  Spaceborne gravitational wave detection mission has a demanding requirement
for the precision of displacement sensing, which is conducted by the
interaction between the laser field and test mass. However, due to the
roughness of the reflecting surface of the test mass, the displacement
measurement along the sensitive axis suffers a coupling error caused by the
residue motion of other degrees of freedom. In this article, we model the
coupling of the test mass residue random motion to the displacement sensing
along the sensitive axis and derived an analytical formula of the required
precision of the surface error for the spaceborne gravitational wave detectors.
Our result shows that this coupling error will not contaminate the picometer
displacement sensing for the test masses in the LISA pathfinder.
