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Abstract
Glare and visual discomfort are important factors that should be taken into account in illumination design. Conventional freeform lenses offer perfect control over the outgoing intensity distribution, thereby allowing optical radiation patterns with sharp cut-offs in order to optimize the unified glare rating index. However, these freeform lenses do not offer control over the near-field luminance distribution. Observing the emitted light distribution from a high-brightness LED through a freeform lens gives a high peak luminance that can result in glare. To reduce this peak luminance, freeform lenses should be used in conjunction with light diffusing structures. However, this diminishes the control over the outgoing intensity distribution what is the main benefit of a freeform lens. Another approach to reduce the observed peak luminance, is by spreading the emitted light over multiple optical channels via freeform lens arrays. This paper proposes a novel method to design luminance spreading freeform lens arrays that offer perfect control over the resulting intensity pattern. The method is based on a non-invertible mapping of a 2D parameter space. This results in a source-target mapping in which multiple ingoing ray directions are mapped onto every position of the target distribution. The case of continuous and discontinuous mappings are both discussed in this paper. Finally, the example of a discontinuous freeform lens array with $7\times 7$ individual lenses is designed and experimentally demonstrated.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Visual discomfort is an important factor that should be taken into account during the design of optical components for general lighting applications. There is still a lot of ongoing research about the perception of discomfort glare and how it must be evaluated [1–3] but all researchers agree that visual discomfort is strongly correlated to high peak luminance values within the field of view.
Freeform lenses can be used to create illumination systems with a specific irradiance or intensity pattern and are therefore of significant interest for many lighting applications. Different successful design strategies exist to generate freeform optical surfaces [4]. The zero-étendue methods can be divided in three groups: ray-mapping methods [5–9], supporting quadrics methods [10–12] and Monge-Ampère methods [13,14]. All these methods make use of the assumption that every ingoing ray from a source distribution $S$ is redirected to a unique position in the target distribution $T$ and, vice versa, that each target position is illuminated by only one ingoing ray that is deviated by the freeform lens (see Fig. 1a). This means that no direct control is offered over the near-field luminance distribution after the lens. When such a freeform lens is used in combination with a high-brightness light source (e.g. a LED- or laser-based light source) high peak luminance and strong visual discomfort can be the result [15]. A solution is to use freeform optics in conjunction with volume- or surface-scattering light diffusers in order to spread the outgoing luminance. However, this always goes at the expense of efficiency and a significant loss in the tailorability of the resulting target distribution.
Fig. 1. (a) A conventional freeform lens redirects every source ray towards a unique position in the target plane. (b) A freeform lens array redirects multiple source rays towards each position in the target plane. This effectively reduces the observed peak luminance from each point in the target plane while maintaining control over the target distribution.
An interesting approach to ensure a significant reduction of the observed peak luminance, is to use an array of freeform lenses (see Fig. 1(b)). By spreading the emitted light towards a specific position over multiple optical channels, i.e. the individual lenses of the freeform lens array, the observed luminance from that position can be effectively reduced. On the other hand, it is still possible to ensure very accurate, tailored illumination distributions with such a freeform lens array.
For a collimated incident beam such a freeform lens array can be achieved by a simple regular tiling of the same freeform lens [16,17]. For non-collimated light sources however, each freeform lens needs to be separately designed because it is illuminated by an incident beam at a specific angle of incidence. Therefore such an array requires the design of many different freeform lenses. Additionally, if the target distribution spans a large angular extent, it is not always possible to generate the full target pattern with each individual lens of the array, because the maximal beam deflection that can be achieved with a single refraction is limited. In this paper an efficient method is presented to design an array of freeform lenses with partially overlapping intensity patterns that combine into the desired target distribution.
