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Measurements for the Optical Parameters of Linear Birefringence Materials by
Using Thermal Light Polarization-Sensitive Optical Coherence Tomography
* a,bYu-Lung Lo, aChia-Chi Liao, and bCheng-Yen Yeh
a
Department of Mechanical Engineering
b
Institute of Micro-Electro-Mechanical System Engineering
National Cheng Kung University
No.1, Ta-Hsueh Road, Tainan 701, Taiwan ROC
*
Corresponding Author: [email protected]
Abstract
The new integrated POSCT with OCT system employing the thermal light source for the higher axial resolution in improving
capability of extracting the extraordinary and ordinary refractive indices and without using the highly precise scanning stage and
stage controller can simultaneously measure the thickness, mean refractive index, apparent phase retardation and optical axis
orientation of a linear birefringence sample. The extraordinary and ordinary refractive indices, therefore, also can be extracted
from above parameters if the order and apparent phase retardation of the multi-order birefringent material is correctly chosen.
The measured data are in good agreements with the theoretical ones. The more error could be found in 3.96% in apparent
phase retardation due to the dispersion effects from the broadband source. As the authors’ knowledge, this is the first
instrument that could extract all the optical parameters including the thickness, mean refractive index, apparent phase
retardation, optical axis orientation, order, and extraordinary and ordinary indices from a linear birefringence material by using
only one system.
Introduction
Developing the technology of OCT has decades, and it is suitable for in-depth cross-sectional images in scattering media based
on optical coherence-domain reflectometry (OCDR) [1]. From then on, many reports in OCT have presented. The subjects of
these research include improving the axial resolution with the thermal light source [2], combing OCT with confocal system [3],
developing various modulation methods in Doppler OCT and wavelength-dependent OCT [4], and developing in full-field or
three dimensional measurement [5].
The typical OCT has axial resolution of 10-20 µm corresponding to the coherence length of the light source and does not
explain the polarization effect or birefringence in the material. A recent and important extension to OCT is the incorporation of
polarized control into the system. Hee et al. [6] had proposed a polarization-sensitive OCDR system that adds the capabilities of
controlling the polarization state of light incident upon the sample and measuring the reflectivity of the light returning in particular
polarization states. Up to the present, many researches using PS-OCT to image the depth resolved optical birefringence of
biological have been published. The feature of PS-OCT is able to provide new contrast mechanisms potentially. de Boer et al. [7]
presented a PS-OCT to record 2D image of the change in the polarization of circularly polarized incoming light backscattered
form a bovine tendon. Hitzenberger et al. [8] presented an improved method of PS-OCT that made the measurement in
backscattered intensity, birefringence and optic axis orientation simultaneously with single A-scan.
In this study, the goal is to integrate our previous OCT system [9, 10] with the PSOCT and adopt the thermal light source to
achieve ultrahigh axial resolution. The new system with no high-precision scanning stage and stage controller would have the
capabilities of simultaneously measuring the thickness, mean refractive index, apparent phase retardation and optical axis
orientation of a single-layer multi-order birefringence sample. The extraordinary and ordinary refractive indices, therefore, also
can be extracted from above parameters. if the order and apparent phase retardation of the multi-order birefringent material is
correctly chosen.
Measurements of the thickness and mean refractive index
The new structure developed in this study is illustrated in Figure 1. The signal received by Detector (1) of the system is
designed to measure the thickness and mean refractive index of a linear birefringent sample. The beam from the thermal light
source is initially split by non-polarizing beam splitter (NPBS(1)) into two beams. One beam namely Beam (A) is defined as the
reference beam and it is reflected by a reference mirror. The other beam can be treated as the light source of the right half part
of the system, which is a conventional PS-OCT system, and then the beam is split into two beams by NPBS(2). Subsequently,
one beam (Beam (C)) passes a quarter-waveplate (QWP1) and reflected by the mirror on the scanning stage for the path-length
scanning. The other beam namely Beam (B) passes another quarter-waveplate (QWP2) and generates a circularly polarized
light onto the birefringence sample.
