## Abstract

Aplanats are imaging optics that completely eliminate both spherical aberration and coma. They can fulfill the practical virtues of permitting sizable gaps between the absorber and the optic, as well as compactness. However, the ability of aplanats to efficiently approach the thermodynamic limit to flux concentration and optical tolerance had remained unrecognized. Both fundamental and applied aspects of dual-mirror aplanats are reviewed and elaborated, motivated by the exigencies of tenable, maximum-performance solar concentrators, including examples from commercial concentrator photovoltaics (CPV). Promising designs for future photovoltaic concentrators are also identified, illustrating how pragmatic constraints translate into devising fundamentally new optics.

©2010 Optical Society of America

## 1. Introduction

#### 1.1 Background and motivation

Aplanats invoke two optical surfaces as degrees of freedom to completely eliminate spherical aberration and coma in the quest for superior image fidelity – originally motivated by and invented for telescopes [1–4]. Dual-mirror devices can additionally be essentially achromatic (important because the realization of high solar concentration in optics with non-planar refractive surfaces is ultimately limited by chromatic aberration). Only recently were aplanatic optics explored as solar concentrators that can approach the thermodynamic limit to flux concentration at high collection efficiency [5–8].

This revival was partly inspired by a paradigm shift in photovoltaic power generation. Millimeter-scale multi-junction commercial solar cells have demonstrated efficiencies exceeding 40%, but only at high concentration, which in turn requires optics that can efficiently deliver high irradiance at liberal optical tolerance. That notwithstanding, the primary motivation for high concentration is economic, because the cell costs then become acceptably small even though they are more expensive on an areal basis than conventional cells. The burden to provide tractable, high-efficiency, affordable solar power is then transferred, in part, to developing maximum-performance optics.

Before portraying the aplanat formalism and a multiplicity of examples, the special motivation that stems from CPV is amplified, along with the relevance of fundamental bounds on flux concentration and optical tolerance.

#### 1.2 Fundamental limits to solar concentration

The fundamental (thermodynamic) bound for flux concentration *C _{max}* is (Eq. (1)) [9]

*NA*is the concentrator's exit numerical aperture (at the absorber), and θ

_{exit}_{s}is the

*effective*solar half-angle comprising the sun's intrinsic value of 4.7 mrad convolved with alignment inaccuracies, errors in mirror shape, and imperfect mirror specularity. (The air-filled optics presented here can incorporate transparent dielectric terminal concentrators without affecting the basic observations or designs.)

Aplanats also allow sizable gaps between the mirrors and the absorber at no loss in concentration or collection efficiency. They fill a fortuitous niche in high-concentration solar applications because their range of validity (to wit, θ_{s} ≤ 20 mrad for delivering maximum concentration at high collection efficiency [5–8]) corresponds to the sweet spot where (a) higher-order aberrations are tolerably small and (b) the flux concentration values of order 10^{3} needed in current CPV and high-temperature solar thermal applications cannot be attained with realistic optics if θ_{s} is noticeably larger. (θ_{s} = 5 mrad has been achieved in customized units [10], and values as low as θ_{s} = 7-8 mrad have been demonstrated in large CPV systems [11].)

#### 1.3 Absorber flux map as a benchmark

One performance metric for evaluating solar concentrators is the absorber flux distribution, which reflects the dilution of absorber power density required to collect a given fraction of incident flux. The flux map of a maximum-performance aplanat identified in Section 4 is presented in Fig. 1 to illustrate the point (where flux concentration is plotted as a function of radial position in the focal plane, with both variables normalized as described in the figure caption). The minimum absorber size corresponds to the uniform core region of the flux map (and hence to the maximum flux concentration that a given optic generates). The maximum absorber size is the value needed to accommodate all the rays reaching the absorber plane (corresponding to the maximum collection efficiency for the particular optic).

