PROCESS FOR CREATING AN OPTICAL COMPONENT FOR GENERATING, FROM A GIVEN LIGHT SOURCE, A GIVEN NEAR-FIELD ILLUMINATION

Abstract

Disclosed is a method for fabricating an optical component that is configured so as to generate on an illumination target in the near-field an illumination that has a determined pattern according to which each point (i) of the illumination target receives a quantity of light (alpha_i) via an illumination generated by an illumination light source that is incident on the optical component, which is placed between the illuminating light source and the illumination target.

Claims

1. Method for fabricating an optical component that is configured so as to generate on a near field illumination target an illumination that has a determined pattern according to which each point (i) of the illumination target receives a quantity of light (alpha_i) via an illumination originating from an illumination light source which is incident on the optical component placed between the illumination light source and the illumination target, wherein the method includes the following steps: (a) positioning an origin point O between the illumination light source and the illumination target; (b) for each point (i) of the illumination target, computing a direction (dir_i) which corresponds to the direction of the vector connecting the origin point O and the point (i) on the illumination target; (c) positioning a reference point A between the origin point O and the illumination target; (d) creating the optical component (Cff) whose surface passes through the reference point A and which, when the illumination originating from an illumination light source is incident on the optical component (Cff), generates an illumination comprised of the all the illuminations of direction (dir_i) and quantity of light (alpha_i); (e) for each point (z_i) of the optical component (Cff) that generates the illumination of direction (dir_i) and quantity of light (alpha_i), computing a corrected direction (dirc_i) which corresponds to the direction of the vector connecting the point (z_i) and the point (i) on the illumination target; (f) determining whether or not, for each point (i), the difference between the direction (dir_i) and the corrected direction (dirc_i) satisfies a predetermined criterion; (g) if, for each point (i), the difference between the direction (dir_) and the corrected direction (dirc_i) satisfies the predetermined criterion, fabricating the optical component that corresponds to the optical component (Cff), (h) if, for each point (i), the difference between the direction (dir_i) and the corrected direction (dirc_i) does not satisfy the predetermined criterion, reiterating the steps (d) to (f) by substituting (dir_i) with (dirc_i).

2. The method according to claim 1, wherein the step (f) further comprises determining whether on each point (i) the mean of the norms of the differences between the direction (dir_i) and the corrected direction (dirc_i) is less than a predetermined value.

3. The method according to claim 1, wherein the optical component is one of a concave optical component and a convex optical component.

4. The method according to claim 1, wherein the optical component is one of a mirror and a lens.

5. The method according to claim 2, wherein the optical component is one of a concave optical component and a convex optical component.

6. The method according to claim 2, wherein the optical component is one of a mirror and a lens.

7. The method according to claim 3, wherein the optical component is one of a mirror and a lens.

8. The method according to claim 5, wherein the optical component is one of a mirror and a lens.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] In order to better illustrate the object of the present invention, a particular embodiment will be described here below, with reference to the accompanying drawings.

[0025] In these drawings:

[0026] FIG. 1 is a diagram illustrating a step of the method according to the invention;

[0027] FIG. 2 is a diagram illustrating another step of the method according to the invention;

[0028] FIG. 3 is a diagram illustrating yet another step of the method according to the invention;

[0029] FIG. 4 is a simulation of different iterations of the method of the invention, for a concave lens and a convex mirror.

DESCRIPTION OF THE PREFERRED EMBODIMENTS'

[0030] As a preamble to the description of the method of the present invention, the method that makes it possible to create an optical component, in order to generate a given illumination from a given light source on a far-field target (at infinity) will be described.

[0031] As previously indicated here above, this method, which corresponds to the method described in the published scientific paper “Light in power: A general and parameter free algorithm for caustic design”, is not the only method that can be used and indeed it is possible for any analogous or equivalent method to be used in the context of the present invention, such that the invention therefore is not in any way limited to this particular method.

[0032] In order to illustrate the method, a plurality of various mirror or lens design problems that occur in non-imaging (anidolic) optics are presented. In all the problems, there is a given light source (either collimated or point source) and a desired illumination on a far-field target, subsequent to reflection or refraction. The goal is to design the geometry of a mirror or a lens that transports the energy emitted by the light source on to the target, with the multiple refractions and reflections not being taken into account. Even though the respective problems considered are different from each other, they share a common structure that corresponds to the equation referred to as generalized Monge-Ampere equation, of which the discrete version is given by Equation (1):

∀i∈{1, . . . ,n}∫.sub.V.sub.i.sub.(ψ)ρ(x)dx=σ.sub.i  (1)

[0033] The method is illustrated for lenses and mirrors, whether concave or convex, with point or collimated light sources.

