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Overview"Freeform lens design has numerous applications in imaging, aerospace, and biomedicine. Due to recent technological advancements in precision cutting and grinding, the manufacturing of freeform optical lenses with very high precision is now possible. However, there is still a significant lack of mathematical literature on the subject, and essentially none related to liquid crystals. Liquid crystals are appealing for use in imaging due to their flexibility and unique electro-optical properties. This book seeks to fill a gap in mathematical literature and attract focus to liquid crystals for freeform lens design. In particular, this book provides a rigorous mathematical perspective on liquid crystal optics, focusing on ray tracing in the geometric optics regime. A mathematical foundation is set to study lens design and ray tracing problems in liquid crystals. As an application, a lens design problem is posed and solved for the case of a simple director field. Anotherimaging topic addressed in this book is that of absolute instruments. Absolute instruments are devices that image stigmatically, i.e., without any optical aberrations. These instruments cannot be designed through transformation optics, and until recently, only a handful of examples were known. Mathematically, this is a largely untapped area of research, yet the applications are profound. This book illustrates the mathematical challenges of obtaining absolute instruments in the context of liquid crystals. As such, we propose weakening the notion of an absolute instrument to allow for a wider class of devices to image ""almost"" stigmatically. Along the way, we make connections between lens design problems and some perhaps unexpected areas of mathematics, including nonlinear partial differential equations, Riemannian geometry, and dynamical systems. Due to remarkable optical phenomena that occur in helical media, such as selective reflection, the electromagnetics of helical media is also discussed. There is a particular focus on the optics of chiral media. This book also shows how various forms of Snell’s Law, a foundational principle seen throughout the text, arise in the context of cholesteric liquid crystals. Finally, the book describes several open directions, revealing the richness of this area which lies at the interface of liquid crystal optics and mathematical analysis. The target audience includes researchers in the field of mathematical optics as well as those interested in liquid crystal theory. Additionally, mathematics graduate students aiming to understand the physical basis of light propagation in liquid crystals would find the text interesting. " Full Product DetailsAuthor: Eric StachuraPublisher: Springer International Publishing AG Imprint: Springer International Publishing AG Edition: 1st ed. 2024 ISBN: 9783031466137ISBN 10: 3031466136 Pages: 190 Publication Date: 14 April 2024 Audience: College/higher education , Postgraduate, Research & Scholarly Format: Hardback Publisher's Status: Active Availability: In Print This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction.- Physical Background.- Fermat Principle and Snell's Law of Refraction.- Ray Equations: Geodesics in Finsler Space.- Aberrations and a Lens Design Problem.- The Challenge of Absolute Instruments.- Conclusions and Outlook.- Appendix.- References.ReviewsAuthor InformationEric Stachura Kennesaw State University Department of Mathematics Marietta, GA United States of America Tab Content 6Author Website:Countries AvailableAll regions |