## Abstract

It has been shown in [ J. A. Hoffnagle and C. M. Jefferson, Opt. Eng. **42**, 3090, (2003)] that a pair of plano-aspheric lenses can be used to transform a radially symmetric, Gaussian beam to a radially symmetric flat-top beam. In this paper it is shown that a pair of plano-freeform lenses can be used to transform a collimated light beam of arbitrary (including, non-radially symmetric) intensity profile to a collimated output beam of constant phase and a priori specified intensity pattern over a given flat region. The curved surfaces of both lenses can be chosen strictly convex which should facilitate fabrication. The required pair of plano-freeform lenses is designed using the supporting quadric method (SQM) [ V. I. Oliker in *Trends in Nonlinear Analysis*, (Springer-Verlag, 2003)] combined with ideas from optimal mass transport [ V. I. Oliker, Arch. Rational Mechanics and Analysis **201**, 1013 (2011)]. Such approach provides a rigorous methodology for designing freeform optics for irradiance redistribution. In this paper, this approach is applied to design of a laser beam shaping system with two lenses.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In many applications an optical system is required to redirect the light from a source and shape its output irradiance distribution so that a prescribed field is generated on a given target. Systems with such capabilities are needed in numerous areas of modern science and technology. Laser technology is one such area in which the beam generated by a laser has to be transformed so that some given region of space is illuminated with an irradiance pattern different from that of the source [1,2]. Similar tasks must be solved in nanotechnology, photonics, photolithography, directed-energy, material processing, astronomical instruments, sensing systems, solar collection, optical engineering metrology, buildings and roads illumination, and many others.

A systematic approach to beam shaping (and to many other optical design problems) is to assume a priori that the solution and the data are rotationally symmetric (RS) [1–6]. In fact, most of the currently used conventional optical devices rely on lenses and mirrors which are rotationally or rectangularly symmetric or can be described by explicit formulas. The principal drawback of this approach is that the inherent limitation of assumed special symmetry excludes or makes highly inefficient utilization of so designed optics in applications requiring control of irradiance produced by nonsymmetric sources and distributed over areas of irregular shape. By contrast, a *freeform* optical surface is a surface designed without any a priori assumptions of available symmetry or explicit formula. Consequently, freeform surfaces provide significantly more degrees of freedom to create optical systems for a wide variety of applications. In addition, such systems can be more compact, energy efficient and light. The goal of this paper is to present a systematic and mathematically rigorous approach to design of a pair of plano-freeform optical surfaces for beam shaping without any a priori symmetry assumptions. At the same time it will be shown that our approach is engineeringly viable and practical.

A typical configuration of a beam shaping optical system is shown schematically on Fig. 1. The spatial coordinates are (*x*, *z*) with *x* = (*x*_{1}, *x*_{2}) ∈ *α*, where *α* is the plane *z* = 0. We work in the geometrical optics (GO) framework and use the associated notions of wavefronts, rays, radiances and irradiances. In analytic formulation of the beam shaping problem (BSP) the lens *R*_{1} is represented as a graph of some function *z*(*x*), *x* ∈ Ω̄, where Ω̄ is the cross section of the input beam. In practice, a lens has the second side, say, to the left of *R*_{1}. We assume that that side is *inactive*, that is, it does not change the directions of incoming light rays; this comment applies also to the lens *R*_{2} with obvious modifications. Everywhere in the paper Ω denotes a bounded open domain on *α* and Ω̃ its closure. The same convention is used to denote the target set *T̄ _{d}* in the plane

*z*=

*d*. Points of the set

*T̄*to be illuminated by the output beam produced by the lens

_{d}*R*

_{2}are denoted by (

*p*,

*d*), where

*p*= (

*p*

_{1},

*p*

_{2}) ∈

*T̄*⊆

*α*and

*T̄*is the projection of

*T̄*on

_{d}*α*. This two-lens system defines a

*ray mapping P*: Ω̄ →

_{d}*T̄*;

_{d}*P*(

_{d}*x*) = (

*p*=

*P*(

*x*),

*d*). The graph of the function

*w*(

*p*),

*p*∈

*T̄*, is the lens

*R*

_{2}. The radiance

*I*

_{1}of the input beam and the irradiance

*I*

_{2}of the output beam are related by Eq. (1) below. The design task is to determine the functions

*z*and

*w*realizing the required optical transformation of rays and radiances.

