Design and Optimization

It is expected that the ability to accurately and efficiently design an imaging system for a specific application will be of increasing importance in the coming decades. Applications of imaging systems range from simple photography to advanced lithography machines. Perhaps the most important way to make an imaging system meet a particular purpose is to engineer the pupil function of the imaging system. This includes designing a pupil surface and often involves the simultaneous design of a post-processing algorithm. Currently these design processes are performed mostly by using numerical optimization methods. Numerical methods in general have many drawbacks including long processing time and no guarantee that one has reached the global optimum. We have developed analytical approaches in designing imaging systems by engineering the pupil function. Two of the most important merit functions that are used for the analysis of imaging systems are the modulation transfer function (MTF) and the point spread function (PSF). These two functions are standard measures for evaluating the performance of an imaging system. Usually during the design process one finds the PSF or MTF for all the possible degrees of freedom and chooses the combination of parameters which best satisfies his/her goals in terms of PSF and MTF. In practice, however, evaluating these functions is computationally expensive; this makes the design and optimization problem hard. In particular, it is often impossible to guarantee that one has reached the global optimum. In this PhD thesis, we have developed approximate analytical expressions for MTF and PSF of an imaging system. We have derived rigorous bounds on the accuracy of these expressions and established their fast convergence. We have also shown that these approximations not only reduce the calculation burden by several orders of magnitude, but also make the analytic optimization of imaging systems possible. We have studied the detailed properties of our approximations. For instance we have shown that the PSF approximation has a complexity which is independent of certain system parameters such as defocus. Our results also help in better understanding the behavior of imaging systems. In particular, using our results we have answered a fundamental question regarding the limit of extension of the depth of field in imaging systems by pupil function engineering. We have derived a theoretic bound and we have established that this bound does not change with change of phase of pupil function. We have also introduced the concept of conservation of spectral signal-to-noise ratio and discussed its implications in imaging systems. Thesis Supervisor: Daniela Pucci de Farias Title: Assistant Professor, Mechanical Engineering Acknowledgments First, I would like to thank my adviser, Prof. Daniela Pucci de Farias. This work would not have been possible without her enthusiasm, guidance and generosity in allowing me to pursue my own ideas and research directions. I will forever be grateful to her for giving me increasing levels of responsibility as my graduate experience progressed. I would also like to thank my informal adviser, Prof. George Barbastathis. I am deeply indebted to him for being an enthusiastic and genius supervisor. His comments and insights were invaluable sources of support for me. I am also grateful to the other member of my thesis committee, Prof. Peter So for his insightful recommendations. The research presented in this dissertation is a result of close collaboration with Prof. Mark A. Neifeld and Drs. Paulo E. X. Silveira, Ramkumar Narayanswamy and Edward Dowski. I would like to thank them all for their valuable comments and suggestions. Special thank goes to Paulo for his continuous support and encouragement and many hours of helpful discussion. I am also grateful to researchers in CDM Optics for two great summers in the research industry. Furthermore, I would like to thank Drs. Michael D. Stenner, Kehan Tian and Wenyang Sun for insightful discussions and exchange of ideas. My education at MIT was enriched by interaction with fellow students and friends. I am especially grateful to Angelina Aessopos, Mohammad-Reza Alam, Husain A. Al-Mohssen, Mohsen Bahramgiri, Lowell L. Baker, Reza Karimi, Hemanth Prakash, Judy Su, Dimitrios Tzeranis and Yunqing Ye for making the courses and research as well as life in MIT much more fruitful. I would like to also thank my other friends, namely Salman Abolfathe Beikidezfuli, Mohammad Araghchini, Mohammad Azmoon, Hoda Bidkhori, Megan Cooper, Eaman Eftekhary, Shaya Famini, Ali Farahanchi, Ghassan Fayad, Julia Greiner, MohammadTaghi Hajiaghayi, Fardad Hashemi, Ali Hosseini, Asadollah Kalantarian, Ali Khakifirooz, Danial Lashkari, Hamed Mamani, Vahab Mirrokni, Ali Parandehgheibi, Mohsen Razavi, Saeed Saremi, Ali Tabaei, Hadi Tavakoli Nia, Luke Waggoner and many others for creating a joyful and unforgettable three years for me. I am also grateful to the staff of Mechanical Engineering Department, specially Leslie Regan, Joan Kravit and Laura E. Kampas for providing a nice academic environment. I would like to use this opportunity to thank my undergraduate advisers, Prof. Ali Amirfazli, Prof. Mikio Horie and Dr. Daiki Kamiya for shaping my career path. Also I am grateful to Seid Hossein Sadat, Hamidreza Alemohammad and Reza Hajiaghaee Khiabani for many helpful discussions. I thank my family for their unconditional support throughout these years. They provided everything I needed to succeed. My parents' everlasting love and prayer has always been and will remain a primary source of support for me and I am forever indebted to them for that. I am grateful to my brother, Vahid for being out there for me and to my sister, Maryam for her love. Last but not least, I thank my wife, Mariam for her understanding, inspiration and endless love. This thesis is dedicated to my family.