Download Report

Refractive-index measurement of bulk materials:
prism coupling method
Hidetoshi Onodera, Ikuo Awal, and Jun-ichi Ikenoue
A simple method of measuring refractive indices of bulk materials using a prism coupling procedure is de-
scribed. Refractive indices are determined from the measurement of the angle incident to the prism at
which total reflection on the prism base breaks. This method is shown to possess the advantages of its simple procedure and sample preparation. The accuracy is comparable with that of minimum deviation method if the prism is well calibrated. Experimental results for several materials are given with an evaluation
of possible errors.
1.
II.
Introduction
Refractive-index measurement is one of the most
essential techniques in optics. The method of minimum deviations 2 is widely used for this purpose and is
regarded as the most reliable standard. However it
requires a portion to be cut out of bulk material and a
high quality prism to be fabricated for the refractiveindex measurement. It means that we need a sample
thick enough (5-10-mm thickness) to cut a prism from.
Furthermore, since the refractive index differs from
place to place in the sample (due to the slight change in
the composition and the amount of impurity), we cannot
know the true refractive-index value of the sample on
which we are going to work, unless it has the shape of a
prism.
In this paper we describe a prism coupling method
which is applicable to bulk materials
of any shape
having only one polished face. The experimental arrangement of this method is similar to that of prism
excitation in thin film waveguides.3 ,4 Refractive indices
can be determined by monitoring the light intensity
reflected at the prism-sample interface.
The procedure is simple and the accuracy is comparable with that of the minimum deviation method if the
apex angle and the refractive index of the prism are well
defined.
Method
The experimental arrangement and the coordinate
system are shown schematically
in Fig. 1. A tent-
shaped prism is tightly mounted on a substrate material
whose refractive index should be measured.
In the case
of an anisotropic substrate, the principal axes should
be oriented to the coordinate axes so that the relative
permittivity tensor becomes diagonal. The refractive
index of the prism np is chosen to be larger than the
refractive indices of the substrates nx, ny, nz.
As illustrated
in Fig. 1, TE(s) or TM(p) polarized
light is launched into the prism and is reflected at the
prism base with an angle 0. Whether the reflection
becomes total or partial is governed by the magnitude
of the propagation constant (=npk,, sind) and the refractive index of the substrate.
First we assume that the light has TE polarization
which senses the refractive index of the y direction ny.
If //k 0 is greater than ny, the incident light reflects totally. Therefore the power reflection coefficient Rp
becomes unity. When the incident angle 0 decreases
and /3/k0 becomes less than ny, some portion of the incident power will be converted into substrate radiation
modes, which results in an abrupt decrease in Rp.
Numerical expression of Rp is given by
ny
/kO,
Rp = 1,
(1)
2
RP -
2
- Se) + (gl + PeSe)
tanh2 ged
2
(Pe + Se)2 + (g2 + PeSe) tanh ged
= g (pe
nig - /3/k0 - ny,,
2
(2)
with
The authors are with Kyoto University, Electronics Department,
Sakyo-ku, Kyoto 606, Japan.
Received 25 October 1982.
0003-6935/83/081194-04$01.00/0.
© 1983 Optical Society of America.
1194
APPLIED OPTICS/ Vol. 22, No. 8 / 15 April 1983
(3)
[ = npko sinO,
Pe = (n2kl ge = (2 -
n
2)1/2,
(4)
(5)
PRISM
Fig. 1. Schematic of experimental arrangement. The coordinate
system used in this paper is also illustrated.
Se = (nik2
[32)1/2,
(6)
where ng is the refractive index of the gap between
prism and substrate, and d represents the gap thickness.
If /3/kois slightly smaller than n, the value of Rp is
approximated in a simpler form:
1-4Se
tanh2gd
1
(7)
2etanh2ged
+
Pe
In practice, monitoring the power of the output light,
the angle of incidence at the prism face a is changed and
the critical angle is found where the relation /3/k = ny
holds and sudden change of the light power occurs.
Then ny can be calculated in terms of a, np, and the
prism angle e:
ny = sina cose + (n
sin 2 a)1/2 sine.
-
(8)
If we turn the prism by 900 on the substrate, n can also
be determined.
