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Mueller matrix ellipsometry study of a circular polarizing filter
Nina Hong, and James N. Hilfiker
Citation: Journal of Vacuum Science & Technology B 38, 014012 (2020); doi: 10.1116/1.5129691
View online: https://doi.org/10.1116/1.5129691
View Table of Contents: https://avs.scitation.org/toc/jvb/38/1
Published by the American Vacuum Society
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Mueller matrix ellipsometry study of a circular
polarizing filter
Cite as: J. Vac. Sci. Technol. B 38, 014012 (2020); doi: 10.1116/1.5129691
View Online Export Citation CrossMar
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Submitted: 30 September 2019 · Accepted: 10 December 2019 ·
Published Online: 26 December 2019
Nina Hong
a)
and James N. Hilfiker
AFFILIATIONS
J.A. Woollam Co., Inc., 645M Street, Suite 102, Lincoln, Nebraska 68508
Note: This paper is part of the Conference Collection: 8th International Conference on Spectroscopic Ellipsometry 2019, ICSE.
a)
Electronic mail: nhong@jawoollam.com
ABSTRACT
The authors introduce an ellipsometric data analysis strategy for a flexible polymeric circular polarizing filter consisting of a thin linear
polarizer and quarter-wave plate sandwich. The circular polarizing filter is an inhomogeneous optical system exhibiting different optical
responses depending on the propagation direction of light. If light enters from the linear polarizer, the transmitted beam is linearly polarized
before entering the quarter-wave plate. The orientation of the quarter-wave plate is rotated 45° from the linear polarizer axis. The emerging
light from the quarter-wave plate is circularly polarized. If light is circularly polarized and enters from the reserve side, the quarter-wave
plate converts it into linearly polarized light. The following linear polarizer either transmits or absorbs the beam depending on the handed-
ness of the original circular polarization. The optical response in the forward direction is utilized in photography to reduce unwanted reflec-
tions in the image. The optical response in the reverse direction is utilized in 3D eyeglasses, which consist of two orthogonal circular
polarizing filters to separate the left and right images. Mueller matrix spectroscopic ellipsometry is used to observe the optical responses of a
circular polarizing filter in both directions. The authors demonstrate data analysis procedures for individual layers to find the optical con-
stants in a wide spectral range from 400 to 1700 nm. The circular polarizing filter measured in the forward direction enables ellipsometry to
determine the included angle between the linear polarizer and the quarter-wave plate. The ellipsometric data analysis result is used to
predict transmitted light intensity versus rotation angle (deg) with respect to any input polarization state.
Published under license by AVS. https://doi.org/10.1116/1.5129691
I. INTRODUCTION
Circular polarizing filters convert unpolarized light to circularly
polarized light by passing light through a linear polarizer followed by
a quarter-wave plate as shown in Fig. 1(a). Circular polarizing filters
are commonly used in front of camera lenses for photography to
filter out unwanted reflections, darken blue skies, or suppress glare
from the surface of the water. Since the light passes through a linear
polarizer first, it passes light polarized in the same linear direction.
This reduces unwanted reflections from nonmetallic surfaces such as
water or glass or scattering of light with particles in the atmosphere.
This improves vividness, color saturation, and contrast of the image.
The quarter-wave plate is used to convert the linearly polarized light
into a circularly polarized light to avoid problems with the polariza-
tion sensitivity of the detector.
The reverse direction of circularly polarized filters is used
for circularly polarized 3D eyeglasses. In the reverse direction,
transmitted light through the entire device is always linearly
polarized regardless of the input polarization state because the
linear polarizer is the last optical element. However, if the input
beam is circularly polarized, it works as a circular polarization
filter, as shown in Fig. 1(b), and the filtration is dependent on the
handedness. The quarter-wave plate converts circularly polarized
light into linearly polarized light. The following linear polarizer,
which is 45° rotated from the quarter-wave plate axis, either trans-
mits or blocks the linearly polarized light depending on the hand-
edness. When two opposite circular polarizing filters are used in
eyewear, each eye sees a different image, and stereoscopic images
are created by the superposition of the two images.
