generated by placing ${N}_{i}$ nodes on each circle with radius $i\mathrm{\Delta}r$ (where $i=1,\dots ,{k}^{\prime}-1$) such that their distance in circumferential direction is closest to $\mathrm{\Delta}r$, i.e., ${N}_{i}=\text{round}[2\pi (i\mathrm{\Delta}r)/\mathrm{\Delta}r]=\text{round}(2\pi i)$. The angular circumferential spacing on that circle then yields $\mathrm{\Delta}{\phi}_{i}=2\pi /{N}_{i}=2\pi /\text{round}(2\pi i)$, and the nodes follow from ${\mathbf{p}}_{\mathcal{D}}(i,j)=i\mathrm{\Delta}r[\mathrm{cos}((j-1)\mathrm{\Delta}{\phi}_{i});\mathrm{sin}((j-1)\mathrm{\Delta}{\phi}_{i})]$, where $j=1,\dots ,{n}_{i}$. The total number of nodes in the grid then is

## APPENDIX B

When presented with the task of optimizing a surface defined by a grid with $N$ nodes, the straightforward approach is to simultaneously solve Eq. (7) at each node ${\mathbf{x}}_{1,n}=[{x}_{n};{y}_{n};{z}_{n}]$, with the desired surface normal at that node ${\widehat{\mathbf{n}}}_{n}=[{x}_{\widehat{\mathbf{n}},n};{y}_{\widehat{\mathbf{n}},n};{z}_{\widehat{\mathbf{n}},n}]$, for ${z}_{n}$. However, it can be challenging to numerically determine the tangent vector to the surface in each node $n$ based on its neighbors, especially if the grid is irregular (where the number of neighbors is not constant). In fact, if a node has more than three neighbors, the resulting system of equations is not linear.

These limitations can be mitigated if the optimization is instead performed on the mid-nodes of the edges connecting two nodes. The edges can be determined by Delaunay triangulation of the nodes, which maximizes the minimum angle of each triangular element spanned by three neighboring nodes [47]. In reasonably uniform grids, such as the ones encountered in this work, this is equivalent to minimizing the length of the edges, i.e., connecting two grid nodes by the shortest possible edge. The edges resulting from Delaunay triangulation are indicated in Fig. 16.

Having determined the edges ${e}_{m}$, i.e., the sets ${\{i,j\}}_{m}$ where ${\mathbf{x}}_{1,i}=[{x}_{i};{y}_{i};{z}_{i}]$ and ${\mathbf{x}}_{1,j}=[{x}_{j};{y}_{j};{z}_{j}]$ are connected nodes and $m=1,\dots ,M$, the approximated tangent vectors to the surface are readily obtained as ${\mathbf{t}}_{ij}=({\mathbf{x}}_{1,j}-{\mathbf{x}}_{1,i})$, as illustrated in Fig. 17. The corresponding approximated normal vectors are ${\widehat{\mathbf{n}}}_{ij}=({\widehat{\mathbf{n}}}_{i}+{\widehat{\mathbf{n}}}_{j})/\Vert {\widehat{\mathbf{n}}}_{i}+{\widehat{\mathbf{n}}}_{j}\Vert $. For a large enough nodal resolution, this approximation leads to accurate results.

Each mid-node must then fulfill Eq. (7), i.e.,

## APPENDIX C

Figure 18 shows the influence of the number of nodes $N$ on the fraction of the irradiance $\gamma $ intercepted within the image area of a hexagonal concentrator with ${C}_{g,\text{design}}=500\times $, ${A}_{1}=1$ and $f=1$. It can be seen that a grid with $N=500$ provides reasonably good results for $\gamma $. The same holds true for other performance metrics, such as average irradiance and efficiencies. For producing smooth, high-resolution irradiance maps, however, a larger number of elements ($N\approx 2500$) is required and used throughout this paper.

## APPENDIX D

Before the question of the variation of the concentrator rim angle with ${C}_{g,\text{design}}$ can be addressed, it is necessary to define the rim angle of a polygonal concentrator. Due to the concentrator’s noncircular perimeter, a line connecting the focal point to a point on the concentrator rim forms a different angle with the optical axis depending on the location of this point. The maximum rim angle is attained in the concentrator corners, while the minimum is reached in the middle of its edges.

