Method for designing a light redirecting surface of a caustic layer, optical security element with a designed light redirecting surface of a caustic layer, marked object, use and method for authenticating an object - Patents.com

Abstract

The present invention provides a method for designing a refractive, transparent or partially transparent, or reflective light redirecting surface of a caustic layer, comprising the steps of providing a discrete representation of an input target image, and determining image pixels p of the target image. i and computing a generalized power diagram for the set of image pixels p i and computing a piecewise light-redirecting surface of the caustic layer based on the calculated optimal set of weights that minimizes a cost function associated with the set of weights. The invention also relates to an optical security element, a marked object, a method for visually authenticating an object, and the use of an optical security element for authentication or to protect against counterfeiting. [Selection diagram] Figure 5

Description

[Technical field]

[0001] The present invention relates to the technical field of designing caustic optical elements, in particular to designing refractive transparent or partially transparent light redirecting surfaces (or reflective light redirecting surfaces) of a caustic layer, and to refractive or reflective optical security elements operable to project a caustic pattern with appropriate illumination. [Background technology]

[0002] There is a need for security features on objects that can be authenticated by so-called "lay people" using commonly available means. These means include using the five senses - often perception and touch - as well as using pervasive tools such as mobile phones.

[0003] Some common examples of security features include forensic fibers, threads, or foils (e.g., incorporated into a substrate such as paper), watermarks, intaglio or microprints (possibly printed on a substrate with optically variable inks) that can be found on banknotes, credit cards, IDs, tickets, certificates, documents, passports, etc. These security features may include optically variable inks, invisible or luminescent inks (fluoresce or phosphoresce under appropriate illumination with specific excitation light), holograms, and / or tactile features. The main aspect of a security feature is that it possesses some physical property (optical effect, magnetic effect, material structure, or chemical composition) that is very difficult to counterfeit, so that an object marked with such a security feature can be reliably considered genuine if this property can be observed or revealed (visually or with a specific device).

[0004] However, these features may not be appropriate if the object is transparent or partially transparent. In fact, transparent objects often require that security elements with the required security features do not change their transparency or their appearance, either for aesthetic or functional reasons. Notable examples include blisters and vials for pharmaceuticals. For example, recently, polymer and hybrid banknotes have incorporated transparent windows into their design, thus creating a demand for security features that are compatible with them. Although the discussion of transparency does not apply to reflective caustics, the approach naturally extends to reflective caustics where, for example, the use case requires the appearance of a polished mirror-like surface to be protected.

[0005] Most existing security features for documents, banknotes, certified tickets, passports, etc. were not specifically developed for transparent objects / areas and, as such, are not suitable for such applications. Other features, such as those achieved with invisible and fluorescent inks, require specific excitation and / or detection tools that may not be readily available to the "average person."

[0006] Translucent optically variable features (e.g., liquid crystal coatings, or latent images from surface structures) are known and can provide this type of functionality. Unfortunately, markings incorporating such security features must generally be viewed against a dark / uniform background for the effect to be fully visible.

[0007] Other known features are diffractive optical elements, such as non-metallized surface holograms. A drawback with these features is that when viewed directly, they exhibit a very low contrast visual effect. Furthermore, when used in combination with a monochromatic light source to project the pattern, they typically require a laser to produce satisfactory results. Furthermore, a very precise relative spatial arrangement of the light source, the diffractive optical element, and the user's eye is required to produce a clearly visible optical effect.

[0008] Laser engraved microtexts and / or microcodes have been used, for example, for glass vials, however they require expensive tools for their application and specific magnifying tools for their detection.

[0009] Therefore, in order to solve the problems existing in the prior art, many further attempts have been made to develop optical security elements suitable for transparent or partially transparent objects.

[0010] One possible approach is to introduce a design methodology that uses a caustic layer having a refractive transparent or partially transparent light redirecting surface, or a reflective light redirecting surface, where the caustic layer has a relief pattern configured to redirect incident light received from a light source to form a projected image that includes the caustic pattern. This approach of course extends to the use of reflective light redirecting surfaces in the case of opaque objects with mirror-like surfaces.

[0011] This approach makes it possible to control the caustic pattern by shaping the surface of the caustic layer. Computational tools based on light transport have been developed to create almost any desired shape by optimizing (calculating) the geometry of the refractive or reflective surfaces of the caustic optics, starting from a target image.

[0012] In an ideal workflow, starting from a target image, calculating a suitable surface should be fast and applicable to a wide range of target images, should not require excessively heavy computational resources, and should not require user intervention beyond selecting and providing the target image.

[0013] Methods for computing a caustic surface starting from a target image have been disclosed in the prior art.

[0014] EP 2711745 A2 discloses discretizing the generated surfaces into meshes, which are subsequently deformed to adjust the brightness of corresponding regions of the image. The normal field associated with the mesh is then determined and integrated to find the corresponding caustic surface. However, given an arbitrary image, there is no guarantee that the corresponding normal field is integrable and additional care must be taken to ensure the same.

[0015] EP 2 963 464 A1 takes a similar approach to determining the optimal transport map (OTM), and similarly requires computing and integrating a normal field which is not guaranteed to be integrable.

[0016] US9188783B2 and US2016041398 divide the generated surface into a set of micropatches, each serving to project a caustic Gaussian kernel, whose superposition approximates the desired image. However, as also noted in EP2711745A2, this method suffers from discretization artifacts and has difficulty resolving low intensity regions. The normal field also needs to be integrated.