1—Mirror Design

1.1—Convex Mirror for a Collimated Light Source

[0034] For this first problem, the light source is collimated: the light source may be encoded by a light intensity function ρ over a 2D domain. For the sake of simplicity, it is assumed that the domain is included in .sup.2×{0}⊂.sup.3 and that all the rays are parallel to the direction z (vertical) and directed in the upward direction. The desired target illumination is in the far field (at infinity) and is described by a set of intensity values σ=(σ.sub.i).sub.1≤i≤N supported on a finite set of directions Y={y.sub.i, . . . , y.sub.n} included in the unit sphere .sup.2. The problem is to find the surface of a mirror that sends the source intensity ρ to the target intensity σ. This problem corresponds to a Monge-Ampére equation in the 2D plane, which corresponds to a quadratic optimal transport problem.

[0035] Given that the number of reflected directions is finite, the mirror surface is composed of a finite number of planar facets. R.sub.ψ is defined as the graph of a convex function of the form xmax.sub.ix|p.sub.i−ψ.sub.i, where x|y denotes the scalar product of x and y; for all i∈{1, . . . , N}, p.sub.i is the orthogonal projection of a unit normal of the plane (referred to as slope in the sections that follow) that reflects according to Snell-Descartes law the vertical ray (0,0,1) towards the direction y.sub.i, and ψ.sub.i is a real number that encodes the elevation of the support plane with the slope p.sub.i.

[0036] ψ:=(ψ.sub.i).sub.1≤i≤N denotes the set of elevations. The Visibility cell V.sub.i(ψ) of y.sub.i is defined as a set of localized points x∈.sup.2×{0} whose rays are reflected towards the direction y.sub.i, which means that the vertical rays hit the ith facet of R.sub.ψ.

[0037] Given the definition of R.sub.ψ, it leads to the following:

V.sub.i(ψ)={x∈.sup.2×{0}|∀j,−x|p.sub.i+ψ.sub.i≤−x|p.sub.j+ψ.sub.j}

[0038] By construction, the vertical ray emanating from the point x∈V.sub.i(ψ) touches the mirror surface at an altitude x|p.sub.i−ψ.sub.i for a given i and is reflected towards the direction y.sub.i, and as a consequence thereof the quantity of light reflected towards the direction y.sub.i is equal to the integral of ρ over V.sub.i(ψ). We also have ∇R.sub.ψ(x)=p.sub.i if x∈V.sub.i(ψ). The problem of the mirror with collimated light source then amounts to finding (ψ.sub.i) such that:

∀i∈{1, . . . ,n}∫.sub.V.sub.i.sub.(ψ)ρ(x)dx=σ.sub.i  (1)

[0039] By construction, a solution to Equation (1) provides a parameterization R.sub.ψ of a convex mirror, which sends the collimated light source to the discrete target σ:

R.sub.ψ:x∈.sup.2(x,max.sub.i(x|p.sub.i−ψ.sub.i) where .sup.2×{0} and .sup.2 are identified.

It should be noted that since the mirror is a graph over .sup.2×{0}, the vectors y.sub.i cannot be directed in the upward direction (upward vertical). In practice, it is assumed that:

y.sub.i∈.sup.2:={x∈.sup.2,x|e.sub.z≤0}.

[0040] In addition, the position of the mirror is localized by considering it only above the support:

X.sub.ρ:={x∈.sup.2×{0},ρ(x)≠0} of ρ.

[0041] The same approach also allows for the construction of concave mirrors using a concave function of the form xmin.sub.ix|p.sub.i+ψ.sub.i This amounts to replacing the Visibility cells by:

V.sub.i(ψ)={x∈.sup.2×{0}|x|p.sub.i+ψ.sub.i≤x|p.sub.j+ψ.sub.j∀.sub.j}

[0042] In this case, a solution to Equation (1) provides a parameterization of a concave mirror R.sub.ψ(x)=(x, min.sub.ix|p.sub.i+ψ.sub.i) that sends the collimated light source ρ to the discrete source σ.

1.2—Concave Mirror for a Point Light Source

[0043] In this second design problem, all of the rays are emitted from a single point in space, situated at the origin, and the light source is described by a function of intensity ρ on the unit sphere .sup.2.

[0044] As in the previous case, the target is in the far field and is described by a set of values σ=(σ.sub.i).sub.1≤i≤N supported on the finite set of directions Y={y.sub.i . . . , y.sub.N}⊂.sup.2. The problem which is being considered is that of finding the surface of a mirror that sends the light intensity ρ to the light intensity σ.

