An arrangement is proposed in which the beam in a microscope traverses the object more than once. This results in the image of a single point for two passes through the object being 2·4 times as sharp as that in a conventional microscope, the side lobes also being extremely small. In the microscope in which the beam passes through the object twice the image amplitude behaves similarly to the image intensity in a conventional partially coherent microscope. Theoretical images of various objects are calculated, and the effects of using annular lenses discussed.