[0017] In all these cases, the actual image projected by the computed caustic surface is finally simulated by ray tracing, and target image adjustments may be required when the caustic pattern does not approximate the target image with sufficient fidelity, which requires extra time and effort and still cannot guarantee perfect accuracy of the obtained caustic pattern.

[0018] It is therefore an object of the present invention to provide a fast, scalable, reliable and accurate method for designing refractive transparent or partially transparent light redirecting surfaces of caustic layers, or reflective light redirecting surfaces, which allows to significantly reduce the total time needed to get from a target image to a corresponding surface, since fewer iterations due to corrections or adjustments are needed and the iterations are faster, which also reduces the overall time needed for the design.

[0019] It is a further object of the present invention to eliminate the step of calculating and integrating the normal field. Eliminating the need to integrate the normal field removes one of the main limitations and sources of inaccuracy of previously known methods.

[0020] It is a further object of the present invention to reduce or completely eliminate user intervention beyond identifying the target image and accepting the resulting surface. Removing the need for user intervention significantly simplifies implementation of the method in the context of a manufacturing workflow where specialized skills are not always available.

[0021] It is a further object of the present invention to provide an optical security element operable to project a caustic pattern in suitable illumination, which is suitable for transparent or partially transparent objects, as well as reflective objects.

[0022] It is a further object of the present invention to provide a marked object comprising an optical security element and selected from the group comprising consumer products, documents of value and banknotes.

[0023] It is a further object of the present invention to provide a method for visually authenticating an object marked with an optical security element using commonly available means.

[0024] A further object of the present invention is to use the optical security element for authenticating and protecting against counterfeiting objects selected from the group including consumer products, documents of value, and banknotes. Summary of the Invention

[0025] According to one aspect, the invention provides a method of designing a refractive, transparent or partially transparent, or reflective light redirecting surface of a caustic layer configured to redirect incident light received from a light source and to form a projected image including a caustic pattern, the method comprising: A computer-implemented step of providing a discrete representation of an input target image, the discrete representation of the input target image comprising associated non-zero target light intensities {I i}, coordinates {(x i ,y i )} for N image pixels p i and Refracted or reflected by the caustic layer, the coordinate (x i ,y i ), i=1,…,N, the constancy of the optical path length of the light rays focused on the image plane points P(i) of the image plane. i a computer-implemented step of computing a piecewise representation of the light redirecting surface z=F(x,y) of the caustic layer with height z above the (x,y) coordinate plane based on the representation of the light redirecting surface using (x,y), i=1,...,N, where each surface piece z=f i (x,y) is the height z i =f i (x i ,y i ), i = 1, …, N, pass through point P(i) and point (x i ,y i ,z i ), and a piecewise representation of the light redirecting surface associated with each of the height values ​​of the N vertices is a surface of revolution about an axis having vertices at z=f ia step formed by the envelope of intersections of (x,y), i=1,...,N; The heights of the vertices of N surface segments z1,…,z N a computer-implemented step of calculating, for each given set of values ​​of I(1), ..., I(N), a corresponding set of values ​​of light intensity I(1), ..., I(N) that is focused at points P(1), ..., P(N), respectively, by a caustic layer that redirects incident light through an associated piecewise light redirecting surface; The respective values ​​of the calculated light intensities I(1),…,I(N) focused on the points P(1),…,P(N) via the associated light redirecting surfaces and the target light intensities I1,…,I N The N heights z1,…,z of the N vertices of the N corresponding surface segments that minimize the difference between N and a computer implemented step of calculating each value of Including, Thus, the present invention relates to a method for obtaining a light redirecting surface having a relief pattern configured to redirect incident light received from a light source to form a projected image that includes a target caustic pattern of a target image.

[0026] According to the invention, for each surface section z=f i (x,y), i=1,...,N, may be approximated, within the paraxial approximation, by taking a Taylor expansion of order k greater than or equal to 2 of the surface section equation that follows from the above constancy of the optical path length.

[0027] In the above method, for i = 1, ..., N, the calculated light intensity I(i) and the corresponding target light intensity I i The height z that minimizes the difference between i The step of calculating can be performed using gradient-free optimization methods. These methods have the advantage of simplicity at the expense of high computational cost.

[0028] For i = 1, ..., N, the calculated light intensity I(i) and the corresponding target light intensity I i The height z that minimizes the difference between iThe step of calculating may preferably be performed using a (capacity-constrained) power map method associated with an optimization method, which may be gradient-free or, preferably, gradient-based (to reduce computational costs).

[0029] In a further aspect of the invention, the designed light redirecting surface is used to generate a machine-ready representation for machining purposes, for example using industry standard formats such as StereoLithography (STL) or Initial Graphics Exchange Specification (IGES). In particular, the machine-ready representation can be used to control a machining tool to machine the light redirecting surface of an optical material substrate, or an intermediate substrate that is further used for mass production of caustic optical elements by replica.

[0030] In a further aspect of the invention, the designed light redirecting surface is a master light redirecting surface to be used to build a replica of the light redirecting surface. In this case, the method may further include creating a replica of the light redirecting surface on a substrate. Creating such a replica may include one of roll-to-roll, foil-to-foil, UV casting, and embossing.

[0031] In another aspect, the present invention provides an optical security element comprising at least one of the refractive or reflective light redirecting surfaces of a caustic layer designed according to the method as described above, said optical security element being capable of marking an object selected from the group comprising consumer products, documents of value, identification cards, tax stamps, and banknotes.

