Patent Publication Number: US-11378725-B2

Title: Apparatus and method for forming a layered diffraction grating and a printed article including a layered diffraction grating

Description:
BACKGROUND 
     Aspects of the exemplary embodiment relate to diffraction gratings and find particular application in connection with an apparatus and method for forming a diffraction grating using xerographic printing. 
     Diffraction gratings are used to diffract light or other incident radiation into several beams and find application in monochromators and spectrometers. Diffraction gratings have a periodic structure, typically formed by parallel ridges or rulings, and can be transmissive or reflective. The directions of the beams depend on the spacing of the grating and the wavelength of the light. See, for example, E. Popov, “Introduction to Diffraction Gratings: Summary of Applications,” in “Gratings: Theory and Numerical Applications,” E. Popov, Editor (2012), http://www.fresnel.fr/files/gratings/Chapter1.pdf. 
     Conventionally, diffraction gratings are often formed on sheets of glass and thus are fairly brittle and costly to produce. It would be desirable to have a transmissive or reflective diffraction grating which is easily produced and is flexible. Such a diffraction grating, in additional to conventional uses, could be incorporated into a book or a document. 
     INCORPORATION BY REFERENCE 
     The following references, the disclosures of which are incorporated herein by reference in their entireties, are mentioned: 
     U.S. Pub. No. 20100046033, published Feb. 25, 2010, entitled METHOD TO IMPROVE IMAGE ON PAPER REGISTRATION MEASUREMENTS, by Kulkarni, et al., describes a method of controlling the placement of images on the output of a printer, including determining a scanner spatial error using an ideal medium having a first two-dimensional array on the ideal medium then determining printer spatial error using a second medium having a second two-dimensional array, and controlling placement of images on the output of the printer based on the scanner spatial error and the printer spatial error. 
     U.S. Pub. No. 20170336716, published Nov. 23, 2017, entitled EUV LITHOGRAPHY SYSTEM FOR DENSE LINE PATTERNING, by Flagello, et al., describes an extreme ultra-violet (EUV) lithography ruling engine configured to print one-dimensional lines on a target workpiece. The device includes a source of EUV radiation, a pattern-source defining a 1D pattern, an illumination unit to irradiate the pattern-source, and projection optics which optically image, with a reduction factor N&gt;1, the 1D pattern on an image surface that is optically-conjugate to the 1D pattern. 
     U.S. Pub. No. 20180134062, published May 17, 2018, entitled METHOD FOR PRODUCING A DOCUMENT AND A DOCUMENT, by Hansen, et al., describes a method for producing a security document. A security feature of the document includes a diffraction relief structure which is molded into the surface of a varnish layer. The diffraction relief structure is formed by regular gratings in which the spacing of the individual structural elements with respect to each other is smaller than a wavelength A in the visible light range. 
     U.S. application Ser. No. 16/129,104, filed Sep. 12, 2018, entitled APPARATUS AND METHOD FOR FORMING A DIFFRACTION GRATING AND PRINTED ARTICLE INCLUDING A DIFFRACTION GRATING, by Chapman, describes forming a grating by printing lines on a transparent sheet. an electronic image of the diffraction grating. The amount of white light that passes can be controlled by the widening the toner mounds which define the lines. One limitation of this is that when trying to limit the amount of light, dot gain may fill in toner between mounds ending the diffraction grating effect. 
     BRIEF DESCRIPTION 
     In accordance with one aspect of the exemplary embodiment, a method of forming an article including a diffraction grating includes forming a periodic structure by printing a first set of parallel lines on a first side of a transparent substrate with a marking material and printing a second set of parallel lines on a second side of the transparent substrate with a marking material, the first and second sets of lines, in combination, defining a grating having a frequency and a spacing between lines which causes incident light to be diffracted into a plurality of beams travelling in different directions. 
     In accordance with another aspect of the exemplary embodiment, an article includes a transparent substrate and a diffraction grating forming a periodic structure. The diffraction grating includes a first set of parallel lines of marking material on a first side of the transparent substrate and a second set of parallel lines of marking material on a second side of the transparent substrate. The diffraction grating has a frequency and a spacing between lines which causes incident light to be diffracted into a plurality of beams travelling in different directions. 
     In accordance with another aspect of the exemplary embodiment, a printing apparatus for forming an article including a diffraction grating includes memory which stores a first vector pattern cell, instructions for combining multiple instances of the first vector pattern cell to form a first array of parallel lines and instructions for applying a transform to the first array of parallel lines to generate a second array of parallel lines, or for applying a transform to the first vector pattern cell to generate a second vector pattern cell and for combining multiple instances of the second vector pattern cell to form a second array of parallel lines. The apparatus further includes a source of a transparent substrate, a first marking engine which prints the first array of parallel lines onto a first side of the transparent substrate, the first marking engine or a second marking engine printing the second array of parallel lines onto a second side of the transparent substrate, and, optionally, an assembly component, which assembles a stack comprising a sheet of glossy paper or card, the printed transparent substrate, and optionally at least one transparent layer and optionally joins the stack together, to form the article. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a first vector pattern cell in accordance with one aspect of the exemplary embodiment; 
         FIG. 2  illustrates a second vector pattern cell in accordance with one aspect of the exemplary embodiment; 
         FIG. 3  illustrates an article including a diffraction grating composed of vector pattern cells according to  FIGS. 1 and 2 ; 
         FIG. 4  is an enlarged view of a portion of the diffraction grating of  FIG. 3 ; 
         FIG. 5  is a side sectional view of one embodiment of the article of  FIG. 3 ; 
         FIG. 6  is a side sectional view of another embodiment of the article of  FIG. 3 ; 
         FIG. 7  is a side sectional view of yet another embodiment of the article of  FIG. 3 ; 
         FIG. 8  is a functional block diagram of an apparatus for forming an article comprising a diffraction grating in accordance with one aspect of the exemplary embodiment; 
         FIG. 9  is a flow chart illustrating a method of forming an article comprising a diffraction grating in accordance with one aspect of the exemplary embodiment; 
         FIG. 10  is a side sectional view of a transparent substrate with lines printed on a first side; 
         FIG. 11  illustrates light transmission by the article of  FIG. 5 ; and 
         FIG. 12  illustrates light reflection by the article of  FIG. 7 . 
     
