/div>
.
Because the upper and lower substrates have the same
refractive index, there are no critical angles and all of the
guided modes will be leaky from both the substrates
(called fully-leaky). We consider that the angle of inci-
dence of polarized light in the upper substrate is
, and
the angle of refraction in the liquid crystal layer is βo for
the ordinary light and βe for the extraordinary light. From
sin sin
g
oo
nn

(1)
sin sin
g
ee
nn

(2)
From the index ellipsoid, it gives
222 2
sin cos
oe
e
oe
nn
n
nn
(3)
where ψ is the angle between the optical axis, the director,
and wave-front normal for the extraordinary light in the
liquid crystal layer, which is given by the follow formula
[20]
1
coscos cossin sinsin
 
 (4)
If the optical axis lies along the x-axis, the TE modes
depend only on no, while TM modes depend on both no
and ne; if the optical axis lies along z-axis, the TE modes
will still sense no, and TM modes will sense no and ne; if
the optical axis is along y-axis, the TE modes are given
by ne while TM modes are given by no. For these three
special cases, the eigenmodes propagating in liquid crys-
tal waveguide geometry are pure TE or pure TM modes.
If the optical axis is rotated out of the y-axis to some ar-
bitrary angle in the x-y plane or y-z plane, or both, e.g.
the optical axis is at a general position as shown in Fig-
ure 3, the eigenmodes are no longer pure TE or TM
modes. It is found when a linearly polarized light (p or s
light) enters such a waveguide, the light will produce
polarization conversion [8], the output light having some
of the orthogonal polarization components present. In
general the polarization conversion signals are specifi-
cally sensitive to the tilt and twist of the director, so these
signals are very useful to study the director distribution.
For either form of incident light, two eigenmodes will be
excited in the liquid crystal layer, one with the E field
along the short semi-axis OB in the ellipse BOF, the oth-
er with the E field along the major semi-axis OF in the
ellipse BOF. So we can decide the polarization conver-
Copyright © 2012 SciRes. JMP
Z. D. ZHANG ET AL.
Copyright © 2012 SciRes. JMP
1588

sion signal from the angle of between -y axis and OB
when either pure s or pure p light enters the liquid crystal
layer. From Figure 3 we have [20]
respectively. Two substrates were separated with 3.1 μm
diameter spacers. The light of pyramid-coupling is
shown in Figure 2, but in our study there are only reflec-
tive signals. The matching fluid (made by CARGILLE
LABS, USA) allows the cell to be rotated with respect to
the pyramid.
sincossin c
cos
1cos cossin
os sin
sin sin





(5)
The experiment arrangement is shown in Figure 4.
Two apertures collimate the light beam from He-Ne laser
(λ = 632.8 nm), and the mechanical chopper modulates
the laser beam at 1.86 kHz to allow the phase-sensitive
detection. The variable attenuator modulates the intensity
of incident light, and the quarter wave plate changes the
polarization state of the laser beam from linear to circular.
The glass is a thick plate to reflect ~5% of the incident
light into a reference detector, the signal from which is
used to compensate for drift in source intensity. Three
rotatable polarizers determine the specific polarization
state of the incident light and reflective light. The com-
puter-controlled rotating table sets the angular position of
the 90˚ MTN cell relative to the incident light and also
positions the reflective light detector so that it collects
the reflective signals; that means the reflectivity detector
must rotate through twice the angle of the 90˚MTN cell
[8]. In order to obtain better reflective polarization con-
version signals which are more sensitive to the twist and
tilt of the director, the cell is rotated against the pyramids
to set the rubbed direction(in x-y plane, see Figure 3)
twisted from the y direction by about 30˚.
Obviously, only when equals 0 or π/2, there is no
polarization conversion. This corresponds to one of three
cases: (1) φ = π/2, (2) θ = 0, (3) φ = 0 and θ = π/2. In a
true liquid crystal cell, the liquid crystal layer is sand-
wiched between two glass substrates with ITO coatings,
alignment layers etc. The liquid crystal layer is further
treated as a multilayer optical system when we use the 4 ×
4 matrix method [21,22] to model the optical property of
the system. Controlling a linearly polarized light (p or s
light) with different incident angles, this will lead to dif-
ferent optical field distributions and optical waveguide
modes, so we can obtain four sets of reflective angle-
dependent signals (Rss, Rpp, Rsp, Rps). Based on the con-
tinuum theory, the director distribution can be predicted,
then we model the angle-dependent optical properties by
multilayer optical theory of the 4 × 4 matrix method.
