## Abstract

We provide a general formulation for the design of any dual-surface aplanatic Fresnel optic (including combinations of refractive and reflective surfaces), with categories of devices that had not previously been recognized. Raytrace simulations for representative Fresnel aplanats in collimation (illumination) mode reveal compact designs with radiative efficiencies close to those of their aplanatic continuous non-Fresnel counterparts, and optical performance approaching the thermodynamic limit for radiative transfer.

© 2017 Optical Society of America

## 1. Introduction

This paper addresses how to design aplanatic Fresnel optics (faceted or segmented, and including combinations of refractive and reflective surfaces), including previously unrecognized categories of Fresnel devices, as well as providing raytrace evaluations of optical performance for representative collimator designs. “Aplanatic” refers to the complete elimination of spherical aberration and coma (the two leading orders of geometric aberration), for which a minimum of two optical contours are required.

Fresnel *lenses* were originally developed almost two centuries ago as alternatives to conventional continuous-surface collimators for lighthouses [1]. The lenses comprise concentric annular sections acting as individual refracting surfaces that collimate wide-angle light emission. This allows a dramatic reduction in lens thickness - and hence in the amount of material required - as well as far easier fabrication, compared to conventional lenses. Fresnel strategies have also been developed for mirror analogs and lens-mirror combinations, for both imaging and nonimaging applications, and for both illuminators and concentrators [1–5].

Aplanats were originally developed as dual-mirror telescopes. However, the general formulation is based on two or more optical surfaces (refractive R, or reflective X) serving as the degrees of freedom to eliminate both spherical aberration and coma. Hence, for the simplest case of dual-contour aplanats, there are four categories: XX, XR, RX and RR [6–8], all of which have recently been explored at basic and applied levels, in some cases discovering fundamentally new classes of aplanats [9–13]. The first and second letter of this nomenclature refer to the aperture and source surfaces, respectively.

By removing both coma and spherical aberration, aplanats offer two major advantages relative to conventional optics. One is superior image fidelity. But the other arguably less recognized benefit is improved radiative transfer [9–13], both as flux concentrators and as illuminators (collimators): the consideration of which prompted this study.

The motivation here is benefitting simultaneously from the nominally distinct virtues of *both* Fresnel *and* aplanatic optics. This specifically relates to the ability of aplanats to approach the thermodynamic limit for radiative transfer [9–13], with the well-established compactness of the Fresnel approach (only rarely developed with the added performance benefit of aplanatism [14–16]). Previously, explicit consideration of Fresnel aplanats had been restricted to a single category of biconvex *lenses* [14–16]. And the evaluation criteria were limited to those of image quality (in contrast to maximum-performance radiative transfer). Furthermore, in [14], it was concluded that the Abbe sine condition, the necessary condition for eliminating coma (Eq. (3) below), can only be fulfilled if the Fresnel facets are formed on a curved surface. We will demonstrate, however, that this requirement is unnecessary. The notion of *nonimaging* Fresnel aplanatic concentrators (designed for extended rather than point sources) that approach the thermodynamic limit was first introduced in [17,18] for 2D (linear) solar concentration.

This study steps beyond prior findings in three significant ways. First, the Fresnel aplanatic formalism is developed and applied not simply to lenses, but also to reflective-refractive and refractive-reflective optics. Second, even within the pure-lens (RR) category, there is a fundamentally new sub-category that was reported in [13], for which the *Fresnel* design had remained unrecognized, and is presented here. And third, rather than appraising image fidelity, the value and assessment of all Fresnel aplanats derived here are for achieving radiative transfer approaching the thermodynamic limit, which is a less commonly appreciated virtue of aplanatic optics in general [6–13]. Consequently, exploring image fidelity and the figures of merit that quantify it (e.g., the point-spread and modulation transfer functions) are not addressed here.

