analytical modeling of surface waves on high-impedance surfacesyakovlev/presentations/... ·...

Analytical Modeling of Surface

Waves on High-Impedance

Surfaces

A. B. Yakovlev, C. R. Simovski, S. A. Tretyakov

O. Luukkonen, G. W. Hanson, S. Paulotto,

P. Baccarelli

NATO Advanced Research Workshop

Metamaterials for Secure Information

And Communication Technologies

Marrakesh, Morocco, 7 – 10 May, 2008

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

2

Outline

Introduction and Motivation Model 1 – Impedance Surface Model 2 – Grounded Dielectric Slab with

Grid Impedance Series-Resonant Grid Model Jerusalem Cross Array Patch Array Mushroom Array

Conclusion

3

Motivation

HIS structures with electrically small FSS elements

Homogenized FSS grids for far-field and near-

field sources

Metamaterial substrates Wire media slabs Slabs with spherical inclusions

Nanotechnology

4

Introduction

Analytical modeling of dense FSS grids Homogenization of impedance surface in terms of effective circuit parameters Homogenization limit of full-wave scattering problem via the averaged impedance boundary condition Parallel resonance of grid and slab surface

impedances

Single unit cell of periodic grid and a single Floquet mode

Babinet principle

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 in the quasi-static limit of 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

dielectric permittivity: 2.7

t

w

d

g

D

h

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

• Quasi-static 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


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


Patch HIS structure


20

Wire Media Slab

z

x

a

02rGround plane z

y

E

H

a k

h 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

h 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

h 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

h 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


25




26



Period: 1.5 mm

27



Period: 2.5 mm

28

Accurate and rapid analysis of surface-wave propagation on dense HIS structures (low frequency approximation)

Analytical modeling is based on the quasi-static 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 HIS structures at low frequencies are due to occurrence of TM backward surface waves associated with wire media slab and capacitive grid

Conclusion

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Mário Silveirinha University of Coimbra, Coimbra, Portugal

Igor Nefedov Helsinki University of Technology,

Helsinki, Finland

Acknowledgment

analytical modeling of surface waves on high-impedance surfacesyakovlev/presentations/... ·...

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