Al-Humaidi, Saleh
(2010)
*FORWARD AND INVERSE PROBLEM FOR NEMATIC LIQUID CRYSTALS.*
Doctoral thesis, University of Manchester.

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## Abstract

This thesis starts with an introduction to liquid crystal properties, which are needed to proceed with this research. From the dielectric tensor which appears in the Maxwell equations, we were able to obtain a relationship between the elements on the main diagonal of the dielectric tensor. This relationship has been discussed and illustrated with some examples for both positive and negative birefringence. By introducing a constrain on the Berreman model, we were able to derive a 2 × 2 differential equation in matrix form which works for both normal and oblique incident. This equation gives us a simple and intuitive means to analyze the evolution of light through all sorts of media i.e. isotropic, anisotropic with a fixed transmission axis and anisotropic with a twisted transmission axis of anisotropy. One of the objectives of this research was to find the right technique to solve the 2 × 2 dynamic equation. Fortunately, the classic Floquet’s theory guarantees the existence of the solution and it gives some of its characteristics. In fact, we were able to solve the 2×2 Schrödinger equation by a new method which we called it in this thesis a rotational frame method. The obtained solution is consistent with Floquet’s theory and agrees totally with the Jones solutions. Also, this solution allows us to test the Berreman approximation.Finally, in this research we were able to encode the orientation of the optical axis inside a liquid crystal sample, into the potential of the Schrödinger equation. As a consequence of that, solving the inverse problem of the Schrödinger equation that is recovering the potential, is indeed recovering the orientation of the director inside the sample. The Berreman inverse problem and its corresponding linearized problem has been considered in this thesis. In these sections, we give a rigorous derivation for the Fréchet derivative.

Item Type: | Thesis (Doctoral) |
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Uncontrolled Keywords: | inverse problems, liquid crystals |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations MSC 2010, the AMS's Mathematics Subject Classification > 45 Integral equations PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 00 GENERAL PHYSICS > 02 Mathematical methods in physics PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 40 ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS > 42 Optics PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 60 CONDENSED MATTER - STRUCTURAL, MECHANICAL AND THERMAL PROPERTIES > 61 Structure of solids and liquids; crystallography |

Divisions: | Manchester Institute for the Mathematical Sciences |

Depositing User: | Prof WRB Lionheart |

Date Deposited: | 18 Dec 2017 05:48 |

Last Modified: | 18 Dec 2017 05:48 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/2605 |

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