    
    
     DETAILED DESCRIPTION 
     Aspects of the exemplary embodiment relate to an article including a diffraction grating which is generated using a vector pattern cell of parallel lines and to an apparatus and method for forming the article. 
     The system and method enable a low-cost 3D diffraction grating to be printed on a 2D printer using standard transparent medias and marking material(s), such as ink(s) or toner(s). 
     The method includes forming a periodic structure by printing lines on first and second sides of a transparent substrate with a marking material. The lines, in combination, have a frequency/pitch and a spacing which causes incident visible light to be diffracted into a plurality of (generally several) beams travelling in different directions. 
     A first vector pattern cell of parallel lines is created with variable angle, frequency and width. A first diffraction grating of size X by Y is created by filling an area of size X by Y with the first vector pattern cell and printing that on a first side of standard transparency media. The first vector pattern cell is transformed to account for front to back registration and desired alignment and an area of size X by Y is printed on the back side of the first diffraction grating. The transformation may include rotation of the first vector pattern cell through 90°, translating the pattern by half the line gap to be centered, and scaling of one of the first and the second vector pattern cells, as needed, to account for registration errors. 
     An advantage of the system and method is that a high frequency of lines is achieved by forming adjacent lines on opposite sides of the substrate, enabling a doubling of the frequency at the same resolution. Another advantage is the ability to provide an improved white light limiter. 
       FIGS. 1 and 2  illustrate example first and second vector pattern cells  10 ,  12 , having horizontal and vertical dimensions (x, y). Dimensions x and y may be, for example, from 5 to 20 pixels. In one embodiment, x=y. Each of the vector pattern cells  10 ,  12  includes “on” pixels (such as pixels  14 ,  16 ) spaced by “off” pixels (such as pixels  18 ,  20 ). The “on” pixels are arranged in parallel lines. For example, the vector pattern cell  10  of  FIG. 1  includes parallel line segments  22 ,  24 , which are spaced at an interval q in the x and/or y direction(s). Similarly, in vector pattern cell  12 , the “on” pixels form parallel line segments  26 ,  28 . The shortest distance (pitch) between midpoints of two adjacent parallel lines is denoted 
             p   (     p   =         q   2       2             
in the illustrated embodiment).
 
     When one or multiple instances of vector pattern cell  10  are printed on a first (recto) side  30  of a transparent substrate  32  ( FIG. 3 ), the line segments  22 ,  24  are connected to form a first array  46  ( FIG. 4 ) of parallel lines  34 ,  36 . When one or multiple instances of vector pattern cell  12  are printed on a second (verso) side  38  of the substrate  32 , the line segments  26 ,  28  are connected to form a second array  48  of parallel lines  40 ,  42 , etc. Parallel lines  40 ,  42  are interspaced with parallel lines  34 ,  36 . As a result, the parallel lines  34 ,  40 ,  36 ,  42  have an (average) pitch, denoted p′, of p/2. 
     In one embodiment, rather than providing a vector pattern cell  12 , a correction matrix or affine transform is applied to the first array  46  of parallel lines generated from vector pattern cell  10  to generate the second array  48 . 
     Each line  34 ,  40 ,  36 ,  42  is constructed from a contiguous sequence of “on” pixels  14 ,  16 ,  18 ,  20 , etc. Each line can have a width w of as little as one pixel (in the X and/or Y direction) although wider lines are contemplated. A pixel represents the smallest dot which can be printed. Pixels are represented by square blocks in  FIGS. 1 and 2 , although it is to be appreciated that they may have other shapes when rendered by printing. An “on” pixel (shown in black) is one which is intended to be rendered with a marking material, such as dry toner particles. In each vector pattern cell  10 ,  12 , the spacing a between adjacent lines in the x and/or y direction can be as little as 1 pixel, although larger spacings are contemplated, such as at least two, or at least three, or at least four “off” pixels  18  between each adjacent pair of on pixels to allow for dot gain (printed pixels being wider than the theoretical width w). On vector pattern cell  10 , each line segment  22 ,  24  is equally angled at angle α to the horizontal, such as at 45° to the horizontal. For example, in  FIG. 1 , the line segments  22 ,  24  are one pixel wide and spaced by three “off” pixels  18  in the x and y directions (z=4), or put another way, the “on” pixels have a frequency of ¼ in the x and y directions. The frequency P of vector pattern cell  10  in the direction perpendicular to the lines 
     
       
         
           
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                 1 
                 p 
               
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                   1 
                   
                     8 
                   
                 
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                     0 
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                     3 
                   
                   ⁢ 
                   
                     5 
                     . 
                   
