Iterative phase retrieval from kinematic rocking curves in CBED patterns

In the two-beam limit, the intensity distribution in the dark field disc of a convergent beam electron diffraction (CBED) pattern represents a rocking curve mapped across the Bragg condition. For kinematic scattering from crystal planes which are bent or displaced within the illuminated column, the diffracted amplitude and phase is the Fourier transform of a phase function with an exponent proportional to the displacement normal to the planes. Recovery of the phase profile is formally equivalent to the one-dimensional phase retrieval problem for an object function with constant modulus within a compact support, given only the diffracted intensity distribution. In simulations using the error reduction algorithm, the computed solution always converged to the original phase profile. As a practical test, the asymmetric rocking curves diffracted from planes inclined to the surfaces in ion-thinned Si specimens were used as the diffraction constraint, with support width equal to the crystal thickness. The calculated displacement curve was S-shaped, interpreted as a dilation of 1% induced by Ar atoms implanted in the surface layers. The factors which limit the accuracy and spatial resolution for phase recovery along the beam direction are discussed.