1. Introduction

The irradiance control problem(ICP) shown in Fig. 1 is a problem of design of an optical element for redirecting and reshaping the irradiance profile of an incident light beam so that a prescribed irradiance distribution is generated on a given target. Such design problems arise in many illumination and nonillumination tasks, for example, in automotive and architectural lighting, indoor wireless communications, beam splitting, focusing, metal cutting, etc. In nonimaging optics, problems of this type form one of the two main groups [1]. When the shapes of the input or output beams and irradiance profiles lack particular symmetries, it is natural to turn to freeform optics whose design is not constrained by a priori assumed symmetries. However, a freeform solution to the ICP is hard to obtain because, in general, even in the geometrical optics(GO) approximation such solution must be a differentiable function satisfying a special nonlinear second order partial differential equation(PDE) of Monge-Ampère type with an unusual boundary condition [2]; see section 2. A mathematically rigorous theory for solving such PDE problems is not yet available. Heuristic (usually iterative) approaches to problems of ICP type have been proposed [3, 4], often, with a posteriori justification by computed examples.

 

Fig. 1 Schematic drawing of a lens to be determined in an ICP. A collimated light beam with cross section $\overline{\mathrm{\Omega }}\subset \alpha$ and irradiance I(x), x ∈ Ω, propagates in direction z. The goal is to determine a refracting plano-freeform lens R which intercepts the beam and redistributes the light over a given region ${\overline{T}}_d$ on a plane α′ parallel to α. In this paper it is assumed that α′ is in the near-field. The irradiance on T_d is a given function L(p, d), p ∈ T, where T is the projection of T_d on α. The lens is defined by the function z(x) to be determined. The refractive index n = const of the lens is given. In the figure the input and output irradiance patterns are not shown. Only the (active) curved side of the lens is shown. Only two complete paths of light are shown.

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Fig. 2 Any horizontal ray hitting $F_{\overline{p},\overline{f}}$ from the left is refracted and passes through the right focus $(\overline{p},d)$.

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The supporting quadric method(SQM) developed in [5, 6] and its more recent extensions [7,8] is a highly versatile, physically transparent and mathematically rigorous methodology for solving optical design problems requiring freeform surfaces. In the SQM framework problems such as the ICP are formulated in geometrical optics(GO) and solved as variational problems with guaranteed convergence to the mathematically rigorous solution. At the same time, dealing directly with the complicated PDEs is avoided. The obtained solution generates a continuous freeform optical surface. In fact, two kinds of surfaces are obtained. The surfaces of the first kind are concave, while the surfaces of the second kind may be neither convex nor concave. When the surface is concave, known results in mathematics guarantee that such a surface is two times differentiable everywhere except, possibly, for a set of zero surface area (sets of “measure zero” in mathematical terms). Additional efforts are usually needed to implement the solution of the variational problem into a numerical procedure [6, 9–11]. Though the SQM has often been used to design freeform optics [9, 12–17], many general questions concerning geometry and diffractive properties of optics designed with the SQM still remain open.

In this paper we provide answers to several of such questions for a refractive plano-freeform lens, designed with the SQM, in the practically important “semi-discrete” case when a collimated beam is transformed into an output beam radiating at a discrete set consisting of an arbitrary number of pixels in a plane with prescribed irradiance concentrations; see Fig. 3(a). Such lens operates as a multifocal lens segmented into subapertures. Each subaperture is a piece of a hyperboloid of revolution with its own focal length determined by the SQM so that accurate control of the irradiance distribution between foci is achieved. It is shown here that the support (=“footprint”) of each subaperture on the planar side of the lens is a convex set whose boundary has a special geometric stucture which is explicitly described. We also obtain a simple expression for the eikonal for each subaperture and of the entire lens. In addition, an explicit relation between the discontinuities in the field of normals across the boundaries of adjacent subapertures and the distance between the foci of these subapertures are established.

 

Fig. 3 (a) The target ${\overline{T}}_d$ is a discrete set of points in the plane α′ with given discrete irradiance distribution L(p), $(p,d)\in {\overline{T}}_d$. (b) The graph of the function z defined by Eq. (7) as a pointwise minimum is the lens R_z. The foci p₁, …, p₅ are not shown.

