## Abstract

We identify and evaluate new categories of dual-contour refractive-reflective aplanatic lenses, some of which can satisfy total internal reflection at the secondary surface. Raytrace simulations for a representative design in both solar concentrator and collimator (illumination) mode reveal high efficiency while approaching the thermodynamic limit for radiative transfer.

© 2015 Optical Society of America

## 1. Introduction

An aplanat is an imaging optic designed to completely eliminate both spherical aberration and coma. Tailoring two optical surfaces (reflective and/or refractive) provides the necessary degrees of freedom. Dual-mirror aplanats were originally developed in the quest for better telescopes [1–4]. The mathematical problem was first solved analytically in [2] and later elaborated in [3,4]. Recently, dual-mirror aplanats were revisited and explored as both solar concentrators and illumination collimators that can approach the thermodynamic limit for radiative transfer at high efficiency [5–8].

Here, we focus on dielectric RX aplanats, where ‘R’ refers to a refractive primary and ‘X’ to a reflective secondary contour (as per the notation of [9–11]). Because the absorber is optically bonded to, or embedded within, the lens, the enhancement factor of *n*^{2} (*n* being the refractive index of the transparent dielectric filling the optic) in the fundamental thermodynamic bound for flux concentration *Cmax* [9] can be realized:

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

_{exit}*NA*is the input numerical aperture,

_{entry}*θ*is the maximum exit half-angle (at the absorber), and

_{exit}*θ*is the effective solar half-angle comprising the convolution of the sun's intrinsic angular radius of 4.7 mrad with imperfections in alignment, mirror shape, and mirror specularity.

_{s}The notion of RX *non*imaging concentrators (designed for extended rather than point sources) that approach the thermodynamic limit was first introduced in [10] as part of the Simultaneous Multiple Surfaces (SMS) method [9], with the point-source limit yielding the corresponding aplanat [9,10,12]. This pioneering work turns out to constitute only one category of RX lenses. The full panorama of 8 possible categories remained unrecognized, and will be elaborated herein. Additional impetuses for exploring aplanats include (a) optical performance being essentially as good as the corresponding SMS devices for *θ _{s}* values of practical interest in solar concentration (< ~20 mrad), at least for dual-mirror designs [13], and (b) analytic solutions for the optical contours (with the associated advantage in design and optimization) whereas the SMS method engenders a point-by-point computational scheme.

RX designs that could achieve reflection at the secondary surface by satisfying total internal reflection (TIR) would be especially attractive not only in avoiding the need for mirrored surfaces, but also in eliminating mirror absorptive losses (typically 4-10% per reflection). They had not been discovered to date, and will be presented for the first time here.

We proceed by presenting new designs for RX aplanats - with a complete classification scheme – and providing illustrative examples. These observations comprise the principal message of this paper, without regard to the pragmatic pros and cons of the new types of RX aplanats reported here, and without attempting to optimize the new aplanats for particular device applications, for which case-specific constraints can be crucial. That notwithstanding, toward illuminating and quantifying the potential for solar concentration and light collimation, one promising representative design is evaluated in detail via raytrace simulation.

## 2. RX aplanat formalism

Aplanatism for RX optics is achieved by satisfying the following conditions (Fig. 1):

- b) Snell’s law of refraction at the primary surface, transition L
_{1}to L_{2}. - d) Snell’s law of reflection at the secondary surface, transition L
_{2}to L_{3}.

*L _{1}*,

*L*and

_{2}*L*are ray trajectories, r is the radial position on the primary entry, and

_{3}*φ*is the angle of ray

*L*with the optic axis at the focus

_{3}*f*.

*F*is the Abbe sphere radius,

*F*=

*R*sin(

_{p}/*φ*), obtained by connecting the focus along

_{max}*L*to the extension of

_{3}*L*(Fig. 1).

_{1}Conditions (a)-(d) yield the following set of equations:

*m*

_{1}=

*Y*/

_{s}*X*(

_{s}, m_{2}=*Y*)/(

_{s}- Y_{p}*X*),

_{s}- X_{p}*c,p,s = ±*1, and the boundary conditions

*Y*(

_{p}*R*)

_{p}*= H*and

_{p}*Y*(

_{s}*R*)

_{s}*= H*.

