## Abstract

LED lighting has been a strongly growing field for the last decade. The outstanding features of LED, like compactness and low operating temperature take the control of light distributions to a new level. Key for this is the development of sophisticated optical elements that distribute the light as intended. The optics design method known as tailoring relies on the point source assumption. This assumption holds as long as the optical element is large compared to the LED chip. With chip sizes of 1 *mm*^{2} this is of no concern if each chip is endowed with its own optic. To increase the power of a luminaire, LED chips are arranged to form light engines that reach several *cm* in diameter. In order to save costs and space it is often desirable to use a single optical element for the light engine. At the same time the scale of the optics must not be increased in order to trivially keep the point source assumption valid. For such design tasks point source algorithms are of limited usefulness. New methods that take into account the extent of the light source have to be developed. We present two such extended source methods. The first method iteratively adapts the target light distribution that is fed into a points source method while the second method employs a full phase space description of the optical system.

© 2014 Optical Society of America

## 1. Introduction

Driven by the rapid development of high power light emitting diodes (LED) as light sources, the need for lenses and mirrors that can distribute the light from LEDs in a predetermined manner has strongly increased. The small size of single chip LEDs enables the realization of very general light distributions at least as long as the size of the optics is large compared to the LED size so that the point source approximation remains valid. Design algorithms that rely on the point source assumption can be considered to be state of the art ([1–5]), although the numerical realization still is a challenging task.

The success of LED light sources triggered the demand for ever higher luminous fluxes in order to enter markets that today are still dominated by conventional light sources. The power output per chip of LED is limited so that multi-chip modules are being developed that now are entering the market with a luminous flux of up to 5000 Lumen. The size of such modules amounts to a few *cm*^{2}. In most cases customers prefer small-sized lighting systems. This forbids scaling the light-forming elements along with the new LED module sizes.

Optical free-form design algorithms thus have to account for the extent of light sources. There have been some enhancements of point source algorithms to account for extended sources. A distribution that is fed into a point source algorithm is interactively adapted so that the distribution that is generated by the resulting optical surface matches the desired distribution as closely as possible. Besides the adaptation of the target distribution, Fournier et.al. [6] optimize the position of a point source in order to get the best results for a stationary extended source.

The SMS method developed by Benítez, Miñano et.al. [7] allows for the simultaneous construction of two optical surfaces that exactly transform two input congruences (zero étendue wave fields) into two prescribed output congruences. With more surfaces more congruences can be transformed. SMS can thus best be described as a multi-étendue-zero-source method. The inclusion of source non-uniformities is not easily possible. The method was originally designed for the 2D case but was extended to approximately handle 3D cases as well. Rabl on the other hand [8] employs the edge ray principle to construct 2D freeform reflector surfaces. Source non-uniformities cannot be taken into account.

In the following section we supply some definitions. In section 2 we give a short overview over existing point source algorithms and in section 3 we outline a new approach to improve optical design for extended sources utilizing phase space concepts.

#### 1.1. Parametrization of optical surfaces

Optical surfaces have to be described or parametrized in some way. Table (1) shows a list of surface parametrization schemes used in the field of optical design methods. The design task consists in determining the parameters of the optical surfaces in an optical system. In the case of free-form surfaces the number of parameters can exceed 10^{5}.

#### 1.2. Light flux conservation

In lossless optical systems the differential fluxes *d*Φ are equal on the source and target side, i.e.:

*L*(

*u⃗*,

*s⃗*) is the radiance (refer to Table (2)). With the assumption of étendue conservation

## 2. Point source algorithms

A lot of design algorithms have been published that exploit the point source assumption or assumption of zero source extent. Table 3 gives an overview over the principal methods. Direct optimization with Monte-Carlo ray tracing has only limited applicability in the field of free-form optics design. The ray mapping methods suffer from the problem of finding ray mappings that meet the integrability condition of the optical surface. In the case of one-dimensional flux density distributions this is of no concern but in the general two dimensional case this can be a severe limitation. The method of ellipsoids is mathematically sound but has some drawbacks. The resulting surface is only *C*^{0} and the computation time is
$O\left(\frac{{K}^{4}}{\gamma}\text{log}\left(\frac{{K}^{2}}{\gamma}\right)\tau \right)$ [12], with *K* the number of ellipsoids and *γ* and *τ* being two characteristic numbers of the problem. This implies a steep increase of the computation time with the number of ellipsoids, so this method is best suited for small numbers of ellipsoids (e.g. [13] employ 100 ellipsoids).