[0032] In another aspect, the present invention provides a marked object comprising an optical security element as described above and selected from the group comprising consumer products, documents of value and banknotes.

[0033] In another aspect, the present invention provides a method for visually authenticating, by a user, an object marked with an optical security element, comprising the steps of: Distance d from the light redirecting surface s - illuminating the light redirecting surface of the optical security element by a point light source located at Distance d from the optical security element i visually observing the caustic pattern projected onto a projection surface at determining that the object is genuine upon assessment by a user that the projected caustic pattern is visually similar to the reference pattern.

[0034] In yet another aspect, the present invention provides the use of an optical security element as described above for authenticating and protecting against counterfeiting an object selected from the group including consumer products, documents of value, identity cards, tax stamps and banknotes.

[0035] The present invention will now be described more fully hereinafter with reference to the accompanying drawings, in which prominent aspects and features of the invention are illustrated. [Brief description of the drawings]

[0036] [Figure 1] FIG. 2 is a schematic diagram of a typical optical setup of a refractive optical security element for the projection of caustic images. [Diagram 2] A diagram illustrating a caustic surface for a single image pixel (image point). [Diagram 3] A diagram illustrating a caustic surface for multiple image pixels (image points). [Figure 4] FIG. 2 illustrates a target image. [Diagram 5] FIG. 1 illustrates a generalized power diagram. [Figure 6] FIG. 2 is a diagram illustrating the appearance of a caustic surface. [Figure 7] FIG. 1 illustrates a side view of an object in front of a detector (ray tracing geometry). [Figure 8] FIG. 13 illustrates a grayscale ray tracing simulation of an image 40 mm from an object. Detailed Description

[0037] In optics, the term "caustic" refers to the envelope of a ray of light refracted or reflected by one or more surfaces, at least one of which is curved, as well as the projection of such a ray onto another surface. More specifically, a caustic is a curve or surface tangent to each ray of light that defines the boundary of the ray's envelope as a curve of convergence. For example, the light pattern formed by the sun's rays on the bottom of a swimming pool is a caustic "image" or pattern formed by a single light redirecting surface (a wavy air-water interface), while light passing through the curved surface of a glass creates a crescent-like pattern on the table the glass rests on as it intersects two or more surfaces that redirect its path (e.g., air-glass, glass-water, air-water).

[0038] In the following, the most commonly found configuration in which the (refractive) caustic layer of an optical security element is bounded by one curved or light redirecting surface and one flat surface shall be used as an example, without limiting the more general case. Herein, the more general "caustic pattern" (or "caustic image") is referred to as the light pattern formed on the screen (projection surface) when a suitably shaped optical surface of the caustic layer (with a light redirecting surface with a suitable relief pattern) redirects light from the source, deflecting it away from some areas of the screen and focusing it in other areas of the screen with a predefined light pattern (i.e. thus forming said "caustic pattern"). Redirection refers to the change in the path of the light rays from the source in the presence of a caustic layer, relative to the path from the source to the screen in the absence of the caustic layer. A caustic layer (refractive or reflective) is thus an optical material having a light redirecting surface with a relief pattern configured to redirect the light received from a light source to form a caustic image. The optical security element according to the invention comprises a caustic layer and may further comprise additional optical element(s) involved in light redirection (eg a lens, or a supporting substrate).

[0039] On the other hand, a curved optical surface is called a "relief pattern" and the optical element bounded by this surface is called a caustic layer. Note that a caustic pattern can be the result of redirecting light by two or more curved surfaces and two or more objects, but possibly at the expense of increased complexity. Furthermore, the relief patterns for generating caustic patterns should not be confused with diffraction patterns (such as those in security holograms).

[0040] The inventive concept can be applied to everyday objects such as, for example, consumer products, ID / credit cards, bank notes, etc. To do so, it is required to dramatically reduce the size of the optical security elements and, in particular, to ensure that the relief depth of the relief pattern is below a tolerance value. To this end, it is particularly useful to have an efficient workflow, since it allows several design iterations until all operational constraints are met.

[0041] In this description, "relief" should be understood as the presence of a height difference between the highest and lowest points of a surface (measured along the optical axis of the optical security element) as well as an elevation difference between the valley floor and the peak (i.e. a "peak to valley" measure). The method according to the invention is not limited to a particular relief, but for many of the contemplated applications the maximum depth of the relief pattern of the optical security element will typically be equal to or less than 250 μm, or more preferably equal to or less than 30 μm, while at the same time exceeding the limit imposed by ultra-precision machining (UPM) and replication processes, i.e. about 0.2 μm.

[0042] According to this description, the height difference between the highest and lowest points in the relief pattern on the light redirecting surface is called the relief depth ε.

[0043] A caustic pattern (image) forming an approximation of a digital image should be understood as the light pattern projected by an optical security element when illuminated by a suitable point-like source. As mentioned above, an optical security element should be understood as a slab of refractive material that serves to create a caustic image.

[0044] The light redirecting surface(s) is the surface(s) (of an optical security element) of a caustic layer responsible for redirecting the incoming light from the source onto the screen, or the (possibly flat) projection surface on which the caustic pattern is formed.

[0045] The optical material substrate used to make the optical (security) elements is a raw material substrate from which a surface is explicitly formed to have a relief pattern and thus to form a light redirecting surface. In the case of a reflective light redirecting surface, the optical material substrate is not necessarily homogeneous or transparent, the same is true for a master surface used only for further replication. For example, the material may be opaque to visible light and reflectivity may be obtained by classical metallization of the formed surface. In the case of a refractive light redirecting surface, the raw material substrate is transparent (or partially transparent) and homogeneous (for photons in the spectrum visible to the human eye) with a refractive index n, and the corresponding light redirecting surface is named "refractive transparent or partially transparent light redirecting surface of refractive index n".

