The optical design presented here has been done in order to achieve superior optical performance in comparison with the state-of-the-art Fresnel CPV systems. The design consists of a Photovoltaic Concentrator (CPV) comprising a Fresnel lens (F) as a Primary Optical Element (POE) and a dielectric solid RXI as a Secondary Optical Element (SOE), both with free-form surfaces (i.e. neither rotational nor linearly symmetric). It is the first time the RXI-type geometry has been applied to a CPV secondary. This concentrator has ultra-high CAP value ready to accommodate more efficient cells eventually to be developed and used commercially in future.
© 2014 Optical Society of America
Present circumstances demand that the CPV concentrators must achieve various goals in order to minimize the energy cost (€/kWh). The high concentration CPV industry is focused on commercializing products based on triple-junction (3J) solar cells, to take advantage of the high efficiency these cells provide through efficient partitioning of the solar spectrum . Even though the 3J cell and assembly cost is currently lower than compared to previous trends, it is still required to have high concentration (>500) and maximum tolerance angle (in CPV it is called acceptance angle) for the systems to be competitive in terrestrial applications. Since these two parameters are reciprocal, it is used a merit function commonly interpreted as the acceptance angle-concentration product (CAP)  in CPV optical design. It is:1]. For a given concentrator architecture, the CAP value is practically constant. The CAP value has the theoretical upper limit derived from the conservation of étendue theorem . Assuming that n is the refractive index of the material surrounding solar cell, it is.
Besides high concentration and wide acceptance angle, the irradiance uniformity and chromatic balance have great importance for the multi-junction (MJ) solar cells. High local differences in flux over the solar cell surface may affect cell efficiency, reliability and durability causing series resistance losses. Previous results  show only slight efficiency decrease even at elevated concentration of thousands of suns with high localized radiation. This behavior is the evidence for the integrity of the tunnel diode in selected cells. In many of the previous findings [4–6] tunnel diodes in MJ concentrator cells did not withstand such high flux levels, so we have to assure they operate in the tunneling region. A fill factor (FF) loss is produced in MJ cells if different wavelengths have different irradiance distribution (known as chromatic aberration ) due to local current mismatch between top and middle cells.
Good irradiance uniformity and chromatic balance on the solar cell is going to be obtained by Köhler integration applied to CPV concentrators. We have used the Simultaneous Multiple Surfaces design method in three dimensions (SMS 3D)  to generate a Köhler integrator array comprising of free-form optical surfaces. It combines the above two functions (concentration and homogenization) using three active optical surfaces, producing both high Cg and high α (i.e. high CAP). Optical free-form surfaces can be manufactured with classical techniques: embossing, compression molding, etc. used for the POE, plastic injection and glass molding (widely known in automotive industry) for the SOE. The production cost is essentially the same as for non-free-form elements; with their superior optical performance.
2. Fresnel-RXI Köhler (FRXI) concentrator design
Concentration and integration in a 4-fold Köhler FRXI is performed with a four-unit Köhler array: Fresnel lens as a POE and RXI as a SOE are divided in four symmetric pairs, with each quarter of the POE corresponding to a quarter of the SOE (shown in Fig. 1).The RXI is a dielectric solid with a small metalized surface on the front (optional) and larger metalized surfaces on the back side. In the case of the design presented here, the front mirror is not illuminated by Fresnel lens so there are no shadowing losses. RXI device was first designed with rotational symmetry [9, 10] and later with free-form surfaces . The descriptor “RXI” is assigning letters to each surface that deflects rays in the order in which they are encountered by a useful light ray passing through the system.
Two optical surfaces to be calculated by the SMS 3D method are the front surface that has two roles: refraction (R) and total internal reflection (I), and back mirror coated reflective surface (X). SMS 3D creates free-form pairs of surfaces by a direct numerical calculation method [8, 9]. We set the initial conditions, which are the size of the receiver (i.e. solar cell), the entry aperture of the concentrator and the refractive index of the dielectric material of which RXI is made. We assume to have a rectangular solar cell located at z = 0.