_{s}Equations (4)-(6) are sufficient to compute the two contours (Eq. (7) is redundant). Solutions can be obtained by solving the system as a differential algebraic equation, or by reducing the system to a single differential equation. The solutions are analytic, albeit not in closed form, and can be readily evaluated numerically with commercial software such as Matlab and Mathematica.

## 3. Categories of RX aplanats

There are 8 basic classes of solutions, corresponding to the parameters {*c*,*p*,*s*} individually assuming the values ± 1. One of these 8 categories is the nonimaging RX SMS concentrator reported in [10], for which the point-source limit yields the corresponding aplanat. The physical significance of parameter ‘*c*’ is whether the optical path crosses the optic axis in the 2D cross-section of Fig. 1 (*c* = 1 for axis crossing). The parameter ‘*p*’ determines whether the slope of the primary monotonically increases (*p* = −1) or decreases (*p* = 1) as *r* approaches the optic axis. The parameter ‘*s*’ determines the sign of the Abbe radius *F* defined as positive when its construction from the focus along *L _{3}* to the extension of

*L*lies below the focus and negative otherwise. Of the 8 possibilities, two do not yield physically admissible solutions: {

_{1}*c*= 1,

*p*= 1,

*s*= 1} and {

*c*= 1,

*p*= 1,

*s*= −1}. The remaining 6 categories possess physically meaningful designs for limited extents of the parameter space {

*R*,

_{p}*H*,

_{p}*R*,

_{s}*H*}.

_{s}Once the mathematically convenient dimensional parameters {*R _{p}*,

*H*,

_{p}*R*,

_{s}*H*} are chosen to describe the RX aplanat, the set {

_{s}*c*,

*p*,

*s*} then characterizes the 8 basic categories. A different choice of dimensional parameters would modify the governing equations and would yield a characterizing set other than {

*c*,

*p*,

*s*}, but the 8 basic categories and the RX aplanatic optical designs remain unaltered. This is analogous to the solutions derived for XX aplanats [8], where a different set of dimensional parameters that were mathematically convenient for the dual-mirror optic yielded a correspondingly different set of 3 characterizing parameters.

Figure 2 depicts 3 solution categories that require a mirrored secondary: RX-1, RX-2 (divided into two sub-classes according to whether the secondary is concave RX-2A or convex RX-2B), and RX-3. The design parameters for Figs. 2-3 are summarized in Table 1.

Figure 3 depicts 3 solution categories that, with available transparent dielectrics for solar concentrators, can satisfy TIR at the secondary: RX-4, RX-5A, RX-5B and RX-6 (where RX-5 is divided into two sub-classes). Unlike the rest of the designs, RX-5B achieves TIR due to a void in the dielectric. There are assorted methods available for fabricating all the RX aplanats (including merging two partially hollowed halves for the unusual RX-5B aplanat), but the issue is ancillary relative to the new observations for identifying and analyzing basic new optical design categories.

The compactness (aspect ratio) of the RX-4, RX-5 and RX-6 designs is limited by the condition for TIR. More compact designs can always be achieved by partially mirroring the secondary in regions where TIR is not fulfilled.

A particularly attractive design is the ‘hybrid’ generated by merging designs RX-4 and RX-5A. Care must be taken to select the design parameters so as to ensure continuity along the full primary and secondary contours. Other types of hybrids that merge different designs are also possible.

Depending on the aplanat category and the input design parameters, the full construction range of 0 ≤ *X _{p}* ≤

*R*may be precluded (see Fig. 3) by (a) the paucity of a physical solution over finite regions as X

_{p}_{p}approaches either the rim of the primary or the optic axis, as in RX-4 and RX-5, respectively, (b) the secondary blocking rays refracted from the primary, as in RX-6, or (c) the slope of the primary diverging, as in RX-3. In these cases, the construction is truncated, i.e., contours are generated only for a limited range of

*X*, as indicated in Table 1. These limitations, and the associated loss of collectible radiation, grow more pronounced as one tries to achieve more compact concentrators. The particular (hybrid) design for which raytrace evaluation is presented here incurs a corresponding central gap loss of 5% - a somewhat arbitrary choice for the maximum acceptable loss, moderated by the requirement of a reasonable lens aspect ratio. However, most of this loss can be obviated, e.g., by filling the gap with a suitable confocal ellipsoidal-cap lens, similar to a comparable solution proposed for the air-filled dual-mirror aplanat in [7]. Moreover, the central gap loss is adjustable subject to concomitant modification of device compactness or creation of assorted hybrid options.

