DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0062] The present invention relates to a lamina and laminae comprising cube corner elements, a tool comprising an assembly of laminae and replicas. There invention further relates to retroreflective sheeting.

[0063] The retroreflective sheeting is preferably prepared from a master mold manufactured with a technique that employs laminae. Accordingly, at least a portion and preferably substantially all the cube corner elements of the lamina(e) and retroreflective sheeting are full cubes that are not truncated. In one aspect, the base of full cube elements in plan view are not triangular. In another aspect, the non-dihedral edges of full cube elements are characteristically not all in the same plane (i.e. not coplanar). Such cube corner elements are preferably “preferred geometry (PG) cube corner elements”.

[0064] A PG cube corner element may be defined in the context of a structured surface of cube corner elements that extends along a reference plane. For the purposes of this application, a PG cube corner element means a cube corner element that has at least one non-dihedral edge that: (1) is nonparallel to the reference plane; and (2) is substantially parallel to an adjacent non-dihedral edge of a neighboring cube corner element. A cube corner element whose three reflective faces comprise rectangles (inclusive of squares), trapezoids or pentagons are examples of PG cube corner elements. “Reference plane” with respect to the definition of a PG cube corner element refers to a plane or other surface that approximates a plane in the vicinity of a group of adjacent cube corner elements or other geometric structures, the cube corner elements or geometric structures being disposed along the plane. In the case of a single lamina, the group of adjacent cube corner elements consists of a single row or pair of rows. In the case of assembled laminae, the group of adjacent cube corner elements includes the cube corner elements of a single lamina and the adjacent contacting laminae. In the case of sheeting, the group of adjacent cube corner elements generally covers an area that is discernible to the human eye (e.g. preferably at least 1 mm ² ) and preferably the entire dimensions of the sheeting.

[0065] “Entrance angle” refers to the angle between the reference axis (i.e. the normal vector to the retroreflective sample) and the axis of the incident light.

[0066] “Orientation” refers to the angle through which the sample may be rotated about the reference axis from the initial zero degree orientation of a datum mark.

[0067] Lamina(e) refers to at least two lamina. “Lamina” refers to a thin plate having length and height at least about 10 times its thickness (preferably at least 100, 200, 300, 400, 500 times its thickness). The invention is not limited to any particular dimensions of lamina(e). In the case of lamina intended for use in the manufacture of retroreflective sheeting, optimal dimensions may be constrained by the optical requirements of the final design (e.g. cube corner structures). In general the lamina has a thickness of less than 0.25 inches (6.35 mm) and preferably less than 0.125 inches (3.175 mm). The thickness of the lamina is preferably less than about 0.020 inches (0.508 mm) and more preferably less than about 0.010 inches (0.254 mm). Typically, the thickness of the lamina is at least about 0.001 inches (0.0254 mm) and more preferably at least about 0.003 inches (0.0762 mm). The lamina ranges in length from about 1 inch (25.4 mm) to about 20 inches (50.8 cm) and is typically less than 6 inches (15.24 cm). The height of the lamina typically ranges from about 0.5 inches (12.7 mm) to about 3 inches (7.62 cm) and is more typically less than about 2 inches (5.08 cm).

[0068] With reference to FIGS. 1-8 lamina 10 includes a first major surface 12 and an opposing second major surface 14 . Lamina 10 further includes a working surface 16 and an opposing bottom surface 18 extending between first major surface 12 and second major surface 14 . Lamina 10 further includes a first end surface 20 and an opposing second end surface 22 . In a preferred embodiment, lamina 10 is a right rectangular polyhedron wherein opposing surfaces are substantially parallel. However, it will be appreciated that opposing surfaces of lamina 10 need not be parallel.

[0069] Lamina 10 can be characterized in three-dimensional space by superimposing a Cartesian coordinate system onto its structure. A first reference plane 24 is centered between major surfaces 12 and 14 . First reference plane 24 , referred to as the x-z plane, has the y-axis as its normal vector. A second reference plane 26 , referred to as the x-y plane, extends substantially coplanar with working surface 16 of lamina 10 and has the z-axis as its normal vector. A third reference plane 28 , referred to as the y-z plane, is centered between first end surface 20 and second end surface 22 and has the x-axis as its normal vector. For the sake of clarity, various geometric attributes of the present invention will be described with reference to the Cartesian reference planes as set forth herein. However, it will be appreciated that such geometric attributes can be described using other coordinate systems or with reference to the structure of the lamina.

[0070] The lamina(e) of the present invention preferably comprise cube corner elements having faces formed from, and thus comprise ,a first groove set, an optional second groove set, and preferably a third primary groove (e.g. primary groove face).

[0071] FIGS. 2-9 illustrate a structured surface comprising a plurality of cube corner elements in the working surface 16 of lamina 10 . In general, a first groove set comprising at least two and preferably a plurality of grooves 30 a , 30 b , 30 c , etc. (collectively referred to as 30 ) are formed in working surface 16 of lamina 10 . The grooves 30 are formed such that the respective groove vertices 33 and the respective first reference edges 36 extend along an axis that intersects the first major surface 12 and the working surface 16 of lamina 10 . Although working surface 16 of the lamina 10 may include a portion that remains unaltered (i.e. unstructured), it is preferred that working surface 16 is substantially free of unstructured surface portions.

[0072] The direction of a particular groove is defined by a vector aligned with the groove vertex. The groove direction vector may be defined by its components in the x, y and z directions, the x-axis being perpendicular to reference plane 28 and the y-axis being perpendicular to reference plane 24 . For example, the groove direction for groove 30 b is defined by a vector aligned with groove vertex 33 b . It is important to note that groove vertices may appear parallel to each other in top plan view even though the grooves are not parallel (i.e. different z-direction component).