Using the GO setting and applying the Snell and energy conservation laws, the relation between the radiance *I*_{1} and irradiance *I*_{2} the plane *z* = *d* can be expressed as

*J*(

*T*) is the Jacobian matrix of the ray mapping

_{d}*T*. Calculating explicitly

_{d}*J*(

*T*), a partial differential equation (PDE) for the function

_{d}*z*[7] can be derived. However, this PDE is strongly nonlinear of second order with second derivatives under the determinant sign [8] and, except for very special cases, such equations can not be solved numerically reliably by available methods. An example of such a special case is when the ray mapping can be represented in the form

*T*(

_{d}*x*) = ∇

*f*(

*x*) where

*f*is a scalar function to be determined and ∇

*f*denotes the gradient of

*f*. Then the Eq. (1) assumes the form

The PDE for design of beam shaping optics with two refractive lenses is not among such special cases. In such circumstances the usual approach to a given design task is to use an ad hoc heuristic technique in each specific case [11,12].

In this paper we avoid direct use of PDE methods. Instead, our approach is based on the mathematically rigorous geometric supporting quadric method (SQM) [9] combined with the variational (optimal transport) theory [7].

It is worthwhile noting that freeform optics can be useful even in the RS case. For example, in [1] the authors designed a two-lens beam shaper under assumption that the input and output radiance patterns are RS. Their approach requires an initial circularization and filtering of the input beam from the laser before it can be accepted by the beam shaper; see Fig. 2. Clearly, a solution unconstrained by RS requirement will provide a useful improvement of the process by eliminating the step (a)→(b) and possibly (b)→(c); see Fig. 2.

## 2. Variational formulation and solution of two-lens beam shaping problem (BSP)

#### 2.1. Statement of BSP

The geometry of the BSP is shown in Fig. 1 and below we refer to this figure. It is assumed that (after a normalization, if necessary) the media to the left of *R*_{1} and to the right of *R*_{2} have constant refractive index *n*, while the media between *R*_{1} and *R*_{2} has refractive index 1. In other words, *n* denotes the *relative* refractive index. Because collimated beams propagate longer distances without dispersion and distortion [1] it is physically natural to assume (and we do so) that the input beam is collimated and propagates in direction **k**. The output beam is also required to be collimated, propagating in the same direction, and of constant phase. It is assumed that the lenses *R*_{1} and *R*_{2} as well as the media are *lossless*, that is, no energy is lost during the propagation of light. An individual light ray of the input beam propagating in the positive *z*-direction **k** is tagged by a point *x* ∈ Ω̄. The light rays of the input beam are refracted by the lens *R*_{1} and produce a beam (not collimated!) which is intercepted and refracted by *R*_{2}, producing the collimated output beam of direction **k**. The output beam, after lens *R*_{2}, in addition to being collimated, must produce a prescribed irradiance distribution *μ* on a given target domain *T̄ _{d}*, which is in the plane

*z*=

*d*parallel to

*α*and located at the distance

*d*> 0 from

*α*. Thus,

*T̄*is the cross section of the output beam by the plane

_{d}*z*=

*d*.

Similar to [1] we consider here the case when the refractive index *n* > 1. The case 0 < *n* < 1 is similar but requires additional considerations. We plan to treat this case in a separate paper. In [1] the authors, in addition to the refractive index *n*, use the distance between the lenses as the given design parameter. In the radially symmetric case such distance is taken as the distance along the common axes of revolution for the two lenses. When radial symmetry is not assumed this parameter is not well defined. Instead, when the lenses are freeform surfaces, we use the parameter *β* = *l* − *nd*, where *l* = *nz*(*x*) + *t*(*x*) + *n*(*d* − *w*(*P*(*x*)) is the optical path length (OPL) [14]. Thus, the refractive index *n*, the distance *d* in Fig. 1 and OPL *l* are constant design parameters. It is known that *β* = 0 is physically inconsistent with *n* ≠ 1 [7]. Physical considerations show that only the case *β* < 0 is of interest and this is assumed in this paper.

