Complementarity, duality and symmetry
Transcript of Complementarity, duality and symmetry
Department of Information Engineering and Mathematical
Science, University of Siena, Siena, Italy
Complementarity, duality and symmetry
Enrica Martini
Exploiting symmetries in artificial materials for antenna applications
About Siena
Applied Electromagnetics group @ UNISI
I. Nadeem L. Passalacqua
M AlbaniIEEE Fellow
A Benini
A ToccafondiIEEE Senior Member
S Maci, IEEE Fellow
Davide RossiTechnician
E MartiniIEEE Senior Member
K. KlionovskyT Paraskevopoulos
M. Faenzi
C. Yepes, L. Smaldore, PhD Project Manager
F. Giusti I Gashi R K Thanikonda E. Addo
F Caminita C Della Giovampaola
G Minatti M Nannetti G Labate N Bartolomei
Postdocs
(PhD) students
AEE Innovation Lab
Metasurface antennas
• From duality theorem to self-complementary antennas
• Homogenized impedance model of MTSs
• Duality for penetrable metasurfaces
• Scattering
• Dispersion
• Self-complementary MTSs
• Duality for impenetrable metasurfaces
• Balanced hybrid condition
• Checkerboard MTS
Outline
Duality theorem
“When two equations that describe the behavior of two different variables are of
the same mathematical form, their solutions will also be identical. The variables
in the two equations that occupy identical positions are known as dual quantities
and a solution of one can be formed by a systematic interchange of symbols to
the other”
Complementary structures
Mutually dual structures
Symmetric electric
currents
Antisymmetric
magnetic currents
2 1E H=
2
2 1H E=
PMC can be taken away
2
01 2
4Z Z
=
1
1
12
b
a
d
c
E d
Z
H d
=
l
l
2
2
22
d
c
b
a
E d
Z
H d
=
l
l
Self-Complementary antennas
Self-complementary antenna are identical to themselves when swapping
empty space with metal except for a rotation or a translation
2
01 2
4Z Z
= 1 2Z Z=
01 2
2Z Z
= =
ZS
Homogenized impedance model
METAMATERIALS
volumetric artificial materials can
be characterized in terms of
effective constitutive parameters
METASURFACES
2D artificial materials can be
characterized in terms of
effective impedance BCs
, μ ε
(k, ) , (k, ) μ ε
ZS=jX
in absence of losses
Impenetrable MTS
Impenetrable (opaque) impedance:
relates the transverse electric field to
the transverse magnetic field
ˆt tjX= E z HGround plane
Patches( ),t tE H
dZ
0Z
hSjX
0Z
jX
2 2
1tan //TM TM
d SjX Z h k k jX = −
Equivalent transmission line model
Penetrable (trasparent) impedance:
relates the transverse electric field to the
discontinuity of the transverse magnetic field
( )ˆt S t tjX + −= −E z H H
( ),t t+ +
E H
( ),t t− +
E H
0Z
SjX
Equivalent transmission line model
0Z
Penetrable MTS
Patches vs. slots
Patch-type metasurface
Slot-type metasurface
• Disconnected metalizations
• Capacitive penetrable reactance
• Disconnected apertures in a metallic sheet
• Inductive penetrable reactanceLjX
CjX
0Z
0Z
0Z
0Z
Patches vs. slots
Patch-type metasurface
Slot-type metasurface
0Z
0Z
0Z
0Z
2
1
slot
slot slot
Lj
L C
−
2
1patch patch
patch
L Cj
C
−
−
XS must be a monotonically increasing function of
frequency (Foster's reactance theorem for passive
lossless 2-port networks*)
Babinet’s principle for complementary MTSs
2
0
4patchslot ZZ
=
Slot-type metasurface Patch-type metasurface
Square patches
Strip grating
0 0 01ln
2 2sin
2
grid
k DZ j j
w
D
= =
2
0 0
4 2patch
grid
Z jZ
= = −
2
0
2
0
4
4
TE
slot
TM
slo
TM
patch
TE
patct h
Z
Z
Z
Z
=
=
Scattering from complementary MTSs
S12 pass-band at resonance
5 10 15 20 25 30 35-40
-35
-30
-25
-20
-15
-10
-5
0
frequency [GHz]
dB
S11
TE,TE,S
11
TM,TM, simulated
S21
TE,TE,S
21
TM,TM, simulated
S21
TE,TE,S
21
TM,TM, eq. circuit
S11
TE,TE,S
11
TM,TM, eq. circuit
S11S12
5 10 15 20 25 30 35-40
-35
-30
-25
-20
-15
-10
-5
0
frequency [GHz]
dB
S11
TE,TE,S
11
TM,TM, simulated
S21
TE,TE,S
21
TM,TM, simulated
S21
TE,TE,S
21
TM,TM, eq. circuit
S11
TE,TE,S
11
TM,TM, eq. circuit
S11
S12
S12 stop-band at resonance
slot patchRT =
slot patchTR =
Combination of complementary MTSs
A planar MTS made of metallic elements closely coupled to its
complementary MTS exhibits a primary pass-band resonance frequency
much lower than that of a single-layer MTS with equal element size
Surface waves
Inductive reactance, XS>0
0 0
0
TM zkZk
=
SjX
0
TMZ
TM pol.
