A variational principle for the scattering amplitude is applied to obtain a relationship, in the Fourier domain, between the scattering potential and the scattered field recorded over a measurement line outside the potential. This relationship leads to a novel iterative reconstruction algorithm for recovering the distribution of the index of refraction from measurements of the scattered field. The proposed technique uses and builds upon the well-knownfiltered backpropagationalgorithm of diffraction tomography. Computer simulations are presented that compare the reconstructions obtained with the proposed algorithm to the ones obtained with Born's approximation.