# Imaging Flow Velocimetry with Laser Mie Scattering
## 4 Fundamental Measurement Limits
### Photon Shot Noise  
Considering the typical illumination with narrow band laser light, the number of detected scattered photons exhibits a Poisson distribution [159]. This phenomenon is known as photon shot noise, which ultimately limits the achievable measurement uncertainty. As a characteristic behavior of shot noise limited measurements, the measurement uncertainty is indirectly proportional to the square root of the number of photons. Shot noise is an important fundamental measurement limit in particular for the measurement of flow velocity images with a high measurement rate, because the limited number of available photons or light energy, respectively, needs to be distributed over space and time. The minimal achievable measurement uncertainty due to photon shot noise is obtained from the square root of the Cramer-Rao bound [160; 161]. The Cramer-Rao bound is explained for instance in [162; 163]. As an alternative approach, [PERSON]'s uncertainty principle [164] can also be applied, which was recently demonstrated by [PERSON] for L2F and LDA [165]. In the following, LDA (difference method) and DGV (without laser frequency modulation) are considered as a representative Doppler technique frequency-based and amplitude-based signal evaluation, respectively. L2F and PTV are selected to represent time-of-flight techniques with time and space measurements, respectively. For DGV, the first error propagation calculation including photon shot noise is from [PERSON] in 1996 [138]. [PERSON] et al. investigated the Cramer-Rao bound due to photon shot noise for DGV with and without laser frequency modulation in 2008 and 2010 [73; 166; 167]. A comprehensive study of the shot noise limits of the different DGV techniques is contained in a topical contribution by [PERSON] [79]. Several studies covered the calculation of the Cramer-Rao bound for LDA with respect to additive white Gaussian noise with constant variance [168; 169; 170; 171]. An error propagation calculation for noise with a Poisson distribution was given by [PERSON] in 1980 [172] and the calculation of the Cramer-Rao bound followed by [PERSON] in 2010 and 2016 [165; 167]. Regarding L2F, an error propagation calculation for Poisson statistics was performed by [PERSON] in 1980 [172]. The calculation of the Cramer-Rao bound was demonstrated by [PERSON] and [PERSON] in 1983 and 1990 [173; 174]. A detailed derivation is also contained in [167] and [165]. The Cramer-Rao bound for the particle location measurement in PTV was investigated for photon shot noise by [PERSON] and [PERSON] in 1993 [175]. The calculation was extended by [PERSON] in 2013 and adapted to evaluate the Cramer-Rao bound for the velocity measurement [176]. It is mentioned for the sake of completeness that similar PIV error considerations were performed under the assumption of white Gaussian noise by [PERSON] in 1997 and 2000 [177; 178]. Regarding photon shot noise, the result of the Cramer-Rao bound for PTV is applicable for PIV measurements with a single scattering particle. The majority of the investigations concern a single measurement technique. A comparison of the minimal achievable uncertainty due to photon shot noise for L2F, LDA and DGV is contained in [167], which is based on the Cramer-Rao bound. The comparison was completed by including PTV in [176]. For L2F and LDA, both the Cramer-Rao bound and Heisenberg's uncertainty principle are applied to compare the minimal achievable measurement uncertainty [165]. In the following, the Cramer-Rao bounds for DGV, LDA, L2F and PTV are extracted from [79; 165; 176]. The measurement uncertainty limits \(u(v)\) for DGV, LDA, L2F and PTV, which follow from the square root of the Cramer-Rao bound, are summarized in Table 2.  
The results in the first column in Table 2 are valid for the case of evaluating the scattered light signal from a single scattering particle.