Uncertainty of Positioning and Displacement Measurements in Quantum and Thermal Regimes

We analyze the performance of position-sensing devices and of distance- or displacement-measuring instruments, and we find that the ultimate uncertainty at the quantum limit of detected signal is given by the same expression in all cases, namely, a characteristic length L_c divided by radicNph, the square root of the number of photons detected in the time interval of the measurement. We derive the expression of the length L_c for well-known position-sensing devices (the quadrant photodiode and the position-sensing detector) and for several measuring instruments (pulsed and sine-wave-modulated rangefinders, triangulation telemeter, laser interferometer, and the optical rule). We also extend the analysis of the uncertainty results to the thermal regime case of detection, i.e., when the detector dark current and preamplifier noises are dominant with respect to quantum noise.