## Abstract

In order to achieve competitive system costs in mass-production, it is essential that CPV concentrators incorporate sufficient manufacturing tolerances. This paper presents an advanced concentrator optic comprising a Fresnel lens and a refractive secondary element, both with broken rotational symmetry, an optic producing both the desired light concentration with high tolerance (high acceptance angle) as well as an excellent light homogenization by Köhler integration. This concentrator compares well with conventional Fresnel-based CPV concentrators.

©2010 Optical Society of America

## 1. Introduction

Minimizing energy cost (€/kWh) is necessary for the success of concentrated photovoltaic energy (CPV). Key to minimizing this cost is an efficient and low cost optical design, goals best met with the fewest elements and the maximum tolerances, but always maintaining the high concentration (>500) that offsets the cost of expensive high-efficiency multi-junction solar cells. A useful merit function for a CPV optic is the concentration-acceptance product [1], which we define as:

Here*C*is the geometric concentration, defined as the ratio of concentrator aperture area to solar cell area,

_{g}*α*is the acceptance angle, defined as the incidence angle at which the concentrator collects 90% of the on-axis power (the optical losses are considered in this computation). The

*CAP*value is limited by thermodynamics to an absolute upper bound [1], given by the value of the refractive index of the material surrounding the solar cell (typically, about 1.4-1.5).

It is remarkable that for a given concentrator architecture, the *CAP* is rather constant with *C _{g}*. This means that if we know the acceptance angle at a given concentration, Eq. (16)H(1) can be used to estimate the acceptance angle that we would obtain if we designed that concentrator architecture for a different concentration.

For a given *C _{g}*, the acceptance angle

*α*measures the total tolerance available to apportion among the different elements of the system: (1) shape errors and roughness of the optical surfaces, (2) concentrator module assembly, (3) array installation, (4) tracker structure finite stiffness, (5) sun-tracking accuracy and (6) solar angular diameter. Each of these items can be expressed as a fraction of the tolerance angle, so that all together comprise the tolerance budget. Alternatively, for a given acceptance angle, a higher

*CAP*allows a higher concentration, consequently reducing cell size and cost.

Besides concentration and tolerance angle, an optical PV concentrator must achieve other goals. One of them is irradiance uniformity on the cell. This is important because the cell efficiency depends on it, but also to assure the long term cell and concentrator reliability. High flux contrast across the cell surface causes series resistance losses, although this is less dramatic in multi-junction solar cells than in silicon cells. Another limitation specific to multi-junction cells, however, comes from tunnel diodes having to be kept operating in their tunneling regime, which limits the irradiance at any point of the cell [2]. Additionally, if different wavelengths have a different irradiance distribution (which has been referred to as chromatic aberration [3] of the irradiance), the cell efficiency can be significantly affected, due to local current mismatch between the top and middle junctions.

There are two candidates for attaining good irradiance uniformity on the solar cell: a kaleidoscope homogenizer and the Köhler integrator. With a kaleidoscope homogenizer, the solar cell is glued to one end and the light enters the other, most of the light reaching the cell after bouncing off the kaleidoscope walls. The light distribution on the cell can be made uniform given sufficient length. Kaleidoscope homogenizers for CPV, however, have manufacturing drawbacks (see the discussion in section 20H5), which encouraged us to further investigate Köhler integration, a well known method in illumination optics [4], usually, comprising multiple elements with separate functions: some that collect and concentrate the radiation (condensers) and others that produce the Köhler integration (fly-eye lens arrays).

In this work we present the application of the LPI’s design methods [5] [6] [7] to generate a Köhler integrator array comprising free-form optical surfaces (*i.e*. neither rotationally nor linearly symmetric). It combines the above two functions (concentration and homogenization) using only two active optical surfaces, producing both high concentration and high acceptance angle (*i.e.*, high *CAP*).

## 2. LPI’s Fresnel Kohler (FK) concentrator design

We have developed a Köhler-based CPV optical device with a flat Fresnel lens as the Primary Optical Element (POE), and a single refractive surface as the Secondary Optical Element (SOE). The concentration and Köhler integration functions are achieved by the lower surface of the Fresnel lens (which has the facets, as per usual in solar applications [8]) together with the upper surface of the SOE. We will perform the integration with a four-unit array: POE and SOE are divided in four symmetric parts, with each quarter of the POE corresponding to a quarter of the SOE.