In previous work, segmented lens arrays with either rotational or longitudinal symmetry were examined [18]. An iterative method was developed to generate partially overlapping light distributions that combine together to a desired target distribution. Similarly, freeform lens array must generate multiple overlapping light distributions in order to achieve a reduction of the observed luminance. However, unlike in the case with rotational or longitudinal symmetry, in which it is straightforward to obtain the lens shape from the known in-going and desired out-going light distribution, this is typically not that straightforward for general freeform lenses. Generating $N\times N$ individual freeform lenses from multiple source and target distributions can therefore be computationally very demanding. In this paper, a different method is proposed that directly constructs $N\times N$ ray mappings, starting from a single source-target mapping. This initial ray mapping is transformed by means of a non-invertible fold mapping which results in multiple ray mappings from which the individual freeform lens elements can be obtained. This folding approach ensures that an array of multiple freeform lens surfaces is obtained which reshape the source distribution $S$ into multiple overlapping beams that form together the desired target distribution $T$.
2. Method
The method that is described in this paper can be seen as an expansion of the ray mapping method for freeform lens design. The ray mapping method starts from a source distribution $S$ that describes the flux density on a certain plane of a source with zero-étendue, and a target distribution $T$ that describes the desired irradiance or illuminance distribution on a chosen target plane. The method makes furthermore use of a source-target mapping that is a diffeomorphism between the two local Cartesian coordinates in the source and target plane such that the source distribution is transformed in the target distribution. When one knows for each incident ray direction, the corresponding outgoing ray direction, the necessary surface normals can be calculated with Snell’s law (for freeform lenses) or the law of reflection (for freeform reflectors). These surface normals can then be used to calculate the freeform surface via numerical integration [6]. The main difficulty of the ray mapping method is finding a source-target mapping such that the resulting normal field is directly integrable into a continuous freeform surface [19].
A natural way to obtain a source-target mapping is by splitting the generation of this mapping in two separate mappings that each start from a uniform square distribution $U$ defined over the domain $\left [0\,,1\right ]^2$: a mapping $\phi$ from $U$ to the source distribution $S$ and a mapping $\psi$ from the same square distribution $U$ to the target distribution $T$ [9]. The source-target mapping is then simply $\psi \circ \phi ^{-1}$ (see Fig. 2 on the left).
Fig. 2. Schematic representation of the proposed algorithm to design freeform lens array with overlapping light intensity distributions.
This method of generating source-target mappings comes with the advantage that one can associate with every ray in the source and target distribution, two parameters $(u,\;v) \in \left [0\,,1\right ]^2$. A uniform rectangular grid $(u_i,\;u_j)$ then automatically generates two equi-flux grids $(x_{s,\;ij},\;y_{s,\;ij}) = \phi (u_i,\;v_j)$ and $(x_{t,\;ij},\;y_{t,\;ij}) = \psi (u_i,\;v_j)$ within the source and target distribution. Furthermore, it is possible to consider an additional mapping $F(u,\;v)=(u',\;v')$ of the domain $\left [0\,,1\right ]^2$ onto itself. If this mapping $F$ leaves the uniform distribution $U$ unchanged, then the mapping $\psi \circ F \circ \phi ^{-1}$ is still a mapping that transforms the source distribution $S$ in the target distribution $T$. To leave the uniform distribution $U$ unchanged, the mapping $F$ should be a symplectic or area-preserving mapping. In a previous paper, we have elaborated how this additional symplectic mapping can be used to derive a source-target mapping that results in an integrable surface normal vector field, for non-paraxial illumination configurations [20]. In this paper, we follow a similar procedure to design luminance spreading freeform lens arrays.
Constructing a freeform lens surface from a one-to-one or diffeomorphic ray mapping results in a lens that redirects every source ray towards a unique position in the target plane. However, in order to reduce glare, multiple ingoing source rays must be redirected towards every single position in the target distribution. This means that the function $\psi \circ F \circ \phi ^{-1}$ has to map multiple source rays $(x_s,\;y_s)_i$ onto every target ray $(x_t,\;y_t)$. Since $\phi$ and $\psi$ are diffeomorphisms which will not be altered, the mapping $F$ must be a non-invertible function that maps a set of $n$ points $(u,\;v)_i$ onto each point $(u',\;v')=F(u,\;v)$ of the domain $\left [0,1\right ]^2$. Also in this case, the additional mapping $F$ should leave the uniform distribution $U$ unchanged, to ensure that $\psi \circ F \circ \phi ^{-1}$ is still a mapping that transforms the source distribution $S$ in the correct target distribution $T$.
