Lens for a lamp and method of producing a die therefor

A lens for a lamp having a non-planar portion in which a large number of Fresnel lens steps or prism steps are formed on the incident surface thereof, yet which produces a very parallel output beam. The lens steps are defined by a tangential vector at an arbitrary point of a refraction boundary surface (Fresnel lens steps) or a total-reflection surface (prism steps) which is in the same direction as an outer product of a normal vector of the refraction boundary surface or total-reflection surface and a normal vector of an exit surface of the lens at a refraction point where a ray refracted by the refraction boundary surface or total-reflection surface is refracted. A method is also disclosed for producing a die for forming such a lens.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention is intended to provide a lens for use in a lamp which 
can provide an even brightness distribution and a good visibility, and a 
method of producing a die for forming that lens. 
2. Description of the Background Art 
Among vehicular lamps is a type having a structure in which an inner lens 
for controlling direct light from a light source and reflection light from 
a reflector is disposed in a lamp space defined by a lamp body and an 
outer lens. The inner lens is a formed product of a transparent synthetic 
resin, and has Fresnel lens steps and prism steps on its one surface. 
As shown in FIG. 16, a lamp 1 is designed to have a curved exit surface of 
an outer lens 2 that conforms to a body shape of a vehicle. 
In the lamp 1, the optical axis x--x of a reflector 3 extends in the 
front-rear direction of the vehicle passing through the center of a 
filament of a bulb 4. An inner lens 5 is disposed between the bulb 4 and 
the outer lens 2. That is, the inner lens 5 is placed immediately inside 
the outer lens 2, and is influenced, like the outer lens 2, by the vehicle 
body shape to have an exit surface 6 assuming a curved shape. 
The inner lens 5 consists of a plate-like portion 5a and a curved portion 
5b that is continuous with the plate-like portion 5a and curved 
increasingly as the position reaches one end in the longitudinal 
direction. Fresnel lens steps 7, 7, . . . are formed on the inner surface 
in the vicinity of the optical axis x--x, and prism steps 8, 8, . . . are 
formed around the Fresnel lens steps 7, 7, . . . 
FIG. 17 is a sectional view showing the main part a of a plate-like inner 
lens 5. 
Fresnel lens steps b, b, . . . are formed on an incident surface of the 
inner lens a in the vicinity of the optical axis x--x of the reflector 3, 
and prism steps c, c, . . . are formed around the Fresnel steps b, b, . . 
. Through the refraction by the Fresnel lens steps b, b, . . . , the 
paraxial rays of the light emitted from a bulb d are controlled to become 
in parallel with the optical axis of the lamp. The outer rays of the light 
from the bulb d that depart from the optical axis of the lamp to go toward 
the peripheral area of the inner lens a are controlled through the total 
reflection by the prism steps c, c, . . . to become in parallel with the 
optical axis. 
This structure is employed because the paraxial rays have small incident 
angles with respect to the inner lens a and can be controlled through the 
refraction phenomenon, but the outer rays departing from the optical axis 
have large incident angles with respect to the inner lens a. Accordingly, 
it is difficult to control the outer rays through refraction. 
In order to accommodate the recent design trend that vehicle bodies are 
rounded or streamlined to improve the aerodynamic characteristics of 
vehicles and to satisfy requirements on design, it is necessary to design 
a lamp shape to have a curve that conforms to the external shape of a 
vehicle body or to have an inclination to the vertical direction. 
Therefore, it is not possible for the inner lens to be limited to a 
plate-like shape, that is, in general the inner lens is required to 
include a curved shape. 
FIG. 18 conceptually shows an example of a method of forming lens steps on 
a curved surface of an inner lens. 
To simplify the description, it is assumed that lens steps are to be formed 
on a spherical surface, as shown in FIG. 18. There may be conceived a 
method in which a plate-like inner lens f on which lens steps are to be 
formed based on concentric reference circles e, e, . . . is employed as a 
reference model of design, and the concentric reference circles e, e, . . 
. are projected onto a spherical surface g. In this case, Fresnel lens 
steps and prism steps are formed on the spherical surface g based on 
reference circles h, h, . . . that are concentric to the optical axis. 
While the above method permits a relatively easy design, it will encounter 
a difficulty in precisely controlling the light paths. As a result, 
parallel rays cannot be obtained over the entire surface of the inner 
lens, and the brightness distribution will be uneven. 
This is a natural result of a fact that fine optical designing is not 
performed on the lens steps in accordance with the surface shape of the 
inner lens. The portion of the inner lens that is not very curved, i.e., 
generally flat portion 5a, will not cause any problems. But the portion 5b 
in which the curvature varies greatly will cause a considerable deviation 
from the desired brightness distribution due to a contribution of 
unexpected rays. 