[0045] Thereafter a concave surface is constructed which is made up of pieces of confocal paraboloids. More precisely, P (y.sub.i, ψ.sub.i) is used to denote the solid paraboloid of which the focal point is at the origin with the focal length ψ.sub.i and with the direction y.sub.i. The surface R.sub.ψ is defined as the boundary of the intersection of the solid paraboloids, that is to say, R.sub.ψ=θ (∩.sub.i P (y.sub.i, ψ.sub.i)). The Visibility cell V.sub.i(ψ) is the set of directions of rays x∈.sup.2 emanating from the light source that are reflected in the direction y.sub.i. Given that each paraboloid ∂P(y.sub.i, ψ.sub.i) is parameterized over the sphere by xψ.sub.i x/(1−x|y.sub.i), it leads to:

[00001]$V_i\left(\psi \right)=\left\{x\in {\mathrm{��}}^2|\forall j,\frac{{\psi }_i}{1-.Math.x|y_i.Math.}\le \frac{{\psi }_j}{1-.Math.x|y_j.Math.}\right\}.$

[0046] The point light source mirror problem then amounts to finding (ψ.sub.p) that satisfies the light energy conservation Equation (1). The mirror surface is then parameterized by:

[00002]$R_{\psi }\text{:}x\in {\mathrm{��}}^2\mapsto {\min }_i\frac{{\psi }_i}{1-.Math.x|y_i.Math.}x.$

[0047] In practice, it is assumed that the target Y is included in .sup.2, that the support X.sub.ρ of ρ is included in .sub.+.sup.2:={x∈.sup.2, x|e.sub.z≥0} and that the mirror is parameterized over X.sub.ρ.

[0048] The mirror surface may also be defined as the boundary of the union (instead of the intersection) of a family of solid paraboloids. The Visibility cell thus then becomes:

[00003]$V_i\left(\psi \right)=\left\{x\in {\mathrm{��}}^2|\forall j,\frac{{\psi }_i}{1-.Math.x|y_i.Math.}\ge \frac{{\psi }_j}{1-.Math.x|y_j.Math.}\right\}$

and a solution of Equation (1) provides a parameterization R.sub.ψ(x)=(xmax.sub.iψ.sub.i/(1−x|y.sub.i)) of the surface of the mirror.

2. Lens Design

[0049] In this section, the goal is to design lenses that refract a given light source intensity to a desired target. In a manner similar to designing of a mirror, collimated or point light sources are considered.

[0050] In this instance n.sub.1 is used to denote the refractive index of the lens, n.sub.2 the refractive index of the ambient space, and K=n.sub.1/n.sub.2 denotes the ratio of the two indices.

[0051] Considered here is a collimated light source that is encoded by a function ρ on a 2D domain and a target illumination supported on a finite set:

Y={y.sub.1, . . . ,y.sub.N}⊂.sup.2,encoded by σ=(σ.sub.i).sub.1≤i≤N.

[0052] The objective is to find the surface of a lens that sends ρ to σ.

[0053] It is assumed that the rays emitted by the light source are vertical and that the base or bottom part of the lens is flat and orthogonal to the vertical axis. There is no angle of refraction when the rays enter the lens, and consequently it is thus only necessary to build the top part of the lens.

[0054] By means of a simple change of variables, it is shown that this problem is equivalent to that of designing a mirror for a collimated light source. More precisely, for every y.sub.i∈Y, now p.sub.i is defined to be the slope of a plane that refracts the vertical ray (0,0,1) to the direction y.sub.i. As well, is defined as the graph of a convex function of the form xmax.sub.ix|p.sub.i−ψ.sub.i, where ψ=(ψ.sub.i).sub.1≤i≤N is the set of elevations. The Visibility cell V.sub.i(ψ) is defined as being the set of points x∈.sup.2×{0} that are refracted to the direction y.sub.i:

V.sub.i(ψ)={x∈.sup.2×{0}|∀j,−x|p.sub.i+ψ.sub.i≤−x|p.sub.j+ψ.sub.j}

[0055] The collimated light source lens design problem thus then amounts to finding the weights ψ=(ψ.sub.i).sub.1≤i≤N that satisfy the Equation (1). In this case, the lens surface is then parameterized by:

R.sub.ψ:x∈.sup.2(x,max.sub.ix|p.sub.i−ψ.sub.i)

[0056] In practice, it is necessary to choose the directions y.sub.i in .sub.+.sup.2 and the mirror to be parameterized over the support X.sub.ρ of ρ.

