Title: Retroreflector with controlled divergence made by the method of groove undulation
Abstract: A ruling of cube corner elements comprising intersecting ruled vee-grooves is characterized in that at least one of the vee-grooves is made such the substrate being ruled and the cutting tool are oscillated with respect to one another during the ruling of the vee-groove. Rulings made in accordance with the instant invention can be designed to produce a cube corner element having broader divergence.
Patent Number: 6,984,047 Issued on 01/10/2006 to Couzin, et al.
| Inventors:
|
Couzin; Dennis I. (Evanston, IL);
Sandler, legal representative; Ronald A. (Chicago, IL)
|
| Assignee:
|
Avery Dennison Corporation (Pasadena, CA)
|
| Appl. No.:
|
167135 |
| Filed:
|
June 11, 2002 |
| Current U.S. Class: |
359/530; 359/900; 264/1.9; 428/167 |
| Current Intern'l Class: |
G02B 5/12.4 (20060101) |
| Field of Search: |
359/529,530,900,831,833-836
264/11,19
428/163,167
|
References Cited [Referenced By]
U.S. Patent Documents
Other References
"Study of Light Deviation Errors in Triple Mirrors and Tetrahedral Prisms", P.R.
Yoder, Jr., J.Optical Soc. Amer., vol. 48, No. 7, pp. 496-499 1958.
|
Primary Examiner: Phan; James
Attorney, Agent or Firm: Renner, Otto, Boisselle & Sklar, LLP
Parent Case Text
This patent application claims the benefit of U.S. Provisional Application No.
60/297,394 filed Jun. 11, 2001.
Claims
What is claimed is:
1. A method of making a pattern of cube corner elements on a substrate surface
comprising the step of ruling a plurality of vee-grooves into the substrate surface
using a cutting tool, said substrate surface lying in the x-y plane of an orthogonal
x-y-z reference system, wherein the cutting tool and the substrate surface are
oscillated with respect to one another during the ruling of at least one of said vee-grooves.
2. The method of claim 1 wherein said cutting tool and said substrate surface
are oscillated with respect to one another in the z direction during said ruling
of said at least one of said vee-grooves.
3. The method of claim 1 wherein said cutting tool and said substrate are oscillated
with respect to one another in the x-y plane sidewise to the direction of the groove
during said ruling of said at least one of said vee-grooves.
4. The method of claim 1 wherein at least one vee-groove is ruled so that the
movement of the tip of the cutting tool defines a straight groove root parallel
to the x-y plane, and the attitude of the cutting tool oscillates within a plane
containing the groove root.
5. The method of claim 1 wherein at least one vee-groove is ruled so that the
movement of the tip of the cutting tool defines a straight groove root parallel
to the x-y plane, and the attitude of the cutting tool oscillates within planes
perpendicular to the groove root.
6. The method of claim 1 wherein at least one vee-groove is ruled so that the
movement of the cutting tool relative to the substrate produces a compound undulation.
7. An article made by the method of claim 1, the article comprising a substrate
surface lying parallel to the x-y plane of the orthogonal x-y-z reference system
and having a pattern of cube corner elements defined by intersecting vee-grooves
formed by the ruling of the cutting tool,
wherein each vee-groove comprises two groove side walls intersecting at a groove
root; and
wherein at least one of the vee-grooves is an undulating groove caused by the
cutting tool and the substrate surface being oscillated with respect to one another
during the ruling of this vee-groove.
8. The article of claim 7 wherein said groove root of said undulating groove
is a curve undulating in a plane perpendicular to the x-y plane.
9. The article of claim 7 wherein said groove root of said undulating groove
is a curve undulating in a plane parallel to the x-y plane.
10. The article of claim 7 wherein said groove root of said undulating groove
is a straight line parallel to the x-y plane and said groove walls of said undulating
groove undulate such that the included groove angle expands and contracts along
the length of the groove.
11. The article of claim 7 wherein said groove root of said undulating groove
is a straight line parallel to the x-y plane and said groove walls of said undulating
groove undulate such that the included groove angle is substantially constant along
the length of the groove.
12. The article of claim 7 wherein said undulating groove has a compound undulation
within the x-y-z reference system.
13. A method of making a retroreflective article comprising
a. making a pattern of cube corner elements on a substrate surface by ruling
a plurality of vee-grooves into the substrate surface using a cutting tool, said
substrate surface lying in the x-y plane of an orthogonal x-y-z reference system,
wherein the cutting tool and the substrate surface are oscillated with respect
to one another during the ruling of at least one of said vee-grooves;
b. making a replica tool from said ruling, said replica tool bearing said pattern
of cube corner elements; and
c. using said tool to form a retroreflective article comprising a replica of
said pattern of cube corner elements.
14. The method of claim 13 wherein said cutting tool and said substrate surface
are oscillated with respect to one another in the z direction during said ruling
of said at least one of said vee-grooves.
15. The method of claim 13 wherein said cutting tool and said substrate are oscillated
with respect to one another in the x-y plane sidewise to the direction of the groove
during said ruling of said at least one of said vee-grooves.
16. The method of claim 13 wherein at least one vee-groove is ruled so that the
movement of the tip of the cutting tool defines a straight groove root parallel
to the x-y plane, and the attitude of the cutting tool oscillates within a plane
containing the groove root.
17. The method of claim 13 wherein at least one vee-groove is ruled so that the
movement of the tip of the cutting tool defines a straight groove root parallel
to the x-y plane, and the attitude of the cutting tool oscillates within planes
perpendicular to the groove root.
18. The method of claim 13 wherein at least one vee-groove is ruled so that the
movement of the cutting tool relative to the substrate produces a compound undulation.
19. A retroreflective article made by the method of claim 13, said retroreflective
article comprising the replica of said pattern of cube corner elements, whereby
said replica comprises intersecting vee-grooves, wherein each vee-groove comprises
two groove side walls intersecting at a groove root, and wherein at least one of
the vee-grooves is an undulating groove.
