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

On the noise effect of test mass surface roughness in spaceborne gravitational wave detectors