[0046] A master light redirecting surface according to this description is the first physical realization of a light redirecting surface from the calculated one. It can be replicated into several copies (tools), which are then used for mass replication.

[0047] A point-like source as used in this description is one whose angular size (from the point of view of the optical security element) is such that the light is a distance d from the light redirecting surface. s Roughly speaking, this is the quantity: (source diameter) × d i / d s is at a distance d from the light redirecting surface. i This means that the resolution of the target caustic pattern on the projected image at the projection surface at d is smaller than the desired resolution (e.g., 0.05-0.1 mm) (see Fig. 1). A screen should be understood as the surface onto which the caustic pattern is projected. The distance between the source and the light redirecting surface is also referred to as the source distance d s and the distance between the light redirecting surface and the screen is called the image distance d i It can be named.

[0048] The term tool (or, when necessary to disambiguate, replication tool) is primarily used for a physical object with a profile of a light redirecting surface that is used for mass replication. A tool may be used, for example, to produce copies of a master light redirecting surface (where the original relief is reproduced by embossing or injection from a master with a corresponding inverse relief). For a tool used to machine the relief pattern of a light redirecting surface, the term machining tool is used for disambiguation.

[0049] Figure 1 provides a schematic diagram of a typical optical setup of a refractive optical security element for the projection of caustic images. An optical security element (1) comprising a caustic layer with a refractive surface redirects light from a point-like source S and projects it onto a suitable screen (3), which can be any surface, such as any object, on which a meaningful image is formed, as shown in Figure 1. A special design of the light redirecting surface can make it possible to project (recognizable) caustic patterns onto curved surfaces. The image can be, for example, a logo, a photo, a number, or any other information that may be relevant in a particular context. The screen is preferably a flat projection surface or a flat part of any object.

[0050] The arrangement of Fig. 1 shows that light from a source S is redirected by a suitably shaped optical surface with a relief pattern (2). This general idea is known, for example, from reflective surfaces of automobile headlights, reflectors and lenses of LED lighting, optical systems in laser optics, projectors and cameras. Usually, however, the goal is to convert a non-homogeneous distribution of light into a homogeneous one. In contrast, the goal of the present invention is to obtain a non-homogeneous light pattern, i.e. a caustic pattern, which (approximately) reproduces some areas of relative brightness of a reference pattern (as represented in a (digital) reference image). If the illuminated relief pattern (2) of the optical element allows the formation of a caustic pattern (4) on a screen (3) to reproduce a known reference pattern with sufficient quality (possibly depending on an overall intensity magnification), then a person visually observing the caustic pattern on the screen can easily see whether it constitutes a valid reproduction of the reference pattern or not, and, if the caustic pattern is sufficiently similar to the reference pattern, considers the object marked with the optical security element to be (with high probability) genuine.

[0051] According to the embodiment of FIG. 1, a light ray from a light source S, which in this example is a point-like source, is directed to a light redirecting surface having a relief pattern (2) at a source distance d s The optical security element is here made of a transparent or partially transparent homogeneous material with a refractive index n. The caustic pattern (4) is located at an image distance d from the light redirecting surface of the optical security element (1). i The authenticity of the optical security element (and therefore of the object marked with this security element) can be assessed directly by visually checking the degree of similarity between the projected caustic pattern and a reference pattern.

[0052] The relief pattern (2) is preferably calculated starting from a specified target digital image. From that calculated relief pattern, a corresponding physical relief pattern can be created on the surface of a suitable optical material substrate (e.g. a transparent or partially transparent material with refractive index n, or a reflective surface of an opaque material) using ultra-precision machining (UPM). When machining a relief on the surface of an opaque optical material substrate to form a reflective surface, good reflectivity is obtained either by suitable properties of the material itself or by the further conventional operation of depositing a thin layer of metal on the relief (metallization). UPM achieves very high precision using diamond machining tools and ultrafine technology tools, resulting in tolerances down to the "sub-micron" or nanoscale level. In contrast, "high precision" in conventional machining refers to tolerances of single digit microns. Other possible suitable techniques for creating a physical relief pattern on a surface are laser ablation and grayscale lithography. As is known in the field of micromachining, each of these techniques has different strengths and limitations in terms of cost, precision, speed, resolution, etc.

[0053] Suitable optical material substrates for refractive light redirecting optics must be optically clear, transparent or at least partially transparent, and mechanically stable. Typically, a transmittance T≧50% is preferred, with T≧90% being most preferred. Also, low haze H≦10% can be used, with H≦3% being preferred, with H≦1% being most preferred. The optical material must also behave correctly during the machining process to provide a smooth, defect-free surface. An example of a suitable substrate is an optically clear sheet of PMMA (also known by trade names such as Plexiglas, Lucite, Perspex, etc.). For reflective caustic light redirecting optics, a suitable optical material substrate must be mechanically stable, and it must be capable of providing a mirror-like finish. Examples of suitable substrates are metals, such as those used for ruled grating masters and laser reflectors, or non-reflective substrates that can be further metallized.

