We report on a new class of scanning optical microscopes with twice the resolution of conventional microscopes. The new microscopes use standard components to produce high intensity images, with large working distances and large depths of focus, without additional mechanical complexity. The microscopes scan a focused laser beam which has passed through a pattern of phase shifts arranged in a spiral staircase. These phase shifts generate a narrow dark spot in the center of the focused beam. The width of this central dark spot is about half the diffraction limit. The signal from this "phase shifted" beam is subtracted from the signal produced by a conventional Gaussian beam which is simultaneously scanned. The difference between the two detected signals corresponds to the signal from a bright beam of width equal to the width of the dark central area. Since only linear operations on observed optical signals are used, there is a useful improvement in resolution of about a factor of 2 This improves both the accuracy with which feature edges may be located, as well as the resolution of small features such as the corners of narrow lines.