Native oxide layer effect on polarization cancellation for mirrors over the visible to near-infrared region

Abstract

The presence of native oxide layers on aluminum mirrors can be a nuisance for precision optical design. As the native oxide thickness varies from mirror to mirror, its effect cannot be completely canceled even in the conventional crossed fold mirror geometry. We show how this effect arises and how it can be mitigated, and provide an experimental demonstration in which the residual linear retardance and linear diattenuation are reduced to <0.14° and <0.001, respectively, over the visible and near-infra-red spectral range.

Full text

Research Article
Suchandra Banerjee*, Russell Chipman, Nathan Hagen and Yukitoshi Otani
Native oxide layer effect on polarization
cancellation for mirrors over the visible to near-
infrared region
https://doi.org/10.1515/aot-2020-0004
Received January 22, 2020; accepted March 10, 2020
Abstract: The presence of native oxide layers on aluminum
mirrors can be a nuisance for precision optical design. As
the native oxide thickness varies from mirror to mirror, its
effect cannot be completely canceled even in the conven-
tional crossed fold mirror geometry. We show how this
effect arises and how it can be mitigated, and provide an
experimental demonstration in which the residual linear
retardance and linear diattenuation are reduced to <0.14°
and <0.001, respectively, over the visible and near-infra-
red spectral range.
Keywords: Mueller matrix polarimeter; oxide layer; polari-
zation; polarization ray tracing.
1 Introduction
Mirrors are considered as ideal reflectors when they reflect
100% of light such that Fresnel amplitude reflection coef-
ficients for s and p polarization rs = rp = 1 for all angles of
incidence. For metal mirrors, however, rs and rp are not
equal. An amplitude and a phase difference between rs
and rp appear as linear diattenuation and linear retard-
ance. These make the reflectance polarization depend-
ent. Breckinridge etal. show that fold mirrors have more
impact on induced polarization than the primary and
secondary mirrors of a Cassegrain telescope have. This
induced polarization varies with the angle of incidence
and the material property of reflecting elements [1, 2].
The induced polarization can be measured with a
polarimeter. Optical instruments like telescopes, spec-
trometers, and imaging systems are combined with
the polarimeters to retrieve polarization information
of various objects. Shamir etal. [3] analyze the effect of
induced polarization in an optical system and show that it
is necessary to eliminate this effect for an accurate polari-
zation measurement.
Clark et al. [4] show how to mitigate the induced
polarization using a spatially varying retardance plate.
However, this method requires a complicated fabrication
process, needs several stacks of birefringent plates, and
is limited to controlling only linear retardance. Mahler
etal. [5] proposed an optical system design where linear
diattenuation is kept below 1% by designing the system’s
mirror coatings.
In an optical system sensitive to polarization effects,
uncapped mirrors are the norm, in order to minimize the
effect of multilayer coatings on the polarization state.
Many laboratory experiments are done over UV to near-
infra-red region, and uncapped aluminum mirrors are
used, which show more than 90% of average reflectance.
Capped mirrors are generally designed for a particular
spectrum band. Lam etal. [6, 7] reported that both of the
linear retardance and linear diattenuation can be bal-
anced simultaneously using a crossed fold mirror config-
uration. Gold mirrors were selected for their simulation,
which showed high performance only in the infra-red
region.
Aluminum coatings have less polarization sensitivity
than silver and gold, and they are common due to their
high reflectivity over UV to near-infra-red wavelength
range and its high thermal resistivity [2, 8]. However,
unlike gold, aluminum reacts with oxygen when exposed
to air and forms a thin native oxide layer on the top of the
surface. The presence of oxide changes the polarization
properties (linear retardance and linear diattenuation)
of the mirror [9]. Harten etal. [10] studied the growth of
*Corresponding author: Suchandra Banerjee, Graduate School of
Engineering, Utsunomiya University, Utsunomiya, Tochigi 321-8585,
Japan, e-mail: banerjee.suchandra@gmail.com
Russell Chipman: College of Optical Sciences, University of Arizona,
Tucson, AZ 85721, USA
Nathan Hagen and Yukitoshi Otani: Department of Optical
Engineering, Center for Optical Research and Education, Utsunomiya
University, Utsunomiya, Tochigi 321-8585, Japan
www.degruyter.com/aot
© 2020 THOSS Media and De Gruyter
Adv. Opt. Techn. 2020; 9(4): 175–181
Published online April 17, 2020

