comparative study of surface waves on high-impedance...
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Comparative Study of Surface
Waves on High-Impedance
Surfaces With and Without Vias
O. Luukkonen, A. B. Yakovlev, C. R. Simovski,
and S. A. Tretyakov
AP-S International Symposium
URSI National Radio Science Meeting
San Diego, California
5 – 11 July, 2008
2
Outline
Introduction and Motivation Surface Waves on HIS Structures
Without Vias Dynamic Models Patch and Jerusalem Cross Arrays
EBG Properties of HIS Structures With
Vias Wire Media Slab and Mushroom Array Mushroom-like Jerusalem Cross Array
Conclusion
3
Introduction
Analytical modeling of dense FSS grids and HIS structures with and without vias
Homogenization of impedance surface in terms of effective circuit parameters Dynamic model obtained from full-wave scattering problem via the averaged impedance boundary condition Homogenization of wire media slab and
mushroom-like HIS structures – ENG approximation
4
Surface Waves on HIS Structures
Without Vias
Dynamic Models
5
Model 1 – Impedance Surface
Zs
Zg Zd ηo g d
s
g d
Z ZZ
Z Z
Transmission Line Model
S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003
Impedance Surface Model
No fields beyond the impedance surface
Ey
Ez
Hx
TMz
Hy
Hz
Ex
TEz
z sZ
h
6
Model 1 – Impedance Surface
S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003
Impedance Surface Model
TEz TMz
Impedance Boundary Condition
at y=0:
ˆsE Z y H
0
2
0 00
1
TE TEz yjk z k yTE
x s z x
TE TE
y zTE TE
s s
E Z H E E e
jk k k
Z Z
0
2
0 0
0
1
TM TMz yjk z k yTM
z s x x
TMTM TM TM sy s z
E Z H H H e
Zk j Z k k
No fields beyond the impedance surface
Ey
Ez
Hx
TMz Hy
Hz
Ex
TEz
z sZ
7
Model 2 -
y
air
slab h
PEC
gZgrid
z
1 1
2 2
Two-sided impedance boundary condition at y=h
1 2 1 2ˆ
gE E Z y H H
Grounded Dielectric Slab with Grid
Impedance on Air-Dielectric Interface
Hy
Hz
Ex
TEz
Ey
Ez
Hx
TMz
8
Dispersion Equations
TEz-odd TMz-even
Two-sided impedance boundary condition at y = h
1 21 2
TE
x x g z zE E Z H H1 2 1 2
TM
z z g x xE E Z H H
Dispersion equations
2 21 2 2
1
coth( )y y y TE
g
jk k k h
Z
2 1
2 2
1 1
tanh( )yTM
y y g TM
g y
j kk k h Z
j Z k
9
Complex Wavenumber Plane
Branch points in the complex -plane at zk1zk k
1Re{ } 0yk - proper modes on the top Riemann sheet
1Re{ } 0yk - improper modes on the bottom Riemann sheet
1Re{ } 0yk - branch cuts condition
Hyperbolic -plane branch cuts: zk1 1
1
Im{ }Re{ }Im{ }
Re{ }
Re{ } Re{ }
z
z
z
k kk
k
k k
2 2
2
2 2
1/ 2
0 0
y z ii
i i
k k k
k nc
c
1Im{ / }zk k
1 -1 1Re{ / }zk k
10
g d
s
g d
Z ZZ
Z Z
Zs
Zg Zd ηo
HIS
S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003
k
θ
Transmission-Line Network Analysis
0
0
cos
cos,
s
sTE
Z
Z
cos
cos,
0
0
s
sTM
Z
Z
Reflection coefficient
Parallel resonance
0dg XX
11
0
2
0
, tan
/
