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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Ptychographic extreme ultraviolet (EUV) diffractive imaging has emerged as a promising candidate for the next generationmetrology solutions in the semiconductor industry, as it can image wafer samples in reflection geometry at the nanoscale. This technique has surged attention recently, owing to the significant progress in high-harmonic generation (HHG) EUV sources and advancements in both hardware and software for computation. In this study, a novel algorithm is introduced and tested, which enables wavelength-multiplexed reconstruction that enhances the measurement throughput and introduces data diversity, allowing the accurate characterisation of sample structures. To tackle the inherent instabilities of the HHG source, a modal approach was adopted, which represents the cross-density function of the illumination by a series of mutually incoherent and independent spatial modes. The proposed algorithm was implemented on a mainstream machine learning platform, which leverages automatic differentiation to manage the drastic growth in model complexity and expedites the computation using GPU acceleration. By optimising over 200 million parameters, we demonstrate the algorithm's capacity to accommodate experimental uncertainties and achieve a resolution approaching the diffraction limit in reflection geometry. The reconstruction of wafer samples with 20-nm high patterned gold structures on a silicon substrate highlights our ability to handle complex physical interrelations involving a multitude of parameters. These results establish ptychography as an efficient and accurate metrology tool.@en

Next-generation metrology solutions in various technology areas require to image sample areas at the nanoscale. Coherent diffractive imaging based on ptychography is the route towards EUV imaging of nanostructures without lenses. A key component in a table-top EUV beamline is a high-brightness high-harmonic generation (HHG) source. Since our research is mainly directed towards wafer metrology for lithography in the semiconductor industry, we adhere to a reflection setup: the EUV light is scattered by the nanostructures at the surface of the sample, and is reflected towards a CCD camera, where a far-field diffraction pattern is recorded. A data-set comprising a multitude of these diffraction patterns is generated for partially overlapping positions of the focused probe on the sample. This provides the necessary redundancy for phase retrieval of the complex-valued field of the sample. Recent advancements in both hardware and software for computation enable the development of advanced algorithms. In particular, the benefits of automatic differentiation are exploited in order to cope with a drastic growth in model complexity. Our computational imaging algorithms realize wavelengthmultiplexed reconstruction and a modal approach for the spatial coherence of the source.@en

Ptychography as a means of lensless imaging is used in wafer metrology applications using Extreme Ultraviolet (EUV) light, where use of high quality optics is out-of-scope. To obtain sufficient diffraction intensity, reflection geometries with shallow (ca. 20 degrees) grazing incidence angles are used, which require re-sampling the diffraction data in a process called tilted plane correction (TPC). The tilt angle used for TPC is conventionally obtained through either experimentally tricky calibration, manual estimation based on diffraction pattern symmetry, although computational approaches are emerging. In this work we offer an improved numerical optimization approach as an alternative to TPC, where we use the flexibility offered by our Automatic Differentiation (AD)-based ptychography approach to include the data resampling into the forward model to learn the tilt angle. We demonstrate convergence of the approach across a range of incidence angles on simulated and experimental data obtained on an EUV beamline with either a high-harmonic generation (HHG)-based or a visible light source.@en

Ptychography in a reflection geometry shows great promise for non-destructive imaging of 3-dimensional nanostructures at the surface of a thick substrate. A major challenge to obtain high quality reflection-ptychographic images under near-grazing conditions has been to calibrate the incidence angle used to straighten the measured curved diffraction patterns in a process referred to as ‘tilted plane correction’ (TPC). In this work, we leverage the flexibility of automatic differentiation (AD)-based modeling to realize an alternative approach, where the tilted propagation is included in the forward model. Use of AD allows us to jointly optimize the tilt angles with the typical probe and object, eliminating the need for accurate calibration or random search optimization. The approach was validated using datasets generated with an extreme ultraviolet (EUV) beamline based on both a tabletop high harmonic generation (HHG) source and a visible laser. We demonstrate that the proposed approach can converge to a precision of ±0.05◦ for probe beams at 70◦ angle of incidence, possibly precise enough for use as a calibration approach. Furthermore, we demonstrate that optimizing for the tilt angles reduces artifacts and increases reconstruction fidelity. Use of AD not only streamlines the current ptychographic reconstruction process, but should also enable optimization of more complex models in other domains, which will likely be useful for future advancements in computational imaging.@en

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.

We present a highly stable, easy-to-use HHG source delivering a record photon flux of >10¹¹ photons/s at 69eV-75eV, being tunable to approx. 100eV which will be used for future photon-hungry applications.

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We demonstrate our beamline using a table-top HHG EUV source for lensless imaging application in reflection m ode. T he s ample r eflection fu nction is reconstructed using an auto-differentiation based ptychographic algorithm built on TensorFlow platform.

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High-harmonic generation (HHG) driven by ultrashort laser pulses is an established process for the generation of coherent extreme ultraviolet (XUV) to soft X-ray radiation, which has found widespread use in various applications [1]. In recent years photon-hungry applications such as coherent diffractive imaging [2] , [3] and applications based on statistical analysis [3] have required more powerful HHG sources, in particular, at high repetition rates. This need can be addressed by using high average power fiber lasers as the HHG drivers [4]. Here, we present a HHG-based XUV source, capable of providing a large photon flux across a wide range between 66 eV and 150 eV. It is driven by a commercial XUV beamline from Active Fiber Systems GmbH consisting of 100-W average power fiber-laser system, delivering up to 300J at <300-fs pulse duration. For HHG this system is operated at 100 W, 600 kHz. A post-compression unit is part of the device to shorten the pulses to ~35 fs, the average power remains at 63W. The turnkey source can provide unprecedented photon fluxes of >10 11 photons/s in each harmonic between 69 eV and 75 eV (HH57-HH63). All fluxes are given at the generation point, i.e. directly after the source.

