Lab
boratory for Applied Electromagnetics and Commun
nications
Electromagnetic Scattering by an
Impedance
p
Sheet with a 1-D Inhomogeneity
g
y
in a Rectangular Waveguide
Keith W. Whites† and Brian B. Glover‡
†Laboratory
for Applied Electromagnetics and Communications,
South Dakota School of Mines and Technology, USA
‡Los Alamos National Laboratory, USA
Metamaterials 2009: 3rd International Congress on Advanced
Electromagnetic Materials in Microwaves and Optics
London, UK, Aug. 30 – Sept. 4, 2009
Sponsored by the National Science Foundation through an EPSCoR Research Infrastructure Improvement (RII) program grant (EPS0554609) and an Integrative, Hybrid & Complex Systems (IHCS) program grant (ECCS-0824034).
Keith W. Whites
Metamaterials 2009
September 2, 2009
1
Adjustable and Tapered R card
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nications
‹
One aspect of this work is to produce lossy films with adjustable
sheet impedance:
– Perhaps isotropic and uniform
– Perhaps spatially varying
– Perhaps anisotropic; etc.
‹
As will be discussed shortly, this will be accomplished by physically
altering commercially available resistive films.
‹
Applications include:
– Reduce backscattering (Tapered R card: Senior and Liepa, 1984;
Haupt and Liepa
Liepa, 1987)
– Adaptive reflector antenna (Haupt, 2006)
– Wu-King taper for ultrawideband antenna:
• Radially inhomogeneous sheet impedance
• Printed conductor onto 370 OPS Kapton XC
(Glover, Kirschenmann, and Whites, “Engineering R-Card Surface
Resistivity with Printed Metallic Patterns, Metamaterials’2007, Oct. 2007.)
Keith W. Whites
Metamaterials 2009
September 2, 2009
2
1
Outline
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This work requires a method for characterizing the scattering by
films that are spatially varying, which is the topic of this talk.
Eventually would like to measure the effective sheet impedance .
z
Creating effective sheet impedance films
z
Mode matching solution
o
scattering by spatially varying sheet impedance films in a rectangular
waveguide
z
Using perforations to create spatial variation
z
Properties of DuPont Kapton 370 XC
z
Measurements for two region perforated Kapton 370 XC
z
Conclusions
o
accuracy of the Maxwell/Maxwell Garnett mixing rule
Keith W. Whites
Metamaterials 2009
September 2, 2009
3
Lab
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Creating Effective Films
Methods to create tapered sheet impedance films include:
„ Printing electrically small, metallic patterns on commercially available films
to reduce the effective surface resistivity. [Films include DuPont's Kapton
XC® or Ohmega Technologies OHMEGA-PLY®
OHMEGA PLY® or 200 Ω/sq stainless steel
coated PET (polyethylene terephthalate), for example.]
Open loops
print quicker
and use less
conductive “ink.”
1 mm
„
Perforating commercially available films to increase the effective surface
resistivity.
i i i
Laser cut square holes
Kapton 370 XC
„
Directly manufacture. Expensive, difficult to realize spatially varying.
Keith W. Whites
Metamaterials 2009
September 2, 2009
4
2
Lab
boratory for Applied Electromagnetics and Commun
nications
Outline
z
Creating effective sheet impedance films
z
Mode matching solution
o
z
scattering
g by
y spatially
p
y varying
y g sheet impedance
p
films in a rectangular
g
waveguide
Using perforations to create spatial variation
o
accuracy of the Maxwell/Maxwell Garnett mixing rule
z
Properties of DuPont Kapton 370 XC
z
Measurements for two region perforated Kapton 370 XC
z
Conclusions
Keith W. Whites
Metamaterials 2009
September 2, 2009
5
Waveguide Characterization
Lab
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nications
‹
‹
‹
‹
‹
‹
Quest is to develop a method for characterizing spatially varying
impedance films. In this work, will assume variation in only one
Cartesian dimension.
Will develop a waveguide technique hoping for higher accuracy in
a confined measurement system.
In actuality, the film is a thin slab of material. Maybe multilayered
in the case of printed film (though not the case with perforated).