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As a proof of concept we present the results of analysis of a plano-freeform lens transforming a uniform circular parallel light beam into an image of high complexity, namely, the photographic monochromatic image of A. Einstein represented by gray values on a grid with ≈ 38K pixels. For the designed lens each pixel is a focus for the corresponding subaperture creating the required irradiance. Explicit determination of each of the subapertures allows us to estimate the diffractive spot sizes and assess diffraction effects on the quality of the image generated by the lens more realistically without calculating Fresnel-Kirchhoff integrals for each pixel. These results extend substantially the preliminary results in [18]. In order to make the presentation reasonably self-contained, we included in section 2 a brief review of the GO formulation of the ICP. Because the results in this paper are very tightly connected with the SQM we felt that a quick introduction into the basic concepts behind this method would make the reading of the paper easier; see section 3. Mathematical details are presented in Appendices A and B.

2. Problem statement and analytic formulation

Consider the ICP of designing a plano-freeform lens with capability to redirect and reshape a collimated light beam so that a pre-specified irradiance distribution is produced on a given target set in a plane; see Fig. 1. In analytic form problems dealing with redirection and redistribution of energy require determination of a lens performing an optical transformation with controlled Jacobian. In the GO approximation, the required transformation (often called ray mapping) is derived using the Snell and the energy conservation laws along tubes of light rays [19] propagating between the source and target regions, $\overline{\mathrm{\Omega }}$ and ${\overline{T}}_d$, respectively, as in Fig. 1. (Everywhere in the paper ovebar over sets means that the sets are closed in the sense of set theory.) Denote this transformation by P_d. We recall the expressions for P_d and its Jacobian [2].

It is assumed that the planar side of the lens is inactive and the light rays change direction only due to the right (active) side. That is, the thin lens approximation is in effect. For brevity, we refer to the active surface of the lens as the lens R. In Cartesian coordinates x = (x₁, x₂), z as in Fig. 1, the lens R is the graph of a smooth (sag) function z(x), $x\in \overline{\mathrm{\Omega }}$. In vector form, P_d = P + dk, where k is a unit vector in direction of the z axis, d = const > 0, $P:\overline{\mathrm{\Omega }}\to \overline{T}$ is the refractor map with $\overline{T}$ being the projection of ${\overline{T}}_d$ on the plane α; thus, P is the projection of P_d on α. Often P is more convenient to use than P_d.

Put M(z) := [1 + (1 − n²)|∇z|²]^1/2, where ∇z := gradz and n is the refractive index of R. It was shown in [2] that

An explicit expression for J(P) in terms of the sag z is given in [2]. It is a strongly nonlinear second order PDE. Since it will not be used it in this paper, we do not reproduce it here.

The problem {Eq. (3), Eq. (4)} is not a standard boundary value problem in PDE’s theory. A mathematically rigorous theory for solving this problem for $z\in C^2(\mathrm{\Omega })\cap C^1(\overline{\mathrm{\Omega }})$ does not exist. Because of strong nonlinearities in Eq. (3) and unusual boundary condition Eq. (4) this problem is hard to analyze directly and, even more so, to solve numerically. The Eq. (3) includes a term with the Hess(z) under determinant, and is referred to as an equation of Monge-Ampère type. The known results on solvability of Monge-Ampère equations are not applicable to {Eq. (3), Eq. (4)} because of the particular nonlinear terms in Eq. (3) and unusual constraint Eq. (4).

If the requirement that $z\in C^2(\mathrm{\Omega })\cap C^1(\overline{\mathrm{\Omega }})$ in Eqs. (3) and (4) is relaxed then a mathematically rigorous theory for solving an appropriately modified variant of the problem {Eq. (3), Eq. (4)} in the class of “weak” solutions (explained in section 3) can be developed and this was done in [16] using the SQM.

3. SQM, semi-discrete ICP and weak solutions

In this section we recall the main steps of the SQM.

