## Abstract

So far, illumination optics design for extended sources (like LEDs) has been mainly treated as perturbations of ideal point sources without any extent. Such approaches work well provided that this ideal case is approximately met. However, the demand for very compact luminaires equipped with modern high-brightness LEDs results in configurations where the actual size of the sources cannot be ignored. Here, we develop a “wavefront tailoring” method producing prescribed illumination patterns while fully encompassing the extended source size. We combine this technique with a wavefront-coupling design scheme, obtaining a powerful tool for creating optics for large Lambertian emitters and prescribed intensity emissions. As an example, we design a highly compact and efficient freeform lens delivering a constant illuminance pattern. Importantly, the presented design strategy can work with various methods for calculating the optical surfaces, such as direct methods or optimization routines.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Shaping the emission of a light source into a prescribed energy distribution is the key task of an illumination system. Several design algorithms exist for the ideal case of point-like sources (zero-étendue emitters) [1–3], when the source size is assumed to be negligible compared to that of the optic. However, designing 3D illumination optics in the more realistic case of extended sources (finite-étendue emitters) remains a challenging problem. The illumination community is currently paying much attention to this topic, given the diffusion of high-brightness extended LEDs. Indeed, the increasing demand for compact LED luminaires calls for configurations where the source lies very close to the optical surfaces and its spatial extent cannot be disregarded. Point-source design algorithms are often not applicable in these cases. Therefore, ongoing research focuses on methods able to deal directly with finite-étendue emitters. Strategies based on feedback optimization use a point source in the initial design of the optic, which is then iteratively modified to adapt to real large emitters [4,5]. However, such approaches do not consider the finite nature of the source from the beginning and are typically less effective for compact configurations [6]. “Extended source” methods do exist, like the ones based on solutions of a differential equation [7,8] or on phase space formulations [9,10]; but they have been applied so far only to 2D and to rotationally symmetric 3D geometries. Also, alternative proposals of 3D optics for extended emitters are currently limited to rotational symmetry [11]. In many illumination problems, though, the source and the desired emission pattern are not rotationally invariant. Therefore, such configurations ultimately require freeform optical surface shapes (that are not restricted to rotational symmetry). A technique suited for designs going beyond rotational symmetry was presented in [12] and is called “wavefront-coupling” design method. It characterizes an extended source in terms of “input” wavefronts and the illumination pattern in terms of “output” wavefronts. The optic is calculated as the one connecting the input wavefronts with the output ones. Over the years, this approach has been extensively applied to the design of freeform optics for prescribed illumination [13]. However, a major limitation lies in the description of the output wavefront rays according to the illumination pattern they produce (such a procedure is called “wavefront tailoring”). In general, the direction of a ray is expressed at a specific 3D point by the optical momentum vector $\mathbf{p}=(p,q,s)$; the components $p$, $q$, $s$ are the three direction cosines of the ray multiplied by the local index of refraction $n$. Since $\Vert \mathbf{p}\Vert =n$, $\mathbf{p}$ has two independent components (it is customary to choose $p$ and $q$), and we write it as $\mathbf{p}=(p,q,{({n}^{2}-{p}^{2}-{q}^{2})}^{1/2})$. In the wavefront tailoring process, the optical momenta of rays are defined over a planar reference surface, spanned by two coordinate parameters (say $x$ and $y)$. So far, wavefront tailoring has been performed assuming that the two independent components of the optical momentum depend on only one of these coordinates each: in particular, $p=p(x)$ and $q=q(y)$. This approximation, called “separation of variables”, reduces the degrees of freedom of the optical momenta, making the wavefront tailoring task much easier. The price to pay is a severe restriction to the generality of the method: the approximated wavefront tailoring technique is effective only for a restricted range of emission angles (up to 20–25° total angular aperture [12,13]), thus largely limiting the applicability of the wavefront-coupling design method.