It is convenient to recall here the expression for the ray mapping in case when the function *z* is differentiable. Decompose the ray mapping *P _{d}* : Ω̄ →

*T̄*as

_{d}*P*(

_{d}*x*) =

*P*(

*x*) +

*d*

**k**, where

*P*: Ω̄ →

*T̄*is the orthogonal projection of

*P*on the plane

_{d}*α*. We refer to

*P*as the

*refractor map*. Put

*p*=

*P*(

*x*) and $M(x)=\sqrt{1+(1-{n}^{2}){\left|\nabla z(x)\right|}^{2}}$, (∇ ≡ grad). It was shown in [7,15] that

*w*defining the lens

*R*

_{2}can be obtained from

*z*[7,16]:

*n*> 1 as it was done by the authors of [1]. In this case, formula (3) implies that if

*β*> 0 then the lens

*R*

_{2}may be in front of lens

*R*

_{1}. To avoid this situation we assume that

*β*< 0.

It is important to note that the factor *M*(*x*) in the denominator of Eq. (2) precludes application of the Monge principle as in [17]. Replacing *M*(*x*) ≈ 1 corresponds to paraxial approximation.

#### 2.2. The supporting quadric method (SQM)

In recent years the SQM has often been applied to design freeform mirrors, lenses and diffractive optical elements [18–23]. It provides a viable and rigorous alternative to the approach requiring solution of the complex nonlinear PDE. With the SQM combined with optimal transport theory the solution of the BSP is found in the class of functions called *weak solutions* (to be defined below) which are continuos, convex but not necessarily two times differentiable everywhere. Consequently, the SQM requires re-formulation of the BSP in geometric terms not involving explicit use of Jacobians. The basic principles of the SQM have been described before [9,22,24]. However, its particular implementation for the BSP requiring two refracting surfaces is presented here for the first time. In order to make our presentation reasonably self-contained we need to recall the geometric/optical properties of two-sheeted hyperboloids. Eventually, the required lenses are obtained as envelopes of families of sheets of such hyperboloids.

The notations here are as in Fig. 1. Put also

**Lemma 2.1**

*(see [7]). For arbitrarily fixed*(

*p*,

*η*) ∈

*α*× ℝ

*the graph of the function*

*is the left branch of a two-sheeted hyperboloid of revolution with axis parallel to the z-axis and eccentricity n. Its foci and center are*

*A horisontal light ray hitting the branch (5) from the left refracts and passes through focus*${F}_{p,\eta}^{r}$

*. An illustration of this refracting property in a plane is shown in Fig. 3.*

*Similarly, for an arbitrary fixed* (*x*, *ξ*) ∈ *α* × ℝ *the graph of the function*

*is the right branch of the two-sheeted hyperboloid of revolution about the axis parallel to z-axis, with eccentricity n, and foci and center*

The freeform lenses for beam shaping designed here are envelopes of families of suitable sheets of hyperboloids of the type (5) and (7). Such envelopes are defined as follows.

**Definition 2.2** *Let* Ω *and T be two bounded domains on the plane α and* Ω̄, *T̄ their closures. Denote by C*(Ω̄) *and C*(*T̄*) *the set of continuous functions on* Ω̄ *and T̄, respectively. Let n* > 1 *and β* < 0*. A pair* (*z*, *w*) ∈ *C*(Ω̄) × *C*(*T̄*) *is called a two-lens (TL) system of type A (or Keplerian) if*

*Similarly, a pair*(

*z*,

*w*) ∈

*C*(Ω̄) ×

*C*(

*T̄*)

*is called a TL system of type B (or Galilean) if*

*The set of TL systems of type A (B) is denoted by*Λ

*(Ω̄,*

_{A}*T̄*) (Λ

*(Ω̄,*

_{B}*T̄*))

*. Below, when a statement is valid for both types of systems,*Λ

*(Ω̄,*

_{A}*T̄*)

*and*Λ

*(Ω̄,*

_{B}*T̄*)

*, we simply write*Λ(Ω̄,

*T̄*).

