On a phase problem of high-resolution Fourier transform X-ray spectroscopy

A phase problem is discussed for high-resolution Fourier transform X-ray spectroscopy where the phase of the interferogram is missing. A numerical iterative method and an analytic logarithmic Hilbert transform method were tested for recovery of the missing phase information from the modulus of the interferogram. These methods were applied to measured data of the Si 14 6 0 back reflection and a calculated interferogram from an X-ray Fabry-Perot interferometer. The iterative method experienced an ambiguity of reconstruction; however, the ambiguity was relatively small and may be acceptable. The logarithmic Hilbert transform method gave a poorer reconstruction for the measured data owing to a lower signal-to-noise ratio, whereas it recovered the original spectrum from the calculated data without noise.