Approximation of electro-optical hysteresis characteristics of ChLC

In order to select proper cholesteric liquid crystal (ChLC) materials and drive schemes for cholesteric liquid crystal displays (ChLCD), it is necessary to make the protracted experimental analysis of electro-optical hysteresis properties of ChLCs. Method for approximation of electro-optical charact...

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Veröffentlicht in:Semiconductor Physics Quantum Electronics & Optoelectronics
Datum:2004
Hauptverfasser: Rybalochka, A., Chumachkova, M., Sorokin, V.
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Veröffentlicht: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2004
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Zitieren:Approximation of electro-optical hysteresis characteristics of ChLC / A. Rybalochka, M. Chumachkova, V. Sorokin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 313-317. — Бібліогр.: 10 назв. — англ.

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spelling Rybalochka, A.
Chumachkova, M.
Sorokin, V.
2017-06-04T16:37:25Z
2017-06-04T16:37:25Z
2004
Approximation of electro-optical hysteresis characteristics of ChLC / A. Rybalochka, M. Chumachkova, V. Sorokin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 313-317. — Бібліогр.: 10 назв. — англ.
1560-8034
PACS: 61.30.-v, 61.66.Hg
https://nasplib.isofts.kiev.ua/handle/123456789/119132
In order to select proper cholesteric liquid crystal (ChLC) materials and drive schemes for cholesteric liquid crystal displays (ChLCD), it is necessary to make the protracted experimental analysis of electro-optical hysteresis properties of ChLCs. Method for approximation of electro-optical characteristics of ChLC offered in this article significantly decreases duration of experimental analysis above. Accuracy of this method is estimated by the comparison of volt-brightness and volt-contrast characteristics of ChLC obtained both by experimental measuring and proposed method of approximation.
The authors would like to gratitude Mr. Р. Titarenko and Mr. Yu. Kolomzarov for their help in preparation of liquid crystal display cells. This work was supported by STCU under the project No. 2025.
en
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
Semiconductor Physics Quantum Electronics & Optoelectronics
Approximation of electro-optical hysteresis characteristics of ChLC
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Approximation of electro-optical hysteresis characteristics of ChLC
spellingShingle Approximation of electro-optical hysteresis characteristics of ChLC
Rybalochka, A.
Chumachkova, M.
Sorokin, V.
title_short Approximation of electro-optical hysteresis characteristics of ChLC
title_full Approximation of electro-optical hysteresis characteristics of ChLC
title_fullStr Approximation of electro-optical hysteresis characteristics of ChLC
title_full_unstemmed Approximation of electro-optical hysteresis characteristics of ChLC
title_sort approximation of electro-optical hysteresis characteristics of chlc
author Rybalochka, A.
Chumachkova, M.
Sorokin, V.
author_facet Rybalochka, A.
Chumachkova, M.
Sorokin, V.
publishDate 2004
language English
container_title Semiconductor Physics Quantum Electronics & Optoelectronics
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
format Article
description In order to select proper cholesteric liquid crystal (ChLC) materials and drive schemes for cholesteric liquid crystal displays (ChLCD), it is necessary to make the protracted experimental analysis of electro-optical hysteresis properties of ChLCs. Method for approximation of electro-optical characteristics of ChLC offered in this article significantly decreases duration of experimental analysis above. Accuracy of this method is estimated by the comparison of volt-brightness and volt-contrast characteristics of ChLC obtained both by experimental measuring and proposed method of approximation.
issn 1560-8034
url https://nasplib.isofts.kiev.ua/handle/123456789/119132
citation_txt Approximation of electro-optical hysteresis characteristics of ChLC / A. Rybalochka, M. Chumachkova, V. Sorokin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 313-317. — Бібліогр.: 10 назв. — англ.
