Electromagnetic inverse problems for nematic liquid crystals and capacitance imaging

Polydorides, Nick (2004) Electromagnetic inverse problems for nematic liquid crystals and capacitance imaging. [MIMS Preprint]

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Abstract

The aim of this study is to formulate and solve the high-frequency electromagnetic problem of wave propagation through nematic liquid crystal cells of arbitrary shape, and subsequently approach the inverse problem of reconstructing the orientational order by means of recovering the dielectric tensors in the interior from a finite set of boundary polarization measurements. The numerical solution of the forward electromagnetic problem is achieved by hybridizing the conventional vector finite elements with a boundary integral method, so that to preserve the necessary continuity conditions for the electromagnetic fields at the boundary of the domain. Combining the finite element equations with a magnetic field integral boundary equation yields surface integrals involving Green s function and its gradient. These integrals have kernels that become asymptotically singular as the distance between the observation and integration points reduces to zero, essentially making the numerical integration process problematic. For their computation a new basis of functions are introduced for the surface current density, the so-called Rao-Wilton-Glisson functions, which e®ectively substitute the tangential components of the magnetic field in the original boundary integrals. The transformed integrals are then treated with the singularity extraction method, essentially separating the smooth from the singular components of the kernels, the former of which are computed using conventional numerical integration and the later using closed form expressions derived for the RWG functions. The forward problem is then linearized with the aid of the Fr´echet derivative of the forward Maxwell operator and subsequently regularized using a Tikhonov type regularization. In regularization we construct a penalty term based on Frank s distortion energy functional, which is known to have a minimum in the neighborhood of stable liquid crystal director configurations. The inverse problem of reconstructing the orientation of the director vector in a uniaxial nematic liquid crystal using a finite set of noise infused boundary polarization measurements is approached as a special case of the inverse permittivity tensor problem, where the dielectric tensors are symmetric and expected to vary most significantly along the directions of their two biggest eigenvalues, which correspond to the associated Euler angles of the director vector.

Item Type:	MIMS Preprint
Additional Information:	This report for a Smith Institute Faraday project, was first published electronically in 2004
Uncontrolled Keywords:	Liquid crystals, inverse problems, permittivity imaging, capacitance tomography
Subjects:	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 > 70 CONDENSED MATTER - ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES > 78 Optical properties, condensed-matter spectroscopy and other interactions of radiation and particles with condensed matter
Depositing User:	Prof WRB Lionheart
Date Deposited:	01 Mar 2007
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/626

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