Optical element and a method of visually authenticating an object

A method of designing a relief pattern of a light-redirecting surface of a caustic layer made of a piece of refractive transparent or partially transparent, or reflective, first optical material, the caustic layer including a mask layer disposed, respectively, on an optical surface of the optical element or within the optical element, the mask layer comprising a mask pattern and having a variable light transmission coefficient, the mask layer being adapted to at least partially transmit incoming light upon illumination of the optical element with a point-like light source, the caustic layer being adapted to redirect incident light received from the point-like light source and form a projected image containing a caustic pattern, the method comprising the computer implemented steps of:

• providing a discrete representation of an input target image of a reference pattern comprising a set P of N image pixels pi of coordinates {(xi,yi)} in the image plane with associated nonzero target light intensities {Ii}, i=1, . . . , N, distributed within a given area of the target image and corresponding to a target caustic pattern of the target image;
• computing a piecewise representation of the light-redirecting surface z=F(x,y) of the caustic layer, with height z above the (x,y) coordinates plane, based on a representation of the light-redirecting surface by means of intersecting pieces of surfaces z=fi(x,y), i=1, . . . , N, respectively obtained from the 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 the 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 the 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 the points P(1), . . . , P(N) by the caustic layer redirecting incident light via the associated piecewise light-redirecting surface according to the variable light transmission coefficient of the mask pattern;
• and
• calculating the respective values of the N heights z1, . . . , zN of the N vertices of the corresponding N pieces of surfaces which minimize the differences between the respective 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,
• thereby obtaining the light-redirecting surface having a relief pattern adapted to redirect incident light received from the light source by the optical element comprising the mask layer and form a projected image containing the target caustic pattern reproducing the reference pattern;
• wherein
• the point-like light source is a source of light whose angular size is sufficiently small that light can be considered to arise from a single point.