The minor rim angle ${\mathrm{\varphi}}_{\mathcal{P}1}^{-}$ can therefore be defined as the angle between the optical axis and the mirror edge in the $x\u2013z$ plane ($y=0$) of a polygonal parabolic concentrator and can be determined by solving on a parabolic dish with focal length $f$ and radius ${a}_{\mathcal{P}1}$

For the same inlet aperture and the same focal length, ${a}_{\mathcal{P}1}<{R}_{\mathcal{D}}<{R}_{\mathcal{P}1}$, and hence ${\mathrm{\varphi}}_{\mathcal{P}1}^{-}<{\mathrm{\varphi}}_{\mathcal{D}}<{\mathrm{\varphi}}_{\mathcal{P}1}^{+}$, where ${\mathrm{\varphi}}_{\mathcal{D}}$ is the rim angle of a disk concentrator of radius ${R}_{\mathcal{D}}$. The difference between ${\mathrm{\varphi}}_{\mathcal{P}1}^{+}$ and ${\mathrm{\varphi}}_{\mathcal{P}1}^{-}$ decreases for increasing ${N}_{\mathcal{P}1}$ of the polygon. Ultimately, in the limit case ${N}_{\mathcal{P}1}\to \infty $ (equivalent to a disk concentrator), ${a}_{\mathcal{P}1}$ and ${R}_{\mathcal{P}1}$ converge to ${R}_{\mathcal{D}}$ and ${\mathrm{\varphi}}_{\mathcal{P}1}^{-}={\mathrm{\varphi}}_{\mathcal{D}}={\mathrm{\varphi}}_{\mathcal{P}1}^{+}$. Table 4 summarizes ${\mathrm{\varphi}}_{\mathcal{P}1}^{-}$ and ${\mathrm{\varphi}}_{\mathcal{P}1}^{+}$ for the tessellating concentrators (${N}_{\mathcal{P}1}=\{3,4,6\}$), having ${A}_{1}=1$ and $f=1$.

The fact that ${\mathrm{\varphi}}_{\mathcal{P}1}^{-}$ and ${\mathrm{\varphi}}_{\mathcal{P}1}^{+}$ are determined on a parabolic dish in Eqs. (D1) and (D2), while the actual concentrator profile is slightly defocused, introduces offsets of the actual rim angles ${\mathrm{\varphi}}_{\mathcal{P}1,m}^{-}$ and ${\mathrm{\varphi}}_{\mathcal{P}1,m}^{+}$ measured on the mirror profile, ${\epsilon}^{-}=({\mathrm{\varphi}}_{\mathcal{P}1}^{-}-{\mathrm{\varphi}}_{\mathcal{P}1,m}^{-})/{\mathrm{\varphi}}_{\mathcal{P}1}^{-}$ and ${\epsilon}^{+}=({\mathrm{\varphi}}_{\mathcal{P}1}^{+}-{\mathrm{\varphi}}_{\mathcal{P}1,m}^{+})/{\mathrm{\varphi}}_{\mathcal{P}1}^{+}$. Table 5 shows the actual rim angles of a hexagonal concentrator for ${C}_{g,\text{design}}=100\times $, $500\times $ and $1000\times $. The offset is negligible (below 1%) for reasonably high ${C}_{g,\text{design}}$, such as those encountered in this paper.

## APPENDIX E: NOMENCLATURE

${a}_{2}$ | Scaled apothem of image (receiver) |

${a}_{\mathcal{P}}$ | Polygon apothem |

$A$ | Area |

$\mathbf{A}$ | Matrix ($M\times N$) containing the |

$\mathbf{b}$ | Column vector ($M\times 1$) containing the $x$ and $y$ coordinates of the desired normal vectors and of the nodes for the surface optimization |

${C}_{g}$ | Geometric concentration, × |

${C}_{g,1,\mathrm{max},3\mathrm{D}}$ | Full-collection geometric concentration limit with one-reflection 3D mirrors, × |

${C}_{g,1,\mathrm{max},\text{square}}$ | Full-collection geometric concentration limit with one-reflection 3D mirrors and a square receiver, × |

${C}_{g,2,\mathrm{max},3\mathrm{D}}$ | Full-collection geometric concentration limit of a 3D secondary mirror, × |

${C}_{g,\text{tot},\mathrm{max},3\mathrm{D}}$ | Full-collection geometric concentration limit of two-stage 3D concentrators, × |

${C}_{g,\text{design}}$ | Geometric design concentration of the primary mirror, × |

$\mathcal{D}$ | Disk with $A=1$ |

$e$ | Edge connecting two nodes in a grid |

$E$ | Local irradiance, $\mathrm{W}/{\mathrm{m}}^{2}$ |

${E}_{\mathrm{av}}$ | Cell-averaged irradiance, $\mathrm{W}/{\mathrm{m}}^{2}$ |

$\u27e8E\u27e9$ | Mean irradiance on the image area, $\mathrm{W}/{\mathrm{m}}^{2}$ |