20. The retroreflective article of claim 19, wherein said groove root of said
undulating groove is a curve undulating in a plane perpendicular to the x-y plane.
21. The retroreflective article of claim 19, wherein said groove root of said
undulating groove is a curve undulating in a plane parallel to the x-y plane.
22. The retroreflective article of claim 19, wherein said groove root of said
undulating groove is a straight line parallel to the x-y plane and said groove
walls of said undulating groove undulate such that the included groove angle expands
and contracts along the length of the groove.
23. The retroreflective article of claim 19, wherein said groove root of said
undulating groove is a straight line parallel to the x-y plane and said groove
walls of said undulating groove undulate such that the included groove angle is
substantially constant along the length of the groove.
24. The retroreflective article of claim 19, wherein said undulating groove has
a compound undulation within the x-y-z reference system.
Description
BACKGROUND OF THE INVENTION
This invention relates to a method of making a retroreflective article having
controlled divergence, and articles made by the method.
It is well known that retroreflective articles can be made from an array of microcube
corner elements. Such an array of microcube corner elements can be made by ruling
a master of "male" cube corners into a planar surface of a plate. This is taught
generally by Stamm U.S. Pat. No. 3,712,706; and also is taught in detail in Pricone
U.S. Pat. No. 4,478,769. Each of these patents is incorporated herein by reference
in its entirety.
U.S. Pat. No. 4,478,769 describes a well-known method of making triangular cube
corner elements, in which the planar surface of a master plate is ruled with a
diamond cutting tool that cuts a series of precise parallel vee-grooves. To rule
equilateral triangular cube corners, three sets of parallel grooves intersecting
one another at angles of 60° are made; each groove also will have an included
angle of substantially 70.53° disposed symmetrically, and will be ruled to
a groove depth determined by the height of the cube corners desired. This method
automatically results in an array of pairs of oppositely oriented equilateral triangular
microcubes on the face of the master. To rule non-equilateral triangle cube corners
the grooves within the parallel sets will contain angles other than 70.53°,
and intersect at angles other than 60°, as disclosed, for example in Rowland
U.S. Pat. No. 3,384,348. Methods for ruling non-triangle cube corners generally
do not use three sets of parallel symmetrically disposed vee-grooves, but the faces
of the cube corners are nevertheless formed from the walls of grooves, as disclosed,
for example in Nelson U.S. Pat. No. 4,938,563.
The ruled master may then be used to make a series of duplicates, such as by
electroforming, and the duplicates are assembled together to form a single "mother"
tool. The assembled "mother" tool is used to electroform molds, which then can
be assembled into a tool capable of providing microcube retroreflective elements
on a web of plastic sheeting material such as by embossing, casting, or other methods
known in the art.
Microcube corner retroreflective sheeting such as made by the method described
above is used in highway safety applications such as highway signs and pavement
markers. In such applications, the microcube corner elements retroreflect light
from a vehicle's headlights back to the driver of the vehicle. This is an inexact
retroreflection in which the divergence angle, α, ranges between approximately
0° and more than 3°. The value of α operative in any given situation
depends on the geometry of the vehicle and the distance from the vehicle to the
retroreflective material. For example, the divergence angle α for a large
truck's right headlight and its driver at a distance of about 40 meters from a
road sign will be approximately 3°, while the divergence angle α for
an automobile's left headlight and its driver at a distance of about 600 meters
from a road sign will be approximately 0.05°.
Also associated with the divergence angle, α, is a rotation angle, ε,
which is a measure of the direction of the divergence. The value of ε will
be different for left and right headlights of a vehicle, and will also depend on
the vehicle geometry and the position of the road sign.
Ideally, microcube corner retroreflective sheeting used in road signs will
produce a pattern of retroreflected light having sufficient intensity over a range
of divergence angle values and rotation angle values. For example, even a non-urban
retroreflective highway sign should retroreflect light through a divergence angle
α of about 1°, which corresponds to the value of α from a large
truck's right headlight back to its driver at a distance of about 120 meters from
the road sign.
Improvements in the precision with which microcube corner elements can
be ruled in a master plate and duplicated by embossing have led to concerns that
such microcube corner retroreflective sheeting may be retroreflective over only
a very narrow range of divergence angle, such as about 0.0-0.5 degrees, as well
as narrow ranges of rotation angle. It would be preferred to provide a ruled array
with cube corners producing the entire desired range of divergence and within very
short distances on the ruled array.
Light that is retroreflected by micro-sized cube corner elements will experience
a certain amount of diffraction because of the very small size of the microcubes.
Such diffraction will result in retroreflection over broader ranges of both divergence
angle and rotation angle. The particular ranges of α and ε will depend
on the particular diffraction pattern of a given microcube, which will depend in
turn upon the cube size, cube shape, the index of refraction of the cube material,
and upon whether or not the cube faces have been metallized. Diffraction, however,
is not a desirable method to enhance retroreflection through broader divergence
and rotational angle, because the very small microcubes that achieve greater diffraction
also cause a substantial quantity of light to be retroreflected with a divergence
angle α of greater than about 3°, where the light is not useful to the
vehicle driver. This is summarized in Table 1.
Table 1 indicates the spreading of retroreflection due to diffraction. Acrylic
equilateral triangle cube corners are used in each case. The millimeter dimension
measures the edge length of the triangle (identically 2.449×the cube depth,
or 1.155×the ruling spacing). The percentages indicate how much of the total
retroreflected flux is within a 1°, 2°, or 3° maximum observation
angle. For example, for the triangle cube corner with side 0.05 mm only 27.9% of
the total retroreflected light arrives between 0° and 1° observation angles.
| TABLE 1 |
|
| Diffraction spreading of retroreflection from different |
| size triangle cube corners |
| |
0.4 mm |
0.2 mm |
0.1 mm |
0.05 mm |
| |
|
| |
0° to 1° |
91.6% |
82.5% |
66.7% |
27.9% |
| |
0° to 2° |
95.7% |
91.6% |
82.4% |
66.6% |
| |
0° to 3° |
97.1% |
94.4% |
88.9% |
79.1% |
| |
|
Diffraction results in idiosyncratic patterns which are unlikely to distribute
the retroreflected light in a manner that will be most useful to a vehicle's driver.
