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 K11, K22 and K33 which are known as Frank elastic constants and are associated with the restoring forces opposing 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 Kii in NLC are in the range 10−12 – 10−11 N . Since they are the main properties exploited in liquid crystal displays, switching devices , 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]:Fig. 1 (in planar or homeotropic configurations), the Euler-Lagrange equation can be reduced to the following form: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 , the reorientation begins for the value of an electric field larger than the threshold value [7,12]. The threshold value Eth can be found from Eq. (2), assuming the solution (where d is the cell thickness and ) and small angles :Eq. (3) since the measurements of for the known cell thickness d and dielectrical anisotropy can determine the value of Kii.
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 K11 [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 K22 and bend K33 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 K11/K22 and K33/K22, and the value of K11 or K33 must be known in advance [29–32]. It is worth noting that the K22 constant is highly important from a technological point of view, since a great deal of NLC devices are in twisted or super-twisted configurations.
In this article, we present an all-optical method to measure K22 and K33 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 Eopt. The electric field in Eqs. (1)-(3) is substituted by an average value which for the monochromatic takes the form: and dielectric anisotropy is replaced by optical anisotropy. For NLCs which are diamagnetic materials the magnetic permeability and at the threshold condition the refractive index is equal to the ordinary refractive index. For a Gaussian beam , where 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]:Eqs. (3) and (4):
The experimental setup is depicted in Fig. 2. A linear polarized light beam 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 , 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) 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 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 is determined at the beam waist (e.g. ) and the accuracy of Ith is calculated according to the formula:
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  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 f1 ∈ <100 ÷ 250> mm. From the obtained graphs as in Fig. 4 the distance can be calculated between the focal point and the position z1 where nonlinear effects starts and the beam intensity corresponds to the value and z2 where the beam intensity is again lower than threshold value. Although, between points z1 and z2 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 .
For the Gaussian beam the width of an illuminated area can be calculated from the equation
In this method we measure the distance around the focal point for which the reorientation occurs. Knowing the exact value of w0 (measured by beam profiler) and the distance we can calculate (using Eq. (9)) the beam waist for which the reorientation starts and further the light intensity 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 is calculated more precisely that previously.
In this case, errors can be calculated from equation: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 w0 was 11.9μm. In the FP method the beam waist was changed in the range and. The results are averaged values of the series of thirty measurements.
E7 is the well-known NLC (Merck Group) with a value of K22 = 4.5pN and K33 = 13pN; 6CHBT (4-trans-4’-n-hexylcyclohexylisothiocyanatobenzene, synthesized at the Military University of Technology) is NLC with typical values of K22 = 3.6pN and K33 ≈ 9.5 pN ; 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 z-scan setup. In our experiments, errors connected with w0 and zth are 2 orders of magnitude smaller than the others and therefore only the errors of power ΔP, thickness Δd and anisotropy Δεa (shown in Tables 1 and 2) have to be taken into account. To compare both methods, the errors in parentheses are connected only with Ith for Δεa = 0 and 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 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 Kii 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 250mm, that gave us the input beam waist in the range and corresponding was changed between for. The difference between errors is connected only with Δw0/w0 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 w0 = 25,6 µm and corresponding was changed in the range for. 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 z1 and z2 planes .
We proposed a new all-optical method which can be used to precise measurements of the elastic constants K22 and K33 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.
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.
References and links
1. C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29(140), 883 (1933). [CrossRef]
2. F. C. Frank, “Liquid crystals. On the theory of liquid crystals,” Discuss. Faraday Soc. 25, 19 (1958). [CrossRef]
3. R. G. Pries, “Theory of the Frank Elastic Constants of Nematic Liquid Crystals,” Phys. Rev. A 7(2), 720–729 (1973). [CrossRef]
4. A. V. Zakharov, M. N. Tsvetkova, and V. G. Korsakov, “Elastic Properties of Liquid Crystals,” Phys. Solid State 44(9), 1795–1801 (2002). [CrossRef]
5. V. Volkmar, Liquid crystals databas, LiqCryst 4.2, (University of Hamburg, 2002).
6. D. K. Yang and S. T. Wu, Fundamentals of liquid crystals devices, (John Wiley & Sons, 2006).
7. I.-Ch. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific Publ., Singapore, 1993).
8. G. Assanto and M. A. Karpierz, “Nematicons: self-localized beams in nematic liquid crystals,” Liq. Cryst. 36(10-11), 1161–1172 (2009). [CrossRef]
9. M. Kwasny, U. A. Laudyn, F. A. Sala, A. Alberucci, M. A. Karpierz, and G. Assanto, “Self-guided beams in low birefringence nematic liquid crystals,” Phys. Rev. A 86(1), 013824 (2012). [CrossRef]
10. J. F. Strömer, E. P. Raynes, and C. V. Brown, “Study of elastic constant ratios in nematic liquid crystals,” Appl. Phys. Lett. 88(5), 051915 (2006). [CrossRef]
11. J. Kędzierski, Z. Raszewski, J. Rutkowska, T. Opara, J. Zieliński, J. Żmija, and R. Dąbrowski, “Optical investigation of the vector field of directors in the L.C,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 249, 199 (1994).
