This dissertation presents a new quantitative phase retrieval algorithm that fully models partially coherent imaging in microscopes. Unlike existing algorithms, our algorithm fully considers the pupil function and illumination by using the Weak Object Transfer Function (WOTF). Using an iterative approach, we extend the applicability of the WOTF beyond weakly scattering objects. This allows almost any measurement to be used during phase retrieval. As an example of how this feature can be used to invent practical new measurement schemes, we present the illumination switched pupil. This measurement uses a phase contrast objective and varied illumination to maximize the sensitivity of the microscope to both the phase and amplitude of the sample. Using only two images, the complex field can be recovered with high sensitivity at almost all spatial frequencies.

A complete model of imaging in the microscope enables self-calibration of the measurements and improved phase retrieval. Since all important characteristics of the microscope can be incorporated, an optimization over critical parameters, such as the best focus position and image alignment, can be performed after the images have been captured. This allows errors in the calibration to be corrected after the measurements have been performed, improving the accuracy of the recovered field while simplifying the experiments.

To verify and apply the algorithm experimentally, we have performed phase retrieval measurements of Extreme Ultraviolet (EUV) photomasks on the zone plate microscope, SHARP, at Lawrence Berkeley National Laboratory (LBNL). Phase retrieval has enabled the quantitative analysis of multilayer roughness and defects. Experiments, comparing the size of defects measured using phase retrieval to measurements performed by AFM, indicate that AFM consistently underestimates the effective height of the buried multilayer defects by 1 nm. Other measurements of defects, comparing the recovered field extracted from standard and from phase contrast images as well as measurements taken under varying illumination, showed consistent results and provide experimental evidence that the algorithm handles the pupil function and partial coherence correctly.