Helicodial periodic structures, such as cholesteric liquid crystals (CLCs) [1], are one-dimensional photonic band gaps (PBG), which are present in the nature in different forms [2]. Those are very promising systems for tunable distributed feedback lasing [3–5], large band reflection [6], tunable filtering [7–9], etc. In addition to their spatial periodic character, CLCs have unique vectorial properties, which are related to their helicoidal nature and which affect significantly the character of modes of the PBG [1]. Thus, for example, the eighenmodes of such a PBG are circularly polarized waves and the momentum conservation requirement prohibits the existence of higher order Bragg reflections at normal incidence [1,10]. This well known phenomenon is demonstrated in Fig. 1 , where we can see the transmission spectra of a mono domain cell filled by a CLC mixture, composed of 46 wt % of CB15 (from Merck) and 54 wt % of a homemade nematic liquid crystal 1756-6: the absence of the second order Bragg reflection is shown by the vertical arrow. Another unique property of those structures is their “spatial adaptivity”. That is, the chemical composition and the molecular chirality ofthe CLC define the natural period P_N of complete (2π) helical molecular rotation for “free” boundary conditions only. In reality, there are always given boundary conditions, which define the “adjusted” period P_A of the helix depending upon the distance L of substrates of the cell that contains the CLC and upon the molecular anchoring conditions on those substrates. Thus, if the alignment angle of the director n (average alignment of molecular axes [1], ) on two opposite substrates (at $z=0$ and $z=L$) of the cell is fixed by strong boundary conditions, say providing a total rotation of θ(L), then the P_A of the CLC will be naturally “self-adjusted” to provide a constant rate of spatial rotation $Q\equiv \partial \theta /\partial z=2\pi /P_A$ of its director in a way to satisfy the above mentioned boundary conditions with $\theta (L)=2\pi L/P_A$. The associated (to this adjustment) energy of director’s deformation will be counter-balanced by the anchoring energy of substrates. This total rotation angle may be split into N_C complete rotations (on 2π) and some additional rotation $\theta (L)=2\pi N_C+\delta \theta$. Thus, we will have $N_C=L/P_A$ for $\delta \theta =0$, which is the case of the present experiment (see later). It must be noticed also that, given the local equivalence of n and - n, the optical periodicity and corresponding resonance wavelength λ_R of the CLC are defined by the half period $P_R=P_A/2$ of the helix (corresponding to director’s rotation on angle π) as ${\lambda }_R=2n_{av}{P}_R$,where $n_{av}=(n_{\parallel }+n_{\perp })/2$ is the average refractive index of the CLC, $n_{\parallel }$ and $n_{\perp }$ being the local extraordinary and ordinary refractive indexes of the CLC, respectively. That is why; the number of those half-helixes (or pitches) $N=2N_C$ is often used for the analyses of the CLC’s behaviour.

 

Fig. 1 Transmission spectra of a CLC mixture, measured with an unpolarized probe beam, demonstrating the key characteristics of a CLC: the absence of the second order Bragg reflection (shown by the vertical arrow) and the large bandwidth of the resonance defined by the local anisotropy of the CLC.

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Finally, the value of P_R depends also upon various external stimuli, such as temperature, electric and magnetic fields [1–12]. This dependence attracted significant interest in view of possible applications in dynamic electro optic modulation devices, see also Ref [13]. In particular, the behaviour of CLC under the action of electric field has been intensively studied for CLCs having positive and negative dielectric anisotropies ∆ε (see Ref [14] and references therein). Various mechanisms of P_R modulation have been considered so far, but finally the electromechanical deformation, induced by DC voltage was clearly identified as a dominating mechanism in CLCs with negative ∆ε [14]. This work reported very interesting data and analyses, including the detailed description of smooth variations of λ_R upon the electric field induced deformation of cell substrates. The possibility of dynamic breaking of the boundary condition, to change the number of pitches by jumps, was briefly mentioned in the stationary excitation regime and also was demonstrated when periodically tightening and loosening the clamp, which held the sample [14].

