Note: Descriptions are shown in the official language in which they were submitted.
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I?LLUfN/11IN_AT~dG LEN'S DuSIGN~D BY
~\~S!C DTl=i F:r~?~ T '
1 1=ia. J
This apptication is a divisional application of Canadian Patent Application
~ No. 2.30 3.007 whicl: was f led on Septennber ] 8, 1998.
BACKGROUND OF THEINVENIT'IQN
Almost all of the prior art of lens design can be subsumed under the
lo imaging category, the purpose of which is an accurate rendering of the
appearance of an object. Lenses have seen little use in the illumination
field,
where reflectors have predominated because of their lower cost and superior
efficiency when used with conventional light sources. 'Thus, there has been
little emphasis.on the design of lenses for illumination, where the purpose is
'-5 the fulfillment of a prescribed pattern of light distribution, and imaging
of the
light source is undesirable. Many of these prescriptions are for rectangular
and other non-circular light patterns.
Because optical lenses have overwhelmingly been formed by
grinding and polishing their surfaces are figures of revolution, such as
2o spheres, tori, and cylinders. In general, figures of revolution are not
suitable
for forming illumination patterns that are not circularly symmetric.
The present invention embodies a different method of lens design
than that of optical imaging lenses. It utilizes shapes that are not figures
of
revolution, but which can be manufactured by molding of plastic or glass.
They are especially suitable for use with light-emitting diodes, the tiny
sizes of
which allow such lenses to be small and easier to injection mold.
The most important lighting prescriptions addressed by the present
invention are for vehicular lamps, by the $ociety of Automotive Engineers, and
for ships, by the United States Coast Guard. Particular embodiments of the
present invention address these prescriptions. These are far-flield
nrescriptions
for an angular dlstr,bution of light, whereas lllumination prescriptlons are
for
some nearby surf ace, such as the walls or ceiiin, of a room..
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The current art of luminaire design utilizes the method of
computerized searching through a number of design variations, with ray tracing
used to evaluate the closeness of a candidate luminaire's output to the
desired
light distribution. This technique is not completely satisfactory because the
vastness of the design space precludes an exact match of luminaire output to
prescribed output, given that the design starting point is only a guess.
An example of traditional design is the lenslet array utilized in
automotive signal lamps. Numerous small lens elements, usually spherical,
cylindrical, or toric, transform the collimated beam from a reflector into a
wide-angle beam shaped to fulfill government standards. Such combinations of
reflector and lenslet arrays, however, typically have poor values of
efficiency,
such as one third. Although the reflector can be blamed for much of this
inefficiency, also at fault is the restriction of lenslet shapes to spheres,
cylinders, and torics (formed by rotating cutting elements), which greatly
limits
the designer's ability to match the shape of the output beam to the prescribed
pattern. Such a match maximizes efficiency, since every point of the
specification can be met with a minimum amount of light.
The general design of rotationally symmetric luminaires uses the
method of matching the cumulative distribution of source intensity with that
of
the desired output. Cumulative intensity runs from 0 to 100%, starting at the
optical axis and going outwards to the edge of the desired output pattern.
Another cumulative distribution is calculated for the intensity of the light
source, over the angular range to be redirected by.the luminaire. Then, any
angle of a ray from the source, liaving a particular percentage of cumulative
source intensity, is redirected into an output angle having the same
percentage
of cumulative output intensity. From these two angles is calculated the angle
the luminaire surface must have to perform the redirection. Then the actual
luminaire surface is derived by radial integration outwards from an initial
starting point. The resultant shape has the slope necessary to redirect the
Iight
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from a rotationally symmetric source into a prescribed rotationally symmetric
output pattern.
This method of matching cumulative distributions is not suitable,
however, for cases where either the source intensity distribution or the
desired
.5 output pattern are not figures of revolution, that is, azimuthally
constant. (An
example of such a source is a light-emitting diode with two bonding pads.)
This is because a cumulative distribution is inherently one-dimensional, while
the two dimensions of angle space prevent a unique line integral from being
used to calculate a meaningful index of the shape of the distribution.
In optical lens design, the conceptually closest design method is that
of anamorphic lenses. These, however, are designed for a prescribed
distortion pattern, a quite different matter than fulfilling a prescribed
variation
in luminous intensity.
SUMMARY OF THE INVENTION
Embodiments of the present invention concern a general class of illlmrination
lenses
that can accurately match a source with a particular desired output, when
either
or both of these are not figures of revolution. No trial and error processes
are
required for their design. The mathematical discipline of differential
geometry
is the basis for the generation of the shapes of particular lenses. As with
the
above-mentioned method of rotational symmetry, there are two basic stages in
the design process:
(1) deriving a distribution of surface slope by matching source and
output intensity patterns;
(2) generating the luminaire shape by integrating the surface slope.
At each stage, however, surface theory requires completely different
design methods than those of the one-dimensional case of rotational symmetry.
When surfaces are studied as curved two-dimensional spaces, intrinsic
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differential geometry is involved, regarding properties unaffected by folding,
but altered by stretching.
