The results of a near-infrared (J H K LP) imaging linear polarimetry survey of 20 young stellar objects (YSOs) in rho Ophiuchi are presented. The majority of the sources are unresolved, with K-band polarizations, P_K < 6 per cent. Several objects are associated with extended reflection nebulae. These objects have centrosymmetric vector patterns with polarization discs over their cores; maximum polarizations of P_K > 20 per cent are seen over their envelopes. Correlations are observed between the degree of core polarization and the evolutionary status inferred from the spectral energy distribution. K-band core polarizations >6 per cent are only observed in Class I YSOs. A 3D Monte Carlo model with oblate grains aligned with a magnetic field is used to investigate the flux distributions and polarization structures of three of the rho Oph YSOs with extended nebulae. A rho proportional to r^(-1.5) power law for the density is applied throughout the envelopes. The large-scale centrosymmetric polarization structures are due to scattering. However, the polarization structure in the bright core of the nebula appears to require dichroic extinction by aligned non-spherical dust grains. The position angle indicates a toroidal magnetic field in the inner part of the envelope. Since the measured polarizations attributed to dichroic extinction are usually <10 per cent, the grains must either be nearly spherical or very weakly aligned. The higher polarizations observed in the outer parts of the reflection nebulae require that the dust grains responsible for scattering have maximum grain sizes <=1.05 microns.
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