To avoid the above problem, it is necessary to alter the method of forming 
the lens steps. However, according to the above method, a proper course of 
designing cannot be obtained easily. Therefore, much time and work are 
needed to design the inner lens, and its final design and performance will 
depend on experiences of a designer. 
SUMMARY OF THE INVENTION 
To solve the above-described problems, according to the present invention, 
a lens for a lamp in which a large number of Fresnel lens steps and/or 
prism steps are formed on an incident surface of the lens having a curved 
portion, is characterized in that a tangential vector at an arbitrary 
point of a refraction boundary surface of the Fresnel lens steps or a 
total-reflection surface of the prism steps coincides with an outer 
product of a normal vector of the refraction boundary surface of the 
Fresnel lens steps or the total-reflection surface of the prism lens steps 
and a normal vector of an exit surface of the lens at a refraction point 
where a ray refracted by the refraction boundary surface of the Fresnel 
lens steps or reflected by the total-reflection surface of the prism steps 
is refracted. 
Further, according to the invention, there is a method of producing a die 
for forming a lens for a lamp in which a large number of Fresnel lens 
steps and/or prism steps are formed on an incident surface of the lens 
having a curved portion. According to that method, a direction of an 
incident ray with respect to an exit surface is first determined according 
to the law of refraction and based on a normal direction at a refraction 
point of the exit surface and a direction of parallel rays so that exit 
rays from the lens become the parallel rays. Then, a refraction boundary 
surface is determined according to the law of refraction in the case of 
the Fresnel lens steps or a total-reflection surface is determined 
according to the law of reflection in the case of the prism steps, based 
on a direction of an incident ray with respect to the incident surface of 
the lens and the direction of the incident ray with respect to the exit 
surface that is obtained above. Next, a vector calculated as an outer 
product of a normal vector of the refraction boundary surface of the 
Fresnel lens steps or the total-reflection surface of the prism steps and 
the normal vector at the refraction point of the exit surface is employed 
as a direction vector for determining a direction of forming the 
refraction boundary surface or total reflection surface. Then, a closed 
curve is generated by connecting these direction vectors as tangential 
vectors using spline approximation. Finally, a V-shaped groove having a 
slanting surface corresponding to the refraction boundary surfaces and/or 
the total-reflection surfaces is formed on a die material along the closed 
curve.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 16, which previously was described, shows a configuration of an inner 
lens of a tail lamp of a vehicle to which the invention may be applied. 
According to the invention, the slope of the refraction boundary surface of 
the Fresnel lens step or the total-reflection surface of the prism step 
that corresponds to the refraction point on the exit surface of the lens 
is determined so as to conform to the curved shape of the exit surface, 
i.e., so that rays refracted by the refraction boundary surface of the 
Fresnel lens step or totally reflected by the total-reflection surface of 
the prism step are refracted by the exit surface to become parallel rays. 
As a result, there can be avoided an unevenness in the brightness 
distribution that would otherwise be caused by light not precisely 
controlled. Further, the Fresnel lens steps and the prism steps can be 
designed according to the procedure that is clear in terms of optics. 
A lens for a lamp and a method of producing a die therefor according to the 
present invention are described hereinafter by way of an embodiment 
accompanied by the drawings. 
FIG. 1 illustrates a, refracting action of the Fresnel lens steps 7, 7, . . 
. , where a light ray is refracted twice while passing through a lens 
step. Reference numeral 9 denotes a curve that represents the exit surface 
6 of the inner lens 5, and is a cross-sectional line obtained when the 
exit surface 6 is cut by a horizontal plane containing the optical axis. 
This line is first given as a shape conforming to the vehicle body shape. 
A broken line Lf in FIG. 1 indicates a light path. A vector V.sub.-- IN is 
a direction vector of an incident ray, and a vector V is a direction 
vector of a refracted ray. 
Reference character A denotes a straight line representing a boundary 
surface S of refraction. A vector N.sub.-- IN is a normal vector of the 
boundary surface S at an incident point PI. 
A vector V.sub.-- OUT is a direction vector of a refracted ray on the exit 
surface 6, and a vector N.sub.-- OUT is a normal vector of the exit 
surface 6 at a point PO on the intersection line 9. 
If it is required that the direction vector V.sub.-- OUT of the finally 
determined ray be in parallel with the optical axis x--x, the path Lf is 
uniquely determined according to Snell's law when the lens thickness is 
specified. 
That is, the direction of the vector V can be determined from the 
parallelism of the vector V.sub.-- OUT and the optical axis x--x, a 
refraction angle formed by the normal vector N.sub.-- OUT and the vector 
V.sub.-- OUT, and a refractive index of the inner lens 5. Further, the 
normal vector N.sub.-- IN and the boundary surface S can be determined 
from the vectors V and V.sub.-- IN. 