TE TE TE
d z ydTE
r z
jZ k k h
k k
2( ) ( )
0/TE TM TE TM
yd zk k krc
2
00
2
0
/, tan 1
/
TM
zTM TM TM
d z ydTM r
r z
k kjZ k k h
k k
Impedance of the grounded dielectric slab “seen” by surface waves
TMz -
Where
is the vertical component of the
wave vector of the refracted wave
Dielectric Impedance
S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003
TEz -
12
The grid impedance is obtained from the full-wave scattering problem via the averaged impedance boundary condition and expressed in terms of effective circuit parameters (effective inductance and effective capacitance)
, ,, , ,TE TE TE TE TE TE
g z g L z g C zZ k Z k Z k
Grid Impedance
Homogenized grid impedance “seen” by surface waves
TEz -
, ,, , ,TM TM TM TM TM TM
g z g L z g C zZ k Z k Z kTMz -
13
Jerusalem Cross Array
D = 4 mm, d = 2.8 mm
t = w = 0.2 mm
h = 6 mm, g = 0.1 mm
dielectric permittivity: 2.7
D
h
t
w
d
g
x
z
r
14
Effective Inductance & Capacitance
Where
Here:
eff
gL2
0F
D
gcsclndC rg
2
10
w
Dlog
kD 22
2
2
4
23
11
udu
uQ
uQF
2
1d
Q
d
gu
2cos 2
k
2
C. R. Simovski, P. de Maagt, and I. V. Melchakova, “High-impedance surfaces having stable resonance with respect to
polarization and incidence angle,” IEEE Trans. Antennas Propagat., Vol. 53, no. 3, pp. 908-914, Mar. 2005
N. Marcuvitz, Waveguide Handbook, Peter Peregrinus Ltd, 1986
S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003
d D
g
w 1TE
g g
g
Z j Lj C
2 1, 1 /TM TM TM
g z g z eff
g
Z k j L k kj C
15
Dispersion Behavior of Surface Waves
P. Baccarelli, S. Paulotto, and C. Di Nallo, “Full-wave analysis of bound and leaky modes propagating
along 2D periodic printed structures with arbitrary metallization in the unit cell,” IET Microwave
Antennas Propagat., Vol. 1, No. 1, pp. 217-225, 2007.
Jerusalem cross HIS structure
Comparison with full-wave results
16
Surface Impedance of HIS
Jerusalem cross HIS structure
Surface impedance of HIS “seen” by surface waves
17
Patch Array
w
w
D
x
D z
h
2,
12 1
2
effTE TE
g zTE
z
eff
Z k j
k
k
2
effTM
gZ j
O. Luukkonen, C. R. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko, A. V. Raisanen, and S. A.
Tretyakov, “Simple and accurate analytical model of planar grids and high-impedance surfaces
comprising metal strips or patches,” http://arxiv.org/abs/0705.3548.
Grid impedance
• Dynamic solution of 2D strip
grid scattering problem
• Averaged impedance boundary
condition
• Approximate Babinet principle ln csc2
effk D w
D
D = 2 mm, w = 0.2 mm, h = 1 mm
dielectric permittivity: 10.2
18
Dispersion Behavior of Surface Waves
P. Baccarelli, S. Paulotto, and C. Di Nallo, “Full-wave analysis of bound and leaky modes propagating
along 2D periodic printed structures with arbitrary metallization in the unit cell,” IET Microwave
Antennas Propagat., Vol. 1, No. 1, pp. 217-225, 2007.