For the collimation perspective (i.e., illumination from a light source at the system focus to collimated far-field radiation, rather than concentration from a far-field source to an absorber at the system focus), the lower bound for the far-field half-angle *θ _{exit}* as a function of the ratio of exit aperture to source (emitter) area is then

*NA*is the exit (collimation) numerical aperture,

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

_{s}*θ*is the source emission half-angle,

_{s}*n*

_{s}is the refractive index of the medium surrounding the source, and

*n*

_{exit}is the refractive index of the medium into which the radiation emerges (see Fig. 1). In concentrator mode, Eq. (1) provides the thermodynamic limit for flux concentration, but with

*NA*referring to that of incident radiation and

_{exit}*NA*to that of radiation at the absorber.

_{s}We proceed by presenting a new recipe for generating any type of Fresnel aplanat, together with illustrative examples and evaluations of representative collimators. First, we review the recent analytical derivations for dual-contour continuous-surface non-Fresnel aplanats [10–13]. The governing equations pertain to *all* aplanats. As such, Section 2.1 reviews them in a form similar to our recent presentations on a variety of dual-surface aplanatic optics [11–13]. Only later in Section 2.2 do we turn to the new aspects of generalizing them to Fresnel designs.

## 2. Design formalism

#### 2.1 Review of the formalism for continuous-surface non-Fresnel aplanats

The conditions that need to be satisfied are [9–13] (see Fig. 1):

*n _{i}* is the refractive index along ray trajectory

*L*.

_{i}*r*denotes radial position at the exit aperture, corresponding to a ray emitted from the focus along

*L*at the arbitrary angle ϕ (between

_{3}*L*and the optic axis). Namely, |ϕ| ranges from 0 to its maximum value at the device’s

_{3}*NA*, with Eqs. (2) and (3) pertaining to each and every value of ϕ. The Abbe sphere radius

*F*is obtained by connecting the extension of any

*L*to that of its associated

_{1}*L*. Also, the subscripts on the constants in Eqs. (2), (3) and (7) are intended to stress that they are not the same constant.

_{3}The combination of Fermat’s principle, Abbe’s sine condition and Snell’s law at one of the surfaces yields three coupled equations:

*and*

_{a}*denote the aperture and source surfaces, respectively. For conciseness, we have defined*

_{s}*m*= (

*Y*–

_{s}*Y*)/(

_{a}*X*–

_{s}*X*). The boundary conditions constitute inputting

_{a}*Y*(

_{a}*R*) =

_{a}*H*and

_{a}*Y*(

_{s}*R*) =

_{s}*H*.

_{s}In Eqs. (4)-(6), the parameters *c, p* and *s* are each either −1 or + 1. The sign of ‘*c*’ is determined by whether ray *L _{2}* at the aperture surface crosses the optic axis (in the cross-section of Fig. 1), with

*c*= 1 when it does. ‘

*p*’ is determined by whether the slope of the aperture surface monotonically decreases (

*p*= 1) or increases (

*p*= −1) as it approaches the optic axis. ‘

*s*’ is determined by the direction of the Abbe radius:

*s*= −1 when the line from the focus along

*L*

_{3}to the extension of

*L*

_{1}is above the focus, and

*s*= 1 otherwise.

The possible combinations of {*c,p,s*} infer 8 basic solution categories, but not all of them correspond to physically admissible solutions. Categories RX and XX both yield six classes of physically admissible solutions. Category XR yields five, and category RR yields three. The solution of Eqs. (4)-(6) provides the coordinates of both the aperture (*X _{a}*,

*Y*) and source (

_{a}*X*,

_{s}*Y*) contours, readily computed with software such as Matlab or Mathematica.

_{s}#### 2.2 Fresnel (faceted) generalization

We now divide both optical surfaces into Fresnel zones. For each zone (index *i*), the aperture and source contours are obtained as an independent aplanatic solution of Eqs. (4)-(6), with each segment having boundary conditions (points) *P _{i}*(

*R*,

_{p,i}*H*) and

_{p,i}*S*(

_{i}*R*,

_{s,i}*H*) (Fig. 2). Each zone must satisfy the Abbe sine condition with its own constant Abbe sphere radius

_{s,i}*F*:

_{i}*i*+ 1 start where zone

*i*ends, although the design formalism can accommodate gaps between zones in order to facilitate manufacturing. The blockage of rays by neighboring zones can be resolved by tuning each zone’s boundary conditions.

