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To be published in Applied Optics:
© 2021 Optical Society of America
Title: A Mueller matrix ellipsometer based on discrete-angle rotating Fresnel rhomb
compensators
Authors: Subiao Bian,Changcai Cui,Oriol Arteaga
Accepted: 12 May 21
Posted 14 May 21
DOI: https://doi.org/10.1364/AO.425899
Research Article Journal of the Optical Society of America A 1
A Mueller matrix ellipsometer based on discrete-angle
rotating Fresnel rhomb compensators
SUB IAO BIAN1,2,3, CHANGCAI CU I1,2 ,AND ORIOL ARTEAG A3,4,5
1Institute of Manufacturing Engineering, Huaqiao University, Xiamen 361021, China
2National and Local Joint Engineering Research Center for Intelligent Manufacturing Technology of Brittle Material Products, Huaqiao University, Xiamen
361021, China
3Dep. Física Aplicada, Feman Group, Universitat de Barcelona, Barcelona 08028, Spain
4Institute of Nanoscience and Nanotechnology (IN2UB), Universitat de Barcelona, Barcelona 08028, Spain
5e-mail:oarteaga@ub.edu
Compiled May 13, 2021
A spectroscopic Mueller matrix ellipsometer based on two rotating Fresnel rhomb compensators with a
nearly achromatic response and optimal retardance is described. In this instrument, the compensators
rotate in a discrete manner instead of being continuously rotating and this allows for a well-conditioned
measurement even for low intensity samples. Moreover, in this configuration, the exposure time of the
CCD detector can be varied within orders of magnitude without interfering with the dynamics of the com-
pensator rotation. An optimization algorithm determines the optimal set of discrete angles that allow the
determination of the Mueller matrix in presence of noise. The calibration of the instrument is discussed
and examples of experimentally determined Mueller matrices are provided. © 2021 Optical Society of America
http://dx.doi.org/10.1364/ao.XX.XXXXXX
1. INTRODUCTION
Mueller matrix (MM) ellipsometry determines the 4
×
4 Mueller
matrix of a sample which is the most general descriptor of the
interaction of a linear medium when it reflects polarized light.
MM ellipsometry is a generalization of standard ellipsometry
which is focused on the particular but highly relevant situation
where the MM to be measured depends on only 2 independent
parameters (for historical reasons often presented as the ellipso-
metric angles
Ψ
and
∆
) and they can be obtained from measuring
two MM elements.
A MM ellipsometer contains a polarization state generator
(PSG) and a polarization state analyzer (PSA) in the path of its
excitation and detection light beams. The roles of the PSG and
PSA are, respectively, to produce and analyze a polarized light,
which can be formally described by a four-component Stokes
vector. For a complete MM measurement, the PSG and PSA
must be complete, in the sense that they must be able to pro-
duce or analyze all four components of the Stokes vector. Many
different MM ellipsometers have been described depending on
their design and optical components (most usually the type of
compensator) used in the PSG and PSA. To mention a few: those
based on the dual rotating compensators [
1
,
2
], those based on
photoelastic modulators [
3
,
4
], those based on liquid crystals
[5,6], and those based on a division of amplitude [7,8].
Among all the possible designs for MM ellipsometers, possi-
bly the most popular implementation is the dual rotating com-
pensator instrument, which historically was also the first one to
be reported [
9
] and that allows for a relatively simple and com-
pact optical system. In the most common optical arrangement,
the two compensators are continuously rotated maintaining a
special speed ratio, and all the 16 elements of the sample’s MM
are related to Fourier coefficients of the detected signal.
