Since lenses exhibit chromatic aberration unlike mirrors, their applicability in solar concentration is limited [1]. Considering a spectrum of 380–1600 nm—a typical range for CPV since out of this range solar flux is low and the external quantum efficiency of a triple junction cell is about 40% [2–5]—the longitudinal chromatic aberration (LCA) of simple Fresnel lenses limits the concentration to about $1000\times$ under normal incidence [6] while the theoretical limit on earth due to the acceptance angle of the solar semi-diameter (${959}^{\prime \prime }$ [7]) is about $46000\times$ [8]. When combining a diverging lens in polycarbonate (PC) and a converging lens of poly(methyl methacrylate) (PMMA), higher concentration ratios can be achieved. Recently, a concentration ratio close to $2000\times$ has been experimentally measured with a flat achromatic Fresnel doublet [9]. However, these lenses are either subject to soiling problems (if the outward surface is textured), or are hard to manufacture (if the interface between the optical materials is textured). Moreover, flat lenses are intolerant to tracking error. The tracking error angle, ${\theta }_{\text{err}}$, causes a displacement of the centroid of the collected flux following $x_C=f\tan {\theta }_{\text{err}}$ (with $f$ the paraxial focal distance), affecting particularly slow systems (i.e., with high $f$-numbers) since the maximum concentration ratio due to the angular size of the source is given by

As for previous publications [6,9,10], PMMA and PC will be used to design the achromatic lenses at 468 and 961 nm: the two wavelengths of achromatization, i.e., the wavelengths focusing exactly on the receiver and minimizing the LCA. Relevant parameters of the achromatic Fresnel prism are sketched in Fig. 1. The receiver is centered at the origin $O(0,0)$ and is perpendicular to the $y$ axis. $\beta$ is the inclination angle of the PC/PMMA-interface and $\alpha$ and $\gamma$ are the prism angle of the PMMA and PC, respectively. Finally, $\delta$ is the deviation angle and ${\theta }_S$ the angular radius of the source.

 

Fig. 1. Sketch of an achromatic prism made with PMMA (purple) and PC (cyan).

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Let’s fix some parameters, say $\delta =30°$ and ${\theta }_S=2.5°$. There is an infinite number of values possible to make one wavelength reach the center of the receiver. For a selected value of the inclination angle of the interface, $\beta$, we require that both 468 and 961 nm impact the receiver, which fixes the values of $\alpha$ and $\gamma$. More precisely, we impose a nonimaging condition in our algorithm: the edges rays are forced to symmetrically impact the center of the receiver, i.e., $|x^{-}+x^{+}|<\epsilon$, with $\epsilon$ close to zero. The minimum size of the receiver is therefore given by

In order to obtain a value that depends only on the deviation angle, we will use the normalized value

As can be deduced from Fig. 2(b), $\mathrm{\Delta }{x}^{\prime }$ decreases when $\beta$ tends to $-45°$. At this value, ${\theta }_1^{+}$ (the angle between the normal of the PMMA and the positive incoming edge-ray) gets close to the limit of 90° [see Fig. 2(a)] and the Fresnel coefficient of reflection gets high. In Fig. 2(c), the evolution of $\mathrm{\Delta }{x}^{\prime }$ with the interface angle is also depicted for a deviation angle of 2°.

 

Fig. 2. Effects of the inclination angle on the incoming angle (a) and on the distance between the edge rays at the receiver level for a deviation angle of 30° (b) and 2° (c).

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Two conclusions can be drawn: (1) curved achromatic segments can highly reduce the size of the receiver for high deviation angles but only a slight improvement is obtained for small angles of deviation, (2) a fully concentration-optimized lens is impossible to design since the incoming angle at the first interface must be 90° along the full lens (and in addition to a high reflection coefficient, this lens would have been intolerant to tracking errors). Finally, these conclusions are simply the same as for prisms made with a single material.

Even if curved lenses cannot improve the concentration factor, it can increase the flux uniformity and the tracking tolerance [1]. In 2003, Leutz and Ries explained that nonimaging (dome-) shaped lenses cannot be designed with commonly available materials [11]. This assertion appears to be valid if the outward surface is perfectly smooth. The idea of a textured surface facing the Sun (grooves-out) is usually immediately rejected in order to avoid soiling problems.

