Evaluation of Kerr constant of blue-phase liquidcrystals by measuring off-axis retardation

in vertical electric field cells

Pao-Ju Hsieh1,2,* and Huang-Ming Philip Chen1

1Department of Photonics, National Chiao Tung University, 1001 University Road, Hsinchu, Taiwan 3002Industrial Technology Research Institute (ITRI), 195, Section 4, Chung Hsing Road, Chutung, Hsinchu, Taiwan 31040

*Corresponding author: pjhsieh@itri.org.tw

Received 2 February 2011; revised 8 July 2011; accepted 11 July 2011;posted 12 July 2011 (Doc. ID 141160); published 15 September 2011

Because of the nonuniform electric field of the in-plane-switching cell in the thickness direction, an ac-curate and efficient way for evaluating the Kerr constant of blue-phase liquid crystal (BPLC) needs to bedeveloped. This study demonstrates amethod for evaluating the Kerr constant bymeasuring the off-axis-induced retardation (Rth) change in normal vertical field cells using a commercial polarimeter. The angle-dependent behavior of theRth change is observed as an electric-tunable positive C retarder. In this paper,a sigmoid fitting model has been chosen for calculating the Kerr constant for considering the very smallintrinsic birefringence of the BPLC. © 2011 Optical Society of AmericaOCIS codes: 160.3710, 160.2100, 160.1190, 120.5050, 120.2130, 350.4800.

Since Kikuchi et al. successfully extended the rangeof temperatures over which the blue phase exists tomore than 60K by polymer stabilization [1–3], blue-phase liquid crystals (BPLCs) have been consideredto be promising for use in next-generation displays.BPLCs have an excellent high-contrast-ratio displayproperty due to their optically isotropic characteris-tics in the voltage-off state, which can show a darkappearance between crossed polarizers. Other at-tractive characteristics include their being align-ment free, having a fast response time, and havingan inherently wide viewing angle. The electro-opticalbehavior of such cubic symmetric BPLCs theoreti-cally exhibits the Kerr effect, described by Eq. (1)[4,5]:

Δnind ¼ λKE2; ð1Þ

where λ is the wavelength of the incident light, K isthe Kerr constant, E is the amplitude of the electric

field, and Δnind is the electric-induced birefringence.Because BPLCs are isotropic materials, this electric-field-induced birefringence (Δnind) in the Kerr effectcan be referred to as an isotropic-to-anisotropic tran-sition. The Kerr constant of the BPLC is related to itselectro-optical performance, because the operatingvoltage is proportional to ð ffiffiffiffi

Kp Þ−1 for obtaining the

same Δnind.The optical axis of the induced refractive index el-

lipsoid is along the electric field vector when an elec-tric field is applied. Only the horizontal electric fieldcontributes to the transmittance of the BPLC cell.Therefore, BPLCs are used to fill in in-plane switch-ing (IPS) cells for observation of their phase changeunder two crossed polarizers. However, the electricfield is not uniform in the thickness direction insidethe IPS cells, as presented in Fig. 1. Moreover thedepth of penetration of the electric field varies withthe conditions. To enhance the transmittance andlower the operating voltage, modified IPS cell de-signs have been presented to solve this problem [6,7].

Without a doubt, the typical vertical electric fieldcells (VF cells) can be regarded with a uniform elec-tric field for measuring accurate Kerr constants [8].

0003-6935/11/275299-04$15.00/0© 2011 Optical Society of America

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As mentioned above, the optical axis of the anisotro-pic ellipsoid is along the applied electric field vector.From Fig. 2, the BPLCs in the VF cell change fromisotropic to anisotropic as the voltage increases, as aresult, the off-axis retardation (Rth) elevates as thetilted angle increased. Such behavior, associatedwith out-of-plane retardation in the thickness direc-tion, can be regarded as that of positive C plate (þCplate) retarder, especially an electrically tunable þCplate retarder. The off-axis retardation is usuallydefined as Eq. (2), adopted to describe the differencebetween in-plane and out-of-plane retardation ofsuch a sample:

Rth ¼ ½nz − ðnx þ nyÞ=2� × d; ð2Þ

where nx, ny, nz are the three refractive indices of theinduced index ellipsoid with respect to the XYZ coor-dinates and d is the thickness of the sample. To cal-culate Rth, measurements were made on the samplethat was inclined at various angles, because the re-tardation of the þC plate is a function of the incidentangle, θ. Equation (3) gives Rth:

Rth ¼ deff ðneff − nxÞ

¼ dcos θ0

�nynzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2ycos2θ0 þ n2

zsin2θ0q − nz

�; ð3Þ

where the refraction angle θ0, can be derived usingSnell’s law from the average refractive index of thesample ni, ðnx þ ny þ nzÞ=3:

θ0 ¼ sin−1

�sin θni

�: ð4Þ

While this birefringence difference is not verylarge observed in the VF cell, it is difficult to performa very precise detection—there is not only a compli-cated optical setup but an accurate light path corre-sponding to a tilted fixture [9,10].

