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

We introduce a novel approach for ptychographic reconstruction, integrating a pre-trained autoencoder within a reconstruction framework based on automatic differentiation. This enables noise-robust imaging and insight into optimization landscapes for applications with prior object knowledge.@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

Computational imaging is increasingly vital for a broad spectrum of applications, ranging from biological to material sciences. This includes applications where the object is known and sufficiently sparse, allowing it to be described with a reduced number of parameters. When no explicit parameterization is available, a deep generative model can be trained to represent an object in a low-dimensional latent space. In this paper, we harness this dimensionality reduction capability of autoencoders to search for the object solution within the latent space rather than the object space. We demonstrate what we believe to be a novel approach to ptychographic image reconstruction by integrating a deep generative model obtained from a pre-trained autoencoder within an automatic differentiation ptychography (ADP) framework. This approach enables the retrieval of objects from highly ill-posed diffraction patterns, offering an effective method for noise-robust latent vector reconstruction in ptychography. Moreover, the mapping into a low-dimensional latent space allows us to visualize the optimization landscape, which provides insight into the convexity and convergence behavior of the inverse problem. With this work, we aim to facilitate new applications for sparse computational imaging such as when low radiation doses or rapid reconstructions are essential.

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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

Optical measurements often exhibit mixed Poisson–Gaussian noise statistics, which hampers the image quality, particularly under low signal-to-noise ratio (SNR) conditions. Computational imaging falls short in such situations when solely Poissonian noise statistics are assumed. In response to this challenge, we define a loss function that explicitly incorporates this mixed noise nature. By using a maximum-likelihood estimation, we devise a practical method to account for a camera readout noise in gradient-based ptychography optimization. Our results, based on both experimental and numerical data, demonstrate that this approach outperforms the conventional one, enabling enhanced image reconstruction quality under challenging noise conditions through a straightforward methodological adjustment.

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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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In this thesis we try to provide novel solutions to key problems related to imaging and imaging system. Imaging is usually referred to as the technique for reproducing the information of the object. In optics, we usually refer the object information to the light field in the object plane due to the interaction of the illumination field and the object. Imaging technique allows the reproduced object information to be recorded by detectors such as human eye, photo resist, or CCD/CMOS sensor. In order to image an object, there must be light. We consider only un polarized quasi-monochromatic light illuminating the object and we approximate the field in the object plane by the multiplication of the transmittance / reflectance of the object and the illumination field (the first Born approximation). However, the results in this thesis can also be generalized to situations beyond the above scope. Here we mainly focus on two aspects: the aberrations, the errors of the imaging system, and the spatial coherence of the light field. In a typical imaging scenario, we consider the object plane field as a source consisting a series of point sources. The field generated by each point source propagates independently to the image plane through the imaging system. Ideally, these fields should all identically have the correct distribution and be centered at the correct location. However, this is not the case in presence of the aberrations. As a result, the image will become blurred. @en

We propose and experimentally demonstrate a noniterative diffractive imaging method for reconstructing the complex-valued transmission function of an object illuminated by spatially partially coherent light from the far-field diffraction pattern. Our method is based on a pinhole array mask, which is specially designed such that the correlation function in the mask plane can be obtained directly by inverse Fourier transforming the diffraction pattern. Compared to the traditional iterative diffractive imaging methods using spatially partially coherent illumination, our method is noniterative and robust to the degradation of the spatial coherence of the illumination. In addition to diffractive imaging, the proposed method can also be applied to spatial coherence property characterization, e.g., free-space optical communication and optical coherence singularity measurement.

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For advanced imaging systems, e.g., projection systems for optical lithography, spatially varying aberration calibration is of utmost importance to achieve uniform imaging performance over the entire field-of-view (FOV). Here we present an efficient, accurate, and robust spatially varying aberration calibration method using a pair of 2-dimensional periodic pinhole array masks: the first mask in the object plane and the second mask in the image plane. Our method divides the entire FOV of the imaging system into partially overlapping subregions by using a measurement system consisting of an additional imaging system and a camera sensor. Each subregion, which covers several mask periods, is imaged onto a distinct camera pixel by the measurement system. Our method measures “Airy disc”-like patterns simultaneously in all subregions by scanning the second mask relative to the first mask over one mask p eriod. The number of subregions is equal to the number of camera pixels, and the sampling number of the measured patterns is equal to the scanning step number. The aberrations can be retrieved from the patterns measured in through-focus planes using an iterative optimization algorithm. In this paper, we performed experimental validation on a realistic lithography machine and demonstrate that our method is capable of retrieving the coefficients of 37 aberration terms, expressed as Zernike polynomials, with a sensitivity at nanometer scale.

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In the theory of partial coherence, coherence singularities can occur in the spectral degree of coherence (SDOC): in case the fields at two different points are completely uncorrelated, the phase of the SDOC is undefined. For a partially coherent vortex beam, the detection of coherence singularities is linked to the measurement of topological charge, whose magnitude equals the number of ring dislocations in its far field amplitude. However, the phase distribution of coherence singularities is rarely mentioned in the literature and the amplitude distribution can hardly reflect the sign of topological charge. In this letter, we present a phase-analysis method for measuring the coherence singularities by introducing a movable perturbation at a certain point in an illumination window of a finite size. Using the proposed method, we measure experimentally the coherence singularities of a partially coherent vortex beam in the focal plane. From the results, the magnitude and sign of the topological charge can be determined simultaneously from the phase distribution of the coherence singularities. Our results can find application in information transmission.

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The complete characterization of spatial coherence is extremely di cult because the mutual coherence function (MCF) is a complex-valued function of four independent Cartesian coordinates. This di culty limits the ability to control and to optimize the spatial coherence in a broad range of key applications. Here we propose an e cient and robust scheme for measuring the complete MCF of an arbitrary partially coherent beam using self-referencing holography, which does not require any prior knowledge or making any assumptions about the MCF. We further apply our method to lensless di ractive imaging, and experimentally demonstrate the reconstruction of a phase object under spatially partially coherent illumination. This application is particularly useful for imaging at short wavelengths, where the illumination sources lack spatial coherence and no high-quality imaging optics are available.

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The increased usage of liquid lenses motivates us to investigate surface waves on the liquid's surface. During fast focal switching, the surface waves decrease the imaging quality. We propose a model that describes the surface modes appearing on a liquid lens and predicts the resonance frequencies. The effects of those surface modes on a laser beam are simulated using Fresnel propagation, and the model is verified experimentally.

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The application of Moiré effect for testing of a lithographic projection lens is reported. The arrangement presented allows measuring magnification, distortion, field curvature and telecentricity of the lens and can be used for its fine tuning. The method is based on two matched two-dimensional gratings, positioned in mutually conjugated planes; one of them can be translated. Visual interpretation of Moiré fringe pattern allows quick diagnostics of position errors exceeding critical dimension, whereas lateral scanning is applied for measuring of smaller magnitude errors. Field curvature and telecentricity are measured by 3D scanning. Presented results are in a good agreement with those obtained elsewhere.

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Wide-angle spectral imaging systems using Fabry-Pérot interferometers face spectral resolution problems, which can only be corrected with high computational effort. A standard lens system is analysed and an alternative telecentric solution is proposed, that solves the issues optically without the use of intensive computation.@en