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Infrared refractive indices of liquid crystals Infrared refractive indices of liquid crystals
Jun Li
University of Central Florida
Shin-Tson Wu
University of Central Florida
Stefano Brugioni
Riccardo Meucci
Sandro Faetti
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Recommended Citation Recommended Citation
Li, Jun; Wu, Shin-Tson; Brugioni, Stefano; Meucci, Riccardo; and Faetti, Sandro, "Infrared refractive indices
of liquid crystals" (2005).
Faculty Bibliography 2000s
. 5399.
https://stars.library.ucf.edu/facultybib2000/5399
J. Appl. Phys. 97, 073501 (2005); https://doi.org/10.1063/1.1877815 97, 073501
© 2005 American Institute of Physics.
Infrared refractive indices of liquid crystals
Cite as: J. Appl. Phys. 97, 073501 (2005); https://doi.org/10.1063/1.1877815
Submitted: 04 January 2005 . Accepted: 01 February 2005 . Published Online: 18 March 2005
Jun Li, Shin-Tson Wu, Stefano Brugioni, Riccardo Meucci, and Sandro Faetti
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Infrared refractive indices of liquid crystals
Jun Li and Shin-Tson Wu
a兲
College of Optics and Photonics, University of Central Florida, Orlando, Florida 32816
Stefano Brugioni and Riccardo Meucci
Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6-50125 Firenze, Italy
Sandro Faetti
Dipartimento di Fisica dell‘Universita’ di Pisa, Via Buonarroti 2, 56127 Pisa, Italy
共Received 4 January 2005; accepted 1 February 2005; published online 18 March 2005兲
The refractive indices of E7 liquid-crystal mixture were measured at six visible and two infrared
共=1.55 and 10.6
m兲 wavelengths at different temperatures, using Abbe and wedged cell
refractometer methods, respectively. The experimental data of the visible wavelengths fit the
extended Cauchy equations well. Using the extended Cauchy equations, we can extrapolate the
refractive indices of E7 to IR. The extrapolated results almost strike through the measured data.
Thus, the extended Cauchy equations can be used to link the visible refractive indices to infrared,
where the refractive index measurements are more difficult. © 2005 American Institute of Physics.
关DOI: 10.1063/1.1877815兴
I. INTRODUCTION
Liquid crystal 共LC兲 possesses a relatively large birefrin-
gence 共⌬n=n
e
−n
o
兲 in the infrared 共IR兲 region
1,2
and has
been used extensively for dynamic scene projectors,
3
laser
beam steering,
4,5
tunable band-gap photonic crystal fibers,
6,7
and millimeter-wave electronic phase shifters.
8,9
Simple
methods, such as voltage- or wavelength-dependent phase
retardation methods, have been developed for measuring the
IR birefringence of LCs.
10
However, only few methods, e.g.,
the Talbot–Rayleigh refractometer
11
and wedged cell
refractometer,
12,13
are available for measuring the individual
extraordinary 共n
e
兲 and ordinary 共n
o
兲 refractive indices in the
IR region. Moreover, these measurements are tedious be-
cause the IR refractometer is harder to align precisely and
most of the moisture-resistant IR substrates are not transpar-
ent in the visible so that visual inspection of LC alignment
quality and cell uniformity is more difficult.
By contrast, in the visible spectral region the LC refrac-
tive indices can be measured quite easily by the commercial
Abbe refractometer. Its accuracy is up to the fourth decimal
point 共±0.0002兲. However, the usable range of Abbe refrac-
tometer is limited to visible and near IR because of the faint
refractive light from the main prism and the transparency of
the prism.
In this paper, we link the visible and IR refractive indi-
ces by the extended Cauchy equations. First, we measured
the refractive indices of E7 LC mixture at six visible wave-
lengths using an Abbe refractometer and two IR wavelengths
共=1.55 and 10.6
m兲 using a wedged cell refractometer.
We fit the visible refractive index data with the extended
Cauchy equation. Once the three Cauchy coefficients are ob-
tained, we use the extended Cauchy equations to extrapolate
the refractive indices to the IR wavelengths. These extrapo-
lated data almost strike through the experimental results. In
Sec. II, we briefly review the extended Cauchy model for
