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1260
J.
Opt.
Soc. Am.
A/Vol.
11, No.
4/April
1994
Tailored
edge-ray
reflectors
for
illumination
Harald
R. Ries*
and Roland
Winston
Enrico
Fermi Institute
and
Department
of Physics,
The
University
of Chicago,
Chicago,
Illinois 60637
Received
May
17, 1993;
revised
manuscript
received
September
28, 1993;
accepted
September
28, 1993
The edge-ray
principle
can be
used to
tailor
a reflector.
However,
one set
of edge
rays already
fully
determines
the reflector
profile.
We
present a
design
method
for tailoring
compact
compound
elliptical
concentrator
(CEC)-type
reflectors
to
a
given
source
and
a desired
angular
power
distribution.
Two
reflected
images
of the
source,
one on
each side
of the
source,
contribute
together
with the
direct radiation
from
the
source to
produce
the desired
power
distribution.
We determine
the
reflector
profile
by numerically
solving
a differential
equation.
No
optimization
is
required.
Beyond
the
angular
region
in which
the
power distribution
can be
strictly
controlled,
the power
drops
to zero
in a finite
decay
range.
This decay
range
becomes
narrower
as
the reflector
increases
in size.
We
show a reflector
for
producing
a strictly
constant
irradiance
from
-43 to
43 deg
from a cylindrical
source of
constant
brightness.
The
reflector
extends
to a maximum
distance
of 8
source
diameters.
No power
is radiated
beyond ± 50
deg.
INTRODUCTION
In a recent
paper we
showed how
a reflector
can be
designed
that produces
a
desired
irradiance
or power
distribution from
a given small
source.' The
idea there
was to translate
the desired
irradiance
distribution
into
an acceptance-angle
function.
The reflector
shape was
then outlined
as a functional
of the acceptance-angle
function.
This
concept,
however,
cannot
readily
be ap-
plied to extended
sources, because
in general
the reflected
image has
its own edge
rays, separated
from those of the
source.
But the total
irradiance in
all cases is strictly
given by the
combined
view factors
of the source
and of
the reflected
image.
Therefore controlling
the edge
rays
of
the reflected image
provides
a means to design
the
reflector
for extended
sources.
It has been shown
that for
secondary concentrators
the reflector
can be
tailored to a
given family
of edge
rays,
2
but that approach
has
generated
only compound
hyperbolic
concentrator
(CHC)-type solutions,
in which
the reflector
approaches
ideality only
in the limit
of being
infinitely
large. In
this paper we show
how the much
more practical
compact compound
elliptical
concentrator
(CEC)-type
reflectors can
be designed to
produce a desired
irradiance
distribution
on a given
target space from
a
given
finite-sized
source. The method
is based
on tai-
loring
the reflector
to a family of
edge rays, but
at the
same time the
full contour of
the reflected source
must
be controlled.
TAILORING
THE REFLECTOR
How
to Tailor Edge
Rays in Two
Dimensions
Let
us assume that
we have a family
of edge rays,
for
example, all left-hand
edge rays
or all right-hand
edge
rays, as produced
by a real
luminaire. Through
each
point in
the space outside
the source there
is precisely
one edge
ray. The
direction of the
edge rays is a
con-
tinuous
and differentiable
(vector)
function of position.
Suppose
that we have
a second, tentative
family of
edge
rays represented
by another
continuous vector
function
in the same
region of space.
Can we design
a reflector
that
precisely reflects
one family onto
the other?
This
indeed
is true: each
point in space
is the intersection
of
precisely one member
of each
family. Therefore
the
inclination of the
desired reflector
in each point
in space
can be calculated.
3
-
5
Thus one
can derive a
differential
equation that
uniquely specifies
the reflector
once the
starting point
has been
chosen.
In this
section we formalize
this line
of reasoning for
the case
in which the
tentative family
of edge rays
is
given
only along a
(not necessarily
straight) reference
line.
This
case suffices for
the usual problems
encountered
in
illumination.
Let
d = a,(x) be the
(two-dimensional)
unit vector
point-
ing toward
the edge of
the source as
seen from
the
point x, and let
k = k(t) be
a parameterization
of the
reference
line according
to some scalar
parameter t.
Let
Cz(t) be the
unit vector pointing
in the direction
of the
edge ray
desired at the
reference location
specified by
t.
We parameterize
the reflector
contour with
respect to the
reference
line by writing
the points
on the reflector
as
R(t) =
k(t) + D (t) .
(1)
Here the scalar
D denotes the
distance from
a point on the
reference
line to the reflector
along the
desired edge ray
through
this point.
The notation is
illustrated in Fig.
1.
Designing
the reflector
shape in
this notation is
equiva-
lent to specifying
the scalar function
D = D(t).
