Fluorescence tomography (FT) is an emerging non-invasive in vivo molecular imaging modality that aims at quantification and three-dimensional (3D) localization of fluorescent tagged inclusions, such as cancer lesions and drug molecules, buried deep in human and animal subjects. Depth-resolved 3D reconstruction of fluorescent inclusions distributed over the volume of optically turbid biological tissue using the diffuse fluorescent photons detected on the skin poses a highly ill-conditioned problem, as depth information must be extracted from boundary data. Due to this ill-posed nature of FT reconstructions, noise and errors in the data can severely impair the accuracy of the 3D reconstructions. Consequently, improvements in the signal-to-noise ratio (SNR) of the data significantly enhance the quality of the FT reconstructions. Furthermore, enhancing the SNR of the FT data can greatly contribute to the speed of FT scans. The pivotal factor in the SNR of the FT data is the power of the radiation illuminating the subject and exciting the administered fluorescent agents. In existing single-point illumination FT systems, the illumination power level is limited by the skin maximum radiation exposure levels. In this research, a multiplexed architecture governed by the Hadamard transform was conceptualized, developed, and experimentally implemented for orders-of-magnitude enhancement of the SNR and the robustness of FT reconstructions. The multiplexed FT system allows for Hadamard-coded multi-point illumination of the subject while maintaining the maximal information content of the FT data. The significant improvements offered by the multiplexed FT system were validated by numerical and experimental studies carried out using a custom-built multiplexed FT system developed exclusively in this work. The studies indicate that Hadamard multiplexing offers significantly enhanced robustness in reconstructing deep fluorescent inclusions from low-SNR FT data.

This thesis is devoted to solving the inverse source problem arising in image reconstruction problems. In general, the solution is non-unique and the problem is severely ill-posed. Therefore, small perturbations, such as the noise in the data, and the modeling error in the forward problem, will cause huge errors in the computations. In practice, the most widely used method to tackle the problem is based on Tikhonov-type regularizations, which minimizes a cost function combining a regularization term and a data fitting term. However, because the two tasks, namely regularization and data fitting, are coupled together in Tikhonov regularization, they are difficult to solve. It happens even if each task can be efficiently solved when they are separate. We propose a method to overcome the major difficulties, namely the non-uniqueness of the solution and noisy data fitting, separately. First we find a particular solution called the orthogonal solution that satisfies the data fitting term. Then we add to it a correction function in the kernel space so that the final solution fulfills the regularization and other physical requirements. The key idea is that the correction function in the kernel has no impact to the data fitting, and the regularization is imposed in a smaller space. Moreover, there is no parameter needed to balance the data fitting and regularization terms. As a case study, we apply the proposed method to Fluorescence Tomography (FT), an emerging imaging technique well known for its ill-posedness and low image resolution in existing reconstruction techniques. We demonstrate by theory and examples that the proposed algorithm can drastically improve the computation speed and the image resolution over existing methods.

Hard x rays come along with a variety of extraordinary properties which make them an excellent probe for investigation in science, technology and medicine. Their large attenuation length in matter opens up the possibility to use hard x-rays for non-destructive investigation of the inner structure of specimens. Medical radiography is one important example of exploiting this feature. Since their discovery by W. C. Röntgen in 1895, a large variety of x-ray analytical techniques have been developed and successfully applied, such as x-ray crystallography, reflectometry, fluorescence spectroscopy, x-ray absorption spectroscopy, small angle x-ray scattering, and many more. Each of those methods reveals information about certain physical properties, but usually, these properties are an average over the complete sample region illuminated by the x rays. In order to obtain the spatial distribution of those properties in inhomogeneous samples, scanning microscopy techniques have to be applied, screening the sample with a small x-ray beam. The spatial resolution is limited by the finite size of the beam. The availability of highly brilliant x-ray sources at third generation synchrotron radiation facilities together with the development of enhanced focusing x-ray optics made it possible to generate increasingly small high intense x-ray beams, pushing the spatial resolution down to the sub-100 nm range. During this thesis the prototype of a hard x-ray scanning microscope utilizing microstructured nanofocusing lenses was designed, built, and