Raytraces were generated with a source distribution over the circular entrance pupil being uniform both spatially and in angle with a sharp cutoff at θ_{s}. The absorber flux map includes all rays reaching the focal plane, i.e., all exit angles are tabulated including those beyond *NA _{exit}*. The optical losses here and in the analyses that follow do not include (a) mirror absorption (1-ρ

^{2}where ρ = reflectivity, since each ray experiences precisely two reflections), or (b) reflections from the absorber and entry glazing, since both are material specific and easily quantified.

#### 1.4 Optical tolerance

An additional criterion is optical tolerance, to wit, the maximum angular deviation θ_{t} up to which 90% of the on-axis collection is realized, as illustrated in Fig. 2
(the 90% figure is the current convention, but is arbitrary).

CPV has evolved primarily to one-optic-one-cell units designed for an averaged flux concentration of at least hundreds of suns (1 sun = 1 mW/mm^{2}), limited chiefly by solar cell series resistance. (*Averaged* flux density refers to the net solar power delivery divided by the total active area of the cell, independent of flux distribution.) The focal spot is commonly smaller than the cell (as in Fig. 2). Measurements from a host of commercial concentrator cells show that even with a localized irradiance of several thousand suns (well above their averaged design irradiance), cell efficiency does not decrease by more than a few percent (relative) [12–14], which allays concerns of the impact of flux inhomogeneities. (Totally passive heat sinks can keep cell temperatures from increasing more than about 20-30 K above ambient, even with acute flux non-uniformities [11,15].)

Larger θ_{t} translates into less expensive optics, tracking and structure. θ_{t} derives from off-axis tracker orientation, array bending, misalignments, etc., bounded by (Eq. (2)) [7]

_{t}) ≈θ

_{t}, where

*C*

_{g}is the geometric concentration (the ratio of entry to absorber area),

*n*is the refractive index of the transparent medium in optical contact with the cell, and θ

_{exit}is the maximum exit half-angle on the cell. The flux concentration is η

_{o}

*C*where η

_{g}_{o}is concentrator optical efficiency at normal incidence. Figure 3 illustrates the variation of optical efficiency with off-axis orientation, along with the fundamental limits, for an aplanat analyzed in Sections 4 and 6.

For economy and clarity of presentation, axisymmetric concentrators with flat one-sided absorbers are analyzed. Correcting for absorber power density dilution when the cell and/or entry are not disks is straightforward and does not basically alter the designs.

## 2. Aplanat formalism

Achieving aplanatism requires satisfying:(a) Fermat's principle of constant optical path (string) length for all on-axis rays, Eq. (3)

(b) Snell's law of reflection at the mirror surfaces, and(c) the Abbe sine condition (constant magnification for all paraxial rays), Eq. (4) where*r*is the radial position on the primary mirror's entry and ϕ is the angle at the focus, as illustrated in Fig. 4 . The length scale is established by the system's focal length

*f*(defined as having unit magnitude).

*f*is also the Abbe sphere radius: one connects the focus along

*L*to the extension of

_{2}*L*. It then follows that the radius of the primary is

_{o}*r*=

_{max}*NA*= sin(ϕ

_{exit}_{max}). The focus is the origin of the coordinate system.

Like essentially any imaging system, the absorber flux map will possess a uniform core region of radius θ_{s}
*f*. But unlike conventional optics such as paraboloidal dishes and standard lenses, both of which incur coma, the constant magnification of the aplanat means that a larger fraction of the incident flux is concentrated into that core region.

The sign of *f* is taken as positive when construction of the Abbe radius from the focus outward *includes L _{2}* (Figs. 5
and 7
-9
), and negative otherwise (Figs. 6
and 10
). The optic axis (

*X*) then goes from negative to positive along incident rays for

*f*= + 1, and from positive to negative for

*f*= −1.

There are two input parameters: (1) *s* = (the vertex separation between the primary and secondary)/*f* = (*x _{po}* –

*x*)/

_{so}*f*, and (2)

*K*= (the separation between the secondary vertex and the focus)/

*f*= -

*x*/

_{so}*f*(see Fig. 4, which illustrates only one of several classes of aplanat, as elaborated below.)