## 3. Examples of Fresnel aplanats

There is considerable flexibility in designing Fresnel aplanats due to the degrees of freedom in selecting the boundary points of each Fresnel zone. The formulation is general for any combination of materials (i.e., refractive indices *n _{1}*,

*n*,

_{2}*n*in Fig. 1). The examples shown here are restricted to the simpler and more practical combination of an air environment and a single dielectric material of refractive index

_{3}*n*. Figure 3 shows several examples. Fresnel contours are plotted with a solid black line. Their corresponding continuous, non-Fresnel designs are plotted with overlay dashed curves. Dielectric sections are colored.

Figure 3(a) presents an RX Fresnel aplanat (with *c* = −1, *p* = −1, *s* = 1, designated as sub-category RX-2A in [11]), with all aperture Fresnel zones constrained to have the same height, for the case {*n _{1}* = 1,

*n*=

_{2}*n*= 1.52}. In designing for high degrees of curvature, while maintaining a constant height of the Fresnel zones, one must be wary of the required widths of the outermost Fresnel zones becoming so small as to create a grating-like dispersion. Moreover, the analyses in this paper are limited to monochromatic radiation. Unfortunately, there are no generalizations for quantifying chromatic aberration - neither as they depend on the fundamental design category of the aplanat, nor as a function of

_{3}*NA*. This has been illustrated by recent evaluations of dispersion losses in radiative efficiency, which vary from only a few percent to tens of percent in continuous aplanats with refractive components [11–13].

Figure 3(b) presents a curved RR Fresnel aplanat (with *c* = −1, *p* = −1, *s* = −1, sub-category RR-1A in [13]), for the case {*n _{1}* =

*n*= 1,

_{3}*n*= 1.52}. The boundary points of the zones of both contours are constrained to reside on circles of different radii centered at the focus - illustrating the ability to tailor Fresnel aplanats on a curved contour.

_{2}Figure 3(c) shows an RR Fresnel aplanat (with *c* = −1, *p* = −1, *s* = −1, sub-category RR-1A in [13]), designed with aperture zones of equal widths, for the case {*n _{1}* =

*n*= 1.52,

_{3}*n*= 1}.

_{2}Figure 3(e) presents a continuous non-Fresnel XR aplanat (with *c* = −1, *p* = 1, *s* = −1, sub-category XR-3 in [12]), with a relatively large aspect ratio of 1.75, for the case {*n _{1}* =

*n*= 1,

_{2}*n*= 1.52}. This device intrinsically contains a central gap region such that 9% of the emitted radiation is not tailored. Figure 3(d) then presents its corresponding Fresnel aplanat, constrained to have equal heights for the facets of the aperture mirror. In the Fresnel aplanat of Fig. 3(d), we introduced a central continuous non-Fresnel RR aplanat of the type of Fig. 3(c) in order to recover the central-gap loss. A similar procedure was applied in [13], which demonstrated that a hybrid aplanatic design - namely, a design that amalgamates distinct aplanatic designs in order to reduce gap losses - can offer superior performance. The side-by-side comparison of the continuous non-Fresnel aplanats to their Fresnel counterparts in Fig. 3 demonstrates the lower aspect ratios and material savings achievable with the Fresnel devices.

_{3}## 4. Optical performance

The axisymmetric RR Fresnel aplanat of Fig. 3(c) and XR Fresnel aplanat of Fig. 3(d) were chosen for raytrace evaluation (with OptiCad^{®} Version 10) as collimators, with a monochromatic Lambertian light source (an approximation for common light-emitting diodes, LEDs) at the focus. The source size chosen corresponds to the value given by the thermodynamic limit of Eq. (1). All dielectric components are assumed to have a refractive index of 1.52. Mirror reflections, and absorption in the dielectric, are not accounted for explicitly, but can readily be assessed. In addition, Fresnel reflections are ignored, since they depend on the particular surface coating used, and, in any event, are straightforward to quantify. The selected designs merely serve as illustrative examples, and are not optimized or tailored to a specific application.