successfully tested. The nanofocusing x-ray lenses were developed by our research group of the Institute of Structural Physics at the Technische Universität Dresden. The prototype instrument was installed at the ESRF beamline ID 13. A wide range of experiments like fluorescence element mapping, fluorescence tomography, x-ray nano-diffraction, coherent x-ray diffraction imaging, and x-ray ptychography were performed as part of this thesis. The hard x-ray scanning microscope provides a stable x-ray beam with a full width at half maximum size of 50-100 nm near the focal plane. The nanoprobe was also used for characterization of nanofocusing lenses, crucial to further improve them. Based on the experiences with the prototype, an advanced version of a hard x-ray scanning microscope is under development and will be installed at the PETRA III beamline P06 dedicated as a user instrument for scanning microscopy. This document is organized as follows. A short introduction motivating the necessity for building a hard x-ray scanning microscope is followed by a brief review of the fundamentals of hard x-ray physics with an emphasis on free-space propagation and interaction with matter. After a discussion of the requirements on the x-ray source for the nanoprobe, the main features of synchrotron radiation from an undulator source are shown. The properties of the nanobeam generated by refractive x-ray lenses are treated as well as a two-stage focusing scheme for tailoring size, flux and the lateral coherence properties of the x-ray focus. The design and realization of the microscope setup is addressed, and a selection of experiments performed with the prototype version is presented, before this thesis is finished with a conclusion and an outlook on prospective plans for an improved microscope setup to be installed at PETRA III. / Aufgrund ihrer hervorragenden Eigenschaften kommt harte Röntgenstrahlung in vielfältiger Weise in der Wissenschaft, Industrie und Medizin zum Einsatz. Vor allem die Fähigkeit, makroskopische Gegenstände zu durchdringen, eröffnet die Möglichkeit, im Innern ausgedehnter Objekte verborgene Strukturen zum Vorschein zu bringen, ohne den Gegenstand zerstören zu müssen. Eine Vielzahl röntgenanalytischer Verfahren wie zum Beispiel Kristallographie, Reflektometrie, Fluoreszenzspektroskopie, Absorptionsspektroskopie oder Kleinwinkelstreuung sind entwickelt und erfolgreich angewendet worden. Jede dieser Methoden liefert gewisse strukturelle, chemische oder physikalische Eigenschaften der Probe zutage, allerdings gemittelt über den von der Röntgenstrahlung beleuchteten Bereich. Um eine ortsaufgelöste Verteilung der durch die Röntgenanalyse gewonnenen Information zu erhalten, bedarf es eines sogenannten Mikrostrahls, durch den die Probe lokal abgetastet werden kann. Die dadurch erreichbare räumliche Auflösung ist durch die Größe des Mikrostrahls begrenzt. Aufgrund der Verfügbarkeit hinreichend brillanter Röntgenquellen in Form von Undulatoren an Synchrotronstrahlungseinrichtungen und des Vorhandenseins verbesserter Röntgenoptiken ist es in den vergangen Jahren gelungen, immer kleinere intensive Röntgenfokusse zu erzeugen und somit das räumliche Auflösungsvermögen der Röntgenrastermikroskope auf unter 100 nm zu verbessern. Gegenstand dieser Arbeit ist der Prototyp eines Rastersondenmikroskops für harte Röntgenstrahlung unter Verwendung refraktiver nanofokussierender Röntgenlinsen, die von unserer Arbeitsgruppe am Institut für Strukturphysik entwickelt und hergestellt werden. Das Rastersondenmikroskop wurde im Rahmen dieser Promotion in Dresden konzipiert und gebaut sowie am Strahlrohr ID 13 des ESRF installiert und erfolgreich getestet. Das Gerät stellt einen hochintensiven Röntgenfokus der Größe 50-100 nm zur Verfügung, mit dem im Verlaufe dieser Doktorarbeit zahlreiche Experimente wie Fluoreszenztomographie, Röntgennanobeugung, Abbildung mittels kohärenter Röntgenbeugung sowie Röntgenptychographie erfolgreich durchgeführt wurden. Das Rastermikroskop dient unter anderem auch dem Charakterisieren der nanofokussierenden Linsen, wobei die dadurch gewonnenen Erkenntnisse in die Herstellung verbesserten Linsen einfließen. Diese Arbeit ist wie folgt strukturiert. Ein kurzes einleitendes Kapitel dient als Motivation für den Bau eines Rastersondenmikroskops für harte Röntgenstrahlung. Es folgt eine Einführung in die Grundlagen der Röntgenphysik mit Hauptaugenmerk auf die Ausbreitung von Röntgenstrahlung im Raum und die Wechselwirkungsmechanismen von Röntgenstrahlung mit Materie. Anschließend werden die Anforderungen an die Röntgenquelle besprochen und die Vorzüge eines Undulators herausgestellt. Wichtige Eigenschaften eines mittels refraktiver Röntgenlinsen erzeugten Röntgenfokus werden behandelt, und das Konzept einer Vorfokussierung zur gezielten Anpassung der transversalen Kohärenzeigenschaften an die Erfordernisse des Experiments wird besprochen. Das Design und die technische Realisierung des Rastermikroskops werden ebenso dargestellt wie eine Auswahl erfolgreicher Experimente, die am Gerät vollzogen wurden. Die Arbeit endet mit einem Ausblick, der mögliche Weiterentwicklungen in Aussicht stellt, unter anderem den Aufbau eines verbesserten Rastermikroskops am PETRA III-Strahlrohr P06.