In the case of enzbod'unents of the present invention, however, the lens
surface
operates in our everyday three-dimensional space, so that extrinsic
differential
aeometry is used to design it. For example, a polyhedron, such as a cube, has
a three-dimensional shape studied by extrinsic differential geometry; but, it
also has such intrinsic properties as those revealed by drawing triangles on
it
that enclose a corner. These triangles will violate the laws of plane
trigonometry (i.e., their interior angles do not add to 180 ), so that this
surface's cubic nature is an intrinsic aspect, independent of it being in
three-
dimensional space.
The particular use of extrinsic differential geometry for the present
invention is in surface synthesis, whereby the lens surface is integrated from
the specification of its tilt at a large number of points. The surface tilt is
calculated according to the laws of optics from knowledge of how the light
from a source must be redirected in order to fulfill a particular
prescription.
When either the source light or the prescription has an intensity distribution
that is not rotationally symmetric, design methods of the prior art are
deficient,
as discussed above. Embodiments of the present invention utilize computer
calculations to
numerically specify a lens surface given the intensity distributions of the
source
and the desired output.
When dealing with an intensity distribution that is not rotationally
symmetric, the arena of expressing this distribution is the surface of a
sphere
of unit radius, known mathematically as the Gaussian sphere, measured in
steradians, with 47r being the solid angle of the entire Gaussian sphere. One
steradian is a circle 65.5 across, or a square 59.3 on a side, in either
case a
total of (180/)' = 3282.8 square degrees. Luminous intensity is simply the
amount of lumens emitted into a solid angle, with a candela = I lumen per
steradian (this has replaced the old term "candlepower", which could mean
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either intensity or a measuring unit of intensity). An
intensity pattern can be graphically presented with
either a two-dimensional map of contours of constant
intensity or a three-dimensional map with height
5 representing intensity.
Central to the design method of embodiments of
the present invention, and an object of embodiments of
the invention, is a third method of displaying an
intensity function: a grid, or mesh, on the Gaussian
sphere, with cells of varying size. The cell size are
inversely proportional to intensity, so that each cell
has the same amount of luminous flux. The particular
grid pattern chosen is called a tessellation, or tiling,
or the sphere. An example is the latitude and longitude
grid of geography, and another is the triangulation based
on the icosahedron. In the case of embodiments of the
present invention, however, the entire sphere need not be
tiled, since ordinarily neither light sources nor
illumination prescriptions cover all directions.
Therefore, there is more freedom to adopt tilings, for
particular designs, that would form what are
mathematically known as incomplete atlases.
It is an object of embodiments of the present
invention to provide an illumination system with a
prescribed output pattern, and which comprises a light
source and an optical lens that redirects source light
into an output beam, the lens having a shape that is not
a surface of revolution.
It is another object of embodiments of the
invention to provide a lens, the surface of which is not
a surface of revolution, and wherein the lens surface
possesses surface normal vectors enabling the lens to
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5a
transform source light into an object beam fulfilling
the prescribed output pattern.
A further object of embodiments of the
invention is to provide a lens, as referred to, and
having overall size relative to the size of the light
source, and obtained by selection of the distance from
an initial starting point of lens surface generated from
the light source, that keeps blurring of the output beam
below a level defined by the angular resolution of the
object pattern prescription.
Yet another object of embodiments of the
invention concerns a lens shape generation method that
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includes the steps:
a) on the Gaussian sphere of directions of the output beam
exiting the surface of the lens, in accordance with the prescribed output
pattern, establish a first grid of equal-flux zones of equal solid angle;
b) on a portion of the Gaussian sphere of directions of the light
emitted from the source into the interior of the lens, establish a grid with
the
same number of zones of equal-flux solid angles as the first grid, and with a
coordinate system topology congruent with that of the first grid, such that
the
zones of the second grid are in one-to-one topological correspondence with the
zones of the first grid, with the flux of each second grid zone in proportion
to
the flux of its corresponding zone of the first grid, according to the local
transmittance of the lens, with either or both of the grids being rotationally
non-symmetric;
c) by use of the correspondence, define a flux-redistributing
directional mapping function from the first Gaussian sphere to the second
Gaussian sphere, whereby any light ray from the source is assigned a direction
in the output beam, according to the zone of the second grid into which the
ray
falls, and so that the redirected ray also falls in the corresponding zone of
the
first grid;
d) by the vector form of Snell's law of refraction, express the
correspondence by establishing on the second grid an overlaying distribution
of
surface normal vectors;
e) from an initial starting point, mathematically generate the.
surface coordinates of the lens by contact-integrating the distribution of the
surface normal vectors, along an initial strip that follows a principal
curvature
of the surface, and then, by successive contact-integrations, orthogonally
outwards from the initial geodesic strip, of the grid surface normals,
generate
adjacent characteristic strips that follow the other principal curvature of
the
lens surface, outward to the boundary of the second grid.
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The method also may include the step of
performing successive integrations of adjacent
characteristic strips in such a way as to fulfill the
integrability condition dictating equality of the
crossed second derivatives of the surface of the lens,
thereby to ensure that the surface of the lens possesses
the surface normals necessary for it to transform the
light from the source into an output beam substantially
fulfilling the prescription.
