Physical bounds on the all-spectrum transmission through periodic arrays: oblique incidence

TLDR: In this paper, the performance of a low-pass screen designed to block electromagnetic waves in a stop band is shown to have an upper bound defined by the static electric and magnetic polarizability per unit area of the screen.

Abstract: The performance of a low-pass screen designed to block electromagnetic waves in a stop band is shown to have an upper bound defined by the static electric and magnetic polarizability per unit area of the screen. The bound is easy to calculate for all angles of incidence and polarizations, and applies regardless of how complicated the screen's microstructure is. For a homogeneous dielectric sheet the bound for TM polarization is more restrictive than the bound for TE, but this is not generally true for a screen with microstructure. The results are verified by measurements and simulations of oblique transmission through an array of split ring resonators, printed on a dielectric substrate.

Figures

Figure 3: Geometry of the metal split ring resonators on a 0.3mm thick FR4 substrate with r = 4.35. The total sheet has 240 × 240 = 57 600 unit cells, and is 60 cm× 60 cm.

Figure 2: The integrated transmission blockage and its bounds (left and right hand side of (3.9) times cos θ) for a 0.3 mm dielectric sheet. Solid lines are for constant (nondispersive) relative permittivity r = 4.35, dashed lines are for frequency dependent (dispersive) relative permittivity r(k) = 1 + (4.35 − 1)/(1 − ik/kc), using kc = 2π/(1µm). The bounds are exactly on top of the simulated transmission blockage. Note the zero transmission blockage for nondispersive case in TM polarization at 64◦, corresponding to the Brewster angle.

Figure 6: The integrated transmission blockage (left hand side of (3.9) times cos θ) for measurements (dashed lines) and simulations (dotted lines) of the sheet in Fig. 3. The bounds (right hand side of (3.9) times cos θ) are solid lines.

Figure 4: Measurement setup for oblique incidence on an array of split ring resonators. The sheet was oriented with the gaps in the split rings along the vertical direction.

Figure 5: The real and imaginary parts of − lnT = − ln |T | − i arg T for θ = 23◦ (upper graph) and 67◦ (lower graph). Solid lines are measurements, dashed lines are simulations.

Figure 1: Geometry of the scattering problem.

Full text

LUND UNIVERSITY
PO Box 117
221 00 Lund
+46 46-222 00 00
Physical bounds on the all-spectrum transmission through periodic arrays: oblique
incidence
Sjöberg, Daniel; Gustafsson, Mats; Larsson, Christer
2010
Link to publication
Citation for published version (APA):
Sjöberg, D., Gustafsson, M., & Larsson, C. (2010).
Physical bounds on the all-spectrum transmission through
periodic arrays: oblique incidence
. (Technical Report LUTEDX/(TEAT-7199)/1-13/(2010); Vol. TEAT-7199).
[Publisher information missing].
Total number of authors:
3
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Electromagnetic Theory
Department of Electrical and Information Technology
Lund University
Sweden
CODEN:LUTEDX/(TEAT-7199)/1-13/(2010)
Physical bounds on the all-spectrum
transmission through periodic arrays:
oblique incidence
Daniel Sj
¨
oberg, Mats Gustafsson, and Christer Larsson

Daniel Sjöberg
Daniel.Sjoberg@eit.lth.se
Department of Electrical and Information Technology
Electromagnetic Theory
Lund University
P.O. Box 118
SE-221 00 Lund
Sweden
Christer Larsson
Christer.Larsson@saabgroup.com and Christer.Larsson@eit.lth.se
Saab Dynamics AB
SE-581 88 Linköping
Sweden
Department of Electrical and Information Technology
Electromagnetic Theory
Lund University
P.O. Box 118
SE-221 00 Lund
Sweden
Mats Gustafsson
Mats.Gustafsson@eit.lth.se
Department of Electrical and Information Technology
Electromagnetic Theory
Lund University
P.O. Box 118
SE-221 00 Lund
Sweden
Editor: Gerhard Kristensson
c

