potential applications of antennas with metamaterial · pdf file1 department of applied...

1

Department of Applied ElectronicsUniversity of Roma TreRome, Italy

Potential Applications of Antennas with Metamaterial Loading

Filiberto Bilotti

2

Road Map

The history of metamaterialsMetamaterial terminologyComplementary metamaterial pairsPatch antennas with metamaterial loadingLeaky wave antennas with metamaterialloadingConclusions

2

3

The history of metamaterials


4

What are metamaterials?Why to use metamaterials?

Metamaterials are artificially engineeredmaterials exhibiting unusual properties that cannot be found in nature.

Metamaterials allows going beyond the classical physical restrictions and limitationsof electrodynamics.

3

5

From natural materialsto complex materials 1/2

First Stage: observation and investigation of the physical phenomena in nature

, ,c c r c rn ε µ= , ,c c r c rn ε µ=

Natural materials

The arrangement of atomes and

molecules determinesthe physical behavior

Optical frequencies

6

From natural materialsto complex materials 2/2

Second Stage: design of artificial materials to imitate the nature at lower frequencies

, ,h h r h rn ε µ=

CompositionAlignment

ArrangementDensity

GeometryHost medium

4

7

From complex materialsto metamaterials

Third Stage: design of artificial materials that exhibit unusual (anomalous, surprising, …) features that cannot be found in nature

8

Microwave applicationsof metamaterials

Fourth Stage: investigate the exciting features of metamaterials to propose novel concepts for microwave components

electronic circuitsradiating components

DPS DNG

5

9

Come back to the nature…Fifth Stage: design of nanostructures to bring back to the nature the unusual properties discovered at microwave frequencies

Naturaloptical

materials

Complexmaterials

Meta materials

Artificialdielectrics

Exoticshapes

Nanotructures

µwaveapplications

of metamaterials

10

Metamaterial terminology


6

11

εRe[ ]

µRe[ ]

DPSk ∈ ℜ

DNG∈ ℜk

ENG

k ∈ ℑMNG

∈ ℑkMNZMNZ

ENZ

ENZ

RegularDielectrics

Metamaterialterminology 1/2

12

Metamaterialterminology 2/2

= ω µε

DPS ENG MNG DNG

k propagation evanescent evanescent propagation

waveβ −jα −jα β

7

13

Complementarymetamaterial pairs


14

1, tan 2, tan

1 2

1, tan 2, tan

1 2

1 1

1 1

∂ ∂=

− ∂ − ∂

∂ ∂=

− ∂ − ∂

Interface Interface

Interface Interface

H Hj n j n

E Ej n j n

ωµ ωµ

ωε ωεDPS DPS

DPS

DNG

SNG

1, tan 2, tan

1 2

1, tan 2, tan

1 2

1 1

1 1

∂ ∂=

− ∂ + ∂

∂ ∂=

− ∂ + ∂

Interface Interface

Interface Interface

H Hj n j n

E Ej n j n

ωµ ω µ

ωε ω ε

Complementary metamaterial pairs 1/6

8

15

0

0

ε εµ µ

= −

= −

DNG DPS DNG

Pendry, PRL, Oct. 2000 Engheta, IEEE AWPL, 1, 10-13, 2002


DPSDPS

d1 d2

16

DPS

d1

k1

DPS

d2

k2

1 1 2 22(k d +k d ) 2m , m 0= π ≠

1 2

1 2

d d m+ , m 02

= ≠λ λ

1 2d d2λ

+ =1 2k k=

The resonance condition for a 1D cavity filled by DPS/DPS pairs imposes a minimum thickness.


9

17

DPS

d1

k1

DNG

d2

k2

1 21 1 2 2

1 2

tan(k d )+ tan(k d ) 0k kµ µ

=

1 1 2 2d d 0µ + µ =

1 2

2 1

d | |d

µ=

µ2 <0µ

i ik d 1


18

-0,8 -0,7 -0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0,0 0,10,0

0,2

0,4

0,6

0,8

1,0

Standard resonator DNG resonator

Nor

mal

ized

ele

ctric

fiel

d

Stratification axis y / λ1

Field distribution in metallic cavities filled by DPS/DPS and DPS/DNG slabs (sizereduction).