The optimization of polarimeters based on rotating compen-
sators has been discussed by many authors [
10
–
20
] and their
results show that there are two critical parameters to consider:
the retardance and the angular increments of the two compen-
sators. For good measurements, optimal values of these pa-
rameters need to be used in order to enhance the stability of
the measurements in the presence of error sources. For spec-
troscopic ellipsometry an added difficulty is that compensators
tend to have a highly chromatic response; for example, the retar-
dation introduced by waveplates based on single crystal plates
approximately has a 1
/λ
dependence. Composite waveplates
consisting of two or more single-waveplates can be designed
as a achromatic or a superachromatic compensator [
21
], but
they tend to be expensive and very difficult to fabricate if a
wide spectral range needs to be covered. Instead of using wave-
plates based on anisotropic crystals, a retarder based on the total
internal reflection phenomenon can be adopted , in a design
that is long-known as a Fresnel rhomb [
22
], and that offers a
much more achromatic retardation response than conventional
waveplates. Compared to other types of compensators, only
few works [
11
,
23
] have reported the use of Fresnel rhombs for
Research Article Journal of the Optical Society of America A 2
ellipsometry or polarimetry systems.
In most of the published works describing MM instruments
with rotating compensators [
2
,
15
,
24
,
25
], the compensators are
continuously rotating at speed ratios of 5:1 or 5:3, which are
two of the ratios that are found to be optimal. In these systems,
the sampled points are taken at evenly spaced angular steps
that depend on the rotation speed of the compensators. The
exposure time of the detector is also constrained by this rotation
speed as it need to be much lower than the period of rotation.
As we will demonstrate, a higher degree of freedom for the opti-
mization is possible if angular constraints are relaxed and one
can freely choose the collection of angles used for measurement.
Because the complete MM has 16 real-valued elements, at least
16 individual measurements are required for a complete MM
measurement; however, usually the number of measurements is
set to higher than 16 to reduce the effect of noise on MM.
In this paper, we describe a MM ellipsometer based on dual
discrete-angle Fresnel rhomb compensators. The Fresnel rhombs
we use have a nearly achromatic retardance over a wide spectral
range (from UV to near IR) and with a value close to the optimal
one. In our system, the compensators rotate in a discrete manner
to an advantageous collection of angles instead of using evenly
distributed angles given by continuous rotation pattern. In this
work we present the experimental implementation of this new
ellipsometer, discussing its optimization and comparing it with
continuously rotating systems. A calibration method as well as
examples of experimental measurements are also provided.
2. WORKING PRINCIPLE
A. General description of the dual rotating compensator tech-
nique
The Stokes-Mueller calculus can be used to represent the polar-
ization effects of a train of optical elements. The polarization of
light incoming on the sample is generated by the polarization
state generator (PSG) composed of a fixed polarizer and a ro-
tating compensator. After interacting with the sample, light is
analyzed by polarization state analyzer (PSA) also composed
of a rotating compensator and a fixed polarizer. After passing
through the PSA the light beam is guided to the detector.
The MM of a rotating compensator, MC, is
MC(θ,δ) =
1 0 0 0
0C2
2θ+S2
2θCδC2θS2θ(1−Cδ)−S2θSδ
0C2θS2θ(1−Cδ)S2
2θ+C2
2θCδC2θSδ
0S2θSδ−C2θSδCδ
,
(1)
where we have employed the short notation
SX≡sin(X),(2a)
CX≡cos(X),(2b)
where
δ
is the retardation introduced by compensator, which
most generally is function of wavelength, and
θ
is the orientation
angle of compensator which changes during the measurement
due to the rotating setup. In a continuous rotation configuration,
the time dependence is typically expressed as
θ(t) = ωt+φ,(3)
where
ω
is the angular speed, and
φ
is the phase, i.e., the ori-
entation of the compensator at
t=
0. For this configuration,
the sampling points typically are evenly distributed across the
complete measuring cycle. However, our system uses a discrete
angle rotation configuration: we use a collection of angles
θi
that are not evenly distributed, nor following a cyclical order.
Therefore Eq. (3) is not applicable and it must be replaced by
θ(i) = θi+φ,(4)
where now
φ
must be understood as an offset angle that indi-
cates the orientation (with respect to the instrument’s coordinate
system, which most usually is defined by the orientation of the
polarizers) of the compensator when
θi=
0. The collection of
angles θiused in our measurements will be discussed below.
