## Abstract

Aplanats refer to inherently imaging optics that wholly eliminate both spherical aberration and coma. They typically comprise two refractive and/or reflective surfaces. For radiative transfer (which is typically nonimaging in nature), aplanats can closely approach the thermodynamic bounds for collimation and concentration, especially significant for light-emitting diodes (LEDs), solar energy, and infrared applications. Recently, we identified previously unrecognized basic categories of aplanats and showed how they can offer powerful new possibilities for LED collimation and for concentrating sunlight. Here, we review and elaborate the full scope of aplanat classifications, with illustrative examples of maximum-performance practical optics for all possible combinations of reflective and refractive contours. These examples subsume the latest invention of faceted (Fresnel) aplanats toward achieving greater compactness and lower mass. We also show how hybrid aplanats that combine the basic categories can improve concentrator and collimator performance.

© 2019 Optical Society of America

## 1. INTRODUCTION

Aplanats are optical devices designed to completely remove the two leading orders of geometric aberration: spherical and comatic. Tailoring two optical contours suffices to solve the problem. Aplanats were invented more than 100 years ago for high-quality imaging in cameras, microscopes, and telescopes. For many decades, aplanats were studied and applied for strictly imaging applications. The classes of aplanats that emerged [1,2] turned out to represent only a fraction of a far richer panorama that was discovered in more recent investigations demonstrating that aplanats should also provide high-performance solutions for nonimaging applications such as light collimation and solar concentration [3–7], prompted by the observation that an optic providing high-fidelity imaging can also approach the fundamental or thermodynamic limit for flux concentration [8]. This paper entails a wide-ranging review and update for the nominally full scope of aplanatic designs, with illustrations for light collimators and solar concentrators.

Since each of the two optical surfaces can be either refractive (denoted by “R”) or reflective (denoted by “X”), it would appear that there are only four categories of dual-contour aplanats: XX, XR, RX, and RR. However, subtler aspects of the solutions point to additional basic subcategories, such that each of the four options contains eight sub-classes [3–6], some of which are not physically meaningful (e.g., due to virtual foci), but most of which comprise realizable optical systems. As elaborated below, aplanats that include a refractive element can also admit a further branching into two distinct classes that differ based on the order in which rays encounter air–dielectric interfaces.

With light collimation and solar concentration motivating this survey—and in the interest of economy of presentation—only far-field solutions are analyzed. It should be noted, though, that the far-field problem constitutes a special limiting case of the general near-field aplanat problem [1,9,10].

Although the original derivations of all these aplanats are cited herein, only the governing equations that permit the user to generate all physically admissible classes of aplanats are explicitly included here. For one novel category of aplanats, we report the design and experimental performance of a light-emitting diode (LED) collimator. Finally, we additionally survey the most recent findings for hybrid and Fresnel (faceted) aplanats.

## 2. DERIVATION OF APLANATIC CONTOURS

The derivations of the two contours comprising aplanats have been presented in [1–6]. One needs to simultaneously impose the conditions of (1) constant optical path length (comprising the sum of each path ${L}_{i}$) from the paraxial wavefront to the focus (in concentrator mode [Fig. 1]), (2) the Abbe sine condition (constant magnification for all rays from a point source to the target), and (3) Snell’s Law. The solutions for a restricted range of classes of aplanats were derived by Head for an RR aplanat [1] and by Lynden–Bell for some XX aplanats [2], the latter conveniently yielding a closed-form solution for the radial ($r$) and axial ($x$) coordinates of the aplanat’s surfaces as parametric solutions in the running angular variable $\varphi $,

The corresponding exercise for the XR aplanat [5] is shown in Fig. 2, with the added input parameter of the dielectric’s refractive index *n*. Although the solution for the aplanatic contours is analytic, it does not yield a closed-form expression, and hence must be evaluated numerically by simultaneously solving,

Figure 3 shows the corresponding construction for RX aplanats [4], with the contours following from the simultaneous solution of

Figure 4 illustrates the RR aplanat construction [6] in collimator mode, with the two contours following from simultaneously solving

## 3. REPRESENTATIVE RESULTS

#### A. Dual-Reflective (XX) Aplanats

The particular category of XX aplanats illustrated in Fig. 1 has most notably been implemented for solar concentration: first for large-scale commercial concentrator photovoltaic systems [11], and, more recently, for ultrahigh concentration (a net flux concentration of 30,000) in a solar furnace used for the synthesis of singular nanomaterials, reaching temperatures exceeding $\sim 3000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{K}$ [12] (Fig. 5). These aplanats exhibit high concentration at high collection efficiency for an effective far-field source angular radius as large as $\sim 20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mrad}$ [3], where “effective” refers to the convolution of the actual source (e.g., the sun’s intrinsic angular radius of 4.7 mrad) with system optical errors. The effective solar angular radius for the concentrator in Fig. 5—a highly specular diamond-turned device—is 5 mrad. Its producing “only” 30,000 suns in our solar furnace [*vis-à-vis* Eq. (5) below] derives from absorptive losses in the two mirrors of the aplanat as well as in an outdoor heliostat of undetermined flatness.