This is shown in FIGS. 4A-D.
It is known in the art to create intentional aberrations in cube corner elements
by causing the dihedral angles of the cube corner elements to deviate slightly
from 90°. The classic paper "Study of Light Deviation Errors in Triple Mirrors
and Tetrahedral Prisms,"
J. Optical Soc. Amer., vol. 48, no. 7, pp. 496-499,
July, 1958 by P. R. Yoder, Jr., describes the well-known spot patterns resulting
from such aberrations.
U.S. Pat. No. 3,833,285 to Heenan, assigned to the common assignee and incorporated
herein by reference in its entirety, teaches that having one dihedral angle of
a macro-sized cube corner element greater than the other two results in extended
observation angularity in macrocubes, and specifically that the retroreflected
light diverges in an elongated pattern.
U.S. Pat. No. 4,775,219 to Appledorn discloses retroreflective articles having
tailored divergence profiles, wherein the cube corner elements are formed by three
intersecting sets of parallel vee-grooves, and wherein at least one of the sets
includes, in a repeating pattern, at least two groove side angles that differ from
one another.
U.S. Pat. No. 6,015,214 to Heenan et al., assigned to the common assignee, teaches
methods of forming microcubes by ruling vee-grooves into the edges of a plurality
of flat plates, and discloses that the tilt angle of a cutting tool with respect
to the surface of the edges being ruled can be adjusted continuously as each groove
is cut as a function of the distance traveled by the cutting tool along the ruled surface.
It is thus one object of the invention to provide an article comprising an array
of retroreflective microcube corner elements having controlled broader divergence.
It is another object of the invention to provide methods for making such an article.
SUMMARY OF THE INVENTION
A retroreflective article having a controlled broader divergence is provided
by
ruling one or more sets of generally parallel vee-grooves to form a plurality of
cube corner elements, each vee-groove having two side walls that intersect at a
groove root, in which ruling non-uniform deviations of the cube dihedral angles
from exactly 90° are intentionally introduced by causing the cutting tool
and the surface of the substrate to oscillate with respect to one another in a
controlled manner during the ruling of at least one of the vee-grooves. The vee-groove
so formed will be an undulating groove. The cube corner element dihedral angles
having at least one face defined by a side wall of an undulating vee-groove will
be non-orthogonal to varying extents, depending on the phase, frequency, and amplitude
of the oscillation during ruling. This introduction of variable, controlled non-orthogonality
of dihedral angles within very short distances of each other on a single ruled
groove will result in a controlled broader divergence of the ultimate retroreflective
article made from such ruled cube corner elements.
DESCRIPTION OF THE FIGURES
FIG. 1 illustrates a substrate oriented in the x-y plane and having a vee-groove
ruled in the y-direction using the method of the prior art; and
FIG. 2 illustrates a small portion of an array of cube corner elements made
in accordance with the first described mode of practicing the invention, in which
the magnitude of the groove undulation and the deviations of the dihedral angles
are greatly exaggerated for the sake of illustration.
FIG. 3 illustrates a small portion of an array of cube corner elements made
in accordance with the third described mode of practicing the invention, in which
the magnitude of the groove undulation and the deviations of the dihedral angles
are greatly exaggerated for the sake of illustration.
FIGS. 4A-D illustrate diffraction patterns from four different prior art unaberrated
cube corners. The patterns show observation angle from 0° to 3° over
all rotation angles. The entrance angle is 0°. The illustrations scale the
pattern logarithmically so that one step corresponds to retroreflectance difference
of approximately 2.5 times. FIG. 4A is for an uncanted acrylic triangle cube, 0.2
mm base dimension, 0.082 mm ruling depth. FIG. 4B is for the identical cube corner
when aluminized. FIG. 4C is for a -9.74° (face more parallel) canted acrylic
triangle cube, 0.256 mm base, 0.091 mm ruling depth. FIG. 4D is for a +11.17°
(edge more parallel) acrylic triangle cube, 0.194 mm base, 0.089 mm ruling depth.
Cube sizes have been chosen having equal optical active areas for purpose of diffraction comparison.
FIG. 5 illustrates the calculated diffraction pattern for the prior art cube
corner of FIG. 4A, but with a simple aberration of +14 arc minutes on each dihedral
angle. The geometric spot pattern has an average divergence of 1.1°.
FIGS. 6A-B illustrate calculated diffraction patterns for the cube corner of
FIG. 4A, but with aberrations in accordance with the first mode of the present
invention. FIG. 6A uses a sinusoidal undulation on each of three grooves sufficient
to give the geometric pattern an average divergence of 1.1°. FIG. 6B uses
a combination of a simple aberration of 9 arc minutes on each dihedral angle with
a sinusoidal undulation on each of three grooves sufficient to give the geometric
pattern an average divergence of 1.1°.
FIGS. 7A-B illustrate calculated diffraction patterns for the cube corner of
FIG. 4A, but with aberrations in accordance with the first mode of the present
invention being applied unequally to the three grooves. For FIG. 7A only one of
the three grooves receives sinusoidal undulation. For FIG. 7B two of the three
grooves receive sinusoidal undulation. For each of these designs, the geometric
light pattern has an average divergence of 1.1°.
FIG. 8 compares of two forms of undulation called "sin" and "sin±½ sin
2".