                 
               
             
           
         
       
     
     In vector pattern cell  12 , the frequency P is the same as for vector pattern cell  10 . However, the angle β of each line segment  26 ,  28  to the horizontal is selected such that the resulting printed lines  34 ,  40 ,  36 ,  42 , all have the same angle to the horizontal when viewed from the same side of the article  32 . In the illustrated example, β=180−α=135°. As will be appreciated, the vector pattern cells  10 ,  12 , for forming parallel lines  34 ,  40 ,  36 ,  42  can be generated with another specified angle, frequency and width, which is variable. In the case where α=90°, β=90°. 
     In combination, the first array  46  and second array  48  form an array  50  of size X by Y with approximately double the number of lines as arrays  46 ,  48 , that can serve as a diffraction grating ( FIG. 3 ). The diffraction grating  50  of size X by Y can be created by filling an area of size X×Y with multiple instances of the vector pattern cell  10  or  12  and printing each array of cells on a respective side  30 ,  38  of the substrate  32  to form a printed article  52 . For example, X is at least 4x or at least 10x, and/or Y is at least 4y or at least 10 y. 
       FIG. 4  shows an enlarged portion of the grating  50  of  FIG. 3 . The frequency P′ of the grating  50  in a direction perpendicular to the lines 
               =     1     p   ′         ,         
where p′ is the pitch (p′=p/2).
 
     In the printed article  52 , the lines  34 ,  40 ,  36 ,  42  may each have a width W (in the X and/or Y direction) of no more than 0.01 cm or no more than 0.005 cm, for example. A distance Q (here, Q=q/2), between the midpoints of adjacent lines  34 ,  40 , in the X and/or Y direction, may be no more than 0.05 cm, or no more than 0.03 cm. In one embodiment, q&gt;2W and Q&gt;W. The ratio of 
             Q   ⁢     :     ⁢           ⁢     W   ⁡     (     ≅       z   2     ⁢     :     ⁢           ⁢   w       )             
may be at least 1 or at least 1.2 or at least 1.5. The width W of the lines depends on the dots per inch (dpi) of the printer. 600 dpi printers can print up to 300 lines per inch, in X and Y directions, with an interval z of two pixels (˜118 lines per cm, which is equivalent to ˜236 lines per cm, when printed on both sides of a sheet), or 75 lines per inch with an interval z of 7 pixels (˜30 lines per cm, which is equivalent to ˜60 lines per cm, when printed on both sides of a sheet). The frequency P′≅84 lines per cm, when printed on both sides of a sheet with an interval z of 7 pixels. Printers with a higher dpi capability, e.g., of 1200 dpi, or 9600 dpi can print a correspondingly larger number of lines per inch with a smaller width W at the same ratio of Q:W.
 