The MM of the sample to be measured is given by
MS=
m00 m01 m02 m03
m10 m11 m12 m13
m20 m21 m22 m23
m30 m31 m32 m33
.(5)
The measurement process can be described by a matrix prod-
uct representing the sequence of optical elements in the instru-
ment. Light entering the PSG, with Stokes vector
Sin
, passes
first through a polarizer,
P0
, and then goes through the first
rotating compensator,
MC0
. After being transmitted through
or reflected from the sample,
MS
, it enters the PSA, in which
it goes through the second rotating compensator,
MC1
, and an
analyzer,
P1
. The Stokes vector at the detector,
Sout
, is given by
the following matrix multiplication:
Sout =P1MC1MSMC0P0Sin ,(6)
where, without loss of generality,
Sin
can be considered as unpo-
larized light. In our setup, the polarizers
P0
and
P1
are oriented
at 90
°
to each other because this crossed arrangement can be
precisely determined from the minimization of the detected in-
tensity during the instrument construction.
The first element of the Stokes vector in Eq. (6) gives the
intensity
I
measured at each wavelength, which is a function of
all sixteen MM elements in Eq. (5). It is useful to write this result
Research Article Journal of the Optical Society of America A 3
as the scalar product of two vectors [25]:
I=
1
C2
2θ0+Cδ0S2
2θ0
C2θ0S2θ0(1−Cδ0)
Sδ0S2θ0
−(C2
2θ1+Cδ1S2
2θ1)
−(C2
2θ0+Cδ0S2
2θ0)(C2
2θ1+Cδ1S2
2θ1)
−C2θ0S2θ0(1−Cδ0)(C2
2θ1+Cδ1S2
2θ1)
−(Sδ0S2θ0)(C2
2θ1+Cδ1S2
2θ1)
−C2θ1S2θ1(1−Cδ1)
−(C2
2θ0+Cδ0S2
2θ0)[C2θ1S2θ1(1−Cδ1)]
−[C2θ0S2θ0(1−Cδ0)][C2θ1S2θ1(1−Cδ1)]
−Sδ0S2θ0[C2θ1S2θ1(1−Cδ1)]
Sδ1S2θ1
Sδ1S2θ1[C2
2θ0+Cδ0S2
2θ0]
Sδ1S2θ1[C2θ0S2θ0(1−Cδ0)]
Sδ0S2θ0Sδ1S2θ1
T
m00
m01
m02
m03
m10
m11
m12
m13
m20
m21
m22
m23
m30
m31
m32
m33
,(7)
where the superscript Tdenotes the transpose.
For a collection of
N
intensity measurements made at dif-
ferent angles (
θ0
and
θ1
) of the compensators, Eq. (7) can be
rewritten as
Ij=BT
j,kAk,(8)
with
j=
1,
. . .
,
N
and
k=
1,
. . .
, 16. Using matrix notation it can
be rewritten as
I=BTA,(9)
where
I
is a vector of
N
components,
A
is a 16-element vector
containing the Mueller matrix elements and
B
is a matrix with
dimension N×16.
If exactly 16 intensity measurements are made (
N=
16),
BT
is a square matrix and by inverting Eq. (9) one can obtain
A
. However this matrix inverse needs to be computed with
good accuracy, and a good conditioning of the matrix
BT
im-
poses strict requirements for the angular orientation of compen-
sators that are rather difficult to realize in continuous rotation
configuration as discussed by Smith [
10
]. On most occasions,
more than 16 measurements are made to overspecify the calcu-
lation and enhance the stability of solution; and in such case,
the Moore–Penrose pseudo-inverse matrix of
BT
can be used to
extract A:
A= (BBT)−1BI =B+I,(10)
where
BBT
is a matrix of dimension 16
×
16, and
B+=
(BBT)−1B
is the pseudo-inverse of
BT
. This calculation allows
the determination of all the MM elements given in
A
from the
Nmeasured intensities given in I.