[0073] As used herein, the term “groove set” refers to grooves formed in working surface 16 of the lamina 10 that range from being nominally parallel to non-parallel to within 10 to the adjacent grooves in the groove set. As used herein “adjacent groove” refers to the closest groove that is nominally parallel or non-parallel to within 1°. Alternatively or in addition thereto, the grooves of a groove set may range from being nominally parallel to non-parallel to within 1° to particular reference planes as will subsequently be described. Accordingly, each characteristic with regard to an individual groove and/or the grooves of a groove set (e.g. perpendicular, angle, etc.) will be understood to have this same degree of potential deviation. Nominally parallel grooves are grooves wherein no purposeful variation has been introduced within the degree of precision of the groove-forming machine. The grooves of the groove set may also comprise small purposeful variations for the purpose of introducing multiple non-orthogonality (MNO) such as included angle errors, and/or skew, and/or inclination as will subsequently be described in greater detail.

[0074] Referring to FIGS. 3-9 , the first groove set comprises grooves 30 a , 30 b , 30 c , etc. (collectively referred to by the reference numeral 30 ) that define first groove surfaces 32 a , 32 b , 32 c , etc. (collectively referred to as 32 ) and second groove surfaces 34 b , 34 c , 34 d , etc. (collectively referred to as 34 ) that intersect at groove vertices 33 b , 33 c , 33 d , etc. (collectively referred to as 33 ). At the edge of the lamina, the groove forming operation may form a single groove surface 32 a.

[0075] In another embodiment depicted in FIG. 4 , lamina 10 may optionally further comprise a second groove set comprising at least two and preferably a plurality of adjacent grooves, collectively referred to as 38 ) also formed in the working surface 16 of lamina 10 . In this embodiment, the first and second groove sets intersect approximately along a first reference plane 24 to form a structured surface including a plurality of alternating peaks and v-shaped valleys. Alternatively, the peaks and v-shaped valleys can be off-set with respect to each other. Grooves 38 define third groove surfaces 40 a , 40 b , etc. (collectively referred to as 40 ) and fourth groove surfaces 42 b , 42 c , etc. (collectively referred to as 42 ) that intersect at groove vertices 41 b , 41 c , etc. (collectively referred to as 41 ) as shown. At the edge of the lamina, the groove forming operation may form a single groove surface 40 a.

[0076] Both these first and second groove sets may also be referred to herein as “side grooves”. As used herein side grooves refer to a groove set wherein the groove(s) range from being nominally parallel to non-parallel to within 1°, per their respective direction vectors, to the adjacent side grooves of the side groove set. Alternatively or in addition thereto, side grooves refers to a groove that range from being nominally parallel to reference plane 28 to nonparallel to reference plane 28 to within 1°. Side grooves are typically perpendicular to reference plane 24 to this same degree of deviation in plan view. Depending on whether the side grooves are nominally parallel or non-parallel within 1°, individual elements in the replicated assembled master typically have the shape of trapezoids, rectangles, parallelograms and pentagons, and hexagons when viewed in plan view with a microscope or by measuring the dihedral angles or parallelism of the side grooves with an interferometer. Suitable interferometers will subsequently be described.

[0077] Although the third face of the elements may comprise working surface 12 or 14 , such as describe in EP 0 844 056 A1 (Mimura et al.), the lamina preferably comprises a primary groove face 50 that extends substantially the full length of the lamina. Regardless of whether the third face is a working surface (i.e. 12 or 14 ) of the lamina or a primary groove face, the third face of each element within a row preferably share a common plane. With reference to FIG. 5-6 and 8 - 9 , primary groove face 50 ranges from being nominally perpendicular to faces 32 , 34 , 40 and 42 to non-perpendicular to within 1°. Formation of primary groove face 50 results in a structured surface that includes a plurality of cube corner elements having three perpendicular or approximately perpendicular optical faces on the lamina. A single lamina may have a single primary groove face, a pair of groove faces on opposing sides and/or a primary groove along the intersection of working surface 16 with reference plane 24 that concurrently provides a pair of primary groove faces (e.g. FIG. 4 ). The primary groove is preferably parallel to reference plane 26 to within 1°.

[0078] A pair of single laminae with opposing orientations and preferably multiple laminae with opposing orientations are typically assembled into a master tool such that their respective primary groove faces form a primary groove. For example, as depicted in FIGS. 6 and 8 - 9 , four laminae (i.e. laminae 100 , 200 , 300 and 400 are preferably assembled such that every other pair of laminae are positioned in opposing orientations (i.e. the cube corner elements of lamina 100 are in opposing orientation with the cube corner elements of lamina 200 and the cube corner elements of lamina 300 are in opposing orientation with the cube corner elements of lamina 400 ). Further, the pairs of laminae having opposing orientation are positioned such that their respective primary groove faces 50 form primary groove 52 . Preferably the opposing laminae are positioned in a configuration (e.g. 34 b aligned with 42 b ) in order to minimize the formation of vertical walls.

[0079] After formation of the groove sets, working surface 16 is microstructured. As used herein, “microstructured” refers to at least one major surface of the sheeting comprising structures having a lateral dimension (e.g. distance between groove vertices of the cube corner structures) of less than 0.25 inches (6.35 mm), preferably less than 0.125 inches (3.175 mm) and more preferably less than 0.04 inches (1 mm). The lateral dimension of cube corner elements, is preferably less than 0.020 inches (0.508 mm) and more preferably less than 0.007 inches (0.1778 mm). Accordingly, the respective groove vertices 33 and 41 are preferably separated by this same distance throughout the groove other than the small variations resulting from non-parallel grooves. The microstructures have an average height ranging from about 0.001 inches (0.0254 mm) to 0.010 inches (0.254 mm), with a height of less than 0.004 inches (0.1016 mm) being most typical. Further, the lateral dimension of a cube corner microstructure is typically at least 0.0005 inches (0.0127 mm). Cube corner microstructures may comprise either cube corner cavities or, preferably, cube corner elements having peaks.

