New approach to construct freeform
surface by numerically differential
formulation
Yu-Lin Tsai
Ming-Chen Chiang
Ray Chang
Chung-Hao Tien
Chin-Tien Wu
New approach to construct freeform surface
by numerically differential formulation
Yu-Lin Tsai
National Chiao Tung University Department of Photonics Hsinchu 30010, Taiwan Ming-Chen Chiang
National Chiao Tung University Department of Applied Mathematics Hsinchu 30010, Taiwan
Ray Chang Chung-Hao Tien
National Chiao Tung University Department of Photonics Hsinchu 30010, Taiwan E-mail:chtien@mail.nctu.edu.tw Chin-Tien Wu
National Chiao Tung University Department of Applied Mathematics Hsinchu 30010, Taiwan
Abstract. We proposed a new design method for single freeform reflective (or refractive) surface tailored to redistribute the radiant flux onto a prescribed illumination pattern. Unlike the conventional optimization approaches based on the grid mapping, in this study we estimated each segmental freeform surface by locally solving a second-order differ-ential equation, which formulates the energy transportation between each domain cell. With finite element method via Hermite element, we validated a series of smooth reflective/refractive surfaces to reallocate the radiant flux from a point source toward a target plane with specific patterns. The proposed technique offers a large flexibility by varying the vectors of each ray with multiple refraction (or reflection), which imposes no restric-tion on the target distriburestric-tion, collective solid angle, or even target topog-raphy. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attri-bution of the original publication, including its DOI. [DOI:10.1117/1.OE.53.3.031307]
Subject terms: freeform design; uniform illumination; nonimaging optics. Paper 131103SS received Jul. 22, 2013; revised manuscript received Oct. 6, 2013; accepted for publication Nov. 7, 2013; published online Dec. 31, 2013.
1 Introduction
Construction of a freeform surface to reallocate the radiant flux is a substantial challenge in terms of nonimaging optics. The basic concept is to convert the radiant flux emitted from a point source onto a prescribed illumination, usually in Cartesian coordinate. Parkyn1firstly pioneered a tessellation
method to map the cell area between a point source in polar grid with a particular rectangular illumination in Cartesian grid. The cells of both grid topologies were sized so that they share common energy. The methodology was based on the assumption that when the polar cell on the light source was exactly mapped to the corresponding rectangular target cell, the flux of the angular cell would be fully transferred to the target cell accordingly. Each corresponding pair of cells on these grids has an associated surface normal vector, and then governs the surface shape using extrinsic differential geometry. The major concern about equal flux grid method lies in a fact that the constructed surface, either reflective or refractive, is unlikely to guar-antee the integrability condition in the overall surface, and thus leads to a discontinuous boundary. The cumulative deviations from the ideal surface normal become more pro-nounced at the far corners of the output grid. The discontinu-ous boundary could lead to a problem. The problem is that a small amount of radiant flux will be distorted by the sharp cliff. Hence, sufficient partition cells shall be numbered to confirm the estimated accuracy for surface construction.
To alleviate the discontinuous issue of freeform, Wang
et al.2 introduced the variable separation mapping strategy
to take into account the normal deviation of the curve below a present value. The surface curve was regenerated
with tangent vectors along the curve perpendicular to the normal vector, thus eliminating the accumulated normal deviation. The proposed scheme constructs a smooth surface by interpolating the discontinuity surface with non-uniform
rational basis spline (NURBS).3On the other hand, the
inte-grability condition η ¼ N · ð∇ × NÞ ¼ 0 is necessary in
order to make a smooth freeform surface, where N is the sur-face normal.4To fulfill the integrability condition, Fournier3
interpolated the ellipsoids, obtained from Oliker’s algorithm
with appropriate grid partition, by using the NURBS where parameters of the NURBS were determined by minimizing
the residual curlη. Although the surface can be made smooth
and the resulted radiant flux can be less distorted via the above mentioned approaches, the amount of energy deviation is still difficult to be estimated before the freeform surface is constructed. Furthermore, the computational costs
of the constrained optimization for the NURBS’ parameters
may rise significantly and the convergence is not guaranteed as the number of grid partition increases.
