Freeform surfaces are optical surfaces without linear or rotational symmetry. Their flexible surface geometry offers high degrees of freedom, which can be employed to avoid restrictions on surface geometry and create compact yet efficient designs with better performance. Therefore, freeform surfaces can endow beam shaping with more new functions and satisfy the ever-growing demand for advanced beam-shaping systems.

The design of freeform beam-shaping optics can be formulated as follows. Given an input (a light source) and an output (a prescribed irradiance/intensity distribution), one or multiple freeform surfaces are to be designed so that the light rays emitted from the source are redirected to produce the prescribed irradiance/intensity distribution. According to the étendue of the light source, the design of freeform beam-shaping optics can be divided into two groups, i.e., zero-étendue algorithms based on ideal source assumption and design algorithms for extended light sources. The zero-étendue algorithms assume that the spatial or angular extent of the light source is zero, which means that only one single ray passes through each ray-piercing point on the optical surface. However, the situation becomes different when the étendue of the light source is non-zero. There are an infinite number of light rays passing through each ray-piercing point on the optical surface. When the influence of the spatial or angular extent of a light source on the performance of the beam-shaping system can be ignored, the light source can be considered as an ideal source (a point source or a parallel beam). Then, the design of freeform beam-shaping optics can be greatly simplified by zero-étendue algorithms. Additionally, as the extended light source can be considered to be composed of an infinite number of ideal light sources, most of the current designs of freeform beam-shaping optics are involved in zero-étendue algorithms.

The zero-étendue algorithms include three typical methods including the ray mapping method, the support quadratic method (SQM), and the Monge-Ampère (MA) method. A key step in the ray mapping method is to find a ray mapping that can satisfy the integrability condition. Finding such an integrable ray mapping may not be a simple task. The SQM is a process of calculating a set of quadric surfaces which are employed to build a freeform surface, which produces a discrete illumination that is an approximation to the prescribed illumination. This method can achieve very complex irradiance/intensity distributions, but it requires tens of thousands of quadratic surfaces to construct smooth and continuous freeform surfaces. In addition, the effectiveness of this method still needs to be further explored when multiple freeform surfaces are needed. The MA method converts the design of freeform beam-shaping optics into an elliptic MA equation with a nonlinear boundary condition. This method reveals the mathematical essence of freeform optics design based on the ideal source assumption. It can satisfy the integrability condition automatically and can be implemented efficiently. Its effectiveness has been proven in a wide variety of applications, and the superiorities are verified in LED and collimated laser beam shaping.

In 1972, Schruben converted a prescribed irradiance design with a freeform reflector into a highly nonlinear partial differential equation of second order and proved that such a differential equation of second order is an MA equation. In 2002, Ries demonstrated that a prescribed irradiance design with a freeform lens can be converted into an MA equation based on the relationship between the power density and the curvature of the wavefront. This method can be adopted to tackle complex designs without any symmetry. Finding the solution to the MA equation is a big challenge. Unfortunately, Ries did not introduce the method leveraged to find the solution and did not disclose any further studies on this method in the subsequent ten years. Thus, other researchers in this field had to explore some new ways to solve this inverse problem. In 2013, Wu converted the design of freeform beam-shaping optics into an MA equation and a nonlinear boundary, and first disclosed a numerical method to solve the MA equation. Over the last ten years, Wu generalized the MA method to achieve freeform and precise irradiance tailoring in arbitrarily oriented planes, and to design freeform optics for flexible and precise control of the intensity and wavefront of a light source.

The MA method relying on the ideal light source assumption is considered as the most advanced point source algorithm that satisfies the integrability condition automatically and can be implemented efficiently. Additionally, it can be generalized to design freeform beam-shaping optics for extended light sources since an extended source can be considered to consist of an infinite number of ideal light sources. The MA method paves a way for the broad application of freeform optics.