According to a further aspect of the present
invention, there is provided in combination, an optical
lens in the form of an asymmetric dome-shaped body
located above an integral support base, and LED light
source means associated with said base, said base
upwardly divergent relatively away from a recess, and
wherein said light source means is embedded in a
transparent mass of material received in the recess,
said base and dome-shaped body being laterally
elongated, said dome-shaped body overhanging said base,
and said body being generally ellipsoidal in lateral
horizontal planes above said base.
According to another aspect of the present
invention, there is provided a luminaire comprising: a)
a lens body having a forwardly dome-shaped inner
portion, and an outer portion extending about and spaced
from said inner portion, said portions being
light-transmitting, and integral, b) the inner portion
extending non-circularly about a forwardly and upward
axis, c) there being a reflector on said outer portion,
whereby a light source in rearward alignment with said
inner portion provides certain light rays that travel
forwardly and are refracted by said dome-shaped inner
portion to travel forwardly from said inner portion, and
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other light rays that travel in said outer portion and
are reflected by said reflector to travel forwardly in
said outer portion, and forwardly from said outer
portion, d) said dome-shaped inner portion located above
an integral base, said light source associated with said
base, said base upwardly divergent relatively away from
a recess, and wherein said light source means is
embedded in a transparent mass of material received in
the recess, said base and dome-shaped body being
laterally elongated, said dome-shaped body overhanging
said base, and said body being generally ellipsoidal in
lateral horizontal planes above said base.
These and other objects and advantages of
embodiments of the invention, as well as the details of
an illustrative embodiment, will be more fully
understood from the following specification and
drawings, in which:
DRAWING DESCRIPTION
Fig. 1 depicts a grid on the Gaussian sphere
of prescribed intensity output;
Fig. 2 depicts a side view of a corresponding
grid on the Gaussian sphere of light-source intensity;
Fig. 3 depicts a top view of the Fig. 2 grid;
Fig. 4 depicts a top view of a lens derived
from these grids;
Fig. 5 depicts a side view of the lens;
Fig. 6 is a schematic diagram of grids and a
generated lens, and light outputs;
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7b
Fig. 7 is a cross section through a lens
generated in accordance with the invention;
Fig. 8 is a top plan view of a lens generated
in accordance with the invention;
Fig. 9 is a perspective view of a lens unit
generated in accordance with the invention, and a
reflector;
Fig. 9a is a section taken through the Fig. 9
lens and reflector;
Fig. 10 is a perspective view of a lens unit
array;
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Fig. 11 is a video or television unit having a screen incorporating
the Fig. 10 lens unit array;
Fig. 12 is a circuit diagram to control LEDs at each of the lens
units in the Fig. 10 array;
Fig. 13 is a diagrammatic view of an audience viewing zone or area
in relation to a large video screen incorporating the Fig. 10 array; and
showing
relative angling of lens units or cells;
Fig. 14 is a plan view of a vessel carrying light sources as will be
referred to; and
Fig. 15 is a top plan view of the Fig. 4 lens.
DETAILED DESCRIPTION
There are several methods for constructing a grid, as referred to,
for a particular specification. A regular tessellation, where all cells are
the
same size, can be warped to fit the prescription. With a large number of
cells,
such a warping can accurately match the prescription.
This warping is accomplished by a coordinate shrinkage of the
regular grid. A typical prescription has a center or a nearby direction of
greatest intensity IMAX. Align a rectangular grid (i.e., an equatorial sector
of
a latitude-longitude grid) on that center, so that it is located on the corner
of a
cell, at the direction (xo,y,,). Use an iteration interval dx that is much
smaller
than the cell width wa of the grid, e.g., dx = wX/1000. Then iterate the
following summation to find the distance x, to the other corner of the warped
cell:
1000
x, = Edx/1(x,yo)
i=0
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This shrinks the cell in the x direction to compensate for the intensity as it
varies with x. In the orthogonal direction y, along a grid line at x;, the
same
method is used:
1000
y, = Edy/I(xj,yo)/I(xj>y)
i=0
Here there is an expansion of the cell as the intensity decreases relative to
its
value on the yo line. By this method, the grid remains orthogonal, with a
different warping pattern in the two directions, but the total warping is a
product of the x and y warpings. This condition is known as separability.
Sometimes the convenience of such separability is not possible, as
when the prescription has a large ratio I(x;,yo)/I(x;,y,), from grid center to
grid
top y, at x and a much smaller one at grid edge x,. This will cause a much
larger number of cells to be generated at x, than at x,). Instead, the x
warping
is redone at each y;, so that the grid is not orthogonal and the grid lines
are
curved.
Alternatively, a polar grid can be used, that is, the polar section of
a latitude-longitude grid, with the pole at the source maximum. Then the
warping can be by making the latitude circles into ovals or by making the
longitude lines at uneven intervals, or both, so that an orthogonal grid would
result. Alternatively, a non-orthogonal grid could be constructed by bending
the longitude lines.
Alternatively, a regular tessellation with a very large number of
cells can be condensed by grouping these small cells into larger ones of
varying size, so that each broup of cells has the same total luminous flux. In
either case, the grid cells are indexed to express position within the grid,
with
either a single number or a pair of numbers, analogous to latitude and
longitude in geography.
