U.S. patent application number 16/484950 was filed with the patent office on 2019-11-28 for end milling metods for making microstructures, especially cube corner elements and articles comprising such microstructures.
The applicant listed for this patent is 3M INNOVATIVE PROPERTIES COMPANY. Invention is credited to Todd A. BALLEN, Christoper S. GRINDE, Nelson D. SEWALL, Kenneth L. SMITH.
Application Number | 20190358918 16/484950 |
Document ID | / |
Family ID | 61244741 |
Filed Date | 2019-11-28 |
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United States Patent
Application |
20190358918 |
Kind Code |
A1 |
SEWALL; Nelson D. ; et
al. |
November 28, 2019 |
END MILLING METODS FOR MAKING MICROSTRUCTURES, ESPECIALLY CUBE
CORNER ELEMENTS AND ARTICLES COMPRISING SUCH MICROSTRUCTURES
Abstract
The present disclosure relates to end milling methods for making
microstructures, and articles containing such microstructures. The
end milling process comprises rotating an end mill (1410r) around
an axis of rotation (1409) and translating the rotating end mill
(1410r) in a working surface (1407) to form a first recess (1420)
such that end mill (1410r) moves along an inclined path (1411)
relative to the working surface (1409), or to a reference plane
along which the working surface (1407) extends. The method forms
flat faces (1441a, 1441b) that meet along a rounded edge (1451),
the edge (1451) being inclined to the reference plane. Both
truncated cube corner elements (see for instance FIGS. 47 and 48)
and full cube corner elements (see for instance FIG. 56A) can be
made in a substrate (4705, 4805, 5605) by forming at least part of
an optical face of the cube corner element (4780, 4880, 5680) by
cutting the substrate (4705, 4805, 5605) with a rotating end mill
(1410r).
Inventors: |
SEWALL; Nelson D.; (Forest
Lake, MN) ; GRINDE; Christoper S.; (Baldwin, WI)
; SMITH; Kenneth L.; (White Bear Lake, MN) ;
BALLEN; Todd A.; (St. Paul, MN) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
3M INNOVATIVE PROPERTIES COMPANY |
St. Paul |
MN |
US |
|
|
Family ID: |
61244741 |
Appl. No.: |
16/484950 |
Filed: |
February 5, 2018 |
PCT Filed: |
February 5, 2018 |
PCT NO: |
PCT/US2018/016839 |
371 Date: |
August 9, 2019 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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62458812 |
Feb 14, 2017 |
|
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|
62515993 |
Jun 6, 2017 |
|
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62527511 |
Jun 30, 2017 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G02B 5/124 20130101;
B29D 11/00605 20130101; B23C 3/10 20130101; B23C 5/10 20130101;
B29L 2011/0091 20130101; B23C 2265/08 20130101 |
International
Class: |
B29D 11/00 20060101
B29D011/00; B23C 3/10 20060101 B23C003/10; B23C 5/10 20060101
B23C005/10; G02B 5/124 20060101 G02B005/124 |
Claims
1-10. (canceled)
11. A method of making a microstructure, comprising: providing a
substrate having a working surface, the working surface extending
along a reference plane; cutting the substrate with a rotating end
mill to form a recess in the substrate; and during the cutting,
causing the rotating end mill to move along a cutting path relative
to the substrate; wherein the cutting path includes a path portion
that is inclined relative to the reference plane further including
forming distinct first and second faces of the recess during the
cutting as the rotating end mill moves along the path portion
wherein the first and second faces are connected by a rounded
edge.
12. The method of claim 11, wherein the rounded edge extends along
the path portion.
13. The method of claim 11, wherein the rotating end mill has a
truncated base, and wherein the rounded edge is an artifact of the
truncated base.
14. The method of claim 11, wherein the recess has a perimeter, and
the rounded edge has a terminus at the perimeter.
15. The method of claim 11, wherein the recess has a base, and the
rounded edge has a terminus at the base.
16. The method of claim 11, wherein the recess has a perimeter and
a base, and the rounded edge has one terminus at the perimeter and
another terminus at the base.
17. The method of claim 11, wherein the rounded edge has a radius
of curvature of at least 10 micrometers as measured in a selected
cross-sectional plane.
18. The method of claim 11, wherein the rounded edge exhibits a
curvature of constant radius in a selected cross-sectional plane
but not in a cross-sectional plane that is perpendicular to both
the first and second faces.
19. The method of claim 11, wherein the rotating end mill is a
first rotating end mill, the cutting path is a first cutting path,
and the path portion is a first path portion, the method further
comprising: further cutting the substrate with a second rotating
end mill to enlarge the recess; during the further cutting, causing
the second rotating end mill to move along a second cutting path
relative to the substrate, the second cutting path including a
second path portion that is inclined relative to the reference
plane; and forming distinct third and fourth faces of the enlarged
recess during the further cutting as the second rotating end mill
moves along the second path portion.
20. The method of claim 19, wherein the third and fourth faces are
nominally flat.
21. The method of claim 19, wherein the third and fourth faces are
nominally mutually perpendicular.
22. The method of claim 19, wherein the second face and the third
face form a nominally flat first compound face.
23. The method of claim 22, wherein the second face and the third
face meet along a break line.
24. The method of claim 23, wherein the enlarged recess has a
perimeter, and the break line extends to the perimeter.
25. The method of claim 23, wherein the enlarged recess has a base,
and the break line extends to the base.
26. The method of claim 23, wherein the enlarged recess has a
perimeter and a base, and the break line extends from the perimeter
to the base.
27. The method of claim 19, wherein the first rotating end mill is
identical to the second rotating end mill.
28. The method of claim 27, wherein the first rotating end mill has
an orientation during the cutting that is the same as that of the
second rotating end mill during the further cutting.
29. The method of claim 27, wherein the first rotating end mill has
an orientation during the cutting that is different than that of
the second rotating end mill during the further cutting.
30. The method of claim 19, wherein the first rotating end mill is
different than the second rotating end mill.
31-252. (canceled)
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit under 35 U.S.C. .sctn.
119(e) of the following pending U.S. provisional applications, each
of which is incorporated herein by reference: U.S. Ser. No.
62/458,812 (Attorney Docket 78684US002), "End Milling Methods for
Making Microstructures, Tools Comprising Microstructures, and
Microstructures", filed Feb. 14, 2017; U.S. Ser. No. 62/515,993
(Attorney Docket 78943US002), "Non-Orthogonal Cube Corner Elements
and Arrays Thereof Made by End Milling", filed Jun. 6, 2017; and
U.S. Ser. No. 62/527,511 (Attorney Docket 78944US002), "Security
Articles Comprising Groups of Microstructures Made by End Milling",
filed Jun. 30, 2017.
FIELD
[0002] The present disclosure relates primarily to microstructures,
including sheeting, films, substrates, tools, and other articles
having at least one surface that contains one or more
microstructures, with particular application to such articles in
which the microstructures are made, or were made in a prior
generation article, at least in part using an end milling method of
material removal. The disclosure also pertains to related methods,
systems, articles, and components.
BACKGROUND
[0003] Articles in which microstructures are formed in a surface of
the article are known and have many uses and applications,
including mechanical applications such as abrasives, and optical
applications such as light directing films and other light
directing materials. One known category of microstructures is the
class of cube corner elements, which are used to make
retroreflective films.
[0004] Simply put, a cube corner element is a structure having a
set of three flat reflective faces or facets, referred to as
optical faces, the optical faces being grouped together and
oriented to be orthogonal to each other. With this geometrical
arrangement, incident light that strikes a first one of the faces
is reflected to a second such face, and then to the third
reflective face. The laws of reflection ensure that the light
reflected by the third face propagates in a direction that is
essentially opposite to the direction of the incident light.
[0005] Retroreflective films and materials are thus characterized
by the ability to redirect light, even light that is obliquely
incident on the film, back toward the originating light source.
Retroreflective sheeting is used widely for a variety of traffic
and personal safety uses. For example, retroreflective sheeting is
commonly employed in road signs, barricades, license plates,
pavement markers, and marking tape, as well as retroreflective
tapes configured for application to vehicles or clothing.
[0006] Microstructured sheeting, such as cube corner
retroreflective sheeting, is commonly produced by first
manufacturing a master mold that has a structured surface. The
structured surface of the master mold corresponds either to the
desired microstructured surface geometry of the finished sheeting
product, or to a negative (inverted) copy thereof, depending upon
whether the finished sheeting is designed to incorporate
microstructured protrusions, such as cube corner pyramids that jut
out of a surface, or microstructured indentations or recesses, such
as cube corner cavities that dip or sink into a surface, or both
microstructured protrusions and recesses. The master mold is then
replicated using any suitable technique such as conventional nickel
electroforming to produce tooling for forming sheeting by processes
such as embossing, extruding, or cast-and-curing. U.S. Pat. No.
5,156,863 (Pricone et al.) provides an illustrative overview of
processes for forming tooling used in the manufacture of cube
corner retroreflective sheeting. Known methods for manufacturing
the original master mold include pin bundling techniques, direct
machining techniques, and techniques that employ laminae.
[0007] In pin bundling techniques, a plurality of pins, each having
a geometric shape such as a cube corner element on one end, are
assembled together to form a master mold. U.S. Pat. No. 1,591,572
(Stimson) and U.S. Pat. No. 3,926,402 (Heenan) provide illustrative
examples. Pin bundling offers the ability to manufacture a wide
variety of cube corner geometries in a single mold, because each
pin is individually machined. However, such techniques are
generally considered to be impractical for making small cube corner
elements--e.g. those having a cube height less than about 1
millimeter--because of the large number of pins and the diminishing
size thereof required to be precisely machined and then arranged in
a bundle to form the mold.
[0008] In direct machining techniques, series of grooves are formed
in the surface of a planar substrate such as a metal plate to form
a master mold comprising microstructures, typically, truncated cube
corner elements. In direct machining, a large number of individual
faces are typically formed along the same groove formed by
continuous motion of a cutting tool. Thus, such individual faces
maintain their alignment throughout the mold fabrication procedure.
For this reason, direct machining techniques offer the ability to
accurately machine very small cube corner elements. A drawback to
direct machining techniques, however, has been reduced design
flexibility in the types of cube corner geometries that can be
produced, due to a reliance on grooves of constant depth to form
the faces of the microstructures, which in turn places restrictions
on the achievable optical characteristics of the finished cube
corner sheeting, such as the "total light return" of the sheeting,
discussed below.
[0009] In techniques that employ laminae, a plurality of thin
sheets or plates, referred to as laminae, are machined to have
grooves or other geometric shapes formed on one longitudinal edge
thereof, and are then assembled to form a master mold. Techniques
that employ laminae are generally less labor intensive than pin
bundling techniques because fewer parts are separately machined.
For example, one lamina can typically have about 400-1000
individual cube corner elements formed therein, in comparison to
one pin having only a single cube corner element. Illustrative
examples of techniques that employ laminae to make cube corner
sheeting and related microstructured articles can be found in EP 0
844 056 A1 (Mimura et al.), U.S. Pat. No. 5,981,032 (Smith et al.),
U.S. Pat. No. 6,015,214 (Heenan et al.), U.S. Pat. No. 6,257,860
(Luttrell et al.), and U.S. Pat. No. 7,156,527 (Smith). However,
techniques employing laminae have less design flexibility compared
to that achievable by pin bundling. For example, the linear or
stacked arrangement of the laminae tends to produce microstructured
elements arranged in rows, which therefore imposes limitations on
the design options available for forming microstructures.
SUMMARY
[0010] Cube corner elements have historically been made by pin
bundling, by direct machining, or by lamina-based fabrication
techniques. We have now developed a completely new technique for
making cube corner elements. The new technique employs a material
removal technology that has been known for many decades, but that
until now has been considered entirely unsuitable for forming cube
corner elements and similar structures. The known material removal
technology we employ is end milling. The new technique is not
limited to the fabrication of cube corner elements, but can be used
to make many other types of structures as well, including many
types of microstructures.
[0011] We disclose herein, among other things, methods of making a
microstructure, the methods including providing a substrate having
a working surface, and cutting the substrate with a rotating end
mill to form a recess in the substrate. During the cutting, the
rotating end mill is caused to move along a cutting path relative
to the substrate, and the cutting path includes a path portion that
is inclined relative to the working surface, or to a reference
plane (such as an x-y plane) along which the working surface
extends.
[0012] We also disclose methods of making a microstructure, the
methods including providing a substrate having a working surface,
and cutting the substrate with a rotating end mill to form a first
recess in the substrate. During the cutting, the rotating end mill
is caused to move relative to the substrate so as to form a first
and second face of the first recess, the first and second faces
defining respective first and second reference planes that
intersect along a line that is inclined relative to the working
surface, or to a an x-y plane along which the working surface
extends.
[0013] We also disclose methods of making a microstructure, the
methods including providing a substrate, cutting the substrate with
a first rotating end mill to form a recess in the substrate, and
further cutting the substrate with a second rotating end mill to
enlarge the recess. The first recess has a first face, and the
further cutting with the second rotating end mill forms a second
face of the enlarged recess. The second rotating end mill may in
some cases be identical to the first rotating end mill. The first
and second faces form a compound face of the enlarged recess.
[0014] We also disclose methods of making a cube corner element in
a substrate, the methods including cutting the substrate with at
least one rotating end mill along at least one cutting path that is
inclined relative to the substrate, or relative to an x-y plane
along which the substrate extends.
[0015] We also disclose methods of making a cube corner element in
a substrate, the methods including forming part of at least a first
optical face of the cube corner element by cutting the substrate
with a rotating end mill.
[0016] We also disclose articles that include a substrate having a
structured surface that extends along a reference plane, and a
feature in the structured surface. The feature, which can be a
recess or a protrusion, has a first face, and a second face
distinct from the first face, that connect to each other by a
rounded edge. The rounded edge extends along an axis that is
inclined at an angle .alpha. relative to the reference plane.
[0017] We also disclose articles that include a cube corner element
defined in a structured surface, the cube corner element including
a rounded edge that is inclined relative to the structured surface,
or to a reference plane along which the structured surface
extends.
[0018] We also disclose articles that include a cube corner element
defined in a structured surface, the cube corner element having a
compound face that includes a first face and a second face that
meet along a break line, the break line being inclined relative to
the structured surface, or to a reference plane along which the
structured surface extends.
[0019] Numerous related methods, systems, and articles are also
disclosed.
[0020] These and many other aspects of the present disclosure will
be apparent from the detailed description below. In no event,
however, should the above summaries be construed as limitations on
the claimed subject matter, which subject matter is defined solely
by the attached claims, as may be amended during prosecution.
BRIEF DESCRIPTION OF THE DRAWINGS
[0021] FIGS. 1, 2, and 3 are perspective views of substrates that
have working surfaces in which one or more microstructures can be,
or are, formed;
[0022] FIGS. 4A-4C depict aspects of fly cutting to contrast with
end milling, where FIG. 4A is a front view of an oversimplified,
stationary cutting tool, FIG. 4B is a front view of the cutting
envelope formed by rapidly rotating the cutting tool of FIG. 4A,
and FIG. 4C is a perspective view of a substrate in which a linear
V-groove has been formed by cutting into the substrate while moving
the rotating cutting tool relative to the substrate;
[0023] FIG. 5A is a front view of an oversimplified, stationary
cutting tool that is adapted for end milling, and FIG. 5B is a
front view of the cutting tool as it rapidly rotates or spins about
its longitudinal axis, forming a cutting envelope in the form of a
truncated cone;
[0024] FIGS. 5C and 5D are schematic side and perspective views,
respectively, of the cutting envelope formed by the rapidly
rotating end mill;
[0025] FIGS. 6A-6C are a sequence of schematic cross-sectional
views showing how a rotating end mill can form a simple blind hole,
cavity, or recess in a substrate;
[0026] FIG. 7 is a schematic view of the path followed by the
rotating end mill in connection with the sequence of FIGS.
6A-6C;
[0027] FIGS. 8 and 9 are schematic top and perspective views,
respectively, of the substrate and blind hole of FIG. 6C;
[0028] FIGS. 10A-10D are a sequence of schematic side views showing
how a rotating end mill can form a simple elongated recess in a
substrate;
[0029] FIG. 11 is a schematic view of the path followed by the
rotating end mill in connection with the sequence of FIGS.
10A-10D;
[0030] FIGS. 12 and 13 are schematic top and perspective views,
respectively, of the substrate and elongated recess FIG. 10D;
[0031] FIGS. 14A-14B are a sequence of schematic cross-sectional
views showing how a rotating end mill can form another type of
recess in a substrate;
[0032] FIG. 15 is a schematic top view of the substrate and cutting
envelope of FIG. 14B, where the cut line 14B-14B shows the
cross-sectional plane of FIG. 14B;
[0033] FIG. 16 is a schematic cross-sectional view of the substrate
through cut line 16-16 in FIG. 15;
[0034] FIG. 17A is a schematic view of the path followed by the
rotating end mill in connection with the sequence of FIGS. 14A-14B,
and FIG. 17B is a schematic view of an alternative path that can be
used to make the very same recess;
[0035] FIG. 18 is a schematic top view similar to FIG. 15, but
omitting the cutting envelope so that the entire recess can be
seen;
[0036] FIG. 19 is a schematic cross-sectional view through the cut
line 19-19 in FIG. 18;
[0037] FIG. 20A is a partial schematic cross-sectional view of the
substrate of FIG. 14B through cut line 20A-20A, and FIG. 20B is a
similar view but through the cut line 20B-20B;
[0038] FIG. 21 is a perspective view of the substrate of FIGS. 14B,
15, 16, 18, 19, 20A, and 20B;
[0039] FIG. 22 is a perspective view of a substrate whose
structured surface is a negative replica of that of FIG. 21, thus
providing a structure that is a protrusion rather than a
cavity;
[0040] FIGS. 23, 24, 25, 26, and 27 are schematic views of
alternative paths of the rotating end mill that can be used to make
alternative recesses;
[0041] FIGS. 28, 29, 30, 31, and 32 are schematic top views of
cavities that can be formed in substrates using the paths of FIGS.
23, 24, 25, 26, and 27, respectively;
[0042] FIG. 33 is a schematic perspective view of a non-tilted and
a tilted cutting envelope of a rotating cutting tool in the context
of a coordinate system, illustrating a polar angle .theta. and an
azimuthal angle (I);
[0043] FIGS. 34, 35, and 36 are schematic side or cross-sectional
views of cutting envelopes similar to that of FIG. 5C but for end
mills of different design;
[0044] FIG. 37 is a schematic view of a path for a rotating end
mill that includes two segments that lie in different vertical
planes, such a path being useful for making other types of
recesses;
[0045] FIG. 38 is a schematic top view of a cavity that can be
formed in a substrate using a path such as that of FIG. 37, the
cavity including two faces that form a compound face;
[0046] FIG. 39 is an isolated front view of the compound face of
FIG. 38;
[0047] FIGS. 40-42 are side or edge views of different compound
faces, illustrating different degrees of alignment between the
constituent faces of the compound face, and the resulting different
break lines between such constituent faces;
[0048] FIG. 43 is a schematic perspective view of two planes with
respective surface normal vectors, the planes representing two
constituent faces of a compound face;
[0049] FIG. 44 is a perspective view of the substrate and recess of
FIG. 38;
[0050] FIG. 45 is a perspective view of a substrate whose
structured surface is a negative replica of that of FIG. 44, thus
providing a structure that is a protrusion rather than a
cavity;
[0051] FIG. 46 is a schematic view of a path for a rotating end
mill that includes three segments that lie in different vertical
planes, such a path being useful for making still other types of
recesses, including cube corner recesses;
[0052] FIG. 47 is a schematic top view of a cavity that can be
formed in a substrate using a path such as that of FIG. 46, the
cavity including three pairs of faces that form three compound
face, the cavity being configured as a truncated cube corner
cavity;
[0053] FIG. 48 is a perspective view of the substrate and recess of
FIG. 47;
[0054] FIG. 49 is a perspective view of a substrate whose
structured surface is a negative replica of that of FIG. 48, thus
providing a structure that is a protrusion rather than a
cavity;
[0055] FIG. 50 is a schematic plan view of a structured surface
comprising two recesses that form a matched pair of truncated cube
corner elements, such recesses being made using the disclosed end
milling techniques;
[0056] FIG. 51 is a schematic plan view of a structured surface
comprising a plurality of matched pairs of truncated cube corner
elements;
[0057] FIG. 52 is a schematic plan view of a structured surface
made using the disclosed end milling techniques, the structured
surface comprising two recesses that form a matched pair of
truncated, backward canted cube corner elements;
[0058] FIG. 53 is a schematic plan view of a structured surface
similar to that of FIG. 52, but where the matched pair of truncated
canted cube corner elements are forward canted;
[0059] FIGS. 54-56A are a group of schematic plan views of
different substrates whose working surfaces are shaped to form
related sets of recesses, the group of figures demonstrating how
the disclosed end milling techniques can be used to form a PG cube
corner element in a substrate;
[0060] FIG. 56B is a schematic plan view of a substrate similar to
that of FIG. 56A, and in which the same PG cube corner element is
formed;
[0061] FIG. 57 is a schematic plan view of a structured surface in
which resides a plurality or array of PG cube corner elements made
by the disclosed end milling techniques;
[0062] FIGS. 58-59 are a pair of schematic plan views similar to
those of FIGS. 54-56A, but demonstrating how to make a canted PG
cube corner element;
[0063] FIG. 60 is a schematic plan view of another PG cube corner
element;
[0064] FIG. 61 is a schematic plan view of a matched pair of PG
cube corner elements like that of FIG. 60, and FIG. 62 is a
perspective view thereof;
[0065] FIGS. 63-67B are a group of schematic plan views that
demonstrate how to make PG cube corner elements like those of FIGS.
60-62; and
[0066] FIGS. 68A, 68B are schematic plan views of structured
surfaces, the figures demonstrating how to make PG cube corner
elements similar to those of FIGS. 60-62, 67A, and 67B (but in the
form of cavities rather than protrusions) by judiciously
rearranging the positions of truncated cube corner elements made by
end milling.
[0067] In the figures, like reference numerals designate like
elements.
DETAILED DESCRIPTION
[0068] We have developed a completely new technique for making cube
corner elements, and the technique is also useful for forming
structures and microstructures other than cube corner elements. The
new technique employs a material removal technology that is known
but that has, until now, been considered entirely unsuitable for
forming cube corner elements and the like. The known material
removal technology we employ is end milling.
[0069] With the new technique, end milling can be used to make
virtually any type of cube corner element, including not only
"truncated" cube corner elements but "full" or "preferred geometry"
(PG) cube corner elements, including those that have a perimeter
shape in plan view that is hexagonal, pentagonal, or quadrilateral,
as discussed further below. Whether truncated or PG, the cube
corner elements can also be canted or uncanted, and symmetrical or
asymmetrical, and large or small, as also discussed below. The new
technique allows the cube corner elements to be individually
tailored and individually cut into the working surface of a
substrate, which allows for almost limitless design flexibility.
For example, in an array or cluster of adjacent microstructures
made by the new technique, one cube corner element can be large and
uncanted, and an adjacent cube corner element can be small and
canted in a first direction, and still another adjacent cube corner
element can be of medium size and canted in a different second
direction; and furthermore, these adjacent, neighboring cube corner
elements need not be constrained to require, for example, that an
optical face of one cube corner element be coplanar with optical
faces of its adjacent cube corner elements. Rather, with the new
technique, a cube corner element may be designed to have no optical
faces that are coplanar (not even nominally coplanar) with any of
the optical faces of its adjacent or neighboring cube corner
element(s).
[0070] The new technique is not limited to the fabrication of cube
corner elements, but can also be used to make other types of
structures and microstructures, some of which are shown or
discussed below, and others of which will be readily apparent to
the reader by simple combinations and extensions of the disclosed
embodiments. Some of these other types of structures may be similar
in appearance to a cube corner element, while others may be very
different. These other structures may be useful in optical
applications, e.g. transparent or reflective optical films, or in
non-optical applications, including but not limited to mechanical
applications such as abrasive articles, non-skid articles, and so
forth. In most cases, the structures include at least one flat face
that has been formed (whether in the finished article itself, or in
a prior generation article related to the finished article by one
or more replication operations) using the new technique by
operation of an end mill. In some cases, a single structure or
microstructure can have two, or three, or four such faces, or more.
The structures, including structures that are cube corner elements,
can also have one or more compound faces, a compound face being
made up of two individual faces that are coplanar or nearly
coplanar, and that meet or come together along a break line.
[0071] The new technique can be used entirely by itself to form one
or more structures or microstructures in a substrate, or it can be
combined with known techniques. For example, some microstructures
can be made with the new technique, while other microstructures in
the same substrate are made with a direct machining technique or
another known technique. Furthermore, a portion, e.g. one face, of
one or more microstructures can be made with the new technique,
while the remainder of such microstructure(s), e.g. another face,
can be made with the direct machining technique.
[0072] In cases where it is desired to form numerous
microstructures in a substrate with the new technique, each
microstructure can be formed individually using end milling.
Depending on the sizes and types of microstructures, and the size
of the substrate, the process of individually forming the entire
array of microstructures may involve many hundreds, or thousands,
or even millions of extremely precise individual cutting steps
using one or more end mills, as described further below. Such an
undertaking, which may be extremely laborious and time consuming if
performed by hand with a human machinist, can be carried out in a
reasonable period of time with high speed, high precision, and high
quality, using modern computer controlled machines, or other
automated cutting systems and equipment.
[0073] Before proceeding further, we pause here to say a few words
about some of the structures, microstructures, and cube corner
elements discussed herein, and related features.
[0074] In most cases, the structures discussed herein, whether a
cube corner element, a microstructure, or both, are protrusions or
cavities that have precisely engineered shapes, and that exist in
the context of a larger surface of which they are a part, the
larger surface often referred to as a structured surface, or as a
working surface in the case of a machinable substrate. In some
cases, the structure can be a protrusion that exists within or
occupies a cavity, see e.g. FIG. 56A or 56B below. The disclosed
structures typically include one or more individual faces or facets
which are flat, and are arranged in a specific, engineered
orientation relative to other faces or surfaces of the
structure.
[0075] In this regard, a face or other surface may be said to be
"flat" if it is (a) precisely flat, e.g., planar to within standard
or applicable manufacturing tolerances, or (b) substantially flat,
e.g., deviating from a best fit reference plane by amounts that are
considered small in the intended application, e.g., deviating from
such a plane by no more than a few micrometers, or by no more than
.+-.1, .+-.0.5, .+-.0.1, .+-.0.05, or .+-.0.005 micrometers, or (c)
functionally flat, e.g., the face functions in a manner that is the
same as or similar to that of a precisely flat or substantially
flat face or surface in the intended application. In some cases, a
face or other surface may be deliberately engineered to have a
small amount of curvature, texture, or other deviation from
flatness in order to provide a given significant optical or
mechanical functionality. In such cases, the face may still be
considered to be nominally flat for purposes of this document if
the face satisfies any of the preceding conditions (a)-(c), or if
the deviation from flatness is small enough that the structure of
which the face is a part still retains a substantial amount of the
functionality it would have had if the face were actually flat. A
nominally flat face may thus refer to, among other things, a face
that is precisely flat, or substantially flat, or functionally
flat.