2.1 Applying a fold mapping
A straightforward way of obtaining such a mapping function $F$ is by a series of folding and stretching operations. Intuitively one can imagine the distribution $U$ as a uniform sheet that is being folded into a smaller rectangle and afterwards stretched to the original size. If the number of layers in the folded sheet is the same everywhere, the total operation will not have altered the total sheet thickness. This means that the uniformity of the distribution $U$ is unchanged by such a mapping function $F$.
This folding operation can be done separately for both parameters $u$ and $v$:
The simplest non-trivial example of such a fold mapping is the tent map which maps the interval $[0,1]$ onto itself twice:For every point $u'$ there are two points $u_1$ and $u_2$ that are mapped onto $u'$. Taking the same tent map for $F_v$ results in a mapping $F = (F_u,F_v)$ that maps four points $\left ((u_1,\;v_1),(u_1,\;v_2),(u_2,\;v_1),(u_2,\;v_2)\right )$ onto every point $(u',\;v')$ in $[0,1]^2$. The condition that the mapping $F$ leaves the uniform rectangular distribution $U$ unchanged, can be mathematically expressed as
The resulting mapping $\psi \circ F \circ \phi ^{-1}$ is a continuous mapping that maps 4 different regions of the source distribution $S$, onto the entire target distribution $T$. In order to ensure that this mapping results in an integrable surface normal vector field, the initial mappings $\phi$ and $\psi$ should be optimal mass transport mappings or curl-free mappings. For the case of non-paraxial illumination configurations however, this is not sufficient [21]. In this case, the resulting mapping $\psi \circ F \circ \phi ^{-1}$ could be made integrable using the approach that was described in [20].
One can distinguish four different regions in the mapping:
- 1. $ \frac{\mathrm{d}F}{\mathrm{d}u} > 0 \quad \& \quad \frac{\mathrm{d}F}{\mathrm{d}v} > 0 \quad \rightarrow $ The lens surface is locally concave
- 2. $ \frac{\mathrm{d}F}{\mathrm{d}u} > 0 \quad \& \quad \frac{\mathrm{d}F}{\mathrm{d}v} < 0 \quad \rightarrow $ The lens surface is locally saddle shaped
- 3. $ \frac{\mathrm{d}F}{\mathrm{d}u} < 0 \quad \& \quad \frac{\mathrm{d}F}{\mathrm{d}v} > 0 \quad \rightarrow $ The lens surface is locally saddle shaped
- 4. $ \frac{\mathrm{d}F}{\mathrm{d}u} < 0 \quad \& \quad \frac{\mathrm{d}F}{\mathrm{d}v} < 0 \quad \rightarrow $ The lens surface is locally convex
There is however one fundamental problem with this approach. It turns out that the ray mapping for the saddle shaped regions of the lens array is not integrable. Even for fairly paraxial configurations in which the convex and concave lens segments give the desired performance, the saddle shaped lens segments do not. This can clearly be seen in Fig. 4, for the case of a collimated ingoing beam that should be converted to a uniform triangular target distribution, but clearly this is not achieved. It also turns out to be very difficult, if not impossible, to adapt the mapping in such a way that it becomes fully integrable, e.g. by using the earlier mentioned method [20]. This problem with saddle shaped lenses seems to be an unsolved issue in freeform optical design [22]. In this case, it prevents the direct design of a continuous luminance spreading freeform lens array.
Fig. 4. (a) The lens surface that is obtained by integrating the folded mapping $\psi \circ F \circ \phi ^{-1}$ in which $F$ is a continuous tent map. The lens surface contains a concave, a convex and two saddle shaped regions. (b) The simulated irradiance distribution at the target plane. (All simulations were performed in LightTools 8.6.0).
