# Limiting Uncertainty Relations in Laser-Based Measurements of Position and Velocity Due to Quantum Shot Noise

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. State of the Art

#### 1.3. Aim and Structure of the Article

## 2. Application of the Cramér-Rao Inequality

#### 2.1. Entropic Uncertainty Principles

#### 2.2. Guide to the Expression of Uncertainty in Measurement

#### 2.3. Beyond the Classical CRB

#### 2.4. CRB for Signals in White Noise

## 3. Position Measurements

#### 3.1. Particle

#### 3.2. Surface

## 4. Displacement, Strain and Velocity Measurements

## 5. Conclusions and Outlook

## Funding

## Conflicts of Interest

## References

- Udem, T.; Holzwarth, R.; Hänsch, T.W. Optical frequency metrology. Nature
**2002**, 416, 233–237. [Google Scholar] [CrossRef] - Trocha, P.; Karpov, M.; Ganin, D.; Pfeiffer, M.H.P.; Kordts, A.; Wolf, S.; Krockenberger, J.; Marin-Palomo, P.; Weimann, C.; Randel, S.; et al. Ultrafast optical ranging using microresonator soliton frequency combs. Science
**2018**, 359, 887–891. [Google Scholar] [CrossRef] [Green Version] - Hell, S.W.; Wichmann, J. Breaking the diffraction resolution limit by stimulated emission: Stimulated- emission-depletion fluorescence microscopy. Opt. Lett.
**1994**, 19, 780–782. [Google Scholar] [CrossRef] - Hao, X. From microscopy to nanoscopy via visible light. Light Sci. Appl.
**2013**, 2, e108. [Google Scholar] [CrossRef] - Aasi, J.; Abadie, J.; Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photonics
**2013**, 7, 613–619. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Berkovic, G.; Shafir, E. Optical methods for distance and displacement measurements. Adv. Opt. Photonics
**2012**, 4, 441–471. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Tu, C.; Yin, Z.; Lin, J.; Bao, F.A. Review of Experimental Techniques for Measuring Micro- to Nano-Particle- Laden Gas Flows. Appl. Sci.
**2017**, 7, 120. [Google Scholar] [CrossRef] - Fischer, A. Imaging flow velocimetry with laser Mie scattering. Appl. Sci.
**2017**, 7, 1298. [Google Scholar] [CrossRef] - Woisetschläger, J.; Göttlich, E. Recent Applications of Particle Image Velocimetry to Flow Research in Thermal Turbomachinery. In Particle Image Velocimetry; Schröder, A., Willert, C.E., Eds.; Springer: Berlin, Germany, 2007; pp. 311–331. [Google Scholar]
- Fischer, A.; König, J.; Czarske, J.; Rakenius, C.; Schmid, G.; Schiffer, H.P. Investigation of the tip leakage flow at turbine rotor blades with squealer. Exp. Fluids
**2013**, 54, 1462. [Google Scholar] [CrossRef] - Candel, S.; Durox, D.; Schuller, T.; Bourgouin, J.F.; Moeck, J.P. Dynamics of Swirling Flames. Annu. Rev. Fluid Mech.
**2014**, 46, 147–173. [Google Scholar] [CrossRef] - Schlüßler, R.; Bermuske, M.; Czarske, J.; Fischer, A. Simultaneous three-component velocity measurements in a swirl-stabilized flame. Exp. Fluids
**2015**, 56, 183. [Google Scholar] [CrossRef] - Fansler, T.D.; Parrish, S.E. Spray measurement technology: A review. Meas. Sci. Technol.
**2015**, 26, 012002. [Google Scholar] [CrossRef] - Gürtler, J.; Schlüßler, R.; Fischer, A.; Czarske, J. High-speed non-intrusive measurements of fuel velocity fields at high-pressure injectors. Opt. Lasers Eng.
**2017**, 90, 91–100. [Google Scholar] [CrossRef] - Kentischer, T.J.; Schmidt, W.; Sigwarth, M.; Uexkuell, M.V. TESOS, a double Fabry-Perot instrument for solar spectroscopy. Astron. Astrophys.
**1998**, 340, 569–578. [Google Scholar] - Werely, S.T.; Meinhardt, C.D. Recent Advances in Micro-Particle Image Velocimetry. Annu. Rev. Fluid Mech.
**2010**, 42, 557–576. [Google Scholar] [CrossRef] - Li, C.H.; Benedick, A.J.; Fendel, P.; Glenday, A.G.; Kärtner, F.X.; Phillips, D.F.; Sasselov, D.; Szentgyorgyi, A.; Walsworth, R.L. A laser frequency comb that enables radial velocity measurements with a precision of 1 cm s
^{−1}. Nature**2008**, 452, 610–612. [Google Scholar] [CrossRef] - Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys.