The key to the design method of the present invention is the
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construction of two such grids: the first (see Fig. 1) on the Gaussian sphere
of
output intensity and the second (see Fig. 2) on the Gaussian sphere of source
intensity. These spheres have no physical location but exist abstractly, with
the purpose of establishing the two grids. The source grid can be thought of
5 having a miniaturized version of the source at its center. The prescription
grid
can be thought of having a miniaturized version of the present invention at
its
center. Both express the far-field behavior of light.
Both grids have the same number of cells and the same indexing
pattern, so that a one-to-one correspondence is established between them. This
10 correspondence requires that both grids have the same topology, so that if
one
is triangular, for example, the other cannot be rectangular, or polar. Another
grid topology is elliptical-parabolic. Thus, the warping method of grid
generation would be preferable, since it could better accommodate the
differences between the source and output distributions of intensity. The
output grid is constructed first because the fulfillment of the prescription
is the
purpose of the lens, and typically, the prescription will have more
irregularities
or idiosyncrasies than the source output.
Fig. 1 depicts one typical grid 100 on the Gaussian sphere of
directions, extending horizontally for a span of 112.5 and vertically 25
above and below the horizontal plane. In the vertical span of 5 above and
below the horizontal plane, cells 110 are half the size of remaining cells
120.
This grid expresses the U.S. Coast Guard specification for navigation lights,
that full intensity be maintained within 5 of the horizontal and half
intensity to
within 25 of the horizontal. Thus, each cell contains the same amount of
light flux. For the sake of clarity of illustration, there are only 22
horizontal
rows of 46 cells in this pattern, although in actual practice there would be
many times more. The grid is centered on axis 130.
Fig. 2 depicts a side view and Fig. 3 a top view of corresponding
light source grid 200, also with 22 rows of 46 cells. This grid is on the
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Gaussian sphere of directions of light coming from a Lambertian source, which
has intensity that falls off with the cosine of angle 210 with axis 220. Cells
230 nearest axis 220 are the smallest, while those further away are larger,
with
corner cell 240 the largest. Horizontal angle 250 is analogous to longitude,
and vertical angle 210 to latitude.
Fig. 4 depicts a top view of a lens 400, a particular embodiment of
the invention, generated froni the correspondence between the grids of Fig. I
and Fig. 2. The optically active top surface 410 is mathematically generated
from the surface normals derived from the corresponding grids 100 and 200.
Below the top surface is conical support base 420. At fhe bottom of the lens
is
an indentation or recess 430 to receive a circuit board 440, upon which are
mounted two rows of light-emitting diodes 450, embedded in a protective,
transparent, epoxy mass 460. Lens 400 is in optical contact with transparent
mass 460, so that there is no air gap. This lens will efficiently transform
the
Lambertian output, represented by grid 200, of absorbing-substrate LEDs 450
into radiation conforming to the Coast Guard Standard, represented by grid
100. This view shows how maximum horizontal source angle 470 is 77 ,
within which is emitted 95% of the output of a Lambertian source. This
illustrates how the weak fringes of a source's output may not be worth
collecting.
In Fig. 4, the generated, curved, surface lines 421 correspond to
generated, curved, grid lines 230 in Fig. 3; and generated, curved, grid lines
422 in Fig. 4 (orthogonal to lines 421) correspond to generated, curved, grid
lines 231 in Fig. 2.
Vertical lines 423 in Fig. 4 represent a side view of the tilted side
of 420. The Fig. 4, Fig. 5 lens is generally dome-shaped and has overall
length L, in the directions indicated by arrows 424 in Fig. 4 that is greater
than its overall width W, in the direction indicated by arrows 427 in Fig. 5
normal to 424 and normal to the page of Fig. 4. The lens is upwardly convex
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along its different lengthwise surface indicated at 426, and upwardly convex
along its uppermost widthwise surface 428.
Fig. 5 depicts a side view of the lens 400, with light-emitting diodes
450. Maximum vertical angle 480 is 60 , the maximum that can be refracted
into the 25 limit of the grid 100. The overall height of a working model of
lens 400 is only about half an inch, far smaller than equivalent light output
incandescent lamps of the prior art. Initial point 500 at the lens crest is
the
starting point for striating lens surface 410. Initial strip 510 extends
horizontally from 500. Characteristic strips 520 extend orthogonally from 510.
The boundary is located at 530.
Usually, the geometry of the source grid is stifficiently different
from that of the output grid that there will be "leftover" source light
outside
the source grid. In this case, the source grid can be overloaded so that some
of the output grids will have more light than that prescribed. Thus, the
overall
light utilization can be maximized, and the extra light will result in the
prescribed intensity being exceeded. In many cases, the prescription is only
for a minimum intensity, so this would be permissible.
This correspondence, between the source grid and the output grid,
specifies the redirection function that the lens must perform in order to
transform the light from ttie source into the desired output beam. The grid
cells must be sufficiently small so that the intensity is nearly constant
within a
cell. They also must be sufficiently numerous that the redirection function
changes slowly from cell to adjacent cell. This enables a smooth construction
of the lens surface.