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We present a parameter retrieval method which incorporates prior knowledge about the object into ptychography. The proposed method is applied to two applications: (1) parameter retrieval of small particles from Fourier ptychographic dark field measurements; (2) parameter retrieval of a rectangular structure with real-space ptychography. The influence of Poisson noise is discussed in the second part of the paper. The Cramér Rao Lower Bound in both applications is computed and Monte Carlo analysis is used to verify the calculated lower bound. With the computation results we report the lower bound for various noise levels and analyze the correlation of particles in application 1. For application 2 the correlation of parameters of the rectangular structure is discussed.

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We investigate the performance of ptychography with noisy data by analyzing the Cramér-Rao lower bound. The lower bound of ptychography is derived and numerically computed for both top-hat plane wave and structured illumination. The influence of Poisson noise on the ptychography reconstruction is discussed. The computation result shows that, if the estimator is unbiased, the minimum variance for Poisson noise is mostly determined by the illumination power and the transmission function of the object. Monte Carlo analysis is conducted to validate our calculation results for different photon flux numbers. Furthermore, the performance of the maximum-likelihood method and the approach of amplitude-based cost-function minimization is studied in the Monte Carlo analysis.

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Diffractive shearing interferometry (DSI) is a method that has recently been developed to perform lensless imaging using extreme ultraviolet radiation generated by high-harmonic generation. In this paper, we investigate the uniqueness of the DSI solution and the requirements for the support constraint size. We find that there can be multiple solutions to the DSI problem that consist of displaced copies of the actual object. These alternative solutions can be eliminated by enforcing a sufficiently tight support constraint, or by introducing additional synthetic constraints. We furthermore propose a new DSI algorithm inspired by the analogy with coherent diffractive imaging (CDI) algorithms: the original DSI algorithm is in a way analogous to the hybrid input-output algorithm as used in CDI, and we propose a new algorithm that is more analogous to the error reduction algorithm as used in CDI. We find that the newly proposed algorithm is suitable for final refinement of the reconstruction.

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The imaging and inspection of extreme ultraviolet (EUV) masks is an important aspect of EUV lithography. The availability of actinic mask inspection tools able to generate highly resolved defect maps of defective EUV layouts is needed to ensure defect-free wafer prints. The technological interest towards phase-shift absorber materials for the next generation of EUV masks, and the associated need for phase metrology at the absorber level, makes phase retrieval methods a particularly interesting option for actinic inspection. In this work we use ptychography as an inspection tool for EUV masks. We show how variational and statistical methods can be employed to include a-priori information in the ptychographic inverse problem and how to cluster different update rules - stemming from the minimization of appropriate cost functionals - to optimally include prior information in ptychography under Poisson noise.

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The development of actinic mask metrology tools represents one of the major challenges to be addressed on the roadmap of extreme ultra violet (EUV) lithography. Technological advancements in EUV lithography result in the possibility to print increasingly fine and highly resolved structures on a silicon wafer, however the presence of fine–scale defects, interspersed in the printable mask layout, may lead to defective wafer prints. Hence the development of
actinic methods for review of potential defect sites becomes paramount. Here, we report on a ptychographic algorithm that makes use of prior information about the object to be retrieved, generated by means of rigorous computations, to improve the detectability of defects whose dimensions are of the order of the wavelength. The comprehensive study demonstrates that the inclusion of prior information as a regularizer in the ptychographic optimization problem
results in a higher reconstruction quality and an improved robustness to noise with respect to the standard ptychographic iterative engine (PIE). We show that the proposed method decreases the number of scan positions necessary to retrieve an high quality image and relaxes requirements in terms of signal to noise ratio (SNR). The results are further compared with the state–of–art total
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We report on a method that allows microscopic image reconstruction from extreme-ultraviolet diffraction patterns without the need for object support constraints or other prior knowledge about the object structure. This is achieved by introducing additional diversity through rotation of an object in a rotationally asymmetric probe beam, produced by the spatial interference between two phase-coherent high-harmonic beams. With this rotational diffractive shearing interferometry method, we demonstrate robust image reconstruction of microscopic objects at wavelengths around 30 nm, using images recorded at only three to five different object rotations.

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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

Scatterometry is an important nonimaging and noncontact method for optical metrology. In scatterometry certain parameters of interest are determined by solving an inverse problem. This is done by minimizing a cost functional that quantifies the discrepancy among measured data and model evaluation. Solving the inverse problem is mathematically challenging owing to the instability of the inversion and to the presence of several local minima that are caused by correlation among parameters. This is a relevant issue, particularly when the inverse problem to be solved requires the retrieval of a high number of parameters. In such cases, methods to reduce the complexity of the problem are to be sought. In this work, we propose an algorithm suitable to automatically determine which subset of the parameters is mostly relevant in the model, and we apply it to the reconstruction of 2D and 3D scatterers. We compare the results with local sensitivity analysis and with the screening method proposed by Morris.

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EUV lithography is the main candidate for patterning of future technology nodes. Its successful implementation depends on many aspects, among which the availability of actinic mask metrology tools able to inspect the patterned absorber in order to control and monitor the lithographic process. In this work, we perform a simulation study to assess the performance of coherent diffractive imaging (CDI) and related phase retrieval methods for the reconstruction of non-Trivially shaped and a-periodic nanostructures from far field intensity data.

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A novel non-iterative phase retrieval method is proposed and demonstrated with a
proof-of-principle experiment. The method uses a fixed specially designed mask and through-focus intensity measurements. It is demonstrated that this method is robust to spatial partial coherence in the illumination, making it suitable for coherent diffractive imaging using spatially partially coherent light, as well as for coherence characterization.@en