oxide layer after evaporation for a long time and showed
the saturation time of the oxidation of a real aluminum
mirror. Depending on environment conditions, the satura-
tion level of thin native oxide layer may vary from mirror
to mirror [10, 11].
A broadband achromatic tunable polarization rotator
was proposed over visible to near-infra-red spectrum using
three uncapped aluminum mirrors to make the whole
system more cost effective than using capped mirrors [12].
The experimental results confirmed that even after polari-
zation cancellation using crossed fold mirror geometry, a
residual amount of linear retardance was present, which
increased the system’s overall linear retardance.
In this manuscript, we show the impact of native
oxide layer thickness on linear retardance and linear
diattenuation by modeling thin films of Al2O3 on the
top of the aluminum. Native oxide layer thicknesses
are determined by fitting the measured linear retard-
ance spectrum to a Fresnel reflection model. In order
to demonstrate the effect of the native oxide layer on
the polarization, we measure the polarization of a beam
reflected by pairs of crossed fold mirrors and compare
the results with no oxide layers, determined by theory.
We verify the effect of native oxide layers on polariza-
tion cancellation by a Mueller matrix spectro-polarime-
ter over the wavelength range 500–800nm. The extent
of polarization cancellation is shown for different oxide
layer thicknesses.
2 Determination of aluminum
mirror’s polarization properties
in the presence of oxide layers
Figure 1 shows the behavior of s and p polarizations
upon reflection from an ideal reflector, a bare aluminum
reflector, and a real aluminum reflector. Light experi-
ences multiple reflections between the thin oxide and
aluminum layers. A relative phase shift φ occurs between
the air-oxide and the oxide-aluminum interfaces and is
expressed as
10
2
cos
,
nd i
π
φλ
=
(1)
with the oxide layer thickness d for the wavelength of light
λ and refractive index n1 [13] of the oxide layer. The ampli-
tude reflection coefficients for s- and p-polarized light
can be found by coherently summing all of the partially
reflected beams’ reflection coefficients and are written as
01 12
01 12
01 12 01 12
,
,
11
i
i
i
ipp p
ss s
ss
pp
ii
ss pp
rre
rre
rere
rr
er
re
φ
φΦ
Φ
φφ
ρρ
+
+
==
==
++
(2)
where r01 and r12 are the Fresnel reflection coefficients for
medium 0 to 1 and for medium 1 to 2, respectively. The
overall amplitude for the s and p components are denoted
by ρs and ρp, and Φs and Φp are the overall phase parts of
Eq. (2). For aluminum, the fast axis is along the direction
of the s-polarized light. Linear retardance δF and linear
diattenuation DF of the real aluminum mirror are calcu-
lated using amplitude reflection coefficients [Eq. (2)] as
22 22
, (| |||)/(|| ||).
FspF sp sp
DrrrrδΦ
Φ
=− =− +
(3)
Using Eqs. (1–3), it is evident that the polarization
parameters δF and DF are functions of thickness of the
oxide layer d. These are plotted in Figure 2 as a function
of wavelength for varying oxide layer thicknesses at 45°
angle of incidence. The linear retardances in Figure 2
include the geometrical transformation effect, which
occurs due to odd number reflections where a ±π phase
shift is introduced at each reflection. If we remove the
geometrical transformation effect (by subtracting 180°)
in Figure 2, we find that the linear retardance actually
increases with the physical thickness of the oxide layer
for the VNIR wavelength range. The effect of the oxide
layer on linear diattenuation is less apparent but cannot
be ignored for high-accuracy polarization measurements.
The induced polarizations of a one-aluminum fold mirror
can be balanced by orienting another aluminum fold
Figure 1:Behavior of s and p polarization components upon reflection from (A) an ideal reflector, (B) a bare aluminum reflector (aluminum
without native oxide layer), and (C) a real aluminum reflector (aluminum with native oxide layer).
176 S. Banerjee et al.: Native oxide layer eect on mirrors

