Complementarity, duality and symmetry

Department of Information Engineering and Mathematical

Science, University of Siena, Siena, Italy

Complementarity, duality and symmetry

Enrica Martini

[email protected]

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



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


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


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


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


Two pole-zero

equivalent

circuit

CST CST

Self-complementary MTS: experimental verification


YX

ഥH

ഥE

TM - probeTE - probe

FeedProbe

11

Zero

padding

FFT2D

Measured field

Isofrequency

contour


TE (𝐻𝑍)

TM (𝐸𝑍)

f = 3.4 GHz (X-canalization)

Re(field) Isofrequency contour

TE

TM

x

y


TE (𝐻𝑍)

TM (𝐸𝑍)

f = 3.5 GHz (Hyperbolic)


TE

TM

x

y


TE (𝐻𝑍)

TM (𝐸𝑍)

f = 4.9 GHz (Y-canalization)


14

TE

TM

Expected from

analytics

x

y


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


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



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


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°


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°


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


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


It possible to electronically depict different paths by acting on micro-switches

at the vertexes


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.

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Complementarity, duality and symmetry