Figure 1. Schematic diagram of the new PS-OCT system
Detector (1) is used to measure the thickness and mean refractive index of the sample that the two parameters are
unconcerned with the polarization state of light. The locations of low-coherence interferometric signals are only considered
while Beam (C) is scanning the optical path of the reflected light. Beside, the new PS-OCT using two cubic NPBSs in common
path, the portion of path-matching signal that sample arm (Beam (B)) and reference arm (Beam (C)) recombined at NPBS(2)
would also be delivered to Detector (1) by NPBS1. In other words, Detector (1) can observe the location of peaks of
interferometric signals of the sample arm and reference arm sequentially. It can directly see the optical path difference by
compared the location of peaks of interferometric signals of sample arm with and without a sample. Thus, the signals recorded
only by Detector (1) can be used for calculating the thickness and mean refractive index of a single-layer sample.
The first and second steps of the setting of the sample and reference planes are shown in Figure 2 for measuring the sample’s
thickness and mean refractive index [10]. The reference plane of the sample stage is denoted as R0. The S and RS denote the
front and rear surfaces of the sample, respectively. The reference plane of the fixed reference mirror by where Beam (A)
reflected is denoted as A.
R0
Tns
RS
S
R0
RS
∆T
Tst
Ts
Figure 2. Steps and the Signals in measuring the thickness and mean refractive index
In the first step, the Beam (B) illuminates the sample stage, and the scanning stage is moved for scanning to record the location
of peaks of the scanning interferometric signal. Two peaks denoted by A and R0 are recorded by Detector (1) and illustrated in
Figure 2(A). In the second step, a sample is put on the sample stage, and the location of peaks of the scanning interferometric
signal is recorded by moving the scanning stage. Three peaks denoted by A, S and RS, as shown in Figure 2(B) could be
obtained.
The location of Peak A is fixed for the reference point at each optical path scanning due to the immobile reference mirror and
the locations of R0, S and RS are sequentially recorded by Detector (1). The time differences denoted Tns and Tst for the location
of A to R0 and the location of A to S can represent the optical path difference with and without the sample. Furthermore, TS is
the time difference for the location of S to RS, which is proportional to the optical path length in the sample. Therefore, the
following equation could be used directly to obtain the thickness of the sample as
d s = ∆T × V = (Tns − Tst ) × V
(1)
where V is the scanning velocity of the scanning stage. For the sample with birefringence, the measured refractive index would
be the mean value of ne and no [11]. As a result, the mean refractive index of the sample could be determined by
n=
Ts × V ne + no
=
ds
2
(2)
Measurements of the apparent phase retardation and optical axis orientation
Assuming that the polarization state of the light reflected from the reference arm is linearly polarized with 45° angle with respect
to the horizon perfectly, and the light has both equal amplitude and phase in two orthogonal polarization directions. As Beam (B)
with circular polarization state illuminates the sample, due to birefringence of the sample, the reflected light from the sample is
an elliptical polarization generally and will interfere with the same optical path from Beam (C). The PBS splits the interferometric
signal into two orthogonal linear polarization states detected by Detectors (2) and (3). The interferometric signals in each
polarization channel are given by [11]
IH =
I0
2
I
IV = 0
2
 
∆L 2 ln 2
Rs ( d ) sin ( β ) exp  − 
 
lc




2
 cos ( 2k ∆L + 2α )


2
 

∆L 2 ln 2  
Rs ( d ) cos ( β ) exp  − 
cos ( 2k ∆L )

 
 
lc
 

(3)
where d is the penetration depth, β is the phase retardation, ∆L = Lr - Ls is the path-length difference of two arms and the Ls is
the path length in the sample arm which includes the optical path in the sample, Ls = Lt + nd , lc is the coherence length of the
light source, and α is the optical axis orientation.