For clarity of illustration, and to confirm the accuracy of the raytrace simulation, we first generated results for a point source - illustrated in Fig. 4. The aim was to check for nominally perfect collimation, along with a first-order estimate of inherent optical losses.

Results with the actual extended source are presented in Figs. 5 and 6, as radiative efficiency (the fraction of radiation emitted within far-field projected solid angle Ω corresponding to polar half-angle θ, Ω = π sin^{2}(*θ*)), as a function of Ω/Ω_{th} where Ω_{th} = π sin^{2}(*θ _{exit}*). Graphs are presented for collimation half-angles 5, 10, 20 and 40 mrad. For an ideal device that achieves the thermodynamic limit, all emitted radiation is contained (at far-field) within Ω

_{th}.

Figure 5 also includes results for the corresponding continuous non-Fresnel aplanats, for comparison. For example, the RR Fresnel aplanat of Fig. 3(c) achieves a radiative performance close to that of its corresponding non-Fresnel aplanat at small *θ _{exit}* (Fig. 5(a)). Figure 5(b) compares the performance of the XR Fresnel aplanat of Fig. 3(d) (but without the addition of a central lens to lessen gap losses) and its corresponding non-Fresnel counterpart, showing almost identical performance for all source sizes. Figure 6 then shows the performance of the same XR Fresnel aplanat of Fig. 5(b) when a central RR aplanatic lens is introduced toward recovering the central gap loss of 9%.

In all instances, optical losses result mainly from emitted rays that (a) miss the secondary, or (b) intersect an adjacent Fresnel zone and are emitted at angles beyond the intended degree of collimation, both of which are a function of source size. Larger values of *θ _{exit}* correspond to larger extended sources, and hence greater deviations from the point-source approximation, in which case higher-order aberrations (above spherical aberration and coma) play a progressively greater role. One manifestation is the worsening of radiative efficiency with

*θ*. For the Fresnel aplanat considered in Fig. 5(a) - where there are essentially no losses due to emitted rays missing the secondary - the approximately 7% and 13% losses for the 5 mrad and 10 mrad cases, respectively, which remain even at large far-field angle, stem from those rays emitted from an

_{exit}*extended*source that do not intersect the intended Fresnel zone. These losses increase with source size.

A more subtle issue is the value of far-field angle at which the efficiency curves in Figs. 5 and 6 asymptote, most notably as *θ _{exit}* increases. As the size of the primary optical element (i.e., the one collecting directly from the emitter) increases relative to the size of the extended source, geometric losses become smaller. This in turn means that the efficiency curves plateau at higher values. This trend can be seen by comparing the asymptotic results from Fig. 5(a) with those from Fig. 5(b) and Fig. 6. Namely, the size of the primary element relative to the source for Fig. 5(a) is considerably greater than for the designs of Fig. 5(b) and Fig. 6.

Finally, we should note that the same types of Fresnel aplanats portrayed here for collimation could equally well be applied to radiation concentration, with the roles of source and target interchanged. Incident radiation covering a small *NA* would be concentrated onto an absorber centered at the focus at large *NA*.

## 5. Conclusions

A new recipe for the design of dual-surface Fresnel aplanats has been introduced. The procedure constitutes a generalization of the design of continuous non-Fresnel dual-surface aplanats described previously in [9–13]. A key step is the independent selection of the boundaries of each Fresnel zone - a degree of freedom that permits the accommodation of constraints such as Fresnel facets all having the same height (Fig. 3(a)), the same width (Fig. 3(c)), or residing on a surface of convenient curvature (such as a spherical cap - Fig. 3(b)).

A comparison of Fresnel aplanats and their corresponding continuous non-Fresnel counterparts demonstrates that significant material savings and superior compactness can be achieved. Raytrace simulations confirm collimation performance approaching the thermodynamic limit, and quantify that acceptably low optical losses can be achieved in compact collimators.

## Funding

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

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