From a manufacturing point of view, Fresnel-Kohler optical surfaces are very similar to a conventional flat Fresnel lens with a glass dome over the cell. This means that they can be manufactured with the same techniques (continuous roll embossing, hot embossing, compression molding, *etc.* for the POE; glass molding for the SOE). Thus their production cost is essentially the same as these well known methods.

Fig. 1 (left) shows a 3D view of the four-unit Fresnel Köhler (FK) concentrator. The Fresnel lens focuses on-axis rays (in maroon) close to the SOE surface. Then the SOE distributes all these rays over the cell surface in a homogeneous way. In addition, rays exiting a certain point of the primary (in green) are approximately imaged on the cell.

Our design procedure is based on the edge-ray theorem of Nonimaging Optics [9], according to which an ideal design must map the edge rays of the input ray bundle onto those of the output ray bundle. In order to illustrate such a mapping and show how the solar homogenization is produced, a simplified 2D version is shown in Fig. 1 (right) and its corresponding phase-space representations in Fig. 2(a) & (b)
(not to scale, in the sense that if drawn to scale, (a) would be a very narrow horizontal stripe and (b) a very narrow vertical stripe). In the phase space at a reference line *z* = constant, a ray crossing that line at coordinate point x is represented as a phase-space point (*x*,*p*), where *p* is the direction cosine with respect to the *x* axis multiplied by the local refractive index [9]. The concentrator transforms the ray bundles from the phase space *x*-*p* at the entry aperture (Fig. 2(a)) to the phase space *x*’-*p*’ at the cell (Fig. 2(b)). The inactive vertical facets of the flat Fresnel lens would introduce gaps in the phase space representation at the cell (a phenomenon usually called “dilution”), but the Figure omits them for the sake of clarity.

For comparison purposes, the edge-ray mapping produced by a conventional imaging-type concentrator is shown in Fig. 2(c) & (d)). At the entry aperture of both systems (Fig. 2(a) and (c)), the sun is represented by a yellow bar (which extends for all the x values of the entry aperture with a vertical width Δ*p*≈2sin*α _{s}* where

*α*is the solar radius,

_{s}*α*≈0.265°). That yellow bar in slightly decentered vertically, which means that the sun is slightly off the on-axis direction. For both concentrators the spatial uniformity across the entry aperture is very good, while the angular uniformity over said aperture is poor within the acceptance angle. The mapping of the imaging concentrator produces a 90° phase-space rotation, so the angular non-uniformity at the entry aperture is reproduced as a spatial non-uniformity of the cell irradiance (in Fig. 2(d) the yellow bar extends along the entire angular range but occupies a small range of

_{s}*x’*values on the cell). In contrast, Köhler integration produces an approximately 180° phase-space rotation, so that spatial uniformity at the entry aperture generates spatial uniformity of cell illumination (in Fig. 2(b) the yellow bar extends along the entire

*x’*range values on the cell). Note that the red rays of the left side of Fig. 1 (right) correspond to the red phase-space sectors in Fig. 2(a) & (b), and the same for the blue rays and sectors of the same two Figures.

Since we only have two surfaces available to design (the SOE surface and the lower surface of the Fresnel lens), there are insufficient degrees of freedom to achieve the ideal mapping of edge rays that is shown in Fig. 2(a) & (b), especially when considering the actual problem is three-dimensional. As a consequence, the design relaxes the edge ray mapping requirement for maximum *CAP*. Additionally, for manufacturability purposes the POE can be composed of four symmetric sectors of a rotationally symmetric Fresnel lens with its symmetry axes parallel to the z axis (see Fig. 1(a)). Fig. 3
shows a close-up rendering of the four-way center of the Fresnel lens on the left and an actual glass-molded SOE on the right.

## 3. Simulation results for the FK concentrator

The various FK concentrator designs and their ray-tracing simulations have the following features: (i) Fresnel lens: made of PMMA (*n*≈1.49), facet draft angle = 2°, vertex radius = 3 μm, facet height<250 μm; (ii) SOE: made of BK7 glass (*n*≈1.51) coupled to the cell with a transparent silicone rubber of n≈1.41 (*e.g.*, Sylgard 182 of Dow Corning); (iii) High efficiency (≈38%) commercial triple-junction cell, with the values of the external quantum efficiency provided by the manufacturer and considered to be independent of the incidence angle on the cell. Absorption in dielectric materials and Fresnel reflections are considered, but surface scattering is neglected. The f-number of the FK concentrator will be varied, and it is defined to be the ratio of the distance between the cell and Fresnel lens to the Fresnel lens diagonal (therefore, it is a purely geometrical value, without the usual optical meaning).