B. Continuous vs discrete rotations
As discussed in previous sections, it is possible to consider two
basic working modes for rotating compensator systems: con-
tinuous and discrete rotation. In continuous rotation, there are
constant angular steps between consecutive acquisitions of the
intensity. Usually, this mode is implemented by rotating the
motors at a constant speed ratio between the PSG and PSA. The
most commonly used speed ratio is 5:1 (meaning that the PSG
rotates 5 times faster than the PSA or vice-versa) because this
leads to a well-conditioned measurement, but there are other
suitable ratios [25].
In a discrete rotation mode, which is the configuration used
in our ellipsometer, there are no constraints in the angular steps
between each consecutive intensity acquisition, meaning that
the angular increments can be chosen freely. This offers an
additional degree of freedom for the system optimization that is
not available in continuously rotating systems.
The rotation angles of the PSG and the PSA need to be chosen
wisely so that they lead to a well-conditioned measurement,
meaning that it has an optimal resilience towards noise and
errors. The most obvious approach is to find the set of angles
that maximize the determinant of
BBT
, so that the calculation
of its matrix inverse, appearing in Eq. (10), is well-conditioned.
In addition to the determinant, one can also study the condition
number of
BT
, which measures how close this matrix is to be a
singular matrix. It is defined :
cond(BT) = kBTkkB+k,(11)
where the notation
k k
denotes the matrix norm. We have
used the 2-norm (which is based on the largest singular value)
as the definition of the matrix norm. The determinant of
BBT
will generally become larger when increasing the number of
angles (number of sampled points), but the condition number
has a limited smallest value of 1, which would be indicative of
perfect conditioning. In practice, however, for a Mueller matrix
measurement based on dual rotating compensators it has been
found that the best possible condition number is around 3 [
17
].
Apart from this practical difference, it has been shown that both
metrics (minimization of the condition number or maximization
of the determinant) are essentially equivalent when designing
an optimization strategy in polarimetry [26].
In order to find the list of angles that offer the best condi-
tion number for our system, we set up an iterative optimization
algorithm using the condition number as a merit function. An-
gles for the PSG and PSA were allowed to vary freely between
0
°
and 180
°
. The results of this optimization are shown in Fig.
1, as a function of the number of sampling points (
N
) which
were varied between 16 (the minimum number possible to deter-
mine the complete Mueller matrix) and 100. The results of our
discrete angle optimization are compared with those of the con-
tinuous rotation when using angular steps of 25
°
:5
°
and 15
°
:3
°
.
We include as supporting data files the list of discrete angles for
the PSG [
27
] and PSA [
28
] that generate the condition numbers
shown in Fig. 1. Note that the list of optimized angles is not
unique, as it is possible to find different sets of angles having an
essentially equivalent performance [10].
Results shown in Fig. 1indicate that when the number of
sampling points is small (e.g.
N<
35) the discrete set of an-
gles has a much better condition number than the continuous
rotation method. Moreover, the condition numbers of the dis-
crete sets are almost flatly distributed for all values of
N
. For a
larger number of sampling points (e.g.
N>
60), continuous and
discrete approaches offer almost comparable results, indicating
that both angular sets can be well-suited for measurements, but
the discrete set always allows for slightly better optimization.
Regardless of the rotation method used, increasing
N
as well as
performing spectrum averaging will be beneficial in most occa-
sions, as it will improve the performance in the presence of noise
at the expense of increasing the duration of the measurement.
Research Article Journal of the Optical Society of America A 4
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
2
3
4
5
6
7
8
9
10
Condition number
N (# sampling intensities)
discrete rotation
continuous rotation (25:5)
continuous rotation (15:3)
Fig. 1.
Comparison between the condition numbers found for
discrete and continuous rotation as a function of the number
(
N
) of sampled intensities. For continuous rotation two differ-
ent cases with a 5:1 speed ratio are included: angle increments
of 25
°
(PSG):5
°
(PSA) and 15
°
(PSG):3
°
(PSA). See as support-
ing data files the underlying list of discrete angles of the PSG
(Data File 1) and PSA (Data File 2) that give these condition
numbers.