An illustration of Eqs. (9)–(10) in 2D and *T̄* consisting of five points *p*_{1}, ..., *p*_{5} is in Fig. 4, where for brevity we put *F*_{pi} = *H*(*x*, *p _{i}*) +

*w*(

*p*),

_{i}*i*= 1, ..., 5. The points

*Q*= (

_{i}*p*,

_{i}*w*(

*p*)) are their second foci lying on the lens

_{i}*R*

_{2}. Because

*β*< 0, by Lemma 2.1 for a fixed

*p*∈

*T̄*the hyperboloid

*H*(

*x*,

*p*) +

*w*(

*p*) transforms the plane front into spherical front with center at (

*p*,

*w*(

*p*)). According to Eq. (9), the lens

*R*

_{1}defined as the graph of

*z*(

*x*),

*x*∈ Ω̄, consists of pieces of hyperboloids

*F*

_{pi,fi},

*i*= 1, ...,

*K*, such that a ray in direction

**k**incident on

*R*

_{1}is intercepted by the “nearest”

*F*

_{pi,fi}and refracted so that it passes through the focus

*Q*. Thus, in this case the lens

_{i}*R*

_{1}is an “envelope” defined by Eq. (9) of a family of suitable hyperboloids and the lens

*R*

_{2}is an envelope defined by Eq. (10).

We call the branches *H* of hyperboloids in the definition (2.2) “supporting” hyperboloids. Obviously, at each point of a TL system of type A and B there exists at least one supporting hyperboloid. In Fig. 4 the supporting hyperboloids of *R*_{2} are not shown.

**Definition 2.3** *For a TL system in* Λ(Ω̄, *T̄*) *the refractor map P*_{z,w} : Ω̄ → *T̄, possibly multivalued, and its inverse* ${P}_{z,w}^{-1}$ : *T̄* → Ω̄ *are defined by equations*

*z*,

*w*) ∈ Λ(Ω̄,

*T̄*). It is clear from Eqs. (9), (10) and (11), (12) that in both cases the functions

*z*and

*w*have continuous extensions

*ẑ*,

*ŵ*to the entire plane

*α*. Further,

*α*the Lipschitz condition with the constant $1/\sqrt{{n}^{2}-1}$. In addition, in any pair (

*z*,

*w*) ∈ Λ

*(Ω̄,*

_{A}*T̄*) the function

*ẑ*is concave and

*ŵ*is convex [7]. For systems of type B convexity/concavity of the

*ẑ*and

*ŵ*, in general, does not hold.

We now formulate precisely the *weak form* of the beam shaping problem. First we define, following [7], the light transfer from Ω̄ to *T̄* for *TL* systems. Let *I*_{1} be a nonnegative integrable function on *α* with spt(*I*_{1}) ⊆ Ω̄, where $\text{spt}({I}_{1}):=\overline{\left\{x\in \overline{\mathrm{\Omega}}\left|{I}_{1}(x)\ne 0\right|\right\}}$. Physically, *I*_{1} is the intensity of the radiance distribution of the input beam. Denote by *ℬ*(*T̄*) the set of subsets of *T̄* measurable with respect to the usual area measure on *α*. Let (*z*, *w*) ∈ Λ(Ω̄, *T̄*) and *P*_{z,w} the refractor map defined by (*z*, *w*). Then for any set *τ* ∈ *ℬ*(*T̄*) the set ${P}_{z,w}^{-1}(\tau )$ is measurable and the function

*ℬ*(

*T̄*). Physically, for each

*τ*∈

*T̄*the value

*G*

_{z,w}(

*τ*) is the total radiance “delivered” by the TL system to the set

*τ*. Moreover, the function

*G*

_{z,w}is weakly continuous, that is, if (

*z*,

_{k}*w*) ∈ Λ(Ω̄,

_{k}*T̄*),

*k*= 1, 2, ..., is a sequence such that

*z*→

_{k}*z*in

*C*(Ω̄) and

*w*→

_{k}*w*in

*C*(

*T̄*) then

**Definition 2.4**

*Let I*

_{1}

*be as above and I*

_{2}≥ 0

*a given irradiance distribution on T̄. A pair*(

*z*,

*w*) ∈ Λ(Ω̄,

*T̄*)

*is a weak solution of the TL beam-shaping problem if*

*and*

*μ*for the irradiance instead of its expression in terms of

*I*

_{2}to emphasize that the irradiance distribution may be a delta-function (or a sum of delta-functions) concentrated, for example, at points. This is physically meaningful, in particular, when we deal with the discrete version of Eq. (17). Thus, a pair (