Ellipsometry is a proven metrology tool to measure thin film
thickness and optical constants.
1
The circular polarizing filter con-
sists of two anisotropic films. This sample structure is well suited
for ellipsometry to determine the thickness and optical constants
for each film. Sample measurement and data analysis are nontrivial
due to direction-dependent optical constants of each film created
ARTICLE avs.scitation.org/journal/jvb
J. Vac. Sci. Technol. B 38(1) Jan/Feb 2020; doi: 10.1116/1.5129691 38, 014012-1
Published under license by AVS.
by anisotropy. We use Mueller matrix (MM) ellipsometry,
2
which is
ideal for anisotropic samples as it measures polarizing and cross-
polarizing effects of the sample as well as depolarization of light.
3
We
demonstrate step-by-step data analysis procedures for these aniso-
tropic films and perform simulations based on the modeling out-
comes to examine the performance of the circular polarizing filter.
For the quarter-wave plate, we limit the data analysis to the
wavelength range where the film is transparent. The phase delay at
normal incidence is created by the index difference between two
orthogonal directions in the sample plane. Thus, our model can
capture the optical anisotropy of this layer as well as the orientation
of the anisotropic optical axes. For the retarding film, analyzing the
transparent region is adequate to extract all parameters of interest.
The details of this data analysis have been discussed in the author’s
previous paper
4
for various polymeric substrates.
Data analysis of a linear polarizer, on the other hand, must
include the absorbing region where linear diattenuation occurs.
Linear diattenuation is the property of a material where the trans-
mission intensity differs between two orthogonal linear polarization
directions.
5
For the linear polarizer, we are mostly interested in
how well the linear polarizer selectively transmits or absorbs light
depending on the input polarization direction. To minimize the
number of free parameters during fitting, data analysis is divided
into multiple steps. First, the transparent region is analyzed, which
provides information regarding film thickness, refractive index, and
the azimuthal orientation. Second, the complex refractive index in
the absorbing region is determined, where one direction remains
transparent while the orthogonal direction absorbs light.
Finally, we apply both models to analyze MM ellipsometry
data acquired from the circular polarizing filter, which consists of
the two films. The included angle of the two films is determined by
measurements performed in the forward direction. This report
shows that ellipsometry can be used as an accurate nondestructive
metrology tool for circular polarizing filters.
II. EXPERIMENT
A dual-rotating spectroscopic ellipsometer (J.A. Woollam RC2®)
is used to measure all data. The instrument specifications and
coordinate system are explained in Ref. 4. Fifteen normalized MM
elements were measured in both reflection and transmission over
a wide spectral range from 193 to 1700 nm. Data analysis was
restricted above 380 or 400 nm. Technically, the optical constants
below 380 nm can be measured using ellipsometry data in reflec-
tion. However, it is unnecessary for this device structure, which
only operates in transmission.
Transmission-mode MM data were acquired for a wide angle
of incidence range from −15° to +75° by 5°. For brevity, only the
angles in steps of 15° are displayed in the figures throughout the
paper. The few negative angles of incidence were included to
ensure proper sample alignment and that films did not exhibit
tilted optical axes relative to the sample normal. Reflection-mode
MM data were measured at multiple angles of incidence from
+55° and +75° by 10°. An automated sample rotation stage with
accuracy better than ±0.02° is used to take azimuthal rotation scans
from 0° to 360° by 2° for both normal incidence transmission and
reflection-mode MM data. Analyzing multiple data at different
azimuthal angles helps reduce parameter correlation for birefrin-
gent samples with absorption.
MM data analysis was performed using the CompleteEASE®
software.