When the incident light has TM polarization, the
critical incidence occurs when /3/ko = n
The power
reflection coefficient Rp for TM polarization has the
form:
nf -
fig S[3/ko
/k ,0 Rp = 1,
taking the structure of a TiO2 (rutile) prism (no = 2.583,
n = 2.865) and a c-cut LiTaO3 substrate (no = 2.176,
n = 2.181) as an example. Light of 0.6328-gm wavelength is assumed. The optic axis of the prism is oriented parallel to the prism edge so that TE polarized
light senses ne and TM polarized light senses no. The
results are shown in Figs. 2 and 3 for TE and TM polarization, respectively. The gap width d is chosen as
a parameter. The curves bend sharply at the points
where /3/k = 2.176 for TE and 2.181 for TM, which
enables us to find the critical angle correctly. Since
field penetration of TM polarized light into the air gap
is much less than that of TE polarized light, the gap
width for TM polarized light should be kept very small
to obtain a large decrease in Rp. However, this difficulty can be obviated to a large extent by the use of
matching oils for the gap medium. Figure 4 shows the
variation of Rp for TM polarization when CH212 (n =
1.74) is used as a matching oil.
The rate of decrease in Rp is also related to the refractive index of the prism np. Equations (4), (7), (11),
and (14) show that it is preferable to use a prism which
has a refractive index close to that of the substrate for
a large decrease in Rp. For example, the decrease in Rp
is nearly twice that of Fig. 2 when we use a SrTiO 3 prism
(n = 2.386) instead of a TiO 2 prism.
Ill.
incidence is calculated for the same structure employed
in Sec. II. The gap is air and the gap thickness is 0.02
gim. For simplicity, we assume that the incident beam
is Gaussian in the z direction but homogeneous in the
(9)-
2
niRg.2(n.'pm
- npsm)2+ (n2n2g, + n4pmsm)
tanh 2gmd
zm - f242Pm2m22tanh
g d
2
g,(n,2p. + flsm + + (nn gm n g
m
m
ix, Rp
Error Estimation
It is often necessary to measure the refractive index
n at some laser wavelengths. When a laser is used as
a light source, diffraction problems may be introduced
because of the limited radius of the Gaussian beam.
This is essentially the same as the diffraction of the
beam from a conventional monochromator. To estimate the amount of this error, Rp for Gaussian beam
(0)
.
with
Pm = (nk 2
[32)1/2
g. = (2 - n2k2)1/2
S = z (n2k2fix
2)1/2.
When /ka is slightly smaller than n, Rp is written
approximately
IRP
as
4nm
e1 - !H~Pm
1 - tanh 2 gd
fi14)
(14)
n,2.1 + np4
g. tanh 2gnd
fig
P.
The refractive index n, can be determined in the same
manner as for the TE polarized case.
To assess quantitatively the variation of Rp around
the critical angle a numerical calculation is carried out
y direction, and the waist of the Gaussian beam is located at the prism base. In the calculation, a Gaussian
beam is decomposed into twenty plane waves having
different angular spectra and amplitudes.
The results for TE polarization are plotted in Fig. 5.
The halfwidth of the Gaussian beam o at the beam
waist is chosen as the parameter. The sharp bend of the
Rp curve becomes rounded and the point where Rp
starts decreasing shifts toward the higher refractiveindex side by the use of a Gaussian beam, which produces some ambiguity in deciding the critical angle.
Hence, a large o Gaussian beam should be used in the
measurement to obtain high accuracy. For example,
if one wants to measure n to the fourth decimal, co
should be larger than 0.8 mm.
15 April 1983 / Vol. 22, No. 8 / APPLIED OPTICS
1195
of permitted error in a and eare comparable with those
in the minimum deviation method which uses a prism
of e = 400 and np = 2. Compared with the Brewster
n/k.
angle method (Abeles method), these values are relaxed
by 1 order of magnitude because a small variation in 0
becomes magnified in a by the refraction at the prism
interface.
f/.
49.45
49.40
4935
e ( )
Fig. 2. Variation of power reflection coefficient Rp for TE polarization vs the angle of incidence 0 (reflection) at the prism base or the
propagation constant [/k0. The parameter at the curves is the gap
thickness.
/k.
nn
I-
2181
2130
2.179
d =0.05jmn
.
2.182
.
57.60
0.02n
e ()
0.95
Fig. 4.
Rp
Variation of power reflection coefficient Rp for TM polar(n = 1.74).
ization. The gap is assumed filled with CH2 I2
0.901
I./,~~~~~~
2.173
0.85
nAr
57.50
-B
we
2174
/k.
2175
2.176
2.177
Or/
57.60
57.55
57.65
e ()
Fig. 3.
Variation
of power reflection
coefficient Rp for TM
polarization.
Other errors may originate in the measurement of the
incident angle a, the prism apex angle e, and the refractive index of the prism np. By differentiating Eq.
