## Abstract

Multi-chip and large size LEDs dominate the lighting market in developed countries these days. Nevertheless, a general optical design method to create prescribed intensity patterns for this type of extended sources does not exist. We present a design strategy in which the source and the target pattern are described by means of “edge wavefronts” of the system. The goal is then finding an optic coupling these wavefronts, which in the current work is a monolithic part comprising up to three freeform surfaces calculated with the simultaneous multiple surface (SMS) method. The resulting optic fully controls, for the first time, three freeform wavefronts, one more than previous SMS designs. Simulations with extended LEDs demonstrate improved intensity tailoring capabilities, confirming the effectiveness of our method and suggesting that enhanced performance features can be achieved by controlling additional wavefronts.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The rapid developments in solid state lighting (SSL) technology are leading to light emitting diode (LED) packages being gradually introduced in illumination fields in which more traditional light sources are still utilized though, like for example the automotive lighting market. Given the relatively low flux of individual chips, LED sources are composed by multichip arrays, as a means to deliver luminous fluxes comparable, for instance, to high intensity discharge lamps (in the order of 1000 lumens). The linear size of the resulting emitters often amounts to several centimeters. When designing dedicated secondary optics, in rare situations in which the space available allows the optical surfaces to be sufficiently far from the LEDs, such sources can be approximately described as point sources. In this case, several design methods exist that allow tailoring the optics to generate prescribed intensity patterns [1–4]. However, in most applications the secondary optics should preferably be compact, and located not too far from the LED, to maximize efficiency and keep the overall size of lamps small. In such situations, the extended nature of LED sources cannot be neglected by the optical designer: this fact has intensified the amount of research on prescribed intensity/irradiance optics for extended sources, where the point-source algorithms fail to provide excellent or even working solutions.

A variety of approaches and design methods have been proposed and are currently being further developed. Some rely on generalizations of point-source algorithms, in which the starting design geometry involves an idealized point source and, subsequently, the prescribed irradiance distribution is iteratively modified to account for a source with finite extent [5]. Other methods are based on a phase space mapping [6,7], and have been applied to 2D but not to 3D. Some 3D solutions have been presented, but are currently limited to translationally or rotationally symmetric optics [7–9]. The 3D prescribed intensity problem for extended sources with no symmetry has no universal solution yet.

In this paper, we use a different approach. The generation of prescribed intensity patterns with extended sources is achieved here by paying attention to wavefront control. Extended sources can be characterized by wavefronts originating from their edges and intensity patterns can be related to these edge wavefronts exiting the optic (to be designed) [10]. The design problem has two challenges: defining the design input and output wavefronts - freeform in general - carefully and calculating the optic which couples these prescribed wavefronts. A design method which allows to achieve such a wavefront coupling procedure is the so-called Simultaneous Multiple Surface (SMS) method [11,12]. This direct method allows the simultaneous calculation of *N* freeform optical surfaces, either refractive or reflective, connecting the input and output components of *N* general-shaped wavefronts.

The SMS method has been widely applied to the design of compact and efficient optics, like collimators and beam shaping devices, for the case *N* = 2, that is, with two simultaneously calculated surfaces controlling two wavefronts [13]. It is intuitive that increasing the number of controlled wavefronts would increase the quality of the intensity pattern generation process. Some advances have been presented for the case of three SMS surfaces controlling three spherical wavefronts, for imaging applications [14]. In this work, we present a first example of SMS with three simultaneously calculated surfaces controlling three freeform wavefronts. In particular, we focus on a specific SMS optic, which is the so-called RXI [15,16]: a very shallow device that was initially conceived as a concentrator and subsequently found many applications as a collimator first, and later as a more general beam shaper. We show here how the RXI can be upgraded from its standard 2-SMS-surface geometry to a new version with 3 SMS surfaces. With three simultaneously calculated surfaces, the optic can perfectly control three wavefronts, as opposed to the two of the standard RXI. Two practical design examples demonstrate that the inclusion of an additional wavefront implies superior control over typical extended LED emissions, and this translates into much better controlled output intensity patterns.

In section 2 we present, in general terms, our method for describing extended sources and intensity patterns by means of wavefronts. In section 3, we describe the 3D RXI optic. The design wavefronts used in the SMS3D calculation of the RXI are indeed generated with the method explained in section 2. We also introduce the simultaneous calculation of three SMS surfaces, in the specific case of the RXI optic. Finally, section 4 shows two practical design examples and compares the performance of standard 2-SMS-surface RXIs with the new, 3-SMS-surface ones.