In additional to the tessellation method, the freeform sur-face design problem is also related to the solution of the well-known Monge-Ampere (MA) equation. In 1972, Schruben first derived the MA equation obtained from reflector design. Beginning in 1980s until the present, Oliker et al. formulated the freeform problem into a highly nonlinear MA type of
partial differential equation (PDE).5–9 The MA PDE also
appears in other research fields, such as differential geom-etry, optimal control and mass transportation, meteorology,
and geostrophic fluid.10–12 Although many works about
existence and regularity of MA PDE have been conducted, numerical computation of freeform surface reconstruction by solving the MA PDE remains a challenge in practice. We are
curious if there exists a systematic way to construct the free-form in which not only the surface is guaranteed to be smooth, but also the correspondence between light source and target plane is ensured under a controllable energy deviation.
In this article, we formulate the ray forward and inverse propagation with negligible energy deviation within the MA PDE framework. By following the works of Caffarelli and
Oliker13, the freeform considered here is assumed to be
con-vex. The model is based on local energy conservation in which the flux correspondence associated with each local domain is prescribed. We take advantage of mathematical analysis of the MA PDE and successfully formulate the free-form design problem into the popular finite element para-digm. The formulation could maintain the ability to control the light beam and guarantee a smooth freeform sur-face. Moreover, this article describes this in detail by going through the different modeling steps of the model. All the main restrictions influencing the target distribution, collec-tive solid angle, or source distribution are discussed, along-side the viability of realizing an optical surface for different specific requests.
2 Freeform Model in Localized PDE
To develop a method for freeform construction subject to the source-target information, we must deduce an adequate, physically realistic model of ray propagation in the free space. The model can then be employed to understand the reflective (refractive) forward process of rendering illumi-nance according to a known optical configuration. Also, the model is applicable to the inverse algorithm of estimating the freeform surface from a given source-target correspond-ing condition. First of all, the ray transportation must satisfy the energy conservation among coordinate transforma-tion.14,15The radiant intensity I (Watts∕sr) of a point source
(in polar domainΩθ) and prescribed illuminance distribution
E (Watts∕m2) of the target region (in Cartesian domainΩ
x),
in one dimension, can be related as Z Ωθ IðθÞdθ ¼ Z Ωx EðxÞdx: (1)
The goal is to design a freeform surface by analyzing the potential and limitation of using the PDE. In this article, the freeform surface is assumed to be continuous and convex. By
partitioning the global domain Ωθ and Ωx into many local
subdomainsΩiθandΩix, so that each corresponding pair sat-isfies the equal flux i ¼ 1; 2; : : : ; n, we are able to ensure the convergence of the integration. As long as the number of partitions is sufficiently large, the freeform can be considered to be a flat surface. As a result, a reduced but simplified representation of a freeform can be derived on each local domain. Hereafter, a flat surface criterion to the free-form in each local domain is called a small local planar approximation.
The representation of the local freeform is indeed a non-linear differential equation and shall be derived in the follow-ing. First, let us introduce some notations. Global coordinate frame is denoted by½0with basis vectorsfe0
1; e02g and origin
point O. The target plane T is expressed by a plane equation C0·½X0− d0¼ 0, where C0¼ ½ c1 c2T, d0, and½X0are
the normal vector, shift and spatial variables, respectively.
Although the laws of reflection (or refraction) are invariant with coordinate system, solution of freeform problem requires that the relations derived from these laws can be expressed in a coordinate system appropriate to the geometry of the given problem. Hence, we define a local coordinate system, by which the i’th local coordinate of any vector
V is denoted by ½Vi with corresponding orthogonal basis
fei
1; ei2g and local origin Oi. Coordinate transformation
from i’th to j’th coordinate system can be conducted via a sequence of operators,ℋji ¼ TjRj
i, whereRji andTj
re-present the rotation of basis vectors and translation: TjðVÞ ¼ ½Vjþ ½Oi− Ojj, respectively. By means of
sub-sequent intermediate local coordinates, the light source O
and the target plane T can be linked as ½Oi¼ ℋi
0½O 0
and C0·ℋ0i½Xi¼ d0, respectively.
