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Broadband free-space characterization of metamaterials
M. H. Belyamoun, O. Dubrunfaut, C. Pareige, Y. Zhu, S. Zouhdi, and F. Ossart
Laboratoire de Genie Electrique de Paris
Introduction
Metamaterials
Artificial periodic structures with exotic electromagnetic properties at themacroscopic scale.
a) Epsilon-negative rods.
b) Split Ring Resonator (negative permeability)
c) High Impedance Surface : Sievenpiper mushrooms.
a b c
2 / 18
Summary
1 Introduction
2 Numerical modelSRR homogenizationHIS homogenization
3 The free space characterization systemA focused systemThru Reflect Line calibrationTime domain analysis
4 Free space measurements of metamaterialsShifted metallic wire rodsHigh Impedance SurfaceSRR characterization
5 Conclusions and outlook
Numerical model SRR homogenization
Reference cell
The SRR acts as an LC resonator.
We want to compute the effective permeability : (~B = µeff~H).
Incident plane wave.
Long wavelength λ a
Perfect conductor : h = 0 inside the ring.4 / 18
Numerical model SRR homogenization
From the split-ring to a closed ring
Cell energy
|C|Hµeff H =
∫Aµ|∇ϕ|2 +
∫∂R
1− i
σωδ|∇Sϕ|2 −
1
Cω2I2
where ϕ stands for the magnetic potential.
5 / 18
Numerical model SRR homogenization
Polaritonic law for 2D-structures
h = h1 over A1 and h = h2 over A2
Faraday : iωΦ + V = 0 and Ampere : I = h1 − h2.
Magnetic field flux : |C|B = µ (|A1|h1 + |A2|h2) andΦ = µh1|A1|Capacitor in the air gap : I = iωCV .
A1
A2
µr
ωω1 ω2
A2
|C|
A1+A2
|C|µeff = µ
|A1|+ |A2||C|
(ωω2
)2
− 1(ωω1
)2
− 1
ω1 = (µC |A1|)−12
ω2 =√
1 + |A1|/|A2| ω1
6 / 18
Numerical model HIS homogenization
Effective impedance computation
a b
Symmetric HIS :
I Impedance extension from 1D → 2D
Asymmetric HIS :
a) Flux method∫C iω(εEiEj − µHiHj ) = |S|E ′js 1
ZijEis
b) Surface impedance method(−H1sy −H2sy
H1sx H2sx
)= −
(Yxx Yxy
Yyx Yyy
)(E1sx E2sx
E1sy E2sy
)7 / 18
The free space characterization system A focused system
The free-space characterization system
2 quad-ridged horn antennas (2-18 GHz), mounted on micrometric stages.
The beam is focused with Rexolite (εr = 2.54) lenses.
Agilent PNA 8364C : S-parameters measurement.
8 / 18
The free space characterization system A focused system
Gaussian beam
Gaussian beam.
We must have an incident planar wave on the sample to compute theeffective parameters.
Negligible diffraction if the sample dimension is 3 times the beam’s radius.
A focused bench is much smaller.9 / 18
The free space characterization system Thru Reflect Line calibration
Thru Reflect Line calibration
Thru
e e
Reflect
metal
e e
Line
10 / 18
The free space characterization system Thru Reflect Line calibration
S parameters of a plexiglas sample (e=5 mm)
2 4 6 8 10 12 14 16 180
0.5
1
1.5
Frequency[GHz]
Mag
nitu
de
S11
S21
2 4 6 8 10 12 14 16 18
−150
−100
−50
0
50
100
150
Frequency[GHz]
Pha
se°
S11
S21
11 / 18
The free space characterization system Time domain analysis
Time domain analysis
0 50 100 150 200 250 300 350 400−160
−140
−120
−100
−80
−60
−40
−20
0
tn
S11
[dB
]
Reflection onthe lens
Sample
reflection on the SMA connector
Wn
tn−p p
RectangularTriangularHanning
The calibrated S parameters are transfered to the time domain.
A window is applied to eliminate the parasite reflections.
Gibbs phenomenon limits the selectivity of the filter.
12 / 18
The free space characterization system Time domain analysis
Influence of the filter type (Plexiglas, e = 5 mm)
2 4 6 8 10 12 14 16 180
0.5
1
1.5
Frequency [GHz]
Mag
S11
S21
Hanning
2 4 6 8 10 12 14 16 180
0.5
1
1.5Mag
Frequency[GHz]
Mag
S11
S21
Window
13 / 18
Free space measurements of metamaterials Shifted metallic wire rods
Shifted metallic wire rods
Characterization in an anechoicchamber.
Time domain with a Hammingwindow.
2 4 6 8 10 12 14 16 18−80
−70
−60
−50
−40
−30
−20
−10
0
10
Fréquences[GHz]
Mag
nitu
de [d
B]
S11
filtrage
S11
anéchoïque
S21
filtrage
S21
anéchoïque
14 / 18
Free space measurements of metamaterials High Impedance Surface
HIS characterization
Epoxy εr = 4, 4h = 2.4 mm
Sievenpiper
Zeff = jωL1−LCω2
2 4 6 8 10 12 14 16 18−2
−1
0
1
2
3
4
Frequency [GHz]
Ze
ff [Ω
]
Z’Z’’
D [mm] g [mm] Measured [GHz] Analytical [GHz]10 4 9,84 9,47 2 10 9,36 2 10,85 10,86
10 5 11,75 11,2
15 / 18
Free space measurements of metamaterials High Impedance Surface
Comparison with the simulations
8 10 12 14 16 18−200
−150
−100
−50
0
50
100
150
200
frequency(GHz)
Pha
se o
f coe
ffici
ent o
f ref
lect
ion
°
Numerical CSTNumerical Poynting fluxNumerical <E>/<H>Measure
Method Frequency [GHz] Sample (D,g)(10,4) (10,5) (7,2) (6,2)
Measurements Resonance 10,35 11,74 10 10,84Bandwidth 9,3-11,37 10,33-12,77 8-11,25 7,06-12,34
〈E〉/〈H〉 Resonance 9,9 11,74 10,3 11,9Bandwidth 8,8-10,8 10,3-12,7 8,9-11,7 10,25-13,75
Poynting flux Resonance 9,4 11,5 10,8 12Bandwidth 8,3-10,3 10,2-12,4 9,3-12,2 10,6-13,75
CST Resonance 9,54 11,23 10,1 11,82Bandwidth 8,56-10,42 10-12,17 8,7-11,42 9,97-13,76
16 / 18
Free space measurements of metamaterials SRR characterization
SRR characterization : rint = 1 mm, rext = 2, 5 mm
d (mm) Resonance frequency [GHz]Measured Analytical Homogenization
0,8 9,4 9,37 9,470,9 9,6 9,94 10,051 10,32 10,47 10,59
6 7 8 9 10 11 12 13 14 15−35
−30
−25
−20
−15
−10
−5
0
Frequency [GHz]
Mag
nitu
de [d
B]
S11
S21
6 7 8 9 10 11 12 13 14 15−180
−160
−140
−120
−100
−80
−60
−40
−20
0
Frequency [GHz]
Ph
ase
°
S11
S21
17 / 18
Conclusions and outlook
Conclusions and outlook
Homogenization of SRR and HIS.
Realization of a compact characterization system.
The S-parameters are filtered in the time domain.
Computation εeff (split-ring).
Homogenization of SRR based structures.
Homogenization of metallic wireds arrays.
Gated Reflect Line calibration.
18 / 18