Fig. 6 shows a lens 80 in cross sectional elevation, and which has
been constructed in the manner described. The lens surface is shown at 81.
The first grid 82 is shown as a section through the expanded Gaussian sphere
of directions of the light output beam exiting from the surface 81 of the
lens.
See for example beam vector V, exiting the lens surface, and corresponding to
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desired beam vector V,' exiting the surface of the grid 82; and beam vector V,
exiting the lens surface, and corresponding to the desired beam vector V2'
exiting the surface of the grid.
The second grid 83 is shown as a section through the reduced
Gaussian sphere of direction of light output from the LED source at 84. See
light output vector V3.
For each pair of corresponding cells on the two grids, there is a
deflection of source ray to output ray that must be carried out by the lens.
This deflection can be produced by refraction, reflection by a thin-film metal
coating, total internal reflection, or a combination of these, if the
deflection is
carried out in stages. Large deflection angles may require such staging, as in
the case of a wide-angle source and a narrow output beam.
For redirection being done by a single refraction, the analysis is as
follows. Snell's law of refraction, at the boundary between two isotropic
1s media of refractive indices n and n', is expressed by the equation
n sin i = n' sin i'
for incidence angles i and i', in the plane of incidence. Rays of light are
represented by three-dimensional, unit-length vectors pointing in the
direction
of propagation. The boundary surface between the two media is represented
by a unit-length vector perpendicular to it, the surface normal vector. For
incident ray I and surface normal vector N, the refracted ray R can be
calculated from the vector form of Snell's law , via the sine vector S and the
cosine vector C:
S = (n'/n) [I - N(N=I)]
C= NV (1 - I SIZ) / I N=I
R=C+S
These equations are for the derivation of the refracted vector R when vectors
I
and N are known. In designing the lens of the present invention, the vector I
is given by position on the source grid, while the vector R is given by the
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corresponding position on the output grid. The surface normal vector N must
be derived for each cell in the source grid. This is done differently
according
to whether n > n' (ray going out of material) or n < n' (going in). In both
cases, a vector M is first calculated, and then unitized to give the surface
normal vector :
N = M/jMj
In the first case, n > n' (going out of material), the vector M is given by:
M = I + [n'/(n-n')] (I - R)
In the case of n < n' (going into material), the vector M is given by:
M = R + [n/(n'-n)] (R - I)
For reflection, either by a metal film or totally internally, the surface
normal
that will reflect incoming ray I into reflected ray Q is given by utilizing
M = Q - I
In one embodiment of the invention shown in Figs. 4 and 5, the
source is in optical contact with the lens, so that the surface normal values
N(i,j) are determined by the case of n > n'.
Fig. 7 shows another embodiment in which the lens 700 has an
entry surface 710 as well as an exit surface 720, so that there is an
intermediate ray 730 inside the lens with vector T(i,j), in the plane defined
by
incident ray 740 with vector I and exiting ray 750 with vector R and
somewhere between them. To minimize aberrations that distort the effects of
finite source size, T is chosen so that the two surface normals, entry N,
(755)
and exit N, (756), have the same angles with rays I and R. This can be done
numerically with a root-bisection procedure. When the deflection angle
between I and R is more than 35 , a refractive index in the 1.5 range, typical
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of injection-moldable plastic, requires two surfaces, and the minimum-
aberration choice of ray T is required. When the biggest angle between the
first and second grids is less than 35 , the entry surface may be hemispheric,
with no net deflection of light. This would be the case for incandescent
lamps,
5 with envelopes having elevated temperatures that require an air gap to
intervene between the lamp and the lens.
The Coast Guard standard for masthead lamps for an anchored ship
has a 360-degree pattern, calling for an incandescent lamp with a vertical
filament. Generally, such filaments come with a vertical supporting post that
10 casts a shadow. The current invention can take a teardrop shape to in
effect
get light around this shadow,
Fig. 8 shows an overhead view with lamp 600, having transparent
envelope 610, vertical oriented cylindrical filament 620, and vertical support
post 630. Lens 640 has inner surface 650 and teardrop exterior 660. Ray 665
15 that just clears post 630 is refracted into its shadow, so the 360-degree
prescription is fulfilled.
More generally, incandescent lamps usually transmit light into a
pattern much bigger than a hemisphere, typically the entire sphere minus a
small portion blocked by the filament support means or envelope sealing
means. If the prescribed illumination pattern is relatively narrow (e.g.,
Society of Automotive Engineers taillight standard with 40 pattern width),
even refraction by two surfaces would be insufficient to redirect a 300 wide
lamp output into such a narrow pattern. In this case, part of the lens would
utilize total internat reflection (TIR) as well as refraction by an entry and
an
exit surface. There are two design strategies, according to whether the TIR
portion of the lens redirects light to the outer portion of the output grid or
has
an output grid of its own with a central portion that is additive to the
output
grid of the all-refractive portion of the lens. In effect, the second strategy
amounts to having two independent sub-lenses acting in parallel to additively
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16
fulfill the illumination prescription.