*z*,

*w*) ∈ Λ(Ω̄,

*T̄*) satisfying Eq. (17) defines two lenses which convert the collimated beam defined by Ω̄ with radiance intensity

*I*

_{1}into a collimated beam irradiating at

*T̄*with distribution

*I*

_{2}. The condition (16) assures the highest possible energy efficiency of the such two-lens. Naturally, the input and output distributions are required to satisfy the necessary condition Because in this definition it is not assumed that

*z*and

*w*are differentiable the pair is a “weak” solution of the BSP. By construction both functions are continuous and the refraction law is built into the definition of the pair (

*z*,

*w*) as envelopes of families of hyperboloids for each of which the law of refraction is satisfied in classical sense; see Lemma 2.1. Furthermore, as it was already mentioned, if (

*z*,

*w*) ∈ Λ

*(Ω̄,*

_{A}*T̄*) then also by construction the function

*z*is convex in the

**k**direction while

*w*is convex in −

**k**direction. According to a well known theorem in convexity theory, these fuctions are twice differentiable everywhere except possibly sets of zero area on their graphs [26].

#### 2.3. Optimal transport and discrete minimization problem

Our next task is to describe an algorithm for solving the discretization of Eq. (17). We will concentrate on the case when (*z*, *w*) ∈ Λ* _{A}*(Ω̄,

*T̄*) since both lenses in this case are convex. First we note that solution of Eq. (17) is equivalent to a solution of an optimal mass transportation problem [7]. In order to outline the connection, recall the formulas (9)–(10) and define the following class of admissible pairs of functions

*(Ω̄,*

_{A}*T̄*) ⊆

*Adm*(Ω̄,

_{A}*T̄*).

**Theorem 2.5** *Suppose n* > 1 *and β* < 0 *and I*_{1} *and μ as above and satisfy Eq. (18). Put*

*Then the problem*

*has a solution*(

*z*

_{min},

*w*

_{min}) ∈ Λ

*(Ω̄,*

_{A}*T̄*)

*. The minimizing pair in Eq. (21) is unique (up to an additive constant) on each connected component of spt*(

*I*

_{1}).

*Furthermore, this pair* (*z*_{min}, *w*_{min}) *is a weak solution of Eq. (17) and conversely any weak solution of Eq. (17) is a minimizing pair of Eq. (21).*

**Remark 2.6** *It is important to note that in Theorem 2.5 it is assumed that Adm _{A}*(Ω̄,

*T̄*) ≠ ∅

*. This imposes a constraint on separation of*Ω̄

*and T̄ in the direction perpendicular to the z axis. For example, if*Ω̄ =

*T̄ then this condition is satisfied. Alternatively, geometric arguments show that condition Adm*(Ω̄,

_{A}*T̄*) ≠ ∅

*can be achieved by adjusting the OPL and the distance d in a way similar to the way it was done in [27]. In practice this property can be verified by a trial run of the code on the given data.*

In view of 2.5, to solve Eq. (17) we need to solve the minimization problem (20)–(21). For that we use a computational algorithm based on linear programming theory [28]. The discrete version of Eqs. (20)–(21) is stated and solved as follows.

**Step 1. Discretization.** Let *M*, *N* ∈ ℕ, *M* ≥ 1, *N* ≥ 2. Subdivide Ω̄ into *M* measurable subsets ${\omega}_{i}^{M}\subset \overline{\mathrm{\Omega}}$, *i* = 1, ..., *M*, and *T̄* into *N* measurable subsets ${\tau}_{j}^{N}\subset \overline{T}$, *j* = 1, ..., *N*, such that for each *i* = 1, ..., *M* diameters ${\omega}_{i}^{M}<1/M$ and for each *j* = 1, ..., *N* the diameters ${\omega}_{j}^{N}<1/N$, and for 1 ≤ *t*, *k* ≤ *M*, 1 ≤ *h*, *l* ≤ *N*,

*i*= 1, 2, ...,

*M*and

*j*= 1, 2, ...,

*N*pick points ${x}_{i}^{M}\in {\omega}_{i}^{M}$ and ${p}_{j}^{N}\in {\tau}_{j}^{N}$, and put

*ω*⊆ Ω̄ and

*τ*⊆

*T̄*we define

**Step 2. Minimization problem and its solution.**Let

*z*(

*x*),

*x*∈ Ω̄ at points

*x*,

_{i}*i*= 1, ...,

*M*giving the curved side of the first lens and the approximate values of the function

*w*(

*p*),

*p*∈

*T̄*at points

*p*,

_{j}*j*= 1, ...,

*N*giving the curved side of the second lens.

