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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