EP3059093B1 - Security element, valuable document comprising such a security element and method for producing such a security element - Google Patents

Description

The present invention relates to a security element for a security paper, document of value or the like, a document of value with such a security element and a method for producing such a security element.

Objects to be protected are often equipped with a security element that allows the authenticity of the object to be checked and at the same time serves as protection against unauthorized reproduction.

Objects to be protected are, for example, security papers, ID and value documents (such as banknotes, chip cards, passports, identification cards, ID cards, stocks, attachments, certificates, vouchers, checks, admission tickets, credit cards, health cards, etc.) as well as product security elements such as labels, Seals, packaging, etc.

A technique that is widespread in the field of security elements and that gives a practically flat film a three-dimensional appearance are various forms of holography. However, these techniques have some disadvantages for the use of a security feature, in particular on bank notes. On the one hand, the quality of the three-dimensional representation of a hologram depends heavily on the lighting conditions. In particular with diffuse lighting, the representations of holograms are often barely recognizable. Furthermore, holograms have the disadvantage that they are now present in many places in everyday life and therefore their special position as a security feature is disappearing.

Proceeding from this, the invention is based on the object of avoiding the disadvantages of the prior art and, in particular, of providing a security element for a security paper, document of value or the like, in which a good three-dimensional appearance is achieved with an extremely flat design of the security element.

According to the invention, the object is achieved by a security element according to claims 1 to 15.

In this way, an extremely flat security element, in which, for example, the maximum height of the facets is not greater than 10 μm, can be provided, which nevertheless creates a very good three-dimensional impression when viewed. It is therefore possible, by means of a (macroscopically) flat surface area, to simulate a surface that appears to be strongly curved for the viewer. In principle, any three-dimensional configurations of the perceptible surface of any shape can be produced in this way. Portraits, objects, motifs or other three-dimensional objects can be reproduced. The three-dimensional impression is always related to the actual spatial shape of the surface area. The surface area can be flat or even curved itself. However, a three-dimensional appearance related to this base surface shape is always achieved, so that the surface area is visible to a viewer then does not appear flat or curved in the same way as the surface area itself.

The surface area that can be perceived as a protruding and / or recessed surface is understood here in particular to mean that the surface area can be perceived as a continuously curved surface. So the area z. B. be perceived as a surface with an apparent curvature that deviates from the curvature or actual spatial shape of the surface area. With the security element according to the invention can correspondingly, for. B. a curved surface can be imitated by adjusting the corresponding reflection behavior.

The surface area is in particular a contiguous surface area. The surface area can, however, also have gaps or even include non-contiguous partial areas. In this way, the surface area can be nested with other security features. The other security features can be, for. B. be a true color hologram, so that a viewer can perceive the true color hologram and the projecting and / or receding surface, which are provided by the surface area according to the invention, together.

The orientation of the facets is selected in particular so that the surface area is perceptible to a viewer as a non-planar surface.

It is also possible that the majority or all of the pixels of the surface area each have a plurality of the optically effective facets with the same orientation.

The optically effective facets can be designed as reflective and / or transmissive facets.

The facets can be formed in a surface of the carrier. It is also possible for the facets to be formed both in the upper side and in the lower side of the carrier and to lie opposite one another. In this case, the facets are preferably designed as transmissive facets with a refractive effect, the carrier itself of course also being transparent or at least translucent. The dimensions and orientations of the facets are then selected in particular so that a surface can be perceived by a viewer in such a way that it jumps forward and / or set back in relation to the actual spatial shape of the upper and / or lower side of the carrier.

The carrier can be designed as a layer composite. In this case, the facets can lie at an interface within the layer composite. So the facets z. B. embossed in an embossing lacquer located on a carrier film, then metallized and embedded in a further lacquer layer (e.g. protective lacquer or adhesive lacquer).

In particular, in the security element according to the invention, the facets can be embodied as embedded facets.

In particular, the optically effective facets are designed in such a way that the pixels have no optically diffractive effect.

The dimensions of the optically effective facets can be between 1 μm and 300 μm, preferably between 3 μm and 100 μm and particularly preferably between 5 μm and 30 μm. In particular, there is preferably an essentially ray-optic reflection behavior or an essentially ray-optic refractive effect.

The dimensions of the pixels are selected such that the area of the pixels is at least one order of magnitude and preferably at least two orders of magnitude smaller than the area of the surface area. The area of the surface area and the area of the pixels are understood here in particular to mean the area when projecting in the direction of the macroscopic surface normal of the surface area onto a plane.

In particular, the dimensions of the pixels can be selected such that the dimensions of the pixels are at least one order of magnitude and preferably at least two orders of magnitude smaller than the dimensions of the area of the surface area at least in one direction.

The maximum dimension of a pixel is preferably between 5 μm and 5 mm, preferably between 10 μm and 300 μm, particularly preferably between 20 μm and 100 μm. The pixel shape and / or the pixel size can, but need not, vary within the security element.

The grating period of the facets per pixel (the facets can form a periodic or aperiodic grating, e.g. a sawtooth grating) is preferably between 1 μm and 300 μm or between 3 μm and 300 μm, preferably between 3 μm and 100 μm or between 5 µm and 100 µm, particularly preferably between 5 µm and 30 µm or between 10 µm and 30 µm. In particular, the grating period is chosen so that at least per pixel two facets of the same orientation are contained and that diffraction effects are practically no longer important for incident light (e.g. from the wavelength range from 380 nm to 750 nm). Since no or no practically relevant diffraction effects occur, the facets can be referred to as achromatic facets or the pixels as achromatic pixels which bring about a directionally achromatic reflection. The security element thus has an achromatic reflectivity with respect to the lattice structure provided by the facets of the pixels.

The facets are preferably designed as essentially flat surface pieces. The chosen formulation, according to which the facets are designed as essentially flat surface pieces, takes into account the fact that in practice, due to production, it is usually never possible to produce perfectly flat surface pieces.

The orientation of the facets is determined in particular by their inclination and / or their azimuth angle. Of course, the orientation of the facets can also be determined by other parameters. In particular, these are two mutually orthogonal parameters, such as B. the two components of the normal vector of the respective facet.

A reflective or reflection-increasing coating (in particular a metallic or highly refractive coating) can be formed on the facets at least in some areas. The reflective or reflection-increasing coating can be a metallic coating that is vapor-deposited, for example. In particular, aluminum, gold, silver, copper, palladium, chromium, nickel and / or tungsten and their alloys can be used as the coating material. Alternatively, the reflective or reflection-increasing coating can be formed by a coating with a material with a high refractive index.

The reflective or reflection-increasing coating can in particular be designed as a partially transparent coating.

A color-shifting coating is formed on the facets at least in some areas. The color-shifting coating can in particular be designed as a thin-layer system or thin-film interference coating. For example, a layer sequence of metal layer - dielectric layer - metal layer or a layer sequence of three dielectric layers, the refractive index of the middle layer being lower than the refractive index of the two other layers, can be implemented. _{ZnS, SiO 2} , TiO ₂ , MgF ₂ , for example, can be used as the dielectric material.

The color-shifting coating can also be designed as an interference filter, a thin, semitransparent metal layer with selective transmission through plasma resonance effects, nanoparticles, etc. The color-shifting layer can in particular also be implemented as a liquid crystal layer, a diffractive relief structure or a sub-wavelength grating. A thin-film system with a structure of reflector, dielectric, absorber (formed on the facets in this order) is also possible.

The thin film system plus facet can be designed not only, as already described, as a facet / reflector / dielectric / absorber, but also as a facet / absorber / dielectric / reflector. The order depends in particular on the side from which the security element is to be viewed. Color shift effects visible on both sides are also possible, if the thin film system plus facet is designed, for example, as an absorber / dielectric / absorber / facet or an absorber / dielectric / reflector / dielectric / absorber / facet.

The color-shifting coating can be designed not only as a thin-film system, but also as a liquid-crystal layer (in particular made of cholesteric liquid-crystalline material).

If a diffusely scattering object is to be reproduced, a scattering coating or surface treatment of the facets can be provided. Such a coating or treatment can scatter according to Lambert's cosine law or there can be a scattered reflection with a directional distribution that deviates from Lambert's cosine law. In particular, scattering with a pronounced preferred direction is of interest here.

When the facets are produced by an embossing process, the embossing surface of the embossing tool, with which the shape of the facets can be embossed in the carrier or in a layer of the carrier, can additionally be provided with a microstructure in order to produce certain effects. For example, the embossing surface of the embossing tool can be provided with a rough surface so that facets with scattered reflection arise in the end product.

In the security element according to the invention, at least two facets can preferably be provided per pixel. It can also be three, four, five or more facets.

In the security element according to the invention, the number of facets per pixel can in particular be selected so that a maximum predetermined Facet height is not exceeded. The maximum facet height can be, for example, 20 μm or 10 μm.