Patch HIS structure
Comparison with full-wave results
19
EBG Properties of Mushroom-Like
HIS Structures
ENG Approximation
20
Wire Media Slab
z
x
a
02rGround plane z
y
E
H
a k
L r
0
eff
r a
a
Anisotropic material characterized by effective permittivity
Quasi-static approximation (ENG approximation)
0ˆ ˆ ˆ ˆ ˆˆ
eff r yyxx yy zz2
2
2
0 0
2 /
ln4 ( )
p
ak
a
r a r
pk is the plasma wavenumber
2
2
0
1p
yy
r
k
k
21
Surface Impedance of Wire Media Slab
Ground plane z
y
E
H
a k
L r
0
eff
r a
a
2
2
0
1p
yy
r
k
k
2
0 2
0
tanˆ1
yd zt t
yd r yy
k h kE j n H
k k
2 2 2
0
0 2 2
0
tan yd r p zTM
d
yd r p
k h k k kZ j
k k k
Impedance boundary condition
at y=h:
Surface impedance 2
2
0z
yd r
yy
kk k
S. A. Tretyakov, Analytical Modeling in Applied Electromagnetics, Boston, MA: Artech House, 2003
22
Mushroom Array
a
Ground plane z
y
E
H
a k
L r
0
eff
r a
g a
a
g
z
x a
02r
g
g
Zs
Zg Zd ηo
g d
s
g d
Z ZZ
Z Z
23
Mushroom Array
a
Ground plane z
y
E
H
a k
L r
0
eff
r a
g a
a
g
z
x a
02r
g
g
Period of vias: 2 mm
Period of patches: 2 mm
Gap: 0.2 mm
Radius of vias: 0.05 mm
Substrate thickness: 1 mm
Dielectric permittivity: 10.2
24
Dispersion Behavior of Surface Waves
Mushroom HIS structure
Comparison with full-wave results
TM proper complex surface wave associated with backward radiation
25
TM Surface Waves
Mushroom HIS structure
ENG approximation Close-up figure
Non-physical higher-order surface-wave modes close to the plasma
frequency, wherein the ENG approximation fails
26
Dispersion Behavior of Surface Waves
Mushroom HIS structure
Transition from backward to forward radiation:
from proper complex to improper complex (leaky wave)
27
Mushroom-Like Jerusalem Cross Array
a
Ground plane z
y
E
H
a k
L r
0
eff
r a
g a
a
g
a
t
w
d
g
x
z
a = 4 mm, d = 2.8 mm
t = w = 0.2 mm,
L = 6 mm, g = 0.1 mm
dielectric permittivity: 2.7
radius of vias: 0.05 mm
28
Dispersion Behavior of Surface Waves
Mushroom-like Jerusalem cross HIS
ENG approximation
Comparison with full-wave results
29
Accurate and rapid analysis of surface-wave propagation on dense HIS structures (low frequency approximation)
Dynamic model is based on the approximation of full-wave scattering problem via the averaged impedance boundary condition. A homogenized surface grid impedance is expressed in terms of effective circuit parameters
It is observed that in dense HIS structures no stopband between TE and TM surface-wave modes occurs at low frequencies. This is in contrast to conventional FSS structures, wherein stopbands occur due to Bragg’s diffraction at resonance frequency
Stopbands in mushroom-like HIS structures at low frequencies are due to occurrence of TM backward surface waves associated with wire media slab and capacitive grid. Proper complex surface waves associated with backward radiation occur in the stopband of proper bound modes
Conclusion
30
Mário Silveirinha University of Coimbra, Coimbra, Portugal
Igor Nefedov Helsinki University of Technology, Helsinki, Finland Paolo Baccarelli University of Rome “La Sapienza”, Rome, Italy Simone Paulotto University of Rome “La Sapienza”, Rome, Italy George Hanson University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, USA
Acknowledgment
31
Surface Impedance of HIS
Patch HIS structure
Surface impedance of HIS “seen” by surface waves
32
Surface Impedance of HIS
Mushroom HIS structure
Surface impedance of HIS “seen” by surface waves
33
Dispersion Behavior of Surface Waves
Mushroom HIS structure
Period: 1.5 mm
34
Dispersion Behavior of Surface Waves
Mushroom HIS structure
Period: 2.5 mm
35
Motivation
HIS structures with electrically small FSS elements
Homogenized FSS grids for far-field and near-
field sources
Metamaterial substrates and EBG structures – Homogenization models
Wire media slabs Slabs with spherical inclusions Mushroom-like EBG structures
Nanotechnology
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