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Tailored freeform optical surfaces

Harald Ries

Optics & Energy Concepts, Paul-Gerhardt-Allee 42, 81245 Munich, Germany,

and Philipps-University, Renthof 5, 35032 Marburg, Germany

Julius Muschaweck

Optics & Energy Concepts, Paul-Gerhardt-Allee 42, 81245 Munich, Germany

Received January 24, 2001; accepted August 31, 2001; revised manuscript received September 4, 2001

Freeform optical surfaces embedded in three-dimensional space, without any symmetry, are tailored so as to

solve the archetypal problem of illumination design: redistribute the radiation of a given small light source

onto a given reference surface, thus achieving a desired irradiance distribution on that surface. The shape of

the optical surface is found by solving a set of partial nonlinear differential equations. For most cases, a few

topologically distinct solutions exist, given suitable boundary conditions. © 2002 Optical Society of America

OCIS codes: 080.2740; 350.4600; 080.2720.

1. INTRODUCTION

The general problem of optical design for illumination can

be stated as follows: Given a speciﬁed light source, de-

sign an optical system that redirects the light from the

source onto a target surface of given shape such as to pro-

duce a given desired irradiance distribution on the target.

In this formulation, for simplicity, multiple optical sur-

faces and gradient index distributions are not considered.

A large amount of work has been devoted to this classical

design problem. Often a simple system consisting of a

single reﬂective or refractive surface is desired. In this

work we also refer to single optical surfaces.

Heuristic approaches are based on the known optical

properties of conics. Imaging optical systems tradition-

ally rely almost exclusively on spherical surfaces. Even

nonimaging systems traditionally involve combinations of

conics. Recently we introduced the idea of tailoring an

optical surface, which consists of translating the desired

optical properties into a suitable differential equation,

which is then solved numerically to determine the shape

of the optical surface.1–5In this way smooth surfaces are

found, which in general are not conic sections. The pre-

vious work on tailoring was applicable to two-dimensional

systems, i.e., systems that possess translational or rota-

tional symmetry. In this paper we follow the same line of

thought for designing freeform three-dimensional sur-

faces. Although the extension from two to three dimen-

sions may seem a minor step, this is by no means true.

Two-dimensional tailoring leads to ordinary differential

equations. Such equations can easily be solved even by

heuristic approaches. Three-dimensional tailoring, in

contrast, leads to nonlinear partial differential equations

that are much more demanding.

Optimization approaches ﬁrst attempt a suitable pa-

rameterization of the surface with a ﬁnite and in general

low number of parameters. The value of these param-

eters are then to be determined by optimization based on

ray tracing and some objective function to measure how

well a surface characterized by certain values of the pa-

rameters performs. Optimization approaches are very

successful for many applications. However, if freeform

optical surfaces are to be designed for illumination tasks,

procedures based on optimization suffer from the fact that

adequate representation of optical surfaces tends to re-

quire a large number of parameters. Even slight varia-

tions of the surface shape can result in large variations in

the produced irradiance distribution. Ray tracing in-

volves a statistical element and therefore cannot assess

the optical performance precisely. At best it can provide

an estimate that becomes better as more rays are used.

For freeform surfaces whose purpose is to produce irradi-

ance distributions that are rich in detail, each evaluation

of the objective function with reasonable accuracy re-

quires a huge computational cost. For these reasons we

do not elaborate further on optimization approaches here.

In this paper we outline a constructive method of de-

sign that translates the desired optical performance into

a differential equation that is solved by numerical meth-

ods to yield the desired shape of the optical surface.

2. TAILORING IN THREE DIMENSIONS

Consider a ray from a small source located at saimed

at a point on the optical surface located at pas shown

in Fig. 1. The laws of reﬂection (refraction) govern

the direction into which this ray is redirected. In effect

these laws establish a relation between the surface-

normal unit vector N, the direction of the incoming ray

In ⫽Unit(p⫺s) , and the direction of the outgoing ray

Out ⫽Unit(t⫺p).