## 2. Design wavefronts for prescribed intensity and extended sources

Nonimaging optical systems transfer light from a source onto a target. There are two main types of problems in nonimaging optics: bundle coupling and prescribed irradiance [10].

In bundle coupling configurations the rays from the edges of the source are directed towards the edges of the target. In that case, the Edge Ray Principle states that all light emitted from the source will fall on the target. However, the illumination of the target cannot be controlled. Ideally, if the source is Lambertian, the target illumination will also be Lambertian. If we take a pinhole on the exit aperture of the optic, the projected image of the source through this pinhole will cover the whole target and no tailoring of the irradiance is possible.

In prescribed irradiance problems the rays from the edges of the source are directed to points inside the target. Taking a pinhole on the exit aperture of the optic, the projected image of the source through this pinhole will now be contained inside the target. We then have the degree of freedom of moving this projected image inside the target to tailor the irradiance pattern. The size, orientation and position of the projected images (through pinholes) inside the target are controlled by freeform wavefronts, which define the irradiance pattern.

We may then consider that edge rays are those redirected from the edges of the source to the edges of the projected images. In the case of bundle coupling the projected images coincide with the target. In the case of prescribed irradiance the projected images are contained inside the target.

In the following, for the sake of simplicity we will refer to rectangular sources, as is often the case with LEDs. We will address the generalization to arbitrary source shapes later in this section. Consider then a square or rectangular source *S* with corners **S**_{1}**S**_{2}**S**_{3}**S**_{4}, an optic *O _{P}* and a target

*T*, as diagrammatic shown in Fig. 1. We take a pinhole

*P*on optic

*O*. The light emitted from

_{P}*S*and crossing

*P*will project an “image”

**I**

_{1}

**I**

_{2}

**I**

_{3}

**I**

_{4}of source

*S*onto the target that will be quadrilateral in shape (although its sides may not be straight). To fully control the projected images one would have to control their entire contour. This, however, is a difficult task. A simplification is to control only the four corners

**I**

_{1}

**I**

_{2}

**I**

_{3}

**I**

_{4}.

Now take ray **S**_{1}*P***I**_{1} from one corner **S**_{1} of source *S* to its corresponding corner **I**_{1} of the projected image through pinhole *P*. As pinhole *P* moves across the optic, corner **I**_{1} will also move across target *T* and this defines a bundle of rays connecting moving *P* and **I**_{1}. All these rays come from point **S**_{1} and, therefore, are perpendicular (that is, they belong) to a spherical wavefront *WF*1* _{i}* centered at

**S**

_{1}before entering optic

*O*. After crossing optic

_{P}*O*these rays

_{P}*P*

**I**

_{1}define another wavefront

*WF*1

*, perpendicular to all of them. Thus, the movement of corner*

_{o}**I**

_{1}in target

*T*is defined (and so can be controlled) by wavefront

*WF*1

*. Accordingly, the location and movements of corners*

_{o}**I**

_{2},

**I**

_{3}and

**I**

_{4}in target

*T*are controlled by wavefronts

*WF*2

*,*

_{o}*WF*3

*and*

_{o}*WF*4

*.*

_{o}Consider now a point **A** inside target *T*. As pinhole *P* moves across the optic, point **A** will be inside some projected pinhole images **I**_{1}**I**_{2}**I**_{3}**I**_{4} and outside others. If point **A** is inside image **I**_{1}**I**_{2}**I**_{3}**I**_{4}, it is illuminated by pinhole *P*. Accordingly, if point **A** is outside image **I**_{1}**I**_{2}**I**_{3}**I**_{4} it is not illuminated by pinhole *P*. The set of all pinholes *P* whose projected images contain point **A** defines an area *A* on optic *O _{P}*. This means that point

**A**on target

*T*will see light coming from an area

*A*on optic

*O*. The larger the area

_{P}*A*is, the more light will reach point

**A**and the higher the irradiance at

**A**. Since wavefronts

*WF*1

*,*

_{o}*WF*2

*,*

_{o}*WF*3

*and*

_{o}*WF*4

*define the corners of projected images*

_{o}**I**

_{1}

**I**

_{2}

**I**

_{3}

**I**

_{4}, these wavefronts also define which projected images contain point

**A**and, therefore, which pinhole positions

*P*illuminate point

**A**, defining area

*A*.