Adjusting the pre-tilt angle and the total twisted angle,
we obtain the best fits between the theory predictions and
the recorded experiment data, so the director distribution
in the liquid crystal cell is determined accurately.
3. Experiment Controlling the cell to rotate through 20˚, at each angle
of incidence, fully-leaky guided modes were set-up in the
liquid crystal layer and the intensity of reflective light at
this angle was recorded. The measurement was repeated
with 1 kHz rms ac voltages of 0 - 7 V applied across the
The experimental 90˚ MTN cell is made by Shenzhen
Live Digital Technology Co., Ltd., which used low-index
(1.52) glass substrate with ITO-coating and the same
(instead of silicon) substrate with aluminum-coating,
Figure 4. Schematic of the experiment installation.
Z. D. ZHANG ET AL. 1589
cell, perpendicular to the substrates. All the measure-
ments were conducted on a monodomain at room tem-
perature 25˚C.
4. Results
In order to obtain the director distribution of the reflec-
tive 90˚ MTN cell and the information about the parame-
ters of different optical layers, the experiment data for Rss,
Rpp, Rsp and Rps are recorded both with and without ap-
plied electric fields. A typical set of experimental data
(crosses) is shown in Figure 5. We note that the polari-
zation-conserving and the polarization-conversion sig-
nals are quite strong, especially the polarization-conver-
sion signals are far stronger than that of the transmissive
TN cell. The reason is that the aluminum coating on the
glass substrate also is a good reflector except for an elec-
trode. According to the numerical modeling of F. Z.
Yang [8], Rsp and Rps will be different unless the director
parallels to the substrate, i.e., θ = 90˚. From Figures 5(c)
and (d), we see that there are differences between Rsp and
Rps, which implies a small tilt angle of the director on
both substrates. The recorded data are fitted to the pre-
dictions produced by the modeling-program, and the fi-
nal fits are given by the full curves in Figure 5.
These parameters for the cell obtained by fitting are as
follows: for the glass substrates n = 1.517 at λ = 632.8
nm; for the ITO εI = 3.25 + i0.079 and d = 30 nm; for the
aluminum εAl = 63.5 + i26.813 and d = 28 nm; for the
alignment layers εPI = 2.05 + i0.005 and d = 35 nm; for
the liquid crystal K11 = 13.1 pN, K22 = 10 pN, K33 = 22.3
pN, ε = 2.1650 + i0.0008, ε// = 2.4359 + i0.001 and d =
3.13 μm, for the tilt angle from z axis θ = 85.5˚ on the
upper substrate and θ = 86.2˚ on the lower substrate, the
corresponding tilt(the tilt angle is between the director
and x-y plane) and twist profile is shown in Figure 6, for
the total twist angle
= 90.3˚ from the upper inner sur-
face to the bottom. The pre-tilt angle on the upper sub-
strate is different from that on the bottom, which sug-
gests that ITO and aluminum coatings have different
effects on the alignment layers.
The all recorded angle-dependent reflectivity data for
each voltage were fitted to the model data produced by
the continuum theory and multilayer optical theory mod-
eling program. For all the voltages, the optical parame-
ters of the different layers used in the fitting are shown in
Table 1. The optical parameter ε = εr + iεi is complex,
where εr is the real part and εi is the imaginary part. The
parameter of aluminum refers to the result of J. R. Sam-
bles [23].
Produced by all the fits, the twist angle φ' = 17.2˚ ±
0.8˚ from the x-axis and the tilt angle θ' = 85.5˚ ± 0.5˚
from z-axis on the upper substrate; the twist angle and tilt
angle on the bottom substrate are φ = 72.5˚ ± 0.5˚ and θ =
(a)
(b)
(c)
(d)
Figure 5. Experimental data (crosses) and fitted theory
(curves) for an applied voltage of 3 V. (a) Rss, (b) Rpp, (c) Rps,
and (d) Rsp.
Copyright © 2012 SciRes. JMP
Z. D. ZHANG ET AL.
1590
Table 1. The optical parameters of 90˚ MTN liquid crystal cell.
Optical layers εr εi Thickness (nm)
ITO 3.24 ± 0.04 0.079 ± 0.015 32 ± 3
Alignment layers 2.025 ± 0.025 0.004 ± 0.001 38 ± 4
Liquid crystal (ε//) 2.430 ± 0.007 0.0011 ± 0.0003
Liquid crystal (ε) 2.172 ± 0.016 0.0010 ± 0.0002
(3.12 ± 0.02) × 103
Aluminum 64 ± 1 26 ± 1 30 ± 2
(a)
(b)
Figure 6. The director profile in 90˚ MTN cell at 3 V. (a)
Tilt angle; (b) Twist angle.