关1⫹n2⫺2n共Out •In兲兴1/2N⫽Out ⫺nIn. (1)

Here Unit denotes the unit vector: Unit(v)⫽v/

冑

v•v. Equation (1) applies for both reﬂection (n

⫽1) and refraction (n⫽1), where ndenotes the index

of refraction of the medium surrounding the source. To

analyze the power density carried by the radiation, we

need to consider an ensemble of adjacent rays. This

leads to the concept of a wave front; see Fig. 2.

590 J. Opt. Soc. Am. A /Vol. 19, No. 3 /March 2002 H. Ries and J. Muschaweck

0740-3232/2002/030590-06$15.00 © 2002 Optical Society of America

A wave front is a family of surfaces, each perpendicular

to the local direction of the radiation. For example, the

undistorted radiation emitted by a point source is de-

scribed by concentric spheres centered at the location of

the source. Wave fronts may be distorted by optical sur-

faces or nonconstant refractive index and thereby have

shapes different from spherical. The power density

transported by the radiation along the direction of propa-

gation is proportional to the absolute value of the Gauss-

ian curvature of the wave front. The curvature tensor Kគ

is the derivative of the unit normal of a surface with re-

spect to the tangent directions t1and t2:

Ki

j⫽

Ni

tj

i,j⫽1, 2. (2)

Since Nis a unit vector, its derivative is perpendicular to

N, i.e., restricted to the tangent plane. The curvature is

thus a two-dimensional tensor. The Gaussian curvature

is the product of the two eigenvalues c1and c2of the cur-

vature tensor deﬁned in Eq. (2). For free propagation,

the curvature of the wave front changes along the ray

such that the two curvature centers (also referred to as

foci) remain ﬁxed. For the radiation of a point source,

the two foci f1and f2coincide with the location of the

source. For a general wave front, the position of the foci

is calculated from the eigenvalues c1and c2of the curva-

ture tensor:

f1⫽p⫺

1

c1

N,f2⫽p⫺

1

c2

N. (3)

Here we adopt the convention that positive curvature im-

plies a convex surface, where the focus is behind. The ir-

radiance of a freely propagating wave front changes in the

direction of propagation according to Ref. 6:

E共p兲⫽E共p0兲

兩

p0⫺f1

兩兩

p0⫺f2

兩

兩

p⫺f1

兩兩

p⫺f2

兩

. (4)

In general, the two foci for each point of the wave front

form two (curved) surfaces in space where the optical sur-

face has imaging properties. Equation (5) diverges if p

coincides with one of the foci. In reality, sources are not

true points and their brightness is not inﬁnite; therefore

the irradiance approaches the brightness of the image of

the source. This phenomenon is illustrated by the bright

lines (caustics) formed on a sunny day at the bottom of a

swimming pool. The two caustic surfaces are the gener-

alization of the line caustic in two dimensions that is use-

ful for illumination design.7The tailoring algorithm as-

sumes a small source, which implies that the source

brightness is much larger than the desired irradiance

anywhere on the target. Therefore the caustic surfaces

must never intersect the target surface. This fact further

illustrates that our approach is rooted in nonimaging op-

tics.

To quantify the irradiance on the target surface, we

ﬁrst need to derive how the curvature of a wave front is

changed by an optical surface. To this end we compute

the derivative of Eq. (2) with respect to displacements of

the point pin the tangent plane of the optical surface.

We are interested only in changes of the normal vectors

conﬁned to the tangent plane of the optical surface:

关1⫹n2⫺2n共Out •In兲兴Kគo

⫽PគOut •KគOut •JគOut ⫺nPគIn •KគIn •JគIn . (5)

Here Kគodenotes the curvature tensor of the optical sur-

face, KគIn the curvature tensor of the incoming optical

beam at the intersection with the optical surface, and

KគOut the curvature tensor of the outgoing optical beam

immediately after reﬂection (refraction) at the optical sur-

face. The tensors PគIn and PគOut are projection operators

from the tangential plane of the incoming or the outgoing

wave front onto the tangential plane of the optical sur-

face. The tensors JគIn and JគOut are the Jacobians, which

project the differentials from the tangent plane of the

wave fronts onto the tangent plane of the optical surface:

JIn i

j⫽

ti

In

tjO,JOut i

j⫽

ti

Out

tjO. (6)

If we choose a coordinate system on each tangent plane

such that the ﬁrst coordinate is sagittal, i.e., co-planar

with the incoming and the outgoing ray, and the second

coordinate is tangential, i.e., parallel on all surfaces, then

the projection operators as well as the Jacobians are di-

agonal and simple:

Fig. 2. Curvature of the optical surface, illustrated here as a re-

ﬂector, and the curvature of the incoming wave front combine to

determine the curvature of the outgoing wave front.

Fig. 1. Sketch of source location s, a point pon the optical sur-

face, and points ton the target surface. Ndenotes the unit nor-

mal to the optical surface.

H. Ries and J. Muschaweck Vol. 19, No. 3/ March 2002 /J. Opt. Soc. Am. A 591

PគIn ⫽JគIn ⫽

冋

N•In 0

01

册

;

PគOut ⫽JគOut ⫽

冋

N•Out 0

01

册

. (7)

In particular, for a reﬂector surface this leads to n⫽1,

Out •In ⫽⫺cos(2

) and N•Out ⫽⫺N•In

⫽cos(

), where

is the incidence angle, and Eq. (5) be-

comes

2共1⫹cos共2

兲兲Kគo⫽

冋

10

0 cos共

兲⫺1

册

•共KគIn ⫺KគOut兲

•

冋

cos共

兲0

01

册

. (8)

Equation (8) can be interpreted as follows: The curva-

ture of the optical surface is the difference between the

projected curvatures of incoming and the outgoing wave

fronts, which is a direct extension of Eq. (1). Projection

essentially increases curvature in the sagittal direction

and decreases it in the tangential direction. This phe-

nomenon was outlined in an earlier paper.8

Given the curvature of the outgoing wave front, we can

calculate the foci of the outgoing beam, g1and g2, and the

irradiance produced at the target surface according to Eq.

(4):

E共t兲⫽

冋

I共In兲

共p⫺s兲•共p⫺s兲

册

⫻

冉

兩

p⫺g1

兩兩

p⫺g2

兩

兩

t⫺g1

兩兩

t⫺g2

兩

冊

Nt•Out. (9)

Here I(In) is the intensity of the source in the direction

In, i.e., the irradiance produced by the source on the unit

sphere in that direction. The ﬁrst fraction in Eq. (9) de-

scribes the irradiance produced by the source at the posi-

tion of the optical surface, measured in the tangent plane

of the incoming radiation. The second fraction describes

propagation of the outgoing beam characterized by the

foci g1and g2between the optical surface at pand the

target point t. Finally, the last scalar product accounts

for the incidence angle with respect to the target surface

normal Nt. Note that the positions of the foci g1and g2

refer to the outgoing wave front. Their relation to the

curvature of the incoming radiation and that of the opti-

cal surface is quantiﬁed by Eq. (5).

If one chooses orthonormal coordinate systems aligned

with a sagittal and a tangential plane on all three sur-

faces, the projection operators are diagonal, which consid-

erably simpliﬁes Eq. (5). But in this coordinate system,

in general the curvature tensor of the optical surface is

not diagonal. One might be tempted to restrict the cur-

vature of the optical surface by requiring that the eigen-

vectors of the curvature tensor also be aligned with the

sagittal and tangential directions, deﬁned by the inci-

dence direction. After all, the distance to these foci is all

that matters for the irradiance. However, for reasons

outlined below, this choice violates the integrability con-

dition and therefore in general rules out a solution of the

differential equations as a smooth, continuous optical sur-

face.

Equation (5) relates the curvature of the optical surface

to its slope. The great achievement of Gauss was to

prove that slope and curvature are inherent properties of

a surface, which do not depend on a particular parameter-

ization. We made use of this fact by choosing an abstract

formulation in the previous derivations in order to better

illustrate the underlying principles.