**1927**, 43, 172–198. [Google Scholar] [CrossRef] - Kennard, E.H. Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys.
**1927**, 44, 326–352. [Google Scholar] [CrossRef] - Saleh, B.E.A.; Teich, M.C. Fundamentals of Photonics; John Wiley & Sons: New York, NY, USA, 2007. [Google Scholar]
- Teich, M.C.; Saleh, B.Q.A. Squeezed states of light. Quantum Opt.
**1989**, 1, 153–191. [Google Scholar] [CrossRef] - Fischer, A.; Czarske, J. Measurement uncertainty limit analysis with the Cramér-Rao bound in case of biased estimators. Measurement
**2014**, 54, 77–82. [Google Scholar] [CrossRef] - Schervish, M.J. Theory of Statistics; Springer: Berlin, Germany, 1997. [Google Scholar]
- Kay, S.M. Fundamentals of Statistical Signal Processing; Prentice Hall: Upper Saddle River, NJ, USA, 1993. [Google Scholar]
- Casella, G.; Berger, R.L. Statistical Inference; Duxbury Press: Belmont, CA, USA, 1990. [Google Scholar]
- Rao, C.R. Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc.
**1945**, 37, 81–91. [Google Scholar] - Cramér, H. Mathematical Methods of Statistics; Princeton University Press: Princeton, NJ, USA, 1946. [Google Scholar]
- Tripathi, G. A matrix extension of the Cauchy-Schwarz inequality. Econ. Lett.
**1999**, 63, 1–3. [Google Scholar] [CrossRef] - Fischer, A.; Czarske, J. Measurement time dependency of asymptotic Cramér-Rao bound for an unknown constant in stationary Gaussian noise. Measurement
**2015**, 68, 182–188. [Google Scholar] [CrossRef] - Joint Committee for Guides in Metrology (JCGM). JCGM 100:2008 Evaluation of Measurement Data—Guide to the Expression Of Uncertainty in Measurement. 2008. Available online: https://www.bipm.org/en/publications/guides/ (accessed on 7 March 2019).
- Fischer, A. Fisher information and Cramér-Rao bound for unknown systematic errors. Measurement
**2018**, 113, 131–136. [Google Scholar] [CrossRef] - Stoica, P. On biased estimators and the unbiased Cramér bound lower bound. Signal Process.
**1990**, 21, 349–350. [Google Scholar] [CrossRef] - Hero, A.O.; Fessler, J.A.; Usman, M. Exploring estimator bias-variance tradeoffs using the uniform CR bound. IEEE Trans. Signal Process.
**1996**, 44, 2026–2041. [Google Scholar] [CrossRef] [Green Version] - Eldar, Y.C. Minimum Variance in Biased Estimation: Bounds and Asymptotically Optimal Estimators. IEEE Trans. Signal Process.
**2004**, 22, 1915–1930. [Google Scholar] [CrossRef] - Eldar, Y.C. Uniformly Improving the Cramér-Rao Bound and Maximum-Likelihood Estimation. IEEE Trans. Signal Process.
**2006**, 54, 2943–2956. [Google Scholar] [CrossRef] - Eldar, Y.C. Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér-Rao Bound. Found. Trends Signal Process.
**2007**, 1, 305–449. [Google Scholar] [CrossRef] - Eldar, Y.C. MSE Bounds With Affine Bias Dominating the Cramér-Rao Bound. IEEE Trans. Signal Process.
**2008**, 56, 3824–3836. [Google Scholar] [CrossRef] - Kay, S.; Eldar, Y.C. Rethinking Biased Estimation [Lecture Notes]. IEEE Signal Process. Mag.
**2008**, 25, 133–136. [Google Scholar] [CrossRef] - Fréchet, M. Sur l’extension de cecertain évaluations statistiques au cas de petits échantillons. Rev. Int. Stat. Inst.
**1943**, 11, 182–205. [Google Scholar] [CrossRef] - Darmois, G. Sur les limites de la dispersion de certaines estimations. Rev. Int. Stat. Inst.
**1945**, 13, 9–15. [Google Scholar] [CrossRef] - Cramér, H. A contribution to the theory of statistical estimation. Scand. Actuar. J.
**1946**, 1946, 85–94. [Google Scholar] [CrossRef] - Wijsman, R.A. On the attainment of the Cramér-Rao lower bound. Ann. Stat.
**1973**, 1, 538–542. [Google Scholar] [CrossRef] - Zeira, A.; Nehorai, A. Frequency Domain Cramer-Rao Bound for Gaussian Processes. IEEE Trans. Acoust. Speech Signal Process.
