Phaseless inverse uniqueness of a three-dimensional scattering problem of second type

In this paper we discuss the phaseless inverse scattering problem in mathematical physics. We measure only the intensity of scattered wave field in far field without phase information. The modulus of the scattered wave field is an analytic function in complex plane. As the parameter of certain analytic function, the traveling time of the scattered wave field is the spectral invariant that controls the behavior of the complex-valued function. Given two sets of identical point-to-point traveling times, we compare the asymptotic behaviors of scattered wave fields in complex plane. Then, we can deduce an inverse uniqueness on the index of refraction from the inverse Radon transform in each 2-dimensional cross section.