An equa-
tion for D is
derived from the
condition that
the reflector
reflect the
desired edge
ray i2(t) into the
actual edge ray
d[R(t)] (and vice versa):
d ()
[R(t) -
(01. dt)
Inserting Eq.
(1) yields
dD
dk/dt (a - i2)
+ D d/dt
et
dt 1 -
a CZ
(2)
(3)
Here
denotes the scalar
product. Equation
(3) is
a
scalar
differential
equation for the
scalar function
D(t).
By solving it we
can determine
the reflector that
tailors
0740-3232/94/041260-05$06.00
©1994 Optical
Society of America
H. R. Ries
and R. Winston
Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1261
then again onto the unit circle tangent to the reference
plane.'
9
The view factor is determined by the contour of
the source
as seen by the observer. In two dimensions
the irradiance E is
E =
B[sin(OR) - sin(OL)],
origin
reference line
Fig. 1. is the unit vector pointing toward the edge of the
source, k is a parameterization of the reference line, and Z is the
unit vector pointing in the direction of the edge ray desired at
the reference location t. The reflector contour is specified by R.
the desired family of edge rays specified by the function
z2 to a given source characterized by the function d.
This approach can also be used to tailor one family of
edge rays onto another with refractive materials rather
than with reflectors. Relation (2) then has to be replaced
by Snell's law. Indeed, Mifiano and Gonzales have re-
cently designed some highly original concentrators with
nonspherical lenses, based on the edge-ray principle.
6
Following the line of reasoning that is presented in this
section, the condition for the existence of a solution is that
each point on the reflector be intersected by precisely one
member of the family of tentative edge rays. For us to
be able to define this family along the reference line, each
point on the reference line also must be intersected by
precisely one tentative edge ray. This condition is both
necessary and sufficient. This is less restrictive than
requiring that the tentative edge rays define the physical
surface that produces them. The family of (for example,
right) edge rays of a physical contour must satisfy the
stronger requirement that precisely one edge ray pass
through each point of the entire space exterior to the
contour. Indeed, our construction can produce families
of edge rays by tailoring, which cannot be produced by
a physical source. This is confirmed by the observation
that curved mirrors produce not only a distorted image
of the source but, furthermore, an image that appears to
move as the observer moves.
The condition mentioned, that each point on the re-
flector as well as each point on the reference line be
intersected by precisely one of the desired edge rays,
implies, however, that the caustic formed by these edge
rays must not intersect either the reflector or the refer-
ence line. The caustic must therefore either be entirely
confined to the region between the reflector and the
reference line or lie entirely outside this region. The first
of these alternatives characterizes CEC-type solutions,
while the second defines CHC-type solutions. We con-
clude therefore that there can be no mixed solutions that
are part CHC and part CEC. In the CEC both foci are
on the same side of the reflector; in the CHC, on the other
hand, the two foci are on opposite sides of the reflector.
7 8
How to Determine the Desired Edge Rays
The irradiance from a Lambertian source of uniform
brightness B is given by its projected solid angle, or view
factor, of the irradiance. The view factor is calculated
by projection of the source first onto the unit sphere sur-
rounding the observer (thus yielding the solid angle) and
(4)
where OR and OL are the angles between the normal to the
reference line and the right and the left edge rays striking
the observer, respectively. If we know the brightness B,
the desired irradiance E, and one edge ray, then Eq. (4)
can be used to determine the desired direction of the other
edge ray.
Suppose that we have a source of a given shape. Then
we know the direction of the edge rays as seen by an
observer as a function of the location of the observer.
Actually the shape of the source can be defined by all
its tangents.
10
," We can now design a reflector so that
it reflects a specified irradiance distribution onto a given
reference line iteratively in the following way:
If an observer proceeds, for example, from right to left
along the reference line, then, for a CEC-type solution, the
perceived reflection moves in the opposite direction. The
right edge ray, as seen by the observer, is the reflection
of the right edge as seen from the reflector and plays
the role of leading edge ray along the reflector. The left
edge ray is just trailing behind. It belongs to an earlier
part of the reflector and has already been accommodated.
This is indicated in Fig. 2. For a CHC-type reflector the
reflected image of the source moves in the same direction
as the observer, and the right edge as seen by the observer
is the reflection of the left edge. If part of the reflector is
known, then the trailing edge ray, reflected by the known
part of the reflector, can be calculated as a function of
location on the reference line. Equation (4) consequently
specifies the direction of the leading edge ray. Then one
can solve Eq. (3) to tailor the next part of the reflector
profile to this leading edge.