FIG. 9 compares the calculated observation angularity of the prior art aberrationless
triangle cube corner for FIG. 4A, with four triangle cube corners aberrated in
accordance with the instant invention, being those for FIG. 5, FIG. 6A, the "sin±½
sin
2" variant of FIG. 6A, and FIG. 6B.
FIG. 10 illustrates the application of equations (1)-(3) to the first described
mode of the invention.
FIG. 11 illustrates the application of equations (4)-(6) to the second described
mode of the invention.
FIG. 12 illustrates the application of equations (10)-(12) to the third and
fourth described modes of the invention.
DETAILED DESCRIPTION OF THE INVENTION
Cube corner retroreflective articles made by a process including the step of
ruling one or more sets of generally parallel vee-grooves in a surface with a vee-shaped
cutting tool can be provided with a controlled broader divergence by oscillating
the cutting tool and the substrate with respect to one another in a controlled
manner as the tool cuts one or more vee-grooves. The controlled oscillation during
ruling will result in an undulating vee-groove producing a controlled variation
in the dihedral angles of the cube corner elements on either side of the undulating
groove, which will controllably broaden the divergence of the ultimate retroreflective
article. Preferably, an entire desired range of divergence will be provided over
very short increments of groove length.
The term "attitude" as used herein shall mean the orientation of a defined axis
of a cutting tool relative to the substrate surface to be ruled. The defined axis
will begin at the tool tip and point generally away from the substrate surface.
The term "cube corner elements" as used herein includes those elements consisting
of three mutually intersecting faces, the dihedral angles of which are generally
on the order of 90°, but not necessarily exactly 90°.
The term "groove root" as used herein means the continuous curve defined by the
motion of the point of the cutting tool under the surface being ruled. A "groove
root" cut in accordance with the method of this invention can be either straight
or wavy, depending on the mode of operation of the instant invention.
The term "groove angle" as used herein means the included angle, measured in
a plane normal to the groove root, between the two walls of the groove cut into
the surface by the cutting tool at any given point along the groove length.
The phrase "the cutting tool and the substrate oscillate with respect to one
another," and substantial equivalents as used herein, shall mean that during the
ruling of a vee-groove either the cutting tool oscillates with respect to the substrate,
or the substrate oscillates with respect to the cutting tool, or both the cutting
tool and the substrate oscillate at the same time, so as to create an undulating vee-groove.
The term "divergence" as used herein is the angle between a light ray that enters
a retroreflecting element (e.g., a cube corner) and the light ray after leaving
that element.
FIG. 1 illustrates a perspective view of a substrate
20 having a planar
surface
22 on which vee-grooves can be ruled with a vee-shaped cutting tool,
such as is known in the art. As illustrated in FIG. 1, the substrate
20
is oriented with the surface
22 to be ruled disposed in the x-y reference
plane of the x-y-z orthogonal reference system, wherein the z direction is perpendicular
to the surface
22. Throughout this patent, the terms "the x-y plane," "the
x-z plane," and "the y-z plane," will mean the x-y reference plane, x-z reference
plane, and y-z reference plane, as defined by the x, y, and z reference axes of
FIG. 1. FIG. 1 further illustrates a typical vee-groove
24 of the prior
art ruled parallel to the y-axis, with the groove root
25 being a straight
line at a constant z-depth relative to the surface
22, and the groove having
constant included angle.
It is to be understood that for rulings on a non-flat substrate, the methods
of
this invention will require small modifications obvious to one skilled in the art.
Such modifications are considered to be within the scope of this invention.
The instant invention will be described in terms of four modes of operation.
It will be understood, however, that these four modes of operation are not necessarily
mutually exclusive, and that two or more such modes of operation can be employed
simultaneously. For ease of understanding, the modes of operation are described
and illustrated herein with respect to the simplest case in which the cube corner
elements being ruled are uncanted equilateral triangles. The applicability of the
instant invention is not so limited, however, and the methods of the invention
can be applied to canted triangular rulings, and hexagonal and rectangular cube
corner elements such as are illustrated and described in U.S. Pat. No. 5,914,813
(Smith et al.); U.S. Pat. No. 5,721,640 (Smith et al.); U.S. Pat. No. 4,938,563
(Nelson et al.); and U.S. Pat. No. 4,895,428 (Nelson et al.). Further, while the
undulations illustrated and described herein are most easily imagined as sinusoidal,
it will be appreciated that such sinusoidal undulations are not a requirement of
the instant invention. There is only the requirement that the undulations must
be piecewise smooth and that the tool accelerations be such that they will not
break the tool or ruin the cutting. A single groove can be made in several consecutive
parts, each of which includes some aspect of undulation.
By way of example and not by way of limitation, FIG. 8 illustrates two forms
of
undulation curves suitable for use in the various modes of operation of the instant
invention. The curve labeled "sin" is a perfect sinusoid. The curve labeled "sin±½
sin
2" follows the function sin-½ sin
2 from 0 to π
and then the function sin±½ sin
2 from π to 2π,
and continues to alternate between the two functions. The sin±½/sin
2
undulation, by having nearly flatted regions, gives more weight to the unaberrated
cubes. In FIG. 8, the sin undulation was adjusted to 0.6*sin so as to produce the
same average geometric divergence as the sin±½/sin
2 undulation,
in order to provide a more appropriate basis of comparison. It will be understood
that the horizontal and vertical scales of FIG. 8 require further adjustment to
the dimensions of the cube corner ruling, such as shown in Table 2.
The two curves in FIG. 9 labeled "20[sin]" and "33[sin±½ sin
2]"
show the difference in observation angularity for the sin and sin±½ sin
2
undulations illustrated in FIG. 8, respectively. Both undulations were chosen
to give average geometric divergence of 1.1°. The bracket notation used in
this application for denoting undulatory designs describes the extent of undulation.