**Step 3.** Refining the subdivisions of Ω̄ and *T̄* one obtains a sequence of solutions of discrete problems guaranteed to converge to a weak solution of Eqs. (20), (21).

**Remark 2.7**
*It was proved in [7], Lemma 4, that the slopes of designed surfaces do not exceed*
$1/\sqrt{{n}^{2}-1}$
*. We also note that the above theory guarantees collimation of the shaped beam for lenses designed exactly. In practice, possible collimation errors depend on the numerical accuracy of the design, fabrication and propagation distance. Some related results are contained in [15]. We hope to further address these issues in a separate publication.*

## 3. Design examples

As a first example, we consider the calculation of two plano-freeform lenses transforming the circular incident beam with a uniform radiance *I*_{1}(*x*) = 1/(*πR*^{2}), $x\in \mathrm{\Omega}=\left\{({x}_{1},{x}_{2})|{x}_{1}^{2}+{x}_{2}^{2}\le {R}^{2}\right\}$ into a rectangular beam with a uniform radiance *I*_{2} (*p*) = 1/(*w*_{1}*w*_{2}), *p* ∈ *T _{d}* = {(

*p*

_{1},

*p*

_{2}) | |

*p*

_{1}| ≤

*w*

_{1}/2, |

*p*

_{2}| ≤

*w*

_{2}/2}. The propagation directions of the incident and of the outgoing beams coincide with the direction of the

*z*axis. The lenses were designed for the following parameters: radius of the incident beam

*R*= 10 mm, the side lengths of the outgoing rectangular beam

*w*

_{1}= 50 mm and

*w*

_{2}= 25 mm, the distance between the tips of the lenses along the

*z*axis

*t*(0) =

*w*(0) −

*z*(0) = 30 mm, and the refractive index of the lens material

*n*= 1.5. Without loss of generality, we assume

*z*(0) = 0.

It follows from the results in Subsections 2.2 and 2.3 that the surfaces of the plano-freeform lenses *z*(*x*) and *w*(*p*) are the solution of a linear programming problem (22)–(23) for the Keplerian configuration. For the Galilean configuration one needs to solve the linear programming problem obtained from Eqs. (22)–(23) by reversing the inequality in Eq. (22) and maximizing the functional in Eq. (23). In both cases, the function *z*(*x*), *x* ∈ Ω was defined on a regular grid with *M* = 10000 points ${\overline{z}}_{i}^{M}$, *i* = 1, . . ., *M*. The function *w*(*p*), *p* ∈ *T _{d}* was defined on a 100 × 100 rectangular grid with

*N*= 10000 points ${w}_{j}^{N}$,

*j*= 1, . . .,

*N*. The corresponding linear programming problems were solved using an in-house implementation of the interior point algorithm taking into account the special sparse form of the constraints matrix given by Eq. (22). The calculation time on a modern PC was about two hours. The calculated two-lens systems are shown in Fig. 5. In the Keplerian configuration, both lenses are convex [Figs. 5(a), 5(b)], whereas for the Galilean configuration [Figs. 5(c), 5(d)], only the second lens is convex [Figs. 5(c), 5(d)].

In order to estimate the performance of the designed two-lens systems, we simulated them in the commercial ray-tracing software TracePro in the following way [31]. First, the calculated optical surfaces represented by the sets of points ${\overline{z}}_{i}^{M}$, *i* = 1, . . ., *M* and ${w}_{j}^{N}$, *j* = 1, . . ., *N* were exported to a computer-aided design software Rhinoceros [32], in which 3D-models of the two plano-freeform lens systems were created (Fig. 5). Then, these 3D-models were used in TracePro for the simulations. Figures 6(a) and 6(b) show the calculated normalized radiance distributions generated by the Keplerian lens system in the planes *z* = 10 cm and *z* = 100 cm. The presented results were obtained by tracing 1 000 000 rays and demonstrate the formation of a rectangle with the required size.