Two pairs of input and output wavefronts (WF1i/WF1o and WF2i/WF2o) are to be coupled in order to calculate two free-form surfaces where one of them passes through one prescribed curve and may be consistent with normal vectors of this curve. This curve may be obtained by the SMS 2D calculation using, for instance, WF1i/WF1o, and a third pair of wavefronts, WF3i/WF3o (see Fig. 2). In our design, normal vectors at the points of the SMS curve are close to the normal vectors of the SMS surface, so we are achieving a partially coupled third pair of wavefronts. As the starting point of the SMS 2D we use a point P placed on the front RI surface whose normal (nP) is chosen. Let WF1o be the spherical output wavefront originated at one corner (-x0, y0, 0) and WF3o its counterpart on the other corner of the cell (-x0, -y0, 0).The rays of the input plane wavefronts WF1i and WF3i make an angle -α/2 and α/2 in the yz-plane, to the direction connecting the center of the cell and center of the POE Fresnel lens unit, respectively. The optical path length (OPL) is constant. We are doing the SMS method sequence and calculating the “seed” curve as the profile of two-mirror (XX) system.
For defining the behavior of the device in perpendicular direction, we use two pairs of normal congruences (WF1i/WF1o and WF2i/WF2o) in a direction perpendicular to the “seed” curve. The WF2o is defined as the spherical wavefront placed at (x0, y0, 0), and the WF2i is a plane wavefront whose rays make an angle –α/2 in the xz-plane to the direction connecting the center of the receiver with the center of the Fresnel lens unit meanwhile the rays of the WF1i make an angle α/2 to the same direction (see Fig. 3). During the SMS 3D calculation, the RXI is considered as a three surface optical device. Input ray bundles are first refracted on the front surface (R), than reflected on the back surface (X) and reflected once more on the front surface (I), without considering the actual surface position (i.e. if rays intercept previously other surface, they are not deflected by it). The initial front refractive surface (R0) is chosen as a surface obtained by extruding the “seed” curve in the xz-direction. Normal congruences WF1i and WF2i are refracted on the R0 surface, perfectly coupled with normal congruences WF1o and WF2o, respectively, and new surfaces X and I are recalculated while maintaining the OPL constant. The calculated I surface is considered as a new R surface (R1) and the described process is repeated N times until the sequence of surfaces RN converges towards the final design. Calculated SMS points may be organized in chains perpendicular to the “seed” curve. Once the surfaces are calculated, the resulting chains are interpolated by means of Non-Uniform Rational B-Spline (NURBS) surfaces that can be analyzed through ray tracing.
After designing one Köhler array unit, our final design is done by making 4 mirror images to make up a 360° solid. Performance of the FRXI is analyzed by ray tracing (see Fig. 1).
3. Simulation results for Köhler RXI concentrators
Two different models for RXI SOEs are calculated: RXI with small front area metalized and RXI without front metallization. We are going to analyze three different 3D Köhler concentrators (all with the RXI as the SOE) labeled as: FRXI_m (POE: 4-fold Fresnel lens, SOE: RXI with frontal metallization), FRXI (POE: 4-fold Fresnel lens, SOE: RXI without metallization), XRXI (POE: 4-fold mirror, SOE: RXI with frontal metallization). These concentrators have an f-number of f/1.4 (where f-number is the ratio of the distance between the cell and Fresnel lens to the Fresnel lens diagonal, i.e. a purely geometrical definition, without the usual optical interpretation). The SOE area is below 4% of the Fresnel lens. SOE size is reduced for smaller f-numbers. RXI without frontal metallization is done in order to make the design cheaper and easier to manufacture.
We consider PMMA (n≈1.49) Fresnel lens with facet draft angle = 2°, SOEs made of B270 glass (n≈1.525) with enhanced second surface silver mirror  where the improvements at shorter wavelengths (top subcell range) are made by the dielectric enhanced layer between the silver and the glass. We use high efficiency (≈38%) commercial 3J cell, whose External Quantum Efficiency (EQE) values are provided by the manufacturer and considered to be independent of the incidence angle on the cell. Fresnel reflections and absorption in materials are considered, no AR coating on refractive surfaces, roughness and dispersion are neglected.