The primary and secondary mirror solutions are given by Eq. (5) [3,4,8]

*p*and

*s*denoting the primary and secondary, respectively. For the

*upward*-facing absorber δ ≤ ϕ ≤ sin

^{−1}(

*NA*) (ϕ is relative to the optic axis), and for the

_{exit}*downward*-facing absorber π – sin

^{−1}(

*NA*) ≤ ϕ ≤ π – δ, with the angle δ determining the degree of reflector truncation that accounts for shading or blocking [8]. These yield the contours on one side of the optic axis; the contours on the other side being their mirror images.

_{exit}## 3. Thermodynamic and historical notes

The Fermat and Abbe conditions (Eqs. (3) and (4)) are commonly viewed as deriving exclusively from optical considerations. They can also be understood from thermodynamics (radiative transfer), given the objective of attaining the thermodynamic limit to concentration with a strictly imaging strategy (i.e., tailoring to paraxial rays only).

A necessary condition for realizing the thermodynamic limit in any optic is constant optical path (string) length for *extreme* incident wavefronts [9] (at ± θ_{s} here), embodied in the method of Hottel's crossed strings [9,16]. As θ_{s} vanishes, the extreme wavefronts converge to the paraxial wavefront, so Fermat's principle (Eq. (3)) becomes a necessary condition for realizing maximum concentration.

A concentrator that reaches the thermodynamic limit must also produce a quasi-lambertian flux map (i.e., uniform in both area and projected solid angle). Imaging optics generate an absorber flux map with a uniform core region of radius θ_{s}
*f*. Hence a second necessary condition for an imaging system to attain the thermodynamic limit is for that core region to constitute the entire uniform flux map at minimum absorber radius, for any entry radius *r* and corresponding exit half-angle ϕ (see Fig. 2), which implies

Fermat's and Abbe's conditions are necessary but not sufficient for an imaging system to reach the thermodynamic limit. This prompts the (unanswered) question of whether conditions for eliminating higher-order aberrations can be derived from thermodynamics, in the quest for maximum-performance imaging optics at progressively larger θ_{s}.

In fact, the first derivation Eq. (4) was not from Abbe's 1873 study of microscope imaging [17]. Rather, a thermodynamic derivation was published by Clausius in his 1864 study of radiative equilibrium [18] (of which Abbe was apparently unaware) and rediscovered by Helmholtz in 1874 [19].

The notion of aplanatism was proposed by Schwarzchild in 1905 as a dual-mirror telescope [1]. Head found an analytic solution (for the *near*-field dual-mirror aplanat) in 1957 [2], but did not explicitly note the formulae for the far-field limit (non-trivial because of nested vanishing and diverging functions). Apparently unaware of Head's solutions, Lynden-Bell derived the closed-form expressions for the far-field dual-mirror aplanat in 2002 [3], and considered an assortment of telescope geometries [3,4]. Finally, a physically transparent classification scheme for dual-mirror aplanats, that identified the full panorama of solutions and evaluated them as concentrators, was proposed by Ostroumov *et al* in 2009 [8].

## 4. Full spectrum of aplanat designs and their performance as concentrators

Equation (4) incorporates 8 fundamentally distinct classes of aplanats. The 8 possibilities derive from both *s* and *K* being either positive or negative, combined with the flat absorber facing upward or downward - recognized and elaborated in [8]. Two of these mathematically valid categories turn out to be physically inadmissible (due to the focus being virtual rather than real) [8]. The remaining 6 groups are illustrated in Figs. 5-10, where each example was designed for *NA _{exit}* = 0.9 and includes traces of the delimiting design rays: one incident at the rim of the primary, and the other being the accepted ray closest to the optic axis. In two instances (Figs. 6 and 10), a single category includes designs that may appear different but stem only from different magnitudes of

*s*and

*K*.

