From 325cd37bdb59b5284bc7aedfbaab441c68ea6232 Mon Sep 17 00:00:00 2001 From: Felix Zimmermann Date: Sat, 6 Nov 2021 00:28:54 +0100 Subject: [PATCH] final --- Tex/conclusion.tex | 2 +- Tex/experiment.tex | 43 ++++++++++++++-------------- Tex/images/sim_foil5umCu_shared.pdf | Bin 106733 -> 128575 bytes Tex/main.tex | 7 +++-- Tex/simulation.tex | 23 +++++++-------- Tex/thanks.tex | 2 +- Tex/theory.tex | 8 ++---- 7 files changed, 41 insertions(+), 44 deletions(-) diff --git a/Tex/conclusion.tex b/Tex/conclusion.tex index 061f730..d74f9b5 100755 --- a/Tex/conclusion.tex +++ b/Tex/conclusion.tex @@ -1,5 +1,5 @@ \chapter{Conclusion and Outlook} -There is still an ongoing discussion about the feasibility of \textit{Incoherent Diffractive Imaging} and about in which areas it can provide advantages about already established methods. It has not yet been experimentally proven that the assumptions made about the statistics of inner shell fluorescence are valid at the atomic level. Under the assumption of chaotic light, classical wave theory can illustrate the basic working principle. In the quantum mechanical description of the underlying two-photon interference, indistinguishability between the quantum paths leading from the emission of two photons to simultaneous detection in two detectors creates the spatial bunching in the fluorescence patterns. Both explanations predict that a successful IDI experiment has to be designed with few distinguishable paths or modes. +There is still an ongoing discussion about the feasibility of \textit{Incoherent Diffractive Imaging} and its potential advantages over already established methods. It has not yet been experimentally proven that the assumptions made about the statistics of inner shell fluorescence are valid at the atomic level. Under the assumption of chaotic light, classical wave theory can illustrate the basic working principle. In the quantum mechanical description of the underlying two-photon interference, indistinguishability between the quantum paths leading from the emission of two photons to simultaneous detection in two detectors creates the spatial bunching in the fluorescence patterns. Both explanations predict that a successful IDI experiment has to be designed with few distinguishable paths or modes. The present work contributes simulations to validate theoretical considerations about the Signal-to-Noise characteristics and to optimize the experimental setup. Furthermore, previously published results indicating that IDI can extract spatial information out of X-ray fluorescence intensity patterns were validated. diff --git a/Tex/experiment.tex b/Tex/experiment.tex index cc97afb..97440fc 100755 --- a/Tex/experiment.tex +++ b/Tex/experiment.tex @@ -92,7 +92,7 @@ \subsection{Magnetite Nanoparticle in Polymer Samples} \begin{figure} \centering - \begin{subfigure}[b]{0.4\textwidth} + \begin{subfigure}[b]{0.42\textwidth} \includegraphics[width=\linewidth]{images/ps5.pdf} \end{subfigure} \begin{subfigure}[b]{0.42\textwidth} @@ -134,7 +134,7 @@ \subsection{Gallium Arsenide Crystal Films} \section{Setup of the Experiment} The setup used at the experimental station EH5 at SACLA is shown in \fref{fig:setup}. -The FEL provided approximately 10\,fs long hard X-ray pulse with $\approx 10^{11}$ photons at 11\,keV, just above the gallium K-edge and below the arsenic K-edge (see \fref{tab:elements}), and 30\,Hz repetition rate \cite{tono2013}. The beam was focused by the \textit{100\,exa} KB nano-focusing system. The focus spot along the beam direction was found by optimizing the strength of (non-linear) amplified spontanous emission (ASE) of a copper foil, resulting in focal FWHM of 200\,nm vertical/100\,nm horizontal as determined by a wirescan \cite{yumoto2020,handa2010,yoneda2015}. +The FEL provided approximately 10\,fs long hard X-ray pulse with $\approx 10^{11}$ photons at 11\,keV, just above the gallium K-edge and below the arsenic K-edge (see \fref{tab:elements}), and 30\,Hz repetition rate \cite{tono2013}. The beam was focused by the \textit{100\,exa} KB nano-focusing system \cite{yumoto2020}. The focus spot along the beam direction was found by optimizing the strength of (non-linear) amplified spontanous emission (ASE) of a copper foil, resulting in focal FWHM of 200\,nm vertical/100\,nm horizontal as determined by a wire scan \cite{handa2010,yoneda2015}. The sample was mounted in a 45° angle to the beam on a stack of two rotational/two translational/two rotational/three translational stages to allow scanning of the sample, to ensure perpendicularity of the scanning directions to the beam to stay within the Rayleigh length (approx. 100\,\textmu m horizontal/200\,\textmu m vertical) while also ensuring a parallel alignment of the sample surface to one of the detectors. @@ -146,11 +146,11 @@ \section{Setup of the Experiment} \begin{figure} \centering \includegraphics[width=0.75\linewidth]{images/setup.pdf} - \caption[Experimental setup at SACLA]{Experimental setup at SACLA: The sample is mounted on a scanning stage and aligned to stay in focus during the scan and be parallel to the octal MPCCD detector, which is in a distance $d_{octal}$. The angle between incoming FEL and Sample is 45°. Behind the sample, a stray light filter, beamstop and (depending on the sample) a filter foil is installed. The dual detector is mounted $d_{dual}$=1\,m away from the sample perpendicular the FEL beam. To reduce air scattering, a vacuum tube is installed in the path from sample to dual.