Analytics of the Gaussian phase object

A Gaussian pure phase object can be expressed as an infinite series of complex Gaussians. In momentum representation, since the Fourier Transform of a Gaussian is another Gaussian, the object wave spectrum is also an infinite series of complex Gaussians. Multiplying by a transfer function that is at most quadratic in spatial frequencies, such as the Fresnel propagator, does not change the structure of the series, which can then be Fourier transformed back to real space analytically. This computational framework provides us with the opportunity of examining the dependence of the image intensity on various key parameters such as defocus distance and Zernike/Hilbert phase plate angles, for the purposes of optimizing contrast and providing guidelines for the design of phase plates for electrons.