## Abstract

The design of freeform lenses and reflectors allows to achieve non-radially symmetric irradiance distributions whilst keeping the optical system compact. In the case of a point-like source, such as an LED, it is often desired to capture a wide angle of source light in order to increase optical efficiency. This generally results in strongly curved optics, requiring both lens surfaces to contribute to the total ray refraction, and thereby minimising Fresnel losses. In this article, we report on a new design algorithm for multiple freeform optical surfaces based on the theory of optimal mass transport that adresses these requirements and give an example of its application to a problem in general lighting.

© 2012 OSA

## 1. Few published algorithms allow flexible irradiance tailoring

In lighting applications, the need for lenses and mirrors that can distribute light in a predetermined manner has strongly increased. To achieve the most general light distributions, the number of degrees of freedom in the design of the optics has to be much higher than for conventional optical elements. This leads to the concept of freeform optical surfaces, for which a multitude of design algorithms has been proposed [1–5]. Refractive optical elements (lenses) suffer from Fresnel reflections at the material-air interfaces and while not all the reflected light is inevitably lost, control over its exact ray path is not easily achieved. As Fresnel reflections increase with the angle of ray deflection, it is desired to use several freeform surfaces to increase system efficiency, even in cases where a single lens surface could suffice to tailor the irradiance distribution.

Of the design algorithms published to date, only the powerful SMS3D method originally proposed by Miñano, Benítez et.al. [5] has been shown to directly tailor multiple smooth surfaces simultaneously to achieve a prescribed irradiance distribution, even taking partially into account extended sources. Yet, to the best of the authors’ knowledge, the only SMS3D implementation from which results have been published is within their group. For the case of a point source, Ries and Muschaweck [2] very elegantly derived a set of partial differential equations that describe a single optical surface, but a generalization to multiple optical surfaces has not yet been proposed.

In this article, using a formulation based on transportation theory [6], we present a flexible algorithm approximating a solution to the general problem of irradiance tailoring with two freeform surfaces and a point source emitter. Further, we demonstrate its viability for a design task from general lighting.

## 2. Ray mapping: relating optical design to transportation theory

In its general formulation, the theory of mass transportation deals with the computation of the optimum path allowing to continuously transfer an initial mass distribution onto a target mass distribution. In the optical case, light flux takes on the role of mass, and hence the source light after projection shall be described as a flux density *μ*_{0} on a 2D domain Ω_{0} (see Fig. 1). Similarly, a target flux density *μ*_{1} is given on a domain Ω_{1}. For the sake of clarity, it shall be assumed that Ω_{0,1} are parallel 2D planes embedded in ℝ^{3} and hence *μ*_{0} and *μ*_{1} are flux densities parametrized by local Cartesian coordinates (*x,y*) in the respective planes (see Fig. 1).

The task of designing a (freeform) optical system then amounts to finding a diffeomorphism (“ray mapping”) so that the transformed irradiance distribution matches the target distribution:

where (*t*,

_{x}*t*) represents the target point in Ω

_{y}_{1}to be reached by a source ray passing by (

*x*,

*y*) in Ω

_{0}. Since the light flux along an infinitesimal light tube from the source to the target is conserved, the irradiance transformation can be written: where D

*u*is the Jacobian of

*u*, representing the compression or dilatation of the tube cross-section along the path, and ○ is the usual composition operator. Integrating this equation on the whole domain Ω

_{0}leads to the global energy conservation relation (∫

_{Ω0}

*μ*

_{0}= ∫

_{Ω1}

*μ*

_{1}).

As the mapping *u* is not unique [7], the main constraint on the optical design task is to find a mapping that leads to a continuously differentiable optical surface. This is expressed in the so-called integrability condition for the surface normal vectors **N**

Computing a ray mapping that exactly enforces Eq. (3) is not an obvious task, as Eq. (2) amounts in general to a non-linear second order partial differential equation through the determinant of the Jacobian (Monge-Ampère-type equation). Dealing with two optical surfaces instead of one makes this problem even more challenging.

## 3. Approximating the optimum ray mapping

The field of surface normal vectors is directly connected to the mapping information (through Snell’s law of refraction). Hence, even if this has not been proven so far, it seems plausible that the curl of the surface normal field can be significantly reduced if the curl of the ray mapping itself is minimized.

Using recent results in the field of mass transportation theory helps to achieve this goal. Most of the work in this field deals with finding an optimal mapping with regards to a prescribed cost function. This is typically a quadratic function of the displacement for each element, weighted by its mass. The general problem, initially described by Monge, was given its modern formulation by Kantorovich [9]. Several methods to solve this problem have been suggested (see e.g. [10] for an overview). In this article, we focus on a first order, parameter-free scheme suggested by Haker in the context of image warping [7].