Furthermore, in the security element according to the invention, the grating period of the facets can be selected to be the same for all pixels. However, it is also possible that individual or several of the pixels have different grating periods. It is also possible that the grating period varies within a pixel and is therefore not constant. Furthermore, phase information can also be impressed in the grating period, which is used to encode further information. In particular, a verification mask with lattice structures can be provided which have the same periods and azimuth angles as the facets in the security element according to the invention. In a partial area of the verification mask, the grids can have the same phase parameter as the security element to be verified and a certain phase difference in other areas. When the verification mask is placed over the security element, the different areas will then appear differently light or dark due to the moiré effect. In particular, the verification mask can be provided on the same object to be protected as the security element according to the invention.

In the security element according to the invention, the surface area can be designed in such a way that it can be perceived by a viewer as an imaginary surface. This is to be understood here in particular as meaning that the security element according to the invention exhibits a reflection behavior that cannot be produced with a real macroscopically curved surface. In particular, the imaginary surface can be perceived as a rotating mirror, which z. B. rotates by 90 °.

Such an imaginary surface and in particular such a rotating mirror is very easy to grasp and verify for a viewer.

In principle, any real curved reflecting or transmitting surface can be modified into an imaginary surface by means of the surface area of the security element according to the invention. This can e.g. B. can be realized in that the azimuth angles of all facets are changed, for example rotated by a certain angle. Interesting effects can be achieved with this. If, for example, one rotates all azimuth angles by 45 ° to the right, the surface area for an observer, if he is illuminated directly from above, is a curved surface that is apparently illuminated from the top right. If you turn all azimuth angles by 90 °, the light reflections move in a direction that is perpendicular to the direction that an observer would expect. This unnatural reflection behavior then also makes it no longer possible for a viewer, for example, to decide whether the curved perceptible surface is to the front or to the rear (in relation to the surface area).

Furthermore, diffraction effects can be suppressed in a targeted manner by means of an aperiodic grating or the introduction of random phase parameters.

It is also possible to "noise" the orientations of the facets (that is to say to change them slightly compared to the optimal shape for the surface to be reproduced) in order, for example, to reproduce surfaces that appear matt. In this way, the surface area not only appears to be jumping forwards and / or backwards in relation to its actual spatial shape, but it can also be given a texture that is precisely positioned in register.

Furthermore, in addition to the surface area, the carrier can have a further surface area, which is preferably nested with the one surface area and in particular is designed as a further security feature. Such training can, for. B. be referred to as nesting or as a multi-channel image. The further surface area can be divided into a plurality of pixels, each comprising at least one optically effective facet, in the same way as the one surface area, the plurality of pixels preferably each having a plurality of the optically effective facets with the same orientation per pixel and the facets are oriented in such a way that a viewer can perceive the further surface area as a surface that is curved or protruding and / or receding in relation to its actual spatial shape. This allows z. B. two different three-dimensional representations can be realized.

By means of the nesting, the one surface area z. B. with additional register-accurate color or grayscale information (combination for example with true color hologram or halftone image, for example on the basis of sub-wavelength gratings) are superimposed.

Furthermore, phase information can be hidden or stored as a further security feature in the arrangement of the facets.

In the security element according to the invention, at least one facet can have a light-scattering microstructure on its surface. Of course, several or all of the facets can also have such a light-scattering microstructure on the facet surface.

For example, the light-scattering microstructure can be designed as a coating. In particular, it is possible to embed the facets and save them as To use an embedding material with which the desired light-scattering microstructure can be realized.

With such a design scattering objects such. B. a marble figure, a plaster model, etc. can be recreated.

Of course, the facets can also be embedded in a colored material in order to create an additional color effect or to simulate a colored object.

In the security element according to the invention, the orientations of several facets can be changed in relation to the orientations for generating the protruding and / or recessed surface so that the protruding and / or recessed surface is still perceptible, but with a surface that appears matt. Thus, the protruding and / or recessed surface can also be presented with a matt surface appearance.

The invention also comprises a method for producing a security element for security papers, documents of value or the like, according to claims 17 and 18.

The production method according to the invention can in particular be developed in such a way that the security element according to the invention and the developments of the security element according to the invention can be produced.

The production method can further include the step of calculating the pixels on the basis of a surface to be adjusted. In this calculation step, the facets (their dimensions and their orientations) are calculated for all pixels. The height modulation of the surface area can then be carried out on the basis of this data.

In the production method according to the invention, the step of coating the facets can also be provided. The facets can be provided with a reflective or reflection-increasing coating. The reflective or reflection-increasing coating can be a complete mirror coating or also a partially transparent mirror coating.

Known microstructuring processes, such as, for example, embossing processes, can be used to produce the height-modulated surface of the carrier. For example, using methods known from semiconductor production (photolithography, electron beam lithography, laser beam lithography, etc.), suitable structures in resist materials can be exposed, possibly refined, shaped and used for the production of embossing tools. Known processes can be used for embossing in thermoplastic films or in films coated with radiation-curing lacquers. The carrier can have multiple layers that successively applied and optionally structured and / or can be composed of several parts.

The security element can in particular be designed as a security thread, tear thread, security tape, security strip, patch or label for application to a security paper, document of value or the like. In particular, the security element can span transparent or at least translucent areas or recesses.

The term security paper is understood here in particular as the not yet fit for circulation preliminary stage to a document of value which, in addition to the security element according to the invention, can also have, for example, further authenticity features (such as luminescent substances provided in the volume). Documents of value are understood here on the one hand to be documents produced from security papers. On the other hand, value documents can also be other documents and objects that can be provided with the security element according to the invention so that the value documents have authenticity features that cannot be copied, whereby an authenticity check is possible and at the same time undesired copying is prevented.

Furthermore, an embossing tool is provided with an embossing surface with which the shape of the facets of a security element according to the invention (including its developments) can be embossed in the carrier or in a layer of the carrier.

The embossing surface preferably has the inverted shape of the surface contour to be embossed, this inverted shape advantageously being produced by the formation of corresponding depressions.

Furthermore, the security element according to the invention can be used as a master for exposing volume holograms or for purely decorative purposes.

In order to expose the volume hologram, a photosensitive layer in which the volume hologram is to be formed can be brought into contact with the front side of the master and thus with the front side of the security element directly or with the interposition of a transparent optical medium.

Then the photosensitive layer and the master are exposed to a coherent light beam, whereby the volume hologram is written in the photosensitive layer. The procedure can be the same or similar to that in the DE 101006 016139 A1 described procedure for generating a volume hologram. The basic procedure is, for example, in sections Nos. 70 to 79 on pages 7 and 8 of the cited publication in connection with Figures 1a, 1b, 2a and 2b described. The entire content of the DE 10 2006 016139 A1 incorporated into the present application with regard to the production of volume holograms.

It goes without saying that the features mentioned above and those yet to be explained below can be used not only in the specified combinations, but also in other combinations or on their own, without departing from the scope of the present invention.

Figur 1

eine Draufsicht einer Banknote mit einem erfindungsgemäßen Sicherheitselement 1;

Figur 2

eine vergrößerte Draufsicht eines Teils der Fläche 3 des Sicherheitselementes 1;

Figur 3

eine Querschnittsansicht entlang der Linie 6 in Figur 2;

Figur 4

eine schematische perspektivische Darstellung des Pixels 4₇ von Figur 2;

Figur 5

eine Schnittansicht einer weiteren Ausführungsform einiger Facetten des Sicherheitselementes 1;

Figur 6

eine Schnittansicht einer weiteren Ausführungsform einiger Facetten des Sicherheitselementes 1;

Figur 7

eine Schnittansicht zur Erläuterung der Berechnung der Facetten;

Figur 8

eine Draufsicht zur Erläuterung eines Quadratrasters zur Berechnung der Pixel;

Figur 9

eine Draufsicht zur Erläuterung eines 60°-Rasters zur Berechnung der Pixel;

Figur 10

eine Draufsicht auf drei Pixel 4 der Fläche 3;

Figur 11

eine Querschnittsansicht der Darstellung von Figur 10;

Figur 12

eine Draufsicht auf drei Pixel 4 der Fläche 3;

Figur 13

eine Querschnittsansicht der Draufsicht von Figur 12;

Figur 14

eine Draufsicht auf drei Pixel 4 der Fläche 3;

Figur 15

eine Schnittansicht der Draufsicht von Figur 14;

Figur 16

eine Draufsicht zur Erläuterung der Berechnung der Pixel gemäß einer weiteren Ausführungsform;

Figur 17

eine Schnittansicht der Anordnung der Facetten der Pixel auf einer zylindrischen Grundfläche;

Figur 18

eine Schnittansicht zur Erläuterung der Herstellung der Pixel für die Anwendung gemäß Figur 17;

Figuren 19 - 21

Darstellungen zur Erläuterung der Winkel bei reflektiven und transmissiven Facetten;

Figur 22

eine Schnittansicht einer nachzustellenden reflektiven Oberfläche;

Figur 23

eine Schnittansicht einer die Oberfläche gemäß Figur 22 nachstellenden Linse 22;

Figur 24

eine Schnittansicht der transmissiven Facetten für die Nachbildung der Linse gemäß Figur 23;

Figur 25

eine Schnittansicht einer nachzustellenden reflektiven Oberfläche;

Figur 26

eine Schnittansicht einer die Oberfläche gemäß Figur 25 nachstellenden Linse 22;

Figur 27

eine Schnittansicht der entsprechenden transmissiven Facetten zur Nachbildung der Linse gemäß Figur 24;

Figur 28

eine Schnittansicht einer Ausführungsform, bei der auf beiden Seiten des Trägers 8 transmissive Facetten ausgebildet sind;

Figur 29

eine Schnittansicht gemäß einer weiteren Ausführungsform, bei der auf beiden Seiten des Trägers 8 transmissive Facetten ausgebildet sind;

Figur 30

eine Darstellung zur Erläuterung der Winkel bei der Ausführungsform, bei der auf beiden Seiten des Trägers 8 transmissive Facetten ausgebildet sind;

Figur 31

eine schematische Schnittansicht eines Prägewerkzeuges zur Herstellung des erfindungsgemäßen Sicherheitselementes gemäß Figur 5.

