Differentiable design of continuous phase plates using multi-level B-splines with smoothness constraints

Continuous phase plates (CPPs) are crucial optical elements in inertial confinement fusion facilities for laser beam smoothing. Traditional design methods based on Gerchberg–Saxton (GS) algorithms face two fundamental limitations: lack of direct control over phase smoothness and susceptibility to local optima convergence. We propose a CPP design method enforcing phase smoothness by modeling the phase distribution using multi-level B-splines and integrating the curvature and power spectral density (PSD) of the surface as the regularization terms. The incorporation of regularization terms can further smooth the phase distribution of the CPP, making it more suitable for practical fabrication. Additionally, we employ a multi-scale optimization strategy to effectively avoid convergence to local optima. The simulation results demonstrate that compared with the modified GS algorithm, our method reduces the root-mean-square deviation (RRMSD) by 59.7%, 51.6%, and 47.4% across three design tasks. Meanwhile, our method exhibits enhanced smoothness, enabling better manufacturability. These improvements substantiate the effectiveness and versatility of our method.