The Fourier transform of a coherent optical image can be evaluated physically by use of a single lens plus free-space propagation, thereby providing the basis for the field of Fourier optics. I point out that one can similarly evaluate the discrete Fourier transform of a sampled or pixelated optical array physically by passing the discrete array amplitudes through a network of single-mode fibers or optical waveguides. A passive optical network that evaluates the fast Fourier transform of a coherent array can be fabricated by use of (N/2)log(2)[N] optical 3-dB couplers plus small added phase shifts. Implementing such networks in fiber or integrated optical form could provide the basis for a possible technology of fiber Fourier optics.