$\u27e8{E}_{\mathrm{av}}\u27e9$ | Mean PV array irradiance, $\mathrm{W}/{\mathrm{m}}^{2}$ |

$f$ | Focal length |

$i$ | Index |

$j$ | Index |

$k$ | Number of nodes along one basis vector for the generation of a gird on a polygon |

${k}^{\prime}$ | Number of nodes along the positive |

$l$ | Mirror length of a V-trough secondary optic |

$m$ | Edge index |

$M$ | Number of edges |

$n$ | Node index |

$\widehat{\mathbf{n}}$ | Surface normal unit vector |

$N$ | Number of nodes |

${N}_{\mathcal{P}}$ | Number of sides of a polygon |

$p$ | Node/point on polygon with $A=1$ |

$\mathcal{P}$ | Polygon with $A=1$ |

$q$ | Index indicating the region within the polygon |

$\dot{Q}$ | Radiative power, W |

${\widehat{\mathbf{r}}}_{i}$ | Inverse incident ray direction, unit vector |

${\widehat{\mathbf{r}}}_{\mathrm{o}}$ | Reflected ray direction, unit vector |

${R}_{\mathcal{D}}$ | Disk radius |

${R}_{\mathcal{P}}$ | Circumradius of polygon |

$\mathcal{S}$ | Square with $A=1$ |

$\widehat{\mathbf{t}}$ | Tangent unit vector |

$\mathbf{u}$ | Basis vector for node generation |

$U$ | Cell-to-cell irradiance uniformity |

$\mathbf{v}$ | Basis vector for node generation |

${w}_{\text{fringe}}$ | Half-width of the fringe region |

$x$ | Spatial coordinate |

$\mathbf{x}$ | Node/point on the mirror/receiver in 3D space |

$y$ | Spatial coordinate |

$z$ | Spatial coordinate |

$\mathbf{z}$ | Column vector ($N\times 1$) containing the unknown $z$-coordinates of the nodes for the surface optimization |

$\alpha $ | Receiver absorptance |

$\gamma $ | Intercept factor |

$\mathrm{\Gamma}$ | Mapping function |

${\delta}_{ij}$ | Kroneker delta |

$\mathrm{\Delta}r$ | Radial distance between nodes along $y=0$, $x>0$ on a disk |

$\mathrm{\Delta}z$ | $={z}_{m}-{z}_{p}$, profile difference between the mirror and an ideal parabola |

$\mathrm{\Delta}{\phi}_{i}$ | Angular circumferential spacing between nodes on a disk, rad |

${\epsilon}^{-}$ | $=({\mathrm{\varphi}}_{\mathcal{P}1}^{-}-{\mathrm{\varphi}}_{\mathcal{P}1,m}^{-})/{\mathrm{\varphi}}_{\mathcal{P}1}^{-}$, offset of the minor rim angle of the mirror from an ideal parabola, ° |

${\epsilon}^{+}$ | $=({\mathrm{\varphi}}_{\mathcal{P}1}^{+}-{\mathrm{\varphi}}_{\mathcal{P}1,m}^{+})/{\mathrm{\varphi}}_{\mathcal{P}1}^{+}$, offset of the minor rim angle of the mirror from an ideal parabola, ° |

${\eta}_{\mathrm{acc}}$ | Acceptance efficiency |

${\eta}_{\mathrm{el}}$ | Electrical receiver efficiency |

${\eta}_{\mathrm{opt}}$ | Optical efficiency |

${\eta}_{\mathrm{tot}}$ | Total system efficiency |

${\theta}_{\text{sun}}$ | Cone-angle of incident solar radiation, rad |

$\rho $ | Mirror reflectance |

$\phi $ | Circumferential polar coordinate, rad |

${\mathrm{\varphi}}_{\mathcal{D}}$ | Rim angle of a disk concentrator, ° |

${\mathrm{\varphi}}_{\mathcal{P}1}^{-}$ | Minor rim angle of a polygonal concentrator, ° |

${\mathrm{\varphi}}_{\mathcal{P}1}^{+}$ | Major rim angle of a polygonal concentrator, ° |

$\psi $ | Vertex angle of a V-trough secondary optic |

## Funding

Kommission für Technologie und Innovation (KTI) (KTI 14048.2 PFIW-IW); Bundesamt für Energie (BFE); Seventh Framework Programme (FP7) (609837, STAGE-STE).

