## Abstract

It is shown how novel solutions for realistic gradient-index lenses create the possibility of nominally stationary solar photovoltaic concentrators capable of daylong averaged flux concentration levels of order 10^{3}.

© 2011 OSA

## 1. Introduction

Can daylong solar concentration levels of the order of 10^{3} (or higher) be realized in nominally stationary systems? Based on the fundamental relation between maximum flux concentration *C _{max}* and acceptance half-angle θ

_{acc}at a given concentrator exit numerical aperture

*NA*[1,2]

_{exit}the answer appears to be no. Indeed, solar concentrators have realized daily-averaged flux concentration of the order of 10^{3} (at θ_{acc} ≈1°) only with precision dual-axis tracking driven from massive pedestals [3–5]. (Commonly, an individual tracker supports dozens of m^{2} of collector, weighing hundreds of kg.) Here, θ_{acc} is the *effective* solar half-angle comprising the sun's intrinsic value of 4.7 mrad convolved with alignment inaccuracies, imperfections in the shapes of optical contours, and deviations of material properties from their design values.

However, Eq. (1) does not prohibit high solar concentration with completely stationary modules and absorbers, *provided* the *optic* tracks the sun [2]. In earlier solar thermal and photovoltaic concentrators, the very nature of the absorbers, or the extensive range of motion to be accommodated, precluded practical consideration of this notion. The evolution of concentrator photovoltaic (CPV) technology to mm-scale solar cells obviates these drawbacks and prompts revisiting the prospect of nominally stationary high-concentration optics, a collateral benefit of which would be rooftop CPV. Furthermore, there are practical micro-mechanical systems capable of the solar tracking with sub-mrad accuracy (small enough not to affect θ_{acc}), over cm dimensions, inside the module [6]. To wit, commercial high-precision linear actuators offer a tracking accuracy better than 0.01 mm, to be implemented here with lenses of cm-scale focal lengths (*vide infra*).

The burden is then transferred to devising a miniature maximum-performance optic for daylong solar beam collection. The inadequacy of conventional lenses and mirrors has long been recognized [1,2]. Even nonimaging designs tailored to nominal collector stationarity succeeded in achieving a daily-averaged flux concentration of only tens of suns at high collection efficiency [7] (1 sun = 1 mW/mm^{2}) – one to two orders of magnitude below the values required for CPV.

The best optical properties for this aim would appear to be perfect imaging - an instance where imaging and nonimaging objectives coalesce because perfect imaging is non-trivially synonymous with attaining the fundamental limit to concentration (Eq. (1)) [1,8]. However, perfect imaging cannot be realized with a finite number of optical elements [9]. While an optic comprising many *reflectors* is impractical, a spherical gradient-index (GRIN) lens can offer a workable solution (because its refractive index distribution is a nominal continuum).

In this work, we identify realizable GRIN lenses for nominally stationary solar concentrators (Fig. 1
) that are amenable to realistic materials and fabrication technologies. First, the solutions for perfect-imaging spherical GRIN lenses are reviewed, including the limiting aspects of the solutions that resulted in their being deemed physically unattainable, mathematical idealizations for *sunlight*. Second, GRIN profiles that preserve perfect imaging but eliminate the need for refractive indices near unity, and markedly reduce the range of refractive indices required, are identified. Third, the influence of lens focal length on the efficiency-concentration tradeoff intrinsic to the constraint of a fully static absorber is analyzed. And fourth, the sensitivities of this optical strategy to dispersion and misalignment are evaluated, and supplemented with a few qualifications for the proposed approach.

## 2. Spherical GRIN lenses

#### 2.1 Classical GRIN solutions

Spherical GRIN lenses offer the possibility of simultaneously satisfying the fundamental limits of *both* imaging *and* nonimaging optics [1,8]. Maxwell initiated the field of GRIN optics in trying to understand the fish eye, and Luneburg was the first to derive the refractive index profile for a far-field source [10] (Figs. 2
-3
).

Although successfully implemented in microwave antennas, Luneburg’s solution remained an esoteric ideal for visible and IR radiation because of its severe constraints:

- (a) transparent materials with refractive indices decreasing to exactly unity,
- (b) a broad range of refractive index,
*Δn*= 0.414, and - (c) the focus resides on the sphere’s exterior.

The last point was resolved in [11] which generalized the solutions to arbitrary focal length *F* (the distance from the focus to the center of the sphere, expressed relative to the sphere’s unit radius), albeit with strictly numerical (rather than analytic) solutions (Figs. 2-3). Nevertheless, the first two limitations remained, and the profiles were restricted to continuous functions.

