Automatic design of off-axis reflective HUD systems based on multi-field SCE and the LM algorithm

Freeform optical elements are increasingly used in practical optical systems, enabled by advances in fabrication and metrology. The automated design of off-axis freeform reflective systems under multi-field constraints remains difficult because aberrations are strongly coupled, and the design space is high-dimensional. We present an automated design workflow that combines multi-field seed-curve extension with constrained nonlinear least-squares refinement. The initial freeform geometry is constructed under global field constraints to embed the desired ray mapping. The freeform coefficients are then optimized using the Levenberg-Marquardt algorithm to improve image quality while controlling distortion. We demonstrate the method by designing a vehicle head-up display (HUD) system and validating it in Zemax OpticStudio. The optimized design achieves a maximum distortion below 3% and an MTF above 0.3 at 6 lp/mm across the full eye box. The results demonstrate that the proposed workflow is effective for the HUD design example studied in this work and is promising for similar off-axis reflective imaging systems.