## REFERENCES AND NOTES

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**22. **The geometric concentration limit for full collection with a one-reflection 3D mirror such as a parabolic dish is given by ${C}_{g,1,\mathrm{max},3\mathrm{D}}=\mathrm{sin}\text{\hspace{0.17em}}{(2\mathrm{\varphi})}^{2}/\mathrm{sin}\text{\hspace{0.17em}}{(2{\theta}_{\text{sun}})}^{2}-1$, where $\mathrm{\varphi}$ is the mirror rim angle [23], with a maximum for $\mathrm{\varphi}=45\xb0$ of $11561\times $ with ${\theta}_{\text{sun}}=4.65\text{\hspace{0.17em}}\mathrm{mrad}$. The geometric concentration limit of a two-stage system is ${C}_{g,\text{tot},\mathrm{max},3\mathrm{D}}={C}_{g,1,\mathrm{max},3\mathrm{D}}{C}_{g,2,\mathrm{max},3\mathrm{D}}$ with ${C}_{g,2,\mathrm{max},3\mathrm{D}}=\mathrm{cos}\text{\hspace{0.17em}}{(\mathrm{\varphi})}^{2}/\mathrm{sin}\text{\hspace{0.17em}}{({\theta}_{\text{sun}})}^{2}$ [24]. It is maximized for $\mathrm{\varphi}=14.86\xb0$ at a value of $43191\times $, which is close to the theoretical limit ${C}_{g,\text{ideal},3\mathrm{D}}=1/\mathrm{sin}\text{\hspace{0.17em}}{({\theta}_{\text{sun}})}^{2}-1=46247\times $. While practical designs, especially if designed for high irradiance uniformity, fall short of these theoretical limits by a considerable margin, the comparison of the limits provides a good concept of the fundamental difference in achievable concentration.

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**32. **For a concentrator design with a circular mirror, the first mapping step (${\mathrm{\Gamma}}_{\mathcal{P}1\to \mathcal{D}}$) can simply be omitted. A nodal grid can be generated directly on the disk and ${\mathrm{\Gamma}}_{\mathcal{D}\to \mathcal{P}2}$ directly produces the image on the receiver. The technique for regular grid generation on a disk used in this paper is outlined in Appendix A.

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**36. **Obstruction by the receiver is neglected throughout this section. Otherwise, the fundamental limit for convex single-reflection concentrators with axial symmetry is ${C}_{g,1,\mathrm{max},3\mathrm{D}}={\mathrm{sin}}^{2}(2{\mathrm{\varphi}}_{\mathcal{P}1}^{+})/{\mathrm{sin}}^{2}(2{\theta}_{\text{sun}})-1$ [37,23].

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**38. **When accounting for obstruction by the receiver, the full-collection concentration ratio with a square receiver is ${C}_{g,1,\mathrm{max},\text{square}}={C}_{g,\text{design}}/{(2{w}_{\text{fringe}}{({C}_{g,\text{design}})}^{1/2}+1)}^{2}-1$.

**39. **The shaded fraction of the inlet area is $1/{C}_{g}$.

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**44. **The secondary optic was modeled as ideal (wall reflectance $\rho =100\%$) to provide results independent of mirror quality. The attenuation by the secondary mirror, however, is negligible. The average number of reflections of rays through the secondary optics was determined to be 9% with the method outlined in [45]. For example, if the mirror reflectance was 90%, then only $1-{0.9}^{0.09}=1\%$ of the rays would be absorbed.

**45. **T. Cooper, F. Dähler, G. Ambrosetti, A. Pedretti, and A. Steinfeld, “Performance of compound parabolic concentrators with polygonal apertures,” Sol. Energy **95**, 308–318 (2013). [CrossRef]

**46. **In reality, with the presented method, it is not possible that all mirrors are perfectly flat simultaneously. Mirrors in the concentrator center will always be slightly less concave than mirrors on the concentrator edge if the same area in the focal plane is to be illuminated. This is due to the inherent coma of focusing concave concentrators, i.e., off-axis rays reflected from the concentrator edge intersect the focal plane further away from the optical axis than rays reflected at the concentrator center. When increasing the number of mirrors, there is a first design point where the innermost mirror is flat while all other mirrors are still concave. Conversely, for a slightly higher number of mirrors, if the outermost mirror becomes flat, all remaining mirrors are convex. However, for an intermediate design, all mirrors can reasonably well be approximated as flat.

**47. **B. Delaunay, “Sur la sphère vide,” Bull. l’Académie des Sci. l’URSS **12**, 793–800 (1934).