[0054] For mass production, further steps of tool creation and mass replication of the optical security element on the target object are required. A suitable process for tool creation from the master is, for example, electroforming. A suitable process for mass replication is, for example, hot embossing of polymer films or UV casting of photopolymers, which can be further carried out either in a roll-to-roll or foil-to-foil process. For the purpose of mass replication, neither the master nor the tool derived therefrom needs to be optically transparent, so that opaque materials (especially metals) can also be used, even when the final product is a refractive optical element. Nevertheless, in some cases, it can be advantageous for the master to be transparent, since this makes it possible to check the quality of the caustic image before proceeding with the tooling and mass replication.

[0055] An important aspect for the use of optical elements (with light redirecting surfaces having relief patterns) as security features is their physical dimensions, which must be compatible with the target object and the optical configuration required to project the caustic image.

[0056] In general, the maximum lateral size is limited by the overall size of the object, which can usually range from a few centimeters to less than 1 cm in less favorable cases. In certain applications, such as for example banknotes, the targeted total thickness is extremely small (on the order of 100 μm or less). Furthermore, the allowable thickness variation (relief) is even smaller for various reasons, including mechanical constraints (weak spots associated with thin areas) and operational constraints (for example, when banknotes are stacked, the stack bulges in response to the thicker parts of the bills, which complicates handling and storage). Typically, for a banknote with a total thickness of about 100 μm, the target thickness of the relief pattern of the optical security element to be included in the banknote can be about 30 μm. For a credit card or ID card with a thickness of about 1 mm, the target thickness of the relief pattern of the optical security element to be included in the credit / ID card will be less than about 400 μm, and preferably about 250 μm or less.

[0057] Furthermore, the source distance and image distance are generally limited to a few tens of centimeters by user comfort. However, obvious exceptions are sunlight or ceiling mounted spotlights, which are not very readily available under certain circumstances. Also, the ratio between the two distances d s / d i is typically greater than 5-10, so as to obtain a clear image (with good contrast) that is easier to recognize. Furthermore, the ratio d s / d i ≧5 and the light source S being preferably point-like (e.g. the lighting LEDs of a conventional mobile phone) allows the light source to be considered practically almost "infinite", and therefore a projection surface only at about the focal distance from the optical security element is suitable for clearly seeing the projected caustic pattern. As a result, the conditions for good visual observation by the user do not require too strict relative spatial arrangement of the light source, the optical security element and the user's eyes.

[0058] In general, thickness and relief are among the most important parameters. Given an arbitrary target image (reference pattern) and optical geometry (i.e., geometric conditions of illumination / observation of the projected caustic pattern), there is no guarantee that the calculated optical surface will have a relief pattern below the stated limits. In fact, in the general case, the opposite is likely to happen, especially with the strictly imposed constraints for optical security elements described above. The proposed method does not automatically take these additional constraints into account. However, since the proposed method can quickly compute the caustic surface corresponding to a given input image, it allows several iterations on the image design until the designed relief corresponds to the constraints. In contrast, computationally expensive conventional methods usually not only do not automatically take the constraints into account, but also impose severe limitations on the iterative design process.

[0059] Although only transmissive caustic optical element configurations are described here, the same arguments may be applied, with slight variations (especially with regard to the application of Fermat's principle), to reflective configurations.

[0060] For the purposes of explanation, it is convenient to define a Cartesian reference system, with the z-axis (perpendicular to the caustic optics) aligned with the optical axis and pointing from the source to the image. To illustrate the concept of the invention, a simple caustic optics of the "plano-convex" type is considered (FIGS. 2-3) and illustrated with a beam of (substantially) parallel light rays. The case where a light source at a finite distance is made understandable by the addition of lens-like optical elements, transforming the finite distance source into a virtual source at infinity. The function of the lens-like elements can finally be directly incorporated into the caustic optics. The x- and y-axes are therefore in the plane of the caustic optical security element (parallel to the input face of the caustic optics). A caustic surface is mathematically described by a scalar function z=F(x,y) that gives the distance z of the surface from the reference plane z=0 at the point of coordinates (x,y) of the caustic optics. For the sake of simplicity in the following description, this plane may be located on the rear surface of the caustic optics, where z=F(x,y) is equal to the thickness of the caustic optics (see FIG. 2). In the example shown in FIG. 2, this plane is parallel to the plane of the caustic image.

[0061] Similarly, a caustic image is described by a scalar function I(x',y') that gives the luminous intensity at a point (or pixel) with coordinates (x',y') on the image plane.

[0062] It should be noted that the use of Cartesian coordinates is for convenience and other systems could be used instead (e.g., where the caustic surface is part of, or supported by, a curved object.) Similarly, the back surface of a caustic optic need not be flat, although this must of course be taken into account.

[0063] The present invention exploits the property of light traveling along paths of fixed optical path length, where the optical path length is an extremum for any small variation in the path (Fermat's principle). For any given point (x0,y0) of the caustic image, the bundle of rays of small cross section that converges there will have traveled paths of the same optical path length. Typically, the relief pattern of the light redirecting surface of the caustic layer has a relief depth ε that is very small compared to the distance d between the caustic layer and the image plane on which the caustic image is formed (see FIG. 2); in fact, typically the resulting value of ε is less than 300 μm, while d is greater than 5 cm (so that ε / d<6 10 -3 ), the relief depth ε is defined as the height difference between the highest and lowest points of the relief pattern. The total thickness of the caustic layer is (e+ε), where e is the thickness of the homogeneous part of the optical material of the caustic layer. In general, the thickness e is also very small compared to the observation distance d, i.e., typically e is less than 1 millimeter (so that e / d≦2 10 -2 , and (e+ε) / d≦2.6 10 -2 ). However, the layer of thickness e, which corresponds to the propagation of the incident light in the caustic layer simply as a parallel light beam, has no effect with respect to the difference in optical path length and is therefore ignored. Considering the caustic layer illustrated in FIG. 2, we consider a light source (s i =∞, d i =d), consider the difference in optical path length Δl between (i) the optical path length l(x0,y0) of a straight ray that enters the plane of the caustic layer (level z=0) at point (x0,y0), passes through a caustic layer (caustic optic) of refractive index n to point (x0,y0) at level z0 of a light redirecting surface with equation z=F(x,y), and reaches the focal point (x0,y0) of the image plane, and (ii) the optical path length l(x,y) that enters the plane of the caustic layer (level z=0) at point (x,y) close to point (x0,y0), passes through the caustic layer to point (x,y) at level z of the light redirecting surface, and is deflected to point (x0,y0) of the image plane. Let r be the distance between points (x0,y0) and (x,y), i.e.