The point source’s flux distribution is projected onto a planar square domain Ω_{0}, thus obtaining a flat flux density *μ*_{0}. A modified stereographic projection is typically used to achieve this, at the same time controlling how much of the source light cone should be captured. The target flux distribution is equally projected onto a parallel square domain Ω_{1}. This is depicted on Fig. 1.

Haker’s procedure [7] consists first in finding an initial mapping between *μ*_{0} and *μ*_{1} (typically by two successive 1D numerical integrations along the Cartesian coordinate axes). This results in a starting mapping denoted *ũ*.

The demonstration established by Haker shows that the family of all mappings transforming *μ*_{0} in *μ*_{1} can be parametrized with a continuous variable *t*. Equivalently, *u* can be regarded as a function of *t* with *u*|_{t}_{=0} = *ũ*, and its derivative with respect to this variable (i.e. the mapping’s flow) can be computed. The flow obeys the following evolution equation:

*u*denotes the mapping’s Jacobian, (

*x*,

*y*)

^{⊥}= (−

*y*,

*x*) represents a rotation by 90 degrees in ℝ

^{2}and Δ

^{−1}div

*u*

^{⊥}denotes the solution

*f*of the Poisson’s equation Δ

*f*= −div

*u*

^{⊥}.

The stationary solution of this equation (when *t* → ∞) is proven by Haker to be the optimal mapping in the sense of the quadratic cost function and, most importantly, to be irrotationnal. In this article, the equation was solved using the same numerical techniques as employed by Haker and using *ũ* as a starting point. Note that the evolution equation itself doesn’t make use of *μ*_{1} as the corresponding information is already contained in the initial mapping *ũ*.

Reducing the mapping’s curl in this way leads to a good approximation of the integrability conditions, as is shown later in Section 5.

## 4. Reconstructing the optical surface from a ray mapping

With the mapping information, the optical surfaces can be computed so that the source rays are deflected to hit the target at the desired positions. The total light deflection is split into several parts (two in the example below), each one realized by a given optical surface. The reconstruction method employed in the present article is a standard least-squares optimization method. The surfaces are each approximated by a triangle mesh with vertices *i* = 1,...,*N* whose positions are given by:

**r**

_{0}(

*i*) is the origin of ray

*i*. This can be the position of a point source or the position of the ray after passing through another optical surface.

**s**(

*i*) is the unit direction vector of the ray and

*λ*(

*i*) is a scalar parameter defining the surface point

*i*.

**s**(*i*) and **N**(*i*) give the direction vector **s*** _{o}*(

*i*) of the ray after refraction (or reflection) at the surface. The normal vectors

**N**(

*i*) at the triangular surface vertex positions are computed as weighted averages of the normals of the vertex’s neighbor faces.

Given **s*** _{o}*(

*i*) and the ray’s position on the optical surface

**r**(

*i*), the point

**T**(

*i*) where the ray intersects the target surface can be computed. The objective function to minimize is thus given by

*T*(

_{x}*i*) and

*T*(

_{y}*i*) are the actual local target coordinates of the ray for a given vector of parameters

**, and**

*λ**t*(

_{x}*i*) and

*t*(

_{y}*i*), respectively, are the desired local target coordinates as computed by the mapping algorithm.

Surface construction based on ray coordinates on the target surface ensures that the integrability condition is always met, as the surface is not reconstructed from a previously calculated field of normal vectors. The algorithm works with ray positions on the target surface, therefore one can distribute the deflection over multiple optical surfaces, introducing an additional degree of freedom into the design not present in algorithms that only consider a single surface.

Using triangular meshes has the advantage of being very flexible for further processing of the resulting surfaces. They can for example be refined or cut very quickly, thus gaining finer control over the resulting lens’s boundary conditions. In combination with a multi-surface design, this also allows to include manufacturing constraints (e.g. to avoid undercuts for injection molding).

## 5. Sample application from architectural lighting: wallwashing

Figure 2 shows a typical and, when high optical efficiency is required, challenging geometry in architectural lighting (“wallwasher configuration”), where a homogeneous light irradiance is desired on a wall located at 80 cm from the ceiling-mounted light source. In a first step, a simplified on-axis setup (also depicted in Fig. 2) is considered to demonstrate the design process and its merits compared to a conventional design using a non-optimized mapping (successive 1D integration along the Cartesian axes). Based on this, the results for the more challenging, tilted setup are presented.

#### 5.1. On-axis example

Following the algorithm outlined above, an initial ray mapping was calculated and corresponding optical surfaces were constructed (assuming the material is PMMA with *n* = 1.49), minimizing objective function (6). To demonstrate the improvement, the initial mapping was optimized as explained above, and a second set of surfaces was subsequently constructed. Figure 3 shows the corresponding mappings, represented as the deformation of a regular grid, and their respective local curl *z* components. The overall curl magnitude has been reduced significantly by a factor of about 200.