#### 3.1 Optical efficiency

Concentrator optical efficiency can be defined as the ratio of power on the cell to the power on the entry aperture when the sun is exactly on-axis. For an FK concentrator with *C _{g}* = 625x and f/1, calculated at a single wavelength (555 nm), the optical efficiency is 85%, without an AR coating on the SOE. In order to estimate the performance limit when there is an AR coating, the optical efficiency for a perfect AR coating (

*i.e.*, zero Fresnel reflection by the SOE) is 89% for f/1 at 555 nm.

Alternatively, an effective polychromatic optical efficiency can be defined as:

*C*, the AM1.5d ASMT G173 spectrum, and the EQE of the sub-cells). The ratio of the currents

_{g}*I*/

_{sc,top}*I*in that FK concentrator under that standard spectrum is nearly the same (1% variation) as that ratio without the concentrator losses. These calculations also show that the difference between the two optical efficiencies (monochromatic @ 555 nm and polychromatic) is negligible (<0.1%) for both the uncoated and perfect AR coated case of that FK concentrator. Therefore, we conclude that for the FK concentrator the monochromatic calculation satisfactorily estimates the optical efficiency.

_{sc}Fig. 4(b) shows the optical efficiency versus the f-number for different FK concentrator designs, all for concentration 625x. Efficiency is slightly less for low f-numbers (at f/0.8 the predicted effective optical efficiency is still 84% without an AR coating on the SOE, and 88% with one), and the monochromatic versus polychromatic cases lead to very similar values, for all f-numbers analyzed.

#### 3.2 Concentration-acceptance angle product (CAP)

In order to compute the *CAP* of a given concentrator design, parallel-ray inputs at various incidence angles will determine the acceptance angle *α* defined in Section 1. An example is an FK concentrator with *C _{g}* = 625x and f/1. Figure 5(a)
shows the monochromatic (@ 555 nm) power impinging the solar cell as a function of the incidence angle of the parallel rays on the entry aperture, relative to on-axis incidence. Two cases are shown: no AR coating on the SOE, and a perfect one on it. The acceptance angle is defined as the 90% values, which means that this concentrator has

*α*= ± 1.43° without an AR coating on the SOE, and

*α*= ± 1.52° with a perfect one. The increased acceptance angle is because the Fresnel losses on the surface of the SOE of the FK concentrator are higher when the sun is off-axis than when it is on-axis. Note that since the system does not have rotational symmetry, the curves in Fig. 5(a) depend on the azimuth at which the rays are traced. In the Figure they are parallel to two sides of the POE, which is the worst case for that FK concentrator (

*i.e.*, any other azimuth leads to a greater or equal acceptance angle).

Figure 6
shows on the left the concentration angle product (*CAP*) of FK concentrator designs, as a function of Fresnel lens f-number, calculated for monochromatic illumination (at 555 nm). All designs have been calculated for a geometrical concentration *C _{g}* = 625x. The

*CAP*is approximately constant in most of the f-number values (

*CAP*≈0.63 with no AR coating on the SOE,

*CAP*≈0.66 with perfect AR coating), dropping slightly for f/0.80. It is remarkable that, as mentioned before, the use of AR coating on the SOE can increase not only the optical efficiency (Fig. 4.(b)) but also the

*CAP*.

Figure 6 shows also a further merit function (blue lines) that we call effective *CAP* (or *CAP**), which is defined by substituting in Eq. (49)H(1) for the acceptance angle *α* by a tracking-error angle *α** (which we can call effective acceptance angle) at which the cell photocurrent reduces to 90% of on-axis (see Fig. 5(b)). Therefore, *CAP** accounts for the finite size of the sun and the comparison of the products of the solar spectrum and the EQE of the three subcell photocurrents to find which one is limiting. The utility of the *CAP** is that it can be experimentally measured. In Fig. 6 we see that the *CAP** of the FK concentrator shows good independence from the f-number (*CAP**≈0.57 with no AR coating; *CAP**≈0.61 with perfect AR coating on the SOE). The good behavior of the FK concentrator for small f-numbers is very interesting when compact CPV modules are desired.