The scenario of our freeform construction is
schemati-cally in Fig. 1. The segmental freeform surface to be
constructed on i’th local coordinate system can be
analyti-cally presented as a function ½Ai¼ ½u; fðuÞ with surface
normal as ½Ni¼ ð−fffiffiffiffiffiffiffiffiffiffiffiffiffiu; 1Þ
f2 uþ 1
p : (2)
As shown in Fig.1(a), while a ray from point source O is
hitting on the point A at the freeform reflector, the ray is guaranteed to be transported toward the target plane T. The refraction law in vector form is
n2 n1 ½XA!i k½XA!ik¼ ½OA!i k½OA!ik− 2 6 4 ½OA !i k½OA!ik·½Ni ! − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 2− n 2 1þ n 2 1 ½OA!i k½OA!ik·½Ni !2 v u u t 3 7 5½Ni; (3)
Fig. 1 (a) Global coordinate system: origin (O) locates at light source with basis vectorsfe01; e02g. Local i’th coordinate system: origin (Oi) locates at freeform surface with local basis vectors fei1; ei2g.The anglesθ and ϕi0are the angle of OO!i with OA!and ei2, respectively. A 1-D freeform model assumes that the global coordinate and local coordinate are coplanar. (b) PartitioningΩθandΩxinton subdomain Ωi
θ and Ωix for i ¼ 1; 2; : : : ; n is based on equal flux theory.
Computation domainΩiuthat unifying variableθ and x into local var-iableu is mapped from subdomain ΩiθandΩix.
where n1 and n2 represent the refraction indexes near the
source side and target plane side, respectively. For the reflec-tion law,7the refraction index can be defined by n1¼ 1.0 and
n2¼ −1.0. Here, we size each cell of subdomain pair (Ωiθ
and Ωi
x) to allow equal flux transfer. In order to derive the
freeform surface via the preceding equations, we must intro-duce an intermediate variable u in the computational domain
(Ωiu) to link the computation in both source and target
domains (Ωi
θ and Ωix). By changing the variable, Eq. (3)
can be rewritten as Z Ωi u IðuÞdθ i dudu ¼ Z Ωi u EðuÞdx i dudu: (4)
Obviously, the integral form of energy conservation, Eq. (1) can be represented by a differential equation in sum-mation:7 EΩi u · dxi du− IΩiu· dθi du ¼ 0 for u ∈ Ω i u: (5)
Letðuih; 0Þ be the coordinate of the hitting point where the incident ray OA!with light source½Oi¼ ðLi
x; LizÞ hits on i’th
local x-axis. The x-coordinate of the hitting point ui
hcan be
expressed as two separate functions of u and θiby
trigono-metric formula: uihðθiÞ ¼ ρ 0 sinðθiÞ cosðθiþ ϕi 0Þ ; (6) uihðuÞ ¼−fðuÞðu − L i xÞ fðuÞ − Li z þ u; (7) where θi and ϕi
0 are the angle of OO
!i
with OA! and ei2,
respectively, and ρ0 is the distance between O and Oi.
