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Fabry-Perot Resonances of Total Transmission in Multilayer Sub-wavelength Partially-Reflecting

Surfaces

C. S. R. Kaipa, A. B. Yakovlev, F. Medina, and F. Mesa

National Radio Science Meeting

University of Colorado at Boulder

6 - 9 January, 2009

Outline

Introduction and Motivation

Analytical Modeling Dynamic Model Circuit Model

Fabry-Perot Resonances

Multilayer Structure Transmission Spectra

Field Distribution

Conclusion

Multilayer Metal-Dielectric Stack

The highest frequency peak is associated with a low field density

inside the metal films,

while the lowest frequency peak corresponds to a situation where

field inside metal is quite strong

Scalora et al., Jour. App. Physics , 83, 2377–2383 (1998).

Feng et al., Phys. Rev. B. , 82, 085117(2005)

General PBG structure Metallic PBG structure

Can a similar effect be observed

at microwaves??

Motivation

The metal films are substituted by perforated

metal layers

Microwave transmissivity of a metamaterial-dielectric stack

Butler et al., Appl. Phys. Lett. , 95, 174101 (2009)

Yakovlev et al., 3rd Int. Congress on Advan. Electromag. Materi. in Microwa. and Optic.,(2009)

Dynamic Model

h r

We use the transfer matrix approach to

characterize the reflected and transmitted fields

Transfer Matrix: Q

Two sided impedance boundary condition

• continuous electric fields across the grid

• discontinuous magnetic fields

Transfer Matrix: P

Propagation through the dielectric region

Global Transfer Matrix Mg = Q.P……….P.Q

z

x y

E

H

Circuit Model

effD 0Z0 0Z

0dZdgZ

h

Typical waveguide problem with discontinuities

pec

pec

pec

pec

m.w m.w hwD

h

hw

δjε

Z; Z

ε

μZ

r

dtan1

0

0

00

δjεβ; βc

ωβ rd tan100

Dc

w

w

D

x

D y

h

Grid impedance

• Dynamic model for effective grid

impedance of fish-net grid

• Averaged impedance boundary

condition

• Approximate Babinet principle

D = 2 mm, w = 0.2 mm, h = 1 mm

dielectric permittivity: 10.2

Luukkonen et al., IEEE Trans. Antennas Propagat., 56, June 2008

Effective Grid Impedance

220

2

1

( ) 2 11 sin

2

effTE

g

eff

pZ j

p g k

k

( ) 2

effTM

g

pZ j

p g

ln csc

2

effk p g

p

sD 3

Incident wave

h

x

z

y z = L

z = 0

Paired Screens of Patches

iE

iH

Substrate thickness: 2 mm

Period: 2 mm

Gap: 0.2 mm

Dielectric permittivity: 10.2

h+∆h h

∆h is the excess lengths associated

with the edge capacitances

8 9 10 11 12 13 14 15 16 1717

-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

|S

21

| d

B

|S21| - Analytical 0

|S21| - Analytical 30

|S21| - Analytical 60

|S21| - HFSS 0

|S21| - HFSS 30

|S21| - HFSS 60

Total Transmission

8 9 10 11 12 13 14 15 16 1717

-30

-25

-20

-15

-10

-5

0

Frequency (GHz)|S

21

| d

B

|S21| - Analytical 0

|S21| - Analytical 30

|S21| - Analytical 60

|S21| - HFSS 0

|S21| - HFSS 30

|S21| - HFSS 60

Fabry-Perot resonances is the underlying mechanism

h

x

z

y

z = L

z = 0

Paired Screens of Fishnets

iE

iH

Incident wave

Substrate thickness: 2 mm

Period: 2 mm

Strip width: 0.2 mm

Dielectric permittivity: 10.2

10 11 12 13 14 15 16 17 18 19 2020

-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

|S

21

| d

B

|S21| - Analytical 0

|S21| - Analytical 30

|S21| - Analytical 60

|S21| - HFSS 0

|S21| - HFSS 30

|S21| - HFSS 60

Total Transmission

10 11 12 13 14 15 16 17 18 19 2020

-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

|S

21

| d

B

|S21| - Analytical 0

|S21| - Analytical 30

|S21| - Analytical 60

|S21| - HFSS 0

|S21| - HFSS 30

|S21| - HFSS 60

Fabry-Perot resonances is the underlying mechanism

Fabry-Perot Type Resonances

Medina, Mesa, Skigin, IEEE Trans. Microw. Theory Tech., 2009

Y

Y

Y Port 2 Port 1

h

gY

hjYYY

hjYYYS

g

ge

tan

tan

gY

hjYYY

hjYYYS

g

go

cot

cot

Even excitation Odd excitation

Total transmission

S

22

0

2

1

1

1

2tan

g

g

YYY

YYjh

Dispersion equation with roots corresponding

to total transmission h

1

1 1 22 2

1 0

2tan 0

g

g

Y CH h h

Y Y C

Dispersion equation for two-sided

patch array (capacitive grid)

1

1 1 22 2

1 0

2tan 0

1/

g

g

Y LH h h

Y Y L

Dispersion equation for two-sided

fishnet array (inductive grid)

