Systematic effects induced by a flat isotropic dielectric slab.

Abstract

The instrumental polarization induced by a flat isotropic dielectric slab in microwave frequencies is discussed. We find that, in spite of its isotropic nature, such a dielectric can produce spurious polarization either by transmitting incoming anisotropic diffuse radiation or emitting when it is thermally inhomogeneous. We present evaluations of instrumental polarization generated by materials usually adopted in radio astronomy, by using the Mueller matrix formalism. As an application, results for different slabs in front of a 32 GHz receiver are discussed. Such results are based on measurements of their complex dielectric constants. We evaluate that a 0.33 cm thick Teflon slab introduces negligible spurious polarization (<2.6 x 10(-5) in transmission and <6 x 10(-7) in emission), even minimizing the leakage (<10(-8) from Q to U Stokes parameters, and vice versa) and the depolarization (approximately 1.3 x 10(-3)).

Full text

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arXiv:astro-ph/0602588v1 27 Feb 2006
Systematic effects induced by a
flat isotropic dielectric slab
Claudio Macculi∗,1, Mario Zannoni♯, Oscar Antonio Peverini‡, Ettore
Carretti∗,2, Riccardo Tascone‡, Stefano Cortiglioni∗
∗INAF-IASF Bologna, Via Gobetti 101, 40129 Bologna, Italy
♯Dip. di Fisica, Univ. di Milano - Bicocca, P.zza della Scienza 3, 20126 Milano, Italy
‡CNR-IEIIT, C.so Duca degli Abruzzi 24, 10129 Torino, Italy
The instrumental polarization induced by a flat isotropic dielectric slab in
microwave frequencies is faced. We find that, in spite of its isotropic nature,
such a dielectric can produce spurious polarization either by transmitting
incoming anisotropic diffuse radiation or emitting when it is thermally inho-
mogeneous. We present evaluations of instrumental polarization generated by
materials usually adopted in Radioastronomy, by using the Mueller matrix
formalism. As an application, results for different slabs in front of a 32
GHz receiver are discussed. Such results are based on measurements of their
complex dielectric constant. We evaluate that a 0.33 cm thick Teflon slab
introduces negligible spurious polarization (< 2.6 × 10−5in transmission and
< 6 × 10−7in emission), even minimizing the leakage (< 10−8from Q to U
Stokes parameters, and viceversa) and the depolarization (∼ 1.3 × 10−3).
2008 Optical Society of America
c ?
OCIS codes: 120.5410, 350.4010, 230.5440, 350.1260, 350.5500.
1.Introduction
The last decade has been characterized by the growing interest into the Cosmic Mi-
crowave Background Polarization (CMBP), that has stimulated the design of po-
larimeters featured by low systematic effects and high sensitivity in microwave fre-
quencies. The CMBP, in fact, is among the most powerful tool to investigate the early
stages of the Universe.1Due to the faint expected signal (a few µK over the ∼ 3 K
unpolarized component), any instrumental effect that can produce spurious polariza-
tion must be analyzed in detail in order to minimize its impact. Weak signals call
for high sensitive radiometers that are usually realized by cooling down the front-end
to cryogenic temperatures by means of cryostats. In Radioastronomy, homogeneous
and low loss dielectric slabs are used to allow the signal to enter the cryostat (e.g. see
experiments in Ref. 2–8). Usually, such materials are considered isotropic and then
not generating instrumental polarization. Hence, no analysis is performed to study
polarization effects related to isotropic flat dielectrics. This work arises from the need
to investigate the linear instrumental polarization introduced by flat vacuum windows
in microwave polarimeters when considered isotropic, thus providing the relationship
1present address: INAF-IASF Roma, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
2present address: INAF-IRA Bologna, Via Gobetti 101, 40129 Bologna, Italy
1

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with the material properties through either the complex dielectric constant or the
complex index of refraction. However, the results reported in this paper are not ex-
haustive as far as the complete design of vacuum windows is concerned. Other effects
may occur to enhance the systematics, such as possible intrinsic or manufacturing
induced birefringence of the sample,9and the present work should be considered as
a first attempting to investigate and minimize the spurious instrumental polarization
that may arise from flat isotropic dielectric slabs. We find that a uniform diffuse ra-
diation does not generate instrumental polarization, which, instead, is generated by
anisotropic components (e.g. CMB). We also estimate the amount of the effects in
the specific case of isotropic dielectric which works in Ka frequency band, centered
at 32 GHz, an interesting band in microwave cosmology.
The paper is organized as follows: in Section (2) we present the theoretical model
used to derive the polarization effects introduced by flat isotropic dielectrics, while in
Section (3) we present the total intensity analysis which provides reflectance, trans-
mittance and absorptance of the materials. Measurements of their complex dielectric
constants are presented in section (4) and, finally, estimates of the spurious effects
predicted by the model for the dielectrics experimentally investigated are reported in
section (5).
2.Instrumental polarization by flat isotropic dielectric slab
The instrumental polarization can be evaluated from the Stokes parameter equa-
tions:10
Ix
Qx
Ux
Vx
∝ ?|Ex
∝ ?|Ex
∝ ?2ℜ{Ex
∝ ?2ℑ{Ex
p|2? + ?|Ex
p|2? − ?|Ex
pEx∗
pEx∗
t|2?
t|2?
(1)
(2)
t}?
t}?
(3)
(4)
where Ex
electric field with respect to the incident plane, ?? denotes the time average (henceforth
we will omit it for clarity), and x = R,T indicates the reflection and transmission
terms, respectively.
The transmission case writes
ET
pand Ex
tare the complex parallel and perpendicular components of the
j= TjEj
(5)
where j = p,t denotes the parallel and perpendicular components of the incoming Ej
electric field and Tj is its complex transmission coefficient due to the flat dielectric
slab.11–14Using Eq. (5) in Eq. (1)–(4), the Stokes parameters of the transmitted wave,
computed in the reference frame defined by the plane of incidence, are given by
ST= MTS0
(6)
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where S0≡ [I0,Q0,U0,V0] and ST≡ [IT,QT,UT,VT] are the Stokes parameters of
the incoming and transmitted radiation respectively, and
MT=