The SMS method allows the control of *N* wavefronts using *N* optical surfaces. This means that, in order to control four wavefronts *WF*1_{o}, *WF*2_{o}, *WF*3_{o} and *WF*4_{o}. (the four corners of the projected images **I**_{1}**I**_{2}**I**_{3}**I**_{4} on target) we need four freeform surfaces. Although theoretically possible, this is a challenging task. For this reason, traditional SMS optics have only two freeform surfaces, which permit to control only two wavefronts (say **S**_{1} and **S**_{2}). These, in turn, allow the control of only two corners of the projected images on target, the extreme points of one edge of the quadrilateral shape of the projected images, in this case **I**_{1}**I**_{2}. The other three sides **I**_{1}**I**_{3}, **I**_{3}**I**_{4} and **I**_{4}**I**_{2} are not properly controlled. A significant improvement should be possible if we were able to fully control, for example, three corners of the source **S _{1}S_{2}S_{3}** and of the projected images, say

**I**

_{1},

**I**

_{2}and

**I**

_{3}. This would give us control of the two sides

**I**

_{1}

**I**

_{2}and

**I**

_{1}

**I**

_{3}of the projected quadrilateral shapes. This is feasible by having three freeform surfaces that control three freeform output wavefronts

*WF*1

*,*

_{o}*WF*2

*,*

_{o}*WF*3

*; with these, the three corners*

_{o}**I**

_{1},

**I**

_{2}and

**I**

_{3}of the projected images of the source on the target can be properly controlled.

The above considerations can be readily extended to source shapes different from rectangles. For example, let us consider a disk source, also common for LEDs. Now, the projected pinhole images are circular in shape (or elliptical, since they may be distorted). We can specify the position and extension of these projected images by controlling several selected source edge points as before. Obviously, the more points we are able to simultaneously control, the more accurate is the approximation of the source edge that we can build. With control over three source edge points, we may for instance choose them to be three vertices of a square (or of a triangle) inscribed in the circle enclosing the source.

Even more generally, the source does not even need to be flat. It may be, for example, a cylinder. In this case, the source wavefronts are perpendicular to rays tangent to the cylinder surface [17].

In a practical design problem, as explained in [12], given a Cartesian coordinate system *x-y-z*, let us consider an optic where the light rays at its exit aperture propagate towards positive *z* values. In this coordinate system, *z* is front, *y* is up and *x* is left (a convention often used in automotive illumination, and based on driver point of view). The rays of the output beams *WFk _{o}* (where

*k*= 1, 2, 3 or 4) can be defined at a reference plane Π

_{0}given by

*z = z*

_{0}, where

*z*

_{0}is selected so that Π

_{0}lies close to the exit aperture of the optic. We then place pinhole

*P*on plane Π

_{0}and move

*P*through the points (

*x*,

*y*) of Π

_{0}to define the wavefronts. Let us then consider the rays perpendicular to a given output wavefront

*WFk*. These rays point in directions

_{o}**p**

*(*

_{k}=*p*(1

_{k}, q_{k},*- p*

_{k}^{2}

*- q*

_{k}^{2})

^{1/2}), where

**p**

*(*

_{k}*x*,

*y*) is a unit vector (optical momentum in air with

*n*= 1). Vector

**p**

*(*

_{k}*x*,

*y*) defines the direction of a ray passing through a pinhole

*P*at a point (

*x,y*) of Π

_{0}, and functions

*p*(

_{k}*x,y*) and

*q*(

_{k}*x,y*) are the projections of vector

**p**

*over axes*

_{k}*x*and

*y*respectively, or

We now make the assumption, for the sake of simplicity, that functions *p _{k}*(

*x,y*) depend solely on

*x*,

*p*(

_{k}*x,y*)

*= p*(

_{k}*x*) and functions

*q*(

_{k}*x,y*) depend solely on

*y*,

*q*(

_{k}*x,y*)

*= q*(

_{k}*y*). This approximation (“separation of variables”) allows us to define the horizontal and vertical acceptances of the exit beam separately, simplifying the generation of wavefronts. We will see later that these wavefronts fit well with the type of beam patterns required, for example, in automotive applications. The expression for

**p**

*then becomes*

_{k}and each vector field **p*** _{k}*(

*x,y*) defines one output wavefront

*WFk*for

_{o}*k*= 1, 2, 3 or 4. For each wavefront

*WFk*, knowledge of the

_{o}*p*(

_{k}*x*) and

*q*(

_{k}*y*) functions at plane Π

_{0}is sufficient for explicitly determining the wavefront. In [12], the established relationship between the wavefronts exiting the optic and the corresponding intensity pattern used the separation of variables approximation to estimate the output intensity connected to specific exit wavefronts.