describing the wavelength and temperature effects of the LC
refractive indices. In Sec. III, we describe the Abbe and
wedged cell refractometers for measuring the visible and IR
refractive indices. In Sec. IV, we link these two sets of ex-
perimental data by the extended Cauchy equations. Excellent
agreement between the model and experiment is found.
II. THEORY
The LC refractive indices are mainly determined by the
molecular structures, wavelength, and temperature. Most of
these effects have been reported previously. Here, we briefly
summarize the wavelength and temperature effects in order
to compare with the experimental results.
A. Wavelength effect
The major absorption of a LC compound occurs in two
spectral regions: ultraviolet 共UV兲 and IR.
12
The
→
*
elec-
tronic transition takes place in the vacuum UV
共100–180 nm兲 region whereas the
→
*
electronic transi-
tion occurs, in the UV 共180–400 nm兲 region. If a LC mol-
ecule has a longer conjugation, its electronic transition wave-
length would extend to a longer UV wavelength. In the near
IR region, some overtone molecular vibration bands
appear.
14
The fundamental molecular vibration bands, such
as CH, CN, and C=C, occur in the mid and long IR regions.
Typically, the oscillator strength of these vibration bands is
about two orders of magnitude weaker than that of the elec-
tronic transitions. Thus, the resonant enhancement of these
bands to the LC birefringence is localized.
15
The three-band model
16
was derived based on the LC
absorption spectra. It takes the three main electronic transi-
tions into consideration: one
→
*
transition 共the
0
-band兲
and two
→
*
transitions 共the
1
-band and
2
-band兲.Inthe
three-band model, the refractive indices 共n
e
and n
o
兲 are ex-
pressed as follows:
16
a兲
Electronic mail: swu@mail.ucf.edu
JOURNAL OF APPLIED PHYSICS 97, 073501 共2005兲
0021-8979/2005/97共7兲/073501/5/$22.50 © 2005 American Institute of Physics97, 073501-1
n
e
⬵ 1+g
0e
2
0
2
2
−
0
2
+ g
1e
2
1
2
2
−
1
2
+ g
2e
2
2
2
2
−
2
2
, 共1a兲
n
o
⬵ 1+g
0o
2
0
2
2
−
0
2
+ g
1o
2
1
2
2
−
1
2
+ g
2o
2
2
2
2
−
2
2
. 共1b兲
The three-band model describes the refractive index origins
of LC compounds. However, a commercial mixture usually
consists of several compounds with different structures. The
individual
i
’s are quite different so that Eq. 共1兲 would have
too many unknowns to describe the refractive indices of a
LC mixture.
To model the refractive indices for LC mixtures, we
could expand Eq. 共1兲 into power series because in the visible
and IR wavelengths, Ⰷ
2
. By keeping up to the
−4
terms,
we derive the extended Cauchy model,
17
n
e
⬵ A
e
+
B
e
2
+
C
e
4
, 共2a兲
n
o
⬵ A
o
+
B
o
2
+
C
o
4
. 共2b兲
Although Eq. 共2兲 is derived based on a LC compound, it can
be extended easily to eutectic mixtures by taking the super-
position of each compound.
17
Equation 共2兲 applies equally
well to both high and low birefringence LC materials. For
low birefringence LC mixtures, the
−4
terms are insignifi-
cant and can be omitted, thus, n
e
and n
o
each has only two
parameters.
18
From Eq. 共2兲, if we measure the refractive indices at
three wavelengths, then the three Cauchy coefficients 共A
e,o
,
B
e,o
, and C
e,o
兲 can be obtained by fitting the experimental
results. Once these coefficients are determined, the refractive
indices at any wavelength can be calculated. From Eq. 共2兲,
the refractive indices and birefringence decrease as the wave-
length increases. In the long-wavelength region, n
e
and n
o
are reduced to A
e
and A
o
, respectively. The coefficients A
e
and A
o
are constants; they are independent of the wave-
length, but dependent on the temperature. That means, in the
IR region the refractive indices are insensitive to the wave-
length, except near the local molecular vibration bands.
B. Temperature effect
The temperature effect of the LC refractive indices can
be expressed by the average refractive index 具n典 and birefrin-
gence ⌬n as
19
n
e
= 具n典 +
2
3
⌬n, 共3a兲
n
o
= 具n典 −
1
3
⌬n. 共3b兲
On the other hand, birefringence is dependent on the order
parameter S. Based on Haller’s approximation, the
temperature-dependent birefringence has the following
form:
20
⌬n共T兲 = 共⌬n兲
o
共1−T/T
c
兲

