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New procedure for measuring refractive indices of uniaxial media in the I.R. spectrum
Hedi Mattoussi
To cite this version:
Hedi Mattoussi. New procedure for measuring refractive indices of uniaxial media in the I.R. spec- trum. Journal de Physique II, EDP Sciences, 1992, 2 (7), pp.1401-1410. �10.1051/jp2:1992208�.
�jpa-00247737�
J. Fhys. II France 2 (1992) 1401-1410 JULY 1992, PAGE 1401 Classification
Physics Abstracts
61.30 78.20D 78.20F
New procedure for measuring refractive indices of uniaxial media in the I.R. spectrum
Hedi Mattoussi (*)
Carnegie Mellon University, Department of Chemistry, 4400 Fifth Avenue, Pittsburgh, PA 15213, U-S-A-
(Received 16 March 1992, accepted 14 April 1992)
R4sumd. Dons cette contribution, nous prdsentons une procddure pour mesurer [es indices de rdfraction et/ou la birdfringence de milieux uniaxiaux et pour [es radiations infrarouge du spectre optique. La prdsente technique est basde sur l'utilisation de l'interfdrence formde par [es ondes ordinaire et extraordinaire crddes aprbs qu'une onde plane entre dons le milieu uniaxial. La
conversion par gdndration d'harmoniques des radiations infrarouge en radiations UV-Visible, facile h ddtecter, est nouvelle. Les donndes expdrimentales sont discutdes en comparaison avec
des donndes prdcddentes obtenues par d'autres techniques.
Abstract. Refractive index measurements of uniaxial media at wavelengths in the I-R- spectrum are described. The experimental technique makes use of the birefringence to generate
interference between the extraordinary and ordinary waves created after an incident plane polarized wave enters the crystal. The use of harmonic generation to convert I-R- radiations to UV-Visible, easy to detect, is novel. The data are compared to previous measurements on
uniaxial crystals using other experimental techniques.
Introduction.
Uniaxial media are optically anisotropic, and their optical characterization requires the
knowledge of two refractive indices, the ordinary n~ and extraordinary n~, defined, respectively, for a plane wave traveling in the medium with its polarization perpendicular or parallel to the optic axis of the system [I]. In addition, measurement of the nonlinear optical (NLO) susceptibilities X ~~~ (i
=
2, 3), using optical harmonics generation, requires accurate
measurement of the refractive index n of the medium [2]. Its dispersion n = n (A must also
be known over a wide range of wavelengths, from UV-Visible to near infrared, for second and third harrnonic generation (SHG and THG) techniques [2]. Therefore, for uniaxial materials,
accurate measurements of the refractive indices n~, n~, and their dispersion relations are
needed for identification purposes, and can be used for NLO data analysis.
In this contribution we present what we believe to be a new procedure for measuring
refractive indices of any uniaxial medium in the infrared spectrum. The experimental
(*) Present address : University of Florida, Dept, of Physics, 215 Williamson Hall, Gainesville FL 32611, U-S-A-
procedure uses light interference between the extraordinary and ordinary waves propagating through the crystal.
General background.
It is known that a plane polarized light wave incident on a uniaxial crystal may be decomposed
into two secondary waves, traveling with speeds c/n" and c/n~, where n" and n' designate the
extraordinary and ordinary refractive indices for the waves traveling in the crystal, respectively, and c is the light speed in vacuum. When these waves recombine at the exit they
have a light path difference 8 (since n" # n'), causing them to interfere [I]. A polarizer (P) and an analyzer (A), set at an angle #i with respect to each other, may be used to define the incident and the outgoing polarizations, respectively. The angle between the optic axis and the polarizer P is 4. A simple case, used in this method, corresponds to crossed polars (#i = ar/2) together with the optic axis n laying in the plane of the sample and at an angle 4 such that 4
= ar/4, as shown in figure I. This allows an enhancement in the oscillation of the transmitted intensity I with 8, which is then given by [1, 3] :
1
= lo sin~ d
(I)
A
where lo is the amplitude of tile transmitted intensity (which depends on the Fresnel coefficients), and A is the incident wavelength [Ii.
~ i
i
Air
j
L-C. ' or Glass walls
n n
Air 9
Normal /
P Incidence
Fig. I. Schematic description for the different waves inside a uniaxial nematic liquid crystal medium
a wedge sample is shown.