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first_indexed 2025-11-24T16:30:36Z
last_indexed 2025-11-24T16:30:36Z
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fulltext 313© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 3. P. 313-317. PACS: 61.30.-v, 61.66.Hg Approximation of electro-optical hysteresis characteristics of ChLC A. Rybalochka, M. Chumachkova, V. Sorokin V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 45, prospect Nauky, 03028 Kyiv, Ukraine Abstract. In order to select proper cholesteric liquid crystal (ChLC) materials and drive schemes for cholesteric liquid crystal displays (ChLCD), it is necessary to make the pro- tracted experimental analysis of electro-optical hysteresis properties of ChLCs. Method for approximation of electro-optical characteristics of ChLC offered in this article significantly decreases duration of experimental analysis above. Accuracy of this method is estimated by the comparison of volt-brightness and volt-contrast characteristics of ChLC obtained both by experimental measuring and proposed method of approximation. Keywords: cholesteric liquid crystal, liquid crystal display. Paper received 08.04.04; accepted for publication 21.10.04. 1. Introduction Cholesteric materials have three main states: planar (P), focal conic (FC), and homeotropic (H). The planar and focal conic states are stable in case of zero external elec- tric and magnetic fields. In the planar state the helical axes are parallel to the display cell surface normal direc- tion. The planar state of ChLC reflects light in accord with the Bragg rule. Therefore, display pixels with ChLC in the planar state look bright [1]. In the focal conic state, the helical axes are distributed almost chaotically. ChLC in this state becomes slightly diffusely scattered. The value of scattered light power in this case is much lower than the Bragg reflection of the planar state and that is why the ChLC cell with absorbed background seems dark when ChLC is in the focal conic state. Thus, the image on bistable ChLCD is formed by pixels with the planar and focal conic states of ChLC. Moreover, it is necessary to note that both the planar and focal conic domains can be contained in the same pixel simultaneously. The reflect- ance of such a pixel has an intermediate value. Gray scale operations for ChLCDs where such states are stable for a long time can be implemented [2�3]. In the homeotropic state, the helical structure in distribution of liquid crystal molecules is absent and all the molecules of ChLC are aligned along one direction. This direction (for ChLC with ∆ε > 0) coincides with the direction of the external electric field. The transition of ChLC to the homeotropic state can be achieved by applying external electric field that exceeds the threshold value EC [4]. In order to rewrite the information on ChLCDs, tran- sitions between the planar and the focal conic states in pixels must be performed. A low voltage pulse can trans- form the planar state into the focal conic one. The value of electric field for this transition is always less than the threshold value EC. The transition of ChLC from the fo- cal conic state to the planar state is more complicated. Firstly, the transition to the homeotropic state must be performed. As the homeotropic state is unstable ChLC transforms into the planar state when external electric field is turned off quickly. It was studied that the transi- tion of ChLC from the homeotropic state to the stable planar state occurs through the transient planar state (P*) [4]. The homeotropic-transient planar transition is very short. Investigation of this transition and its application in driving ideology for ChLCD allow creation of fast dynamic drive schemes with addressing speed of about milliseconds per row [5�8]. If the electric field is turned off slowly ChLC transforms from the homeotropic state into the stable focal conic one. Thus, from the homeo- tropic state ChLC can be addressed both in the planar and focal conic states. Cholesteric material can be addressed in different sta- ble states by the same voltage pulse depending on its ini- tial state. For the fixed pulse duration there is a voltage range in which the field-induced homeotropic state is held and the planar, the focal