**1990**, 38, 1063–1066. [Google Scholar] [CrossRef] - Bhattachyryya, A. On some analogues to the amount of information and their uses in statistical estimation. Synkhya
**1946**, 8, 1–14. [Google Scholar] - Wolfowitz, J. The Efficiency of Sequential Estimates and Wald’s Equation for Sequential Processes. Ann. Math. Stat.
**1947**, 18, 165–308. [Google Scholar] [CrossRef] - Gosh, J.K.; Purkayastha, S. Sequential Cramér-Rao and Bhattacharyya Bounds: Work of G. R. Seth and Afterwards. J. Indian Soc. Agric. Stat.
**2010**, 64, 137–144. [Google Scholar] - Barankin, E.W. Locally Best Unbiased Estimates. Ann. Math. Stat.
**1949**, 20, 477–500. [Google Scholar] [CrossRef] - McAulay, R.J.; Hofstetter, E.M. Barankin Bounds on Parameter Estimation. IEEE Trans. Inf. Theory
**1971**, 17, 669–676. [Google Scholar] [CrossRef] - Chapman, D.G.; Robbins, H. Minimum variance estimation without regularity assumptions. Ann. Math. Stat.
**1951**, 22, 581–586. [Google Scholar] [CrossRef] - Kiefer, J. On Minimum Variance Estimators. Ann. Math. Stat.
**1952**, 23, 493–655. [Google Scholar] [CrossRef] - Fraser, D.A.S.; Guttman, I. Bhattacharyya without regularity assumptions. Ann. Math. Stat.
**1952**, 23, 493–655. [Google Scholar] [CrossRef] - Kullback, S.; Leibler, R.A. On Information and Sufficiency. Ann. Math. Stat.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Kullback, S. Certain Inequalities in infomation theory and the Cramér-Rao inequalitity. Ann. Math. Stat.
**1954**, 25, 745–751. [Google Scholar] [CrossRef] - Blyth, C.R.; Roberts, D.M. On inequalitites of Cramér-Rao type and admissibility proofs. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Theory of Statistics; University of California Press: Berkeley, CA, USA, 1972; pp. 17–30. [Google Scholar]
- Abel, J.S. A Bound on Mean-Square-Estimate Error. IEEE Trans. Inf. Theory
**1993**, 39, 1675–1680. [Google Scholar] [CrossRef] - Arndt, C. Information Measures: Information And Its Description In Science And Engineering (Signals and Communication Technology); Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Gart, J.J. An Extension of the Cramer-Rao Inequality. Ann. Math. Stat.
**1959**, 30, 271–640. [Google Scholar] [CrossRef] - Simonov, A.N. Cramer-Rao bounds in functional form: theory and application to passive optical ranging. J. Opt. Soc. Am. A
**2014**, 31, 2680–2693. [Google Scholar] [CrossRef] - Wernet, M.P.; Pline, A. Particle displacement tracking technique and Cramer-Rao lower bound error in centroid estimates from CCD imagery. Exp. Fluids
**1993**, 15, 295–307. [Google Scholar] [CrossRef] - Fischer, A. Messbarkeitsgrenzen Optischer Strömungsmessverfahren: Theorie und Anwendungen. In Dresdner Berichte zur Messsystemtechnik; Shaker: Aachen, Germany, 2013; Volume 8. [Google Scholar]
- Westerweel, J. Fundamentals of digital particle image velocimetry. Meas. Sci. Technol.
**1997**, 8, 1379–1392. [Google Scholar] [CrossRef] - Westerweel, J. Theoretical analysis of the measurement precision in particle image velocimetry. Exp. Fluids
**2000**, 29, S3–S12. [Google Scholar] [CrossRef] - Falconi, O. Maximum Sensitivities of Optical Direction and Twist Measuring Instruments. J. Opt. Soc. Am.
**1964**, 54, 1315–1320. [Google Scholar] [CrossRef] - Lindegren, L. Photoelectric Astrometry—A Comparison of Methods for Precise Image Location. In Modern Astrometry; Institut für Astronomie (Universitäts-Sternwarte Wien): Vienna, Austria, 1978; pp. 197–217. [Google Scholar]
- Lindegren, L. Observing Photons in Space. In ISSI Scientific Report Series: High-Accuracy Positioning: Astrometry; Springer: New York, NY, USA, 2013; Volume 9, pp. 299–311. [Google Scholar]
- Fischer, A. Fundamental uncertainty limit for speckle displacement measurements. Appl. Opt.