10

19

DPSENG

d1

k1

MNG

d2

k2

1 21 1 2 2

1 2

| |tanh( d )- tanh( d ) 0µ µα α =

α α

1 1 2 2d | |d 0µ − µ =

1 2

2 1

d | |d

µ=

µ

i id 1α

A metallic cavity filled by a DPS(or ENG)/MNG pair works as a cavity filled by a DPS/DNG pair.


20

Compact scatterersand compact antennas 1/2

DPS

incE

incHDPS

incE

incH

incE

incHDPS

SNG

incE

incHDPS

SNG

incE

incH

incE

incH

DPSDPS

DPSSNG

DPSSNG

Resonantcompact bi-

layer scatterers

Resonantcompact bi-

layer antennas

RECIPROCITY

11

21

DPS

ENG

Ziolkowski’s groupresonant sub-λ

dipole antennas

Roma Tre – UPenn resonant sub-λ

patch and leakywave antennas

Compact scatterersand compact antennas 2/2

DPS DNG

22

Patch Antennas with Metamaterial Loading


12

23

patch

ground plane

substrate

Patch antenna: standard configuration

Brief introduction on Patch Antennas 1/4

24

Patch antennas: feeding techniques

coaxial cable microstrip line aperture coupling


13

25

Patch antennas: pros and contra

TolerancesEasy integration withprinted circuits

Low radiation efficiencyEasy fabricationLow gainConformabilityLow powerSmall volumeSpurious radiationLow weightLow polarzation purityLow profileNarrow bandwidthLow cost

ContraPros


26

Patch antennas: applications


14

27

Standard dielectric

z

X

y

W

d

Radiation mechanism and design 1/3

Standard rectangular patchSurface – wave contribution (degraded radiation pattern and poor efficiency)

Substrate thickness:λ/20 – λ/100

L

28


The electric field may be assumed vertically directedThe magnetic field does not have the vertical component (TMz modes)

15

29


Fringing effect is responsible for the radiationThe electric field must be out of phase at the two radiating edges of the patch.

L=λ/2

30

Cavity model for analyzingpatch antennas 1/2

Cavity model (since the substrate is very thin, only TMz modes are present)

PEC

PMC

The modes of the patch may becalculated as the modes of the

PEC-PMC cavity

2 2[m,n ,0] 0TM

r r

c1 m nf2 L W

π π⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟π µ ε ⎝ ⎠ ⎝ ⎠

LW

Imposing the boundary conditionsthe calculation of the resonantfrequencies is straightforward

16

31

The dominant mode along x is the TM100

The magnetic currents at theradiating edges are responsible

for the radiation

Electric current densitydistribution of the dominant mode

on the patch surface

Cavity model for analyzingpatch antennas 2/2

32

DPS DPS

DPS DNG

z

xy

W

Ld

Rectangular patch antennas with metamaterial loading

DPS DNG

DNG DPS

Is it possible to apply the same concept to microstrip antennas?

xy

xy

z z

17

33

z

xW

L

d1 1,ε µ 2 2,ε µ

TMm00

( )1-η LηL

[ ] ( )[ ]1 21 2

1 2

k tan L k tan 1 L kk

ωεη = − − η

ωµ

η ε−

− η ε2

11L 0→

≤ η ≤0 1

Filling Factor

Dispersion Equation for TMm00 modes

Cavity model for patch antennas with MTMs 1/6

The dispersion equation may be written with the explicit presenceof the filling factor η.

34


When L is very small compared to λ, if the two materials have Re[ε]>0, the dispersion equation cannot be satisfied for any value of η.

As in the 1D-cavity, when L is small compared to λ, the total length L is not relevant for the dispersion equation to be satisfied: the only relevant quantities are the filling factor η and the permittivities of the two materials.