} + \caption[Experimental setup at SACLA]{Experimental setup at SACLA: The sample is mounted on a scanning stage and aligned to stay in focus during the scan and be parallel to the octal MPCCD detector, which is in a distance $d_{octal}$. The angle between incoming FEL and Sample is 45°. Behind the sample, a stray light filter, beamstop and (depending on the sample) a filter foil is installed. The dual detector is mounted $d_{dual}$=1\,m away from the sample perpendicular the FEL beam. To reduce air scattering and absorption, a vacuum tube is installed in the path from sample to the dual detector.} \label{fig:setup} \end{figure} \paragraph{Imaging the Focus} -To image the focus, one of the following foils with purity >99.99\% was mounted on the sample holder: 500\,nm iron (Goodfellow), 10\,\textmu m iron (Nilaco), 5\,\textmu m (Nilaco) and 20\,\textmu m (Goodfellow). The fluorescence was recorded using the dual detector at a distance of $d_\text{dual}$=1\,m. +To image the focus, one of the following foils with purity >99.99\% was mounted on the sample holder: 500\,nm iron (on polymer support, Goodfellow), 10\,\textmu m iron (Nilaco), 5\,\textmu m (Nilaco) and 20\,\textmu m (Goodfellow). The fluorescence was recorded using the dual detector at a distance of $d_\text{dual}$=1\,m. The X-ray beam was attenuated by 0.4\,mm silicon, resulting in 20-30\,\textmu J energy at the sample for pulse energies at the upstream energy monitors of 400\,\textmu J \cite{yabashi2015,tono2013}. \paragraph{Imaging Nanoparticles} For imaging the prepared iron nanoparticle and reference samples, the octal detector, which was the main imaging detector for these samples was moved to $d_\text{octal}\approx$43\,cm. The vacuum tube in front of the dual detector was partially removed to avoid mechanical interference. The sample was slightly moved out of focus along the beam direction by 400\,\textmu m instead of using any attenuation to increase the number of fluorescence photons. In between the nanoparticle samples, a single 10\,\textmu m iron foil measurement using the dual detector was taken to confirm the beam conditions. @@ -193,7 +193,7 @@ \subsection{Preprocessing} \end{figure} \paragraph{Detector Artifacts} -During the experiment, an unnoticed failure of the electronics of the octal detector occurred, manifesting as periodic noise on most of the detector tiles at a much higher level than expected (see \fref{fig:octalissue} for an illustration). Not all columns of a tile or read-out port were affected, in contrast to common-mode effects more commonly encountered with CCD detectors. As the overall affected area of the detector is substantial and the noise leaks through the photon counting scheme by slightly influencing the probability of a pixel being considered a photon hit, causing correlations and therefore artifacts in the reconstruction in one direction, a correction was applied to the affected columns of affected tiles of the detector. For the correction, affected columns were first identified by filtering on the Fourier transform, neighboring effected columns are considered as a block and the medians over those blocks were calculated and subtracted. This scheme was chosen instead of the more commonly used portwise common mode correction due to the uneven strength of the artifact (see \fref{fig:app_correction} for a comparison). Using this correction reduced the artifact, but did not completely remove it. As the remaining artifact in the reconstructions is limited to low $q$ compared to the expected positions of the features of the samples and only two orthogonal lines, it can be masked out. +During the experiment, an unnoticed failure of the electronics of the octal detector occurred, manifesting as periodic noise on most of the detector tiles at a much higher level than expected (see \fref{fig:octalissue} for an illustration). Not all columns of a tile or read-out port were affected, in contrast to common-mode effects more commonly encountered with CCD detectors. The overall affected area of the detector is substantial and the noise leaks through the photon counting scheme by slightly influencing the probability of a pixel being considered a photon hit. As this causes correlations between affected pixels and therefore artifacts in the reconstruction, a correction was applied to the affected columns of affected tiles of the detector. For the correction, affected columns were first identified by filtering on the Fourier transform. Neighboring effected columns were considered as a block and the medians over those blocks were calculated and subtracted. This scheme was chosen instead of the more commonly used portwise common mode correction due to the uneven strength of the artifact (see \fref{fig:app_correction} for a comparison). Using this correction reduced the artifact, but did not completely remove it. As the remaining artifact in the reconstructions is limited to low $q$ compared to the expected positions of the features of the samples and only two orthogonal lines, it can be masked out. \begin{figure} @@ -216,11 +216,11 @@ \subsection{Preprocessing} \begin{equation} w (x)= (1-p)\delta(x)+\rect\left(\frac{x-1/2}{E_{signal}}\right)\frac{p}{E_{signal}} \end{equation} -was included, thus smearing the peaks to lower energies with a strength parameter $p$ (initialized as 25\%). The mean squared difference the logarithmic values was minimized. With this regression, an estimate for the number of modes of the binomial distribution as well as for the mean number of signal and scattering photons was found. +was included, thus smearing the peaks to lower energies with a strength parameter $p$ (initialized as 25\%). The mean squared difference of the logarithmic values was minimized. With this regression, an estimate for the number