_{p}## 4. Optical performance

The axisymmetric hybrid RX aplanat in Fig. 3 was chosen for evaluation as a solar concentrator based on its qualifying as a TIR lens and its relative compactness. It is offered as one representative example suitable for solar concentration and light collimation, without the presumption of a broad comparison of all RX aplanats for a variety of applications. Absorption and scattering in the dielectric, as well as Fresnel reflective losses of incident radiation at the air-dielectric entry are not included since they can be case-specific, can vary with lens dimensions and surface processing, and are readily quantified. We note, however, that the Fresnel reflective loss at the entry to this particular hybrid concentrator turns out to be essentially the same as for a planar-entry optic, i.e., less than a 1% incremental difference. (It should be noted that the raytrace simulation can neglect Fresnel reflections at the air-dielectric interface of concentrator entry without eliminating their essential role for TIR at the dielectric-air interface of the secondary’s profile.)

In order to distinguish the inherent impact of geometric (higher-order) vs chromatic aberration, and using the software package OptiCad^{®} Version 10 for all the evaluations that follow, we raytraced the concentrator for monochromatic radiation, with *θ _{s}* = 5, 10 and 20 mrad - revealing that ray rejection did not exceed 1%. The specific impact of chromatic aberration was characterized from raytracing a point source (collimated light at normal incidence) for the full solar spectrum and the dispersion relation for BK7 glass - revealing an equivalent non-negligible optical error of ~2 mrad. This is why there are progressively smaller losses as

*θ*increases (Fig. 4), and why the trend for efficiency varying with

_{s}*θ*in Fig. 4(b) reverses for monochromatic vs solar input. The point-source input was limited to this single instance of isolating dispersion losses. All raytrace results summarized in Figs. 4 and 5 were based on extended solar and emitter sources, respectively, as described below.

_{s}#### A. Solar concentration

Flux maps (Fig. 4) were generated by tracing ~10^{7} rays distributed uniformly in area and projected solid angle, with a top-hat solar angular distribution, at optical tolerances *θ _{s}* = 5, 10 and 20 mrad.

*θ*refers to the angular extent of the radiative input, and not to the actual optical tolerance of a concentrator, which can be evaluated from plots of the type presented in Fig. 4a. The flux at the concentrator exit falls mainly within (but is not exclusively limited to) the design angle

_{s}*θ*in the basic bound of Eq. (1). Because the spatial (rather than the angular) distribution at the absorber is typically of greatest interest, and for economy of presentation, exit flux maps are presented as local irradiance vs. radial position.

_{exit}The flux maps in Fig. 4(a) quantify target flux uniformity. Note that the logarithmic ordinate scale (adopted in order to clearly capture the wide range of concentration) may appear to exaggerate flux inhomogeneities that, in reality, are modest. In any event, flux non-uniformities of these magnitudes exert a negligible influence on the conversion efficiency of current commercial concentrator photovoltaics, as confirmed experimentally in [14–16].

In Fig. 4(a), an ideal concentrator would exhibit step-function behavior with a cutoff at the radius corresponding to the thermodynamic limit for a given θ_{s}. In Fig. 4(b), an ideal concentrator would exhibit efficiency (the fraction of incident radiation concentrated onto a given focal spot area) proportional to area up to the minimal absorber area corresponding to the thermodynamic limit (and a constant maximum value beyond that).

For applications in the mid-to-far infrared, RX aplanats would produce considerably higher concentration (or optical tolerance) at high efficiency, due to *n* being ~2.5-4 with a far weaker dependence on wavelength than candidate solar dielectrics have in the visible to near infrared, and the devices could be more compact as a consequence of higher *n*.