Step 1

The use of SQM begins with the following basic fact crucial to our construction. Consider a parallel light beam with cross section $\overline{\mathrm{\Omega }}$ propagating in direction k of the z-axis. Let the refractive index of the desired lens be n > 1. Let, as in Fig. 1, α and α′ be two parallel planes at a distance d > 0 from each other. Fix a number $\overline{f}<0$. Assume that the target ${\overline{T}}_d$ on the plane α′ consists of only one arbitrarily picked point $(\overline{p},d)$ with $\overline{p}=(x_1(\overline{p})$, $x_2(\overline{p}))\in \alpha$. Consider a refracting conic surface $F_{\overline{p},\overline{f}}$ which is a graph of the sag function

However, we can define $P_d^{-1}$ and P⁻¹ as set valued maps by setting

Clearly, $P_d^{-1}(\overline{p},d)=P^{-1}(\overline{p})=\overline{\mathrm{\Omega }}$ and below we discuss only the map P⁻¹. For any p ∈ α put

Step 2

Suppose now that $\overline{\mathrm{\Omega }}$ is as before, $\overline{T}=\{p_1,\dots ,p_K\}$, where p_i, i = 1, …K, K ∈ ℕ, K ≥ 1, are distinct points on α, and ${\overline{T}}_d=\{(p_1,d),\dots ,(p_K,d)\}$ as in Fig. 3(a). For K hyperboloids with sag functions $z_{p_i,f_i}$, f_i < 0, i = 1, …, K, defined as in Eq. (5) with $\overline{p},\overline{f}$ replaced by p_i, f_i, define

The lens R_z defines the map $P_z:\overline{\mathrm{\Omega }}\to \overline{T}$ and its inverse $P_z^{-1}:\overline{T}\to \overline{\mathrm{\Omega }}$. Namely,

Let μ₁ ≥ 0, …, μ_K ≥ 0 be such that

Step 3

The last step of the SQM is the proof that a solution to the ICP for a given distributed irradiance pattern μ on $\overline{T}$ is a limit of a sequence of solutions to semi-discrete problems when discrete distributions converge to μ; see [16] for details. In practice, it is the semi-discrete version of the ICP defined by Eqs. (9) that has to be actually solved.

In summary, at the end of step 2 the described methodology provides a framework for designing a freeform multifocal lens solving the semi-discrete ICP. This lens is defined by Eq. (7) and consists of subapertures $F_i=(x,z(x))\equiv (x,z_{p_i,f_i}(x))$, $x\in {\overline{\mathrm{\Omega }}}_i$, i = 1, …, K, each of which is a part of R_z over ${\overline{\mathrm{\Omega }}}_i$. The subapertures “split” the entire beam into parts B_i, i = 1, …, K, with cross sections ${\overline{\mathrm{\Omega }}}_i$ and ${\cup }_{i=1}^K{\overline{\mathrm{\Omega }}}_i=\overline{\mathrm{\Omega }}$. For each $p_i\in \overline{T}$ the rays of B_i emitted by x ∈ Ω_i, where Ω_i is the interior of ${\overline{\mathrm{\Omega }}}_i$, are refracted by F_i so that they are focused at (p_i, d). The behavior of the subaperture “boundary” rays emitted by $x\in \partial {\overline{\mathrm{\Omega }}}_i$ is discussed in section 4, subsection 4. It follows from the discussion above that the SQM produces a continuous in $\overline{\mathrm{\Omega }}$ solution z of the semi-discrete ICP. Because z is not necessarily differentiable, it is called a “weak” solution.

4. Properties of weak solutions to semi-discrete ICP

Let z be a weak solution defined by Eq. (7) of the ICP Eqs. (9). Assume that $\overline{\mathrm{\Omega }}$ is convex.

1. Concavity and differentiability

By construction, z is continuous in $\overline{\mathrm{\Omega }}$. Because the minimum in Eq. (7) is taken over a set of concave functions, the function z is also concave [21]. This is the Keplerian configuration when refracted rays crossover [22]. Replacing min by max in Eq. (7) we obtain new solutions of ICP which are continuous and piece-wise concave but globally may be neither convex nor concave (the refracted rays in this case don’t crossover). Most of the results below are valid for both types of solutions. For definiteness, for the rest of the paper and without further reminding we discuss only the concave solutions defined by Eq. (7).

