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Liquid Crystals Beyond Displays: Chemistry, Physics, and Applications by Quan Li

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Chapter 1

Liquid Crystal Lasers

Hideo Takezoe

Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Tokyo, Japan

1.1 Introduction

Liquid crystals (LCs) have fluidity and a long-range orientational order. These properties enable us to use LCs as display materials. Another important property is a positional order. The periodicity is in the range not only of the molecular length periodicity like in smectic LCs but also of visible light wavelength. The latter generally arises from chirality, and in many cases results in helical structures. The most well-known example is cholesteric LCs (CLCs), in which the local structure is nematic and the director rotates to form a helical structure with the helical axis perpendicular to the director. The media that have periodic structures in the optical wavelength are called photonic crystals. Hence, we can call CLC a one-dimensional (1D) photonic crystal. Like an energy gap for electrons propagating in periodic crystal structures, a stop band emerges at the edges of the first Brillouin zone in CLCs. Within the stop band, light dampens and cannot propagate. When the light propagation is limited along any direction, we call it the photonic bandgap (PBG) [1, 2]. In this chapter, the stop band is called PBG in a broad sense.

The dispersion relation between angular frequency ω and wavenumber k in vacuo is given by ω = ck, where c is the velocity of light (Figure 1.1a). In CLCs, the refractive index changes periodically, so ...

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