‘Negative’ World of Metamaterialsor

How Ambiguity of Square Root Reversed Electromagnetics

“Negativan” svijet metamaterijalaili ili

kako je dvoznačnost kvadratnog korijena okrenula elektromagnetizam naglavačke

Silvio Hrabar

Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia

Outline

•Electromagnetics of Classical (“Positive”) Materials

•Mathematics versus Physics and Engineering

•Are “Negative” Materials Possible?

•New Physical Phenomena - Fundamental research

•Potential Engineering Applications – Applied Research

•Conclusions

•Are “Negative” Materials Possible?

Mathematics versus Physics and Engineering Mathematics versus Physics and Engineering

PhysicsMathematics

Service ?

Engineering

What is the difference among engineers , physicists and mathematicians ?physicists and mathematicians ?

Reality

Observation

ModelPrediction

Model gives non-physical results – Why ?

Project

Is the basic theory correct?

Scientist Engineer

PhysicsMathematics

Engineering

Electromagnetics of Classical (“Positive”) Materials Electromagnetics of Classical (“Positive”) Materials

QQ rr

Q2

Fr

The Story of Electricity

•Ancient Greece – Why does amber attract papyrus ?

•Charles Augustin de Coulomb (1736-1806) and concept of Electric field

Q4 Q5 Qn

12202

20 aR

QQKF

rr=02F

r

Q0

nn

n aR

QQKa

R

QQKa

R

QQKF 12

0

0132

03

30122

02

20 ...rrrr

++=

000

limQ

FE

Q

rr

→=

Q3

EQaR

Qa

R

Qa

R

QKQF n

n

nrrrrr

0120

13203

3122

02

20 ... =

+=

EQFrr

≠

Q2

02Fr

( ) .

),,(

0

0200

BEBvEQ

RQvfEQFrrrrr

rrr

⊥×+=

+=

The Story of Magnetism

•Ancient Greece – a strange rock from island of Magnesia

Hendrik Antoon Lorentz

Q0 vr

Andre Marie Ampere (1775-1836)

Concept of B field

Wedding of Electricity and Magnetism –Electromagnetism (Electromagnetics)

•Michael Farady (1791-1867)

Are theE and B Vectors

t

BV

∂∂−=r

Are theE and B VectorsEnough?

Yes in principle, but whatabout the influence of matter?…atomic dimensions are on order of 10-11 m!

field Magnetic1

densityfluxElectric

BH

EDrr

rr

⋅=

⋅=

µ

ε

t

EJH

∂∂∂+=×∇r

r

rrr

ε

James Clerk Maxwell (1831-1879)

t

HE

∂∂−=×∇r

rµ

Solution of these equations is a wave!

{ }rr

What is EM Harmonic Wave ?

02

akrr

λπ=

{ }zkjtj eeAzktAtxerrωω Re)cos(),( =⋅−=

Wave vector

From Theory towards Applications…

•Heinrich Rudolf Hertz (1857-1894)

•Nikola Tesla (1856-1943)

•Guglielmo Marconi (1874-1937)

…. and here we are !

.,,, µεHErr

Are “Negative” Materials Possible?Are “Negative” Materials Possible?

(Are some solutions of Maxwell equations meaningless?)

• A plane wave solution of Maxwell equations: E

H

Pk

Forward wave

What would happen if µµµµ and εεεε werenegative numbers? (Veselago, 1968)

H

Right-handedmaterial

E

H

P

k Left-handedmaterial

HEP

uk

AeE rkj

vvv

rr

r rr

×=

=

= −

µεω0

Backward wave

Sign ambiguity

One-dimensional Wave Propagation in Material with

µµµµ >0 and εεεε >0 (forward wave propagation )

E

source

Vp>0Vg>0

0

X

+

One-dimensional Wave Propagation in Material withµµµµ <0 and εεεε <0 (backward wave propagation )

E

source

Vp<0Vg>0

0

X

+

• P is oppositeto k !

• This causes ‘reversion’ of many basic electromagnetic phenomena such as Snell law, Doppler effect e.t.c.

‘Reversion’ of Snell Law

air (ε µr r=1, =1)

εε µµ rr rr >0,<0, >0<0

Right-handedmaterial

Left-handedmaterial

φ−φ

Material withMaterial with

Concave and Convex Lenses using material with

µµµµ <0 and εεεε <0

The Concept of Metamaterial

scatterer (implant)

host material

a

• a<<λ, the structure behaves as a homogenous material with some new µ and ε ⇒metamaterial

incoming plane wave

E

H

P

Classification of Metamaterials •Single-

negative

materials

(SNG)

•Double-

positive

materials

(DPS)

•Double-

negative

materials

(DNG)

•Single-

negative

materials

(SNG)

Thin-wire Plasma-like Structure with ε <0( Rotman 1962, Pendry 1998)

12

>>

<<

−=

a

a

p

r

λω

ωε

E

H

P

1>>r

a

εreffz

fpf

1

0

real part

imaginary part

The ‘Split-ring Resonator’ Structure with µ<0 (Pendry 1999)

