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Fresnel edge fringes observed in a lensless point projection field-emission electron microscope operating at 90 eV have been studied and found to be formally equivalent to the fringes observed in transmission electron microscopy (TEM) under weak scattering conditions at the edge of an opaque object. The tip-to-spectrum distance z1 plays the role of the objective lens defocus setting Δf in conventional TEM. The image magnification, effective source size, transverse coherence width, instrumental resolution, and source brightness are all obtained from an analysis of the fringe spacings and intensity. The quantum mechanical upper limit on source brightness, as well as relationships among beam brightness, coherence parameters, and degeneracy, are discussed, and the degeneracy measured from experimental Fresnel fringes.

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Dark matter comprising a Bose-Einstein condensate (BEC) forms structures of the size of its coherence length, as determined by equilibrium between quantum pressure and gravitational attraction. This is also the core size of quantum vortices in a BEC superfluid. From the density and rotation curve of the Andromeda Galaxy (M31) we estimate the particle mass, particle density, coherence length, critical temperature, critical angular frequency, and vortex line density of the dark matter condensate composing the halo.