0 02
TM
S
ZjX+ =
2 2
00
02S
k kX
k
−=
2
0
0
21 SXk k
= +
2 2
0zk j k k= − −
capacitive
0
TMZ
00 0
TE
z
kZ
k=
SjX
0
TEZ
TE pol.
0 02
TE
S
ZjX+ =
0 0
2 2
02
S
kX
k k
= −
−
No solutions!
inductive
0
TEZ
Surface waves
Inductive reactance, XS>0
Frequency
k
Radiating waves
SW region
(bound waves)
SSW
Xk
= +
22
1k k =
An inductive reactance supports a TM surface wave (SW), or "slow wave"
(k>k0 →vp=/k<c)
Surface waves on uniform impenetrable MTSs
Capacitive reactance, XS<0
0 0
0
TM zkZk
=
SjX
00 0
TE
z
kZ
k=
SjX
0
TEZ
0
TMZ
TM pol.
TE pol.
0 02
TM
S
ZjX+ =
0 02
TE
S
ZjX+ =
2 2
00
02S
k kX
k
−=
2
00 1
2 S
k kX
= +
0 0
2 2
02
S
kX
k k
= −
−
2 2
0zk j k k= − −
No solutions!0
TMZ
0
TEZ
capacitive
inductive
Surface waves on uniform impenetrable MTSs
0.4pFC =
Capacitive reactance, XS<0
2
00 1
2
Ck k
= +
Strip grating
Is this structure inductive or capacitive?
All depends on polarization!
• inductive if the electric field is parallel to the strip
• capcitive if the electric field is orthogonal to the strips
wD
Self-complementary MTSs
Self-complementary metasurfaces are identical to themselves when
swapping empty space with metal (Babinet’s inversion) except for a rotation or a
translation
Babinet’sinversion
Babinet’sinversion
90° rotation translation
The strip grating becomes
self-complementary if w=D/2
wD
TE and TM SWs have the
same dispersion equation
Self-complementary MTSs
conj L
TMjX−
TMjX−
TEjX
TEjX
con1/ j C
disj L
dis1 j C
wLcon
1-w 2 (Lcon
Ccon
)
æ
èçç
ö
ø÷÷
1-w 2 (Ldis
Cdis
)
wCdis
æ
èçç
ö
ø÷÷ =
z 2
4
Babinet principle
( ) ( )con dis/ 2 2 /B L C = =
2
,
2 2 2
,
0
( / )1
(1 / ) B
SW
T L TE C
SW S
M
Wk k k k
= = = +
−
( )
2
, con
2 2
0
21
1
TM L
SW
Lk k
= +
− ( )
2
, dis
2 2
0
12 1
TE C
SW
Ck k
= + −
dis con con dis / 2L C L C = =
dis dis con con 01/ 1/L C L C = =
Equal dispersion equation
for TE and TM cases
Self-complementary MTSs
Low frequency
y
z
x
E
z
H
z
After the first resonance
E
z
H
z
• TE-TM degeneracy
• All-frequencies hyperbolicity
• Dual-Directional Canalization
slots
dipoles
Self-complementary MTSs
8
x
y
a =
7 m
ma = 7 mm
7 m
m
7 mm
1 mm
Copper
FR-4 ε = 3.9, tanδ
= 0.025
Self-complementary MTSs
Two pole-zero
equivalent
circuit
CST CST
Self-complementary MTS: experimental verification
Self-complementary MTS: experimental verification
YX
ഥH
ഥE
TM - probeTE - probe
FeedProbe
11
Zero
padding
FFT2D
Measured field
Isofrequency
contour
Self-complementary MTS: experimental verification
TE (𝐻𝑍)
TM (𝐸𝑍)
f = 3.4 GHz (X-canalization)
Re(field) Isofrequency contour
TE
TM
x
y
Self-complementary MTS: experimental verification
TE (𝐻𝑍)
TM (𝐸𝑍)
f = 3.5 GHz (Hyperbolic)
Re(field) Isofrequency contour
TE
TM
x
y
Self-complementary MTS: experimental verification
TE (𝐻𝑍)
TM (𝐸𝑍)
