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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 10563–10571
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High resolution irradiance tailoring using multiple freeform surfaces

Adrien Bruneton, Axel Bäuerle, Rolf Wester, Jochen Stollenwerk, and Peter Loosen  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 10563-10571 (2013)
http://dx.doi.org/10.1364/OE.21.010563


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Abstract

More and more lighting applications require the design of dedicated optics to achieve a given radiant intensity or irradiance distribution. Freeform optics has the advantage of providing such a functionality with a compact design. It was previously demonstrated in [Bäuerle et al., Opt. Exp. 20, 14477–14485 (2012)] that the up-front computation of the light path through the optical system (ray mapping) provides a satisfactory approximation to the problem, and allows the design of multiple freeform surfaces in transmission or in reflection. This article presents one natural extension of this work by introducing an efficient optimization procedure based on the physics of the system. The procedure allows the design of multiple freeform surfaces and can render high resolution irradiance patterns, as demonstrated by several examples, in particular by a lens made of two freeform surfaces projecting a high resolution logo (530 × 160 pixels).

© 2013 OSA

1. Fine irradiance tailoring remains a challenging task

The design of freeform optics is becoming the preferred route to achieve compact optical systems producing a prescribed irradiance or radiant intensity distribution. They are used in a wide range of applications ranging from general lighting (street lighting for example) to automotive lighting and also more specialized applications like laser beam shaping.

In the first part of this article, the principle of the ray mapping computation as described in [1

1. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012) [CrossRef] [PubMed] .

] is briefly reviewed. The main contribution of this article is then presented in the subsequent section: the parametrization chosen for the surface representation allows to readily compute a localized light flux traversing the optical elements. A comparison of the flux hitting the target with the prescribed irradiance allows the optics to be directly optimized to fine-tune the results obtained from the mapping. Important numerical aspects have to be taken into account during calculations and will be highlighted.

2. Ray mapping computation to efficiently find a starting point

The overall design task amounts to finding optical surfaces that can be coupled to a zero-étendue source (for example a point source or a perfectly collimated source) so that after traversing the optical system, the light forms a prescribed irradiance distribution on a target plane. To this end, each source ray is uniquely associated with a point in a 2D plane Ω0 perpendicular to the optical axis via a suitable projection. The target irradiance is equally projected in a consistent way onto a target plane Ω1. The task of designing a (freeform) optical system can then be described as calculating a diffeomorphism (“ray mapping”) so that the transformed irradiance distribution matches the target distribution:
u:Ω0Ω1,(x,y)(tx,ty)
(1)
where (tx, ty) represents the target point in Ω1 to be reached by a source ray identified by (x, y) in Ω0. Figure 1 recalls this principle and shows how a point from the source plane Ω0 is uniquely mapped to a point on the target plane Ω1.

Fig. 1 Point source case: the mapping u is computed between the projected source plane Ω0 and the target plane Ω1. The freeform surfaces are positioned between Ω0 and Ω1. They implement the mapping u which associates each point (x, y) ∈ Ω0 to its counterpart (tx, ty) ∈ Ω1. Here, two examples of such an association are represented by arrows.

An initial mapping ũ between these two planes can easily be computed using successive linear integrations along the Cartesian coordinate axes [2

2. W. A. Parkyn, “Design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3428, 154–162 (1998) [CrossRef] .

]. In a second step, the mapping itself is optimized to make it irrotational. This ensures better adherence of the field of surface normal vectors N to the integrability condition
N(×N)=0
(2)
required to build a smooth optical element [3

3. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18, 5295–5304 (2010) [CrossRef] [PubMed] .

, 4

4. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002) [CrossRef] .

]. For the mapping optimization, Haker [5

5. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. of Comput. Vision 60, 225–240 (2004) [CrossRef] .

] proposed an evolution equation over a pseudo-time variable t converging towards a stationary irrotational solution (t → ∞):
ut=1μ0Du(Δ1divu)withu|t=0=u˜
(3)
where Du denotes the mapping’s Jacobian, (x, y) = (−y, x) represents a rotation by 90 degrees in ℝ2 and Δ−1 divu denotes the solution f of Poisson’s equation Δf = −divu with Dirichlet boundary conditions.