#### 2.2 Generalized GRIN solutions amenable to realistic materials for solar concentration

Morgan [12] introduced the extra degree of freedom of an outer shell of constant index *n _{constant}*, with a continuum interior profile. Permitting this type of discontinuity surmounts the first two constraints noted above (see Figs. 2-3). While the existence of such solutions had been recognized mathematically, they had not been explored for solar concentration.

Consider the far-field solution with a single discontinuity: a continuum core distribution up to radius *a* and a uniform outer shell. The governing integral equation (Eq. (2)) is [12]:

where *κ* (skewness [1,8]) is constant for a particular ray along its entire trajectory, and *r** is the smallest radius along that trajectory. The solution for the continuum profile (0 ≤ *r* ≤ *a*) is *n* = (1/*a*) exp(ω(ρ,*F*) – Ω(ρ)), where

(0 ≤ ρ ≤ 1, *F* ≥ 1). The integrals in Eq. (3) must be computed numerically. Solutions exist provided Eq. (4) is satisfied [12]:

Figure 4
summarizes how *n _{constant}* affects that layer’s allowed thickness and

*F*.

We find that solutions with constant-index shells can (a) significantly raise the minimum refractive index to well above unity, e.g., to values above 1.2, and, simultaneously, (b) reduce *Δn* from more than 0.4 (for the Luneburg lens) to less than 0.2. Suitable off-the-shelf solar-transparent materials – commonly available plastics and glasses that are also apposite for GRIN lens fabrication processes [13,14] - typically have refractive indices from ~1.3 to ~2, which can be accommodated by the generalized solutions illustrated above.

Lower minimum *n* values create greater design flexibility. The lowest value reported to date is ~1.05 for laboratory SiO_{2} thin films [15]. Question: With fully continuum solutions (which require *n*(1) = 1), can collection losses be limited to only a few percent by omitting the outermost layers and building the sphere from *n* = 1.05 inward? Raytracing (with lenses tracking solar motion over 0-60° for a static absorber) reveals that this compromise results in >60% ray rejection when designing for only 10% of *C _{max}* (by oversizing the absorber).

## 3. Selecting lens focal length

The focal spot projected onto the cell will vary from a minimal disc at normal incidence to an elliptical area that increases with solar incidence angle (Fig. 5
), and engenders a tradeoff between collection efficiency and concentration. For given *F* and θ_{acc}, these results are independent of the particular *n*(*r*) because they all yield the same imaging properties. (The target flux maps are non-uniform, but the inhomogeneities are of minor consequence for current concentrator cells, as elaborated in Section 5.)

Equation (1) can be recast here as *C _{max}* = (

*F*sin(θ

_{acc}))

^{−2}. The dependence of collection efficiency on

*F*is less obvious. At short

*F*, a sizable fraction of collectible radiation strikes the underside of the absorber and hence is unutilizable. Completely avoiding this loss requires

*F*≥ √3 (see Fig. 6 ). (

*F*values of at least ~1.74 are also mandated in order to avoid the lens trajectory not intersecting the static plane of the absorber).

Daylong collection was evaluated by averaging over incidence angles from 0 to 60°, with the largest value - corresponding to ~8 hr/day of collection – chosen based on considerations of excessive mutual shading inside the module. (We also energy-weighted the solar input at each incidence angle and averaged over the year, based on typical clear-day mid-latitude solar beam radiation [2], and found negligible changes relative to taking the simple time-weighted average.) Lens design and performance evaluation are based on monochromatic radiation at mid-spectrum. Representative dispersion losses (somewhat case-specific based on the materials chosen for lens fabrication) are quantified below in Section 4A, Fig. 8 .

Characteristic curves of collection efficiency vs geometric concentration *C* (relative to *C _{max}*) were generated by raytrace simulation for a range of

*F*. (Collection efficiency here omits Fresnel reflections and absorption, for the lens and module cover glazing, which are readily quantified, and depend on whether anti-reflective coatings are applied.) θ

_{acc}= 5 mrad was adopted, based on its being achievable in miniaturized solar concentrators [16]. (θ

_{acc}= 7 mrad has been realized in large-scale CPV systems with massive dual-axis trackers [3,5].) The plots in Fig. 7 were also found to be insensitive to θ

_{acc}as large as 10 mrad (provided the abscissa remains

*relative*concentration

*C*/

*C*), which effectively incorporates non-negligible random errors in the thickness and exact refractive index of the spherical shells in the GRIN lens.