18

35

-2.5 -2.4 -2.3 -2.2 -2.1 -2.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Res

onan

t Fre

quen

cy [

GH

z ]

εENG / ε0

W

0 02 ,ε µL/2

d0,ENGε µL/2

L = 50 mm


Also in this case there is no need for a DNG material: an ENG medium is enough.

η ε−

− η ε2

11f0 = 2.44 GHzεr = 2.2

36

0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.200.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

µENG

= µ0

µENG = 3µ0

µDNG

= -µ0

µDNG

= -3µ0

Res

onan

t Fre

quen

cy [

GH

z ]

Plasma Frequency [GHz]

⎛ ω ⎞ε = − ε⎜ ⎟⎜ ⎟ω⎝ ⎠

2p

021

W

0 02 ,ε µL/2

d0,ENGε µL/2

L = 50 mm


Permeability variations do not affect the resonant frequency.

Drudedispersion

model

19

37


Ez component Hx component

0.00 0.01 0.02 0.03 0.04 0.05-300

-150

0

150

300

450

600

750

Ele

ctric

Fie

ld E

z [ V

/ m

]

y [ m ]

ε2 = 2.2, f = 2.44 GHz ε2 = -2.2, f = 0.50 GHz

0.00 0.01 0.02 0.03 0.04 0.050.0

0.2

0.4

0.6

0.8

1.0

Mag

netic

Fie

ld H

x [ A

/ m

]

y [ m ]

ε2 = 2.2, f = 2.44 GHz ε2 = -2.2, f = 0.50 GHz

38


DPS ENG DPS DPS

Radiation from this kind of antenna is very poor.

f = 0.50 GHz f = 2.44 GHz

20

39

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-15

-10

-5

0

5

10

15

20

25

30

Rel

ativ

e P

erm

ittiv

ity

Frequency [GHz]

Re[ε2]

Im[ε2]

Full wave simulations for the rectangular patch 1/7

0 02 ,ε µW/2

d = 1.5 mm0,ENGε µW/2

W = 50 mm

L = 40 mm

Lorentz Modelfor the permittivity

Probe Impedance: 125 Ohm

Probe Location: xp = – W/4, yp = 0

Probe Radius: 0.3 mm

40

0.0 0.5 1.0 1.5 2.0 2.5 3.0-30 dB

-25 dB

-20 dB

-15 dB

-10 dB

-5 dB

0 dB

Ret

urn

Loss

[dB

]

Frequency [GHz]0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0

-150

-100

-50

0

50

100

150

200

Inpu

t Im

peda

nce

[Ohm

]

Frequency [GHz]

Input Reactance Input Resistance


Return Lossas a function of frequency

Input Impedanceas a function of frequency

21

41


f = 0.48 GHz f = 2.44 GHz

Ez component

42


f = 0.48 GHz f = 2.44 GHz

Ez

Ex

22

43

f = 0.48 GHz f = 2.44 GHz

Directivity


44


The variation of the electric field under the patch is responsible for the poor radiation of a rectangular patch loaded with DPS-ENG media.

23

45


Plasmonic Resonances

46

a1

a

y

xx

y

z

DPS

DPSDPS

DPS


Geometry of a circular patch antenna with DPS-DPS loading.

