# Chiral Optical Tamm States at the Interface between an All-Dielectric Polarization-Preserving Anisotropic Mirror and a Cholesteric Liquid Crystal

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## Abstract

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## 1. Introduction

## 2. Description of the Model

## 3. Eigenmode Phase Matching Condition

## 4. Temporal Coupled-Mode Theory

## 5. Results and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OTS | Optical Tamm State |

CLC | Cholesteric Liquid Crystal |

PPAM | Polarization-Preserving Anisotropic Mirror |

COTS | Chiral Optical Tamm State |

## References

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**Figure 1.**Schematic representation of the interface between a dielectric polarization preserving anisotropic mirror and a cholesteric liquid crystal.

**Figure 2.**(

**a**) The effective reflective index (green) depending on the depth z of the layered structure and normalized local intensity ${\left|E\right|}^{2}$ of the chiral optical Tamm state (red). (

**b**) The transmission spectra for circular polarizations coinciding (red) with and opposite (blue) to the handedness of the cholesteric: The refractive indices of the ordinary and extraordinary beam for polarization-preserving anisotropic mirror (PPAM) and cholesteric liquid crystal (CLC) coincide and correspond to common values for nematic liquid crystals of cyanobiphenyl series: ${n}_{o}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$1.45, ${n}_{e}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$1.7. The wavelength in the center of the bandgaps of both crystals is ${\lambda}_{g}^{c}$ = ${\lambda}_{g}^{p}\phantom{\rule{3.33333pt}{0ex}}=$ 1 $\mathsf{\mu}$m. PPAM consists of 30 layers with the total thickness 30/2(${n}_{o}^{p}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{n}_{e}^{p})\phantom{\rule{3.33333pt}{0ex}}\approx \phantom{\rule{3.33333pt}{0ex}}$4.76 $\mathsf{\mu}$m. A CLC layer has 8 helix pitches with the total thickness 8·2/(${n}_{o}^{c}\phantom{\rule{3.33333pt}{0ex}}+\phantom{\rule{3.33333pt}{0ex}}{n}_{e}^{c})\phantom{\rule{3.33333pt}{0ex}}\approx \phantom{\rule{3.33333pt}{0ex}}$5.08 $\mathsf{\mu}$m.

**Figure 3.**(

**a**) The transmission spectra upon changing the pitch of the CLC helix and frequency shift of the center of its bandgap ${\omega}_{g}^{c}$: The frequency of the center of the PPAM bandgap is ${\omega}_{g}^{p}=const$. At ${\omega}_{g}^{c}/{\omega}_{g}^{p}=1$, the parameters correspond to Figure 1. The angle between the optical axes of PPAM and CLC at the interface has the values within the range $\pi /4\le \phi \le 5\pi /4$ with the step $\pi /10$. (

**b**) Dispersion curves of chiral optical Tamm state (COTS): Axes are the same as in Figure 3a. In the overlapping area of the bandgaps, the solutions of Equation (8) are given taking into account Equations (1)–(5) for finite media—red color, and Equations (6) and (7) for semi-infinite media—blue color. The straight green lines denote the bandgap boundaries.

**Figure 4.**The normalized spectral line-width $\Delta \omega /\omega $ of COTS upon changing (

**a**) the contrast ${n}_{o}/{n}_{e}$ and (

**b**) the number of periods N for both PPAM and CLC: The coupled-mode theory is confirmed by the Berreman transfer matrix method. In Figure 4a, the number of periods is $N={N}^{p}={N}^{c}=15$, and in Figure 4b, the contrast is ${n}_{o}^{p}/{n}_{e}^{p}={n}_{o}^{c}/{n}_{e}^{c}=0.86\approx 1.45/1.7$. In both cases, the centers of the PPAM and CLC bandgaps coincide: ${\omega}_{g}^{p}={\omega}_{g}^{c}$.

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**MDPI and ACS Style**

Rudakova, N.V.; Timofeev, I.V.; Bikbaev, R.G.; Pyatnov, M.V.; Vetrov, S.Y.; Lee, W.
Chiral Optical Tamm States at the Interface between an All-Dielectric Polarization-Preserving Anisotropic Mirror and a Cholesteric Liquid Crystal. *Crystals* **2019**, *9*, 502.
https://doi.org/10.3390/cryst9100502

**AMA Style**

Rudakova NV, Timofeev IV, Bikbaev RG, Pyatnov MV, Vetrov SY, Lee W.
Chiral Optical Tamm States at the Interface between an All-Dielectric Polarization-Preserving Anisotropic Mirror and a Cholesteric Liquid Crystal. *Crystals*. 2019; 9(10):502.
https://doi.org/10.3390/cryst9100502

**Chicago/Turabian Style**

Rudakova, Natalya V., Ivan V. Timofeev, Rashid G. Bikbaev, Maxim V. Pyatnov, Stepan Ya. Vetrov, and Wei Lee.
2019. "Chiral Optical Tamm States at the Interface between an All-Dielectric Polarization-Preserving Anisotropic Mirror and a Cholesteric Liquid Crystal" *Crystals* 9, no. 10: 502.
https://doi.org/10.3390/cryst9100502