Extended homeomorphic Fourier transform framework for non-bijective phase-to-frequency mappings in focused optical field simulation

The homeomorphic Fourier transform (HFT) offers an efficient approach for modeling the diffraction of focused optical fields by exploiting the stationary phase approximation and establishing a bijective mapping between spatial phase and local angular frequency. However, this bijective assumption breaks down for complex or highly aberrated wavefronts, where phase gradients lead to overlapping mappings. In this work, we develop an extended homeomorphic Fourier transform (EHFT) framework that generalizes the original HFT to accurately handle non-bijective phase-to-frequency mappings. The framework introduces an energy-conserving correction to derive a discrete local angular spectrum under the stationary phase approximation, incorporates a local angular spectrum rearrangement strategy that merges redundant frequency components into a compact and quasi-uniform spectral grid, and employs a matrix triple product formulation to enable flexible and accurate inverse Fourier transforms. The EHFT framework preserves the high computational efficiency of the original HFT while substantially enhancing robustness against mapping degeneracies. Its accuracy and efficiency are validated through comprehensive simulations and experimental measurements on aberrated focused fields, demonstrating its potential as a versatile and powerful tool for modeling complex optical field propagation in practical systems.