Equation (3) indicates that amplitude implies the information of the phase retardation and the phase of the interference term
with Doppler shift frequency implies the information of the optical axis orientation. The results are obvious that amplitude of
signals is sensitive to phase retardation and independent of the axis orientation of the sample. The optical axis orientation just
affects the phase of the carrier frequency. An algorithm, Hilbert transform, previously reported in the context of Doppler OCT [12]
and differential phase contrast OCT [13] is used to determine the amplitude and phase of the oscillating term of each
interferometric signal at any depth. Consequently, the apparent phase retardation and optical axis orientation can be
determined from the processed value of two orthogonal signals by the following equations as [11]
 IH (d ) 

 IV ( d ) 


β = kd ∆n = arctan 
α=
∆φ φH − φV
=
2
2
(4)
(5)
Therefore, using Hilbert transform to solve the amplitude and apparent phase of the measured signals is a convenient and
simple method to obtain the birefringence properties. However, it should be noticed in Equation (4) that it is restricted by the
definition of the tangent function and a positive number in the amplitude of the Hilbert transform, which means the dynamic
range of the apparent phase measurements is 0° ~ 90°.
Measurements of the extraordinary and ordinary refractive indices
The waveplates can be manufactured either as multi-order or zero-order plates. The phase retardation of a multi-order
waveplate, therefore, can be expressed as
β + 2mπ =
2π
λ
d ( ne − no )
(6)
where β is the apparent phase retardation (0 to 2π), (ne-no) is the birefringence of optical materials at λ, m is the order of a
multi-order waveplate, and d is the thickness of a waveplate. For the defined thickness, mean refractive index, apparent phase
retardation, and the assumed order (i.e. m = 1~10), the birefringence is calculated by Equation (6), and, therefore, the extra
ordinary and ordinary refractive indices calculated by Equation (7) could be obtained as
λβ mλ
+
4π d 2d
λβ mλ
no = n −
−
4π d 2d
ne = n +
(7)
Since the thickness, mean refractive index, and apparent phase retardation are all defined, the extra-ordinary and ordinary
refractive indices of the linear birefringence sample can be calculated by Equation (7) if the order and the exactly apparent
phase retardation are correctly chosen.
It should be noticed that the dynamic range in measuring the apparent phase retardation is restricted in the range of 0~90°.
Thus, the measured apparent phase retardation is needed to be verified and modified by the above reason and the further
discussion is introduced in Section “Calculation of the extraordinary and ordinary refractive indices”.
Axial resolution in the new PS-OCT
The axial resolution is one of the important subjects in this study. The thermal light with center wavelength at 590 nm and
spectral bandwidth ∆λ = 160 nm is used for improving the axial resolution, and the calculated coherence length is 0.96µm. After
comparing with the experimental results, the actual coherence length of the thermal light is known about 1.5 µm. It is obvious
that using thermal light truly improves the axial resolution.
The dispersion effect has to be considered while using such a broadband light source like the halogen lamp. The group
dispersion will increase the coherence length and decrease the amplitude of an interferometric signal [14]. It means that the
dispersion will degrade the axial resolution and fringe visibility of the low coherence interferometry. In order to estimate the
influence of the dispersion in the interferometric signal, the width and maximum amplitude of the interferometric signal are
measured while inserting four different thicknesses of samples into the sample arm. The D263T glass plate is measured and
the measured coherence envelopes of five different cases are shown in Figure 3. The experimental results show that the thicker
sample results in the poorer axial resolution and the lower amplitude of the interferometric signals. This phenomenon can be
compensated by inserting a dispersion compensating plate into the reference arm of the low coherence interferometer [15].
1
No sample
150um
300um
450um
600um
0.9
Normalized amplitude
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-20
-15
-10
-5
0
5
Optical path difference (um)
10
15
20
Figure 3. The measured coherence envelope with different thicknesses of D263T glass plate
Measured results of the thickness and the mean refractive index
There are two kinds of samples for demonstrating the capability of measuring the thickness and refractive index. One is the
sample made up of D263T glass, and the other is a multi-order QWP (CVI, QWPM-633-10-4-R15). Before the measurement,
the location of the fixed reference mirror A and reference plane R0 of the scanning interferometric signal have to be recorded.