24

47

PEP

PEP

PMP


x

y

z

DPS

DPS

[ ][ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

′ ′−µ = µ

′ ′ ′ ′ ′−n 1 1 n 2 1 n 2 n 2 1 n 2

1 2n 1 1 n 2 1 n 2 n 2 1 n 2

J k a J k a Y k a Y k a J k aJ k a J k a Y k a Y k a J k a

Dispersion Equation for TMmn0 modes

48


DPS

DPS

x

y

z

DPS DPS a1 = 12 mm

a = 20 mm

[m,n ,0] 0mnTM

r r

c1f2 a

′χ=

π µ ε

Resonance Frequencies

r 2.33ε =

[1,1,0] 011TM

r r

c1f2 a

′χ= =

π µ ε2.88GHz

[1,2 ,0] 012TM

r r

c1f2 a

′χ= =

π µ ε4.77GHz

[0 ,1,0] 01 0TM

r r

c1f2 a

′χ= =

π µ ε5.99GHz

TM110

TM120

TM100

d = 1.5 mm

25

49

[ ][ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

′ ′−µ = µ

′ ′ ′ ′ ′−n 1 1 n 2 1 n 2 n 2 1 n 2

1 2n 1 1 n 2 1 n 2 n 2 1 n 2

J k a J k a Y k a Y k a J k aJ k a J k a Y k a Y k a J k a

22

21

2 n2

2 n1

n 011 n 01

η ε− =

− η ε

− η µ− >

+ η µ

→a 0Dispersion Equation for TMmn0 modes

1aa

η=Fillingfactor

When the patch radius is smaller compared to λ, the dispersion equation can be written in terms of the:

filling factor ηmode order npermittivities or permeabilities


50

Depending on the materials we use to load the antenna, we may choose the dominant mode of the cylindrical patch resonator.


22

21

2 n2

2 n1

n 011 n 01

η ε− =

− η ε

− η µ− >

+ η µ

26

51


Ez EzEz

n = 0 n = 1 n = 2

52


f = 0.50 GHz f = 2.88 GHz

27

53

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-15

-10

-5

0

5

10

15

20

25

30

Rel

ativ

e P

erm

ittiv

ity

Frequency [GHz]

Re[ε2]

Im[ε2]

Full wave simulations for the circular patch 1/6

Lorentz Modelfor the permittivity

Probe Impedance: 50 Ohm

Probe Location: ap = 0.75 a, φp = -π

Probe Radius: 0.3 mm

a1 = 12 mm

a = 20 mmMNG

DPS

ap = 15 mm

54

0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600-40 dB

-30 dB

-20 dB

-10 dB

0 dB

Ret

urn

Loss

[dB

]

Frequency [GHz]


0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

80

Inpu

t Im

peda

nce

[Ohm

]

Frequency [GHz]

Input Reactance Input Resistance

Return Lossas a function of frequency

Input Impedanceas a function of frequency

28

55


Ez @ f = 2.88 GHz

56


Current Density and Directivity @ f = 2.88 GHz

29

57


Ez and Ex @ f = 0.473 GHz

58


Current Density and Directivity @ f = 0.473 GHz

30

59

Leaky wave antennas with metamaterial loading


60

Natural Modes of a Grounded Slab 1/2

( ) ( )( ) ( )

0 0

0 0

TE: cos sin 0

TM: cos sin 0

y y y y

y y y y

k k d j k k d

k k d j k k d

µ µ

ε ε

+ =

+ =

y

xd ε, µ

( )0I: max ,k kβ > 0, ∈ ℑy yk k

0II: k kβ< < 0,∈ℜ ∈ ℑy yk k

Suraface waves (only with negativeconstitutive parameters)

Regular surface waves

31

61

Natural Modes of a Grounded Slab 2/2

( ) ( )( ) ( )

0 0

0 0

TE: cos sin 0

TM: cos sin 0

y y y y

y y y y

k k d j k k d

k k d j k k d

µ µ

ε ε

+ =

+ =

y

xd ε, µ

Leaky waves (only with anomalousconstitutive parameters) with high leakagefactor. Low directivity.

Leaky waves (only with anomalousconstitutive parameters) with low leakagefactor. High directivity.

0III: Re[ ]< <k kβ 0 0Re[ ] , Re[ ]> <y yk k k k

( )0IV: Re[ ] min ,< k kβ 0 0Re[ ] , Re[ ]< <y yk k k k1

0sin (Re[ ]/ )− kθ β

62

ENZ-MNZ metamaterials for high directivity LW radiators

( )0IV: Re[ ] min ,< k kβ 0 0Re[ ] , Re[ ]< <y yk k k k

( ) ( )( ) ( )

0 0

0 0

TE: cos sin 0

TM: cos sin 0

y y y y

y y y y

k k d j k k d

k k d j k k d

µ µ

ε ε

+ =

+ =

( )0 2 2

0 2 2

2 1,

2

,

Nd

kNd

k

πµ µ

βπε ε

β

−≅

−

≅−

An almost real solution may be found if the two terms of each equation become sufficiently small. By inspection, it is easy to derive the conditions for both the constitutive parameters and d.