In the present work, we further investigate this phenomenon but in much stronger excitation regime. This is achieved by means of the use of a very thin cell substrate subjected to AC voltage. We show that those CLCs may demonstrate multiple abrupt transient pitch changes even at stationary regime of excitation, at predetermined values of fixed electric field (without tightening and loosening the clamp) and that this “jumping” process is happening via the formation of a transient disclination (line of abrupt change of director orientation), while maintaining its bulk helical characteristics.

The material composition used in the present work was the mixture CB15, purchased from Merck and used without modifications. It is a CLC with negative ∆ε, thus, the application of an electric field is stabilizing the helix. The cell was built by using an ITO coated glass substrate of thickness 0.7 mm, which was additionally coated by a uniformly rubbed planar alignment Polyimide (PI 150, from Nissan). An adhesive wall was dispensed on the periphery of this substrate providing a square shaped working area (optical window) of ≈6.5 x 6.5 mm² size. The adhesive contained spacers to provide the desired thickness of the cell L=5 μm (±0.5 μm). Then a second substrate, similar to the first one, but with thickness of 0.1 mm, was pressed on the first one and the peripheral adhesive was photo polymerized by using a UV lamp exposure. The CLC mixture was then injected into the obtained sandwich-like cell by capillary action at room temperature. All our experiments have been done at room temperature (the isotropic phase transition of CB15 happening approximately at 37.5°C±0.5°C).

We have started our experiments by spectral studies using a Scienctech spectrophotometer with THI Halogen lamp and LDA 2000 monochromator. It is interesting to note that the transmission spectra (Fig. 2 ) of obtained cells were not perfectly symmetric, which could be related to the imperfections of the helix (tilted domains generating effective “blue shift” of reflection) as well as to the non ideal collimation of the probe beam (which creates an artificial “apodization” effect of the grating structure). However, it is also important to mention that even in perfect experimental conditions the resonance of the CLC has certain width ∆λ, which is defined as $\mathrm{\Delta }\lambda =\mathrm{\Delta }n{P}_A$where the $\mathrm{\Delta }n\equiv n_{\parallel }-n_{\perp }$is the local optical anisotropy of the CLC [1,11].

 

Fig. 2 The transmission spectra of the CLC cell for unpolarized input light at different RMS voltages applied to the cell up to the first jump of the pitch (voltages are growing from 0 to 18 RMS Volts; low excitation regime).

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We have analyzed the spectral characteristics of the cell for different voltages applied to ITOs (waiting 5-10 minutes after each voltage change, before taking the data) by means of an Avtech power supply (AV-151 B-C) generating sin shaped AC signal of 1 kHz frequency.

As one can see in the Fig. 2 and Fig. 3 , the moderate increase of the voltage brings to smooth changes of λ_R, as already reported in Ref [14]. One of the noticeable differences (apart of the material and AC excitation) here is the slope of this variation, which is very high in our case; approximately 18 nm for 17 V (applied to the 5 μm thick cell), providing thus a shift coefficient ≈5.3 nm/V/μm, which is more than twice the maximum value reported so far [15], thanks to the thin cell substrate used here.

 

Fig. 3 The dependence of the resonance wavelength of the CLC upon the RMS voltage applied to the cell (growing voltage: open squares and decreasing voltage: filled circles).

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As we can see also in the Fig. 2, the value of λ_R of the CLC is approximately equal to 0.542 μm in the ground state (U=0 volt; the resonance shifting towards shorter wavelengths with the temperature increase). Surprisingly, in spite of the rather large utilisation of the CB15 mixture, to the best of our knowledge, there is no information about its optical properties. We have done measurements of its average refractive index, which gave us the following value: $n_{av}\approx 1.567\pm 0.002$. We can thus estimate the room temperature ($T\approx 21°C$) value of $P_A=0.542\mu m/1.567\approx 0.35\mu m$. Thus, for an initial thickness of $L=5\mu m$, the number of helixes would then be $N_C=5/0.35\approx 14$ (so, 28 half helixes). In the same time, we used also the corresponding width of the resonance $\mathrm{\Delta }\lambda =40nm$ (from Fig. 2) to evaluate the local anisotropy of that compound $\mathrm{\Delta }n=\mathrm{\Delta }\lambda /P_A$≈$40nm/0.35\mu m=0.114$, supposing an ideal helix which is a rather high value for CLC materials.