After that, D263T glass plate whose refractive index and thickness, 1.5116 (at 1550 nm wavelength) and 150 µm (supplied by
the manufacturer), is put on the sample stage and the scanning stage is moved for matching the optical path length of the
sample.
Finally, three time differences, Tns, Tst and Ts can be calculated after the peak detection. Therefore, the thickness, ds, and
refractive index, n, can be obtained by Equations (1) and (2), and the experimental results are illustrated in Table 1. It should be
noted that due to the center wavelength of the thermal light source is around 590 nm, the refractive index which is the function
of wavelength could not be the same as specification: 1.5116 (at 1550 nm wavelength). Therefore, the theoretical refractive
index value at 590 nm is calculated by Sellmeier equation [16], and the theoretical refractive index of the glass plate at 590 nm
is calculated about 1.5167. From the average value of the measured refractive index, the measured refractive index is very
close to the theoretical value. From Table 1, the standard deviations of the refractive index and thickness of D263T glass plate
are 0.00185 and 0.310 µm respectively. The measured refractive index compared with the theoretical value obtained from
Sellmeier equation has an error of 0.23%, and the measured thickness compared with 150 µm has an error of 0.76%. The
values express that the measured result of a 150µm glass plate is good, which also means the influence of dispersion on the
thickness and refractive index is slight.
Table 1. The results of the single-layer sample (D263T glass plate)
New structure PSOCT
n
1
2
3
4
5
6
7
8
9
10
AVG
STD
1.51785
1.51623
1.51555
1.51974
1.51979
1.51661
1.51641
1.51963
1.51458
1.51764
1.51704
0.00185
ds (µm)
151.250
150.950
151.150
150.750
150.975
151.275
150.850
151.550
151.725
150.975
151.145
0.310
Next, the same procedure for measuring the multi-order QWP (CVI, QWPM-633-10-4-R15) at 633 nm wavelength is proceeded,
and the experimental results are illustrated in Table 2. For the same reason mentioned above, the theoretical value of mean
refractive index of the crystal quartz QWP can be obtained by averaging the ne and no, which can be calculated by a Laurent
series formula [16]. The theoretical value of the mean refractive index of a multi-order QWP is 1.54877 at 590 nm obtained by
averaging the theoretical values on ne = 1.55333 and no = 1.54421. In Table 2, the obtained thickness and mean refractive
index are 582.555 µm and 1.5504 respectively. The theoretical value of thickness and mean refractive index of a multi-order
QWP (CVI, QWPM-633-10-4-R15) are 575.917 µm and 1.54877 at 590 nm wavelength respectively, as illustrated later in Table
5, which is somewhat different from our measurement values. It is noticed that although the D263T is used to compensate for
the dispersion effects, the dispersion of the sample could not be perfectly compensated. It is believed that the error of the mean
refractive index and thickness of a birefringence sample mainly results from this imperfect dispersion compensation. The
measured thickness and mean refractive index have error of 1.14 % and 0.11% respectively and it is believed that the
experimental results are still acceptable.
Table 2. The result of the single layer birefringent sample (QWPM-633-10-4-R14)
New structure PS-OCT
1
2
3
4
5
6
7
8
9
10
AVG
STD
n
ds (µm)
1.54849
1.54933
1.55216
1.55250
1.55039
1.54885
1.55159
1.54956
1.55103
1.55014
1.55040
0.00132
583.375
583.300
581.850
582.125
582.725
582.625
582.275
582.875
582.225
582.175
582.555
0.488
Measured results of the apparent phase retardation and optical axis orientation
Before measuring the whole optical parameters of the birefringent sample with our new structure, the capability of the apparent
phase retardation and optical axis orientation measurements needs to be demonstrated. Two different types of
quarter-waveplate, zero-order QWP (CVI, QWP0-633-04-4-R10) and multi-order QWP (CVI, QWPM-633-10-4-R15), are used
as the samples in our measurement system.