[0076] A microstructure may refer generally to a structure that is
small in the context of the particular application. For example, a
microstructure may be a structure that has at least one
characteristic dimension, such as a depth, height, length, width,
or diameter (as broadly defined, not limited to circular features),
that is on the order of 1 millimeter or less, e.g., 2 mm or less,
or 1 mm or less, or 0.5 mm or less, or even 200 micrometers or
less, or 100 micrometers or less, and typically also at least 5,
10, 15, or 20 micrometers in size. In some cases, two, three, or
more of these characteristic dimensions of a structure may satisfy
any of these size conditions, and any combination of these size
conditions.
[0077] A microstructure may also, or alternatively, be as described
in U.S. Pat. No. 4,576,850 (Martens). For example, the
microstructures may generally be discontinuities such as
projections and indentations in the surface of an article that
deviate in profile from an average center line drawn through the
microstructure such that the sum of the areas embraced by the
surface profile above the center line is equal to the sum of the
areas below the line, the line being essentially parallel to the
nominal surface (bearing the microstructure) of the article. The
heights of the deviations may typically be about .+-.0.005 to
.+-.750 micrometers, as measured by an optical or electron
microscope, through a representative characteristic length of the
surface, e.g., 1 to 30 cm. The average center line can be plano
(i.e., planar or straight), or concave or convex (whether or not
aspheric), or combinations thereof. Articles where the deviations
are of low order, e.g., from .+-.0.005 to 0.1 micrometer or,
preferably, to .+-.0.05 micrometers, and where the deviations are
of infrequent or minimal occurrence, i.e., where the surface is
free of any significant discontinuities, can be considered to have
an essentially "smooth" surface, as well as a "flat" surface if the
average center line is plano. Other articles may have deviations of
high order, e.g., from .+-.0.1 to .+-.750 micrometers, and
attributable to microstructures comprising a plurality of
utilitarian discontinuities which may be the same or different, and
spaced apart or contiguous in a random or ordered manner.
[0078] As mentioned above, a cube corner element can be simply
described as a structure having a set of three flat reflective
faces or facets, referred to as optical faces, the optical faces
being grouped together and oriented to be orthogonal to each other,
such that incident light that strikes a first one of the faces is
reflected to a second such face, and then to the third reflective
face. Following the laws of reflection, the light reflected by the
third face propagates in a direction that is essentially opposite
to (anti-parallel to) the direction of the incident light. Such a
structure generally resembles the corner of a cube, hence the term
"cube corner" element. The term cube corner element also applies to
a structure having a set of three faces that, though not themselves
reflective, can be made reflective, or that can be replicated in a
positive or negative replica to provide the three reflective faces
referred to above, and in that case the three original
(non-reflective) faces are still considered to be optical faces for
purposes of the present disclosure. The term cube corner element
also applies to structures in which the three optical faces are not
precisely mutually perpendicular, but where small deviations from
orthogonality, typically less than 1 degree, or less than 0.5
degrees, or less than 0.2 degrees, or less than 0.1 degree, or on
the order of 1 minute of arc (0.0167 degrees), exist between any
two of the three faces such that the retroreflected light is
slightly spread out angularly relative to a straight line that
connects the light source to the cube corner element. Such a
deviation from precise orthogonality is referred to as a dihedral
angle error, or more briefly as an angle error. Faces that are
described herein as being nominally orthogonal (or substantially
orthogonal) refer to faces that are either precisely orthogonal, or
that deviate from orthogonality by a dihedral angle error.
[0079] The angle errors are large enough or substantial enough to
produce a commercially useful change in the divergence profile or
other light return pattern created by the cube corner element(s),
but not so large that the cube corner element(s) cease to
effectively function as retroreflector(s). In cases of most
commercial interest, the angle errors fall within a range whose
lower limit is 1 or 2 minutes of arc, and whose upper limit is 40
minutes of arc (useful in highway or traffic sign sheeting,
producing up to a 2 degree deviation of light away from the
direction of perfect retroreflection) or 70 minutes of arc (useful
in shorter-viewing distance applications such as license plates or
security features, producing up to a 4 degree deviation of light)
or 120 minutes of arc (2 degrees of arc) (useful in certain display
applications, producing up to an 8 degree deviation of light).
[0080] Some cube corner elements include, in addition to the set of
three optical faces, one or more other faces that have a very
different orientation from any of the optical faces. Such other
faces are referred to as non-optical faces of the cube corner
element.
[0081] Discussions of cube corner elements often make reference to
dihedral edges, non-dihedral edges, and dihedral angles. A dihedral
edge of a cube corner element is an edge formed at the intersection
of any two of the optical faces of that cube corner element. A
non-dihedral edge is any other edge of the cube corner element,
including for example an edge formed by an optical face of the
given cube corner element and an optical face of an adjacent cube
corner element, or an edge formed by an optical face and a
non-optical face of the given cube corner element. Edges--both
dihedral edges and non-dihedral edges--of conventional cube corner
elements are classically shown or described as being sharp in
nature. In the present disclosure, some edges of a cube corner
element or other structure may be sharp, while other edges may be
curved or rounded.
[0082] A dihedral angle is the internal angle between two given
non-parallel planes, which planes may represent two given faces of
a disclosed structure or structured surface, such internal angle
being measured in a reference plane that is perpendicular to both
of the given planes. Stated differently, the internal angle
representing the dihedral angle between the given planes is
measured between a first and second line that intersect at a point,
where the first line lies in one of the given planes, the second
line lies in the other of the given planes, and where the first and
second lines are each perpendicular to a line of intersection of
the given planes. If two planes (or faces) are parallel, the
dihedral angle between them is zero. The angle errors mentioned
above in relation to non-orthogonal cube corner elements generally
refer to small deviations of a dihedral angle from 90 degrees,
unless otherwise clearly indicated.
[0083] The three dihedral edges of a cube corner element come
together or intersect to form a vertex of the cube corner element.
Depending on the nature of the edges and other design factors, the
vertex may be a sharp point, while in other cases it may be blunt,
for example, it may be a small flat area or other small area.
Furthermore, the vertex may be a local maximum or a local minimum
of the structured surface, depending on whether the cube corner
element juts out of the surface, or sinks into the surface. The
vertex is a local maximum, e.g. an apex or peak, when the cube
corner element is a protrusion, i.e., where solid material exists
in the space between the three optical faces. The vertex is a local
minimum, e.g. a base or nadir, when the cube corner element is a
cavity or recess, i.e., where vacuum, air, or a non-solid material
exists in the space between the three optical faces.
[0084] Each cube corner element defines an axis, commonly referred
to as a symmetry axis or a cube axis of the cube corner element.
The symmetry axis passes through the vertex and trisects the
structure, forming an equal angle with all three optical faces of
the cube corner element. A cube corner element whose symmetry axis
is perpendicular to the structured surface of which the cube corner
element is a part is referred to as an "uncanted" cube corner
element. Otherwise, if the symmetry axis is tilted relative to the
structured surface, the cube corner element is said to be
"canted".
[0085] Cube corner elements are also traditionally classified in
terms of the configuration of the cube corner element's
non-dihedral base edges, as well as the orientation of the cube's
symmetry axis. A "truncated" cube corner element is one in which
(a) the base edges of all three optical faces are coplanar
(including substantially coplanar), typically lying in a plane that
is parallel to the structured surface, or (b) such base edges of
the three optical faces are all parallel to such plane, or (c) both
(a) and (b). These base edges form a substantially
triangular-shaped perimeter in plan view, which perimeter is then
referred to as a "base triangle" of the truncated cube corner
element. A truncated cube corner element that is uncanted produces
a base triangle that is equilateral, i.e., each corner of the base
triangle has an interior angle of precisely 60 degrees. See for
example U.S. Pat. No. 3,712,706 (Stamm). A cube corner element
whose symmetry axis is tilted in one way is said to be "forward
canted" or "positive canted", if the base triangle associated with
such element has only one interior angle greater than 60 degrees.
See for example U.S. Pat. No. 4,588,258 (Hoopman). A cube corner
element whose symmetry axis is tilted in the opposite way is said
to be "backward canted" or "negative canted", and has an associated
base triangle in which two of the three interior angles are greater
than 60 degrees. See for example U.S. Pat. No. 5,565,151
(Nilsen).
[0086] In contrast to truncated cube corner elements, other cube
corner elements, sometimes referred to as "full cube corner
elements" (or "full cubes") or "preferred geometry (PG) cube corner
elements" (or "PG cubes") may be designed such that at least one
non-dihedral edge of one optical face is not coplanar with
non-dihedral edges of the other two optical faces. A PG cube corner
element that is part of a structured surface of cube corner
elements may alternatively or additionally be described as a cube
corner element having at least one non-dihedral edge that: (1) is
nonparallel to a reference plane along which the structured surface
extends; and (2) is substantially parallel to an adjacent
non-dihedral edge of a neighboring cube corner element. A full cube
may refer to a cube corner element that (1) is substantially free
of vertical or near-vertical optical and non-optical faces, and (2)
has a percent active area greater than 90% at normal incidence or
at another design angle of incidence (and where we assume the
illumination axis and the observation axis are collinear). A
truncated cube corner element cannot satisfy this combination of
conditions.
[0087] Some such PG and full cube corner elements have a perimeter
in plan view that is hexagonal in shape, while others are
pentagonal, and still others are quadrilateral, including but not
limited to those having the shape of a parallelogram, rectangle, or
square. Other PG and full cubes can have still other plan view
perimeter shapes. The shapes of one or more of the optical faces
(each such shape defined in the plane of its respective optical
face) can also be used to identify PG and full cube corner
elements, or to distinguish them from truncated cube corner
elements which have triangular faces. Some PG and full cube corner
elements may have at least one optical face that is rectangular,
square, trapezoidal, or pentagonal, for example. In most cases, a
given PG cube corner element is also a full cube corner element,
and vice versa, but exceptions may exist. In the discussion that
follows, cube corner element(s) referred to as a PG cube corner
element(s) is or are assumed to also be full cube corner
element(s), and vice versa, unless otherwise indicated.
[0088] Regardless of which of these perimeter shapes or face shapes
are selected, suitably designed and proportioned PG and full cube
corner elements are known to exhibit a higher total light return,
e.g. when used in cube corner retroreflective sheeting, in
comparison to truncated cube corner elements. Full cubes or PG
cubes can be uncanted or canted in analogous fashion to truncated
cube corner elements. See for example U.S. Pat. No. 6,015,214
(Heenan et al.).
[0089] With regard to light that is initially incident on a given
cube corner element or array of such elements along an axis of
incidence, the "total light return" for such element(s) is the
fraction of such incident light that is reflected back along the
axis of incidence within a specified half angle thereof, e.g. 4
degrees. The total light return can also be considered as the
product of the percent active area of the cube corner element(s)
and the normalized retroreflected ray intensity, for normally
incident light, or for another specified orientation of the
incident light. By way of example, in the limit where the
normalized retroreflected ray intensity is 1.0, the maximum
theoretical total light return for the truncated cube corner
geometry depicted in U.S. Pat. No. 3,712,706 (Stamm) for normally
incident light is 67%, whereas the maximum theoretical value is
100% for some PG cube corner elements, discussed below.
[0090] Returning now to the discussion of our new approach for
making cube corner elements and other structures, FIGS. 1, 2, and 3
show perspective views of different representative substrates 105,
205, 305 in which the disclosed structures or microstructures can
be, or are, formed. Each of these substrates has various outer
surfaces, one of which may be designated a working surface (or
structured surface) insofar as it is configured, or at least
suitable, for precision material removal machining operations
including at least end milling. The substrate 105 of FIG. 1 has
side surfaces 106, one of which is a major surface and is
designated a working surface 107. The substrate 205 of FIG. 2 has
side surfaces 206, one of which is a minor surface and is
designated a working surface 207. The substrate 305 of FIG. 3 has
the form of a right circular cylinder having a longitudinal
rotational axis 309, and as such has end surfaces and a
circumferential surface, the circumferential surface being
designated a working surface 307. The substrates 105, 205, 305 are
made of machinable materials, or, at least a layer or other portion
of such substrates located at the working surfaces thereof is made
of machinable materials. Desirable machinable materials include
those capable of being machined with an end mill cleanly without
burr formation, with low ductility and low graininess, and the
ability to maintain dimensional accuracy after the material removal
operation. Depending on the desired feature size and other factors,
a variety of machinable plastics or metals may be used. Suitable
plastics include thermoplastic or thermoset materials such as
acrylics or other materials. Machinable metals include aluminum,
brass, copper, electroless nickel, and alloys thereof. Preferred
metals include non-ferrous metals. The selected metal may be formed
into sheets or layers by for example rolling, casting chemical
deposition, electro-deposition, or forging. Preferred machining
materials are typically chosen to minimize wear of the end mill
cutting tool.
[0091] In many of the figures that follow, microstructures or other
features are shown in the context of a Cartesian xyz coordinate
system with mutually orthogonal x-, y-, and z-axes. Such Cartesian
coordinate systems are likewise included in FIGS. 1 and 2. However,
any other suitable coordinates systems, or no coordinate system,
may also be used as desired. The depicted coordinate systems are
not intended to be limiting in any way, but are included for ease
of reference, for visualization, and for comparison. In most if not
all cases, the coordinate system is oriented such that the z-axis
is perpendicular to (and pointing directly away from) the working
surface, or perpendicular to a reference plane along which the
working surface or the structured surface extends, such reference
plane then corresponding to an x-y plane defined by the x- and
y-axes of the coordinate system. Again, this orientation is for
reference and ease of description only, and should not be construed
as limiting. In many of the figures, the x-axis is oriented such
that, in cases where a tool path has an inclined path portion, the
path portion lies in the x-z plane. But other orientations can of
course also be used.
[0092] The z-axis of the coordinate system, as well as the
substrate, may have any desired orientation relative to the
direction of gravity, for example, the z-axis may point in the
opposite direction as gravity (e.g., "upwards"), or in the same
direction as gravity (e.g., "downwards"), or in any other
direction. Further in this regard, in the description below,
reference is made to one or more "vertical planes". Unless clearly
indicated to the contrary, such a vertical plane is to be
understood in relation to the working surface or structured surface
at issue, not necessarily to gravity; a vertical plane thus refers
to a plane that contains the normal vector of the working surface
or structured surface (hence also containing the z-axis when
oriented parallel to such normal vector), regardless of the
direction of gravity relative to such surface, even though, in many
cases, the normal vector and the z-axis are aligned with the
gravitational axis. Terms such as "up", "down", and the like should
also be interpreted flexibly and in a non-limiting way.
[0093] In the case of a non-flat working surface or structured
surface, such as the simply curved working surface 307 of FIG. 3,
such a surface can be conceptually subdivided into a multitude or
grid of small localized regions. Each localized region is a closer
approximation of a flat surface, and a local Cartesian xyz
coordinate system may then be assigned to any such region by
orienting the z-axis to be normal or perpendicular to the working
surface at that position.
[0094] Material can be selectively and precisely removed from the
working surfaces of the substrates 107, 207, 307 using one or more
rotating end mills as described further below to create in such
substrates any or all of the cube corner elements or other
microstructures disclosed herein.
[0095] FIGS. 4A-4C show, in simplified schematic form, aspects of
fly cutting, in order to draw a distinction between fly cutting and
end milling. FIG. 4A shows a front view of a stationary cutting
tool 408 made for fly cutting. The fly cutting tool 408 includes a
cutting head 410 attached to a shaft, the tool mounted in such a
way as to allow for rotation about a transverse rotational axis
409. The cutting head 410 is V-shaped with a sharp point or vertex.
A motor (not shown) engages the fly cutting tool 408 and spins it
rapidly about the axis 409. The rapidly rotating cutting tool 408r
includes a rapidly rotating, ring-shaped cutting head or cutting
envelope 410r as shown in FIG. 4B. The rotating cutting tool can be
used to cut a V-groove into the working surface of a substrate as
shown in FIG. 4C. There, a machinable substrate 405 has outer
surfaces 406, one of which is designated a working surface 407. A
straight V-groove 416 is cut into the working surface 407 by the
cutting envelope 410r as the rotating cutting tool 408r is caused
to move relative to the substrate 405 along a straight path 415 as
shown. The V-shaped profile of the cutting head 410, and of the
cutting envelope 410r, is copied in the V-groove 416. Direct
machining of (truncated) cube corner elements involves repeating
this fly cutting process to form a whole set of parallel V-grooves
in the substrate 405, and then forming in similar fashion two other
sets of parallel V-grooves that intersect the first set such that
the resulting faces (remnants of V-groove side surfaces) formed by
the fly cutting produce an array of cube corner elements.
[0096] Here and elsewhere in the present disclosure, whenever a
cutting tool, whether a fly cutting tool or an end mill tool, is
described as being moved or caused to move, e.g., caused to move
relative to a substrate, the reader will understand that such
motion may be carried out by (a) holding the substrate stationary
and translating the (rotating) cutting tool, or (b) holding the
(rotating) cutting tool stationary and translating the substrate,
or (c) translating both the (rotating) cutting tool and the
substrate, such that a relative motion is set up between the
cutting tool and the substrate along the designated path.
[0097] FIGS. 5A-5D show, in simplified schematic form, aspects of
end milling. FIG. 5A shows a front view of a stationary cutting
tool 508 made for end milling. The end milling tool 508 includes a
cutting head 510 attached to a shaft, the tool mounted in such a
way as to allow for rotation about a longitudinal rotational axis
509. The cutting head 510 has a cutting edge that is tapered, but,
for reasons that will become apparent, the taper is truncated at
the end rather than terminating at a sharp point. The taper can be
characterized by an angle, referred to herein as a tool half angle
(THA), between the tapered cutting edge and the rotational axis
509. A motor (not shown) engages the end milling tool 508 and spins
it rapidly about the axis 509. The rapidly rotating cutting tool
508r includes a rapidly rotating, cone-shaped cutting head or
cutting envelope 510r as shown in FIG. 5B. Enlarged views of the
cutting envelope 510r are shown in FIG. 5C (side or cross-sectional
view) and FIG. 5D (perspective view).
[0098] The enlarged FIGS. 5C and 5D illustrate more clearly that
the rotating end mill cutting envelope 510r actually has the shape
of a cone whose end or tip is truncated. In an end milling
operation, the rotating cutting tool needs to be capable of being
forced down onto, and into, a substrate to remove any material that
exists in the path of the cutting envelope 510r. One can readily
appreciate from this that, in order to make the cutting envelope
sharp and not truncated, the tip of the cutting head 510 would have
to be an extremely sharp, fine point, similar to the tip of a
needle. But such a delicate pointed tip would be quickly damaged or
destroyed in use as the rotating tool contacts and cuts into the
substrate. Therefore, for practical reasons, the end mill cutting
head 510 is truncated, and the cutting envelope 510r formed by the
rotating cutting head is similarly truncated at an end or bottom
510rb of the envelope. Besides the bottom 510rb, the cutting
envelope 510r also has a top 510rt of diameter Dtop, and a side
510rs in the form of a portion of a cone. The bottom 510rb has a
diameter Dbot. The side 510rs makes an angle THA (tool half angle)
relative to the axis 509, THA also being the half angle of the
stationary cutting head 510. The side 510rs makes an angle THAC
(tool half angle complement, such that THAC+THA=90 degrees)
relative to the transverse plane or line perpendicular to the axis
509. Because the envelope 510r is formed by rotation about the axis
509, the envelope 510r exhibits rotational symmetry about the axis
509, and the top 510rt and the bottom 510rb are both circular in
shape, i.e., of constant radius as measured from the axis 509.
[0099] The end milling tool 508 and its dimensions and other
features can be selected as desired from any of the numerous end
milling tools currently available in the marketplace, or it can be
specially ordered to exacting specifications, within limits, from
existing or future tool making companies, or otherwise specially
made. The dimensions Dtop, Dbot, THA, and THAC of the cutting
envelope 510r can therefore also be selected or tailored within
reasonable limits, by appropriate selection of the end milling
tool. There are practical limits to how small the diameter Dbot can
be made. We are aware of milling tools for which Dbot is on the
order of 20 micrometers, which corresponds to a radius (Dbot/2) of
10 micrometers; however, specialized end milling tools with even
smaller values of Dbot may exist, and future advances in cutting
technology may also allow for smaller values of Dbot.
[0100] We have explained why the end of the cutting envelope of the
rotating end mill is truncated rather than being a sharp point. In
some cases, the truncated end may be a small, flat, circular area
that meets the side 510rs at a sharp edge, substantially as shown
in FIGS. 5B and 5C. In other cases, the truncated end may be
rounded, curved, or have some other non-flat (but still truncated)
shape, and it may or may not form a sharp edge with the side 510rs.
Whatever the non-flat shape is, it exhibits rotational symmetry
with regard to the rotational axis 509. For simplicity and
consistency in the description that follows, we assume the
truncated end is small, flat, and circular, forming a sharp edge
with the side 510rs, as shown in FIGS. 5B and 5C. The reader will
understand, however, that the truncated end may instead have any
suitable non-flat shape, and such an alternative truncated shape
will produce a corresponding change in the shape of surfaces and
features that are formed by the cutting action of such truncated
end, in particular, features referred to as rounded edges in the
structures discussed and illustrated further below.
[0101] The rotating cutting tool 508r can be used to cut a simple
blind hole, cavity, or recess into the working surface of a
substrate, as shown in the sequence of schematic cross-sectional
views of FIGS. 6A-6C. There, a machinable substrate 605, which may
be the same as or similar to any of the substrates discussed
herein, including any of the substrates of FIG. 1-3 or 4C, has a
working surface 607. Cartesian coordinates are defined such that
the working surface 607 lies in, or extends parallel to, the x-y
plane, and the z-axis represents a normal vector of the surface
607. A rotating cutting tool poised above the substrate provides a
rotating end mill cutting envelope 610r, as discussed above in
connection with cutting envelope 510r. The cutting envelope 610r is
symmetric about a rotational axis 609, and has the shape of a
truncated cone, with a conical side 610rs and an end or bottom
610rb. The rotational axis 609 is parallel to the z-axis, and
perpendicular to the working surface 607.
[0102] To make a simple blind hole in the substrate 605 by end
milling, the rotating tool, and the cutting envelope 610r, is moved
towards the working surface 607 as indicated by the
downward-pointing arrow in FIG. 6A, such motion advancing beyond
the working surface 607 and into the substrate 605 by the cutting
action of the envelope 610r, until the envelope (and cutting tool)
is at least partially lodged within the substrate 605, as shown in
FIG. 6B. Thereafter, the cutting envelope 610r is drawn out of the
substrate 605 along the same path that it entered, as indicated by
the upward-pointing arrow in FIG. 6C. The withdrawal of the cutting
envelope reveals a cavity or recess 620 that was formed by the
milling action of the cutting tool as it penetrated the substrate,
such recess 620 having a curved side surface 631 and a base 620b.
The motion of the cutting envelope into and out of the substrate
605 (see the arrows in FIGS. 6A and 6C) is assumed to be parallel
to the z-axis. (The reader will also keep in mind, in view of the
preceding discussion about relative motion, that the motion of the
cutting tool or cutting envelope 610r may be accomplished in any
desired manner, e.g. holding the substrate 605 stationary and
translating the cutting envelope 610r (according to the arrows in
FIGS. 6A, 6C), or holding the cutting envelope 610r stationary and
translating the substrate 605 (according to arrows opposite those
of FIGS. 6A, 6C), or any combination thereof, and this same
understanding should be applied to all other cutting motions
disclosed herein.) The side surface 631 of the recess 620
corresponds to, and has the same conical shape as, the conical side
610rs of the cutting envelope 610r, and the base 620b of the recess
620 corresponds to, and has the same shape as, the bottom 610rb of
the cutting envelope, i.e., small, flat, and round. Of course, if
the bottom of the cutting envelope is a truncated shape other than
flat, e.g., rounded, then the base 620b of the recess will take on
the same non-flat shape as the bottom of the cutting envelope.
[0103] FIG. 7 is a schematic representation of the path followed by
the rotating end mill in connection with the sequence of FIGS.
6A-6C. In FIG. 7, E represents the plane of the working surface 607
of substrate 605, and the path of the rotating end mill is
represented by line 715. The surface plane E is parallel to the x-y
plane, and the path 715 is straight, and parallel to the z-axis.
The rotating end mill or cutting envelope 610r: begins at a point
S, poised above the plane E; then moves along path 715 to point A,
where contact is first made between the rotating end mill and the
substrate, and where cutting begins; then continues cutting the
substrate as it moves along path 715 to a deepest point B; then
moves back along path 715 from point B to point A; then withdraws
entirely from the recess 620 by moving from point A back to point
S. Of the various identifiable portions or segments of the path
715, cutting of the substrate occurs only in the segment directed
from point A to point B.
[0104] FIGS. 8 and 9 are schematic top and perspective views,
respectively, of the substrate 605 and recess 620 of FIG. 6C, where
like elements have like reference numbers and need no further
explanation. In these views, a perimeter 620p of the recess 620 is
also visible. The perimeter 620p is circular in shape, like the
base 620b, because of the way the rotating end mill plunges into
the substrate to form the recess, and because of the rotational
symmetry of the cutting envelope 610r.
[0105] Substrate removal by end milling can also be used to make
other types of recesses, cavities, and holes. The sequence of FIGS.
10A-10D shows how end milling can be used to cut a simple elongated
recess into the working surface of a substrate. In these figures, a
machinable substrate 1005, which may be the same as or similar to
substrate 605, has a working surface 1007. Cartesian coordinates
are defined such that the working surface 1007 lies in, or extends
parallel to, the x-y plane, and the z-axis represents a normal
vector of the surface 1007. A rotating cutting tool poised above
the substrate provides a rotating end mill cutting envelope 1010r,
as discussed above. The cutting envelope 1010r is symmetric about a
rotational axis 1009, and has the shape of a truncated cone, with a
conical side 1010rs and an end or bottom 1010rb. The rotational
axis 1009 is parallel to the z-axis, and perpendicular to the
working surface 1007.
[0106] To make a simple elongated recess in the substrate 1005 by
end milling, the rotating tool, and the cutting envelope 1010r, is
moved towards the working surface 1007 as indicated by the
downward-pointing arrow in FIG. 10A, such motion advancing beyond
the working surface 1007 and into the substrate 1005 by the cutting
action of the envelope 1010r, until the envelope (and cutting tool)
is at least partially lodged within the substrate 1005, as shown in
FIG. 10B. Next, instead of being immediately withdrawn from the
substrate as in FIG. 6C, the cutting envelope 1010r is moved
laterally through the substrate, cutting as it goes, as indicated
by the right-facing arrow in FIG. 10C. The lateral motion is
parallel to the x-axis. In the view of FIG. 10C, one can already
identify a recess 1020 that has been formed, even though the recess
is partially occupied by the cutting envelope 1010r. Also visible
in that figure are certain elements or features of the recess 1020:
a side surface 1031a, another side surface 1041a, a transition line
1061a1, and a base 1020b. (We will see shortly that the surface
1031a is a curved surface, while the surface 1041a is a flat face.)
Finally, after the lateral cut of FIG. 10C, the cutting envelope
1010r is drawn vertically out of the substrate 1005, as indicated
by the upward-pointing arrow in FIG. 10D. The withdrawal of the
cutting envelope reveals the full elongated cavity or recess 1020
that was formed by the milling action of the cutting tool as it cut
through the substrate 1005. In addition to the elements 1031a,
1041a, 1061a1, and 1020b mentioned above, the recess 1020 can also
be seen in FIG. 10D to have another side surface 1031b, and another
transition line 1061a2. These and other features of the recess 1020
are discussed further below in connection with FIGS. 12 and 13.