of modes of the binomial distribution as well as for the mean number of signal and scattering photons was found. \paragraph{Photon counting} -The mean number of signal and scattering photons found from the spectra as well as the peak width and the charge sharing point spread function for the detectors were used to find the thresholds for a classification of the pixel values into signal photon numbers for each of the groups as described in \fref{sec:chargesharing}. +The mean number of signal and scattering photons found from the spectra as well as the peak width and the charge sharing point spread function for the detectors were used to find the thresholds for the classification of the pixel values into signal photon numbers for each of the groups as described in \fref{sec:chargesharing}. With the determined K$_\alpha$, K$_\beta$, and scattering intensities, 1000 images with 4096x4096 pixels were simulated with Gaussian charge sharing. The determined peak widths were used as the FWHM of the assumed to be Gaussian detector noise. The number of signal photons corresponding with the highest probability to each energy was determined for 10\,eV bins. \paragraph{Final shot filtering} @@ -231,9 +231,9 @@ \subsection{Preprocessing} \paragraph{Crystal orientation} -For determining the relative orientation of the crystal with regards to the detector, Kossel lines as described in \fref{sec:kossel} can be used. This can either be done by reprojecting the planar detector onto a sphere by an inverse gnomonic projection and determining circle center and radius, eg. by an Hugh-Transform, or by a fitting conic sections to the lines on the planar detector \cite{morris1968,morawiec2016,faigel2016,herron2018}. As the observable curvature of the Kossel lines on the detector in an IDI setup is small and therefore the circles are difficult to determine, the second method was chosen. A semi-automatic alignment program was developed (\fref{fig:kosselfit}) to aid with the procedure. +For determining the relative orientation of the crystal with regards to the detector, Kossel lines as described in \fref{sec:kossel} can be used. This can either be done by reprojecting the planar detector onto a sphere by an inverse gnomonic projection and determining circle center and radius a Hugh-Transform, or by fitting conic sections to the lines on the planar detector \cite{morris1968,morawiec2016,faigel2016,herron2018}. The observable curvature of the Kossel lines on the detector in an IDI setup is small. Hence, the centers of the circles would be challenging to determine, and the second method was chosen. A semi-automatic alignment program was developed (\fref{fig:kosselfit}) to aid with the procedure. -For each sample, the images were split into sets of 5000 shots. After filtering for hot pixels and cutting bellow a noise threshold of 2\,keV, the mean was taken. To better distinguish the Kossel lines from the uniform fluorescence, a Gaussian blurred version ($\sigma$=20\, px) was subtracted (see \fref{fig:kosselgaasmean} for examples). The Kossel lines were roughly identified manually and the local maxima in the 20x20\,px surrounding block of an identified area were considered as points on the Kossel line. A least-square regression was performed, +For each sample, the images were split into sets of 5000 shots. The mean was taken after filtering for hot pixels and cutting below a noise threshold of 2\,keV. To better distinguish the Kossel lines from the uniform fluorescence, a Gaussian blurred version ($\sigma$=20\, px) was subtracted (see \fref{fig:kosselgaasmean} for examples). The Kossel lines were roughly identified manually, and the local maxima in the 20x20\,px surrounding block of an identified area were considered as points $\vec{r}$ on a Kossel line. A least-square regression was performed, \begin{equation} \arg\!\min_{R,t} \sum_{r} \min_{q\in Q} \left\Vert \frac{2 d \left( \vec{r} - \vec{t} \right) \cdot \vec{Rq}}{\left|\vec{q}\right| \left| \left(\vec{r}-\vec{t}\right)\right|} -\lambda \right\Vert_2^2 \,, @@ -247,7 +247,7 @@ \subsection{Preprocessing} \begin{figure} \centering \includegraphics[width=0.8\linewidth]{images/kosselfit.png} - \caption[User Interface for Kossel line based alignment]{User interface created for performing a regression of the detector orientation and position to the Kossel lines visible in experimental data. The lines can be automatically identified, be manually modified afterwards and the reflections considered selected. This tool is used to perform the alignment for the GaAs single crystal samples.} + \caption[User Interface for Kossel line based alignment]{User interface created for performing a regression of the detector orientation and position to the Kossel lines visible in experimental data. The lines can be automatically identified, be manually modified afterwards and the indices of potential lines to be considered selected. This tool was used to perform the alignment for the GaAs single crystal samples.