µµµµr

real part

Imaginarypart

λλ

<<<<

a

rfmpf0f

0

1<0

Imaginarypart

The first reported Backward-wave Material (Smith et al. 2000)

Split-ring resonator

Wires only

SRRs only

SRRs +Wires

µεω=k

Thin-wire mediaSRRs +Wires

Wires only : µµµµ >0 εεεε <0 , k becomes imaginary number ⇒⇒⇒⇒ no propagation

SRR’s only : µµµµ <0 εεεε >0, k is again imaginary number ⇒⇒⇒⇒ no propagation

SRR’s and wires: µµµµ <0 εεεε <0, k becomes positive real number ⇒⇒⇒⇒ backward-wave propagation

Capacitivelly loaded loops and thin-wire structure (Hrabar, Barbaric; 2000 –1/3)

Capacitivelly loaded loops and thin-wire structure (Hrabar, Barbaric 2000; –2/3)

-20

0

Capacitivelly loaded loops and thin-wire structure (Hrabar, Barbaric 2000; –3/3)

8 9 10 11 12 13 14 15 16-140

-120

-100

-80

-60

-40

frequency [Ghz]

tra

nsm

issi

on

[dB

]

LHM wire medium capacitively loaded loops

⋅

−+−=

CjLjR

KjHH S

i

ωω

ωµ01

si HHH +=

3

4

5real part imaginary part

Calculated effective permeability

The SRR behaves as capacitivelly loaded loop(Hrabar, Bartolic, Eres; 2002)

ZIHi Hs

Z

ZaV

+

Io

r

7 7.5 8 8.5 9 9.5 10

-4

-3

-2

-1

0

1

2

Frequency [GHz]

Schelkunoff, Friis –’Antennas, Theory and Practice’, 1952

>>

<<

−=

r

a

a

pr

λω

ωε

2

1

Wire Medium as Transmission Line (Hrabar 2003,2006)

E

H

P

1 ( )( )EjjEE εωµωµεω =−=∇ 22

ε

0

)l

0 εr<0

εreffz

fpf

1

0

real part

imaginary part

Free-spaceWires

0

)l

εr<0

f<fp0

)l

f>fp

0>εr>1

( )( )( )( )VYjXjVZY

x

V

EjjEE

shuntseries ωω

εωµωµεω

=−=∂∂

=−=∇

2

2

)l

Z

Y

I(H)

V(E)

New Physical Phenomena -Fundamental research Fundamental research

Experimental Verification of Negative Refraction (Shelby et al. 2001)

Experimental Verification of Negative Refraction (Shelby et al. 2001)

Experimental Verification of Negative Refraction (Hrabar, Bartolic; 2003-1/2)

1

23

1 2

3

Experimental Verification of Negative Refraction (Hrabar, Bartolic; 2003-2/2)

Simulation of Negative Refraction at Plan-parallel Slab with n=-1

(Ziolkowski; 2003)

a)b)d

1

1 1

11

1

1 1

2 2

) )x x

Verification of BW propagation by measurement of Negative Refraction (Hrabar, Bartolic, Sipus, 2004)

metallic plate

absorber

double rings

EH

P

a) A case of ‘forward-wave’ slab (ε>0,µ>0)b) A case of ‘backward-wave’ slab (ε<0,µ<0)

Super Lens’ with no Resolution Limit (Pendry, 2000)

sourcefocus

εµ

=1=1

εµ

=1

=1εµ

=-1

=-1rr

rr

rr

source

Propagating part of Fourier spectrum

Evanescent part of Fourier spectrum

Experimental verification (Grbic et al. 2004 –1/2)

Figure taken from G. Eleftheriades (University of Toronto)presentation slides

Experimental verification (Grbic et al. 2004 – 2/2)

Figure taken from G. Eleftheriades (University of Toronto)presentation slides

Potential Engineering ApplicationsI – Guiding of EM energyI – Guiding of EM energy

Classical rectangular waveguide– Electrical field distribution (TE case)

εr>0 µr >0

f/fc 1

Tra

nsm

issi

onForward waves

ISOTROPIC DPS MATERIAL

εr

•High-pass behavior

f/fc

•f > fc, PropagatingE field•f < fc, EvanescentE field

•P, k •P, k

Miniaturized Waveguide based on Anisotropic Meta-material (Hrabar, Bartolic, Sipus; 2003)

d

d

Miniaturized Waveguide based on Negative Permeability Metamaterial

(transversal dimension reduced down to 25%)

Miniaturized waveguide (a=15 mm) filled with capacitively loaded rings – the transversal width

reduced down to 4% !

copper ringchip capacitor

Potential Engineering ApplicationsII – AntennasII – Antennas

Does an open-ended SNG-filled miniaturized waveguideradiate into a free space ? (Hrabar, Zivkovic; 2005)

The simulated radiation (BW mode, k<0)

