1. Introduction

Chiral liquid crystals, which have a helical structure formed by self-assembly, have recently attracted much attention because their helical structure can be regarded as a one-dimensional pseudo-photonic crystal [1]. Photonic crystals are dielectric periodic structures with an interval of submicron length. Their periodicity opens up a photonic band gap which results in many unique optical properties [2, 3]. It is expected that photonic crystals will enable the development of next-generation optical devices such as low-threshold lasers and microoptical circuits. However, the fabrication of a periodic structure with a periodicity equivalent to the visible wavelength generally requires a high-cost process. Therefore, a self-assembled nano-helical structure of chiral liquid crystals would be advantageous for nanostructure fabrication. Furthermore, the helical pitch of chiral liquid crystals is easily controlled by external fields, such as temperature, mechanical stress, or electric field [4–6]. Due to such large tunabilities, chiral liquid crystals have been regarded as a potential candidate of tunable photonic crystals.

It is well known that cholesteric and chiral smectic C phases consist of a helical structure. There have been many studies on cholesteric liquid crystals (CLCs) as a chiral photonic crystal [7–16]. In contrast, relatively few studies have investigated a chiral smectic C phase of ferroelectric liquid crystals (FLCs) as a photonic crystal [5, 6, 17]. The reason is that an effective birefringence of FLCs is much smaller than that of CLCs. In general, a typical effective birefringence of CLCs is 0.2, whereas that of FLCs is 0.01 to 0.03 due to a small tilt angle of molecules. The small birefringence of FLCs brings a narrow photonic stopband and a weak optical confinement.

To improve an optical confinement of chiral photonic crystals, many researchers have studied the designs of a cavity structure by introducing photonic crystal defects or multilayer structures [10–17]. For example, an FLC sandwiched between two dielectric multilayers has a much higher optical confinement than a simple FLC [17]. A cholesteric blue phase, which has a three-dimensional optical confinement, has also been studied extensively [18, 19]. On the other hand, since scattering at orientational defects weakens an optical confinement of a cavity, there are several reports of improving molecular orientation [7, 20].

We here focus on an optical stopband of an FLC at highly oblique incidence because chiral liquid crystals exhibit unique optical properties at highly oblique incidence. It is well known that CLCs reflect circularly polarized light at normal incidence. However, at highly oblique incidence, CLCs show polarization independent reflection [21]. Such basic optical properties were studied at an early stage of the development of liquid crystal optics in the 1970s and 1980s [21–25]. Angular dependence of FLCs was also studied in early 1980s to reveal optical properties of full-pitch and half-pitch bands [24, 25]. According to the earlier experiments, more than 50% of light are reflected by a full-pitch band at highly oblique incidence. Insofar as we know, there has not been optical applications or devices developed with a full-pitch band at highly oblique incidence. In this study, we have demonstrated a wavelength and bandwidth tunable photonic stopband in an FLC using oblique incidence.

2. Experimental

We used an FLC (CS1017, Chiso) in experiment which has a chiral smectic C phase from 20 to 55 ^◦C. The helical pitch of the FLC is 475 nm at 20 ^◦C and the tilt angle is 26^◦. The birefringence of the FLC is 0.16 at 546 nm. A 30-μm homeotropic cell is prepared by two glass substrates coated with a polyimide (JALS- 2021-R2, JSR). The FLC is filled into the cell at an isotropic phase, and then a uniform chiral smectic C phase was obtained by slowly cooling down the cell. In this configuration, the helical axis of the FLC is perpendicular to the glass surface. To measure transmission properties of the cell, light from a halogen lamp with or without a polarizer was used as an incident light and the transmitted light was measured using a CCD spectrometer (Bluewave, StellaNet). All experiments were carried out in a thermostatic chamber to control temperature.

Transmission spectra of the FLC cell at highly oblique incidence were measured using two equilateral-triangular prisms which were two of N-SF11 (n = 1.78 at 589.6 nm) or two of BK7 (n = 1.51 at 632.8 nm). The FLC cell was sandwiched between the two prisms, as shown in Fig. 1 . Here, a glycerol-water solution (glycerol 90% - water 10%) whose refractive index is 1.48 at 600 nm was filled in the gaps between the cell and prisms to suppress reflection loss. In Fig. 1, θ₁ is the incident angle into the prism from air and θ₂ is the propagation angle in the glass substrate of the FLC cell. If light beam enters a simple cell without the prisms, the maximum angle of θ₂ is approximately 41.5^◦ owing to refraction at the boundary between air and glass substrate. That is, the propagation angle in the glass substrate is limited by refraction in a simple cell. However, the use of the prisms allows us to measure transmittance at highly oblique incidence. In this study, we measured transmission spectra with θ₂ from 41^◦ to 68^◦.