1
2(|Tp|2+ |Tt|2)
1
2(|Tp|2− |Tt|2)
0
0
1
2(|Tp|2− |Tt|2)
1
2(|Tp|2+ |Tt|2)
0
0
0
0
0
0
ℜ{TpT∗
−ℑ{TpT∗
t}
t} ℜ{TpT∗
ℑ{TpT∗
t}
t}





(7)
is the Mueller matrix.10Since generally Tp?= Ttand TpT∗
slab generates cross-contamination between I and Q, and U and V parameters.
An interesting case is that of unpolarized incoming radiation, defined by I0
case, the outgoing radiation is featured by
t?= 0, a flat isotropic dielectric
un. In this
ST=





1
2(|Tp|2+ |Tt|2)I0
1
2(|Tp|2− |Tt|2)I0
0
0
un
un





(8)
Therefore, an instrumental polarization is generated and the contamination affects
only Q. Hereafter, we call it spurious polarization.
Similar considerations hold for the reflection case, by replacing Tjby Rj.
It is worth noting that MT(and MR) is the product of the Mueller matrices of
polarizer and retarder,10allowing the description of a flat isotropic dielectric slab as
a combination of these two polarizing devices.
About the effects generated by the emitted radiation, we adopt an approach based
on the radiation power rather than on the electric field. The signal emitted by the
slab is thermal noise characterized by a continuum spectrum related to the physical
temperature of the dielectric Tph. In the microwave frequency domain, by adopting
the Rayleigh-Jeans approximation, the brightness temperature of a thermal source
is proportional to its physical temperature.15Thus, the two intensity components
(parallel and perpendicular) of the emitted signal can be computed as
|Eε
j|2∝ εjTph
(9)
where εjis the Emittance for the j-component. Assuming that the slab is in thermal
equilibrium, the equivalence between Emittance εjand Absorptance Ajholds16
εj(ν,θi) ≡ Aj(ν,θi) = 1 − |Rj(ν,θi)|2− |Tj(ν,θi)|2
allowing us the computation of the emission coefficients from those of reflection and
transmission. Due to the thermal noise nature of the emitted components, these can
be considered uncorrelated,16so that the Stokes parameters write
(10)
Sε=





1
2(εp+ εt)Tph
1
2(εp− εt)Tph
0
0





(11)
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giving the interesting result that the thermal noise injected by the flat slab is partially
polarized. Once again, the contamination affects only Q.
In case of unpolarized incoming radiation, it is convenient to describe the instrumental
polarization through the equations
QR
QT
Qε
SP
= SPRI0
= SPTI0
= SPεTph
un
(12)
(13)
(14)
SPun
SP
with the spurious polarization coefficients SP given by
SPR
=
1
2(|Rp|2− |Rt|2)
1
2(|Tp|2− |Tt|2)
1
2(εp− εt)
(15)
SPT
=
(16)
SPε
=
(17)
While |Rj|2and |Tj|2can be measured, it is generally hard to measure εjand, in turn,
the coefficient SPε. However, it is straightforward that the relation
SPR+ SPT+ SPε= 0(18)
holds, providing thus a way to evaluate SPε.
Interesting special case is that of perpendicular incident radiation (θi= 0 in Fig. 1),
for which |Rp|2= |Rt|2and |Tp|2= |Tt|2, then
SPx(θi= 0) = 0(19)
that is no spurious polarization is generated.
Frequently, the dielectric slab is placed in front of a collecting system (e.g. a feed-
horn which we assume aligned with the slab axis), thus it is important to evaluate the
instrumental polarization propagating inside the receiver. Contributions come from
the radiation transmitted and emitted by the dielectric. In fact, a further signal can be
generated by reflection due to the signal emitted by the receiver toward the free space
and backscattered by the slab. However, this occurs at angles of incidence θi ∼ 0,
thus implying negligible SPR.
The Eq. (6) and (11) provide the instrumental polarization due to the radiation
coming from a given direction and in the reference frame defined by the plane of
incidence and its normal (see Fig. 1).
Such signals must be evaluated in the laboratory reference frame (e.g. the Antenna
Reference Frame: ARF), by accounting for the rotation of the azimuthal angle β. By
referring to Fig. 1, if S0
the ARF, the polarization state of the transmitted component can be described as:10
ARFdefines the Stokes parameters of the incoming radiation in
ST
ARF(2β) = R(−2β)MTR(2β) S0
ARF= HT(2β) S0
ARF
(20)
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where the HT(2β) matrix is:





TSPTcos(2β)SPTsin(2β)
33)sin(2β)cos(2β) −LTsin(2β)
33cos2(2β)
−LTcos(2β)
0
SPTcos(2β) T cos2(2β) + MT
SPTsin(2β) (T − MT
0
33sin2(2β) (T − MT
33)sin(2β)cos(2β) T sin2(2β) + MT
LTsin(2β)
LTcos(2β)
MT
33





(21)
and
T=
1
2(|Tp|2+ |Tt|2)
= ℑ{TpT∗
= ℜ{TpT∗
(22)
LT
MT
t}
t}
(23)
(24)
33
where R(2β) is the Mueller matrix for rotation10and T is the Transmittance of the
slab. Given an input Stokes parameter Y0, we define its contamination over the X
output as XY0: such a cross-term is the leakage. Thus, by definition, the transmission
function of the X0parameter is XX0.
About the emitted component, in the ARF the signal expected is:
Sε
ARF(2β) =





εTph
SPεcos(2β)Tph
SPεsin(2β)Tph
0





(25)
where ε =1
It is worth noting that each quantity hitherto discussed is well-defined for values of
the polar angle up toπ
The effect on the spurious polarization due to the incoming unpolarized diffuse signal
characterized by a brightness temperature distribution15Tb(ν,θ,β) can be computed
by integrating the spurious components (SPT) of Eq. (20) all over the directions. Such
integrations lead to the following outputs for Q and U in antenna temperature:15
2(εp+ εt) is the Emittance of the slab.
2rad.
QT
SP(d) =
1
ΩFF
A∆ν
?
∆ν
? ?π/2
0
dθSPT(d,ν,θ′)PFF
n(θ)sin(θ)
?2π
0
dβTb(ν,θ,β)cos(2β)
?
dν
(26)
UT
SP(d) =
1
ΩFF
A∆ν
?
∆ν
? ?π/2
0
dθSPT(d,ν,θ′)PFF
n(θ)sin(θ)
?2π
0
dβTb(ν,θ,β)sin(2β)
?
dν
(27)
where ∆ν is the frequency bandwidth, PFF
tern and the antenna solid angle in far field regime15respectively; θ′(θ) is the angle
of incidence on the slab of the radiation coming from the (far field) direction θ: if
the slab is in the near field of the antenna, then θ′< θ (flat slab and antenna are
assumed coaxial, so θi≡ θ), otherwise in far field θ′≡ θ. In order to emphasize the
effect introduced by the slab, in Eq. (26) and (27) we assume an ideal feed-horn fea-
tured by a β-symmetric co-polar pattern (i.e. null cross-polarization), the feed horn
n
and ΩFF
Aare the normalized co-polar pat-
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spectral transfer function constant all over the frequency band (Pn(ν,θ,β) ≡ Pn(θ))
and negligible edge effects between the slab and the feed aperture in the case of near
field position.
Writing the brightness temperature with respect to its mean and anisotropy compo-
nents
Tb(ν,θ,β) = Tb,0(ν) + ∆Tb(ν,θ,β)
makes null the contribution of the mean term in the Eq. (26)–(27), which thus become
(28)
QT
SP(d) =
1
ΩFF
A∆ν
?
∆νdν
?π/2
0
dθSPT(θ′)PFF
n(θ)sin(θ)
?2π
0
dβ∆Tb(ν,θ,β)cos(2β)
(29)
UT
SP(d) =
1
ΩFF
A∆ν
?
∆νdν
?π/2
0
dθSPT(θ′)PFF
n(θ)sin(θ)
?2π
0
dβ∆Tb(ν,θ,β)sin(2β)
(30)
This gives the important result that a flat isotropic dielectric slab can generate spu-
rious polarization only in presence of anisotropic incoming signal. Similarly, in ax-
isymmetric optics17,18the instrumental polarization is generated only by incoming
anisotropic radiation. This is particularly relevant for the CMB, whose anisotropy is
low (∆Tb/Tb∼ 10−5).19,20
The spurious polarization due to the emission, can be computed by replacing Tbwith
the physical temperature and, from Eq. (25), the correlated component of the thermal
noise collected by the feed is
1
ΩNF
0
Qε
SP(d) =
A∆ν
?
∆νdν
?π/2
dθSPε(d,ν,θ)PNF
n(θ)sin(θ)
?2π
0
dβ ∆Tph(θ,β)cos(2β)
(31)
Uε
SP(d) =
1
ΩNF
A∆ν
?
∆νdν
?π/2
0
dθSPε(d,ν,θ)PNF
n(θ)sin(θ)
?2π
0
dβ ∆Tph(θ,β)sin(2β)
(32)
where PNF
placed in the near field of the antenna. Even in this case, the contribution of the mean
component is null and the spurious polarization equations are like for Eq. (29)–(30),
but with the anisotropic thermal component ∆Tphinstead of ∆Tb, and PFF
by PNF
Since both the transmitted and emitted signals are not correlated, due to the additive
property of the Stokes parameters10the total spurious polarizations is
n
is the normalized co-polar pattern in near field at the slab position if it is
n
replaced
n.
QSP
USP
= QT
= UT
SP+ Qε
SP+ Uε
SP
(33)
(34)
SP
From the matrix HT(2β), the depolarization of Q and U can be estimated as:
DQT
=
Q0− QT
Q0
U0− UT
U0
= (1 − T)cos2(2β) + (1 − MT
33)sin2(2β)(35)
DUT