An alternative way of looking at this problem is to consider the functions *p _{min}*(

*x*),

*p*(

_{max}*x*),

*q*(

_{min}*y*) and

*q*(

_{max}*y*) which represent the minimum and maximum values of

*p*and

*q*for the rays crossing plane Π

_{0}. These functions may now be combined in four options:

*WFk*for

_{o}*k*= 1, 2, 3 or 4.

As seen from expression (2), momentum **p** is fully defined by functions (*p,q*). Let us now consider the area *A _{I}* of the optic that contains all points emitting in a direction

**p**

*defined by (*

_{I}*p*,

_{I}*q*). In order to determine

_{I}*A*, let us first consider what happens along direction

_{I}*x*, as illustrated in Fig. 2. Along this direction the emission is bond by functions

*p*(

_{min}*x*) and

*p*(

_{max}*x*) defined for the whole aperture of the optic contained between

*x*and

_{L}*x*. In particular, for a position

_{R}*x*, the emission is bound by

_{P}*p*<

_{m}*p*<

*p*. If

_{M}*p*<

_{m}*p*<

_{I}*p*then position

_{M}*x*emits in a set of directions including

_{P}*p*. Moving

_{I}*x*laterally inside

_{P}*x*one may find the portion of the optic

_{L}x_{R}*x*comprising points emitting in direction

_{min}x_{max}*p*[10]. Acting accordingly for the

_{I}*y*direction, we may determine that a portion

*y*of axis

_{min}y_{max}*y*emits in direction

*q*. Area

_{I}*A*lit in direction (

_{I}*p*,

_{I}*q*) is then defined by

_{I}*x*<

_{min}*x*<

*x*and

_{max}*y*<

_{min}*y*<

*y*and is therefore a rectangle on plane Π

_{max}_{0}.

The intensity of light emitted in direction (*p _{I}*,

*q*) is proportional the projection of area

_{I}*A*in that direction and is given by [10]

_{I}*L*is the luminance of the light source (optical losses are not considered here) and the square root factor is the cosine of the angle between the normal to

*A*and the emission direction (

_{I}*p*,

_{I}*q*). The projected area

_{I}*A*is then directly related to functions

_{I}*p*(

_{min}*x*),

*p*(

_{max}*x*),

*q*(

_{min}*y*) and

*q*(

_{max}*y*) which, in turn, form the four output wavefronts

*WFk*as indicated by Eq. (3). This means that we have a connection between the output wavefronts

_{o}*WFk*and the intensity that they generate. We can then adjust these output wavefronts according to the desired intensity distribution

_{o}*I*(

*p, q*).

As stated above, a design method which is particularly suited for wavefront-coupling problems is the Simultaneous Multiple Surface method (SMS), in its 3D version (SMS3D). In the next section we describe the application of the SMS method to the generation of the RXI optic. First we review the standard version with 2 SMS surfaces, and then we present the novel RXI with three simultaneously calculated surfaces controlling three design wavefronts.

## 3. RXI optic: extension from 2 to 3 freeform surfaces

#### 3.1 Standard RXI with 2 coupled wavefronts

The original RXI optic was presented in 1995 as a two-dimensional SMS design [15], and in 2004 it was generalized to three dimensions [13]. We will refer here to the 3D version, intended as a beam shaper for an extended LED source. Its optically active surfaces are three: the fixed cavity and the SMS freeform RI and X surfaces (Fig. 3(a)). They work as follows:

- - the cavity is the surface enclosing the LED. It is given as an initial parameter of the system and does not change during the SMS calculation. It may have different shapes (particularly, it can be a plane as Fig. 3(a) shows), prescribed by the designer. Its form influences the way the light enters the optic, hence it has an impact on the design feasibility and geometry. Typically, the distance between the LED (whose width is in the order of 1 to 6 millimeters) and the cavity is between 1 and 5 millimeters.
- - The front RI surface is a freeform surface calculated by means of the SMS method. It performs two optical functions: total internal reflection (TIR) of the light rays refracted by the cavity, and refraction (R) of the rays reflected by the X surface. The region of the RI surface closest to the cavity may need to be metallized if TIR fails.
- - The back freeform X mirror is the other SMS surface of the system. It operates reflection (X) on the rays reflected by TIR at the RI surface, and redirects them to the RI again for the final refraction.

The RI and X surfaces are computed simultaneously via SMS. The optical path of a ray emitted by the LED which enters the RXI is: (a) refraction at the cavity, (b) total internal reflection at the RI surface, (c) reflection at the X mirror and (d) refraction at the RI surface.