86.5˚ ± 0.3˚, respectively. The diagram is shown in Fig-
ure 7.
5. Conclusion
The reflective-mode 90˚ MTN liquid crystal cell has been
studied by use of the fully-leaky guided wave technique.
Figure 7. The diagram of twist angle and tilt angle on both
upper and bottom substrates.
The p or s light with different incident angles is coupled
into the cell by a pyramid, and the angle-dependent re-
flective signals are recorded. Then we used the modeling
program based on both the continuum theory and the 4 ×
4 matrix multilayer optical theory to fit the recorded data
for all the applied voltages (0 - 7 Vrms). Finally, we ob-
tained the information about the director distribution and
the parameters of different optical layers (see Table 1),
and we found that the pre-tilt angle on the upper sub-
strate is different from that on the bottom, which sug-
gests that the ITO and the aluminum coatings have dif-
ferent effects on the alignment layers.
6. Acknowledgements
This research was supported by Natural Science Founda-
tion of Hebei Province under Grant No. A2010000004,
Research Project of Hebei Education Department under
Grant No. Z2012061, and Key Subject Construction
Project of Hebei Province University. The authors thank
Professor Fuzi YANG for his helpful discussions.
REFERENCES
[1] S. T. Wu and C. S. Wu, “Mixedmode Twisted Nematic
Liquid Crystal Cells for Reflective Displays,” Applied
Physics Letters, Vol. 68, No. 11, 1996, pp. 1455-1457.
doi:10.1063/1.116252
[2] K. H. Fan Chiang, S. H. Chen and S. T. Wu, “Diffraction
Effect on High-Resolution Liquid-Crystal-on-Silicon De-
vices,” Japanese Journal of Applied Physics, Vol. 44, No.
5A, 2005, pp. 3068-3072. doi:10.1143/JJAP.44.3068
Copyright © 2012 SciRes. JMP
Z. D. ZHANG ET AL. 1591
[3] J. Chen, S. M. Morris, T. D. Wilkinson, J. P. Freeman,
and H. J. Coles, “High Speed Liquid Crystal over Silicon
Display Based on the Flexoelectro-Optic Effect,” Optics
Express, Vol. 17, No. 9, 2009, pp. 7130-7137.
doi:10.1364/OE.17.007130
[4] F. Z. Yang, L. Z. Ruan, and J. R. Sambles, “Polarization-
Conversion Guided Mode (PCGM) Technique for Ex-
ploring Thin Anisotropic Surface Layers,” Optics Express,
Vol. 15, No. 18, 2007, pp. 11234-11240.
doi:10.1364/OE.15.011234
[5] F. Z. Yang, L. Z. Ruan and J. R. Sambles, “Exploration of
the Surface Director Profile in a Liquid Crystal Cell Us-
ing Coupling Between the Surface Plasmon and Half-
Leaky Optical Guided Modes,” Applied Physics Letters,
Vol. 92, No. 15, 2008, pp. 1-3. doi:10.1063/1.2908224
[6] F. Z. Yang, L. Z. Ruan, S. A. Jewell and J. R. Sambles,
“Coupled Surface Plasmons and Optical Guided Wave
Exploration of Near-Surface Director Profile,” New Jour-
nal of Physics, Vol. 9, No. 49, 2007, pp. 1-11.
[7] F. Z. Yang, S. A. Jewell, L. Z. Ruan and J. R. Sambles,
“ Complex Permittivities of a Nematic Liquid Crystal in a
Hybrid-Aligned Cell,” Journal of the Optic Society of
America B, Vol. 24, No. 3, 2007, pp. 527-531.
doi:10.1364/JOSAB.24.000527
[8] F. Z. Yang and J. R. Sambles, “Optical Fully Leaky Mode
Characterization of Standard Liquid-Crystal Cell,” Jour-
nal of the Optic Society of America B, Vol. 16, No. 3,
1999, pp. 488-497. doi:10.1364/JOSAB.16.000488
[9] F. Z. Yang, H. J. Gao and J. R. Sambles, “Fully Leaky
Guided Wave Determination of the Original Alignment
Direction for the Directors at the Walls in a Twisted Ne-
matic Liquid Crystal Cell,” Journal of Applied Physics,
Vol. 92, No. 4, 2002, pp. 1744-1751.
doi:10.1063/1.1491025
[10] B. Hodder, F. Yang and J. R. Sambles, “Optical Charac-
terization of the Director Profile in a Ferroelectric Liquid
Crystal Cell with Homeotropic Alignment,” Journal of
Applied Physics, Vol. 89, No. 1, 2001, pp. 5-9.
doi:10.1063/1.1330549
[11] F. Z. Yang, J. R. Sambles, Y. M. Dong and H. J. Gao,
“Fully Leaky Guided Wave Determination of the Polar
Anchoring Energy of a Homogeneously Aligned Nematic
Liquid Crystal,” Journal of Applied Physics, Vol. 87, No.