 

Fig. 1 Schematic of experimental setup for measurement of transmission spectrum at highly oblique incidence. (a) an FLC cell sandwiched between two triangular prisms. (b) an enlarged view of the FLC cell.

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3. Results and discussion

The temperature dependence of a selective reflection from the FLC at normal incidence is shown in Fig. 2 . Figure 2(a) shows measured transmission spectra of the FLC in which the incident light is non-polarized. Due to small birefringence, the stopband is narrow and weak. The full width at half depth of the stopband is 23 nm at 21.6 ^◦C. The pitches of the FLC at 17.1, 21.6 and 25.3 ^◦C are 443, 492 and 540 nm, respectively. These pitches were determined by the center wavelengths of the selective reflections. The temperature dependence of the helical pitch is plotted in Fig. 2(b). It is clear that the pitch of the helical structure shortens with decreasing temperature. We also calculated transmission spectra by using Berreman’s 4 × 4 matrix method [26] in which we did not take account of the refractive index dispersion of the FLC. The ordinary refractive index n_o and extraordinary refractive index n_e of the FLC were assumed to be 1.49 and 1.65, respectively. The pitches of the FLC in the calculations were based on the experimental results. Figure 2(c) shows the calculated spectra in which thelegend indicates the helical pitch of the FLC. The depths of the calculated stopbands are deeper than these of measured stopbands. This is because experimental results include non-uniformities of cell thickness and helical pitch.

 

Fig. 2 (a) Measured transmission spectra of the FLC as a function of temperature. (b) Temperature dependence of helical pitch of the FLC. (c) Calculated transmission spectra as a function of helical pitch.

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The measured and calculated oblique-incidence transmission spectra for P-polarization light are shown in Fig. 3 . We here used two prisms of N-SF11 and these measurements were carried out in the thermostatic chamber at 24 ^◦C. In the calculations, we took account of the refraction index dispersions of the prism, glycerol-water solution, and glass substrate. Figure 3(a) show the experimentally obtained transmission spectrum at θ₁ = 50^◦ at which θ₂ becomes 41.8^◦. Although θ₂ varies depending on wavelength due to refractive index dispersion of the materials, we simply express θ₂ at 600 nm as θ₂ here. In Fig. 3(a), there is a stopband splitting in two dips around 600 nm which is a half-pitch band of the FLC. Figure 3(b) shows the calculated transmission spectrum at θ₂ = 41.4^◦ (θ₁ = 50.8^◦). The calculated spectrum also has a stopband splitting in dips around 600 nm. The angles in calculation are slightly different with those in experiment but the calculation approximately agrees with the experimental result. We consider that the error is caused by frequency dispersion of the FLC and experimental error. This is because that the propagation angle in the FLC is affected by the refractive index of the FLC. The information of the angle is important for the complete reproduction of the spectrum. Figure 3(c) shows the measured transmission spectrum at θ₂ = 67.2^◦ (θ₁ = 15^◦). In the spectrum, a wide stopband is observed from 670 to 790 nm. The full width at half depth of the stopband is 176 nm which is much larger than the stopbands at normal incidence. This band is not a half-pitch band but a full-pitch band. Note that the full-pitch band shows total reflections from 670 to 790 nm despite the incident light having only P-polarization. The calculation spectrum in Fig. 3(d) also shows total reflections. In the both measured and calculation spectra in Figs. 3(c) and 3(d), the maximum transmittances are less than 80%. This is caused by the Fresnel reflection between the optical glass N-SF11 and glycerol-water solution. Further, the measured transmittances gradually decrease at shorter wavelengths because N-SF11 has absorption in the region. In Fig. 3(d), the spectrum has oscillation caused by multiple reflection between the boundaries of the helical structure to the glass surfaces [21]. Inthe measured spectrum in Fig. 3(c), the oscillation was smoothed by the non-uniformity of a cell thickness.

 

Fig. 3 Measured transmission spectra of the FLC cell at highly oblique incidence for P-polarization light at 24 ^◦C; (a) θ₁ = 50^◦ and θ₂ = 41.8^◦, (c) θ₁ = 15^◦ and θ₂ = 67.2^◦. Calculated transmission spectra; (b) θ₁ = 50.8^◦ and θ₂ = 41.4^◦, (d) θ₁ = 15.8^◦ and θ₂ = 66.4^◦. In the calculations, the helical pitch is set to be 521 nm.

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The transmission characteristics of the FLC for S-polarization were also investigated. Figure 4 shows measured and calculated oblique-incidence transmission spectra for S-polarization light. As is the case with P-polarization, a half-pitch band of the FLC is observed in Figs. 4(a) and 4(b). A full-pitch band of the FLC appears from 670 to 790 nm in Figs. 4(c) and 4(d). It is particularly important that the S-polarized light is also totally reflected at the range from 670 to 790 nm. That is, both P- and S-polarized light are totally reflected at the frequency range. These results suggest that the full-pitch band of the FLC at highly oblique incidence has no polarization dependence.

