We have constructed a correlation microscope based on the Mirau interferometer configuration using a thin silicon nitride film beam splitter. This microscope provides the amplitude and phase information for the reflected signal from a sample located on the microscope-object plane. The device is remarkably insensitive to vibrations and is self-correcting for spherical and chromatic range aberrations of the objective. An imaging theory for the correlation microscope has been derived, which predicts accurately both the transverse resolution at a sharp edge and the range resolution for a perfect plane reflector. The range resolution is slightly better than that for a scanning optical microscope using a lens with the same aperture.