[0057] It should be noted that it is also possible to construct concave lenses by taking into consideration the parameterizations with convex functions of the form xmin.sub.ix|p.sub.i+ψ.sub.i.

2.2 Convex Lens for a Point Light Source

[0058] The same problem is considered, except that the collimated light source is replaced by a point light source. As in the configuration of the collimated light source, the base or bottom part of the lens is fixed. A piece of sphere centered at the source is chosen, such that the rays are not deviated. The lens is composed of pieces of ellipsoids of constant eccentricities K>1, where K is the ratio of indices of refraction. Each ellipsoid ∂E(y.sub.i, ψ.sub.i) can be parameterized over the sphere by xψ.sub.ix/(1−Kx|y.sub.i)

[0059] The Visibility cell is then:

[00004]$V_i\left(\psi \right)=\left\{x\in {\mathrm{��}}^2|\forall j,\frac{{\psi }_i}{1-K.Math.x|y_i.Math.}\le \frac{{\psi }_j}{1-K.Math.x|y_j.Math.}\right\}$

[0060] The point light source lens problem thus then amounts to finding the weights (ψ.sub.i).sub.1≤i≤N that satisfy the Equation (1).

[0061] The top surface of the lens is then parameterized by:

[00005]$R_{\psi }\text{:}x\in {\mathrm{��}}^2\mapsto {\min }_i\frac{{\psi }_i}{1-K.Math.x|y_i.Math.}x.$

[0062] In practice, it is necessary to choose the set of directions y.sub.i so as to belong to .sub.+.sup.2 and the lens to be parameterized over the support X.sub.ρ⊂.sub.+.sup.2 of ρ.

[0063] It is also possible to choose to define the lens surface as the boundary of the union (instead of the intersection) of a family of solid ellipsoids. In this case, the Visibility cells are given by:

[00006]$V_i\left(\psi \right)=\left\{x\in {\mathrm{��}}^2|\forall j,\frac{{\psi }_i}{1-K.Math.x|y_i.Math.}\ge \frac{{\psi }_j}{1-K.Math.x|y_j.Math.}\right\}$

[0064] and a solution to the Equation (1) provides a parameterization R.sub.ψ(x)=x max.sub.iψ.sub.i/(1−Kx|y.sub.i) of the lens surface.

3. General Formulation

[0065] Let X be a domain either of the plane .sup.2×{0}, or of the unit sphere .sup.2; ρ: X.fwdarw. a probability density; and Y={y.sub.1 . . . , y.sub.N}⊂.sup.2 a set of N points. The function G: .sup.N.fwdarw..sup.N g is defined by:

G.sub.i(ψ)=∫.sub.V.sub.i.sub.(104)ρ(x)dx

Where G(ψ)=(G.sub.i(ω)).sub.1≤i≤N and V.sub.i(ψ)⊂X is the Visibility cell of y.sub.i, whose definition depends on the non-imaging problem. The use of this notation enables the reformulating of Equation (1) so as to find the weights ψ=(ψ.sub.i).sub.1≤i≤N such that:

∀i∈{1, . . . ,N},G.sub.i(ψ)=σ.sub.i(2)

4. Visibility and Power Cells

[0066] It is therefore necessary to compute the Visibility cells V.sub.i(ψ) associated with each optical modelling. The Visibility cells always have the same structure, making it possible to build a generic algorithm, as detailed here below. In all of the non-imaging optics problems, the Visibility cells are of the following form:

V.sub.i(ψ)=Pow.sub.i(ψ)#X  (3)

[0067] For a collimated light source, X denotes the plane .sup.2×{0} and for a point light source, X is the unit sphere .sup.2. The sets Pow.sub.i(P) are the usual Power cells of a weighted point cloud P={(p.sub.u, ω.sub.i)}⊂.sub.3×:

Pow.sub.i(P):={x∈.sup.3|∀j,∥x−p.sub.i∥.sup.2+ω.sub.i≤∥x−p.sub.j∥.sup.2+ω.sub.j}.

[0068] The expression of the weighted point cloud P={(p.sub.i, ω.sub.i))} depends on the problem. The deduction of the expression for p.sub.i and w in the collimated light source mirror case is explained, with the other formulas being set out in Table 1 for the other cases. In the collimated light source mirror case, the light source is collimated and p.sub.i∈.sup.2×{0} is the slope of the plane that reflects (according to Snell-Descartes law) the vertical ray upwards e.sub.z:=(0, 0, 1) towards the direction y.sub.i. A calculation shows that p.sub.i=P.sup.2 (y.sub.i−e.sub.z) y.sub.i|e.sub.z|e.sub.z, where denotes the orthogonal projection onto .sup.2×{0}.