conic and transient planar states 314 SQO, 7(3), 2004 A. Rybalochka et al.: Approximation of electro-optical hysteresis characteristics of ChLC are addressed into the focal conic state by the same volt- age pulse. Such behavior demonstrates hysteresis prop- erties of ChLC that are widely used in addressing ChLCDs [8�9]. One of the problems that can be solved by the detailed experimental analysis of the electro-optical hysteresis characteristics of ChLC is the problem to select ChLC material and a drive scheme for ChLCDs. Unfortunately, direct measurements of all required hysteresis character- istics can take a lot of time. Therefore, the search of new ways to decrease the duration of such experimental analy- sis is actual and important problem. In our first work devoted to solving the problem mentioned above, the method for approximation of electro-optical character- istics of ChLC that describes a behavior of cholesteric material in transitions from the field-induced homeotropic state into the stable planar and focal conic states was examined [10]. This method allows to obtain the suffi- cient quantity of approximated characteristics of ChLC only due to measurements of four special electro-optical characteristics. In this article, we propose the possibility of adaptation of proposed earlier method for an approxi- mation of electro-optical characteristics that describe electro-optical response of ChLC to voltage pulses from the initial stable planar state. Also, the comparison of volt-contrast characteristics of ChLC that were obtained both by experimental measurements and theoretical ap- proximations are presented. We performed all our measurements on the display cell filled with the liquid crystal mixture BL126 (Merck), cell thickness d = 4.4 µm at the room temperature. Dis- play Measuring System SV-200 was used to measure all the experimental characteristics. 2. Experiment We performed experimental measurements of electro- optical characteristics that describe electro-optical re- sponse of the ChLC to voltage pulses from the initial sta- ble planar state. We used the sequences of voltage pulses with the waveform that is demonstrated in Fig. 1. A high voltage Ur1 = 50V is applied to the display cell during the time interval Tr1=0.5 s in order to transform the ChLC into the homeotropic state. Then, during the time inter- val Tr2 = 2 s ChLC transforms into the stable planar state. Thus, during the time interval Treset = Tr1 + Tr2 ChLC is transformed into the stable planar state irrespectively to its initial state. Then, the bias voltage pulse (Ub, Tb) is applied to analyze the response of cholesteric material. The final reflectance of ChLC is measured in two sec- onds (Twait = 2 s) after the bias pulse, when the external field is absent (Uwait = 0 V) and when the reflectance does not change more in time. Ten dependencies of the stabi- lized reflectance of the cholesteric display cell vs. the amplitude of the bias voltage Ub for different values of the bias time Tb (0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 1, 3, 5 and 10 s) were measured. In Fig. 2 five of them are presented. The way of analytical approximation of these dependen- cies will be presented below. 3. Method of approximation The stable state of ChLC in a display cell is a compli- cated domain structure with a different preferred direc- tion of the helical axis orientation in different domains. In our simple model of stable cholesteric state, we as- sume that this state consists of two type of domains: the �planar� domains in which helical axes are strongly par- allel and the �focal conic� domains in which helical axes are strongly perpendicular to the display cell surface normal direction. It is possible to define the parameter ν as a part of the �planar� domains in a stable state of ChLC (0 ≤ ν ≤ 1). So, the reflectivity (R) of the stable state of ChLC is determined through the parameter in accord to the following equation: ( ) ( ) νν ×−+= minmaxmin RRRR (1) where Rmin and Rmax are reflectivities of the focal conic state and the planar one, correspondingly. However, the approximation of electro-optical characteristics from Fig. 2 by formula (1) is very simplified. Therefore, we propose the following dependency R(ν) that gives us good coincidence