The reflecting surface requires its own exit surface, which must not
block significant amounts of light from the source. Therefore, the exit
surface
should take the form of a conicoid with its apex at the light source. A
conicoid is a warped cone, a developable surface consisting of lines radiating
from its apex. The outer part of the first output grid must be refracted
through
this conicoid to become a transformed grid that is the prescription for the
reflecting surface. In Fig. 9, lens 900 has central body 910 that is similar
to
the lens of Fig. 4 & Fig. 5, with light source 920. Surrounding it and
extending from its periphery is conicoid surface 930 with its apex at source
920. Reflecting surface 940 extends downward from conicoid 930.
The light-emitting diode (LED), in contrast, typically emits light
into only a hemisphere, because of being mounted on larger objects, such as
circuit boards. Also, its low operating temperature allows it to be in optical
contact with a plastic lens, in the case of a relatively wide illumination
prescription, such as the Coast Guard navigation lights. For the narrower-
angle automotive lights, an entry surface may be necessary to redirect the
light
from the edge of the second grid to the edge of the first grid.
The object of the present invention is to construct one or more lens
surfaces. The first surface encountered by light from the source is given by
r(i,j) = r(i,j) I(i,j)
where r is the distance from the source to -the lens surface, in the direction
of
ray vector I. A second surface would be specified by
r2(i,j)= ri(i,j) + r1z(i,j) T(i,j)
where r1z is the distance from the first surface, along refracted vector T. If
the
second surface is reflecting, then there would be a third surface, specified
by
r3 (i,J)= r2(iJ) + r23 (iJ) Q (i>J)
where r23 is the distance from the second surface, along reflected vector Q.
Each lens surface must be shaped so that each ray vector
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encountering it is properly redirected: I into R(single surface) or I into T
and T into R,or T into Q and Q into R. Each redirection happens when a ray
encounters the proper surface normal vector N. Surfaces can be expressed in
a coordinate system having its origin at the light source, with the z axis
aligned
with the output beam, so that a point (x,y,z) on the surface is specified by
the
function z(x,y), having derivatives p=(az/ax) and q=(az/ay). Then the
surface normal is given by N = (-pi - qj + k)//(l + p2 + q'-), where i, j, and
k are the Cartesian uilit vectors defining the x, y, and z coordinate axes,
respectively.
In seeking to calculate a lens surface from knowledge of its surface
normals, expressed as an array N(i,j), a numerical iteration will be required,
beginning with an initial point and moving outwards This iteration should
neither move away from a correct solution or converge to an incorrect
solution. Unlike many problems of surface generation, that of the present
invention is greatly helped by the knowledge that each surface point z(i,j)
must
lie on a ray vector, such as I(i,j) in the case of a single-surface lens.
Thus, a
trial point can be moved along this ray until the best point is found for a
fit of
the desired surface normal to the adjacent surface points and to their surface
normals. A fundamental property of surfaces in space is smoothness,
expressed by the equality of crossed second derivatives:
aZz /axay = aZz/ayax
It can also be expressed as an integrability constraint
ap/ay = aq/ax
This constraint can be utilized to check the fit of the generated surface to
the
array of surface normals.
Surface curvature is measured by how fast the normal vector
rotates, due to motion tangent to the surface. Except for a very few surfaces,
such as spheres that have constant curvature, the surface curvature at a point
varies with the tangent direction, according to a function known as the shape
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operator. The maximum and minimum values of the surface curvature define
the principal curvatures, primary and secondary respectively. They lie in
directions perpendicular from each other. The two orthogonal tangent vectors
aligned to the principal curvatures, along with the surface normal vector,
define the principal frame field. This is a triad of vectors that is defined
for
every surface point. Also known as the Darboux frame, it is very convenient
for surface generation.
Accordingly, the generation of the lens surface would typically
begin at some initial point, r(0,0), and proceed outwards, generating an
initial
strip of surface (using the terminology of partial differential equations).
The
most accurate integration is obtained when the outward direction is chosen to
coincide with the primary principal curvature (for example as seen in Fig. 4),
so that successive changes of the normal vectors align with the initial strip.
Along this principal curvature, the integration is one-dimensional and hence
easier to perform. Another initial strip would be generated along the
secondary principal curvature (for example as seen in Fig. 5). Then the
surface is completed by successively generating characteristic strips parallel
to
the primary initial strip, beginning on successive sites on the secondary
initial
strip.
Numerical accuracy is critical to a successful lens design, so that the
derived surface has everywhere the proper surface normal vector. The method
of initial and characteristic strips enables the use of the principal frame
field,
rather than the triple vector cross product. The latter method is abstractly
suitable for keeping a characteristic strip parallel to a prior strip. For
actual
computation, however, the small angles between successive grid points causes
the triple vector product, with its squaring of these small angles, to have
extreme sensitivity to machine precision (i.e., the number of decimal points
utilized by the particular computer). This sensitivity typically results in
large
surface generation errors and a failure of the generation process.
