# Freeform Optics: Optimal Transport, Minkowski Method, and Monge–Ampère-Type Equations

Communicated by *Notices* Associate Editor Reza Malek-Madani

## 1. Description and Background

A freeform optical surface, simply stated, refers to a surface whose shape lacks rotational symmetry. The use of such surfaces allows generation of complex, compact, and highly efficient imaging systems. Ever since lenses without symmetries were used in World War I in periscopes, the engineering and design of freeform optical surfaces have gone through remarkable evolution, with applications in a wide range of areas, including medical devices, clean energy technology, military surveillance equipments, mobile displays, remote sensing, and several other areas of imaging and nonimaging optics that can benefit from distributing light from a source to a target in a controlled fashion using spatially and energy-efficient systems. See 17 and the references therein.

Mathematically, the design of freeform optical surfaces is an inverse problem related to optimal transportation theory and leads to a class of nonlinear partial differential equations (PDE) called generated Jacobian equations for which the Monge–Ampère equation is a prototype. The modeling of the problem is based on the systematic application of the laws of reflection and refraction in geometric optics along with energy conservation principles.

For completeness we recap the laws of geometric optics. Consider a reflecting or refracting surface and suppose a unit vector in the direction of is incident on the surface at a point where the normal to is given by the unit vector Reflection and refraction happen in tandem. (Fig. .1). However, for simplicity of the models we treat them separately. If is a unit vector in the direction of the reflected ray, the law of reflection (*angle of incidence is equal to angle of reflection*) in vector form says:

If medium *Snell’s law*), which states *angle of incidence*) and *angle of refraction*) in vector form, states that:

where *relative refractive index*,

When medium I is optically denser than medium II or equivalently

Similar physical restriction applies for

By using equation 1.1 or 1.2 we can define a map

Generally, in an optical surface design problem the data consists of two bounded domains

The objective is then to find an optical surface (*lens/mirror*)

for all

In practice, *ray-tracing map* of

The physical meaning of equation 1.5 is that the prescribed intensity of reflected or refracted light on the set *reflector/refractor* problem.

The models used to obtain numerical and analytical solutions for *reflector/refractor* problems are developed by using one of the following three approaches. One of the approaches is to use variational method where the problem is cast as an optimal mass transportation problem from

In the discussion below, we will focus on refractor problems and briefly describe the aforementioned approaches by using prototype refractor problems. We will state the problems explicitly and show how the available methods are used to solve the corresponding design problems. We will also mention problems that are worth studying in the future. We point out to the reader that unless mentioned otherwise, we consider

## 2. Optimal Transport: The Far Field Refractor Problem

Several freeform lens design problems have been successfully modeled by using optimal transport framework on the sphere. See 41020. The models have also been adopted by the optics community. Among these problems is the far field refractor problem.

### 2.1. Statement of the far field refractor problem with point source

In this problem, we are given two domains

The objective is to find a refracting surface parametrized as

In particular, if we let *far field refractor problem* is defined as an interface

We recall for *push forward measure*. Because of 1.3, we assume

By using Snell’s law 1.2 it can be shown that the semi-ellipsoid of revolution (ellipse in 2D case) with one focus at

for some

### 2.2. Review of optimal transport problems

Let *cost function*. Denote by *transport map* from

among all transport maps from

among all *transport plans*

Here

The

Also

A function

Problem 2.4 is a linear optimization problem with convex constraints and thus, it admits a dual problem of maximizing

over

One of the basic results in optimal transport theory is that if *Kantorovich potential*, is a

### 2.3. The far field refractor problem as optimal transport

To transform the far field refractor problem 2.2 to an optimal transport problem, we introduce the logarithmic cost function

there exists 10 a

then for each

That is, at each point

If

and hence

Furthermore if

The optimal transport approach can also be used in the case of anisotropic materials where optical properties vary according to the direction of propagation of light 6. In this type of media modeling the design problem is more complicated due to issues such as *birefringence*, a situation where the incident rays may be refracted into two rays. It is also used to develop numerical methods to approximate solutions 4 to freeform design problems. Recently, a more efficient method called entropic regularization is applied to the the reflector problem 2 and it is likely that this approach can be applied to refractor problems.

Finally, it is worth noting that the far field problem has not been studied in its full generality using optimal transport method. In practice, if a ray of light hits an interface between two media which have different optical properties, the ray is partly reflected back and partly transmitted (Fig. 1). It is an open problem whether or not this energy imbalance between the source and target can be modeled as a variational problem by using the optimal transport approach.

## 3. Minkowski Approach: The Near Field Refractor Problem

The Minkowski problem involves finding a closed convex surface whose Gaussian curvature at a point with exterior normal

### 3.1. Statement of the near field refractor problem with point source

For this problem we are given

The *near field refractor problem* involves finding an interface

holds for all Borel sets

Similar to the semi-ellipsoids in the far field problem, there are surfaces that have uniform refracting property in the near field case. The Descartes ovoid refracts all rays to a single point (Fig. 2). Let

where

If the relative refractive index of the material inside the oval

The Descartes ovoids will be used as the building blocks of the near field refractors and we look for a solution for the near field refractor problem among surfaces which are supported from above by an ovoid at each point.

It is known that the near field refractor problem can’t be cast as an optimal mass transport problem. However, Minkowski’s approach can be used to solve the problem.

### 3.2. The near field refractor problem using Minkowski’s approach

Once again let