In inhomogeneous materials, difficult to obtain mode expansion of
electromagnetic fields.
However, here we have very thin, high contrast films. Model these
with sheet impedance boundary condition.
Effect sheet impedance Zs(x) will vary across the surface.
Keith W. Whites
Metamaterials 2009
September 2, 2009
6
3
Mode match solution - 1
„
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„
Solve with mode match approach.
Incident TE10 mode:
Top View
⎛ π x ⎞ − jβ z ,10 z
E = sin ⎜
⎟e
⎝ a ⎠
β
⎛ π x ⎞ − jβ z ,10 z
H xi = − z ,10 sin ⎜
⎟e
ωμ
⎝ a ⎠
i
y
⎛ mπ ⎞
β z2,m 0 = ω 2 με − ⎜
⎟
⎝ a ⎠
„
„
n̂
2
With no y variation in the incident fields or the specimen, then only TEm0
modes (and no TM modes) will be scattered by the specimen.
S
Subsequently,
non-zero components off the reflected
f
fields
f
include
β z ,m 0
⎛ mπ x ⎞ jβ
Am sin ⎜
⎟e
⎝ a ⎠
m=1 ωμ
∞
⎛ mπ x ⎞ jβ z ,m 0 z
E yr = ∑ Am sin ⎜
⎟e
⎝ a ⎠
m=1
„
∞
H xr = ∑
z ,m 0 z
Non-zero components of the transmitted fields include
β z ,m 0
⎛ mπ x ⎞ − jβ
Bm sin ⎜
⎟e
⎝ a ⎠
m=1 ωμ
∞
⎛ mπ x ⎞ − jβ z ,m 0 z
E yt = ∑ Bm sin ⎜
⎟e
⎝ a ⎠
m=1
∞
H xt = −∑
Keith W. Whites
z ,m 0 z
September 2, 2009
Metamaterials 2009
7
Mode match solution - 2
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z
Employing the impedance sheet boundary condition at z = 0 in
which the tangential electric field is continuous across the sheet
while the tangential magnetic field is discontinuous:
nˆ × E = Z s ( x ) nˆ × ⎡⎣ nˆ × ( H + − H − ) ⎤⎦
z
Leads to a matrix equation for the vector of transmitted field
amplitudes Bi
Q⋅B = P
where
2β
a
Qij = δ ij + z , j 0 I R ,ij
2
ωμ
Pi =
2β z ,10
10
ωμ
I R ,i1
⎛ iπ x ⎞ ⎛ jπ x ⎞
I R ,ij ≡ ∫ Z s ( x ) sin ⎜
⎟ sin ⎜
⎟ dx
⎝ a ⎠ ⎝ a ⎠
0
a
and δij is the Kronecker delta function.
Keith W. Whites
Metamaterials 2009
September 2, 2009
8
4
Compare with CST MWS Simulation
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‹
‹
To help validate the accuracy of this mode matching solution,
comparisons were made with the frequency domain solver of CST
Microwave Studio.
Considering a two region resistive film:
333.33 OPS
‹
‹
833.33 OPS
Within the frequency domain solver, CST MWS provides for an
ideal impedance sheet boundary condition, with infinitesimal
thickness, in an integral equation solution.
No such boundary condition is available with the time domain solver.
Keith W. Whites
September 2, 2009
Metamaterials 2009
9
Two Region Specimen Results
Used 20 modes in the mode matching solution (both transmitted
and reflected), and ~70k mesh cells in the CST MWS solution.
0.8
S parameter magnitude
0.7
178.8
0.6
CST: Mag S11
CST: Mag S21
0.5
Mode match: Mag S11
Mode match: Mag S21
0.4
0.3
1.4
178.6
178.4
CST: Ang S11
178.2
Mode match: Ang S11
1.3
0.2
8
9
10
11
12
13
1.2
f (GHz)
178.0
Angle of S21
Angle of S11
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„
177.8
177.6
177.4
177.2
1.1
1.0
CST: Ang S21
Mode match: Ang S21
0.9
177.0
0.8
176.8
176.6
8
9
10
11
12
0.7
13
8
f (GHz)
„
„
9
10
11
12
13
f (GHz)
Effects of higher order modes directly observed in phase angle
data: would be 180º or 0º for uniform resistive sheets.