"20[sin]" denotes sinusoidal undulation of such amplitude and pitch that the single
such groove contributes a maximum of 20 arc minutes of dihedral angle error in
the ruling. "33[sin±½ sin
2]" denotes sin±½ sin
2
undulation of such amplitude and pitch that the single such groove contributes
a maximum of 33 arc minutes of dihedral angle error in the ruling. That both rulings
produce the same 1.1° average geometric divergence in the acrylic product
is explained by the sin±½ sin
2 undulation having nearly flatted
regions and thereby producing a greater proportion of nearly unaberrated cubes
in the population. Undulations useful for the present invention do not have to
be perfectly periodic and do not have to follow any explicit mathematical functions.
The four simplest modes of operation of the invention are explained below. In
the following discussion, the notation {dot over (δ)} shall mean the change
in the rise or fall of each groove root introduced by the first mode of operation
of the instant invention; the notation {umlaut over (δ)} shall mean the change
in the direction of the groove root within the x-y plane introduced by the second
mode of operation of the instant invention; and the notation
##EQU1##
shall mean the change in the half angles of the respective grooves as ruled
in accordance with both the third and fourth modes of operation of the instant invention.
First Mode—Vertical Undulation
In a first mode of operation of the instant invention, the cutting tool is maintained
at a constant attitude with respect to the substrate, and during the ruling of
at least one vee-groove the cutting tool and the substrate oscillate with respect
to one another in a vertical direction, i.e., in a direction parallel to the z-axis
of FIG. 1. This will result in a vee-groove having a non-constant groove angle.
The resulting groove root is a vertically undulating curve. The two-dimensional
projection of the groove root in the x-y plane is a straight line. However, the
intersection of the groove wall with the x-y plane is a horizontally undulating
curve. In accordance with the definition of the phrase "the cutting tool and the
substrate oscillate with respect to one another" as set forth above, it will be
appreciated that the same effect can be achieved either by holding the substrate
in a fixed position, and moving the tip of the cutting tool in a vertically undulating
curve, while maintaining the attitude of the cutting tool constant, or by vertical
oscillation of the substrate while the cutting tool is moving in a straight line,
or by simultaneous out-of-phase movement in the z-direction of both the cutting
tool and the substrate. The choice of whether to move the cutting tool, or the
substrate, or both during the cutting of the vee-groove will depend upon the design
and functionality of the graver that controls the cutting tool and the fixture
that holds the substrate.
A first, inconsequential, effect of this first mode of operation of the instant
invention is the introduction of intersection errors in a pattern of ruled triangular
cube corner elements. That is, even if the other two groove sets are formed entirely
of straight grooves of the prior art, the vertically undulating groove root made
in accordance with the first mode of operation of the instant invention will not
always intersect the vertices of the other two groove sets at their exact points
of intersection.
A second, consequential, effect of this first mode of operation of the invention
is the purposeful introduction of variations or "errors" in the dihedral angles
of cubes having a cube face formed by a side wall of an undulating groove. The
frequency of the oscillation will be such that one period of oscillation spans
several cube corner widths. Thus, for a single triangular cube described in part
by a segment of an undulating groove root, the groove root segment is essentially
descending, essentially ascending, or essentially level. The cube dihedral angle
terminating at or near the lower end of a groove root segment will be slightly
more obtuse than had the groove root been level. Similarly, the cube dihedral angle
terminating at or near the raised end of a groove root segment will be slightly
more acute, than had the groove root segment been level. For substantially equilateral
triangle cube corners, the change to the cube dihedral angle is the groove slope
angle divided by √6. Generally, where all three grooves that define a cube
corner element are vertically undulating grooves, then for each such cube corner
element, the deviation in depth of each groove will affect the two dihedral angles
defined in part by that groove side wall, with the total effect on the dihedral
angles being very nearly additive.
FIG. 10 summarizes the aberrations due to the first mode when the undulation
is long enough that curvature within a single cube is insignificant. A male equilateral
triangle cube corner is illustrated as formed by three grooves. Each of the grooves
g
1, g
2, g
3 are shown rising by a corresponding
angular amount {dot over (δ)}
1, {dot over (δ)}
2,
{dot over (δ)}
3 in the direction indicated. If a {dot over (δ)}
value is negative, then the groove is instead falling. The three dihedral edges
are labeled with their angle errors, e
1; e
2; e
3,
that is, their deviations from perfect 90°. The dihedral angle errors are
given by approximate equations (1)-(3).
##EQU2##
It will be appreciated that these equations require some adjustment for non-equilateral
triangle cube corners.
FIG. 2 roughly represents a portion of a single groove g ruled in accordance
with the first mode of operation of the invention, viewed as a projection in the
x-y plane, wherein the depth of the cutting tool is varied with respect to the
plane of ruling. In this case, the tip of the cutting tool is rising as the tool
moves along the groove g from left to right in the Figure. In each cube, dihedral
angle d
1 terminates at or near the lower end of the groove root segment,
and dihedral angle d
2 terminates at or near the raised end of a groove
root segment. Then in all ten cubes illustrated the dihedral angles d
1 will
be slightly greater than 90°, and the dihedral angles d
2 will be
slightly less than 90°. It may be seen in FIG. 2 that the dihedral angles
d
1 and d
2 need not terminate exactly at groove root g, but
can terminate near the groove root. It also may be seen that the dihedral angle
d
3 is unaffected by the undulation of groove root g.
Second Mode—Horizontal Undulation
In a second mode of operation, the cutting tool is maintained at a constant attitude
and a constant depth with respect to the substrate, and during the ruling of at
least one vee-groove the cutting tool and the substrate oscillate with respect
to one another in a horizontal direction sidewise to the direction of ruling. This
will result in a groove of constant depth along the z-axis and substantially constant
groove angle, and wherein the groove root is an undulating curve in a plane parallel
to the x-y plane. It will be appreciated that the same effect can be achieved by
holding the substrate in a fixed position, and moving the tip of the cutting tool
in an undulating curve in the x-y plane, while maintaining the attitude of the
cutting tool constant, or by horizontal oscillation of the substrate while the
cutting tool moves in a straight line, or by simultaneous out-of-phase horizontal
movement of both the cutting tool and the substrate. The choice of whether to move
the cutting tool or the substrate, or both, will depend upon the design and functionality
of the graver that controls the cutting tool and the fixture that holds the substrate.