Fig. 32a -32 c

Darstellungen zur Erläuterung eingebetteter Facetten, wobei die Facetten als reflektive Facetten ausgebildet sind;

Fig. 33a + 33b

Darstellungen zur Erläuterung eingebetteter Facetten, wobei die Facetten als transmissive Facetten ausgebildet sind;

Figur 34

eine Darstellung zur Erläuterung eingebetteter streuender Facetten, und

Figur 35

eine Darstellung zur Erläuterung eingebetteter matt glänzender Facetten.

The invention is illustrated below by way of example with reference to the accompanying drawings, which also disclose features essential to the invention explained in more detail. For the sake of clarity, a representation true to scale and proportion is dispensed with in the figures. Show it:

Figure 1

a plan view of a bank note with a security element 1 according to the invention;

Figure 2

an enlarged plan view of part of the surface 3 of the security element 1;

Figure 3

FIG. 6 is a cross-sectional view taken along line 6 in FIG Figure 2 ;

Figure 4

FIG. 4 is a schematic perspective illustration of the pixel 4 ₇ of FIG Figure 2 ;

Figure 5

a sectional view of a further embodiment of some facets of the security element 1;

Figure 6

a sectional view of a further embodiment of some facets of the security element 1;

Figure 7

a sectional view for explaining the calculation of the facets;

Figure 8

a plan view for explaining a square grid for calculating the pixels;

Figure 9

a plan view for explaining a 60 ° raster for calculating the pixels;

Figure 10

a plan view of three pixels 4 of the area 3;

Figure 11

FIG. 3 is a cross-sectional view of the illustration of Figure 10 ;

Figure 12

a plan view of three pixels 4 of the area 3;

Figure 13

Figure 13 is a cross-sectional view of the top view of Figure 13 Figure 12 ;

Figure 14

a plan view of three pixels 4 of the area 3;

Figure 15

FIG. 3 is a sectional view of the top view of FIG Figure 14 ;

Figure 16

a plan view for explaining the calculation of the pixels according to a further embodiment;

Figure 17

a sectional view of the arrangement of the facets of the pixels on a cylindrical base;

Figure 18

FIG. 13 is a sectional view for explaining the manufacture of pixels for the application of FIG Figure 17 ;

Figures 19-21

Representations to explain the angles in reflective and transmissive facets;

Figure 22

a sectional view of a reflective surface to be adjusted;

Figure 23

a sectional view of the surface according to Figure 22 adjusting lens 22;

Figure 24

a sectional view of the transmissive facets for the replica of the lens according to FIG Figure 23 ;

Figure 25

a sectional view of a reflective surface to be adjusted;

Figure 26

a sectional view of the surface according to Figure 25 adjusting lens 22;

Figure 27

a sectional view of the corresponding transmissive facets for emulating the lens according to FIG Figure 24 ;

Figure 28

a sectional view of an embodiment in which transmissive facets are formed on both sides of the carrier 8;

Figure 29

a sectional view according to a further embodiment, in which transmissive facets are formed on both sides of the carrier 8;

Figure 30

an illustration to explain the angles in the embodiment in which transmissive facets are formed on both sides of the carrier 8;

Figure 31

a schematic sectional view of an embossing tool for producing the security element according to the invention according to FIG Figure 5 .

Figures 32a-32c

Representations to explain embedded facets, the facets being designed as reflective facets;

Figures 33a + 33b

Representations to explain embedded facets, the facets being designed as transmissive facets;

Figure 34

an illustration for explaining embedded scattering facets, and

Figure 35

a representation to explain embedded matt glossy facets.

At the in Figure 1 The embodiment shown, the security element 1 according to the invention is integrated in a bank note 2 in such a way that the security element 1 differs from the one shown in FIG Figure 1 The front side of the banknote 2 shown is visible.

The security element 1 is designed as a reflective security element 1 with a rectangular outer contour, the area 3 delimited by the rectangular outer contour being divided into a plurality of reflective pixels 4, of which a small part is enlarged into Figure 2 are shown as a top view.

The pixels 4 are square here and have an edge length in the range from 10 to several 100 μm. The edge length is preferably not greater than 300 μm. In particular, it can be in the range between 20 and 100 μm.

The edge length of the pixels 4 is selected in particular such that the area of each pixel 4 is at least one order of magnitude, preferably two orders of magnitude smaller than the area 3.

The majority of the pixels 4 each have a plurality of reflective facets 5 of the same orientation, the facets 5 being the optically effective surfaces of a reflective sawtooth grid.

In Figure 3 the sectional view along the line 6 for six adjacent pixels 4 ₁ , 4 ₂ , 43, 44, 4 ₅ and 4 _{6 is} shown, the illustration in FIG Figure 3 and also in the other figures is partly not true to scale for better illustration. Furthermore, in order to simplify the illustration in FIGS. 1 to 3 and also in FIG Figure 4 the reflective coating on the facets 5 is not shown.

The sawtooth grid of the pixels 4 is formed here in a surface 7 of a carrier 8, the surface 7 structured in this way preferably having a reflective coating (in Figure 3 not shown) is coated. The carrier 8 can be, for example, a radiation-curing plastic (UV resin) which is applied to a carrier film (not shown) (for example a PET film).

How Figure 3 As can be seen, the pixels 4 ₁ , 4 ₂ , 4 ₄ , 4 ₅ and 4 ₆ each have three facets 5, the orientation of which is the same _{per pixel 4 1} , 4 ₂ , 4 ₄ , 4 ₅ and 4 _6. The sawtooth grids and thus also the facets 5 of these pixels are the _{same except for their different inclinations σ 1} , σ ₄ (to simplify the illustration, only the inclination angles σ ₁ and σ ₄ of one facet 5 of the pixels 4 ₁ and 4 _{4 are} shown ). The pixel 4 ₃ here has only a single facet 5.

Seen in plan view ( Figure 2 ) the facets 5 of the pixels 4 ₁ - 4 _{6 are} strip-shaped mirror surfaces that are aligned parallel to one another. The orientation of the facets 5 is selected in such a way that a viewer can perceive the surface 3 as a projecting and / or receding surface compared to its actual (macroscopic) spatial shape, which here is the shape of a flat surface. Here a viewer takes the in Figure 3 Surface 9 shown in section is true when looking at the facets 5. This is achieved by choosing the orientations of the facets 5, which reflect the incident light L1 as if it were on a surface according to the direction indicated by line 9 in FIG Figure 3 indicated three-dimensional shape falls, as is shown schematically by the incident light L2. The reflection generated by the facets 5 of a pixel 4 corresponds to the mean reflection of the area of the surface 9 converted or adjusted by the corresponding pixel 4.

In the case of the security element 1 according to the invention, a height profile that appears three-dimensional is thus simulated by an arrangement of reflective sawtooth structures (facets 5 per pixel 4), which imitate the reflection behavior of the height profile, screened here. Any three-dimensionally perceptible motifs can thus be generated with the surface 3, such as a person, parts of a person, a number or other objects.

In addition to the slope σ of the individual facets 5, the azimuth angle α must also be adapted to the following surface. For pixels 4 ₁ - 4 ₆ , the azimuth angle α is relative to the direction according to arrow P1 ( Figure 2 ) 0 °. For the pixel 4 ₇ , the azimuth angle α is approximately 170 °, for example. The sawtooth grid of pixel 4 ₇ is in Figure 4 shown schematically in three-dimensional representation.

To produce the security element 1, the reflective sawtooth structures can be written into a photoresist, for example by means of gray-scale lithography, then developed, galvanically molded, embossed in UV lacquer (carrier) and mirrored. The mirroring can be implemented, for example, by means of an applied metal layer (for example vapor-deposited). Typically, an aluminum layer with a thickness of 50 nm, for example, is applied. Of course, other metals such as silver, copper, chromium, iron, etc. or alloys thereof can also be used. As an alternative to metals, high-index coatings can also be applied, for example ZnS or TiO ₂ . The vapor deposition can be applied over the entire surface. However, it is also possible to carry out a coating only in areas or in the form of a grid, so that the security element 1 is partially transparent or translucent.

The period A of the facets 5 is the same for all pixels 4 in the simplest case. However, it is also possible to vary the period A of the facets 5 per pixel 4. For example, the pixel 4 _{7 has} a smaller period A than the pixels 4 ₁ - 4 ₆ (FIG. 2). In particular, the period A of the facets 5 can be chosen randomly for each pixel. By varying the choice of the period Λ of the sawtooth grating for the facets 5, any visibility of a diffraction image going back to the sawtooth grating can be minimized.