In this paper, we overcome this problem and present a generalized procedure for performing wavefront tailoring. Using the conservation properties of the fundamental quantity called étendue 2D [14], we develop a theoretical framework in which the components of the optical momentum $\mathbf{p}$ depend simultaneously on both the coordinates of the reference plane: $p=p(x,y)$ and $q=q(x,y)$. This gives the utmost flexibility in specifying individual ray directions in an output wavefront: contrary to the previous approximated approach, at every point of the reference plane rays are now defined freely with the only condition that they must be perpendicular to a wavefront. This fact generalizes the applicability of the wavefront tailoring technique: the method can now model more general emission patterns, without restrictions on the emission angle range. We incorporate this new wavefront tailoring tool into a wavefront-coupling design strategy, building a flexible method for illumination optical design with extended Lambertian sources. As a demonstrative example, we design an efficient and compact freeform lens coupled to an extended LED, providing a square illuminance pattern subtending a maximum angular aperture of about 80°.

Notice that we have not mentioned any method for calculating the optical surfaces. Indeed, one remarkable feature of our design approach is modularity. The proposed wavefront tailoring method provides the output wavefronts realizing a prescribed illumination, independently of the technique employed for calculating the optic. Any surface calculation method able to couple input wavefronts (obtained from the geometry of the source, [12]) with output ones can be employed in a wavefront-coupling problem. Here, we use the Simultaneous Multiple Surface method (SMS, [15]). Another option is to adopt optimization routines.

In the following sections, we first present the theory and necessary background of the generalized wavefront tailoring method. We then describe the design example based on wavefront tailoring coupled to the SMS method, and finish with the conclusion and an outlook.

## 2. GENERALIZED WAVEFRONT TAILORING

We consider a system comprising an extended emitter, an optic and a planar target. The emitter is represented by wavefronts stemming from some of its edge points. These “input” wavefronts, travelling from emitter to optic, carry information about the source shape and size. The “output” wavefronts define the irradiance (power per unit area on the target surface) or intensity (power per unit solid angle) pattern that the optic produces; this process can be described in terms of “pinholes”. For instance, in the case of prescribed irradiance in 2D geometry (as detailed in [14]), a moving pinhole is placed at the target. The edge rays (rays representing, locally, the edge directions of light propagation) through this moving pinhole define the illumination cone at each point of the target. These edge rays may be described by two output wavefronts. For the case of prescribed intensity, instead, the moving pinhole is placed at the exit aperture of the optic. The edge rays through this pinhole define the emission cone at each point of the aperture and thus, which directions it illuminates. Also in this case, these edge rays can be described by two output wavefronts. The optic must then couple the input and output wavefronts, either in the case of prescribed irradiance or prescribed intensity.

Here, we deal with the generation of intensity patterns in 3D. We then have a moving pinhole placed at the exit aperture of the optic. The edge rays crossing this pinhole leave the optic and eventually hit a very distant target: there, they define the position and size of the source images projected by infinitesimal areas of the optic (“pinhole images”). The superimposition of all these pinhole images gives the total illumination pattern. Again, the optic is required to transform the input wavefronts into the output ones. We assume that the optic simultaneously controls four wavefronts. In particular, we consider four corner points of a source with rectangular shape and four corners of the pinhole images: the pinhole images are then quadrilateral in shape, even though their sides might not be straight (see Fig. 1).

The generalized wavefront tailoring method uses two main concepts: the relation between intensity and area of the optic illuminating a given direction (“lit area”), and the conservation of étendue 2D. We describe them in the following.

#### A. Relation between Wavefronts, Lit Area, and Generated Intensity

The notion of wavefront is related to the concept of normal congruence. A set of rays whose trajectories are normal to a theoretical one-parameter family of surfaces is a *normal ray congruence* [15]. These (abstract) surfaces are the wavefronts, and the family parameter is the optical path length. One wavefront is sufficient for identifying a normal congruence. We fix the optical momentum of rays belonging to an output wavefront (i.e., to an output normal congruence) over a reference plane ${\mathrm{\Pi}}_{0}$. In practical problems, ${\mathrm{\Pi}}_{0}$ is placed close to the expected position of the optic exit aperture. Without losing generality, we identify ${\mathrm{\Pi}}_{0}$ with the $z=0$ plane in a $x-y-z$ Cartesian coordinate system, like in the system schematized in Fig. 1. If every point ($x$, $y)$ of ${\mathrm{\Pi}}_{0}$ is crossed by only one ray of the normal congruence, the momentum $\mathbf{p}$ of the ray set is described over ${\mathrm{\Pi}}_{0}$ by the momentum function