Daniel Sjöberg, Mats Gustafsson, and Christer Larsson, Lund, August 4, 2010

1
Abstract
The performance of a low-pass screen designed to block electromagnetic waves
in a stop-band is shown to have an upper bound dened by the static electric
and magnetic polarizability per unit area of the screen. The bound is easy to
calculate for all angles of incidence and polarizations, and applies regardless of
how complicated the screen's microstructure is. For a homogeneous dielectric
sheet the bound for TM polarization is more restrictive than the bound for
TE, but this is not generally true for a screen with microstructure. The results
are veried by measurements and simulations of oblique transmission through
an array of split ring resonators, printed on a dielectric substrate.
1 Introduction
In applications like radomes, spatial lters, polarizers, energy saving windows etc it is
important to understand the transmission of electromagnetic waves through a planar
screen of nite thickness, often having some microstructure. For normal incidence, it
was shown in [11] that the blockage in transmission for a given nonmagnetic screen,
integrated over all wavelengths, has an upper bound determined by the electric
polarizability per unit area of the screen. In this paper, we generalize this result to
include oblique incidence and magnetic materials, and investigate the dependence
on polarization and angle of incidence for the incident wave.
Similar physical bounds restricting the performance of passive structures have
been presented in several papers treating matching methods [5], nite size scatter-
ers [10, 2628], antennas [8, 9], absorbers [18], articial magnetic ground planes [4],
and metamaterials [7, 25,29]. A common factor for all these bounds, is that the inte-
grated electromagnetic interaction of the scatterer, antenna, or material, is bounded
by the static properties of the system. This result is a consequence of assuming the
system to be linear, causal, time translational invariant, and passive. Often, the
static properties (such as polarizability) can be directly associated with properties
such as the volume of the scatterer or similar. For instance, variational principles
can be used to show that the electric polarizability of a given body, with or without
inhomogeneous microstructure, is bounded above by the electric polarizability of a
circumscribed metal body [12,21].
The scattering of electromagnetic waves by an isotropic slab is well known and
documented in many text books, see for instance [3, 17,19]. The results have been
generalized to homogeneous slabs of arbitrary bianisotropic materials [14,23], but
when the slab is inhomogeneous, for instance by loading it with metal inclusions, it
is often necessary to resort to numerical methods to calculate reection and trans-
mission. The low-frequency limit for arbitrary slabs has been derived in [21], where
it is seen that the low-frequency asymptotic is given by the electric and magnetic
polarizability per unit area of the screen.

2
x
y
z
E
E
i
r
E
t
d
a
x
a
y
µ
Figure 1
: Geometry of the scattering problem.
2 Analyticity of the transmission coecient
We start by investigating the analyticity of the transmission coecient. A typical
geometry of the problem is depicted in Fig. 1, where a plane wave is incident on a
periodic structure with thickness
d
. The incident eld has constant polarization
E
(i)
0
and unit propagation direction
ˆ
k
, and can hence be written
E
(i)
(r, t) = E
(i)
0
f(t −
c
−1
0
ˆ
k · r)
, where the time dependence satises
f(t) = 0
for
t < 0
. Due to causality,
the total eld is zero until the plane wave has arrived,
i.e.
,
E(r, t) = 0
when
t − c
−1
0
ˆ
k · r < 0
(2.1)
Since the geometry is periodic, we also have the translational invariance
E(r + r
n
, t) = E(r, t − c
−1
0
ˆ
k · r
n
)
(2.2)
where
r
n
= n
1
a
1
+ n
2
a
2
,
n
1
and
n
2
taking integer values. The vectors
a
1
and
a
2
are lattice vectors in the
xy
-plane, forming the sides of the unit cell
U
, which has
area
A = |
ˆ
z ·(a
1
×a
2
)|
. Using the causality property (2.1), we can write the Fourier
transform of the eld as (where
k = ω/c
0
is the wave number in vacuum,
c
0
being
the speed of light in vacuum and
ω
the angular frequency)
E(r, k) =
Z
∞
−∞
E(r, t)e
ikc
0
t
dt =
∞
Z
c
−1
0
ˆ
k·r
E(r, t)e
ikc
0
t
dt
= e
ik
ˆ
k·r
Z
∞
0
E(r, t + c
−1
0
ˆ
k · r)e
ikc
0
t
dt
| {z }
˜
E(r,k)
(2.3)
The function
E(r, t + c
−1
0
ˆ
k · r)
is periodic in
r
due to the translational invariance
(2.2). This property is inherited by
˜
E(r, k)
, which is also analytic in
k
for
k
in the

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Physical bounds on the all-spectrum transmission through periodic arrays: oblique incidence" ?