If, however, one proceeds to actually construct an opti-

cal surface, one cannot avoid resorting to a particular rep-

resentation based on some coordinate system. As we

have seen, it is not possible to diagonalize the projection

and the curvature tensors simultaneously. Equations

(1), (4), and (5) translate to a set of two elliptical nonlin-

ear partial differential equations. Equivalent equations

were derived by Monge and Ampe

`re more than 200 years

ago.9The general problem for a point source was pre-

sented in the form of an elliptical equation of the Monge–

Ampe

`re type by Komissarov and Boldyrev.10 The prob-

lem that we have treated has also been analyzed in detail

by Schruben.11 This author used an equivalent but nev-

ertheless different notation and also showed that the

problem is governed by an elliptic Monge–Ampe

`re differ-

ential equation. However, the earlier authors give no

hint about actually generating a solution for a particular

desired distribution. The reason for us to use a different

formulation is that our primary goal is to actually ﬁnd an

algorithm that generates a numerical solution in the gen-

eral case. We did not attempt a closed-form solution be-

cause we did not wish to restrict the design input to func-

tions represented as closed-form expressions. Indeed,

although the numerical problem is quite demanding and

challenging, we were able to devise a numerical algorithm

for ﬁnding a numerical solution following standard nu-

merical procedures outlined in Ref. 12. On the basis of

the examples we have tackled so far, we are conﬁdent that

we have developed a powerful tool applicable to an ex-

tremely large class of problems.

Suppose we have integrated the second-order equation

and determined the normal to the surface. Is it possible

to determine the reﬂector surface? It has been shown13

that a necessary and sufﬁcient condition for the existence

of a surface everywhere perpendicular to a vector ﬁeld N,

i.e., for the existence of a quasipotential, is that this vec-

tor ﬁeld satisfy

N•curl共N兲⫽0. (10)

This constitutes an additional condition to be imposed on

the on the second derivative of the surface parameteriza-

tion. An equivalent condition was found by Parkyn.14

One is free to choose boundary conditions for a particular

solution; however, one cannot freely choose a complete

boundary curve for the optical surface. In general, such

a choice would ﬁx at least one component of the normal

vector and therefore violate either Eq. (1) or, most prob-

ably, the integrability condition. Therefore we chose

to require the boundary of the optical surface to lie on an

a priori deﬁned two-dimensional manifold such as a cyl-

592 J. Opt. Soc. Am. A /Vol. 19, No. 3 /March 2002 H. Ries and J. Muschaweck

inder or a cone. We found that with this boundary con-

dition the solution is unique up to a scaling factor.

3. NUMERICAL SOLUTION AND

BOUNDARY CONDITIONS

In order to determine the shape of the optical surface, we

have to solve the set of differential equations. Let us re-

view them in turn:

• Equation (1), which speciﬁes which point on the tar-

get surface is illuminated by a given point on the optical

surface.

• Equation (9), which connects the desired irradiance

on the target to the focal points of the outgoing wave

front.

• Equation (5), which connects the curvature of the

outgoing wave front to the curvature of the incoming

wave front and the curvature of the optical surface.

• Equation (10), the integrability condition.

Equation (10) has a key role. It may seem that the de-

sired irradiance distribution on the target may be

achieved with an entire family of curvature tensors, and

one may simply choose a convenient representative. In-

deed, solutions based on such an arbitrary choice have

been shown to work in particular, usually highly symmet-

ric, conﬁgurations.15 However, we found that an arbi-

trary choice in the general case violates the integrability

condition. Therefore we have added the integrability

condition explicitly to the set of determining equations.

The set of equations is nonlinear and elliptical. As out-

lined in Ref. 12, equations of this type need special care.

We have been able to devise an algorithm by discretiza-

tion of the optical surface to a grid of points and subse-

quently identifying derivatives with ﬁnite differences.

