Abstract: 4Pi microscopy has been invented by Stefan Hell to improve the resolution of confocal fluorescence microscopy along the optical axis. The convolution kernel (or point spread function, short psf) of a 4Pi microscope differs from the psf of a corresponding confocal microscope approximately by a squared cosine factor resulting in a main peak of much smaller band-width and several (typically at least two) side lobes. The size and position of the main peak and the side lobes of the 4Pi psf depend on a phase parameter, which has been assumed to be space-invariant so far, yielding standard deconvolution problems with positivity constraint. However, this assumption is violated e.g. for inhomogeneous refractive indices, and the space dependence of the psf is considered one of the main problems in 4Pi microscopy. In this project the joint recovery of the three-dimensional density of fluorescent markers and the slowly varying phase function is considered as a nonlinear inverse problem, which is tackled by a regularized Newton method. We study the convergence of an iteratively regularized Gauss-Newton method incorporating nonnegativity constraints and present reconstruction results both for synthetic and real data.