The fact that the size of the rectangle and the radiance distribution remain nearly unchanged at different distances from the element confirms that the wavefront of the beam generated by the lenses is nearly plane. Indeed, the calculated root-mean-square deviation of the optical path length from a constant value at *z* = 10 cm does not exceed 1 nm. The normalized root-mean-square deviations (NRMSD) of the numerically simulated radiance distributions from a constant value amount to 12.1% and 13.8% at *z* = 10 cm and *z* = 100 cm, respectively.

Note that the radiance smoothly decreases toward the boundaries of the rectangle [see the cross-sections in Figs. 6(a) and 6(b)]. This decrease is due to Fresnel losses, which were neglected when designing the lens surfaces. Indeed, in the Keplerian configuration the deflection angles of the rays upon refraction by the freeform surfaces of the lenses vary from zero for the central ray going to the center of the generated rectangle to 35.8° for the rays going to the corners of the rectangle. Since the Fresnel losses increase with an increase in the deflection angle [33], the generated radiance smoothly decreases toward the boundaries of the rectangle in Figs. 6(a) and 6(b) Because of such large deflection angles, the total Fresnel losses on the freeform surfaces of the lenses amount to 24.6% of the incident beam flux.

Figures 6(c) and 6(d) show the calculated normalized radiance distributions generated by the Galilean lens system at the same distances *z* = 10 cm and *z* = 100 cm. The radiance distributions generated in this case are near-uniform and have better quality comparing to the previous case. The NRMSD of the numerically simulated radiance distributions from a constant value amount to 3.9% and 4.1% at *z* = 10 cm and *z* = 100 cm, respectively. The better uniformity of the generated rectangular radiance distribution can be explained by decreased deflection angles leading to lower Fresnel losses. Indeed, in the Galilean configuration, the deflection angles of the rays do not exceed 26°. In this case, the total Fresnel losses on the freeform surfaces of the lenses amount to 10.1% of the incident beam flux.

As a second, more sophisticated example, we considered the design of a two-lens system transforming an incident circular beam with uniform radiance into a star-shaped beam with uniform radiance. The vertices of the star (of the region *T _{d}*) lie on a circle with radius

*R*

_{st}= 24.2 mm. We assume the distance between the tips of the lenses along the z axis

*t*(0) =

*w*(0) −

*z*(0) = 50 mm, with the rest geometric and simulation parameters (

*R*,

*n*,

*M*,

*N*) remaining the same as in the previous case. In this example, we consider only the Galilean configuration since it provides a higher quality of the generated distribution. The plano-freeform lenses calculated by solving a linear programming problem Eqs. (22)–(23) at

*M*=

*N*= 10000 are shown in Fig. 7. Figure 8 shows the calculated normalized radiance distributions generated by the two-lens system at

*z*= 10 cm and

*z*= 100 cm. The simulation results confirm the formation of a star-shaped beam with the required size. The NRMSD of the calculated distributions from a constant value amount to 5.1% and 8.4% at

*z*= 10 cm and

*z*= 100 cm, respectively. As in the previous example, the estimated root-mean-square deviation of the optical path length from a constant value at

*z*= 10 cm does not exceed 1 nm.

## 4. Summary

It is shown theoretically and by computed simulation examples that beam shaping with two plano-freeform refractive lenses is feasible even when the lenses are not required to be rotationally symmetric. Utilization of freeform lenses improves dramatically the efficiency of a laser output applied for illumination of a given area with a prescribed irradiation distribution. Since our approach is not constrained by the requirement of rotational symmetry of any component of the designed system, the number of possible applications is also substantially increased in comparison with designs relying on systems with all components rotationally symmetric [1]. In this paper such capabilities are demonstrated for nonisotropic parallel input and output beams. Spherical input with parallel output beams are also possible and will be considered in separate publications.

## Funding

US AFOSR FA9550-18-1-0189 (V. Oliker); Russian Science Foundation 18-19-00326 (L. L. Doskolovich, D. A. Bykov).

## Acknowledgments

The theoretical study concerning solution of the two-lens beam shaping problem is based upon work supported by the US Air Force Office of Scientific Research (AFOSR) under award number FA9550-18-1-0189; the work on design and simulation of the considered examples of two-lens systems was supported by the Russian Science Foundation.