Starting from a surface represented as a triangular mesh and using the vertex positions along the source rays as scalar variables, it is a key result of [1

1. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012) [CrossRef] [PubMed] .

] that a multi-surface optical element can be constructed by performing a least-squares optimization. The corresponding merit function directly compares the actual position of rays traced through the optical system with their position prescribed by the irrotational mapping and thus significantly reduces the computation effort over optimization schemes that compare irradiance estimates derived from Monte-Carlo analyses (brute-force ray-tracing). It was demonstrated that this design procedure performs well in cases where homogeneous irradiance of a square area is required, and that the procedure is equally well suited for more complex cases found for example in the automotive industry. The design of a two-sided fog light lens demonstrates the latter [6

6. A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012) [CrossRef] .

].

3. High-resolution tailoring of the irradiance distribution

The main advantage of this method is that the optical surface is readily represented by a smooth function, as the field of surface normals automatically conforms to the integrability condition Eq. (2). However, a disadvantage is the high computational cost which is partly due to the sensitivity of the target irradiance to the surface’s second derivatives. In addition, this procedure has, to the best of the authors’ knowledge, yet only been published and proven for a single optical surface, which is a major difference to the work presented in [1

1. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012) [CrossRef] [PubMed] .

] and here.

We observe that, on the one hand, the flux emitted by the source ΦS can be written as follows:
  • for a point source:
    ΦS=d2ΦS=I(θ,ϕ)d2Ω
    (4)
    where d2ΦS is the elementary light flux, d2Ω the elementary solid angle surrounding the direction of emission (θ, ϕ), and I(θ, ϕ) is the radiant intensity (in W/sr) of the point source. The elementary solid angle d2Ω is approximated by the solid angle encompassed by the three source rays around a triangular face (see Fig. 2). This solid angle remains constant throughout the construction.
  • for a collimated source:
    ΦS=d2ΦS=BS(x,y)dxdy
    (5)
    where BS(x, y) is the source (radiant) irradiance (in W/m2). In this case, the source rays are all directed towards the same direction (e.g. along the z-axis) and the cross section of a triangular tube remains constant between the source and the first optical surface.

Fig. 2 Elementary light flux tube (prism-shaped) delimited by the rays passing through the vertices of a triangular surface element (blue). The vertex rays are represented in dark, the face ray in green. If the vertex ray directions (e.g. the red vector s) drawn from the source are held constant, the light flux within the triangular tube also remains constant.

In both cases, the flux from the source within a triangular tube remains constant along its path through the optical system, as illustrated by Fig. 2.

On the other hand, the flux reaching the target ΦT can be written as a function of the irradiance (in W/m2) on the target. A triangular flux tube is projected as a triangular area on the target as illustrated in Fig. 2. The flux in this tube can be computed as:
ΦT(j)=AjBT(x,y)dxdy
where Aj is the area of the target triangle corresponding to the j-th face of the surface, and BT(x, y) the desired irradiance at the target point (x, y). This integral is approximated with the simplest quadrature:
ΦT(j)=BT(xj,yj)Aj
(6)
with (xj, yj) being the center of the triangular face on the target as illustrated in Fig. 2.

3.1. Objective function components

Similar to the purely mapping-based approach in [1

1. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012) [CrossRef] [PubMed] .

], a least-squares optimization procedure is proposed. Taking into account the description of the light flux above, the merit function being optimized can now be detailed as a collection of three components:
  • light flux difference: this component aims to equalize the light flux in Eqs. (4) (or (5)) and (6). It ensures that the desired irradiance BT(x, y) is achieved on the target;
  • smoothing component: in order to avoid local minima corresponding to triangular mesh components with abnormally high or low radiant intensity values (distorted triangles), a smoothing component is added. It enforces that face rays intersect the target near the center of the respective triangular area on the target. A face ray is closely correlated to the normal vector of the corresponding triangular face on the optical surface. Thus this component gives an advantage to nearly co-planar neighboring triangles on the optical surface;
  • distance to target’s boundary: the source and the target total light flux are both normalized to unity, so that all the source rays going through the system must eventually reach the target. This is explicitly enforced by penalizing edge rays which deviate too far from the target boundary.