Such choices will be understood by those skilled in the ruling arts. Further, where
the aim of the cutting tool is constant, rather than tangent to the undulating
curve, there will be a very slight variation in groove angle, but this variation
will not have a significant effect on divergence for the amplitudes of groove undulations
applicable to the instant invention.
A first, inconsequential, effect of this second mode of operation of the instant
invention is the introduction of intersection errors in a pattern of ruled triangular
cube corner elements. That is, even if the other two groove sets are formed entirely
of straight grooves of the prior art, the horizontally undulating groove root made
in accordance with the second mode of operation of the instant invention will not
always intersect the vertices of the other two groove sets at their exact points
of intersection.
A second, consequential, effect of this second mode of operation of the invention
is that the three corner angles of a triangular cube corner are modified. The dihedral
angle of a triangle cube corner is made acute if it meets a triangle corner at
which the angle has been modified to be smaller. Similarly the dihedral angle of
a triangle cube corner is made obtuse if it meets a triangle corner at which the
angle has been modified to be larger. For a substantially equilateral triangle
cube corner, the change to a dihedral angle is approximately equal to the change
to the corresponding triangle corner angle divided by √3. These changes
to the dihedral angles of the affected cube corners result in a cube corner article
having broader divergence.
FIG. 11 summarizes aberrations resulting from the second mode when the undulation
is long enough that curvature within a single cube is insignificant. A male equilateral
triangle cube corner is illustrated as formed by three grooves. Each of the grooves
forming the triangle is shown rotated by a corresponding angular amount {umlaut
over (δ)}
1, {umlaut over (δ)}
2, {umlaut over
(δ)}
3 in the direction indicated. If a {umlaut over (δ)}
value is negative, then the groove is rotated oppositely. The three dihedral edges
are labeled with their angle errors, e
1, e
2, e
3,
that is, their deviations from perfect 90°: The dihedral angle errors are
then given by approximate equations (4)-(6).
##EQU3##
It will be appreciated that these equations require some adjustment for non-equilateral
triangle cube corners.
Third Mode—Rocking To and Fro
In a third mode of operation of the invention, at least one vee-groove is ruled
so that the movement of the tip of the cutting tool defines a straight groove root
parallel to the x-y plane, and the attitude of the cutting tool oscillates within
a plane containing the groove root. That is, the upper end of the cutting tool
oscillates to and fro parallel to the direction of the groove. The attitude of
the tool with respect to the substrate must be controlled as a function of the
position of the cutting tool along the groove. In this third mode of operation,
the center of oscillation is preferably the tip of the cutting tool. This mode
produces a groove having groove walls that undulate such that the included groove
angle expands and contracts along the length of the groove.
As the attitude of the cutting tool oscillates in accordance with this third
mode
of the invention, the included groove angle will vary continuously according to
the well-known adjustment technique for groove angle. A vee cutter with included
half-angle h, raked to angle R, cuts a groove of half-angle H, slightly larger
than h, as given by equation (8).
##EQU4##
Regardless of the direction of rake, consider R positive. Consider a small
variation, of magnitude much less than the rake itself, applied to R. The variation
ΔR may be positive or negative. When the rake angle changes to R+ΔR,
the included half-angle H of the groove changes by an amount ΔH given by
approximate equation (9).
Δ
H≈sin
h×cos
h×sin
R×ΔR (9)
Thus a rake undulation in the vee cutter produces an approximately proportional
undulation in the half-groove angle.
FIG. 3 illustrates ten cube corners along a portion of a groove that narrows
from left to right. The groove walls produced in accordance with this third mode
of the invention will intersect the x-y plane in undulatory curves corresponding
to the variations in included groove angle. These variations of the groove angle
along the length of the groove will result in variations in the dihedral angles
of the cube corner elements defined in part by the groove side walls. For each
of the cubes illustrated in FIG. 3, the two dihedral angles d
1 and d
2
are approximately equal, while their averaged value decreases from cube to
cube from left to right along the illustrated portion of groove g. The dihedral
angles d
3 remain unaffected. For substantially equilateral triangle
cube corners, increasing a groove's included angle has the effect of increasing
the cube dihedral angles formed in part from that groove by an amount equal to
the groove's increase divided by 2√2. Similarly, reducing a groove's included
angle reduces corresponding cube dihedral angles by the same 2√2 factor.
For a substantially equilateral triangle cube corner formed from two or more modified
grooves, the modifications to the cube dihedral angles are very nearly sums of
the effects of the separate groove modifications. The purposely introduced variations
in cube dihedral angles will result in a cube corner retroreflective article of
broader divergence.
FIG. 12 summarizes the aberrations due to the third mode when the undulation
is long enough that curvature within a single cube is insignificant. A male equilateral
triangle cube corner is illustrated as formed by three grooves. Each of the grooves
forming the triangle are designated by their half-groove angle errors,
##EQU5##
that is, their deviations from perfect 35.26° The three dihedral edges
are labeled with their angle errors, e
1, e
2, e
3,
that is, their deviations from perfect 90°. The dihedral angle errors are
then given by approximate equations (10)-(12).
##EQU6##
It will be appreciated that these equations require some adjustment for non-equilateral
triangle cube corners.