Although Fig. 6 has been calculated for *C _{g}* = 625x, it is remarkable that for a given f/number, the

*CAP*and

*CAP**of the FK concentrator design remain approximately constant when the concentration is changed. For instance, for f/1 the FK concentrator with no AR coating on the SOE has

*C*= 625x and

_{g}*α*= ± 1.43° (which corresponds to the values

*CAP*≈0.63 mentioned before). If the FK concentrator is designed for

*C*= 1000x, the resulting acceptance angle can be well estimated to be

_{g}*α*= ± 1.43°(625/1000)

^{1/2}= ± 1.13°. Table 1 summarizes other examples of such calculations. (This approximation is not accurate for

*α**< ± 0.7° because the sun radius and POE chromatic aberration do not scale, but those values are out of the range shown in Table 1).

#### 3.3 Irradiance and intensity on the cell

As mentioned in the Introduction, the spatial non-uniformity of cell irradiance reduces the I-V curve fill factor, due to higher series resistance losses, and possibly also due to the current density being limited by a tunnel junction. This limitation may not occur for specific cells [1], but it is not guaranteed by the manufacturer to never happen. Moreover, chromatic non-uniformity of cell irradiance can produce an additional fill factor loss [3] due to a local current mismatch between the top and middle junctions. Again, quantification of the efficiency losses due to these non-uniformities depends on the specific cell design and manufacturer.

The FK concentrator produces excellent irradiance uniformity on the cell, with negligible chromatic aberration of that irradiance. Figure 7
shows two irradiance distributions of an FK concentrator of Cg = 625x, f/1, no AR coating of SOE, when the sun is on-axis with Direct Normal Irradiance (DNI) of 850W/m^{2}. The distribution on the left is the irradiance when the solar spectrum is restricted to the top subcell spectrum (360-690 nm), while the distribution on the right corresponds to the middle subcell spectral range (690-900 nm).

This result may seem surprising, since it is known that a conventional Fresnel lens has considerable chromatic aberration (which certainly does affect the *CAP**, as already discussed). In the FK concentrator, however, the chromatic aberration of the Fresnel lens barely affects the irradiance, thanks to its Kohler integration basis: White light that is split by the Fresnel lens into a range of different focal points will be refocused at the same point of the cell, after refraction by the SOE.

Another interesting feature is the angular intensity distribution of the light reaching the cell. Figure 8 shows the cumulative distribution of light power on the cell with off-normal angle. When the sun is perfectly tracked, the cell is entirely illuminated within a cone of ± 30° or less from normal, and within a cone of ± 40° for a tracking error of 1°. Since solar cell reflectivity is usually higher at large incidence angles (>50-60°) than at normal incidence, the intensity profile of the FK concentrators guarantees optimum light coupling into the cell.

## 4. Comparison with other Fresnel-based designs

Comparing the FK concentrator with other five more conventional CPV concentrators designs that use flat Fresnel lens as a primary, as shown schematically in Fig. 9
. All the concentrators will have the same POE entry aperture area (625 cm^{2}), and concentration ratio will be varied (and thus cell size will vary accordingly). Both the Fresnel lens and the solar cell are square.

From left to right in Fig. 9, first is a Fresnel lens concentrator with no secondary, which is a type of system being used for instance by Concentrix Solar [10]. This concentrator, if designed for maximum *CAP* at acceptance angles significantly higher that the angular radius of the solar disc, would produce a highly non-uniform illumination (it performs approximately as an imaging concentrator; see Fig. 2(c)&(d)). Therefore, this concentrator must sacrifice *CAP* to reduce the peak concentration to moderate enough values.

Second is a hemispherical glass dome centered on the cell surface and in optical contact with it. The purpose of the hemispherical centered dome is to increase the *CAP*, since it produces an optical linear magnification about 1.5 (the refractive index of the SOE), *i.e.*, the cell is optically made 1.5 times bigger. Higher magnifications are possible by decentering the spherical dome, but then the optical efficiency will be lower due to higher Fresnel reflections. In any case, the compromise between *CAP* and peak concentration also apply for these domed designs. This type of dome concentrator has been considered as a candidate for improving the *CAP* of Concentrix design [11].

Third, we selected for the comparison the SILO secondary [12], which was proposed by Sandia Labs in the late 80’s and later commercialized by Alpha Solarco Inc [13]. It comprises a conventional rotational Fresnel lens as the primary and a single-surface rotationally symmetric lens as a secondary which images the primary onto the cell. Thus it resembles the Kohler integration idea, but it is made of a single unit (instead of four). The fourth SOE of Fig. 8 is a hollow reflective truncated pyramid (XTP), which is the type of SOE being used by Amonix [15] and Guascor Foton [16]. And finally, the fifth is the dielectric-filled truncated pyramid (RTP), which works by total internal reflection. This type of SOE is used in several commercial products [17].