[0107] FIG. 11 is a schematic representation of the path followed
by the rotating end mill in connection with the sequence of FIGS.
10A-10D. In FIG. 11, .SIGMA. represents the plane of the working
surface 1007 of substrate 1005, and the path of the rotating end
mill is represented by line 1115. The surface plane .SIGMA. is
parallel to the x-y plane. The path 1115 has a segment from S1 to
B, and a segment from C to S2, that are straight, and parallel to
the z-axis. The path 1115 also has a segment from B to C which is
straight and parallel to the x-axis. The rotating end mill or
cutting envelope 1010r: begins at a point S1, poised above the
surface plane .SIGMA.; then moves along path 1115 to point A, where
contact is first made between the rotating end mill and the
substrate, and where cutting begins; then continues cutting the
substrate as it moves along path 1115 to a deepest point B; then
moves laterally, parallel to the x-axis, from point B to point C;
then moves along path 1115 vertically from point C to point D; then
withdraws entirely from the recess 1020 by moving vertically from
point D to point S2 above the plane .SIGMA.. Throughout its travel
along the path 1115, the rotating end mill or cutting envelope
1010r maintains its rotational axis 1009 parallel to the z-axis. Of
the various identifiable portions or segments of the path 1115,
actual cutting of the substrate 1005 occurs only in two segments:
the segment from point A to point B, and the segment from point B
to point C.
[0108] FIGS. 12 and 13 are schematic top and perspective views,
respectively, of the substrate 1005 and recess 1020 of FIG. 10D,
where like elements have like reference numbers. In these views,
the recess 1020 can be seen to be bounded by a perimeter 1020p,
portions of which are straight and portions of which are curved.
The views of FIGS. 12 and 13 show more clearly that elements 1031a
and 1031b are curved side surfaces of the recess 1020. Indeed, each
curved side surface 1031a, 1031b corresponds to, and has the same
conical shape as, a half section of the conical side 1010rs of the
cutting envelope 1010r. In contrast, elements 1041a, 1041b are flat
side faces of the recess 1020. Each face 1041a, 1041b is inclined
in the same amount as the conical side 1010rs of the cutting
envelope 1010r. Stated differently, the angle formed between face
1041a and the x-z plane (as well as the angle between face 1041b
and the x-z plane) equals the tool half angle (THA) of the cutting
envelope 1010r. The illustrated transition lines 1061a1, 1061a2,
1061b1, 1061b2 are not physical edges, but are provided in the
figures to show where the transition from a curved surface (such as
surface 1031a) to a flat face (such as face 1041a) occurs. Note
that although the flat faces 1041a, 1041b are themselves each
inclined relative to the working surface 1007, neither of the flat
faces 1041a, 1041b has an edge that is inclined relative to the
working surface 1007. Also shown in FIGS. 12 and 13 is the base
1020b of the recess 1020.
[0109] The rotating cutting tool of an end mill can also be used to
cut other recesses, including recesses that can form the basis of
the newly disclosed techniques for making cube corner elements and
other structures. One such recess is made with an end milling
procedure shown in the sequence of schematic cross-sectional views
of FIGS. 14A-14B. There, a machinable substrate 1405, which may be
the same as or similar to substrate 1005, has a working surface
1407. Cartesian coordinates are defined such that the working
surface 1407 lies in, or extends parallel to, the x-y plane, and
the z-axis represents a normal vector of the surface 1407. A
rotating cutting tool, initially poised above the substrate,
provides a rotating end mill cutting envelope 1410r, as discussed
above. The cutting envelope 1410r is symmetric about a rotational
axis 1409, and has the shape of a truncated cone, with a conical
side 1410rs and a flat, round end or bottom 1410rb. The rotational
axis 1409 is parallel to the z-axis, and perpendicular to the
working surface 1407.
[0110] To begin making the recess in the substrate 1405 by end
milling, the rotating tool, represented by the cutting envelope
1410r, is moved towards, and into, the working surface 1407, in
similar fashion to FIGS. 6A-6B and FIGS. 10A-10B. Thus, just as in
those other figures, the motion of the cutting envelope 1410r
advances beyond the working surface 1407 and into the substrate
1405 by the cutting action of the envelope 1410r, until the
envelope (and cutting tool) is at least partially lodged within the
substrate 1405. This is shown in FIG. 14A, which is analogous to
FIGS. 6B and 10B. Thereafter, the cutting envelope 1410r is drawn
out of the substrate 1405. But instead of drawing the cutting
envelope 1410r out along a purely vertical path like in FIG. 6C,
and instead of moving the cutting envelope along a purely lateral
(horizontal) path like in FIG. 10C, the cutting envelope 1410r is
drawn out along a path that is inclined relative to the working
surface 1407, while maintaining the vertical orientation of the
rotational axis 1409. The inclined path: has a nonzero lateral
component and a nonzero vertical component; lies in the x-z plane
of FIG. 14B; and is parallel to an inclined reference axis 1411.
The reference axis 1411 is inclined at an inclination angle .alpha.
(alpha) measured relative to the x-y plane, or to the working
surface 1407. FIG. 14B shows the substrate 1405 at the point where
the rotating end mill tool, or cutting envelope 1410r, has just
emerged from the recess 1420 which it formed in the substrate. The
withdrawal of the cutting envelope 1410r reveals the cavity or
recess 1420 that was formed by the milling action of the cutting
tool through the substrate. The conical side 1410rs of the cutting
envelope 1410r makes an angle THAC relative to the x-y plane or
surface 1407.
[0111] Some, but not all, of the features of the recess 1420 are
visible in the cross-sectional view of FIG. 14B. A number of other
views of the substrate 1405 and recess 1420 are also provided so
the reader can more readily understand the topography of the
working surface 1407 and the recess 1420 and related features. In
particular: FIG. 15 shows a top view of the setup of FIG. 14B; FIG.
16 shows a cross-sectional view through cut line 16-16 in FIG. 15;
FIG. 18 shows a top view like FIG. 15 but without the cutting
envelope; FIG. 19 shows a cross-sectional view through cut line
19-19 in FIG. 18; FIG. 20A shows a partial cross-sectional view
along cut line 20A-20A in FIG. 14B; FIG. 20B is similar to FIG. 20A
but taken along cut line 20B-20B in FIG. 14B; and FIG. 21 shows a
perspective view of the substrate 1405. (FIGS. 17A and 17B will be
discussed further below.) The reader is invited to review and
compare all of these figures to gain a clear understanding of the
recess 1420. With reference to this group of figures, and keeping
in mind that like reference numerals designate like elements, and
that some elements are not visible or not labeled in some figures,
we will now discuss the following features associated with the
recess 1420: base 1420b; curved side surface 1431; flat face 1441a
and flat face 1441b; transition lines 1461a, 1461b; rounded edge
1451; transition lines 1451a, 1451b; terminus 1451t of the rounded
edge; and a periphery or perimeter 1420p.
[0112] The base 1420b of the recess 1420 corresponds to, and has
the same shape as, the bottom 1410rb of the cutting envelope, i.e.,
small, flat, and round in those cases where the bottom 1410rb is
also small, flat, and round. Though the base 1420b is small, it
lies in a plane parallel to the x-y plane due to the fact that the
rotational axis 1409 of the cutting envelope 1410r was parallel to
the z-axis at the deepest point of the cutting.
[0113] The curved side surface 1431 of the recess 1420 corresponds
to, and has the same conical shape as, a portion of the conical
side 1410rs of the cutting envelope 1410r.
[0114] The flat faces 1441a and 1441b are formed together as a pair
while the cutting envelope 1410r cuts its way through the substrate
1405 along the inclined cutting path. The faces 1441a, 1441b are
not parallel to each other, but come together or meet along a
rounded edge 1451, discussed in more detail below. A true measure
of the full included angle (dihedral angle) between the faces
1441a, 1441b can be taken in any cross-sectional plane that is
perpendicular to both faces, and therefore also perpendicular to
the edge 1451. Measuring the angle between the faces 1441a, 1441b
in other cross-sectional planes will yield values for the included
angle (referred to as apparent included angles) other than the true
dihedral angle.
[0115] Cut lines 20A-20A and 20B-20B (see FIG. 14B), for example,
represent cross-sectional planes that are perpendicular to both
faces 1441a, 1441b. A secondary Cartesian coordinate system, x'y'z'
(see FIGS. 14B, 20A, and 20B) is defined such that the y'-z' plane
is parallel to these cross-sectional planes, and the x' axis is
parallel to the rounded edge 1451 (and also to the axis 1411).
Thus, the cross-sectional views of FIGS. 20A and 20B show the true
included angle (dihedral angle) between the faces 1441a, 1441b,
which included angle is labeled as 180-2.OMEGA., and where .OMEGA.
(upper case omega) is the angle each such face makes with the x'-y'
plane. FIG. 20B may be contrasted with FIG. 19, which is a similar
cross-sectional view but in the y-z plane of the primary xyz
coordinate system along cut line 19-19 (see FIG. 18). The y-z plane
is not perpendicular to the faces 1441a, 1441b, nor is it
perpendicular to the edge 1451. In that FIG. 19, an apparent
included angle is measured, 180-2.omega., where .omega. (lower case
omega) is the apparent angle each face 1441a, 1441b makes with the
x-y plane in this cross-sectional plane. Geometric principles can
be used to show that w is greater than .OMEGA., and that the
apparent included angle 180-2.omega. of FIG. 19 is therefore less
than the true dihedral angle 180-2.OMEGA. of FIG. 20B. The apparent
included angle 180-2.omega. between the faces is also greater than
the full angle of the cutting envelope 1410r, see 2*THA in FIG. 16.
(It can also be shown that the true dihedral angle 180-2.OMEGA.
between the faces 1441a, 1441b, and the dihedral angle between
similar faces disclosed herein whenever the rotational axis of the
end mill cutting envelope is not perpendicular to the axis of the
inclined cutting path during the formation of those faces, is
greater than the cutting envelope full angle 2*THA.) With
appropriate selection of the tool half angle THA and the
inclination angle .alpha. of the inclined cutting path, as well as
other factors discussed below, the faces 1441a, 1441b can be made
to have any desired dihedral angle 180-2.OMEGA., within practical
limits. In cases involving the fabrication of cube corner elements,
where the end milling procedures of FIGS. 14A-14B are just two of
many other end milling cutting steps that are performed in the
substrate (see e.g. FIG. 47 below), the THA and a parameters are
chosen so the dihedral angle 180-2.OMEGA. between the faces equals
90 degrees. For example, in the end milling procedure of FIG. 14B,
selecting THA to equal 35.264 degrees (i.e., THAC equals 54.736
degrees) and selecting the inclination angle .alpha. to equal
35.264 degrees ensures the faces 1441a, 1441b will have a 90 degree
dihedral angle, i.e., they will be mutually orthogonal. Other
combinations of the THA, a parameters can be used to make mutually
orthogonal faces of different orientations relative to the x-y
plane, as well as faces that are not mutually orthogonal.
[0116] The faces 1441a, 1441b can also be said to define respective
reference planes that intersect along a line that is inclined
relative to the working surface, such line being parallel to the
axis 1411 and therefore, inclined at the same angle .alpha..
[0117] Also seen in the group of FIGS. 14B, 15, 16, 18, 19, 20A,
20B, and 21 are transition lines 1461a, 1461b. These features are
not physical edges, but are provided in the figures to show where
the transition from a curved surface, i.e., curved side surface
1431, to a flat face, i.e., face 1441a and face 1441b, occurs.
[0118] Also seen in the group of figures is a rounded edge 1451,
located at the place where faces 1441a, 1441b come together. As
best seen in FIG. 14B, the rounded edge 1451 extends in a direction
that is parallel to the inclined cutting path and reference axis
1411. Stated differently, the edge 1451 extends along an axis
parallel to the inclined cutting path and axis 1411. FIG. 14B shows
that the edge 1451 is inclined at the same angle .alpha. relative
to the x-y plane as the axis 1411. This is because the edge 1451 is
formed by the motion of the base 1410rb (or, of the outer circular
cutting edge of the base 1410rb) while the cutting envelope moves
through the solid material of the substrate along the inclined
cutting path.
[0119] Since the edge 1451 is formed by the relative motion of the
base 1410rb, the roundedness or curvature of the edge 1451 is a
direct consequence of the fact that the cutting envelope 1410r is
truncated to have a base 1410rb which is circular (rotationally
symmetric) about the axis 1409 of the rotating end mill. That is,
if it were possible to modify the cutting envelope 1410r to have a
sharp, pointed cutting tip rather than the blunt base, then the
faces 1441a, 1441b would meet to form a sharp edge. As it is, the
tip of the cutting envelope is truncated, and as a result, the base
has a circular cutting edge, and the edge 1451 is rounded. The
reader will understand that deviations of the truncated base of the
cutting envelope from perfect flatness, as discussed above, can
affect the shape of the rounded edge 1451, as well as similar
rounded edges on the many structures discussed below. But in
general the edge 1451, and the similar edges, will still be rounded
due to the fact that the cutting envelope is truncated (whether
flat or non-flat) and has circular symmetry about the rotational
axis of the end mill cutting tool.
[0120] The rounded edge 1451 is not rounded or curved in an
arbitrary fashion, but in a particular way as a result of the
orientation of the circular base 1410rb while the rounded edge was
being formed by the end mill. This can be best understood with
reference to FIG. 14B. In that figure, due to the fact that the
rotational axis 1409 is parallel to the z-axis (and was parallel to
the z-axis while the cutting envelope moved along the inclined
cutting path), the base 1410rb of the cutting envelope lies in the
x-y plane. In this orientation of the cutting envelope 1410r, the
base 1410rb cuts circular shapes in the x-y plane. Therefore, a
cross-section of the rounded edge 1451 in a plane parallel to the
x-y plane will be a circular arc, i.e., it will exhibit a curvature
of constant radius in that plane. This can be seen in FIG. 18 at
the terminus 1451t of the rounded edge 1451, where the radius of
curvature of the terminus 1451t is constant and equal to the radius
of curvature of the base 1420b. However, in other cross-sectional
planes, such as the y-z plane (see the cross-sectional view of FIG.
19), or a plane perpendicular to both faces 1441a, 1441b (see the
cross-sectional view of FIG. 20B), the projection of the circular
arc in such other plane produces an elliptical shape, which thus
exhibits a variable curvature rather than a constant curvature.
This is opposite of what one would expect for faces made (e.g. in
the side of a lamina) by fly cutting using a dulled cutting tool
having a rounded tip of constant curvature. In that case, the round
tip would produce a rounded edge between the first and second face,
but the curvature of the rounded edge would be circular (constant
radius) in a cross-sectional plane that is perpendicular to both
the first and second face, and would have a non-constant radius
(non-circular arc) in other cross-sectional planes.
[0121] The foregoing discussion about the rounded edge 1451
exhibiting a curvature of constant radius in a plane other than a
plane that is perpendicular to both faces 1441a, 1441b is also
applicable to cases in which the bottom of the cutting envelope is
not flat.
[0122] The curvature or radius, and the concomitant width, of the
rounded edge 1451 can be selected as desired, within limits, by
suitable selection of the end milling tool, which in turn controls
the dimension Dbot (see FIG. 5C), and other dimensions, of the
cutting envelope 1410r. In many applications, including many cases
in which the disclosed end milling techniques are used to make cube
corner elements, it will be desirable to have as small a radius of
curvature for the rounded edge 1451 as possible, such that the
rounded edge 1451 is as close to a sharp edge as possible. This is
because rounded surfaces on a cube corner element are usually
associated with inactive areas of retroeflection, and reductions in
total light return. As mentioned above, the dimension Dbot may be
as small as roughly 20 micrometers, or possibly smaller, which
means the radius of curvature of the rounded edge 1451, in the
above-described cross-sectional plane that yields constant
curvature, may be as small as roughly 10 micrometers, or possibly
smaller. However, in some cases it may actually be desirable for
the rounded edge to have the 10 micrometer radius of curvature, or
to have an even greater radius of curvature (i.e., more rounded,
less sharp), whether for reasons of, for example, tool cost, or
mold separation issues, or performance characteristics of the
structured article.
[0123] It is apparent from the above discussion that the rounded
edge 1451--as well as the other surfaces and features of the recess
1420--is an artifact of the rotating end mill (cutting envelope
1410r) that was used to form it. This fact may be manifested in the
machined article itself (as well as in later generation articles
whose structured surface is made by one or more replication steps
of the working surface of the machined article) by small,
nano-scale traces or features that result from imperfections or the
scalloping action of the cutting tool as it works its way through
the substrate material to form the edge. The rounded edge of a
given disclosed substrate or article may thus be capable of being
identified as an artifact of a rotating end mill, or of a truncated
base thereof, solely on the basis of a close inspection of the
article itself, e.g. by electron microscope or other advanced
inspection techniques. This is true regardless of whether the
bottom of the cutting envelope is flat or non-flat.
[0124] Also seen in the group of FIGS. 14B, 15, 16, 18, 19, 20A,
20B, and 21 are transition lines 1451a, 1451b. Like the transition
lines 1461a, 1461b, the lines 1451a, 1451b are not physical edges,
but are provided in the figures to show where the transitions from
the rounded edge 1451 to the flat faces 1441a, 1441b occur.
[0125] The terminus 1451t of the rounded edge 1451 has already been
referenced above. We only note here that the two-ended rounded edge
1451 also has another terminus, opposite the terminus 1451t, such
other terminus being disposed at the base 1420b.
[0126] The terminus 1451t also forms part of the larger periphery
or perimeter 1420p of the recess 1420. For the particular recess
1420, the perimeter 1420p has the general form of a teardrop. The
perimeter 1420p has straight portions at the upper edges of the
faces 1441a, 1441b, and a curved portion (circular arc of large
radius of curvature) at the upper edge of the curved surface 1431,
and an opposed curved portion (circular arc of small radius of
curvature) corresponding to the terminus 1451t of the rounded edge
1451.
[0127] Having now described the recess 1420 in detail, we return
briefly to FIGS. 17A and 17B to discuss the end milling process by
which the recess was formed, or by which it can be formed.
According to the sequence of FIGS. 14A, 14B and the associated
discussion, the rotating end mill (cutting envelope) 1410r begins
by plunging directly downward into the substrate, and then later
withdrawing along a straight inclined path at the angle .alpha..
This fabrication sequence is represented by the path of FIG.
17A.
[0128] In FIG. 17A, .SIGMA. represents the plane of the working
surface 1407 of substrate 1405, and the path of the rotating end
mill is represented by line 1715a. The surface plane .SIGMA. is
parallel to the x-y plane. The path 1715a lies entirely in the x-z
plane. The path has a segment from S1 to A, and a segment from A to
B, that are straight, and parallel to the z-axis. The path also has
a segment from B to C, and a segment from C to S2, that are
straight but inclined at the angle .alpha. relative to the x-axis,
the same angle .alpha. as in FIG. 14B. The rotating end mill or
cutting envelope 1410r: begins at the point S1 above the plane
.SIGMA.; then moves along path 1715a to point A, where the rotating
end mill first makes contact with the substrate; then continues
cutting the substrate as it moves along path 1715a to a deepest
point B; then moves along an incline at the angle .alpha. from
point B to point C; then continues in the same inclined direction
while withdrawing from the recess 1420 as it moves along path 1715a
from point C to point S2 above the plane .SIGMA.. Throughout its
travel along the path 1715a, the rotating end mill or cutting
envelope 1410r maintains its rotational axis 1409 parallel to the
z-axis. Of the various identifiable portions or segments of the
path 1715a, actual cutting of the substrate 1405 occurs in two
segments: the segment from A to B, and the segment from B to C.
Thus, the path portions above the plane .SIGMA., i.e., the path
segments from St to A, and from C to S2, can be changed as desired
with no effect on the shape of the recess formed, e.g. by
repositioning points S1 or S2 to any other locations above the
plane .SIGMA., since those path portions involve no cutting of the
substrate.
[0129] FIG. 17B depicts a path 1715b that is similar to path 1715a,
in fact, path 1715b is simply the path 1715a in reverse order.
Thus, in the process of FIG. 17B, the rotating end mill or cutting
envelope 1410r: begins at the point S1 above the plane .SIGMA.;
then moves at an incline (inclination angle .alpha.) along path
1715b to point A, where the rotating end mill first makes contact
with the substrate; then continues cutting the substrate as it
moves at the same incline (inclination angle .alpha.) along path
1715b to a deepest point B (note, at this point B on path 1715b the
formation of the recess 1420 is complete); then begins to withdraw
from the recess as it moves vertically from point B to point C;
then completely withdraws from the recess 1420 as it moves along
path 1715b from point C to point S2 above the plane .SIGMA..
Throughout its travel along the path 1715b, the rotating end mill
or cutting envelope 1410r maintains its rotational axis 1409
parallel to the z-axis. Of the various identifiable portions or
segments of the path 1715b, actual cutting of the substrate 1405
occurs in only one segment: the segment from A to B.
[0130] Cutting the substrate 1405 with the cutting envelope 1410r
according to the motion of path 1715a (FIG. 17A) produces the
recess 1420 as shown in FIGS. 14B, 15, 16, 18, 19, 20A, 20B, and
21. But cutting the same substrate with the same cutting envelope
according to the motion of path 1715b (FIG. 17B) produces the same
recess 1420. There are some differences in the process by which the
recess is formed, but the end result is the same recess. One
process difference is the number of cutting path segments--the path
1715a has two such segments (A to B and B to C in FIG. 17A),
whereas path 1715b has only one cutting path segment (A to B in
FIG. 17B). Another process difference is the order in which the
various features of the recess 1420 are formed: with path 1715a,
the curved side surface 1431 is formed before the faces 1441a,
1441b, and the last feature formed is the terminus 1451t of the
rounded edge 1451; but with path 1715b, the terminus 1451t is
formed first, followed by the faces 1441a, 1441b, and then the
curved side surface 1431.
[0131] Any of the structures disclosed herein are capable of being
replicated by using a first substrate as a master mold, or tool, or
stamper, from which a structured surface can be created in a second
substrate by known replication techniques such as molding,
embossing, extruding, stamping, or cast-and-curing. A single or odd
number of replication procedures produce a structured surface in
the replica that is a negative or inverted copy of the original
structured surface; two or another even number of replication
procedures produce a structured surface in the replica that is a
positive or non-inverted copy of the original structured surface,
i.e., it is substantially the same as the original structured
surface.
[0132] In one known replication procedure, a fluid resin
composition is cast onto the structured surface of a tool (first
substrate) and allowed to harden to form a sheet (second
substrate), the sheet then having an inverted structured surface
relative to the tool. A preferred method for casting fluid resin
onto a tool is described in U.S. Pat. No. 7,410,604 (Erickson et
al.). A variety of suitable resin compositions can be used so long
as the resin can be formed into the desired configuration. In the
case of cube corner sheeting or other sheeting that is intended for
outdoor use, the resin is typically a transparent material that is
dimensionally stable, durable, weatherable, and readily formable
into the desired configuration. Examples of suitable materials
include: acrylics, which have an index of refraction of about 1.5,
such as Plexiglas brand resin manufactured by Rohm and Haas
Company; polycarbonates, which have an index of refraction of about
1.59; reactive materials such as thermoset acrylates and epoxy
acrylates; polyethylene based ionomers, such as those marketed
under the brand name of SURLYN by E. I. Dupont de Nemours and Co.,
Inc.; (poly)ethylene-co-acrylic acid; polyesters; polyurethanes;
and cellulose acetate butyrates. Polycarbonates are particularly
suitable for cube corner sheeting because of their toughness and
relatively higher refractive index, which generally contributes to
improved retroreflective performance over a wider range of entrance
angles. These materials may also include dyes, colorants, pigments,
UV stabilizers, or other additives.
[0133] In FIG. 22, a substrate 2205 is shown that has a structured
surface 2207 which is a negative replica of the working surface
(structured surface) 1407 of the substrate 1405 of FIGS. 14B, 15,
16, 18, 19, 20A, 20B, and 21. The substrate 2205 may thus be made
by replication from the substrate 1405. The structured surface 2207
therefore comprises a protrusion 2220Z, which juts out of the
structured surface 2207, in place of the recess 1420 of substrate
1405. We introduce a Cartesian coordinate system in FIG. 22 such
that the x-y plane corresponds to the plane along which the
structured surface 2207 generally extends, the z-axis represents a
normal vector of the surface 2207, and the x-axis is oriented so
that a rounded edge 2251Z, which corresponds inversely to the
rounded edge 1451 of substrate 1405, lies in the x-z plane.
[0134] Features of the protrusion 2220Z have a one-to-one (and
inverted) relationship with corresponding features of the recess
1420. The reader will appreciate that aspects and characteristics
that were described in connection with the recess 1420 apply in
corresponding fashion to the protrusion 2220Z. With that
understanding, and wishing to avoid unnecessary repetition, we now
briefly summarize the following features associated with the
protrusion 2220Z: peak or apex 2220Za; curved side surface 2231Z;
flat face 2241Za and flat face 2241Zb; transition lines 2261Za,
2261Zb; rounded edge 2251Z; transition lines 2251Za, 2251Zb;
terminus 2251Zt of the rounded edge; and a periphery or perimeter
2220Zp.
[0135] The peak or apex 2220Za of the protrusion 2220Z has a small,
flat, and round shape, and it lies in a plane parallel to the x-y
plane.
[0136] The curved side surface 2231Z of the protrusion 2220Z has a
conical shape.
[0137] The flat faces 2241Za and 2241Zb come together or meet along
a rounded edge 2251Z. The dihedral angle between the faces 2241Za,
2241Zb (which we may refer to again as 180-2.OMEGA.) can be
measured in any cross-sectional plane that is perpendicular to both
faces, and therefore also perpendicular to the edge 2251Z. The
faces 2241Za, 2241Zb can be made to have any desired dihedral angle
180-2.OMEGA., within practical limits. In cases involving cube
corner elements, the dihedral angle 180-2.OMEGA. between the faces
equals 90 degrees. The faces 2241Za, 2241Zb can be said to define
respective reference planes that intersect along a line that is
inclined relative to the structured surface 2207, such line being
inclined at an angle .alpha..
[0138] Transition lines 2261Za, 2261Zb are not physical edges, but
are provided in the figure to show where the transitions from the
curved side surface 2231Z to the flat faces 2241Za, 2241Zb
occur.
[0139] The rounded edge 2251Z is located at the place where faces
2241Za, 2241Zb come together. The rounded edge 2251Z extends along
an axis that is inclined at the angle .alpha. relative to the x-y
plane. The rounded edge 2251Z is not rounded or curved in an
arbitrary fashion. A cross-section of the rounded edge 2251Z in a
plane parallel to the x-y plane will be a circular arc, i.e., it
will exhibit a curvature of constant radius in that plane. However,
in other cross-sectional planes, such as the y-z plane, or a plane
perpendicular to both faces 2241Za, 2241Zb, the shape of the
rounded edge 2251Z is elliptical, thus exhibiting a variable
curvature rather than a constant curvature. The radius of
curvature, and concomitant width, of the rounded edge 1451 can be
within desired limits, but is generally at least 10 micrometers, as
measured in the cross-sectional plane that yields constant
curvature, and in some cases it can be substantially greater. As a
result of being replicated from an article that was made by end
milling, the rounded edge 2251Z is also an artifact of a rotating
end mill, as manifested for example by small, nano-scale traces or
features that resulted from imperfections or the scalloping action
of a cutting tool as it worked its way through the previous
generation substrate material, and were carried over by replication
to the edge 2251Z.