Excellent agreement: good confidence in mode match solution.
Keith W. Whites
Metamaterials 2009
September 2, 2009
10
5
Lab
boratory for Applied Electromagnetics and Commun
nications
Outline
z
Creating effective sheet impedance films
z
Mode matching solution
o
z
scattering
g by
y spatially
p
y varying
y g sheet impedance
p
films in a rectangular
g
waveguide
Using perforations to create spatial variation
o
accuracy of the Maxwell/Maxwell Garnett mixing rule
z
Properties of DuPont Kapton 370 XC
z
Measurements for two region perforated Kapton 370 XC
z
Conclusions
Keith W. Whites
Metamaterials 2009
September 2, 2009
11
Spatially Varying Sheet Impedance
Lab
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z
z
z
Ideal resistive sheets with constant sheet resistance were used in
the previous simulations.
How can one create physically realizable films with varying sheet
impedance?
One approach is to use the concept of “effective film,” as
mentioned at the beginning of this presentation, and print metallic
particles onto a resistive film, or else perforate the film:
Resistive sheet
Constant effective Zs
Perforations
Increasing effective Zs
z
Perforations in a host resistive film more closely matches an
effective film than printing particles on top of the film.
Keith W. Whites
Metamaterials 2009
September 2, 2009
12
6
Accuracy of Maxwell Garnett
The effective sheet resistance of square holes in a SC lattice are
extremely well predicted by the simple Maxwell Garnett formula:
Rs ,eff = Rs ,0
‹
1 +ν
1 −ν
ν = perforation area fraction
Rs,eff extracted for 228 OPS ideal resistive film in X band waveguide
with uniform square perforations on a SC lattice at 10 GHz. Using
CST MWS (frequency domain solver) and averaging Rs,eff from S11
and S21, though no difference.
10000
Effective Rs (OPS)
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‹
Extracted from CST MWS simulation
Maxwell Garnett
No more than 2
2.4%
4% difference! MG
slightly over-predicts Rs (because
it’s a lower bound on σ).
1000
100
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Area fraction of square perforations
‹
We first observed this behavior with effective quasistatic permittivity
(conductivity) of SC lattice of cubes (Whites and Wu, JAP, 2000).
Keith W. Whites
September 2, 2009
Metamaterials 2009
13
Linearly Tapered Resistive Sheet
Comparing mode match solution to CST MWS (frequency domain)
with ideal resistive sheet: Linear taper Rs from 228 OPS to 684 OPS.
Four modes in mode match solution: has converged to four decimal
places in magnitude and two decimal places in phase (compared to
20 modes).
„
5
0.70
180
0.44
0.68
0.42
4
179
0.66
0.36
177
|S21|
0.38
0.64
Ang(S11)
178
3
0.62
CST MWS
Mode match
2
Ang(S21)
0.40
|S11|
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„
0.60
0.34
CST MWS
Mode match
0.32
176
0.30
Hole sizes NOT
linearlyy tapered!
p
8
9
10
11
1
0.56
0
8
12
9
10
11
12
Frequency (GHz)
Frequency (GHz)
„
0.58
175
From these results, conclude:
–
–
–
–
Have created effective linear tapering in the effective sheet resistance.
Maxwell Garnett formula accurately predicting effective sheet resistance.
This spatial rate of change in the effective sheet resistance is acceptable.
While geometry changing in vertical direction, it is electromagnetically uniform.
Keith W. Whites
Metamaterials 2009
September 2, 2009
14
7
Lab
boratory for Applied Electromagnetics and Commun
nications
Outline
z
Creating effective sheet impedance films
z
Mode matching solution
o
z
scattering
g by
y spatially
p
y varying
y g sheet impedance
p
films in a rectangular
g
waveguide
Using perforations to create spatial variation
o
accuracy of the Maxwell/Maxwell Garnett mixing rule
z
Properties of DuPont Kapton 370 XC
z
Measurements for two region perforated Kapton 370 XC
z
Conclusions
Keith W. Whites
Metamaterials 2009
September 2, 2009
15
Two Region Perforated Specimen
Lab
boratory for Applied Electromagnetics and Commun
nications
z
z
z
z
z
The last set of data we will show involves the measurement of a two
region specimen in a WR-90 rectangular waveguide.