} \label{fig:kosselfit} \end{figure} @@ -256,7 +256,7 @@ \subsection{Preprocessing} \subsection{Reconstruction} -For the foil data, the small-angle approximation is valid, and each pixel approximately covers the same solid angle, permitting the use of a pixel-wise correlation after normalization as described in \fref{sec:normal} for the reconstruction of $|S(q)|^2$. For the nanoparticle data, the maximum recorded angles are higher and at the limit of the small-angle regime, resulting in $\delta q_x,_y$ between neighboring pixels being lower for pixels further off-axis. However, as the features in the structure factor are at relatively low $q$, the difference is less than the $q$ resolution, and the same reconstruction can be used. +For the foil data, the small-angle approximation is valid, and each pixel approximately covers the same solid angle, permitting the use of a pixel-wise correlation after normalization as described in \fref{sec:normal} for the reconstruction of $|S(\vec{q})|^2$. For the nanoparticle data, the maximum recorded angles are higher and at the limit of the small-angle regime, resulting in $\delta q_x,_y$ between neighboring pixels being lower for pixels further off-axis. However, as the features in the structure factor are at relatively low $\left|\vec{q}\right|$, the difference is less than the $\vec{q}$ resolution, and the same reconstruction can be used. For the wide-angle crystal data, a full three-dimensional correlation of the rotation and offset corrected data was performed using the resampled FFT-based approach described in \fref{sec:corr}. This takes the varying solid angles fully into account. @@ -264,7 +264,7 @@ \section{Results of the Experiment} Applying the aforementioned preprocessing and analysis steps to the recorded fluorescence images for all three classes of samples leads to results as follows. \subsection{Imaging the Focus} -The regression of the spectra (\fref{fig:spectrafoil}) resulted in a mean K$_{\alpha}$ photon count per pixel of 0.004$\pm$0.001 (500\,nm iron foil) to 0.041$\pm$0.001 (20\,\textmu m copper foil). The mode number could not be reasonably estimated for the 500\,nm iron sample due to the influence of air scattering and fluorescence stemming from the experimental setup and low number of signal photons. For the other samples, the regression provided estimates in the range of 15-25 modes. The charge sharing approximation parameter $p$ was found to be 0.43$\pm$0.01, meaning about 40\% of the photon hits were affected. The simplifications of assuming a uniform charge cloud by using a convolution windows with a flat side lobe led to disagreement between the slightly curved histograms and the flat regressions in the valleys between integer multiples of the single-photon energy. This is no longer present in the spectra simulated with a Gaussian charge cloud (see \fref{fig:thresholdsfoil}) for an example. +The regression of the spectra (\fref{fig:spectrafoil}) shows a mean K$_{\alpha}$ photon count per pixel of 0.004$\pm$0.001 (500\,nm iron foil) to 0.041$\pm$0.001 (20\,\textmu m copper foil). The mode number cannot be reasonably estimated for the 500\,nm iron sample due to the influence of air scattering and fluorescence stemming from the experimental setup and low number of signal photons. For the other samples, the regression gives estimates in the range of 15-25 modes. The charge sharing approximation parameter $p$ is found to be 0.43$\pm$0.01, meaning about 40\% of the photon hits were affected. The simplifications of assuming a uniform charge cloud by using a convolution windows with a flat side lobe leads to disagreement between the slightly curved histograms and the flat regressions in the valleys between integer multiples of the single-photon energy. This is no longer present in the spectra simulated with a Gaussian charge cloud (see \fref{fig:thresholdsfoil}) for an example. The reconstructions were performed once without discretization and only applying a threshold of 1500\,eV and once with discretization into photons using the thresholds found in the simulation of the spectra. Without discretization, strong artifacts were visible in the reconstruction, which were mostly reduced by the applied photon counting (see \fref{fig:foil_photonmodes} for a comparison). @@ -281,17 +281,16 @@ \subsection{Imaging the Focus} \end{figure} -The excited volume can be approximated in the horizontal direction as a rectangular function with a width of $\sqrt{2}$-times the thickness of the foil used, due to the 45° angle, as the horizontal FWHM of the focus is much smaller than the thickness of the foil and ignoring absorption. In the vertical direction, the volume is limited by the FWHM of the focus and can be roughly approximated to be Gaussian. -In a small-angle approximation, the reconstruction is the magnitude squared of the 2D Fourier transform of the volume. As the magnitude squared of the Fourier transform of a Gaussian with FWHM $f$ is another Gaussian with FWHM $\frac{4\sqrt{2}\log{2}}{f}$ and of a rectangular function\footnote{If the foil were thicker, the absorption length $a$ could not be ignored and the more complicated $\left|\mathscr{F}\left(e^{-\frac{x}{a}} \Pi \left(\frac{x}{w}\right)\right)\right|^2 \propto \frac{\cosh \left(\frac{w}{a}\right)-\cos (q w)}{a^2 q^2+1}$ - could be used instead.} with width $w$ is proportional to $\sinc^2{\left(\frac{q w}{2}\right)}$, a 2D least-squares regression of the product of those functions is performed on the reconstructed image to estimate the focal FWHM \cite{butz2015}. As the horizontal direction is under-sampled, the horizontal profile will not be resolved in the reconstruction. The function used for the regression is sampled at 25x the resolution in both directions and averaged. This corresponds to a convolution with a rectangular sampling kernel and reduces the effect of undersampling on the values determined by the regression. Furthermore, a rotation parameter is introduced to account for a slight misalignment of the detector. +The excited volume can be approximated in the horizontal direction as a rectangular function with a width of $\sqrt{2}$-times the thickness of the foil used, due to the 45° angle, as the horizontal FWHM of the focus is much smaller than the thickness of the foil and ignoring absorption. In the vertical direction, the volume is limited by the FWHM of the focus and can be approximated to be Gaussian. +In a small-angle approximation, the reconstruction is the squared magnitude of the 2D Fourier transform of the volume. As the squared magnitude of the Fourier transform of a Gaussian with FWHM $f$ is another Gaussian with FWHM $\frac{4\sqrt{2}\log{2}}{f}$ and of a rectangular function\footnote{If the foil were thicker, the absorption length $a$ could not be ignored and the better approximation $\left|\mathscr{F}\left(e^{-\frac{x}{a}} \Pi \left(\frac{x}{w}\right)\right)\right|^2 \propto \frac{\cosh \left(\frac{w}{a}\right)-\cos (q w)}{a^2 q^2+1}$ + could be used instead.