     In one embodiment, the frequency P′ is greater than 30 lines/cm in order to bend the light into different spectra/colors, such as at least 50 lines/cm, or at least 80 lines/cm, or at least 120 lines/cm, or at least 160 lines/cm, or at least 320 lines/cm, such as up to 2000 lines/cm. Put another way, in various embodiments, the pitch p′ of the lines  34 ,  40 ,  36 ,  42  may be no greater than 0.35 mm, or no greater than 0.3 mm, or no greater than 0.2 mm, or no greater than 0.15 mm, or no greater than 0.10 mm, or no greater than 0.06 mm, or no greater than 0.03 mm. 
     In one embodiment, the substrate  32  includes or consists of a single transparent layer  60  (e.g., a sheet or roll) of transparent media, as illustrated in side sectional view in  FIG. 5 . The first set  46  of parallel lines  34 ,  36 , etc., is printed on the upper (first) surface  30  of the transparent layer  60  with a toner or other marking material. The second set  48  of parallel lines  42 ,  44 , etc., is printed on the lower (second) surface  38  of the transparent layer  60  with a toner or other marking material. The lines  34 ,  36  are in the form of ridges with a height h, in the Z direction, a spacing a, and an interval q. Similarly, lines  42 ,  44  and are in the form of ridges with a height h, a spacing a, and an interval q. The lines  42 ,  44  are each offset from a nearest line  34 ,  36 , in the X and or Y direction, by k, so that in combination, adjacent pairs of the lines  34 ,  42 , and  36 ,  44  are spaced by an interval Q, where Q=q/2. 
     As will be appreciated, rather than being rectangles, the ridges defined by the toner lines  34 ,  36 ,  42 ,  44  when viewed under a microscope, appear more like rolling hills than rectangles. The grating  50  passes much of the light and acts as a transmission grating. 
     While in one embodiment, the edges of adjacent lines are colinear (where, k=Q=a=w), in other embodiments, spaces  150  may exist between the adjacent lines, as illustrated in  FIG. 5  (where k&gt;w, k&gt;Q). The width s of the spacing can be used to control the amount of light that passes through the grating. The spacing s between a sequential pair of lines  34 ,  40  or  36 ,  42  may be ≥0 mm. When s=0, the spacing is closed as much as possible (assuming no dot gain issue), limiting white light transmission. To compensate for dot gain, the nominal gap a between printed pixels on each side may be at least two pixels wide so that even with a slight dot gain, there is still a space s&gt;0 between adjacent lines. Where the size of printed pixels can be controlled so that the printed lines do not exceed the theoretical pixel width (e.g., less than the theoretical pixel width), the nominal gap a can be as little as 1 pixel and s may be less than 1 pixel. 
       FIG. 6  is a side sectional view of another embodiment of the article of  FIG. 3 , in which adjacent lines  34 ,  40  are partially, but not completely, overlapping in the Z direction. In this embodiment, the offset k&lt;w. A maximum spacing s 1  between lines is greater than 0, in this example. 
     In some embodiments, the diffraction grating  52  includes at least 10, or at least 20, or at least 50 lines  34 ,  36  on the first, e.g., upper surface of the substrate  60  and the same, or approximately the same, number of lines  40 ,  42  on the lower surface. 
     The toner (sometimes referred to as dry ink) used to form the lines may be a conventional toner suited to xerographic printing. Suitable toners generally include particles, generally composed of one or more colored pigments embedded inside polymer beads. In one embodiment, the toner is black (K) in color or is a color which generally absorbs incident light. When applied to the sheet  60 , the toner particles carry a charge. For example, during printing, a negatively-charged toner is attracted to a positive latent image on a photoreceptor and the toner is then attracted to the transparent layer  60 , which also positively charged (or vice versa). The same toner, or a different toner, may be used for printing each side  30 ,  38  of the sheet  60 . 
     Alternatively, the lines  34 ,  36 ,  42 ,  44  are formed from inkjet ink, which may include a curable polymer resin to increase the height of the lines. The resin may be cured with heat, UV or other suitable radiation, combination thereof, or the like. While inkjet printers tend to be less accurate in placement of dots, they suffer less from dot gain and thus the lines may be printed closer. 
     In one embodiment, the article includes more than one layer, as illustrated in  FIG. 7 . In the embodiment of  FIG. 7 , the printed article, denoted  52 ′, includes or consists of a first transparent layer  60 , as for  FIGS. 5 and 6 , and a light-reflective layer  64 , having a first (upper) surface  66  and an opposed lower surface  68 , which defines a lower surface of the printed article  52 ′. Additionally, the printed article  52 ′ of  FIG. 7  (and optionally also that of  FIGS. 5 and 6 ) may include at least one additional transparent layer, such as one or more upper transparent layer(s)  70 ,  72 , positioned above the layer  60  (in the Z direction), on the opposite side of the substrate  32  to the reflective layer  64 . In one embodiment, layers  60  and  64  are adjacent, with the reflective layer contacting the second set of lines  40 ,  42 , etc. Additionally, or alternatively, there may be one or more lower transparent layers (not shown), which is/are positioned intermediate the reflective layer  64  and the transparent substrate  60 . 
     Each of the additional transparent layers  70 ,  72 , etc. may be formed from the same material as layer  60  or from a different transparent material and may be of the same or different thickness(es). An uppermost one of the upper transparent layers  70  defines an upper surface  76  of the printed article  52 ′. A lowermost one of the upper transparent layers  72  may have a lower surface  74  which is in contact with the upper set of printed lines  34 ,  36 , etc. The layer  72  may extend (at least partially) into gaps  78  defined between the printed lines  34 ,  36  or the gaps  78  may be filled with air or with a separate transparent toner material (or polymer resin suitable for inkjet printing). Similarly, gaps  80  between the lower lines  40 ,  42  may be filled with air or with a transparent toner material (or polymer resin suitable for inkjet printing). While two upper transparent layers  70 ,  72  are shown, it is to be appreciated that there may be zero, one, two, or at least three, or at least four upper and/or lower transparent layers, such as up to ten. 
     In one embodiment, rather than using multiple top layers  70 ,  72 , a single layer may be employed with a thickness greater than t 1 . 
