FULL-FIELD MAPPING OF THE STRESS-INDUCED BIREFRINGENCE ON
THE INTERNAL INTERFACES USING A POLARIZED LOW COHERENCE
LIGHT INTERFEENCE MICROSCOPE
Jenq-Shyong Chen & Yung-Kuo Huang,
Department of Mechanical Engineering, National Chung-Cheng University
168, University Rd., Ming-Hsiung, Chia-Yi 621, Taiwan, ROC
Email: [email protected],
ABSTRACT
A polarization-sensitive optical coherence microscope (PS-OCM) based on the Linnik interference architecture has been
developed to non-destructively measure birefringence distribution at the interfaces of a multi-layer structure.
PS-OCM
utilizes the low coherence interference principle to enable the depth-resolved mapping of the birefringence distribution inside
the materials. By simultaneous detection of interference fringes in two orthogonal polarization states allows determination of
the Strokes parameters of light. Comparison of the Strokes parameters of the incident state to that reflected from the sample
can yield a depth-resolved map of optical properties such as birefringence and refractive index.
1. INTRODUCTION
An important demand in semiconductor and optoelectronic industries is a non-invasive method to evaluate the defects and
residual stress distribution inside the semiconductor and optoelectronic devices. These devices are fabricated by a series of
planar process and formed as a multiple layer structure. The defect and residual stress existed in the interfaces can
significantly affect the functionality, characteristics and reliability of these devices.
Because most semiconductor and optic
materials such as ceramic, silicon, polymer and glass, are either intrinsic or stress-induced birefringence, therefore, the
birefringence distribution may, for instance, indicate the signatures of the defects and strain distribution. The inspection
method for this purpose should be non-invasive, fast and high spatial resolution.
The Rayleigh scattered light method [1-3] has been developed for a non-invasive evaluation the three dimensional stress
field of glasses and polymers. The sensitivity of this method, however, is poor due to the weak Rayleigh scattering and the the
measurement throughput is low due to the point scanning measurement. Another problem is the spatial resolution in the depth
and lateral resolutions of the previous method is not good enough for the semiconductor and optoelectronic devices. In this
work, we report a polarization-sensitive optical coherence microscope (PS-OCM) which is developed for overcoming the
problems of scattered light method.
Optical coherence tomography (OCT) [4-7] is a noninvasive depth-resolved tomograhical method used to inspect the
internal structures of semi-transparent materials such as biological materials by measuring backscattered light from
intra-interfaces inside the materials. OCT uses the short coherence gating characteristics of the low coherence interference by
broadband light source. Depth resolution of 1-5　m or even submicron [8] has been achieved at mm depth range with
broadband femtosecond laser based OCT technology. Because the OCT is based on the light interference principle, the
dynamic range of the sensitivity is more than 100dB. Polarization sensitive OCT (PS-OCT)[9-11] is an extended embodiment
of the OCT technology that enables the polarization state of backscattered light to be detected and quantified. By
simultaneous detection of interference fringes in two orthogonal polarization states allows determination of the Strokes
parameters of light [12]. Comparison of the Strokes parameters of the incident state to that reflected from the sample can yield
a depth-resolved map of optical properties such as birefringence, refractive index and optical axis [13-14]. Because many
semiconductor and optic materials such as ceramic/wafer/polymer/glass are stress-induced birefringence materials, changes in
birefringence distribution may, for instance, indicate changes in material uniformity and stress inside the materials. For the
reason, the PS-OCT was applied to nondestructively test of for local defect detection and spatial stress distribution
measurement of the stress-induced birefringence materials [15-16].
Most OCT systems are developed for the medical applications based on low NA fiber-optic Michelson interferometers to
provide a long depth of focus greater than the Z-dimension (depth direction) of the samples to enable cross-sectional imaging.