Because of the anisotropy of the medium, the resulting extraordinary and ordinary waves
created inside the crystal, after a beam enters the medium with an incident angle
o, have different refraction angles o( and o(, respectively (Fig. I). These waves have different
optical path lengths, d( and d(, through the crystal. However, since the birefringence of the
crystal is always small ((n~-n~( «n~, n~), one can approximate the refraction angles o( and o( by an average value o', and write : d( =d(
= d'= d/cos o' [3]. An analytical expression for 8 is reached [1, 3] :
d(n~ n~) sin~ p
8
= (2)
cos o'
N° 7 BIREFRINGENCE OF UNIAXIAL MEDIA IN THE I-R- 1403
where p is the angle between the direction of propagation of the light beam and the optic axis of the medium, n (Fig, I). The angles o and o'obey Snell's law : (n~) sin o'
= n~,~ sin o, where (n~) is the effective refractive index for the average refracted wave inside the crystal.
In addition, the angles o' and p may be related trigonometrically [I]. For the particular configurations where the rays are propagating in a plane either parallel or perpendicular to the optic axis n, the relation becomes simple : p = ar/2 o' and p
= ar/2, respectively.
These cases are of great interest and they are also very simple (Fig, I) [1, 3].
Two, a priori different, experimental procedures for varying the parameter 8, and thus
generating interference in the transmitted intensity, are described. They involve monitoring
the physical thickness over which the light beam travels inside the crystal.
ROTATION oF A PLANE PARALLEL CRYSTAL. In the first procedure, 8 is monitored as the
propagation angle in a flat plane parallel crystal is varied (a
m 0 in Fig, I). For the case where
n is perpendicular to the plane of incidence (4
= ar/4 and p
= ar/2), the expressions for 8 and (n~) may be simplified to :
d(n~ n~)
~
cos o' ~~~
and
(~)~ l~/ ~ ~j ~~~
The variation in 8 reflects that of the optical path d'(o'). For the case where
n is within tile plane of incidence (4
= ar/4 and p = ar/2 o'), 8 and (n~) become :
8
= d(n~ n~) cos o' (5)
and
(ns(1'))~ i
Ii
~ ~~l/~' ~
~l~~'
~~~
The use of the property n~ n~ « n~, n~ may lead to a simple solution for 8. In fact, within these conditions (n~) can be approximated by the refractive index fi of the average isotropic
medium. This approximation is accurate for both configurations. The resulting error is very small (less than 5 fb) for materials with relatively high birefringence [3].
A scan of the transmitted intensity as function of the angle of incidence o provides an oscillating function. The minima occur at values of o~ for which 8 is an integer multiple
of A,
sin f 1"
8 m A d Ii
<n~>
~ne no) ~7)
where v
= -1/2 and 1/2 for the cases where 4 = + ar/4 and 4
= ar/4, respectively, and m~ = m~(v). In either case, for two minima I and j, 8 obeys
8~~ = 8~ 8~ =
j I A. The two configurations above can provide values for the birefringence (n~ -n~( if (n~) is
replaced by the constant n in equation (7). However, a fit for the minima in the I vs. o~ curve, using the full expression for (n~), allows one to extract accurate values for the
individual refractive indices, provided that d is known [3].
In practice, an estimate of (n~-n~( is deduced from experiment in either case
(4 = ar/4 or ar/4) and assuming (n~) (or (n~(o')) )
= n. This value of (n~ n~ is then used to fit the data for either case, n I and I to the plane of incidence, using equation (7) for 8~. Nevertheless, the use of only one configuration (for example 4 =ar/4, since
(n~) has a simpler expression) provides values for the individual refractive indices if the
previous estimate for (n~ n~[ is then used to numerically fit tile experimental data.
In the experimental procedure for this method, the angle o' is varied by rotation of a planar sample with respect to the incident beam, where the axis of rotation is kept perpendicular to the plane of incidence (Fig. I).
TRANSLATION OF A WEDGED CRYSTAL. The use of a wedged sample with a fixed wedge
angle a, in the second method, allows one to monitor the optical path in the Sample while
keeping the propagation angle constant. The variation of the light path difference 8 is monitored by translating the cell along an axis perpendicular to the incident beam [4]. At
norrnal incidence, where the angles inside the crystal obey the relationships:
o(
= o(
= o'm 0 (Fig. I), and for the conditions outlined above ( 4 = arm and p
= ar/2)
8 becomes :
8
= [n~ n~ (d~ + x tan a (8)
where x is the translational distance along the sample plane and do is the initial thickness (at
x = 0). As discussed above, two minima I and j, located at x, and x~, respectively, obey the condition :
8~~ = [8~ 8~[ = [j -ii
= [n~-n~[ ([xi -x~[ tan a) (9) Knowing the distance between the corresponding displacements [x, -x~[ and the wedge angle a, a straightforward analysis of the experimental data provides accurate value for [n~ n~[.
Applying the two methods for the same material allows one to reach an accurate measurement for the individual refractive indices. From the wedge sample metllod, a value for [n~ n~ is obtained. This value is then used to fit the experimental data for the rotation of a plane parallel sample, and the case where n is norrnal to the plane of incidence.
Experimental section.