[0080] In one embodiment, as depicted in FIG. 3-9 , the present invention relates to a lamina and laminae comprising a side groove set wherein adjacent grooves comprise different included angles. “Included angle” refers to the angle formed between the two faces of a V-shaped groove that intersect at the groove vertex. The included angle is typically a function of the geometry of the diamond-cutting tool and its position relative to the direction of cut. Hence, a different diamond tool is typically employed for each different included angle. Alternatively, yet more time consuming, different included angles may be formed by making multiple cuts within the same groove. The included angles of a first groove (e.g. side groove) differs from an adjacent groove (e.g. second side groove) by at least 2° (e.g. 3°, 4°, 5°, 6°, 7°, 8°, 9°) preferably at least 10° (e.g. 11°, 12°, 13°, 14°), and more preferably by at least 15° (e.g. 16°, 17°, 18°, 19°, 20°). Accordingly, the difference in included angle is substantially larger than differences in included angles that would arise from purposeful angle errors introduced for the purpose of non-orthogonality. Further, the difference in included angles is typically less than 70° (e.g. 65°, 60°, 50°), preferably less than 55°, more preferably less than 50°, and even more preferably less than 40°.

[0081] In one aspect, the differing included angles (e.g. of adjacent side grooves) are arranged in a repeating pattern to minimize the number of different diamond cutting tools needed. In such embodiment, the sum of adjacent side groove angles is about 180°. In a preferred embodiment, the lamina comprises a first sub-set of side grooves having an included angle greater than 90° alternated with second sub-set of side grooves having an included angle less than 90°. In doing so, the included angle of a first groove is typically greater than 90° by an amount of at least about 5°, and preferably by an amount ranging from about 10° to about 20°; whereas the included angle of the adjacent groove is less than 90° by about the same amount.

[0082] Although, the lamina may further comprise more than two sub-sets and/or side grooves having included angles of nominally 90°, the lamina is preferably substantially free of side grooves having an included angle of nominally 90°. In a preferred embodiment, the lamina comprises an alternating pair of side grooves (e.g. 75.226° and 104.774°) and thus, only necessitates the use of two different diamonds to form the totality of side grooves. Accordingly, with reference to FIGS. 6-9 , every other side grooves, i.e. 30 a , 30 c , 30 e , etc. has an included angle of 75.226° alternated with the remaining side grooves, i.e. 30 b , 30 d , etc. having an included angle of 104.774°. As will subsequently be described in further detail, employing differing included angles in this manner improves the uniformity index.

[0083] In another aspect, alternatively or in combination with the differing included angles (e.g. of adjacent side grooves) being arranged in a repeating pattern, the resulting cube corner elements have faces that intersect at a common peak height, meaning that cube peaks (e.g. 36 ) are within the same plane to within 3-4 microns. It is surmised that a common peak height contribute to improved durability when handling the tooling or sheeting by evenly distributing the load.

[0084] Alternatively or in combination thereof, the lamina comprises sideways canted cube corner elements. For cube corner elements that are solely canted forward or backward, the symmetry axes are canted or tilted in a cant plane parallel with reference plane 28 . The cant plane for a cube corner element is the plane that is both normal to reference plane 26 and contains the symmetry axis of the cube. Accordingly, the normal vector defining the cant plane has a y component of substantially zero for cube corner elements that are solely canted forward or backward. In the case of cube corner elements that are solely canted sideways, the symmetry axes of the cubes are canted in a plane that is substantially parallel to reference plane 24 and thus, the normal vector defining the cant plane has an x component of substantially zero.

[0085] The projection of the symmetry axis in the x-y plane may alternatively be used to characterize the direction of cant. The symmetry axis is defined as the vector that trisects the three cube corner faces forming an equal angle with each of these three faces. FIGS. 10 a - 10 c depict three different cube corner geometries in plan view that are solely backward canted, solely sideways canted, and solely forward canted, respectively. In these figures the cube peak extends out of the page and the projection of the symmetry axis (extending into the page from the cube peak) in the x-y plane is shown by the arrow. The alignment angle is measured counterclockwise in this view from the dihedral edge 11 (e.g. dihedral 2-3) that is approximately normal to a side of the cube in plan view. In the case of backward canting in the absence of sideways canting, the alignment angle is 0 degrees, whereas in the case of forward canting in the absence of sideways canting the alignment angle is 180 degrees. For a cube that has been canted sideways in the absence of forward or backward canting, the alignment angle is either 90° (as shown in FIG. 10 b ) or 270°. Alignment angle is 90° when the projection of the symmetry axis points to the right ( FIG. 10 b ) and 270° when the projection of the symmetry axis points to the left.

[0086] Alternatively, the cube may be canted such that the cant plane normal vector comprises both an x-component and y-component (i.e. x-component and y-component are each not equal to zero). At an alignment angle between 0° and 45° or between 0° and 315° the backward cant component is predominant with the backward cant component and sideways cant component being equal at an alignment angle of 45° or 315°. Further at an alignment angle between 135° and 225°, the forward cant component is predominant with the forward cant component and sideways cant component being equal at 135° and at 225°. Accordingly, cant planes comprising a predominant sideways cant component have an alignment angle between 45° and 135° or between 225° and 315°. Hence, a cube corner element is predominantly sideways canting when the absolute value of the y-component of the cant plane normal vector is greater than the absolute value of the x-component of the cant plane normal vector.

