How should handle the circular convolution and the linear convolution in the angular spectrum method?

[MiZhou22]

Hi, there.
Firstly, thank you for your great job for the fresh men like me! It's really helpful!
So here is the thing, I read the angular spectrum method part to implement the wave propagation, and I noticed you just used two fft2s and ifft. So this is a circular convolution, right? I am wondering what should we choose between circular convolution and linear convolution?
Thanks in advance!

[rafael-fuente]

Hi nophy,
Circular convolutions require periodic functions, so the angular spectrum method is really a linear convolution.
In the implementation, it's not computed directly as a linear convolution since we already know analytically the expression for the Transfer function of free space H = exp(1j * kz * z), and by using it we save the computation of one FFT.

[MiZhou22]

Thank you for your reply! @rafael-fuente

I think we can achieve linear convolution by padding the wavefield to twice its size and double sampling the transfer function H in the frequency domain. I implemented this in python, and I got some artifacts ring in the edge of the diffraction pattern. I got the idea from this paper: "A Band-Limited Angle Spectral Method for Numerical Simulation of Free-Space Propagation in Near and Far Fields". Am I saying it right?

I'm still a little confused about the implement of the holography. I'm working on computer-generated holography and I've seen some work using ASM to perform the Gerchberg-Saxton algorithm. But I think we should use FFT instead of ASM cause we could use the fourier lens between the cammera and the SLM.

Anyway, I implement the GS algorithm based on ASM with linear convolution, and I found that the result is much worse than the result based on ASM with circular convolution. So I am confused now!!!

My question might be annoying to ansower cause I am new to this field, and I am not a native English speaker. So thank you very much!!!!

[rafael-fuente]

The ASM doesn't seem to work very well with a direct Gerchberg-Saxton, since the generated phase mask usually have a resolution inferior to the pixel pitch or it simply doesn't converge to a decent solution.

Instead, I'm using other CGH methods when working with ASM. I uploaded an example in the repository of a CGH generated with ASM and Adam optimization algorithm: https://github.com/rafael-fuente/diffractsim/blob/main/examples/usaf_30cm_phase_hologram_generation_and_reconstruction.py