1. Introduction

Quasicrystals are a class of systems presenting long-range order but no periodicity. They have fascinating physical properties like the formation of multiple photonic band gaps (PBGs), the presence of fractal transmission resonance, and the occurrence of critically localized states[1, 2, 3, 4, 5, 6, 7]. Among the various one-dimensional (1D) quasicrystals described by the sequences named Fibonacci[1, 2, 3, 4, 5], Thue-Morse[6], or Rudin-Shapiro[7], a Fibonacci system has been studied in various material systems theoretically and experimentally. Most experimental reports were limited to inorganic multilayered systems such as TiO₂/SiO₂[2, 3] and porous silicon structures[4, 5]. Recently, as a different Fibonacci system from the dielectric multilayers, Fibonacci number patterns in Ag core/SiO_X shell microstructures were reported[8]. So far, however, no organic Fibonacci systems have been even proposed.

A cholesteric liquid crystal (CLC) used in this study is an 1D self-assembled photonic crystal showing a PBG for circularly polarized light with the same handedness as a cholesteric helix[9]. These characteristics such as selective reflection in a specific wavelength region and circular polarization of the reflected light are attractive for reflective color-displays without the need for back-lightening, polarizers, or color filters[10, 11, 12].

In this paper, we introduce Fibonaccian defects into organic CLC systems and simulate reflection spectra from the proposed system numerically. A quasi-periodic system is considered as a sample to exemplify its intrinsic property such as multiple PBGs. Among several sequences mentioned above, we employ the simplest system, the Fibonacci sequence, for the purpose of future experimental demonstrations: that is, the number of building units of a Rudin-Shapiro sequence is larger than that of the other sequences[7] and, at the same order, total number of layers of the Thue-Morse sequence is larger than that of the Fibonacci sequence[6]. The helical CLC has a single PBG or one reflection color, in which the PBG width of CLC is proportional to the anisotropy of refractive indices and is in the range of 50—100nm. By using the Fibonaccian defects, however, we can achieve simultaneous red, green, and blue reflections or multiple PBGs using single-pitched CLC systems. Without introducing any other materials, a series of conventional CLC films with Fibonaccian phase defects can give rise to such optical characteristics. This Fibonaccian CLC system can in practice be achieved by a polymeric cholesteric liquid crystal (PCLC)[13].

2. Fibonaccian defects in CLCs

A Fibonacci sequence is based on the recursive relation F ₀={A}, F ₁={B}, and F_j=F _j-2 F _j-1 for j≥2. The lower order sequences are F ₂={AB}, F ₃={BAB}, F ₄={ABBAB}, etc. In general, a 1D quasi-periodic system has been formed by stacking two different dielectric materials, A and B, according to the Fibonacci generation scheme[1, 2, 3]. In CLCs, however, the periodicity comes from the structure formed by rodlike molecules including chiral molecules arranging themselves in a helical fashion[9]. Hence, the controllable factors for forming Fibonacci structures are different from those in dielectric multilayered systems. Since the origin of the periodicity in cholesteric structures is a spontaneous orientational order of molecules, it is difficult to control a fundamental unit of the periodicity and make a quasi-periodic Fibonacci system with CLCs. Therefore we propose the Fibonacci structure by introducing defects into the CLCs theoretically.

Defects of PCs have been generated by inserting isotropic materials with different refractive index or thickness in periodic structures. In addition to such a defect, a different type of defect is possible in CLCs; introduction of a phase jump (orientational defect) into a cholesteric helix[14, 15, 16, 17]. To make a quasi-periodic structure with a CLC, we employed the second approach to photonic defects and introduced Fibonaccian defects into CLCs as shown in Fig. 1.

 

Fig. 1. (Color online) Schematic view of Fibonaccian defects in periodic cholesteric structures: F ₂={AB} and F ₃={BAB} with ϕ=0 and ϕ=π/2. A (blue) and B (red) are two units of the quasi-periodic Fibonacci system, i.e., F ₀={A} and F ₁={B}. ϕ is an orientational defect angle between the molecular director in A layer and that in B layer at the interface (yellow).