[0087] For embodiments wherein the sideways canted cubes are formed from an alternating pair of side grooves having different included angle cubes where the cant plane is parallel to reference plane 24 the adjacent cubes within a given lamina (e.g. α-β or α′-β′) are canted in the same or parallel planes. However, in general, if there is an x component to the cant plane normal vector, then adjacent cubes within a particular lamina are not canted in the same plane. Rather, the cube corner matched pairs are canted in the same or parallel planes (i.e. α-α′ or β-β′). Preferably, the cube corner elements of any given lamina have only two different alignment angles, e.g. derived from adjacent side grooves comprising different included angles. The alignment angle for the sideways canting example in FIG. 10 b is 90°, corresponding to the β-β′ cubes in FIG. 6 . Similarly, the alignment angle for the α-α′ sideways canted cubes in FIG. 6 is 270° (not shown).

[0088] FIG. 11 depicts laminae wherein the cubes are canted forward; whereas FIG. 12 depicts laminae wherein the cubes are canted backward. Cube designs canted in this manner result in a single type of matched opposing cube pairs. The cube 54 a of FIG. 11 with faces 64 a and 62 b is the same as the cube 54 b with faces 64 b and 62 c that is the same as cube 54 c with faces 64 c and 62 d , etc. Accordingly, all the cubes within the same row are the same, the row being parallel to reference plane 24 . The cube 54 a is a matched opposing pair with cube 56 a having faces 66 e and 68 d . Further, the cube 54 b is a matched opposing pair with cube 56 b . Likewise, cube 54 c is a matched opposing pair with cube 56 c . Similarly, cube 57 of FIG. 12 is a matched opposing pair with cube 58 . Matched pairs result when 180° rotation of a first cube about an axis normal to the plane of the structured surface will result in a cube that is super-imposable onto a second cube. Matched pairs need not necessarily be directly adjacent or contacting within the group of cube corner elements. Matched pairs typically provide retroreflective performance that is symmetric with respect to positive or negative changes in entrance angle for opposing orientations.

[0089] In contrast, sideways canting results in a cube design comprising two different cube orientations within the same row and thus created by the same side groove set. For a single lamina comprising both the first and second set of side grooves or a pair of adjacent laminae assembled in opposing orientations, the laminae comprise four distinctly different cubes and two different matched pairs, as depicted in FIGS. 6 , 8 - 9 . The four distinctly different cubes are identified as cubes alpha (α) having faces 32 b and 34 c , beta (β) having faces 32 c and 34 d , alpha prime (α′) having faces 40 c and 42 d , and beta prime (β′) having faces 40 b and 42 c . Cubes (α, α′) are a matched pair with the same geometry when rotated 180° such that the cube faces are parallel, as are cubes (β, β′). Further, the cubes on adjacent laminae (e.g. 100 , 200 ) are configured in opposing orientations. Although the symmetry axis of the cubes is tipped sideways, the bisector plane of the side grooves (i.e. the plane that divides the groove into two equal parts) preferably ranges from being nominally parallel to the bisector plane of an adjacent side groove (i.e. mutually parallel) to being nonparallel within 1°. Further, each bisector plane ranges from being nominally parallel to reference plane 28 to being nonparallel to reference plane 28 within 1°.

[0090] FIGS. 13-14 are isobrightness contour graphs illustrating the predicted total light return for a retroreflective cube corner element matched pair formed from a material having an index of refraction of 1.59 at varying entrance angles and orientation angles. In FIG. 13 the matched pair is forward canted 9.74° (e.g. cube corner elements 54 , 56 of FIG. 11 ). In FIG. 14 , the matched pair is backward canted 7.47° (e.g. cube corner elements 57 , 58 of FIG. 12 ). FIGS. 15-19 are isobrightness contour graph illustrating the predicted total light return for laminae comprising retroreflective cube corner elements formed from a material having an index of refraction of 1.59 at varying entrance angles and orientation angles where the cube corner elements are canted sideways 4.41°, 5.23°, 6.03°, 7.33°, and 9.74°, respectively for alignment angles of 90° and 270°. An alternating pair of side grooves (i.e. 75.226° and 104.774°) is utilized for FIG. 17 to produce cube corner elements that are sideways canted by 6.03°. The sideways canted arrays have two types of matched pairs, the β-β′ and α-α′ as depicted in FIG. 6 . These matched pairs have alignment angles of 90° and 270° respectively. In each of FIGS. 15-19 , the isobrightness contour graph is for laminae having the same (i.e. vertical) orientation as depicted in FIGS. 6, 11 and 12 .

[0091] Predicted total light return for a cube corner matched pair array may be calculated from a knowledge of percent active area and ray intensity. Total light return is defined as the product of percent active area and ray intensity. Total light return for directly machined cube corner arrays is described by Stamm U.S. Pat. No. 3,812,706.

