Theory and algorithm of the homeomorphic Fourier transform for optical simulations

Area of Science:

• Optics and Photonics
• Computational Physics
• Applied Mathematics

Background:

• The fast Fourier transform (FFT) accelerated physio-optics modeling but struggles with strong wavefront phases.
• Numerical complexity increases significantly with phase variations, requiring dense sampling for FFT accuracy.

Purpose of the Study:

• To develop an approximated algorithm for computing Fourier transforms of fields with strong wavefront phases.
• To introduce a computationally efficient alternative to the FFT in specific optical scenarios.

Main Methods:

• Proposed an approximated algorithm based on the bijective mapping behavior of amplitude distributions in strong phase fields.
• Named the approximation 'homeomorphic Fourier transform' due to this characteristic.
• Derived the mathematical approximation for the Fourier integral.

Main Results:

• Demonstrated that the Fourier transform of fields with strong wavefront phases can be approximated by a bijective mapping of the amplitude distribution.
• The homeomorphic Fourier transform simplifies the computational process for such optical fields.
• Numerical applications confirmed the advantages of the proposed method in computing processes.

Conclusions:

• The homeomorphic Fourier transform offers a significant computational advantage for physio-optics modeling involving strong wavefront phases.
• This method addresses the limitations of the FFT in scenarios with complex phase variations.
• The approach provides a valuable tool for efficient optical design and simulation.