## Abstract

In this article we present a new all-optical method to measure elastic constants connected with twist and bend deformations. The method is based on the optical Freedericksz threshold effect induced by the linearly polarized electro-magnetic wave. In the experiment elastic constants are measured of commonly used liquid crystals 6CHBT and E7 and two new nematic mixtures with low birefringence. The proposed method is neither very sensitive on the variation of cell thickness, beam waist or the power of a light beam nor does it need any special design of a liquid crystal cell. The experimental results are in good agreement with the values obtain by other methods based on an electro-optical effect.

© 2014 Optical Society of America

The theory describing the distortions in NLCs, namely the continuum theory, has been well developed by Oseen and Frank [1,2]. According to the above, the bulk elastic properties of NLCs can be described by three invariants *K _{11}*,

*K*

_{2}_{2}and

*K*which are known as Frank elastic constants and are associated with the restoring forces opposing

_{33}*splay, twist and bend*distortions, respectively [3,4]. They determine the extent to which liquid crystals distort and respond to the applied fields. Typical values of

*K*in NLC are in the range 10

_{ii}^{−12}– 10

^{−11}N [5]. Since they are the main properties exploited in liquid crystal displays, switching devices [6], conducting research concerning among others self-focusing, self-phase modulation and light guiding [7–10] thorough knowledge of their value is indispensable. Since all deformations can be described by a superposition of the three basic deformations (

*splay*,

*twist*and

*bend*shown in Fig. 1). They are usually determined by applying an external field (electric or magnetic) to the NLC cell in a direction perpendicular to the director orientation fixed by surface anchoring forces. The density of free energy in NLCs deformed by an electric field can be described by the equation [3,4,7, 10,11]:

*θ*is the angle between the electric field and the director (see Fig. 1). Equation (2) contains only one elastic constant (i = 1,2,3) which corresponds to the configurations from Figs. 1(a)-(c) respectively, and is valid for small reorientation angle. In the case where $\overrightarrow{E}\perp \overrightarrow{n}$, the reorientation begins for the value of an electric field larger than the threshold value $E\ge {E}_{th}$ [7,12]. The threshold value

*E*can be found from Eq. (2), assuming the solution $\theta ={\theta}_{0}\mathrm{cos}\left(\raisebox{1ex}{$\pi z$}\!\left/ \!\raisebox{-1ex}{$d$}\right.\right)$

_{th}*(*where

*d*is the cell thickness and $-d/2<z<d/2$) and small angles $\mathrm{sin}2\theta \approx 2\theta $

*:*The elastic constants can be designated directly from Eq. (3) since the measurements of ${E}_{th}$ for the known cell thickness d and dielectrical anisotropy ${\epsilon}_{a}$ can determine the value of

*K*.

_{ii}This threshold effect known as a Freedericksz transition [7,12] occurs when the external field is large enough to overcome the elastic energetic barrier and is the most commonly used method of measuring the Frank elastic constants [13–26], since the threshold transition can be easily detected by a birefringence or capacitance change. The configuration depicted in Fig. 1(a) is very useful for determining the splay elastic constant *K _{11}* [27,28], provided the NLC dielectrical anisotropy is positive and the surface anchoring is strong. However, configurations from Figs. 1(b) and 1(c) are not very practical and easy to obtain, since it is difficult to obtain a uniform transverse electric field. Moreover, in geometry from Fig. 1(b) the capacitance change does not occur and the optical retardation measurement is quite difficult, since the polarization direction of a light beam adiabatically follows the director. This is the reason why the determination of twist

*K*and bend

_{22}*K*elastic constants are complicated. Measurements of those elastic constants can be done by applying magnetic field oblique to the cell but still normal to the initial direction. However, the latter gives only the ration

_{33}*K*and

_{11}/K_{22}*K*, and the value of

_{33}/K_{22}*K*or

_{11}*K*must be known in advance [29–32]. It is worth noting that the

_{33}*K*constant is highly important from a technological point of view, since a great deal of NLC devices are in twisted or super-twisted configurations.