== (1 − T)sin2(2β) + (1 − MT
33)cos2(2β)(36)
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The loss of the Q-signal, DQT, could be evaluated in percentage term with respect to
the measured Q0
min absence of the slab. The result is:
ηQT(d) =
1
ΩF F
A∆ν
?
∆νdν
?π/2
?
0
dθ PFF
n(θ)sin(θ)?2π
∆νdν?π/2
0 dβ DQT(d,ν,θ′,β)Q0(ν,θ,β)
1
ΩF F
A∆ν
0
dθ PFF
n(θ)sin(θ)
?2π
0 dβQ0(ν,θ,β)
(37)
The same for U. Assuming a constant incoming signal (Q0(ν,θ,β) = Q0), a zero-order
estimate of Eq. (37) is given by:
ηQT(d) =
2π
ΩFF
A
1
2π
1
2
?π/2
0
?2π
0
˜DQT(d,θ′)PFF
n(θ)sin(θ) dθ(38)
˜DQT(d,θ′) =
?DQT(d,ν,θ′,β)?νdβ =
1 − ?T(d,ν,θ′)?ν+ 1 − ?MT
=
?
33(d,ν,θ′)?ν
?
(39)
where ??νis the in-band average.
The QT
U0 leakage term results:
QT
U0(d) =
1
ΩFF
?2π
0
A∆ν
?
∆νdν
?π/2
0
dθ
?
T(d,ν,θ′) − MT
33(d,ν,θ′)
?
PFF
n(θ)sin(θ)
×
dβ sin(2β)cos(2β)∆U0(ν,θ,β)(40)
The same for UT
incoming signal is anisotropic.
Finally, the leakage due to V0is given by:
Q0 by replacing Q with U. Note that a non null result arises if the
QT
V0(d) = −
1
ΩFF
A∆ν
?
∆νdν
?π/2
0
dθLT(d,ν,θ′)PFF
n(θ)sin(θ)
?2π
0
dβ sin(2β)∆V0(ν,θ,β)
(41)
UT
V0(d) =
1
ΩFF
A∆ν
?
∆νdν
?π/2
0
dθLT(d,ν,θ′)PFF
n(θ)sin(θ)
?2π
0
dβ cos(2β)∆V0(ν,θ,β)
(42)
Once again, a leakage is generated only in case of anisotropic incoming signal.
3.Total intensity analysis
The effects on the total intensity signal can be evaluated by estimating both trans-
mission and reflection properties of the slab (see for details Ref. 11–14).
The incoming signal collected by the system slab-feed is given by (in antenna tem-
perature):
TT
A(d) =
1
ΩFF
A∆ν
?π/2
0
dθ
?2π
0
dβ
?
∆νdν T(d,ν,θ′)Tb(ν,θ,β) PFF
n(θ)sin(θ)(43)
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where T is the Transmittance of the slab.
An estimate of the effect is given by the relative transmitted signal Λ(d) = TT
in the simple case of an isotropic input signal Tb(ν,θ,β) = T0:
A(d)/T0
Λ(d) =
2π
ΩFF
A
?π/2
0
?T(d,ν,θ′)?νPFF
n(θ)sin(θ)dθ(44)
If the feed horn directivity is high, most of the signal is collected close to 0◦. In the limit
of low loss dielectrics, the maxima of Λ(d) will be identified by integer multiples of
the well known thickness d = d0≡ λ/2n, which identifies the Transmittance maxima
for null incidence,11where n is the real part of the complex index of refraction.
Similarly, the thermal noise injected by the dielectric can be computed when it is in
thermal equilibrium at the physical temperature Tph. Its emission is that of a greybody
at temperature Tphfeatured by an Emittance ε(θi) ≡ A(θi) = 1 − [R(θi) + T(θi)],
where R =
the microwave frequency domain, the Rayleigh-Jeans approximation can be adopted.
Hence, in term of brightness temperature,15the thermal noise emitted by the slab is
simply ε(θi)Tph. Thus, the signal collected by the antenna is:
1
2(|Rp|2+ |Rt|2) is the Reflectance of the slab. Since here we consider
Tε
A(d) =
1
ΩNF
A∆ν
?π/2
0
dθ
?2π
0
dβ
?
∆νdν ε(d,ν,θ) Tph(θ,β)PNF
n (θ)sin(θ)(45)
where PNF
placed at its near field, and a thermally inhomogeneous but thermally stabilized slab
has been considered.
In a conservative approach, the antenna noise temperature can be estimated as:
n (θ) is the Near Field normalized co-polar pattern of the feed if the slab is
Tε
A(d) ≤ ?ε(d,ν,θ)?ν|MAX(θ)× Tph(θ,β)|MAX(θ,β)
where MAX(y) stands for the maximum value over the quantity y.
(46)
4.Microwave Tests
In order to provide realistic estimates of the analysis hitherto carried out, we per-
formed measurements on several samples to determine their complex dielectric con-
stant ǫ (ǫ = ǫr[1 − itan(δe)], where tan(δe) is the electric tangent loss), which entries
in the determination of the Rj, Tj and εj quantities11–14by means of the complex
index of refraction n (n = n− iκ, where κ is the extinction coefficient). In fact, they
are related as follow (for µr= 1):
n =
?ǫr
2
??
1 + tan2(δe) + 1
?1/2????tan(δe)<<1≃√ǫr
?1/2????tan(δe)<<1≃1
(47)
κ =
?ǫr
2
??
1 + tan2(δe) − 1
2
√ǫrtan(δe) (48)
Since electromagnetic properties of polymers can vary with the composition, history
and temperature of the specimen,21measurements are mandatory. As a matter of
8

