Analysis and simulation of the operation of a Kelvin probe

Abstract

Experiments that measure extremely small gravitational forces are often hampered by the presence of non-gravitational forces that can neither be calculated nor separately measured. Among these spurious forces is electrostatic attraction between a test mass and its surroundings due to the presence of spatially varying surface potential known as the "patch effect." In order to make surfaces with small surface potential variation, it is necessary to be able to measure it. A Kelvin probe (KP) measures contact potential difference (CPD), using the time-varying capacitance between the sample and a vibrating tip that is biased with a backing potential. Assuming that the tip remains constant, this measures the sample's surface potential variation. We examine the operation of the KP from the perspective of parameter estimation in the presence of noise. We show that, when the CPD is estimated from measurements at two separate backing potentials, the standard deviation of the optimal estimate depends on the total observing time. Further, the observing time may be unevenly divided between the two backing potentials, provided the values of those potentials are correspondingly set. We simulate a two-stage KP data analysis, including a sub-optimal estimator with advantages for real-time operation. Based on the real-time version, we present a novel approach to stabilizing the average distance of the tip from the sample. We also present the results of a series of covariance analyses that validate and bound the applicability of the suboptimal estimator, make a comparison with the results of an optimal estimator and guide the user. We discuss the application of the KP to the LISA and to a test of the weak equivalence principle.