typos

Tex/simulation.tex

@@ -173,7 +173,7 @@ \subsection{Undersampling and  Sample Size}
 
 
 A simulation of a cubic single crystal with a simple cubic lattice of varying size (from 20$^3$ to 200$^3$\,atoms) is performed. The lattice constant is chosen as 5.7\,\AA, the fluorescence energy as 8\,keV. The simulated detector has 1024x1024 50\,µm sized pixels and is placed 8\,cm from the  sample. 
-The simulation of the fluorescence patterns is performed with 4x4 oversampling (4096x4096 pixel) and rebinning to the detector size. Only a single coherence mode is simulated. The number of photons emitted by the sample in 4$\pi$ is chosen as equal the number of atoms in the sample. Therefor, the mean number of photons per pixel is (especially for small crystal sizes) very small, but experimentally feasible.
+The simulation of the fluorescence patterns is performed with 4x4 oversampling (4096x4096 pixel) and rebinning to the detector size. Only a single coherence mode is simulated. The number of photons emitted by the sample in 4$\pi$ is chosen as equal the number of atoms in the sample. Hence, the mean number of photons per pixel is (especially for small crystal sizes) very small, but at the upper bound of an achievable photon yield in an experiment.
 
 The peak signal-to-noise ratio is calculated by simulating 20-2000 independent images  and taking the mean intensity at the visible Bragg peaks positions as signal and the standard deviations at those positions over the independent simulations as noise, resulting in an estimated of the peak SNR of a single image.
 Due to the low photon numbers, the Poisson noise is dominating the noise characteristic and with an increase in atoms in the focus, the SNR increases linear (as shown in \fref{fig:SNRNatoms}), up to the point where a the peaks are no longer fully sampled and the signal decreases linear with a further increase in the number of atoms in the crystal., resulting in a nearly constant peak SNR.
@@ -214,7 +214,7 @@ \subsection{Multiple Samples}
 The structure factor of this ensemble of spherical particles is determined by three factors: The structure factor of the focus, the structure factor of points following an Poisson Sphere Distribution and the structure factor of a single sphere.(see \fref{fig:multisphere1}). As the number of spheres is increased, the influence of the focal volume increases and the distribution of the spheres's centers increases (see  \fref{fig:multisphere3})
 For spheres with 20\, nm radius, a spacing layer of 5\,nm around each sphere, with  50000 excited atoms per sphere on average, a focus of 200nm (FWHM), the fluorescence on a 1024x1024 pixel (pixelsize 100\,um) detector placed 30\,cm is simulated. In this geometry, assuming constant distance to the sample for each detector pixel, approximately 1\% of the emitted photons reach the detector. 
 For each number of spheres 5000 images are used for an IDI reconstruction (see  \fref{fig:multisphere2}).
-As the number of photons sphere is kept constant, opposing effects occur:  With increasing number of particles and therefor photons recorded, the Poisson noise is reduced, but the signal strength in low scattering angles is decreased as the structure factor changes from a single sphere to a hard-sphere model. Therefor, for the chosen simulation parameters an optimum in the detectebility of a correlation effect can be found for a medium number of samples in the focus, below
+As the number of photons sphere is kept constant, opposing effects occur:  With increasing number of particles, more are photons recorded and the Poisson noise is reduced, but the signal strength in low scattering angles is decreased as the structure factor changes from a single sphere to a hard-sphere model. Therefore, for the chosen simulation parameters an optimum in the detectebility of a correlation effect can be found for a medium number of samples in the focus.
 
 
 As the number of particles inside the focus is increased, in the low $q$ region of the IDI reconstruction, the focal volume becomes more visible. For high numbers of particles, the structure factor of the distribution of the centers of the particles causes a reduction in the scattering factor in the low $q$ region. For low numbers of particles inside the focus, the Poisson noise casued by the low photon numbers dominates the error.
@@ -274,13 +274,13 @@ \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 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 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.
+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. Therefore, 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.
 
 
 
 
- As a thicker sample gives more photons and therefore less Poisson noise, but a higher number of temporal and spatial modes and therefore a lower expected lower speckle visibility, a simulation close to experimentally feasible parameters is desirable.
+ As a thicker sample gives more photons and therefore less Poisson noise, but a higher number of temporal and spatial modes, thus a lower expected lower speckle visibility, a simulation close to experimentally feasible parameters is desirable.
 
 
 
@@ -334,7 +334,7 @@ \section{Implications for an experimental design}
 	\centering
 	\includegraphics[width=0.5\linewidth]{images/sim_foil5umCu_shared.pdf}
 	\label{fig:simfoil}
-	\caption{Simulation result for a metal foil placed in a 200\,nm focus viewed perpendicular to the incoming FEL beam. No correction for charge sharing is applied. The center pixel is masked, as $g^2(0)-1\approx 
+	\caption[Simulation of a metal foil with similar parameters as used in the experiment]{Simulation result for a metal foil placed in a 200\,nm focus viewed perpendicular to the incoming FEL beam. No correction for charge sharing is applied. The center pixel is masked, as $g^2(0)-1\approx 
 		\frac{1}{M}+\frac{1}{\mu}$ is dominated by the mean count and does 
 		not carry spatial information.}
 \end{figure}

Tex/theory.tex

@@ -355,7 +355,7 @@ \subsection{Signal}
 
 
 Path differences longer than the coherence length give additional modes $M_S$, therefore the speckle visibility will be reduced for large samples:
-This is for example important, if the exciting beam is not perpendicular to the detector (such as in an off-axis setup), as the excitation time difference between different parts of the a thick sample is not compensated for by the difference in flight time to the detector or in a wide angle experiment. In a small angle setup with a detector perpendicular to the beam, fluorescence from different emitters along the sample thickness will arrive at approximately equal time, therefor the sample thickness does not introduce additional modes)
+This is for example important, if the exciting beam is not perpendicular to the detector (such as in an off-axis setup), as the excitation time difference between different parts of the a thick sample is not compensated for by the difference in flight time to the detector or in a wide angle experiment. In a small angle setup with a detector perpendicular to the beam, fluorescence from different emitters along the sample thickness will arrive at approximately equal time, hence the sample thickness does not introduce additional modes)
 
 
 If the speckle pattern is not spatially resolved by the detector because its resolution is to low compared to the change in intensity, undersampling will occur, the meassured signal will be spatially averaged giving independent spatial sampling modes $M_D$ \cite{goodman2007}.