In our discrete-angle ellipsometer, we typically use values
of
N
between 20 and 60, and
N=
40 is our most common con-
figuration. With this number of acquired intensities, a typical
spectroscopic MM measurement takes less than 40s. In a con-
tinuously rotating MM ellipsometer, as motors usually rotate at
a relatively fast speed, the number of sampled point
N
tends
to considerably larger than 100 but, at the expense that, little or
no spectrum averaging is possible because the sampling time
needs to be much smaller than the rotation period of the mo-
tors. Speed-wise, a continuously rotating system might be able
to offer overall faster measurements that our system because
it collects intensity data at all times. However, in our discrete
system, as the motors are not moving while the spectrum is be-
ing measured, one can effectively increase the sampling time as
much as desired. A long sampling time permits a large number
of spectrum averages or using a long exposure time. This last
case is especially beneficial when examining samples with low
reflectance or transmittance that might not be measurable in a
continuously rotating system. An additional advantage of our
discrete rotation system is that the measurement is, to a large
extend, unaffected by mechanical vibrations as motors do not
move during data collection for each angle of the PSG and PSA.
3. INSTRUMENT DESCRIPTION
The ellipsometer has a design that allows the sample to be
mounted horizontally. The angle of incidence (AOI) can be
adjusted in a very wide interval range, from
∼
20
◦
up to trans-
mission configuration (AOI=90
◦
). The PSG and PSA arms are
mounted on motorized rotators (Newport) that permit adjusting
the angle of incidence with an accuracy better than
∼
0.2
◦
. The
PSG and PSA include a polarizer (Glan-Taylor calcite polarizing
prism) and a compensator (a Fresnel rhomb, described in detail
in the next section). Each compensator is mounted on a lens tube
specific for Fresnel rhombs (Thorlabs) to minimize any applied
stress on the rhombs. The mounted Fresnel rhombs attached
to resonant piezoelectric motors (Thorlabs ELL14) which are
not designed for continuous operation but for discrete angle
positioning. A scheme of the basic optical design and photos of
the instrument are shown in Fig. 2.
The light source is a deuterium halogen lamp (Ocean Optics
DH2000) which is connected to a reflective collimator (Thorlabs)
by an optical fiber. Collimated light is sent through the PSG-
Sample-PSA until another reflective collimator is used to focus
the light onto a second optical fiber that is connected to a CCD
spectrometer (Avantes AvaSpec-ULS2048XL-EVO) that collects
the intensity of all wavelengths at the same time. The spectral re-
gion that the instrument covers expands from 240nm to 1150nm.
The low wavelength limit is established by the calcite polarizing
prisms, that do not transmit light in the far UV, while the high
wavelength limit is given by the detection range of the CCD
spectrometer.
Fig. 2.
Photos (a), (b) and scheme (c) of the MM ellipsometer
based on discrete-angle rotating Fresnel rhomb compensators.
The optical configuration is P1MC1MSMC0P0.
A. The Fresnel rhomb achromatic compensator
Spectroscopic MM ellipsometers based on a dual rotating com-
pensator are benefited by the use of achromatic compensators
offering an optimal value of the retardance. For a complete
MM measurement this value is around
∼
132
◦
for both compen-
sators, as it has been found independently by various authors
[15,16,29].
A Fresnel rhomb is the most achromatic design of a compen-
sator available, and thus it is well suited for MM ellipsometry
on wide spectral range. Commercially available Fresnel rhombs
most usually introduce a retardation of 90
◦
(quarter-wave) or
180
◦
(half-wave). A quarter-wave compensator is compatible
with full MM measurement, but this retardation is somewhat
Research Article Journal of the Optical Society of America A 5
far from the optimum value of
∼
132
°
. Therefore, we custom
designed the Fresnel rhomb compensator used in our system
to adjust the retardation to this optimal value instead of relying
on commercial quarter-wave rhombs. The customized design
slightly increases the cost and lead time with respect to using
off the shelf retarders, but it allows for an optimal polarimetric
performance over the whole spectral range.