[0140] Transition lines 2251Za, 2251Zb, like the transition lines
2261Za, 2261Zb, are not physical edges, but are provided in the
figure to show where the transitions from the rounded edge 2251Z to
the flat faces 2241Za, 2241Zb occur.
[0141] The terminus 2251Zt is one of two ends of the rounded edge
2251Z. The other end or terminus of the rounded edge 2251Z is
disposed at the apex 2220Za. The terminus 2251Zt also forms part of
the larger periphery or perimeter 2220Zp of the protrusion 2220Z.
The perimeter 2220Zp has the general form of a teardrop, with
straight portions at the lower edges of the faces 2241Za, 2241Zb,
and a curved portion (circular arc of large radius of curvature) at
the lower edge of the curved surface 2231Z, and an opposed curved
portion (circular arc of small radius of curvature) corresponding
to the terminus 2251Zt of the rounded edge 2251Z.
[0142] The protrusion 2220Z is made by replication from the
teardrop-shaped recess 1420. And the process of forming the recess
1420, whether by the path of FIG. 17A or the path of FIG. 17B,
involves cutting a substrate with a rotating end mill that moves
through the substrate along a cutting path, where the cutting path
includes a path portion that is inclined relative to the working
surface of the substrate. In the path 1715a of FIG. 17A, the path
segment from B to C is a cutting path inclined at the angle
.alpha., and in the path 1715b of FIG. 17B, the path segment from A
to B is a cutting path inclined at the angle .alpha..
[0143] The teardrop-shaped recess 1420 and protrusion 2220Z both
bear characteristic marks of being made, directly or indirectly, by
an end milling method. And although neither the recess 1420 nor the
protrusion 2220Z is a cube corner element, they may nevertheless be
useful--whether individually or as one member of an array or group
of identical or different recesses or protrusions--when
incorporated into the structured surfaces of films, sheeting, or
other articles in optical, mechanical, or other applications.
[0144] Furthermore, the end milling methods for forming cube corner
elements as disclosed herein spring from a recognition that the
same end milling process that was used to make the flat faces of
the recess 1420 and the protrusion 2220Z--namely, cutting a
substrate with a rotating end mill that moves through the substrate
along a cutting path that includes a portion inclined relative to
the working surface of the substrate--can be repeated to form
additional flat faces, and at least some of the original flat faces
and the additional flat faces can then be combined to form a cube
corner element in the substrate. As explained further in the
discussion and figures that follow, such combination relies on
appropriate control of the orientation of the flat faces relative
to each other and to the working surface (or structured surface),
which in turn is controlled by judicious selection of the end
milling cutting parameters, such as the tool half angle THA of the
cutting envelope for each cutting step, the inclination angle
(.alpha., or .beta., .gamma., etc. as discussed below) of the
inclined cutting path portion for each cutting step, the
orientation of the cutting tool rotational axis relative to the
z-axis and to the vertical plane of the inclined cutting path
portion for each cutting step, and the relative orientations of the
cutting path portions for the various cutting steps.
[0145] But before describing cube corner elements made by the end
milling method, we wish to point out that numerous variations of
the teardrop-shaped recess 1420, and its corresponding protrusion
2220Z, can be made that are not cube corner elements. A few of
these numerous variations are described with reference to
alternative paths of motion of the rotating end mill, as shown in
FIGS. 23-27. Recesses that correspond to these alternative paths
are shown in FIGS. 28-32, with the recess of FIG. 28 corresponding
to the path of FIG. 23, the recess of FIG. 29 corresponding to the
path of FIG. 24, and so forth.
[0146] The reader needs to be cautioned here that a given path of
motion for the end mill, such as even the path of FIG. 17A or 17B
(or any of the paths of FIGS. 23-27), can produce a whole family of
different recesses (and corresponding protrusions), of which the
cavity 1420 shown in FIGS. 14B, 15, 16, 18, 19, 20A, 20B, and 21 is
only one member. The family of such different recesses uses the
same path (e.g., FIG. 17A or 17B), but different end mill
parameters, for example, different tool half angles THA (see FIG.
5C and FIGS. 34-36 below), different values for Dbot and Dtop (see
FIG. 5C), and different orientations of the cutting tool rotational
axis relative to the z-axis and to the vertical plane of the
cutting path portion (see e.g. angles .theta. and .PHI. in FIG. 33
below). Stated differently, a given path of motion for the end
mill, by itself, is insufficient to specify the shape or
configuration of a particular recess or corresponding protrusion,
unless the other end mill parameters are also provided.
[0147] With this in mind, we turn our attention to FIGS. 23-27 and
corresponding FIGS. 28-32. For simplicity, the entire path of
motion of the end mill in each of FIGS. 23-27 is assumed to lie in
a single vertical plane, which we designate the x-z plane. In each
of these figures, .SIGMA. represents the plane of the working
surface (or structured surface) of the substrate being cut, and
.SIGMA. is deemed to be parallel to the x-y plane of the coordinate
system. Each of the paths of FIGS. 23-27 will be described as
beginning at a point S1 and ending at a point S2, but the reader
will understand that the rotating end mill may follow the reverse
path, beginning at S2 and ending at S1, to make the same recess.
The recesses of FIGS. 28-32 correspond respectively to the paths of
FIGS. 23-27 (keeping in mind that each of the illustrated recesses
is one member of a whole family of possible recesses that may
correspond to those paths) and assume that the rotational axis of
the rotating cutting tool (or cutting envelope) is at all times
parallel to the z-axis as it follows the corresponding path. The
recesses of FIGS. 28-32 also assume these additional rotating end
mill parameters (refer to FIG. 5C): THA=35.264 degrees, and Dbot=30
micrometers. The xyz coordinate axes in FIGS. 28-32 are consistent
with the xyz axes in their corresponding FIGS. 23-27.
[0148] In FIG. 23, a path 2315 of the rotating end mill may begin
at point S1 above the surface, proceed to point A located on the
surface plane .SIGMA., then proceed to a deepest point B, then to a
point C, then to a point D located on the surface plane .SIGMA.,
and finish at point S2 above the surface. The segments of the path
2315 from S1 to A, and from A to B, and from C to D, and from D to
S2 are straight and parallel to the z-axis. The segment of the path
2315 from B to C is straight but inclined at an angle .alpha.
relative to the x-axis. When following the path 2315 in this
manner, actual cutting of the substrate occurs only in the segment
from A to B, and in the segment from B to C. (In a reverse path,
cutting occurs only from D to C, and from C to B.)
[0149] FIG. 28 depicts a substrate 2805, which may be the same as
or similar to substrates discussed above, having a working surface
2807 in which is formed a recess 2820, the recess being formed by a
rotating end mill as described above which moves along the path
2315 of FIG. 23. The recess 2820 has a perimeter 2820p and a base
2820b. The recess 2820 also has a large curved side surface 2831a,
and a smaller curved side surface 2831b. The recess 2820 also has
flat faces 2841a, 2841b which come together or meet along a rounded
edge 2851. The dihedral angle between these flat faces is 90
degrees. The rounded edge extends along an axis that is inclined
relative to the x-y plane at the same inclination angle .alpha. as
in FIG. 23. In FIG. 28, a is assumed to be 35.264 degrees. The
rounded edge 2851 has a terminus 2851t, and another terminus at the
base 2820b. Transition lines 2861a, 2861b, 2861c, 2861d mark the
boundaries between flat faces and curved side surfaces. Transition
lines 2851a, 2851b mark the boundaries between the rounded edge and
the flat faces. The curved side surfaces, flat faces, rounded edge,
and transition lines have the same or similar characteristics and
properties as corresponding features of the cavity 1420 described
in detail above. With reference to the path 2315, the flat faces
2841a, 2841b of the recess are formed while the cutting envelope
moves along path segment B-C, the curved side surface 2831a is
formed while the cutting envelope is positioned at point B, and the
curved side surface 2831b is formed while the cutting envelope is
positioned at point C.
[0150] In FIG. 24, a path 2415 of the rotating end mill may begin
at point S1 above the surface, proceed to point A located on the
surface plane .SIGMA., then proceed to a deepest point B, then to
another deepest point C, then to a point D located on the surface
plane .SIGMA., and finish at point S2 above the surface. The
segments of the path 2415 from S1 to A, and from A to B are
straight and parallel to the z-axis. The segment of the path 2415
from B to C is straight and parallel to the x-axis. The segments of
the path 2415 from C to D, and from D to S2 are straight but
inclined at an angle .alpha. relative to the x-axis. When following
the path 2415 in this manner, actual cutting of the substrate
occurs only in the segments from A to B, and from B to C, and from
C to D. (In a reverse path, cutting occurs only from D to C, and
from C to B.)
[0151] FIG. 29 depicts a substrate 2905, which may be the same as
or similar to substrates discussed above, having a working surface
2907 in which is formed a recess 2920, the recess being formed by a
rotating end mill as described above which moves along the path
2415 of FIG. 24. The recess 2920 has a perimeter 2920p and a base
2920b. The recess 2920 also has a large curved side surface 2931,
and two smaller curved side surfaces 2932a, 2932b. The recess 2920
also has flat faces 2941a, 2941b which both intersect the base
2920b. The dihedral angle between these flat faces is the same as
the full cone angle of the cutting envelope, i.e., 2*THA. The
recess 2920 also has flat faces 2941c, 2941d which come together or
meet along a rounded edge 2951. The dihedral angle between these
flat faces is 90 degrees. The rounded edge extends along an axis
that is inclined relative to the x-y plane at the same inclination
angle .alpha. as in FIG. 24. In FIG. 29, a is assumed to be 35.264
degrees. The rounded edge 2951 has a terminus 2951t, and another
terminus at its intersection with the base 2920b. Transition lines
2961a, 2961b, 2961c, 2961d, 2961e, 2961f mark the boundaries
between flat faces and curved side surfaces. Transition lines
2951a, 2951b mark the boundaries between the rounded edge and the
flat faces. The curved side surfaces, flat faces, rounded edge, and
transition lines have the same or similar characteristics and
properties as corresponding features of the cavity 1420 described
in detail above. With reference to the path 2415, the flat faces
2941a, 2941b of the recess are formed while the cutting envelope
moves along path segment B-C, the flat faces 2941c, 2941d are
formed while the cutting envelope moves along path segment C-D, the
curved side surface 2931a is formed while the cutting envelope is
positioned at point B, and the curved side surfaces 2932a, 2932b
are formed while the cutting envelope is positioned at point C.
[0152] In FIG. 25, a path 2515 of the rotating end mill may begin
at point S1 above the surface, proceed to point A located on the
surface plane .SIGMA., then proceed to a deepest point B, then to a
point C, then to a point D located on the surface plane .SIGMA.,
and finish at point S2 above the surface. The segments of the path
2515 from S1 to A, and from A to B are straight and parallel to the
z-axis. The segment of the path 2515 from B to C is straight but
inclined at an angle .alpha.1 relative to the x-axis. The segments
of the path 2515 from C to D, and from D to S2 are straight but
inclined at an angle .alpha.2 (where .alpha.2>.alpha.1) relative
to the x-axis. When following the path 2515 in this manner, actual
cutting of the substrate occurs only in the segments from A to B,
and from B to C, and from C to D. (In a reverse path, cutting
occurs only from D to C, and from C to B.) FIG. 30 depicts a
substrate 3005, which may be the same as or similar to substrates
discussed above, having a working surface 3007 in which is formed a
recess 3020, the recess being formed by a rotating end mill as
described above which moves along the path 2515 of FIG. 25. The
recess 3020 has a perimeter 3020p and a base 3020b. The recess 3020
also has a large curved side surface 3031, and two smaller curved
side surfaces 3032a, 3032b. The recess 3020 also has flat faces
3041a, 3041b which come together or meet along a rounded edge 3051.
The dihedral angle between these flat faces is 71.783 degrees. The
rounded edge 3051 extends along an axis that is inclined relative
to the x-y plane at the same inclination angle .alpha.1 as in FIG.
25. In FIG. 30, .alpha.1 is assumed to be 10 degrees. The rounded
edge 3051 has a terminus 3051t, and another terminus at the base
3020b. The recess 3020 also has flat faces 3042a, 3042b which come
together or meet along a rounded edge 3052. The dihedral angle
between these flat faces is 90 degrees. The rounded edge 3052
extends along an axis that is inclined relative to the x-y plane at
the same inclination angle .alpha.2 as in FIG. 25. In FIG. 30,
.alpha.2 is assumed to be 35.264 degrees. The rounded edge 3052 has
a terminus 3052t, and another terminus 3051t which it shares with
the other rounded edge 3051. Transition lines 3061a, 3061b, 3061c,
3061d, 3062a, 3062b mark the boundaries between flat faces and
curved side surfaces. Transition lines 3051a, 3051b, 3052a, 3052b
mark the boundaries between the rounded edges and the flat faces.
The curved side surfaces, flat faces, rounded edges, and transition
lines have the same or similar characteristics and properties as
corresponding features of the cavity 1420 described in detail
above. With reference to the path 2515, the flat faces 3041a, 3041b
of the recess are formed while the cutting envelope moves along
path segment B-C, the flat faces 3042a, 3042b are formed while the
cutting envelope moves along path segment C-D, the curved side
surface 3031 is formed while the cutting envelope is positioned at
point B, and the curved side surfaces 3032a, 3032b are formed while
the cutting envelope is positioned at point C.
[0153] In FIG. 26, a path 2615 of the rotating end mill may begin
at point S1 above the surface, proceed to point A located on the
surface plane .SIGMA., then proceed to a deepest point B, then to a
point C located on the surface plane .SIGMA., and finish at point
S2 above the surface. The segments of the path 2615 from S1 to A,
and from A to B are straight but inclined at an angle .alpha.1
relative to the x-axis. The segments of the path 2615 from B to C,
and from C to S2 are straight but inclined at an angle .alpha.1
(where .alpha.2 is about equal to .alpha.1, but they are inclined
in opposite senses as shown in the figure) relative to the x-axis.
When following the path 2615 in this manner, actual cutting of the
substrate occurs only in the segments from A to B, and from B to C.
(In a reverse path, cutting occurs only from C to B, and from B to
A.)
[0154] FIG. 31 depicts a substrate 3105, which may be the same as
or similar to substrates discussed above, having a working surface
3107 in which is formed a recess 3120, the recess being formed by a
rotating end mill as described above which moves along the path
2615 of FIG. 26. The recess 3120 has a perimeter 3120p and a base
3120b. The recess 3120 also has two curved side surfaces 3131a,
3131b. The recess 3120 also has flat faces 3141a, 3141b which come
together or meet along a rounded edge 3151. The dihedral angle
between these flat faces is 90 degrees. The rounded edge 3151
extends along an axis that is inclined relative to the x-y plane at
the same inclination angle .alpha.1 as in FIG. 26. In FIG. 31,
.alpha.1 is assumed to be 35.264 degrees. The rounded edge 3151 has
a terminus 3151t, and another terminus at the base 3120b. The
recess 3120 also has flat faces 3142a, 3142b which come together or
meet along a rounded edge 3152. The dihedral angle between these
flat faces is 90 degrees. The rounded edge 3152 extends along an
axis that is inclined relative to the x-y plane at the same
inclination angle .alpha.2 as in FIG. 26. In FIG. 31, .alpha.2 is
assumed to be 35.264 degrees. The rounded edge 3152 has a terminus
3152t, and another terminus at the base 3120b. Transition lines
3161a, 3161b, 3162a, 3162b mark the boundaries between flat faces
and curved side surfaces. Transition lines 3151a, 3151b, 3152a,
3152b mark the boundaries between the rounded edges and the flat
faces. The curved side surfaces, flat faces, rounded edges, and
transition lines have the same or similar characteristics and
properties as corresponding features of the cavity 1420 described
in detail above. With reference to the path 2615, the flat faces
3141a, 3141b of the recess are formed while the cutting envelope
moves along path segment A-B, the flat faces 3142a, 3142b are
formed while the cutting envelope moves along path segment B-C, and
the curved side surfaces 3131a, 3131b (as well as the base 3120b)
are formed while the cutting envelope is positioned at point B.
[0155] In FIG. 27, a path 2715 of the rotating end mill may begin
at point S1 above the surface, proceed to point A located on the
surface plane .SIGMA., then proceed to a deepest point B, then to a
point C located on the surface plane .SIGMA., and finish at point
S2 above the surface. The segments of the path 2715 from S1 to A,
and from A to B are straight but inclined at an angle .alpha.1
relative to the x-axis. The segments of the path 2715 from B to C,
and from C to S2 are straight but inclined at an angle .alpha.2
(where .alpha.2<.alpha.1, and these angles are inclined in
opposite senses as shown in the figure) relative to the x-axis.
When following the path 2715 in this manner, actual cutting of the
substrate occurs only in the segments from A to B, and from B to C.
(In a reverse path, cutting occurs only from C to B, and from B to
A.)
[0156] FIG. 32 depicts a substrate 3205, which may be the same as
or similar to substrates discussed above, having a working surface
3207 in which is formed a recess 3220, the recess being formed by a
rotating end mill as described above which moves along the path
2715 of FIG. 27. The recess 3220 has a perimeter 3220p and a base
3220b. The recess 3220 also has two curved side surfaces 3231a,
3231b. The recess 3220 also has flat faces 3241a, 3241b which come
together or meet along a rounded edge 3251. The dihedral angle
between these flat faces is 110.988 degrees. The rounded edge 3251
extends along an axis that is inclined relative to the x-y plane at
the same inclination angle .alpha.1 as in FIG. 27. In FIG. 32,
.alpha.1 is assumed to be 45.526 degrees. The rounded edge 3251 has
a terminus 3251t, and another terminus at the base 3220b. The
recess 3220 also has flat faces 3242a, 3242b which come together or
meet along a rounded edge 3252. The dihedral angle between these
flat faces is 90 degrees. The rounded edge 3252 extends along an
axis that is inclined relative to the x-y plane at the same
inclination angle .alpha.2 as in FIG. 27. In FIG. 32, .alpha.2 is
assumed to be 35.264 degrees. The rounded edge 3252 has a terminus
3252t, and another terminus at the base 3220b. Transition lines
3261a, 3261b, 3262a, 3262b mark the boundaries between flat faces
and curved side surfaces. Transition lines 3251a, 3251b, 3252a,
3252b mark the boundaries between the rounded edges and the flat
faces. The curved side surfaces, flat faces, rounded edges, and
transition lines have the same or similar characteristics and
properties as corresponding features of the cavity 1420 described
in detail above. With reference to the path 2715, the flat faces
3241a, 3241b of the recess are formed while the cutting envelope
moves along path segment A-B, the flat faces 3242a, 3242b are
formed while the cutting envelope moves along path segment B-C, and
the curved side surfaces 3231a, 3231b (as well as the base 3220b)
are formed while the cutting envelope is positioned at point B.
[0157] The recesses of FIGS. 28-32, as well as all other recesses
disclosed herein, can be formed individually or in groups or arrays
in the structured surface of an article, and such individual or
groups of recesses can also be replicated to form individual or
groups of corresponding protrusions in the structured surface of an
article.
[0158] As discussed above, additional design flexibility for the
disclosed recesses and protrusions can be realized by--in addition
to tailoring the angle of inclination (e.g. .alpha., .alpha.1,
.alpha.2, .beta., .gamma., etc.) of the inclined cutting path
portion--tailoring the end mill parameters such as THA, Dbot, and
Dtop (see FIG. 5C), as well as the orientation of the cutting tool
rotational axis relative to the z-axis and to the vertical plane of
the cutting path portion.
[0159] The orientation of the cutting tool rotational axis can be
expressed in terms of a polar angle .theta. and an azimuthal angle
.PHI. as shown in the schematic diagram of FIG. 33. In that figure,
a cutting envelope 3310r is shown that may be the same as or
similar to any of the end mill cutting envelopes described above.
Thus, the cutting envelope 3310r includes a cone-shaped side
3310rs, a truncated end or bottom 3310rb, and a rotational axis
3309. The cutting envelope 3310r is shown in the context of a
Cartesian xyz coordinate system, in similar fashion to other
figures herein. For purposes of FIG. 33, we assume the z-axis
corresponds to a normal vector of the working surface or structured
surface being cut by the end mill, and that such surface extends
parallel to the x-y plane. We further assume that to the extent the
cutting envelope moves along a path portion that is not parallel to
the z-axis--i.e., either inclined to the x-y plane or parallel to
the x-y plane--such path portion lies in the x-z plane. In this
way, the reference by which we measure the azimuthal angle .PHI. of
a tilted cutting envelope is a vertical plane in which the path
portion lies.
[0160] The cutting envelope 3310r is shown for reference purposes
as having no tilt, i.e., its rotational axis 3309 is parallel to
the z-axis. This can be contrasted with a generalized tilted
cutting envelope 3310r' shown in the figure. Like the cutting
envelope 3310r, the cutting envelope 3310r' has a cone-shaped side
3310rs', a truncated end or bottom 3310rb', and a rotational axis
3309'. The cutting envelope 3310r' however is tilted to have a
nonzero polar angle .theta. and a nonzero azimuthal angle .PHI.,
i.e., the rotational axis 3309' makes a nonzero angle .theta.
relative to the z-axis, and the vertical plane containing the
rotational axis 3309' (which we may refer to as the rotational axis
vertical plane) makes a nonzero angle .PHI. relative to the x-z
plane. FIG. 33 also shows the cutting envelope 3310r' as being
separated from (located outside of) the x-z plane. This separation
is only for purposes of better illustrating the nonzero polar and
azimuthal angles .theta., .PHI., and the reader will understand
that the cutting envelope 3310r' (e.g. at least the bottom 3310rb'
thereof) would normally reside in the x-z plane as it cuts along a
path portion which we have assumed lies in the x-z plane. For
purposes of some of the embodiments described below, we can assign
positive or negative values to .PHI. depending on the direction of
rotation of the rotational axis vertical plane relative to the x-z
plane (in which the inclined path portion resides): a
counterclockwise rotation about the z-axis, as depicted in the
cutting envelope orientation 3310r' shown in FIG. 33 and in
accordance with the so-called right hand rule, is considered
positive, whereas a clockwise rotation about the z-axis is
considered negative.
[0161] For a given cutting path or path portion, the cutting
envelope of the rotating end mill can be tilted as desired, whether
with a nonzero polar angle .theta. and a zero azimuthal angle .PHI.
(i.e. tilted in the vertical plane of an inclined cutting path
portion), or with nonzero polar and azimuthal angles (i.e. tilted
outside the vertical plane of an inclined cutting path portion), or
not tilted at all (vertically oriented in the vertical plane of an
inclined cutting path portion), to achieve the desired orientation
of the various curved surfaces, flat faces, or other features of a
given recess. For example, for an inclined cutting path lying in
the x-z plane which is used to cut flat faces of a recess, by
adjusting the polar angle .theta. of the rotational axis of the
cutting envelope, the dihedral angle between such faces can be
adjusted, for a given tool half angle THA. Selecting the polar
angle .theta. such that the rotational axis is perpendicular to the
direction or axis of the cutting path produces a maximum dihedral
angle (equal to 2*THA), whereas adjusting the polar angle .theta.
to reduce the angle between the rotational axis and the axis of the
cutting path produces a larger dihedral angle between the flat
faces (greater than 2*THA). The azimuthal angle .PHI. can also be
adjusted (for nonzero polar angles .theta.), for example, to orient
the flat faces in an asymmetric fashion relative to the vertical
plane in which the cutting path resides. Note that nonzero values
for the angles .theta. and .PHI. have ramifications on the
curvature characteristics of the rounded edge that connects the two
flat faces. That is, changing the angles .theta. and .PHI. will in
general change the orientation of the cross-sectional plane in
which the rounded edge exhibits a curvature of constant radius.
[0162] Besides tailoring the orientation of the cutting tool
rotational axis, design flexibility can also be provided by
appropriate selection of end mill parameters such as THA and Dbot.
A comparison of different values of THA is provided in the group of
FIGS. 34, 35, and 36. In FIG. 34, a first rotating end mill
provides a first cutting envelope 3410r, with a conical side
3410rs, a truncated end or bottom 3410rb, and a tool half angle
THA1. THAC1 is the geometric complement of THA1. In FIG. 35, a
second rotating end mill provides a second cutting envelope 3510r,
with a conical side 3510rs, a truncated end or bottom 3510rb, and a
tool half angle THA2. THAC2 is the geometric complement of THA2. In
FIG. 36, a third rotating end mill provides a third cutting
envelope 3610r, with a conical side 3610rs, a truncated end or
bottom 3610rb, and a tool half angle THA3. THAC3 is the geometric
complement of THA3. Any one of the cutting envelopes 3410r, 3510r,
or 3610r may be the same as or similar to the cutting envelope 510r
discussed above, however, the cutting envelopes have different half
(and full) cone angles, with THA2 being greater than THA1, and THA3
being greater than THA2. Selecting an end mill tool that provides
an increased or decreased value of THA allows the designer to
increase or decrease, respectively, the dihedral angle between flat
faces of a recess formed by the disclosed methods, and to achieve a
target dihedral angle.
[0163] FIG. 37 illustrates a path 3715 for a rotating end mill that
is similar in most respects to the paths of FIGS. 17A, 17B, and
23-27, except that the path of FIG. 37 does not lie entirely within
a single vertical plane. Instead, the path 3715 includes two
segments that lie in different vertical planes. The path 3715 is
shown in the context of a Cartesian xyz coordinate system, in
similar fashion to other figures herein. For purposes of FIG. 37,
we assume the z-axis corresponds to a normal vector of the working
surface or structured surface of a substrate being cut by the end
mill, and that such surface extends parallel to the x-y plane.
However, in FIG. 37 we assume the actual plane of the working
surface (see e.g. .SIGMA. in FIGS. 17A, 17B, and 23-27) lies above
the x-y plane such that it intersects the z-axis at a point which
we label as both Az and Cz. We also assume the plane of the working
surface contains points A and C in FIG. 37. The point Az is the
projection of point A onto the z-axis, and the point Cz (the same
point as Az) is the projection of the point C onto the z-axis. A
point Axy is the projection of the point A onto the x-y plane, and
a point Cxy is the projection of the point C onto the x-y plane.
Another point, B, is located on the path 3715 and lies beneath the
plane of the working surface. The point B is in fact the deepest
(within the substrate) of any point on the path 3715.
[0164] The path 3715 has two distinct segments or portions, one
from point A to point B (or vice versa), and one from point B to
point C (or vice versa). The point B is common to, or shared by,
both segments. The A-B segment of the path 3715 lies in the x-z
vertical plane, and is inclined relative to the x-y plane (and to
the working surface of the substrate) at an angle .alpha.. The B-C
segment of the path 3715 lies in a different vertical plane, and is
inclined relative to the x-y plane (and to the working surface of
the substrate) at an angle .beta.. The vertical plane of the B-C
segment contains the z-axis and point C, and it intersects the
other vertical plane (the x-z plane) at an angle .PHI. (upper case
phi). A secondary Cartesian coordinate system x'y'z' may be defined
in which the z'-axis is identical to the primary z-axis, and such
that the vertical plane of the B-C segment is the x'-z' plane of
the secondary coordinate system. The x'-axis is shown in FIG. 37,
but the y'-axis is not shown to avoid overly complicating the
figure. The intersection angle .PHI. between non-parallel vertical
planes can in general be selected as desired, but in many of the
disclosed embodiments the intersection angle .PHI. is at least 10
degrees.