To circumvent this fundamental issue, the requirement for continuity is given up. The condition that $F$ leaves the uniform rectangular distribution $U$ unchanged (Eq. 3) clearly does not depend on the sign of the slope value. One can thus simply change the sign of the slope of a line segment by switching the function values of the start and end points. For the tent map, this results e.g. in the discontinuous mapping that is shown in Fig. 5. The lens surface that is obtained by integrating the folded mapping $\psi \circ F \circ \phi ^{-1}$ in which $F = (F_u,F_v)$ is the combination of two such discontinuous mappings, will be a discontinuous surface consisting of four lens segments. If the slope values of both $F_u$ and $F_v$ are the same (positive for concave lenses, negative for convex lenses), no saddle shaped segments will be present in the resulting lens surface. In Fig. 6 the resulting lens array and the obtained irradiance distribution are shown for such a discontinuous mapping. In this case, the lens array does achieve the desired target distribution with high accuracy.
Fig. 6. (a) The lens surface that is obtained by integrating the folded mapping $\psi \circ F \circ \phi ^{-1}$ in which $F$ is a discontinuous function. The lens surface consists of multiple convex lenses. (b) The simulated irradiance distribution at the target plane.
2.2 Freeform lens arrays with partial light mixing
The application of the discontinuous tent map for illumination configurations with a collimated light beam is not very interesting because it results in a regular tiling of four identical freeform lenses which could be obtained in a more simple manner. For illumination configurations with non-collimated light sources and target beam patterns with large angular extent, the design of a luminance spreading freeform lens array is much more demanding. In this case, it is no longer possible that all individual freeform lens segments illuminate the entire target pattern, because of the maximal ray-deflection that can be achieved with a single refraction. This implies that the lenses need to illuminate different overlapping parts of the target distribution (= partial light mixing), such that the combination of all contributions results in the desired target distribution. It is for these cases that the approach shows its full potential.
In this paper, the design of an array of $7\times 7\ (=N\times N)$ freeform lenses is considered as an example. This lens array reshapes the light from a point source with a Lambertian intensity pattern and full opening angle of $120^\circ$, into a uniform equilateral triangular distribution that is inscribed in a circle of radius $1000$ mm at a distance of $1000$ mm. The redirected light towards each position in this triangular target distribution is spread over $3\times 3\ (=n\times n)$ lenses. Spreading the light over $3\times 3$ lenses is in this case the maximal luminance spreading that can be achieved without having high Fresnel reflection losses.
Equation (3) can be easily generalised to multimodal mappings:
For the considered example, the domain $[0,1]$ for parameters $u$ and $v$ must be divided into $N = 7$ parts, while every value of $u'$ and $v'$ must correspond to $n=3$ points out of this domain.
Fig. 7. (a) Continuous folding function for a freeform lens array with $7\times 7$ lenses in which the light towards each position comes out of $3\times 3$ lenses. (b) Discontinuous version of the same folding function.
Just as in the previous case, this continuous piecewise linear functions $F_u$ and $F_v$ will result in a ray mapping that generates saddle shaped regions on the corresponding lens surface. Therefore, similar integratibility issues will be encountered. In order to eliminate these saddle shaped surface regions, the continuity of the function $F$ is again given up. Also for this case, the statement is valid that if the slope values of both $F_u$ and $F_v$ are the same for every point (u, v), then no saddle shaped surface segments will be obtained. This condition can simply be achieved by changing the sign of the slope of certain line segments. In order to obtain only convex lenses, all slope values should be negative. The resulting function $F_u$ for convex lenses can be seen in Fig. 7(b). The combined function $F(u,\;v)=(F_u(u),F_v(v))$ will result in a folded mapping $\psi \circ F \circ \phi ^{-1}$ that maps 9 points of the source distribution $S$, onto each point of the target distribution $T$.