Illustrations of other classes of XX aplanats are presented in Fig. 6 [3], with device parameters chosen so as to approach maximum solar concentration [8] (hence with a high but pragmatic NA value of 0.9),

with ${\mathrm{\Theta}}_{s}$ denoting the effective angular radius of the solar disk (i.e., of the far-field source). Practical optimized designs must also account for losses that can stem from (1) shading of the primary by the secondary, (2) blocking of the secondary by the absorber, (3) incident rays that miss the primary and hence are not focused, and (4) ray rejection as a consequence of higher order aberrations (e.g., astigmatism and distortion).#### B. Reflective–Refractive (XR) Aplanats

For solar (and usually infrared) concentrators with refractive elements, chromatic aberration can substantially reduce collection efficiency (and thereby flux concentration). As elaborated below, this is especially pertinent for RR optics and, to a lesser degree, for RX devices. It is less severe for XR aplanats because (a) most of the optical path length can occur in air (from the first reflection at the “X” surface), thereby significantly reducing the optical path length inside the dielectric, and (b) concentration can be enhanced by a factor of ${n}^{2}$ (where $n$ is the dielectric’s refractive index), provided the absorber is optically coupled to the secondary dielectric element. Moreover, the mass of dielectric required for the XR aplanats can be noticeably less than for their RX and RR counterparts.

Figure 7 presents examples for the five types of XR aplanats that are physically meaningful [5]. When Fresnel reflections are accounted for, designs XR-1 and XR-3 have the practical advantage of incurring lower losses, as a consequence of having smaller incidence angles on the dielectric.

#### C. Refractive–Reflective (RX) Aplanats

The six physically admissible solutions for RX aplanats [4] are graphed in Figs. 8 and 9. There are eight mathematical classes, but two are not physically meaningful: $\{c,p,s\}=\{1,1,1\}$ and $\{c,p,s\}=\{1,1,-1\}$. Figure 8 shows examples of solutions where one surface of the dielectric must be mirrored, and Fig. 9 illustrates solutions where the reflection (“X”) is the total internal reflection at the dielectric–air interface. The latter offers the advantage of avoiding the absorption loss of a mirror. With the absorber embedded in—and optically coupled to—the dielectric element, the enhancement in concentration by a factor of ${n}^{2}$ can be realized [because, in Eq. (1), NA is actually proportional to the refractive index of the medium in which the absorber is situated [8]].

#### D. Refractive–Refractive (RR) Aplanats

RR aplanats are commonly referred to as aplanatic lenses. Only three of the eight basic classes of mathematical solutions are physically meaningful [6], and are illustrated in Fig. 10 in collimator mode, in part because chromatic aberration in these devices is significant over the broad solar spectrum, compromising concentrator efficiency. However, chromatic aberration is essentially irrelevant for the collimation of LED emission. These designs have their focus in air, which precludes the potential improvement (tightening) of the collimation angle by a factor of $n$ [8]. However, as demonstrated below in Section 3.E.3, an exchange of the dielectric and air at the interfaces can yield physically meaningful solutions where the light source to be collimated (e.g., the LED) can be in optical contact with the dielectric.

#### E. Complementary Aplanatic Designs

With the exception of the XX aplanats, all of the optics depicted above admit complementary designs where the order at the dielectric–air interfaces is reversed (Figs. 11–13). In mathematical terms, it means using $1/n$ wherever $n$ appears in Eqs. (2)–(4). The assertion that these basic solutions are not obvious is based on aplanats of these types not having been reported previously (to the best of our knowledge). We review these complementary solutions not so much in the spirit of practical devices for immediate application, but rather of fundamental solutions that apparently had remained unrecognized and might be found to have pragmatic functions.

### 1. Complementary XR Aplanats

Comparable to the XR aplanats portrayed above, there are five categories of physically admissible solutions—illustrated in Fig. 11 in concentrator mode [5]. In the XR-3 and XR-4 designs, the reflection “X” is based on total internal reflection. In some designs, optical losses due to rays missing the primary or, after a reflection or refraction, missing the secondary, can be sizable.