 

Fig. 4 Measured transmission spectra of the FLC cell at highly oblique incidence for S-polarization light at 24 ^◦C; (a) θ₁ = 50^◦ and θ₂ = 41.8^◦, (c) θ₁ = 15^◦ and θ₂ = 67.2^◦. Calculated transmission spectra; (b) θ₁ = 50.8^◦ and θ₂ = 41.4^◦, (d) θ₁ = 15.8^◦ and θ₂ = 66.4^◦. In the calculations, the helical pitch is set to be 521 nm.

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It is well known that a stopband of chiral liquid crystals exhibits reflection more than 50% at oblique incidence. This can be easily explained by the fact that the helical periodicity of a chiral liquid crystal is no longer a pure sinusoidal pattern in oblique directions. The phenomenon becomes conspicuous at highly oblique incidence. On the other hand, why is the full-pitch band of the FLC at highly oblique incidence much wider than that of at normal incidence? This can be qualitatively explained by the use of angular dependence of Fresnel reflection at a boundary. Let us consider the FLC has an effective birefringence of 0.03. In addition, for simplicity, we suppose the FLC is a simple multilayer. In this case, the stopband is formed by multiple reflection at boundaries between n₁ = 1.49 and n₂ = 1.52. Incident angle dependences of a single Fresnel reflection at the boundary are easily obtained, as shown in Fig. 5(a) . Figure 5(b) shows an enlarged view of the reflection curve for small incident angles. Since the birefringence is small, the reflectance at normal incidence is only 0.01%. Such a weak reflection is not suited for forming a wide reflection band. However, reflections at high incident angles are much stronger than those at small incident angles. For example, the reflectances for P- and S-polarizations at 75^◦ are 0.97 and 1.39%, respectively, which are approximately 100 times larger than that at normal incidence. A reflectance of 1% by one Fresnel reflection is equivalent to at normal incidence with Δn = 0.33. The increase in reflectance can be explained by the assuming that the index contrast becomes large effectively. Therefore, it is easy to understand that the large effective contrast results in the wide stopband at highly oblique incidence. On the other hand, this can be also explained by Fourier analysis. As mentioned above, a stopband is formed by multiple reflection. When reflection is weak at each layer, many periods are required to form a stopband. In real space, the many periods means a long distance. According to the Fourier theory, frequency space is defined as the reciprocal of real space. Therefore, in the case of weak reflection, the frequency range of the stopband becomes narrow. In contrast, when reflection is strong, a stopband can be formed by multiple reflection in fewer periods. In this case, the frequency range of the stopband becomes wider due to the shorter distance. The most important point here is that the incident angle is associated with the width of the stopband. That is, we can control thebandwidth by changing the incident angle. Further, if temperature or electric field is controlled, the helical pitch can be tuned independently of the bandwidth.

 

Fig. 5 (a) Incident angle dependences of a single Fresnel reflection at the boundary between ordinary index of 1.49 and effective extraordinary index of 1.52. (b) an enlarged view of the reflection curve for small incident angles.

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We have experimentally demonstrated the bandwidth and wavelength tuning of the stopband by controlling incident angle and temperature. In the experiment, we first set the incident angle to control the bandwidth, and then temperature was varied to tune the center wavelength of the stopband. Figure 6(a) shows the measured transmission spectra of the FLC for P-polarization as functions of θ₂ and temperature. We here used two prisms which were made of BK7. Note that three stopbands have the same center wavelength of 700 nm, but the bandwidths are not same. Each of the full width at half depth at 14.7, 21.5, and 26.8 ^◦C are 111, 140, and 172 nm, respectively. Figure 6(b) shows the calculated transmission spectra in which the parameter of wavelength is based on the experimental results shown in Fig. 2(b). The angles are slightly different but the calculations approximately agree with the experimental results. Although the maximum transmittance decreased in the experimental results, we consider that the decrements are caused by the Fresnel reflection between the boundaries. The loss could be improved by matching the refraction indices of the materials. Finally, we emphasize that these phenomena occur in only the full-pitch band. A half-pitch band of CLCs or FLCs is not appropriate for the wavelength and bandwidth tuning because the optical properties of a half-pitch band are more complicated [21].

 

Fig. 6 (a) Measured transmission spectra of the FLC as functions of θ₂ and temperature. (b) Calculated transmission spectra of the FLC as functions of θ₂ and helical pitch p.

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4. Conclusions

We demonstrated the wavelength and bandwidth tunable photonic stopband using the full-pitch band of the FLC. The wide stopband was observed at highly oblique incidence which did not have polarization dependence despite using a FLC. Furthermore, the bandwidth was controlled independently of the center wavelength of the stopband by changing incident angle. We expect that the FLC stopband contributes for the development of optical filters and new photonic crystal applications.