The further increase of the voltage shows an abrupt “return jump” of λ_R, at $U\approx 17V_{RMS}$, see Fig. 2 (curve for 18 V) and Fig. 3 (open squares). This jump happens within approximately 3 min at fixed voltage (observed with a probe beam of diameter ≈1 mm). In fact, with the continued increase of the voltage, we observe multiple zones of such smooth modulations followed by abrupt changes of the value of λ_R. We estimate that the cell thickness is reduced (from initial 5μm) to approximately 4.4μm for the voltage value of 30 V. The decrease of the voltage then brings to the inverse transformation (Fig. 3, filled circles) with however some variations of the positions of those jumps and also of the depth of modulation of the values of λ_R. This hysteresis-like behaviour may be related to the mechanical movement of the substrate, which in addition, is coupled with the mechanical movement of the CLC and with the deformation of its director. Thus, the starting point during our experiments was ${\lambda }_R=0.528\mu m$ or ${\lambda }_R=0.542\mu m$ in a non regular manner. We would need more studies to better understand the origins of such bistability, while similar phenomena have been also described in [16]).

It is clear that the use of a thin (0.1 mm) substrate allowed us to obtain significantly stronger (more deformation is obtained at much lower voltages) electro-mechanical effect compared to Ref [14]. As we can see in the Fig. 4 , the bending of the thin substrate is easily observed (at room temperature) in the reflected, from the cell, light after filling it by the CLC (the cell is illuminated by the broad band light of the ambient illumination, and its reflection is observed at the optimal angle to visualize the periodic transverse modulation of reflection).

 

Fig. 4 Ring structure observed in the reflected, from the CLC-filled cell, for various voltages applied 20 V (left picture) and 10 V (right picture). Vertically aligned half ellipsoidal white zones (on right and left sides of each picture) are the conductive adhesive zones with vertical wires in the bottom zones of the figure.

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In fact, the electro-mechanical modulation of the cell thickness L is so strong that the energy of director deformation (needed to adapt the value of P_R) is overcoming the energy necessary to create a transient disclination of the director. This disclination is the transition mechanism, which allows the re-adjustment of the P_R for the given value of L, while preserving the overall helical structure of the CLC, as demonstrated in the Fig. 5 . In fact, the spectral monitoring of the cell during this transition was done when the applied voltage was set just above the critical voltage U_C (that is necessary to cause the jump of λ_R). The consecutive moments of the same jump are represented by the curves 1 (squares), 2 (circles) and 3 (triangles), taken with approximately 1 min of delay. This monitoring confirms that the spectral and vectorial characteristics of the resonance do not change significantly (obviously, averaged on the area covered by the probe beam, approximately 1 mm²) during those jumps, the second order Bragg reflection being always non observable (not shown here).

 

Fig. 5 Spectral modifications of the cell during the evolution of the disclination (at a fixed voltage, slightly above the jumping threshold voltage) allowing the re-adjustment of the pitch of the helix. Consecutive spectra (labelled 1, 2 and 3) are taken with approximately 1 min of delay.

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Observations of the same transition, using Zeiss polarizing microscope, allow the confirmation of the above mentioned scenario. As it can be seen in the Fig. 6 (bottom left), in this particular case, the disclination is generated almost simultaneously in the top right corner and the left side of the picture, when the voltage exceeds the threshold value 20V (which, by the way, differs from cell to cell, but also can be slightly different for the same cell from experiment to experiment, on approximately ±10%). Then, for a fixed voltage (20 V), the left-side disclination wall propagates faster and merges with the top right corner zone providing a stabilized CLC with a new P_R. It must be emphasized however that this picture shows only the central part of the cell where the disclination walls usually appear. From cell to cell, various surface or volume defects were at the origin of those disclinations.

 

Fig. 6 Microscope observation of the transient propagation of the disclination wall allowing the establishement of a self-adjusted (to the new value of L) period of director rotation. Consecutive pictures (at 20 V) are taken with approximately 0.5 min of delay.