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Figure 1 shows a schematic illustration of Fibonaccian defects, F ₂ and F ₃ with ϕ=0 and ϕ=π/2, in periodic cholesteric structure, where A (blue) and B (red) are two units of the Fibonacci system, F ₀ and F ₁, and ϕ is an orientational defect angle between the molecular director in A layer and that in B layer at the interface (yellow). The case of ϕ=0 in Fig. 1 corresponds to the unmodified structure without a defect or a conventional cholesteric structure.

3. Multiple PBGs from the single-pitched CLCs

In order to investigate the optical properties of these quasi-periodic systems, we performed a theoretical calculation of the light propagation in the single pitch CLCs with the Fibonaccian defects using a 4×4 Berreman matrix[18].

The refractive indices of the birefringent planes are taken to be n_o=1.56 and n_e=1.78, which are typical values for a PCLC [13]. We considered the right-handed helical CLC (R-CLC) with a cholesteric pitch P and an optical pitch λ_P=n̅_P=532nm, where n̅ is an average refractive index and is given by n̅=[(n ² _o+n ² _e)/2]^1/2. Here, λ_P was so chosen that the center of multiple PBGs was located at a wavelength of green between primary colors such as red, green, and blue. The incident light was fixed in a right circularly polarized (RCP) light, since RCP light cannot propagate through the R-CLCs in a spectral range corresponding to the PBG. Under these fixed conditions, we calculated reflection spectra of samples changing the orientational defect angle, the order of the Fibonacci sequence, and a unit length U_L of A or B in the Fibonacci system.

Figure 2 shows the dependence of reflection spectra on a defect angle when the order of the Fibonacci system is F ₄={ABBAB} with 5 layers and a unit length of the sequence is U_L=1P. As a reference, the reflection spectrum of a conventional CLC without a defect, i.e., ϕ=0 was calculated as shown in Fig. 2(a). It shows a single green-colored reflection, placing the center of the PBG at the spectral position of λ_P.

 

Fig. 2. (Color online) Reflection spectra for the Fibonacci system F ₄={ABBAB} with orientational defect angles (a) ϕ=0, (b) ϕ=π/3, (c) ϕ=π/2, and (d) ϕ=π/6. For all simulations, we have considered a RCP incident light and the R-CLC with refractive indices n_e=1.78 and n_o=1.56, an optical pitch λ_P=532nm, and an unit length of A or B layer U_L=1P.

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As shown in Fig. 2(b), multiple PBGs with three red-, green-, and blue-colored reflections are generated by introducing the Fibonaccian defect with ϕ=π/3 into a single green-colored CLC. The PBG width of a conventional CLC, depending on the difference between refractive indices n_e and n_o, is about 50nm or 100nm and is limited to a single-colored region. In this sense, it is important to note that, without introducing any other materials or with the only single-pitched CLC, multiple reflections covering full colors could be obtained only by the structural change of the system.

The colors of such multiple PBGs could be controlled by changing an orientational defect angle. For example, Fig. 2(c) exhibits the reflection spectrum of the Fibonacci system with ϕ=π/2, where, as compared with Fig. 2(b), the reflectance due to the green-colored PBG at the center diminishes and that due to the red- and blue-colored PBGs increases. On the contrary, in the case of Fig. 2(d) with ϕ=π/6, a green-colored reflection becomes dominant. This optical behavior of multiple PBGs depending on an orientational defect angle has not been found in other Fibonacci systems[1, 2, 3, 4, 5]. The effect of tunable colors is similar to that discussed in the periodic CLC structure with orientational defects[14, 15], where the introduction of the phase jump in the CLC helix gives the phase difference between the reflections from the two parts separated by the phase jump. Hence, varying the orientational defect angle from ϕ=0 to ϕ=π adjusts the spectral position of defects from the low to the high wavelength band edge.