This design leads in practice to images with lateral resolution of the order of 10~30 m. Conventional OCT is a kind of
point-scanning microscope that will take time to complete a 3D profiling. One variant of OCT, which exploits the advantage of
high NA objective lens and full-field measurement capability of conventional optical microscope, is called full-field OCM(optical
coherence microscope)[17-19]. The OCM preserves the en face full-field cross-sectioning capability by sacrificing the
sensitivity due to the cross-talk effect. Full-field birefringence imaging by PS-OCM has also been reported for the en face
birefringence measurement applications [20-21]. However, the proposed PS-OCM requires two-step en face measurements
for each cross-section birefringence measurement. Another problem of the previous work is the high NA lens has very short
depth of focus of less than 10　m. The visibility of the interference fringe is quickly decreased as increasing imaging depth
because the refractive index mismatch between air and the sample.
In this work, we propose a novel full-field PS-OCM using two CCD image sensors that simultaneously detect the two
polarization states at one shot.
In order to maintain a good fringe visibility when the tomographic depth is larger than the
depth of focus of the objective lens, a motion stage is used to adjust the reference mirror position in order to compensate for the
refractive index mismatch between air and the sample.
2. OPTIC SYSTEM
The proposed PS-OCM of our work is sketched in Fig. 1. The instrument is based on a Linnik interference microscope.
Two important advantages of the Linnik configuration are: 1) it is much flexible to adjust the optical path, focus, and phase
modulation of reference arm; and 2) the strong aberrations in high NA wide-aperture lens can be compensated for due to the
symmetrical and identical microscope objectives at the sample and reference arms.
Figure 1: The optical architecture of the PS-OCM
The polarizer (5) is used to create the incident linear polarized light. The polarized light beam is further separated into two
light beams and redirected to the sample arm and reference arm directions. Their corresponding Jones matrices (Er of
reference arm and Es of sample arm) are described as:
Er = Es =
⎡1⎤
1
E0 ⎢ ⎥
2 ⎣0⎦
(1)
Where Er and Es represent the reference and sample light beams respectively. The reference reflection mirror is mounted
on a PZT piezoelectric actuator which is purposed for high-speed reference arm phase modulation control. The reference
objective lens, reference reflection mirror and PZT actuator are all fixed on a same stage driven by a step motor. The step
motor is used for two purposes: 1) to fine tune for the optical path difference between the reference and sample arms due to
manufacturing tolerance and optical alignment; and 2) to real-time adjustment the position overlapping of the confocal gate and
coherence gate due to the refractive index mismatch between the air and sample.
o
An achromatic quarter wave plate (QWP1) is inserted on the reference arm optical path with a 22.5 angle inclination of its
fast axis relative to the incoming reference beam. Because the reflected light from the reference will pass through the QWP1
o
again, the total phase change of the reference beam is 45 that corresponds to the a Jones matrix of
M QWP1 ⋅ M QWP1
⎡ sin(45o ) cos(45o ) ⎤
1 ⎡1 1 ⎤
=⎢
=
⎢
⎥
o
o ⎥
2 ⎣1 − 1⎦
⎣cos(−45 ) sin(−45 )⎦
(2)
o
The sample light beam passes through the achromatic quarter waveplate(QWP2) which has a 45 inclination of its fast axis
relative to the incoming beam. The QWP2 changes the light beam from linear polarization to right-circular polarization. If the
sample is a birefringence material, the reflected sample light becomes as elliptically polarized light. Because the reflected
sample light beam will pass through the QWP2 again, the elliptical polarization is returned to linear polarization but associated
with a phase delay corresponding to the birefringence state of the sample. The birefringence-encoded Jones matrix of the
reflected sample light beam is
J s = exp( i
δ ⎡ s11
)
2 ⎢⎣ s21
s12 ⎤
s22 ⎥⎦
(3)
s11 = cos 2 (θ ) + sin 2 (θ ) × exp( −iδ )
s12 = cos(θ ) × sin(θ ) × (1 − exp( −iδ ))
s 21 = cos(θ ) × sin(θ ) × (1 − exp( − iδ ))
s22 = sin 2 (θ ) + cos2 (θ ) × exp(−iδ )
(4)
(5)
(6)
(7)
δ is the birefringence-induced phase delay and θ is the inclination angle of fast axis of the QWP.