A fixed period A is provided within a pixel 4. In principle, however, it is also possible to vary the period A within a pixel 4, so that aperiodic sawtooth grids are present per pixel 4.

The period A of the facets 5 is on the one hand to avoid undesirable diffraction effects and to minimize the necessary film thickness (thickness of the carrier 8), on the other hand, preferably between 3 μm and 300 μm. In particular, the distance is between 5 μm and 100 μm, a distance between 10 μm and 30 μm being particularly preferably selected.

In the exemplary embodiment described here, the pixels 4 are square. However, it is also possible to design the pixels 4 to be rectangular. Other pixel shapes can also be used, such as a parallelogram or hexagonal pixel shape. The pixels 4 preferably have dimensions that are on the one hand larger than the distance between the facets 5 and on the other hand are so small that the individual pixels 4 do not attract the naked eye. The size range resulting from these requirements is between about 10 and a few 100 µm.

Gradients σ and azimuth angle α of the facets 5 within a pixel 4 then result from the gradient of the adjusted height profile 9.

In addition to the slope σ and the azimuth angle α, a phase parameter p _i can also optionally be introduced for each pixel 4. The surface relief of the security element 1 can then be described in the i-th pixel 4i by the following height function hi (x, y): $$H_i\left(x,y\right)={\mathrm{A.}}_i\left[\left(-x\cdot {\mathit{sin\alpha }}_i+y\cdot {\mathit{cos\alpha }}_i+p_i\right)\mathit{mod}{\mathrm{\Lambda }}_i\right]$$

Ai is the amplitude of the sawtooth grid, α _i the azimuth angle and Ai the grid period. "mod" stands for the modulo operation and returns the positive remainder in the case of division. The amplitude factor Ai results from the slope of the adjusted surface profile 9.

By changing the phase parameter p _i , the sawtooth grids or the facets 5 of different pixels 4 can be shifted relative to one another. Random values or other values varying per pixel 4 can be used for the parameters p _i. As a result, any diffraction pattern of the sawtooth grid (the facets 5 per pixel 4) or the grid grid of the pixels 4 that may still be visible can be eliminated, which can otherwise cause undesirable color effects. Furthermore, due to the varied phase parameters p _{i, there are} also no particular directions in which the sawtooth grids of adjacent pixels 4 fit one another particularly well or particularly poorly, which prevents visible anisotropy.

In the security element 1 according to the invention, the azimuth angle α and the gradients σ of the facets 5 per pixel 4 can be selected such that they do not correspond as closely as possible to the reproduced surface 9, but rather deviate from it somewhat. For this purpose, a (preferably random) component can be added for each pixel 4 to the optimal value for the adjustment of the surface 9 in accordance with a suitable distribution. Depending on the size of the pixel 4 and the strength of the noise (standard deviation of the distribution), different interesting effects can be achieved in this way. With very fine pixels 4 (around 20 µm), the otherwise glossy surface appears increasingly matt as the noise increases. Larger pixels (around 50 µm) give an appearance comparable to that of a metallic coating. In the case of very large pixels (several 100 µm), the individual pixels 4 are resolved by the naked eye. They then appear like rough but smooth sections that light up brightly from different viewing angles.

The strength of the noise can be selected differently for different pixels 4, which means that the curved surface appears on different ones Places can appear differently smooth or matt. In this way, for example, the effect can be generated that the viewer perceives the surface 3 as a smooth projecting and / or recessed surface that has a matt lettering or texture.

It is also possible to apply a color-changing coating, in particular a thin film system, to the facets 5. The thin-film system can for example have a first, a second and a third dielectric layer which are formed on top of one another, the first and third layers having a higher refractive index than the second layer. Due to the different inclinations of the facets 5, different colors can be perceived by a viewer without having to rotate the security element 1. The perceptible surface thus has a certain color spectrum.

The security element 1 can in particular be designed as a multi-channel image that has different sub-areas nested in one another, at least one of the sub-areas being designed in the manner according to the invention, so that this sub-area can be perceived by the viewer as a spatial sub-area. Of course, the other partial areas can also be formed in the manner described by means of pixels 4 with at least one facet 5. The other partial areas can, but do not have to, be perceptible as areas projecting and / or receding in relation to the actual spatial shape. The nesting can be designed, for example, in the manner of a chessboard or also of strips. Interesting effects can be achieved by nesting several partial areas. If, for example, the adjustment of a spherical surface is nested with the representation of a number, this can be done in such a way that for the viewer the impression is created that the number is inside a glass ball with a semi-reflective surface.

In addition to the already described use of color-shifting coatings, it is also possible to additionally provide the security element 1 according to the invention with color information. For example, color can be printed on the facets 5 (either transparent or thin) or provided below an at least partially transparent or translucent sawtooth structure. For example, a motif represented by means of the pixels 4 can be colored in this way. If, for example, a portrait is being recreated, the layer of paint can provide the color of the face.

A combination with a true color hologram or kinegram, in particular nesting with a true color hologram that shows a colored representation of the surface 9 simulated with the pixels 4, is also possible. This means that the achromatic three-dimensional image of an object appears colored at certain angles.

A combination with a subwavelength grating is also possible. In particular, the interlaced display of the same motif using both techniques is advantageous, in which the three-dimensional effect of the sawtooth structures is combined with the color information of the subwavelength gratings.

The surface 9 reproduced with the pixels 4 can in particular be a so-called imaginary surface. This is understood here to mean the formation of a reflection or transmission behavior that cannot be produced with a real curved reflecting or transmitting surface.

To further explain the concept of the imaginary surface, a mathematical criterion for delimiting real surfaces is introduced below and explained using the example of a rotating mirror.

When simulating a real curved surface, this can be described by a height function h (x, y). It can be assumed here that the function h (x, y) is differentiable (non-differentiable functions could be approximated by differentiable functions, which would ultimately produce the same effect on the observer). If one now integrates the gradient of h (x, y) along an arbitrarily closed curve C, the integral vanishes: $${\displaystyle {\oint }_{\mathrm{C.}}\nabla H\left(x,y\right)\overrightarrow{\mathit{ds}}}=0$$

Figuratively speaking, this means that you walk up the same height differences as you walk down along a closed path and arrive at the same height again at the end. The sum of the height differences overcome in this way must therefore be zero.

In the security element 1 according to the invention, the slope and azimuth of the facets 5 correspond to the gradient of the height function. Cases can now be constructed in which the slope and azimuth of the facets 5 merge practically continuously into one another, but no height function can be found with which the above integral disappears. In this case, we are talking about the adjustment of an imaginary surface.

A special design is, for example, a rotating mirror. To do this, we first consider the reproduction of a real convex mirror with a parabolic profile. The height function is given by $$H\left(x,y\right)=-\mathrm{<figure-callout\; id="2"\; label="c\; x"\; filenames="imgf0001.png"\; state="\{\{state\}\}">c</figure-callout>}\left(x^{}\right)\left({}^2+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}^2\right)$$

und $$A\left(x,y\right)=2c\sqrt{\left(x^2+y^2\right)}$$

Where c> 0 is a constant and determines the curvature of the mirror. In such a mirror, the viewer can see an upright, reduced reflection of himself. The parameters of the sawtooth structures are then given by $$\alpha \left(x,y\right)=\arctan \left(x,y\right)$$

and $$\mathrm{A.}\left(x,y\right)=2c\sqrt{\left(x^2+{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}^2\right)}$$

If a constant angle δ is now added to the azimuth angle α, the mirror image is rotated by precisely this angle. If δ is not an integral multiple of 180 °, an imaginary surface is created. If, for example, δ = 90 ° is selected, the mirror image is rotated by 90 ° and a mirror image is obtained that cannot be achieved with a smooth, curved real surface. If one sets the gradient of h equal to the slope of the sawtooth structures, one can now find closed curves in which the above integral does not vanish. For example, a curve K along a circle around the center point with radius R> 0 $${\displaystyle {\oint }_K\nabla H\left(x,y\right)\overrightarrow{\mathit{ds}}}={\displaystyle {\oint }_{K}2c\cdot \mathit{ds}}=\mathrm{4th}\mathit{\pi cR}\ne 0$$

Figuratively speaking, this rotating mirror simulates a surface where you walk continuously uphill along a circle, but at the end arrive back at the same height at which you started. Such a real surface obviously cannot exist.

In the case of the security elements 1 described so far, it was assumed that the surface is designed as a reflective surface. However, the same effects of the three-dimensional effect can essentially also be achieved in transmission if the sawtooth structures or the pixels 4 with the facets 5 (including the carrier 8) are at least partially transparent. The sawtooth structures are preferably located between two layers with different refractive indices. In this case, the security element 1 then appears to the observer like a glass body with a curved surface.

The advantageous configurations described can also be used for the transmissive design of the security element 1. For example, the rotating mirror of an imaginary surface can rotate the image when looking through it.