**2017**, 56, 7013–7019. [Google Scholar] [CrossRef] [PubMed] - Born, M.; Wolf, E. Principles of Optics; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Willert, C. Stereoscopic digital particle image velocimetry for application in wind tunnel flows. Meas. Sci. Technol.
**1997**, 8, 1465–1479. [Google Scholar] [CrossRef] - Prasad, A.K. Stereoscopic particle image velocimetry. Exp. Fluids
**2000**, 29, 103–116. [Google Scholar] [CrossRef] - Goodman, J.W. Speckle Phenomena in Optics; Roberts and Company: Greenwood Village, CO, USA, 2007. [Google Scholar]
- Tausendfreund, A.; Alexe, G.; Stöbener, D.; Fischer, A. Application limits of digital speckle photography for in-process measurements in manufacturing processes. In Proceedings of the 2018 European Optical Society Biennial Meeting (EOSAM), Delft, The Netherlands, 8–12 October 2018; pp. 255–256. [Google Scholar]
- Chao, J.; Ward, E.S.; Ober, R.J. Fisher information theory for parameter estimation in single molecule microscopy: tutorial. J. Opt. Soc. Am. A
**2016**, 33, B36–B57. [Google Scholar] [CrossRef] - Dorsch, R.G.; Häusler, D.; Herrmann, J.M. Laser triangulation: Fundamental uncertainty in distance measurement. Appl. Opt.
**1994**, 33, 1306–1314. [Google Scholar] [CrossRef] - Pavliček, P.; Hýbl, O. White-light interferometry on rough surfaces—Measurement uncertainty caused by noise. Appl. Opt.
**2012**, 51, 465–473. [Google Scholar] [CrossRef] - Häusler, G. Encyclopedia of Modern Optics; Elsevier: Oxford, UK, 2005; Volume 1, pp. 114–123. [Google Scholar]
- Ingelstam, E. Problems Related to the Accurate Interpretation of Microinterferograms; Interferometry, National Physical Laboratory Symposium No. 11; Her Majesty’s Stationery Office: London, UK, 1960; pp. 141–163. [Google Scholar]
- Pavliček, P.; Häusler, G. Methods for optical shape measurements and their measurement uncertainty. Int. J. Optomech.
**2014**, 8, 292–303. [Google Scholar] [CrossRef] - Pavliček, P.; Pech, M. Shot noise limit of the optical 3D measurement methods for smooth surfaces. Meas. Sci. Technol.
**2016**, 27, 035205. [Google Scholar] [CrossRef] - Tausendfreund, A.; Stöbener, D.; Fischer, A. Precise In-Process Strain Measurements for the Investigation of Surface Modification Mechanisms. J. Manuf. Mater. Process.
**2018**, 2, 9. [Google Scholar] [CrossRef] - Eman, J.; Sundin, K.G.; Oldenburg, M. Spatially resolved observations of strain fields at necking and fracture of anisotropic hardened steel sheet material. Int. J. Solids Struct.
**2009**, 46, 2750–2756. [Google Scholar] [CrossRef] [Green Version] - Tausendfreund, A.; Borchers, F.; Kohls, E.; Kuschel, S.; Stöbener, D.; Heinzel, C.; Fischer, A. Investigations on material loads during grinding by speckle photography. J. Manuf. Mater. Process.
**2018**, 2, 71. [Google Scholar] [CrossRef] - Adrian, R.J. Particle-Imaging Techniques for Experimental Fluid Mechanics. Annu. Rev. Fluid Mech.
**1991**, 23, 261–304. [Google Scholar] [CrossRef] - Maas, H.G.; Gruen, A.; Papantoniou, D. Particle tracking velocimetry in three-dimensional flows—Part 1: Photogrammetric determination of particle coordinates. Exp. Fluids
**1993**, 15, 133–146. [Google Scholar] [CrossRef] - Adrian, R.J. Scattering particle characteristics and their effect on pulsed laser measurements of fluid flow: Speckle velocimetry vs. particle image velocimetry. Appl. Opt.
**1984**, 23, 1690–1691. [Google Scholar] [CrossRef] - Adrian, R.J. Twenty years of particle image velocimetry. Exp. Fluids
**2005**, 39, 159–169. [Google Scholar] [CrossRef] [Green Version] - Thompson, D.H. A tracer-particle fluid velocity meter incorporating a laser. J. Phys. E Sci. Instrum.
**1968**, 1, 929–932. [Google Scholar] [CrossRef] - Tanner, L. A particle timing laser velocity meter. Opt. Laser Technol.
**1973**, 5, 108–110. [Google Scholar] [CrossRef] - Ator, J.T. Image-velocity sensing with parallel-slit reticles. J. Opt. Soc. Am.
**1963**, 53, 1416–1419. [Google Scholar] [CrossRef] - Aizu, Y.; Asakura, T. Principles and development of spatial filtering velocimetry. Appl. Phys. B Lasers Opt.