_{22}In this article, we present an all-optical method to measure *K _{22}* and

*K*elastic constants in NLCs based on the induce optical nonlinearity i.e. molecular reorientation in liquid crystals. Hence, this method does not need any special design of the NLC cell besides only optical anisotropy must be known beforehand. As the laser beam intensity exceeds the optical Freedericksz transition threshold, molecular directors are reoriented by an electrical field of electro-magnetic wave

_{33}*E*. The electric field in Eqs. (1)-(3) is substituted by an average value which for the monochromatic takes the form: ${E}^{2}={E}_{opt}^{2}/2$ and dielectric anisotropy is replaced by optical anisotropy. For NLCs which are diamagnetic materials the magnetic permeability $\mu \approx 1$ and at the threshold condition the refractive index is equal to the ordinary refractive index${n}_{0}$. For a Gaussian beam ${E}_{opt}={E}_{0}\mathrm{exp}\left(-{r}^{2}/{w}^{2}\right)$, where

_{opt}*r*– radial component and w is the beam width, the relation between the electric field and the power of the beam is given by equation [33,34]:

The experimental setup is depicted in Fig. 2. A linear polarized light beam $\lambda =532nm$ passes through a polarizer and a half-wave plate (used to control light beam power and polarization) and is splitted by a beamsplitter. The reflected beam is incident on the detector 1 (Det 1) that controls the input beam power. The transmitted beam is gently focused to a waist of more than $10\mu m$, using a lens with properly chosen focusing length. The beam waist was measured using beam profiler Thorlabs BP209-VIS/M). NLC is sandwiched between two parallel glass plates (spaced$d=50\mu m$) with specially design internal surfaces to provide molecular anchoring (planar or homeotropic). The beam is normally incident on the sample along the z-axis and the sample is placed in the focal point of the lens. The outgoing beam is measured using the second detector Det 2.

The positive optical anisotropy of used NLC indicates, that the long axis of molecules tends to aligned parallel to the electric field. If a high enough light intensity $I>{I}_{th}$ illuminates the sample, the orientation inside the cell is changing. Light propagation through a cell is accompanied by an intensity-dependent phase shift, resulting from the intensity-dependent refractive index distribution which leads in turn to the modulation in reorientation angle distribution inside the cell. As a results, the transmitted light exhibits diffraction rings in the far field, as depicted in Fig. 3(a). For registration of reorientation, we use a precise method based on measuring the on-axis intensity. The central part of the beam (an inner part of the diffraction pattern) is isolated using the aperture located in front of the detector Det 2. Changes in the transmitted intensity detected in the center of the outgoing beam give us an evidence of reorientation of the molecules. As can be seen in Fig. 3(b), the output power increase proportional to the input power, at the input power of about 5mW the first peak appears, which indicates the threshold value for reorientation. Further increasing of the input power leads to decreasing the output one causes by the appearance of the first diffraction ring. The measurements depicted in Fig. 3(b) differ slightly each other as a results of a different position of the sample vs focal point of the beam, thus longitudinal shift of the input beam.

In this method, called a fixed beam waist method (FW), the threshold intensity ${I}_{th}$ is determined at the beam waist ${w}_{0}$ (e.g. ${I}_{th}=2{P}_{th}/\pi {w}_{0}^{2}$) and the accuracy of I_{th} is calculated according to the formula:

*w*, hence any imperfections in the experimental setting (i.e. position of the sample relative to the focal point of the beam) can lead to less accurate calculations of elastic constants.

_{0}To increase the accuracy of our measurements the method with fixed power (FP) of the beam and changing beam size is proposed. For this method a conventional z-scan setup [35] was adopted. The z-scan technique is performed by translating a sample (using a motorized translation stage) through the beam waist of a focused beam and then measuring the power transmitted through the sample. In experiments, we used lenses with focal lengths *f _{1} ∈ <100 ÷ 250> mm*. From the obtained graphs as in Fig. 4 the distance ${z}_{th}=1/2\left({z}_{1}-{z}_{2}\right)$ can be calculated between the focal point and the position z

_{1}where nonlinear effects starts and the beam intensity corresponds to the ${I}_{th}$ value and z

_{2}where the beam intensity is again lower than threshold value. Although, between points z

_{1}and z

_{2}many effects can be observed, e.g. nonlinear self-focusing, nonlinear self-diffraction, intensity dependent light scattering and losses, in this case it is important only to determine the threshold value for reorientation ${I}_{th}$.

