# Chiral Optical Tamm States: Temporal Coupled-Mode Theory

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## Abstract

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## 1. Introduction

#### A Method to Describe Spectral Peaks

## 2. Model

#### Maxwell Equations in the Basis Associated with the Cholesteric Director

## 3. Solution without the Low-Anisotropy Approximation

#### 3.1. The Case of Equal Anisotropies, ${\delta}_{\u03f5}={\delta}_{\mu}$

#### 3.2. The Case of Unequal Anisotropies, ${\delta}_{\u03f5}\ne {\delta}_{\mu}$

## 4. Relaxation Time and Spectral Manifestation

#### 4.1. Temporal Coupled-Mode Theory

#### 4.2. TCMT Applicability Limits

#### 4.3. Numerical Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

COTS | chiral optical Tamm state |

TCMT | temporal coupled-mode theory |

HPM | handedness-preserving mirror |

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**Figure 1.**Circular Bragg diffraction forming COTS. The cholesteric director is shown in blue and green, and the electric field is in red and yellow. The angle between them does not change with depth.

**Figure 2.**HPM and a conventional mirror. The x-polarized light reflection is different. For HPM, the reflected electric strength ${E}_{x}$ preserves its phase and the phase of the magnetic strength ${H}_{y}$ alters instead.

**Figure 3.**Inversion symmetry of dispersion curves, Equation (12). (

**a**) wavelength $\tilde{\lambda}$ as a function of the refractive index $|{n}_{f,s}|$. (

**b**) frequency $\tilde{\omega}=1/\tilde{\lambda}$ as a function of the wave number $|{\tilde{k}}_{f,s}|$. The blue curve is the fast wave, the purple curve – slow wave. The solid curve is drawn for ${\u03f5}_{e}={\mu}_{e}=3/2,{\u03f5}_{o}={\mu}_{o}=2/3$, ${\delta}_{\u03f5}={\delta}_{\mu}$. The semicircle refers to a diffracting wave for which the refractive index acquires purely imaginary values $|{n}_{f}|=Im\left({n}_{f}\right)$. The Mauguin regime, $\tilde{\lambda}\ll 1$, is equivalent to homogenization $\tilde{\omega}\ll 2\pi $ according to the inversion symmetry of Equation (12). The symmetry is violated for the dashed line with ${\u03f5}_{o}=2/3,{\u03f5}_{e}=3/2,{\mu}_{e}={\mu}_{o}=1$.

**Figure 4.**The Kopp–Genack effect. The linewidth saturates with increasing cholesteric layer thickness. Polarization reversal of the optimal exciting light. (

**a**) reflection spectrum, Equation (28). (

**b**) spectral dip width, Equation (30). The cholesteric helix pitch is $p=1\mu m$, electric anisotropy is ${\delta}_{\u03f5}=0.1$; there is no magnetic anisotropy, ${\delta}_{\mu}=0$. (

**c**) reflection at the resonance frequency $\omega ={\omega}_{0}$, Equation (31). (

**d**) state amplitude at the resonance frequency $\omega ={\omega}_{0}$, Equation (27).

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**MDPI and ACS Style**

Timofeev, I.V.; Pankin, P.S.; Vetrov, S.Y.; Arkhipkin, V.G.; Lee, W.; Zyryanov, V.Y. Chiral Optical Tamm States: Temporal Coupled-Mode Theory. *Crystals* **2017**, *7*, 113.
https://doi.org/10.3390/cryst7040113

**AMA Style**

Timofeev IV, Pankin PS, Vetrov SY, Arkhipkin VG, Lee W, Zyryanov VY. Chiral Optical Tamm States: Temporal Coupled-Mode Theory. *Crystals*. 2017; 7(4):113.
https://doi.org/10.3390/cryst7040113

**Chicago/Turabian Style**

Timofeev, Ivan V., Pavel S. Pankin, Stepan Ya. Vetrov, Vasily G. Arkhipkin, Wei Lee, and Victor Ya. Zyryanov. 2017. "Chiral Optical Tamm States: Temporal Coupled-Mode Theory" *Crystals* 7, no. 4: 113.
https://doi.org/10.3390/cryst7040113