# Imaging Flow Velocimetry with Laser Mie Scattering
## 4 Fundamental Measurement Limits
### Photon Shot Noise
In this case, the temporal resolution is determined by the particle transit time \(T_{\rm t}=w/|v|\) with the particle velocity component \(v\) along the sensitivity direction and the respective spatial resolution \(w\), which is the dimension of the measurement volume. As a result, all measurement techniques are indirectly proportional to the square root of the spatial resolution \(w\) and the square root of the rate \(\dot{N}_{\rm photon}\) of the observed, scattered photons. Note that the photon rate is defined as an average photon rate by \(\dot{N}_{\rm photon}=N_{\rm photon}/T_{\rm t}\), where \(N_{\rm photon}\) is the total number of observed, scattered photons per particle. Note also that the product \(\dot{N}_{\rm photon}w\) represents the integral \begin{table}
\begin{tabular}{l l l} \hline \hline  & \(\mathbf{u(v)}\) **for a Single Particle** & \(\mathbf{u(\varphi)}\) **for Multiple Particles** \\ \hline DGV & \(c_{1}\cdot\dfrac{|v|^{1/2}\cdot\dfrac{\lambda}{|\tau^{\prime}|}}{\sqrt{\dot{N }_{\rm photon}w}}\) & \(c_{1}\cdot\dfrac{1}{\sqrt{M}}\cdot\dfrac{\dfrac{\lambda}{|\tau^{ \prime}|}}{\sqrt{\dot{N}_{\rm photon}T}}\) & \(c_{1}=\dfrac{\sqrt{\tau+\tau^{2}}}{|\tilde{q}-\tilde{i}|/\sqrt{2}}\) \\ LDA & \(c_{2}\cdot\dfrac{|v|}{\sqrt{\dot{N}_{\rm photon}tw}}\) & \(c_{2}\cdot\dfrac{|v|}{\sqrt{\dot{N}_{\rm photon}T}}\) & \(c_{2}=\sqrt{\dfrac{3}{\pi}}\cdot\dfrac{d}{w}\) \\ L2F & \(c_{3}\cdot\dfrac{|v|^{3/2}}{\sqrt{\dot{N}_{\rm photon}tw}}\) & \(c_{3}\cdot\dfrac{|v|}{\sqrt{\dot{N}_{\rm photon}T}}\) & \(c_{3}=\dfrac{\tilde{b}}{w}\) \\ PTV & \(c_{4}\cdot\dfrac{|v|^{3/2}}{\sqrt{\dot{N}_{\rm photon}tw}}\) & \(c_{4}\cdot\dfrac{|v|}{\sqrt{\dot{N}_{\rm photon}T}}\) & \(c_{4}=\dfrac{0.94}{w_{\rm px}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Square root of the Cramer-Rao bound due to photon shot noise for Doppler and time-of-flight measurement techniques, which is the minimal achievable measurement uncertainty \(u(v)\) for a single particle and \(u(\varphi)\) for multiple particles, respectively. DGV, Doppler global velocimetry; LDA, laser Doppler anemometer; L2F, laser-2-focus anemometry; PTV, particle tracking velocimetry. of the available photon rate over space and is proportional to the available light power. Furthermore, the uncertainty limit is directly proportional to the particle velocity \(v\) to the power of \(\frac{3}{2}\) for LDA, L2F and PTV and to the power of \(\frac{1}{2}\) for DGV. Concerning DGV, the ratio \(\frac{\lambda}{|\tau^{\prime}|}\) occurs instead of the factor \(v\). Assuming the typical values for the laser wavelength \(\lambda=532\,\)nm and the maximal slope \(|\tau^{\prime}|=2\,\)GHz\({}^{-1}\) of the spectral transmission curve of the molecular filter [79], the ratio \(\frac{\lambda}{|\tau^{\prime}|}\) is larger for velocities \(v<266\,\)m/s. Hence, DGV has a lower sensitivity for sub-sonic flows. In the uncertainty limit, the factors \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\) occur. For DGV, the factor \(c_{1}\) results from the transmission \(\tau\) of the molecular filter and the length of the sensitivity vector \(\vec{\sigma}-\vec{i}\). With \(\tau=0.5\) and a perpendicular arrangement of the observation and illumination direction (\(|\vec{\sigma}-\vec{i}|=\sqrt{2}\)), it is \(c_{1}\approx 0.9\). For LDA, the factor \(c_{2}\) is mainly determined by the ratio \(\frac{d}{v}\) between the distance \(d\) of the interference fringes and the dimension \(w\) of the measurement volume perpendicular to the fringes (\(1/\mathrm{e}^{2}\) width). This ratio is the reciprocal of the number of fringes. For 10 fringes, it is for instance \(c_{2}\approx 0.1\). For L2F, the factor \(c_{3}\) is the ratio \(\frac{\tilde{b}}{w}\) between the diameter \(\tilde{b}\) (\(1/\mathrm{e}^{2}\) width) of the two beams and the distance \(w\) of the parallel beams. Here, it is assumed \(c_{3}\approx 0.1\) as an example.