For the Gaussian beam the width of an illuminated area can be calculated from the equation

In this method we measure the distance $z={z}_{th}$ around the focal point for which the reorientation occurs. Knowing the exact value of w_{0} (measured by beam profiler) and the distance ${z}_{th}$ we can calculate (using Eq. (9)) the beam waist for which the reorientation starts and further the light intensity ${I}_{th}=2{P}_{IN}/\pi {w}^{2}\left({z}_{th}\right)$ needed to induce this reorientation. In this method the input power remains constant and we do not need to place the sample exactly in the focal point (actually this point is obtained experimentally), as a result the light intensity ${I}_{th}$ is calculated more precisely that previously.

In this case, errors can be calculated from equation:

*z*distance. In Tables 1 and 2 we summarize results obtained by method with fixed beam waist and fixed power. In the FW method the input beam waist w

_{th}_{0}was 11.9μm. In the FP method the beam waist $w\left({z}_{th}\right)$was changed in the range $\left(30\xf750\right)\mu m$ and${z}_{th}\in \left(2\xf76\right)mm$. The results are averaged values of the series of thirty measurements.

E7 is the well-known NLC (Merck Group) with a value of *K _{22}* = 4.5pN and

*K*= 13pN; 6CHBT (4-trans-4’-n-hexylcyclohexylisothiocyanatobenzene, synthesized at the Military University of Technology) is NLC with typical values of

_{33}*K*= 3.6pN and

_{22}*K*9.5 pN [5]; 903 and 1110 are NLCs synthesized at the Military University of Technology by Prof. Dąbrowski group. The total error is a function of power, width and polarization of the beam, cell thickness, and transfer step in the

_{33}≈*z-scan*setup. In our experiments, errors connected with

*w*and

_{0}*z*are 2 orders of magnitude smaller than the others and therefore only the errors of power

_{th}*ΔP*, thickness

*Δd*and anisotropy

*Δε*(shown in Tables 1 and 2) have to be taken into account. To compare both methods, the errors in parentheses are connected only with

_{a}*I*for

_{th}*Δε*and

_{a}= 0*Δd = 0*. As it can be seen, the method with fixed power is much more precise.

In Ref 33 authors shows the role played by the beam waist in the computation of the threshold values in the case of a Gaussian beam collimated inside the cell. This effect is important for narrow beams. Following this, we made a series of measurements to address the role of the finite size of the beam in calculating the intensity threshold and finding the elastic constant values. In both methods, for the used $50\mu m$ in thickness cell we obtain that for beam size larger than 10μm, the presented methods are not sensitive to the beam waist. To prove the above, experiments with the fixed power of a beam were made for a few lens with a different focal length. The results are summarized in Fig. 5 and Table 3. In Fig. 5 we plot the K_{ii} as a function of focal lens for two NLCs, namely 6CHBT and 1110. We used 5 different lenses, with different focal length in the range from 100 to 250*mm*, that gave us the input beam waist in the range $\left(18.3\xf740.9\right)\mu m$and corresponding $w\left({z}_{th}\right)$was changed between $\left(29\xf747\right)\mu m$ for${z}_{th}\in \left(2\xf710\right)mm$. The difference between errors is connected only with *Δw _{0}/w_{0}* and for a wider beam waist it decreases.

We also test the influence of input beam power on calculated elastic constant values. The results presented in Table 4 were obtain for *w _{0} = 25,6 µm* and corresponding $w\left({z}_{th}\right)$was changed in the range $\left(30\xf750\right)\mu m$ for${z}_{th}\in \left(2.6\xf77\right)mm$. The measured values do not significantly depend on used beam power. We can expect that thermal effects (for high powers) could be important near the beam waist but not in the z

_{1}and z

_{2}planes [36].

## Conclusions

We proposed a new all-optical method which can be used to precise measurements of the elastic constants *K _{22}* and

*K*in nematic liquid crystals based on nonlinear effects. The obtained results are in good agreement with other experiments. The proposed all-optical method with fixed power is more precise and less sensitive to experimental conditions than the previously used ones. Among others, it does not need any precise placing of the sample in the setup.

_{33}## Acknowledgment

The authors thank Prof. R. Dąbrowski for providing nematic liquid crystals and Dr. E. Nowinowski-Kruszelnicki for sample preparation. This work was supported by the National Science Centre under the grant agreement DEC-2012/06/M/ST2/00479.

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