Because for each i the function $z(x)\equiv z_{p_i,f_i}(x)$, $x\in {\overline{\mathrm{\Omega }}}_i$, and $z_{p_i,f_i}$ is infinitely differentiable on the entire plane α, the function z is infinitely differentiable in Ω_i. Denote by ∂Ω_i the boundary of ${\overline{\mathrm{\Omega }}}_i$. For x ∈ ∂Ω_i the “one-sided” partial derivatives are defined but the usual derivatives of z of the first and higher orders may not be defined on ${\mathrm{\Omega }}_{ij}:={\overline{\mathrm{\Omega }}}_i\cap {\overline{\mathrm{\Omega }}}_j$ for i ≠ j and Ω_ij ≠ ∅.

2. Focal function

It is convenient to introduce the “focal” function of the lens defined by Eq. (7). Let $f:\overline{T}\to (-\infty ,0)$ and f_i:= f (p_i). The pair (z, f) is called a refractor pair if (7) and the relations

Fix an i ∈ {1, …, K}. A direct check shows that for any $x\in {\overline{\mathrm{\Omega }}}_i$ the expression in square brackets in Eq. (10) is equal to nf_i/(n² − 1), that is, the supremum in Eq. (10) is attained at each $x\in {\overline{\mathrm{\Omega }}}_i$. By Eq. (7) for any j ∈ {1, …, K}, j ≠ i and x ∈ Ω_j we have $z(x)=z_{p_j,f_j}(x)<z_{p_i,f_i}(x)$. This implies that the expression in square brackets evaluated at such x is strictly less than nf_i/(n² − 1), that is, for a given i the supremum in Eq. (10) is attained only in ${\overline{\mathrm{\Omega }}}_i$. Since z is differentiable in Ω_i, Eq. (10) implies that P(x) = p_i at any x ∈ Ω_i, and

3. Normals on boundaries of subapertures

In Appendix A, it is shown that if the points p_i and p_j in the target $\overline{T}$ are sufficiently close to each other and F_i ∩ F_j ≠ ∅ then for $x\in {\overline{\mathrm{\Omega }}}_{ij}$ the normals n_i(x) to F_i and n_j(x) to F_j at any x ∈ Ω_ij satisfy the inequality

4. Boundaries of subapertures

For p_i ≠ p_j the set Ω_ij is the projection of Γ_ij:= F_i ∩ F_j. Assume Γ_ij ≠ ∅ and not a single point. We claim: 4(a) Γ_ij is a hyperbola in a plane parallel to k; 4(b) each ${\overline{\mathrm{\Omega }}}_i$ is a convex set; 4(c) the boundary $\partial {\overline{\mathrm{\Omega }}}_i$ is a union of straight line segments and possibly pieces of $\partial {\overline{\mathrm{\Omega }}}_i\cap \partial \overline{\mathrm{\Omega }}$. Proofs of these properties are in Appendix B. A simple example of a segmented beam is shown in Fig. 4. In this example, the freeform lens splits a uniform circular beam shown in Fig. 4(a) so that the p₁, …, p₇ in the plane α′:= {z = 150mm} are illuminated with irradiances indicated in Fig. 4(b). This lens consists of seven subapertures.

 

Fig. 4 The designed lens realizes a Keplerian configuration [22] in which the refracted rays crossover and the images of the segments in (a) are switched in (b) relative to the origin in (a) (indicated by a small bright square). The outer rectangle in (a) is a graphical artifact.

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5. Eikonal and paraxial approximation

1. Derivation of the eikonal

The eikonal (=phase function) does not explicitly enter the strong or the weak GO formulations of the ICP. However, the sag function z of the lens in the GO solution can be converted to the phase shift due to the lens by multiplying it by (n − 1)k, where k = 2π/λ is the wave number [23]. For the rotationally symmetric case and when the input beam is Gaussian and output beam is uniform, Hoffnagle [23] has shown that in paraxial limit the eikonal of the GO solution is identical to the one derived using Fourier optics. Here we derive the eikonal of a GO solution to the ICP without any symmetry assumptions.