     In one embodiment, the light-reflective layer  64  is formed of a flexible material, such as a glossy paper or card. The paper or card may be formed from fibers derived primarily from wood, cotton, hemp, or combination thereof. A coating  82  on the paper may be formed from inorganic materials and/or an organic polymer, such as polyethylene. Example inorganic materials include clay (kaolin), chalk (calcium carbonate), bentonite, and talc, and mixtures thereof. The chalk or china clay may be bound to the paper or card with synthetic viscosifiers, such as styrene-butadiene latexes and/or natural organic binders, such as starch. The coating formulation used to form the coating  82  may also contain dispersants, resins, and other additives. 
     The light-reflective layer  64  has a thickness t 2  which may be less than the thickness t 1  of the transparent layer  60 , depending on the type of glossy paper or card used. t 2  may be, for example, at least 0.07 mm and in some cases, up to 0.1 mm, or up to 0.2 mm, or higher in the case of card stock. In some embodiments, the reflective layer  64  may be printed, e.g., using one or more marking materials, such as ink(s) or toner(s). 
     The reflective layer  64  reflects light in the visible range of the spectrum (about 400 to 700 nm). Reflection can be measured in terms of specular gloss according to TAPPI Test Method T 653 om-07: Specular gloss of paper and paperboard at 20 degrees (70° from the plane of the paper), and may be at least 60, or at least 70. 
     The printed article  52 ′ of  FIG. 7  can serve as a partial mirror. 
     In another embodiment, one or more of the transparent layer(s)  70 ,  72  may be omitted, and optionally added later. 
     In the embodiment of  FIG. 7 , the transparent layer(s)  60 ,  72 ,  70  may consist of or comprise transparent sheets that are stacked one on top of the other. The transparent layer(s)  60 ,  72 ,  70  transmit light in the visible range of the spectrum, e.g., have a transmission of at least 80% of the visible light which is incident normal (90°) to one surface. As will be appreciated, the layer  60  becomes semi-transparent (i.e., lower transmittance) when printed with the lines  34 ,  36 ,  40 ,  42 , etc. 
     Examples of polymeric materials suitable for use as layers  60 ,  72 , and/or  70  include polyvinyl alcohol (PVOH); polystyrene (PS) and styrene copolymers, such as acrylonitrile-butadiene-styrene (ABS); polyvinylidene fluoride; polyvinyl acetals, such as polyvinyl butyral; unsaturated and saturated polycarbonates; polyvinylpyrrolidone (PVP); polyoxymethylene (also known as acetal, polyacetal, and polyformaldehyde); vinyl imidazole copolymers, such as 1-vinylimidazole; polyamides, such as aliphatic polyamides, polyphthalamides, and aramids; polyethers and polyesters, e.g., polyethylene terephthalate (PET), polybutylene terephthalate, poly(lactic acid) (PLA); cyclic olefin copolymers (COC), such as ethylene-norbornene copolymer; polyaryletherketones, such as polyetheretherketone (PEEK); polyetherimides (PEI) (e.g., ULTEM); polyimides (PI); polyolefins, such as polypropylene (PP) and polyethylene (PE); polyacrylates and methacrylates, such as poly(methyl methacrylate) (PMMA), poly(vinyl chloride) (PVC); polyacrylonitrile (PAN); and copolymers and mixtures thereof. The layers  60 ,  72 ,  70  may be formed from the same or different materials. 
     In one embodiment, the transparent layer(s)  60 ,  72 ,  70  may be a standard transparency media, such as polyester (polyethylene terephthalate), cellulose acetate, acrylic, or the like, and may be coated with a thin coating to improve printability. 
     A suitable thickness of the layers  60 ,  70 ,  72  depends, in part, on the type of polymer, which affects the refractive index, strength, and flexibility of the layer. Sheets of polymeric material  60  should be sufficiently flexible yet strong enough to pass through a xerographic printer without breaking or causing jams. Additionally, if upper and/or lower layers  70 ,  72 , etc., are to pass through the printer, they should also have suitable flexibility and strength properties. The thickness of the layers  60 ,  70 ,  72  may be, for example, at least 0.02 mm (20 μm), or at least 0.04 mm and/or up to 1.0 mm, or up to 0.5 mm, or up to 0.3 mm, or up to 0.2 mm, or up to 0.175 mm, or up to 0.15 mm. Where only a single upper layer  70  is used, it may have a thickness of at least 0.1 mm and/or up to 0.5 mm, or greater. 
     In some embodiments, the layers  60 ,  70 ,  72  may all have the same thickness. In some embodiments, the upper layer or layers  70 ,  72  may each have a thickness t 3  which is greater than the thickness t 1  of the substrate layer  60 . For example, t 3  may be greater than 0.3 mm in some cases, such as at least 0.4 mm or at least 0.5 mm. 
       FIG. 8  illustrates a printing apparatus  90  for forming the printed article  52 ,  52 ′ of  FIG. 5, 6 , or  7 . The apparatus includes a controller  92 , a substrate source  94 , a sheet feeder  96 , a transport mechanism  98 , at least one xerographic (electrophotographic) marking engine  100 ,  102 , optionally, an assembly component  104 , and an output device  106 , such as a tray. The term “printer” or “printing apparatus,” as used herein, encompass any apparatus, such as a digital copier, bookmaking machine, facsimile machine, multi-function machine, etc., which performs a print outputting function for any purpose. In one embodiment, the printer includes two marking engines  100 ,  102 , arranged for duplex printing. An inverter  108 , positioned between the two marking engines, inverts sheets printed by the first marking engine, so they can be printed, on the second side, by the second marking engine  102 . In another embodiment, an inversion path (not shown) returns the recto-printed sheets to the first (only) marking engine  100  for printing the verso side. 
     The controller  92  includes memory  110 , which stores the digital patterns  10  and  12  from which a diffraction grating  50  can be generated by printing multiple instances of each pattern on a respective side  30 ,  38  of the substrate  60 . The memory  110  further includes software instructions  112  for rendering the diffraction grating  50  on the transparent sheet  60 . A processor  114 , in communication with the memory, executes the instructions. The controller may include an input device  116  for receiving the patterns  10 ,  12 , and/or information for generating the patterns, from a user, e.g., via a graphical user interface (GUI)  120 . The received information may include one or more of the width w, interval z and angle α, β. The controller includes an output device  122  for sending instructions to the marking engine(s)  100 ,  102  for rendering the diffraction grating  50  on the transparent layer  60 . Hardware components  110 ,  114 ,  116 ,  122  of the controller may be communicatively connected by a data/control bus  124 . As will be appreciated, the controller, or parts thereof, may be remote from the rest of the printing apparatus  90 , e.g., on a remote server that is communicatively connected with the other parts of the printing apparatus by wired or wireless links. 
     The memory  110  may represent any type of non-transitory computer readable medium such as random access memory (RAM), read only memory (ROM), magnetic disk or tape, optical disk, flash memory, or holographic memory. In one embodiment, the memory  110  comprises a combination of random access memory and read only memory. 
