  We analyse the response of a spatially extended direction-dependent local
quantum system, a detector, moving on the Rindler trajectory of uniform linear
acceleration in Minkowski spacetime, and coupled linearly to a quantum scalar
field. We consider two spatial profiles: (i) a profile defined in the
Fermi-Walker frame of an arbitrarily-accelerating trajectory, generalising the
isotropic Lorentz-function profile introduced by Schlicht to include
directional dependence; and (ii) a profile defined only for a Rindler
trajectory, utilising the associated frame, and confined to a Rindler wedge,
but again allowing arbitrary directional dependence. For (i), we find that the
transition rate on a Rindler trajectory is non-thermal, and dependent on the
direction, but thermality is restored in the low and high frequency regimes,
with a direction-dependent temperature, and also in the regime of high
acceleration compared with the detector's inverse size. For (ii), the
transition rate is isotropic, and thermal in the usual Unruh temperature. We
attribute the non-thermality and anisotropy found in (i) to the leaking of the
Lorentz-function profile past the Rindler horizon.