[0092] For an initial unitary light ray intensity, losses may result from two pass transmissions through the front surface of the sheeting and from reflection losses at each of the three cube surfaces. Front surface transmission losses for near normal incidence-and a sheeting refractive index of about 1.59 are roughly 0.10 (roughly 0.90 transmission). Reflection losses for cubes that have been reflectively coated depend for example on the type of coating and the angle of incidence relative to the cube surface normal. Typical reflection coefficients for aluminum reflectively coated cube surfaces are roughly 0.85 to 0.9 at each of the cube surfaces. Reflection losses for cubes that rely on total internal reflection are essentially zero (essentially 100% reflection). However, if the angle of incidence of a light ray relative to the cube surface normal is less than the critical angle, then total internal reflection can break down and a significant amount of light may pass through the cube surface. Critical angle is a function of the refractive index of the cube material and of the index of the material behind the cube (typically air). Standard optics texts such as Hecht, “Optics”, 2nd edition, Addison Wesley, 1987 explain front surface transmission losses and total internal reflection. Effective area for a single or individual cube corner element may be determined by, and is equal to, the topological intersection of the projection of the three cube corner surfaces on a plane normal to the refracted incident ray with the projection of the image surfaces of the third reflection on the same plane. One procedure for determining effective aperture is discussed for example by Eckhardt, Applied Optics, v. 10, n. 7, July 1971, pg. 1559-1566. Straubel U.S. Pat. No. 835,648 also discusses the concept of effective area or aperture. Percent active area for a single cube corner element is then defined as the effective area divided by the total area of the projection of the cube corner surfaces. Percent active area may be calculated using optical modeling techniques known to those of ordinary skill in the optical arts or may be determined numerically using conventional ray tracing techniques. Percent active area for a cube corner matched pair array may be calculated by averaging the percent active area of the two individual cube corner elements in the matched pair. Alternatively stated, percent active aperture equals the area of a cube corner array that is retroreflecting light divided by the total area of the array. Percent active area is affected for example by cube geometry, refractive index, angle of incidence, and sheeting orientation.

[0093] Referring to FIG. 13 vector V ₁ represents the plane that is normal to reference plane 26 and includes the symmetry axes of cube corner elements 54 , 56 in FIG. 11 . For example, V ₁ lies in a plane substantially normal to the primary groove vertex 51 formed by the intersection of the primary groove faces 50 . The concentric isobrightness curves represent the predicted total light return of the array of cube corner elements 54 , 56 at various combinations of entrance angles and orientation angles. Radial movement from the center of the plot represents increasing entrance angles, while circumferential movement represents changing the orientation of the cube corner element with respect to the light source. The innermost isobrightness curve demarcates the set of entrance angles at which a matched pair of cube corner elements 54 , 56 exhibit 70% total light return. Successively outlying isobrightness curves demarcate entrance and orientation angles with successively lower percentages.

[0094] A single matched pair of forward or backward canted cubes typically have two planes (i.e. V ₁ and V ₂ ) of broad entrance angularity that are substantially perpendicular to one another. Forward canting results in the principle planes of entrance angularity being horizontal and vertical as shown in FIG. 13 . The amount of light returned at higher entrance angles varies considerably with orientation and is particularly low between the planes of best entrance angularity. Similarly, backward canting results in the principle planes of entrance angularity (i.e. V ₃ and V ₄ ) oriented at roughly 45° to the edge of the lamina as shown in FIG. 14 . Similarly, the amount of light returned at higher entrance angles varies considerably with orientation and is particularly low between the planes of best entrance angularity.

[0095] FIGS. 15-19 depict the predicted total light return (TLR) isointensity contours for a pair of opposing laminae having sideways canted cubes. The light return is more uniform as indicated by the shape of the plot approaching circular, in comparison to the isointensity plots of forward and backward canted cubes of FIGS. 13 and 14 . FIGS. 15-19 depict substantially higher light return at the locations of low light return evident in FIGS. 13 and 14 . Accordingly, good retention of TLR at higher entrance angles (up to at least 45° entrance) is predicted. This improved orientation performance is desirable for signing products since the signs are commonly positioned at orientations of 0°, 45° and 90°. Prior to the present invention, a common method for improving the uniformity of total light return with respect to orientation has been tiling, i.e. placing a multiplicity of small tooling sections in more than one orientation, such as described for example in U.S. Pat. No. 5,936,770. Sideways canted cube corner arrays can improve the uniformity of total light return, without the need for tiling and thus the array is substantially free of tiling in more than one orientation.

[0096] In order to compare the uniformity of total light return (TLR) of various optical designs, the average TLR at orientations of 0°, 45° and 90° may be divided by the range of TLR at orientations of 0°, 45° and 90°, i.e. the difference between the maximum and minimum TLR at these angles, all at a fixed entrance angle. The entrance angle is preferably at least 30° or greater, and more preferably 40° or greater. Preferred designs exhibit the maximum ratio of average TLR relative to TLR range. This ratio, i.e. “uniformity index (UI)” was calculated for a 40° entrance angle for the forward and backward canted cubes of FIGS. 13 and 14 , respectively as well as for the sideways canted cubes having various degrees of tilt of FIGS. 15-19 . For Table 1 the spacing of the side grooves is equal to the thickness of the lamina (i.e. aspect ratio=1). The calculated uniformity index is summarized in Table 1 as follows: 1

	TABLE 1
	Forward	Backward	Sideways (alignment angle = 90°)
Amount of cant	9.74	7.47	4.41	5.23	6.03	7.33	9.74
(arc minutes)
Avg. TLR	0.210	0.133	0.160	0.184	0.209	0.180	0.166
(0/45/90)
TLR Range	0.294	0.154	0.090	0.023	0.034	0.167	0.190
(0/45/90)
UI	0.71	0.87	1.79	8.02	6.23	1.08	0.88
1 $\mathrm{Uniformity}\mathrm{Index}\left(\mathrm{UI}\right)=\frac{\mathrm{Average}\mathrm{TLR}\mathrm{of}0°,45°,90°}{\mathrm{Range}\mathrm{at}0°,45°\mathrm{and}90°}$

[0097] Improved orientation uniformity results when the uniformity index is greater than 1. Preferably, the uniformity index is greater than 3 (e.g. 4), and more preferably greater than 5 (e.g. 6, 7, 8). Uniformity index will vary as a function of variables such as cube geometry (e.g. amount and type of cant, type of cube, cube shape in plan view, location of cube peak within aperture, cube dimensions), entrance angle, and refractive index.