EXPERIMENTAL SETUP. The light source for the I-R- radiation is a pulsed Nd : YAG laser
(Fig. 2). The short pulse duration (about 5 ns), the incident radiation wavelength, and for the most part the quantum efficiency of the photomultiplier tube causes problems in the detection of the signal. To circumvent these limitations the I-R- beam exiting the crystal, I-e- S in figure 2, is used as a source to generate optical harrnonic(s) from a suitable material
(component Sj in Fig. 2) [2, 5]. The signal at frequency 2w or 3w (for 2nd and 3rd harmonic
generation processes) falls within the UV-Visible region and is therefore easy to detect.
The I-R- Nd: YAG laser source provides a signal at A =1064nm. The other I-R-
wavelengths are obtained through the use of a Raman shifter [5]. The Raman shifter, pumped
with the incident YAG signal, provides signals at =1542nm for methane gas and
A
=
1907 nm for hydrogen gas : Stokes components. The signal at 2w (or 3w) is detected
using a photomultiplier tube, and then digitized using an ADC system. The sample signal is norrnalized by the signal from a reference in order to take account of fluctuations in the
source intensity (Fig. 2). Since the harrnonic signals are proportional to a power of the
incident intensity, e,g., I~~ cc (I~ )~, and I~~ cc (I~)~, [2, 5] these signals reproduce exactly
N° 7 BIREFRINGENCE OF UNIAXIAL MEDIA IN THE I.R, 1405
Reference Branch
F~ L~ L~ F~
,', ~2 Fnw
,
P3 ~~" P4
F~ L~ P A L~ F~ L~ L~ F~
i w
~~ ~i S
'
f~'C°( Prism
~2 S~
SarnPle Branch
~nw
~
= l 064 nm
~~= yAG Raman Cell
Laser P~
Fig. 2. Schematic description for the experimental setup Harmonic Generation (H.G.) apparatus for the incident I-R- radiation. L, and L~ are weakly converging lenses, D, and D~ are diaphragms their
use allows a better resolution of the angle o and the displacement x. The filter F, is similar to
F~. The sample S could be a flat or a wedge sample.
the periodicity of I~ as function of the rotation angle o and the translation distance
x for method I and 2, respectively. Consequently, the data analysis for the variation of the harmonic intensity with either o or x provides information pertaining to the fundamental
wavelength, and the above forrnulation for the data analysis can be used.
A schematic description of the experimental arrangement is provided in figure 2. The set of
prisms Pi, ~,~ are needed to select and direct the desired I-R- wavelengths from the other Stokes components, whereas the prism P~ is used to split the incident beam into 2 secondary
beams directed to the sample and reference (the two branches in Fig. 2). A Rhomb rotater (Karl Lambrecht Inc.) was used to select the incident polarization (P). A Nicol prism (CVI
Laser Corp.), most effective in the I-R- spectrum, was used to define the outgoing
polarization (analyzer A). Beside these elements, an I-R- bandpass filter (F~) selects only the
signal at the frequency w after the sample, eliminating harrnonic signal generated from the
sample which also oscillates with o, or x (though with different dependences, I-e-, Maker
Fringes) [2, 5]. Two visible bandpass filters (F~) and two interference filters (F~~), inserted
on each branch, are used to select the signal at the frequency 2w (or 3w) after the beam passes through the harmonic sources (Fig. 2).
MATERIALS. The materials used for the present study were a nematic lyotropic liquid
crystal polymer and conventional planar uniaxial crystals, quartz and potassium dihydrogen phosphate KDP (provided by Cleveland Crystals Inc). Planar configuration (n parallel to the sample walls) was used for all materials in the present study. The polymer liquid crystal we
used is a poly(p-phenylene benzobisthiazole), PBT, provided by SRI Intemational (Stanford, CA) through the courtesy of Dr. J. F. Wolfe. We used PBT macromolecules with molecular
weight M
=
34 000, an index of polymerization of 140, and an average rod length of 160 nm [6, 7]. Flat cells (provided by Hellma Cells Inc.) of 375
~L thickness with good optical quality (Am flatness) and wall parallelism, and wedge cell with wedge angle a
= 0.95 ± 0.01°, allowing a continuous thickness scan from 200 to 800 ~L, were used. Good quality and stable
monodomains are obtained by applying a strong magnetic field (H=4Tesla) to freshly prepared samples for several hours, with H parallel to the sample walls and to the direction of
the flow. The alignment quality of the planar monodomains thus obtained is checked by
looking at the extinction properties of transmitted light between crossed polars in polarized light microscopy, and also by looking at the conoscopic figures provided by these
monodomains. For quartz, we used a uniaxial flat slab with thickness 4 190 ± 2
~L and a wedge sample with an angle a
= 3 ± 0.02°. For KDP, only a flat uniaxial slab with thickness d
= 3 968 ± 2
~L was used. The thicknesses were measured using a micrometer with 2
~L
accuracy, while the wedge angle was checked by looking at the reflected light spot at a
distance of I m from the sample.
