# Imaging Flow Velocimetry with Laser Mie Scattering
## 4 Fundamental Measurement Limits
### Photon Shot Noise
For PTV, the factor \(c_{4}\) is the reciprocal of the particle displacement \(w_{\mathrm{px}}\) between the two light pulse in unit pixels and can also be understood as a measure of the ratio between an optimal size of the particle image and the particle displacement. For the typical setting \(w_{\mathrm{px}}\approx 10\), it is \(c_{4}\approx 0.1\). As a result, the uncertainty limits for LDA, L2F and PTV are equal and usually lower than for DGV. Assuming a spatial resolution of \(w=100\,\upmu\)m and an averaged scattered light power of \(0.1\,\)nW with the wavelength \(\lambda=532\,\)nm, i.e., an average photon rate \(\dot{N}_{\mathrm{photon}}=2.7\times 10^{8}\,\)s\({}^{-1}\), the measurement uncertainty limit \(u(v)\) over the velocity is shown in Figure 9a for all four measurement techniques. Note that the same average photon rate and the same spatial resolution are assumed for the different measurement techniques. The same spatial resolution in all dimensions means for L2F that two light sheets instead of two light beams are considered. When using light beams, the average photon rate, as well as the spatial resolution perpendicular to the sensitivity direction is usually higher for L2F than for LDA, DGV and PTV. The flow velocity measurement can be performed for multiple scattering particles, which pass the measurement volume during a given measurement time \(T\) (temporal resolution). The measurement result is then the mean value of the particle velocity. Considering a measurement time \(T\) that is significantly larger than the transit time \(T_{\mathrm{t}}\) of a single particle through the measurement volume,  
Figure 9: Minimal achievable measurement uncertainty due to photon shot noise over the flow velocity \(v\) for an average photon rate per particle of \(\dot{N}_{\mathrm{photon}}=2.7\times 10^{8}\,\)s\({}^{-1}\) (**a**) for a single particle (that passes the measurement volume with the respective dimension \(w=100\,\upmu\)m) and (**b**) for multiple particles (that occur during the measurement time \(T=0.1\,\)s). The formulas are from Table 2 and \(c_{1}=0.9\), \(c_{2}=c_{3}=c_{4}=0.1\), \(M=1\), \(\lambda=532\,\)nm, \(\tau^{\prime}=2\,\)GHz\({}^{-1}\). the maximal number of measured particles is \(T/T_{\rm t}=T/(w/v)\). Hence, the measurement uncertainty is reduced by the factor \(\frac{1}{\sqrt{T/(w/v)}}\). The resulting uncertainty limit \(u(\tilde{v})\) is given in the second column in Table 2. In addition, DGV is not restricted to having one single particle in the measurement volume. The higher the number \(M\) of scattering particles in the measurement volume, the larger is the total number of photons that are scattered and observed. Indeed, DGV is known to require a higher seeding particle concentration than other techniques. In order to take this effect into account, \(M\) is included in the respective equation and \(M=100\) is considered in the numerical example. The resulting measurement uncertainty limit for DGV is independent of the particle velocity, whereas the measurement uncertainty limit for LDA, L2F and PTV is directly proportional to the particle velocity, cf. Figure 9b for a temporal resolution of \(T=0.1\,\mathrm{ms}\). Below \(239\,\mathrm{m/s}\), the minimal achievable measurement uncertainty \(u(\tilde{v})\) due to photon shot noise for DGV is higher than for LDA, L2F and PTV. For the example of an averaged scattered light power of \(0.1\,\mathrm{nW}\), the measurement uncertainty limit is \(0.15\,\mathrm{m/s}\) for DGV and \(0.06\%\) for LDA, L2F and PTV, respectively.