The transmissive design of the security element is described in more detail below in connection with the Figures 19 to 29 described.

The security against forgery of the security element 1 according to the invention can be increased by further features which are only visible with aids and which can also be referred to as hidden features.

For example, additional information can be encoded in the phase parameters of the individual pixels 4. In particular, a verification mask can be produced with lattice structures that have the same periods and azimuth angles as the security element 1 according to the invention. In a partial area of the area, the lattices of the verification mask can have the same phase parameters as the security element to be verified, and a certain phase difference in other areas . These different areas will appear differently light or dark due to moiré effects when the security element 1 and the verification mask are placed on top of one another.

In particular, the verification mask can also be provided in the bank note 2 or the other element provided with the security element 1.

In addition to the outline shapes described, the pixels 4 can also have other outlines. These outlines can then be recognized with a magnifying glass or a microscope.

Furthermore, instead of the corresponding saw teeth or facets 5, any other desired structure can also be embossed or inscribed in a small portion of the pixels 4 without this being noticed by the naked eye. In this case, these pixels are not part of the area 3, so that the area 3 is interlaced with the differently designed pixels. These other formed pixels can be, for example, every 100th pixel in comparison to the pixels 4 of the area 3. A micro-font or a logo can be incorporated into these pixels, for example 10 µm letters in a 40 µm pixel.

In the exemplary embodiments described so far, the facets in the surface 7 of the carrier 8 are formed in such a way that the deepest points or the minimum height values of all facets 5 ( Figure 3 ) lie in one plane. However, it is also possible to form the facets 5 in such a way that the mean values of the heights of all the facets 5 are at the same height, as in FIG Figure 5 is shown schematically. It is also possible to design the facets 5 in such a way that the peak values or the maximum height values of all facets 5 of the pixels 4 are at the same height, as in FIG Figure 6 is indicated schematically.

In Figure 7 FIG. 13 is a sectional view in the same manner as in FIG Figure 3 however, a mirror surface 10 is drawn in for the pixel 4 _4, which mirrors the surface 9 _{in the region of the pixel 4 4.} With a pixel size of, for example, 20 μm to 100 μm, such a mirror surface 10 would lead to undesirably large heights d being present. With a mirror inclination of 45 °, the corresponding mirror surface 10 would protrude from the xy plane by 20 µm to 100 µm. However, maximum heights d of 10 μm are preferred. The mirror surface 10 is therefore still subjected to a modulo d operation, so that the in Figure 7 Facets 5 drawn are formed, the normal vectors n of the facets 5 corresponding to the normal vector n of the mirror surface 10.

The surface 9 to be readjusted can be present, for example, as a set of x, y values, each with an associated height h in the z direction (3D bitmap). A defined square or 60 ° grid ( Figures 8, 9 ) being constructed. The grid points are connected in such a way that there is an area coverage in the xy plane with triangular tiles, as shown in Figures 8 and 9 is shown schematically.

The h values are taken from the 3D bitmap at the three corner points of each tile. The smallest of these h-values is subtracted from the h-values of the three corner points of the tiles. With these new h values at the corner points, a sawtooth surface is built up from inclined triangles (triangular plane pieces). The plane pieces protruding too far from the x-y plane are replaced by the facets 5. The surface description for the facets 5 is thus obtained and the security element 1 according to the invention can be produced.

The surface 9 to be adjusted can be given by a mathematical formula f (x, y, z) = h (x, y) −z = 0. The facets 5 or their orientations are obtained from tangential planes of the surface 9 to be simulated. These can be determined from the mathematical derivation of the function f (x, y, z). The facet 5 placed at a point x ₀ , yo is described by the normal vector: $$\stackrel{\rightharpoonup }{n}=\left(\begin{array}{c}{\mathrm{n}}_{\mathrm{x}}\\ {\mathrm{n}}_{\mathrm{y}}\\ {\mathrm{n}}_{\mathrm{z}}\end{array}\right)=\left(\begin{array}{c}\frac{\partial f}{\partial x}\left(x_0,y_0,z_0\right)\\ \frac{\partial f}{\partial y}\left(x_0,y_0,z_0\right)\\ \frac{\partial f}{\partial z}\left(x_0,y_0,z_0\right)\end{array}\right)/\sqrt{{\left(\frac{\partial f}{\partial x}\left(x_0,y_0,z_0\right)\right)}^2+{\left(\frac{\partial f}{\partial y}\left(x_0,y_0,z_0\right)\right)}^2+{\left(\frac{\partial f}{\partial z}\left(x_0,y_0,z_0\right)\right)}^2}$$

The azimuth angle α of the tangential plane is arctan (n _{y /} n _x ) and the slope angle σ of the tangential plane is arccos n _z . The surface f (x, y, z) = can be arbitrarily curved and (xo, yo, zo) is the point on the surface for which the calculation is currently being carried out. The calculation is carried out one after the other for all the points selected for the sawtooth structure.

Regions are cut out from the inclined planes with the normal vectors calculated in this way, which are to be attached to the selected points in the xy plane, so that at neighboring xy points Overlapping of the associated elements can be avoided. The inclined plane pieces that protrude too far out of the xy plane are divided into smaller facets 5, as in connection with Figure 7 has been described.

dabei sind x_i, yi, zi die Dreiecks-Eckpunkte.The surface to be readjusted can be described by triangular patches, the flat triangular pieces being spanned between selected points which lie within and on the edge of the surface to be readjusted. The triangles can be described as plane pieces by the following mathematical function f (x, y, z) $$\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\left|\begin{array}{ccc}\mathrm{x}-{\mathrm{x}}_1& \mathrm{y}-{\mathrm{y}}_1& \mathrm{z}-{\mathrm{z}}_1\\ {\mathrm{x}}_2-{\mathrm{x}}_1& {\mathrm{y}}_2-{\mathrm{y}}_1& {\mathrm{z}}_2-{\mathrm{z}}_1\\ {\mathrm{x}}_3-{\mathrm{x}}_1& {\mathrm{y}}_3-{\mathrm{y}}_1& {\mathrm{z}}_3-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_1\end{array}\right|=0,$$

where x _i , yi, zi are the triangle corner points.

In this case, the surface can be projected into the xy plane and the individual triangles can be tilted according to their normal vector. The inclined plane pieces form the facets and if they protrude too far from the xy plane, as in connection with Figure 7 has been described, divided into smaller facets 5.

If the surface to be adjusted is given by triangular patches, you can also proceed as follows. The entire surface to be reproduced is subjected at once (or parts of each surface) to a Fresnel construction modulo d (or modulo di). Since the surface to be simulated consists of plane pieces, triangles that are filled with the facets 5 are automatically created on the xy plane.

The construction of the facets can also be carried out as follows. In the x-y plane, over which the surface 9 to be reproduced is defined, suitable x-y points are selected and they are connected in such a way that the x-y plane is covered with polygon tiles. The normal vector is determined from the surface 9 to be reproduced above a randomly selected point (e.g. a corner point) in each tile. A Fresnel mirror (pixel 4 with several facets 5) corresponding to the normal vector is now attached to each tile.

Square tiles or pixels 4 are preferably used. In principle, however, any (irregular) tiling is possible. The tiles can be contiguous (which is preferred for greater efficiency) or there can be joints between the tiles (for example, with circular tiles).

The angle of inclination σ of the plane can be represented as follows $$\mathrm{\sigma }=\arccos {\mathrm{n}}_{\mathrm{z}}=\arccos \raisebox{1ex}{$\frac{\partial f}{\partial z}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{{\left(\frac{\partial f}{\partial x}\right)}^2+{\left(\frac{\partial f}{\partial \mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}\right)}^2+{\left(\frac{\partial f}{\partial \mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}\right)}^2}$}\right.$$

wobei α = 0° bis 180° für n_y > 0 und α = 180° bis 360° für n_y < 0.The azimuth angle α of the slope can be represented as follows $$\mathrm{\alpha }=\arctan \left({\mathrm{n}}_{\mathrm{y}}/{\mathrm{n}}_{\mathrm{x}}\right)=\arctan \raisebox{1ex}{$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}$}\!\left/ \!\raisebox{-1ex}{$\frac{\partial \mathrm{f}}{\partial \mathrm{x}}$}\right.,$$

where α = 0 ° to 180 ° for n _y > 0 and α = 180 ° to 360 ° for n _y <0.

The inventive determination of the facets 5 including their orientations can be carried out in two fundamentally different ways. The x-y plane can thus be divided into pixels 4 (or tiles) and the normal vector for the reflective flat surface is determined for each pixel 4, which is then converted into several facets 5 with the same orientation. Alternatively, it is possible to approximate the surface 9 to be simulated by plane pieces, if it is not already given by plane pieces, and then to subdivide the plane pieces into the individual facets 5.

In the first procedure, tiling is therefore initially determined in the x-y plane. The tiling can be created in any way you like. However, it is also possible for the tiling to consist of nothing but equal squares with side length a, where a is preferably in the range from 10 to 100 μm. The tiling can, however, also consist of different shaped tiles that fit together exactly or in which joints occur. The tiles can be shaped differently and contain coding or hidden information. In particular, the tiles can be adapted to the projection of the surface to be reproduced in the x-y plane.