[0165] The path 3715 disregards and omits any motion of the cutting
envelope of the rotating end mill that occurs above the surface of
the substrate, and that therefore does not involve any cutting,
such as the segments from S1 to A and from C to S2 in FIG. 17A.
Rather, the path 3715 only includes segments that involve cutting
of the substrate. When a cutting envelope, which has a tool half
angle THA and truncated bottom as discussed above, moves along the
inclined segment A-B of the path, it forms a first and second flat
face. In fact, by the time the cutting envelope reaches the deepest
point B, it has already formed a teardrop-shaped recess such as
that shown in FIG. 21 above. If we then cause the cutting envelope
to follow the other inclined segment from B to C, the cutting
envelope enlarges the recess, and in doing so, forms third and
fourth flat faces that are distinct from the first and second flat
faces. (In general, a first end mill (and first cutting envelope)
can be used for the segment A-B, and a second end mill (and second
cutting envelope) can be used for the segment B-C, and the first
and second end mills need not be, but in some cases can be, the
same. Furthermore, such first and second end mills need not have,
but in some cases can have, the same orientation in terms of polar
angle .theta. and azimuthal angle .PHI. (see FIG. 33) for each of
the respective cutting paths. In some cases where the end mill
cutting tool is tilted at a nonzero polar angle .theta., the
cutting envelope can be tilted in the same orientation relative to
a fixed reference mark (or coordinate system) on the structured
surface for both cutting path segments, and in that case the
azimuthal angles .PHI. for the different cutting paths will be
different, since we assume the angles .PHI. are measured relative
to the vertical plane of each respective cutting path segment,
rather than to a single fixed vertical plane.)
[0166] The third and fourth flat faces associated with the path
segment B-C can have any desired configuration, and any degree of
alignment, or misalignment, relative to the first and second flat
faces associated with the path segment A-B. However, we have found
that particularly useful structures, including cube corner
elements, can be formed by configuring the third and fourth flat
faces so that one of these faces is sufficiently aligned with one
of the first and second faces such that the two aligned faces form
a single compound face. Such an embodiment is shown in FIG. 38.
[0167] FIG. 38 depicts a substrate 3805, which may be the same as
or similar to substrates discussed above, having a working surface
3807 in which is formed a recess 3820, the recess being formed by a
rotating end mill (or by two rotating end mills) as described above
which moves along a path having two different inclined segments
with a shared deepest endpoint, similar to the path 3715 of FIG.
37. (The reader is cautioned that the placement of the secondary
(primed) coordinate axes relative to the primary (unprimed)
coordinate axes is different in FIG. 38 than in FIG. 37;
nevertheless, the recess of FIG. 38 is cut with a two-segment path
very much like the path 3715 of FIG. 37, i.e., with two inclined
paths or path portions that lie in different vertical planes.)
[0168] The recess 3820 has a perimeter 3820p and a base 3820b, the
base corresponding to the deepest (and shared) point of the cutting
path segments. The recess 3820 has a curved side surface 3831. The
recess 3820 also has two flat faces 3841a, 3841b which come
together or meet along a rounded edge 3851. The dihedral angle
between these flat faces can be tailored as desired, but in the
illustrated embodiment it is 90 degrees. The rounded edge 3851
extends along an axis that is inclined at an angle .alpha. relative
to the x-y plane, see e.g. FIG. 37. In FIG. 38, .alpha. is assumed
to be 35.264 degrees. At such an inclination angle, the faces
3841a, 3841b can be made to have a 90 degree dihedral angle if a
cutting envelope is used in which THA=35.264 degrees, with a polar
angle .theta.=0 and an azimuthal angle .PHI.=0. The rounded edge
3851 has a terminus 3851t, and another terminus at the base
3820b.
[0169] The recess 3820 also has two other flat faces 3842a, 3842b
which come together or meet along a second rounded edge 3852. The
dihedral angle between these flat faces can be tailored as desired,
but in the illustrated embodiment it is 90 degrees. The rounded
edge 3852 extends along an axis that is inclined at an angle .beta.
relative to the x-y plane, see e.g. FIG. 37. In FIG. 38, .beta. is
assumed to be 35.264 degrees. At such an inclination angle, the
faces 3842a, 3842b can be made to have a 90 degree dihedral angle
if a cutting envelope is used in which THA=35.264 degrees, with a
polar angle .theta.=0 and an azimuthal angle .PHI.=0. The rounded
edge 3852 has a terminus 3852t, and another terminus at the base
3820b. The base 3820b can also be considered a vertex, at least
because it is located at the junction of the rounded edges 3851,
3852. Transition lines 3861, 3862 mark the boundaries between flat
faces and the curved side surface 3831. Transition lines 3851a,
3851b, 3852a, 3852b mark the boundaries between the rounded edges
and the flat faces. The curved side surface, flat faces, rounded
edges, and transition lines have the same or similar
characteristics and properties as corresponding features of
cavities described in detail above.
[0170] By comparing the recess 3820 of FIG. 38 with the
teardrop-shaped recess 1420 shown in FIG. 18, we can appreciate
that the recess 3820 can be made in two steps, corresponding to the
two inclined segments or portions of the cutting path such as are
shown in FIG. 37. In a first step, the cutting envelope of the
rotating end mill cuts into the substrate while moving along an
inclined cutting path located in the x-z plane (see e.g. segment
A-B in FIG. 37). At the end of this first step, a teardrop-shaped
recess is formed similar to, or the same as, that of FIG. 18. In a
second step, the same cutting envelope withdraws from the substrate
not vertically, but along a second inclined cutting path that is
located in a different vertical plane, see the x'-z' plane of FIG.
38. During the second step the cutting envelope enlarges the recess
and forms two additional flat faces, resulting in the recess 3820
of FIG. 38. In an alternative embodiment, the first cutting
envelope can be withdrawn from the teardrop-shaped cavity in such a
way as to preserve the teardrop-shaped cavity, and then in an
alternative second step, a different second cutting envelope (e.g.
having a different THA) can be moved along the second inclined
cutting path so as to form two alternative flat faces.
[0171] The two different inclined cutting paths or path segments
lie in different vertical planes, namely: the x-z plane of FIG. 38,
which is responsible for forming the rounded edge 3851 and the flat
faces 3841a and 3841b; and the x'-z' plane of FIG. 38, which is
responsible for forming the rounded edge 3852 and the flat faces
3842a and 3842b. These different vertical planes intersect at an
angle .PHI. (not labeled in FIG. 38 but analogous to the angle
.PHI. shown in FIG. 37). The maker of the recess may control the
motion of the rotating end mill in any desired way, and may thus
make the intersection angle .PHI. any desired value. In the
embodiment of FIG. 38 we have selected an intersection angle
.PHI.=120 degrees. Given our selected geometry for the faces 3842a,
3842b, this intersection angle ensures that the face 3842a is in
substantial alignment with the face 3841b, such that these faces
together form a compound face. By this we mean that the individual
faces are so closely aligned, and so near each other, that the
structure of which they are a part functions in a manner that is
the same as or similar to that of a replacement structure, the
replacement structure being otherwise identical to the first
structure except that the two faces of the first structure are
replaced by a single, unitary face. The two constituent faces of
the compound face thus function substantially as a single face. In
general, the constituent faces, having been formed in different
machining steps, are distinguishable from each other, and they meet
along a feature referred to herein as a break line. The break line
in a given embodiment may be easily detected, if the constituent
faces have a slight misalignment, or, where alignment is perfect or
nearly perfect, the break line may be difficult to detect, e.g.,
detectable only with advanced inspection equipment such as an
electron microscope. In the case of the recess 3820, the
constituent faces 3842a, 3841b of the compound face meet along a
break line 3871. The break line 3871 extends from the base 3820b to
the perimeter 3820p of the recess 3820. In FIG. 38 and the other
figures below, break lines are distinguished from transition lines
by use of a different broken line pattern.
[0172] An isolated front view of the compound face of FIG. 38 is
shown in FIG. 39. The compound face is labeled CF in FIG. 39. Other
reference labels in FIG. 39 are the same as those of FIG. 38,
requiring no further explanation.
[0173] FIGS. 40-42 illustrate different degrees of alignment or
misalignment that may be realized between the two constituent faces
of a compound face. These figures are drawn in side or edge view,
such that the flat faces appear as line segments. In FIG. 40,
constituent faces 4041, 4042 of a compound face CF are in perfect
or near perfect alignment, such that they come together or meet
along a break line 4071 that is difficult to detect, e.g.,
detectable only with advanced inspection equipment. In FIG. 41,
constituent faces 4141, 4142 of a compound face CF have a small
translational misalignment, such that they come together or meet
along a small step-like break line 4171, which is less difficult to
detect than the break line 4071. In FIG. 42, constituent faces
4241, 4242 of a compound face CF have a small angular misalignment,
such that they come together or meet along a subtle edge-like break
line 4271, which is also less difficult to detect than the break
line 4071.
[0174] Further in regard to the angular misalignment of FIG. 42,
FIG. 43 schematically illustrates two planes PLN1, PLN2, which may
represent the constituent faces of a compound face such as that of
FIG. 42. The planes have respective normal vectors, N1 and N2,
which are perpendicular to the respective plane. Angular
misalignment as depicted schematically in FIG. 42 occurs when N1
and N2 deviate slightly from perfect parallelism. The angular
deviation may occur along any direction or plane. In exemplary
embodiments, precision machining techniques can be used to ensure
the angular deviation is small, e.g., less than 10 degrees, or less
than 5 degrees, or less than 1 degree.
[0175] A perspective view of the substrate 3805 and recess 3820 of
FIG. 38 is shown in FIG. 44. Like reference numbers designate like
elements, with no further explanation being necessary.
[0176] In FIG. 45, a substrate 4505 is shown that has a structured
surface 4507 which is a negative replica of the working surface
(structured surface) 3807 of the substrate 3805 of FIGS. 38 and 44.
The substrate 4505 may thus be made by replication from the
substrate 3805. The structured surface 4507 therefore comprises a
protrusion 4520Z, which juts out of the structured surface 4507, in
place of the recess 3820 of substrate 3805. We introduce a
Cartesian coordinate system in FIG. 45 such that the x-y plane
corresponds to the plane along which the structured surface 4507
generally extends, the z-axis represents a normal vector of the
surface 4507, and the x-axis is oriented so that a rounded edge
4551Z, which corresponds inversely to a rounded edge of the recess
3820, lies in the x-z plane.
[0177] Features of the protrusion 4520Z have a one-to-one (and
inverted) relationship with corresponding features of the recess
3820. The reader will appreciate that aspects and characteristics
disclosed in connection with the recess 3820 apply in corresponding
fashion to the protrusion 4520Z. With that understanding, and
wishing to avoid unnecessary repetition, we now briefly summarize
the following features associated with the protrusion 4520Z: peak
or apex 4520Za; curved side surface 4531Z; flat faces 4541Za,
4541Zb, 4542Za, and 4542Zb; break line 4571Z between the
constituent flat faces 4541Zb and 4542Za of the compound face;
transition lines 4561Z, 4562Z; rounded edges 4551Z, and 4552Z;
terminus 4551Zt of one rounded edge and terminus 4552Zt of the
other rounded edge; and a periphery or perimeter 4520Zp.
(Transition lines that mark the boundaries between the rounded
edges and the adjacent flat surfaces are shown in the figure but
not labeled.)
[0178] The peak or apex 4520Za of the protrusion 4520Z has a small,
flat, and round shape, and it lies in a plane parallel to the x-y
plane. The apex 4520Za can be considered a vertex of the
protrusion, at least because it is located at the junction of the
rounded edges 4551Z, 4552Z.
[0179] The curved side surface 4531Z of the protrusion 4520Z has a
conical shape.
[0180] The flat faces 4541Za and 4541Zb come together or meet along
a rounded edge 4551Z. The dihedral angle between the faces 4541Za,
4541Zb (which we may refer to as 180-2*.OMEGA.1) can be measured in
any cross-sectional plane that is perpendicular to both faces, and
therefore also perpendicular to the edge 4551Z. The faces 4541Za,
4541Zb can be made to have any desired dihedral angle
180-2*.OMEGA.1, within practical limits. In cases involving cube
corner elements, the dihedral angle 180-2*.OMEGA.1 between the
faces equals 90 degrees. The faces 4541Za, 4541Zb can be said to
define respective reference planes that intersect along a line that
is inclined relative to the structured surface 4507, such line
being inclined at an angle .alpha..
[0181] The flat faces 4542Za and 4542Zb come together or meet along
a rounded edge 4552Z. The dihedral angle between the faces 4542Za,
4542Zb (which we may refer to as 180-2*.OMEGA.2) can be measured in
any cross-sectional plane that is perpendicular to both faces, and
therefore also perpendicular to the edge 4552Z. The faces 4542Za,
4542Zb can be made to have any desired dihedral angle
180-2*.OMEGA.2, within practical limits. In cases involving cube
corner elements, the dihedral angle 180-2*.OMEGA.2 between the
faces equals 90 degrees. The faces 4542Za, 4542Zb can be said to
define respective reference planes that intersect along a line that
is inclined relative to the structured surface 4507, such line
being inclined at an angle .beta..
[0182] Transition lines 4561Z, 4562Z are not physical edges, but
are provided in the figure to show where the transitions from the
curved side surface 4531Z to the flat faces 4541Za, 4542Zb
occur.
[0183] The rounded edge 4551Z is located at the place where faces
4541Za, 4541Zb come together. The rounded edge 4551Z extends along
an axis that is inclined at the angle .alpha. relative to the x-y
plane. The rounded edge 4551Z is not rounded or curved in an
arbitrary fashion, but in a particular manner as described above in
connection with FIGS. 21 and 22. Thus, a cross-section of the
rounded edge 4551Z in a plane parallel to the x-y plane is a
circular arc, i.e., it exhibits a curvature of constant radius in
that plane. In other cross-sectional planes, including a plane
perpendicular to both faces 4541Za, 4541Zb, the shape of the
rounded edge 4551Z is elliptical, thus having a variable curvature
rather than a constant curvature. The radius of curvature of the
rounded edge 4551 is generally at least 10 micrometers. The rounded
edge 4551Z is also an artifact of a rotating end mill.
[0184] Similarly, the rounded edge 4552Z is located at the place
where faces 4542Za, 4542Zb come together. The rounded edge 4552Z
extends along an axis that is inclined at the angle .beta. relative
to the x-y plane. The rounded edge 4552Z is not rounded or curved
in an arbitrary fashion, but in a particular manner as described
above in connection with FIGS. 21 and 22. Thus, a cross-section of
the rounded edge 4552Z in a plane parallel to the x-y plane is a
circular arc, i.e., it exhibits a curvature of constant radius in
that plane. In other cross-sectional planes, including a plane
perpendicular to both faces 4542Za, 4542Zb, the shape of the
rounded edge 4552Z is elliptical, thus having a variable curvature
rather than a constant curvature. The radius of curvature of the
rounded edge 4552 is generally at least 10 micrometers. The rounded
edge 4552Z is also an artifact of a rotating end mill.
[0185] The terminus 4551Zt is one of two ends of the rounded edge
4551Z. The other end or terminus of the rounded edge 4551Z is
disposed at the apex 4520Za. Similarly, the terminus 4552Zt is one
of two ends of the rounded edge 4552Z, the other end being disposed
at the apex 4520Za. Both the terminus 4551Zt and the terminus
4552Zt form part of the larger periphery or perimeter 4520Zp of the
protrusion 4520Z. The perimeter 4520Zp includes straight portions
at the lower edges of the faces 4541Za, 4541Zb, 4542Za, 4542Zb, and
a curved portion (circular arc of large radius of curvature) at the
lower edge of the curved surface 4531Z, and other curved portions
(circular arcs of small radius of curvature) corresponding to the
terminus 4551Zt and the terminus 4552Zt.
[0186] The recess 3820 and protrusion 4520Z both bear
characteristic marks of being made, directly or indirectly, by an
end milling method. Each of these structures may be useful--whether
individually or as one member of an array or group of identical or
different recesses or protrusions--when incorporated into the
structured surfaces of films, sheeting, or other articles in
optical, mechanical, or other applications.
[0187] FIG. 46 illustrates a path 4615 for a rotating end mill that
is similar in most respects to the path of FIG. 37, except that the
path of FIG. 46 is not limited to only two different vertical
planes. Instead, the path 4615 includes three segments, each of
which lies in its own, different, vertical plane. The path 4615 may
be considered in the context of one or more Cartesian coordinate
systems, in similar fashion to other figures herein. However, for
simplicity and ease of illustration, only one z-axis is included in
the figure. We assume the z-axis corresponds to a normal vector of
the working surface or structured surface of a substrate being cut
by the end mill, and that such surface extends parallel to an x-y
plane. The actual plane of the working surface (see e.g. .SIGMA. in
FIGS. 17A, 17B, and 23-27) is assumed to intersect the z-axis at a
point which we label alternatively as Az, Cz, or Dz. We also assume
the plane of the working surface contains points A, C, and D in
FIG. 46. The point Az, Cz, Dz is the projection of point A onto the
z-axis, and also the projection of the point C onto the z-axis, and
also the projection of the point D onto the z-axis. Another point,
B, is located on the path 4615 and lies beneath the plane of the
working surface. The point B is in fact the deepest (within the
substrate) of any point on the path 4615.
[0188] The path 4615 has three distinct segments or portions, one
from point A to point B (or vice versa), one from point B to point
C (or vice versa), and one from point B to point D (or vice versa).
The point B is common to, or shared by, all three segments. The A-B
segment of the path 4615 lies in a first vertical plane, and is
inclined relative to the working surface of the substrate at an
angle .alpha. (alpha, not shown in FIG. 46). The first vertical
plane contains the point A and the z-axis. A primary xyz coordinate
system may be defined in which the z-axis is as shown in FIG. 46,
and the x-axis is oriented so that the x-z plane is the first
vertical plane.
[0189] The B-C segment of the path 4615 lies in a different second
vertical plane, and is inclined relative to the working surface of
the substrate at an angle .beta. (beta, not shown in FIG. 46). The
second vertical plane contains the point C and the z-axis. A
secondary x'y'z' coordinate system may be defined in which the
z'-axis is the same as the z-axis, and the x'-axis is oriented so
that the x'-z' plane is the second vertical plane.
[0190] The B-D segment of the path 4615 lies in a different third
vertical plane, and is inclined relative to the working surface of
the substrate at an angle .gamma. (gamma, not shown in FIG. 46).
The third vertical plane contains the point D and the z-axis. A
tertiary x''y''z'' coordinate system may be defined in which the
z''-axis is the same as the z-axis, and the x''-axis is oriented so
that the x''-z'' plane is the third vertical plane.
[0191] The first and second vertical planes intersect at an angle
.PHI.1 (not shown in FIG. 46). The second and third vertical planes
intersect at an angle .PHI.2 (also not shown in FIG. 46). The first
and third vertical planes intersect at an angle .PHI.3 (also not
shown in FIG. 46). The intersection angles .PHI.1, .PHI.2, .PHI.3
can in general be selected as desired, but in many of the disclosed
embodiments the intersection angle .PHI. is at least 10 degrees. In
some cases, all three angles .PHI.1, .PHI.2, .PHI.3 are the same,
while in other cases only two of the three angles are the same,
while in still other cases all of the three angles are different
from each other.
[0192] The path 4615 disregards and omits any motion of the cutting
envelope of the rotating end mill that occurs above the surface of
the substrate. Rather, the path 4615 only includes segments that
involve cutting of the substrate. When a cutting envelope, which
has a tool half angle THA and truncated bottom as discussed above,
moves along the inclined segment A-B of the path, it forms a first
and second flat face. In fact, by the time the cutting envelope
reaches the deepest point B, it has already formed a
teardrop-shaped recess such as that shown in FIG. 21 above. If we
then cause the cutting envelope to follow the inclined segment from
B to C, the cutting envelope enlarges the recess, and in doing so,
forms third and fourth flat faces that are distinct from the first
and second flat faces, such as to provide a cavity like that of
FIG. 38 above. We may then further enlarge the substrate by causing
the cutting envelope to follow the remaining segment B-D, and in
doing so, the cutting envelope forms fifth and sixth flat faces
that are distinct from the first through fourth faces. (In general,
a first end mill (and first cutting envelope) can be used for the
segment A-B, and a second end mill (and second cutting envelope)
can be used for the segment B-C, and a third end mill (and third
cutting envelope) can be used for the segment B-D, and the first,
second and third end mills need not be, but in some cases can be,
the same. Furthermore, such first, second, and third end mills need
not have, but in some cases can have, the same orientation in terms
of polar angle .theta. and azimuthal angle .PHI. for each of the
respective cutting paths. In some cases where the end mill cutting
tool is tilted at a nonzero polar angle .theta., the cutting
envelope can be tilted in the same orientation relative to a fixed
reference mark (or coordinate system) on the structured surface for
all three cutting path segments, and in that case the azimuthal
angles .PHI. for the different cutting paths will be different,
since we assume the angles .PHI. are measured relative to the
vertical plane of each respective cutting path segment, rather than
to a single fixed vertical plane.)
[0193] The third and fourth flat faces associated with the path
segment B-C can have any desired configuration, and any degree of
alignment, or misalignment, relative to the first and second flat
faces associated with the path segment A-B, and furthermore, the
fifth and sixth flat faces associated with the path segment B-D can
have any desired configuration, and any degree of alignment or
misalignment, relative to any of the first through fourth flat
faces. However, we have found that particularly useful structures,
including cube corner elements, can be formed by configuring the
faces such that two of the faces (e.g. the second face and the
third face) form a nominally flat first compound face, and two
other faces (e.g. the fourth face and the fifth face) form a
nominally flat second compound face, and the remaining two faces
(e.g. the first face and the sixth face) form a nominally flat
third compound face. Such an embodiment is shown in FIG. 47.
[0194] FIG. 47 depicts a substrate 4705, which may be the same as
or similar to substrates discussed above, having a working surface
4707 in which is formed a recess 4720, the recess being formed by a
rotating end mill (or by two or three different rotating end mills)
as described above which moves along a path having three different
inclined segments with a shared deepest endpoint, similar to the
path 4615 of FIG. 46. (The reader is cautioned that the schematic
depiction of FIG. 46 may not correspond exactly to features of the
recess 4720 in FIG. 47; nevertheless, the recess of FIG. 47 is cut
with a three-segment path very much like the path 4615 of FIG. 46,
i.e., with three inclined paths or path portions that lie in
different vertical planes.)
[0195] The recess 4720 has a perimeter 4720p and a base 4720b, the
base corresponding to the deepest (and shared) point of the cutting
path segments. The recess 4720 has two flat faces 4741a, 4741b
which come together or meet along a rounded edge 4751. The dihedral
angle between these flat faces can be tailored as desired, but in
the illustrated embodiment it is 90 degrees. The rounded edge 4751
extends along an axis that is inclined at an angle .alpha. relative
to the x-y plane. In FIG. 47, a is assumed to be 35.264 degrees. At
such an inclination angle, the faces 4741a, 4741b can be made to
have a 90 degree dihedral angle if a cutting envelope is used in
which THA=35.264 degrees, with a polar angle .theta.=0 and an
azimuthal angle .PHI.=0. The rounded edge 4751 has a terminus
4751t, and another terminus at the base 4720b.
[0196] The recess 4720 also has two other flat faces 4742a, 4742b
which come together or meet along a second rounded edge 4752. The
dihedral angle between these flat faces can be tailored as desired,
but in the illustrated embodiment it is 90 degrees. The rounded
edge 4752 extends along an axis that is inclined at an angle .beta.
relative to the x-y plane. In FIG. 47, .beta. is assumed to be
35.264 degrees. At such an inclination angle, the faces 4742a,
4742b can be made to have a 90 degree dihedral angle if a cutting
envelope is used in which THA=35.264 degrees, with a polar angle
.theta.=0 and an azimuthal angle .PHI.=0. The rounded edge 4752 has
a terminus 4752t, and another terminus at the base 4720b.
[0197] The recess 4720 also has two other flat faces 4743a, 4743b
which come together or meet along a second rounded edge 4753. The
dihedral angle between these flat faces can be tailored as desired,
but in the illustrated embodiment it is 90 degrees. The rounded
edge 4753 extends along an axis that is inclined at an angle
.gamma. relative to the x-y plane. In FIG. 47, .gamma. is assumed
to be 35.264 degrees. At such an inclination angle, the faces
4743a, 4743b can be made to have a 90 degree dihedral angle if a
cutting envelope is used in which THA=35.264 degrees, with a polar
angle .theta.=0 and an azimuthal angle .PHI.=0. The rounded edge
4753 has a terminus 4753t, and another terminus at the base 4720b.
The base 4720 can also be considered a vertex, at least because it
is located at the junction of the rounded edges 4751, 4752, 4753.
Transition lines 4751a, 4751b, 4752a, 4752b, 4753a, 4753b mark the
boundaries between the rounded edges and the flat faces. The flat
faces, rounded edges, and transition lines have the same or similar
characteristics and properties as corresponding features of
cavities described in detail above.
[0198] By comparing the recess 4720 of FIG. 47 with the
teardrop-shaped recess 1420 shown in FIG. 18, we can appreciate
that the recess 4720 can be made in three steps, corresponding to
three inclined segments or portions of the cutting path such as are
shown in FIG. 46. In a first step, the cutting envelope of the
rotating end mill cuts into the substrate while moving along an
inclined cutting path located in the x-z plane (see e.g. segment
A-B in FIG. 46). At the end of this first step, a teardrop-shaped
recess is formed similar to, or the same as, that of FIG. 18. In a
second step, the same cutting envelope withdraws from the substrate
not vertically, but along a second inclined cutting path that is
located in a different vertical plane, see e.g. the segment B-C in
FIG. 46. During the second step, the cutting envelope enlarges the
recess and forms two additional flat faces, resulting in an
enlarged recess that may be the same as or similar to the recess
3820 of FIG. 38. In a third step, the same cutting envelope may cut
through some remaining portions of the substrate by moving the
cutting envelope along a third inclined cutting path located in
still another vertical plane, see e.g. the segment B-D in FIG. 46.
During the third step the cutting envelope further enlarges the
recess, and forms two additional flat faces, resulting in a further
enlarged recess such as the one shown in FIG. 47. In an alternative
embodiment, the first cutting envelope can be withdrawn from the
teardrop-shaped cavity in such a way as to preserve the
teardrop-shaped cavity, and then in an alternative second step, a
different second cutting envelope (e.g. having a different THA) can
be moved along the second inclined cutting path so as to form two
more (third and fourth) flat faces, and in an alternative third
step, a different third cutting envelope (e.g. having still another
THA) can be moved along the third inclined cutting path so as to
form the two final (fifth and sixth) flat faces.
