Abstract
With the ongoing progress of optoelectronic components, laser-based measurement systems allow measurements of position as well as displacement, strain and velocity with unbeatable speed and low measurement uncertainty. The performance limit is often studied for a single measurement setup, but a fundamental comparison of different measurement principles with respect to the ultimate limit due to quantum shot noise is rare. For this purpose, the Cramér-Rao bound is described as a universal information theoretic tool to calculate the minimal achievable measurement uncertainty for different measurement techniques, and a review of the respective lower bounds for laser-based measurements of position, displacement, strain and velocity at particles and surfaces is presented. As a result, the calculated Cramér-Rao bounds of different measurement principles have similar forms for each measurand including an indirect proportionality with respect to the number of photons and, in case of the position measurement for instance, the wave number squared. Furthermore, an uncertainty principle between the position uncertainty and the wave vector uncertainty was identified, i.e., the measurement uncertainty is minimized by maximizing the wave vector uncertainty. Additionally, physically complementary measurement approaches such as interferometry and time-of-flight positions measurements as well as time-of-flight and Doppler particle velocity measurements are shown to attain the same fundamental limit. Since most of the laser-based measurements perform similar with respect to the quantum shot noise, the realized measurement systems behave differently only due to the available optoelectronic components for the concrete measurement task.
Highlights
The Cramér-Rao bound (CRB) is the inverse of the Fisher information and the Cramér-Rao inequality represents an entropic uncertainty relation
Yields the minimal achievable measurement uncertainty squared, the CRB is derived for light signals superposed by white Poissonian noise, and the specific CRB results for the different particle and surface position measurement techniques are reviewed
Many measurement techniques exhibit a similar or even the same CRB, which is indirectly proportional to the total number of detected photons and indirectly proportional to the wave number squared
Summary
Optical measurement systems use light to carry and transport information with the fastest possible speed—with light speed! to measure (or to quantify) means to compare with a unit, and for optical measurements the measurand is compared with units originating from photon characteristics, which enables fast and precise measurements (to a certain extent). Considering dimensional measurements with visible light, the reference is the photon wavelength from 380 nm to 750 nm that enables measurement resolutions in the micro- or nanometer range and even below. Optical position or distance measurements have driven science forward, e.g., using the frequency comb technique [1] (Nobel prize in physics 2005) allowing 100 million measurements per second with sub-millimeter resolution [2], using the stimulated emission depletion. Entropy 2019, 21, 264 technique [3] (Nobel prize in chemistry 2014) allowing microscopy with visible light below Abbe’s diffraction limit in the nanometer range [4] as well as using interferometry and squeezed light techniques [5] that enabled the detection of gravitational waves [6] (Nobel prize in physics 2017) by measuring distance variations in the order of attometer in milliseconds. The occurring question for laser-based measurements in particular regarding position (or distance) and velocity information is: Does a natural, fundamental limit of measurability exist and how to describe it?
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