_{max}To gauge realistic design conditions and to evaluate CPV optical performance, we first note that today’s concentrator cells exhibit efficiencies that peak at irradiance values not exceeding ~10^{3} suns [4,17–21]. At θ_{acc} = 5 mrad and *F* = 1.74, *C _{max}* ≈13,000, so

*C*= 1,300 corresponds to

*C*/

*C*= 0.1. The geometric collection efficiency is then 98% (Fig. 7). In addition, achieving substantial optical tolerance to off-axis orientation (e.g., a tolerance half-angle θ

_{max}_{t}≥ 1°) invariably mandates designing for θ

_{acc}well below θ

_{t}, which in turn means

*C/C*values substantially below unity [22].

_{max}Concentration can be increased to ~4,000 (conceivably germane for future ultra-efficient ultra-small solar cells) at only 5% ray rejection. Even in the extreme case of θ_{acc} = 10 mrad (so that *C* = 1,300 corresponds to *C*/*C _{max}* ≈0.4), the geometric losses are only 7%. The results summarized in Fig. 7 allow the estimation of collection efficiency for essentially all concentration values of interest, and sharpen the high-collection potential of spherical GRIN lenses even in the nominally stationary strategy portrayed here.

As an example of the key dimensions becoming viable when commercial miniature cells are used, consider cell and lens diameters of 1 and 36 mm, respectively (*C* = 1,300) in a square 2.5 m^{2} module ~65-70 mm deep (glazing, heat sink and internal micro-tracker included). The internal lens tracking requires a clearance of ~1.5 lens radii: a dead space of ~27 mm on each edge of the module’s periphery, corresponding to ~3% of the module’s gross area (in addition to the packing loss of ~11% for spheres in a plane). The lenses (polymeric layers of density ~1 g/cm^{3} [13,14]), would then comprise a mass of ~15 kg per m^{2} of module aperture (to which an extruded aluminum micro-tracker assembly would add ~5 kg).

## 4. Secondary losses

#### 4.1 Dispersion

Chromatic aberration (dispersion loss) was evaluated using an AM1.5D solar spectrum and a Cauchy-type dispersion relation based on the measured properties of representative materials [15] - plotted in Fig. 8. Not unexpectedly, dispersion losses increase with focal length and with concentration, but are far lower for GRIN lenses than for conventional homogeneous lenses. For example, at *F* = 1.74 and *C* = 1,300 the dispersion loss is only 1%. Unlike conventional lenses where chromatic aberration amplifies an intrinsically aberrated optic, the spherical GRIN lens starts aberration-free (geometrically) so that dispersion imposes a near-negligible loss (unless concentration approaching *C _{max}* is required).

#### 4.2 Sensitivity to misalignment

Figure 9
quantifies sensitivity to absorber misalignment (or equivalently, a systematic error in the internal tracking motion of the lenses). Efficiency-concentration curves were generated for a given translation of the absorber from its intended position in a *F* = 1.74 system. For perspective, consider C ≈1,300 (*C*/*C _{max}* ≈0.1) with a solar cell 1 mm in diameter (so the lens diameter is ~36 mm). The focal spot is noticeably smaller than the cell for most of the daily collection period. As a consequence, even sizable displacements result in a ray rejection of only ~1-2%. Given the tolerance and robustness of current high-efficiency concentrator solar cells to markedly inhomogeneous flux maps (see Section 5), this result augurs favorably for exceptional tolerance to optical errors and sufficing with tractable internal tracking elements.

## 5. Caveats

Nominal stationarity comes at the price of a yearly-averaged cosine of the incidence angle less than unity, to wit, ~0.7 for a clear mid-latitude location [2] – an unavoidable compromise unrelated to the optics. Close-packed lenses would incur ~30% shading within the module, or one could markedly reduce shading by spacing them and accepting that ~30% of the intercepted radiation misses the lenses as in Fig. 1b-c (or intermediate arrangements).

In addition, high concentration forgoes acceptance of diffuse radiation [1,2]. Stationary non-concentrating solar collectors benefit from diffuse collection, whereas the concentrators envisioned here collect essentially none (common to all CPV).

