Design of freeform phase diffractive optical elements based on the quadratic assignment problem

The design of freeform phase diffractive optical elements is a challenging task, typically necessitating the use of complex differential equations or a large number of iterative calculations. This paper proposes what we believe to be a novel approach to address this problem. In this strategy, we introduce overall comparison optimization (OCO) to ensure the fast convergence of the cost function. The quadratic assignment problem (QAP) is used as the mathematical framework for designing freeform phase diffraction optics. Specifically, the ray mapping calculation problem in geometric optics is simplified as a QAP. To solve this problem, we apply the OCO method, which ensures that the cost function rapidly progresses in the "non-negative" direction, thereby facilitating fast convergence in each optimization iteration. In this manner, the proposed approach alleviates the computational burden associated with repeated evaluations of the cost function and accelerates convergence in the design process. We construct holographic masks using the OCO method and perform simulations to demonstrate the potential of the proposed method in swiftly realizing complex illumination patterns. The results show that the design model has good performance when dealing with complex illumination tasks. The conclusions obtained in this paper can be extended to the realization of phase-only holography and the solution of freeform surfaces illumination design.