With the exception of SILO, we have selected concentrators that are being used in current commercial products, or that have been considered as next-generation designs. We have included the SILO in spite of it not being commercialized already, because it is conceptually related to our FK concentrator.

The secondaries and cells shown in Fig. 9 are not to scale, but the concentrator ratio of system height to POE diagonal (f-number) are, with Table 2
listing their numerical values. The true-scale relative size comparison among the different SOE’s can be seen in the cross-sectional drawings of Fig. 10
. All these concentrators have the same POE entry aperture area (625 cm^{2}) and the same acceptance angle (*α* = ± 1°).

In the next sections, ray-trace simulations of these concentrators will be presented. We have performed them for all but the SILO, for which the data has been calculated from the information in [12]. All the concentrators have been given the following parameters: (i) Fresnel lens: area = 625 cm^{2}, made of PMMA (*n*≈1.49), facet draft angle = 2°, vertex radius = 3 μm, facet height<250 μm; (ii) Dielectric SOE: made of BK7 glass (*n*≈1.51) [14] with edge radii 300 microns; (iii) Reflective SOE: made of first-surface silver coating on an aluminum sheet; (iv) conformal coating on the solar cell of transparent silicone rubber of n≈1.41, which protects the cell and adheres it to the dielectric SOE (when this applies); (v) High-efficiency (≈38%) commercial triple-junction cell, with the external quantum efficiencies provided by the manufacturer and considered to be independent of the incidence angle on the cell. Absorption in dielectric materials and Fresnel reflections are considered, but not surface scattering.

#### 4.1 Optical efficiency

The optical efficiency (both monochromatic and polychromatic) of the types of concentrator under comparison is very similar, except when the f-number is very low. This occurs in the RTP, which has approximately a 4% lower absolute efficiency than the others. This calculation neglects the fact that when no dielectric SOE is used (*i.e.*, when no SOE is used and the XTP) the conformal coating may reduce the apparent reflectance of the cell grid-lines, due to light trapping by total internal reflection on the conformal coating-air interface. The amount of light trapping depends on the grid-lines’ surface properties and geometry, and its quantification therefore depends on the specific grid line technology.

There are still some aspects not accounted for by the ray traces, which may affect optical efficiencies in more refined models. First, total internal reflection (TIR) is sensitive to surface scattering, dirt and surface corrosion of the glass of the RTP secondary (with otherwise 100% efficiency). Second, perfect application of the silicone coupler is assumed, but imperfections are difficult to avoid in the case of the RTP secondary in low-cost high-volume production (a further discussion is given in Section 73H5).

It is also noticeable that the average ray path length inside the RTP secondary is about twice as long as in the tolerance-equivalent FK concentrator SOE. This reduces the material candidates to be used for the RTP to a few highly transparent glasses (as BK7), while a wider variety of materials is available for the SOE of the FK concentrator.

#### 4.2 Concentration-acceptance angle product

Figure 11
shows the effective concentration acceptance angle product *CAP** for the selected concentrators versus the f-number in the case that no AR coating is applied to the SOE’s. We can see that the FK concentrator is superior to all of them, being the RTP the second concentrator in the ranking.

Since we have stipulated that the entry aperture of the Fresnel lens is the same for all concentrators (625 cm^{2}), we will say that two concentrators are tolerance-equivalent if they have the same effective acceptance angle *α**. Table 3
shows that the FK’s superior *CAP** makes it able to concentrate significantly more than the any other configuration with the same tolerance. This was previously illustrated when representing the cells size in Fig. 10, which corresponds to the first row in Table 3.

#### 4.3 Cost

Table 3 lists the concentration for effective acceptance angle *α** of ± 1° of the different configurations of Fig. 9. Since the different configurations have the same aperture area and acceptance angle, it is safe to assume that cost savings will come only from reduced cell area and reduced SOE costs. The key question for the FK concentrator is to evaluate whether the reduced cell cost allowed by its higher concentration capability is balanced by the cost of its secondary. This question also arises for the other concentrators, so it is interesting to compare all of them in terms of cost per POE unit area (which is representative if neglecting their differences in optical efficiencies; see Section 4.1). For the FK concentrator and SILO, the f-numbers were fixed to 1.0 and 1.2, respectively, for this cost calculation.

