MATLAB Implementation of Gerchberg-Saxton Algorithm

The Gerchberg-Saxton (GS) algorithm is an iterative beam shaping algorithm commonly used in optics to transform Gaussian-distributed beams into ring-shaped uniform distributions. This technology finds important applications in laser processing, holographic imaging, and optical computing. Basic Principles of GS Algorithm The GS algorithm is based on iterative Fourier transforms, achieving beam shaping by repeatedly converting beams between spatial and frequency domains while applying target intensity constraints. Specific steps include: first, input the initial phase and amplitude distribution of the Gaussian beam; then perform Fourier transform to the frequency domain and apply amplitude constraints for the target ring-shaped intensity; next, inverse transform back to the spatial domain and apply phase constraints again. This process iterates repeatedly until the beam's intensity distribution approaches the target ring-shaped distribution. MATLAB Implementation Approach Parameter Initialization: Define basic parameters such as beam wavelength, spot size, sampling points, and generate initial Gaussian beam amplitude with random phase distribution using functions like randn() for phase initialization. Iterative Optimization: Forward Fourier Transform: Convert spatial domain beam to frequency domain using fft2() function. Apply Amplitude Constraints: In frequency domain, replace beam amplitude with target ring distribution while preserving phase information using conditional indexing operations. Inverse Fourier Transform: Return to spatial domain using ifft2() and check convergence conditions. Phase Update: Preserve new phase distribution using angle() function for next iteration. Convergence Judgment: Determine iteration termination by comparing current intensity distribution with target distribution using error metrics like RMS difference. Optimization and Extensions Convergence Acceleration: Implement weighted update strategies by mixing old and new phase distributions using linear combination to accelerate convergence. Multi-mode Shaping: For more complex target intensity distributions (multiple rings or custom shapes), adjust amplitude constraint conditions with logical mask operations. Noise Suppression: Incorporate low-pass filtering during iteration using filter functions like fspecial() to reduce high-frequency noise effects. Although computationally intensive, the GS algorithm can be efficiently implemented in MATLAB utilizing matrix operations and FFT (Fast Fourier Transform), making it suitable for optical simulations and practical beam shaping system designs.