A connection between the wavefronts exiting the optic and the output intensity distribution is established with the following construction. We take four functions ${p}_{\mathrm{min}}(x,y)$, ${p}_{\mathrm{max}}(x,y)$, ${q}_{\mathrm{min}}(x,y)$, and ${q}_{\mathrm{max}}(x,y)$, representing the minimum and maximum values of the $p$ and $q$ components of $\mathbf{p}(x,y)$. We arrange them into four combinations, each one giving the momentum function of an output normal congruence:

In the most general case, we would need eight functions to define the four ${\mathbf{p}}_{k}$ (two independent components for each ${\mathbf{p}}_{k})$. However, in Eq. (2) we have used just four functions ${p}_{\mathrm{min}}$, ${p}_{\mathrm{max}}$, ${q}_{\mathrm{min}}$, and ${q}_{\mathrm{max}}$, which correspond to rectangular pinhole images. This assumption works well in design cases with plane symmetries, like the example that we will discuss in Section 3. Situations lacking symmetry or in which the shape of the source projected images is very different from that of the desired pattern may be tackled by enlarging the optic size; in this case, the size of the pinhole images decreases and the influence of their shape on the total pattern becomes less important.

The concept of “lit area” provides the connection between functions ${p}_{\mathrm{min}}$, ${p}_{\mathrm{max}}$, ${q}_{\mathrm{min}}$, ${q}_{\mathrm{max}}$, and the intensity pattern generated by wavefronts $\mathrm{WF}{k}_{o}$. Let the optical momentum be defined over a rectangular region ($[-{x}_{L},+{x}_{L}]$, $[-{y}_{L},+{y}_{L}])$ of the reference plane; we call this area “reference exit aperture”. Consider the two functions ${p}_{\mathrm{min}}(x,y)$, ${p}_{\mathrm{max}}(x,y)$ first. They may be thought of as surfaces in $(x,y,p)$ space.

In Fig. 2(a) an example is shown with ${p}_{\mathrm{min}}(x,y)<{p}_{\mathrm{max}}(x,y)$. At each point ($x$, $y$) of the reference exit aperture, the $p$ component of a generic emission direction ($p$, $q$) is limited by surfaces ${p}_{\mathrm{min}}$ and ${p}_{\mathrm{max}}$. The two surfaces enclose a volume in $(x,y,p)$ space: cutting this volume with a horizontal plane $p={p}_{I}$ gives a surface ${s}_{I}$. The projection of ${s}_{I}$ over plane ($x$, $y$) is the lit area $A({p}_{I})$: the set of points of the reference exit aperture emitting toward directions with the $p$ component of the optical momentum equal to ${p}_{I}$. The same argument is applied to two functions ${q}_{\mathrm{min}}(x,y)$, ${q}_{\mathrm{max}}(x,y)$ in $(x,y,q)$ space to determine the lit area $A({q}_{I})$: the portion of the reference exit aperture emitting into directions whose $q$ component is a specific ${q}_{I}$ [see Fig. 2(b)]. Finally, the area $A({p}_{I},{q}_{I})$ of the reference exit aperture emitting into direction $({p}_{I},{q}_{I})$ is the intersection of $A({p}_{I})$ and $A({q}_{I})$ [Fig. 2(c)]. Rigorously, $A({p}_{I},{q}_{I})$ is formed by the points of the exit aperture for which

