comparative study of surface waves on high-impedance...

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

http://www.tkk.fi/en/index.html

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


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


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


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


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


18


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


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


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


Mushroom HIS structure


TM proper complex surface wave associated with backward radiation

25

TM Surface Waves


ENG approximation Close-up figure

Non-physical higher-order surface-wave modes close to the plasma

frequency, wherein the ENG approximation fails

26



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


radius of vias: 0.05 mm

28


Mushroom-like Jerusalem cross HIS

ENG approximation


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

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Patch HIS structure


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33



Period: 1.5 mm

34



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

comparative study of surface waves on high-impedance...

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