3.2. Numerical considerations

A consistent and homogeneous scaling of the three merit function components allows a better numerical stability, the ability to easily scale up or down the resolution of the surface (number of triangles), or the possibility to change the geometry of the problem without re-adjusting from scratch the convergence criteria of the optimizer.

All three components of the objective function above are hence chosen to be homogeneous to unity and to remain of the same order of magnitude, regardless of the geometry or resolution of the problem:
  • the flux difference for a face is multiplied by the number of faces. Since the source flux is already normalized to unity, this ensures the scalability with regards to the number of triangles in the mesh. The effective light flux going through a face hence always has the order of magnitude unity;
  • the smoothing component is scaled by the typical size of a triangle area on the target which varies as the square root of the total number of triangles;
  • the distance to the target boundary is scaled by the target size.

The optimization procedure also involves the computation of a numerical gradient in the form of a standard forward difference using a step size h. It can be shown [8

8. D. Bindel and J. Goodman, “Principles of scientific error computing,” tech. rep., New York University - Computer Science (2009).

] that on a finite-precision machine with machine precision εm such a computation leads to the following final relative error:
f˜(x)=f(x)(1+εm2f(x)hf(x)+f(x)2f(x)h)
(7)
with ′(x) the numerical approximation of the derivative and εm the machine round-off precision. The last two terms are competing one with another and dictate an optimum value for h. Noting εm = 10m, h = 10n, 2f/(hf′) = 10u and f″(x)/(2f′(x)) = 10v the optimum value of n satisfies v + n = −n + u + m, hence:
n=(uv+m)/2
Typically, with the scaling considerations mentioned above (u, v, m) = (−1, −1, −15) giving a step size h ≈ 10−7 and an overall relative computation precision around ε = 10v+n ≈ 10−8. However, the computation above discards the error induced by the evaluation of f itself (propagation error): a slightly bigger step size (around h ≈ 10−6) gives in practice better results and has been retained in the final implementation.

Finally, the computation time using this new objective function is about a factor 4 longer than the mapping-only approach, but this remains within minutes on a standard PC for typical applications (general lighting, automotive, etc.).

4. Sample applications

The procedure presented above allows the design of various optical configurations with a wide range of applications. We present two complex cases. For each design below the improvement between the initial mapping approach and the flux optimized approach is presented, and the examples are given in order of increasing complexity.

4.1. Double sided freeform lens

First, a refractive case with two freeform surfaces is presented. The target plane is positioned perpendicular to the light axis and is centered on it at a distance of 100 mm. The prescribed irradiance is the letter “B”.

The flux optimization provides a sharper cut-off at the boundaries of the pattern as can be seen in the two central holes, and on the upper and lower tips of the left part of the distribution. The two freeform surfaces forming the lens were designed using 4,000 triangles for each optical element. The final surfaces consist of two NURBS fitted to the two triangular meshes. The design assumes that the lens is made of glass (n = 1.5) and that a light cone with a half-angle of 45 degrees is captured from the point source.

4.2. Logo lens

Fig. 3 Double sided freeform lens projecting the letter “B” (lengths in mm, irradiance in a.u.); (a) with mapping optimization only; (b) with mapping and flux optimization; (c) illustration of the geometry.
Fig. 4 Ray-tracing of high-resolution logo generated with 16 million rays (lengths in mm, irradiance in a.u.); (a) prescribed irradiance distribution (passed to the algorithm in black and white); (b) one freeform surface with mapping optimization only; (c) one freeform surface with mapping and flux optimization; (d) two freeform surfaces with mapping and flux optimization.

Similar to what has been done previously by Bortz and Shatz [9

9. J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007) [CrossRef] .

], the following fractional RMS measure was computed to assess the quality of the design:
RMS=(IcIpIref)2
(8)
where Ic is the computed irradiance as given by the ray tracing, Ip is the prescribed irradiance and Iref is a reference irradiance. Bortz and Shatz restricted their computation to an area where the irradiance was sufficiently high and hence took Iref = Ip. In our case, given the contrast at hand (about 4:1) and the fact that the analysis should not be restrained to the letter cores (areas of high irradiance), we computed the RMS with both Iref = Iavg, the (arithmetical) average of the light irradiance, and Iref = Imax, the peak irradiance found at the core of a letter, for example.