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Discussion

One can notice (from the Fig. 3) that the slope of the dependence λ_R(U) in the first zone (for voltages below 17 V) is relatively small and the corresponding voltage range of smooth changes (before the first jump) is rather broad. In contrast, the slope increases and the smooth zone’s width decreases for further zones. This is consistent with the hypothesis of periodic generation of transient disclination lines allowing the jumps of P_A. In fact, for a given value of cell thickness L, its reduction ΔL must be redistributed equally on the number of pitches and the director’s rotation rate Q (between jumps) must then be increased consequently $Q+\mathrm{\Delta }Q=2\pi /(P_{A0}-\mathrm{\Delta }L/N_C)$ to reduce the initial period P_A0 of the helix on the amount $\mathrm{\Delta }L/N_C$. In the same time, the appearance and propagation of the disclination wall is a rather complex phenomenon [1,16]. The situation here is further complicated due to the presence of the mechanical movement and stabilizing electrical field (since Δε<0). Qualitatively, we can imagine the process of observed pitch jumps as the transient analogy of structures observed in the stationary Cano wedge cell [17,18]. This analogy however is limited to the role of disclination wall spliting two neighboring zones with clearly distinguishable pitches. Closer analyses of the disclination wall, dynamically propagating from the left corner, shows that its width is at the order of l ~3 μm with some “fine structure” (not shown here) of light transmission modulation (across the wall) with characteristic sizes of ≤ 0.5 μm. In the same time, the character (and structure) of the disclination line, which started from the top right corner, is different, which perhaps could explain its lower spatial mobility. Further studies must be conducted to understand this difference, but the possibility of generating singularity lines of different forces has been already discussed in the literature [1]. For the moment, in a very rough approximation, we could imagine that the additional density of free energy (due to the additional rotation rate ΔQ allowing the pitch adaptation) could be estimated to be at the order of $F_A\sim 0.5K_2\mathrm{\Delta }{Q}^2$, where K₂ is the twist elasticity constant, typically at the order of 10⁻⁶ erg/cm [1]. Given the complexity of the disclination wall (including various types of deformation), we could consider the so called one-constant elasticity approximation ($K_1=K_2=K_3\equiv K$) to evaluate very roughly the corresponding energy. Thus, with the scale of this deformation l, the corresponding energy F_d might be estimated by the following expression $F_d\approx K/l^2$ [1]. Thus, the generation of the disclination wall will be energetically justified for voltage values (and corresponding deformations) satisfying the condition $F_A\ge F_d+\mathrm{\Delta }{F}_S$, where ΔF_S is the difference in anchoring energy before and after the generation of the disclination. In the framework of present approximation, we shall consider the ΔF_S to be independent upon the thickness of the cell (it could be interesting to use the described here experimental technique to estimate the surface anchoring potential [19]). Thus, for approximately the same value of F_d, the pitch jumps should happen when the thickness reduction ΔL would generate the critical change of rotation rate $\mathrm{\Delta }{Q}^2\approx 2(K{l}^2+\mathrm{\Delta }{F}_S)/K_2$. Given the definition of $\mathrm{\Delta }Q=2\pi \mathrm{\Delta }L/[P_{A0}(N_C{P}_{A0}-\mathrm{\Delta }L)]$, we obtain, for small deformations (ΔL<<L), that those critical values of ΔQ (and thus the corresponding pitch jumps) would be achieved for larger values of ΔL if the number of pitches N_C was initially high. In the same way, the slopes of both dependences ΔQ(ΔL) and λ_R(ΔL), will be higher for lower values of N_C. Both above mentioned hypotheses are in agreement with our experimental observations (Fig. 3).

In conclusion, we believe that the present work allowed us to observe significantly higher electro-optic effect by using a thin cell substrate. The combination of two conditions (more flexibility and helix stabilizing field) allowed us to generate multiple pitch jumps. The spectral and polarizing microscope observations confirmed that the key features of the helix are preserved and the self-adaptation proceeds via a transient disclination propagation. This work however shows also that the modulation range of λ_R in those self-adaptive CLCs is limited (in our case, to approximately ± 5%) due to the self-adaptivity of the CLC, which generates transient disclination and re-adjusts its period when the deformation of the pitch becomes energetically “unfavorable”.