32

63

( )( ) ( )

2 2 22 0 1 2 0 2 2 0 1 2

2 2 20 2 1 2 0 2 1 2 2

TE:

TM:

− = − +

+ = − −

TE TE TE TEy y

TM TM TM TMy y y

f f k j k f f

k f f j k f f k

µ µ µ µ

µ ε

( )( )

cot /

tan /

=

=

TEi yi yi i i

TMi yi yi i i

f k k d

f k k d

µ

ε

Groundedmetamaterial

bi-layer

Groundedmetamaterial bi-layerdispersion equations

Grounded bi-layers planar uniform LW antennas 1/7

64

[ ]( )( )

1 2 0

1 2 0

Im 0

TE: max ,

TM: max ,

β

µ µ µ

ε ε ε

( )( )

21 2 2 2 1

21 2 1 1 2

TE: /

TM: /

y

y

d d k

d d k

µ µ

ε ε

1 1 2 2max , 1y yk d k d⎡ ⎤⎣ ⎦Sub-λ thickness

condition

High directivityconditions

Retardation effects are not significant: depending on the polarization, onlyone constitutive parameter is involved.


33

65

1 2 0

1 03

2 0

1 0

2 0

0.06

10d / 50d / 35.5

−

ε = ε = εµ = µ

µ = − µ

= λ= λ

020

40

60

80280

300

320

340

40dB

30dB

20dB

10dB

0dB

55θ = °


1 1,ε µ

y

1d2 2,ε µ2d

x

0 0,ε µ

m

( )21 2 2 y2 1TE : d d / kµ µ

0D 22dB=

66


020

40

60

80280

300

320

340

-40dB

-30dB

-20dB

-10dB

0dB

ω = 0.998 ω0

ω = 0.999 ω0

ω = ω0

ω = 1.001 ω0

ω = 1.002 ω0

ω = 1.003 ω0

( )2

0 2 20

1⎛ ⎞

= −⎜ ⎟−⎝ ⎠m

F ωµ ω µω ω

Material dispersion

1 1,ε µ

y

1d2 2,ε µ2d

x

0 0,ε µ

m

34

67

020

40

60

80280

300

320

340

-40dB

-30dB

-20dB

-10dB

0dB γm = 0

γm = ω0 / 105

γm = ω0 / ( 5 104 )

γm = ω

0 / ( 2 104 )

γm = ω

0 / 104

γm = ω0 / ( 5 103 )

( )2

0 2 20

1m m

Fj

ωµ ω µω ω ωγ

⎛ ⎞= −⎜ ⎟− −⎝ ⎠

Material losses

1 1,ε µ

y

1d2 2,ε µ2d

x

0 0,ε µ

m


68

0.0 5.0x10-5 1.0x10-4 1.5x10-4 2.0x10-4 2.5x10-4 3.0x10-4 3.5x10-4

0.808

0.810

0.812

0.814

0.816

0.818

0.820

0.822

0.824

0.826

Re

[ β ]

γm / ω0

0.0 5.0x10-5 1.0x10-4 1.5x10-4 2.0x10-4 2.5x10-4 3.0x10-4 3.5x10-40.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Im [

β ]

γm / ω0

( )2

0 2 20

1m m

Fj

ωµ ω µω ω ωγ

⎛ ⎞= −⎜ ⎟− −⎝ ⎠

Material losses1 1,ε µ

y

1d2 2,ε µ2d

x

0 0,ε µ

m


35

69

020

40

60

80280

300

320

340

40dB

30dB

20dB

10dB

0dB1 0

42 0

1 2 0

1 0

2 0

0.025

10

d / 27d /30

−

ε = − ε

ε = − εµ = µ = µ

= λ

= λ


( )21 2 1 y1 2TM : d d / kε ε

1 1,ε µ

y

1d2 2,ε µ2d

x

0 0,ε µ

p0θ = °

0D 17dB=

70

Compact cylindricalleaky wave antennas 1/15

( )( ) ( )