**1987**, 43, 209–224. [Google Scholar] [CrossRef] - Oliver, C.J. Accuracy in laser anemometry. J. Phys. D Appl. Phys.
**1980**, 13, 1145–1159. [Google Scholar] [CrossRef] - Lading, L. Estimating time and time-lag in time-of-flight velocimetry. Appl. Opt.
**1983**, 22, 3637–3643. [Google Scholar] [CrossRef] [PubMed] - Lading, L.; Jørgensen, T.M. Maximizing the information transfer in a quantum-limited light-scattering system. J. Opt. Soc. Am. A
**1990**, 7, 1324–1331. [Google Scholar] [CrossRef] - Fischer, A.; Pfister, T.; Czarske, J. Derivation and comparison of fundamental uncertainty limits for laser-two-focus velocimetry, laser Doppler anemometry and Doppler global velocimetry. Measurement
**2010**, 43, 1556–1574. [Google Scholar] [CrossRef] - Fischer, A. Fundamental uncertainty limit of optical flow velocimetry according to Heisenberg’s uncertainty principle. Appl. Opt.
**2016**, 55, 8787–8795. [Google Scholar] [CrossRef] - Yeh, Y.; Cummins, H.Z. Localized Fluid Flow Measurements with an He-Ne Laser Spectrometer. Appl. Phys. Lett.
**1964**, 4, 176–178. [Google Scholar] [CrossRef] - Tropea, C. Laser Doppler anemometry: Recent developments and future challenges. Meas. Sci. Technol.
**1995**, 6, 605–619. [Google Scholar] [CrossRef] - Rife, D.C.; Boorstyn, R.R. Single-tone parameter estimation from discrete-time observations. IEEE Trans. Inf. Theory
**1974**, IT-20, 591–598. [Google Scholar] [CrossRef] - Besson, O.; Galtier, F. Estimating Particles Velocity from Laser Measurements: Maximum Likelihood and Cramér-Rao Bounds. IEEE Trans. Signal Process.
**1996**, 12, 3056–3068. [Google Scholar] [CrossRef] - Shu, W.Q. Cramér-Rao Bound of Laser Doppler Anemometer. IEEE Trans. Instrum. Meas.
**2001**, 50, 1770–1772. [Google Scholar] - Sobolev, V.S.; Feshenko, A.A. Accurate Cramer-Rao Bounds for a Laser Doppler Anemometer. IEEE Trans. Instrum. Meas.
**2006**, 55, 659–665. [Google Scholar] [CrossRef] - Meyers, J.F. Development of Doppler global velocimetry as a flow diagnostic tool. Meas. Sci. Technol.
**1995**, 6, 769–783. [Google Scholar] [CrossRef] - Charrett, T.O.H.; Ford, H.D.; Nobes, D.S.; Tatam, R.P. Two-Frequency Planar Doppler Velocimetry (2-ν-PDV). Rev. Sci. Instrum.
**2004**, 75, 4487–4496. [Google Scholar] [CrossRef] - Fischer, A.; Büttner, L.; Czarske, J.; Eggert, M.; Grosche, G.; Müller, H. Investigation of time-resolved single detector Doppler global velocimetry using sinusoidal laser frequency modulation. Meas. Sci. Technol.
**2007**, 18, 2529–2545. [Google Scholar] [CrossRef] - Landolt, A.; Rösgen, T. Global Doppler frequency shift detection with near-resonant interferometry. Exp. Fluids
**2009**, 47, 733–743. [Google Scholar] [CrossRef] - Lu, Z.H.; Charett, T.O.H.; Tatam, R.P. Three-component planar velocity measurements using Mach-Zehnder interferometric filter-based planar Doppler velocimetry (MZI-PDV). Meas. Sci. Technol.
**2009**, 20, 034019. [Google Scholar] [CrossRef] - McKenzie, R.L. Measurement capabilities of planar Doppler velocimetry using pulsed lasers. Appl. Opt.
**1996**, 35, 948–964. [Google Scholar] [CrossRef] - Fischer, A.; Büttner, L.; Czarske, J.; Eggert, M.; Müller, H. Measurement uncertainty and temporal resolution of Doppler global velocimetry using laser frequency modulation. Appl. Opt.
**2008**, 47, 3941–3953. [Google Scholar] [CrossRef] [PubMed] - Fischer, A.; Czarske, J. Signal processing efficiency of Doppler global velocimetry with laser frequency modulation. Optik
**2010**, 121, 1891–1899. [Google Scholar] [CrossRef] - Fischer, A. Model-based review of Doppler global velocimetry techniques with laser frequency modulation. Opt. Lasers Eng.
**2017**, 93, 19–35. [Google Scholar] [CrossRef] - Pfister, T.; Fischer, A.; Czarske, J. Cramér-Rao lower bound of laser Doppler measurements at moving rough surfaces. Meas. Sci. Technol.
**2011**, 22, 055301. [Google Scholar] [CrossRef]

