Advanced Fourier Transform for Optics

Comparison of three Fourier transform techniques based on a truncated spherical wave with varying NA. In computational optics, it is a crucial question to sample the phase of electromagnetic fields. For the standard fast Fourier transform (FFT), the required sampling points increase dramatically with the NA; by using the semi-analytical Fourier transform with the quadratic phase part handled analytically, the required sampling points remains small in medium NA cases; for high-NA cases, the homeomorphic Fourier transform becomes applicable and can then take over the job.
Comparison of three Fourier transform techniques based on a truncated spherical wave with varying NA. In computational optics, it is a crucial question to sample the phase of electromagnetic fields. For the standard fast Fourier transform (FFT), the required sampling points increase dramatically with the NA; by using the semi-analytical Fourier transform with the quadratic phase part handled analytically, the required sampling points remains small in medium NA cases; for high-NA cases, the homeomorphic Fourier transform becomes applicable and can then take over the job.
Picture: IAP, F.Wyrowski

Fourier transform, as one of the most influential mathematical method in various scientific fields, plays an indispensable role in computational optics. It is a key technology that connects different modeling domains in field tracing and, especially, it also determines the computational efficiency. Advanced Fourier transform techniques, namely the semi-analytical Fourier transform and the homeomorphic Fourier transform, are included in the field tracing concept, and that brings the boost in the simulation efficiency.