Page 9

fact, it is convenient to extract the slab of interest from the same bulk of material
used to cut the samples under test.
We have investigated High Density Polyethylene (HDPE), Teflon, Polypropylene and
Nylon, which are dielectrics commonly used from microwave to far infrared frequen-
cies. The tests have been performed in the frequency band 27 < ν < 37 GHz, inter-
esting for microwave cosmology.22The measurements have been performed by means
of a vector network analyzer (Model HP8510). A waveguide device has been real-
ized in standard WR-28. It consists of two shells cut along the E-plane (Fig. (2))
which embeds the test sample. The test is based on the comparison of the reflection
and transmission parameters S11and S21of the global device23(waveguide plus sam-
ple), by making two measurements with and without the dielectric test sample. The
measured insertion loss and return loss can be related to impedance and propagation
constants characterizing the equivalent transmission line of the fundamental mode. In
turn, these constants are related to ǫr(one of the wanted quantity), the resistivity of
the material ρmand the size of the sample, so allowing the computation of the other
relevant quantity, tan(δe). Estimates of ǫrand tan(δe) are carried out by performing
a best fit on the data and assuming a second order polynomial behavior for ǫrand
ρm. Such estimates will be specialized by computing their in-band (30.4–33.6 GHz)
average values, which are those of interest to the BaR-SPOrt project. An example
of our results is reported in Fig. (3), which shows the measurements and the best fit
model of the scattering parameters in the case of a Teflon sample. The main error
source for the fitting procedure comes from the knowledge of the sample sizes (re-
ported in Table (1); see also Fig. (4) for parameter definitions). Such errors are taken
into account when computing the in-band average values. In Fig. (5) the complex
constants are shown while Table (2) and (3) report the in–band average values and
variations. For sake of completeness, we compare our data with those reported in the
literature (see Table (4)). There is a good consistency for the estimated dielectric
constants. Large differences, instead, are found for the electric tangent loss of Teflon
and Polypropylene. The poor precision is due to the experimental technique adopted,
which is not ideal for low loss dielectrics. In this case, a more accurate evaluation of
the electric tangent loss could be obtained by applying the cavity technique.25These
considerations look supported by the high precision data of both ǫrand tan(δe) ob-
tained for the Nylon sample. Although we provide just upper limits for tan(δe), no
higher precision measurements seem necessary since the contamination computed in
Section (5) are already negligible. Due to our interest in tiny polarization signal, we
performed extensive measurements on Teflon to investigate its optical anisotropy. In
fact, it seems to be more promising among the materials selected. The manufacturing,
in fact, can introduce in these polymers anisotropic electromagnetic and structural
properties, since the molecular chains (structural units) will be preferentially aligned
in certain directions,26,27thus transforming the dielectric into a sort of polarizer.
Hence, we preferred to use a Teflon block that has been casted and not extruded to
minimize such manufacturing effects. We assume such a block as homogeneous. By
considering the vibration of the electric field propagating in the rectangular waveg-
uide, it is possible to investigate the optical anisotropy of dielectrics by cutting the
sample as shown in Fig (4). Since typical feeds for microwave cosmology are charac-
9

