Estimating depolarization with the Jones matrix
quality factor
James N. Hilfiker, Jeffrey S. Hale, Craig M. Herzinger, Tom Tiwald, Nina Hong,
Stefan Schoche and Hans Arwin
The self-archived postprint version of this journal article is available at Linköping
University Institutional Repository (DiVA):
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-141108
N.B.: When citing this work, cite the original publication.
Hilfiker, J. N., Hale, J. S., Herzinger, C. M., Tiwald, T., Hong, N., Schoche, S., Arwin, H., (2017), Estimating depolarization with the Jones matrix quality factor, Applied Surface Science, 421, 494-499. https://doi.org/10.1016/j.apsusc.2016.08.139
Original publication available at:
https://doi.org/10.1016/j.apsusc.2016.08.139
Copyright: Elsevier
Estimating Depolarization with the Jones Matrix Quality Factor
James N. Hilfiker1,*, Jeffrey S. Hale1, Craig M. Herzinger1, Tom Tiwald1, Nina Hong1, Stefan Schöche1, and Hans Arwin2
1. J.A. Woollam Co. Inc., 645 M Street, Lincoln, NE 68508, USA
2. Department of Physics, Chemistry, and Biology, Linköping University, SE 581 83 Linköping, Sweden *Corresponding author: jhilfiker@jawoollam.com, TEL: +1-402-477-7501
Abstract
Mueller matrix (MM) measurements offer the ability to quantify the depolarization capability of a sample. Depolarization can be estimated using terms such as the depolarization index or the average degree of polarization. However, these calculations require measurement of the complete MM. We propose an alternate depolarization metric, termed the Jones matrix quality factor, QJM, which does not require the complete MM. This metric provides a measure of how close, in a
least-squares sense, a Jones matrix can be found to the measured Mueller matrix. We demonstrate and compare the use of
QJM to other traditional calculations of depolarization for both isotropic and anisotropic depolarizing samples; including
non-uniform coatings, anisotropic crystal substrates, and beetle cuticles that exhibit both depolarization and circular diattenuation.
1. Introduction
Spectroscopic ellipsometry measurements use polarized light to characterize thin films and bulk materials. During the measurement, the polarized measurement beam may transform into partially polarized light. This reduction in the degree of polarization for the measurement beam is referred to as depolarization. Depolarization is a feature of the sample or measurement caused by non-uniform or patterned films, finite bandwidth, angular beam spread, scattering, or a collection of multiple, incoherent beams such as from front and back of a thick substrate [1]. Correctly modelling these effects can improve accuracy of thin film characterization.
Mueller matrix (MM) measurements offer a complete description of the polarization-transformation of a sample or optic, including its depolarization capability. There are nine degrees of freedom within the MM associated with depolarization. Unfortunately, it is difficult to visualize whether a MM is depolarizing simply from examining its elements. To help, various single-valued metrics have been developed to estimate the depolarization capability from the full MM [2, 3].
We propose an alternate depolarization metric, termed the Jones matrix quality factor, QJM. This term provides a measure
of how close a best-fit Jones matrix is to the measured Mueller matrix. Since the Jones matrix is intrinsically non-depolarizing, the difference between the best-fit Jones matrix and the measured Mueller matrix is a figure of merit for the amount of depolarization. A key advantage of QJM is the ability to estimate depolarization even from an incomplete MM.
2. Theoretical background
For ellipsometry measurements of isotropic samples, it is common to estimate depolarization as [4]:
(
)
[
2 2 2]
1
100
%
Depol
=
−
N
+
C
+
S
(1)where N, C, and S are elements of the normalized Mueller matrix defined as:
−
−
−
=
C
S
S
C
N
N
isotropic0
0
0
0
0
0
1
0
0
1
M
(2)The value of %Depol ranges from 0% for non-depolarizing samples to 100% for a completely depolarizing ellipsometry measurement. While only three elements of the normalized MM are required for this calculation, it is only valid for isotropic
samples. Depolarization can be estimated for anisotropic, cross-polarizing samples, when the complete MM is measured, using terms such as the quadratic depolarization index, PD, [2]:
2 11 2 11 2
3m
m
m
P
ij ij D∑
−
=
(3)The value of PD ranges from 0 for completely depolarizing samples to 1 for non-depolarizing samples.
