Structure of the dielectric tensor in nematic liquid crystals with topological charge

P. Vaveliuk; F. Moraes; S. Fumeron; O. Martinez Matos; M. L. Calvo

doi:10.1364/JOSAA.27.001466

Journal of the Optical Society of America A
Vol. 27,
Issue 6,
pp. 1466-1472
(2010)
•https://doi.org/10.1364/JOSAA.27.001466

Structure of the dielectric tensor in nematic liquid crystals with topological charge

P. Vaveliuk, F. Moraes, S. Fumeron, O. Martinez Matos, and M. L. Calvo

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P. Vaveliuk, F. Moraes, S. Fumeron, O. Martinez Matos, and M. L. Calvo, "Structure of the dielectric tensor in nematic liquid crystals with topological charge," J. Opt. Soc. Am. A 27, 1466-1472 (2010)

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Abstract

A q-plate is a stratified medium composed of a uniaxial nematic liquid crystal with an inhomogeneous orientation of the optical axis possessing a topological charge q , which looks promising as a switching device. This work reports an approach to diagonalizing the dielectric tensor in q-plates, giving a detailed study of nontrivial tensor coordinate transformations. The relationship between the diagonal dielectric tensor and the topological charge of the plate is established. From that, the dielectric tensor for a given q-plate can be explicitly calculated in a convenient frame, and hence the field propagation equations can be derived. These studies establish a theoretical basis that could further deepen the research on these particular structures for nonconventional manipulation of the light.

© 2010 Optical Society of America
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q-plate ^b	Rectangular $(x, y, z)$	Circular-cylindrical $(r, φ, z)$	Parabolic-cylindrical $(u, v, z)$
$[0, \frac{π}{2}]$	$(\begin{matrix} n_{o}^{2} & 0 & 0 \\ 0 & n_{e}^{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$	$(\begin{matrix} \frac{n_{+}^{2} - n_{-}^{2} \cos (2 φ)}{2} & \frac{- n_{-}^{2} \sin (2 φ)}{2} & 0 \\ \frac{- n_{-}^{2} \sin (2 φ)}{2} & \frac{n_{+}^{2} + n_{-}^{2} \cos (2 φ)}{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$	$(\begin{matrix} \frac{n_{o}^{2} u^{2} + n_{e}^{2} v^{2}}{u^{2} + v^{2}} & \frac{- n_{-}^{2} u v}{u^{2} + v^{2}} & 0 \\ \frac{- n_{-}^{2} u v}{u^{2} + v^{2}} & \frac{n_{e}^{2} u^{2} + n_{o}^{2} v^{2}}{u^{2} + v^{2}} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$
$[1, \frac{π}{2}]$	$(\begin{matrix} \frac{n_{o}^{2} x^{2} + n_{e}^{2} y^{2}}{x^{2} + y^{2}} & \frac{n_{-}^{2} x y}{x^{2} + y^{2}} & 0 \\ \frac{n_{-}^{2} x y}{x^{2} + y^{2}} & \frac{n_{e}^{2} x^{2} + n_{o}^{2} y^{2}}{x^{2} + y^{2}} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$	$(\begin{matrix} n_{o}^{2} & 0 & 0 \\ 0 & n_{e}^{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$	$(\begin{matrix} \frac{n_{o}^{2} u^{2} + n_{e}^{2} v^{2}}{u^{2} + v^{2}} & \frac{n_{-}^{2} u v}{u^{2} + v^{2}} & 0 \\ \frac{n_{-}^{2} u v}{u^{2} + v^{2}} & \frac{n_{e}^{2} u^{2} + n_{o}^{2} v^{2}}{u^{2} + v^{2}} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$
$[\frac{1}{2}, 0]$	$(\begin{matrix} \frac{n_{+}^{2}}{2} + \frac{n_{-}^{2} x}{2 \sqrt{x^{2} + y^{2}}} & \frac{- n_{-}^{2} y}{2 \sqrt{x^{2} + y^{2}}} & 0 \\ \frac{- n_{-}^{2} y}{2 \sqrt{x^{2} + y^{2}}} & \frac{n_{+}^{2}}{2} - \frac{n_{-}^{2} x}{2 \sqrt{x^{2} + y^{2}}} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$	$(\begin{matrix} \frac{n_{+}^{2} + n_{-}^{2} \cos (φ)}{2} & \frac{n_{-}^{2} \sin (φ)}{2} & 0 \\ \frac{n_{-}^{2} \sin (φ)}{2} & \frac{n_{+}^{2} - n_{-}^{2} \cos (φ)}{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$	$(\begin{matrix} n_{e}^{2} & 0 & 0 \\ 0 & n_{o}^{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix})$