[0199] The three different inclined cutting paths or path segments
lie in different vertical planes, namely: the x-z plane of FIG. 47,
which is responsible for forming the rounded edge 4751 and the flat
faces 4741a and 4741b; the x'-z' plane of FIG. 47, which is
responsible for forming the rounded edge 4752 and the flat faces
4742a and 4742b; and the x''-z'' plane of FIG. 47, which is
responsible for forming the rounded edge 4753 and the flat faces
4743a and 4743b. The x-z plane intersects the x'-z' plane at an
angle .PHI.1, the x'-z' plane intersects the x''-z'' plane at an
angle .PHI.2, and the x''-z'' plane intersects the x-z plane at an
angle .PHI.3 (these angles not labeled in FIG. 47). The maker of
the recess may control the motion of the rotating end mill in any
desired way, and may thus make the intersection angles .PHI.1,
.PHI.2, .PHI.3 any desired values, with the constraint that
.PHI.1+.PHI.2+.PHI.3=360 degrees. In the embodiment of FIG. 47 we
have selected .PHI.1=.PHI.2=.PHI.3=120 degrees. Given our selected
geometries for the various flat faces 4741a, 4741b, 4742a, 4742b,
4743a, 4743b, these intersection angles ensure that: the face 4742a
is in substantial alignment with the face 4741b to form a first
compound face; the face 4742b is in substantial alignment with the
face 4743a to form a second compound face; and the face 4743b is in
substantial alignment with the face 4741a to form a third compound
face, where each of these compound faces is similar to or the same
as the compound face described above in connection with FIGS.
39-43. The constituent faces 4742a and 4741b of the first compound
face meet along a first break line 4771, the faces 4742b and 4743a
of the second compound face meet along a second break line 4772,
and the faces 4743b and 4741a of the third compound face meet along
a third break line 4773. Each of the break lines 4771, 4772, and
4773 extends from the base 4720b to the perimeter 4720p of the
recess 4720.
[0200] A perspective view of the substrate and recess of FIG. 47 is
provided in FIG. 48, where like reference numbers designate like
elements, and no further explanation being necessary.
[0201] The reader who is familiar with cube corner literature will
immediately recognize that the recess 4720 is substantially similar
in appearance to an uncanted, truncated cube corner element. And
since in the embodiment of FIGS. 47-48 we have designed the various
pairs of flat faces to intersect each other at 90 degree dihedral
angles, the first, second, and third compound faces are mutually
perpendicular, and the recess 4720 is therefore also a cube corner
element 4780. The first, second, and third compound faces are the
optical faces of the cube corner element 4780. The rounded edges
4751, 4752, 4753 are the dihedral edges of the cube corner element,
and the non-dihedral edges are the three sides of the
triangular-shaped perimeter 4720p, which constitutes the base
triangle of the cube corner element 4780. Note that the interior
angles of the base triangle are all 60 degrees, in accordance with
an uncanted cube corner element.
[0202] We have thus demonstrated a technique for forming a cube
corner element in a substrate using nothing more than a rotating
end mill having a conical cutting envelope. Moreover, all flat
faces (both the constituent faces and the compound faces) of the
cube corner element can be made with a single end mill cutting
tool, and with cutting steps in which the rotational axis of the
end mill is at all times perpendicular to the working surface of
the substrate. Of course, these statements should not be
interpreted in an unduly restrictive manner, since other
embodiments disclosed herein may have only some, or only one flat
face of the cube corner element being made with a rotating end
mill, and some embodiments may have one or some flat faces formed
with a first rotating end mill (having a first cutting envelope),
and another or other flat faces formed with a different second
rotating end mill (having a different second cutting envelope), and
so forth, and some embodiments can use rotating end mills in which
the axis of rotation is tilted relative to the working surface of
the substrate.
[0203] The fact that the dihedral edges of the cube corner element
4780 are rounded rather than sharp, and that the vertex or base
4720b is truncated rather than pointed, will have ramifications on
the optical performance of the cube corner element. In particular,
these deviations from sharpness will reduce the total light return
of the cube corner element 4780 compared to another cube corner
element that is the same except for the deviations from sharpness.
However, the reduction in total light return can be managed by
keeping the dimension Dbot (see FIG. 5C), and the associated radius
of curvature of the rounded edges, as small as possible relative to
the other dimensions of the cube corner element. Furthermore, as
shown further below, the same end milling techniques used to make
the cube corner element 4780 can be used to make alternative cube
corner elements, including PG cube corner elements, in which the
rounded edges occur at the non-dihedral edges, rather than at the
dihedral edges, of the cube corner element.
[0204] The reader will appreciate that the cube corner element 4780
can be made in any desired size (e.g. as measured by the depth of
the base 4720b, or by a lateral dimension such as a diameter of the
perimeter 4720p), limited only by the size or thickness of the
substrate, and the size of available end mill cutting tools. The
reader will also appreciate that numerous cube corner elements,
whether of the same size or different sizes, can easily be made in
the same working surface or substrate, simply by executing
additional cutting operations with the rotating end mill.
Furthermore, any one, or some, or all of the additional cube corner
elements formed in the same substrate may have the same orientation
as cube corner element 4780 relative to the xyz coordinate system
of FIGS. 47-48, or they may be oriented differently, e.g. as
associated with any arbitrary amount of rotation about the z-axis
in FIGS. 47-48. Stated differently, the disclosed techniques give
the designer the freedom to form additional cube corner elements
that are adjacent to the cube corner element 4780, but at different
orientations (e.g. by selecting vertical planes that are different
from the set of vertical planes used to make the element 4780),
such that none of the optical faces (compound faces) of the cube
corner element 4780 are coplanar (not even nominally coplanar) with
any of the optical faces of its adjacent or neighboring cube corner
element(s). Furthermore, it will be apparent from the teachings
herein that one, some, or all of the recesses or structures need
not have triangular base perimeters, and need not be cube corner
elements. The base of a given structure may for example have a
square, hexagonal, or pentagonal shape, or a more complex shape
such as a star shape.
[0205] The reader will also appreciate that by changing one or some
of the cutting parameters, e.g., by changing the THA of the cutting
envelope, or by changing the inclination angle of the cutting path
segments, the same technique that is used to make the cube corner
element 4780 can be used to make other prismatic structures that
are not cube corner elements but that may resemble cube corner
elements, insofar as they may comprise a triangular base perimeter
and three triangular compound faces, but whose compound faces
deviate greatly from being mutually perpendicular. Such alternative
prismatic structures may be useful in light management films other
than cube corner sheeting, or in mechanical applications such as
abrasive or non-stick films, for example.
[0206] A perspective view of a substrate 4905 whose structured
surface 4907 is a negative replica of that of FIG. 48 is shown in
FIG. 49. The substrate 4905 may thus be made by replication from
the substrate 4705. The structured surface 4907 therefore comprises
a protrusion 4920Z, which juts out of the structured surface 4907,
in place of the recess 4720 of substrate 4705. Since the recess
4720 is a cube corner element (cavity) 4780, the protrusion 4920Z
is also a cube corner element (protrusion) 4980.
[0207] We introduce a Cartesian coordinate system in FIG. 49 such
that the x-y plane corresponds to the plane along which the
structured surface 4907 generally extends, the z-axis represents a
normal vector of the surface 4907, and the x-axis is oriented so
that a rounded edge 4951Z, which corresponds inversely to a rounded
edge of the recess 4720, lies in the x-z plane.
[0208] Features of the protrusion 4920Z have a one-to-one (and
inverted) relationship with corresponding features of the recess
4720. The reader will appreciate that aspects and characteristics
disclosed in connection with the recess 4720 apply in corresponding
fashion to the protrusion 4920Z. With that understanding, and
wishing to avoid unnecessary repetition, we now briefly summarize
the following features associated with the protrusion 4920Z: peak
or apex 4920Za; flat faces 4941Zb and 4942Za (which form a first
compound face), 4942Zb and 4943Za (which form a second compound
face), and 4943Zb and 4941Za (which form a third compound face);
break lines 4971Z, 4972Z, and 4973Z between the constituent flat
faces of the three compound faces; rounded edges 4951Z, 4952Z, and
4953Z; terminuses 4951Zt, 4952Zt, and 4953Zt of the respective
rounded edges; and a periphery or perimeter 4920Zp. Transition
lines that mark the boundaries between the rounded edges and the
adjacent flat surfaces are shown in the figure but not labeled.
[0209] The peak or apex 4920Za of the protrusion 4920Z has a small,
flat, and round shape, and it lies in a plane parallel to the x-y
plane. The apex 4920Za can be considered a vertex of the
protrusion, at least because it is located at the junction of the
rounded edges 4951Z, 4952Z, 4953Z.
[0210] The flat faces 4941Za and 4941Zb come together or meet along
a rounded edge 4951Z. The dihedral angle between the faces 4941Za,
4941Zb (which we may refer to as 180-2*.OMEGA.1) can be measured in
any cross-sectional plane that is perpendicular to both faces. The
faces 4941Za, 4941Zb can be made to have any desired dihedral angle
180-2*.OMEGA.1, but in the depicted embodiment in which the
protrusion 4920z is a cube corner element, the dihedral angle
equals 90 degrees. The faces 4941Za, 4941Zb can be said to define
respective reference planes that intersect along a line that is
inclined relative to the structured surface 4907, such line being
inclined at an angle .alpha..
[0211] The flat faces 4942Za and 4942Zb come together or meet along
a rounded edge 4952Z. The dihedral angle between the faces 4942Za,
4942Zb (which we may refer to as 180-2*.OMEGA.2) can be measured in
any cross-sectional plane that is perpendicular to both faces. The
faces 4942Za, 4942Zb can be made to have any desired dihedral angle
180-2*.OMEGA.2, but in the depicted embodiment in which the
protrusion 4920z is a cube corner element, the dihedral angle
equals 90 degrees. The faces 4942Za, 4942Zb can be said to define
respective reference planes that intersect along a line that is
inclined relative to the structured surface 4907, such line being
inclined at an angle .beta..
[0212] The flat faces 4943Za and 4943Zb come together or meet along
a rounded edge 4953Z. The dihedral angle between the faces 4943Za,
4943Zb (which we may refer to as 180-2*.OMEGA.3) can be measured in
any cross-sectional plane that is perpendicular to both faces. The
faces 4943Za, 4943Zb can be made to have any desired dihedral angle
180-2*.OMEGA.3, but in the depicted embodiment in which the
protrusion 4920z is a cube corner element, the dihedral angle
equals 90 degrees. The faces 4943Za, 4943Zb can be said to define
respective reference planes that intersect along a line that is
inclined relative to the structured surface 4907, such line being
inclined at an angle .gamma..
[0213] The rounded edge 4951Z is located at the place where faces
4941Za, 4941Zb come together. The rounded edge 4951Z extends along
an axis that is inclined at the angle .alpha. relative to the x-y
plane. The rounded edge 4951Z is not rounded or curved in an
arbitrary fashion, but in a particular manner as described above in
connection with FIGS. 21 and 22. The radius of curvature of the
rounded edge 4951 is generally at least 10 micrometers. The rounded
edge 4951Z is also an artifact of a rotating end mill.
[0214] Similarly, the rounded edge 4952Z is located at the place
where faces 4942Za, 4942Zb come together. The rounded edge 4952Z
extends along an axis that is inclined at the angle .beta. relative
to the x-y plane. The rounded edge 4952Z is not rounded or curved
in an arbitrary fashion, but in a particular manner as described
above in connection with FIGS. 21 and 22. The radius of curvature
of the rounded edge 4952 is generally at least 10 micrometers. The
rounded edge 4952Z is also an artifact of a rotating end mill.
[0215] Similarly, the rounded edge 4953Z is located at the place
where faces 4943Za, 4943Zb come together. The rounded edge 4953Z
extends along an axis that is inclined at the angle .gamma.
relative to the x-y plane. The rounded edge 4953Z is not rounded or
curved in an arbitrary fashion, but in a particular manner as
described above in connection with FIGS. 21 and 22. The radius of
curvature of the rounded edge 4953 is generally at least 2
micrometers. The rounded edge 4953Z is also an artifact of a
rotating end mill.
[0216] The terminus 4951Zt is one of two ends of the rounded edge
4951Z, the other end being disposed at the apex 4920Za. The
terminus 4952Zt is similarly one of two ends of the rounded edge
4952Z, the other end being disposed at the apex 4920Za. The
terminus 4953Zt is one of two ends of the rounded edge 4953Z, the
other end being disposed at the apex 4920Za. All three terminuses
4951Zt, 4952Zt, and 4953Zt form part of the larger periphery or
perimeter 4920Zp of the protrusion 4920Z. The perimeter 4920Zp
includes straight portions at the lower edges of the compound
faces, and curved portions (circular arcs of small radius of
curvature) at the terminuses of the rounded edges.
[0217] The recess 4720 and protrusion 4920Z both bear
characteristic marks of being made, directly or indirectly, by an
end milling method. Each of these structures may be useful--whether
individually or as one member of an array or group of identical or
different recesses or protrusions--when incorporated into the
structured surfaces of films, sheeting, or other articles in
optical, mechanical, or other applications.
[0218] As mentioned above, the disclosed end milling techniques can
be used to form numerous cube corner elements and other structures
in the working surface of a substrate. FIG. 50 is a view of a
structured surface in which two recesses 5020, 5021 have been
formed adjacent to each other using the disclosed techniques. We
assume the faces of these recesses are configured to be mutually
perpendicular to each other such that the recess 5020 is a cube
corner element 5080 and the recess 5021 is a cube corner element
5081, but alternative embodiments are also contemplated in which
the mutual perpendicularity condition is not satisfied, and the
recesses are not cube corner elements. The structured surface
extends along an x-y plane, and has a normal vector which we can
associate with a z-axis of a coordinate system.
[0219] The cube corner element 5080 is assumed to be the same as,
or similar to, the cube corner element 4780. A direct comparison
can be made, for example, between the plan view of element 5080 in
FIG. 50 and the plan view of element 4780 in FIG. 47. The labeled
elements of the cube corner element 5080--the perimeter 5020p,
vertex or base 5020b, flat faces 5041a, 5041b, 5042a, 5042b, 5043a,
5043b which form compound faces with break lines 5071, 5072, and
5073, and rounded edges 5051, 5052, 5053--all have direct
counterparts in the cube corner element 4780, and need no further
explanation.
[0220] The cube corner element 5081 is assumed to be a cube corner
element that is substantially the same as cube corner element 5080,
but having an orientation that is rotated 180 degrees about the
z-axis. Stated differently, the cube corner elements 5080, 5081
have corresponding features, and those features are related to each
other by a 180 degree rotation about an axis parallel to the
z-axis. Such structures are referred to as a matched pair of cube
corner elements. Therefore, the labeled elements of the cube corner
element 5081--the perimeter 5021p, vertex or base 5021b, flat faces
5044a, 5044b, 5045a, 5045b, 5046a, 5046b which form compound faces
with break lines 5074, 5075, and 5076, and rounded edges 5054,
5055, 5056--all have direct counterparts in the cube corner element
5080 (and cube corner element 4780), and need no further
explanation.
[0221] The cube corner elements 5080, 5081 are shown to be spaced
apart slightly such that a small narrow strip (flat area) appears
between two straight sides of their respective perimeters. In
alternative embodiments, the elements 5080, 5081 can be formed
closer together such that the compound face formed by faces 5042b,
5043a (in cube corner element 5080) and the compound face formed by
the faces 5045b, 5046a (in cube corner element 5081) intersect each
other to form a sharp edge.
[0222] FIG. 51 is a view of a structured surface in which numerous
recesses 5120, 5121, 5122, 5123, 5124, 5125, and so forth have been
formed adjacent to each other using the disclosed techniques, e.g.,
as describe in connection with FIG. 47. The structured surface
extends along an x-y plane, and has a normal vector which we can
associate with a z-axis of a coordinate system. We assume the faces
of the recesses are configured to be mutually perpendicular to each
other such that the recesses form respective cube corner elements
5180, 5181, 5182, 5183, 5184, 5185, and so forth, but alternative
embodiments are also contemplated in which the mutual
perpendicularity condition is not satisfied, and the recesses are
not cube corner elements. The cube corner elements can be
configured as matched pairs of cube corner elements as described in
connection with FIG. 50, e.g. with cube corner elements 5180, 5181
forming a first matched pair, cube corner elements 5182, 5183
forming a second matched pair, cube corner elements 5184, 5185
forming a third matched pair, and so forth. Adjacent cube corner
elements are shown to be spaced apart slightly such that a small
narrow strip (flat area) appears between two straight sides of
their respective perimeters. In alternative embodiments, some or
all adjacent pairs of the cube corner elements can be formed closer
together such that the compound faces of adjacent cube corner
elements intersect each other to form sharp edges.
[0223] The disclosed techniques for forming cube corner elements
and similar structures are readily amenable to tailoring the shapes
and orientations of the various flat faces--by tailoring the shape
of the cutting envelope of the rotating end mill, or the geometry
of the inclined cutting paths, or both--so as to produce
reconfigured or modified structures. We can use such design tools
to create canted cube corner elements, and PG cube corner
elements.
[0224] FIG. 52 is a schematic plan view of some recesses 5220, 5221
formed in a structured surface using the disclosed end milling
techniques, the recesses being a matched pair of backward canted,
truncated cube corner elements 5280, 5281. The structured surface
extends along an x-y plane, and has a normal vector which we can
associate with a z-axis of a coordinate system.
[0225] By comparing FIG. 52 with FIG. 50, and with the benefit of
knowing how the cube corner elements of FIG. 50 are made using as a
basic building block the teardrop-shaped structure shown in FIG.
18, we can immediately understand the process by which the modified
configuration of FIG. 52 can be made using the disclosed end
milling techniques.
[0226] The cube corner element 5280 of FIG. 52 has the following
labeled elements: a perimeter 5220p; a vertex or base 5220b; flat
faces 5241a, 5241b, 5242a, 5242b, 5243a, 5243b, which form first,
second, and third compound faces with respective break lines 5271,
5272, and 5273; and rounded edges 5251, 5252, 5253. All of these
elements have counterparts in the cube corner element 5080 of FIG.
50, even though the counterpart elements are not identical. Having
the benefit of the prior description, the reader will recognize
that the rounded edge 5251 and the faces 5241a, 5241b are formed in
a first cutting step by moving a first end mill cutting envelope
along a first inclined path segment (disposed in a first vertical
plane) at an inclination angle .alpha., and the rounded edge 5252
and the faces 5242a, 5242b are formed in a second cutting step by
moving a second end mill cutting envelope along a second inclined
path segment (disposed in a second vertical plane) at an
inclination angle .beta., and the rounded edge 5253 and the faces
5243a, 5243b are formed in a third cutting step by moving a third
end mill cutting envelope along a third inclined path segment
(disposed in a third vertical plane) at an inclination angle
.gamma., where the first, second, and third inclined path segments
share a common deepest point as illustrated in FIG. 46, the deepest
point corresponding to the base 5220b.
[0227] The only remaining question is to determine what values to
assign to the various cutting parameters that will cause the first,
second, and third compound faces to form a cube corner element in
which the symmetry axis of the cube is tilted in the backward
direction, i.e., such that the base triangle or perimeter of the
cube corner element has two interior angles greater than 60
degrees. This is a straightforward geometrical exercise. We arrive
at the following answers to provide the cube corner element 5280
with an 8.826 degree cant in the backward direction, such a cube
corner element having a base triangle with interior angles of (66,
66, 48) degrees:
[0228] First cutting step in the first vertical plane: [0229]
THA=35.264 degrees; [0230] Cutting envelope tilt: .theta.=8.826
degrees, .PHI.=0 degrees; [0231] Inclination angle .alpha.=26.438
degrees;
[0232] Second cutting step in the second vertical plane: [0233]
THA=35.264 degrees; [0234] Cutting envelope tilt: .theta.=8.826
degrees, .PHI.=114 degrees; [0235] Inclination angle .beta.=39.284
degrees; [0236] Intersection angle .PHI.1 between 1.sup.st and
2.sup.nd vertical planes: 114 degrees;
[0237] Third cutting step in the third vertical plane: [0238]
THA=35.264 degrees; [0239] Cutting envelope tilt: .theta.=8.826
degrees, .PHI.=-114 degrees; [0240] Inclination angle
.gamma.=39.284 degrees; [0241] Intersection angle .PHI.2 between
2.sup.nd and 3.sup.rd vertical planes: 132 degrees;
[0242] Consistent with the discussion in relation to FIG. 33, the
values given for the polar and azimuthal angles (.theta., .PHI.),
for each of these three cutting steps used to form the cube corner
element 5280, are measured relative to the local vertical plane
(e.g. an x-y plane, or an x'-y' plane, or an x''-y'' plane, similar
to those of FIG. 47) for that particular cutting step. Despite the
fact that different azimuthal angles are given for the three
cutting steps, the reader can verify based on the other angles
involved (.PHI.1, .PHI.2), and inspection of the figures, that the
orientation of the cutting envelope is in this case the same
relative to a fixed reference mark (or coordinate system) on the
structured surface for all three cutting steps and their associated
inclined cutting paths.
[0243] Using these cutting parameters, we produce the backward
canted cube corner element 5280 as described. The same cutting
steps, but carried out with a 180 degree rotation about the z-axis,
produces the other backward canted cube corner element 5281 along
with its labeled elements: a perimeter 5221p; a vertex or base
5221b; flat faces 5244a, 5244b, 5245a, 5245b, 5246a, 5246b, which
form fourth, fifth, and sixth compound faces with respective break
lines 5274, 5275, and 5276; and rounded edges 5254, 5255, 5256. The
cube corner element 5281 is a matched pair to the cube corner
element 5280, which means the element 5281 has the same degree of
cant in the backward direction and a base triangle having the same
set of interior angles as those mentioned above for element
5280.
[0244] FIG. 53 is a schematic plan view of some recesses 5320, 5321
formed in a structured surface using the disclosed end milling
techniques, the recesses being a matched pair of cube corner
elements 5380, 5381 similar to those of FIG. 52 except that they
are forward canted, rather than backward canted. The structured
surface extends along an x-y plane, and has a normal vector which
we can associate with a z-axis of a coordinate system.
[0245] Similar to FIG. 52, upon inspection of FIG. 53, and with the
benefit of the above teachings, we can immediately understand the
process by which the modified configuration of FIG. 53 can be made
using the disclosed end milling techniques.
[0246] The cube corner element 5380 of FIG. 53 has the following
elements: a perimeter 5320p; a vertex or base (not labeled); flat
faces (not labeled) that form first, second, and third compound
faces CF1, CF2, CF3 with respective break lines (not labeled); and
rounded edges (not labeled). All of these elements have
counterparts in the cube corner element 5080 of FIG. 50, and in the
cube corner element 5280 of FIG. 52, even though the counterpart
elements are not identical. Having the benefit of the prior
description, the reader will recognize that a first rounded edge
and associated faces are formed in a first cutting step by moving a
first end mill cutting envelope along a first inclined path segment
(disposed in a first vertical plane) at an inclination angle
.alpha., and a second rounded edge and associated faces are formed
in a second cutting step by moving a second end mill cutting
envelope along a second inclined path segment (disposed in a second
vertical plane) at an inclination angle .beta., and a third rounded
edge and associated faces are formed in a third cutting step by
moving a third end mill cutting envelope along a third inclined
path segment (disposed in a third vertical plane) at an inclination
angle .gamma., where the first, second, and third inclined path
segments share a common deepest point as illustrated in FIG. 46,
the deepest point corresponding to the base of the cube corner
element 5380. Faces from the first, second, and third cutting steps
meet to form the first, second, and third compound faces CF1, CF2,
CF3.
[0247] The only remaining question is to determine what values to
assign to the various cutting parameters that will cause the first,
second, and third compound faces to form a cube corner element in
which the symmetry axis of the cube is tilted in the forward
direction, i.e., such that the base triangle or perimeter of the
cube corner element has only one interior angle greater than 60
degrees. This is a straightforward geometrical exercise. We arrive
at the following answers to provide the cube corner element 5380
with a 21.781 degree cant in the forward direction, such a cube
corner element having a base triangle with interior angles of (50,
50, 80) degrees:
[0248] First cutting step in the first vertical plane: [0249]
THA=35.264 degrees; [0250] Cutting envelope tilt: .theta.=21.781
degrees, .PHI.=180 degrees; [0251] Inclination angle .alpha.=57.045
degrees;
[0252] Second cutting step in the second vertical plane: [0253]
THA=35.264 degrees; [0254] Cutting envelope tilt: .theta.=21.781
degrees, .PHI.=-50 degrees; [0255] Inclination angle .beta.=22.622
degrees; [0256] Intersection angle .PHI.1 between 1.sup.st and
2.sup.nd vertical planes: 130 degrees;
[0257] Third cutting step in the third vertical plane: [0258]
THA=35.264 degrees; [0259] Cutting envelope tilt: .theta.=21.781
degrees, .PHI.=50 degrees; [0260] Inclination angle .gamma.=22.622
degrees; [0261] Intersection angle .PHI.2 between 2.sup.nd and
3.sup.rd vertical planes: 100 degrees;
[0262] Consistent with the discussion in relation to FIG. 33, the
values given for the polar and azimuthal angles (.theta., .PHI.),
for each of these three cutting steps used to form the cube corner
element 5380, are measured relative to the local vertical plane
(e.g. an x-y plane, or an x'-y' plane, or an x''-y'' plane, similar
to those of FIG. 47) for that particular cutting step. Despite the
fact that different azimuthal angles are given for the three
cutting steps, the reader can verify based on the other angles
involved (.PHI.1, .PHI.2), and inspection of the figures, that the
orientation of the cutting envelope is in this case the same
relative to a fixed reference mark (or coordinate system) on the
structured surface for all three cutting steps and their associated
inclined cutting paths.
[0263] Using these cutting parameters, we produce the forward
canted cube corner element 5380 as described. The same cutting
steps, but carried out with a 180 degree rotation about the z-axis,
produces the other forward canted cube corner element 5381 along
with the following elements: a perimeter 5321p; a vertex or base
(not labeled); flat faces (not labeled) that form fourth, fifth,
and sixth compound faces CF4, CF5, CF6 with respective break lines
(not labeled); and rounded edges (not labeled). The cube corner
element 5381 is a matched pair to the cube corner element 5380,
which means the element 5381 has the same degree of cant in the
forward direction and a base triangle having the same set of
interior angles as those mentioned above for element 5380.
[0264] The cube corner elements of FIGS. 47-53 are all truncated
cube corner elements. The disclosed end milling techniques can be
extended to fabricate PG cube corner elements, and similar
structures. The group of FIGS. 54-56A demonstrates one way this can
be done.
[0265] We begin with FIG. 54, which illustrates a substrate 5405
having a working or structured surface 5407 that extends parallel
to an x-y plane, the working surface also having a normal vector
corresponding to the z-axis of the coordinate system. Three
distinct recesses 5420, 5421, and 5422 are already formed in the
working surface by the disclosed end milling cutting steps.
[0266] The recess 5420 may be the same as or similar to the recess
3820 described previously, as can be appreciated by a comparison of
FIG. 38 to the recess 5420 of FIG. 54. Thus, we need only point out
selected elements of recess 5420-a perimeter 5420p, a vertex or
base 5420b, a curved side surface 5431, flat faces 5441a, 5441b,
5442a, 5442b, rounded edges 5451, 5452 inclined at respective
inclination angles .alpha.1, .beta.1, transition lines 5461, 5462,
and a break line 5471 along which the faces 5441b, 5442a meet to
form a compound face CF1--and note that they have direct
counterparts in the elements of the previously described recess
3820.
