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Formulation of a Reflector-Design Problem for a Lighting Fixture

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J. S. Schruben
J. S. Schruben
J. S. Schruben
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Abstract and Figures

The problem of designing a reflector to distribute the illumination of a nonisotropic point source on a plane aperture according to a pre-assigned pattern is analyzed. An integral equation and equivalent partial differential equation are derived. The form of the latter reveals this reflector-design problem to be a singular elliptic Monge–Ampère boundary-value problem.
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JOURNAL OF THE OPTICAL SOCIETY OF AMERICA DECEMBER 1972
Formulation of a Reflector-Design Problem for a Lighting Fixture
J. S. SCHRUBEN
Westinghouse Research Laboratories, Pittsburgh, Pennsylvania 15235
(Received 3 June 1972)
The problem of designing a reflector to distribute the illumination of a nonisotropic point source on a
plane aperture according to a pre-assigned pattern is analyzed. An integral equation and equivalent partial
differential equation are derived. The form of the latter reveals this reflector-design problem to be a singular
elliptic Monge-Ampere boundary-value problem.
INDEX HEADINGS:
Sources; Mirror.
Spreading the light of a small high-intensity disc
source (e.g., 175-W BOC mercury arc) over a cor
tively large aperture (e.g., a square aperture
length 0.6 m) in a ceiling illuminating fixture re
the resulting glare and veiling reflections in an in
lighting situation.' Two design goals for the refleci
such a fixture are, therefore, to ensure that the
aperture is uniformly illuminated and that it
light with appropriate directionality. We focus
attention on the first goal since the directional rec
ments on the emitted light may often be met by .
a diffuser or prismatic surface in the ceiling
plane.
U
xi
ZI
Plane PI
Ceiling Plane P
x
FIG. 1. Point source of light above a ceiling plane.
harge
para-
with
duces
trisnr
we introduce this design problem for specular reflectors
in terms of integral and differential equations.
INTEGRAL EQUATION
torv The light source is assumed to have some arbitrary
large directional intensity distribution I and dimensions that
emits are negligible compared to the fixture size. Distances
emour are normalized such that the distance from the source
ouire to the ceiling plane P is unity (see Fig. 1). The aperture
use
of is an area R on P that is to be illuminated.
Here Since the intensity of the source is directional, I may
be defined as a function of position on the unit sphere
centered at the source. Spherical coordinates could be
used, but it is preferable to employ parametric co-
ordinates (uv) of the unit sphere. These may be ob-
tained as stereographic coordinates, as illustrated in
Fig. 2, by projecting the unit sphere from its point of
tangency to P onto the plane P' parallel to P and also
tangent to the sphere. The stereographic coordinates of
a point on the sphere so projected are the rectangular
v u, v coordinates of the corresponding point on P'. The
origin of this coordinate system is centered at the point
of tangency of the sphere with P'.
Figure 2 also shows a rectangular coordinate system
x, y defined on the ceiling plane P and centered at the
point of tangency of the sphere with P, where the x and
y axes are parallel to the u and v axes, respectively. It is
assumed that the origin of the x, y system on P is an
y interior point of the aperture R. On R is defined a func-
tion L =L(x,y), which is the desired pattern of reflected
illumination on the aperture. This is defined as the
desired pattern of total illumination at a point (x,y)
on the aperture from which has been subtracted the
direct illumination of the source at (xy) which can be
obtained directly from the intensity distribution I.
Thus, we have L(x,y)=desired illumination at (x,y)
-direct illumination at (xy). The choice of the pattern
y of illumination on the aperture is arbitrary except for
certain restrictions to be imposed shortly.
For a reflecting surface to produce the required
pattern of reflected light L(x,y) on the aperture, it must
reflect all solid angles Q of light rays from the source
towards the reflector onto areas v(fQ)
on the aperture
such that the energy contained in a solid angle of rays
Q equals the energy specified by L over the area P(U)
1498
VOLUME 62, NUMBER 12
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