The lens R_z designed with the SQM is defined by Eq. (7) and the wavefront is generated by subapertures $F_{p_i,f_i}$, i = 1, 2, …, K. From the point of view of GO each $F_{p_i,f_i}$ focuses the light into the focus (p_i, d). Because the eikonal is determined up to a constant, we may assume that the region $\overline{\mathrm{\Omega }}$ in the plane α : {z = 0} is the planar side of the lens R_z. The material of the lens is assumed to have refracting index n which is also the eccentricity of each $z_{p_i,f_i}$, i = 1, 2, …, K. Thus the geometry of the lens R_z is consistent with the refractive properties of the material.

Fix an i ∈ {1, …, K}. We construct the eikonal ${\mathrm{\Psi }}_{p_i,f_i}$ for the subaperture F_i. The reference phase is taken as $n{z}_{p_i,f_i}(p_i)=n\left(\frac{f_i}{n-1}+d\right)$. The phase shift at x due to the interval between z = 0 and $z=\frac{f_i}{n-1}+d$ which “includes” the lens, is calculated at the plane $z=\frac{f_i}{n-1}+d$ as

The sag z is continuous in $\overline{\mathrm{\Omega }}$. Hence the discontinuities in ${\mathrm{\Psi }}_{R_z}$ are due to discontinuities in the focal function f (p), $p\in \overline{T}$. On the other hand, by Eq. (5)z(p_k)−d = − f_k/(n−1), k = i, j, and then |f_i − f_j| = (n − 1)|z(p_i) − z(p_j)|. By Lemma 6.1 in [16], for any p_i, $p_i\in \overline{T}$ we have $|z(p_i)-z(p_j)|\le |p_i-p_j|/\sqrt{n^2-1}$. Thus,

For future reference we record here an expression for the optical path length(OPL) of a lens designed with SQM. Clearly, for each i = 1, …, K and all $x\in {\overline{\mathrm{\Omega }}}_i$ we have

2. Paraxial approximation of the eikonal

Applying the standard approximation for the square root in z_p,f and taking into account that f < 0 we obtain for each i = 1, 2, …, K

6. Specific computational design - image of A. Einstein

The photographic image of A. Einstein in Fig. 5(c) from the internet is given as a set $\overline{T}=\{p_1,\dots ,p_K\}$ of K = 37, 904 pixels with gray values μ₁, …, μ_K representing brightness of each pixel in the range from 0 (dark) to 255 (maximal brightness). The pixels form a rectangular 184 × 206 array. In our example the given gray values were treated as prescribed irradiances. The goal was to determine a plano-freeform lens which intercepts, redirects and reshapes a circular, uniform, collimated light beam into illumination pattern in the plane α′= {z = 150mm} producing the rectangular array with gray values μ₁, …, μ_K. The schematic in Fig. 3(a) applies in this example. The results are shown graphically in Fig. 5 (the pictures are not to scale).

 

Fig. 5 (a) synthetic image of A. Einstein produced with the designed freeform lens shown in (b) (planar side is not shown); (c) original photo; (d) plot of the gray values of the photo.

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Overview of parameters and results

This ICP was formulated as a semi-discrete problem in the form given by Eqs. (9) and solved numerically in the SQM framework on a standard desktop computer. The numerical results were obtained using a system of algorithms and computational codes developed jointly by the present author and Dr. B. Cerkasskiy.

The dimensions of the image $\overline{T}:61\text{mm}\times 68.3333\text{mm}$ with distance of 0.3333mm between adjacent pixels in x₁ and x₂ directions; refracting index n = 1.4607 (fused silica at wavelength λ = 532nm); the input beam with cross section $\overline{\mathrm{\Omega }}$ was modeled with 515, 431 point sources of equal strength distributed uniformly (in radial direction and on each azimuthal circle) on a disk of radius 20.05mm centered at (0, 0) on the reference plane α := {z = 0}; the target set is in the plane α′; the curved side of the designed lens, denoted, again, by R_z is concave. Its peak-to-valley width (in z− direction) is 10.397666mm; the distance from the plane α′ to the apex of the lens is 105.055008mm. No despecling of the synthetic image and no smoothing by interpolation or any other method was performed on the synthetic image nor on the lens.