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The use of principal curvatures to align the initial strip highlights an
important factor in choosing the output and source grids: they should be
aligned so that their axes correspond to the maximum and minimum amounts
of redirection of the source light into the output light. Then the array
N(i,j) of
surface normals will be prealigned with the principal curvatures of the lens
surface. The placement of the grids in the source and output light
distributions
should typically be either the maxima or the centroids of said distributions,
which would respectively be the ray vectors 1(0,0) and R(0,0). The primary
and secondary initial strips would be formed by integrations from initial
point
r(0,0) through r(1,0), r(2,0), etc., and from r(0,0) through r(0,1), r(0,2),
etc.,
respectively. The first characteristic strip would begin at r(0,1) through
r(1,1),
r(1,2), etc., and at each point r(i,j) = r(i,j) I(i,j), the value of r(i,j) is
a
function of prior values r(i,j-1), preceding it on the characteristic strip,
and
r(i-l,j) on the adjacent prior strip.
Once each strip has been generated, the prior strip, if it is not an
initial strip, can be checked point-by-point for fit to both sides, and
adjusted
according to the integrability constraints. Then, the characteristic strip
could
be regenerated from this new prior strip. This is known as a relaxation
method.
For implementations of the present invention with multiple surfaces,
the surfaces are generated with the innermost surface.first, and outwards in
succession. The initial points for successive surfaces would be chosen so that
the iteration does not collide with a prior surface. This may require several
trials. In the case of Fig. 9, the initial strip of reflecting surface 940 can
be
either upper rim 950 or lower rim 960.
Finally, when a lens design is completed and checked for
performance, it may need to be adjusted if its transmittance is anywhere
significantly reduced by large angles of refraction. For example, at
refractive
index 1.5, the largest useable incidence angle from air into the material is
75 ,
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where the transmittance is 75 %, rather than the 96% of normal incidence.
This is equivalent to a reduction in source intensity. The source grid would
have its cells enlarged to accommodate this effect, and the lens would be
recalculated with the modified grid. Similar adjustments can be made for
5 scattering from surface roughness or from metal-film reflectivity of less
than
unity.
Accordingly, the invention contemplates an illumination system with
a prescribed output pattern comprising a light source and an optical lens
redirecting the light of the source into an output beam, the lens with
multiple
10 surfaces, at least one of which has a shape that is not a surface of
revolution,
the shape generated by the following method:
a) on the Gaussian sphere of directions of the output beam
exiting the surface of the lens, in accordance with the prescribed output
pattern, establish a first grid of equal-flux zones of solid angle;
15 b) on a portion of the Gaussian sphere of directions of the light
emitted from the source into the interior of the lens, establish a second grid
with the same number of equal-flux zones of solid angles as the first grid,
with
a coordinate-system topology congruent with that of the first grid, such that
the
zones of the second grid are in one-to-one correspondence with the zones of
20 the first grid, with the flux of each zone in proportion to its
corresponding
zone of the first grid, according to the local transmittance of the lens, with
either of both of the grids being rotationally non-symmetric;
c) by the correspondence define a flux-redistributing directional
mapping function from the first Gaussian sphere to the second Gaussian
sphere, whereby any light ray from the source can be assigned a direction in
the output beam, according to the zone of the second grid into which the ray
falls, so that the redirected ray falls in the corresponding zone of the first
grid; -
d) establish one or more lens surfaces to redirect the source
rays to the output rays, using the vector laws of refraction or reflection to
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derive a distribution of normal vectors for each surface; and
e) from the distributions of normal vectors, successively
generate each lens surface, beginning with that nearest source and progressing
outwards.
Further, the lens surfaces are generated from the distributions of
normal vectors by the following method:
f) from an initial starting point, calculate the surface
coordinates of each surface of the lens by contact-integrating the
distribution of
surface normal vectors, along an initial strip that follows one principal
curvature of the surface and then, by successive contact-integrations,
orthogonally outwards from the initial strip, of the grid of surface normals,
generate adjacent characteristic geodesic strips outward to the boundary of
the
second grid;
g) to ensure that the surface of the lens possesses the surface
normals necessary for it to transform the light from the source into an output
beam substantially fulfilling the prescription, perform the successive
integrations of adjacent characteristic geodesic strips so as to fulfill the
integrability condition dictating equality of the crossed second derivatives
of
the surface of the lens;
h) and determine the overall size of the lens relative to the size
of the light source by selecting the distance of the initial point from the
source
to keep the blurring of the output beam below a level defined by the angular
resolution of the prescription.
In addition, the source is typically provided to be in optical contact.
with the leiis, and the redirection is from within the lens material outwards
into
the environment, by refraction, with the distribution of surface normals
determined by the vector form of Snell's law of refraction.
In this regard, the lens is provided to have an entry surface
admitting light from the source into its interior and an exit surface for
light
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leaving the interior; and the entry surface performs negligible redirection,
and
substantially the entirety of the redirection is performed by the external
surface. The entry surface redirects the source light, so that a new system of
interior rays is used to generate the surface normal distributions of both of
the
surfaces, with redirection allocated between the entry and exit surface
normals.
Typically, a reflecting surface may be provided, and wherein large
redirection angles are implemented with the reflecting surface. Also, the
reflecting surface typically utilizes total internal reflection, which is
preferable
to thin film metal coating. The coating would only be necessary when the
lo incidence angle is less than the critical angle arcsin (1/n) for refractive
index n.