For the cost model we have assumed the same Fresnel lens price for all the designs (their area is the same, 625 cm^{2}, and all can be made with the same technology). We have also considered that the encapsulation cost is common to all designs, and that the rest of the module (heat spreader, by-pass diode, electrical connection, heat sink, housing, *etc*.) is also common to all concentrators. Since all of them have the same acceptance angle we have considered that the cost of module assembly, array installation, structure and tracker design are the same for all the cases. Therefore, the module cost will differ only in the cost of the triple-junction cell plus the cost of the SOE. We have not used real quotations from suppliers for such costs, but rather estimations of high-volume market prices in volume.

For the triple-junction cell, all the concentrators use the same cell technology and the cell cost is proportional to its active area, with areal cost $7/cm^{2}. This proportionality probably is not true for small enough cells (area below 30-50 mm^{2}), because the non-active area of the cell becomes relatively larger and because the higher cost to test and bin the individual cells. Since, however, we are considering in this exercise a POE of 625 cm^{2}, cells smaller than 50 mm^{2} imply geometrical concentrations above 1250x, which is outside the range considered here.

Among the different SOEs there is a clear distinction between reflective and refractive elements. The cost of the XTP reflexive SOE has been considered proportional to its reflective area, with an areal cost of $0.004/cm^{2}. For the dielectric SOEs, the cost assumes no AR coating, and the estimated values are shown in Fig. 12
as a function the SOE height *h*. Dome-type SOEs refers to FK, SILO and spherical-dome configurations (for this cost estimation, the outer part of the spherical dome, which is not active, is assumed to be truncated to make its volume comparable to that of the FK and SILO).

Figure 13
shows the addition of the SOE and solar cell costs per unit area of the Fresnel lens as a function of the effective acceptance angle *α**. The comparison between different concentrators should be made at the same effective acceptance angle *α** (*i.e.*, they are tolerance-equivalent).

83HFor instance, if we specify *α** = ± 1°, the FK concentrator has an SOE + cell cost around $110/m^{2}, while for the RTP, the XTP, and the SILO the costs are $150/m^{2}, $200/m^{2}, and $340/m^{2}, respectively. Note that this is because the cell in the FK is smaller (thanks to its higher *CAP**). It is remarkable that if no SOE is used the cell cost is $670/m^{2} (outside the graph!) for *α** = ± 1°, because to achieve such an *α** the C_{g} must be as low as 104 (for this reason systems with no SOE usually are designed for *α** values smaller than ± 1° [10], which increase the costs of assembly, installation, *etc*.).

Figure 14
shows the SOE and active area of the cells in the two most cost-effective cases with *α** = ± 1°: the FK and RTP concentrators. Both concentrators have the same POE area (625 cm^{2}) and the cost advantage of the FK is mainly due to the smaller cell active area required (shown as blue squares): 92 mm^{2} in the RTP versus 59 mm^{2} in the FK concentrator.

The previous cost calculations considered that the refractive surfaces of the dielectric SOEs are not AR coated. Since both optical efficiencies (Section 3.1 and 4.1) and *CAP** (Sections 3.2 and 4.2) are improved in the FK, it is expected that the AR coating application will be cost effective, but this calculation will not be considered here. Note that since not only the *CAP** but efficiency is improved, its cost effectiveness cannot be computed just per unit area of the POE, but the total system cost ($/W) needs to be known as well.

#### 4.4 Irradiance

When comparing the irradiance distribution of the different selected concentrators, we can group the FK concentrator, RTP, and SILO as devices producing good homogenization, and the XTP, spherical dome, and the case without SOE as devices producing poor homogenization, especially the last two when their design is done to maximize *CAP**. This is the case illustrated for the spherical dome in Fig. 15(a)
, which is the irradiance distribution obtained for the *C _{g}* = 625x centered spherical dome designed for maximum

*CAP**(that gives 0.266) referred to in Fig. 11. The peak irradiance is ≈6,000 suns (@DNI = 850 W/m

^{2}), which seems rather high but might be acceptable [2]. It will be probably be somewhat lower in practice, however, (this ray-trace model does not include scattering and optical manufacturing errors), but is instructive to compare it with the FK concentrator of the same

*C*= 625x (Fig. 15(b)), with an irradiance plateau of ≈460 suns (@DNI = 850 W/m

_{g}^{2}). Note that it is also quite remarkable that their effective acceptance angles are very different:

*α** = ± 0.61° for the spherical dome and

*α** = ± 1.30° for the FK concentrator.

