Mathematical Methods in Liquid Crystal Optics and Lens Design

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.