of experimental and theoretical curves: U U U TTT T T wait r1 r1 r2 reset b b t Fig. 1. The voltage waveform to measure the electro-optical response of the ChLC. Fig. 2. Experimental dependencies R(Ub) at different values of the bias time Tb. 10 s 1 s 3 s 0.5 s 0.1 s U , V R, a.u. b 0.10 0 3 6 9 12 15 18 21 24 0.15 0.20 0.25 0.30 0.35 0.40 0.50 0.55 A. Rybalochka et al.: Approximation of electro-optical hysteresis characteristics of ChLC 315SQO, 7(3), 2004 ( ) ( )      ××−+= νπν 2 sin2 minmaxmin RRRR (2) Each experimental curve in Fig. 2 and our model dis- tribution of the �planar� and �focal conic� domains can be presented as it is shown in Fig. 3. For each curve in Fig. 3, four threshold voltages can be determined: Up c, Up�fc max, Up�fc min and Up�p. The voltage Up c is the thresh- old voltage that transforms ChLC into the homeotropic state during the bias time Tb. The voltages Up�fc max and Up�fc min are the maximal and the minimal voltages that address the cholesteric material from the stable planar state into the final stable focal conic state with the mini- mal reflectance. Finally, the voltage Up�p � is the maxi- mal voltage that does not change reflectance from the stable planar state of ChLC during the bias time Tb. As we can see from the formula (2), for the approxi- mation of electro-optical characteristics of ChLC, it is necessary to set the value of the maximal reflectance Rmax, the minimal reflectance Rmin and the law of the parameter change. The values of Rmax and Rmin can be easily determined from the experimental data of Fig. 2: Rmax = 0.51 and Rmin = 0.11. For each curve in Fig. 2 there are two ranges of voltage Ub changing for which reflectance differs from Rmax and Rmin: in the range )()()( max bfcpb c pbI TUTUT −−=∆ reflectance increases from Rmin to Rmax and in the range )()( min bfcpbII TUT − −=∆ )() bpp TU −− reflectance decreases from Rmax to Rmin with an increase of the voltage Ub. In order to define the pa- rameter ν as a function ν = F(Ub, Tb), the following de- pendencies should be determined (see Fig. 3): )( b c p TU , )(max bfcp TU − , )(min bfcp TU − and )( bpp TU − . Then, the func- tion ν = F(Ub, Tb) can be expressed in the following way: ( )             ≥ ≤≤ ∆ − ≤≤ ≤≤ ∆ − ≤≤ = − − −− −− − − )(,1 )()(, )( )( )()(,0 )()(, )( )( )(0,1 , max max maxmin min min b c pb b c pbbfcp bI bfchb bfcpbbfcp bfcpbbpp bII bbfcp bppb bb TUU TUUTU T TUU TUUTU TUUTU T UTU TUU TUν As we can see from Figs 2 and 3, states of ChLC with any values of reflectance between Rmin and Rmax are both in the range ∆I and in the range ∆II. In our model, these states have the same domain distribution and differ only in a sequence of textural transitions of ChLC during applyication of the voltage waveform from Fig. 1. In the ranges ∆I and ∆II, the �focal conic� domains are formed by reorientation of the initial �planar� domains during the bias time by the voltage Ub. These ranges differ in the formation of the final �planar� domains. In the range ∆I, the planar state in the �planar� domains is formed from the homeotropic state of ChLC when the bias voltage is turned off. The �planar� domains in the range ∆II are the initial �planar� domains that do not reorient by the bias voltage. As we can see from the formula (3), in order to define the parameter ν in the range ∆I it is necessary to be aware of the behavior of functions )(max bfcp TU − and )( b c p TU . It is possible to suppose that )(max bfcp TU − = const = 22.8 V (see Fig. 2). In Fig. 4 experimental and approximated de- pendencies )( b c p TU are presented. We approximate the function )( b c h TU as follows:       −+×+×= )11(1)( * 1 )( b STc pb c p T TaUTU , (4) where )(STc pU = )(∞c pU , a1 is a coefficient and *T is some characteristic time that we set to be equal to the charac- teristic time for this ChLC [10]: *T = TFC = 1 s. The coefficient a1 can be determined directly from the for- mula (4) using experimental data from Fig. 4 (curve Up c). From the experimental curve Up c, we can determine )(STc pU = )s10(c pU = 23.8 V and )s1(c pU = 24.15 V. Sub- stituting these data into the formula (4), we obtain the value of the coefficient a1 = 0.035 . In the range ∆II, in order to define the parameter ν it is necessary to know behavior of functions )(min bfcp TU − , )( bpp TU − and )( bII T∆ .