**Figure 1.**Measurement arrangement to illustrate the (u,v)-coordinates in the image plane, the (x,y)-coordinates in the object plane as well as the distances ${z}_{\mathrm{i}}$, ${z}_{\mathrm{o}}$ to define the absolute value of the image magnification $M=\frac{u}{x}=\frac{{z}_{\mathrm{i}}}{{z}_{\mathrm{o}}}$.

**Figure 2.**Normalized CRB for (

**a**) WPN and (

**b**) AWGN of the lateral particle position x as a function of the normalized particle image radius $(d/2)/{w}_{\mathrm{pixel}}$ ($1/{\mathrm{e}}^{2}$ intensity radius) and for the particle positions $\left(\right)open="("\; close=")">\frac{x}{{w}_{\mathrm{pixel}}/M},\frac{y}{{w}_{\mathrm{pixel}}/M}$ where $(0,0)$ is the pixel center. The legends indicate the normalized x-values.

**Figure 3.**Symmetric measurement arrangement to measure the 3d particle position with a stereoscopic approach (triangulation). Please note that the y-position, which is perpendicular to the z- and the x-axis, is neglected here.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fischer, A.
Limiting Uncertainty Relations in Laser-Based Measurements of Position and Velocity Due to Quantum Shot Noise. *Entropy* **2019**, *21*, 264.
https://doi.org/10.3390/e21030264

**AMA Style**

Fischer A.
Limiting Uncertainty Relations in Laser-Based Measurements of Position and Velocity Due to Quantum Shot Noise. *Entropy*. 2019; 21(3):264.
https://doi.org/10.3390/e21030264

**Chicago/Turabian Style**

Fischer, Andreas.
2019. "Limiting Uncertainty Relations in Laser-Based Measurements of Position and Velocity Due to Quantum Shot Noise" *Entropy* 21, no. 3: 264.
https://doi.org/10.3390/e21030264