If the initial mappings $\phi$ and $\psi$ are curl-free mappings, the mapping $\psi \circ F \circ \phi ^{-1}$ can be directly integrated to obtain the corresponding discontinuous lens array. For non-paraxial illumination configurations, the sub-mappings corresponding with each individual lenssegment should be further optimized to make them integrable [20]. Once this is done, the full discontinuous lens array can be directly calculated from these integrable mappings. For this example, the lens array is implemented as the exit surface of a lens component, of which the entrance surface is a planar interface situated at $10$ mm from a point source. The refractive index of the lens material is chosen equal to 1.54, in correspondence with the material of the fabricated prototype. The resulting lens component can be seen in Fig. 8(a). The simulated irradiance distribution matches the desired target distribution very well (see Fig. 8(c)).
Fig. 8. (a) The resulting discontinuous freeform lens array. (b) A fan of source rays is refracted by the freeform lens array into multiple overlapping fans of outgoing rays. (c) The simulated irradiance pattern that results from the lens array with a point source.
In order to explain more in detail how the luminance spreading by the designed freeform lens array is exactly happening, both the source distribution $S$, the target distribution $T$ and the discontinuous function $F_u$ are schematically shown in Fig. 9. The discontinuous function $F_v$ is the same as $F_u$.
Fig. 9. (a) The source distribution divided into different source patches. (b) The discontinuous folding function $F_u$ in which three line segments are highlighted. (c) The target distribution divided into different target patches.
Every combination of a line segment in the function $F_u$ and a line segment in the function $F_v$, corresponds to a rectangular patch in the $u,\;v$-parameter space. This rectangular patch is mapped by $\phi$ on a certain source patch in the source distribution and is mapped by $\psi$ on a certain target patch in the target distribution. The mapping $\psi \circ F \circ \phi ^{-1}$ thus maps this source patch onto the corresponding target patch. This means that each ingoing ray of the source patch is mapped onto an outgoing ray of the target patch. The knowledge of the ingoing and outgoing ray directions, allows to calculate the necessary surface normals of the freeform lens using Snell’s law. The resulting surface normals can then be used to calculate the shape of a single lens segment of the full array via numerical integration. A source patch thus corresponds with a single freeform lens of the full array, while the target patch corresponds with a certain part of the target distribution that is illuminated by that specific freeform lens.
The spreading of light by different lenses towards multiple overlapping areas in the target distribution is clearly illustrated in Fig. 9. Three line segments of the function $F_u$ are highlighted. The combination of each of these highlighted line segments with one light segment (= segment 3) of the function $F_v$, results in three different source patches and three different target patches, which are also highlighted with the corresponding pattern or color. As can be seen, the middle area of the triangular target distribution is already illuminated by three different lenses. If we would complete this analysis for all lenses, we would see that all areas of the complete target distribution are illuminated by exactly $3\times 3$ lenses. This also implies that when an observer is located at a certain position within the illuminated area and looks at the lens array which is illuminated by the source distribution, will observe that light is being emitted in his direction from multiple individual lenses. This effect is illustrated in Fig. 10(a), which shows a rendering of the corresponding single-channel freeform lens compared to the freeform lens array.
Fig. 10. (a) A rendered visualisation of a single freeform lens and a freeform lens array that are illuminated by a small light source. (b) The simulated irradiance pattern that results from a single freeform lens (left) and the freeform lens array (right) when illuminated by a lambertian disk source with diameter = $2$ mm.
To understand why this luminance spreading leads to a possible reduction of the observed peak luminance, one should take the limited resolution of the human eye into account. Spreading the luminance over multiple lens segments does not automatically lead to a reduction of the emitted peak luminance values. In fact, the peak luminance will stay the same. However, when the source extent or the lens element size is sufficiently small, the luminance spot from an individual lens will be smaller than the eye resolution spot and should thus be convoluted with the eye resolution spot in order to calculate the observed luminance spot. Taking this additional luminance spreading of the human eye into account, will imply in most cases that the observed peak luminance from a lens array is significantly lower than from a single-channel freeform lens. A detailed analysis of the reduction of discomfort glare due to the spreading of the observed luminance and reduction of its peak value however, is beyond the scope of the current paper.