f = 4.9 GHz (Y-canalization)
Re(field) Isofrequency contour
14
TE
TM
Expected from
analytics
x
y
Self-complementary MTS: experimental verification
CST Simulation (field) Experiments (field) FFT Spectrum from measurements
(dashed line from CST)
Impenetrable MTSs
TM
dZ
0
TMZ
h
SjX
h
ˆt S tjX= E z H
er
( ) 2 2
1, tanTM
r d SWX h Z h ke = −
0 0TM
SX X+ =
sw is the solution of
the dispersion equation
Example:
1.5h mm=
3re =
1 0
11
2 2
1 1
11
1
r
r
z SW
TM zd
k k
k k
kZ
k
e
e
=
=
= −
=
Free space light line
Dielectric light line
SW on grounded slab
Patterned metallic cladding on a grounded slab
ˆ( )t tjX k= E z H
Equivalent circuit
( )TM
dZ k
0 ( )TMZ k 0 ( )TMZ k
( )0
0tanzd
r
kTM
cc zdkZ j hk
e=
0 ( )TMZ k
( ) / /op TM TMS ccSjX k jX Z =
h
opSX
f
Higher inductive
reactances for thicker
and denser slabs
SX
Patch-type metasurface
SjX
Surface waves on impenetrable MTSs
Inductive impenetrable reactance, XS>0
0 0
0
TM zkZk
=
jX
0
TMZ
TM pol.
0 0TMZ jX+ =
2 2
0
0
0
k kX
k
−=
2
0
0
1X
k k
= +
2 2
0zk j k k= − −
capacitive
k k =
rk k e=
f
Slower waves for denser and thinner
substrates and for larger patches
Patterned metallic cladding on a grounded slab
Slot-type metasurface
ˆ( )t tZ k= E z H
Equivalent circuit
( )TM
dZ k
0 ( )TMZ k
( )1 0
0tanz
r
kTM
cc zdkZ j hk
e=
0 ( )TMZ k
( )Z k
SX
Patterned metallic cladding on a grounded slab
Slot-type metasurface( )1 0
1tanz
r
kcc
TM zkZ j hk
e=
k k =
rk k e=
( )0
1
tanr
z
cc
TM k
k
Z j h
e
=
=
( )0
1
tanhr
z
cc
TM k
k j
Z j h
e
= −
= −
0 ( )TMZ k0 ( )TMZ k
Higher inductive reactances for thinner and less dense slabs
k
f
Complementarity in impenetrable MTSs
For impenetrable MTSs realized on a grounded slab, capacitive impedance
is only obtained after the first resonance
In that region, MTSs can emulate PMC behaviour
reflection with a
phase of 0°
Complementarity in impenetrable MTSs
Application to obtain perfect pattern symmetry and high polarization purity in MTS-
loaded horns
2
0
TE TMX X = −Balanced hybrid condition
Metahorn with balanced hybrid condition
11.9 GHz
Φ=0°
11.9 GHz
Φ=90°
Complementarity in impenetrable MTSs
Application to design dual-polarized MTS antennas
2
0
TE TMX X = −Balanced hybrid condition
XP=-15.5dB
29.1dB
CO(φ=45°) XP(φ=45°)
Checkerboard MTS
What about a self-complementary checkerboard-type MTS?
?All depends on the status of the connections at the vertices
Checkerboard MTS
Disconnected
vertices
CjX
TEY
TEY
Connected
vertices
Checkerboard MTS
TE-Disconnected (J-loop) TM-connected (E-star)
x
y
x
y
E E
Checkerboard MTS on a grounded slab
The propagation of the SW can be transversely confined along specific
propagation paths constituted by an L-type or C-type MTS road in an
environment of complementary MTS.
E-field
H-field
Periodic b.c.