Note that the lit area $A({p}_{I},{q}_{I})$ may have any shape. By taking all directions $({p}_{I},{q}_{I})$ in the emission range of interest, the lit area function $A(p,q)$ is built. Then, the associated intensity distribution $I(p,q)$ follows from the known formula

where $L$ is the luminance of the emitter (see for example [14]). The relation between lit area and intensity takes the simple form of Eq. (5) when luminance is constant. The following conditions hold: the luminance $L$ is independent of the angular coordinates (as it happens in Lambertian sources) and independent of the spatial coordinates (i.e., the emitter is uniform). These assumptions match well the characteristics of LED chips. In addition, we assume that the optic (to be calculated) is lossless ($L$ is conserved). If in a design problem the above assumptions are not met, spatial or angular non-uniformities of luminance can be tackled by adopting an iterative scheme: the optic calculated using Eq. (5) would serve as the starting configuration of the design process. In subsequent design steps, the initially prescribed intensity distribution may be modified to compensate the effects of deviations from uniform and Lambertian emitter luminance; the optic would then be re-calculated based on the modified target intensity distribution.Equation (5) is the cornerstone of the wavefront tailoring method. In a standard illumination problem, a prescribed intensity distribution $I(p,q)$ is assigned: the corresponding lit area $A(p,q)$ follows from Eq. (5). The problem is then to find the set of functions ${p}_{\mathrm{min}}(x,y)$, ${p}_{\mathrm{max}}(x,y)$, ${q}_{\mathrm{min}}(x,y)$, and ${q}_{\mathrm{max}}(x,y)$ generating the desired lit area. In general, manipulating four functions of two variables is often unpractical, given the amount of degrees of freedom involved. In the following, we show how to reduce four two-variable functions to a set of one two-variable and three one-variable functions. This is achieved by applying the concept of étendue 2D to a normal congruence.

#### B. Consequences of the Conservation of étendue 2D

In three dimensions, the optical invariant known as étendue 2D or Lagrange invariant can be written as (see [14])

Now, the fundamental relation described by Eq. (7) has to hold for the momentum components of the output wavefronts $\mathrm{WF}{k}_{o}$ built in Eq. (2). Applying Eq. (7) to the four optical momenta of Eq. (2) yields four equations:

The crucial information contained in Eq. (12) is that the four two-variable functions ${p}_{\mathrm{min}}(x,y)$, ${p}_{\mathrm{max}}(x,y)$, ${q}_{\mathrm{min}}(x,y)$, ${q}_{\mathrm{max}}(x,y)$ can be written in terms of ${p}_{\mathrm{min}}(x,y)$ plus three functions of one variable: ${g}_{1}(y)$, ${g}_{2}(x)$, and ${g}_{3}(y)$. Thus, to do wavefront tailoring we have to fix one two-variable and three univariate functions. For example, we start by defining ${p}_{\mathrm{min}}(x,y)$; then ${q}_{\mathrm{min}}(x,y)$ is found by fixing ${g}_{1}(y)$. As we will show, information on ${g}_{1}(y)$ can be inferred from symmetries in the pattern geometry, if present. The remaining functions ${g}_{2}(x)$ and ${g}_{3}(y)$, defining ${p}_{\mathrm{max}}(x,y)$ and ${q}_{\mathrm{max}}(x,y)$, can be chosen by a comparison between the intensity pattern resulting from Eq. (5) and the desired intensity distribution.

Notice that, in the previous “separation of variables” approach, wavefront tailoring was required to manipulate four functions of one variable. In the new version presented here, the supplementary complication is limited to replacing a function of one variable with one of two variables. The added value is a method able to tackle generalized intensity configurations with wider emission angles.

## 3. DESIGN EXAMPLE

As a demonstrative application, we assign a target illumination pattern and use the wavefront tailoring scheme to generate the associated output wavefronts. With them, we calculate an SMS lens achieving the corresponding illumination functionality, when applied to an extended source. The chosen target pattern presents a square extended region with constant illuminance and steep falloffs at its four edges [Fig. 3(a)]; the illuminance distribution is evaluated over a far-field plane $z={z}_{\text{target}}$ normal to the $z$ axis (we set ${z}_{\text{target}}=1000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm})$. We identified this pattern as a good combination of a large uniform central region and a border with a fast yet gradual decreasing illuminance. Generating extremely steep illumination cutoffs with extended sources is indeed very demanding and may influence an important fraction of the optic that we want to calculate, with the risk of not having enough degrees of freedom left for controlling the constant-illuminance section (see for instance [8,18]). The pattern is oriented as in Fig. 3(b), and symmetric with respect to planes $x=\pm y$, $x=0$, and $y=0$; when seen from the origin, the region with constant illuminance subtends a 30° half-angle on the $x=0$ and $y=0$ planes [point ${\mathbf{A}}^{\prime}$ in Fig. 3(b)]. The maximum emission half-angle is about 39.2°, along the pattern diagonals (planes $x=\pm y$), subtended by corner points like ${\mathbf{B}}^{\prime}$ in Fig. 3(b).