Total Transmission

Frequency of total transmission

for two-sided patch structure

Excess length associated with edge capacitance

h

(mm) (GHz)

Calculated

(GHz)

Excess length

1 17.2105 13.0606

2 11.425 10.2174

4 7.279 7.1182

6 5.458 5.4616

8 4.392 4.4304

10 3.686 3.7268

TTf

TTf

TE/TM

r

TE/TM

TT

hhθε

cf

Δsin

θε

Ccηh

2

r

TE

g0TE

sinΔ

r

TM

g0TM

ε

Ccηh

Δ

1 1.5 2 2.5 3 3.5-15

-10

-5

0

5

10

15

1h, rad

H(

1h

)

h = 1 mm

h = 2 mm

h = 4 mm

h = 6 mm

h = 8 mm

h = 10 mm

Total Transmission

h (mm)

(GHz

Calculated (GHz)

Excess length

1 24.595 21.5196

2 15.455 14.7546

4 9.165 9.0589

6 6.566 6.5359

8 5.124 5.1121

10 4.203 4.1977

TTf

TTf

Frequency of total transmission

for two-sided fishnet structure

Excess length associated with edge inductance

TE/TM

r

TE/TM

TT

hhθε

cf

Δsin

η

cLh

TE

gTEΔ 2

0

2

sin

TM

r gTM

r

c Lh

1 1.5 2 2.5 3 3.5-15

-10

-5

0

5

10

15

1h, rad

H(

1h

)

h = 1 mm

h = 2 mm

h = 4 mm

h = 6 mm

h = 8 mm

h = 10 mm

Geometry

Substrate thickness (h): 6.35 mm

Period (D): 5 mm

Strip width (w): 0.15 mm

Dielectric permittivity: 3

Loss tangent = 0.0018

Thickness of grid = 18 µm

D

D w

w

Power Transmission Spectra

5 6 7 8 9 10 11 12 13 14 15 160

0.2

0.4

0.6

0.8

11

Frequency (GHz)

|S21|2

|S21|2 FEM model

|S21|2 Experimental

|S21|2 Analytical

A

B C D

Butler et al., Appl. Phys. Lett. , 95, 174101 (2009)

Pass Band

What is the nature of these

resonances ??

can we tune them

can we predict the pass band

The number of transmission

peaks are equal to the number

of layers

Electric Field Distributions

06.3512.719.0525.425.4-3

-2

-1

0

1

2

33

Distance along Z (mm)

Ey V

/m

Analytical

FEM model

06.3512.719.0525.425.4-3

-2

-1

0

1

2

3

Distance along Z (mm)

Ey V

/m

Analytical

FEM model

06.3512.719.0525.4-3

-2

-1

0

1

2

3

Distance along Z (mm)

Ey V

/m

Analytical

FEM model

06.3512.719.0525.4-3

-2

-1

0

1

2

33

Distance along Z (mm)

Ey V

/m

Analytical

FEM model

Butler et al., Appl. Phys. Lett. , 95, 174101 (2009)

Mode A Mode B

Mode C Mode D

Frequencies of interest are the ones

corresponding to the lower and the

upper band edges

Electric Field Distributions

06.3512.719.0525.431.7538.144.4550.857.1557.15-3

-2

-1

0

1

2

33

Distance along Z (mm)

Ey V

/m

06.3512.719.0525.431.7538.144.4550.857.1557.15-3

-2

-1

0

1

2

33

Distance along Z (mm)

Ey V

/m

The field pattern for the first and the last resonance is of the same

qualitative behavior

The phase shift from cell to cell along z is close to zero for the first

resonance and close to π for the last resonance.

The value of the upper limit is close to the resonance frequency of

the single dielectric layer ignoring the grids.

Limiting Case

No of Layers (GHz) (GHz)

4 7.004 11.61

5 6.78 12.2

6 6.664 12.56

10 6.468 13.19

18 6.38 13.49

36 6.38 13.6

LBfUBf

Increasing the number of layers

Resonance peaks

Transmission band

Lower and upper band edges

5 7 9 11 13 15 16160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11

Frequency GHz

|S2

1|2

|S21|2

Can be analyzed using periodic structures

Propagation Characteristics

d/2 d/2

)sin(2

coscosh1

kdZY

jkddg

Upper Limit Condition

Lower Limit Condition

1cosh d 0)sin(,1cos kdkd

kd

1)sin(2

cos1

kdZY

jkdg

Brillouin Diagram

Transmission band coincides

with the finite structure

Lower limit is influenced

on grid impedance

Upper limit is solely controlled

by the slab thickness

The observed resonances of transmission at low frequencies in multilayered sub-wavelength grids correspond to Fabry-Perot type resonances of a dielectric slab loaded with effective grid admittances

Analytical formulas for frequencies of transmission are obtained for two-sided patch arrays and wire grids in terms of an excess length associated with effective edge capacitance and inductance

The range of frequencies where the peaks are expected for a finite stacked structure can be analytically and accurately estimated from the Bloch analysis using the proposed circuit model

Conclusion