6
The Levenberg–Marquardt multivariate regression algo-
rithm was used to find the best-fit parameters. Dispersion functions
were created using either Kramers–Kronig consistent b-spline
7
or
generalized oscillator layer
8
to describe complex refractive index
versus wavelength (or energy).
III. MODELING
This section introduces MM ellipsometry data features and
procedures for ellipsometry modeling. The subsections show the
components of our modeling strategy: (A) a quarter-wave plate,
which creates a 90° phase shift; (B) a linear polarizer, which behaves
as a linear retarder in the near-infrared and a linear diattenuator in
the visible; and (C) a circular polarizing filter, a combination of the
two optical systems.
A. Quarter-wave plate
Quarter-wave plate is a polarization element that creates a 90°
phase shift between two orthogonal polarizations. The MM of a
horizontal quarter-wave plate and Stokes vectors, s, for input and
output light in transmission are shown in Eq. (1). Transmission of
light through a quarter-wave plate swaps the last two Stokes param-
eters along with a sign-change for the fourth parameter. This indi-
cates that a quarter-wave plate converts ±45° linearly polarized
light into circularly polarized light and vice versa,
1000
0100
0001
0010
2
6
6
4
3
7
7
5
s0
s1
s2
s3
2
6
6
4
3
7
7
5
¼
s0
s1
s3
s2
2
6
6
4
3
7
7
5
:(1)
FIG. 1. (a) Forward direction and (b) reverse direction operation of a circular
polarizing filter consisting of a linear polarizer and quarter-wave plate.
ARTICLE avs.scitation.org/journal/jvb
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Published under license by AVS.
A phase shift caused by the optical path difference due to different
indices of refraction is given as
δ(rad) ¼2π
λt(jnynxj):(2)
Here, tis the film thickness, λis the wavelength, and njis the
index of refraction in the j-direction. The quarter-wave plate studied
in this report is a thin polymeric substrate with biaxial anisotropy.
As the retardance is wavelength dependent, a quarter-wave shift
occurs only for a single wavelength. The wavelength where a quarter-
wave shift occurs is tunable by adjusting film thickness. Equation (3)
shows the MM of a rotated retarder where the phase shift (or retard-
ance), δ, is a wavelength-dependent parameter and θis the angle
between the retarder and the ellipsometer coordinate system,
10 0 0
0cos
22θþcosδsin22θ(1 cosδ) cos 2θsin 2θsin δsin 2θ
0(1cos δ)cos2θsin2θsin22θþcos δcos22θsin δcos 2θ
0sinδsin2θsinδcos2θcos δ
2
6
6
4
3
7
7
5
:
(3)
Note that the last MM element, m44, contains information about the
retardance while remaining independent of retarder orientation. It is
convenient to limit the analysis to normal incidence and use m44 to
fit only for the index difference between the x- and y-directions,
which determines the retardance. Once the retardance is estimated,
we can expand our analysis to the remaining MM elements and
match the data to find the azimuthal orientation, θ. A full characteri-
zation to find the refractive indices in the x-,y-,andz-directions can
be accomplished by analyzing all data including oblique angles.
Ellipsometry data at high angles give great sensitivity to the
out-of-plane index as the z-component of the electric field increases
as the angle increases.
Figure 2 shows an excellent agreement between the MM
ellipsometry data and generated curves from the biaxial model. The
y-axis is scaled to show the shape of the curves. The high-frequency
oscillations in the data are from coherent thickness interference
between the beams reflected from the top and bottom surfaces of
FIG. 2. MM ellipsometry measurement (solid lines) and model generated (dotted line) data of quarter-wave plate in the transmission mode from −15° to +75°.
FIG. 3. Thickness interference oscillations are well resolved in the selected MM
elements (m12 and m44) from 1400 to 1700 nm. Film thickness is optically deter-
mined by matching these interference oscillations.