Here we describe an alternate quantity called the Jones matrix quality factor, QJM. This term provides a measure of how
close a best-fit Jones matrix is to the measured Mueller matrix. The calculation finds a normalized Jones matrix, J, that best corresponds to the measured normalized Mueller matrix, M, even if M does not correspond exactly to a Jones matrix. The conversion between Jones and Mueller matrices can be found in the review by Chipman [5]. To evaluate the closeness of J to M, the corresponding normalized Mueller matrix for J (referred to here as N) is calculated by minimizing the difference between M and N, and we define:
(
)
∑
−
−
=
ij ij ij fit JMm
n
x
x
Q
2 exp1
(4) with1
11 11= n
≡
m
(5)Here, xexp is the number of measured MM values and xfit is the number of real-valued Jones matrix parameters that were
adjusted (real and imaginary parts count as separate fit parameters). All ellipsometer configurations we will discuss incorporate at least one compensator and xfit = 6. Ellipsometers without at least one compensating element, such as rotating
polarizer and rotating analyzer configurations, collect fewer MM elements and are incapable of characterizing depolarization. If the measured MM is perfectly matched by an equivalent Jones matrix, then QJM = 0 and the MM does not depolarize. In fact, when QJM = 0 the MM is both non-depolarizing and physically realizable since it is matched perfectly by an equivalent Jones matrix. The presence of depolarization is indicated by QJM > 0. It should be noted that QJM is not an
additional measurement parameter, but simply an indication of the depolarizing capability from the measured MM parameters. All information is contained within the MM parameters themselves.
We show MM measurements or simulated results for three instrument types: dual-rotating compensator ellipsometers (dual-RCE), polarizer-compensator-sample-analyzer (PCSA) and polarizer-sample-compensator-analyzer (PSCA) ellipsometers [6-8]. Equations 6-8 show the normalized MM elements that can be measured by each ellipsometer configuration.
=
− 44 43 42 41 34 33 32 31 24 23 22 21 14 13 121
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
RCE DualM
(6)
⋅
⋅
⋅
⋅
=
34 33 32 31 24 23 22 21 14 13 121
m
m
m
m
m
m
m
m
m
m
m
PCSAM
(7)
⋅
⋅
⋅
⋅
=
43 42 41 33 32 31 23 22 21 13 121
m
m
m
m
m
m
m
m
m
m
m
PSCAM
(8)While not specifically addressed here, similar results can be found for phase modulation ellipsometers, based entirely on the total number of MM elements measured, which depends on the number of phase modulators used and the optical configuration during measurement. Thus, the QJM calculation is related to the number of MM elements that are measured, and not specific to the ellipsometer technology that allows measurement of these elements.
As defined by Eqn. 4, the maximum range for QJM depends on the number of measured normalized MM elements. For
PCSA and PSCA systems, xexp = 11 and a fully depolarizing MM (a = 0 in Eqn. 9) results in a maximum value of QJM =
5
1
~ 0.447. When the full MM is measured, xexp = 15 and the maximum value of QJM =1
3
~ 0.577 for fullydepolarizing MM.
Figure 1 compares values of QJM and 1-PD for a uniform, partial depolarizer, represented as [9]:
a
a
a
0
0
0
0
0
0
0
0
0
0
0
0
1
(9) When the full MM is measured (dual-RCE), 1-PD has the same shape as QJM although with different scaling. QJM is alsocalculated for PCSA and PSCA configurations, which result in a different shape and range.
Fig. 1. Comparison of metrics for a uniform, partial depolarizer where a (Eqn. 9) defines the diagonal MM elements. The QJM metric can be rescaled to values between 0 and 1, but specific to certain ellipsometer configurations. For
example, QJM16 can be formulated to scale from 0 to 1 for measurements of the complete MM from dual-RCE instruments,
with:
JM
JM
Q
Q
16= 3
⋅
(10)3. Experimental
A dual-RCE instrument (Woollam RC2®) was used to measure the complete MM for energies from 0.73 eV to 6.46 eV (192 nm to 1690 nm). This instrument collects light on a silicon CCD for energies greater than 1.24 eV and an InGaAs linear diode array for energies less than 1.24 eV. The measurements are equally spaced in wavelength with 1 nm and 2.5 nm resolution on the Si CCD and the InGaAs array, respectively.