We use a “V”-shaped biprism design with the two prisms dis-
posed symmetrically and optically contacted. The prisms have
been built in fused silica because this material is transparent
over a wide spectral range in the UV, visible, and near IR. This
“V”-shaped Fresnel rhomb induces four total internal reflections
and prevents the beam emerging from the biprism to displace
from its original direction. The total retardance introduced by
such biprism in contact with the air is given by [30]
δ=8 tan−1 cos φ(n2sin2φ−1)1/2
nsin2φ!,(12)
where
n
is the refractive index of fused silica and
φ
in the angle
of incidence in each internal reflection (around 66
◦
in our design,
as indicated in Fig. 3). This retardation is highly achromatic, but
as the refractive index
n
depends slightly on the wavelength,
the calibration still needs to take care of a small but unavoidable
variation of the retardation with the wavelength.
Fig. 3. Fresnel rhomb compensator used.
4. CALIBRATION
The calibration is performed in the straight-through configura-
tion without any sample (or “air sample”). The detected inten-
sity in this calibration configuration is
I=1−(C2
2θ0+Cδ0S2
2θ0)(C2
2θ1+Cδ1S2
2θ1)
−C2θ0C2θ1S2θ0S2θ1(1−Cδ0)(1−Cδ1) + Sδ0Sδ1S2θ0S2θ1.
(13)
In a calibrated system, this intensity, when analyzed according
to Eq. (10), should lead to an identity Mueller matrix. Therefore,
a systematic failure in obtaining an identity matrix is a clear
symptom of a miscalibrated system.
There are four parameters that play a major role in the cali-
bration of this instrument: the offset angle for each compensator,
φ0
and
φ1
, and the retardances,
δ0
and
δ1
. As discussed before,
the retardation of Fresnel rhomb compensators still has some de-
pendence on the wavelength (i.e. we consider
δ0(λ)
and
δ1(λ)
)
that needs to be determined. The calibrated values of all these
parameters can be obtained by studying how their uncertainty
propagates on the measured Mueller matrix elements. Let
δ0
0
,
δ0
1
,
φ0
0
,
φ0
1
be the calibrated parameters that relate to the uncalibrated
ones by
δ0=δ0
0+∆δ0,δ1=δ0
1+∆δ1,(14a)
φ0=φ0
0+∆φ0,φ1=φ0
1+∆φ1,(14b)
where
∆i
are the calibration errors that can be generally consid-
ered to be small (meaning that
cos(∆i)'
1 and
sin(∆i)'∆i
)
because of the prior knowledge we have of the design and ori-
entation of our compensators.
By substituting Eq. (14) in Eq. (13) it is possible to study how
the errors
∆i
affect the MM measured in the calibration config-
uration. Because the errors are small, one can neglect terms
that are of second or higher order with ∆i. With this procedure,
one finds the following linear relations between certain MM
elements and the errors:
m01 'Sδ1∆δ1,(15a)
m10 'Sδ0∆δ0,(15b)
m02 '(1+Cδ1)∆φ1,(15c)
m20 ' −(1+Cδ0)∆φ0,(15d)
m12 '2(∆φ1−∆φ0),(15e)
m21 ' −2(∆φ1−∆φ0).(15f)
Given these linear relationships, it is straightforward to set two
algorithms (one for the angular offsets and another for the re-
tardances) that minimize these particular MM elements to find
the calibrated parameters
δ0
0
,
δ0
1
,
φ0
0
,
φ0
1
. For a collection of in-
tensity measurements made in the calibration configuration, the
algorithms iteratively calculate the Mueller matrix (according to
Eq. (10)) assuming different values of the calibration parameters
from a given starting value until they find a minimum for the
corresponding matrix elements given in Eq. (15).
The algorithm that determines the angular offsets uses all
the spectral data simultaneously, as the angles
φ0
0
,
φ0
1
should be
wavelength-independent, while the algorithm that determines
the retardances δ0
0,δ0
1is executed wavelength-by-wavelength.
Fig. 4.