Boundary Conditions
If the reflector is terminated, then the reflected radiation
does not terminate where the leading edge from the end
of the reflector strikes the reference line but rather where
the trailing edge from the end of the reflector strikes the
reference line; see Fig. 3. Thus there is a decay zone
on the reference line that subtends an equal angle as
the source as seen from the end of the reflector. In this
region the previously leading edge is at the end of the
CEC reflector CHC reflector
leading eaing
suc trailing edge soreedge
edge
trailing
edge
reference line reference line
Fig. 2. As the observer proceeds along the reference line, the
desired irradiance translates into a desired leading edge. This
in turn determines the reflector profile. The arrows indicate
the directions of the iterations.
H. R. Ries and R. Winston
1262 J. Opt. Soc. Am. A/Vol.
11, No. 4/April 1994
start
reference ine
Fig. 3. The start and the
end of the reflector cause a rise zone
and a decay zone,
respectively, each subtending an angle equal
to that of the source.
symmetry
plane),
then we must
require that,
as seen
from the symmetry
plane,
the two perceived
reflections
be equal. For
all other points
in the target domain
we now
have the additional
freedom of
choosing the
relative contributions
of each reflector
side. In CEC-
type solutions both
reflections appear
to be situated be-
tween the target space
and the reflector. Thus,
as the
observer moves, both
reflection images move
in the op-
posite direction. Toward the
end, when the observer
approaches the outermost part of the illuminated
target
region,
first the reflection
on the same side
disappears at
the cusp in the
center. Thereafter
the reflection
opposite
to the observer
starts to
disappear past
the outer edge of
the opposite
reflector, while the source itself is shaded
by
the outer edge
of the reflector
on the observer's
side.
These
events determine the end point of the reflector,
because
now the total
radiation in the
target region equals
the total
radiation emitted
by the source, as
pointed out
in Ref. 1.
SOLVED EXAMPLE:
CONSTANT
IRRADIANCE
We design
a CEC-type reflector
to produce
a constant ir-
radiance on a distant plane
from a finite-sized cylindrical
6
reference line
Fig. 4. The reflector is designed
so that each side illuminates
both sides of the
target region. Thus an observer sees two
reflections in addition
to the source S.
reflector, while the trailing edge gradually closes in. An
analogous rise zone exists
at the other end of the reflector,
where
the trailing edge is
initially fixed to the start
of
the reflector. However, there is an
important conceptual
difference
between these two regions,
in that the rise
of
the irradiance can be modeled by
tailoring of the reflector
to the leading edge, while the decay
cannot be influenced
once the reflector is terminated. Therefore
the way in
which we proceed in the iterative tailoring of
the reflector
makes a difference.
Determining
the Reflector Profile
If the source radiates
in all directions and we want
to avoid trapped radiation, that
is, radiation reflected
back onto the source, then we propose that
the reflected
radiation
from each side of the reflector cover the whole
target domain.
At the same time the normal to the
reflector surface should
never intersect the source. This
means that left- and right-side
reflectors are joined in
a cusp.
Note
that this is different from the approach presented
in Ref.
1, in which each side of the reflector illuminated
only one side
of the target domain (the same side for
the CHC-type solution,
the opposite side for the CEC-
type solution). Thus, according
to the scheme proposed
here, an observer in the target domain
perceives radiation
from two distinct reflections of the
source, one from each
reflector
side, in addition to the direct radiation from
the
source. This is depicted in Fig.
4. Similar irradiation
patterns
have been described by Elmer for line sources.'
2
If we
assume symmetry and want the reflector to
be
continuous and
differentiable (except for the cusp in the
5
4
3
2
° |- I
/ I I.
W. MU
.I
=i 1 ~ Lt
-60 -45 -30 -15 0
15 30 45 60
Fig. 5. Power from the source
and from the reflections on each
side add up to produce an
angular power distribution (vertical
axis) proportional to 1/ cos
2
(0) and thus a constant irradiance on
a distant plane from a finite-sized
cylindrical source of uniform
brightness. Angles A-E correspond
to the edge rays marked in
Fig. 7.
4 -
0
-4
- I - I I
-8 -4 0 4
8
Fig. 6. CEC-type reflector profile, which
produces a constant
irradiance on a distant plane from a cylindrical
source of constant
brightness and unit diameter.
l -
--
..l..{
H. R. Ries and R. Winston
1
Vol. 11, No.
4/April 1994/J. Opt. Soc. Am.
A 1263
R2
CD
B C
A D A
Fig. 7. Particular edge rays that correspond to the angles desig-
nated in Fig. 5. At the largest angle for which constant irradi-
ance occurs, the radiated power comes from the source's reflection
between R1 and R2.