A reference point can then be defined in any way in each tile. The normal vectors in the points of the surface to be simulated, which are perpendicular to the reference points in the tiles, are assigned to the corresponding tiles. If several normal vectors are assigned to the reference point in the surface to be readjusted above the reference point (e.g. at an edge or corner where several surface pieces butt against each other), one can determine an averaged normal vector from these normal vectors.

A subdivision is defined in each tile in the x-y plane. This subdivision can be any. The azimuth angle α and the slope angle σ are then calculated from the normal vector. Optionally, you can also define an offset system that assigns an offset (height value) to each facet 5. The offset can be arbitrary in any area of the subdivision. However, it is also possible to apply the offset in such a way that the mean values of the facets 5 are all at the same level or that the maximum values of all the facets 5 are at the same level.

In the subdivisions in the assigned tiles, inclined plane pieces with the normal vector assigned to the tile are then applied computationally as facets 5, taking into account the offset system. The surface shape calculated in this way is then formed in the surface 7 of the carrier 8.

However, it is not only possible to define any subdivision in each tile in the xy plane. For example, you can also define grid lines that are approximately or exactly perpendicular to the projection of the normal vector in the xy plane. The grid lines can be at any distance from one another. However, it is also possible for the spacing of the grid lines to follow a certain scheme. For example, grid lines cannot be provided exactly parallel to one another in order to avoid interference, for example. However, it is also possible that the grid lines are parallel to one another, but have different distances. The different distances between the grid lines can include a coding. It is also possible that the grid lines of all facets 5 in each pixel 4 have the same spacing. The distance can be in the range from 1 µm to 20 µm.

The grid lines can also have the same spacing within each tile or within each pixel 4, but vary per pixel 4. The grid line spacing Ai and the angle of inclination σ _{i of} the associated facet 5 determine the structure thickness d _i = _{t i} · tan σ _i , where d _{i is} preferably 1 to 10 μm.

The facets 5 can also all have the same height d. Then the lattice constant is determined in some areas by the inclination angle σ _{i of} the associated facet i: Λ _i = d / tan σ _i .

The azimuth angle α and the slope angle σ are then in turn determined from the normal vector. The sawtooth grid defined by grid lines, azimuth angle and angle of inclination is attached to the associated tile by calculation, taking into account the offset system.

One can also start from a surface 9 to be reproduced, which is made up of plane pieces i (or which is processed in such a way that it is made up of plane pieces i), the structural depth of the surface to be reproduced and the dimensions of the plane pieces being somewhat greater than di .

For example, the plane pieces i are each given by three corner points x _1i , y _1i , z _1i ; x _2i , y _2i , z _2i; x _3i , y _3i , Z _3i .

The relief from plane pieces is represented by z = f (x, y), where $$\left(\mathrm{x}-{\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">x</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|-\left(\mathrm{y}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|+\left(\mathrm{z}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\right)\left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|=0$$

This results in resolved after z $$z={\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}+\frac{\left(\mathrm{y}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|-\left(\mathrm{x}-{\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">x</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|}{\left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|}$$

The sawtooth surface, whose structure thickness in the areas i is smaller than di, results from z modulo di, where z is calculated from the above formula and where the x and y values in the calculation are each within the range given by x _1i , y _1i ; x _2i , y _2i ; x _3i , y _3i given triangle lie in the xy-plane.

The sawtooth surface calculated in this way is automatically composed of the facets 5. This results in the lattice constants Ai in the areas i $${\mathrm{\Lambda }}_{\mathrm{i}}={\mathrm{d}}_{\mathrm{i}}/\tan {\mathrm{\sigma }}_{\mathrm{i}}$$

wobei σ_i der Steigungswinkel des durch x_1i, y_1i, z_1i; x_2i, y_2i, z_2i; X_3i, y_3i, z_3i gegebenen Dreiecks ist.If a lattice constant Λ that is the same everywhere is desired, the following d _{i should be used} $${\mathrm{d}}_{\mathrm{i}}=\mathrm{\Lambda }\tan {\mathrm{\sigma }}_{\mathrm{i}}$$

where σ _{i is the pitch angle of} the through x 1i, y _1i , z _1i ; x _2i , y _2i , z _2i ; X _3i , y _3i , z _3i given triangle.

The following alternative procedure is possible. In the following formula A, a surface 9 to be readjusted lying above the xy plane is described by triangular plane pieces $$z={\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}+\frac{\left(\mathrm{y}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|-\left(\mathrm{x}-{\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">x</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|}{\left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|}$$

_{The plane pieces} i are each given by three corner points x 1i, y _1i , z _1i ; x _2i , y _2i , z _2i ; x _3i , y _3i , z _3i .

The corner points are numbered so that z _{1i is} the smallest value among the three values z _1i , z _2i , z _3i (z _1i = min (z _1i , z _2i , z _3i )).

The following formula B represents a sawtooth surface which simulates the three-dimensional impression of the surface 9 to be readjusted given by the formula A $$z=\frac{\left(\mathrm{y}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|-\left(\mathrm{x}-{\mathrm{<figure-callout\; id="1"\; label="x"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">x</figure-callout>}}_{1,\mathrm{i}}\right)\cdot \left|\begin{array}{cc}{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">z</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">z</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|}{\left|\begin{array}{cc}{\mathrm{x}}_{2,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{2,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\\ {\mathrm{x}}_{3,\mathrm{i}}-{\mathrm{x}}_{1,\mathrm{i}}& {\mathrm{<figure-callout\; id="3"\; label="y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">y</figure-callout>}}_{3,\mathrm{i}}-{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">y</figure-callout>}}_{1,\mathrm{i}}\end{array}\right|}$$

As you can see, the sawtooth surface according to formula B differs from the surface to be _adjusted according to formula A in that the minimum value z 1i in area i is subtracted from the value z. The sawtooth surface according to formula B consists of inclined triangles attached to the xy plane.

If a maximum thickness di is specified for the structure depth, it is possible that the maximum thickness for the sawtooth surface according to formula B is exceeded. On the other hand, the formation of the individual facets with the same normal vector according to z modulo d _i helps, where z is calculated from the above formula B and the x and y values in the calculation are each within the range given by x _1i , y _1i ; x _2i , y _2i ; x _3i , y _3i given triangle lie in the xy-plane.

The sawtooth surface calculated in this way is composed of the triangular areas which are filled with the facets 5, the lattice constants A in the areas i being Λ _i = d _i / tan σ _i . The angle σ _i is the slope _{angle of the through x 1i} , y _1i , z _1i ; x _2i , y _2i , z _2i ; x _3i , y _3i , z _3i given triangle.

The procedures shown here for surfaces to be adjusted, which are described by triangles and which, according to the invention, are converted into pixels 4 with a plurality of facets 5, are to be understood as examples. In general, the following procedure is followed according to the invention in the case of surfaces to be adjusted, which are described by plane pieces. The plane pieces are divided into sections. In the case of the subdivisions, a value (for example the minimum value of z in the section) is subtracted. According to the invention, a sawtooth grid is thus obtained that is flatter than the surface 9 to be readjusted and that has the same normal vectors in the sections in each case.

This sawtooth grid imitates the original surface 9 to be reproduced, including its three-dimensional impression. This sawtooth grid is flatter than a sawtooth grid created using the same procedure without subdividing the pixels 4 into several facets 5 according to the invention.

In Figure 10 a top view of three pixels 4 of the area 3 according to a further embodiment is shown, the pixels 4 being irregular (solid lines) with irregular subdivisions or facets 5 (dashed lines). The pixel edges and the subdivisions are straight lines here, but they can also be curved.

In Figure 11 the corresponding cross-sectional view is shown, the normal vectors of the facets 5 being shown schematically. The normal vectors of all facets 5 are the same per pixel 4, while they differ from pixel 4 to pixel 4. The normal vectors lie obliquely in space and generally not in the plane of the drawing, as in Figure 11 is shown for simplicity.

In Figure 12 FIG. 11 is a top view with the same division of the pixels 4 as shown in FIG. 11, but the division (facets 5) per pixel 4 being different. In the exemplary embodiment shown, the grating period A of the facets 5 in each pixel 4 is constant, but different from pixel 4 to pixel 4.

Figure 13 shows the corresponding cross-sectional view.

In Figure 14 a further modification is shown in which the pixel shape is the same as in FIG Figure 10 . However, the division is coded 4 per pixel. Every second grid line spacing is twice as large as the previous grid line spacing. In Figure 15 the corresponding cross-sectional view is shown.