number

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[0064] According to Fermat's principle, we must have Δl=0, so solving the quadratic equation in z gives

number

number

[0065] As a result, instead of a focal point (x0,y0), we have a focal point (x i ,y i ), i = 1,…,N, we consider any one of

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number

[0066] The present invention relates to the paraxial approximation, i.e., when r ≪ d, and therefore

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[0067] For example, the point (x i ,y i ) local approximation f(x,y) around i If we consider (x,y) and only consider the first non-zero term in the Taylor expansion, we obtain a simplified approximation of the local representation,

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[0068] In the Taylor expansion to the next non-zero order (k=4), the point (x i ,y i ) Approximation of the local representation around

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[0069] Local expression f i Consider a piecewise parabolic approximation of F(x,y) obtained by approximating (x,y) up to the first nonzero order. i ,y i ) above height z i and the adjacent points (x j ,y j ) above height z j The intersection of two such (circular) paraboloids with x i ,y i ) and (x j ,y j ) in the image plane. i ,y i ), i = 1, …, N}, and the heights of the vertices of the paraboloids associated with each of the points {z i For the corresponding set of {k,i=1,…,N}, the resulting (outer) envelope of the intersection of these paraboloids (defining the piecewise light redirecting surface) is formed by the portion of the paraboloid bounded by sharp parabolic curves. These curves can be calculated by solving a simple order 2 algebraic equation. For the Taylor expansion of order k=4 and above, the corresponding “basic shape function” z=f i (x,y) is more complicated than just a paraboloid, and computing the intersections of the surface segments (when setting different heights for their vertices) becomes more tedious.

[0070] In the example shown in FIG. 3, an incoming parallel ray of light illuminates the planar (input) face z=0 of the caustic layer with uniform intensity I0, and thus for a given piecewise approximation of the light redirecting surface z=F(x,y), i.e., N vertices (x i ,y i ,z i ) and the corresponding basic shape functions f i For (x,y), i=1,…,N, we define the point (x j ,y j ) to the intensity I(j),

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[0071] The formula for a piecewise approximation of the representation of a caustic surface z=F(x,y) for a given number N of image points (x i ,y i ), for i = 1,…,N, is therefore

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[0072] Once a piecewise approximation of the light redirecting surface z=F(x,y) has been obtained (for a given set of N vertices), we apply a piecewise approximation to each selected point (x i ,y i ), the corresponding distribution of light intensities I(i) at i = 1, ..., N, and the relationship between I(i) and a given (target) intensity I i For each target point (x i ,y i ) is needed to estimate the difference in the height of the vertex z i , i=1,…,N is the sum

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[0073] For example, surface f j If a local section of (x,y) is approximated by the leading term of a Taylor expansion, i.e., by a paraboloid, then a point (x j ,y j ) is the nonzero intensity I(j) at each vertex (x i ,y i ,z i ), after intersection with the remaining paraboloids forming a piecewise surface F with i ≠ j, i ∈ {1,…,N} (and possibly with the boundaries of the caustic layer window), the vertices (x j ,y j ,z j ), i.e., paraboloid (j). If paraboloid (j) is completely masked by at least one paraboloid (i) (i.e., z i Z j , the intensity I(j) is zero. As mentioned above, the contour of the intersection of two paraboloids (i) and (j) is given by two points (x i ,y i ) and (x j ,y j), which is parallel to the optical axis along z, and the intersection of this plane with the (x,y) plane at z=0 defines a line segment. When considering the intersection of paraboloid (j) with a neighboring paraboloid (i), the corresponding line segment on the plane z=0 is defined as a convex polygonal cell Ω j Obviously, we draw a point (x j ,y j ) is the light intensity I(j) delivered to the cell Ω j and therefore the delivered intensity I(j) arises only from the incident (uniform) parallel light rays collected by the cell Ω j is proportional to the area a(j) of the cell a(j). Naturally, the sum of all the areas of the cells associated with the envelopes of all intersecting paraboloids must be equal to the total area A (on the plane z=0) of the collection (window) of incident rays.