     The digital processor device  114  can be variously embodied, such as by a single core processor, a dual core processor (or more generally by a multiple core processor), a digital processor and cooperating math coprocessor, a digital controller, or the like. The digital processor  114 , in addition to executing instructions  112  may also control the operation of the printer. 
     The term “software,” as used herein, is intended to encompass any collection or set of instructions executable by a computer or other digital system so as to configure the computer or other digital system to perform the task that is the intent of the software. The term “software” as used herein is intended to encompass such instructions stored in storage medium such as RAM, a hard disk, optical disk, or the like, and is also intended to encompass so called “firmware” that is software stored on a ROM or the like. Such software may be organized in various ways, and may include software components organized as libraries, Internet based programs stored on a remote server or so forth, source code, interpretive code, object code, directly executable code, and so forth. It is contemplated that the software may invoke system level code or calls to other software residing on a server or other location to perform certain functions. 
     The substrate source  94  holds a supply of transparent sheets to serve as the transparent layer(s)  60 ,  70 ,  72 , etc. (and/or reflective layer  64 ) and may include one or more trays. In some embodiments, a first of the trays holds transparent sheets  60  and a second of the trays holds reflective sheets  64 . The feeder  96  supplies the transparent sheets  60  to the transport system  98 , which may be composed of rollers, belts, or the like. The transport system  98  conveys the transparent sheets  60 , in the direction of arrow A, along a print media path  130  to the first xerographic marking engine  100 , where the first set of lines  34 ,  36  is printed on the print media  60  using toner particles to form a first semi-transparent diffraction grating layer. The printed article is conveyed, by the transport mechanism  98 , from the first marking engine  100 , via the inverter  108 , to the second xerographic marking engine  102 , where the second set of lines  40 ,  42  is printed on the print media  60  using toner particles to form a second semi-transparent diffraction grating layer. The printed assembly  52  is conveyed from the second marking engine  102  to the assembly component  104  and/or output device  106 . 
     The exemplary marking engines  100 ,  102  may both be monochrome (single color) marking devices, employing a single toner type, such as black toner. However, color marking engine(s) may alternatively be used. The marking engine  100  may include suitable hardware elements employed in the creation of desired images by electrophotographic processes. In one embodiment, each marking engine includes a charge retentive surface, such as a rotating photoreceptor  132  in the form of a belt or drum. The lines  34 ,  36 , etc., are created on a surface of the photoreceptor. Disposed at various points around the circumference of the photoreceptor  100  are xerographic subsystems which may include a charging station  134 , such as a charging corotron, for each of the toner colors to be applied (one in the case of a monochrome printer, four in the case of a CMYK printer), an exposure station  136 , which forms a latent image on the photoreceptor (e.g., with a laser or LED light source) corresponding to the lines  34 ,  36 , a developer unit  138 , associated with each charging station for developing the latent image formed on the surface of the photoreceptor by applying a toner to obtain a toner image. A transferring unit  140 , such as a transfer corotron and/or an intermediate transfer belt, transfers the toner image (diffraction grating) thus formed to the surface of the transparent sheet  60  as it passes by the photoreceptor. Optionally, a fuser  142  fuses the diffraction grating image  50  to the sheet. The fuser generally applies at least one of heat and pressure to the sheet  60  to physically attach the toner to the sheet. The marking engine  102  is similarly configured to marking engine  100  to apply lines  40 ,  42 , etc. 
     In other embodiments, each marking engine  100 ,  102  is an inkjet marking engine which employs liquid ink(s) rather than dry toner. 
     The assembly component  104  may be used to form the printed article  52 ′ from the printed article  52  output by the marking engine  102 . In one embodiment, the assembly component  104  assembles the layers  64 ,  60 ,  72 ,  70  in sequence and joins them together, e.g., with one or more of heat, an adhesive, binding, clamps, and stapling. The assembled layers only need to be joined around a perimeter, outside the area to be used as the diffraction grating  50 . Thus, one or more of the four sides can be stapled, glued, bound into a spine of a book, etc., as illustrated in  FIG. 7  at  144 . The printed article  52 ′ is output. In other embodiments, the assembly component  104  may be a separate device/omitted. 
       FIG. 9  illustrates a method of printing a diffraction grating in accordance with the exemplary embodiment, which may be performed with the apparatus of  FIG. 8 . The method begins at S 100 . 
     At S 102 , a first vector pattern cell  10  is defined (e.g., as illustrated in  FIG. 1 ). This step includes defining a frequency 1/q (or 1/p), angle α, and width w (in pixels) of the vector. The vector pattern cell  10  is stored in memory, such as printer memory  110 . 
     At S 104 , a transform T, such as a correction matrix or affine transform, is defined to account for front to back registration. The front to back transform may account for position, scaling, rotation and shear. In particular, the transform T may account for the inversion of the first side array of lines when the substrate  60  is inverted. A desired overlap correction s is optionally applied for controlling the amount of light that passes through the grating. Each individual pixel may print larger than the electronic pixel due to dot gain. Accordingly, this may be taken into consideration in defining the vector pattern cells. The definition of the transform T may include passing a registration transparency through the printer, printing the parallel lines on front and back sides and measuring the extent to which the lines are not equally spaced (the spacing s should ideally be the same between each pair of printed lines  34 ,  40  and  40 ,  36 , etc.) and in general, s&gt;0. 
     Methods for computing a transformation are described, for example, in above-mentioned U.S. Pub. No. 20100046033. 
     In one embodiment, at S 106 , a second vector pattern cell  12  is defined (e.g., as illustrated in  FIG. 2 ). This step includes defining an angle β of the vector. The frequency 1/q (or 1/p) and width w of the vector may be the same as for pattern  10 . The vector pattern cell  12  may be generated by applying the transformation T to the vector pattern cell  10 , to account for front to back registration and desired alignment. The vector pattern cell  12  is stored in memory, such as printer memory  110 . 