In addition to the differential equations, one can

specify boundary conditions. The boundary conditions

also need special consideration as indicated by

Schruben.11 If we were simply to specify the outer

boundary of the optical surface as a curve in space, this

would also specify one tangent at the rim and therefore in

general violate Eq. (1). Therefore we can require only

that the boundary of the optical surface lie on some pre-

deﬁned surface. This allows us, for example, to require

that the boundary of the optics be on a cylinder but not

that it be on a circle.

In addition, we found that because in the limit of a

small source the optical laws scale, we can require a cer-

tain average size of the optical surface (or average dis-

tance from the source). However, this condition may vio-

late energy conservation, because the entire optical

surface must intercept a total power from the source that

precisely matches the integral of the desired irradiance

on the target.

An additional restriction applies to refractive optics.

The maximum angle by which a ray can be redirected by

a lens is limited by total internal reﬂection. When tailor-

ing refractive optics, we did encounter occasionally un-

physical solutions, which attempt to change the direction

of the rays by too large an angle, meaning that in reality,

total internal reﬂection would occur for part of or the en-

tire optical surface.

The two caustic surfaces are the reason for the exis-

tence of three topologically distinct solutions. We have

explained that the caustic surfaces must never intersect

the target surface. Therefore in general there are dis-

tinct solutions: one in which both caustic surfaces lie be-

tween source and target, one in which both caustic sur-

faces lie outside the region between source and target,

and a third in which one caustic surface lies between

source and target and one does not. Apart from these

distinctions and the restrictions outlined above, we found

that if a suitable set of boundary conditions is speciﬁed, a

solution can be found by tailoring, and this solution ap-

pears to be unique.

Fig. 3. Sketch of the setting: The lens is made to collect light

from an isotropic point source and redistribute it onto a target

perpendicular to the axis. Length units are arbitrary; the draw-

ing is not to scale.

Fig. 4. Shape of the freeform lens that casts the letters OEC on

a square background, as it would appear under oblique illumina-

tion. A slight bulging is visible, which remotely resembles the

letter shape. Note that the rim lies on a cone as required by the

boundary condition.

H. Ries and J. Muschaweck Vol. 19, No. 3/ March 2002 /J. Opt. Soc. Am. A 593

4. SOLVED EXAMPLES

As a demanding example and proof of concept, we speci-

ﬁed a desired irradiance distribution that is constant over

a square target except for the contours of the letters

‘‘OEC,’’ for which we speciﬁed an intensity three times

that of the background. All rays incident on the optical

surface are to be redistributed on the target with no rays

outside. For this case we designed a refractive (lens) sur-

face to perform the task in a setting sketched in Fig. 3.

Of course, it would be a simple task to create any de-

sired irradiance distribution in this setting with a simple

slide projector. However, such an approach wastes radia-

tion by absorption in the slide. In contrast, the method

that we outline here creates the desired irradiance distri-

bution purely by redirecting the radiation. Thereby all

intercepted radiation is redirected to the target.

The shape of the lens is depicted in Figs. 4 and 5. By

careful inspection, one has the impression of recognizing

regions on the lens surface singled out by slightly differ-

ent curvatures, which resemble the letters on the target

although the shape is distorted. In retrospect, this is

easy to understand since the regions of the lens contrib-

uting to the letters have to be three times as large in area

as a result of energy conservation. The rim of the lens

was made to lie on a cone, oriented with its axis toward

the center of the desired distribution and a tip half-angle

of roughly 33 deg. The energy balance determines the

exact value of the tip angle.

We veriﬁed the optical performance of the tailored free-

form lens by importing the shape into a commercial ray-

tracing program16 and performed an independent evalua-

tion by tracing 9 ⫻106rays from a point source through

the lens and computing their intersection with the target

plane. The result is shown in Fig. 6. Figure 7 plots the

intensity along the horizontal cross section through the

center in order to quantitatively verify the intensity ratio

of zero (outside) to 1 to 3.