ARTICLE avs.scitation.org/journal/jvb
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Published under license by AVS.
the quarter-wave plate. The oscillations are more prominent at high
angles and long wavelengths due to the longer coherent length of
light. These oscillations can be ignored in the biaxial modeling as
shown in Fig. 2 by assuming completely incoherent interference
between the multiple combined beams and fitting through the
baseline of the data. We can also use these oscillations to optically
determine the film thickness. For this purpose, the model needs to
be modified to account for the coherent interference of multiple
beams and the influence of the system bandwidth. Film thickness
becomes a fit parameter in the modified model. Figure 3 shows two
of the MM elements at long wavelengths. The interference oscilla-
tions are well resolved and determine the thickness of the quarter-
wave plate to be 77.75 μm.
The complex refractive indices, nx,ny, and nz, of the quarter-
wave plate are shown in Fig. 4(a).Figure 4(b) shows the in-plane
retardance calculated by Eq. (2). The result indicates that the tested
polymeric film is a zero-order wave plate and works as a quarter-
wave plate at 580 nm. If the retardance is not exactly a quarter-wave,
polarization conversion between linear and circular polarization is
imperfect. The amount of imperfection can be estimated using the
spectroscopic ellipsometry result.
B. Linear polarizer
Linear polarizers transmit light of a specific linear polariza-
tion and absorb light polarization of the orthogonal direction.
The difference in transmission intensities is called linear
FIG. 4. Refractive index (n) vs wavelength (nm) in the x-,y-, and z-directions
(a) and in-plane retardance (deg) vs wavelength (nm) (b) are obtained from MM
ellipsometry data analysis for a quarter-wave plate.
FIG. 5. Rotation scan of normalized 15MM elements at normal incidence for linear polarizer from 400 to 1700 nm.
ARTICLE avs.scitation.org/journal/jvb
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Published under license by AVS.
FIG. 6. MM spectroscopic ellipsometry measurement (solid line) and model generated (dotted line) data for a linear polarizer in the transmission mode at 0° (a) and 45°
(b) orientations.
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Published under license by AVS.
FIG. 7. MM spectroscopic ellipsometry measurement (solid line) and model generated (dotted line) data for a linear polarizer in the reflection mode at 0° (a) and 45° (b)
orientations.
ARTICLE avs.scitation.org/journal/jvb
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diattenuation. Modeling a linear polarizer is more complex due
to the added parameters to describe direction-dependent absorp-
tion. The ellipsometer used for this work has rotating compensa-
tors before and after the sample. The dual-rotating compensator
system avoids issues with low signal intensity for linearly polar-
ized light whose transmission axis is nonideal for the analyzer
axis. Thus, the dual-rotating compensator configuration is ideal
for determining the direction-dependent refractive index and the
extinction coefficient of a linear polarizer.
The MM of an ideal linear polarizer with a horizontal trans-
mission axis and the Stokes vectors, s, for input and output light in
transmission are given as
1
2
1100
1100
0000
0000
2
6
6
4
3
7
7
5
s0
s1
s2
s3
2
6
6
4
3
7
7
5
¼1
2
s0þs1
s0þs1
0
0
2
6
6
4
3
7
7
5
:(4)
Equation (4) tells us that an ideal linear polarizer transmits linear
polarization parallel to its transmission axis regardless of the input
polarization state. The MM of an ideal linear polarizer with an
arbitrary transmission axis, θ, is given as
1
2
1 cos 2θsin 2θ0
cos 2θcos22θcos 2θsin 2θ0
sin 2θcos 2θsin 2θsin22θ0
00 00
2
6
6
4
3
7
7
5
:(5)
Here, varying the transmission axis of the linear polarizer is
equivalent to varying the angle between the linear polarizer and the
ellipsometer coordinate system.