To generate a uniform far-field illuminance, an optic has to deliver an intensity distribution of the form $I={I}_{0}/{\mathrm{cos}}^{3}\text{\hspace{0.17em}}\theta $, where ${I}_{0}$ is the intensity value at $\theta =0\xb0$ (see [14]). Using the fact that $\mathrm{cos}\text{\hspace{0.17em}}\theta =\sqrt{1-{p}^{2}-{q}^{2}}$, by means of Eq. (5) we can find the lit area function of interest:

assuming for simplicity that the source luminance is $L=1$. We must model functions ${p}_{\mathrm{min}}(x,y)$, ${p}_{\mathrm{max}}(x,y)$, ${q}_{\mathrm{min}}(x,y)$, ${q}_{\mathrm{max}}(x,y)$ to generate the assigned lit area function Eq. (13). We define them over a square region of the reference plane ${\mathrm{\Pi}}_{0}$, formed by points $(x,y,0)$ for which $-5<x,y<5$ (in millimeters). This choice influences the shape and size of the real exit aperture of the optic. A way of fixing functions like $p(x,y)$ is to choose “control points” in space $(x,y,p)$ and fit them with appropriate polynomial functions. This allows us to exploit the symmetries of the pattern. For instance, function ${p}_{\mathrm{min}}(x,y)$ may be written asFirst, we work on the $x$ axis and tailor ${p}_{\mathrm{min}}(x,0)$ and ${g}_{2}(x)$(or, equivalently, ${p}_{\mathrm{max}}(x,0)$) as follows. We choose ${p}_{\mathrm{min}}(x,0)$ and ${p}_{\mathrm{max}}(x,0)$ so that the $x$-side ${A}_{x}$ of the lit area behaves like ${A}_{x}(p)\propto {(1-{p}^{2})}^{-2}$ [see Fig. 4(a)], since a uniform illumination in a two-dimensional case is given by $I={I}_{0}/{\mathrm{cos}}^{2}\text{\hspace{0.17em}}\theta $ (see [14]). In this way, the prescribed lit area function Eq. (13) is matched along the $x$ axis for directions $(p,0)$. Given the pattern symmetry, it must be ${A}_{x}(p)={A}_{x}(-p)$: this is achieved by requiring that ${p}_{\mathrm{max}}(x,0)=-{p}_{\mathrm{min}}(-x,0)$. Function ${g}_{2}(x)$ is thus entirely determined.

Second, we specify ${p}_{\mathrm{min}}$ along the edge of the reference exit aperture. We impose that border points of the aperture emit toward the edge of the light pattern, as in Fig. 4(b). For instance, point $\mathbf{A}=(5,0,0)$ of the aperture illuminates point ${\mathbf{A}}^{\prime}$ on the target plane, corresponding to direction ($\phi =0\xb0$, $\theta =30\xb0$). Corner point $\mathbf{B}=(5,5,0)$, instead, emits into direction ${\mathbf{B}}^{\prime}$ defined by ($\phi =45\xb0$, $\theta \approx 39\xb0$). Intermediate points of the exit aperture edge emit into intermediate positions of the pattern edge, building up the border of the illuminated region.

Finally, ${p}_{\mathrm{min}}$ is finely tuned by looking for the best match between the related lit area function and the target one, Eq. (13). Indeed, knowledge of ${p}_{\mathrm{min}}$ and ${g}_{2}$ automatically fixes functions ${g}_{1}$ and ${g}_{3}$, and so the full set ${p}_{\mathrm{min}}$, ${p}_{\mathrm{max}}$, ${q}_{\mathrm{min}}$, ${q}_{\mathrm{max}}$: from this, the “effective” lit area is calculated and compared with the target one. Additional control points in $(x,y,p)$ space are used in this phase, to adjust ${p}_{\mathrm{min}}$ over regions not already covered in the previous steps. The final set (${p}_{\mathrm{min}}$, ${p}_{\mathrm{max}}$, ${q}_{\mathrm{min}}$, ${q}_{\mathrm{max}}$) defines four output wavefronts to be used in the SMS calculation, according to Eq. (2).