-3
-6
edegrees]
--
-50 -25 0 25 50
Fig. 8. Closest approach distance of edge rays as a function of
the angle 0. In this representation, the distance between edge
rays at a fixed angle is a measure of the respective contributions
to total radiated power.
source of uniform brightness. This requires the angular
power distribution to be proportional to 1/ cos
2
(0) (see
Ref. 1). In Fig. 5 we show the necessary power from both
reflections so that the total power is as required. The re-
flector is depicted in Fig. 6. It is designed starting from
the cusp in the symmetry axes. Note that each reflection
irradiates mostly the opposite side but is visible from the
same side, too. Some angles have been designated by the
letters A-E in Fig. 5. The corresponding edge rays are
indicated in Fig. 7.
The way in which these reflections fit together to make
up the total radiated power is illustrated in Fig. 8, in
which we plot the closest approach distance of edge rays
as a function of the angle . In this representation the
distance between edge rays at a fixed angle is a measure
of the respective contributions to total radiated power.
Between -A and A the reflections are immediately
adjacent to the source. The cusp in the center is not
visible. Between A and B the reflection from the same
side as the observer slowly disappears at the cusp, while
the other increases in size for compensation. Starting
with C the source is gradually eclipsed by the end of
the reflector. The largest angle for which a constant
irradiance can be achieved is labeled D. The source
is not visible. The power is produced exclusively by
the opposite-side reflection extending from R1 to R2 as
shown in Fig. 7. The reflector is truncated at R2 so that
between D and E the reflection gradually disappears at
the end of the reflector.
The inner part of the reflector, which irradiates the
same side, is somewhat arbitrary. In the example shown
we have designed it as an involute, because this avoids
trapped radiation and at the same time yields the most
compact design. At the center the power from each re-
flection is nearly equal to that of the source itself. Once
the power radiated to the same side has been
determined,
the reflector is designed so that the sum of the contribu-
tions of the two reflections and the source matches the
desired distribution. If we proceed outward, the eclipse
of the source by the reflector is not known at first, because
it depends on the end point. This problem is solved by
iteration of the whole design procedure several times.
The point of truncation is determined by the criterion
that the reflector intersect the edge rays marked B from
the cusp. This is because the present design is based on
a maximum of one reflection. This criterion is also the
reason for designing the inner part as an involute.
The decay range D to E depends only on the distance
from the end point to the source. Depending on the
starting distance from the cusp to the source, the device
can be designed to be either more compact, but with
a broader decay zone, or larger, and with a narrower
decay zone. The reflector shown has a cusp distance of
2.85 source diameters. The end point is at a distance
of 8.5 source diameters. This ensures that a constant
irradiance is produced between -43 and 43 deg. The
decay zone is only 7 deg. This design was chosen in
order that-the source be eclipsed just before the angle
of truncation.
The reflector cannot be made much more compact as
long as one designs for a maximum of one reflection.
At the angle D the opening is nearly completely filled
with radiation, as shown in Fig. 7. The distance over
which the reflector extends downward from the source
is also determined by the maximum power that must be
produced at D, nearly six times the power radiated by the
source alone along the axis (see Fig. 5). The distance of
the cusp also cannot be diminished, because otherwise the
criterion for the end of the reflector is reached sooner, the
reflector has to be truncated, and the maximum power
produced is also less.
CONCLUSIONS
The solution presented in this paper involves at most
one reflection. In principle, however, systems based on
multiple reflections can be designed, as well, following the
method of tailoring described. As more reflections con-
tribute, the freedom of the designer increases. This free-
dom can be used to adapt the reflector to other criteria,
such as compactness. Multiple reflections, on the other
hand, imply higher losses, and such systems might gen-
erally be more difficult to manufacture and might require
higher precision.
H. R. Ries and R. Winston
6
3
1264
J.
Opt.
Soc.
Am.
A/Vol.
11,
No.
4/April
1994
In any
case,
independent
of the
number
of
reflections,
once
the general
architecture
has
been
decided,
tailoring
the reflector
to
one set
of edge
rays
determines
its shape,
without
the
need for
approximations
or
optimizations.
ACKNOWLEDGMENTS
We are
grateful
to Vladimir
Orlov
for many
helpful
dis-
cussions
and
to Ari
Rabl, whose
presentation
at
the Sede
Boqer
Workshop
inspired
the
development
of the
distinct
geometries
of
CEC and
CHC
leading
edge
rays.
This
research
was supported
in part
by
the U.S.
Department
of
Energy
Office
of Basic
Energy
Sciences
under
contract
DE FG02-87ER
13726
and by
the
Swiss
Federal
Office
of
Science
and
Education.
*Present
address,
Paul
Scherrer
Institute,
CH-5232,
Villigen,
Switzerland.
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H. R.
Ries
and R.
Winston
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