The transformation between variables u and θican be linked
by the joint point uh and the Jacobian chain rule becomes
dθi du¼ dθi duih dui h du : (8)
Similarly, suppose the incident ray OA!is reflected onto
the point½Xi¼ ðxi; ziÞ at the target plane T. The variables xi and u related by the law of reflection can be expressed as
k½OA!ik
k½XA!ikðu − xiÞ ¼ 2ð½OA
!i
·½NiÞ½−f0ðuÞ − ðu − LixÞ; (9)
k½OA!ik
k½XA!ik½fðuÞ − zi ¼ −2ð½OA
!i
·½NiÞ½−f0ðuÞ − ½fðuÞ − Li
z: (10)
Dividing Eqs. (9) by (10), the equation that defines a local freeform surface can be obtained:
u − xi fðuÞ − zi¼ 2ð½OA!i·½NiÞ − ðu − Li xÞ −2ð½OA!i ·½NiÞ½−f0ðuÞ − ½fðuÞ − Lizi: (11) If we fix the source and target plane position, the freeform surface defined in Eq. (11) can also be considered as a para-meterized surface where the parameter is the x-coordinate of the i’th coordinate system. The freeform parameters depend on the external geometry at which each optical route O-A-X constitutes an epipolar plane. In order to transform the
under-determined Eq. (11) to a determined problem, we replace zi
by xi through the equation C0·ℋ0i½Xi ¼ d0:
xiðuÞ ¼Ci0f0ðuÞ þ Ci1fðuÞ þ Ci2u þ Ci3
Ci
4f0ðuÞ þ Ci5fðuÞ þ Ci6u þ Ci7
þ Rðu; fÞ; (12)
where Rðu; fÞ ¼ Oðu2; uf; uf0; f2; ff0; f02Þ is the high
order nonlinear term that can be neglected under the small
local planar approximation, and the coefficients, Cij,
j ¼ 1; 2; : : : ; 7, are constants with respect to the i’th local
coordinate system. For the refraction case, the variables xi
and u can be related by the law of refraction following a
sim-ilar derivation. After differentiating Eq. (12) to obtain
Jacobian dxi∕du and inserting Eqs. (8) and (12) into
Eq. (5), we can devise the local reflection (refraction) free-form surface to a nonlinear differential equation:
αi
1f0 0þ αi2f0þ αi3f þ αi4¼ 0; (13)
whereαi
1,αi2,αi3, andαi4are coefficient functions subject to
low order derivatives of f. This equation describes the ray-tracing map and the energy redistribution. The total energy is
conserved as ∫Ωi
xEðuÞxðuÞdu in each local domain.
Therefore, the energy deviation is mainly contributed by the neglected term Rðu; fÞ and can be controlled when the integral∫Ωi
xEðuÞRðu; fÞdu can be estimated.
The construction of freeform surface by numerically dif-ferential formulation has an advantage in that we can regu-larize the smoothness tolerance by setting the partition number (Ωiθ and Ωi
x) in a tradeoff between computational
efficiency and surface smoothness. In order to ensure the numerically estimated accuracy of freeform surface, suffi-cient partition cells shall be numbered in both domains.
Here, we impose the edge rays Ri−1 and Ri in the polar
cell toward the corresponding boundary ½bi−1; bi in the
Cartesian cell, so the radiant flux is fully transferred without stray loss. The edge ray,Ri, reflected by the freeform surface
is propagating toward the corresponding point bi. We shall
set a reasonable right end point of the intermediate cellΩi
u,
by which the freeform possesses convexity in the local
domain. To solve the nonlinear PDE in Eq. (13), we employ
the vanished moment method proposed by Feng and Neilan
and use Newton’s method to find the adequate root function.
The linearized equation at each iteration step can be solved by the finite element method. Here, we skip some
straight-forward but tedious expansion and rearrangement. Figure2
demonstrates the key steps of the freeform construction proc-ess. The algorithm of freeform construction in this session can be summarized as following:
3 Freeform Surface via Local Reconstruction Algorithm: Error Evaluation
In order to validate the proposed freeform scheme, we first constructed a surface by the previous algorithm and com-pared it with a known parabolic reflector; both are aimed to
deliver a collimation beam as shown in Fig. 3. A point
source was placed at the focal point ðx; yÞ ¼ ð0; 10Þ of
an ideal parabolic reflector y ¼ −x2∕12 þ 13. Here, we
limit the emitted angle from the point source within the
range of Ωθ ¼ ½−119 deg; 119 deg, and the radiant flux
reflected by the ideal parabolic reflector was directed
toward the given target planeΩx¼ ½−10; 10 with a uniform
and parallel propagation. Figure 3(a) exhibits the surface
comparison between the curve constructed via proposed
algorithm and ideal parabolic curve. The surface error es¼
maxx∈Ω