### 2. Complementary RX Aplanats

Complementary RX aplanats (Fig. 12) [4] are distinct from the RX aplanats elaborated in Section 3.C above by (a) having a planar entry aperture, (b) rays exiting from the dielectric component into air *before* undergoing reflection, and (c) the focus being in air. One of their main disadvantages is chromatic aberration for broadband (e.g., solar) input, stemming in part from the relatively long optical path length of rays after exiting the dielectric element.

### 3. Complementary RR Aplanats

The RR aplanats detailed in Section 3.D have three classes of physically admissible solutions for corresponding complementary RR devices (Fig. 13) [6]. Each optic comprises two dielectric elements separated by air. The exit (in collimator mode, or, equivalently, the entry in concentrator mode) is planar. These designs benefit from the possibility of the light source (at the focus in collimator mode) being optically coupled to the dielectric.

#### F. Hybrid and Fresnel Aplanats

Combining solutions from different classes of aplanats [6] (e.g., XR + RR)—nominally hybrid designs—offers an additional degree of freedom, and hence can offer devices better suited to practical constraints such as compactness and lower optical losses. For illustration, one such hybrid of different classes from the RX category is shown in Fig. 14, conflating the RX-4 and RX-5A solutions of Fig. 9. Although illustrated in Fig. 14 in concentrator mode, it was designed as a collimator for a $1.6\times 1.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$ white LED and manufactured from acrylic (refractive index $n=1.49$), to which the LED was optically bonded. The full assembly and collimated beam are shown in Fig. 15. Given the collimator exit aperture diameter of 50 mm and the dimensions of the square LED, the emitted beam divergence should be $\sim 3\xb0$, consistent with the measured collimation.

Faceted optical surfaces have the potential to improve compactness—most notable in the development of traditional Fresnel lenses. It also motivates the exploration of Fresnel-type aplanats, where a refractive contour can be divided into faceted zones, with each zone comprising a nominally autonomous aplanatic optic [7]. A boundary condition for each aplanatic facet is its starting point being chosen as the end point of the preceding facet. In addition—and analogous to conventional Fresnel lenses—the possibility of rays being blocked by adjacent facets can be averted via the judicious selection of boundary conditions [7]. The design principle for one such Fresnel RR aplanat is shown in Fig. 16. Other classes of Fresnel RR aplanats are drawn in Fig. 17.

## 4. CONCLUSION

The aim of this review is to concisely summarize the full range of dual-contour aplanatic optics in a single paper. The coarsest classification scheme implies four categories based on whether a given contour is reflective (X) or refractive (R): XX, RX, XR, and RR. However, more careful analysis of the solutions admitted by the governing equations reveals a far richer classification scheme, and hence a broader assortment of aplanats, many of which are suitable for solar concentration and LED collimation, for which illustrative examples have been presented.

First, the mathematical solutions for each nominally basic category (XX, RX, XR, and RR) divide into eight fundamental classes, although not all of them turn out to be physically admissible. The solutions also depend on the choice of boundary conditions, and, for aplanats that include a refractive element, on the dielectric’s refractive index.

Second, additional aplanatic solutions emerge by interchanging the order in which rays encounter air and dielectric at the refractive interfaces.

The expansive scope of aplanatic solutions provides the user with a panoply of options that, in addition to providing optical performance that approaches the thermodynamic limit, can also facilitate satisfying practical constraints such as compactness, minimal optical loss, and ease of manufacture.

The geometrical vector flux approach based on which a variety of nonimaging devices have been predicated [8] has also been applied to a few nonimaging simultaneous multiple surface (SMS) designs where aplanats represent their point source limit [8,14,15].

Aplanats are inherently imaging optics. Their design is predicated on point sources, although they completely remove coma (i.e., the first order of aberration beyond spherical aberration, hence starting to accommodate the extended nature of the light source). Aplanats have been shown to yield concentration and collimation performance approaching their respective thermodynamic limits [3–7] for sufficiently small extended sources, to wit, source angular radii no larger than about 20–40 mrad (depending on the class of aplanat)—a range that is fortuitously adequate for many solar concentration and LED collimation applications. For larger source sizes (hence for lower concentration and wider collimation angles), solutions inherently rooted in nonimaging optics [8,13] provide superior solutions.

Results have been restricted to the far-field problem, which must be a limiting case of the generalized near-field problem. Hence the same approaches presented here provide equations that can be solved for the corresponding near-field problems with identical classifications, some of which have already been investigated [9,10].

## Acknowledgment

Heylal Mashaal was the recipient of a Howard and Lisa Wenger graduate scholarship during the period of this research.

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