We study the ensemble of linear polarization measurement in the optical afterglows of long-duration gamma-ray bursts. We assume a non sideways-expanding top-hat jet geometry and use the relatively large number of measurements under the assumption that they represent a statistically unbiased sample. This allows us to constrain the ratio between the maximum predicted polarization and the measured one, which is an indicator of the geometry of the magnetic field in the downstream region of the external shock. We find that the measured polarization is substantially suppressed with respect to the maximum possible for either a completely ordered magnetic field parallel to the shock normal or to a field that is entirely contained in the shock plane. The measured polarization is limited, on average, to between 25 and 30% of the maximum theoretically possible value. This reduction requires the perpendicular component of the magnetic field to be dominant in energy with respect to the component parallel to the shock front, as expected for a shock generated and/or shock compressed field. We find, however, that the data only marginally support the assumption of a simple top-hat jet, pointing towards a more complex geometry for the outflow.
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