} with width $w$ is proportional to $\sinc^2{\left(\frac{q w}{2}\right)}$, a 2D least-squares regression of the product of those functions is performed on the reconstructed image to estimate the focal FWHM \cite{butz2015}. As the horizontal direction is under-sampled, the horizontal profile cannot be resolved in the reconstruction. The function used for the regression is sampled at 25x the resolution in both directions and averaged. This corresponds to a convolution with a rectangular sampling kernel and reduces the effect of undersampling on the values determined by the regression. Furthermore, a rotation parameter is introduced to account for a slight misalignment of the detector. -For a 5\,\textmu m Copper foil placed at three different positions, at the nominal focus and two positions further upstream, results of the auto-correlation reconstruction are shown in \fref{fig:Cu5umreco2d}. -The regression in these results estimates of the focal width in the vertical direction at the nominal focus position along the beam as (240$\pm$20)\,nm and while defocused by 250\,\textmu m as (380$\pm$40)\, nm. The stated errors only include regression uncertainty. The determined width at the focal position is in good agreement with the width determined by the wire scan, (210$\pm$ 10)\,nm. In \fref{fig:fereco2d} the results for 10\,\textmu m and 500\,nm iron foils as well as an example of the result of the regression is shown. As the 5\,\textmu m and 10\,\textmu m foils suffer from severe undersampling along the horizontal direction, only for the 500\,nm iron foil (\fref{fig:fe500nmreco2d}) the regression of the width in this axis can be performed. The regression estimates the width of the emitting volume in focal position as (500$\pm$250)\,nm in the horizontal direction, agreeing with the expected value of (500$\cdot\sqrt{2})$\,nm. +For the 5\,\textmu m Copper foil placed at three different positions -- at the nominal focus and two positions further upstream -- results of the auto-correlation reconstruction are shown in \fref{fig:Cu5umreco2d}. +The regression in these results estimates of focal width in the vertical direction at the nominal focus position along the beam as (240$\pm$20)\,nm and while defocused by 250\,\textmu m as (380$\pm$40)\, nm. The stated errors only include the regression uncertainty. The determined width at the focal position is in good agreement with the width determined by the wire scan, (210$\pm$ 10)\,nm. In \fref{fig:fereco2d} the intensity correlation for the 10\,\textmu m and 500\,nm iron foils as well as an example of the result of the regression is shown. As the 5\,\textmu m and 10\,\textmu m foils suffer from severe undersampling along the horizontal direction, only for the 500\,nm iron foil (\fref{fig:fe500nmreco2d}) the regression of the width in this axis can be performed. The regression estimates the width of the emitting volume in focal position as (500$\pm$250)\,nm in the horizontal direction, agreeing with the expected value of (500$\cdot\sqrt{2})$\,nm. A comparison of the intensity correlations of shots recorded at different pulse energies, as reported by the upstream energy monitors, shows that for low reported energies the amplitude of the correlations is lower (see \fref{fig:energy_binned}). This confirms the decision to use the energy measurement as a surrogate parameter for pulse length and as a filtering criterion \cite{inoue2019}. -Nevertheless, for all three foils in focal position the signal amplitude can be extrapolated by the regression as 0.004$\pm$0.001 with no significant difference between the samples. -The noise levels as estimated by the standard deviation over pixels outside the feature containing region of interest are 0.00013 for the copper foil, 0.0002 for the iron 10\,\textmu m foil, and 0.002 for the iron 500\,nm foil. The mean over the regions of interest of standard errors of the mean over the different shots used in the reconstruction are equal to these as well. +Nevertheless, for all three foils in focal position the signal amplitude is extrapolated by the regression as 0.004$\pm$0.001 with no significant difference between the samples. The noise levels as estimated by the standard deviation over pixels outside the feature containing region of interest are 0.00013 for the copper foil, 0.0002 for the 10\,\textmu m iron foil, and 0.002 for the 500\,nm iron foil. The standard errors of the mean over the different shots used in the reconstruction averaged over the region of interests are equal to these within the stated bounds. \vspace*{1cm} \begin{figure} \centering @@ -347,9 +346,9 @@ \subsection{Imaging Nanoparticles} The mean number of counted signal photons per pixel after filtering is $0.03\pm0.01$ for the nanoparticle samples as well as the iron chloride reference sample. The number of images analyzed has been limited to 30,000 for each sample, except for the \enquote{10\,nm in PS} sample only having 25,000 