Hard x rays come along with a variety of extraordinary properties which make them an excellent probe for investigation in science, technology and medicine. Their large attenuation length in matter opens up the possibility to use hard x-rays for non-destructive investigation of the inner structure of specimens. Medical radiography is one important example of exploiting this feature. Since their discovery by W. C. Röntgen in 1895, a large variety of x-ray analytical techniques have been developed and successfully applied, such as x-ray crystallography, reflectometry, fluorescence spectroscopy, x-ray absorption spectroscopy, small angle x-ray scattering, and many more. Each of those methods reveals information about certain physical properties, but usually, these properties are an average over the complete sample region illuminated by the x rays. In order to obtain the spatial distribution of those properties in inhomogeneous samples, scanning microscopy techniques have to be applied, screening the sample with a small x-ray beam. The spatial resolution is limited by the finite size of the beam. The availability of highly brilliant x-ray sources at third generation synchrotron radiation facilities together with the development of enhanced focusing x-ray optics made it possible to generate increasingly small high intense x-ray beams, pushing the spatial resolution down to the sub-100 nm range. During this thesis the prototype of a hard x-ray scanning microscope utilizing microstructured nanofocusing lenses was designed, built, and successfully tested. The nanofocusing x-ray lenses were developed by our research group of the Institute of Structural Physics at the Technische Universität Dresden. The prototype instrument was installed at the ESRF beamline ID 13. A wide range of experiments like fluorescence element mapping, fluorescence tomography, x-ray nano-diffraction, coherent x-ray diffraction imaging, and x-ray ptychography were performed as part of this thesis. The hard x-ray scanning microscope provides a stable x-ray beam with a full width at half maximum size of 50-100 nm near the focal plane. The nanoprobe was also used for characterization of nanofocusing lenses, crucial to further improve them. Based on the experiences with the prototype, an advanced version of a hard x-ray scanning microscope is under development and will be installed at the PETRA III beamline P06 dedicated as a user instrument for scanning microscopy. This document is organized as follows. A short introduction motivating the necessity for building a hard x-ray scanning microscope is followed by a brief review of the fundamentals of hard x-ray physics with an emphasis on free-space propagation and interaction with matter. After a discussion of the requirements on the x-ray source for the nanoprobe, the main features of synchrotron radiation from an undulator source are shown. The properties of the nanobeam generated by refractive x-ray lenses are treated as well as a two-stage focusing scheme for tailoring size, flux and the lateral coherence properties of the x-ray focus. The design and realization of the microscope setup is addressed, and a selection of experiments performed with the prototype version is presented, before this thesis is finished with a conclusion and an outlook on prospective plans for an improved microscope setup to be installed at PETRA III.:1 Introduction ............................................... 1 2 Basic Properties of Hard X Rays ............................ 3 2.1 Free Propagation of X Rays ............................... 3 2.1.1 The Helmholtz Equation ................................. 4 2.1.2 Integral Theorem of Helmholtz and Kirchhoff ............ 6 2.1.3 Fresnel-Kirchhoff's Diffraction Formula ................ 8 2.1.4 Fresnel-Kirchhoff Propagation .......................... 11 2.2 Interaction of X Rays with Matter ........................ 13 2.2.1 Complex Index of Refraction ............................ 13 2.2.2 Attenuation ............................................ 15 2.2.3 Refraction ............................................. 18 3 The X-Ray Source ........................................... 21 3.1 Requirements ............................................. 21 3.1.1 Energy and Energy Bandwidth ............................ 21 3.1.2 Source Size and Divergence ............................. 23 3.1.3 Brilliance ............................................. 23 3.2 Synchrotron Radiation .................................... 