The importance of inverse problems is paramount in science and physics because their solution provides information about parameters that cannot be directly observed. This thesis discusses and details the application of a few inverse methods in optical imaging, metrology and inspection of lithographic targets, particularly patterned structures on top of an extreme ultraviolet (EUV) lithographic mask...@en

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
variation based ptychographic imaging@en

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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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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The effect of conductor surface roughness on electro-thermally (ET) induced passive intermodulation (PIM) products is analysed and discussed. The analytical model combined with the commercial simulator SIMBEOR is used to examine the products of ET nonlinearity, which is significantly influenced by the surface topography of the conductor. It is further shown that, in spite of the rise of ohmic losses with frequency, conductor roughness may somewhat allay the ET-PIM growth due to concurrent increase of additional heat-sinking provided by the interiors of dendrites and conductor cladding.

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