A formula for the image intensity of phase objects in Zernike mode

I present an analytical expression for the image intensity of a phase object visualized in Zernike phase contrast mode. The formula is valid for periodic and non-periodic weak and strong objects, and accounts for the effects of finite illumination. The expression provided is intended as a generalization of the standard reference formula given in the Born and Wolf [Principles of Optics, sixth ed., Pergamon Press, New York, 1980, p. 427] textbook as well as of the formalism employed to evaluate imaging doses in Zernike mode [M. Malac, M. Beleggia, R. Egerton, Y. Zhu, Ultramicroscopy 108 (2008) 126]. I illustrate the usefulness of the improved expression by means of three examples: a sinusoidal phase grating, a Gaussian object, and a phase step. The optimal Zernike phase angle yielding maximum image contrast is found to be object-dependent and not necessarily equal to π/2π/2. Phase plate optimization criteria are derived and presented for two of the examples considered.