[0069] The Visibility cell of y.sub.i is then given by:

V.sub.i(ψ)={x∈.sup.2×{0}|∀j,−|p.sub.i+ψ.sub.i≤−x|p.sub.j+ψ.sub.j}=Pow.sub.i(P)∩(.sup.2×{0}),

[0070] where ω.sub.i=2ψ.sub.i−∥p.sub.i∥.sup.2.

[0071] It may thus be concluded therefrom that the Visibility cells for a convex mirror of the point light source mirror problem are given by the Equation (3), in which the weighted point cloud is given by the first row of Table 1, which gives the formulas for the weighted points used to define the Power cells in the Equation (3) for various different non-imaging optics problems. In the lens design problem, K>0 is the ratio of the indices of refraction, K>1 in the point source and lens configuration. Ccv signifies concave, Cvx signifies convex, signifies that the optical component converges to a concave component when the discretization tends to infinity, CS signifies collimated light source, PS point light source:

TABLE-US-00001 TABLE 1 Type Points Weights Cvx (CS/miroir) [00007]$p_i=\text{?}$ ω.sub.i = 2ψ.sub.i − ||p.sub.i||.sup.2  Ccv (Cs/miroir) [00008]$p_i=-\text{?}$ ω.sub.i = 2ψ.sub.i − ||p.sub.i||.sup.2  Cvx (PS/miroir) [00009]$p_i=-\text{?}$ [00010]${\omega }_i=-\text{?}-\text{?}$  (PS/miroir) p.sub.i = y.sub.i/(2 ln(ψ.sub.i)) [00011]${\omega }_i=\text{?}-\text{?}$ Cvx (CS/lentille) [00012]$p_i=-\text{?}$ ω.sub.i = 2ψ.sub.i − ||p.sub.i||.sup.2 Ccv (Cs/lentille) [00013]$p_i=\text{?}$ ω.sub.i = 2ψ.sub.i − ||p.sub.i||.sup.2 Cvx (PS/lentille) [00014]$p_i=-\kappa \text{?}$ [00015]${\omega }_i=-\text{?}-\text{?}$  (PS/lentille) p.sub.i = κy.sub.i/(2 ln(ψ.sub.i)) [00016]${\omega }_i=\text{?}-\text{?}$ indicates data missing or illegible when filed

5. Generic Algorithm

[0072] For each optical design problem, given a light source intensity function, a target light intensity function and an error parameter, Algorithm 1 (here below) provides a triangulation of a mirror or a lens that satisfies the light energy conservation Equation (1).

[0073] The main problem is to find weights ψ such that G(ψ)=σ. This is achieved by using a damped Newton algorithm which has a quadratic local convergence rate for optimal transport problems or for Monge-Ampere equations in the plane.

[0074] The algorithm comprises three steps: [0075] Initialization: the source density is discretized into a piecewise affine density and the target one into a finitely supported measure. This is then followed by constructing the initial weights ψ.sup.0 satisfying the condition ∀i, G.sub.L(ψ.sup.(0))>0. [0076] Damped Newton: a sequence ψ.sup.k is constructed following Algorithm 2 until ∥G(ψ.sup.k−σ)∥.sub.∞≤η. [0077] Surface Construction: finally the solution ψ.sup.k∈.sup.N is converted into a triangulation. Depending on the non-imaging optics problem, this amounts to approximating a union (or an intersection) of half-spaces (or solid paraboloids, or ellipsoids) by a triangulation.

Initialization:

[0078] Discretization of Light Intensity Functions: The framework of the method makes it possible to support any type of collimated or point light source, or target light intensity functions. It may be for example a positive function on the plane or the sphere (depending on the problem) or a greyscale image, which is seen as a piecewise affine function. First the support of the source density ρ is approached by a triangulation T and it is assumed that the density ρ: T.sup.+ is affine on each triangle. Then p is normalized by dividing it by the total integral ∫.sub.T ρ(x)dx.

[0079] In a similar manner, the target light intensity function can also be any discrete probability measure. If the user provides an image, it can be transformed into a discrete measure on the form σ=Σ.sub.iσ.sub.iδ.sub.y.sub.i by making use of Lloyd's algorithm or more simply, by taking one Dirac measure per pixel, the latter approach being adopted in this instance. The target measure is also normalized by dividing by the discrete integral σ=Σ.sub.iσ.sub.i. Furthermore, min.sub.iσ.sub.i>0 is needed for the damped Newton algorithm, however this is not a restriction: if σ.sub.i=0, this simply leads to removal of the corresponding Dirac measure δ.sub.y.sub.i, thus ensuring that no light is sent to y.sub.i.