FIGS. 2-7 show, step-by step, a method of producing a die for the Fresnel 
lens steps 7, 7, . . . As is apparent from the fact that a cross-section 
obtained by cutting the Fresnel lens step 7 by a plane including the 
optical axis has a triangular shape, a die for making the lens can be 
produced by forming, by NC machining, V-shaped grooves corresponding to 
the respective steps on a die material. 
First, as shown in FIG. 2, on a design exit surface K, the vector V is 
determined according to the Snell's law from the normal vector N.sub.-- 
OUT at the exit point PO and the direction vector V.sub.-- OUT of a 
refracted ray that passes the point PO and is in parallel with the optical 
axis x--x. In general, the exit surface K is a free surface that cannot be 
expressed by an analytical function. 
Next, as shown in FIG. 3, the boundary surface S of refraction and the 
normal vector N.sub.-- IN thereof are determined according to the Snell's 
law from the vector V and the direction vector V.sub.-- IN of the incident 
ray. 
Then, as shown in FIG. 4, an outer product (vector product) of the vectors 
N.sub.-- OUT and N.sub.-- IN is calculated as a vector W, which is 
contained in the boundary surface S and has a direction indicating a 
forming direction of the boundary surface S. 
FIG. 5 shows a closed curve 10, which is a spline curve having vectors W 
sequentially obtained at the respective varying points PO as tangential 
vectors. The closed curve 10 has the optical axis x--x as its center line 
and is located on the light source side of the exit surface K, and serves 
as a reference line for machining the die. 
In general, the closed line 10 is not circular when viewed along the 
optical axis, which is understood by considering that it is a very special 
case that the boundary surfaces S at the respective points are included in 
a single sphere. 
As shown in FIG. 6, the incident ray is refracted by the very small surface 
S formed under the exit surface K, and further refracted by the exit 
surface K to exit as a ray in parallel with the optical axis. By 
connecting the very small surfaces S along the closed curve 10, a 
continuous boundary surface relating to one Fresnel step 7 is formed. 
FIG. 7 shows how a V-shaped groove 11 is formed on a die material M by 
controlling the movement of a cutting tool along the closed curve 10. An 
outside slanting surface 11a of the V-shaped groove 11 relates to the 
formation of the incident surface of the Fresnel lens step 7. An angle of 
an inside slanting surface of the V-shaped groove 11 with respect to the 
optical axis is set at a constant value for convenience of the die 
extraction. 
Next, the formation of the prism steps 8, 8, . . . is described. 
FIG. 8 illustrates total reflecting and refracting actions of the prism 
steps 8, 8, . . . While passing through the lens step 8, the light ray is 
first refracted, then totally reflected, and again refracted. 
Reference numeral 12 denotes a curve that represents the exit surface 6 of 
the inner lens 5, and that is a cross-sectional line obtained by cutting 
the exit surface 6 by a horizontal plane including the optical axis. This 
curve is first given as a shape conforming to the vehicle body shape. 
A broken line Lp in FIG. 8 indicates a light path. A vector v.sub.-- IN is 
a direction vector of an incident ray, and a vector v is a direction 
vector of a refracted ray. 
Reference character B denotes a straight line that represents a 
total-reflection surface R, and a vector n.sub.-- IN is a normal vector of 
the total-reflection surface R at an incident point QI. 
A vector v.sub.-- OUT is a direction vector of a refracted ray on the exit 
surface 6, and a vector n.sub.-- OUT is a normal vector of the exit 
surface 6 at a point QO on the intersection line 12. 
If it is required that the direction vector v.sub.-- OUT of a finally 
determined ray be in parallel with the optical axis x--x, the path Lp is 
uniquely determined according to the Snell's law and the total-reflection 
law when the lens thickness is specified. 
That is, the direction of the vector v can be determined from the 
parallelism of the vector v.sub.-- OUT and the optical axis x--x, a 
refraction angle formed by the normal vector n.sub.-- OUT and the vector 
v.sub.-- OUT, and a refractive index of the inner lens 5. Further, the 
normal vector n--IN and the total-reflection surface R can be determined 
from the vectors v and v.sub.-- IN. 
It is noted that in the above calculation an approximation is used that the 
direction of the incident ray is not changed through the first refraction, 
or a direction change is negligibly small. 
FIGS. 9-14 show, step-by step, a method of producing a die for the prism 
steps 8, 8, . . . The die is produced by forming, by NC machining, 
V-shaped grooves corresponding to the respective steps on a die material. 
As shown in FIG. 9, on the exit surface K, the vector v is determined 
according to the Snell's law from the normal vector n.sub.-- OUT at the 
exit point QO and the direction vector v.sub.-- OUT of a refracted ray 
that passes the point QO and is in parallel with the optical axis x--x. 