Fourth Mode—Rocking Side to Side
Whereas the foregoing third mode of operation can be understood as "forward-backward"
oscillation of the attitude of the cutting tool, the fourth mode of operation of
the invention uses "side-to-side" oscillation of the cutting tool. That is, the
tip of the cutting tool still defines a straight groove root parallel to the x-y
plane, while the attitude of the cutting tool oscillates within planes perpendicular
to the groove root. That is, the attitude oscillates transverse to the direction
of the ruling. In this method, while the magnitude of the groove angle itself will
not change along the length of the groove, the progressive change in attitude of
the cutting tool along the groove will result in variations in the angle between
the x-y plane and the groove walls that produce the cube corner elements on either
side of the groove. The dihedral angles of the cube corner elements defined in
part by the undulating groove surfaces will thereby be modified, similarly to the
third mode of operation of the invention. The identical equations (10)-(12) apply
to this fourth mode. These errors in the dihedral angles will in turn result in
a cube corner retroreflective article of broader divergence.
Curvature
It is contemplated that the undulatory method of the instant invention can give
deviations from flatness of the ruled cube corner faces on the order of 0.01°.
For the sizes of cube corners contemplated, this will be just a small fraction
of a wavelength of visible light. This effect will be insignificant, particularly
when the retroreflective article made from the cube corner array is sheeting, retroreflective
fabric, or traffic control devices.
A Difference Among the Modes
For either equations (1)-(3) (i.e., the first mode of operation) or equations
(4)-(6) (i.e., the second mode of operation) it is seen that equation (7) must hold.
e1+e2+e3=0 (7)
Thus, the first and second modes, applied as described, result in cube corners
with no net dihedral angle error. It is sometimes desireable to have net dihedral
angle error, as shown in Example 4 below. To produce net error by the first or
second modes, it is a simple matter to incorporate a bias error beneath the errors
produced by undulation. For example, the groove angles can be chosen different
from the perfect 70.53°, and these imperfect grooves can be subjected to vertical
(first mode) or horizontal (second mode) undulation. To determine the imperfect
groove angles desired for the bias, equations (10)-(12) may be solved for the groove
errors in terms of the dihedral errors. This gives new equations (13)-(15).
##EQU7##
Again, it will be appreciated that these equations require some adjustment
for non-equilateral triangle cube corners. The
##EQU8##
from equations (13)-(15) will serve as a biasing adjustment to the groove angles
which will then be undulated to produce the additional errors according to the
first or second mode. When using the third or fourth modes, no separate biasing
is necessary, since the biases discovered from equations (13)-(15) can be parts
of the
##EQU9##
of the undulation.
In each of the foregoing modes of operation of the invention, the result will
be that the dihedral angles of the cube corner elements formed by the grooves cut
by the inventive method will differ from one cube corner element to the next. This
non-equivalency of the cube corner elements will broaden, in a controlled manner,
the divergence of a retroreflective structure made from an array including the
non-equivalent cube corner elements.
It will be appreciated that the modes of operation described above can be used
in any combination. Where the cube corner elements are made from three intersecting
sets of approximately parallel grooves, the groove variations can be made in any
number of grooves in a set, and in one, two, or three sets. If grooves in more
than one set are varied, the result will be, in general, a randomized combination
of three dihedral angles in each cube.
Also each undulation can be a compound undulation, produced by a combination
of any number of the basic four modes, or any other mode, of oscillation. For example
the graver can be oscillating both vertically and horizontally at once (mode 1+mode
2), or it can be rocking to-and-fro and side-to-side at once (mode 3+mode 4), or
any of the nine other logical combinations. In compound modes, the amplitude and
length of undulations need not agree. In all such cases, the equation sets (1)-(3),
(4)-(6), and (10)-(12) can be separately applied to obtain the dihedral angle errors
for one cube, and the separate results added to determine the dihedral angle errors
of the compound mode undulation.
FIGS. 10-12 and corresponding equation sets (1)-(3), (4)-(6), and (10)-(12)
show the effect of the different modes of the invention on the aberration of a
single cube corner. The purpose of this invention is to enable the optical designer
to produce an array with a great variety of aberrated cube corners. Each triangle
cube corner has six contiguous neighbors (two at each of its vertices) with its
same orientation. Each of these neighbors of the first cube corner will in general
have quite different aberrations from the first, because these neighbors will be
formed from one groove in common with the first, which will have changed slightly,
and two grooves not in common with the first, which can have undulations out of
phase with the undulations of the corresponding grooves forming the first cube.
Those skilled in the art will recognize that not all geometrical light patterns
are achievable by the present invention. For one thing, as shown in the Yoder paper,
there are limits on the geometrical light distributions achievable by dihedral
aberrations. For another, not all distributions of dihedral aberration are achievable
by the present invention. Any technique that randomly combines three groove variations
must include all "cross terms", that is, for any part of the groove
1 variation,
any part of the groove
2 variation, and any part of the groove
3
variation, these three parts will occur together in forming a cube corner somewhere
in the array. This makes the invention more suitable for smooth spreadings of the
retroreflection than for precise arrangements of the light, as illustrated in Examples
3-6 described below.
COMPARATIVE EXAMPLES
All of the following Examples 1-6 will be based on an uncanted acrylic triangle
cube corner having 0.2 mm base dimension corresponding to 0.082 mm ruling depth.
Observation angles from 0° to 3° will be considered, while entrance angle
is fixed at 0°. The light source is assumed to have the spectral power distribution
of CIE Illuminant A, corresponding to an incandescent lamp, and the detector is
assumed to have the spectral sensitivity function CIE V(λ), corresponding
to human photopic vision.
All of the example outcomes are results of optical calculation. Undulatory modelings
are based on 1000 randomly generated cube corners. A shorthand has been employed
in describing sinusoidal undulations: N[sin] will be understood as undulation of
degree such that the single such groove contributes up to N arc minutes to dihedral
angle error.
Example 1
Prior Art
The first example is the perfect, unaberrated cube corner with retroreflected
light pattern shown in FIG. 4A. The geometric light pattern must have zero divergence,
but the cube size introduces noticeable diffraction, as summarized in Table 1.