The effect of the dispersion on measuring the optical path difference related to the thickness and mean refractive index is
slighter than that on measuring the apparent phase retardation of a birefringent sample. It is because the dispersion easily
affects the amplitude of signals and results in the more error in extracting the apparent phase retardation. Therefore, the
dispersion has to be compensated well while measuring the apparent phase retardation. In this study, it is discovered that the
D263T glass plate can still be used for compensation in experiments. According to the obtained amplitude and phase, the
apparent phase retardation and optical axis orientation of the sample can be calculated by Equations (4) and (5).
Figure 4 shows the experimental data in three different cases in examining the zero-order QWP for ten times. One of the three
cases is without using the compensating plate, another is using a compensating plate, and the other is using a compensating
plate and a 633 nm band-pass filter. The average apparent phase retardation of the measuring results without compensation is
81.9° and has a difference of 9 % as compared to 90°, and that with compensation is 86.31° and has a difference of 4.09 %.
The quarter-waveplate provides 90° phase retardation only when the wavelength of light is 633 nm. As filtering the beam from
the light source by a band-pass filter at 633 nm with 10 nm spectral bandwidth, the amplitude of the interferometric signal at the
horizontal direction is too small to be observed. Thus, it could be said that the measured phase retardation extremely approach
90°. It indeed displays that the result with both dispersion compensation and band-pass filter at 633 nm is closest to 90° phase
retardation since the incident light most fits the specification of the zero-order QWP (CVI, QWP0-633-04-4-R10).
92
Apparent phase retardation (deg)
90
88
86
84
82
80
78
76
Zero-order
Zero-order-comp
Zero-order-comp-filter
74
72
70
1
2
3
4
5
6
7
8
9
10
Measurements
Figure 4. Apparent phase retardation measurements of zero-order QWP (CVI, QWP0-633-04-4-R10)
Following the same experimental steps to measure the apparent phase retardation in the multi-order QWP, the experimental
data in Figure 5 also show three different cases. The average apparent phase retardation of the multi-order QWP with and
without 633 nm band-pass filter are 83.41° and 47.863° respectively, and the worst is the situation without neither the
compensating plate nor the 633 nm band-pass filter. Obviously, the error of the phase retardation from the zero-order QWP is
less than the multi-order one at the wide range of wavelength and it is attributed to the zero-order QWP offers substantially
lower dependence on temperature and wavelength than multi-order waveplates [17]. Furthermore, because the dynamic range
of the extracted apparent phase retardation is limited from 0° to 90°, a lot of difference between the cases with and without 633
nm band-pass filter is found. The further explanation in the modified apparent phase retardation could be explained in next
Section by correctly defining the order of the multi-order sample.
Apparent phase retardation (deg)
90
85
80
75
70
Multi-order
65
Multi-order-comp
60
Multi-order-comp-filter
55
50
45
40
35
30
1
2
3
4
5
6
7
8
9
10
Measurements
Figure 5. Apparent phase retardation measurements of multi-order QWP (CVI, QWPM-633-10-4-R15)
100
90
80
70
60
50
40
30
20
10
0
-10
y = 1.0396x - 0.5634
Theoretical
Measured
0
10
20 30 40 50 60 70 80
Set optical axis orientation (deg)
90 100
Measured optical axis orientation (deg)
Measured optical axis orientation (deg)
The measurement of the optical axis orientation is according to the phase difference between the two orthogonal polarization
states of interferometric signals. The phase difference between two orthogonal interferometric signals can be obtained by
Hilbert transform. The measurements are conducted for rotating the sample in the increment of 10° step by step, and each
optical axis orientation is obtained from the average value of ten time measurements. Figure 6(A) shows the experimental
results of the optical axis orientation in the zero-order QWP and the mean standard deviation is 2.9°. Figure 6(B) is the result of
multi-order QWP and the mean standard deviation is 3.1°. It is observed that the measured results are not very good and the
one possible reason for the worse measured result is that the vibration of a moving scanning stage will disturb the scanning
interferometric signals of PS-OCT system.