mirror’s surface normal in such a way that the fast polari-
zation axis of the first mirror is transformed into the slow
polarization axis of the second mirror and vice versa. This
is called the crossed fold mirror (CFM) geometry [6]. Here,
the linear diattenuation of the second fold mirror takes an
opposite sign of the first mirror’s, so that linear diattenua-
tion and linear retardance both cancel out [7].
For the CFM geometry, let us define the propagation
vectors as
123
{0, 0, 1}, {0, 1, 0}, {1, 0, 0}.===kkk
(4)
Incident light propagates along the z direction until the
first mirror M1 and gets reflected along the y direction until
it meets the second mirror M2. After reflection from M2, the
light exits along the x direction. The s, p, and k are mutually
orthogonal to each other, therefore for each mirror surface,
s and p polarization vectors can be found as
,in,out11
,in,in ,i
n,
out
1,out1,out
()/| |,
()/| |,
()/| |,
qq qq
qq
qqqq
qq
qq qq
++
++
==××
=× ×
=× ×
ss kn kn
pk
sksp
ks ks (5)
where
{1,2}q∈
represents the mirror’s number. For
qth mirror, the phase φq is associated with oxide layer
thickness dq. Polarization transformation through
CFM geometry is shown in Figure 3. The direction kq of
ray propagation is determined by the mirror’s surface
normal vector nq as
11
()/| |.
qqqqq++
=− −nkkkk
(6)
Using Eqs. (4–6), it is seen in Table 1 that the fast axis
of the first mirror along s1,in is transformed into the slow
axis of the second mirror along p2,in, and the slow axis of
Figure 2:The effect of oxide layer thickness on polarization parameters: linear retardance and diattenuation. Both δF and DF are plotted as a
function of wavelength in the VNIR region for oxide layer thicknesses from 1 to 6nm.
k1k1
k2
k2
k3
p1in M1
x
y
z
x
y
z
M1
M
2
S1in S2in
p1out
p2in
S2out
p2out
S1out
Figure 3:The s and p polarization transformations through (A) the first mirror M1 along light direction k1 and k2 and (B) the crossed fold
mirror geometry.
Table 1:Crossed fold mirror geometry: mirror surface orientation
(nq), input s-polarization direction (sq,in), and incident and exiting
p-polarization directions (pq,in and pq,out).
nS,in p,in p,out
{,
−1/ 2,
1/ 2
}{−, , } {, −, }{, , }
ns,in p,in p,out
{
−1/ 2,
1/ 2,
}{, , −} {−, , }{, , }
S. Banerjee et al.: Native oxide layer eect on mirrors 177

the first mirror, which is along p1,out, becomes the fast axis
of the second mirror along s2,in.
To quantify the linear retardance and the linear diatten-
uation after polarization cancellation, we use a polarization
ray tracing (PRT) matrix calculus model of the CFM config-
uration [14]. The PRT matrix Pq is a modified form of Jones
matrix in local coordinates {s, p}, projected into the global {s,
p, k} coordinate system. The linear retardance and linear dia-
ttenuation of the CFM configuration cannot be determined
by simply cascading individual Jones matrices of reflecting
surfaces. Hence, the PRT matrix calculation method is used
to determine the polarization behavior for such a system. As
s, p, and k are orthogonal to each other, they form an orthog-
onal matrix Oq. By relating the augmented Jones matrix of
the mirror with the input and output orthogonal matrices
Oq,in and Oq,out, the PRT matrix Pq is formed as
1
,out ,in
1
,1,1
,,,,
,1,1
,,,,
,1,1,,,,
00
00
,
001
qqqq
qx qx qx qx qx qx
s
qy qy qy pqyqyqy
qz qz qz qz qz qz
sp
ks
pk
r
sp krspk
sp
ks
pk
−
−
++
++
++
=






=







POJO
(7)
where sq and pq are given in Eq. (5). The matrix Pq includes
the geometrical transformation effect, which makes
an incident right-handed coordinate system into a left-
handed coordinate system upon reflection. The Qq matrix
does not contain any polarization information and purely
depends on the {s, p, k} coordinate system. By multiplying
the reflection matrix I with the input and output orthogo-
nal matrices (Oq,in and Oq,out), the geometrical transforma-
tion matrix Q for surface number q is
1
,out ,in
1
,1,1
,,,,
,1,1
,,,,
,1,1,,,,
100
01
0.
001
qqq
qx qx qx qx qx qx
qy qy qy qy qy qy
qz qz qz qz qz qz
sp
ks
pk
sp
ks
pk
sp
ks
pk
−
−
++
++
++
=