The irradiance homogeneity in XTP, spherical dome, and the case without SOE can be improved by an adequate Fresnel lens design, but that improvement is done at the expense of reducing the *CAP**.

Regarding spectral inhomogeneity of the irradiance, the same group is weak: XTP, spherical dome, and the case without SOE (especially for low f-numbers). Even though they can be designed to minimize that aberration at the nominal POE to cell distance, the balance is rapidly lost when that distance is modified (as occurs if the Fresnel lens parquet shows warping or bowing). On the other hand, the FK concentrator, RTP, and SILO are not that sensitive to the variation of that distance.

To complete this comparison, another irradiance related issue occurs inside the SOE. For the cases of the RTP and SILO, the Fresnel lens produces a region of high solar radiation density near the top of the SOE (see schematic ray-trace of Fig. 9). In the early 90’s Alpha Solarco reported problems of degradation of the SILO glass due to UV absorption by undesired particle traces (solarization) [13]. By now glass technology has improved, and this risk may be already solved, but even so the 4-fold FK concentrator divides that risk by four (since the high radiation is split in the four units of the SOE).

## 5. Practical considerations for mass-production

There are several features of the SOE of the FK concentrator that make it advantageous with respect to other dielectric SOE’s that use flow-line type [9] total internal reflectors, as for instance the RTP.

First, all the surfaces of the RTP are optically active (see Fig. 16 ). This makes it complex to hold it without introducing optical losses and compromising the mechanical stability (some CPV manufacturers hold the RTP from the joint to the cell, which compromises the requirements for the RTP-cell coupling material). In contrast, a large portion of the SOE surface has no optical function, and consequently can be used to make that mechanical fixing and to introduce alignment features.

Another problematic issue is that the optical coupling of the cell and the SOE is very critical for the RTP, because lateral spillage of the silicone rubber causes significant optical losses from leakage through it. If to avoid spillage the joint is under-filled, the resulting air gap produces optical losses too. These losses cannot be quantified until full production is achieved. This particularly discourages the use of RTP for small cells (<5mm side). In the FK concentrator, however, the overflow does not affect the optical performance, which greatly simplifies the joint coupling. Particularly, the optical coupling and mechanical fixturing functions can be totally separated in the FK concentrator.

## 6. Conclusions

The FK concentrator is a reliable and robust high-performance device that allows mass production thanks to its high tolerance and practical manufacturability aspects. The FK optical surfaces, from the manufacturing point of view, are very similar to a conventional flat Fresnel lens and a conventional dome. This means that they can be manufactured with the same techniques (continuous roll embossing, hot embossing, compression molding, *etc*. for the POE; glass molding for the SOE) and that their production cost is essentially the same, but their optical performance (*CAP**) is much better.

The FK concentration acceptance angle products, *CAP* and *CAP**, are the highest among the concentrators based on flat Fresnel lenses considered here, and are also superior to other designs reported in [19]. However, two designs using mirrors as POE have been reported with a higher *CAP* than the FK [18]. The irradiance uniformity obtained by the FK concentrator is excellent, without the chromatic aberration so typical of other Fresnel concentrators. This minimizes the losses associated with such inhomogeneities, while diminishing the reliability risks associated with elevated local concentrations. It is also remarkable that the FK concentrator maintains its high *CAP** for rather small f-numbers, which is very interesting if compactness of the CPV module is desired.

The FK concentrator is in our opinion an excellent candidate to make low-cost high-concentration CPV modules that combine very high electrical efficiency at the array level (and not only at the single cell level) with cost-effective assembly and installation.

The free-form Köhler array solutions are not limited to the present FK concentrator. Even for the flat Fresnel lens as a primary, multiple options for the secondary are envisaged for future improvements. The true innovation in these new designs is that they are free-form Köhler integrating arrays. This degree of freedom enables the design of optical surfaces that can perform different functions at the same time (improving the device performance without affecting its cost). This allows good irradiance uniformity and high tolerance angle at high concentration values.

## Acknowledgments

The LPI’s FK device presented in this paper is protected under US and International patents pending by LPI, LLC, 2400 Lincoln Avenue, Altadena, CA 91001 USA http://www.lpi-llc.com/. The authors thank the Spanish Ministry MCI (project SIGMASOLES PSE-440000-2009-8) and the Madrid Regional Agency IMADE (project PIE/469/2009) for their partial support. The authors also thank Bill Parkyn for his help in editing the paper.