The reflected sample and reference beams are combined, interfered and then pass thought the polarized beam splitter
which separates the interfered light into two beams with polarization states orthogonal to each other. Two CCD image sensors
are used to detect the interfered light intensity of these two polarization states, named as vertical polarized interference fringe
and horizontal polarized interference fringe. The Jones vector of the horizontal polarized
EH
and vertical polarized
expressed as
⎡ EH ⎤
⎢ E ⎥ = ES 1 + ER1
⎣ V⎦
(8)
Where
⎡1 ⎤
1
E0 ⎢ ⎥ exp( −i 2kl r )
2 ⎣0 ⎦
⎡1 ⎤
1
= M QWP 2 J S J S M QWP 2
E 0 ⎢ ⎥ exp( −i 2 kl s )
2 ⎣0 ⎦
E R1 = Rr M QWP1 M QWP1
(9)
E S1
(10)
Therefore,
EV is
⎡EH ⎤
⎡exp(i 2θ ) sin(δ ) ⎤
⎡1⎤
i
⎢ E ⎥ = Rr 2 E 0 ⎢1⎥ exp( −i 2kl r ) + Rs iE 0 ⎢
⎥ exp( −i 2kl s )
cos(δ )
⎣
⎦
⎣⎦
⎣ V⎦
Where
lr
and
spectral density.
ls are the optical length of reference arm and sample arm.
(11)
We assumed the light source has Gaussian
The signal density of the CCD was shown
2 1
= A0 [ Rr + Rs sin 2 (δ ) + Rs Rr sin(δ )G (∆l )cos( 2k∆l + 2θ )]
(12)
4
2
2 1
IV = Ev = A0 [ Rr + Rs cos 2 (δ ) + Rs Rr cos(δ )G (∆l ) cos(2k∆l )]
(13)
4
⎡ ⎛ 2 ∆ l ln 2 ⎞ 2 ⎤
⎟ ⎥
G ( ∆ l ) = exp ⎢ − ⎜⎜
(14)
⎟ ⎥
lw
⎢ ⎝
⎠
⎣
⎦
Where ∆l is the optical path difference between reference and sample beams. l w is the width of wave length spectrum.
Rr and Rs are the reflection ratios of the sample and reference mirrors.
Figure 2 shows a simulation result of Equations (12), (13) and (14) when the δ is 0.1*π and θ is 10 degree.
I H = EH
2
Figure 2:
A simulation of the vertical and horizontal interferences
There are DC terms existed in the intensity measurements. We estimated the DC terms in Equations (12) and (13) by
averaging method and then, removed the DC terms from the measured intensity by subtraction.
The AC terms in Equations
(12) and (13) after subtraction of the DC terms can be determined as:
R s R r sin( δ ) G (∆ l ) cos( 2 k ∆ l + 2θ )
I ACH =
(15)
R s R r cos( δ ) cos( 2 k ∆ l )
I ACV =
(16)
Appling the Hilbert transform to the AC term of the interference signals, we can obtain
U V = H ( I ACV ) .
H represents the Hilbert transform operator. The birefringence
δ
UH = H(I ACH)
and fast axis angle
determined as
θ=
1
(∠U H − ∠U V )
2
δ = tan
−1
(U
H
/UV
(17)
)
(18)
and
θ , then, can be
3. COMPENSATION OF THE REFRACTIVE INDEX MISMATCH
In this PS-OCM, there are two optical gates inherent in the tomograpic depth of the system, the confocal gate and the
coherence gate. Optical gate means the fringe visibility will be deteriorated when the interference signal is outside the optical
gate. The confocal gate (or called depth of focus) is determined by the NA of the MO that is approximately about 10　m of the
10x MO lens. The coherence gate (or called the coherence length) is determined the spectrum distribution of the light source
that is about 2 m of the halogen light.
In this Linnik PS-OCM, best visibility is achieved when the focal gate and coherence gate are set at the same position.