=−








QOIO
(8)
To eliminate the inversion effect due to an odd
number of reflections, an inverse geometrical transforma-
tion matrix Qq is applied on Pq, and
1
qq
−
QP
is obtained. The
cumulative P and Q matrices for the CFM system (Figure3)
are given by
21 21
, and .==PPPQQQ
(9)
We determine the linear retardance δ and linear diat-
tenuation D of pairs of mirrors in the CFM geometry from a
singular value decomposition of Q−1P, which gives Unitary
and diagonal matrices in the form
∑
UV.
Linear retard-
ance is the difference between fast and slow polarization
axes λf and λs of the Unitary matrix, which is written as
arg[ ]arg[].
fs
δλ
λ
=−
(10)
The diagonal matrix has singular values (tf and ts),
which correspond to the magnitude of reflection coeffi-
cients rs and rp. Thus, D can be formulated as
22
22
.
fs
fs
tt
Dtt
−
=+
(11)
The amount of linear retardance and linear diattenua-
tion [Eqs. (10) and (11)] for the CFM configuration depends
on the oxide layer thicknesses d1 and d2. If the oxide layer
thicknesses are not perfectly equal to each other, there
will be a residual amount of linear retardance and diatten-
uation even after cancellation. This is shown in Figure4
where the polarization cancellation is not perfectly bal-
anced due to having d1≠d2 and leaves a residue.
3 System and methods
For our experiments, we employ a dual rotating retarder-
based spectroscopic Mueller matrix polarimeter (Axo-
metrics, Huntsville, AL, USA) [https://www.axometrics.
com/products/axoscan]. The polarimeter configuration is
based on Azzam’s model [15]. The Axometrics polarimeter
(ASMP) has mainly four units: light source, polarization
state generator (PSG), polarization state analyzer (PSA),
and detector. The light source is a tunable visible Xenon
arc lamp (500–800nm) in ASMP. The PSG and PSA are free
to move and can be set manually to any desired position.
We retrieve sample polarization characteristics, namely,
retardance, diattenuation, and depolarization by decom-
posing the full Mueller matrix using the Lu-Chipman
Figure 4:Effect of oxide layer thickness difference on residual
linear retardance and diattenuation of the crossed fold mirror
configuration.
178 S. Banerjee et al.: Native oxide layer eect on mirrors

method [16, 17]. The system’s accuracy is verified first by
measuring air as a sample. The ASMP system has an accu-
racy of 0.1% and has a precision of 0.01% in the Mueller
matrix elements for 40 points of averaging data together.
After decomposing the Mueller matrix, the maximum
error present in linear retardance is 0.01° and in diattenu-
ation is 0.001 in the VNIR range.
4 Experimental results and
discussion
We measure the polarization characteristics of three aluminum flat
mirrors M1, M2, and M3 (TFAN-10C03-10, Global Optosigma) at 45°
angle of incidence by ASMP. Aluminum mirrors without thin film
coating are selected for the measurement. All of the mirrors are kept
in room temperature for more than 180days such that the oxide layer
thickness has stabilized. The surface normal of the mirrors are set at
45° and are placed in a mechanical stage having a precision of 0.01°.
Three mirrors are then placed one at a time in between the PSG and
PSA of the ASMP, and the Mueller matrix is measured to retrieve the
complete polarization characteristics.
Figure 5 shows the measured linear retardance and diattenua-
tion of the mirrors over the VNIR. It is observed from Figure 5 that the
linear retardance values are dierent for the three mirrors, although
the mirrors have the same aluminum coating [18]. Also, the linear
retardance values of M1, M2, and M3 are not equal with the simulated
linear retardance of a bare aluminum mirror. The measurements are
repeated five times with 40 average points, and the maximum error
present in linear retardance is ±0.08° (Figure 5A). The mismatch in
linear retardance between bare aluminum and the measured real alu-
minum mirrors is because of the native oxide layer. The oxide layer
thicknesses are determined by fitting the measured linear retardance
data using Eqs. (1–3). After fitting the linear retardance data spec-
trally over 500–800nm, the amount of native oxide layer thicknesses
are determined as 4.40 ± 0.15nm, 5.61 ± 0.19nm, and 4.14 ± 0.15nm,
respectively, for M1, M2, and M3. With these native oxide layer thick-
nesses, diattenuation is also fitted over the spectrum and is shown
in Figure 5B. Next, we perform polarization cancellation by choosing
two dierent sets of mirrors depending on their native oxide layer
thickness separation. The first pair M1-M2 and second pair M1-M3
have native oxide layer thickness dierences of 1.21nm and 0.26nm,
respectively. Each pair of mirrors is placed in the CFM configuration
in between the PSG and PSA of the ASMP following the propagation
directions given in Eq. (4). Figure 6 shows the experimental layout.
The Mueller matrix of the mirror pair is measured over a spectrum.
The residual linear retardance and linear diattenuation are retrieved
using the Lu-Chipman decomposition method, with results shown in
Figure 7. We perform five successive measurements to determine the
error and are shown in Figure 7. It is observed that both of the polari-
zation parameters (δ and D) are close to perfectly balanced when the
native oxide layer thickness dierence is minimized. For the first pair
of mirrors having 1.21nm of native oxide layer thickness dierence,
there is a residual linear retardance of 0.65 ± 0.01° and linear diat-
tenuation of 0.003 at 650 nm. For the second pair of mirrors having
almost equal oxide layer thicknesses (dierent by only 0.26 nm),
the residual linear retardance and linear diattenuation reduce to
0.14 ± 0.02° and 0.001 at 650nm, as shown in Figure 7.
If polarization cancellation is exactly balanced, the magnitudes
of the diagonal elements of the Mueller matrix for the mirror pair
become one, while the non-diagonal elements are zero. For a non-
ideal mirror pair, the polarization cancellation, which is shown over
the spectral range 500–800nm (Figure 7), the polarization compen-
sation is almost constant over the specified spectrum region. This is
Figure 5:Spectroscopic measurement of polarization of three aluminum mirrors at a 45° angle of incidence: (A) measured linear retardance
and (B) linear diattenuation are fitted with oxide layer thicknesses using Eqs. (1–3).
Figure 6:The experimental setup for measurement of polarization
cancellation: P, polarizer; R, retarder; A, analyzer are used in a
Mueller matrix polarimeter. Rotation angles of retarders are θ and
5θ, respectively, in the PSG and PSA. M1-M2 is arranged according
to propagation directions (k). The laboratory coordinate system is
shown.
S. Banerjee et al.: Native oxide layer eect on mirrors 179

