New methods and applications of ptychography

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.