"Symmetrical iterative Fourier-transform algorithm using both phase and amplitude freedoms", by Jinsong Liu, A. J. Caley and M. R. Taghizadeh, Optics Communications, , 267: 347-355, 2006.
A modified Gerchberg–Saxton algorithm, which we call symmetrical iterative Fourier transform algorithm, is presented and employed to a series of super-Gaussian beam shaping problems and arbitrary beam shaping problems. The main features of this algorithm include (1) using an amplitude-symmetric function of the failed result about the Fourier-domain constraint; (2) using both phase and amplitude freedoms in the iteration process, i.e. the noise produced outside the signal window by each iteration is kept. A formula is given to determine a critical constant in the search for the amplitude-symmetric function. The new method results in highly efficient solutions with highly precise beam profiles in the signal window for all the test problems.
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