The presence of two SMS surfaces means that the SMS calculation involves two wavefronts as input parameters, named WF1 and WF2. In general, an SMS3D calculation requires an initial geometrical configuration, in the form of a curve in 3D space, from which the computation of the SMS surfaces starts. This curve (“seed rib”, [13]) may be described by an analytical function, or it could be itself the result of an SMS calculation. For the RXI, the seed rib is built as an “SMS chain” calculated with WF1 or WF2 and one additional wavefront, called WF3. In total, we have three design wavefronts: WF1, WF2 and WF3, shown schematically in Fig. 4. Normally, the input wavefront components are spherical and stem from some of the LED corner points. The output components WF1_{o} and WF2_{o} shape the horizontal distribution of the exit beam; WF3_{o} has horizontal distribution similar to WF1_{o} and WF2_{o}, but vertically it changes so as to partially control the vertical intensity profile.

The SMS design process begins with the calculation of the seed rib using wavefronts WF1 and WF3. From this initial 2D geometry, the full SMS3D computation of the RI and X surfaces is performed using WF1 and WF2. A detailed explanation of the full standard RXI calculation can be find in [13]. The resulting optic has perfect control over WF1 and WF2, that is, the input components WF1_{i}, WF2_{i} are exactly transformed into the output components WF1_{o}, WF2_{o}. The control over WF3 is only partial, mostly in the vicinity of the seed curve, since this wavefront is not directly used in the calculation of SMS3D chains forming the main body of the optic. This will be shown in the next section, when comparing this device with the novel 3-SMS-surface RXI.

The coupling of the third wavefront, which is only assured in the vicinity of the seed curve, varies in different SMS devices [18]. When the design geometry and wavefronts are symmetrical, and the SMS chains remain not far from the seed curve, it might turn out that WF3 is coupled fairly well [13]. This is not the case in the 2-SMS-surface designs presented here, as described below.

As explained in the introduction, we are interested in increasing the number of perfectly controlled wavefronts; this will enhance the RXI capability of generating prescribed patterns from extended sources. Notice that in the standard RXI we are not using all the optically relevant surfaces to their full potential, since the cavity is kept constant during the SMS calculation. Releasing this degree of freedom and including the cavity into the SMS procedure would lead to the simultaneous calculation of three optical surfaces; correspondingly, the resulting optic will simultaneously control three wavefronts. This is presented in the next subsection.

#### 3.2 Enhanced 3-SMS-surface RXI

Including the cavity surface into the SMS calculation requires an additional design wavefront, which we call WF4. A viable strategy for the design of the 3-SMS-surface RXI is to start with a standard 2-SMS-surface RXI optic (calculated in the aforementioned fashion, with a prescribed cavity) and use it as the starting point for the three-surface computation. Therefore, a total of four wavefronts are employed: WF1, WF2, WF3 (for determining the seed rib of the initial RXI) and WF4. The characteristics of WF4 will depend on the specific performance sought. One may want to obtain a 3-SMS-surface RXI that perfectly controls the wavefronts WF1, WF2 and WF3: in this case, WF4 may be chosen to coincide with WF3. In the examples presented here we will show other possibilities for WF4.

There are various options for the simultaneous calculation of the RI, X and cavity surfaces. Here we describe an iterative approach:

- 1) A first standard 2-SMS-surface RXI is designed, using a prescribed plane cavity and specifically prepared wavefronts WF1, WF2 and WF3.
- 2) Using the additional design wavefront WF4, a new cavity is calculated: it is given by the Cartesian oval coupling the input component WF4
_{i}(emitted by some specific edge point of the source) with the output WF4_{o}(which has been back propagated from the infinite towards the RXI, refracted at the RI surface, reflected at the X mirror and reflected again at the RI surface). This new cavity replaces the one considered in 1). The resulting “intermediate” RXI will perfectly control WF4, but not anymore WF1 and WF2. - 3) Using as starting cavity the one computed in 2), new RI and X surfaces are SMS-calculated, by means of WF1 and WF2. The resulting RXI will perfectly control WF1 and WF2, but the coupling of the components of WF4 may degrade.
- 4) Points 2) and 3) are repeated in sequence; at each iteration, the calculated cavity is compared with the previously computed one. If the average distance between their points is below some prescribed tolerance value, usually of the order of 5 microns (matching state of the art manufacturing form factor accuracy), the two surfaces are considered equal and the process stops; the last calculated RI, X and cavity form the final RXI. If this is not the case, points 2) and 3) are repeated until convergence is reached.