( ) ( )

( ) ( )( ) ( )

10 1 1 0

12 1 3 1

21 2 20 4 1 0

11 1 0 0 1 0 0

2 1 3 1

ˆ

ˆ

ˆ

ˆ ˆ

− −

− −

− −

− − −

−

⎧<⎪

⎪ ⎡ ⎤= + < <⎨ ⎣ ⎦⎪⎪ − >⎩

+ <

⎡ ⎤+⎣ ⎦=

TM j zt in

TM TM j zTM t t in out

TM j zout

TM j z TM j zt t t in

TM TM j zt t

TM

j c J k e a

j c J k c Y k e a a

j c H k e a

j c J k e k c J k e a

j c J k c Y k e

β

β

β

β β

β

ωε β ρ ρ

ωε β ρ ρ ρ

ωε β β ρ ρ

ρ β ρ ρ

ρ ρ

φ

H φ

φ

ρ z

E( ) ( )

( ) ( ) ( ) ( )

12 1 3 1

2 214 1 0 4 0 0 0

ˆ

ˆ

ˆ ˆ

− −

− − −

⎧⎪

+⎪⎪⎨

⎡ ⎤+ + < <⎪ ⎣ ⎦⎪

+ >⎪⎩

TM TM j zt t t in out

TM j z TM j zt t t out

k c J k c Y k e a a

j c H k e c k H k e a

β

β β

β ρ ρ ρ

ρ β ρ ρ

ρ

z

ρ z

We are interested here in the modal solutions that do not

exhibit field variations along φ.

36

71

( ) ( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 0 0 1 0 1

0 0 0 0 0

20 1 0 1 1 0

20 0 0 0

/ / 00

00 / /

0

t in t in t in

t t in t t in t t in

t out t out t in

t t out t t out t out

J k a J k a Y k ak J k a k J k a k Y k a

J k a Y k a H k a

k J k a k Y k a H k a

ε ε ε ε

ε ε ε ε

− −− −

=− −

− −

0

11

t out

t out

k ak a

Sub-λ thicknesscondition

High directivityconditions

[ ]0

Im 0β

ε ε

[ ] ( )2 0

2

2 /Reln /in out ina a a

ε εβ −


72

04

0

/100

1060

−

=

= −

= °

ina λ

ε εθ

1.223out ina a=[ ] ( )2 0

2


ε εβ −

( )40 0.5 5.764 10−= + ⋅k jβExact value of the propagation

constant of the leaky mode

Electric and magnetic field

distribution of the leaky mode


37

73

30

60

90

120

150

180

210

240

270

300

330

dB

dB

dB

dB

dB

dB

Elevation Azimuth

0

30

60

90

120

150

180

210

240

270

300

330

0dB

20dB

40dB

0dB

20dB

40dB

f = 1 GHz f = 1.5 GHz f = 2 GHz f = 3 GHz f = 4 GHz

40

2

10−

=

= −of GHz

ε ε1.83

69.15=

=outa mm

D dBThe beam angle scans with frequency in a very smooth way (quasi-static resonance)


74

0

30

60

90

120

150180

210

240

270

300

330

0dB

20dB

40dB

0dB

20dB

40dB

ωp = 0.9999 ω* ωp = 0.99995 ω* ωp = ω* ωp = 1.00002 ω*

2

021 pωε ε

ω⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

Material dispersion isadded through Drude

dispersion formula

The result is a fine tuning of the beamdirection with the

frequency


38

75

Material losses are added in Drude

dispersion formula

The result is the reduction of the

directivity, while the beam direction is

almost not affected

( )

2

01⎛ ⎞

= −⎜ ⎟⎜ ⎟−⎝ ⎠

p

j τ

ωε ε

ω ω ω

90 105 120 135 150 165 180

-10dB

0dB

10dB

20dB

30dB

40dB

50dB

60dB

ωτ = 10-5 ω*

ωτ = 10-4 ω*

ωτ = 0


76

0

20

/10

1060

−

=

= −

= °

ina λ

ε εθ

1.187=out ina a[ ] ( )2 0

2


ε εβ −

( )20 0.5 4.560 10−= + ⋅k jβExact value of the propagation

constant of the leaky mode

Electric and magnetic field

distribution of the leaky mode


39

77

20

2

10−

=

= −of GHz

ε ε17.8

17.46=

=outa mm

D dB

The transverse dimensions are larger thanin the previous case and there is a

stronger dependence on the frequency.

