We describe a quantitative method for measuring the phase of a propagating wave field in three dimensions by use of a scanning optical-fiber interferometer. Because phase modulation in the reference arm is exploited, this technique is insensitive to large variations in the intensity of the field being studied and is therefore highly suitable for measurement of phase within spatially confined optical beams. It uses only a single detector and is not reliant on lock-in electronics. The technique is applied to the measurement of the near field of a cleaved optical fiber and is shown to produce results in good agreement with theory.