The resulting 3-SMS-surface RXI will have perfect control over WF1, WF2 and WF4. In some cases, partial control over WF3 is also retained. In general, convergence of the method to a well-defined RXI geometry depends largely on the chosen design wavefronts: it is not guaranteed for every collection of normal congruences. In the next section, we present some examples in which a 3-SMS-surface RXI has been calculated and compare its performances with the standard, 2-SMS-surface RXI.

## 4. Design examples with the 3-SMS-surface RXI

The algorithm for the generation of a 3-SMS-surface RXI has been first validated with a simple proof-of-concept design, in which all design wavefronts (input and output) are spherical. Subsequently, it has been applied to more complicated design situations, with freeform wavefronts and asymmetric configurations. We present first the spherical wavefronts example and then move to a freeform example motivated by automotive lighting. In the following, we describe output wavefronts and intensity patterns using angular coordinates H (horizontal axis, that goes from left L to right R, in degrees) and V (vertical, going from Down D to Up U, in degrees). Their relationship with the optical momenta *p* and *q* (see section 2) are H = -arcsin(*p*) and V = arcsin(*q*). Such coordinates are common in automotive lighting, as anticipated above.

#### 4.1 Proof of concept: spherical wavefronts

We consider a design problem in which an RXI is required to connect spherical input wavefronts with planar output ones (which are a particular case of spherical wavefronts with infinite radius). As shown in Fig. 4, the source is a 1-mm-side square rotated 45 degrees around the x axis, simulating a typical extended LED. Input wavefronts WF1_{i}, WF2_{i} and WF3_{i} are spherical and centered at the source corner points specified in Fig. 4 (*x _{s}*-

*y*coordinates, in millimeters):

_{s}These are the input wavefronts commonly used for designing a standard RXI with two SMS surfaces; this choice generally allows good control of the overall source emission. The output components of WF1, WF2 and WF3 are planar and point, in this case, towards the directions

The output wavefronts are defined over a square reference plane, which defines a maximum possible aperture for the RXI exit surface (the RI surface). The structure of WF1_{o}, WF2_{o} and WF3_{o} aims to project a well-defined, static image of the 1-mm chip having 3.2 degrees in horizontal and 3.5 degrees in vertical. These three wavefronts already allow the calculation of a standard 2-SMS-surface RXI. We may now define the new WF4, to be included in the extended 3-SMS-surface calculation. This additional design wavefront will have a strong impact on the SMS calculation and on the performance of the RXI. Here, we are mainly interested in enhancing control over the full intensity pattern, with a particular emphasis on the definition of the vertical intensity distribution all over the horizontal emission interval. Therefore, we choose WF4 as follows:

- - WF4
_{i}is spherical and centered at the point (0, 0.5) on the source plane (midpoint of the LED top edge). This adds control over the LED emission coming from the*y*> 0 region._{s} - - WF4
_{o}is planar and points to (0 R, 1.8 D), the intermediate horizontal direction between WF1_{o}and WF2_{o}; vertically it is aimed the same as WF3_{o}. We will see that this choice improves the definition of the intensity pattern for negative values of V.

A standard 2-surface RXI and a new 3-SMS-surface one have been designed, according to the explained procedures. In Fig. 5 the two optics are shown. The starting cavity is a plane parallel to the LED plane; the material is Polycarbonate, *n* = 1.5848 at λ = 0.587 µm, no dispersion considered. Both optics are 40 mm wide and 40 mm high. The distance between the LED center and the cavity, measured along the normal to the source plane, is 2.828 mm for the standard 2-SMS-surface optic and 1.854 mm for the 3-SMS-surface RXI. The maximum light collection angle (the angle formed by the normal to the source plane and the LED rays collected by the cavity surface) is comprised between 68 and 80 degrees for the 2-SMS-surface RXI; between 58 and 90 degrees for the 3-SMS-surface RXI.

To verify how the resulting RXIs control the design wavefronts, the rays belonging to WF1_{o}, WF2_{o}, WF3_{o} and WF4_{o} are traced back towards the optics and propagated through them until reaching the source plane, where an irradiance detector is placed. Figure 6 shows the result of such ray tracing simulations. With the original 2-SMS-surface RXI, WF1 and WF2 are both well focused onto the correct points on the LED border: this means that the RXI is able to perfectly couple their input and output components. This has to be expected, since WF1 and WF2 are the normal congruences used in the 3D SMS calculation. The blurry focusing of WF3 indicates that the control over this wavefront is only approximate. In fact, in general the coupling of WF3_{i} with WF3_{o} is achieved only for the rays crossing the RI surface in the vicinity of the seed rib, while it degrades in areas far from this central strip.