usable shots after filtering. At similar signal photon counts on the octal as observed for the foils on the dual detector, the peak in the spectrum at the energy of the FEL beam is more pronounced, leading to difficulties distinguishing higher numbers of photons. Hence, only one and two-photon hits are used in the photon discretization (see \fref{fig:spectrum_nano} for an example). -Using a sample with just the polymer, without any added iron (neither nanoparticles nor atomic), the influence of air scattering on the correlations can be examined. In the results of the iron chloride sample having similar numbers of signal photons, the effect of uncorrected detector artifacts could be seen. Neither result show a strong correlation signal (see \fref{fig:resnano2d}). The strength of the noise as quantified by the standard deviation over pixels outside the center area of the 2d-reconstruction of the samples is (1.4$\pm$0.1)$\cdot$10$^{-4}$. +Using a sample with just the polymer, without any added iron (neither nanoparticles nor atomic), the influence of air scattering on the correlations can be examined. As the iron chloride sample resulted in similar numbers of iron fluorescence photons, but has no structure at the observed $q$-range, in its autocorrelation the effect of uncorrected detector artifacts would be visible. Neither result shows a strong correlation signal (see \fref{fig:resnano2d}). The strength of the noise as quantified by the standard deviation over pixels outside the center area of the 2D-reconstruction of the samples is (1.4$\pm$0.1)$\cdot$10$^{-4}$. -In the radially averaged $g^2$ of the four different samples (shown in \fref{fig:resnanorad}) neither a feature corresponding to the distance of aggregated particles, as visible in the SAXS measurements, nor a minimum at $q\approx4.5/r$, indicative of spherical samples, can be distinguished. +In the radially averaged $g^2$ of the four different samples (shown in \fref{fig:resnanorad}) neither a feature corresponding to the distance of aggregated particles, as visible in the SAXS measurements (see \fref{fig:saxsps}), nor a minimum at $q\approx4.5/r$, indicative of spherical samples, can be distinguished. \FloatBarrier \subsection{Imaging Crystals} @@ -388,7 +387,7 @@ \subsection{Imaging Crystals} \label{tab:kosselfit} \end{table} -Using the determined rotation and translation to determine for each pixel the corresponding $\vec{q}$ and performing the corrected reconstruction (averaging over all images left after filtering) results in a volume of 3750\,x\,4032\,x\,500 pixels$^3$ with maximum extent of approx. $\pm$3\,x\,$\pm$3\,x\,0.5 \AA$^{-3}$. The non-trivial shape of the reconstructed volume is influenced by the placement of the eight individual tiles of the detector, the applied mask to each tile, and the overall detector offset and slight rotation. +Using the determined rotation and translation to determine for each pixel the corresponding $\vec{q}$ and performing the corrected reconstruction (averaging over all images left after filtering) results in a volume of 3750\,x\,4032\,x\,500 pixels$^3$, corresponding to approximately $\pm$3\,x\,$\pm$3\,x\,0.5 \AA$^{-3}$. The non-trivial shape of the reconstructed volume is influenced by the placement of the eight individual tiles of the detector, the applied mask to each tile, and the overall detector offset and slight rotation. In slices in the $q_z=$\,0\,nm$^{-1}$ plane, Bragg peaks with a Laue index $h=0$ should be visible. As the determined detector rotation has an estimated uncertainty of approximately $\pm$0.2° in each direction, maximum intensity projections along 10 $q_z$ slices (up to $q_z\approx$\,0.15\,nm$^{-1}$) are shown in \fref{fig:resgaas1} to ensure Bragg peaks slightly rotated out the of the plane would still be visible. Marked areas in the overview on the left are shown in detail on the right. The noise level in the regions of interest around the expected Bragg peak positions estimated from the standard deviation over all pixels in the regions is 0.0002$\pm$0.0001. No Bragg peaks are clearly distinguishable, i.e., higher than 4$\sigma$, in the result. Similar results for the second, 45° rotated, sample are shown in \fref{fig:resgaas2}. 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Correlated noise by systematic noise sources does not follow this relation, and should be minimized, for example, by masking out affected areas of the detector. +As shown in \fref{fig:SNRNimages} if only considering shot and phase noise, uncorrelated between different images, the SNR scales with the square root of the number of images. This can be used to estimate the number of shots necessary to achieve an SNR\,$>5$ (Rose Criterion) to be able to verify IDI as an imaging method experimentally \cite{rose}. Correlated noise by systematic noise sources does not follow this relation, and should be minimized, for example, by masking out affected areas of the detector. \paragraph{Number of Modes} To Illustrate the effect of the number of modes on the signal and the SNR, simulations were performed by averaging the intensity over $M$ realizations of the random phases. In the results of the simulation of a crystalline sample, (\fref{fig:modes}) the predicted $1/M$ scaling of the signal can be seen. Furthermore, as expected, in \fref{fig:SNRNimages} the SNR shows the same scaling. @@ -287,9 +286,9 @@ \subsection{Parameters Influencing the Signal and Noise Characteristics} A simulation of a cubic single crystal with a simple cubic lattice of varying size from 20$^3$ to 200$^3$\,atoms was 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 was performed with 