## References and links

**1. **P. Benitez, and J. C. Miñano, “Concentrator Optics for the next generation photovoltaics”. Chap. 13 of A. Marti & A. Luque. *Next Generation Photovoltaics: High Efficiency through Full Spectrum Utilization*, (Taylor & Francis, CRC Press, London, 2004).

**2. **A. Braun, B. Hirsch, E. A. Katz, J. M. Gordon, W. Guter, and A. W. Bett, “Localized radiation effects on tunnel diode transitions in multi-junction concentrator solar cells,” Sol. Energy Mater. Sol. Cells **93**(9), 1692–1695 (2009). [CrossRef]

**3. **S. Kurtz, and M. J. O’Neill, “Estimating and controlling chromatic aberration losses for two-junction, two-terminal devices in refractive concentrator systems”, 25th PVSC; pp.361–367, (1996).

**4. **W. Cassarly, “Nonimaging Optics: Concentration and Illumination”, in the *Handbook of Optics*, 2nd ed., pp. 2.23–2.42, (McGraw-Hill, New York, 2001)

**5. **P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. **43**(7), 1489–1502 (2004). [CrossRef]

**6. **J. C. Miñano, M. Hernandez, P. Benítez, J. Blen, O. Dross, R. Mohedano, and A. Santamaría, “Free-form integrator array optics”, in *Nonimaging Optics and Efficient Illumination Systems II*, SPIE Proc., R. Winston & T.J. Koshel ed. Vol. 5942–12, (2005).

**7. **US and International patents pending by LPI, LLC, 2400 Lincoln Avenue, Altadena, CA 91001 USA http://www.lpi-llc.com/.

**8. **R. Leutz, and A. Suzuki, *Nonimaging Fresnel Lenses*, (Springer-Verlag, Berlin, 2001).

**9. **R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, “Nonimaging Optics”, (Elsevier-Academic Press, New York, 2005).

**10. **http://www.concentrix-solar.de/fileadmin/user_upload/Download/Technical_Data_Sheets_Q3-2009.pdf.

**11. **G. Peharz, J. Jaus, P. Nitz, T. Schmidt, T. Schult, and A. W. Bett, “Development of refractive secondary optics for flatcon^{®} modules”, 23rd European Photovoltaic Solar Energy Conference, 1DV.3.34, (2008). Note that in this reference Cg is defined using a circular active area instead of square, so the geometrical concentration 385x in it corresponds to 302x here.

**12. **L. W. James, Contractor Report SAND89–7029, (1989).

**13. **D. Anderson, B. Bailor, D. Carroll, E. Schmidt, P. Tyjewski, M. Uroshevich, “Alpha Solarco’s Photovoltaic Development Concentrator Program”, Contractor report SAND95–1557, (1995).

**14. **The same BK7 glass has been considered for all the SOE’s under comparison. Though BK7 can be molded (see for instance, http://www.rpoptics.com/index.php?page=rpo-moldable-glass-data) is more common the use of, for instance, B270. The light absorption in B270 is slightly higher than in BK7, which causes that if the comparison is done using B270, the efficient of the RTP (whose optical path is longer) is penalized the most.

**15. **http://www.amonix.com/technology/index.html.

**16. **http://www.guascorfoton.com/home_en.php.

**17. **See, for instance: www.sol3g.com, http://www.solfocus.com/, and K. Araki et al., “Development of a new 550X concentrator module with 3J cells-Performance and. Reliability-”, Proc. 31st IEEE PVSC, (2005).

**18. **P. Zamora, A. Cvetkovic, M. Buljan, M. Hernández, P. Benítez, J.C. Miñano, O. Dross, R. Alvarez, A. Santamaría, “Advanced PV Concentrators”, 34th IEEE PVSC, (2009).

**19. **M. Victoria, C. Domínguez, I. Antón, G. Sala, “Comparative analysis of different secondary optical elements for aspheric primary lenses,” Opt. Express **17**, 6487–6492 (2009). The design with highest *CAP** in this reference has a rotationally-symmetric CPC-type TIR-based secondary. It achieved *CAP** = 0.54 for a square aperture and square cell (for which the FK has a higher value of *CAP** = 0.57–0.61). Moreover, it produces a poor irradiance uniformity. It also has the encapsulation problems discussed in Section 5.