(3) Fig. 3. Schematic form of experimental curves and model distri- bution of domains. Fig. 4. Experimental and approximated dependencies of the voltage c pU vs. the bias time Tb. T , s U ,V b 23.5 0 1 2 3 4 5 10 24.0 24.5 25.0 25.5 26.5 27.0 26.0 V V R , a. u. R R R U T ( ) U T ( ) U T ( ) D T ( ) D T ( ) U U U p�p b 0 1 316 SQO, 7(3), 2004 A. Rybalochka et al.: Approximation of electro-optical hysteresis characteristics of ChLC In Fig. 5, experimental and approximated dependences )(min bfcp TU − , )( bpp TU − and )( bII T∆ are presented. We ap- proximate functions )(min bfcp TU − and )( bpp TU − by expres- sions similar to presented earlier in the formula (4):       −+×+×= −− )11(1)( 2 )min(min b FCST fcpbfcp T TaUTU (5)       −+×+×= −− )11(1)( 3 )( b FCST ppbpp T TaUTU (6) where )min(ST fcpU − is the minimal voltage that address chol- esteric material from the stable planar state into the final stable focal conic state with the minimal reflectance dur- ing the bias time Tb = 10 s, )(ST ppU − is the maximal voltage level that does not change reflectance of the stable planar state of ChLC during the bias time Tb = 10 s, a2 and a3 � coefficients. The coefficients a2 and a3 can be determined directly from the formulae (5) and (6) using the experi- mental data from Fig. 5 (curves min fcpU − and ppU − ): a2 = = 0.244 and a3 = 0.403. Substituting functions )( b c p TU , )(min bfcp TU − , )( bpp TU − and )(max bfcp TU − = const = 22.8 V in the formula (3), we can obtain the final expression for ν = F(Ub, Tb). Substi- tuting this expression into the formula (2), we get the final expression for reflectance vs. the bias time: R(Tb). Approximated dependencies R(Ub) at different values of the bias time Tb are presented in Fig. 6. In Figs 7 and 8, experimental and approximated volt-contrast characte- ristics of ChLC are presented. The comparison of these characteristics is carried out by calculations for each volt- contrast curve of the voltage range (∆U) where contrast has a value not less than Ñ1 = 4.2:1 and Ñ2 = 3.5:1 (see Fig. 9). This comparison shows quite good coincidence of these voltage ranges both for experimental and approxi- mated volt-contrast characteristics. It means that proposed simple method for the approximation of electro-optical characteristics of ChLC can be effectively used for the analysis of electro-optical hysteresis properties of ChLC. As a result for approximation of electro-optical volt- brightness and volt-contrast characteristics of ChLC, we need to measure six electro-optical characteristics of ChLC. Four electro-optical characteristics have been proposed earlier [10]: 1. Measurements of the quasi-static electro-optical re- sponse of ChLC from the initial field induced homeotropic state. In this case, the sequence of volt- age pulses with the voltage waveform from Fig. 1 with Tr2 = 0 s and Tb > 10 s should be used. From these curves, we determine several parameters of ChLC in- cluding the voltage level UD as the minimal ampli- tude of a voltage pulse that can address ChLC from the homeotropic state to the stable focal conic state with the minimal reflectivity; 2. Measurements of the special dependency Rmin(Tb). For this measuring, also the sequence of voltage pulses with the voltage waveform from Fig.1 with Tr2 = 0 s is used. But in this case, the bias voltage level is fixed (Ub = UD) and the bias time is changed. As a result, we obtain the experimental dependence Rmin(Tb), and than from this curve we determine the value of the characteristic time TFC; Fig. 5. Experimental and approximated dependencies of voltages min fcpU − , ppU − and the voltage range ∆II vs. the bias time Tb. U , Vb 0 3 6 9 12 15 18 21 24 R, a.u. 