Figure 5 shows a rotation scan of MM data from 0° to 360° for
the linear polarizer, which reveals wavelength-dependent optical
responses. Between 700 and 1700 nm, the optical response is similar
to a linear retarder. The last MM element, m44 ,doesnotchangewith
azimuthal orientation. The wavelength dependence of m44 is due to
the cosine function of retardance as shown in Eq. (3) for linear
FIG. 8. Complex refractive index vs wavelength (nm) in the transmission and
extinction axes for linear polarizer.
FIG. 9. Rotation scan of normalized 15 MM elements at normal incidence for circular polarizing filter in the forward direction from 400 to 1700 nm.
ARTICLE avs.scitation.org/journal/jvb
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retarders and the wavelength dependence of retardance as shown
in Eq. (2). The remaining MM elements in the bottom-right 3 3
submatrix show repeated values every 90° or 180° as predicted by the
trigonometric functions of 2θin Eq. (3). Analysis of the transparent
region is performed as explained in Sec. III A. The linear polarizer is
found to exhibit uniaxial anisotropy. Between 400 and 700 nm, the
main optical response is found in the top-left 3 3submatrix.The
repeated values in this submatrix correspond to the trigonometric
functions of 2θin Eq. (5). The rotation scan of MM data tells us that
the linear retarder shows linear diattenuation in the visible wave-
length range.
Reflection MM data is analyzed in addition to transmission
MM data to obtain the extinction coefficient. Figures 6 and 7
show the MM data in transmission and reflection in two different
orientations. Multiple orientations were analyzed simultaneously
to break parameter correlation, which occurs for anisotropic
absorbing samples as explained in Ref. 4. Although the linear
polarizer axis could not be perfectly aligned to the ellipsometer
coordinate system by hand, the MM data at 0° orientation
show very little cross-polarization.Thisisconfirmed as the
off-diagonal 2 2 submatrix elements are close to zero in both
transmission and reflection. The modeling result indicates that
the misorientation is just 1.3°. The −1valuesinthem12 and m21
elements in the visible show that the transmission axis is in the
vertical direction. Accordingly, the transmission axis is close to
−45° for Figs. 6(b) and 7(b),whichleadstosignificant
cross-polarization.
A uniaxial model is established to fit all data shown in
Figs. 6 and 7and one more MM data in reflection rotated by 90°.
The resulting complex refractive index over a wide wavelength
range from 400 to 1700 nm is shown in Fig. 8. At long wavelengths,
the refractive index shows a normal dispersion in both transmission
and extinction directions with no absorption. Thus, the sample
behaves as a linear retarder. For visible wavelengths, the extinction
coefficient in the extinction axis shows strong absorption due to
organic absorption. Thus, an electric field oscillating along this
direction is strongly absorbed through the linear polarizer.
The opposite direction remains transparent in the visible with an
extinction coefficient of less than 0.0001. The refractive index for
the transmission axis shows very little dispersion as the wave-
length changes.
C. Circular polarizing filter
The last sample consists of a linear combination of the previ-
ously studied two layers. This optical system is inhomogeneous
since the optical response is different depending on whether the
beam enters from one side or the other, as demonstrated in Fig. 1.
The forward direction converts unpolarized light into circularly
polarized light. The reverse direction can pass or block circularly
polarized light depending on its handedness.
Figure 9 shows a rotation scan of MM data from 0° to 360° for
the circular polarizing filter in the forward direction. Visually, we
can divide the optical response into two wavelength ranges. Above
FIG. 10. MM spectroscopic ellipsometry measurement (solid line) and model generated (dotted line) data for circular polarizers in the forward direction transmission mode.
ARTICLE avs.scitation.org/journal/jvb
J. Vac. Sci. Technol. B 38(1) Jan/Feb 2020; doi: 10.1116/1.5129691 38, 014012-8
Published under license by AVS.
700 nm, the bottom-right 3 3 submatrix elements show interest-
ing patterns with repeated values as the circular polarizer rotates.