xkSFFðxÞ − SPðxÞk, maximum deviation between
the constructed freeform surface (SFF) and the ideal
para-bolic curve (SP), is on the order of 10−4to 10−6, which is
extremely small and quadratically inverse to the number of partition cells. The results are in agreement with the theo-retical prediction that the error comes from the finite
element method with Hermite element.16–18
4 Construction Algorithm: Two Design Types 4.1 Uniform Illuminance
After validating our freeform algorithm with a comparative parabolic surface by forward testing, we now aim to achieve a uniform illuminance with a geometric correspondence by the freeform surface via local reconstruction algorithm
(FS-LRA). As shown in Fig. 4, a 20-m-wide target plane was
placed beneath a point source with 6-m distance. The emitted angle from the point source was confined for reflection and
refraction withinΩθ ¼ ½−30 deg; 30 deg. With a mere 80
partition cells and 960 partition cells in both source-target
subdomains, a freeform reflector and refractor can be gener-ated in about 1.1 and 10 s, with 2.3 GHz Intel® dual core computer. The study employed a ray tracing simulation on the constructed freeform surface with a commercial tool LightTools®, in which light rays from the source were inci-dental to the target plane. The root-mean-square (RMS) non-uniformity is merely 1.05% (reflector) and 2.10% (refractor), respectively. It is noted that the accuracy or partition number of FS-LRA is highly dependent on the tolerance of the free-form surface. For unifree-form illuminance request in both
Algorithm Freeform surface via local reconstruction (FS-LRA). Input:Ωθ: Domain of light source:Ωx: Domain of target; O: Light
source; T; Target plane equation; Global coordinate frame½0; O1: initial point of freeform surface:
Output:f ðxÞ: freeform surface function;
1. Partition the source and target domain intoΩiθandΩix for i ¼ 1; 2; : : : ; n;
2. Setup position and orientation of the freeform (i.e., determine global and the first local coordinate frame½1):
3. fori ¼ 1 to n do 4. DetermineΩiu;
5. Solve equation (13) onΩiu;
6. Setup thei þ 1-th coordinate frame ½iþ1;
7. Apply transformation operatorHiþ1i to the light source and target plane;
8. end for
Fig. 2 Overview of the freeform construction process. Step 1: The source domain and target domain are divided into n subdomain pairsðΩiθ; ΩixÞ, i ¼ 1; 2; : : : ; n. Step 2: The global information including the location of light source and the target plane is transformed intoi’th coordinate. Step 3: The computational domainΩiuis determined by the corresponding subdomain pairðΩiθ; ΩixÞ, the refraction or reflection laws and the convexity of the freeform. Step 4: The freeform surface is calculated by solving Eq. (13) on Ωiu using Hermite finite element method. Step 5: The (i þ 1)’th coordinate system is defined by the rightmost boundary pointOiþ1and the normal and tangent vectors atOiþ1. Ifi is not equal to n, go to step 2. Step 6: The freeform surface is assembled from local surfaces.
reflect/refractive cases, a monotonic curve is accessible with the prescribed simple target surface. Because the mathemati-cal form in the refraction case is more complex than that in reflection case, 960 partitions are required for convergence to achieve the design uniform illumination pattern, but only 80 partitions are required for the constructed reflector to achieve the same uniformity.
4.2 Linear Varying Illuminance
In this example, we extended our technique to create a pre-scribed color illumination by a pair of independent sources
with different colors. Figure5shows the schematic diagrams
of the system layout for both reflect/refractive cases. With a 20-m target plane for both cases, we aimed to create a lin-early varying international commission on illumination
(CIE) chromatic ðx; yÞ along the x-direction associated
with two color sources. The distance between source and tar-get plane is 10 and 70 m, respectively. The light sources leave the center at 10 m and 15 m, respectively. The different configuration conditions of reflection and refraction design
are to achieve desired illuminance performance. As shown in Figs.5(c)and5(d), the illuminances feature a high linearity along the x-direction. There are only about 5.58% and 1.52% RMS nonuniformity with the perfect chromatic line for reflection and refraction cases, respectively.
