When two superconducting contacts are made on either side of a mesoscopic normal wire, the electrical conductance is a periodic function of the phase difference between the superconductors. For this structure, the oscillation at zero voltage and zero temperature is a small mesoscopic effect, with an amplitude of order e 2 /h. In contrast, we predict that a finite bias voltage V will induce giant oscillations associated with the classical proximity effect. These are a finite fraction of the overall conductance, exhibit a maximum when eV equals the Thouless energy, and decrease at higher voltages. This effect may account for the largeamplitude oscillations measured in recent experiments by Petrashov et al.

We examine equivalent forms of the current-voltage relationship applicable to mesoscopic superconductors and use one of these to derive general conditions for the sign of the conductance in a four-probe measurement. For disordered systems, with well separated voltage probes, where the n o d-s t a t e conductance is positive, we predict that the sign of the 1ongiNdinal four-probe conductance of superconducting wires can be reversed by varying an applied magnetic field, and that at certain critical values of the field. the conductance passes through a singularity.

We analyse the electrical conductance G(φ) of a two-dimensional, phase-coherent structure in contact with two superconductors, which is known to be an oscillatory function of the phase difference φ between the superconductors. It is predicted that for a ballistic sample, the amplitude of oscillation will be enhanced by placing a normal barrier at the normalsuperconducting interface, and that by tuning the strength of the barrier, it can be made orders of magnitude greater than values observed in recent experiments. Giant oscillations can also be obtained without a barrier, provided that a crucial sum rule is broken. This can be achieved by disorder-induced normal scattering. In the absence of zero-phase inter-channel scattering, the conductance possesses a zero-phase minimum and a maximum at φ = π .

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