The absorber flux maps are non-uniform - most pronounced at normal incidence when the projected focal spot is smallest (Fig. 5). In principle, flux inhomogeneity augments solar cell series resistance losses. However, current commercial concentrator cells have exhibited efficiency decreases not exceeding measurement uncertainties at flux localization as acute as a few hundred percent [19–21].

For square (as opposed to circular) cells the simplest approach is inscribing the design focal spot within the square and reducing averaged concentration by a factor of π/4 – especially given the relative ease of attaining high irradiance such that avoiding a further loss in collection efficiency is paramount. The slight worsening in flux uniformity does not noticeably diminish the efficiency of the cells, and realizing concentration values of order 10^{3} would not be compromised. When flux uniformity is critical, kaleidoscopes [19] and Köhler integrators [23] can be added. Moreover, passive heat sinks can maintain cell temperatures at no more than ~20-30 K above ambient temperature, at irradiance levels of thousands of suns, even with marked flux non-uniformities [19–21,24].

Cost projections for the eventual mass production of the specific optical and internal tracking components portrayed here are precarious, but at least coarse estimates are in order. The processed polymeric materials typically cost no more than a few US$ per kg – hence below US$100 per m^{2} of aperture for modules of the type described at the end of Section 3. Automated fabrication costs of the order of US$0.1 per lens when billions are required (for GW-level power generation) are not unreasonable, amounting to roughly US$100 per m^{2} of module aperture. Working experience with the types of precision delta robotic systems that would comprise the internal micro-tracker indicates that large-volume production at ~US$100 per m^{2} of module aperture is not unfeasible.

## 6. Conclusions

A basic shift in solar concentrator strategy has been portrayed: transferring the burden of accurate solar tracking *from* conventional massive units on which a multiplicity of solar modules are mounted, *to* miniaturized mechanical components inside modules that are completely stationary. Whereas the dimensions of earlier solar thermal and photovoltaic systems precluded this approach, the evolution of CPV technology to mm-scale cells and hence cm-scale optics creates new possibilities. This is a necessary, but not a sufficient, condition: an optic capable of performing near the thermodynamic limit for concentration and acceptance angle must also be identified, must be amenable to existing materials and affordable mass fabrication, and must be operable with feasible micro-mechanical systems.

Thanks to (1) recent advances in GRIN fabrication techniques, (2) the evolution of new classes of solutions to the classic problem of perfect imaging, (3) the miniaturization of high efficiency solar cells and (4) the availability of adequate micro-mechanical drivers, spherical GRIN lenses offer an unprecedented solution to this problem for CPV – even accounting for chromatic aberration and misalignment.

The stationary high-concentration modules described here incur an unavoidable loss in the averaged incidence angle cosine, as does any stationary aperture. Eliminating massive precision tracking of large arrays in favor of precision cm-scale lens tracking inside the modules may render the tradeoff worth considering, and opens the possibility of rooftop CPV.

The high performance potential of the optical strategy portrayed here applies equally well to 2D systems, i.e., line-focus cylindrical GRIN lenses, albeit with attainable concentration being roughly the square root of the 3D values. The efficiency-concentration characteristics are even slightly better because the dilution of absorber power density is less pronounced in 2D. With current and projected CPV applications focusing on concentration levels from hundreds to the order of 10^{3} suns, detailed analyses were restricted here to 3D systems.

The perception that high concentration is inseparably linked to massive trackers is supplanted here by a different paradigm. Furthermore, if the unorthodox strategy of stationary modules is relaxed, and conventional CPV tracking with constant normal incidence is considered, then spherical GRIN lenses represent a solution that is ostensibly superior to other optical systems by virtue of basically attaining the fundamental limit to concentration and acceptance angle.

Finally, although this study focused on photovoltaic concentrators, it can also be extended to areas that require high-performance wide-angle imaging, e.g., photography and infrared imaging – applications where collection efficiency is typically less critical than for solar, and near-maximum flux concentration is especially valuable for high signal-to-noise ratios.

## References and links

**1. **R. Winston, P. Benítez, and J. C. Miñano, with contributions from N. Shatz and J. Bortz, *Nonimaging Optics* (Elsevier, Oxford, 2005).

**2. **A. Rabl, *Active Solar Collectors and Their Applications* (Oxford U. Press, NY, 1985).

**3. **S. Horne, SolFocus Inc., 510 Logue Ave., Mountain View, CA 94043, technical report and private communication (2008).