Infrared (IR) measurements were collected with a PSCA instrument (Woollam IR-VASE®). This instrument uses a Fourier transform infrared (FTIR) source with resolution set to 4 wavenumbers for the measurements. All calculations of
QJM were performed in the commercial software included with the ellipsometer (Woollam CompleteEASE® and WVASE®).
4. Results and discussion
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1
1
-P
DQ
JMa
QJM, Dual-RCE QJM, PCSA QJM, PSCA 1-PD, Dual-RCE4.1 Non-uniform oxide on silicon
An oxide layer deposited on silicon substrate with chemical vapor deposition (CVD) was characterized with MM spectroscopic ellipsometry (MM-SE). The film thickness is non-uniform and the measurement consists of values averaged over the thickness distribution (〈N〉, 〈C〉, and 〈S〉) within the measured spot, with normalized MM form similar to Eqn. 2 [10]:
−
−
−
C
S
S
C
N
N
0
0
0
0
0
0
1
0
0
1
(11)Figure 2 shows the measured MM from this non-uniform CVD oxide for photon energies from 1 to 5 eV at 75° angle of incidence. The film remains isotropic, which is evident from off-diagonal block elements near zero. Note that m22 remains
close to 1 for all measured angles and energies, which is consistent with the nature of the film: light that is linearly polarized along the p- or s- axes will maintain its polarization upon reflection from the sample, regardless of variations in film thickness.
Fig. 2. MM measurement from a CVD oxide film on silicon with non-uniform thickness.
The film is very non-uniform, which at some wavelengths leads to measured averages 〈N〉, 〈C〉, 〈S〉 → 0. This produces the maximum depolarization from such an isotropic sample, given as:
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
(12)Here, the sample conserves p and s polarized light, while totally depolarizing linearly polarized light oriented at ±45°. This condition produces %Depol = 100% per Eqn. 1, and this value is reached at energies near 4.35 and 4.8 eV as seen in Fig. 3. Fig. 3 also shows QJM and 1-PD for the same measured MM. At the point where %Depol = 100%, QJM =
2
3
~0.471 while 1-PD =
1
−
1
3
~ 0.423. Neither represents the maximum amount of depolarization from a MM, which is dueto the fact that incoming light with pure p and s linear polarizations are not depolarized.
Fig. 3. Calculated depolarization metrics from the non-uniform CVD oxide measurement, along with model results that yielded a 12% thickness non-uniformity.
The CVD film was modelled as a transparent layer, using Sellmeier dispersion [11], resulting in a layer thickness of 894.8 nm. A non-uniform thickness calculation was used to match the data and corresponding depolarization. This is accomplished by calculating the MM for a series of models with slightly different thickness, with special weighting for the different thickness values determined from a “convolution” profile shape. To match the measured data and depolarization for this sample required thickness non-uniformity calculations of 12% using a Gaussian distribution of thicknesses around the average thickness. The model results are shown as dashed lines in both Fig. 2 and Fig. 3.
4.2 Sapphire substrate
Full MM measurements in both transmission and reflection were collected from a 0.55 mm thick, a-plane sapphire substrate. The transmission MM measurements were collected from 0° to 75° angle of incidence, while the reflected MM measurements were collected from 20° to 75° and included collection of multiple reflections from within the substrate. Both transmitted and reflected measurements at 75° are shown in Fig. 4. The sample exhibits strong retardance with interference between the ordinary and extraordinary rays traveling through the thick, anisotropic substrate. At oblique angles, there are also Fresnel-like effects at the interfaces, which appear as diattenuation. Finally, there is a small amount of diattenuation within the sapphire optical constants near 6 eV. All of these optical properties combine in various degrees to affect every element of the MM, as evident from Fig. 4. This makes it difficult to examine the MM and determine the existence of depolarization. Thus, the depolarization metrics discussed are very useful.