100
90
y = 0.9812x - 0.2999
80
70
60
50
40
30
20
10
0
-10 0
Theoretical
Measured
10 20 30 40 50 60 70 80 90 100
Set optical axis orientation (deg)
(A) Zero-Order QWP
(B) Multi-Order QWP
Figure 6. The measured optical axis orientation
Calculation of the extraordinary and ordinary refractive indices
Up to the present, the thickness, mean refractive index, apparent phase retardation, and optical axis orientation of the sample
have been obtained. It is also been discussed that the ne and no could be calculated by Equation (7) if the order of the sample is
known correctly.
For the measured thickness, mean refractive index, apparent phase retardation, and the assumed order (i.e. m = 1~10), the
possible birefringence calculated by Equation (6) and the possible extra-ordinary and ordinary refractive indices calculated by
Equation (7) are obtained and listed in Table 3. It is should be noticed that due to the limited dynamic range of the measuring
apparent phase retardation in Equation (4), the possible apparent phase retardations in four quadrants are considered. In Table
3, the estimating values in the condition of m = 1~10 and the possible apparent phase retardations only in the first and fourth
quadrants are listed. It could be observed that the estimating extraordinary refractive index is in the range of 1.55047~1.55591,
and the ordinary one is in the range of 1.54490~1.55034.
Table 3. The calculated birefringence for the assumed orders at 590 nm wavelength
β = 47.863°
(The First Quadrant)
m
0
1
2
3
4
5
6
7
8
9
10
∆n=ne-no
0.00013
0.00115
0.00216
0.00317
0.00419
0.00520
0.00621
0.00722
0.00824
0.00925
0.01026
ne
1.55047
1.55098
1.55148
1.55199
1.55250
1.55300
1.55351
1.55402
1.55452
1.55503
1.55554
β = 360°-47.863° = 312.137°
(The Fourth Quadrant)
no
1.55034
1.54983
1.54932
1.54882
1.54831
1.54780
1.54730
1.54679
1.54629
1.54578
1.54527
∆n=ne-no
0.00088
0.00189
0.00290
0.00392
0.00493
0.00594
0.00695
0.00797
0.00898
0.00999
0.01101
ne
1.55084
1.55135
1.55186
1.55236
1.55287
1.55338
1.55388
1.55439
1.55489
1.55540
1.55591
no
1.54996
1.54946
1.54895
1.54845
1.54794
1.54743
1.54693
1.54642
1.54591
1.54541
1.54490
By comparing the possible birefringence with the common uniaxial birefringent crystals [18], the only material with the
extra-ordinary and ordinary refractive indices corresponding to the estimating ones is quartz. Thus, the material of the sample
(CVI, QWPM-633-10-4-R15) is confirmed as quartz. From the birefringence of quartz (∆n = 0.009), the possible order, 8 and 9,
the corresponding apparent phase retardation, 312.137° and 47.863°, and the corresponding birefringence, 0.00898 and
0.00925, of a sample are chosen from Table 3 for the further confirmation.
Based on the dispersion equation [17], the difference of birefringence between 590 and 633 nm wavelengths is only 0.00005.
Therefore, for affirming which order of the sample is, the measured apparent phase retardation at 633nm wavelength with
633nm band-pass filter, as illustrated in Figure 5, could be reconsidered for confirmation. Again, substituting the measured
thickness, the possible apparent phase retardation in four quadrants (83.412°, 96.858°, 263.412°, and 276.588° at 633 nm
wavelength), and the assumed order, 1~10, into Equation (6), all the possible birefringence at 633 nm wavelength could be
estimated. It is found that only the estimated birefringence at 633 nm wavelength, 0.00894, with the order 8 and the apparent
phase retardation 83.412° is closest to the birefringence at 590 nm wavelength, 0.00898, with the order 8 and the apparent
phase retardation 312.137°. Also, the difference of birefringence between 590 and 633 nm wavelengths is 0.00005, and it
complies with the previous statement. In final, the order of a sample is confirmed to be 8 and therefore, the real measured
apparent phase retardation is 312.137° at 590 nm wavelength.