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.50 0.55 10 s 1 s 3 s 0.5 s 0.1 s Fig. 6. Approximated dependencies R(Ub) at different values of the bias time Tb. C U , Vb6 8 10 12 14 16 18 20 22 24 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10 s 1 s 3 s 0.5 s 0.1 s Fig. 7. Experimental volt-contrast characteristics of the ChLC. T , s U, V U U U U D D b 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 16 14 12 10 8 6 4 2 18 A. Rybalochka et al.: Approximation of electro-optical hysteresis characteristics of ChLC 317SQO, 7(3), 2004 3. Measurements of the electro-optical response of a cho- lesteric display cell as in the point 1 with Tb = TFC; 4. Measurements of the electro-optical response of a cho- lesteric display cell as in the point 1 with Tb > TFC. And in this paper we offer two electro-optical charac- teristics more: 5. The measurement of the quasi-static electro-optical response of ChLC from the initial planar state. In this case, the sequence of voltage pulses with the voltage waveform from Fig. 1 and Tb > 10 s must be used. 6. The measurement of the electro-optical response of a cholesteric display cell as in the point 5 at Tb = TFC. The duration of experimental measurements of these six characteristics is about two hours that is signifi- cantly less than duration of experimental analysis necessary for investigation of electro-optical proper- ties of ChLC using the standard way. Moreover, us- ing the proposed method of approximation, it is pos- sible to obtain approximated electro-optical charac- teristics of ChLC, similar to characteristics in a Figs 6 and 8, for any value of the parameter Tb. 4. Conclusions In this article, the evolution of the proposed earlier method for an approximation of electro-optical characteristics of ChLC that describes a behavior of cholesteric material at transitions from the field induced homeotropic state to the stable planar and focal conic states was demonstrated. This method can also be implemented for an approxima- tion of electro-optical characteristics that describe electro- optical response of ChLC on voltage pulses from the sta- ble planar state. The volt-contrast characteristics can be obtained, too. As a result, this method allows to obtain approximated characteristics of ChLC only due to meas- urements of only six special electro-optical characteris- tics. It allows to essentially decrease the duration of ex- perimental analysis of hysteresis properties of ChLC in comparison with the standard procedure of measuring electro-optical characteristics of ChLC. Acknowledgements The authors would like to gratitude Mr. P. Titarenko and Mr. Yu. Kolomzarov for their help in preparation of liquid crystal display cells. This work was supported by STCU under the project No. 2025. References 1. P. de Jen, Physics of liquid crystalls, �Mir�, Moscow (1977) (in Russian). 2. J. Gadhi, D. K. Yang, Gray Scale Drive Schemes for Bistable Cholesteric Reflective Displays // Asia Display 1998, pp. 127- 130 (1998). 3. A. Rybalochka, V. Sorokin, A. Kozachenko, A. Sorokin, Electronic principle of gray scale realization for Cholesteric LCD // Proc. of the 8th Int. Symposium �Advanced Display Technologies�, p. 59-64 (1999). 4. D.K. Yang, Z.J. Lu, Switching Mechanism for Bistable Re- flective Cholesteric Displays: A Rapid Addressing Scheme // SID 95 Digest, p. 351-354 (1995). 5. X.Y. Huang, D.K. Yang, P. Bos and J.W. Doane, Dynamic Drive for Bistable Reflective Cholesteric Displays: A Rapid Addressing Scheme // SID 95 Digest, pp. 347-350 (1995). 6. X.Y. Huang, D.K. Yang, P.J. Boss and J.W. Doane, Dy- namic Drive for Bistable Reflective Cholesteric Displays: A Rapid Addressing Scheme // SID 95 Digest, pp. 347-350 (1995). 7. A. Kozachenko, P. Oleksenko, V. Sorokin, V. Nazarenko, Histeresis as a Key Factor for the Fast Control of Reflectivity in Cholesteric LCDs // Conference Record of the IDRC 97, p. 148-151 (1997). 8. A. Rybalochka, V. Sorokin, S. Valyukh, A. Sorokin, V.Nazarenko, Simple Drive Scheme for Bistable Cholesteric LCDs // SID 2001 Digest, pp. 882-885 (2001). 9. M. Kawachi, O. Kogure, Hysteresis Behavior of Texture in the Field-Induced Nematic-Cholesteric Relaxation // SID 2001 Digest Japan J. Appl. Phys., 16(9), pp. 1673-1678 (1977). 10. A. Rybalochka, M. Chumachkova, V. Sorokin, Approxima- tion of electro-optical characteristics of a ChLC at transitions from the homeotropic texture // Semiconductor physics, quan- tum electronics & optoelectronics, 6(3), pp. 411-416 (2003). C U , Vb66 8 10 12 14 16 18 20 22 24 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 10 s 1 s 3 s 0.5 s 0.1 s DU, V T , sb 0 0 1 2 2 3 4 6 8 4 5 10 12 10 C = 4,2:1 C = 3,5:1 1 2 Fig. 8. Approximated volt-contrast characteristics of the ChLC. Fig. 9. The voltage range ∆U vs. the bias time Tb for experimen- tal and approximated volt-contrast characteristics.