In this paper, the authors derived a bound on the all-spectrum transmission blockage through a low-pass periodic screen for oblique incidence in ( 3.9 ). 

Q2. Why is the longitudinal polarizability ezz smaller than the transverse series?

For TM polarization the longitudinal polarizability γezz can be associated with the low frequency series inductance of the sheet [23], implying that further gain in bandwidth for TM polarization might be achieved by increasing the equivalent series inductance of the sheet. 

Q3. What is the importance of understanding the transmission of electromagnetic waves through a planar screen?

In applications like radomes, spatial lters, polarizers, energy saving windows etc it is important to understand the transmission of electromagnetic waves through a planar screen of nite thickness, often having some microstructure. 

Q4. What is the generalization of the results of the electromagnetic scattering of a slab?

The results have been generalized to homogeneous slabs of arbitrary bianisotropic materials [14, 23], but when the slab is inhomogeneous, for instance by loading it with metal inclusions, it is often necessary to resort to numerical methods to calculate re ection and transmission. 

Q5. What can be used to show that the electric polarizability of a given body?

For instance, variational principles can be used to show that the electric polarizability of a given body, with or without inhomogeneous microstructure, is bounded above by the electric polarizability of a circumscribed metal body [12, 21]. 

Q6. Why is the TE result independent of the frequency dependent material?

It is further seen that the overall level of transmission blockage is increased for the dispersive material, particularly for normal incidence, and the TE result is independent of the angle of incidence (except for the scaling with cos θ), due to the high frequency response being tuned to vacuum. 

Q7. What is the common factor for all these bounds?

A common factor for all these bounds, is that the integrated electromagnetic interaction of the scatterer, antenna, or material, is bounded by the static properties of the system. 

Q8. What is the polarizability of the tefl?

The static polarizability factors are [21]γexx = γeyy = Ad( r(0)− 1), γezz = Ad(1− r(0)−1) (5.2)whereas the high frequency refractive index is n∞ = √ ∞, where ∞ = limk→∞ r(k) is the high frequency limit of the relative permittivity. 

Q9. What is the polarizability of the screen?

1. The electric and magnetic polarizability matrices γe and γm give the total electric and magnetic dipole moment per unit area induced in the screen when subjected to homogeneous elds E0 and H0 as p/A = 0γe · E0/A and m/A = γm ·H0/A, respectively. 

Q10. What is the transmission coe cient for a dielectric sheet?

T (k) = (1− r0(k)2)ei(β(k)−β0(k))d1− r0(k)2ei2β(k)d (5.1)where the wave numbers in the material and in the surrounding free space are given by β(k)2 = k2( r(k) − sin2 θ) and β0(k) = k cos θ, respectively, and the interface re ection coe cient is r0 = (Z − Z0)/(Z + Z0) with Z = η0k/β and Z0 = η0/ cos θ for TE polarization, and Z = η0β/( r(k)k) and Z0 = η0 cos θ for TM polarization. 

Q11. What can be the common property of a static electromagnetic system?

the static properties (such as polarizability) can be directly associated with properties such as the volume of the scatterer or similar. 

Q12. What is the polarization of the dielectric substrate?

The bounds11were demonstrated theoretically and experimentally using nonmagnetic structures, and it was seen that a composite structure with metal patterns like split ring resonators on a dielectric substrate can increase the transmission blockage by an order of magnitude compared to the pure dielectric substrate. 

Q13. What is the low frequency limit for arbitrary slabs?

The low-frequency limit for arbitrary slabs has been derived in [21], where it is seen that the low-frequency asymptotic is given by the electric and magnetic polarizability per unit area of the screen. 

Q14. What is the polarization of the tm?

For the nondispersive permittivity, the Brewster angle phenomenon is clearly seen in Fig. 2: at the Brewster angle θB = arctan( √ r) = 64◦, the interface re ection coe cient r0 for TM polarization is exactly zero, and the transmission coe cient has unit amplitude |T | = 1.