[0098] FIG. 20 illustrates the uniformity index plotted versus alignment angle for canted cube corner arrays with varying amounts of cant and varying x and y components for their cant plane normal vectors. The arrays have two types of matched pairs, similar to the β-β′ and α-α′ as depicted in FIG. 6 , although these cubes and/or pairs may not be mutually adjacent. The cubes in each matched pair have substantially the same alignment angle. Alignment angles for the two types of matched pairs differ from 0° or 180° by the same amount. For example, for an alignment angle of 100° (differing from 180° by 80°) for a first cube or first matched pair the second (e.g. adjacent) cube or second matched pair would have an alignment angle of 260° (also differing from 180° by 80°).

[0099] FIG. 20 shows that the average TLR for polycarbonate (having an index of refraction of 1.59) as well as the uniformity index are maximized for cube geometries having a predominant sideways canting component, i.e. the range roughly centered about alignment angles of 90° and 270°. Note that alignment angles between 0° and 180° are presented on the X or horizontal axis of FIG. 20 from left to right. Alignment angles increasing from 180° to 360° degrees are plotted from right to left.

[0100] Preferably, the alignment angle is greater than 50° (e.g. 51°, 52°, 53°, 54°), more preferably greater than 55° (e.g. 56°, 57°, 58°, 59°), and even more preferably greater than 60°. Further the alignment angle is preferably less than 130° (e.g. 129°, 128°, 127°, 126°) and more preferably less than 125° (e.g. 124°, 123°, 122°, 121°), and even more preferably less than 120°. Likewise the alignment angle is preferably greater than 230° (e.g. 231°, 232°, 233°, 234°), and more preferably greater than 235° (e.g. 236°, 237°, 238°, 239°), and even more preferably greater than 240° . Further the alignment angle is preferably less than 310° (e.g. 309°, 308°, 307°, 306°) and more preferably less than 305° (e.g. 304°, 303°, 302°, 301°) and even more preferably less than 300°.

[0101] The amount of tilt of the cube symmetry axes relative to a vector perpendicular to the plane of the cubes is at least 2° and preferably greater than 3°. Further, the amount of tilt is preferably less than 90°. Accordingly, the most preferred amount of tilt ranges from about 3.5° to about 8.5° including any interval having end points selected from 3.6°, 3.7°, 3.8°, 3.9°, 4.0°, 4.1°, 4.2°, 4.3°, 4.4° and 4.5° combined with end points selected from 7.5°, 7.6°, 7.7°, 7.8°, 7.9°, 8.0°, 8.1°, 8.2°, 8.30° and 8.4°. Cube geometries that may be employed to produce these differing amounts of sideways cant are summarized in Table 2. The alignment angle may be 90° or 270° for each amount of cant. 2

[0102] Although differing included angles alone or in combination with the previously described sideways canting provide improved brightness uniformity in TLR with respect to changes in orientation angle over a range of entrance angles, it is also preferred to improve the observation angularity or divergence profile of the sheeting. This involves improving the spread of the retroreflected light relative to the source (typically, vehicle headlights). As previously described retroreflected light from cube corners spreads due to effects such as diffraction (controlled by cube size), polarization (important in cubes which have not been coated with a specular reflector), and non-orthogonality (deviation of the cube corner dihedral angles from 90° by amounts less than 1°). Spread of light due to non-orthogonality is particularly important in (e.g. PG) cubes produced using laminae since relatively thin laminae would be required to fabricate cubes where the spreading of the return light was dominated by diffraction. Such thin laminae are particularly difficult to handle during fabrication.

[0103] Alternatively, or in addition to the features previously described, in another embodiment the present invention relates to an individual lamina, a master tool comprising the assembled laminae, as well as replicas thereof including retroreflective replicas, comprising side grooves wherein the side grooves comprise “skew” and/or “inclination”. Skew and/or inclination provides cubes with a variety of controlled dihedral angle errors or multiple non-orthogonality (MNO) and thus improves the divergence profile of the finished product. As used herein “skew” refers to the deviation from parallel with reference to reference plane 28 .

[0104] FIG. 21 shows an exaggerated example in plan view of a single lamina with one row of cube corner elements comprising skewed grooves. Side grooves 80 a and 80 b are cut with positive skew, grooves 80 c and 80 e without skew, and groove 80 d with negative skew. The path of the side grooves 80 has been extended in FIG. 21 for clarity. Provided side grooves 80 a , 80 c , and 80 e have the same included angle (e.g. 75°, first groove sub-set), the same cutting tool or diamond can be used to form all of these grooves. Further, if the alternating grooves, namely 80 b and 80 d have the same included angle (e.g. 105°, second groove sub-set) 80 b and 80 d can be cut with a second diamond. The primary groove face 50 may also be cut with one of these diamonds if the primary groove half angle is sufficiently close to the side groove half angle for the first or second sub-sets. Optionally, one of the cutting tools may be rotated during cutting of the primary groove face in order to achieve the correct primary groove half angle. The primary groove face is preferably aligned with the side of the lamina. Thus, the entire lamina can be cut incorporating MNO with the use of only two diamonds. Within each groove set skew can be easily varied during machining to produce a range of dihedral angles. Cube corners in general have three dihedral angles attributed to the intersections of the three cube faces. The deviation of these angles from 90° is commonly termed the dihedral angle error and may be designated by dihedral 1-2, dihedral 1-3, and dihedral 2-3. In one naming convention, as depicted in FIG. 22 , cube dihedral angle 1-3 is formed by the intersection of groove surface 82 with primary groove face 50 , cube dihedral 1-2 is formed by the intersection of groove surface 84 with primary groove face 50 , and cube dihedral 2-3 is formed by the intersection of groove surface 84 with groove surface 82 . Alternatively, the same naming convention may be employed wherein the third face is working surface 12 or 14 rather than a primary groove face. For a given groove, positive skew ( 80 a , 80 b ) results in decreasing dihedral 1-3 and increasing dihedral 1-2 while negative skew results in increasing dihedral 1-3 and decreasing dihedral 1-2.