This study first demonstrates a method for evalu-ating the Kerr effect of the BPLC by a setup incorpor-ating a commercial Mueller matrix polarimeter(MMP), which is usually used to evaluate the filmretardance. As shown in Fig. 3, a commercial MMP(from Axometrics), equipped with a wave functiongenerator, WFG600 (from FLC), was used to deter-mine directly the Rth of the sample at different vol-tages. The light source for the polarimeter was axenon arc lamp equipped by a scanning diffraction-grating monochromator; the 550nm wavelengthwas chosen for the measurements for comparison.

The polymer-stabilized BPLC sample was a mix-ture of homologous LC cyanobiphenyl and flourobi-phenyl compounds (68.6%) with chiral dopants(5.3% R1011 and 14% CB15, from Merck) and mono-mers (5.6% TMPTA and 6.4% RM257, from Aldrich),and the measured average refractive index (ni) was1.58 (Abbe Refractometer from Atago). The samplefor our measurement was filled into a vertical fieldcell with a 4:8 μm cell, then cured in an LC blue-phase range. To prevent disturbance of the align-ment layer, the cell was assembled from only twobare indium tin oxide (ITO) glass substrates withoutthe application of polyimide. After that, the samplewas automatically measured by being mounted ona rotatable and tiltable fixture in the above setup, in-clined at an angle θ that varied from −45° to 45° insteps of 5°.

Figure 4 plots the angle-dependent retardationcurves induced at various voltages, and the mea-suredRth in various electric fields is depicted in Fig. 5(black squares). As shown in Fig. 4, the “smilingcurve” behavior of the induced Rth at various vol-tages implies BPLC in the VF cell as þC plate char-acteristic description. It is notable that the curve at0V shows that it is slightly raising in both largerviewing angles; it is realized the BPLCs still possessa very small intrinsic birefringence, ∼0:01. The re-sulting data follow the Kerr’s law at low E very welland show a saturation effect at high E, in a mannervery similar to the extended Kerr effect described ina previous report [9]. In this paper, a well-fitted sig-moid curve model was used for the retrieving data(red line), and it is written as Eq. (5):

Rth;ind ¼ Rs

1þ 10ðE2i −E

2Þ�p ; ð5Þ

where Rs is the induced saturated retardation value,Ei denotes the value of the electric field at the pointof inflection in the middle of the sigmoid curve, and prepresents a slope parameter.

Glass substrate

ITO electrode

BPLC

Fig. 1. (Color online) Schematic IPS cell structure of a conven-tional device with an interdigital electrode structure wherein anonuniform electric field is applied in the thickness direction.

Glass substrate

ITO electrode

Optical

index ellipsoid

of BPLC

x

y

z

x

y

z

x

y

z

V1=0

V2

V3

Fig. 2. (Color online) Effective optical index ellipsoids of BPLCunder various electric fields: V3 > V2 > V1.

5300 APPLIED OPTICS / Vol. 50, No. 27 / 20 September 2011

The intrinsic birefringence at E ¼ 0 is Rs=ð1þ10E

2i �pÞ, from Eq. (5), induced to 0:5Rs at E ¼ Ei,

and at high E, the induced Rth will reach Rs. Theresulting slope of Eq. (5) is easily calculated as 1

4Rs ·p · lnð10Þ by using a differential method, as Eapproaches Ei. Recalling the physical meaning ofthe Kerr constant, it has been found to approxi-mately represent the slope; that is, K ∼

14Rs · p·

lnð10Þ. Therefore, to combine Eqs. (1) and (2), theKerr constant is given as follows:

K ¼ Rs · p · lnð10Þ4 · λ · d : ð6Þ

The parameters of the curve in Fig. 5 are obtained:Rs ¼ 310:285nm, Δns ¼ 0:0646, p ¼ 0:015, andEi ¼ 6:29V=μm. The Kerr constant is therefore cal-culated as 1:09nm=V2. Because the slope value is re-trieved from the middle high point of the curve(E ∼ Ei), it is believed that Eq. (6) describes a goodfit for the Kerr constant.

As mentioned in Eqs. (4) and (5), the average re-fractive index ni of the sample affects directly neff,as well as Rth, so several different refractive indexvalues are set to realize the tolerance range. Table 1presents the variation in the Kerr constant, which isonly about 2%–3%, as the refractive index variesby �0:02.

In this study, a method for evaluating the accurateKerr constants of BPLC is demonstrated. By using asetup with a commercial polarimeter, the angle-dependent behavior of the off-axis retardation

Fig. 4. (Color online) Linear variation of measured retardancewith the angle of incidence at various electric fields.

Fig. 5. (Color online) Rth obtained at different electric fields(black squares) and curve fitted using model (red curve).

Fig. 3. (Color online) Measurement fixture of AxoScan MMP and schematic diagram of the setup.

Table 1. Calculated Kerr Constants with Various Assumed Index, ni,and Corresponding Deviations

Average Refractive Index, ni 1.56 1.58 1.6

Kerr constant (m=V2) 1:06E − 9 1:09E − 9 1:12E − 9Deviation (%) −3:18 0 2.63

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changes in normal vertical field cells is observed asan electric-tunable positive C retarder. A small in-trinsic birefringence of BPLC is also observed. In thispaper, a well-fitted sigmoid model is used to retrievethe Kerr constant of the BPLC.

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