[0080] Selection of the Initial Family of Weights ψ.sup.0: As previously mentioned here above, it is necessary to ensure that at each iteration all Visibility cells have non-empty interiors. In particular, it is necessary to choose a set of initial weights ψ.sup.0=(ψ.sub.i.sup.0).sub.1≤i≤N such that the initial Visibility cells are not empty.

[0081] For the collimated light sources cases (with mirror or lens), it is noted that if ψ.sub.i.sup.0=∥p.sub.i∥.sup.2/2 is chosen, then ω.sub.i=0, where p.sub.i is obtained by using the formulas of Section 4 (Visibility and Power Cells). The Visibility diagram then becomes a Voronoi diagram, and consequently p.sub.i∈V.sub.i(ψ.sup.o).

[0082] For the point light source mirror case, a calculation shows that if one were to choose ψ.sub.i.sup.0=1, then −y.sub.i∈V.sub.i(ψ.sup.o).

[0083] For the point light source lens case, it can be shown that if one were to also choose ψ.sub.i.sup.0=1, then y.sub.i∈V.sub.i(ψ.sup.o).

[0084] It should be noted that the previous expressions for ψ.sup.o ensure that G.sub.i(ψ.sup.0)=ρ(V.sub.i(ψ.sup.0))>0 only when the support X.sub.ρ of the light source is sufficiently large. By way of example in the case of a point source mirror, if y.sub.i.Math.X.sub.ρ, then G.sub.i(ψ.sup.0)=0 may be obtained. In order to deal with this difficulty, use is made of a linear interpolation between ρ and a constant density supported on a set that contains the (−y.sub.i)'s. This strategy also works for the collimated source lens case, the point source lens and collimated source lens cases.

[0085] Damped Newton Algorithm: When the light source is collimated (that is X=.sup.2×{0}), the problem is known to be an optimal transport problem in the plane for the quadratic cost, the function G is the gradient of a concave function, its Jacobian matrix DG is symmetric and DG≤0. Moreover, if G.sub.i(ψ)>0 for all i and if X.sub.ρ is connected, then the kernel of DG spans over ψ=cst. This ensures the convergence of the damped Newton algorithm presented in algorithm 2 here below, where A.sup.+ denotes the pseudo-inverse of the matrix A. In practice, taking the pseudo-inverse matrix of D{tilde over (G)}() ensures that the mean of the remains constant. Still in practice, one row and one column are removed from the matrix in order to make it full rank.

[0086] When the light source is a point source, a change is effected in respect of the variables {tilde over (ψ)}=ln (ψ) and {tilde over (G)}=G.Math.exp, such that G(ψ)=σ. This change in variable transforms the optical component design problem into an optimal transport problem, ensuring that {tilde over (G)} is the gradient of a concave function and that D{tilde over (G)} is symmetric negative, and therefore easily invertible. In the point light source mirror problem with convex mirrors, the damped Newton algorithm has been proven to be converging.

[0087] Computation of G and DG: According to Section 4, the Visibility cells V.sub.i(ψ) may be computed by the intersecting of a certain 3D power diagram with a triangulation T of the support X.sub.ρ of ρ. Such an intersection may for example be computed by the algorithm described in “A numerical algorithm for L.sup.2 semi-discrete optimal transport in 3D”, Bruno Lévy, arXiv preprint arXiv: 1409.1279 (2014). Then G.sub.i(ψ)=∫.sub.V.sub.i.sub.(ψ) ρ(x)dx may be computed by using first order quadrature formulas. The computation of DG is done by using forward-mode automatic differentiation, where the gradient of G.sub.i(ψ) is stored as a sparse vector. It should be noted that this works quite efficiently since all the numbers that occur in the computation of G.sub.i(ψ) depend only on the values of ψ.sub.j where j is such that (i, j) are neighbours in the Visibility diagram, i.e. V.sub.i(ψ)∩V.sub.j(ψ)≠Ø.

[0088] Linear system: Given that D{tilde over (G)} is sparse and symmetric negative, the solving of the linear systems is done using the preconditioned conjugate gradient.

[0089] Surface construction: In the last step of Algorithm 1, a triangulation of the mirror or lens surface is constructed. The input is a family of weights solving Equation (2) and the parameterization function R.sub.ψ whose formula is given here above and depends on the different cases. Each Visibility cell is triangulated by taking the convex envelope of the vertices of its boundary. A vertex of the triangulation will belong to at least one Visibility cell. For each vertex, it is possible to compute exactly the normal to the continuous surface using the Snell-Descartes law since the incident ray and the corresponding reflected/refracted direction y.sub.i are known.