5 Conclusion
In this article, we proposed a freeform model based on local energy conservation associated with a series of coordinate transformation. The major advantage of this method is to exploit the well-proven existence and continuity of an elliptic nonlinear differential equation of the Monge-Ampere type. We successfully formulated the freeform design into a finite
element paradigm with Newton’s iteration. The key
maneu-ver is to unify the variables (θ and x) in both domains (Ωi
θ
andΩi
x) into a common intermediate variable u in the
com-putational domain (Ωi
u). After imposing the small local
pla-nar approximation, which is absolutely valid with a sufficient number of partition cells, a complete freeform reconstruction algorithm can be developed. In addition, the segmental dif-ferential formulation is capable of taking numerical error into account in each step. The proposed surface parameterization guarantees the surface smoothness with no restriction on the desired target distribution, collective solid angle, or source distribution, respectively. This technique still leaves many opportunities open and clearly more research must be carried
Fig. 3 (a) S: point light source. The ideal parabolic curve is y ¼ −x2∕12 þ 13. (b) The maximum surface difference between
ideal parabolic curve and numerical surface. The surface error with respect to subdomain numerical approximates a function SðxÞ≈ 1∕x2. The amount of error reduces quadratically as we increase
the number of partition cells.
Fig. 4 (a) and (b) Layout of the freeform surface reflector and refractor is designed for uniform illuminance, respectively. (c) Illuminance dis-tribution on the target plane. Inset is the magnification of illuminance plateau pattern. The nonuniformity of reflection case was well con-trolled within 1.05% RMS and 1.89% peak-to-valley. (d) The nonun-iformity of refraction case was well controlled within 2.10% RMS and 4.01% peak-to-valley.
Fig. 5 (a) and (b) Layout of the freeform surface reflectors and refrac-tor is designed for chromatic varying illuminance. The CIE chromatic of two point sources is redðx; yÞ ¼ ð0.73; 0.27Þ and green ðx; yÞ ¼ ð0.38; 0.62Þ. (c) and (d) Comparison of illuminance pattern by the con-structed freeform surface (dash line) and that with ideally linear shape (solid line). The nonuniformity with the ideal linear case was well con-trolled within 5.58% RMS and 1.52% RMS in reflection case and refraction case, respectively. (e) and (f) Five sampling points along x-direction (−10, −5, 0, 5, and 10 m) exhibit a linearly chromatic shift in CIE 1931x, y chromaticity coordinate in both case.
out to explore its potential in full. First, at this moment, the issue was tackled in a one-dimensional case; complete treat-ment in consideration of twist deformation in freeform sur-face is underway. Second, point source approximation is another subject to be addressed when the freeform structure was placed in proximity of the light source such as light emitting diodes applications. The preliminary results pre-sented in this article, however, indicate that parameterizing the freeform surface via local freeform PDE may create another route in the field of freeform optics, as it has the potential to take freeform design into many nonimaging applications.
Acknowledgments
This work was financial supported by the National Science Council of Taiwan government under contract NSC101-2622-E-009-001-CC2 and 100-2115-M-009-001. The first author and second author contributed to this work equally. References
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Yu-Lin Tsai is a PhD candidate in the Department of Photonics at National Chiao Tung University. He received his MA degree in the Institute of Mathematical Modeling and Scientific Computing from National Chiao Tung University, Hsinchu, Taiwan, in 2010. His research interests include nonimaging optical design and freeform design problems.
Ming-Chen Chiang is an engineer in the Department of Wireless Product of Universal Scientific Industrial. He received his MA degrees both in IMMSC (Institute of Mathematical Modeling and Scientific Computing) and Institute of Communication Engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2011. His inter-ests include the freeform design problems and communication chan-nel analysis.
Ray Chang is a master student in the Institute of Display at National Chiao Tung University. He received his MA degree from the Institute of Display from National Chiao Tung University, Hsinchu, Taiwan, in 2013. His research includes nonimaging freeform design problems. Chung-Hao Tien is an associate professor in the Department of Photonics at the National Chiao Tung University (NCTU) where he has been a faculty member since 2004. He received his PhD degree in the Institute of Electro-Optical Engineering from NCTU in 2004. His research interests include computational imaging, free form nonimag-ing, display and\ lighting optics.
Chin-Tien Wu is an associate professor in the Department of Applied Mathematics at National Chiao Tung University. He received his PhD degree in Scientific Computation and Mathematical Modeling from the University of Maryland in 2003. His research interests include scien-tific computing and freeform design problems.
