A tractor beam is a travelling wave that transports illuminated objects back to its source, opposite to the wave's direction of propagation, along its entire length. The requisite retrograde force arises when an object scatters the wave's momentum density downstream into the direction of propagation, and then recoils upstream by conservation of momentum. Achieving this condition imposes constraints on the structure of the wave, which we elucidate in the Rayleigh limit, when the wavelength

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... s the size of the object. Continuously propagation-invariant modes such Bessel beams do not satisfy these conditions at dipole order in the multipole expansion, and so cannot serve as general-purpose long-ranged tractor beams. Modes with discrete propagation invariance, however, can act as first-order tractor beams. We demonstrate this by introducing a class of minimal solenoidal waves together with a set of design criteria that distinguish tractor beams that pull objects from repulsor beams that push them.