Next, as shown in FIG. 10, the total-reflection surface R and the normal 
vector n.sub.-- IN thereof are determined according to the reflection law 
from the vector v and the direction vector v.sub.-- IN of the incident 
ray. 
Then, as shown in FIG. 11, an outer product (vector product) of the vectors 
n.sub.-- OUT and n.sub.-- IN is calculated as a vector w, which is 
contained in the total-reflection surface R and indicates a forming 
direction of the total-reflection surface R. 
FIG. 12 shows a closed curve 13 that is obtained as a spline curve, which 
is a spline curve having vectors w sequentially obtained at the respective 
varying points QO as tangential vectors. The closed curve 13 is a 
machining line having the optical axis x--x as its center line and located 
on the light source side of the exit surface K. 
It is noted that in general the closed curve 13 is not circular when viewed 
along the optical axis. 
As shown in FIG. 13, the incident light is first refracted by a very small 
incident surface I formed under the exit surface K, then reflected by the 
very small total-reflection surface R, and again refracted by the exit 
surface K, to finally exit in parallel with the optical axis. A continuous 
total-reflection surface relating to one prism step 8 is formed by 
connecting the very small total-reflection surfaces R along the closed 
curve 13. 
FIG. 14 shows a V-shaped groove 14 formed by a cutting tool along the 
closed curve 13. An inside slanting surface 14a of the V-shaped groove 14 
relates to the formation of the incident surface I of the prism step 8, 
and an outside slanting surface 14b of the groove 14 relates to the 
formation of the total-reflection surface R of the prism step 8. An angle 
of the slanting surface 14a with respect to the optical axis is set at a 
constant value for convenience of the die extraction. 
In FIG. 15, the part, in the vicinity of the optical axis, of the die 
(including the closed curves 10, 13) for the inner lens 5 is enlarged. 
As described above, it is rare that a lens for a vehicular lamp has a 
complicated surface. That is, in general, it consists of a plate-like main 
portion and an increasingly curved portion that is continuous with the 
main portion. 
Reference numerals 15, 15, . . . in FIG. 15 denote closed curves serving as 
reference in forming V-shaped grooves on the die. The part of the closed 
curves 15, 15, located on the right side of the V--V line are related to 
the steps to be formed on the plate-like portion 5a of the inner lens 5, 
and have the same intervals on the H--H line. 
Points on the H--H line are selected as the origins of the closed curves 
15, 15, . . . 
The remaining part of the closed curves 15, 15, . . . located on the left 
side of the V--V line are related to the steps to be formed on the curved 
portion 5b of the inner lens 5. One can find a tendency that the interval 
of the closed curves 15, 15, . . . gradually increases as the position 
goes along the closed curve from its intersection with the V--V line 
toward its intersection with the H--H line. 
That is, the closed curves 15, 15, . . . are obtained by first arranging, 
at regular intervals, the origins for forming those curves on an 
intersection line obtained by cutting the flat portion of the reference 
surface of the die by a horizontal plane including the optical axis, and 
then performing the spline approximation as described above in connection 
with FIGS. 5 and 12. One closed curve generally assumes a circular arc 
around the flat portion of the die, and assumes a shape expanded outward 
from a circular arc around the curved portion of the die. 
Therefore, the inner lens 5 has such features in configuration that the 
boundary lines between the adjacent steps formed on the plate-like portion 
5a are concentric circular arcs, and that the boundary lines between the 
adjacent steps formed on the curved portion 5b are curves gradually 
deviating from concentric circular arcs. 
Thus, as is apparent from the process of forming the Fresnel lens steps 7, 
7, . . . and the prism steps 8, 8, . . . , the boundary surfaces of 
refraction and the total-reflection surfaces are defined with the exit 
surface 6 of the inner lens 5 as the reference so that at each position 
the refracted ray is directed in parallel with the optical axis x--x. 
Therefore, the precise light path control can be performed in accordance 
with the surface shape of the inner lens 5. 
As is apparent from the above description, according to the lens for use in 
a lamp and the method of producing the die therefor of the invention, the 
steps can be designed by defining the slopes of the boundary surfaces of 
refraction of the Fresnel lens steps or those of the total-reflection 
surfaces of the prism steps in accordance with the curved shape of the 
exit surface of the lens so that the refracted rays are always output from 
the exit surface as parallel rays. As a result, precise light distribution 
control can be performed in accordance with the laws of optics. 
Although the above embodiment is directed to the case in which the 
invention is applied to the inner lens of the vehicular lamp, it is 
apparent that the invention is not limited to such a case but can 
generally be applied to a wide variety of lenses for lamps.