Since the measured thickness, the mean refractive index, the order, the birefringence, and the apparent phase retardation are
all defined, the extra-ordinary and ordinary refractive indices of the linear birefringence sample also are able to be calculated by
Equation (7). Therefore, the calculated ne and no are obtained as shown in Table 4. The calculated ne and no are 1.55489 and
1.54591 respectively on average, and the difference in refractive indices between ne and no could be calculated as 0.00898.
They are in good agreements with the optical parameters in the crystal quartz at 590 nm wavelength.
Times
1
2
3
4
5
6
7
8
9
10
AVG
STD
Table 4. The calculated ne and no of the multi-order QWP (QWPM-633-10-4-R14)
Thickness (µm)
Mean refractive index
no
ne
583.375
1.54849
1.55297
1.54401
583.300
1.54933
1.55382
1.54485
581.850
1.55216
1.55666
1.54767
582.125
1.55250
1.55700
1.54801
582.725
1.55039
1.55488
1.54590
582.625
1.54885
1.55334
1.54436
582.275
1.55159
1.55608
1.54709
582.875
1.54956
1.55405
1.54507
582.225
1.55103
1.55553
1.54654
582.175
1.55014
1.55463
1.54564
582.555
1.55040
1.554894
1.545914
0.513944
0.001386
0.001389
0.001383
∆n
0.00897
0.00897
0.00899
0.00899
0.00898
0.00898
0.00899
0.00898
0.00899
0.00899
0.00898
7.92×10-6
The all measured parameters of the multi-order quarter waveplate (CVI, QWPM-633-10-4-R15) at 590 nm are all listed in Table
5. From the comparison between the measured values and the theoretical ones, the measurement in the thickness and mean
refractive index are accurately enough and that in optical axis orientation is acceptable. However, the measured apparent
phase retardation after modification has more error in 3.96 % because of the dispersion effects by using a broadband source in
the new structure and also the imperfect dispersion compensation.
Table 5. Measured results and theoretical value of multi-order QWP (QWPM-633-10-4-R14) at 590 nm wavelength
Measured value
Theoretical value
Error
Thickness
582.555 µm
575.917 µm
1.14 %
Mean refractive index
1.55040
1.54877
0.11 %
Apparent
360° - 47.863°
325.018°
3.96%
Phase retardation (β)
= 312.137°
Birefringence (∆n)
0.00898
0.00912
1.54 %
ne
1.55489
1.55333
0.1 %
no
1.54591
1.54421
0.11 %
Optical axis orientation (α)
Figure. 6
NA.
NA.
Conclusions and Discussions
The capability of a new PS-OCT without using the highly precise scanning stage and stage controller in measuring the
thickness, mean refractive index, apparent phase retardation and optical axis orientation of a single-layer multi-order
birefringence sample has been demonstrated successfully. Surprisingly, the extraordinary and ordinary refractive indices also
could be precisely extracted. In general, the traditional PS-OCT has been applied many years in tissue sectioning, however only
the case with the phase retardation in range from 0 to 900 with zero order is without further modification. Therefore, to develop a
new PS-OCT that is suitable for the multiple-order linear birefringence materials with the full range of the phase retardation
measurement is the future work.
In summary, the measured data are in good agreements with the theoretical ones. As the authors’ knowledge, this is the first
instrument that could extract all the optical parameters of a linear birefringence material including the thickness, mean refractive
index, apparent phase retardation, optical axis orientation, order, and extraordinary and ordinary indices by using only a
thermal-light PS-OCT system.
Acknowledgements
The authors gratefully acknowledge the financial support provided to this study by the National Science Council under Grant No.
NSC 94-2212-E-006-021 in Taiwan.
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