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[0074] The above argument using the example of a parabolic surface remains true even if the equation for the surface section, derived directly from the constancy of the optical path length, is not approximated or is approximated by its Taylor expansion to any (even) order k>2 (the resulting equation still describes a surface of revolution), and in iteration step n of the minimization operation, the value {z i (n) ,i=1,…,N} is the set of

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[0075] Function (i.e., cost function)

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[0076] According to the present invention, and with the piecewise representation of light redirecting surfaces described above, the technical problem of computing a light redirecting surface of a caustic layer configured to redirect incident light received from a light source to form a projected image comprising a given caustic pattern (i.e., a given distribution of non-zero light intensities) of a target image is solved by: providing a discrete representation of an input target image, the discrete representation of the input target image including associated non-zero target light intensities {I i} in the image plane, i ,y i )}, i = 1, ... N image pixels p i providing a set P of Refracted or reflected by the caustic layer, the coordinate (x i ,y i ), i=1,…,N, the constancy of the optical path length of a ray focused on a point P(i) in the image plane. i (x,y), i=1,...,N, based on the representation of the light redirecting surface, computing a piecewise light redirecting surface z=F(x,y) of the caustic layer with height z above the (x,y) coordinate plane, i (x,y) is the height z i =f i (x i ,y i ), i = 1, …, N, pass through point P(i) and point (x i ,y i ,z i ), where a piecewise light redirecting surface associated with each of the height values ​​of the N vertices is formed by the envelope of the intersections of the corresponding N surface segments; The heights of the vertices of N surface segments z1,…,z N For a given set of values ​​of each of p, p = p ... The respective values ​​of the calculated light intensities I(1),…,I(N) focused on the points P(1),…,P(N) via the associated light redirecting surfaces and the target light intensities I1,…,I N The N heights z1,…,z of the N vertices of the N corresponding surface segments that minimize the difference between N This is solved by calculating the respective values ​​of

[0077] For example, in minimizing a cost function Σ using the Nelder-Mead simplex method, the optimization is done by minimizing the vertices of a non-degenerate simplex S in an optimization space of N dimensions (i.e., N heights z1,…,z N ), and the set of N+1 points Q(1),…,Q(N+1) located at

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[0078] According to a preferred mode of the invention, the optimal light redirecting surface is advantageously obtained using a (generalized) power projection (also known as the Voronoi projection or Laguerre / Voronoi projection (see F. de Goes et al., “Blue Noise through Optimal Transport”, CAN Transactions on Graphics, vol. 31(6), (SIGGRAPH Asia) 2012) (source code available, see also the website http: / / www.geometry.caltech.edu / BlueNoise / ). In fact, this method is powerful and, in the case corresponding to the optimization problem of the invention, it is possible to obtain a light redirecting surface with weights w i Here, the height z i corresponds to and capacity m i Here we prove the power diagram as the only solution "for any given capacity constraint", which minimizes a concave function of the weights corresponding to the cell region a(i) (see especially the appendix of the above-cited paper by Goes et al.).

[0079] Since any image can be approximated by a finite collection of pixels, the caustic surface can be approximated by a composition of corresponding surface segments (e.g., paraboloids). Thus, the target image I t Given (x',y') (see Figure 4), the problem of computing the caustic surface that generates it is I t Weights {w i} to reduce the number of iterations required to find a suitable set of iterations.

[0080] Under the optimal transport hypothesis (see the article by Goes et al. mentioned above), this means that i ,y i )} for the power diagram weights {w i}(here the height {z i}), which results in a capacity {mi} (here cell area {a(i)}) is the target image intensity {I t (x i ,y i )}. i , i=1,...,n} and the corresponding cell boundary ∂Ω i (Cell Ω in region a(i) i Once the heights of the surfaces (Ω) have been obtained by the power projection, a piecewise surface is reconstructed by considering the intersections of cylinders, constructed along the axis z, whose base is formed by the boundaries of the cells, with each surface segment having a vertex at the height obtained above. In the preferred mode, the surface segments are approximated by paraboloids, in which case the cell Ω i The boundary of ∂Ω i is a polygon, and the computation of distances of points to the boundary and gradients is greatly simplified. In the more general case (i.e., the surface partition is not approximated or is approximated by a Taylor expansion of order greater than two), the cell Ω i The boundary of ∂Ω i is still a closed curve, but it is made up of curves and the calculation of the distance and slope of a point to the boundary above is more complicated.

[0081] The results obtained for the target image depicted in Figure 4 are shown in Figures 5-8. The target image has 100x100 pixels, 900 of which are non-zero (i.e., representing areas of the caustic image that have non-zero luminosity). Figure 5 shows the calculated power diagram, Figure 6 illustrates the corresponding caustic surface, Figure 7 shows the ray tracing geometry, and Figure 8 depicts the results of the ray tracing.

[0082] w i By minimizing over the function Σ i │m i -I i │ 2 can be solved by a simple gradient descent algorithm. This process is i} (in most cases by taking all values ​​equal), then we define an optimal set of corresponding partitions, {w i} towards capacity m i Cell Ω i Then the resulting optimal set {w i}, the parabolic element {z i} is obtained and the resulting polygon cell Ω i The boundary of ∂Ω i From the basis ∂Ω with paraboloids i The final piecewise caustic surface is constructed by the intersection of the vertical (along z) cylinders of

[0083] A caustic layer with a light redirecting surface computed and designed according to the present invention forms a projected image with a caustic pattern that reproduces a reference pattern that is easily recognizable by a person without the use of further means (i.e. with the naked eye) or using common readily available means, such that an object marked with this optical security element can be easily visually authenticated by a person. The transparent aspect of the refractive optical security element makes it particularly suitable for making at least partially transparent substrates (e.g. glass or plastic bottles, bottle caps, watch glasses, jewelry, gemstones, etc.).

[0084] The disclosed method for designing refractive, transparent or partially transparent, or reflective light redirecting surfaces of caustic layers is fast, scalable, reliable, and accurate. It allows for a significant reduction in the number of iterations required to get from a target image to a corresponding surface, since no corrections or adjustments are required. This also reduces the overall time required for the design.