Page 10

terized by high directivity,4,28,29most of the signal will be collected close to 0◦. Thus,
two samples have been cut along the z–axis, but rotated by 90◦each other, to probe
the material for the incoming electric field vibrating in x and y directions.
The results in the two directions are consistent each other within the error (see Fig.
(6) and Table (5)), and possible anisotropy of the complex index of refraction between
0◦and 90◦of rotation is summarized by the following data:
∆?n?ν
∆?κ?ν
= ?n?1−cut
= ?κ?1−cut
ν
− ?n?2−cut
− ?κ?2−cut
ν
= 0.01 ± 0.01
= (4.2 ± 5.8) × 10−5
(49)
(50)
νν
which means that we can set at 1σ of Confidence Level the following upper limit:
∆?n?ν = 0.02 and ∆?κ?ν = 10−4. These measurements allow us to consider our
Teflon sample optically isotropic at least for our purposes.
5.Estimates of the systematic effects.
The analysis performed in Section (2) includes the far and near field regime, even
though for our purpose most estimates will be provided for a flat slab placed in near
field position. However one case of far field will be also considered. We take into
account an instrument featured by a 10% bandwidth, typical of recent microwave po-
larimeters, so that all the relevant quantities of Sections (2) and (3) are evaluated as
in-band average. Furthermore, here we replace the angle θ′with θ (i.e. the incoming
Far Field direction) making easier the computation of all the interesting quantities.
This approximation does not prevent the aim of this work. In fact, if the slab is close
to the feed aperture then θ′< θ, and, in turn, this substitution provides conserva-
tive estimates. The materials considered here are Teflon, HDPE and Polypropylene,
disregarding the Nylon due to its high values of ǫrand tan(δe), since we are inter-
ested in the minimization analysis of the systematic effects. For the Teflon, we take
into account the 1–cut, since its complex index of refraction is more precise than the
other cut.
In Fig. (7) the in-band average reflectance, transmittance and emittance of the di-
electrics versus θiare shown for 3 thickness values. For each panel the central value
is the thickness which maximizes the transmission at θi= 0 (see the Teflon plots): it
corresponds to d = d0≡ λ/2n where λ ∼ 0.94 cm. As expected, these plots show that
either increasing κ or the thickness generates an increase of the emittance. Similarly,
the reflectance increases by increasing n. Close to the axis, transmittance variations
are very small (for θi< 15◦is lower than 10−3), then T(d,ν,θ′) ≃ T(d,ν,θ), so al-
lowing the substitution θ′with θ. In these frequencies such dielectrics show very low
losses (∼ 10−3÷10−4, see the emittance plot), then the thickness d0provides a good
estimate of the thickness that maximizes the transmittance.
A first estimate of the thermal noise (Tnoise) injected by the slab can be given by Eq.
(46) using the parameter ?ε(d,ν,θ)?ν|MAX(θ) (see bottom-right plot of Fig. (7) and
Table (6) for the estimates).
A better estimate of the signal transmission and of the thermal noise collected by the
feed (Fig. (8)) can be given using Eq. (44) and (45) where, for the emission, the near
field pattern has to be used. The adopted patterns are shown in Fig. (9), by using the
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Page 11

BaR-SPOrt far field one as realistic example,29and by assuming a Gaussian beam
approximation for the near field.31In the last case, it is necessary to set a geomet-
rical configuration of the slab in front of the feed aperture to produce the near field
pattern. Then, for our purpose, a circular flat slab will be adopted. Hereafter, we will
refer to the estimates of quantities related to antenna integrals as “BaR-SPOrt case”
but, as we will show in this section, the core idea can be applicable in the same way
to other cases. The plot in Fig. (8) shows, as expected, that the noise due to the slab
increases with the thickness. The plot of the relative transmitted signal Λ shows the
typical interference trend. As highlighted in the magnified frame, the value of the first
maximum correspond to a thickness of 0.33 cm for the Teflon and 0.31 cm for HDPE
and Polypropylene, that is the ideal value d0in case of the low absorbing material
approximation. It is due to the narrow beam of the feed adopted which favors angles
very close to 0◦. Thus, in case of narrow far field pattern, such a thickness provides
the best size for maximizing the transmittance of the slab. By choosing as optimal
thickness d0, the residuals with respect to 1 (i.e. Λ(d) − 1) give an estimate of the
signal loss (∼ 10−3). Besides the slab thickness optimization, this analysis allows us
to choose even the best material, which in the band adopted is Teflon.
Table (6) shows the parameters ?ε(d,ν,θ)?ν|MAX(θ)and the quantity Tε
tained by increasing the antenna integral in Eq. (45) that are used to evaluate the
noise temperature injected by the slab (Tnoise). Between the two estimates, the beam
integration gives a correction of ∼ 10% reducing the value provided by the Eq. (46):
the estimates are the same within 10% variation. Thus, the Eq. (46) is an easy way
to estimate the thermal noise. As shown, the Teflon is the material which injects the
lowest noise (162 mK at TMAX
ph
= 300 K), even though HDPE is just slightly worst
and can be considered as well.
The spurious polarization coefficients are shown in Fig. (10). Oscillations in the spuri-
ous response are generated by increasing the optical path inside the slab as is typical
for interference phenomena. Once again, the approximation for the angle can be ap-
plied since, close to the feed axis, the relation |?SPT(d,ν,θ′)?ν| < |?SPT(d,ν,θ)?ν|
holds. Then a conservative estimate of Eq. (29)–(32) can be represented by:
A(d)/TMAX
ph
ob-
Qx
SP|MAX= Ux
SP|MAX= |?SPx(d,ν,θ)?ν|MAX(θ)× |∆T(ν,θ,β)|MAX(ν,θ,β)
where x is either T or ε, and ∆T is the variation of either the brightness or the physical
temperature.
The upper limit of the spurious polarized transmission and emission are plotted in
Fig. (11) for the BaR-SPOrt case by conservatively approximating the Eq. (29) as
follows:
(51)
QT
SP(d)=
1
ΩFF
?2π
0
A∆ν
?
∆νdν
?π/2
0
dθSPT(d,ν,θ′)PFF
n(θ)sin(θ) ×
×
dβ∆Tb(ν,θ,β)cos(2β) (52)
<|∆Tb(ν,θ,β)|MAX
2π
ΩFF
A
?π/2
0
|?SPT(d,ν,θ)?ν|PFF
n(θ)sin(θ) dθ(53)
11