Intrinsic losses refer to shading, blocking and ray rejection in the point-source limit (θ_{s} = 0). Additional losses that derive from an extended source (θ_{s} > 0) were assessed by raytrace simulation. The results revealed only three aplanat classes (Figs. 5-7) with performance near the thermodynamic limit [8]. Specifically, at θ_{s} = 10 mrad, the optics in Figs. 5-7 attain ~90% of the thermodynamic limit when the étendue of the entry and absorber are equal (~10% loss from shading, blocking and ray rejection), and ~95% if the absorber diameter is oversized by 10%. Even at θ_{s} = 20 mrad, 80-90% of the thermodynamic limit is achieved in the étendue-matched design, with a 5% improvement for the oversized absorber [8]. Flux maps from which points of this nature can be quantified are plotted in Fig. 1 for the aplanat of Fig. 6(a).

Concentrator performance worsens rapidly as θ_{s} is increased further due to higher-order aberrations (e.g., astigmatism, field curvature, etc.). As noted in Section 1.2, this is not a deterrent for high-concentration solar applications where the very possibility of achieving high irradiance is in any event inherently linked to small θ_{s} values.

Although several of the basic classes in Figs. 5-10 may be relevant for solar thermal applications, or for CPV with active cooling (namely, where downward-facing absorbers are tenable – Figs. 7, 8 and 10), only those in Figs. 5 and 6 are feasible for one-cell-one-optic photovoltaic concentrators with passive heat rejection. (The aplanat of Fig. 9 has a upward-facing absorber but incurs excessive inherent losses.)

## 5. Aplanats in commercial CPV

When the focus must reside inside the optic, and the preferred siting of the solar cell (bonded to an extensive heat sink) is outside the optic, one can design for an ostensibly lower *NA _{exit}* in the aplanat's focal plane and introduce a terminal concentrator between the focus and the actual absorber. This strategy was adopted for Generation One SolFocus concentrators developed for a square 100 mm

^{2}concentrator solar cell (Fig. 11(a) ).

The measured net solar power delivery onto the cell is 50 W/cm^{2} at peak solar beam radiation [11]. Field measurements revealed an effective θ_{s} of 7-8 mrad with an optical tolerance ≈80% of the fundamental limit (Eq. (2)) [11].

An analogous Generation Two glass-filled planar dual-mirror aplanat was designed [6] and prototyped for a square 1 mm^{2} cell optically bonded to the vertex of the primary (Fig. 11(b)) with comparable performance [11]. Because θ_{s} « 1 and no optical transfer relies on refraction, this design is essentially achromatic [6]. (The planar all-glass terminal concentrator in Generation One also incurs negligible dispersion losses.)

## 6. Wineglass aplanats

The limitations of photovoltaic concentrators of the type portrayed in Section 5 – learned during their production and field monitoring [11] – prompted the exploration of different classes of optics that could satisfy additional requirements deemed vital to future CPV systems, e.g., (a) obviating the need for an optical bond between the solar cell and any optical element, (b) mitigating multiple-element alignments, (c) avoiding damage at off-axis orientation, (d) allowing solar cell placement outside the optic *without* the need for an optical channel extractor, (e) low-mass units and (f) being conducive to miniaturization and precision high-volume production, all while maintaining at least as high an optical tolerance as measured for the concentrators of Fig. 11.

The wineglass-shaped aplanat (Figs. 6(b) and 12
) is an appealing candidate (provided mm-scale cells are deployed, *vide infra*). Its focal plane is external to the optic, with no perceivable problems in designing for high *NA _{exit}*. The role of a terminal concentrator as a means to locate the cell outside the optic is thereby eliminated. The high aspect ratios (relative to the fundamental limit of 1/4) are not a deterrent with mm cells because concentrator depth is then only a few cm: amenable to large-volume precision fabrication, with low mass per unit area. For example, to achieve a flux concentration of ~500 suns on a 1 mm

^{2}cell at high optical efficiency, concentrator depth in Fig. 12(a) would be only ~50 mm (and about half that value in the lens-enhanced version of Fig. 12(b)).