Next, Fig. 3 shows reflection spectra of the Fibonacci systems F ₅={BABABBAB} with 8 layers, F ₆={ABBABBABABBAB} with 13 layers, and F ₇={BABABBABABBABBABABBAB} with 21 layers when orientational defect angles are (a) ϕ=π/3 and (b) ϕ=π/2. Colors in Fig. 3 represent the reflectance calculated from the quasi-periodic Fibonacci system. In Fig. 3, the whole reflectance and the number of multiple PBGs are proportional to the order of the Fibonacci system, and the increase in the Fibonacci order from F ₅ to F ₇ results in narrower widths of the central PBGs located in the green-colored region. These behaviors are similar to those of a general PBG material, i.e., the increasing of reflectance and the narrowing of the PBG width with increasing sample thickness. Particularly, the reflectance of the central PBG located in the green-colored region of Fig. 3(b) with ϕ=π/2 displays a pronounced dramatic dependence on the Fibonaccian order.

 

Fig. 3. (Color online) Reflection spectra for the Fibonacci systems F ₅={BABABBAB}, F ₆={ABBABBABABBAB}, and F ₇={BABABBABABBABBABABBAB} with orientational defect angles (a) ϕ=π/3 and (b) π/2. Simulation conditions for CLCs and incident lights are the same as those of Fig. 2. Colors represent the reflectance calculated from the Fibonacci systems.

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Fig. 4. (Color online) Reflection spectra as a function of the unit length U_L for the Fibonacci systems (a)F ₄, (b)F ₅, (c)F ₆, and (d)F ₇ with an orientational defect angle ϕ=π/3. U_L is normalized with a cholesteric pitch P. Simulation conditions are fixed as those employed for calculations in Fig. 2 and colors refer to reflectance of the systems.

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In the Fibonacci system of CLCs, we also investigated the dependence of reflection spectra on the unit length of the quasi-periodicity. Figure 4 shows reflection spectra as a function of U_L for the Fibonacci systems (a) F ₄, (b) F ₅, (c) F ₆, and (d) F ₇ with ϕ=π/3, where UL is normalized with P. Increasing U_L from 2P to 5P, the width of the principal PBGs near λ_P becomes narrow. For example, in Fig. 4(a) the principal PBGs between 430 nm and 650 nm for U_L=2P become narrower by 90 nm, resulting in those between 460 nm and 590 nm for U_L=5P. It is also noticed that all principal PBGs in Fig. 2 and Fig. 3 always exist between 380 nm and 700 nm due to a constant U_L=1P, although other simulation conditions such as the order of the Fibonacci sequence and an orientational defect angle are different from each other. Hence, we can conclude that broader multiple PBGs covering full colors are obtained by use of the Fibonaccian CLCs with shorter U_L. With the increase of U_L, transmission regions between principal PBGs (shown by dips in reflection spectra) become narrow and the reflectance of whole PBGs increases, as shown in Fig. 4. Considering finer PBG structures around the principal PBGs, the number and reflectance of the fine structures are proportional to U_L of the quasi-periodic Fibonacci system (See Fig. 4). Finally, we should note that only negligible reflectance arises for left circularly polarized (LCP) light at least for small orders of Fibonacci system with R-CLCs such as F ₂, F ₃, and F ₄.

4. Conclusions

In summary, by introducing Fibonaccian defects into organic CLC systems, we demonstrated simultaneous red-, green-, and blue-colored reflections or multiple PBGs from single-pitched CLCs theoretically. The color of reflections calculated from the quasi-periodic Fibonacci system could be controlled by an orientational defect angle. Moreover we investigated the dependence of reflection spectra on the order and the unit length of the Fibonacci system and found that the PBG width decreases and reflectance increases with increasing the order and the unit length of the Fibonacci system. This Fibonaccian CLC system could be fabricated using PCLCs and provide a range of possibilities for reflective color-displays. The experiments are undertaken in our lab and will be reported in the future.