Page 12

while the Eq. (31), computed in cylindrical coordinates, has been approximated as:
Qε
SP(d,z) < ∆Tph(ν,ρ,β)|MAX
2π
ΩNF
A
?R
0
|?SPε(d,ν,ρ)?ν| PNF
n(ρ,z) ρ dρ (54)
where R is the radius of the circular slab and z its position with respect to the
feed horn waist. Such plots show that the low relative spurious polarization of the
transmitted component (10−3÷ 10−5) is minimized by the same thickness d0which
maximizes the transmittance. For the emitted component, the integrated spurious
term is very low (10−6÷ 10−8).
Once again, it could be useful to compare the two methods to evaluate the spurious
polarization. The first one is defined by the Eq. (51), while the second one by Eq.
(53) and (54), which even accounts for the antenna pattern. Results, shown in Tables
(7) and (8), are computed by taking into account as example the known anisotropic
signal of the CMB (∆TMAX
b
= 100µK) and a guess value of ∆TMAX
the thermal gradient of the slab.
Taking into account the antenna pattern, in the BaR-SPOrt case such values are
∼ 200 times lower than the rough estimate provided by the Eq. (51), showing how
critical it is to consider the antenna pattern in this case. This is due to the high
directivity of the feed horn adopted and to the rapid decrease of the |SP| functions
close to 0◦. Teflon is the material which introduces the lowest spurious polarizations
(∼ 0.6µK).
For sake of completeness, we insert an analysis of the spurious polarization generated
in transmission regime as the beam of the collecting system increases (Fig. (12))
when the slab is placed in far field (i.e. θ′≡ θ). Here a Gaussian pattern has been
considered, thus performing a general but optimistic analysis, since in case of co-polar
pattern of real feed the expected spurious signal is greater than the level generated
by a Gaussian one. In fact, the Gaussian pattern shows a rapid decrease out of the
beam, thus producing lowest response as shown for comparison between the minima
of Teflon at 7◦in the left-plot of Fig. (12) and in Fig. (11) (the difference is a factor
3, ∼ 5 dB). The plots in Fig. (12) show that the broader the beam the higher the
spurious level generated by the slab. As expected, this is due to the rapid increase of
the spurious coefficient |SPT| out of the null incidence (see Fig. (10)). Moreover, the
thickness setting positions of minima varies with the beam due to the low directivity of
the collecting system as the beam increases. In particular, such positions are different
from those expected for narrow beam around null incidence (e.g. see the vertical lines
across the left-plot of Fig. (12) and Section (3)). For comparison, the level of spurious
effects produced by a good feed featured by low cross-polarization (e.g. - 40 dB of
BaR-SPOrt17,18), even though not optimised to minimize such a systematic, is ∼ - 25
dB, as represented by the horizontal line traced across the right-plot of Fig. (12). Such
a level matches the requirements for CMBP experiments. The systematics generated
by flat dielectric slabs prevail on that produced by good feeds for beams greater than
∼ 15◦, thus requiring either thickness optimization analysis or the choice survey to
control spurious polarizations.
The estimates of the depolarization effects introduced by the dielectrics are shown in
Fig. (13).
ph
= 1 K related to
12

