Phaseless inverse scattering and the phase problem in optics

Two related problems are considered: (i) the inverse scattering problem for a potential V(x) supported on the half-line {x≥0}, when the given data is ‖R−(k)‖, the amplitude of the reflection coefficient and (ii) determination of a function g(t) supported on the half-line {t≥0} when the given data is ‖ĝ(k)‖, the amplitude of the Fourier transform of g. Under certain conditions on V or g, uniqueness theorems are proved and computational methods are developed. A numerical example of recovery of V(x) from ‖R−(k)‖ is given.