[0085] Also, the step of computing and integrating the normal field is eliminated, providing an efficient optimization technique through minimization of the volume constraint.

[0086] Moreover, user intervention beyond specifying the target image and accepting the resulting surface is completely eliminated. Removing the need for user intervention significantly simplifies implementation of the method in manufacturing contexts where specialized skills are not always available.

[0087] The subject matter disclosed above should be considered illustrative rather than limiting and serves to provide a better understanding of the invention defined by the independent claims.

Claims

1. 1. A method of designing a caustic layer, a refractive transparent or partially transparent light redirecting surface, or a reflective light redirecting surface, the caustic layer being a piece of optical material, having a relief pattern, and configured to redirect incident light received from a light source and to form a projected image including the caustic pattern, the method comprising: A computer-implemented step of providing a discrete representation of an input target image, the discrete representation of the input target image including associated non-zero target light intensities {I ｉ }, coordinates in the image plane {(x ｉ , y ｉ )) for N image pixels p ｉ and The caustic layer refracts or reflects the coordinates (x ｉ , y ｉ ), i=1,...,N of the image plane, a plurality of intersecting surface segments z=f ｉ a computer-implemented step of computing a piecewise representation of the light redirecting surface z=F(x,y) of the caustic layer with height z above the (x,y) coordinate plane based on the representation of the light redirecting surface using (x,y), i=1,...,N, where each surface segment z=f ｉ (x, y) is the height z ｉ = f ｉ (x ｉ , y ｉ ), i = 1, ..., N, pass through point P(i) and point (x ｉ , y ｉ , z ｉ ) and the piecewise representation of the light redirecting surface associated with respective values ​​of height of N vertices is: [0010] and the function H[X] satisfies the following: [0025] , N, where j=1, . . . , N; The height z of the vertices of the N surface segments １ , …, z Ｎ For a given set of respective values of each of these, a computer-implemented step of calculating a corresponding set of values of the light intensities I(1), …, I(N) focused on the points P(1), …, P(N), respectively, by the caustic layer that redirects incident light through the associated segmented light-direction-changing surface the respective values ​​of calculated light intensities I(1), ..., I(N) focused to the points P(1), ..., P(N) via the associated light redirecting surface, and the target light intensity I １ , …, I Ｎ , which minimizes the difference between the N heights z of the N vertices of the corresponding N surface sections and the respective corresponding values ​​of １ , ..., z Ｎ a computer implemented step of calculating said respective values ​​of Including, Thus, obtaining a light redirecting surface having a relief pattern configured to redirect incident light received from the light source to form a projected image including the target caustic pattern of the target image.

2. Each surface section z=f ｉ (x,y), i=1,...,N are approximated, within a paraxial approximation, by taking a Taylor expansion of order k greater than or equal to 2 of the equation for the surface section that results from the constancy of the optical path length; [0030] where d is the distance between the caustic layer and the image plane on which the projected image is formed, n is the refractive index of the caustic layer, and z ｉ = f ｉ (x ｉ , y ｉ ) and [0045] The method of claim 1, wherein

3. For i=1, . . . , N, the calculated light intensity I(i) and the corresponding target light intensity I ｉ The height z that minimizes the difference between ｉ The method of claim 1 or 2, wherein the step of calculating is performed using a gradientless optimization method.

4. For i=1, . . . , N, the calculated light intensity I(i) and the corresponding target light intensity I ｉ The height z that minimizes the difference between ｉ 3. The method of claim 1, wherein the step of calculating is performed using an optimization method that relies on power diagrams for the calculation of an associated cost function and its derivatives.

5. The method according to any one of claims 1 to 4, wherein the designed light redirecting surface is used to generate a machine-ready representation for controlling a machining tool.

6. The method of any one of claims 1 to 5, wherein the designed light redirecting surface is a master light redirecting surface to be used to build replicas of the light redirecting surface.

7. The method of claim 5 further comprising replicating the light redirecting surface on a substrate.

8. The method of claim 6 , wherein creating the replica comprises one of roll-to-roll, foil-to-foil, UV casting, and embossing.

9. A method for the manufacture of an optical security element comprising the step of designing at least one refractive or reflective light redirecting surface of a caustic layer designed according to a method according to any one of claims 1 to 7.

10. 10. A method for manufacturing an optical security element according to claim 9, for marking an object selected from the group comprising consumer products, documents of value, identity cards, tax stamps and banknotes.

11. - producing an optical security element by a method according to claim 9 or 10; A method of producing a security marked object, comprising the step of marking an object selected from the group consisting of consumer products, documents of value, identification cards, tax stamps, and banknotes with said optical security element.

12. A method for producing an optical security element according to claim 9 or 10, comprising the steps of: Distance d from the light redirecting surface ｓ illuminating the light redirecting surface of the optical security element with a point light source located at From said optical security element a distance d ｉ visually observing the caustic pattern projected onto a projection surface at 13. A method for producing an optical security element, characterized in that upon assessment by a user that the projected caustic pattern is visually similar to a reference pattern it can be determined that an object marked with said optical security element is genuine.

13. A method for producing an optical security element according to claim 9 or 10, comprising the steps of:

1. A method for manufacturing an optical security element, characterized in that the optical security element is for authenticating and protecting against counterfeiting objects selected from the group including consumer products, documents of value, identity cards, tax stamps, and banknotes.

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