Retardations of the two Fresnel rhomb compensators as
determined from calibration. The fits provide the retardation
values used for the basic calculus in Eq. (10)
Fig. 4shows the results of our wavelength-by-wavelength
retardance calibration routine. The spectral values of
δ0
0(λ)
and
δ0
1(λ)
are respectively found from minimizing
m10
and
m01
, as
suggested by Eq. (15b) and Eq. (15a). The values of
δ0
0(λ)
and
δ0
1(λ)obtained from calibration can be fit by a Sellmeier disper-
sion function, as shown in Fig. 4. A retardation of 131
◦
was used
as initial guess for the fits.
Research Article Journal of the Optical Society of America A 6
Fig. 5.
Mean values (left) and standard deviations (right) of the normalized MM elements for two normal-incidence transmission
measurements: air (no sample) shown by a solid blue line, and a nickel sulphate hexahydrate crystal shown by a dashed red line.
5. EXAMPLES
The measurement process of this MM ellipsometer is mainly
controlled by three parameters: the exposure time of the CCD
detector in the spectrometer, the number of averaged spectra
(i.e. the number of spectral acquisitions that are collected and
averaged at every angle to reduce noise), and the number of
sampling intensities
N
. Thanks to the discrete angle approach,
the exposure time and the number of spectrum averages are not
limited by the dynamics of the acquisition in case continuously
rotating motors were used. Hence, if necessary, one can arbi-
trarily increase the exposure time and the number of spectrum
averages without any affectation in the motor rotation. The only
limitation usually is the intensity saturation of the CCD, which
needs to be avoided. The possibility of a free adjustment of
the exposure time and the number of spectrum averages can
be especially useful when the reflectance or the transmittance
of the sample under study is very low. Another form of noise
reduction in our instrument is the averaging between complete,
consecutive measurements which also allows the determination
of the standard deviations of each MM elements.
Fig. 5shows the mean values and standard deviations (la-
beled as errors) of two transmission normal incidence measure-
ments. They were made with
N=
50 and each measurement
was repeated 4 times in order to calculate the standard devia-
tions. The first measurement was made without any sample
(or an “air sample”), which is the same configuration used for
calibration and that should produce a diagonal MM matrix. The
second measurement corresponds to a single crystal of nickel sul-
phate hexahydrate (
α
NiSO
4
.6H
2
O or NSH). NSH forms tetrag-
onal uniaxial crystals belonging to the enantiomorphous space
groups P4
3
2
1
2 and P4
1
2
1
2 that show circular dichroism (seen by
the nonvanishing and equal values of
m03
and
m30
) and circular
birefringence (seen by
m12 ' −m21 6=
0
)
in the visible and near
infrared. This data was collected from a crystal of NSH with a
thickness of
∼
0.35 mm and cleaved along (001). The standard
deviations are in most of the spectral range around or below
0.005, only having significant increases above 1000 nm, where
the efficiency of our CCD detector drops or in the strong ab-
sorption band of NSH around 400nm. From the figure, it is also
evident that some elements (particularly
m02
,
m12
and
m21
) sys-
tematically show larger standard deviations than the rest of MM
elements. According to the error propagation given in Eq. (15),
it seems that these larger standard deviations are most likely
produced by some repeatability problems in the angular posi-
tioning of the piezoelectric motor used in the PSA, which makes
the offset angle
φ1
not being sharply defined. In the future, we
will check if replacing this motor with a new unit improves the
standard deviations of these MM elements.
Fig. 6.
Measured MM (dashed blue line) and fit (solid red line)
for a Si wafer illuminated at an angle of incidence of 60◦.