Denote by $\widehat{G}\left(z,p_i\right)$ the irradiance at p_i obtained by numerically simulated ray tracing through the designed lens. Put $\widehat{G}:=(\widehat{G}\left(z,p_1\right),\dots ,\widehat{G}\left(z,p_K\right))$ and μ := (μ₁, …, μ_K). Tables 1 and 2 below shows values of several validation parameters for our example. The notations are: the symbol ║ · ║ indicates the RMS value and $\overline{\mu -\widehat{G}}$ is the mean value. The small value of Δ/║μ║ shows the high accuracy of performance of the designed lens.

Table 1. Comparison of prescribed with computed irradiances

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Table 2. RMS value of OPL and its variation

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In the next table we present results of calculated optical path values OPL_i, i = 1, …, K of the subapertures. Note that our computational algorithm finds the solution satisfying the irradiance requirements on the target and the OPL_i are defined by the computed sag but not constrained a priori; the values in the table are obtained from the numerical solution. In this table: OPL := (OPL₁, …, OPL_K), δ := ║OPL║, β := ║(OPL₁ − δ, …, OPL_K − δ)║. Table 2 shows that the differences in the OPL_i for different subapertures is relatively small.

The values $\widehat{G}\left(z,p_i\right)$, i = 1, …, K, were also used to generate the Fig. 5(a) to allow visual comparison with the original photograph. All light rays incident on the lens were refracted and redistributed on the target rectangle $\overline{T}$. Thus, the theoretical energy efficiency of the designed lens is ≈ 100%. The fine details in Fig. 5 (a) demonstrate that our approach to ICPs can be used in problems in which high resolution of the reconstructed image is required.

Diffraction effects

Since a lens designed with the SQM functions as a multifocal lens, it is natural to attempt to evaluate the mutual diffraction effects due to subapertures. The SQM allows to determine each subaperture of a designed lens and, in principle, one can calculate the Fresnel-Kirchhoff integral for each subaperture. However, for a freeform lens with a large number of subapertures this approach may not be practical. The supports of the subapertures while convex, don’t have special symmetries and efficient and accurate analysis of diffraction effects deserves a separate investigation. Below, we present some observations in the case of the image of A. Einstein.

We are aware of only two papers discussing diffraction effects for freeform optics designed with the SQM. In [17] the sizes of diffraction spots were estimated for diffractive optical elements in the Fresnel regime under a priori assumption that diffraction spots are identical squares and in [18] a refracting lens with five foci was considered in the Fraunghofer regime under assumption that supports of subapertures and diffraction spots are circular. Their radii were estimated by the prescribed irradiances using energy conservation. Fig. 4(a) shows that the supports of subapertures may be neither square not circular. Diffraction effects, mainly, for freeform mirrors producing one-dimensional patterns have also been discussed in [24].

Here, we use the diameter and widths in the x₁ and x₂ coordinate directions of a subaperture support to generate “one-number” estimates allowing us to identify subapertures with diffraction spots overlapping with adjacent spots in all or in coordinate directions. Such estimates are certainly approximate but may be useful for initial analysis of designs. For pixels in the image of A. Einstein our estimates of diffraction spot sizes are constructed as follows. Recall that ${\overline{\mathrm{\Omega }}}_i$ denotes the planar support of the subaperture F_i with focus (p_i, d), i = 1, …, K. Denote by τ_i the diffraction spot formed by F_i at (p_i, d) on the focal plane α′. In the SQM for a given solution we determine for each i = 1, …, K the supports ${\overline{\mathrm{\Omega }}}_i$, their diameters D_i, and widths $D_{x_{1}i}$ and $D_{x_{2}i}$ in coordinate directions. By analogy with the one-dimensional Fresnel diffraction [25] the diffraction spot diameter for each τ_i is calculated as DSD_i:= f_i λ/D_i, where f_i is the focal length of F_i and λ is the wavelength immediately after the lens. The latter is calculated from the given wavelength of the input beam and refractive index of the lens. Similarly, the diffraction spot widths in x₁ and x₂ directions are calculated as $f_i\lambda /D{S}_{x_{1}i}$ and $f_i\lambda /D{S}_{x_{2}i}$.

