# US 12,214,612 B2 · Method of designing a light-redirecting surface of a caustic layer, an optical security element comprising the designed light-redirecting surface of the caustic layer, a marked object, use and method of authenticating the object

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- Number: US 12,214,612 B2
- Document ID: US12214612B2
- Status: Granted
- Kind: B2
- Application number: 17/282,983
- Filed: 2019-10-04
- Published: 2021-11-04
- Granted: 2025-02-04
- Earliest claimed: 2018-10-05
- Assignees: Sicpa Holding SA
- Applicants: Sicpa Holding SA
- Inventors: Andrea Callegari; Mathieu Gillieron; Oscar De Feo
- Web page: https://patentgrep.com/patent/US12214612B2

## CPC

- B42D 25/324: Reliefs
- B42D 25/351: Translucent or partly translucent parts, e.g. windows
- B42D 25/36: comprising special materials
- G02B 27/0012: {Optical design, e.g. procedures, algorithms, optimisation routines}
- G02B 27/0927: {Systems for changing the beam intensity distribution, e.g. Gaussian to top-hat}
- B42D: BOOKS; BOOK COVERS; LOOSE LEAVES; PRINTED MATTER CHARACTERISED BY IDENTIFICATION OR SECURITY FEATURES; PRINTED MATTER OF SPECIAL FORMAT OR STYLE NOT OTHERWISE PROVIDED FOR; DEVICES FOR USE THEREWITH AND NOT OTHERWISE PROVIDED FOR; MOVABLE-STRIP WRITING OR READING APPARATUS
- G02B: OPTICAL ELEMENTS, SYSTEMS OR APPARATUS

## Abstract

The invention relates to a method of designing a refractive transparent or partially transparent light-redirecting surface, or a reflective light-redirecting surface, of a caustic layer comprising providing a discrete representation of an input target image, calculating a generalized power diagram for a set of image pixels pi of the target image and computing a piecewise light-redirecting surface of the caustic layer based on a calculated optimal set of weights minimizing a cost function associated with the set of image pixels pi. The invention also relates to an optical security element, a marked object, a method of visually authenticating an object and use of the optical security element for authenticating or securing against counterfeiting.

## Claims

### Claim 1

A method of forming a caustic layer having a relief pattern formed on a lens element and adapted to redirect incident light received from a light source and to form a projected image containing a caustic pattern, the method comprising computer implemented steps of:
providing a discrete representation of an input target image comprising a set P of N image pixels pi of coordinates {(xi,yi)} in an image plane with associated nonzero target light intensities {Ii}, i=1, . . . , N, distributed within a given area of the input target image and corresponding to a target caustic pattern of the input target image;
computing a piecewise representation of a light-redirecting surface z=F(x,y) of the caustic layer, with height z above an (x,y) coordinates plane, based on a representation of the light-redirecting surface by intersecting pieces of surfaces z=fi(x,y), i=1, . . . , N, respectively obtained from stationarity of an optical path length of rays refracted, or reflected, by the caustic layer and focused on points P(i) of the image plane of coordinates (xi,yi), i=1, . . . , N, wherein each piece of surface z=fi(x,y) is a surface of revolution around an axis passing through a point P(i) and having a vertex at point (xi,yi,zi), with height zi=fi(xi,yi), i=1, . . . , N, the piecewise representation of the light-redirecting surface associated with respective values of the heights of the N vertices being formed by an envelope of intersections of the corresponding N pieces of surfaces z=fi(x,y), i=1, . . . , N;
for a given set of respective values of heights z1, . . . , zN of the vertices of the N pieces of surfaces, calculating a corresponding set of values of light intensities I(1), . . . , I(N) which are respectively focused on points P(1), . . . , P(N) by the caustic layer redirecting incident light via the associated piecewise light-redirecting surface; and
calculating the respective values of the N heights z1, . . . , zN of the N vertices of the corresponding N pieces of surfaces that causes the differences between the values of calculated light intensities I(1), . . . , I(N) focused on the points P(1), . . . , P(N) via the associated light-redirecting surface and the respective corresponding values of the target light intensities I1, . . . , IN to be minimized to a least value as a result of calculating the respective values of the N heights z1, . . . , zN; and
forming the caustic layer having the light-redirecting surface with the calculated values of the N heights z1, . . . , zN of the N vertices of the corresponding N pieces of surfaces on an optical material substrate.

### Claim 2 (depends on 1)

The method according to claim 1, wherein each piece of surface z=fi(x,y), i=1, . . . , N, is approximated by taking, within the paraxial approximation, a Taylor expansion of order k greater or equal than two of the expression of the piece of surface obtained from the stationarity of the optical path length.

### Claim 3 (depends on 1)

The method according to claim 1, wherein the step of calculating the heights zi minimizing the differences between the calculated light intensities I(i) and the corresponding target light intensities Ii, for i=1, . . . , N, is performed by means of a gradient-free optimization method.

### Claim 4 (depends on 1)

The method according to claim 1, wherein the step of calculating the heights zi minimizing the differences between the values of the calculated light intensities I(i) and the corresponding value of the target light intensities Ii, for i=1, . . . , N, to zero is performed by means of an optimization method resorting to a power diagram for the computation of an associated cost function and its derivatives.

### Claim 5 (depends on 1)

The method according to claim 1, wherein the designed light-redirecting surface is used to generate a machine-compatible representation for controlling a machining tool.

### Claim 6 (depends on 1)

The method according to claim 1, wherein the designed light-redirecting surface is a master light-redirecting surface to be used to build a replica of the light-redirecting surface.

### Claim 7 (depends on 5)

The method according to claim 5, further comprising replicating the light-redirecting surface on a substrate.

### Claim 8 (depends on 6)

The method according to claim 6, wherein replicating the light-redirecting surface on a substrate comprises one of roll-to-roll, foil-to-foil, UV casting, and embossing.