#### B. Collimation (illumination mode)

Because (a) light-emitting diodes (LEDs) are inherently monochromatic, (b) the spectral emission of ostensibly white LEDs can vary considerably among available models, (c) LEDs have relatively wide-angle emission, and (d) the purpose of this section is to provide quantitative results for a representative system, the analysis was limited to a monochromatic quasi-lambertian emitter (a coarse approximation for common LEDs), for which we raytraced for the far-field flux map. The LED can either be bonded to, or embedded within, monolithic RX lenses.

Equation (1) still pertains as the basic bound relating the degree of collimation *θ _{s}* given the ratio of aperture-to-source area

*C*, corresponding to

*d*=

_{LED,min}*D*sin(

*θ*)/(

_{s}*n*sin(

*θ*)), where

_{exit}*D*is the diameter of the refractive (primary) aperture, and

*θ*is the maximum emission angle inside the dielectric.

_{exit}Raytrace results are presented in Fig. 5 as efficiency (defined as the fraction of emitted radiation within a given far-field projected solid angle Ω corresponding to polar half-angle *θ*, *Ω* = π sin^{2}(*θ*)) against *Ω* relative to its value at the thermodynamic limit *Ω _{th}* = π sin

^{2}(

*θ*). The monochromatic emission of light-emitting diodes (LEDs) eliminates dispersion losses and accounts for especially good performance.

_{s}There are optical losses that derive from a fraction of emitted rays missing the secondary and thereby exiting the aplanat at angles beyond the design *θ _{exit}*. This is the main reason the illuminator performance curves plotted in Fig. 5 fall below the ideal limit.

## 5. Conclusions

New categories of RX aplanatic lenses have been identified (refractive primary and reflective secondary surfaces comprising a single monolithic device where the absorber is optically bonded to, or embedded in, the lens dielectric), analyzed and solved analytically. There are 8 distinct solution categories, 6 of which provide physically admissible solutions, all of which have been illustrated. Three of these categories yield especially attractive solutions where the secondary surface can satisfy TIR, thereby obviating the need for a mirrored surface and the associated absorptive losses. This approach allows a rapid and systematic exploration of parameter space for the identification of valid and feasible designs.

Raytrace simulations reveal concentrator performance approaching the thermodynamic limit, even at high exit *NA*, and quantify the separate losses from geometric vs chromatic aberration. RX concentrators can realize the *n*^{2} enhancement factor in flux concentration at fixed optical tolerance (or, equivalently, an improvement of a factor of *n* in optical tolerance at fixed concentration). When used in illumination mode, these RX aplanats closely approach the corresponding thermodynamic limit for collimation as a function of light source size.

A few nonimaging TIR concentrators that nominally can approach the thermodynamic limit preceded the SMS strategy, most notably dielectric compound parabolic concentrators [17], all-dielectric lens-profile designs [18], and dielectric tailored edge-ray concentrators (DTERCs) [19]. In order to avoid large ray rejection, excessive ray leakage, or unwieldy aspect ratios, the first two are limited to nominal acceptance half-angles (*θ _{s}*) above ~200 mrad (even for monochromatic radiation) which severely restricts either concentration or optical tolerance. At high concentration, the DTERC incurs ray rejection losses of ~10% or higher for monochromatic radiation but, more critically, suffers from excessive dispersion losses for solar input, to wit, of the order of tens of percent. Furthermore, unlike the SMS design strategy, these 3 types of nonimaging TIR concentrators do

*not*reduce to aplanats in the point-source limit.

The same methodology could also be applied to other types of RX designs, e.g., *near-field* RX aplanats, SMS RX devices [9], and dual-surface functional method RX optics [20].

Even the pragmatic constraint of the absorber residing at the exit of (i.e., nominally external to) the lens (e.g., to facilitate passive cooling for photovoltaics or LEDs) can be accommodated by designs RX4, RX5 and RX6 (Fig. 3), with the added benefit of being TIR lenses. The rich spectrum of optical designs uncovered here could be useful for high performance (1) photovoltaic concentrators, (2) LED collimators, and (3) infrared optics.

## Acknowledgment

Heylal Mashaal is the recipient of a Howard and Lisa Wenger graduate scholarship.

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