This problem of soiling exists in the case of dome-shaped lenses: two different dome-shaped lenses are shown in Fig. 3 with the PMMA component either above or below the interface. In both cases, the dome-shaped achromatic lenses were optimized in order to have the edge-rays impacting symmetrically the plane of the absorber.

 

Fig. 3. Dome-shaped lenses, i.e., with the interface, in blue, situated on a perfect circle. On the left, the top material is the PMMA (in purple) and on the right, the top material is the PC (in cyan).

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In the first case, when the PMMA is above the interface, the distance between $x^{+}$ and $x^{-}$ decreases with the deviation angle (which corresponds to the previous analysis), meaning that the flux reaching the receiver will be highly nonuniform. Moreover, this design suffers from an important shadowing effect at the air/PMMA interface.

Concerning the second design, with the PC above the interface, the distance between $x^{+}$ and $x^{-}$ increases with the deviation angle, reducing drastically the concentration ratio and negatively affecting the uniformity of the flux reaching the receiver. The shadowing effect is however much lower than that of the first achromatic dome-shaped Fresnel lens.

To overcome the problem of uniformity, we have to move to free-shape lenses. We will force the slope of the interface to be such that $\mathrm{\Delta }x$ remains constant whatever the radial distance of the incoming edge-rays. As previously deduced from Fig. 2, the inclination angle of the left design of Fig. 3 must be increased in order to increase the distance between $x^{+}$ and $x^{-}$, meaning that the radius of curvature of the interface must increase too. To the contrary, regarding the right design of Fig. 3, the inclination angle must decrease. Therefore the radius of curvature of the interface must also be decreased.

The optimized designs are shown in Figs. 4(a) and 4(b), both have been optimized for an angular radius of the source of 5° and has been achromatized for 468 and 961 nm. On the one hand, the first design [Fig. 4(a), with the PMMA on top] is prone to soiling and suffers from an important shadowing effect and therefore it will be no longer discussed. On the other hand, the second design [Fig. 4(b)] does not suffer from any shadowing effect and the soiling problem is avoided in spite of the texturization of the outer surface.

 

Fig. 4. Shaped Fresnel lenses optimized and illuminated with an angular diameter of 5°. Left and center: achromatic. Right: singlet.

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The performance of this latest design will be compared to a nonimaging shaped Fresnel singlet [see Fig. 4(c)], only made with PMMA [1]. The comparison has been performed with a ray-tracing program with a simulated source whose characteristics match those of Sun. At the design wavelength(s), the (achromatic) shaped lens has a detector diameter given by $\mathrm{\Delta }{x}_0=2f\tan {\theta }_S$. We will investigate the energy enclosed in this radius as a fraction of the incoming energy (in other words the optical efficiency ${\eta }_{\text{opt}}$), to which corresponds an optical concentration ratio

Table 1 presents relevant results for different design angles $({\theta }_{\text{dsn}})$ and different illumination angles $({\theta }_S)$. Fresnel reflexion losses have been simulated, but the roughness has been neglected.

Table 1. Nonimaging Shaped Lenses: Comparison Between Singlets and Achromatic Doublets

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As can be expected, achromatic-shaped lenses concentrate much more energy in $\mathrm{\Delta }{x}_0$ than singlets, and the optical efficiency is always higher with doublets despite the supplementary interface. The advantage of the doublets is particularly pronounced for small angles of incidence where the chromatic aberration highly enlarges the focal spot produced by the singlet. In this condition, the flux reaching the receiver is almost perfectly uniform with the doublet while the distribution is Gaussian with the singlet, as shown on Fig. 5.

In conclusion, nonimaging achromatic shaped Fresnel doublets allow for very high concentration, up to $\sim 8500\times$ for an angle of incidence of 0.26°, with a flux spectrally uniform (no chromatic aberration) and spatially uniform too (flat shape irradiance). To achieve such performance, a texturization of the outward surface was necessary while avoiding dust accumulation thanks to a profile without local minimum.

 

Fig. 5. Comparison of the concentration distribution for the singlet (left) and the doublet (right) with ${\theta }_{\text{dsn}}={\theta }_S=0.26°$. The red and purple circles have the same diameter of $\mathrm{\Delta }{x}_0=2f\tan {\theta }_S$. They contain, respectively, 17.1% and 84.3% of the incoming irradiance (both squares receive about 88%), corresponding to the last line of Table 1.

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