The source chosen for the design is a $1\times 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{mm}}^{2}$ Lambertian emitter, normal to the $z$ axis and emitting into a hemisphere. The SMS calculation is detailed in Supplement 1. Close-up views of the resulting lens, with two freeform surfaces, are shown in Fig. 5(a). The material used is Polycarbonate, common in LED lighting. The 1.41-mm source diagonal and the total height of 2.36 mm yield an extremely compact system with height-to-source ratio less than 1.7. The illuminance pattern on a 1-m-away detector is shown in Fig. 5(b) and compared with the target pattern. The extent of the square flat top along the $x$ and $y$ axes is very close to the desired $\pm 30\xb0$ emission. The uniformity in the flat-top area is ${E}_{\mathrm{min}}/{E}_{\text{average}}\cong 0.9$, where $E$ is the illuminance (well above the typical 0.5 minimum value required by European standards [19]). The lens has high efficiency: 81% of the source emission reaches the target plane (considering only geometrical losses).

Some deviations of the pattern in Fig. 5(b) from the prescribed distribution appear in the corner regions. These differences come from the use of an SMS lens with two surfaces to couple the four input wavefronts from the source corners with the four output wavefronts generated by the wavefront tailoring method. A “perfect” wavefront coupling would require calculating four SMS surfaces—a challenging task which has no solution yet. Nevertheless, four-surface systems tend to be too complicated and bulky for illumination applications, which instead favor compact solutions with fewer surfaces. Additional details on these points are provided in Supplement 1. As a final comment, we notice that ideal, total control of the light emitted by the source border and of the intensity pattern produced by the optic can be achieved only with an infinite number of optical surfaces controlling the same infinite number of wavefronts.

## 4. CONCLUSION

The presented generalized wavefront tailoring method constitutes a major advance in illumination design for extended sources. It allows us to directly tackle prescribed emission patterns more general than rotationally symmetric ones. The resulting design strategy, based on the coupling of wavefronts describing the optical features of a system, is particularly suited for freeform illumination designs where the extension of the source must be fully taken into account. We demonstrated the capabilities of our wavefront tailoring technique for the case of a target illuminance pattern with no rotational symmetry and maximum total emission angle close to 80°. The designed system is extremely compact, with total height over source-size ratio of just 1.7, and efficient (optical efficiency is higher than 80%). To our knowledge, this is the first time that such a compact and efficient freeform lens is proposed which achieves a near-ideal target illumination pattern.

At the current stage of development, the wavefront tailoring method can be immediately applied to extended sources with constant luminance (uniform Lambertian emitters), which match most single-chip LEDs very well. In situations where multi-chip LEDs are used, the method can be embedded in an iterative routine to account for possible non-uniformities of luminance. Another foreseen extension of our method could be to increase the number of controlled wavefronts (currently, four), which determines the shape of the pinhole images used in constructing the lit area function. Increasing this number would allow an even better control over the optic emission at pinhole level; plus, it would further enhance the description of the source emission, especially for source shapes departing from a quadrilateral geometry.

It is important to highlight that the wavefront tailoring method is fully independent of the technique used for practically calculating the optic. Thus, various design options are possible for determining the optical surfaces, either direct methods (as in our SMS example) or through optimization routines. Due to this compatibility and the excellent optical performance, we expect that the wavefront tailoring method will find broad application in a wide range of modern illumination design problems.

## Funding

FP7 People: Marie-Curie Actions (608082 - ADOPSYS); Fonds Wetenschappelijk Onderzoek (FWO); Industrial Research Fund; Methusalem; Onderzoeksraad of the Vrije Universiteit Brussel.

## Acknowledgment

We thank Vrije Universiteit Brussel for providing equipment and software licenses; EU Marie Curie ITN “ADOPSYS”; FWO-Vlaanderen for providing a post-doctoral grant to Fabian Duerr.

See Supplement 1 for supporting content.

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