The observation angle aspect of the diffraction light pattern is also shown in
FIG. 9 as the curve labelled "aberrationless". The curve in FIG. 9 is derived from
FIG. 4A.
The geometrical light pattern from a single cube corner consists of six punctal
spots, as explained in the cited work of P. R. Yoder, Jr. An unaberrated cube corner
produces its six spots in coincidence. In the following Examples 2-6 different
techniques are used to aberrate cube corners. For purposes of comparison, the aberrations
in each of these examples are chosen so that the geometrical light pattern in each
case has average divergence approximately 1.1°, where the divergence is measured
from the centroid of all the punctal spots. All illustrations show diffraction
light patterns rather than geometrical patterns, as this is what small cube corners
really produce.
Example 2
Prior Art
Example 2 represents the simplest way known to the prior art to make an array
of aberrated cube corners having average geometric divergence of 1.1°. Each
dihedral angle of each cube corner is made 14 minutes greater than the perfect
90°. FIG. 5 shows the resulting diffraction pattern. Most of the energy is
near the characteristic six punctal spots as described above, which diffraction
joins into a ring. FIG. 9 shows the ring as a hump in the curve labeled 14,14,14,
peaking at about 1.1°. The average intensity at 1.1° observation angle
is about eight times the intensity at 0° observation angle. A retroreflector
of this kind would not have road applications since it only functions well over
such a short range of observation angles that, at any distance, the full range
of vehicles (from trucks to cars) could not benefit. It could have specialized
applications for instrumentation.
Examples 3-6 illustrate rulings that can be made in accordance with the
instant invention. The illustrative rulings have planned aberrations that result
in broadened divergence profiles. In all these examples, the undulations are adjusted
to give the same average geometric divergence, 1.1°, as in prior art Example
2. FIGS. 6A and 6B illustrate smooth divergence spreadings achieved with the rulings
of Examples 3 and 4, while FIGS. 7A and 7B show some limited aiming capabilities
achieved with the rulings of Examples 5 and 6. Table 2 gives the ruling details
corresponding to Examples 3-6.
| TABLE 2 |
|
| Ruling details using vertical undulation |
| EXAMPLE |
SHORTHAND |
FIGURE |
FORM |
PITCH |
AMPLITUDE |
START |
|
| |
33[sin ± ½ sin2] |
|
sin ± ½ sin2 |
4 mm |
0.0097 mm |
random |
| 3 |
20[sin] |
6A |
sin |
4 mm |
0.0116 mm |
random |
| 4 |
9 + 15[sin] |
6B |
sin |
4 mm |
0.0087 mm |
random |
| 5 |
35[sin] on G3 |
7A |
sin |
4 mm |
0.0204 mm |
random |
| 6 |
24[sin] on G1 & G2 |
7B |
sin |
4 mm |
0.0135 mm |
random |
|
FIG. 9 illustrates the observation angularity achieved with the rulings of Examples
1-4, as well as the 33[sin±½ sin
2]
Example 3
For Example 3 sinusoidal vertical undulation is assumed according to Mode 1
of this invention. Ruling dimensions are shown in Table 2 under 20[sin]. FIG. 6A
shows the resulting calculated diffraction pattern. It has a gentle central peak
and is uncluttered with diffraction and aberration artifacts out to about 2°
observation angle. This is the beneficial result of the cube corners not being
alike as they were in Examples 1 and 2. FIG. 9 shows the observation angularity
of Example 3 as curve 20[sin]. Its intensity at 1.1° is only 37% that of prior
art Example 2 at 1.1°. However its intensity at 0° is 16 times that of
prior art Example 2, and its intensity at 2° is 2.2 times that of prior art
Example 2. Unlike Example 2, a retroreflector having the aberrated cube corners
of Example 3 is not limited to close applications, but performs well over a broad
distance range.
Example 4
For Example 4 sinusoidal vertical undulation according to Mode 1 is assumed
in combination with a bias of 9 arc minutes error on each dihedral angle. That
is, all the grooves are initially planned to make 9 arc minutes of dihedral error
and a sinusoidal undulation 15/20 as great as that of Example 3 is superimposed
on this. Ruling dimensions are given in Table 2 under 9+15[sin]. FIG. 6B shows
the resulting diffraction pattern, practically flat to about 1.5°. It must
be appreciated that the diffraction pattern illustrations are logarithmically scaled
and one step corresponds to difference in retroreflectance of approximately 2.5
times. FIG. 9 shows that the observation angularity of Example 4 is nearly flat
to about 1.5°. Example 4 could function as a close distance retroreflector
in road applications.
Example 5
Example 5 shows the effect of sinusoidal vertical undulation according to
Mode 1 but assumed for only one set of grooves, G3, of the three
sets of grooves G1, G2, G3 ruled to make triangular cube corners.
Ruling dimensions are given in Table 2 under "35[sin] on G3." Equations
(1)-(3) show how this affects two of the three dihedral angles. FIG. 7A shows the
diffraction pattern. This example illustrates how the method can produce patterns
with directed observation angularity. This is the reason this Example 5 is not
included among the curves of FIG. 9, which are averages over all directions.
Example 6
Example 6 shows the effect of sinusoidal vertical undulation according to
Mode 1 assumed for two of the three groove sets. Ruling dimensions are given
in Table 2 under "24[sin] on G1&G2." Equations (1)-(3) show how this
affects two of the three dihedral angles. FIG. 7B shows the diffraction pattern.
This example illustrates a different type of directed observation angularity that
can be achieved with the method of the instant invention.
It will be appreciated that while Examples 3-6 use Mode 1, using the dimensions
in Table 2, Mode 2 could have been used with different dimensions to give
identical results and Modes 3 or 4 to give almost identical results.
Preferably, the entire desired range of divergence will be provided over
a short increment of groove length, in order to avoid a spotty appearance. For
retroreflective road sign applications there should be no patches larger than about
4 mm diameter visibly different from adjacent patches. Practical un