2.3 Fabrication issues and first prototype
Since $\phi$ is a mapping from a uniform distribution to the source distribution, the areas of the individual lenses scale proportional to the source distribution. This means that source areas with high light flux contain more lens segments than areas with a lower light flux. If lenses of a specific size are needed, one can change the start and end points of the line segments in $F_u$ and $F_v$, although condition (4) should always be satisfied. One reason to influence the relative size of the individual lenses via the fold mapping, is to satisfy certain manufacturing requirements. Still, manufacturing such discontinuous freeform lens arrays with high accuracy is very demanding. No manufacturing process today can realise the infinite curvatures that are required by the discontinuous transitions between the individual lenses. These discontinuous lens borders will therefore always generate some stray light in practice.
In Fig. 10(b) the simulated irradiance distributions are shown for both the single-channel freeform lens and corresponding freeform lens array, when combined with an extended disk source of $2$ mm in diameter. It is clear that the impact of the source extent on the obtained target distribution is larger in the case of the freeform lens array. This is also a direct consequence of the discontinuous lens borders.
An advantage of discontinuous lens arrays on the other hand is that they can be made quite thin, similar to a Fresnel lens. In the case of a continuous freeform lens array, the optical component can become quite bulky, as was demonstrated in an earlier publication for the case with rotational or longitudinal symmetry [15].
A prototype of the discontinuous lens array has been printed with a Formlabs 2 printer with a layer resolution of 25 $\mu$m. Although the optical quality of the resulting component is not sufficient to achieve a very accurate target distribution, the qualitative comparison of the obtained irradiance distribution (Fig. 11(a)) with the one shown in Fig. 10(b) is quite good. The effect of the luminance spreading over 9 different lens segments can also be clearly seen in Fig. 11(b).
Fig. 11. (a) Obtained irradiance distribution with the 3D printed prototype. (b) A picture of the illuminated freeform lens array by a small LED light source illustrates the spreading of the luminance towards the observer position, over 9 different lens segments.
3. Conclusion and future work
Control of the observed peak luminance and radiation pattern of a lighting luminaire is essential to develop glare free lighting systems. In this paper, a novel method is presented to design a luminance spreading freeform lens array that offers perfect control over the resulting radiation pattern.
The method is based on a non-invertible mapping of a 2D parameter space which results in a source-target mapping in which multiple ingoing ray directions are mapped onto every position of the target distribution. For the non-invertible mapping, a certain folding function is used that can adopt many forms depending on the application. The functions that are considered in this paper are piece-wise linear continuous or discontinuous functions.
Because of integrability issues with continuous folding functions, the discontinuous case was further explored and elaborated. This approach results in a discontinuous source-target mapping which can be used to calculate the individual lenses of a discontinuous freeform lens array. The peak luminance reduction of such a freeform lens array is due to the fact that multiple lenses are illuminating overlapping areas in the target distribution. This effectively reduces the observed peak luminance compared to a single freeform optic. An example of such a freeform lens array with $7\times 7$ individual lenses was designed, the performance was simulated and also experimentally demonstrated using a 3D-printed prototype.
With a continuous fold mapping it is possible to generate $C^2$ smooth freeform lens array. Such smooth freeform lens arrays would be easier to manufacture than their discontinuous counterparts and give better performance. At this moment however there seems to be a fundamental issue with the saddle shaped lens regions that are necessary in that case. Existing ray mapping algorithms can create or optimize convex or concave freeform surfaces, but no algorithm allows the creation of integrable ray mappings for saddle shaped lenses or reflectors. This is certainly an interesting pathway for future research in the domain of freeform optical design.
Funding
Agentschap Innoveren en Ondernemen (145023).
Disclosures
The authors declare no conflicts of interest.
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