The chart in Fig. 6 shows the cumulative distribution of pixels vs. DSD_i for the image in Fig. 5(a) produced by the lens in Fig. 5(b). It allows to estimate the number of pixels with DSD_i ≥ γ, where γ ∈ (0, 0.76). For example, for γ = 0.3333 there are ≈ 6, 000 pixels (≈ 16%) with DSD_i > 0.3333. As noted earlier, the adjacent pixels in the image Fig. 5(a) are ≈ 0.3333mm from each other in the x₁ and x₂ directions. Thus, diffraction spots of pixels with DSD_i larger than 0.3333 may overlap with diffraction spots of neighboring pixels. Similarly, overlaps may occur in x₁ and x₂ directions for pixels with $D{S}_{x_1}$, $D{S}_{x_2}\ge 0.3333$. The inter-relations between diffraction spots of pixels in the image can be seen in the color maps in Fig. 7 showing the distributions of DSD, $D{S}_{x_1}$, $D{S}_{x_2}$ for all pixels in the image Fig. 5(a) (left, middle and right images in Fig. 7, respectively).

 

Fig. 6 Distribution of pixels as a function of DSD_i.

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Fig. 7 Color coded distributions of DSD, $D{S}_{x_1}$ and $D{S}_{x_2}$ for the image in Fig. 5(a); cyan: DSD, $D{S}_{x_1}$, $D{S}_{x_2}<0.3333\text{mm}$; magenta: 0.3333mm ≤ DSD, $D{S}_{x_1}$, $D{S}_{x_2}\le 0.6666\text{mm}$; black: DSD, $D{S}_{x_1}$, $D{S}_{x_2}>0.6666\text{mm}$.

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Because the irradiance of the input beam is nearly uniform, the subapertures illuminating pixels with small prescribed irradiances have small areas and consequently large diffraction spots. For example, for the image in Fig. 5(c) the gray values for pixels with DSD_i > 0.3333 are in the range between 1 and 34. These pixels are shown in black in Fig. 5(a) (and in Fig. 5(c)) and in magenta and black in Fig. 7. On the average, the gray values in that range are on the order of 0.08% of the overall maximum. The majority of such pixels are near the boundary of the image; cf. with Fig. 5(a)). Thus, diffraction spots of large size have limited effect on the image quality.

7. Summary

The irradiance control problem arising in many applications is considered in the case when the freeform refracting lens transforms a collimated light beam into a beam generating a prescribed irradiance distribution at a given set of pixels in the focal plane. The analytical GO formulation of such design problem leads to a second order nonlinear PDE and boundary condition which can not be solved directly and reliably by presently available methods. By contrast, the supporting quadric method(SQM) provides a mathematically rigorous and physically transparent framework for determination of weak solutions defining freeform lenses producing the prescribed irradiance redistribution, while avoiding the difficulties connected with the approach involving PDEs. New geometric and diffractive properties of freeform refracting lenses designed with the SQM are established and investigated in this paper. These properties, combined with the fact that such lenses are segmented into subapertures which are pieces of hyperboloids of revolution, show that their geometry is very simple and they operate as multifocal lenses. This facilitates their performance evaluation. The particularly simple structure of subapertures of such lenses should be also useful at the fabrication stage. Explicit expressions for the eikonal of such a lens and of its subapertures are also derived. In an illustrative example, we analyze numerically the properties of a freeform lens designed with the SQM for transforming the input irradiance of a uniform collimated beam into an irradiance distribution representing the photographic image of A. Einstein. In this example the image consists of 37, 904 pixels with given gray values which were used as prescribed irradiances. Performance of the lens was validated by simulated ray tracing, comparison of prescribed irradiances with approximate irradiances determined by ray tracing, and by approximate estimates of diffraction effects due to the subapertures of the lens at each pixel; see Fig. 5, Tables 1, 2 and Fig. 6, 7. Most of results extend to one- and two-component optics designed with the SQM for applications with targets in the near- and far-field.