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 with a focus on ray tracing in the geometric optics regime. A mathematical foundation, especially involving variational methods, is set in order 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.

Another imaging topic addressed in this book is absolute instruments. Absolute instruments are devices which 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. In particular, this book illustrates the mathematical challenges of obtaining absolute instruments in the context of liquid crystals. As such we also propose a weakening of the notion of 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.

Finally, the book describes a number of open directions (of which there are many), revealing the richness of this area which lies at the interface of geometric optics and variational methods for liquid crystals. The target audience is mathematicians with a background in analysis or differential equations, but not necessarily electromagnetism or geometric optics. Additionally, mathematics students interested in expanding beyond "classical" variational problems in liquid crystals will be interested in the plethora of new research directions.