3.1. Concentration-acceptance angle product (CAP)
Table 1 shows the acceptance angles and CAP values of our designs. A merit function called effective CAP (CAP*) is defined by substituting in Eq. (1) angle α by the angle α* that is the minimum sun’s tilting angle from the on-axis position at which the cell photocurrent reduces to 90% of its on-axis value. The value of CAP* can be experimentally measured .
Changing the 4-fold primary Fresnel lens (FRXI_m) for a 4-fold primary mirror of the same f-number the XRXI design is obtained. The angular acceptance increased from ± 1.02° to ± 1.24° (see Table 1). This design has strong theoretical importance with the CAP value as one of two highest ever obtained in a CPV, and among Köhler designs being the highest .
3.2. Optical efficiency
The optical efficiency is defined as a power on the solar cell surface over the power of a perfectly-tracked sun beam. Both monochromatic (555nm) and polychromatic optical efficiencies are shown in Table 2.Polychromatic optical efficiency is calculated in a 350nm-1800nm range of wavelengths and weighted by the AM1.5d ASTM G173 spectrum. Table 2 contains the effective optical efficiency calculated by weighting the polychromatic optical efficiency by the EQEs of a “standard” 3J solar cell receiver .
Optical efficiency (Table 2) for the XRXI includes the PMMA cover transmission of 0.91. The AR coating on the SOE would provide the optical efficiency increase of 2% absolute.
3.3. Irradiance and intensity on the cell
Due to Köhler integration, good irradiance uniformity and chromatic balance on the solar cell is obtained. Figure 4 shows the irradiance distributions for top, middle and bottom junctions of the 3J solar cell for the FRXI_m concentrator.
These are calculated by separating the solar spectrum wavelength range for each junction (deduced from the EQEs) and dividing each subrange by 5-10 discrete wavelengths. Each wavelength will have its own weight obtained from the irradiance density of the AM1.5d ASTM G173 spectrum so that the integral value for each subrange equals unity.
The FRXI Köhler optics has theoretically proven to have the highest concentration capability among all flat Fresnel based systems. It is suitable for self-supporting systems. FRXI concentrators compares well with other conventional concentrators with flat Fresnel lens as a primary [13–15]. The ratio of system height to POE diagonal (f-number) is listed together with the Cg of different configurations (Table 3). This table corrects for some mistakes found in Fig. 8 of Ref . Both the Fresnel lens and the solar cell are square.
Although flat Fresnel lenses have smaller concentration capability than mirrors or curved Fresnel lenses, the use of an advanced secondary as the RXI can compensate for it. FRXI concentrators will use solar cell with the smallest area, so material and cost savings will come from reduced cell area and reduced SOE material costs (Fig. 5(Left)). It is necessary to highlight the superior CAP value in comparison with other concentrators (Fig. 5 (Right)).
Two different HCPV concentrators designed with the SMS 3D method and built-in two-directional Köhler integration are presented in this paper. By superimposing the Köhler integration on the SMS optical surfaces we obtained optical designs capable of performing different functions at the same time, so no additional element is necessary. The FRXI CAP value outperforms the conventional Fresnel-based HCPV concentrators. Good irradiance uniformity and chromatic balance with a high tolerance angle at high concentration values is obtained and that leads to the advanced features of these systems.
The devices presented in this paper are the subject of a US and international patent application. Authors thank the European Commission (SMETHODS: FP7-ICT-2009-7 Grant Agreement No. 288526, NGCPV: FP7-ENERGY.2011.1.1 Grant Agreement No. 283798), the Spanish Ministries (ENGINEERING METAMATERIALS: CSD2008-00066, SEM: TSI-020302-2010-65 SUPERRESOLUCION: TEC2011-24019, SIGMAMODULOS: IPT-2011-1441-920000, PMEL: IPT-2011-1212-920000), and UPM (Q090935C59) for the support given to the research activity of the UPM-Optics Engineering Group, making the present work possible. M. Buljan would like to thank the Spanish Ministry for providing her an FPU grant.
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