5. CONCLUSIONS

We have presented a method for designing freeform opti-

cal surfaces tailored to speciﬁc illumination tasks. Given

a small source with known intensity distribution, given

also a desired irradiance distribution on a target surface,

we have devised a constructive procedure to design the

shape of an optical surface, either reﬂective or refractive,

that redistributes the radiation from the source onto the

target, precisely matching the desired distribution. Tai-

loring draws on principles of nonimaging optics, applied,

as far as we know for the ﬁrst time, to problems without

Fig. 5. Shape of the freeform lens in an isosag representation.

Lines indicate equal sag. Spacing is 0.01 unit. The ragged rim

is an artifact of the representation. The rim of the lens was

made to lie on a cone oriented with its axis toward the center of

the desired distribution.

Fig. 6. Irradiance distribution produced by the freeform tailored

lens based on Monte Carlo ray tracing of 9 ⫻106rays. Inter-

section of the rays with a plane at the position of the nominal tar-

get was binned by 129 ⫻129 pixels. Intensity variations are

entirely compatible with shot noise.

Fig. 7. Horizontal cross section through the irradiance distribu-

tion in Fig. 5 at vertical position 0. Irradiance is in arbitrary

units. Note the contrast of 0/1/3. Noise is completely within

statistical bounds.

594 J. Opt. Soc. Am. A /Vol. 19, No. 3 /March 2002 H. Ries and J. Muschaweck

any symmetry. Because of its generality this approach is

very promising for a large variety of applications.

We have presented idealized results in which we delib-

erately excluded many real-world effects such as ﬁnite

source size, Fresnel reﬂections and, above all, manufac-

turing errors such as slope errors of the surface. We did

this in order to show the strength and limitations of the

mathematical procedure in the sense of a proof of concept.

We found that the numerical procedure is extremely fast

and efﬁcient. For the example presented, in which the

surface was rendered as a grid of 129 ⫻129 points, the

computation required roughly one hour on a standard

contemporary personal computer. The numerical preci-

sion was far beyond manufacturing capabilities.

In fact, we found it more difﬁcult to assess the perfor-

mance of the tailored optical surface with ray tracing

than to design it. For an illustration, imagine an ambi-

tious claim that within a target area of, say, 1 m2there is

not a single squared millimeter on which the irradiance is

more than 1% off the nominal value. In fact, we believe

that apart from manufacturing tolerances, the pure com-

putational procedure outlined here is quite capable of

such outstanding performance because we have veriﬁed

that the tolerance in selected outgoing rays is of the order

of 1

m. To substantiate such a claim by a Monte Carlo

ray-tracing technique, one would need more than 104rays

per bin in order to reach a statistical error below 1%. On

the other hand, to match the spatial resolution, one would

need at least 106pixels. These requirements combine to

a requirement of 1010 rays. This is beyond the capability

of today’s computers, which typically require of the order

of 10⫺3s per ray.

We are well aware that our example is of no practical

value. The reason we chose it was, again, to demonstrate

a capability that other known design methods can hardly

match. The desired irradiance distribution is what we

would call rich in detail. It is hard to imagine that a pa-

rameterization of a surface based on few parameters for

such a task would be feasible.

The example chosen deliberately involves a discontinu-

ous desired irradiance distribution that features steps in

irradiance by a factor of 3. Because irradiance on the

target essentially is linked to the curvature of the tailored

surface, the optical surface designed is continuous and

smooth. Only the curvature exhibits steps. The tailored

surface is geometrically smooth (G1) , even for disconti-

nous desired intensity distribution. Since intensity is re-

lated to the curvature of the optical surface, in general, if

the desired irradiance is cncontinuous, the tailored opti-

cal surface will feature Gn⫹2geometric continuity. This

facilitates manufacturing.

ACKNOWLEDGMENTS

We thank Wolfgang Spirkl and Andreas Timinger for

valuable suggestions and discussions. We are grateful to

the reviewer for pointing out Refs. 10, 11, and 13.

Harald Ries can be reached by e-mail at harald.ries

@oec.net.

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