thickness

Tex/simulation.tex

@@ -265,10 +265,11 @@ \section{Time Dependent Simulations}
 This procedure is efficient in regards of discrete time steps that need to be calculated and can easily be run on a GPU.
 
 \subsection{Influence of the Pulse Length}
-For a sphere with 10\,nm radius consisting of $2*10^5$ atoms emitting 6.4\,keV fluorescence captured by an 256x256@50\,um detector in 20\,cm distance, the speckle SNR of a series of simulations with different decay times $\tau$ of the emission and different FWHM of the exciting Gaussian pulse are shown in \fref{fig:tdpshere_specke}.  These follow the $1/\sqrt{erfcx(2\sigma/\tau})$ relation as predicted by the theory.  For each simulation, a reconstruction can be performed, resulting in radial profiles as shown for one $\tau$ in \fref{fig:tdpshere_recon}. The visibility of the reconstructions (\fref{fig:tdsphere_vis}) shows the for long pulses a reciprocal relationship.
+For a sphere with 10\,nm radius consisting of $2*10^5$ atoms emitting 6.4\,keV fluorescence captured by an 256x256@50\,um detector in 20\,cm distance, the speckle strength (calculated as the standard deviation of the speckle pattern over the mean) of a series of simulations with different decay times $\tau$ of the emission and different FWHM of the exciting Gaussian pulse are shown in \fref{fig:tdpshere_specke}.  These follow the $1/\sqrt{erfcx(2\sigma/\tau})$ relation as predicted by the theory.  For each simulation, a reconstruction can be performed, resulting in radial profiles as shown for one $\tau$ in \fref{fig:tdpshere_recon}. The visibility of the reconstructions (\fref{fig:tdsphere_vis}) shows the for long pulses a reciprocal relationship.
 
 \subsection{Influence of the Sample Thickness}
-To investigate the influence of the sample thickness, the speckle SNR of a second simulation is shown in \fref{fig:thickness}. In this simulation, $10^6$ 8\,keV emitters are placed inside a 200\,nm x 200\,nm (FWHM) Gaussian volume with varying thickness. To ensure sufficient sampling of the speckle pattern, the axis of varying thickness is always set perpendicular to the detector. The 64x64 pixel (pixelsize 100\,um) is placed in 1\,m distance, the simulation is performed with 4x oversampling and rebinning in both directions. The varying angle $\alpha$ influences only the mean (for each emitter) of the 1\,fs FWHM Gaussian from which the emission is sampled without influencing the overall volume in which the emitters are placed. Therefor, the change in SNR is only caused by the finite coherence time $\tau=1\,fs$, not by a change of the speckle size. This simulations show that in high angles the limited coherence length of the fluorescence reduces the speckle SNR for thick samples, whereas in small angles, the sample thickness does not influence the SNR. In the 0° limit, the thickness is in beam direction and the position of an emitter along this axis does not change the arrival time of its contribution the the speckle pattern.
+To investigate the influence of the sample thickness, the speckle strength in a second simulation is shown in \fref{fig:thickness}. In this simulation, a constant number ($10^6$) of 8\,keV emitters are placed inside a 200\,nm x 200\,nm (FWHM) Gaussian volume with varying thickness. To ensure sufficient sampling of the speckle pattern, the axis of varying thickness is always set perpendicular to the detector. The 64x64 pixel (pixelsize 100\,um) detector is placed in 1\,m distance. The simulation is performed with 4x oversampling and rebinning in both directions. The varying angle $\alpha$ influences only the mean (for each emitter) of the 1\,fs FWHM Gaussian from which the emission is sampled without influencing the overall volume in which the emitters are placed. Therefor, the change in SNR is only caused by the finite coherence time $\tau=1\,fs$, not by a change of the speckle size. Furthermore, no shot noise is considered in the simulation.
+This simulations show that in high angles the limited coherence length of the fluorescence reduces the speckle SNR for thick samples, whereas in small angles, the sample thickness does not influence the SNR: In the 0° limit, the thickness is in beam direction and the position of an emitter along this axis does not change the arrival time of its contribution the the speckle pattern.
 
 
 
@@ -296,7 +297,7 @@ \subsection{Influence of the Sample Thickness}
 		\includegraphics[width=\linewidth]{images/thickness.pdf}
 		\caption{Speckle SNR for different sample thicknesses and angles}
 	\end{subfigure}
-	\caption[Speckle SNR in Time Dependent Simulation]{ In a) the SNR of the simulated speckle pattern for different pulse length and decay times $\tau$ of a spherical object is compared with the theoretical dependence on the ratio of pulse length and $\tau$ (details in text), showing good agreement. In b) the influence of sample thickness on the speckle SNR under different angles is show (using the mean of 5 independent simulations and the standard deviation as errors). For small angles, the sample thickness does not influence the SNR.}
+	\caption[Speckle SNR in Time Dependent Simulation]{ In a) the SNR of the simulated speckle pattern for different pulse length and decay times $\tau$ of a spherical object is compared with the theoretical dependence on the ratio of pulse length and $\tau$ (details in text), showing good agreement. In b) the influence of sample thickness on the speckle SNR under different angles is show (using the mean of 5 independent simulations and the standard deviation as errors). The dashed lines are $y=c/\left(1+(\sfrac{\sin^2 \alpha}{4})\right)$ regressions. For small angles, the sample thickness does not influence the SNR.}
 \end{figure}
 \begin{figure}
 	\centering