The new 3-SMS-surface RXI, instead, is able to perfectly focus WF1, WF2 and WF4 onto the relevant prescribed source points: it has perfect control over the three of them. Furthermore, focusing of WF3 is improved with respect to the standard RXI, even though WF3 is not directly involved in the extended 3-surface SMS calculation. This holds because the more points of the extended source we fix, the less freedom the rest of points have to move freely throughout the pattern. Thus the 3-SMS-surface RXI has perfect control over three design wavefronts and good control over a fourth one.

The corresponding improvement in the control over the full LED emission is clear from the far-field intensity diagrams in Fig. 7. The standard RXI creates a well-defined upper cut-off and, in general, has good control over the upper region of the pattern. In the lower illuminated areas, though, the control over the beam is only approximate: the bottom corners clearly show lateral distortion of the bottom side of the source pinhole images and the pattern expands horizontally. With the new 3-SMS-surface RXI, the control over the intensity pattern is uniform all over the lit area: the cut-offs are now well defined also in the lower and lateral sides of the pattern; the prescribed vertical intensity distribution is maintained all over the horizontal span of the emission. The result is a uniformly illuminated, flat-top, rectangular distribution. This example therefore proves that perfect control over an additional design wavefront, by means of an additional SMS surface, can greatly enhance the beam control potential of the RXI.

To conclude this section, the optical efficiency of both RXIs has been measured with ray-tracing, as the fraction of the source energy which reaches the intensity detector: it is 65% for the standard 2-SMS-surface RXI and 54% for the 3-SMS-surface one.

#### 4.2 Automotive “mid-beam” headlights: freeform wavefronts

In this advanced example we consider cylindrical and freeform output wavefronts creating an intensity pattern inspired by low-beam headlights. A typical low-beam pattern has some distinctive features: an upper cut-off shaped as an angled step, a hotspot region very close to the cut-off, on top and centered horizontally, where required intensities are higher, and finally a wide horizontal angular aperture (up to 130 degrees full angle in some cases). One method for generating such light distribution is to superimpose the patterns created by different optics. For instance, one could use three optics: one creates the hotspot, another one takes care of the medium-high intensity emission around the hot-spot angular aperture and one last optic is in charge of beams with the widest spread. Of course, the number of optics employed may be higher and the total pattern would have a finer composition. Here, we study the application of the RXI to create the so-called “mid-beam” pattern defining the illumination around the hot-spot, which requires medium angular spread. The desired output pattern is shown in Fig. 8: it horizontally extends from 12 degrees left to 12 degrees right, and vertically from the horizon (0 degrees) to 2.7 degrees down, with a nearly flat vertical profile.

With the 3-SMS-surface RXI the focus is, also in this example, on improving the overall control over the pattern, especially on the definition of its vertical profile. The emitting source is the same of the previous example and we select the same edge points as centers of the input wavefronts WF1_{i}, WF2_{i}, WF3_{i} and WF4_{i} (Fig. 9 left). The corresponding prescribed structure for the output wavefronts is depicted in Fig. 9 right. WF1_{o} and WF2_{o} are symmetrical with respect to the *y-z* plane. Horizontally, they span the interval from 12L to 12R, keeping a constant angular acceptance. Vertically, they point at 0 degrees: with this choice, the whole light beam is forced to propagate with V < 0 and the upper cut-off is generated. The vertical structure of WF3_{o} is tailored to produce a region of constant intensity comprised between 2.7D and 0 degrees. Its horizontal composition, instead, is taken to be the same as WF1_{o}. The additional WF4_{o} has vertical V distribution identical to WF3_{o}; its horizontal H orientation is intermediate between WF1_{o} and WF2_{o}, for each point of the exit aperture. As in the previous example, the output wavefronts are defined over a square reference plane. Again, with this configuration we expect to force enhanced control over the target region with lower V coordinates. With these prescriptions, WF1_{o} and WF2_{o} are cylindrical wavefronts, while WF3_{o} and WF4_{o} are purely freeform.