If the surface to be simulated is given as a contour line image, the normal vectors can be determined as follows. One chooses discrete points on the contour lines 15 (in Figure 16 a schematic top view is shown) and connects these points in such a way that a triangular tiling is created. The calculation of the normal vector for the triangles is carried out as already described.

$${\mathrm{x}}_{\mathrm{trans}}=2\mathrm{\pi }\mathrm{r}\mathrm{\Phi }/360,\mathrm{\Phi }=360{\mathrm{x}}_{\mathrm{trans}}/2\mathrm{\pi r}$$

In the previous embodiments, the normal vector was always calculated relative to the xy plane. However, it is also possible to calculate the normal vector in relation to a curved base surface, such as a cylinder surface. In this case, the security element can be provided on a bottle label (for example on the bottle neck) in such a way that the following surface can then be perceived spatially undistorted by a viewer. For this purpose, it is only necessary to convert the normal vector n related to the cylinder _{surface into the normal vector n trans} related to a plane, so that the manufacturing methods described above can be used. When the security element according to the invention is then applied as a bottle label to the bottle neck (with the cylindrical curvature), the following surface 9 can then be perceived undistorted in a three-dimensional manner. The conversion to be carried out results from the following formulas $$\mathrm{x}=\mathrm{r}\sin \mathrm{\Phi },\mathrm{\Phi }=\arcsin \mathrm{x}/\mathrm{r}$$

$${\mathrm{x}}_{\mathrm{trans}}=2\mathrm{\pi }\mathrm{r}\mathrm{\Phi }/360,\mathrm{\Phi }=360{\mathrm{x}}_{\mathrm{trans}}/2\mathrm{\pi r}$$

wobei n = Normalenvektor über (x,y).The normal vector n _trans at the point (x _trans , y) can be calculated as follows. $${\overrightarrow{\mathrm{n}}}_{\mathrm{trans}}=\left(\begin{array}{ccc}\cos \varphi & 0& \sin \varphi \\ 0& 1& 0\\ -\sin \varphi & 0& \cos \varphi \end{array}\right)\cdot \overrightarrow{\mathrm{n}}$$

in which n = Normal vector over (x, y).

The security element 1 according to the invention can be designed not only as a reflective security element 1, but also as a transmissive security element 1, as has already been mentioned. In this case, the facets 5 are not mirrored and the carrier 8 consists of a transparent or at least translucent material, with the view taking place in transparency. In the case of illumination from behind, a user should perceive the reproduced surface 9 as if a reflective security element 1 according to the invention that is illuminated from the front is present.

The facets 5 calculated for a reflective security element 1 are replaced by data for microprisms 16, the corresponding angles in the case of reflection ( Figure 19 ) and for transmissive prisms 16 in Figures 20 and 21 are shown. Figure 20 shows the incidence on the inclined facets 5, whereas Figure 21 shows the incidence on the smooth side, which is preferred because of the possible larger angles of incidence of light.

The azimuth angle of the reflective facet 5 is referred to as α _s and the slope angle of the facet 5 is referred to as σ _s . The refractive index of the microprism 16 is n, the azimuth angle of the microprism 16 is α _p = 180 ° + α _s . The angle of inclination of the microprism 16 according to FIG Figure 20 is sin (σ _p + 2σ _s ) = n sin σ _p , where for small angles 2 σ _s = (n -1) σ _p and 4 σ _s = σ _p (for n = 1.5) applies.

The angle of inclination of the microprism 16 according to Figure 21 is sin (2σ _s ) = n sin β; sin (σ _p ) = n sin (σ _p - β), where for small angles 4 σ _s = σ _P (for n = 1.5) applies.

$${\mathrm{n}}_{\mathrm{x}}=\cos \alpha \cdot \sqrt{1-{\cos }^2\sigma },{\mathrm{n}}_{\mathrm{y}}=\sin \alpha \cdot \sqrt{1-{\cos }^2\sigma }$$

If α and σ are known, the components of the normal vector are: $${\mathrm{n}}_{\mathrm{z}}=\cos \mathrm{\sigma },{\mathrm{n}}_{\mathrm{y}}/{\mathrm{n}}_{\mathrm{x}}=\sin \mathrm{\alpha }/\cos \mathrm{\alpha },{{\mathrm{n}}_{\mathrm{x}}}^2+{{\mathrm{n}}_{\mathrm{y}}}^2+{{\mathrm{<figure-callout\; id="2"\; label="n\; z"\; filenames="imgf0001.png"\; state="\{\{state\}\}">n</figure-callout>}}_{\mathrm{z}}}^2=1$$

$${\mathrm{n}}_{\mathrm{x}}=\cos \alpha \cdot \sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0001.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\sigma },{\mathrm{n}}_{\mathrm{y}}=\sin \alpha \cdot \sqrt{1-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="imgf0001.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\sigma }$$

In Figure 22 a reflective surface 9 to be adjusted with a hill 20 and a depression 21 is shown schematically. The negative focal length -f of the reflective hill 20 is r / 2 and the positive focal length f of the reflective trough 21 is r / 2.

In Figure 23 a lens 22 is shown schematically, which has a transparent concave section 23 and a transparent convex section 24. The concave section 23 simulates the reflecting hill 20, the negative focal length -f of the concave section 23 being 2r. The transparent convex section 24 simulates the reflective trough 21 and has a positive focal length f = 2r.

The lens 22 according to Figure 23 can be replaced by the sawtooth arrangement according to FIG.

The arrows in Figures 20 to 23 show schematically the beam path for incident light L. From these beam paths it can be seen that the lens 22 in transmission adjusts the surface 9 as desired.

In the Figures 25 to 27 an example is shown in which the sawtooth side is on the incident side. Otherwise, the illustration in FIG. 25 corresponds to the illustration in FIG Figure 22 , corresponds to the representation of Figure 26 the representation in Figure 23 and corresponds to the representation of Figure 27 the representation in Figure 24 .

The methods described above can be used to calculate the transmissive sawtooth structures.

In the Figure 27 The transparent sawtooth structure shown corresponds essentially to a cast of a corresponding reflective sawtooth structure for the adjustment of the surface 9 according to FIG Figure 25 . In this case, however, the following surface appears much flatter when viewed through (with a refractive index of 1.5) than when viewed in reflection. The height of the sawtooth structure is therefore preferably increased or the number of facets 5 per pixel 4 is increased.

Of course, it is also possible to provide the described sawtooth structures with a semitransparent mirror coating. In this case, the following surface 9 generally appears more deeply structured in reflection than in transparency.

Furthermore, it is possible to provide both sides of a transparent or at least translucent carrier 8 with a sawtooth structure which has the multiplicity of microprisms 16, as shown in FIG Figures 28 and 29 is indicated. At Figure 28 the sawtooth structures 25, 26 are mirror-symmetrical on both sides. At Figure 29 the two sawtooth structures 25, 27 are not designed to be mirror-symmetrical.

To calculate a sawtooth structure 25 and 27 according to Figures 28 and 29 it can be assumed that the sawtooth structure 25, 27 is composed of a prismatic surface 28 with a pitch angle σ _p and an auxiliary prism 29 with a pitch angle σ _h attached below it, as in FIG Figure 30 is shown schematically. Thus σ _p + σ _{h is} the effective total prism angle.

$${\mathrm{\sigma }}_{\mathrm{p}}+{\mathrm{\sigma }}_{\mathrm{h}}=\mathrm{\beta }1+\mathrm{\beta }2,$$

If the relief slope angle to be imitated is denoted by σ _s , the following applies, since the angle sum in the triangle is 180 °: $$90°-\mathrm{<figure-callout\; id="1"\; label="\beta "\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">\beta </figure-callout>}1+90°-\mathrm{<figure-callout\; id="2"\; label="\beta "\; filenames="imgf0001.png"\; state="\{\{state\}\}">\beta </figure-callout>}2+{\mathrm{\sigma }}_{\mathrm{p}}+{\mathrm{\sigma }}_{\mathrm{H}}=180°$$

$${\mathrm{\sigma }}_{\mathrm{p}}+{\mathrm{\sigma }}_{\mathrm{H}}=\mathrm{<figure-callout\; id="1"\; label="\beta "\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">\beta </figure-callout>}1+\mathrm{<figure-callout\; id="2"\; label="\beta "\; filenames="imgf0001.png"\; state="\{\{state\}\}">\beta </figure-callout>}2,$$

ergibt sich für: $${\mathrm{\sigma }}_{\mathrm{p}}-\arcsin \left(\left(\sin {\mathrm{\sigma }}_{\mathrm{p}}\right)/\mathrm{n}\right)=\arcsin \left(\left(\sin \left(2{\mathrm{\sigma }}_{\mathrm{s}}+{\mathrm{\sigma }}_{\mathrm{h}}\right)\right)/\mathrm{n}\right)-{\mathrm{\sigma }}_{\mathrm{h}}$$

Due to the law of refraction $$\sin {\mathrm{\sigma }}_{\mathrm{p}}=\mathrm{<figure-callout\; id="1"\; label="n\; sin\; \beta "\; filenames="imgf0001.png,imgf0010.png"\; state="\{\{state\}\}">n</figure-callout>}\sin \mathrm{\beta }\mathrm{}1,\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="imgf0001.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(2{\mathrm{\sigma }}_{\mathrm{s}}+{\mathrm{\sigma }}_{\mathrm{H}}\right)=\mathrm{<figure-callout\; id="2"\; label="n\; sin\; \beta "\; filenames="imgf0001.png"\; state="\{\{state\}\}">n</figure-callout>}\sin {\mathrm{\beta }}_{}{\mathrm{}}_2$$

results for: $${\mathrm{\sigma }}_{\mathrm{p}}-\arcsin \left(\left(\sin {\mathrm{\sigma }}_{\mathrm{p}}\right)/\mathrm{n}\right)=\mathrm{<figure-callout\; id="2"\; label="arcsin\; sin"\; filenames="imgf0001.png"\; state="\{\{state\}\}">arcsin</figure-callout>}\left(\left(\sin \right)\right)\left(\left(\left(2{\mathrm{\sigma }}_{\mathrm{s}}+{\mathrm{\sigma }}_{\mathrm{H}}\right)\right)/\mathrm{n}\right)-{\mathrm{\sigma }}_{\mathrm{H}}$$

Thus, based on the relief slope angle σ _s to be imitated at z. B. given auxiliary prism pitch _{angle σ h} easily the sought pitch _{angle σ p of} the prismatic surface 28 can be calculated.