     At S 108 , a diffraction grating size is defined (e.g., by a user via the GUI  120 ), which is used to determine the number of cells  10  (and correspondingly cells  12 ) to be composed into an array. 
     At S 110 , a first array  46  is generated by combining multiple instances of vector pattern cell  10 . In another embodiment, rather than defining the vector pattern cell  10  and forming an array from multiple instances, a respective single X×Y array may be defined for the first side of the layer  60 . 
     At S 112 , a second array  48  is generated. In one embodiment, array  48  is generated by combining multiple instances of vector pattern cell  12  generated at S 106 . Alternatively, the second array  48  is generated by applying the transform T, such as a correction matrix or affine transform, to the first array  46 . 
     At S 114 , a semi-transparent partial diffraction grating  144  ( FIG. 9 ) is formed by printing lines  34 ,  36 , etc., corresponding to the array  46  of vector pattern cells  10 , onto a first side  30  of a transparent sheet  60 , using the xerographic marking engine  100 , to form a pattern of equally-spaced ridges having a length l which is substantially greater than the height h. While this type of printing is often considered 2-dimensional (2D), there is a 3D aspect due to the toner height h. 
     At S 116 , a semi-transparent diffraction grating  50  ( FIG. 5 ) is formed by printing lines  40 ,  42  corresponding to the array  48  of vector pattern cells  12  onto the partial diffraction grating  144 , i.e., on the second side  38  of the transparent sheet  60 , using the xerographic marking engine  102 , to form a pattern of equally-spaced ridges having a length l which is substantially greater than the height h. A first edge of each line  40 ,  42 , etc. is offset from the first edge of the nearest line  34 ,  36 , etc. on the upper surface by an offset k, such that midpoints of pairs of lines  34 ,  40  lines are equally spaced. For example, k&lt;q, k≥Q and/or k≥w. 
     At S 118 , for forming a mirror as in  FIG. 7 , a reflective layer  64  may be provided, such as a sheet of coated white paper or card, which may be printed with an image. In some embodiments, the reflective layer  64  may be passed through the printing apparatus, e.g., without printing, using feeder  96  and transport mechanism  98 . In other embodiments, a stack of reflective layers  64  is provided downstream of the marking engine(s)  100 ,  102 . 
     At S 120 , one or more transparent sheets for layering over the diffraction grating may be provided. For example, one, two, three, or more fully-transparent sheets  70 ,  72 , may be passed through the printer from the substrate source  94 , without printing or with only limited printing in an area outside the region serving as the diffraction grating  50 . In other embodiments, a stack of transparent sheets is provided downstream of the marking engine  102 . 
     At S 122 , a layered stack of medias is created, e.g., by stacking the layers from back to front. The first (optional) layer is the reflective layer  64 . The second layer is the printed semi-transparent diffraction grating layer  52 . The third and subsequent layers (optional) are fully-transparent sheets  70 ,  72 , etc. 
     At S 124  the layers  64 ,  60 ,  72 ,  70  in the stack are joined together ( FIG. 7 ), e.g., by binding, stapling, adhesive, and/or heat, or a subset of the layers is assembled, for inclusion of additional layers later. The resulting mirror  52 ′ can be used as a standalone mirror or formed into another article. For example, it may be used as a page of a book (S 126 ). This step may be incorporated into step S 124 . 
     The method ends at S 128 . 
     As will be appreciated, steps S 118 , S 120 , S 122 , and S 124  may be omitted to form a transmission grating article  52  rather than a reflective grating article  52 ′. The transmission grating article  52  may be formed into another article. For example, it may be used as a page of a book (S 126 ). 
     The method illustrated in  FIG. 9  may be implemented, in part, in a computer program product that may be executed by the controller  92 . The computer program product may comprise a non-transitory computer-readable recording medium on which a control program is recorded (stored), such as a disk, hard drive, or the like. Common forms of non-transitory computer-readable media include, for example, floppy disks, flexible disks, hard disks, magnetic tape, or any other magnetic storage medium, CD-ROM, DVD, or any other optical medium, a RAM, a PROM, an EPROM, a FLASH-EPROM, or other memory chip or cartridge, or any other non-transitory medium from which a computer can read and use. The computer program product may be integral with the controller  92  (for example, an internal hard drive of RAM), or may be separate (for example, an external hard drive operatively connected with the printer), or may be separate and accessed via a digital data network such as a local area network (LAN) or the Internet (for example, as a redundant array of inexpensive or independent disks (RAID) or other network server storage that is indirectly accessed by the controller, via a digital network). 
     Alternatively, the method may be implemented in transitory media, such as a transmittable carrier wave in which the control program is embodied as a data signal using transmission media, such as acoustic or light waves, such as those generated during radio wave and infrared data communications, and the like. 
     As will be appreciated, the steps of the method need not all proceed in the order illustrated and fewer, more, or different steps may be performed. 
     In use, the incident light may be natural or “white” light, having a range of wavelengths in the visible range of the spectrum, or may be monochromatic light. The transmissive diffraction grating article  52  (and similarly reflective diffraction grating article  52 ′) causes the incident light to be diffracted into several beams travelling in different directions. As is known in the art, diffraction gratings, either transmissive or reflective, can separate different wavelengths of light using a repetitive structure, here a sequence of parallel lines  34 ,  36 ,  40 ,  42 , which have a similar height and shape that extend above and below the layer  60 . The structure of the grating affects the amplitude and/or phase of the incident wave  160 , causing interference in the output wave  162  ( FIG. 11 ). In the transmissive case, the repetitive structure can be thought of as many tightly spaced, openings, where light can scatter. Solving for the irradiance as a function wavelength and position of the openings, an expression for transmission diffractive gratings when the angle of incidence, relative to normal to the surface, of light on the upper side  30  is θ i , the spacing is s and the wavelength is λ can be defined as in the following simplified grating equation:
 