expected from the nature of the measured linear retardance data of
the three mirrors (Figure 5).
When polarized light reflects from a rough surface, light gets
scattered, and the degree of polarization of the incident polarized
light reduces. So to confirm the roughness of the aluminum thin
film, we measure the depolarization parameter from a 4 × 4 Mueller
matrix, which shows a value almost equal to the System’s error
level (0.001). Thus, it ensures the roughness of the top surface of
the mirror. The roughness influences the depolarization parameter
only and does not have any impact on the linear retardance and
diattenuation.
5 Conclusion
We presented the polarization behavior of environment-
sensitive aluminum mirrors when a natural oxidation
forms, and then, we showed how this native oxide layer
causes artifacts on polarization balancing using various
sets of the CFM configurations. First, we determine the
amount of native oxide layer thickness from the polariza-
tion properties of the aluminum mirrors over the VNIR.
The oxide layer thickness varies from mirror to mirror, as
we confirmed by measuring linear retardance of three alu-
minum fold mirrors (M1, M2, and M3). Next, we performed
polarization cancellation using the CFM geometry. We
showed and demonstrated experimentally using an ASMP
how the oxide layer thickness differences determine a
system’s polarization sensitivity. We showed that polari-
zation cancellation is getting closer to perfectly balanced
when cross fold mirrors have almost equal native oxide
layer thicknesses.
A system’s sensitivity determines how much residual
δ and D can be tolerated. High-precision optical systems
like micro-lithography, lenses, coronagraphs, and space-
borne telescopes require high accuracy measurements, so
that the polarization needs to be controlled to the toler-
ance level of a system.
In theory, a batch of mirrors are all processed in the
same way, exposed to the same coating source for the
same length of time, and then to the same oxygenated
atmosphere, giving them identical coating properties. In
real manufacturing environments, there exist variations
in mirror properties even within the same batch. The
measurement of the oxide layer, thus, can be useful for
chirped mirrors or optical elements in telescopes to cancel
unwanted effects of manufacturing uncertainty.
For real telescopic and other optical systems, polari-
zation cancellation needs to perform across a set of rays,
and analysis is necessary to do over the exit pupil. In such
cases, the native oxide layer thickness variation over the
surface may effect on the predicted behavior, and thus, we
are extending this analysis from point detection measure-
ment to imaging across the full diameter of the fold mirror
and will be shown in a future publication.
Acknowledgment: S. Banerjee is grateful to the Ministry of
Education Culture, Sports, Science and Technology—Gov-
ernment of Japan (MEXT) for a supporting scholarship.
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