[0090] Algorithms 1 and 2 are described here below:

TABLE-US-00002 Algorithm 1: Mirror/Lens Construction Input: A light source intensity function ρ.sub.in A target light intensity function σ.sub.in A tolerance η Output: A triangulation R.sub.T of a mirror or a lens Step 1: Initialization T, ρ ← DISCRETISATION_SOURCE (ρ.sub.in) Y, σ ← DISCRETISATION_TARGET (σ.sub.in) ψ.sup.0 ← INITIAL_VECTORS (Y) Step 2: Solve Equation (2): G(ψ) = σ ψ ← DAMPED_NEWTON (T, ρ, Y, σ, ψ.sup.0, η) Step 3: Construct a triangulation R.sub.T of R R.sub.T ← SURFACE_CONSTRUCTION (ψ, R.sub.ψ)

TABLE-US-00003 Algorithm 2: Damped Newton Method for G(ψ) = σ Input: The source ρ : T    .sup.+ and a target σ = Σ.sub.iσ.sub.iδ.sub.y.sub.i; an initial vector ψ.sup.0 and a tolerance η > 0. Step 1: Transformation to an Optimal Transport Problem IF X =  .sup.2 × {0} then   = ψ.sup.0 (et {tilde over (G)} = G) IF X =  .sup.2, then   = (ln(ψ.sub.i.sup.0)).sub.1≤i≤N (and {tilde over (G)} = (G.sub.i ∘ exp).sub.1≤i≤N) Step 2: Solve the equation {tilde over (G)}({tilde over (ψ)}) = σ Initialization: ε.sub.0 : = [min.sub.i G.sub.i(ψ.sup.0), min.sub.iσ.sub.i] > 0, k: = 0 While ||{tilde over (G)}(   ) − σ||.sub.∞ ≥ η - Compute d.sub.k = −D{tilde over (G)}(   ) + ({tilde over (G)}(   ) − σ) - Find the smallest   ∈   provided that {tilde over (ψ)}.sup.k,l : = {tilde over (ψ)}.sup.k + 2.sup.−ld.sub.k satisfies:   [00017]$\{\begin{array}{c}{\min }_i{\tilde{G}}_i\left({\tilde{\psi }}^{k,\ell }\right)\ge {\mathrm{.Math.}}_0\\ {.Math.\tilde{G}\left({\tilde{\psi }}^{k,\ell }\right)-\sigma .Math.}_{\infty }\ge \left(1-2^{-\left(\ell +1\right)}\right){.Math.\tilde{G}\left({\tilde{\psi }}^k\right)-\sigma .Math.}_{\infty }\end{array}\hspace{1em}$ - Set {tilde over (ψ)}.sup.k + .sup.1 = {tilde over (ψ)}.sup.k + 2.sup.−ld.sub.k and k ← k + 1 Return ψ := ({tilde over (ψ)}.sub.i.sup.k).sub.1≤i≤N si X =  .sup.2 × {0} or  ψ := (exp({tilde over (ψ)}.sub.i.sup.k)).sub.1≤i≤N si X =  .sup.2

[0091] Although this value is given only by way of an example, in practice, it is possible to choose η=10.sup.−8.

[0092] The foregoing sections therefore serve to explain one of the methods for solving far-field non-imaging (anidolic) optics problems, that is to say, constructing a lens or a mirror, whether concave or convex, from a given source of light, which is either a collimated or point source, in order to achieve a given illumination on a target at infinity.

[0093] Although this is not the only method that it is possible to use in the context of the invention, the method described here above can be used in the context of the step (d) of the method of the invention described here below.

[0094] The problem solved in the far field by the above method can also be solved in the near field.

[0095] The method of the invention for solving the near-field problem is detailed here below, in conjunction with FIGS. 1-3.

[0096] In the method according to the invention, there is an illumination target T and an illumination light source S, the illumination target T being in the near field and therefore at a finite distance from the illumination light source S.

[0097] From this illumination light source S, it is sought to obtain a given illumination on the target T, by incidence of the light coming from the illumination light source S on an optical component Cnf to be designed.

[0098] The illumination on the target T is characterized by a quantity of light received alpha_i at any point i of the target T.