Results and discussion.
Typical intensity I~~ vs. rotation angle o and displacement x for rotating a plane parallel sample or translating a wedge crystal are shown in figure 3. Each point on these plots corresponds to a value of the angle o or displacement x, and to an average over 10 pulses. For both methods oscillating functions, I
=
I (o) and I
=
I (x), are easily observed as predicted by equation (I). The minima location as well as their number, for a given rotation or
translation intervals, depend on the geometry (n parallel or perpendicular to the plane of incidence), the wavelength A, the sample thickness d or wedge angle a, and the medium
birefringence. The resolution for the minima location is better defined when the 2nd harrnonic generation technique (SHG) is used [3]. This is primarily attributed to the fact that
I~~ involves a lower power of the fundamental intensity than I~~ (I~~ cclj whereas
I~~ cc I(, and sharper minima for I~~ vs. o and x curves result).
For thick samples and/or fairly large birefringence [n~ n~[, one may encounter problems
associated with the emergence of more than one order in the interfering transmitted ligth
beam at the detector. It is therefore necessary to use diaphragms before and after the sample
as well as weakly converging lenses. This allows us to improve the resolution of the parameters o and x inside the crystal, and allows the selection of a single order in the
interfering intensity at the detector (Fig. 2). These precautions are very important (and sometimes crucial) for smaller wavelengths [3].
Values for the birefringence [n~ n~[, using the wedge sample analysis for the PBT and the quartz crystals are listed in tables I and II. The corresponding values n~ n~ using the
aP.
approximation (n~)
= fl to fit the minima off vs, o curves, obtained using the first method
(plane parallel sample with 4 = ar/4), are also reported in the second column of tables I, II, and first column in table III for the different wavelengths and materials used. In addition, the values for the individual refractive indices n~ and n~ together with the corresponding birefringence using the numerical fit for I vs. o and starting from either [n~ n~[~
,
or P.
[n~ n~ using the wedge sample when possible, are also given in tables I, II and III. The agreement between the values for [n~ n~[ from the wedge sample case and [n~ n~[~~
from tile flat crystal is very satisfactory. However, a much better agreement between the
birefringence values extracted from the wedge sample case and the numerical fit for the flat
crystal is found ; the values differing by less than I fb for the quartz crystal and less than 5 fb for the PBT solutions. Another important point is the excellent agreement of the present
values of n~ and n~ with those listed in the literature for classic uniaxial quartz and KDP crystals [8]. Moreover, the present measurements actually provide values at wavelengths
which have not been scanned before [8]. The slight discrepancy between the flat cell and the
wedge case for PBT solutions may be attributed to the modest quality of the wedge
monodomain compared to that in the plane parallel case. In fact, the path length,
N° 7 BIREFRINGENCE OF UNIAXIAL MEDIA IN THE I-R- 1407
(al
_
( j§ >li It ,(
j
"1 :1:I (
j j 'I Q'
c ,,, ., , , =. ;
~ ~ If j j
£ _'.
: %
< ,
.,, -
s ,' .' ;.
m .'
,
: ..
.; ..
.' I ; ',
.. I .,
.,
.,.; jj j; .. ;,
o k I U 'd I
-50 -25 0 25 50
Rotation Angle
(bi
( l
j
1 ( (
4
' '
' ..
-
~
' fi o
j. ~~
'l I 1. # i
"
l '
O'
' '
i #' :
~
i '
j
i '
° ~' ~ ~ ~ ii '
0 3 6 9 12 15
X(mm)
Fig. 3. Typical plots for the intensity I~~ vs. o or x from PBT nematic monodomains (c
= 8.13 x 10~~g/cm~) : (a) flat sample (d
= 375 ~c) at A
=
064 nm with ~
=
arm ; (hi Wedge sample («
= 0.95°) at A
= 1542 nm.
d, in the wedge cell reaches values (up to 800 ~L) slightly too large to guarantee a stable nematic monodomain for the present PBT solutions [6, 7]. Cells with a smaller wedge angle
and light paths d varying between values smaller than 500 ~L, for instance, could provide a
better quality, and more stable, monodomains. In addition, such cells may help to avoid the
problems associated with the resolution in the parameter x for the wedge monodomains. The
dispersion of the individual refractive indices for these materials over a wide range of
wavelengths from the near I.R, to the blue region of the optical spectrum will be studied independently [3].
The technique described here provides a good and accurate tool to measure the
birefringence and the refractive indices for uniaxial materials. It may also be used to extract the corresponding values for biaxial media [9]. The use of the harmonic generation signal
(I2w or I3w) to convert the I-R- radiation to visible or U-V- where the quantum efficiency of