6, 2000, pp. 2726-2735. doi:10.1063/1.372247
[12] S. A. Jewell and J. R. Sambles, “Fully Leaky Guided
Mode Study of the Flexoelectric Effect and Surface Po-
larization in Hybrid Aligned Nematic Cells,” Journal of
Applied Physics, Vol. 92, No. 1, 2002, pp. 19-24.
doi:10.1063/1.1483392
[13] L. Z. Ruan, F. Z. Yang and J. R. Sambles, “Voltage De-
pendent Director of a Homeotropic Negative Liquid
Crystal Cell,” Applied Physics Letters, Vol. 93, No. 3,
2008, Article ID: 031909.
[14] S. A. Jewell and J. R. Sambles, “Fully-Leaky Guided
Mode Measurement of the Flexoelectric Constant (e11+
e33) of a Nematic Liquid Crystal,” Crystals and Liquid
Crystals, Vol. 401, No. 1, 2003, pp. 67-73.
doi:10.1080/744815187
[15] S. L. Cornford, T. S. Taphouse, C. J. P. Newton and J. R.
Sambles, “Determination of the Director Profile in a Ne-
matic Cell from Guided Wave Data: An Inverse Prob-
lem,” New Journal of Physics, Vol. 9, No. 166, 2007, pp.
1-15.
[16] S. A. Jewell and J. R. Sambles, “Optical Characterization
of a Dual-Frequency Hybrid Aligned Nematic Liquid
Crystal Cell,” Optics Express, Vol. 13, No. 7, 2005, pp.
2627-2633. doi:10.1364/OPEX.13.002627
[17] H. Y. Xing , W. J. Ye , N. F. Wu , Z. D. Zhang and L.
Xuan, “Exploration of the Sum of Flexoelectric Coef-
fcients of Nematic Liquid Crystals,” Chinese Optics Let-
ters, Vol. 10, No. 5, 2012, Article ID: 052301.
[18] H. Y. Xing, W. J. Ye, N. F. Wu, L. B. Si and Z. D. Zhang,
“Determination of Director Profile in the Vertical Align-
ment Nematic Liquid Crystal Cell by the Full Leaky
Guided Mode Technique,” Solid State Phenomena, Vol.
181-182, 2012, pp. 265-268.
doi:10.4028/www.scientific.net/SSP.181-182.265
[19] F. Z. Yang, J. R. Samble and G. W. Bradberry, “Guided
Modes and Related Optical Techniques Liquid Crystal
Alignment Studies,” In: S. J. Elston and J. R. Sambles,
Eds., The Optics Thermotropic Liquid Crystals, Taloy
and Francis Ltd., London, 1998, pp. 85-95.
[20] F. Z. Yang and J. R. Samble, “Guided Mode Studies of
Liquid Crystal Layers,” In: M. Iwamoto, K. Kaneto and S.
Mashiko, Eds., Nanotechnology and Nano-Interface Con-
trolled Electronic Devices, Taylor and Francis Ltd., Lon-
don, 2003, pp. 271-280.
doi:10.1016/B978-044451091-4/50016-9
[21] D. W. Berreman, “Optics in Stratified and Anisotropic
Media: 4 × 4 Matrix Formulation,” Journal of the Optic
Society of America A, Vol. 62, No. 4, 1972, pp. 502-510.
doi:10.1364/JOSA.62.000502
[22] D. Y. K. Ko and J. R. Sambles, “Scattering Matrix Me-
thod for Propagation of Radiation in Stratified Media:
Attenuated Total Reflection Studies of Liquid Crystals,”
Journal of the Optic Society of America A, Vol. 5, No. 11,
1988, pp. 1863-1866. doi:10.1364/JOSAA.5.001863
[23] J. R. Sambles, G. W. Bradbery and F. Z. Yang, “Optical
Excitation of Surface Plasmons: An Introduction,” Con-
temporary Physics, Vol. 32, No. 3, 1991, pp. 173-183.
doi:10.1080/00107519108211048
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