We propose an approach to deal with the problem of freeform surface illumination design without assuming any symmetry based on the concept that this problem is similar to the problem of optimal mass transport. With this approach, the freeform design is converted into a nonlinear boundary problem for the elliptic Monge–Ampére equation. The theory and numerical method are given for solving this boundary problem. Experimental results show the feasibility of this approach in tackling this freeform design problem.

A method using a freeform surface lens for LED secondary optic design is proposed in this paper. By Snell’s Law, the differential equations are given to build the relationship between the normal direction of a freeform surface and its input/output ray vectors. Runge–Kutta formulas are used to calculate the differential equations to design the freeform surface. Moreover, the optical model for uniform illumination is simulated and optical performance is analyzed. A practical freeform surface lens for LED uniform illumination is fabricated using an injection molding method. By the process, our system demonstrates a uniform illumination with a divergence half-angle of $6\mathrm{°}$ and an efficiency of 78.6%.

Incoherent collimated beam has a wide application, and reshaping the collimated beam with freeform optics has become a popular and challenging topic of noniamging design. In this paper, we address this issue, embedded in three-dimensional space without any symmetry, with a freeform surface from a new perspective. A mathematical model is established for achieving the one-freeform surface design based on the problem of optimal mass transport. A numerical technique for solving this design model is disclosed for the first time, and boundary conditions for balancing light are presented. Besides, some key issues in achieving complex illuminations are addressed, and the influence of caustic surface on this design model is also discussed. Design examples are given to verify these theories. The results show elegance of the design model in tackling complex illumination tasks. The conclusions obtained in this paper can be generalized to achieve LED illumination and tackle multiple freeform surfaces illumination design.

An off-axis optical system with an $x-y$ polynomial surface for a see-through head-mounted display (HMD) is presented. With the use of an $x-y$ polynomial surface, an off-axis see-through HMD system with one tilted spherical combiner and six relay optical lenses is designed. An off-axis see-through HMD with a $15\mathrm{mm}$ exit pupil, a $40°\times 30°$ FOV, and $70\mathrm{mm}$ eye relief is achieved. Furthermore, a binocular off-axis see-through HMD prototype is fabricated and the performance of the system is experimentally tested; $972\mathrm{cd}/{\mathrm{m}}^2$ of luminance is measured at the position of the exit pupil of the HMD. A typical augmented reality image captured outdoors with the HMD prototype is also demonstrated.