Concentrator aspect ratio can be lessened – and the inherent loss of rays that miss the primary (traverse the waist and are not focused) can be recovered - by incorporating a lens of diameter equal to that of the waist as an integral part of the glazing (Fig. 12(b)) [7]. Considerations of cleaning and maintenance prompt a plano-convex lens that eliminates spherical aberration (hence a hyperboloidal contour, which follows from Fermat's strings). For designs akin to the *f*/4 lens in Fig. 12(b), chromatic aberration is negligible [7].

Both the pure-mirror and lens-enhanced versions approach maximum performance. Namely, they achieve flux concentration within a few percent of the thermodynamic limit, and about 80% of the fundamental limit for optical tolerance (see Fig. 3), confirmed for θ_{s} ≈10 mrad at a flux concentration of ~500 suns [7].

Some concentrators incur damage at off-axis orientation because of an intense caustic on optical elements. But even at large off-axis angles, caustics do not endanger optical surfaces in the wineglass, e.g., the highest irradiance on the mirror in Fig. 12(b) is about two orders of magnitude below the absorber irradiance. The optic may then be produced from polymers rather than by metal or glass shaping, which reduces mass and cost.

## 7. Conclusions

Aplanatic solar concentrators can approach the thermodynamic limit to both concentration and optical tolerance. With two mirror contours as design degrees of freedom, both spherical aberration and coma can be eliminated in essentially achromatic devices. Dual-mirror aplanats can feasibly accommodate high *NA _{exit}* values, at θ

_{s}values commensurate with high absorber irradiance.

The reflector shapes have closed-form solutions, which allow the extensive exploration of a multitude of configurations, and facilitate both optimization studies and affordable manufacturing techniques. Aplanats also allow a substantial gap between the reflectors and the absorber at no loss in efficiency or concentration.

Although the investigation of dual-mirror aplanats as concentrators was initially motivated by CPV with one-optic-one-cell configurations and passive cooling, the insights gained in tackling the problem revealed a rich spectrum of designs that could also be of value for (a) CPV with large multi-cell absorbers (and therefore active cooling), (b) solar thermal concentrators, (c) infrared concentration, and even (d) light collimation from quasi-lambertian sources where the roles of source and target are reversed [5,8]. These are cases where downward-facing absorbers may be acceptable and device aspect ratio may not be critical.

The classification scheme for dual-mirror aplanats provides a fundamental, physically transparent categorization that allows the identification of a spectrum of distinct optics. The specific examples here are not claimed to represent optimized configurations, since optimizations are invariably case specific. Rather, the intention was to depict the full landscape of potentially feasible aplanats, and to highlight the potential for realistic maximum-performance solar concentrators, with special emphasis on CPV.

These subsume aplanats for what are currently the highest irradiance commercial large-scale CPV systems (Fig. 11). They also spawned wineglass designs that can (a) obviate the need for optical bonds, (b) exhibit optical tolerance approaching the fundamental limit, (c) mitigate damage at off-axis orientation and (d) allow convenient location of the solar cell and heat sink for any desired *NA _{exit}*. Lens-enhanced wineglass aplanats enjoy improved compactness and, with mm cells, a concentrator depth not exceeding a few cm: amenable to precision large-volume fabrication.

Finally, the far-field solutions are actually a limiting case of the general near-field dual-mirror aplanat, solved analytically by Head [2], and, to our knowledge, unexplored for maximum-performance near-field optics until quite recently [20], with ultrahigh irradiance systems now realized for biomedical and chemical reactor applications [21,22].

## Acknowledgments

I am grateful to my colleagues Daniel Feuermann and Roland Winston, and to my MSc students Natalia Ostroumov and Alex Goldstein, for their intellectual companionship in forging the path to maximum-performance aplanats. I also appreciatively acknowledge Gary Conley and Steve Horne of SolFocus Inc. for their stimulus and support in stewarding these concepts into tangible photovoltaic concentrators.

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