(8), we find the amount these errors contribute to the
final result. Since these quantities depend on the values of np, e, and a, we consider one example of np =
2.865, e = 50°, and a = -16° (n - 2). When each error
is assumed to exist separately, we must know a to better
than 30 sec, e to better than 10 sec, and np to better than
0.00013to determine n to within 10-4.
1196
The amounts
APPLIED OPTICS/ Vol. 22, No. 8 / 15 April 1983
49.40
()
Fig. 5. Variation of power reflection coefficient R, for TE polarization for Gaussian beam incidence. The gap thickness is taken to
be 0.02ym. The parameter at the curves is the halfwidth of the incident Gaussian beam.
2173
fl/k.
r174 2.175 2.176 2.177 2.178 2179 2.180
P
same substrate as in (1), but the measurement was
performed another day. The results show the high reproducibility of this method. The substrate used in
measurement (3) was cut from the same wafer from
which the substrate (1) was cut and annealed at 12000C
for 9 h in air. Increases in the refractive index (especially in ne) may be caused by the outdiffusion of Li2 O.
,
,
In measurements (4) and (5), identical refractive indices
were measured at different polarizations. Both results
agree well within the experimental errors.
At these measurements, about two-thirds of the rms
errors originated in the use of a Gaussian beam whose
halfwidth was 0.42 mm. Hence, the measurement becomes more accurate if a larger Gaussian beam is used.
i
l
-4 00
-350'
For example, if we use an 0.8-mm halfwidth Gaussian
beam, the amount of the rms errors will be reduced to
half.
-330'
- 340'
a'
Fig. 6. Measured output light intensity vs incident angle: TiO2
prism; c-cut LiTaO3 sample; TE polarization.
Table 1. Measured Refractive Indicesfor Several Materials at 0.6328 lum
by the Prism CouplingMethod;These Values are BelievedAccurate to
+0.0003
Measurement
Substrate
1
2
3
4
5
c-cut LiTaO 3a
c-cut LiTaOQa
c-cut LiTaO 3 b
a -cut LiTaO3c
Corning 7059
c-cut LiNbO 3
c-cut A12 03
6
7
TE
n =
=
n =
n =
2.1765
2.1765
2.1771
2.1765
n = 1.5283
n
= 2.2865
n = 1.7659
fn
TM
ne=
2.1815
n = 2.1816
ne = 2.1840
no = 2.1765
n = 1.5286
n = 2.2012
ne = 1.7582
a Same substrate was used in measurements (1) and (2).
b Substrate was annealed at 1200'C for 9 h in air.
cOptical axis was oriented parallel to the propagation direction.
IV.
Experimental Results
V.
In the experiment, a He-Ne laser of 0.6328-gm
wavelength was used as the light source. Halfwidth of
the Gaussian beam was 0.42 mm. The output light
intensity was monitored by a Si photocell.
All the
measurements were performed at 250 i 1C.
The
prism was TiO 2 and its optical axis was oriented parallel
to the prism edge. The refractive indices and the angle
of this prism were measured to be ne = 2.8641 0.0001,
nr = 2.5830
0.0001, and = 50.773 0.002° in ad-
vance by the minimum deviation method.
Figure 6 shows the variation of the output light intensity when a c-cut LiTaO 3 was used as the substrate.
The polarization was TE so that
n
was to be measured.
The curve agrees well with the theoretical curve illustrated in Fig. 5.
Summary
We have described a new method of refractive-index
measurement utilizing a prism coupling technique
widely used in guided wave optics. This method has the
advantage of its simple procedure and sample preparation. It is applicable to samples of any shape which
have a polished area small enough for prism coupling.
Since this method uses an excitation prism as the
standard, accurate calibration of the prism is necessary
in terms of the refractive index and the apex angle.
Another factor limiting the precision is the angular
spectrum of the incident beam. Numerical analysis
shows that the halfwidth of the Gaussian beam should
be more than 0.8 mm to determine the refractive index
to within the fourth decimal.
The measured refractive indices of LiTaO3 and those
References
1. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,
errors in these results are 0.0003. Since a detectable
amount of decrease in Rp was obtained even at TM
polarization, we did not use matching oils throughout
the measurements. In measurement (2) we used the
1959), pp. 177-180.
2. W. L. Bond, J. Appl. Phys. 36, 1674 (1965).
3. P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 14, 291
(1969).
4. R. Ulrich and R. Torge, Appl. Opt. 12, 2901 (1973).
of several other materials are listed in Table I. The rms
15 April 1983 / Vol. 22, No. 8 / APPLIED OPTICS
1197