The last two methods in the above list are the most advanced point source algorithms that satisfy the integrability condition automatically and can be implemented efficiently. A drawback of these methods is that highly nonlinear equations have to be solved which requires starting configurations that are close enough to the solution in order to converge.

Fig. 1 shows the general set up of a point source, an optical element with free-form surfaces and a target plane.

Flux conservation yields:

In Eq. (4)*s*,

_{x}*s*are the direction vector components parallel to the source surface,

_{y}*u*,

_{x}*u*are the Cartesian coordinates in the target plane,

_{y}*I*(

_{s}*s*,

_{x}*s*) is the direction dependent intensity of the source and

_{y}*E*(

_{t}*u*,

_{x}*u*) the irradiance on the target plane.

_{y}#### 2.1. Monge-Ampère transport problem

The surface design task is to find a bijective coordinate mapping:

that realizes the prescribed irradiance distribution on the target. Inserting into Eq. (4) yields:*J*:

## 3. Algorithms for extended sources

Besides the SMS method [7] extended source algorithms mostly rely on point source algorithms wrapped in an additional optimizing loop.

#### 3.1. Irradiance adaptation

One method to deal with extended sources is to adapt the target irradiance distribution in such a way that the resulting optical surface creates the desired distribution for an extended source. There are different adaption strategies described in literature. The simplest method is to adapt the irradiance distribution point-wise [16]. Fig. 2 schematically shows a method that employs a Monge-Ampère equation to compute the adapted distribution [17].

#### 3.2. A phase space method

A more rigorous approach has to deal with flux conservation formulated in the optical phase space (see Eq. (1)). Multiplying Eq. (3) by cos(*θ _{t}*) and integrating both sides over target side solid angles yields:

Fig. 3 shows an optical system with an extended source, a spherical lens and a target plane. On the left and right side, two-dimensional slices of the optical phase space are shown. Whereas the phase space on the source side is densely occupied, this does not hold for the target side. In order to avoid tracing significantly more rays than necessary, the target phase space over which is integrated can be limited to the relevant parts.

One possibility to constrain it is by coarsely forward ray tracing from the light source and assessing which parts of the target phase space are subsequently relevant for the integral. Another possibility is to limit the integral to those rays that, from the considered target point, intersect the optics. This does not assess which portion of the source phase space is relevant, but we will consider the amount of additional rays that are traced, while not contributing to the irradiance, to be negligible.

The following section will give an example of an approximate evaluation of the integral Eq. (12) and application to the optimization of one free-form surface.

#### 3.3. Example computation

In the following example, the integral Eq. (12) is evaluated over the portion of the target phase space that originates from the optics, for each considered target point. To achieve this, rays are traced backwards from the respective target points through the optics onto the source plane, where their radiance information is extracted. In the present example we treat a 2-dimensional case. Each of the two surfaces of a lens are described by polynomials of degree 8. The coefficients of the two polynomials are optimized simultaneously. The integral is discretized with respect to the angle *θ _{t}*:

_{t,i,i+1}is the solid angle between rays

*i*and

*i*+ 1. Evaluating the summation numerically for a set of target points yields an approximation of the irradiance pattern across these target points.

For the optimization of the free-form surfaces, a gradient method together with an appropriate merit function is employed. The merit function is defined as the sum over the squares of the differences between prescribed and calculated target irradiances. The power of the prescribed and achieved distributions on the target is normalized to 1 *W*, as in this example, the light flux passing through the optics is enclosed in the analyzed target area. Fig. 4 shows the set up and the shapes of the two free-form surfaces of the lens. The resulting target distribution is shown in Fig. 5.

## 4. Conclusion

With the advent of high power LED sources, designing general free-form optical elements to realize arbitrary irradiance distributions has attracted much attention within the last decade. Point source algorithms are quite mature today and have been used to design different optical systems for automotive and general illumination, image projection and other applications. Most of these algorithms rely on the fact that ray directions are unique, i.e. the sources have zero étendue. Real sources have finite étendue, the point source assumption thus is only an approximation more or less justified. As a rule of thumb the dimensions of the optical elements have to be at least five times larger compared to the source extent in order for the point source assumption to be valid. For smaller ratios the source extent has to be taken into account. Besides the target distribution adaptation method, that is an approximation as well, phase space methods are the only approach to deal with extended sources without employing approximations and thus being limited to large optic to source ratios. The 2D phase space method presented in the present article demonstrates good performance even in case of a small optic to source extend ratio of about two. It can handle non-uniform sources and two surfaces simultaneously, its extension to 3D surfaces is straightforward and currently underway.

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