A standard 2-SMS-surface RXI has been calculated using wavefronts WF1, WF2 and WF3; from this, using also wavefront WF4, a 3-SMS-surface RXI has been generated. The two optics are shown in Fig. 10. The LED-cavity distance is again 2.828 mm for the 2-SMS-surface RXI, with a maximum light collection angle going from 68 to 78 degrees; it is 0.785 mm for the new 3-SMS-surface one, with a maximum collection angle comprised between 53 and about 90 degrees. The efficiency is 68% for the 2-SMS-surface optic and 59% in the 3-SMS-surface case. The 2-SMS-surface RXI is 30 mm wide and 40 mm high; width and height of the 3-SMS-surface optic are 26 and 40 mm.

In Fig. 11 we show the comparison between the wavefront coupling performances of the two optics, using reverse ray tracing from output design wavefronts towards the emitter. As in the previous case, the standard RXI controls well WF1 and WF2, and partially WF3. The new RXI instead fully controls WF1, WF2 and WF4 and provides an improved coupling for WF3 too.

The corresponding improvements in handling the full LED emission are evident from Fig. 12. With the standard RXI, the vertical intensity profile behaves according to prescriptions only in the vicinity of H = 0. Also, it appears that, moving towards the lower part of the pattern, the control over the pinhole source projections is progressively lost: for V = 0, the pattern extends from H = 12L to H = 12R, while for V = 2.7D, the extension is [6L, 6R]. With the new 3-SMS-surface RXI the vertical intensity profile is more accurately defined, over a larger horizontal interval. The asymmetry between the upper and lower part of the pattern, caused by lateral distortion of the bottom side of the LED pinhole images, is corrected: the intensity distribution has a horizontal extension close to [12L, 12R], for every value of the V angular coordinate in the desired interval.

Very low intensity values close to H = 12L and 12R are due to the small areas of the RXI illuminating these extremal regions. It is anyway clear that the control over an additional wavefront is strongly beneficial for the optic ability to deal with extended source emission and to create illumination patterns which are closer to the prescribed ones.

Residual differences between the intensity produced by the 3-SMS-surface RXI and the target pattern are due to the fact that, with a finite number of surfaces, we control a finite number of source edge points. Perfect match between target and produced pattern would imply perfect control of the source border, which could be achieved only with an infinite number of surfaces.

When analysing the response of the RXI calculation to changes in the initial conditions, we verified that the position of WF4_{i} on the source plane is crucial to the performance attained by the resulting optic. In both the examples shown here, the left-right symmetry of the design configurations allowed us to place WF4_{i} at point (0, 0.5) on the source plane (corresponding to the midpoint of the LED top edge), in order to force additional control on the radiation coming from the center-top part of the LED. In general, an effective choice for WF4_{i} should be based on the presence of symmetries and on the analysis of which parts of the source need enhanced control; at the same time, it should take into account the full extension of the source.

## 5. Conclusion

We have presented an alternative method to generate prescribed intensity patterns for extended sources, such as LEDs. With this approach, the source and the output intensity pattern are described by specific edge rays grouped in input/output wavefront pairs. As described in section 2, the optics that couple these wavefront pairs then creates the prescribed illumination pattern. Such a wavefront-matched optical design has been achieved with the SMS method. In this context, we have also introduced the first example of an SMS optic with three simultaneously calculated surfaces controlling three freeform wavefronts. This fact is of significant theoretical importance, given the novelty of the result, and is furthermore substantial from a practical standpoint. In the case of the so-called RXI optic, we have shown that adding full control of a third design wavefront clearly enhances the accuracy in creating prescribed intensity patterns. In a practical example, the source mimics a typical extended LED and the prescribed intensity distribution is close to an automotive low-beam headlight component. The 3-SMS-surface RXI optic, with full control over three design wavefronts, allows superior precision in the definition of the exit beam compared to the standard RXI with only two design wavefronts. These results support the validity of the wavefront-coupling approach as a powerful method for generating prescribed illumination with extended sources.

## Funding

European Community’s Seventh Framework Programme (FP7) (619912, 608082); Fonds Wetenschappelijk Onderzoek (FWO-Vlaanderen); Interuniversity Attraction Poles of the Belgian Science Policy Office (IAPBELSPO) (IAP P7-35); Industrial Research Fund (IOF); Methusalem; Onderzoeksraad (OZR) of the Vrije Universiteit Brussel.

## Acknowledgments

We thank Vrije Universiteit Brussel and Universidad Politécnica de Madrid for providing necessary equipment and software licenses; projects HI-LED (619912) and ADOPSYS (608082) of the People programme (Marie Curie Actions) of European Community’s Seventh Framework Programme (FP7); Fonds Wetenschappelijk Onderzoek (FWO-Vlaanderen) for providing a post-doctoral grant to Fabian Duerr.

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