It should be noted that the calculations listed for the imitation of a mirror relief by prisms are based on a perpendicular view. When viewed at an angle, distortions can occur and when viewed in white light, colored edges can appear shown motif, since the refractive index n used in the calculation depends on the wavelength.

The ones in the Figures 1 to 30 The reflective or refractive security elements shown can also be embedded in transparent material or provided with a protective layer.

Embedding takes place in particular to protect the micro-optical elements from contamination and abrasion and to prevent unauthorized readjustment by embossing the surface structure.

Example: Embedded Mirrors

When embedding or applying a protective layer, the properties of the micro-optical layer with the facets 5. In change Figures 32a-c this behavior is illustrated for embedded mirrors (the facets 5 are designed as mirrors), where Figure 32a shows the arrangement before embedding.

When the mirrors are embedded in a transparent layer 40, the direction in which a mirror image appears changes as Figure 32b shows. If the original reflection effect is to be achieved with a relief reproduced by embedded micromirrors 5, this must be taken into account in the inclination angle of the micromirrors, see Figure 32c .

Example: embedded prisms

In the case of embedded prisms 16, a difference in the refractive index between the prism material and the embedding material 40 is necessary and must be taken into account when calculating the light beam deflection.

Figure 33b shows schematically the adjustment of the reflective arrangement of FIG Figure 32a by a transmitting prism arrangement with exposed prisms 16, as already z. B. at the Figures 19-27 discussed.

Figure 33b shows schematically a possible re-creation of the reflective arrangement of FIG Figure 32a by embedded prisms 16, the refractive indices of prism material and embedding material 40 having to differ.

Example: Embedded Scattering Facets

In the two previous examples, the reproduction of reflective objects was demonstrated. Scattering facets can be used to reproduce scattering objects (e.g. marble figure, plaster model), here is an example (see Figure 34 ):
The following structure is implemented on a film 41 as a carrier material: The embossed facets 5, which simulate the object surface, are located on the back of the film. The facets 5 have dimensions of, for example, 10 μm to 20 μm. A lacquer 42 pigmented with titanium oxide (particle size approx. 1 μm) is applied to the facets 5, so that the facets 5 are filled with this scattering material. The viewing side is indicated by the arrow P2.

Example: Embedded matt, glossy facets

A matt reflective object can be reproduced in the following way (see Figure 35 ):
The following structure is implemented on a film 41 as a carrier material: The embossed facets 5, which simulate the object surface, are located on the rear side of the film. The facets 5 have dimensions of, for example, 10 μm to 20 μm. The embossed layer is provided with a semitransparent mirror coating 43 and a lacquer 42 pigmented with titanium oxide (particle size approx. 1 μm) is applied to it, so that the facets are filled with this scattering material. When viewed from the viewing side, the item being traced appears matt-glossy. The viewing side is indicated by the arrow P2.

Colored facets:

The embedding of the facets in the Figures 32b, 32c, 33b , 34 or 35 with colored material (also material colored differently in areas).

The security element 1 according to the invention can be used as a security thread 19 ( Figure 1 ) be trained. Furthermore, the security element 1 can not only be formed, as described, on a carrier film from which it can be transferred to the value document in a known manner. It is also possible to form the security element 1 directly on the value document. Direct printing with subsequent embossing of the security element on a polymer substrate can thus be carried out in order to form a security element according to the invention, for example in the case of plastic banknotes. The security element according to the invention can be formed in the most varied of substrates. In particular, it can be in or on a paper substrate, a paper with synthetic fibers, ie paper with a proportion of x polymeric material in the range of 0 <x <100% by weight, a plastic film, e.g. B. a film made of polyethylene (PE), polyethylene terephthalate (PET), polybutylene terephthalate (PBT), polyethylene naphthalate (PEN), polypropylene (PP) or polyamide (PA), or a multilayer composite, in particular a composite of several different films (composite composite) or a paper-film composite (film / paper / film or paper / film / Paper), wherein the security element can be provided in or on or between each of the layers of such a multilayer composite.

In Figure 31 an embossing tool 30 is shown schematically, with which the facets 5 in the carrier 8 according to FIG Figure 5 can be embossed. For this purpose, the embossing tool 30 has an embossing surface 31 in which the inverted shape of the surface structure to be embossed is formed.

Of course, not only for the embodiment according to Figure 5 a corresponding embossing tool can be provided. An embossing tool can also be made available in the same way for the other described embodiments.

List of reference symbols

1

Security element

2

Banknote

3

area

4th

pixel

5

Facets

6th

line

7th

surface

8th

carrier

9

trailing surface

10

Mirror surface

15th

Contour line

16

Microprism

19th

Security thread

20th

hill

21

trough

22nd

lens

23

concave section

24

convex section

25th

Sawtooth structure

26th

Sawtooth structure

27

Sawtooth structure

28

prismatic surface

29

Auxiliary prism

30th

Embossing tool

31

Embossing area

40

transparent layer

41

foil

42

pigmented varnish

43

semi-transparent mirror coating

L.

incident light

L1

incident light

L2

incident light

P1

arrow

P2

arrow

Claims (18)

1. A security element for a security paper, value document or the like, having
   a carrier which has an areal region which is divided into a multiplicity of pixels which respectively comprise at least one optically active facet (5), wherein the facets are so oriented that the areal region is perceptible to a viewer as an area that protrudes and/or recedes relative to its actual spatial form, and wherein a color-shifting coating is configured on the facets, at least regionally, wherein the oriented facets reflect incident light in such a way, as if it falls on an implemented or simulated surface, wherein the reflection generated by the facets of the pixel corresponds to the average reflection of the surface region implemented or simulated by the corresponding pixel.
2. The security element according to claim 1, wherein the orientation of the facets is so chosen that the areal region is perceptible to a viewer as a non-planar area.
3. The security element according to claim 1 or 2, wherein the color-shifting coating is realized as a subwavelength grating or diffractive relief structure.
4. The security element according to any of the above claims, wherein the optically active facets are configured as reflective facets.
5. The security element according to any of the above claims, wherein the optically active facets are configured as transmissive facets with a refractive effect.
6. The security element according to any of the above claims, wherein the optically active facets are so configured that the pixels have no optically diffractive effect.
7. The security element according to any of the above claims, wherein the maximum height of the optically active facets is not greater than 10 µm.
8. The security element according to any of the above claims, wherein the area of each pixel is smaller than the area of the areal region by at least one order of magnitude.
9. The security element according to any of the above claims, wherein the facets are configured in a surface of the carrier.
10. The security element according to any of claims 1 to 8, wherein the facets are configured as embedded facets.
11. The security element according to any of the above claims, wherein the orientation of the facets is determined by their inclination and/or their azimuth angle.
12. The security element according to any of the above claims, wherein there is configured on the facets at least in certain regions a reflective or reflection-enhancing coating, in particular by a coating with a material having a high refractive index.
13. The security element according to any of the above claims, wherein the maximum extension of a pixel is between 5 µm and 5 mm, preferably between 10 µm and 300 µm, particularly preferably between 20 µm and 100 µm.
14. The security element according to any of the above claims, wherein the areal region is perceptible to a viewer as an imaginary area whose reflection behavior or transmission behavior cannot be produced with a real bulged reflective or transmissive surface, wherein the areal region is perceptible in particular as a rotating mirror.
15. The security element according to any of the above claims, wherein the orientations of several facets are so changed relative to the orientations for producing the protruding and/or receding area that the protruding and/or receding area is still perceptible but with a surface of matt appearance.
16. A value document having a security element according to any of the above claims.
17. A manufacturing method for a security element for security papers, value documents or the like, wherein the surface of a carrier is so height-modulated in an areal region that the areal region is divided into a multiplicity of pixels respectively having at least one optically active facet, wherein the facets are so oriented that the areal region is perceptible to a viewer of the manufactured security element as an area that protrudes and/or recedes relative to its actual spatial form, and wherein a color-shifting coating is configured on the facets, at least regionally, wherein the oriented facets reflect incident light in such a way, as if it falls on an implemented or simulated surface, wherein the reflection generated by the facets of the pixel corresponds to the average reflection of the surface region implemented or simulated by the corresponding pixel.
18. The manufacturing method according to claim 17, wherein the surface of the carrier in the areal region is divided into plane elements and the surface given by the plane elements is subjected overall to a Fresnel construction Modulo d.

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