. 共4兲
In Eq. 共4兲, 共⌬n兲
o
is the LC birefringence in the crystalline
state 共or T=0 K兲, the exponent

is a material constant, and
T
c
is the clearing temperature of the LC material under in-
vestigation. The average refractive index decreases linearly
with increasing temperature as
具n典 = A − BT. 共5兲
Substituting Eqs. 共4兲 and 共5兲 back to Eqs. 共3a兲 and 共3b兲,we
derive the four-parameter model for describing the tempera-
ture dependence of the LC refractive indices,
21
n
e
共T兲⬇A − BT +
2共⌬n兲
o
3
冉
1−
T
T
c
冊

, 共6a兲
n
o
共T兲⬇A − BT −
共⌬n兲
o
3
冉
1−
T
T
c
冊

. 共6b兲
Although Eq. 共6兲 has four parameters, we can get 关A,B兴 and
关共⌬n兲
o
,

兴, respectively, by two-stage fittings. To obtain
关A,B兴, we fit the average refractive index 具n典=共n
e
+2n
o
兲/3as
a function of temperature using Eq. 共5兲. To find 关共⌬n兲
o
,

兴,
we fit the birefringence data as a function of temperature
using Eq. 共4兲. Therefore, these two sets of parameters can be
obtained independently from the same set of refractive indi-
ces but at different forms.
III. EXPERIMENT
We measured the refractive indices of E7 in the visible
spectral region using a multiwavelength Abbe refractometer
and measured the refractive indices at =1.55 and 10.6
m
using the wedged LC cell refractometer method, respec-
tively.
A. Measurements at the visible-light spectrum
We measured the refractive indices of E7 using a multi-
wavelength Abbe refractometer 共Atago DR-M4兲 at =450,
486, 546, 589, 633, and 656 nm. The accuracy of the Abbe
refractometer is up to the fourth decimal. For a given wave-
length, we measured the refractive indices of E7 from 15 to
50 °C with a 5 °C interval. The temperature of the Abbe
refractometer is controlled by a circulating constant-
temperature bath 共Atago Model 60-C3兲. The LC molecules
are aligned perpendicular to the main and secondary prism
surfaces of the Abbe refractometer by coating these two sur-
faces with a surfactant comprising of 0.294 wt %
hexadecyletri-methyle-ammonium bromide 共HMAB兲 in
methanol solution. Both n
e
and n
o
are obtained through a
polarizing eyepiece.
B. Measurements at =1.55 and 10.6
m
Figure 1 depicts the experimental apparatus for measur-
ing the refractive indices of E7 at =1.55
m. A laser diode
operating at the fundamental Gaussian mode was used as the
light source. The laser was especially designed to be injected
into an optical fiber. In order to obtain a free-propagating
laser beam a collimator was connected to the output side of
073501-2 Li
et al.
J. Appl. Phys. 97, 073501 共2005兲
the optical fiber. The laser beam was linearly polarized by
means of a polarizer and the polarization axis was 45° with
respect to the LC directors. The beam diameter was con-
densed to ⬃1 mm by two confocal lenses 共L
1
and L
2
兲 shown
in Fig. 1. To form a wedged LC cell, two ITO 共indium-tin-
oxide兲 glass substrates were separated by two spacers having
different thicknesses. The wedge angle was measured by an
optical method to be
=0.040 17 rad with a precision of 6
⫻10
−5
rad.
The alignment of the LC inside the cell is planar which
was obtained by buffing the spin-coated polyvinyl alcohol
layer. Good LC alignment was obtained by inspecting the
cell under a polarizing optical microscope. During experi-
ments, the front substrate was arranged to be normal to the
incoming infrared laser beam. The wedged cell was enclosed
inside a thermostat and the sample temperature was con-
trolled within 0.1 °C accuracy. A pointlike detector designed
for operation at =1.55
m was mounted on a micrometric
track. Under such condition, the detector can be moved up
and down with a very precise control of its position. The
position of the refracted laser beam is easily detected by
moving the detector on the track. The detector signal is dis-
played on a LeCroy digital oscilloscope. The measurement
principle consists of the evaluation of the deviation angle
experienced by the laser beam due to the refraction of the
beams by the LC material. In fact, when the laser beam
passes through the wedged LC cell it undergoes a splitting
process, and two beams 共ordinary and extraordinary兲 emerge
from the cell. The refractive indices are then retrieved by
using the Snell law and simple geometrical calculations. The
little shifts due to the presence of the glass substrates are also
taken into account. The accuracy on the measurement of the
refractive indices is estimated to be ⬃0.7%.
The experimental apparatus used for the measurement at
10.6-
m wavelength is analogous to the aforementioned
one. The laser source is a continuous-wave 共CW兲 CO
2
laser
that operates on the fundamental Gaussian mode at
=10.6
m 共line P20兲. The detector used is the infrared pyro-
electric video camera 共Spiricon Pyrocam III, model PY-III-
C-A兲. The two glass substrates of the wedged cell are now
replaced by two ZnSe plates that allow the transmission of
the CO
2
laser beam. More details about the experimental
method and the experimental apparatus at 10.6-
m wave-
length can be found in Ref. 18. In this case the estimated
accuracy on the refractive indices is 0.5%.
IV. RESULTS AND DISCUSSIONS
The refractive indices of E7 were measured at =450,
486, 546, 589, 633, and 656 nm in the temperature range
from 15 to 50 °C with a 5 °C interval and at =1.55 and
10.6
m, respectively. Figure 2 depicts the temperature-
dependent refractive indices of E7 at =589 nm. The open
squares, circles, upward triangles represent n
e
, n
o
, and aver-
age refractive index 具n典 for E7, respectively. The solid curves
are the fitting results using Eqs. 共6a兲 and 共6b兲. The fitting
parameters 关A,B,共⌬n兲
o
,