 s [sin(θ m )−sin(θ i )]= mλ   (1)
 
     where m is the order of principal maxima, such as 0 or 1, and θ m  is the angle, relative to normal to the lower surface  38 , of light exiting the grating. θ i  and θ m  are both positive if on the same side of the surface normal, otherwise θ m  is negative. For a given order m, different wavelengths of light will exit the grating at different angles. For white light sources, this corresponds to a continuous, angle-dependent spectrum. 
     As an example, when illuminated with a beam  160  of light, the transmission grating  52  splits the beam into two (or more) beams with different angles, as illustrated in  FIG. 11 . Light striking the toner lines  34 ,  36 ,  40 ,  42 , tends to be substantially absorbed rather than transmitted or reflected, particularly when the toner is black in color. 
     In the case of a reflective grating article  52 ′, where incident  160  and reflected light  162  are on the same side  30  of the grating (see  FIG. 12 ), a simplified grating equation can be defined as follows:
 
 s [sin(θ m )+sin(θ i )]= mλ   (2)
 
     where θ i  is positive and θ m  is negative if the incident and diffracted light are on opposite sides of the surface normal, otherwise both are positive. 
     For further details on the theory of diffraction gratings, see, for example Thorlabs Grating Tutorial, available at https://www.thorlabs.com/tutorials. 
     As an example, when illuminated with a beam  160  of light, the reflective grating  52 ′ reflects the beam as two (or more) beams with different angles, as illustrated in  FIG. 12 . 
     EXAMPLE 
     A partial mirror is created by printing a diffraction grating at 75 lines per inch on both sides of a transparent plastic transparency material. The transparency material is placed on top of a sheet of glossy paper. Three sheets of the same transparency material are placed on top of the printed diffraction grating and the assembly bound together. The resulting mirror is sufficiently reflective for a user to see features of his face. Color rendering and visibility are improved as compared with a similar partial mirror created by printing a diffraction grating at 75 lines per inch on only one side of a transparent plastic transparency material with three sheets of the same transparency material placed on top. 
     It will be appreciated that variants of the above-disclosed and other features and functions, or alternatives thereof, may be combined into many other different systems or applications. Various presently unforeseen or unanticipated alternatives, modifications, variations or improvements therein may be subsequently made by those skilled in the art which are also intended to be encompassed by the following claims.