24 3.3 Layout of a Synchrotron Radiation Facility ............... 27 3.4 Liénard-Wiechert Fields .................................. 29 3.5 Dipole Magnets ........................................... 31 3.6 Insertion Devices ........................................ 36 3.6.1 Multipole Wigglers ..................................... 36 3.6.2 Undulators ............................................. 37 4 X-Ray Optics ............................................... 39 4.1 Refractive X-Ray Lenses .................................. 40 4.2 Compound Parabolic Refractive Lenses (CRLs) .............. 41 4.3 Nanofocusing Lenses (NFLs) ............................... 43 4.4 Adiabatically Focusing Lenses (AFLs) ..................... 45 4.5 Focal Distance ........................................... 46 4.6 Transverse Focus Size .................................... 50 4.7 Beam Caustic ............................................. 52 4.8 Depth of Focus ........................................... 53 4.9 Beam Divergence .......................................... 53 4.10 Chromaticity ............................................ 54 4.11 Transmission and Cross Section .......................... 55 4.12 Transverse Coherence .................................... 56 4.12.1 Mutual Intensity Function ............................. 57 4.12.2 Free Propagation of Mutual Intensity .................. 57 4.12.3 Mutual Intensity In The Focal Plane ................... 58 4.12.4 Diffraction Limited Focus ............................. 59 4.13 Coherent Flux ........................................... 60 4.14 Two-Stage Focusing ...................................... 64 4.14.1 The Prefocusing Parameter ............................. 65 4.14.2 Required Refractive Power ............................. 67 4.14.3 Flux Considerations ................................... 70 4.14.4 Astigmatic Prefocusing ................................ 75 5 Nanoprobe Setup ............................................ 77 5.1 X-Ray Optics ............................................. 78 5.1.1 Nanofocusing Lenses .................................... 79 5.1.2 Entry Slits ............................................ 82 5.1.3 Pinhole ................................................ 82 5.1.4 Additional Shielding ................................... 83 5.1.5 Vacuum and Helium Tubes ................................ 83 5.2 Sample Stages ............................................ 84 5.2.1 High Resolution Scanner ................................ 84 5.2.2 High Precision Rotational Stage ........................ 85 5.2.3 Coarse Linear Stages ................................... 85 5.2.4 Goniometer Head ........................................ 85 5.3 Detectors ................................................ 86 5.3.1 High Resolution X-Ray Camera ........................... 86 5.3.2 Diffraction Cameras .................................... 89 5.3.3 Energy Dispersive Detectors ............................ 91 5.3.4 Photodiodes ............................................ 93 5.4 Control Software ......................................... 94 6 Experiments ................................................ 97 6.1 Lens Alignment ........................................... 97 6.2 Focus Characterization ................................... 99 6.2.1 Knife-Edge Scans ....................................... 100 6.2.2 Far-Field Measurements ................................. 102 6.2.3 X-Ray Ptychography ..................................... 103 6.3 Fluorescence Spectroscopy ................................ 105 6.3.1 Fluorescence Element Mapping ........................... 107 6.3.2 Fluorescence Tomography ................................ 110 6.4 Diffraction Experiments .................................. 111 6.4.1 Microdiffraction on Phase Change Media ................. 112 6.4.2 Microdiffraction on Stranski-Krastanow Islands ......... 113 6.4.3 Coherent X-Ray Diffraction Imaging of Gold Particles ... 115 6.4.4 X-Ray Ptychography of a Nano-Structured Microchip ...... 117 7 Conclusion and Outlook ..................................... 121 Bibliography ................................................. 125 List of Figures .............................................. 139 List of Publications ......................................... 141 Danksagung ................................................... 145 Curriculum Vitae ............................................. 149 Erklärung .................................................... 