· · · Institutional Repository

Poster presentation with Philips Research Laboratories

This study presents an experimental method to determine the resist parameters that are at the origin of a general blurring of the projected aerial image. The resist model includes the effects of diffusion in the horizontal plane and a second cause for image blur that originates from a stochastic variation of the focus parameter. The used mathematical framework is the so-called Extended Nijboer-Zernike (ENZ) theory. The experimental procedure to extract the model parameters is demonstrated for several 193 nm resists under various conditions of post exposure baking temperature and baking time. The advantage of our approach is a clear separation between the optical parameters, such as feature size, projection lens aberrations and the illuminator setting on the one hand and process parameters introducing blur on the other.

Phase Measurement Interferometers (PMI) are widely used during the manufacturing process of high quality lenses. Although they have an excellent reproducibility and sensitivity, the set-up is expensive and the accuracy of the measurement needs to be checked frequently. This paper discusses an alternative lens metrology method that is based on an aerial image measurement. We discuss the Extended Nijboer-Zernike (ENZ) method and its application to aberration measurement of a high-NA optical system of a wafer stepper. ENZ is based on the observation of the through-focus intensity point-spread function of the projection lens. The advantage of ENZ is a simple set-up that is easy to run and maintain and provides good accuracy. Therefore the method is useful during lens assembly in the factory. The mathematical framework of ENZ is shown and the experimental procedure to extract aberrations for a high-NA lens is demonstrated on a high-NA DUV lithographic lens. PMI data is given as reference data. It is shown that ENZ provides an attractive alternative to the interferometer.

Data from call centres at two municipalities were analysed in order to quantify flooding frequencies and associated flood risks for three main failure mechanisms causing urban flooding. The aim was to find out whether current operational strategies are efficient for flood prevention and if directions for improvement could be found. The results show that quantified flood risk for the two cases is well above the standard which is defined in sewer management plans. The analysis pointed out that gully pot blockages are the main cause of flooding and handling gully pot blockages should therefore be a priority for sewer operators. Reactive handling of calls, as is currently applied, is inefficient if all calls are reacted upon since a small portion of all calls report serious consequences like flooding in buildings or wastewater flooding. Preventive cleaning of sewer pipes proves to be an efficient strategy to reduce flooding due to sewer blockages as flood risk associated with sewer blockages is lower in case of higher cleaning sewer frequencies. Sewer blockages often have serious consequences, thus preventive handling is to be preferred to reactive cleaning. According to the results of this analysis, reduction of flooding sewer overloading is not of primary concern, because serious consequences for this failure mechanism are rare compared to other failure mechanisms.

This thesis is dedicated to the study of a class of probabilistic inequalities, called Rosenthal inequalities. These inequalities provide two-sided estimates for the p-th moments of the sum of a sequence of independent, mean zero random variables in terms of a suitable norm on the sequence itself. Rosenthal inequalities are named after H.P. Rosenthal, who first discovered them for scalar-valued random variables around 1970. The main results of this thesis extend Rosenthal's inequalities in two different directions. In the first part we consider random variables taking values in a Banach space. The main results give Rosenthal-type inequalities in the case where the Banach space is either a Hilbert space or an Lp-space. The inequalities developed in this setting are principally designed to prove a novel Ito isomorphism for vector-valued stochastic integrals with respect to a compensated Poisson random measure. These kind of isomorphisms are a key tool in the analysis of stochastic partial differential equations. The Rosenthal-type inequalities are further extended to apply to random variables taking values in a noncommutative Lp-space associated with a von Neumann algebra. By specializing this result to von Neumann algebras of square matrices, quantitative bounds are found for the moments of the largest singular value of a random matrix in terms of its entries. In the second part of this thesis Rosenthal's original inequalities are generalized to sequences of noncommutative random variables, given by elements of a noncommutative symmetric space. As is the case in the first part, these noncommutative Rosenthal inequalities are applied to obtain Ito isomorphisms for stochastic integrals. For the proof of the noncommutative Rosenthal inequalities several new tools are developed which are interesting in their own right. Novel results are found for other probabilistic inequalities in noncommutative symmetric spaces, such as Khintchine and Burkholder-Gundy inequalities, as well as results in the interpolation theory for such spaces.

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer–Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on pointby-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer–Zernike approach into the Ignatowsky–Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the “vectorial” pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.