Page 13

Close to the axis we find that ?˜DT
estimate Eq. (38) approximating ?˜DT
selected thickness and materials, the loss of the polarized signal is marginal (∼ 10−3).
It is worth noting that the thickness d0maximizing the transmittance also minimizes
the depolarization (see the magnified frame of right panel in Fig. (13)).
Finally, the leakages are estimated. In a conservative approach, the QT
(Eq. (40)) can be estimated as:
Q(d,ν,θ′)?ν∼ ?˜DT
Q(d,ν,θ′)?ν with ?˜DT
Q(d,ν,θ)?ν, then it is possible to
Q(d,ν,θ)?ν. Thanks to the
U0 leakage term
QT
U0(d) < |∆U0(ν,θ,β)|MAX(ν,θ,β)× fT,MT
fT,MT
ΩFF
A
0
33(d)(55)
33(d) =
2π
?π/2
[?T(d,ν,θ)?ν− ?MT
33(d,ν,θ)?ν] PFF
n(θ)sin(θ) dθ (56)
The approximation holds because the term (?T?ν−?MT
trend for θicomparable with the BaR-SPOrt beam (∼ 7◦), as shown in top-left panel
of Fig. (14). For the selected thickness and materials, the maximum leakage from U0to
Q in the 30.4–33.6 GHz band is about 0.07×|∆U0|MAX. A more rigorous computation,
which takes into account the beam pattern, provides the result shown in Fig. (14)
(bottom-left panel). The minimum leakage is realized with the thickness d0for which
the values drop down to negligible values (10−8÷ 10−9with respect to |∆U0|MAX).
Same results hold also for UT
Similarly, the QT
33?ν) has a monotone growing
Q0.
V0 term (Eq. (41)) can be estimated as:
QT
V0(d) < |∆V0(ν,θ,β)|MAX(ν,θ,β)× fLT(d)
fLT(d) =
ΩFF
A
0
(57)
2π
?π/2
?|LT(d,ν,θ)|?νPFF
n(θ)sin(θ) dθ(58)
where the considerations done for (?T?ν−?MT
HDPE and Polypropylene introduce in the 30.4–33.6 GHz band a 0.16 maximum
leakage of V0into Q and U. If such a function is smoothed by the beam pattern,
then, for the selected thickness d0which also here minimize the effect, the leakage
becomes negligible (10−4÷ 10−5with respect to |∆V0|MAX).
Again, among the selected materials, Teflon introduces the lowest leakage and depo-
larization effects.
Note that the depolarization and leakage in transmission are signal losses which can
be recovered by an overall instrument calibration.
33?ν) can be extended to ?|LT|?ν. Teflon,
6.Conclusion
In this work we presented the systematic effects introduced by a flat slab of isotropic
dielectric.
We presented an overall analysis of the interaction between electromagnetic radia-
tion and isotropic dielectric at microwave frequencies, by analyzing transmittance,
reflectance, absorptance, spurious polarization, leakage and depolarization by means
of the Mueller formalism.
The important result is that spurious polarization, and leakage between the Stokes
parameters are produced even by optically isotropic dielectrics, but only when they
13

Page 14

are thermally inhomogeneous or the incident radiation is anisotropic. In particular,
it has been provided an estimate of the expected systematic effects introduced by
Teflon, Polypropylene and HDPE, together with algorithms for their thickness opti-
mization to minimize the effects.
Measurements of dielectric constant and electric tangent loss of Teflon, HDPE, Polypropy-
lene and Nylon have been provided between 27 GHz and 37 GHz at 300 K of physical
temperature. The Teflon sample analyzed is featured by the lowest ?ǫr?ν∼ 2.04 and
?tan(δe)?ν ∼ 1.6 × 10−4averaged in the 30.4–33.6 GHz band. Moreover, we found
that no optical anisotropy at level of 1% has been measured from our Teflon sample
about the index of refraction.
The analysis shows that Teflon, among the selected materials, is the best material in
the investigated band which minimizes the systematic effects.
As discussed in Section (5), the optimal thickness to maximize transmission (∼ 0.999)
and reduce emission (∼ 0.00054) is d = 0.33 cm. The maximum thermal noise in-
troduced by this slab is 162 mK at a physical temperature of Tph = 300 K. This
thickness, which minimizes also the transmitted spurious polarization (< 2.6 × 10−5,
thus producing in emission a spurious level < 6 × 10−7), the leakage (< 10−8from
Q0to U, or U0to Q, ∼ 5 × 10−5from V0to Q or U) and the depolarization (∼
1.3 × 10−3), corresponds to d = λ/2n. Broadly speaking, in the approximation of
low absorbing material and high feed horn directivity, such a thickness maximizes the
transmission and reduces all the other effects.
We have also shown that for dielectrics in far field regime, the transmitted spurious
polarization prevails on the one produced by good feeds (cross-polarization ∼ – 40 dB)
for beams greater than ∼ 15◦, thus showing the need of either thickness optimization
analysis or the choice survey of flat dielectrics to control spurious polarizations. The
position of such thicknesses, which set maxima and minima of the spurious response,
depends on the beam adopted.
Acknowledgments
Authors wish to thanks Renzo Nesti and Vincenzo Natale for useful discussion, and
the anonymous referee for the useful comments and the encouragement to improve
the paper. This work is inserted in the BaR-SPOrt program, an experiment aimed at
detecting the CMBP, which is funded by ASI (Italian Space Agency).
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