Fig. 5 Zoom on the lower right part of the logo for RMS analysis. Ray-tracing with 2 million rays on a grid of 66×39 points (lengths in mm, irradiance in a.u.). (a) one freeform surface with mapping optimization only; (b) one freeform surface with mapping and flux optimization; (c) two freeform surfaces with mapping and flux optimization.

Table 1. Fractional RMS for a sub-area of the logo projection. RMSavg (resp. RMSmax) denotes the RMS computed with Iref being the average irradiance (resp. with Iref being the peak irradiance)

table-icon
View This Table

Finally, the overall optical efficiency in this case is close to its maximum theoretical value, since all the rays of the collimated source are captured by the optical system.

5. Conclusion and future work

Acknowledgments

The authors would like to thank the German Research Foundation (DFG) for its support within the Cluster of Excellence “Integrative Production Technology for High-Wage Countries” at RWTH Aachen University. This work has partly been funded by the German Federal Ministry of Education and Research (BMBF, grant numbers 13N10832, 13N10833).

References and links

1.

A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20, 14477–14485 (2012) [CrossRef] [PubMed] .

2.

W. A. Parkyn, “Design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3428, 154–162 (1998) [CrossRef] .

3.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18, 5295–5304 (2010) [CrossRef] [PubMed] .

4.

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590–595 (2002) [CrossRef] .

5.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. of Comput. Vision 60, 225–240 (2004) [CrossRef] .

6.

A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, fresnel-type freeform optics,” Proc. SPIE 8485, 84850H (2012) [CrossRef] .

7.

A. Bäuerle, R. Wester, C. Schnitzler, and P. Loosen, “Design of efficient freeform lenses for mass-market illumination applications using hybrid algorithms,” Proc. SPIE 7788, 778807 (2010) [CrossRef] .

8.

D. Bindel and J. Goodman, “Principles of scientific error computing,” tech. rep., New York University - Computer Science (2009).

9.

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007) [CrossRef] .

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(220.2740) Optical design and fabrication : Geometric optical design
(080.1753) Geometric optics : Computation methods
(080.4225) Geometric optics : Nonspherical lens design
(080.4298) Geometric optics : Nonimaging optics

ToC Category:
Geometric Optics

History
Original Manuscript: March 1, 2013
Revised Manuscript: April 13, 2013
Manuscript Accepted: April 16, 2013
Published: April 23, 2013

Citation
Adrien Bruneton, Axel Bäuerle, Rolf Wester, Jochen Stollenwerk, and Peter Loosen, "High resolution irradiance tailoring using multiple freeform surfaces," Opt. Express 21, 10563-10571 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10563


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References

  1. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express20, 14477–14485 (2012). [CrossRef] [PubMed]
  2. W. A. Parkyn, “Design of illumination lenses via extrinsic differential geometry,” Proc. SPIE3428, 154–162 (1998). [CrossRef]
  3. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express18, 5295–5304 (2010). [CrossRef] [PubMed]
  4. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19, 590–595 (2002). [CrossRef]
  5. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. of Comput. Vision60, 225–240 (2004). [CrossRef]
  6. A. Bruneton, A. Bäuerle, M. Traub, R. Wester, and P. Loosen, “Irradiance tailoring with two-sided, fresnel-type freeform optics,” Proc. SPIE8485, 84850H (2012). [CrossRef]
  7. A. Bäuerle, R. Wester, C. Schnitzler, and P. Loosen, “Design of efficient freeform lenses for mass-market illumination applications using hybrid algorithms,” Proc. SPIE7788, 778807 (2010). [CrossRef]
  8. D. Bindel and J. Goodman, “Principles of scientific error computing,” tech. rep., New York University - Computer Science (2009).
  9. J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE6670, 66700A (2007). [CrossRef]

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