Periodic b.c.
y
x
z
y
Connected
patches
Disconnected
patches
b=0.84mm; h=0.635mm
Connection
s
Checkerboard MTS on a grounded slab
A
B
C D
S21
(dB
)
Frequency (GHz)
-0.5
-1.0
-1.5
-2.0
-2.52 3 4 5 6
A
B
C
D
A and B are 22 cm long.
Insertion losses for a conventional microstrip transmission line compared to a single-
row CBMS transmission line for the prototypes shown on the right
Performance similar to conventional
devices
D. Gonzalez
Checkerboard MTS on a grounded slab
It possible to electronically depict different paths by acting on micro-switches
at the vertexes
Checkerboard MTS on a grounded slab
Experimental setup for optical control of CBMS transmission line based on a
33 mW laser source at 800 nm, focused on a spot of size 30 µm. Two gaps
are investigated with 15 mm and 30 mm, respectively.
Laser beam
Advantages:
• no bias lines are required
• only a small area needs to
be illuminated
• easy integration with
active devices grown in
the semiconductor
substrate
Optically reconfigurable checkerboard MTS
Line
With
1 gap
Continuous
Line
30 mm
15 mm
15 mm
30 mm gap
535mm
30 mm 15 mm
535mm
“Invisible” patch: measurements
Laboratorio elettromagnetismo applicato
2
Measurements set-upMeasurements carried out at UNISI's anechoic chamber
Gain
10
5
0
-5
References 1/2
1. Y. Mushiake, “Self-complementary antennas,” in IEEE Antennas and Propagation Magazine, vol. 34, no.
6, pp. 23-29, Dec. 1992, doi.
2. E. Martini, M. Mencagli, S. Maci, “Metasurface transformation for surface wave control," Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 373, no.
2049.
3. E. Martini, F. Caminita, S. Maci, “Double-scale homogenized impedance models for periodically
modulated metasurfaces,” EPJ Applied Metamaterials, December 2020.
4. Foster, R. M., "A reactance theorem", Bell System Technical Journal, vol.3, no. 2, pp. 259–267, Nov.
1924.
5. D. González-Ovejero, E. Martini and S. Maci, “Surface Waves Supported by Metasurfaces With Self-
Complementary Geometries,” in IEEE Transactions on Antennas and Propagation, vol. 63, no. 1, pp.
250-260, Jan. 2015.
6. O. Luukkonen et al., “Simple and Accurate Analytical Model of Planar Grids and High-Impedance
Surfaces Comprising Metal Strips or Patches,” in IEEE Transactions on Antennas and Propagation, vol.
56, no. 6, pp. 1624-1632, June 2008,.
7. D. S. Lockyer, J. Vardaxoglou, and R. A. Simpkin, “Complementary frequency selective surfaces,” IEE
Proc. Microw., Antennas Propag., vol. 147, no. 6, pp. 501–507, Dec. 2000.
8. Oleh Yermakov, Vladimir Lenets, Andrey Sayanskiy, Juan Baena, Enrica Martini, Stanislav Glybovski,
and Stefano Maci, “Surface Waves on Self-Complementary Metasurfaces: All-Frequency Hyperbolicity,
Extreme Canalization, and TE-TM Polarization Degeneracy,” Phys. Rev. X, 11, 031038 – Published 18
August 2021.
References 2/2
9. V. Sozio et al., “Design and Realization of a Low Cross-Polarization Conical Horn With Thin
Metasurface Walls,” in IEEE Transactions on Antennas and Propagation, vol. 68, no. 5, pp. 3477-
3486, May 2020.
10. A. Tellechea, F.Caminita, E. Martini, I. Ederra, J.C. Iriarte, R.Gonzalo, S. Maci, “Dual circularly-
polarized broadside beam metasurface antenna,” IEEE Trans Antennas Propagat., vol. 64, no. 7, July
2016.
11. A. Tellechea Pereda, F. Caminita, E. Martini, I. Ederra, J. Teniente, J. C. Iriarte, R. Gonzalo, S. Maci,
“Experimental Validation of a Ku-Band Dual-Circularly Polarized Metasurface Antenna,” in IEEE
Transactions on Antennas and Propagation, vol. 66, no. 3, pp. 1153-1159, March 2018.
12. D. González-Ovejero et al., “Basic Properties of Checkerboard Metasurfaces,“ in IEEE Antennas and
Wireless Propagation Letters, vol. 14, pp. 406-409, 2015.
Questions?