4x4 oversampling and rebinning to the detector size. Only a single coherence mode was simulated. The number of photons emitted by the sample in 4$\pi$ was chosen as half the number of atoms in the sample. Hence, the mean number of photons per pixel is especially for small crystal sizes very small, still higher than the expected photon yield in an experiment. -The peak signal-to-noise ratio was calculated by simulating 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}). This holds up to the point where the Bragg 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. -If instead of considering the peak value of the reconstruction as the signal, the three-dimensional integral over the Bragg peak would be considered as the signal, the decreasing width of the Bragg peak with increasing size of the crystal would result in a nearly constant SNR under these low photon count conditions. +The peak signal-to-noise ratio was calculated by simulating 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 estimation 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 linearly (as shown in \fref{fig:SNRNatoms}). This holds up to the point where the Bragg peaks are no longer fully sampled and the signal decreases linearly with a further increase in the number of atoms in the crystal, resulting in a nearly constant peak SNR. +If instead of considering the peak value of the reconstruction as the signal, the three-dimensional integral over the Bragg peak was considered the signal, the decreasing width of the Bragg peak with increasing size of the crystal would result in a nearly constant SNR under these low photon count conditions. @@ -306,7 +305,7 @@ \subsection{Parameters Influencing the Signal and Noise Characteristics} \caption{SNR dependence on detector size} \label{fig:SNRdetsize} \end{subfigure} - \caption[SNR dependence on crystal size and detector size]{SNR dependence on the crystal size (a): As the crystal size increases, the SNR increases linearly (as shown by the linear regression) with the number of atoms in the sample as long as the Bragg peaks is sufficiently sampled (blue points), in undersampling conditions the signal strength decreases linear with the number of atoms resulting in a near constant SNR (right side of the dashed line). The error bars show the standard deviation of the SNR calculated for the 12 visible Bragg peaks. For details on the simulation parameters, see main text. The SNR dependence on the detector size for a simulated foil sample (b) shows a linear relationship between the square root of the number of pixels and the SNR.} + \caption[SNR dependence on crystal size and detector size]{SNR dependence on the crystal size (a): As the crystal size increases, the SNR increases linearly (as shown by the linear regression) with the number of atoms in the sample as long as the Bragg peaks is sufficiently sampled (blue points), in undersampling conditions the signal strength decreases linearly with the number of atoms resulting in a near constant SNR (right side of the dashed line). The error bars show the standard deviation of the SNR calculated for the 12 visible Bragg peaks. For details on the simulation parameters, see main text. The SNR dependence on the detector size for a simulated foil sample (b) shows a linear relationship between the square root of the number of pixels and the SNR.} \end{figure} @@ -324,9 +323,9 @@ \subsection{Parameters Influencing the Signal and Noise Characteristics} A simulation with a varying number of spheres placed randomly inside the focal volume while ensuring a minimum distance between neighboring spheres with no other interaction was performed to decide if measuring the fluorescence of more than one nanoparticle in one shot is advantageous. The minimum distance was chosen according to the typical size of nanoparticle capping agents; the number of spheres was increased up to a simulated Poisson sphere distribution (see appendix \fref{algo:bridson} for details) as densest random placement of particles, leading to 25\% of the volume filled by possible emitters. -Three factors determine the structure factor of these samples: The structure factor of the focus, the structure factor of points following the poison sphere distribution, and the structure factor of a single sphere (see \fref{fig:multisphere1} and \fref{fig:multisphere3}). +Three factors determine the structure factor of these samples: The structure factor of the focus, the structure factor of points following a Poisson sphere distribution, and the structure factor of a single sphere (see \fref{fig:multisphere1} and \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 200\,nm FWHM, the fluorescence on a 1024x1024 pixel (pixel size 100\,\textmu m) detector placed 30\,cm was 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 were used for a radial reconstruction using the direct method (see \fref{fig:multisphere2}). -As the number of photons emitted by each sphere was kept constant, opposing effects occur: With an increasing number of particles, more are photons are emitted, and the Poisson noise is reduced, but the signal strength in low scattering angles decreases as the structure factor changes from a single sphere to a hard-sphere model and the influence of the focal volume and the distribution of the spheres on the structure factor increases, +As the number of photons emitted by each sphere was kept constant, opposing effects occur: With an increasing number of particles, more photons are emitted, and the Poisson noise is reduced. On the other hand, the signal strength in low scattering angles decreases as the structure factor changes from a