An important application of the present invention is in large-scale
video displays utilizing groups of individually controlled, light-emitting
diodes:
red, green, blue, and possibly yellow. Each group would have its own lens,
similar to that of Figs. 4 & 5, but possibly somewhat smaller. A full video
display, as for a sports stadium, would have 525 rows of 800 lenses, each with
a group of LEDs. As shown in Fig. 10, adjacent lenses 1101 and 1102 lie
vertically staggered so that they do not obstruct each other's horizontally
directed output. Lens 1101 is positioned over array 1110 of six LEDs: 2 red,
2 green, 1 blue, and 1 yellow. The lens has a horizontal swath of 120 and a
vertical swath of 30 . An advantage over the prior art of large-scale
television
is that the curved surfaces of the lens disperse reflections from glare
sources
such as the sun. In addition, back surface 1120 would have a low reflectance
means, such as a black matte coating.
Fig. 12 shows multiple LEDs at 850-855, controlled as at their
luminous intensities, as by a master control 856. They have different
wavelengths, as for example two red-emitting LEDs 850 and 851, two green-
emitting LEDs, as at 852 and 853, one blue-emitting LED 854, and one
yellow-emitting LED 855. Each of the lenses in Fig. 10 typically has a light
source consisting of the Fig. 12 LED cluster; and the LEDs are controlled, as
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to provide a viewable colored picture, changing with time. Bus 857 represents
control signal path to other LEDs in the array.
In Fig. 10, the multiple lenses or units indicated at 1150 are in an
array, as shown (arranged in rows and columns), and may comprise a video
screen for display of television or computer imagery, so that each of the
lenses
comprises a pixel of the display. See Fig. 11 showing a video unit, same as a
flat TV set 859 having a viewable screen 860 made up of the lens array or
raster of Fig. 10.
Fig. 13 shows the lens array 860 like that of Fig. 10, for example,
which is shallowly curved to be effectively aimed at a pecific audience
location indicated at 861. Lenses at opposite ends of the screen are
differentially angled to provide light outputs subtending the audience zone.
The lenses in the array in Fig. 10 are mounted as on a surface 1120 of low
reflectance, to provide the effective "screen".
Fig. 14 shows a vessel 870 in plan view, incorporating lens units as
described. The light sources shown may include incandescent lamps, at the
LED locations described above, with a white light prescription for lenses.
Lens 871 is a Coast Guard 360-degree masthead navigation light, as seen in
Fig. 8; and the lens has a horizontal cross section with a teardrop shape that
eliminates shadowing of the filament of the lamp by support posts of the
filament. The lamps 872 may have a substantially cylindrical filament in a
vertical orientation. The prescription is the Coast Guard 135-degree stern
navigation light; and the lens utilizes a reflective surface to redirect a
portion
of the output of the lamp into the prescribed output.
Bow lenses are indicated at 873 for red starboard, and 874, each
with the 112 side marker prescription.
Fig. 9a shows a luminaire, in accordance with the invention, and
characterized by the following:
a) a lens body 880 having a forwardly dome-shaped inner
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portion 880a, and an outer portion 880b extending about and spaced from the
inner portion, the portions being light-transmittin(Y and integral;
b) the inner portion extending non-circularly about a forwardly
extending axis 882;
c) there being a reflector 883 on the outer portion, whereby a
light source at 884 in rearward alignment with the inner portion provides
certain light rays 885 that travel forwardly and are refracted by the dome-
shaped inner portion to travel forwardly from the inner portion, and other
light
rays 886 that travel in the outer portion and are reflected by the reflector
to
travel forwardly in the outer portion and forwardly from the outer portion.
The Fig. 9a luminaire may have the configuration of the Fig. 9 lens
unit.
The reflecting surface of reflector 883 is typically generated by the
method that includes:
a) partitioning the first grid into an inner refraction-only portion
and an outer reflection-assisted zone, according to the maximum practical
redirection angle of the refractive index of the material of the lens, and
effect a
corresponding partition of the third grid of light interior to the lens,
b) surrounding the exterior surface of the central refraction-only
portion of the lens, establish an outer conicoid portion of the exterior
surface
of the lens, the conicoid tilted so as to minimize the impingement upon it of
rays coming directly from the source,
c) via backwards ray tracing of refraction through the tilted
surface, transform the outer portion of the first grid into a grid upon the
Gaussian sphere of directions of light interior to the lens,
d) establish a one-to-one correspondence between the partial
grid of backwards-ray-traced light interior to the lens and the outer portion
of
the third grid,
e) by the vector form of the law of reflection, express the
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correspondence by establishing on the outer portion of the third grid an
overlaying distribution of surface normal vectors,
f) utilizing the outer perimeter of the tilted exterior surface as
an initial strip, mathematically generate the surface coordinates of the
reflective
5 surface by the method of contact integration of orthogonal characteristic
strips,
g) and extending the conical surface sufficiently far so that the
characteristic strips will not irnpinge upon the light source or the inner
lens
surface.
The lens is further characterized as having
lo a) TIR surface,
b) mirror surface.
Fig. 15 shows the generally ellipsoidal (elongated) top plan view outline
shape of the Fig. 4 lens.
