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Photonics Research
Vol. 1, Issue 1, 58 (2013)

Analytic solutions of the normal modes and light transmission of a cholesteric liquid crystal cell

Sabrina Relaix, Mykhailo Pevnyi^*, Wenyi Cao, and and Peter Palffy-Muhoray

Author Affiliations

Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA

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DOI: 10.1364/PRJ.1.000058 Cite this Article Set citation alerts

Sabrina Relaix, Mykhailo Pevnyi, Wenyi Cao, and Peter Palffy-Muhoray. Analytic solutions of the normal modes and light transmission of a cholesteric liquid crystal cell[J]. Photonics Research, 2013, 1(1): 58 Copy Citation Text

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Abstract

Cholesteric liquid crystals, consisting of chiral molecules, form self-assembled periodic structures exhibiting a photonic bandgap. Their selective reflectivity makes them well suited for a variety of applications; their optical response is therefore of considerable interest. The reflectance and transmittance of finite cholesteric cells is usually calculated numerically. Evanescent modes in the bandgap make the calculations challenging; existing matrix propagation methods cannot describe the reflection and transmission coefficients of thick cholesteric cells accurately. Here we present analytic solutions for the electromagnetic fields in cholesteric cells of finite thickness, and use them to calculate the transmission and reflection spectra. The use of analytic solutions allows for the accurate description of arbitrarily thick cholesteric cells, which would not be possible with only direct numerical methods.

Keywords

Anisotropic optical materials Birefringence Chiral media Liquid crystals Photonic bandgap materials

\hat{η} = \cos (q z) \hat{x} + \sin (q z) \hat{y} \hat{ξ} = - \sin (q z) \hat{x} + \cos (q z) \hat{y} .

(1)

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E^{''} (z) + 2 i k E^{'} - k^{2} E (z) = - \frac{ω^{2}}{c^{2}} {\bar{\bar{ε}}}_{r} E (z),

(2)

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(α^{2} - ε_{∥} + n^{2}) E_{∥} = - 2 i n α E_{⊥} (α^{2} - ε_{⊥} + n^{2}) E_{⊥} = 2 i n α E_{∥},

(3)

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n^{2} = \frac{ε_{∥} + ε_{⊥}}{2} + α^{2} \pm {[\frac{{(ε_{∥} - ε_{⊥})}^{2}}{4} + 2 (ε_{∥} + ε_{⊥}) α^{2}]}^{\frac{1}{2}} .

(4)

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n_{1}^{\pm} = \pm {[\bar{ε} + α^{2} + {(δ^{2} + 4 \bar{ε} α^{2})}^{\frac{1}{2}}]}^{\frac{1}{2}},

(5)

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n_{2}^{\pm} = \pm {[\bar{ε} + α^{2} - {(δ^{2} + 4 \bar{ε} α^{2})}^{\frac{1}{2}}]}^{\frac{1}{2}},

(6)

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E_{1}^{\pm} = E_{∥}^{\pm} [\begin{matrix} 1 \\ \pm i A \end{matrix}] e^{i (\pm k_{1} z - ω t)},

(7)

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E_{2}^{\pm} = E_{⊥}^{\pm} [\begin{matrix} \mp i B \\ 1 \end{matrix}] e^{i (\pm k_{2} z - ω t)} .

(8)

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A = \frac{2 α^{2} - \frac{ε_{∥} - ε_{⊥}}{2} + {[\frac{{(ε_{∥} - ε_{⊥})}^{2}}{4} + 2 (ε_{∥} + ε_{⊥}) α^{2}]}^{\frac{1}{2}}}{2 α {\frac{ε_{∥} + ε_{⊥}}{2} + α^{2} + {[\frac{{(ε_{∥} - ε_{⊥})}^{2}}{4} + 2 (ε_{∥} + ε_{⊥}) α^{2}]}^{\frac{1}{2}}}^{\frac{1}{2}}},

(9)

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B = \frac{2 α^{2} + \frac{ε_{∥} - ε_{⊥}}{2} - {[\frac{{(ε_{∥} - ε_{⊥})}^{2}}{4} + 2 (ε_{∥} + ε_{⊥}) α^{2}]}^{\frac{1}{2}}}{2 α {\frac{ε_{∥} + ε_{⊥}}{2} + α^{2} - {[\frac{{(ε_{∥} - ε_{⊥})}^{2}}{4} + 2 (ε_{∥} + ε_{⊥}) α^{2}]}^{\frac{1}{2}}}^{\frac{1}{2}}},

(10)

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A = \frac{n_{1}^{2} + α^{2} - ε_{∥}}{2 n_{1} α}, B = \frac{n_{2}^{2} + α^{2} - ε_{⊥}}{2 n_{2} α} .

(11)

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E_{1}^{+} = E_{1}^{+} e^{- i ω t} ((1 + A) [\begin{matrix} 1 \\ i \end{matrix}] e^{i (k_{1} z + q z)} + (1 - A) [\begin{matrix} 1 \\ - i \end{matrix}] e^{i (k_{1} z - q z)}),

(12)

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E_{1}^{-} = E_{1}^{-} e^{- i ω t} ((1 + A) [\begin{matrix} 1 \\ - i \end{matrix}] e^{- i (k_{1} z + q z)} + (1 - A) [\begin{matrix} 1 \\ i \end{matrix}] e^{- i (k_{1} z - q z)}),

(13)

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E_{2}^{+} = E_{2}^{+} e^{- i ω t} ((1 + B) [\begin{matrix} - i \\ 1 \end{matrix}] e^{i (k_{2} z + q z)} + (1 - B) [\begin{matrix} i \\ 1 \end{matrix}] e^{i (k_{2} z - q z)}),

(14)

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E_{2}^{-} = E_{2}^{-} e^{- i ω t} ((1 + B) [\begin{matrix} i \\ 1 \end{matrix}] e^{- i (k_{2} z + q z)} + (1 - B) [\begin{matrix} - i \\ 1 \end{matrix}] e^{- i (k_{2} z - q z)}),

(15)

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E_{i} = [\begin{matrix} E_{i x} \\ E_{i y} \end{matrix}] e^{i (k_{s} z - ω t)} E_{r} = [\begin{matrix} E_{r x} \\ E_{r y} \end{matrix}] e^{i (- k_{s} z - ω t)} E_{t} = [\begin{matrix} E_{t x} \\ E_{t y} \end{matrix}] e^{i (k_{s} (z - d p) - ω t)} .

(16)

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[\begin{matrix} E_{i x} \\ E_{i y} \\ E_{r x} \\ E_{r y} \end{matrix}] = [\begin{matrix} \bar{\bar{P}} & \bar{\bar{0}} \\ \bar{\bar{Q}} & \bar{\bar{0}} \end{matrix}] [\begin{matrix} E_{t x} \\ E_{t y} \\ 0 \\ 0 \end{matrix}] .

(17)

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[\begin{matrix} E_{t x} \\ E_{t y} \end{matrix}] = {\bar{\bar{P}}}^{- 1} [\begin{matrix} E_{i x} \\ E_{i y} \end{matrix}],

(18)

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[\begin{matrix} E_{r x} \\ E_{r y} \end{matrix}] = \bar{\bar{Q}} {\bar{\bar{P}}}^{- 1} [\begin{matrix} E_{i x} \\ E_{i y} \end{matrix}] .

(19)

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Sabrina Relaix, Mykhailo Pevnyi, Wenyi Cao, and Peter Palffy-Muhoray. Analytic solutions of the normal modes and light transmission of a cholesteric liquid crystal cell[J]. Photonics Research, 2013, 1(1): 58

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