Astigmatic intensity equation for electron microscopy based phase retrieval

Phase retrieval, in principle, can be performed in a transmission electron microscope (TEM) using arbitrary aberrations of electron waves; provided that the aberrations are well-characterised and known. For example, the transport of intensity equation (TIE) can be used to infer the phase from a through-focus series of images. In this work an "astigmatic intensity equation" (AIE) is considered, which relates phase gradients to intensity variations caused by TEM objective lens focus and astigmatism variations. Within the paraxial approximation, it is shown that an exact solution of the AIE for the phase can be obtained using efficient Fourier transform methods. Experimental requirements for using the AIE are the measurement of a through-focus derivative and another intensity derivative, which is taken with respect to objective lens astigmatism variation. Two quasi-experimental investigations are conducted to test the validity of the solution.