That means during the depth tomographic scanning, the focus plane of sample objective and imaging plane must be maintained
always in the conjugated image condition. This conjugated image condition is set by adjusting the positions of the two MO
lenses of the sample and reference paths. The conjugated image condition is determined by the material refractive indexes in
the sample beam and reference beam. Initially, the two MO lenses have the same material refractive indexes (i.e. both in the
air). However, when the tomographic depth is moved inside the sample, the refractive index mismatch between the sample
and reference (i.e. the air) will create a distance deviation between the confocal gate and coherence gate positions. After a
certain depth, the coherence gate will move out of the range of the focus depth of the MO lens. The visibility of the fringe
visibility is spoiled when the coherence gate and confocal gate are not overlapped anymore. Figure 3 shows a measurement
example of a glass when the refractive index mismatch effect had not been compensated for. Figure 3(a) is the measurement
result when the tomographic position is on the top surface of the sample. The coherence gate was inside the confocal gate.
Good fringe visibility was obtained in this case. However, when the tomographic plane was targeted at the bottom surface of
the sample (200　m depth from the top surface) as shown in Fig. 3(b), the coherence gate and confocus gate were separated
from a distance. In this case, the fringe visibility was deteriorated. The problem was solved by using a stepping motor to
continuously adjust the optical path length of the reference beam as shown in the Figure 1.
(a) Fringe on the top surface
(b) Fringe on the bottom surface
Figure 3: The fringe visibility
4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 4 shows an example of the birefringence distribution on the top surface of a glass. We scribed the glass surface
using a diamond tip. The stress-induced birefringence variation around a scribed groove was clearly identified and quantified.
High birefringence variation was concentrated on the near-by of the scribed groove. However, the birefringence property
became less in the region that far away from the scribed grooved.
Birefringence(degree)
µm
600
80
70
500
60
400
50
300
40
200
30
20
100
10
100
Figure 4:
200
300
400
500
600
700
800 µ m
The birefringence distribution on the top surface of a grooved glass
Figure 5 shows another example that a glass of 1mm thickness is compressed by a load. We had measured the
interference components of the two polarized states on the bottom face of the glass. We successfully demonstrated that the
PS-OCM can inspection the stress-induced birefringence inside the internal interface of a layered material. Before loading, the
interfered vertical and horizontal polarization light intensity were 5340 and 1963 on gray unit as shown in Figures 6(a) and 6(b).
However, after loading, these values became as 3168 and 3637 on gray unit as shown in Figures 6(c) and 6(d).
The measured birefringence value can be linked to the stress using the following relationship:
∆n = CS
(19)
∆n is the stress-induced refractive index variation, S is the stress, and C is called the Brewster’s constant or
−6
2
stress-optical coefficient. For the Na-Ca glass the Brewster’s coefficient is about 2.6x 10 mm / N . Because the
Where
variations in the interfered polarized intensities are 1774 and 2172 value in the horizontal and vertical directions respectively, we
can estimate the birefringence-induced rotation angle was 390 based on the Equation (18). Therefore, the stress-induced
refractive index variation is estimated as
∆n =
δλ0 n
360 × d
= 9.58 x10 − 5
The stress can then be estimated as
(20)
36.8 N / mm 2 .
Figure 5: Loading on a glass
Figure 6: The interfered intensities on the bottom surface of glass. (a) Vertical polarization before loading, (b) horizontal
polarization before loading, (c) vertical polarization after loading, (d) horizontal polarization after loading
5. CONCLUSIONS
We have demonstrated that the proposed PS-OCM can do the full-field depth-resolved mapping of the birefringence
distribution inside the materials. For the case of the stress-induced birefringence sample, we can correlate the measure
birefringence to the refractive index variation. And, finally the stress field can be estimated from the birefringence distribution.
From the preliminary experimental results, successful measurement of the birefringence distribution on the bottom surface of a
glass sample has been achieved. Because the bottom surface exists a discontinuous optical refractive index from the glass to
the air, the reflected light has fringe intensity large enough detected the CCD image sensor. The future challenge will be
measurement of the spatially distribution of the birefringence inside the homogeneous materials such as the silicon wafer and
-6
glass. In the case of homogenous materials, the Rayleigh scattered light is weak (usually 10 of the incident light). Therefore,
the dynamic range of the measured interfered intensity will be very poor because the interfered fringe is buried by a strong DC
term. Other means such as optical method or software image processing method must be applied to enhance the dynamic
range of the signal.
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