One possible way of proving the famous and important Riemann hypothesis would be to realize the Riemann zeta zeroes as the eigenvalues of some self-adjoint operator, using self-adjointness to show that the non-trivial zeroes all lie on the critical line. This is known as the Hilbert-Polya conjecture. Interpreting such a self-adjoint operator as an observable of some quantum system, one can try to describe the Riemann zeta zeroes in physical terms, making the Hilbert-Polya conjecture an interesting border case between mathematics and physics. Major work on the Hilbert-Polya conjecture was done by Alain Connes. In his framework, the prolate spheroidal differential operator plays an important but auxiliary role. Before Connes’ work, this second-order differential operator was mainly known for its use in signal analysis, specifically in Slepian’s work on signals which are both time-limited and band-limited. In a more recent article, Connes and Moscovici show that the relation between the prolate spheroidal operator and the Riemann zeta function might be deeper than it seems. They show that a certain self-adjoint realization of the prolate spheroidal operator on the entire real line has discrete spectrum that is asymptotically similar to the squares of the Riemann zeta zeroes. This surprising discovery allows them to construct an operator which approximately solves the Hilbert-Polya conjecture. Though Connes and Moscovici’s article is wonderful and inspiring, it is tersely written. Hence, in this bachelor thesis, details have been provided to the reasoning of the article. In addition, the physical interpretations of the Riemann zeta zeroes, which support the Hilbert-Polya conjecture, have also been presented.

As demand for chips increases and critical dimension keeps shrinking, the inspection of wafer becomes one of the critical challenges in the high volume production of chips. Coherent Fourier scatterometry (CFS) is a scanning bright field technique that is capable to detect nanoparticles on Si surfaces. The optical readout of CFS consists of a two pixel split detector that gives a voltage signal based on the intensity difference between the two halves. While an area of the wafer is raster scanned, the flat surface with no particles gives a zero voltage and on the particle an asymmetrical (non-zero) signal will be detected. In this thesis, we demonstrate the use of synthetic optical holography (SOH) as a new method to improve the sensitivity of CFS. By adding a reference mirror with a piezo stage to the setup, we can interfere the scattered field with a plane reference wave. And by moving the mirror in steps, we can change the phase of the reference wave for every line of the raster scan, such that the 2D scan represents a digital off-axis hologram. Applying the standard digital holography reconstruction process, we retrieve a signal that equals the complex far field difference scaled by the reference field amplitude. Overall, we see consistent signal-to-noise ratio (SNR) improvement over the conventional CFS. For the detection of a polystyrene latex (PSL) particle with a diameter of 60 nm (λ/10) on a silicon wafer, this new implementation leads to a SNR of 10 dB, which is about 4 dB over the filtered conventional CFS signal. For measurements of a dust particle using a very low amount of incident power (0.0014 mW on wafer), a SNR gain of more than 6 dB is achieved compared to filtered conventional CFS, due to the attenuation of low frequency electronic noises. Therefore, the implementation of SOH improves the sensitivity for detecting small nanoparticles and allows low power applications, such as biological imaging.

Scatterometry is a non-destructive metrology technique widely used in the semiconductor industry for the reconstruction of periodic structures from diffraction measurements. This involves solving a so-called inverse problem, which can be done by tuning the geometry parameters of a forward model such that the discrepancy between the measured diffraction pattern and the diffraction pattern computed using a Maxwell solver, are minimized. In order to meet with current semiconductor metrology demands, soft x-ray (SXR) scatterometry has been introduced. In SXR a short wavelength and broad band illumination source is used, allowing for the reconstruction of smaller and more complex structures. However, this does require the use of a computationally expensive forward model. This also complicates the assessment of the sensitivity of the measurement setup to the various grating parameters and whether the parameters can be determined independently. The current approach to this problem is strictly local. In order to address these issues, the use of a surrogate model for the Maxwell solver based on Polynomial Chaos Expansion (PCE) is investigated in this report. The performance of the PCE based surrogate model is accessed for the SXR metrology application on a simple 1D line grating.

This thesis addresses new methods and applications of ptychography which is a scanning Coherent Diffraction Imaging(CDI) method for reconstructing a complex valued object function from intensity measurements recorded in the Fraunhofer or Fresnel diffr- action region. The technique provides a solution to the so-called 'phase problem' and is found to be very suitable for Extreme Ultraviolet (EUV) and X-ray imaging applications due to its high fidelity and its minimum requirement on optical imaging elements. Moreover, abundant studies show that ptychography is able to provide a wide field-of-view and retrieve the illumination probe also. During the last two decades, ptychography has been successfully demonstrated with X-ray radiation sources, electron beams and visible light sources. Chapter 1 is an introductory chapter which gives an overview of CDI techniques. The goal is to provide the necessary knowledge so that readers with different background can easily understand the following chapters. This chapter contains three parts. For the first part we introduce the problem statement of CDI, the approximations that are commonly used in CDI, i.e. the projection approximation, the Fraunhofer approximation, and the required conditions of these approximations. This part also includes the introduction about the discrete Fourier transform, the chirp-Z transform, the issue of sampling and the coherence requirements. The second part of this chapter gives a brief introduction about iterative and non-iterative phase retrieval methods in CDI. For the final part of this chapter, we discuss the fundamental of ptychography which is the main topic of this thesis. We first derive an iterative ptychographic algorithm based on the steepest descent method, then explain the extended field-of-view and the ambiguities in ptychography. Some of the recent developments of ptychography are included in this part as well. For performing phase retrieval in the EUV regime more efficiently, developing polychromatic ptychography is desirable. As an alternative to the existing ptychographic information multiplexing method, we present in Chapter 2 an another scheme where all monochromatic exit waves are expressed in terms of the amplitude of the transmission function and the thickness function of the object. Our proposed algorithm is a gradient based method and its validity is studied numerically. In addition, the sampling issue which appears in the polychromatic ptychography scheme and its influence to the reconstruction quality are discussed. In Chapter 3 we investigate the performance of ptychography with noisy data by analyzing the Cram'{e}r Rao Lower Bound (CRLB). 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. In Chapter 4 we present a parameter retrieval method which combines ptychography and additional prior knowledge about the object. 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 chapter. The CRLB 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 the correlation of particles in application 1. For application 2 the correlation of parameters of the rectangular structure is discussed. The thesis is concluded with Chapter 5 where the main contribution of this thesis is listed. Furthermore, the unfinished work during my PhD and the possible extensions of the topics discussed in this thesis are addressed in this last chapter.@en