Fig. 6shows the normalized spectroscopic MM of a <100> Si
wafer measured at an angle of incidence (AOI) of
∼
60
◦
. The well-
known block-diagonal matrix symmetry, typical of ellipsometry
measurements in isotropic materials, such as cubic silicon, is
measured [
31
]. Silicon is an indirect gap semiconductor with
well-known optical functions and it is usually used as reference
material for ellipsometry measurements. A fit of the experi-
mental data using the known values of the optical functions of
silicon [
32
], varying only the thickness of the oxide overlayer
and the angle of incidence is included in Fig 6. The fit values
for the elements out from the block-diagonal, shown in the fig-
ure on an expanded scale, are always zero in layered models of
isotropic materials. The fit determined the oxide thickness to be
Research Article Journal of the Optical Society of America A 7
1.81
±
0.03 nm and the angle of incidence to be 60.11
◦±
0.02
◦
,
where the χ2of the fit was 0.67.
Fig. 7.
Optical absorption coefficient
α
of silicon as determined
from direct numerical inversion of ellipsometry data.
Fig. 7shows the spectroscopic absorption coefficient
α
of
silicon as directly determined from numerical inversion of the
ellipsometry measurements (assuming the SiO
2
overlayer thick-
ness determined from the fit). In the considered spectral range,
the absorption coefficient varies from various orders of mag-
nitude and it is a good indicator of how well the ellipsometer
measurements small values in the MM elements m13 and m31.
A final example is included in Fig. 8to illustrate the possibil-
ity that offers the discrete-angle MM ellipsometer of studying
samples providing very low light levels. This measurement
was made on a forest of vertically aligned carbon nanotubes
(VACNT) grown by plasma-enhanced chemical vapor deposi-
tion (see in Fig. 8a a photo of the sample and the details of
the fabrication are given in [
33
]). VACNT consists of carbon
nanotubes oriented with their longitudinal axis perpendicular
to a substrate surface. These structures are considered to be-
have almost like an ideal black-body: its sparse nanostructure
gives an effective refractive index very close to that of air. The
impedance-matched interface allows light to enter the nanotube
forest without significant reflection or scattering, and the con-
ductive nature of the nanotubes causes complete absorption of
the entered light [
34
]. However, the reflectance has a step in-
crease at grazing incidence as it is shown in Fig. 8b. At an AOI
of 60
◦
, the reflectance is well below 1% in most of the studied
spectral range, while at 80
◦
it rises by more than one order of
magnitude. Fig. 8c shows that, despite the very low light levels
at an AOI of 60
◦
, it is still possible to measure the MM for the
specularly reflected light. For this measurement, an integration
time of 500 ms was used in the CCD detector, while 5 ms were
enough for the measurement at 80
◦
. The integration time used
for the measurements shown in Figs. 5and 6varied between 0.5
ms and 3 ms.
6. CONCLUSIONS
We have presented a novel spectroscopic MM ellipsometer built
in our laboratory at the University of Barcelona. The instrument
uses two discretely rotating Fresnel rhomb compensators to
obtain simultaneously the 16 elements of a Mueller matrix. Some
of the features that we highlight from this ellipsometer are:
•
The exposure time of the CCD detector can be varied within
orders of magnitude without altering the dynamics of the
compensator rotation. This makes it ideal for measurements
involving low light levels.
•
The calibration is completely self-consistent and can be run
in the transmission configuration without requiring any
reference sample.
•
A well-conditioned measurement can be obtained for any
number of sampled intensities
N
(
N=
16 is the minimum
possible) and in the whole covered spectral range.
•
A precision better than 0.5% is achieved for all elements
of the Mueller matrix with relatively fast measurements (a
typical spectroscopic measurement lasts less than 1 minute,
with all wavelengths measured at the same time).
•
It has a lower cost than other MM ellipsometers because
affordable piezoelectric motors are used for the rotation of
the compensators.
FUNDING
Ministerio de Ciencia Innovacin y Universidades RYC2018-
024997-I and RTI2018-098410-J-I00 (MCIU/AEI/FEDER, UE).
Opening project of National and Local Joint Engineering Re-
search Center for Intelligent Manufacturing Technology of Brittle
Material Products (2020IME-I001).
ACKNOWLEDGMENTS
Authors thank Dr. S. Hussain for providing the nanotube forest
sample.
DISCLOSURES
The authors declare no conflicts of interest.
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