With the continuously growing transistor density in the semiconductor industry, where smaller features and tighter tolerances in the nanometer scale are the norm, there is a growing demand for more advanced metrology tools during manufacturing. While optical metrology in the visible range is the main workhorse, such as scatterometry, its performance is limited by the wavelength. Moving beyond the visible spectrum to extreme ultraviolet (EUV) and soft X-ray (SXR) wavelengths promises higher resolutions, but not without challenges. In this wavelength regime traditional microscopy struggles due to the non-availability of high numerical aperture optics for EUV and SXR at reasonable costs, failing to fully make use of the potential benefits of the shorter wavelength.

This thesis covers an alternative route to traditional microscopy through advanced lensless imaging or coherent diffractive imaging (CDI) techniques, particularly focusing on ptychography within the EUV and soft X-ray ranges demonstrated on a designed EUV beamline. Before we go into details about the setup we start with the basics of diffraction, coherent imaging, and ptychography in chapter 2. In chapter 3 we cover the generation of coherent EUV and SXR light from various sources, ranging from large scale facilities like synchrotrons to table-top High Harmonic Generation (HHG) setups.

From this point we cover the design and construction of a table-top EUV lensless imaging beamline in chapter 4, which has been designed and built from scratch at the Optics Research Group at the Delft University of Technology as part of this thesis. This section covers the design, optimization, and assembly of the beamline tailored for lensless imaging of industry relevant samples in a reflection configuration using EUV and soft X-rays from HHG sources. The beamline is split into several subsystems, a high intensity femtosecond fiber laser, the HHG EUV source, optical components for spectral filtering and illumination, and optomechanical systems required for ptychography which are individually covered in the chapter.

In chapter 5 we demonstrate one of the first ptychography reconstructions obtained on the EUV beamline, as presented in chapter 4, by illuminating an object with EUV light at 17.3 nm and 17.9 nm in a grazing reflection orientation of 20 degrees. The object, a silicon substrate which has been patterned with a 20 nanometer thick gold layer with feature sizes from a few microns down to 15 nanometers, has been reconstructed. This reconstruction has been generated with an in-house developed ptychography algorithm, based on automatic differentiation, for the ptychography reconstruction process. The experimental results demonstrate the feasibility of achieving high resolution reconstructions.

Line features down to 50 nanometers were retrieved with this method, close to the diffraction limit given an imaging NA of 0.17 at 18 nm wavelength. A reconstructed structure height of 23.6 ± 0.62 nm agrees well with the 20 nm nominal design value and the height retrieved from an atomic force microscope (AFM) measurement of 22.8 ± 1.45 nm.

The results in chapter 5 are quite promising, though there are certain challenges which need to be solved to further improve the reconstruction quality. One of these challenges are intensity stability issues which are typically associated with EUV sources based on high harmonic generation. Chapter 6 proposes a simple computational method to mitigate intensity fluctuations during ptychography scans by introducing a scanning position dependent multiplication factor. The algorithm effectively corrects for power fluctuations, enabling object reconstruction even in the presence of significant intensity variations up to 50 percent during the overall scan.

Chapter 7 presents a compact schlieren (from the German word ’streak’) imaging system integrated within the HHG EUV source, enabling quantitative density retrieval of the gas jet used to drive the high harmonic generation process. Schlieren imaging provides a straightforward alternative to vibration sensitive techniques like interferometry and can be used as a standardized tool for HHG sources, allowing for a better comparison among different HHG setups and for the optimization of HHG light sources.

In summary, this thesis does not only cover the field of lensless EUV microscopy but also covers the design process of such a beamline and the challenges associated with HHG EUV sources. This work presents a starting point for experimental EUV metrology within the Optics Research Group at the Delft University of Technology and enables future academic research relevant for the semiconductor industry.
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Because of the growing demand for more advanced electric devices, an exponential growth of the number of transistors are supposed to be integrated into a single chip. To manufacture devices in the scale of nanometer cost-effectively, an accurate measurement for lithography process calibrating is necessary. Ptychography is a computational imaging technique which has the potential to serve as a metrology solution for semiconductor devices. It can reconstruct complex-valued permittivity function of an object from an extensive set of measured intensities of the diffraction patterns in the far field. It should be noted that also 2D and 3D a-periodic objects can be measured with this technique. Currently, this technique has been widely used in reconstructing thin and weakly scattering (satisfying the first Born approximation) objects in a transmission geometry (detectors and sources are placed on different sides of the objects).We propose an accurate 3D ptychography multi-layer Born model, and apply it to reconstruct refractive index distributions of semi-conductors. This model slices the sample into thin layers and first Born approximations are applied to each layer sequentially. For a transmission geometry, this model considers both forward scattering effects and backward scattering effects \cite{Chen2020Multi-layerMicroscopy}, as opposed to normal beam propagation models. For reflection geometry (detectors and sources are placed on the same side of the object), this model collects reflected fields on each layer of the sample and these fields will interact with the object again later together with other upwards fields. This procedure enables detectors arranged in a reflection geometry to gather complete information about the object. We implement the model on the existing ptychography platform based on the auto-differentiation (AD) solvers and manage to reconstruct high-resolution images of 3D objects.

The scientific community recognizes the critical role played by ptychography in nanoscale imaging. Compared with the conventional imaging, which has high requirements on the manufacturing of optical elements, ptychography, as a computational imaging technique, uses a set of measured intensities of the diffraction patterns to reconstruct the image of the object and hence no imaging system is needed. This technique is especially useful in the short wavelength, e.g. EUV, regime, where manufacturing high quality optical elements such as mirrors is extremely expensive.

Most of the present ptychographic algorithms require the illumination of the object to be both spatially and temporally coherent so that the diffraction pattern can be interpreted as the intensity of the Fourier transform of the field exiting the object. However, the coherence of the sources that produce the EUV radiation often cannot be guaranteed. Therefore, it is crucial to extend the ptychography method to consider partial coherence effects. This requires the use of a flexible propagator which depends on the wavelength to deal with the temporal partial coherence and a modal representation for the spatially partially coherent field. Also, the ambiguity of the reconstructed modes of the probe will be solved by an orthogonalization approach, which could enhance the reproducibility of the results. These methods will be implemented on an existing ptychography platform based on automatic-differentiation and will be validated using both simulation data and experimental data.

In this thesis, several phase retrieval methods are discussed. Since the focus will mainly be on theory rather than experiment, the structure has been determined by the similarities and differences of the mathematics of these methods. For example, a distinction is made between non-iterative and iterative methods, and between single-shot iterative phase retrieval and multiple-shot iterative phase retrieval (ptychography). However, it must be noted that phase retrieval methods that are mathematically similar, are suitable for fundamentally different experimental setups. For example, one can consider setups for lensless imaging, of which an interesting application is metrology using Extreme Ultraviolet (EUV) radiation. In such setups, no focusing optics are used, and one typically computes an image from far-field intensity patterns. On the other hand, there are setups for aberrated imaging. In these setups, one does use focusing optics to form images, but by introducing some sort of variations or perturbations, one can generate a set of images from which a complex-valued field can be computed. For example, regular ptychography and Fourier ptychography are mathematically the same, but the former is used for lensless imaging, while the latter is used for aberrated imaging. Mathematically, the only difference between these two ptychographic approaches is that the roles of object space and Fourier space are interchanged.@en