[0099] According to the method, the following steps are carried out: [0100] (a) positioning an origin point O between the illumination light source S and the illumination target T; [0101] (b) for each point i of the illumination target T, computing a direction dir_i which corresponds to the direction of the vector connecting the origin point O and the point i on the illumination target T (FIG. 1); [0102] (c) positioning a reference point A between the origin point O and the illumination target T; [0103] (d) creating the optical component Cff whose surface passes through the reference point A and which, when the illumination originating from an illumination light source S is incident on the optical component Cff, generates an illumination comprised of all the illuminations of direction dir_i and quantity of light alpha_i (FIG. 2); [0104] (e) for each point z_i of the optical component Cff that generates the directional illumination dir_i and quantity of light alpha_i, computing a corrected direction dirc_i which corresponds to the direction of the vector connecting the point z_i and the point i on the illumination target T (FIG. 3); [0105] (f) determining whether or not, for each point i, the difference between the direction dir_i and the corrected direction dirc_i satisfies a predetermined criterion; (g) if, for each point i, the difference between the direction dir_i and the corrected direction dirc_i satisfies the predetermined criterion, creating the optical component C that corresponds to the optical component Cff, [0106] (h) if, for each point i, the difference between the direction dir_i and the corrected direction dirc_i does not satisfy the predetermined criterion, reiterating the steps (d) to (f) by substituting dir_i with dirc_i.

[0107] The method of the invention here above is applicable to any non-imaging (anidolic) optics problem, in particular for a fabricated optical component that is a concave or convex mirror or a lens.

[0108] The optical component created in the step (g) is created according to the conventional means for creating an optical component, whether this be a mirror or a lens, for example by milling on a 3-axis numerical control (CNC) machine tool after prior milling of the blank, this being preferably computer assisted or computer controlled so as to improve the finish of the optical component thus produced, optionally followed by sanding and/or polishing. For a lens, it is possible to use, for example, though not exclusively, poly(methyl methacrylate) (PMMA), while for a mirror, aluminum may be used.

[0109] The method of the invention may be translated into algorithmic form as in the Algorithm 3 here below, in order to provide for a better understanding thereof and possibly an automated implementation of the same, in a manner analogous to the method set out in the preamble to the detailed description for a far field target.

[0110] The method described here above, with a target illumination σ=Σ.sub.i=1.sup.Nσ.sub.iδ.sub.z.sub.i supported on a set of points Z={z.sub.1, . . . , x.sub.N)⊂.sup.3, is, as indicated in the method, tantamount to iteratively solving a far-field problem, namely Equation (1), which rapidly converges to the near field solution.

[0111] In the Algorithm 3, SOLVE_FF (T, ρ, Y.sup.k, σ, η) denotes an algorithm that solves the far field problem between a source ρ: T.sup.+ and a target σ=Σ.sub.iσ.sub.iδ.sub.y.sub.i supported on Y⊂.sup.2 for a numerical error η. The Step 2 of Algorithm 1 may for example be implemented in order to solve this problem.

TABLE-US-00004 Algorithm 3: Optical Component Design for a Near-Field Target Input: The source ρ : T   .sup.+ and a target σ = Σ.sub.i=1.sup.Nσ.sub.iδ.sub.z.sub.i ; an initial vector ψ.sup.0 and two tolerances η, η.sub.NF > 0. Initialization: ∀i, c.sub.i.sup.0 = 0 While ∥c.sup.k+1 − c.sup.k∥.sub.1/N > η.sub.NF - Compute v.sub.i.sup.k = R.sub.ψ.sub.k(c.sub.i.sup.k) - Set y.sub.i.sup.k = (z.sub.i − v.sub.i.sup.k)/∥z.sub.i − v.sub.i.sup.k∥ - Solve ψ.sup.k+1 ← SOLVE_FF (T, ρ, Y.sup.k, σ, η) - Update c.sup.k+1 to be the barycenter of V.sub.i(ψ.sup.k+1).

[0112] It is clear that when a fixed point is reached in Algorithm 3, which corresponds algorithmically to the method of the invention, the corresponding weight vector ψ is a near-field solution.

[0113] The Applicant has observed in practice that the process converges very quickly. Over several attempts, an error η.sub.NF of less than 10.sup.−6 was obtained after only 6 iterations.

[0114] The convergence of the method of the invention is illustrated in FIG. 4, which represents, on the example of the image of the train on the first line, a rendering of the target image, the first iteration, the second iteration and the sixth iteration, for a collimated source concave lens configuration; while the second line represents a rendering of the target image, the first iteration, the second iteration and the sixth iteration, for a collimated source convex mirror configuration. It can therefore be observed that after six iterations, the image is almost identical to the target image.

[0115] The method according to the invention may be applied to any non-imaging optics problem, regardless of whether the light source is a point or collimated source, whether the optical component is a mirror or a lens, whether the geometry of the optical component is concave or convex. The method of the invention works equally well even if the light source is not uniform.