兴 are 关1.7546,5.36
⫻10
−4
,0.3768,0.2391兴. The solid straight lines are the fit-
ting results using Eq. 共5兲. The fitting parameters 关A,B兴 are
关1.7546,5.36⫻10
−4
兴. Clearly, the average refractive index
decreases linearly as the temperature increases.
Next, we use the extended Cauchy model to fit the re-
fractive indices n
e
and n
o
measured at the above-mentioned
visible wavelengths and temperature range. Then we get the
six Cauchy coefficients 关A
e
,B
e
,C
e
兴 and 关A
o
,B
o
,C
o
兴 for n
e
and n
o
, respectively. In Table I, we show the fitting param-
eters 关A
e
,B
e
,C
e
兴 and 关A
o
,B
o
,C
o
兴 for the extended Cauchy
model 关Eqs. 共2a兲 and 共2b兲兴 in the temperature range from 15
to 50 °C by using the experimental data measured in the
visible spectrum.
Figure 3 depicts the wavelength-dependent refractive in-
dices of E7 at T=25 °C. The open squares and circles rep-
resent the n
e
and n
o
of E7 in the visible region while the
downward and upward triangles stand for the measured data
at =1.55 and 10.6
m, respectively. The solid curves are
the fittings to the experimental n
e
and n
o
data in the visible
spectrum by using the extended Cauchy model 关Eqs. 共2a兲
and 共2b兲兴. The fitting parameters are listed in Table I. In Fig.
3, we extrapolate the extended Cauchy model to the near-
and far-infrared spectra. The extrapolated lines almost strike
through the center of the experimental data measured at
=1.55 and 10.6
m. The largest difference between the ex-
trapolated and experimental data is only 0.4%. Considering
the experimental error in the wedged cell refractometer
FIG. 1. The experimental apparatus for measuring the refractive indices at
IR laser wavelengths. 共1兲 Laser source. 共2兲 Optical fiber. 共3兲 Beam collima-
tor. 共4兲 Polarizer. 共5兲 Thermostat containing the wedged LC cell. 共6兲 Extraor-
dinary beam. 共7兲 Ordinary beam. 共8兲 Micrometric track. 共9兲 Pointlike detec-
tor. 共10兲 Digital oscilloscope. 共11兲 Temperature controller. L
1
and L
2
are two
confocal lenses acting as a beam condenser.
FIG. 2. Temperature-dependent refractive indices of E7 at =589 nm. The
open squares and circles represent the n
e
and n
o
measured at =589 nm
using the multiwavelength Abbe refractometer, respectively. The open up-
ward triangles are the average refractive index 具n典 calculated by the experi-
mental data. The solid curves are the fittings using the four-parameter model
关Eqs. 共6a兲 and 共6b兲兴. The fitting parameters 关A,B,共⌬n兲
o,

兴 are
关1.7546,5.36⫻10
−4
,0.3768,0.2391兴. The solid straight line is the fitting
using Eq. 共5兲. The fitting parameters 关A,B兴 are 关1.7456,5.36⫻ 10
−4
兴.
073501-3 Li
et al.
J. Appl. Phys. 97, 073501 共2005兲