## References

**1. **J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. , **42**(11), 3090–3099 (2003). [CrossRef]

**2. **F. M. Dickey, ed., *Laser Beam Shaping: Theory and Techniques*, 2nd ed. (CRC Press, 2014). [CrossRef]

**3. **B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform radiation,” Appl. Opt. **4**(11), 1400–1403 (1965). [CrossRef]

**4. **J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (1969).

**5. **P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. **19**(20), 3545–3553 (1980). [CrossRef] [PubMed]

**6. **J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. **39**(30), 5488–5499 (2000). [CrossRef]

**7. **V. I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mechanics and Analysis **201**, 1013–1045 (2011). [CrossRef]

**8. **D. Gilbarg and N. S. Trudinger, *Elliptic Partial Differential Equations of Second Order*, 2-nd ed. (Springer-Verlag, 1983). [CrossRef]

**9. **V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis, M. Kirkilionis, S. Krömker, R. Rannacher, and F. Tomi, eds., (Springer-Verlag, 2003).

**10. **Z. Feng, B. D. Froese, C.-Y. Huang, D. Ma, and R. Liang, “Creating unconventional geometric beams with large depth of field using double freeform-surface optics,” Appl. Opt. **54**, 6277–6281 (2015). [CrossRef] [PubMed]

**11. **R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. **38**(2), 229–231 (2013). [CrossRef] [PubMed]

**12. **C. Bösel and H. Gross, “Double freeform illumination design form prescribed wavefronts and irradiances,” J. Opt. Soc. Am. A **35**(2), 236–243, (2018). [CrossRef]

**13. **J. A. Hoffnagle and M. Jefferson, “Transformation of the transverse intensity profile of a laser beam by aspheric lenses,” presented at SLAC, Oct . 2012.

**14. **M. Born and E. Wolf, *Principle of Optics*, 7th ed. (Cambridge University, 1999). [CrossRef]

**15. **V. I. Oliker, “On design of freeform refractive beam shapers, sensitivity to figure error and convexity of lenses,” J. Opt. Soc. Am. A **25**(12), 3067–3076 (2008). [CrossRef]

**16. **D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Taylor & Francis, 2006).

**17. **V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. of Photonics for Energy **3**, 035599 (2013). [CrossRef]

**18. **S. V. Kravchenko, E. V. Byzov, M. C. Moiseev, and L. L. Doskolovich, “Development of multiple-surface optical elements for road lighting,” Opt. Express **25**(4), A23–A35 (2017). [CrossRef] [PubMed]

**19. **F. Fournier, “Freeform reflector design with extended sources,” Ph. D. Dissertation, (University of Central Florida, 2010).

**20. **C. Canavesi, “Subaperture conics and geometric concepts applied to freeform reflector design for illumination,” Ph. D. Dissertation, (University of Rochester, 2014).

**21. **D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. **36**(6), 918–920 (2011). [CrossRef] [PubMed]

**22. **V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express **25**(4), A58–A72 (2017). [CrossRef] [PubMed]

**23. **L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express **23**(15), 19605–19617 (2015). [CrossRef] [PubMed]

**24. **V. I. Oliker, A rigorous method for synthesis of offset shaped reflector antennas, Computing Lett. **2**(1–2), 29–49 (2006). [CrossRef]

**25. **R. Luneburg, *Mathematical Theory of Optics* (University of California, 1964).

**26. **R. Schneider, *Convex Bodies: The Brunn–Minkowski Theory*, 2-nd ed., (Cambridge University, 2014).

**27. **V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Advances in Appl. Math. **62**, 160–183 (2015). [CrossRef]

**28. **P. R. Thie and G. E. Keough, *An Introduction to Linear Programming and Game Theory*, 3-rd ed. (J. Wiley & Sons, 2008). [CrossRef]

**29. **L. Rúshendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika **51**(3), 245–258 (2000). [CrossRef]

**30. **Q. Mérigot and É. Oudset, “Discrete optimal transport: complexity, geometry and applications,” Discrete & Computational Geometry **55**(2), 263–283 (2016). [CrossRef]

**31. ** Opto-mechanical software TracePro.https://www.lambdares.com/tracepro/

**32. ** Rhinoceros.http://www.rhino3d.com

**33. **M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express **23**, A1140–A1148 (2015). [CrossRef] [PubMed]