151 / Aufgrund ihrer hervorragenden Eigenschaften kommt harte Röntgenstrahlung in vielfältiger Weise in der Wissenschaft, Industrie und Medizin zum Einsatz. Vor allem die Fähigkeit, makroskopische Gegenstände zu durchdringen, eröffnet die Möglichkeit, im Innern ausgedehnter Objekte verborgene Strukturen zum Vorschein zu bringen, ohne den Gegenstand zerstören zu müssen. Eine Vielzahl röntgenanalytischer Verfahren wie zum Beispiel Kristallographie, Reflektometrie, Fluoreszenzspektroskopie, Absorptionsspektroskopie oder Kleinwinkelstreuung sind entwickelt und erfolgreich angewendet worden. Jede dieser Methoden liefert gewisse strukturelle, chemische oder physikalische Eigenschaften der Probe zutage, allerdings gemittelt über den von der Röntgenstrahlung beleuchteten Bereich. Um eine ortsaufgelöste Verteilung der durch die Röntgenanalyse gewonnenen Information zu erhalten, bedarf es eines sogenannten Mikrostrahls, durch den die Probe lokal abgetastet werden kann. Die dadurch erreichbare räumliche Auflösung ist durch die Größe des Mikrostrahls begrenzt. Aufgrund der Verfügbarkeit hinreichend brillanter Röntgenquellen in Form von Undulatoren an Synchrotronstrahlungseinrichtungen und des Vorhandenseins verbesserter Röntgenoptiken ist es in den vergangen Jahren gelungen, immer kleinere intensive Röntgenfokusse zu erzeugen und somit das räumliche Auflösungsvermögen der Röntgenrastermikroskope auf unter 100 nm zu verbessern. Gegenstand dieser Arbeit ist der Prototyp eines Rastersondenmikroskops für harte Röntgenstrahlung unter Verwendung refraktiver nanofokussierender Röntgenlinsen, die von unserer Arbeitsgruppe am Institut für Strukturphysik entwickelt und hergestellt werden. Das Rastersondenmikroskop wurde im Rahmen dieser Promotion in Dresden konzipiert und gebaut sowie am Strahlrohr ID 13 des ESRF installiert und erfolgreich getestet. Das Gerät stellt einen hochintensiven Röntgenfokus der Größe 50-100 nm zur Verfügung, mit dem im Verlaufe dieser Doktorarbeit zahlreiche Experimente wie Fluoreszenztomographie, Röntgennanobeugung, Abbildung mittels kohärenter Röntgenbeugung sowie Röntgenptychographie erfolgreich durchgeführt wurden. Das Rastermikroskop dient unter anderem auch dem Charakterisieren der nanofokussierenden Linsen, wobei die dadurch gewonnenen Erkenntnisse in die Herstellung verbesserten Linsen einfließen. Diese Arbeit ist wie folgt strukturiert. Ein kurzes einleitendes Kapitel dient als Motivation für den Bau eines Rastersondenmikroskops für harte Röntgenstrahlung. Es folgt eine Einführung in die Grundlagen der Röntgenphysik mit Hauptaugenmerk auf die Ausbreitung von Röntgenstrahlung im Raum und die Wechselwirkungsmechanismen von Röntgenstrahlung mit Materie. Anschließend werden die Anforderungen an die Röntgenquelle besprochen und die Vorzüge eines Undulators herausgestellt. Wichtige Eigenschaften eines mittels refraktiver Röntgenlinsen erzeugten Röntgenfokus werden behandelt, und das Konzept einer Vorfokussierung zur gezielten Anpassung der transversalen Kohärenzeigenschaften an die Erfordernisse des Experiments wird besprochen. Das Design und die technische Realisierung des Rastermikroskops werden ebenso dargestellt wie eine Auswahl erfolgreicher Experimente, die am Gerät vollzogen wurden. Die Arbeit endet mit einem Ausblick, der mögliche Weiterentwicklungen in Aussicht stellt, unter anderem den Aufbau eines verbesserten Rastermikroskops am PETRA III-Strahlrohr P06.:1 Introduction ............................................... 1 2 Basic Properties of Hard X Rays ............................ 3 2.1 Free Propagation of X Rays ............................... 3 2.1.1 The Helmholtz Equation ................................. 4 2.1.2 Integral Theorem of Helmholtz and Kirchhoff ............ 6 2.1.3 Fresnel-Kirchhoff's Diffraction Formula ................ 8 2.1.4 Fresnel-Kirchhoff Propagation .......................... 11 2.2 Interaction of X Rays with Matter ........................ 13 2.2.1 Complex Index of Refraction ............................ 13 2.2.2 Attenuation ............................................ 15 2.2.3 Refraction ............................................. 18 3 The X-Ray Source ........................................... 21 3.1 Requirements ............................................. 21 3.1.1 Energy and Energy Bandwidth ............................ 21 3.1.2 Source Size and Divergence ............................. 23 3.1.3 Brilliance ............................................. 23 3.2 Synchrotron Radiation .................................... 