[0267] Two other recesses 5421, 5422, which are identical to the
recess 5420 except for rotations about the z-axis, are also
provided. Elements of recess 5421 include: a perimeter 5421p, a
vertex or base 5421b, a curved side surface 5432, flat faces 5443a,
5443b, 5444a, 5444b, rounded edges 5453, 5454 inclined at
respective inclination angles .alpha.2, .beta.2, transition lines
5463, 5464, and a break line 5472 along which the faces 5443b,
5444a meet to form a compound face CF2. Elements of recess 5422
include: a perimeter 5422p, a vertex or base 5422b, a curved side
surface 5433, flat faces 5445a, 5445b, 5446a, 5446b, rounded edges
5455, 5456 inclined at respective inclination angles .alpha.3,
.beta.3, transition lines 5465, 5466, and a break line 5473 along
which the faces 5445b, 5446a meet to form a compound face CF3. All
of these elements have direct counterparts to elements in the
recess 5420 (as well as to elements in the recess 3820 of FIG. 38),
and thus no further explanation of these elements is necessary.
[0268] The recesses 5420, 5421, 5422 are substantially identical to
each other except for their relative rotations about the z-axis.
The recess 5421 is rotated 120 degrees clockwise relative to recess
5420, and the recess 5422 is rotated 120 degrees clockwise relative
to recess 5421. With these relative orientations, the compound
faces CF1, CF2, and CF3 are mutually orthogonal to each other, but
are separated from each other by a central triangular region
bounded by straight portions of the three perimeters 5420p, 5421p,
5422p. The three compound faces can be used to form a PG cube
corner element if the three recesses are brought closer
together.
[0269] The substrate 5505 of FIG. 55 has a working surface 5507
similar to that of the substrate 5405, but where the recesses have
been brought closer together so the compound faces contact each
other in some places, but are separated from each other in other
places by a (smaller) central triangular region. The working
surface 5507 extends parallel to an x-y plane, and has a normal
vector corresponding to the z-axis of the coordinate system. The
same three recesses 5420, 5421, and 5422 that were formed as
distinct in FIG. 54 have been formed again in the substrate 5505 by
the end milling technique, except that the recesses have been
brought or formed closer together such that they partially overlap,
while maintaining their relative rotational orientations.
[0270] When the three recesses are formed closer to each other, the
cutting of the end mill partially destroys or obliterates some
features of a first recess as it moves along cutting paths to form
a later second recess, and other portions of the first and second
recesses are destroyed or obliterated as the rotating end mill
moves along still other cutting paths to form the third and final
recess. The destruction or obliteration results in the formation of
new, sharp edges at places where the overlapping recesses
intersect.
[0271] This can be seen in FIG. 55. The recesses have been formed
closer together such that the bases 5420b, 5421b, 5422b are closer
together, and the central triangular region of separation is
smaller. A remnant of the recess 5420, labeled 5420', is still
intact, as is a remnant of the recess 5421, labeled 5421', and a
remnant of the recess 5422, labeled 5422'. These partial recesses
5420', 5421', and 5422' combine to form a single larger recess
5520. Portions of the compound faces CF1, CF2, CF3 have been
destroyed. At some places, the compound faces now intersect each
other to form sharp edges SE12a, SE23a, and SE13a. The rounded
edges also intersect each other to form sharp edges SE12b, SE23b,
and SE13b. Break lines 5574, 5575, 5576 also form as outer compound
faces (not separately labeled) are created from other faces of the
now-overlapping recesses.
[0272] The substrate of FIG. 56A represents the final stage of
bringing the previously separate recesses together. The substrate
5605 of FIG. 56A has a working surface 5607 similar to those of the
substrates 5405 and 5505, the working surface 5607 extending
parallel to the x-y plane, and having a normal vector corresponding
to the z-axis. The same three recesses 5420, 5421, and 5422 that
were formed as distinct in FIG. 54, and partially overlapping in
FIG. 55, have been formed again in the substrate 5605 by the end
milling technique, except that the recesses have been brought or
formed even closer together so they partially overlap even more,
while still maintaining their relative rotational orientations.
[0273] In the working surface 5607, by bringing the recesses even
closer together, the compound faces CF1, CF2, CF3 come into full
contact with each other, with no longer any central triangular
region of separation between them. Instead, the three compound
faces, and their respective break lines 5471, 5472, 5473 (see e.g.
FIG. 54 or 55), come together at a central point P, the point P
being a local maxima of the structured surface. The compound faces
CF1, CF2, CF3 also intersect each other along three sharp edges,
which also come together at the point P.
[0274] The bases 5420b, 5421b, 5422b are closer together in FIG.
56A than in FIG. 54 or 55. Remnants of the recesses 5420, 5421, and
5422 combine to form a single larger recess 5620. Break lines 5674,
5675, 5676 appear in outer compound faces (not separately labeled)
that are created from other faces of the original three recesses.
Portions of the original compound faces have been destroyed, but
the compound faces CF1, CF2, CF3 that remain are each 4-sided
(quadrilateral), are mutually orthogonal, and intersect each other
along sharp, dihedral edges. Together, the compound faces CF1, CF2,
CF3 of FIG. 56A form a cube corner element 5680. The six inclined
rounded edges form non-dihedral edges of the cube corner element,
and form a boundary or perimeter in plan view that is hexagonal, as
seen in the figure. Such a cube corner element 5680 is a PG cube
corner element. The cube corner element has a symmetry axis that
passes through the point P and is parallel to the z-axis, i.e., the
cube corner element 5680 is uncanted. Given that the point P is a
relative maximum in the structured surface, i.e., it is a peak or
apex relative to its immediate surroundings, the cube corner
element 5680 is thus a cube corner protrusion, despite the fact
that it resides at the center of a recess 5620, and despite the
fact that it can be formed entirely by material removal steps of
the rotating end mill, without any replication, or any negative or
inverted copies.
[0275] The substrate 5605' of FIG. 56B has a working surface 5607'
that is the same as or similar to that of the substrate 5605 of
FIG. 56A, except for the orientation of three break lines located
between the perimeter of the recess and the perimeter of the cube
corner element 5680. In FIG. 56A, the recess is labeled 5620, and
the break lines are labeled 5674, 5675, 5676. These break lines
assume the rotating cutting tool does not completely withdraw from
the substrate as it moves along the inclined cutting paths used to
form the rounded edges. In contrast, the alternative break lines
5674', 5675', 5676' of FIG. 56B (which provide an alternative
recess 5620') do assume the rotating cutting tool completely
withdraws from the substrate as it moves along the inclined cutting
paths used to form the rounded edges. In all other respects, the
working surfaces 5607, 5607' are the same.
[0276] In a straightforward fashion, the end mill cutting steps
used to form the PG cube corner element 5680 of FIG. 56A or 56B can
be stepped and repeated in a given substrate to form a structured
surface comprising an array of closely-packed PG cube corner
elements. Such a structured surface is shown in FIG. 57. In that
figure, rounded edges and flat faces have been formed by the
disclosed end milling techniques in the same manner as described in
connection with FIGS. 54-56A, but where the cutting steps are
repeated in an orderly fashion to form an array of the resulting PG
cube corner elements. Thus, the structured surface includes a PG
cube corner element 5780 comprising compound faces CF1, CF2, CF3
which meet at a point P1 (an apex), another PG cube corner element
5781 comprising compound faces CF4, CF5, CF6 which meet at a point
P2 (an apex), another PG cube corner element 5782 comprising
compound faces CF7, CF8, CF9 which meet at a point P3 (an apex),
and another PG cube corner element 5783 comprising compound faces
CF10, CF11, CF12 which meet at a point P4 (an apex). The number of
such PG cube corner elements that can be combined in an array is
not limited.
[0277] The PG cube corner elements of FIGS. 54-57 are all uncanted
PG cube corner elements. The disclosed end milling techniques can
be extended to fabricate canted PG cube corner elements, and
similar structures. The pair of FIGS. 58-59 demonstrates one way
this can be done.
[0278] FIG. 58 illustrates a substrate 5805 having a working or
structured surface 5807 that extends parallel to an x-y plane, the
working surface also having a normal vector corresponding to the
z-axis of the coordinate system. Three distinct recesses 5820,
5821, and 5822 are formed in the working surface by the disclosed
end milling cutting steps. In a similar manner to the way in which
the individual recesses of FIG. 54 were brought closer together to
form the PG cube corner element of FIG. 56A, the recesses 5820,
5821, and 5822 of FIG. 58 are configured with compound faces,
rounded edges, and other elements in such a way that, when the
recesses are brought together and made to overlap as was done in
FIG. 56A, the overlapping (and interfering) features produce a
canted PG cube corner element.
[0279] The recess 5820 of FIG. 58 includes: a perimeter 5820p; a
vertex or base 5820b; a curved side surface 5831; flat faces 5841a,
5841b, 5842a, 5842b; rounded edges 5851, 5852 inclined at
respective inclination angles .alpha.1, .beta.1; transition lines
5861, 5862; and a break line 5871 along which the faces 5841b,
5842a meet to form a compound face CF1.
[0280] The recess 5821 of FIG. 58 includes: a perimeter 5821p; a
vertex or base 5821b; a curved side surface 5832; flat faces 5843a,
5843b, 5844a, 5844b; rounded edges 5853, 5854 inclined at
respective inclination angles .alpha.2, .beta.2; transition lines
5863, 5864; and a break line 5872 along which the faces 5843b,
5844a meet to form a compound face CF2.
[0281] The recess 5822 of FIG. 58 includes: a perimeter 5822p; a
vertex or base 5822b; a curved side surface 5833; flat faces 5845a,
5845b, 5846a, 5846b; rounded edges 5855, 5856 inclined at
respective inclination angles .alpha.3, .beta.3; transition lines
5865, 5866; and a break line 5873 along which the faces 5845b,
5846a meet to form a compound face CF3.
[0282] The recesses 5820, 5821, and 5822 are similar to the
recesses 5420, 5421, 5422, respectively, and have elements that are
counterparts, even though the counterpart elements are not
identical. The reader will recognize that it is a straightforward
geometrical problem to determine how to modify the cutting
parameters of the recesses so as to reorient the compound faces
appropriately to form a (PG) cube corner element that is suitably
canted. We have determined the following cutting parameters will
provide compound faces and other features of the recesses such that
when they are brought together, they form a PG cube corner element
with a 9.985 degree cant (which cant angle would produce a base
triangle of (49, 61, 70) in a corresponding truncated cube corner
element):
[0283] Recess 5820--first cutting step in a first vertical plane to
form rounded edge 5851: [0284] THA=35.264 degrees; [0285] Cutting
envelope tilt: .theta.=9.985 degrees, .PHI.=29.386 degrees; and
[0286] Inclination angle .alpha.=26.690 degrees.
[0287] Recess 5820--second cutting step in a second vertical plane
to form rounded edge 5852: [0288] THA=35.264 degrees; [0289]
Cutting envelope tilt: .theta.=9.985 degrees, .PHI.=148.386
degrees; [0290] Inclination angle .beta.=43.960 degrees; and [0291]
Intersection angle .PHI.1 between 1.sup.st and 2.sup.nd vertical
planes: 119 degrees.
[0292] Recess 5821--first cutting step in a first vertical plane to
form rounded edge 5853: [0293] THA=35.264 degrees; [0294] Cutting
envelope tilt: .theta.=9.985 degrees, .PHI.=148.386 degrees; and
[0295] Inclination angle .alpha.=43.960 degrees.
[0296] Recess 5821--second cutting step in a second vertical plane
to form rounded edge 5854: [0297] THA=35.264 degrees; [0298]
Cutting envelope tilt: .theta.=9.985 degrees, .PHI.=-80.614
degrees; [0299] Inclination angle .beta.=34.228 degrees; and [0300]
Intersection angle .PHI.1 between 1.sup.st and 2.sup.nd vertical
planes: 131 degrees. The orientation of the recess 5821 relative to
the recess 5820 is such that the first vertical plane of recess
5821 (in which edge 5853 resides) is parallel to the second
vertical plane of recess 5820 (in which edge 5852 resides).
[0301] Recess 5822--first cutting step in a first vertical plane to
form rounded edge 5856: [0302] THA=35.264 degrees; [0303] Cutting
envelope tilt: .theta.=9.985 degrees, .PHI.=29.386 degrees; and
[0304] Inclination angle .alpha.=26.690 degrees.
[0305] Recess 5822--second cutting step in a second vertical plane
to form rounded edge 5855: [0306] THA=35.264 degrees; [0307]
Cutting envelope tilt: .theta.=9.985 degrees, .PHI.=-80.614
degrees; [0308] Inclination angle .beta.=34.228 degrees; and [0309]
Intersection angle .PHI.1 between 1.sup.st and 2.sup.nd vertical
planes: 110 degrees. The orientation of the recess 5822 relative to
the recess 5820 is such that the first vertical plane of recess
5822 (in which edge 5856 resides) is parallel to the first vertical
plane of recess 5820 (in which edge 5851 resides).
[0310] Consistent with the discussion in relation to FIG. 33, the
values given for the polar and azimuthal angles (.theta., .PHI.),
for each of the above six cutting steps used to form the cube
corner element 5980, are measured relative to the local vertical
plane (e.g. an x-y plane, or an x'-y' plane, or an x''-y'' plane,
similar to those of FIG. 47) for that particular cutting step.
Despite the fact that different azimuthal angles are given for the
six cutting steps, the reader can verify based on the other angles
involved, and inspection of the figures, that the orientation of
the cutting envelope is in this case the same relative to a fixed
reference mark (or coordinate system) on the structured surface for
all six cutting steps and their associated inclined cutting
paths.
[0311] These cutting parameters provide the recesses 5820, 5821,
5822 with suitably oriented compound faces such that, when combined
by overlapping, the result is a canted PG cube corner element. The
substrate 5905 of FIG. 59 has a working surface 5907 extending
parallel to the x-y plane, and having a normal vector corresponding
to the z-axis. The same three recesses 5820, 5821, and 5822 that
were formed as distinct in FIG. 58 have been formed again in the
substrate 5905 by the end milling technique, except that the
recesses have been brought or formed closer together so they
partially overlap, while maintaining their relative rotational
orientations. Remnants of the recesses 5820, 5821, and 5822 combine
to form a single larger recess 5920.
[0312] In the working surface 5907, the compound faces CF1, CF2,
CF3 come into full contact with each other. The three compound
faces come together at a point P, the point P being a local maxima
of the structured surface. The compound faces CF1, CF2, CF3 also
intersect each other along three sharp edges, which also come
together at the point P.
[0313] Portions of the original compound faces have been destroyed,
but the compound faces CF1, CF2, CF3 that remain are each 4-sided
(quadrilateral), are mutually orthogonal, and intersect each other
along sharp, dihedral edges. Together, the compound faces CF1, CF2,
CF3 of FIG. 59 form a cube corner element 5980. The six inclined
rounded edges form non-dihedral edges of the cube corner element,
and form a boundary or perimeter in plan view that is hexagonal, as
seen in the figure. Such a cube corner element 5980 is a PG cube
corner element. The cube corner element has a symmetry axis that
passes through the point P and is tilted to the z-axis, i.e., the
cube corner element 5980 is canted. Given that the point P is a
relative maximum in the structured surface, i.e., it is a peak or
apex relative to its immediate surroundings, the cube corner
element 5980 is thus a cube corner protrusion, despite the fact
that it resides at the center of a recess 5920, and despite the
fact that it can be formed entirely by material removal steps of
the rotating end mill, without any replication, or any negative or
inverted copies.
[0314] FIGS. 60-62 illustrate other types of PG cube corner
elements. In FIG. 60, a PG cube corner element 6081 has first,
second, and third optical faces F1, F2, F3, the faces intersecting
along sharp edges SE12, SE23, SE13 as shown. The sharp edges
intersect at a point P1, which is a local peak or apex.
[0315] In FIG. 61, the cube corner element 6081 is combined with a
similar PG cube corner element 6181 to form a matched pair of cube
corner elements. The cube corner element 6181 has fourth, fifth,
and sixth optical faces F4, F5, F6, the faces intersecting along
sharp edges SE45, SE56, SE46 as shown. The sharp edges intersect at
a point P2, which is a local peak or apex.
[0316] FIG. 62 shows a perspective view of the matched pair of FIG.
61. In the view of FIG. 62, some non-optical faces of the cube
corner elements 6081, 6181 can also be seen.
[0317] FIGS. 63 through 67B illustrate how PG cube corner elements
such as those of FIGS. 60-62 can be made with the disclosed end
milling techniques. We begin with a recess 6320 as shown in FIG.
63. The recess 6320 includes: a perimeter 6320p; and a base 6320b;
curved side surfaces 6331a, 6331b; flat faces 6341a, 6341b which
meet along a rounded edge 6351, and flat faces 6342a, 6342b which
meet along a rounded edge 6352; a terminus 6351t of the rounded
edge 6351, and a terminus 6352t of the rounded edge 6352;
transition lines 6361a, 6361b, 6362a, 6362b; and transition lines
6351a, 6351b, 6352a, 6352b. These features may be the same as or
similar to corresponding features of the recess 3120 of FIG. 31,
with no further explanation being necessary. The inclination angle
of the rounded edge 6351 is 35.264 degrees, and the inclination
angle of the rounded edge 6352 is 35.264 degrees. These are the
same inclination angles of the inclined cutting paths followed by
the cutting envelope, which is assumed to have a THA=35.264 degrees
and no tilt (.theta.=0). The dihedral angle between the faces
6341a, 6341b is 90 degrees, and the dihedral angle between the
faces 6342a, 6342b is also 90 degrees.
[0318] Next, we perform another cutting operation with another
rotating end mill, as shown in FIG. 64.
[0319] We follow an inclined cutting path that corresponds to the
rounded edge 6451, at an inclination angle of 50.768 degrees, with
a cutting envelope having THA=35.264 degrees and no tilt
(.theta.=0). This cutting operation forms, in addition to the
rounded edge 6451, flat faces 6441a, 6441b, transition lines 6451a,
6451b, and terminus 6451t. The flat face 6441a forms a compound
face CF1 (labeled in FIG. 65) with face 6341a, meeting at a break
line 6471a. The flat face 6441b forms a compound face CF2 (labeled
in FIG. 65) with face 6342a, meeting at a break line 6471b. The
dihedral angle between the faces 6441a, 6441b is 131.808
degrees.
[0320] Next, we perform another cutting operation with another
rotating end mill, as shown in FIG. 65. We follow an inclined
cutting path that corresponds to the rounded edge 6551, at an
inclination angle of 50.768 degrees, with a cutting envelope having
THA=35.264 degrees and no tilt (.theta.=0). This cutting operation
forms, in addition to the rounded edge 6551, flat faces 6541a,
6541b, transition lines 6551a, 6551b, and terminus 6551t. The flat
face 6541a forms a compound face CF3 with face 6341b, meeting at a
break line 6571a. The flat face 6541b forms a compound face CF4
with face 6342b, meeting at a break line 6571b. The dihedral angle
between the faces 6541a, 6541b is 131.808 degrees.
[0321] Next, we repeat these operations to form overlapping
recesses of the same design, as shown in FIG. 66. In that figure, a
substrate 6605 has a working surface 6607 in the context of an xyz
coordinate system. First, the recess 6520 is cut into the working
surface, just as in FIG. 65. Next, substantially similar recesses
6621, 6622, and 6623 are cut so as to overlap with each other and
with the recess 6520, as shown in FIG. 66. The cutting of the
recess 6621 obliterates part of the original recess 6520 (see e.g.
FIG. 65), to provide the modified recess 6520' seen in FIG. 66.
Sharp edges are formed at the intersection of adjacent compound
faces. The recess 6621 has a base 6621b and compound faces CF5,
CF6, CF7, CF8. The recess 6622 has a base 6622b and compound faces
CF9, CF10, CF11, CF12. The recess 6623 has a base 6623b and
compound faces CF13, CF14, CF15, CF16.
[0322] Finally, we machine straight, constant depth V-grooves into
the substrate as shown in FIG. 67A. In the figure, the substrate is
labeled 6605' and the working surface is labeled 6607' to reflect
the changes made by the new V-grooves. Primes are also added to the
cavity labels and to the compound face labels to reflect their
modifications by the V-grooves. A V-groove 6716 has a flat bottom
6716b and angled side surfaces 6716a, 6716c. Another V-groove 6717,
parallel to the groove 6716 and to the y-axis, has a flat bottom
6717b and angled side surfaces 6717a, 6717c. (In alternative
embodiments, the flat bottom of either or both of V-grooves 6716,
6717 may be omitted and replaced with a sharp vertex. One or both
of the V-grooves 6716, 6717 may in this regard be made by fly
cutting rather than by end milling.) The V-groove 6716 is cut with
a THA=35.264 degrees such that the side surface 6716c is
perpendicular to the compound faces CF3', CF5', and so forth, and
forms sharp edges with such compound faces. The V-groove 6717 is
cut with a THA=35.264 degrees such that the side surface 6717a is
perpendicular to the compound faces CF4', CF6', and so forth, and
forms sharp edges with such compound faces. The structured surface
of FIG. 67A provides PG cube corner elements of the type shown in
FIGS. 60-62, the vertexes or peaks of such cube corner elements
labeled in FIG. 67A as P1, P2, P3, . . . P6. For clarity, FIG. 67B
reproduces the substrate 6605' and working surface 6607' with fewer
reference labels, and with dark outlines to identify a matched pair
of PG cube corner elements 6781, 6782.
[0323] Another way to make PG cube corner elements having
rectangular (or square) perimeters like those of FIGS. 60-62, 67A,
and 67B is illustrated by the pair of FIGS. 68A, 68B, except that
the PG cube corner elements are in the form of cavities or recesses
rather than protrusions as in FIGS. 60-62, etc., and the rounded
edges are located at dihedral edges rather than non-dihedral edges
of the cube corner elements. Thus, in FIG. 68A, a structured
surface 6807 has a plurality of recesses 6820 through 6829 formed
therein which correspond respectively to (truncated) cube corner
elements 6880 through 6889. Each of the recesses 6820, 6822, 6824,
6826, 6828 may be the same as or similar to the recess 4720
described above in connection with FIG. 47. To avoid unnecessary
repetition, reference numbers for all the various transition lines,
break lines, terminuses, and faces of such recesses are not
provided in FIG. 68A, and descriptions are also omitted as
unnecessary. Recesses 6821, 6823, 6825, 6827, and 6829
(corresponding respectively to cube corner elements 6881, 6883,
6885, 6887, and 6889) are formed in similar fashion such that cube
corner elements 6880, 6881 form a matched pair, and cube corner
elements 6882, 6883 form a matched pair, and so forth. The bases of
the various cavities, 6820b, 6821b, 6829b are labeled for
convenience to allow us to describe the arrangement or locations of
the illustrated cube corner elements. The cube corner elements
6880, 6882, etc., and their bases, are arranged in one row parallel
to the y-axis, and the cube corner elements 6881, 6883, etc., and
their bases, are arranged in another row parallel to the y-axis, as
shown.
[0324] Using the cube corner array of FIG. 68A as a reference
point, PG cube corner elements having rectangular perimeters in
plan view may be made by a combination of (1) reducing the spacing
between cube corner elements in each row an appropriate amount (and
the same amount), and (2) reducing the spacing between the two rows
of cube corner elements an appropriate amount. When these
guidelines are followed to form a new (but related) structured
surface 6807', the cutting of the end mill partially destroys or
obliterates some features of previously formed recess(es) as it
moves along cutting paths to form new recess(es). The destruction
or obliteration results in the formation of new, sharp edges at
places where the overlapping recesses intersect. The result can be
seen in the structured surface 6807' of FIG. 68B, where
corresponding features to those in FIG. 68A can be seen. The bases
6820b, 6821b, 6822b, etc. of the cavities can be seen to be more
closely spaced in both the x- and y-directions, and new sharp edges
and new break lines can be seen to define the rectangular perimeter
of each cube corner element. Two of the cube corner elements 6886',
6887' are outlined in bold, and can be seen to form a matched pair
of PG cube corner elements. One, some, or all of the cube corner
elements in an array such as is shown in FIG. 68B may be uncanted,
or forward canted, or backwards canted, or symmetrical about the
x-z plane, or asymmetrical about such plane.
[0325] Additional Discussion
[0326] The inclined cutting paths and cutting path segments and
portions that produce the rounded edges and flat faces discussed
herein can have any number of different configurations. As
illustrated and discussed above, such an inclined cutting path can
be straight, and can be segmented, with different inclined segments
lying in the same vertical plane in some cases, and lying in
different vertical planes in other cases. The inclined cutting path
can also be curved. Such a curved, inclined cutting path may have a
curvature that lies in a single vertical plane, or it may not, e.g.
as in the case of a corkscrew-shaped path or portion thereof, and
the curvature may be of constant radius or variable radius. In most
cases, redesigning an initially straight inclined cutting path by
adding a controlled amount of curvature to the path has the effect
of causing the associated surfaces formed by the cutting
envelope--which surfaces would otherwise be flat faces if a
straight cutting path were followed--to deviate from perfect
flatness and have a controlled amount of curvature, the curvature
of the surfaces being a function of how strongly or weakly curved
the inclined cutting path is. When substantially flat surfaces are
desired to be formed, any curvature of the inclined cutting path is
preferably small, i.e., having a large radius of curvature. Any
given one of the disclosed inclined straight, segmented, or curved
cutting paths can be approximated by, and replaced with, a
multitude of small individual straight or curved pieces strung
together, the collection of which approximates the given cutting
path and produces a structure having the same or similar optical,
mechanical, or other relevant properties as the original structure.
Combinations of all the foregoing inclined cutting paths, whether
curved, straight, segmented, or otherwise, are also contemplated
herein.
[0327] The angles of inclination relative to the plane of the
working surface that are discussed herein, e.g., .alpha., .beta.,
.gamma., .alpha.1, .alpha.2, etc., may be in any suitable range,
e.g. from 1 to 89 degrees, or 10 to 70 degrees, or from 20 to 60
degrees. The polar angle of tilt .theta. of the rotational axis of
the cutting envelope relative to the z-axis may in many cases be
zero. A particularly useful range of .theta. when designing and
fabricating cube corner elements, ranging from uncanted to canted,
is from 0 to 55 degrees.
[0328] It is understood that the disclosed end milling process is
typically used to make master tooling with the structured surface
on the working surface of the substrate of the master tooling. The
structured surface corresponding either to the desired
microstructured geometry in the finished article or to a negative
(inverted) copy thereof, depending upon whether the finished
article is to have protrusions or cavities (or both). It is
understood that the end mill process may be used along with other
direct machining processes to form the various microstructures on
the substrate.
[0329] To form a master tool of suitable size for forming an
article such as retroreflective sheeting, a plurality of toolings
(also referred to as tiles) are formed by electroplating the
surface of the master tool to form negative copies, subsequently
electroplating the negative copies to form positive copies,
electroplating the positive copies to form second generation
negative copies, etc. The positive copy has the same cube corner
element structure as the master tool, whereas the negative copy is
the cube cavity replica. Accordingly, a negative copy tool is
employed to make a positive copy (i.e., cube corner element)
sheeting whereas, a positive copy tool is employed to make a
negative copy (i.e., cube corner cavity) sheeting. Further,
retroreflective sheeting may comprise combination of cube corner
elements and cube corner cavity microstructures. Electroforming
techniques such as described in U.S. Pat. No. 4,478,769 (Pricone et
al.) and U.S. Pat. No. 6,159,407 (Krinke et al.) are known. Tiling
such toolings together can then assemble a master tool of the
desired size.
[0330] The tooling is then used in an embossing, molding,
extruding, cast-and-curing process to form articles. The formed
tooling may be desired, or an inverse of the formed tooling. The
articles made from molding processes using the tooling will result
in the microstructures from the tooling forming on to the
articles.