12. H. Mada, “Wall Effect of Frederiks Transition in Nematic Liquid Crystals,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 82(2), 53–59 (1982). [CrossRef]
13. S. Dasgupta and S. K. Roy, “Effect of a rigid, non-polar solute on the dielectric anisotropy, the splay and bend elastic constants, and on the rotational viscosity coefficient of 4-4′-n-heptylcyanobiphenyl,” J. Liq. Cryst. 30(1), 31–37 (2003). [CrossRef]
14. S. W. Morris, P. P. Muhoray, and D. A. Balzarini, “Measurements of the Bend and Splay Elastic Constants of Octyl-Cyanobiphenyl,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 139(3-4), 263–280 (1986). [CrossRef]
15. M. Ginovska, G. Czechowski, A. Andonovski, and J. Jadzyn, “Dielectric, viscous and elastic properties of nematogenic 1-(4- trans -propylcyclohexyl)-2-(4-cyanophenyl)ethane,” Liq. Cryst. 29(9), 1201–1207 (2002). [CrossRef]
16. M. L. Dark, M. H. Moore, D. K. Shenoy, and R. Shashidhar, “Rotational viscosity and molecular structure of nematic liquid crystals,” Liq. Cryst. 33(1), 67–73 (2006). [CrossRef]
17. S. Faetti and B. Cocciaro, “Elastic, dielectric and optical constans of the nematic mixture E49,” Liq. Cryst. 36(2), 147–156 (2009). [CrossRef]
18. Y. Zhou and S. Sato, “A method for determining elastic constants of nematic liquid crystals at high electric fields,” Jpn. J. Appl. Phys. 36(Part 1, No. 7A), 4397–4400 (1997). [CrossRef]
19. H. Ishikawa, A. Toda, H. Okada, and H. Onnagawa, “Relationship between order parameter and physical constants in fluorinated liquid crystals,” Liq. Cryst. 22(6), 743–747 (1997). [CrossRef]
20. S. DasGupta and S. K. Roy, “Splay and bend elastic constants and rotational viscosity coefficient in a mixture of 4–4-n-pentyl- yanobiphenyl and 4–4-n-decyl-cyanobiphenyl,” Phys. Lett. A 306(4), 235–242 (2003). [CrossRef]
21. L. A. Parry-Jones and M. A. Geday, “Measurement of Twist Elastic Constant in Nematic Liquid Crystals using Conoscopic Illumination,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 436(1), 259 (2005). [CrossRef]
22. K. Koyama, M. Kawaida, and T. Akahane, “A method for determination of elastic constants K1, K2, K3 of a nematic liquid crystal only using a homogeneously aligned cel,” Jpn. J. Appl. Phys. 28(8), 1412–1416 (1989). [CrossRef]
23. A. V. Dubtsov, S. V. Pasechnik, D. V. Shmeliova, V. A. Tsvetkov, and V. G. Chigrinov, “Special optical geometry for measuring twist module K22 and rotation viscosity of nematic liquid crystals,” Appl. Phys. Lett. 94, 181910 (2009). [CrossRef]
24. N. V. Madhusudana and R. Pratibha, “Elasticity and Orientational Order in Some Cyanobiphenyls: Part IV. Reanalysis of the Data,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 89(1-4), 249–257 (1982). [CrossRef]
25. S. Singh, Liquid Crystal: Fundamental, Chapter 4, (World Scientific, 2002).
26. H. L. Ong, “Measurement of nematic liquid crystal splay elastic constants with obliquely incident light,” J. Appl. Phys. 70(4), 2023 (1991). [CrossRef]
27. A. Kumar, “On the Dielectric and Splay Elastic Constants of Nematic Liquid crystals with Positive Dielectric Anisotropy,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 575(1), 30–39 (2013). [CrossRef]
28. A. Srivastava and S. I. Singh, “Elastic constants of nematic liquid crystals of uniaxial symmetry,” J. Phys. Condens. Matter 16(41), 7169–7182 (2004). [CrossRef]
30. M. P. Allen, M. A. Warren, M. R. Wilson, A. Sauron, and W. Smith, “Molecular dynamics calculation of elastic constants in Gay–Berne nematic liquid crystals,” J. Chem. Phys. 105(7), 2850 (1996). [CrossRef]
31. P. I. C. Teixeira, V. M. Pergamenshchik, and T. J. Sluckin, “A model calculation of the surface elastic constants of a nematic liquid crystal,” Mol. Phys. 80(6), 1339–1357 (1993). [CrossRef]
32. T. Toyooka, G. Chen, H. Takezoe, and A. Fukuda, “Deteremination of twist elastic constant in 5CB by four independent light-scattering techniques,” Jpn. J. Appl. Phys. 26(12), 1959–1966 (1987). [CrossRef]
33. W. K. Bajdecki and M. A. Karpierz, “Nonlinear optical measurements of elastic constants in nematic liquid crystals,” Acta Physica Polonica A. 95, 793–800 (1999).
34. L. Calero, W. K. Bajdecki, and R. Meucci, “Reorientation effect induced by a CW CO2 laser in nematic liquid crystal,” Opt. Commun. 168(1-4), 201–206 (1999). [CrossRef]
35. R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. 18(3), 194 (1993). [CrossRef] [PubMed]
36. M. H. Majles Ara, S. H. Mousav, M. Rafiee, and M. S. Zakerhamidi, “Determination of Temperature Dependence of Kerr Constant for Nematic Liquid Crystal,” Mol. Cryst. Liq. Cryst. (Phila. Pa.) 544, 227 (2011).