While many metrics could be applied to the full MM, QJM can also be applied to a partial MM measurement. To test
this, we calculated QJM from the full MM and compare to calculated QJM when excluding the final row to simulate the effect
of a PCSA ellipsometer configuration. These calculations are shown in Fig. 5a and Fig. 5b for the transmitted and reflected MM measurements, respectively.
Fig. 4. MM measurements from an a-plane sapphire in both transmission and reflection at 75°.
Fig. 5. QJM for (a) transmitted and (b) reflected MM measurements of a-plane sapphire substrate.
Two primary sources of depolarization are considered for each measurement: incoherent multiple beams and the finite bandwidth of the detection system. The incoherent multiple beams play a much larger role in the reflected data, where front and back reflected contributions have similar intensities. The finite bandwidth effects were modelled using a Gaussian wavelength distribution with 2.7 nm full-width, half-maximum. These effects become more pronounced at higher energies and are the cause of the increase in the average transmitted QJM as well as the suppression of the oscillations in QJM for the
reflection measurement. Above 5.5 eV, the reflected QJM oscillations are further suppressed by the small absorption and
dichroism of the sapphire substrate.
Reflected MM data from the sapphire substrate were also measured using a PSCA instrument at IR wavelengths (Fig. 6.). Interference features are visible for wavelengths below 6 µm due to multiple incoherent beams traveling through the substrate and recombining in the detected light. Figure 7 shows the extinction coefficient determined with a basis-spline analysis [12] of the sapphire optical constants, which shows the suppression of the backside reflections above 6 µm due to material absorption. Although this PSCA instrument does not measure the last column of the MM, it is still possible to calculate QJM,
as shown in Fig. 7. The depolarization due to multiple coherent reflections within the substrate is suppressed when the backside reflection is no longer detected due to absorption in the substrate.
Fig. 6. IR MM measurement of a-plane sapphire in reflection using a PSCA instrument.
Fig. 7. Calculated QJM for the IR MM measurement shown in Fig. 6.
4.3 Cetonia aurata
MM data from the cuticle of a specimen of the scarab beetle Cetonia aurata (Linneaus, 1782), shown in Fig. 8, show a narrow spectral region with significant circular diattenuation [13, 14]. The optical features are modelled as a twisted anisotropic layer where the total amount of twist is “smeared” to better reproduce the data features and qualitatively match
Fig. 8. Measured MM data from Cetonia aurata at angles of incidence 25º, 40º and 60º. The dotted lines show model calculated MM-data in an electromagnetic model [13].
5. Conclusions
MM measurements can be used to quantify depolarizing capability of a sample. However, it can be difficult to estimate depolarization by visual inspection of the MM. Single-value metrics are used to estimate the depolarization from MM, but rely on measurement of the complete MM (all 15 normalized values). We have introduced a new figure of merit, QJM that
can be calculated from the complete MM or from an incomplete MM, such as measured with PCSA or PSCA instruments. The value of QJM ranges from 0 for non-depolarizing MM to 0.577 for completely depolarizing MM when complete MM is
measured. When QJM ~ 0, Jones matrix measurements are adequate to describe the polarization transformation of the optic or
sample. However, as QJM becomes large the depolarization capability of the sample should be measured and described in order to improve the accuracy of spectroscopic ellipsometry characterization.
We have demonstrated the use of QJM for both isotropic and anisotropic samples with both complete and incomplete MM
measurements. For an isotropic CVD oxide with non-uniform thickness, QJM approaches 0.471 when the %Depol for an
isotropic calculation equals 100%. This is due to the nature of the isotropic sample with non-uniform thickness, which preserves the polarization of linear p- and s- polarizations. Here, QJM describes the depolarization capability of the overall
MM. For anisotropic sapphire substrates, QJM was calculated for both dual-RCE measurements and PSCA infrared
measurements. For this sample, two primary sources of depolarization were identified. Both the multiple incoherent beams traveling through the anisotropic substrate and the finite bandwidth of the detector were modelled to match the measured
QJM. For MM measurements of a beetle cuticle, depolarization results from the smeared “twisting” of the anisotropic optical
axis within the measured surface. The depolarization from this smearing is identified with QJM in the presence of circular
diattenuation resulting from this specimen.
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