single sphere to a hard-sphere model and the influence of the focal volume and the distribution of the spheres on the structure factor increases. In the case of 16 randomly positioned spheres, the mean distance to the nearest neighbor in the simulation is approximately three times the diameter. Compared to a single sample, the noise as measured by the standard error in the reconstruction is reduced by a factor of 10. At the same time, the signal strength at the first maximum decreases by a factor of only 3, leading to an overall gain in SNR. The dense random packing, with a mean nearest simulated neighbor distance of 1.4 times the diameter, reduces the noise further by an additional factor of 5, but reduces the signal by a factor of 10. Hence, the medium-dense sample shows the best SNR in this simulation. diff --git a/Tex/thanks.tex b/Tex/thanks.tex index 6952c3f..f77d927 100644 --- a/Tex/thanks.tex +++ b/Tex/thanks.tex @@ -1,6 +1,6 @@ \chapter*{Acknowledgements} \onehalfspacing -I would like to thank my supervisors, Tais Gorkhover and Thomas Möller, for their continued support; the Gorkhover group at SLAC, i.e., Stephan Kuschel, Matt Ware, and Siqi Li, for the warm welcome to California, for allowing me to learn from them (not only about the life of Selachimorpha), for joined discussions, planning, preparation, and execution of the experiments. \textcolor{white}{Baby shark, do do, do do do do} +I would like to thank my supervisors, Tais Gorkhover and Thomas Möller, for their continued support; the Gorkhover group at SLAC, i.e., Stephan Kuschel, Matt Ware, and Siqi Li, for the warm welcome to California, for allowing me to learn from them (not only about the life of Selachimorpha\footnote{\textcolor{white}{Baby shark, do do, do do do do}}), for joined discussions, planning, preparation, and execution of the experiments. Additionally, I would like to thank An Marshal for the help in getting the TEM images, Ben Reeves for his preparation of the GaAs samples, Carina Hedrich for the AAO sample preparation, Amy Cordones-Hahn for guidance in chemistry, and Cindy Patty for general awesomeness. diff --git a/Tex/theory.tex b/Tex/theory.tex index c0032d7..d5ed882 100755 --- a/Tex/theory.tex +++ b/Tex/theory.tex @@ -242,7 +242,7 @@ \section{Incoherent Diffractive Imaging} \caption[Main concept of IDI]{In IDI, an X-ray pulse is used to excite fluorescence inside the sample. The resulting speckle pattern is recorded on a integrating pixelated detector. Similar to the intensity correlations in the HBT experiment, the spatial correlation of pixels within the recorded speckle pattern is calculated. The need for fast detectors to resolve the fluctuating speckle pattern is replaced by the ultra-short excitation. Within each shot, only a finite number of random realizations of the speckle pattern is recorded. Therefore, the speckle pattern has contrast. To increase statistics in a few photon measurement, the average of the intensity correlations of many shots is taken.} \label{fig:idi} \end{figure} - The main idea leading from Hanbury Brown and Twiss's intensity correlations to IDI is replacing the need for detectors measuring the intensity fluctuations much faster than the variation of the random phases by using ultra-short pulses for fluorescence excitation. Each excitation then only creates finite realizations of the random phases, thus a speckle pattern. The basic setup used is shown in \fref{fig:idi}. + The main idea leading from Hanbury Brown and Twiss's intensity correlations to IDI is solving the need for detectors measuring the intensity fluctuations much faster than the variation of the random phases by using ultra-short pulses for fluorescence excitation. Each excitation then only creates finite realizations of the random phases, thus a speckle pattern. The basic setup used is shown in \fref{fig:idi}. A simplified explanation of IDI's ability to extract spatial information from these patterns can easily be made for chaotic light sources in a classical view by application of the Siegert relation. \paragraph{Siegert Relation} @@ -254,11 +254,11 @@ \section{Incoherent Diffractive Imaging} \end{equation} This relation of the first and second order correlation function is known as the \textit{Siegert Relation} \cite{ou2017}. -As, as according to the van Cittert–Zernike theorem (\fref{eq:vcz}), $g^1$ encodes structural information, +As $g^1$ encodes structural information according to the van Cittert–Zernike theorem (\fref{eq:vcz}), \begin{equation} g^1(\vec{k_1},\vec{k_2}) \propto \mathscr{F}S(\vec{r}) = S(\vec{q}) \,, \end{equation} -the difference of the intensity-intensity correlation from unity is proportional to the Fourier transform of the arrangement of emitters with $\vec{q}=\vec{k_1}-\vec{k_2}$ (see \fref{fig:scatteringvectors} for the definition of the involved vectors). +the difference of the intensity-intensity correlation from unity is proportional to the squared magnitude of the Fourier transform of the arrangement of emitters with $\vec{q}=\vec{k_1}-\vec{k_2}$ (see \fref{fig:scatteringvectors} for the definition of the involved vectors). This is the basic principle used in IDI to recover three-dimensional spatial information from the intensity correlation. \begin{figure} \centering @@ -339,8 +339,6 @@ \section{Signal to Noise Considerations} \begin{align*} M_{t,\text{rect}} =T \left[\int_{-\infty}^{\infty} \Lambda\left(\frac{t}{T}\right) \left|\gamma(t)\right|^2 \diff t \right]^{-1} \,, \numberthis \\ \label{eq:modesapprox} -\end{align*} -\begin{align*} \text{with }\Lambda(x) = \begin{cases} 1-\left|x\right|& \textit{if }|x|<1\\ 0 & \textit{otherwise}\\