[0105] For example, with reference to FIG. 21 four different cubes are formed. The first cube 86 a has groove surfaces (i.e. faces) 82 a and 84 b , the second cube 86 b groove surfaces 82 b and 84 c , the third cube 86 c groove surfaces 82 c and 84 d , and the fourth cube 86 d has groove surfaces 82 d and 84 e . The intersection of groove surfaces 82 a , 82 b , and 84 d with groove face 50 are less than 90°, whereas the intersection of groove surfaces 84 b and 82 d with groove face 50 are greater than 90°. The intersection of groove surfaces 82 c , 84 c , and 84 e with groove face 50 are 90° since grooves 80 c and 80 e are without skew. The preceding discussion assumes that the inclination (as will subsequently be defined) is the same for all the side grooves in FIG. 21 and equals zero. The (e.g. PG) cube corner elements are trapezoids or parallelograms in plan view shape as a result of using skewed grooves during machining.

[0106] Alternatively, or in addition to the features previously described, the side grooves may comprise positive or negative inclination. “Inclination” refers to the deviation in slope in reference plane 28 of a particular side groove from the nominal orthogonal slope (i.e. the slope of the vector normal to the primary groove surface). The direction of a side groove is defined by a vector aligned with the vertex of said groove. Orthogonal slope is defined as the slope in which the vertex 90 of a groove would be parallel to the normal vector of groove face 50 (normal to groove face 50 ) for skew equal to zero. In one possible naming convention, positive inclination 92 results in decreasing both dihedral 1-3 and dihedral 1-2 for a given side groove while negative inclination 94 results in increasing both dihedral 1-3 and dihedral 1-2.

[0107] Combining skew and/or inclination during machining provides significant flexibility in varying the dihedral angle errors of the cube corner elements on a given lamina. Such flexibility is independent of cant. Accordingly skew and/or inclination may be employed with uncanted cubes, forward canted cubes, backward canted cubes, as well as sideways canted cubes. The use of skew and/or inclination provides a distinct advantage as it can be introduced during the machining of individual lamina without changing the tool (e.g. diamond) used to cut the side grooves. This can significantly reduce machining time as it typically can take hours to correctly set angles after a tool change. Furthermore, dihedral 1-2 and dihedral 1-3 may be varied in opposition using skew and/or inclination. “Varied in opposition” as used herein is defined as intentionally providing within a given cube corner on a lamina dihedral 1-2 and 1-3 errors (differences from 90°) that differ in magnitude and/or sign. The difference in magnitude is typically at least ¼ arc minutes, more preferably at least ½ arc minutes, and most preferably at least 1 arc minutes. Hence the grooves are nonparallel by amount greater than nominally parallel. Further, the skew and/or inclination is such that the magnitude is no more than about 1° (i.e. 60 arc minutes). Further, the (e.g. side) grooves may comprise a variety of different components of skew and/or inclination along a single lamina.

[0108] Dihedral angle errors may also be varied by changing the half angles of the primary or side grooves during machining. Half angle for side grooves is defined as the acute angle formed by the groove face and a plane normal to reference plane 26 that contains the groove vertex. Half angle for primary grooves or groove faces is defined as the acute angle formed by the groove face and reference plane 24 . Changing the half angle for the primary groove results in a change in slope of groove face 50 via rotation about the x-axis. Changing the half angle for a side groove may be accomplished via either changing the included angle of the groove (the angle formed by opposing groove faces e.g. 82 c and 84 c ) or by rotating a groove about its vertex. For example, changing the angle of the primary groove face 50 will either increase or decrease all of the dihedral 1-2 and dihedral 1-3 errors along a given lamina. This contrasts to changes in inclination where the dihedral 1-2 and dihedral 1-3 errors can be varied differently in each groove along a given lamina. Similarly, the half angle for the side grooves may vary, resulting in a corresponding change in dihedral 2-3. Note that for side grooves that are orthogonal or nearly orthogonal (within about 1°) to the primary groove face, dihedral 1-2 and dihedral 1-3 are very insensitive to changes in side groove half angle. As a result, varying the half angles of the primary or side grooves during machining will not allow dihedral 1-2 and dihedral 1-3 to vary in opposition within a given cube corner. Varying the half angles of the primary or side grooves during machining may be used in combination with skew and/or inclination to provide the broadest possible control over cube corner dihedral angle errors with a minimum number of tool changes. While the magnitude of any one of half angle errors, skew, or inclination can ranges up to about 1°, cumulatively for any given cube the resulting dihedral angle error is no more than about 1°.

[0109] For simplicity during fabrication, skew and/or inclination are preferably introduced such that the dihedral angle errors are arranged in patterns. Preferably, the pattern comprises dihedral angle errors 1-2 and 1-3 that are varied in opposition within a given cube corner.