Monte-Carlo ray tracing of both sets of surfaces was performed, and Fig. 4 shows the final irradiance distributions for each case. All analyses were done using FRED optical software [11] with 5 million rays and a detector resolution of 71x71 points on the target plane. The source was a standard Lambertian point emitter, with an angular distribution spanning a half cone angle of 70 degrees, and directed towards positive *z*.

Although the shape of the lens is very similar, the light distribution in part (d) of Fig. 4 shows significantly less deviation from a homogeneous irradiance distribution and the square target geometry is also adhered to much better. This confirms that the curl reduction leads to a satisfactory ray mapping, i.e. better compatible with the integrability condition (3) than the initial mapping, whilst correctly approximating the desired flux transformation.

#### 5.2. Off-axis configuration

Figure 5 shows a 3D CAD model of the relevant lens surfaces for the final wallwasher configu-ration resulting from the optimized ray mapping and the irradiance distribution as predicted by ray tracing.

Given the challenging configuration of the problem, i.e. the high inclination of the target with respect to the optical axis and the short optic-to-target distance, the target geometry (rectangular area of 2.8 × 2.8 m^{2}) as well as the desired homogeneity of the irradiance distribution on the target are respected rather well. In the square area centered on the target and covering 2 m × 2 m, the intensity variation is below 10% on any cross section along the *y* axis, and below 30% on any cross section along the *x* axis. The imbalance between these two values arises from the inclination of the optical element with respect to the target plane. Finally, the Monte Carlo simulation shows that over 78% of the source light power is transfered to the target, including all Fresnel losses, thus illustrating the high optical efficiency of the resulting lens. A similar simulation with only one freeform surface leads to greater Fresnel losses (overall optical efficiency reduced to 70%), and fails to acceptably achieve the target distribution as the source light cone is too wide for all the rays to be correctly deflected towards the target.

## 6. Comparing Fresnel losses for designs with one and two freeform surfaces

In order to estimate the Fresnel reflection losses of a lens, the mapping of source rays to target points has to be known, as well as a lens geometry as outlined above. In a setup with a single active surface, the first lens surface is chosen as a hemisphere with its center at the location of the point source, as depicted in Fig. 6. Hence, it does not provide any ray deflection and the second surface has to enforce the entire ray mapping. Conversely, the deflection can be distributed among the surfaces if using a setup with two freeform surfaces, also depicted in Fig. 6. For the purpose of this analysis, a ray of light shall be considered a Fresnel loss whenever it is reflected off a surface that is mainly intended to be transmissive.

In the case of an off-axis wallwasher-like setup as in Fig. 2(b), the surfaces have been computed under the assumption of equal distribution of the deflection angles across the two surfaces. The Fresnel losses are then given as a sum over the reflection losses to all rays which in turn are given by the Fresnel formulae [12]. Figure 7 shows the computed Fresnel losses for systems that employ a single freeform surface and two freeform surfaces, respectively.

The total Fresnel losses of the lens also depend on the angle of source rays that is captured, and high optical efficiency demands high capturing angles. The two-surface system shows almost constant Fresnel losses as a function of the capturing angle. This is due to the fact that with two freeform surfaces, the angles of incidence on each surface do not exceed a certain threshold value beyond which Fresnel reflection increases significantly (cf. Fig. 7(b)). In the present case, the maximum angles of incidence are typically below 30 degrees. This is in strong contrast to the case of a single freeform surface, for which the maximum angles of incidence on the second lens surface come close to the angle of total internal reflection and thus show much higher Fresnel reflection losses.

The total Fresnel losses in the case of two active surfaces is around 8 to 10% which is close to the losses for normal incidence, whereas in the case of a single active surface the minimum value is about 15%. The strong increase in Fresnel reflections for the case of a single-surface lens for small capturing angles is due to comparatively large deflections necessary to direct the rays to the outer regions of the target plane. The Fresnel losses found for the application shown in Section 5 are in good agreement with the estimations presented here.

## 7. Conclusion

We have demonstrated a novel two-step optical design algorithm that provides first results to directly tailor irradiance distributions using multiple freeform surfaces. This algorithm is fast (computing time of a few minutes on today’s desktop computers) and its applicability and practical relevance was verified in a general lighting application. The first step consists in computing a ray mapping from source rays to target points that is subsequently optimized to make it irro-tationnal, and in a second step, the optical surfaces are computed using ray positions extracted from the mapping on a target plane. This procedure can equally be applied to reflective optics without further adaptation.

Using the proposed algorithm, it is possible to directly tailor a multi-surface optical system (in this case, a double-sided freeform lens) to achieve a close approximation of a predefined irradiance distribution whilst at the same time capturing a significant part of the light emitted by the source. The additional degrees of freedom arising from having multiple surfaces can be used to significantly reduce Fresnel losses, include manufacturing constraints and reduce overall part dimensions.

## Acknowledgments

This work has partly been funded by the German Federal Ministry of Education and Research (BMBF) within its program on freeform optics (grant numbers 13N10832, 13N10833).

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