[0331] With the disclosed end milling technique, microstructures
having edges can be formed into substrates. In particular,
microstructures with sharp edges meeting at a corner can be formed.
Microstructures with three edges meeting at a corner to form a cube
corner element are useful in making retroreflective materials. The
end milling techniques described herein demonstrate the ability to
make truncated and PG cube corner elements.
[0332] Using the techniques described, numerous types of
microstructures, including both truncated and non-truncated cube
corner elements, can be fabricated with a variety of other shapes
and configurations. The microstructures may have outer perimeter
shapes in plan view selected from, for example, triangles,
rhombuses, trapezoids, rectangles, parallelograms, pentagons, and
hexagons, and combinations of these shapes. Many other perimeter
shapes, including but not limited to N-pointed star shapes, where N
may be 3, 4, 5, 6, etc., are also contemplated.
[0333] The term "sheeting" may refer to a thin piece of polymeric
(e.g., synthetic) material upon which microstructures have been
formed. Sheeting with cube-corner microstructure is used as a
retroreflective sheeting. A retroreflective sheeting is useful for
a variety of uses such as traffic signs, pavement markings, vehicle
markings, and personal safety articles, in view of its high
retroreflected brightness.
[0334] The sheeting may be of any width and length, such dimensions
only being limited by the equipment (e.g., width of the tool, width
of the slot die orifice, etc.) from which the sheeting was made.
The thickness of sheeting typically ranges from about 0.004 inches
(0.1016 mm) to about 0.10 inches (2.54 mm). Preferably the
thickness of sheeting is less than about 0.020 inches (0.508 mm)
and more preferably less than about 0.014 inches (0.3556 mm). The
sheeting may further include surface layers such as seal films or
overlays. In the case of retroreflective sheeting, the width is
typically at least 30 inches (122 cm) and preferably at least 48
inches (76 cm). The sheeting is typically continuous in its length
for up to about 50 yards (45.5 m) to 100 yards (91 m) such that the
sheeting is provided in a conveniently handled roll-good.
Alternatively, however, the sheeting may be manufactured as
individual sheets rather than as a roll-good. In such embodiments,
the sheets preferably correspond in dimensions to the finished
article. For example, the retroreflective sheeting may have the
dimensions of a standard U.S. sign (e.g., 30 inches by 30 inches
(76 cm by 76 cm)) and thus the microstructured tool employed to
prepare the sheeting may have about the same dimensions. Smaller
articles such as license plates or reflective buttons may employ
sheeting having correspondingly smaller dimensions.
[0335] The retroreflective sheet is preferably manufactured as an
integral material, i.e., wherein the cube-corner microstructure
elements are interconnected in a continuous layer throughout the
dimension of the mold, the individual elements and connections
therebetween comprising the same material. The surface of the
sheeting opposing the microprismatic surface is typically smooth
and planar, also being referred to as the "land layer". The
thickness of the land layer (i.e., the thickness excluding that
portion resulting from the replicated microstructure) is between
0.001 and 0.100 inches and preferably between 0.003 and 0.010
inches. Manufacture of such sheeting is typically achieved by
casting a fluid resin composition onto the tool and allowing the
composition to harden to form a sheet. A preferred method for
casting fluid resin onto the tool is described in U.S. Pat. No.
7,410,604 (Erickson et al.).
[0336] Optionally, however, the tool can be employed as an
embossing tool to form retroreflective articles, such as described
in U.S. Pat. No. 4,601,861 (Pricone et al.). Alternatively, the
sheeting can be manufactured as a layered product by casting the
cube-corner elements against a preformed film as taught in PCT
publication WO95/11464 (Benson Jr. et al.) and U.S. Pat. No.
3,684,348 (Rowland), or by laminating a preformed film to preformed
cube-corner elements. In doing so the individual cube-corner
elements are interconnected by the preformed film. Further, the
elements and film are typically comprised of different
materials.
[0337] In the manufacture of the sheeting, it is preferred that the
channels of the tool are roughly aligned with the direction of the
advancing tool as further described in U.S. Pat. No. 6,884,371
(Smith). Accordingly, prior to any further manufacturing steps, the
primary groove of the sheeting would be substantially parallel to
the edge of the roll of the sheeting.
[0338] A variety of suitable resin compositions for the sheeting
can be used so long as the resin can be formable into the desired
configuration. For retroreflective sheeting that will be used
outdoors, the resin is typically a transparent material that is
dimensionally stable, durable, weatherable, and readily formable
into the desired configuration. Examples of suitable materials
include acrylics, which have an index of refraction of about 1.5,
such as Plexiglas brand resin manufactured by Rohm and Haas
Company; polycarbonates, which have an index of refraction of about
1.59; reactive materials such as thermoset acrylates and epoxy
acrylates; polyethylene based ionomers, such as those marketed
under the brand name of SURLYN by E. I. Dupont de Nemours and Co.,
Inc.; (poly)ethylene-co-acrylic acid; polyesters; polyurethanes;
and cellulose acetate butyrates. Polycarbonates are particularly
suitable because of their toughness and relatively higher
refractive index, which generally contributes to improved
retroreflective performance over a wider range of entrance angles.
These materials may also include dyes, colorants, pigments, UV
stabilizers, or other additives.
[0339] When making a retroreflective sheeting, a specular
reflective coating such as a metallic coating can be placed on the
backside of the cube-corner elements. The metallic coating can be
applied by known techniques such as vapor depositing or chemically
depositing a metal such as aluminum, silver, or nickel. A primer
layer may be applied to the backside of the cube-corner elements to
promote the adherence of the metallic coating. In addition to or in
lieu of a metallic coating, a seal film can be applied to the
backside of the cube-corner elements; see, for example, U.S. Pat.
No. 4,025,159 (McGrath) and U.S. Pat. No. 5,117,304 (Huang et al.).
The seal film maintains an air interface at the backside of the
cubes that enables total internal reflection at the interface and
inhibits the entry of contaminants such as soil and/or moisture. A
layer of barrier elements may be used on a pressure sensitive
adhesive layer to form the air interface at the backside of the
cubes that enables total internal reflection at the interface and
inhibits the entry of contaminants, such as disclosed in U.S.
Patent Application Publication 2013/0034682 (Free et al.).
[0340] A separate overlay film may be utilized on the viewing
surface of the sheeting for improved (e.g., outdoor) durability or
to provide an image receptive surface. Indicative of such outdoor
durability is maintaining sufficient brightness specifications such
as called out in ASTM D4956-16b after extended durations of
weathering (e.g., 1 year, 3 years). Further, the daytime luminance
factor (CapY) is preferably greater than 30 before and after
weathering.
[0341] An adhesive layer also can be disposed behind the
cube-corner elements or the seal film to enable the cube-corner
retroreflective sheeting to be secured to a substrate. Suitable
substrates include wood, aluminum sheeting, galvanized steel,
polymeric materials such as polymethyl methacrylates, polyesters,
polyamids, polyvinyl fluorides, polycarbonates, polyvinyl
chlorides, polyurethanes, and a wide variety of laminates made from
these and other materials.
[0342] For articles formed from a substrate comprising a structured
surface made using the disclosed end milling process (or an inverse
of structured surface), the formed article will take the shape of
the structured surface of the substrate and will include the
features of the structured surface of the substrate as described
above.
[0343] For example, when using the end milling process where cuts
of different lateral direction, thickness direction, or varying
angles of the end mill axis, varying orientations of the resulting
microstructure can be formed. For example, when making a
cube-corner microstructure, dihedral angle errors can be
introduced. It may be desirable to introduce such errors to improve
the uniformity of total light return (TLR).
[0344] Therefore, it is possible to form truncated cubes that are
symmetrical or are tilted, for example forward or backward canted.
See U.S. Pat. No. 5,565,151 (Nilsen) and U.S. Pat. No. 4,588,258
(Hoopman). Canting of PG cube corner elements is described in U.S.
Pat. No. 6,015,214 (Heenan et al.).
[0345] Therefore, it is possible to form full cube corner element
or PG cube corner elements with skew and inclination. See U.S. Pat.
No. 9,470,822 (Smith).
[0346] Articles containing microstructures may be used for a
variety of applications where microstructures having generally
planar surfaces and edges are useful. For example, articles
containing microstructures can be used as abrasive materials or for
directing light or noise.
[0347] Further Applications
[0348] The above teachings can be used to make articles comprising
cube corner elements, and articles comprising arrays of cube corner
elements, that incorporate dihedral angle errors between at least
two optical faces of a given cube corner element, or between
portions of such at least two optical faces. These dihedral angle
errors can be applied to any of the cube corner elements disclosed
herein. In some cases, different dihedral angle errors can be
applied to different constituent faces of one or more compound
faces of one or more cube corner elements. In some cases, the
dihedral angle errors can be applied equally to the constituent
faces of one or more compound faces of one or more cube corner
elements. In some cases, dihedral angle errors can be applied in
patterns other than rows of cube corner elements. The above
teachings can also be used to make security articles with embedded
security features.
[0349] The security articles, which may be or comprise films or
sheetings, incorporate structured surfaces having an array of
microstructures, including microstructures made by the disclosed
end milling techniques. In some cases, the structured surface of
the security article may contain only microstructures made by the
disclosed end milling techniques. All, some, or none of the
microstructures in the array may be or comprise cube corner
elements, and the security article may be retroreflective or
otherwise optical in nature. The security feature of the security
article may thus be capable of detection by visual or optical
means. As demonstrated by the above teachings, the disclosed end
milling techniques allow for the fabrication of a structured
surface comprising an array of tens, hundreds, or thousands (or
more) of closely packed or adjacent microstructures. Due to the
fact that each microstructure in such an array can be made
individually and independently of any neighboring microstructures
(within certain limits, e.g., the formation of one recess may
overlap with or impinge upon a previously formed, neighboring
recess), the array can incorporate differences from one
microstructure to the next which are visually detectable under some
set of conditions, and the differences can be tailored to be
distributed spatially in the array to define optically detectable
indicia, the indicia thus serving as a security feature. In cases
where the microstructures are cube corner elements, the differences
may involve one or more of: different cube corner types, such as
truncated versus PG cube corner elements, or hexagonal PG cube
corner elements versus rectangular PG cube corner elements;
different degrees of cant, such as uncanted versus canted cube
corner elements, or forward canted versus backward canted cube
corner elements; different dihedral angle errors, such as cube
corner elements with no dihedral angle errors versus cube corner
elements with some dihedral angle errors, or cube corner elements
with only one or a small number of dihedral angle errors versus
cube corner elements with three or many dihedral angle errors; cube
corner elements of different sizes; and cube corner elements having
different orientations with regard to rotation about the z-axis
(surface normal direction) of the structured surface. Any of these
differences can be incorporated into a structured surface of
microstructures in a spatially patterned manner to define optically
detectable indicia serving as a security feature, such feature
being embedded in the structured surface of the article. The
differences may be designed to be small or subtle, such that the
security feature is difficult to detect, or the differences may be
large, making the security feature easier to detect.
EXAMPLES
Example 1--Uncanted Hex Cube
[0350] A full cube corner element similar to that shown in FIG. 56A
was prepared according to the following procedure: six cuts were
formed on a hard copper substrate (approximately 25.4 mm thick)
using a diamond end mill such as shown schematically in FIGS.
5A-5D, the end mill having a 36.264.degree. taper angle per side
(THA=36.264 degrees) and a tip diameter of 0.06 mm (Dbot=60
micrometers). The substrate was of the type shown in FIG. 1 and
included a working surface 107. The end mill was positioned
relative to the substrate such that the axis of rotation of the end
mill was normal to the working surface as shown in FIG. 6A. That
is, the cutting envelope was not tilted, and .theta.=0.
[0351] Once the flat tip of the end mill was set to the top surface
of the substrate, the end mill was moved away from the substrate a
calculated distance of approximately 0.042 mm above the surface.
The Z position was then set to be zero at this location. The
resulting full cube corner element was approximately 0.121 mm thick
(high).
[0352] The cuts were formed by moving the rotating end mill in an
end mill path following the Cartesian coordinates provided in Table
1 below, wherein the initial (X,Y) position (0, 0) was arbitrarily
assigned to a location of the substrate as starting point. X and Y
values follow standard Cartesian coordinate system directions. A
positive Z value was used to indicate that the end mill was moved
away from the working surface.
TABLE-US-00001 TABLE 1 Positions X Y Z 1 0 0 0.01 2 0 0 -0.081649 3
0 -0.11547 -0.163299 4 0.1 -0.173205 -0.081649 5 0.2 -0.11547
-0.163299 6 0.2 -0.11547 0.01 7 0.1 +0.173205 0.01 8 0.1 +0.173205
-0.081649 9 0.1 +0.057735 -0.163299 10 0.2 0 -0.081649 11 0.3
+0.057735 -0.163299 12 0.3 +0.057735 0.01
[0353] A first cut was formed when the end mill moved from
positions 2 to 3. A second cut was formed when the end mill moved
from positions 3 to 4. A third cut was formed when the end mill
moved from positions 4 to 5. A fourth cut was formed when the end
mill moved from positions 8 to 9. A fifth cut was formed when the
end mill moved from positions 9 to 10. A sixth cut was formed when
the end mill moved from positions 10 to 11.
[0354] A local coordinate offset was then applied so to reposition
the end mill to a new (0, 0) (X,Y) location. An array of full cube
elements was produced by repeating the described pattern 76 times
in the X direction at a pitch spacing of 0.2 mm, and 44 times in
the Y direction at a pitch spacing of -0.346410162 mm, producing an
array of microstructures having an area of approximately 15 mm by
15 mm.
Example 2--Sideways Canted Hex Cubes
[0355] An array of closely packed, sideways canted, hexagonal full
cube corner elements, each one similar to that shown in FIG. 59,
was prepared according to the following procedure: three cuts
corresponding to three non-dihedral edges of a hexagonal cube
corner element were formed on a hard copper substrate
(approximately 25.4 mm thick) using a diamond end mill such as
shown in FIG. 5D, the end mill having a 35.264.degree. THA, and a
tip diameter (Dbot) of 0.06 mm, the three cuts forming three
rounded edges and six flat faces associated with the rounded edges;
the same group of three cuts was then made at other positions in
the working surface in a step-and-repeat fashion such that adjacent
groups of the flat faces formed three compound faces arranged as
the optical faces of canted, hexagonal full cube corner elements in
a close-packed array. The substrate was similar to that shown in
FIG. 1, with a working surface 107. The rotating end mill was
tilted relative to the substrate such that the axis of rotation of
the end mill was set at a polar angle .theta. of 9.9854 degrees
relative to the surface normal of the working surface.
[0356] We designated a machine working surface coordinate system
(X,Y,Z), this coordinate system being fixed relative to the machine
working surface. The Z-axis was parallel to the surface normal of
the working surface. The X-axis was oriented such that the
rotational axis vertical plane of the end mill (i.e. the vertical
plane containing the rotational axis of the cutting envelope) was
parallel to the X-Z plane. This tilt orientation of the cutting
envelope relative to the fixed XYZ working surface coordinate
system was the same, and the end mill cutting tool was the same,
for all three cuts performed.
[0357] Once the center of the cutting tool tip was set to the top
surface of the substrate, the end mill was moved away from the
substrate a calculated distance of approximately 0.041 mm above the
surface in the Z direction. The Z position was then set to be zero
at this location.
[0358] The cuts were formed by moving the end mill in a path
following the Cartesian coordinates provided in Table 2 below while
maintaining the end mill's same tilt orientation, wherein the
initial (X,Y) position (0, 0) was arbitrarily assigned to a
location over the substrate as a starting point. X and Y values
follow Cartesian coordinate directions. A positive Z value was used
to indicate that the end mill was moved away from the working
surface.
TABLE-US-00002 TABLE 2 Position X Y Z 1 0 0 0.01 2 0 0 -0.150398 3
-0.123982 -0.076316 -0.290795 4 -0.157639 -0.279923 -0.150398 5
0.085695 -0.416956 -0.290795 6 0.085695 -0.416956 0.01
[0359] As the rotating end mill moved from position 1 to position
2, the cutting envelope burrowed partially into the substrate, for
example as seen in FIGS. 6B, 10B, and 14A. A first cut was then
made, forming a first rounded edge and two flat faces, by moving
the end mill from position 2 to a deepest point at position 3. A
second cut was then made, forming a second rounded edge and two
more flat faces, by moving the end mill from position 3 to (less
deep) position 4. A third cut was then made, forming a third
rounded edge and two more flat faces, by moving the end mill from
position 4 to another deepest point at position 5 (same depth as
position 3). The cutting envelope was then retracted from the
substrate by moving from position 5 to position 6.
[0360] This pattern of three cuts was then repeated at a different,
precisely defined positions on the working surface of the
substrate. To form a first row of cuts, the X axis was moved
relative to its initial position by 0.367316 mm, and the Y axis was
moved relative to its initial position by -0.060717 mm. A temporary
working surface coordinate system was then defined such that the
shifted (X, Y) position was deemed to be (0, 0). The cutting
envelope (with the same tilt orientation as before) was then moved
to the positions described in Table 2, forming three additional
rounded edges and six additional flat faces. The same procedure of
shifting the X and Y axes, redefining the (X, Y) origin, and making
the cuts associated with Table 2 was repeated until a first row or
string of cuts (and rounded edges) was complete.
[0361] After the first row was complete, a second row of cuts was
started, the second row having the same geometry as the first row
but shifted laterally in X and Y position. The X and Y axes were
moved relative to the initial position at the start of cutting. The
X axis was then moved relative to the initial position by 0.157639
mm and the Y axis was moved relative to the initial position by
0.279923. At this point, another temporary working surface
coordinate system was defined such that the shifted (X, Y) position
was deemed to be (0, 0), and the process of cutting another full
row of cuts (and corresponding rounded edges, etc.) was carried out
as described in the previous paragraphs. After the second row of
cuts was completed, a third row of substantially similar cuts, but
shifted in position in similar fashion, was made in the same way.
The result was an array of closely packed, sideways canted,
hexagonal full cube corner elements, each one similar to that shown
in FIG. 59.
Example 3--Non-Canted Full Cubes with Pseudo-Randomized MNO
[0362] An array of full (PG) cube corner elements similar to that
of FIG. 68B, but containing thousands of cube corner elements, and
where the cube corner elements all had rectangular perimeters whose
aspect ratio was close to 1.7, was cut in the surface of a hard
copper substrate using only a rotating end mill, whose motion was
computer controlled to make all the cuts necessary to form all the
faces in the array. The hard copper substrate was nominally 25.4 mm
thick. A primary set of six cuts was used to form six rounded edges
and six compound faces, such compound faces forming a matched pair
of cube corner elements analogous to cube corner elements 6886',
6887' in FIG. 68B. The end mill used was a diamond end mill
substantially as shown in FIGS. 5A-5D, with a taper angle per side
(THA) equal to 36.264 degrees, and a tip diameter (Dbot) equal to
60 micrometers. The substrate was of a type shown in FIG. 1 and
included a working surface 107. For all faces of the array, the end
mill was positioned relative to the substrate such that the axis of
rotation of the end mill was perpendicular to the working surface
as shown in FIG. 6A. That is, the cutting envelope was not tilted,
and the polar angle .theta. was zero.
[0363] In preparation for the cutting, the flat tip of the end mill
was placed at the top, flat surface of the substrate, and was then
moved away from the substrate a vertical distance of approximately
0.042 mm above the surface. This vertical position (Z coordinate)
was then set to be zero at this location.
[0364] A sequence of cuts was then made in the working surface to
form the first two non-orthogonal cube corner elements in a closely
packed array of non-orthogonal cube corner elements, by moving the
rotating end mill along a path defined by the Cartesian coordinates
provided in Table 3 below. The initial (X,Y) coordinate of (0,0)
was assigned to a particular location on the substrate. The X, Y,
and Z values in Table 3 follow a standard Cartesian coordinate
system, with the plane defined by Z=-0.042 mm corresponding to the
initially smooth, flat working surface of the substrate, and
positive Z values located above the working surface. In comparison
to the x, y, z axes shown in FIG. 68B, the X, Y, Z axes of Table 3
correspond to a 90 degree counterclockwise rotation of the x, y, z
axes about the z-axis.
[0365] Before executing the sequence of cuts, a total of nine
random numbers between -0.005 mm and +0.005 mm were selected. For
instance, three random values were selected and assigned to RandX1,
RandX2, RandX3, three more random values were selected and assigned
to RandY1, RandY2, RandY3, and a final three random values were
selected and assigned to RandZ1, RandZ2, RandZ3. As shown in Table
3, these random numbers were applied as offsets to nominal X, Y, Z
coordinates, the nominal coordinates being those that produce
strictly orthogonal cube corner elements and 90 degree dihedral
angles. The random numbers were not applied to coordinates that
correspond to the bases of the cavities (positions 3 and 10 for one
cube corner element, positions 10 and 12 for the other) so that the
base positions throughout the array would produce a well-defined,
regular grid, but in alternative embodiments offsets can also, or
instead, be applied to the base positions.
TABLE-US-00003 TABLE 3 Path of Rotating End Mill Position X Y Z 1 0
0 0.01 2 .sup. 0 + RandX1 0 + RandY1 -0.1633 + RandZ1 3 0.2 0.1155
-0.3266 4 0.2 + RandX2 0.2309 + RandY2 -0.2449 + RandZ2 5 0.2
0.3464 -0.3266 6 .sup. 0 + RandX3 0.4619 + RandY3 -0.1633 + RandZ3
7 .sup. 0 + RandX3 0.4619 + RandY3 0.01 8 0.4 + RandX1 0 + RandY1
0.01 9 0.4 + RandX1 0 + RandY1 -0.1633 + RandZ1 10 0.2 0.1155
-0.3266 11 0.2 + RandX2 0.2309 + RandY2 -0.2449 + RandZ2 12 0.2
0.3464 -0.3266 13 0.4 + RandX3 0.4619 + RandY3 -0.1633 + RandZ3 14
0.4 + RandX3 0.4619 + RandY3 0.01
[0366] The rotating end mill formed a first cut (and a first
rounded edge and first pair of flat faces) in the substrate as the
end mill moved from position 2 to 3. A second cut was formed (along
with a second rounded edge and a second pair of flat faces) when
the end mill moved from position 3 to 4. A third cut was formed
(and a third rounded edge and a third pair of flat faces) when the
end mill moved from position 4 to 5. A fourth cut was formed (and a
fourth rounded edge and a fourth pair of flat faces) when the end
mill moved from position 5 to 6. A fifth cut was formed (and a
fifth rounded edge and a fifth pair of flat faces) when the end
mill moved from position 9 to 10. A sixth cut was formed (and a
sixth rounded edge and a sixth pair of flat faces) when the end
mill moved from position 12 to 13. Having finished these cuts, the
first, second, and fifth rounded edges formed dihedral edges of a
first cube corner element, and the third, fourth, and sixth rounded
edges formed dihedral edges of a second (matched pair) cube corner
element, the three optical faces of each cube corner element being
compound faces. As a result of the random offsets provided by the
random numbers, the constituent faces of each compound face were in
general not precisely parallel to each other, and the nominally
orthogonal dihedral angles between optical faces (and portions
thereof) contained angle errors as discussed above. Each cube
corner element had a depth (below the initial flat surface of the
substrate) of 284.6 micrometers.
[0367] Having formed the two non-orthogonal cube corner elements,
the same procedure was carried out in a step-and-repeat fashion to
form more pairs of cube corner elements, by applying a local
coordinate offset to reposition the (0,0) (X,Y) coordinate to a new
location on the substrate. For each new repetition of the sequence
of six cuts, a new set of nine random values between -0.005 mm and
+0.005 mm were selected, such that sequentially formed pairs of
cube corner elements would have different orientations of their
respective faces and different sets of dihedral angle errors.
(However, the random number generator used a technique based on a
fractal pattern, and as a result, only a limited number--on the
order of several dozen--of unique sets of nine random values were
used, and after the last such unique set was used, the same sets of
random numbers were used again, in the same order, to form the next
pairs of cube corner elements during the step-and-repeat process.
This re-use of the random value sets, combined with the methodical
progression of the step-and-repeat process along a given direction,
produced a pattern that was substantially larger than the size of
the cube corner elements and visually apparent to an observer of
the machined substrate.) In all, the six cut sequence was stepped
and repeated 76 times in the X direction at a pitch spacing of 0.4
mm, and 38 times in the Y direction at a pitch spacing of 0.4619
mm. This produced an array of well over 5,000 non-orthogonal cube
corner elements, each having a rectangular perimeter in plan view
with side dimensions of 400 micrometers in the X direction and 231
micrometers in the Y direction, the array filling a rectangular
area on the substrate about 30 mm in the X direction and 18 mm in
the Y direction.
[0368] Alongside this first rectangular array of non-orthogonal PG
cube corner elements on the same substrate, a substantially similar
second array of cube corner elements was formed by the same end
milling procedure and the same step-and-repeat procedure, except
that the random number values were all zero so that constituent
faces of every cube corner element in this second array were
parallel to each other, and the nominally orthogonal dihedral
angles between optical faces (and portions thereof) did not contain
any significant angle errors as discussed above. Thus,
substantially all the cube corner elements in the second array were
strictly orthogonal PG cube corner elements.
[0369] In the resulting machined substrate, the working surface was
divided into a first portion containing the first array of
(non-orthogonal) PG cube corner elements, and a second portion
containing the second array of (orthogonal) PG cube corner
elements. The machined substrate was observed under normal
illumination, i.e., illumination along an illumination axis
substantially parallel to the Z-axis of the substrate, and at both
(a) a small or zero observation angle (i.e., the angle between the
illumination axis and an observation axis (a line drawn from the
substrate to the camera lens) was substantially zero), and (b) a
substantially larger observation angle. At the larger observation
angle, the array of orthogonal cube corner elements lost
brightness, while the array of non-orthogonal cube corner elements
gained brightness (relative to zero observation angle) as a
consequence of the dihedral angle errors. The pattern discussed
above relating to the re-use of the random number sets could be
seen at both the small and the larger observation angles in the
first portion of the working surface.
[0370] Unless otherwise indicated, all numbers expressing
quantities, measurement of properties, and so forth used in the
specification and claims are to be understood as being modified by
the term "about". Accordingly, unless indicated to the contrary,
the numerical parameters set forth in the specification and claims
are approximations that can vary depending on the desired
properties sought to be obtained by those skilled in the art
utilizing the teachings of the present application. Not as an
attempt to limit the application of the doctrine of equivalents to
the scope of the claims, each numerical parameter should at least
be construed in light of the number of reported significant digits
and by applying ordinary rounding techniques. Notwithstanding that
the numerical ranges and parameters setting forth the broad scope
of the invention are approximations, to the extent any numerical
values are set forth in specific examples described herein, they
are reported as precisely as reasonably possible. Any numerical
value, however, may well contain errors associated with testing or
measurement limitations.
[0371] Various modifications and alterations of this invention will
be apparent to those skilled in the art without departing from the
spirit and scope of this invention, and it should be understood
that this invention is not limited to the illustrative embodiments
set forth herein. The reader should assume that features of one
disclosed embodiment can also be applied to all other disclosed
embodiments unless otherwise indicated. It should also be
understood that all U.S. patents, patent application publications,
and other patent and non-patent documents referred to herein are
incorporated by reference, to the extent they do not contradict the
foregoing disclosure.
* * * * *