Using Kapton 370 XC as the resistive sheet (~40 μm thick).
Specimen was laser cut to produce a SC lattice of square holes:
Perforation area fraction = 0.340. Using Maxwell Garnett, expect Zs
to increase by 2.03 [=(1+0.340)/(1-0.340)] over neat R card.
Some discoloration around edges of perforations. Localized heating
during laser cutting? Heating will change properties of Kapton XC.
Keith W. Whites
Metamaterials 2009
September 2, 2009
16
8
Properties of Kapton XC
‹
Kapton XC is a fairly complex material. Carbon black dispersed in
polyimide in a multilayered structure.
Complex εr from 5 specimens measured in a flanged waveguide:
80
0
εr'eff
60
51.44
40
-100
20
εr"eff
0
εr"eff
-50
εr'eff
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‹
-150
-20
-79.56/ωε0
-200
9
10
11
12
Frequency (GHz)
‹
‹
Appreciable variation between specimens: a well-known property
of Kapton XC. (Also known to be somewhat anisotropic.)
A parallel RC model with εr,eff = 51.44-j79.56/ωε0 approximately
describes the measured complex permittivity of Kapton XC within
the X-band, though a more sophisticated model is likely necessary
especially for broadband modeling.
Keith W. Whites
Metamaterials 2009
September 2, 2009
17
Lab
boratory for Applied Electromagnetics and Commun
nications
Outline
z
Creating effective sheet impedance films
z
Mode matching solution
o
z
scattering
g by
y spatially
p
y varying
y g sheet impedance
p
films in a rectangular
g
waveguide
Using perforations to create spatial variation
o
accuracy of the Maxwell/Maxwell Garnett mixing rule
z
Properties of DuPont Kapton 370 XC
z
Measurements for two region perforated Kapton 370 XC
z
Conclusions
Keith W. Whites
Metamaterials 2009
September 2, 2009
18
9
Two Region Specimen Measurement
For mode match calculations, extracted Zs
from waveguide measurement of uniform
specimens (neat R card, then perforated
specimen).
i
)
Used these measured Zs values at each
frequency to calculate scattering for the two
region specimen.
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z
z
0.80
-164
0.55
-6
-166
0.75
0.50
-168
|S21|
-172
Ang(S11)
|S11|
Measured
Mode match solution
0.40
0.70
-174
-10
0.65
Ang(S21)
-8
-170
170
0.45
-12
-176
Measured
Mode match solution
0.60
0.35
-178
0.30
-180
8
9
10
11
Frequency (GHz)
Keith W. Whites
-14
0.55
8
12
9
Measurements by T. Amert
10
11
12
Frequency (GHz)
Metamaterials 2009
September 2, 2009
19
Conclusions
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nications
9 Overall quest of this work is to develop a method for characterizing
the scattering by spatially varying impedance films.
9 Presented a rectangular waveguide mode matched-based solution
for characterizing films with a 1-D variation in sheet impedance.
9 Showed a method for spatially tailoring such films by using
perforations of varying size on a uniform grid.
9 Presented evidence that the change in sheet impedance is
extremely well predicted by the Maxwell Garnett formula.
9 Measured results with a two region specimen of neat and
perforated Kapton XC were shown.
Keith W. Whites
Metamaterials 2009
September 2, 2009
20
10
Lab
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Thank You
Keith W. Whites and Brian B. Glover
Laboratory for Applied Electromagnetics and Communications
Department of Electrical and Computer Engineering
South Dakota School of Mines and Technology
501 East Saint
i Josephh Street, Rapid
id City,
i SD 57701 USA
†Voice: +1-605-394-6861, E-mail: whites@sdsmt.edu
Sponsored by the National Science Foundation through an EPSCoR Research Infrastructure Improvement (RII) program grant (EPS0554609) and an Integrative, Hybrid & Complex Systems (IHCS) program grant (ECCS-0824034).
Keith W. Whites
Metamaterials 2009
September 2, 2009
21
11