This is because the system is simply a linear combination of two
linear retarders in this wavelength range. Between 400 and 700 nm,
m12 and m13 values change repeatedly between −1 and +1. The
m12 (or m13) element decides the linear diattenuation between hori-
zontal (or +45°) and vertical (or −45°) linear polarization. Thus,
these two MM elements are related to how well linearly polarized
light transmits depending on the polarization direction. On the
other hand, the circular polarizer equally transmits circularly polar-
ized light regardless of its handedness as found from m14 ¼0.
Another interesting MM element is m41. This MM element is
related to the capability of the optical system to convert unpolar-
ized light into circularly polarized light, referred to as circular
polarizance.
9,10
Since the measured value is +1, we can infer that
the device works as a right-circular polarizing filter by converting
unpolarized light into right-circularly polarized light. When the cir-
cular polarizer is measured in the reverse direction (the data are
not shown here), the m41 element is 0 and the m14 element is +1.
In this case, the sample shows circular diattenuation,
9,10
which
transmits right-circularly polarized light and blocks left-circularly
polarized light. Thus, the device works as a right-circular polarizer.
For the circular polarizing filter, we build a model with fixed
optical constants determined from the individual films. The fit
parameters include each layer’s thickness and orientation. The
best-fit parameters are determined from transmission MM data at a
single orientation. Figure 10 shows that the measured data (solid
line) match the model generated data (dotted line). To ensure
model validity, we further fitreflection-mode MM data at three
different orientations (0°, 45°, and 90°). The best-fit thickness of
the quarter-wave plate is 78.64 μm and that of the linear polarizer
is 80.46 μm. The included angle shows a small variation from 43°
to 46° between different data sets.
IV. RESULTS AND DISCUSSION
In this section, we evaluate the performance of the device in
both directions. In the forward direction, detected intensity
versus rotation angle (deg) of the device is predicted for unpolar-
ized light and four linear polarizations. Figure 11 shows the simu-
lated results for a wavelength of 580 nm. The sinusoidal functions
show the orientations for the minimum and maximum intensities.
The device equally transmits unpolarized light at all rotation
angles. Due to the different Fresnel reflection coefficients between
the p- and s-waves, reflected light upon a material is polarized.
When the device is used in photography, the filter can be rotated
to reduce unwanted reflections. We also simulate normalized
transmittance through the circular polarizing filter in the reserve
direction. In this direction, the input light is either right- or left-
circularly polarized (Fig. 12). The simulation shows that the
device transmits right-circularly polarized light and blocks left-
circularly polarized light in the visible. This function is utilized in
3D eyeglasses. As the wavelength increases, the device no longer
filters circular polarization.
V. SUMMARY AND CONCLUSIONS
MM spectroscopic ellipsometry was used to measure three
different optical systems: a quarter-wave plate, a linear polarizer,
and a circular polarizing filter. We demonstrated an ellipsometric
data analysis strategy for a complete description of the samples.
The optical constants of each layer were measured in terms of the
complex refractive index over a wide spectral range from visible to
near-infrared. Anisotropic modeling provided the fast axis
FIG. 11. Simulated detected intensity (%) vs rotation angle (deg) for unpolarized
and linearly polarized light through a circular polarizing filter in the forward direction.
FIG. 12. Simulated normalized transmittance vs wavelength (nm) for right- and
left-circularly polarized light through a circular polarizing filter in the reverse
direction.
ARTICLE avs.scitation.org/journal/jvb
J. Vac. Sci. Technol. B 38(1) Jan/Feb 2020; doi: 10.1116/1.5129691 38, 014012-9
Published under license by AVS.
orientation of the quarter-wave plate, the transmission axis of the
linear retarder, and the included angle of the circularly polarized
filter. The optical response of the circular polarizing filter was
dependent on the direction of light propagation. We simulated the
transmission intensities in both directions with respect to various
polarization states. The MM ellipsometry data analysis method
developed in this report can potentially be used for nondestructive
quality control of these optical devices.
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Published under license by AVS.
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