**4. **K. Araki, T. Yano, and Y. Kuroda, “30 kW concentrator photovoltaic system using dome-shaped Fresnel lenses,” Opt. Express **18**(S1), A53–A63 (2010). [CrossRef] [PubMed]

**5. **W. Nishikawa, E. Green, and S. Crowley, “Energy production of CPV power plants at ISFOC,” presented at International Conference on Solar Concentrators for the Generation of Electricity (ICSC – 5), Nov. 16‐19, 2008, Palm Desert, CA.

**6. **Edmund Optics Inc, 101 E. Gloucester Pike, Barrington, NJ 08007, “High precision linear actuators” (2010).

**7. **P. Kotsidas, E. Chatzi, and V. Modi, “Stationary nonimaging lenses for solar concentration,” Appl. Opt. **49**(27), 5183–5191 (2010). [CrossRef] [PubMed]

**8. **J. M. Gordon, “Spherical gradient-index lenses as perfect imaging and maximum power transfer devices,” Appl. Opt. **39**(22), 3825–3832 (2000). [CrossRef]

**9. **J. C. Maxwell, “On the general laws of optical instruments,” Q. J. Pure Appl. Math. **2**, 233–247 (1854).

**10. **R. K. Luneburg, *The Mathematical Theory of Optics* (U. California Press, Berkeley, 1964).

**11. **A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A **223**(1153), 216–225 (1954). [CrossRef]

**12. **S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. **29**(9), 1358–1368 (1958). [CrossRef]

**13. **G. Beadie, J. S. Shirk, A. Rosenberg, P. A. Lane, E. Fleet, A. R. Kamdar, Y. Jin, M. Ponting, T. Kazmierczak, Y. Yang, A. Hiltner, and E. Baer, “Optical properties of a bio-inspired gradient refractive index polymer lens,” Opt. Express **16**(15), 11540–11547 (2008). [PubMed]

**14. **C. Ye and R. R. McLeod, “GRIN lens and lens array fabrication with diffusion-driven photopolymer,” Opt. Lett. **33**(22), 2575–2577 (2008). [CrossRef] [PubMed]

**15. **J. Q. Xi and F. Martin, “Schubert, J.K. Kim, E.F. Schubert, M. Chen, S.Y. Lin, W. Liu, and J.A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics **1**, 176–179 (2007).

**16. **D. Feuermann, J. M. Gordon, and M. Huleihil, “Solar fiber-optic mini-dish concentrators: first experimental results and field experience,” Sol. Energy **72**(6), 459–472 (2002). [CrossRef]

**17. **A. W. Bett, and H. Lerchenmüller, “The Flatcon system from Concentrix Solar,” in: *Concentrator Photovoltaics*, A. Luque and V.M. Andreev, eds. (Springer, Berlin, 2007), Ch. 14, pp. 301–320.

**18. **M. Yamaguchi, K. Araki, and T. Takamoto, “Concentrator solar cell modules and systems developed in Japan,” in: *Concentrator Photovoltaics*, A. Luque and V.M. Andreev, eds. (Springer, Berlin, 2007), Ch. 15, pp. 321–340.

**19. **J. M. Gordon, E. A. Katz, D. Feuermann, and M. Huleihil, “Toward ultra-high-flux photovoltaic concentration,” Appl. Phys. Lett. **84**(18), 3642–3644 (2004). [CrossRef]

**20. **E. A. Katz, J. M. Gordon, W. Tassew, and D. Feuermann, “Photovoltaic characterization of concentrator solar cells by localized irradiation,” J. Appl. Phys. **100**(4), 044514 (2006). [CrossRef]

**21. **O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultra-small multi-junction concentrator solar cells,” Appl. Phys. Lett. **91**(6), 064101 (2007). [CrossRef]

**22. **A. Goldstein and J. M. Gordon, “Tailored solar optics for maximal optical tolerance and concentration,” Sol. Energy Mater. Sol. Cells . in press.

**23. **P. Benítez, J. C. Miñano, P. Zamora, R. Mohedano, A. Cvetkovic, M. Buljan, J. Chaves, and M. Hernández, “High performance Fresnel-based photovoltaic concentrator,” Opt. Express **18**(S1), A25–A40 (2010). [CrossRef] [PubMed]

**24. **J. Sun, T. Israeli, T. A. Reddy, K. Scoles, J. M. Gordon, and D. Feuermann, “Modeling and experimental evaluation of passive heat sinks for miniature high-flux photovoltaic concentrators,” J. Sol. Energy Eng. **127**(1), 138–145 (2005). [CrossRef]