[0110] Spot diagrams are one useful method based on geometric optics of illustrating the spread in the retroreflected light resulting from non-orthogonality from a cube corner array. Cube corners are known to split the incoming light ray into up to six distinct return spots associated with the six possible sequences for a ray to reflect from the three cube faces. The radial spread of the return spots from the source beam as well as the circumferential position about the source beam may be calculated once the three cube dihedral errors are defined (see e.g. Eckhardt, “Simple Model of Cube Corner Reflection”, Applied Optics, V10, N7, July 1971). Radial spread of the return spots is related to observation angle while circumferential position of the return spots is related to presentation angle as further described in US Federal Test Method Standard 370 (Mar. 1, 1977). A non-orthogonal cube corner can be defined by the surface normal vectors of its three faces. Return spot positions are determined by sequentially tracking a ray as it strikes and reflects from each of the three cubes faces. If the refractive index of the cube material is greater than 1, then refraction in and out of the front surface cube must also be taken into account. Numerous authors have described the equations related to front surface reflection and refraction (e.g. Hecht and Zajac, “Optics”, 2 ^nd edition, Addison Wesley 1987). Note that spot diagrams are based on geometric optics and hence neglect diffraction. Accordingly, cube size and shape is not considered in spot diagrams.

[0111] The return spot diagram for five different cubes that are backward canted by 7.47 degrees (e.g. FIG. 12 ) with errors in the primary groove half angle of five consecutive grooves of +2, +4, +6, +8, and +10 arc minutes is depicted in FIG. 24 . The half angle errors for the side grooves are zero (dihedral 2-3=0) in this example, as are skew and inclination. All the side groove included angles are 90°. The vertical and horizontal axes in FIG. 24 correspond to reference planes 28 and 24 , respectively. Note that changes in the slope of the primary groove face result in return spots located primarily along the vertical and horizontal axes.

[0112] The dihedral errors as a function of primary groove half angle errors are presented in Table 3 for the same errors used to produce FIG. 24 . Note that dihedral 1-2 and dihedral 1-3 have the same magnitude and sign and thus, do not vary in opposition. 3

[0113] The return spot diagram for the same type of backward canted cubes with dihedral 2-3 errors of −20, −15, −10, −5, and 0 arc minutes is depicted in FIG. 25 . The half angle errors for the primary groove are zero (dihedral 1-3=dihedral 1-2=0) in this example, as are skew and inclination. As stated previously, variations in the half angles for the side grooves may be used to produce the dihedral 2-3 errors. The dihedral 2-3 errors result in return spots located primarily along the horizontal axis.

[0114] FIG. 26 depicts a return spot diagram resulting from combining primary groove half angle errors with variations in the half angles for the side grooves for the same type of backward canted cubes as described with reference to FIGS. 24-25 . In this example, the primary groove half angle error is 10 arc minutes while dihedral 2-3 error is 0, 2, 4, and 6 arc minutes respectively for four different adjacent cubes on the lamina. A constant included angle error of +3 arc minutes could be used to produce these side grooves, with the opposing half angle errors as shown in Table 4. The return spots are again located primarily along the vertical and horizontal axes, with some spreading in the horizontal plane due to the nonzero values for dihedral 2-3. Overall the return spot diagram is localized and non-uniform.

[0115] The dihedral errors as a function of primary groove half angle errors are presented in Table 4 for the errors used to produce FIG. 26 . Note that dihedral 1-2 and dihedral 1-3 have the same magnitude and sign and hence do not vary in opposition (i.e. are substantially free of varying in opposition). Note that a given cube corner is formed by two adjacent side grooves and preferably a primary groove surface. The upper side groove in FIG. 22 forms dihedral 1-3 while the lower side groove forms dihedral 1-2. The intersection of the upper and lower side grooves forms dihedral 2-3. Side groove included angle is the sum of the upper and lower half angle errors for a groove that forms adjacent cubes (e.g. with reference to Table 4 the total included angle is +3 arc minutes and results from adding the upper half angle of a first cube with the lower half angle of the adjacent cube). 4

[0116] The preceding examples (i.e. FIGS. 24-26 ) were for backward canted cubes with varying half angle errors. In an analogous manner, forward canted cubes can be shown to have qualitatively similar return spot diagramss, i.e. substantially non-uniform with spots localized especially along the horizontal and vertical axes. Dihedral 1-2 and dihedral 1-3 of forward canted cubes also will have the same magnitude and sign and thus are substantially free of varying in opposition. In consideration of the uses of cube corner retroreflective sheeting, it is apparent that localized, non-uniform spot diagramss (e.g. FIGS. 24-26 ) are generally undesirable. Sheeting may be placed on signs in a wide variety of orientations, both as the background color as well as in some cases as cut out letters. Furthermore, signs may typically be positioned on the right, on the left, or above the road. In the case of vehicles marked with retroreflective sheeting for conspicuity, the position of the vehicle relative to the viewer is constantly changing. Both the left and right headlights of a vehicle illuminate a retroreflective target, and the position of the driver is quite different with respect to these headlights (differing observation and presentation angles). Vehicles such as motorcycles, where the driver is directly above the headlight, are commonly used. Finally, all of the relevant angles defining the viewing geometry change with distance of the driver/observer to the retroreflective sheeting or target. All of these factors make it clear that a relatively uniform spread of return spots is highly desirable in retroreflective sheeting. Because of the flexibility to easily introduce a wide range of dihedral angle errors, including dihedral 1-2 and dihedral 1-3 that vary in opposition, skew and/or inclination may be utilized to provide a relatively uniform spot return diagram.

[0117] FIG. 27 presents a return spot diagram resulting from variations in only inclination on a single lamina with the same backward canted cubes used in FIGS. 24-26 . Half angle errors for the side grooves are +1.5 arc minutes on each side (dihedral 2-3 and side groove angle error of +3 arc minutes) and primary groove error is zero. Skew is constant in this example at +7 arc minutes. Inclination is varied in a repeating pattern over every four grooves (i.e. two grooves +5 arc minutes, then two grooves −1 arc minute). The spot pattern is much more uniformly distributed both radially (observation) and circumferentially (presentation) in comparison with FIGS. 24-26 .

[0118] The dihedral errors for this example of varying inclination are presented in Table 5. The order of machining of the inclinations (arc minutes) is �