24 3.3 Layout of a Synchrotron Radiation Facility ............... 27 3.4 Liénard-Wiechert Fields .................................. 29 3.5 Dipole Magnets ........................................... 31 3.6 Insertion Devices ........................................ 36 3.6.1 Multipole Wigglers ..................................... 36 3.6.2 Undulators ............................................. 37 4 X-Ray Optics ............................................... 39 4.1 Refractive X-Ray Lenses .................................. 40 4.2 Compound Parabolic Refractive Lenses (CRLs) .............. 41 4.3 Nanofocusing Lenses (NFLs) ............................... 43 4.4 Adiabatically Focusing Lenses (AFLs) ..................... 45 4.5 Focal Distance ........................................... 46 4.6 Transverse Focus Size .................................... 50 4.7 Beam Caustic ............................................. 52 4.8 Depth of Focus ........................................... 53 4.9 Beam Divergence .......................................... 53 4.10 Chromaticity ............................................ 54 4.11 Transmission and Cross Section .......................... 55 4.12 Transverse Coherence .................................... 56 4.12.1 Mutual Intensity Function ............................. 57 4.12.2 Free Propagation of Mutual Intensity .................. 57 4.12.3 Mutual Intensity In The Focal Plane ................... 58 4.12.4 Diffraction Limited Focus ............................. 59 4.13 Coherent Flux ........................................... 60 4.14 Two-Stage Focusing ...................................... 64 4.14.1 The Prefocusing Parameter ............................. 65 4.14.2 Required Refractive Power ............................. 67 4.14.3 Flux Considerations ................................... 70 4.14.4 Astigmatic Prefocusing ................................ 75 5 Nanoprobe Setup ............................................ 77 5.1 X-Ray Optics ............................................. 78 5.1.1 Nanofocusing Lenses .................................... 79 5.1.2 Entry Slits ............................................ 82 5.1.3 Pinhole ................................................ 82 5.1.4 Additional Shielding ................................... 83 5.1.5 Vacuum and Helium Tubes ................................ 83 5.2 Sample Stages ............................................ 84 5.2.1 High Resolution Scanner ................................ 84 5.2.2 High Precision Rotational Stage ........................ 85 5.2.3 Coarse Linear Stages ................................... 85 5.2.4 Goniometer Head ........................................ 85 5.3 Detectors ................................................ 86 5.3.1 High Resolution X-Ray Camera ........................... 86 5.3.2 Diffraction Cameras .................................... 89 5.3.3 Energy Dispersive Detectors ............................ 91 5.3.4 Photodiodes ............................................ 93 5.4 Control Software ......................................... 94 6 Experiments ................................................ 97 6.1 Lens Alignment ........................................... 97 6.2 Focus Characterization ................................... 99 6.2.1 Knife-Edge Scans ....................................... 100 6.2.2 Far-Field Measurements ................................. 102 6.2.3 X-Ray Ptychography ..................................... 103 6.3 Fluorescence Spectroscopy ................................ 105 6.3.1 Fluorescence Element Mapping ........................... 107 6.3.2 Fluorescence Tomography ................................ 110 6.4 Diffraction Experiments .................................. 111 6.4.1 Microdiffraction on Phase Change Media ................. 112 6.4.2 Microdiffraction on Stranski-Krastanow Islands ......... 113 6.4.3 Coherent X-Ray Diffraction Imaging of Gold Particles ... 115 6.4.4 X-Ray Ptychography of a Nano-Structured Microchip ...... 117 7 Conclusion and Outlook ..................................... 121 Bibliography ................................................. 125 List of Figures .............................................. 139 List of Publications ......................................... 141 Danksagung ................................................... 145 Curriculum Vitae ............................................. 149 Erklärung .................................................... 151