0

30

60

90

120

150

180

210

240

270

300

330

-10dB

0dB

10dB

20dB

-10dB

0dB

10dB

20dB

Elevation Azimuth

0

30

60

90

120

150

180

210

240

270

300

330

-10dB

0dB

10dB

20dB

-10dB

0dB

10dB

20dB

f = 1 GHz f = 1.5 GHz f = 2 GHz f = 3 GHz f = 4 GHz


78

2

021 pωε ε

ω⎛ ⎞

= −⎜ ⎟⎜ ⎟⎝ ⎠

Material dispersion isadded through Drude

dispersion formula

The result is a fine tuning of the beamdirection with the

frequency

0

30

60

90

120

150180

210

240

270

300

330

-10dB

0dB

10dB

20dB

-10dB

0dB

10dB

20dB

ωp = 0.99 ω* ωp = 0.995 ω* ωp = ω* ωp = 1.002 ω* ωp = 1.005 ω*


40

79

Material losses are added in Drude

dispersion formula

The result is the reduction of the

directivity, while the beam direction is

almost not affected

( )

2

01⎛ ⎞

= −⎜ ⎟⎜ ⎟−⎝ ⎠

p

j τ

ωε ε

ω ω ω

90 105 120 135 150 165 180-15dB

-10dB

-5dB

0dB

5dB

10dB

15dB

20dB

ωτ = 10-2 ω*

ωτ = 10-1 ω*

ωτ = 0


80

CST Microwave Studio FW simulations

The near field is dominated by the TM LW whose E field is almost

radially directed. This is a good hintfor both feed and inclusion design.

L = 75 cmDrude dispersion for ε


41

81

The amplitude of the Poynting vectordecays along the antenna axis

The electric field is radially directed

f = 1.975 GHzf = 1.975 GHz


82

1.960 1.965 1.970 1.975 1.980 1.985105°

110°

115°

120°

125°

130°

135°

140°

145°

150°

155°

160°

Bea

m D

irect

ion

[deg

rees

]

Frequency [GHz]

Scanning features of the cylindrical leaky wave antenna

as a function of frequency

f = 1.975 GHz


42

83

The 3D radiation patterns show that the structure is long enough not to have back-radiation.

f = 1.975 GHz f = 1.985 GHzf = 1.960 GHz


84

A smaller structure gives reduceddirectivity while the back-radiation is

increased, due to the reflections at the no-feeding end.

f = 1.975 GHz f = 1.985 GHzf = 1.960 GHz

L = 25 cm


43

85

Conclusions


86

Metamaterial complementary pairs are able to overcome the diffraction limit in the design of microwave components.Sub-wavelength cavities, waveguides, scatterers, and antennas may be obtained.Patch antennas and leaky wave antennaswith sub-wavelength resonant dimensions have been presented in details.

Conclusions

44

87

Acknowledgements

Prof. Lucio Vegni (University of Roma Tre)

Dr. Andrea Alù (University of Roma Tre)

Prof. Nader Engheta (University of Pennsylvania)

88

ReferencesAlù, Bilotti, Engheta, Vegni, IEEE IMS 2005, Long Beach, USA, June 2005Bilotti, Alù, 1st EU Ph.D. School on Metamaterials, San Sebastian, Spain, July 2005Alù, Bilotti, Engheta, Vegni, IEEE AP/URSI Symp., Washington, USA, July 2005Alù, Bilotti, Engheta, Vegni, ICEAA’05, Turin, Italy, September 2005Alù, Bilotti, Engheta, Vegni, ICECom’05, Dubrovnik, Croatia, October 2005

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