RF feed

the double ring inclusions

the aperture (0.25 x 0.25 ) λ λ

Experiemental Miniaturized Open-ended MNG Waveguide Radiator (Hrabar, Jankovic 2005)

coaxial feed the double ring inclusionslattice constant of 6 mm

•εr1 •εr2

•θ•θ2

Phenomenon of ultra-refraction

1r

2r

2

1

sin

sin

εε

θθ =

•θ1

•θ1

•θ2

Antenna •If medium 2 is air

( )( ).00if

sinsin

2

111

2

=⇒== −

θεθεθ

r

r

0°

-40

-30

-20

-10

0

10

201

23

4

5

6

7

8

9

1 0

1 1

1 2

1 3

1 4

1 5

1 6

1 7

1 8

1 9

2 0

2 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

3 0

3 1

3 2

3 3

3 4

3 53 6

3 7

3 83 9

4 0

4 1

4 2

4 3

4 4

4 5

4 6

4 7

4 8

4 9

5 0

5 1

5 2

5 3

5 4

5 5

5 6

5 7

5 8

5 9

6 0

6 1

6 2

6 3

6 4

6 5

6 6

6 7

6 8

6 9

7 0

7 17 2

10.25 GHz

10.50 GHz

[dB]

0°

90°–90°

180°

•Figures copied from the original article

Simulated magnitude of the y-componnet of electric

field of the shortened horn

Shortened Horn with ENZ Metamaterial (Hrabar, Bonefacic, Muha, 2008)

•Simulated magnitude of the y-componnet of electric field of the shortened horn with ENZ slab

•Comparison of prototyped shortened horns

• (length of 33% of the length of optimal horn)

• (length of 52% of the length of optimal horn)

Potential Engineering ApplicationsIII – ResonatorsIII – Resonators

Sub-wavelength Resonator(Engheta, 2001)

slab 1slab 2ε

µ1>0

>01εµ2<0

<02

P1 P2

•The resonance frequency depends on d2/d1instead of d2+d1 which would hold in ordinary materials

metallic plate metallic plate

k1k2

waveguide part(backward-wave line)

the slot

coaxial part(forward-wave line)

coupling loop

Experimental verification of Engheta’s resonator (Hrabar et al., 2004)

slab 1slab 2 ε

µ1>0

>01εµ2<0

<02

P1P2

k1k2

the inclusion N connector

x0

N connector

overall length

~ λ/30

(f=350 MHz)

metallic platemetallic plate

1

Potential Engineering ApplicationsIV – Going optical!IV – Going optical!

Going Optical - Approach IScaling Down the Size of Inclusions

Figure taken from V. Shalaev (Purdue University )presentation slides

Going Optical –Approach II Make Use of Surface Plasmons

The Lycurgus Cup (glass; British Museum; 4th

century A. D.)

Going Optical – A ‘Soft’Approach Make Use of Surface Plasmons

Permittivity of silver

Nano-transmission-lines

Forward-wave

Backward-wave

From Plasmonic Spheres to Nano-circuit Elements in Optical Domain (Alu, Engheta; 2005)

Nano-antenna

Backward-wave

Nano-circuit

New Idea - Scaled RF Replicas of Plasmonic Structures Plasmonic Structures

Prototyping of ‘RF replicas’ of plasmonic spheres(Hrabar, Eres; Kumric, 2007)

Best’s spherical resonator

Simulated E-field

distribution of ideal

plasmonic nanosphere

Best’s spherical resonator

Prototyping of “Plasmonicc” spheres

•diameter = 5.8 cm

-400

-300

-200

-100

0

Pha

se[d

egre

es]•1 •20•L

Measurement of E-field phase distribution along four-sphere chains (RF replicas of plasmonic WG)•Measuring points

solid: simulated; dashed: measured

•FW

0 5 10 15 20-500

Measuring point

•1 •20

•T

•Measuring points0 5 10 15 20

0

100

200

300

400

Pha

se[d

egre

es]

Measuring point

solid: simulated; dashed: measured

•BW

Prototyping of ‘RF analog of ‘plasmonic circular cluster ‘ (Hrabar, Muha, Zaluski, Mlakar, 2010)

Figure taken from N. Engheta (University of Pennsylvania)presentation slides

Figure taken from N. Engheta (University of Pennsylvania)presentation slides

Figure taken from N. Engheta (University of Pennsylvania)presentation slides

Scaled prototype of optical D-dot wire (Muha, Hrabar et al. 2011)

Near-field Superlens

Figure taken from V. Shalaev (Purdue University )presentation slides

Experimentally achieved images

written object (top)

N. Fang, H. Lee, C. Sun, X. Zhang, Science 308, 534, (2005).

optical image (center)

optical image without

super- lens

Potential Engineering ApplicationsV – Is it feasible to be invisible?V – Is it feasible to be invisible?

Figure taken from V. Shalaev (Purdue University )presentation slides

‘Transformation Electromagnetics’(Dollin 1961, Pendry 2006)

•Experimental verification already reported

First Experimental Results (Schurig 2006)

Simulations Measurements

Problem 1 All these structures are again extremely narrowband!

Figure taken from V. Shalaev (PurdueUniversity ) presentation slides

Problem 2 All these structures have significant loss !Figure taken from V. Shalaev (Purdue

University) presentation slides

Dispersion models of passive materials (metamaterials) with εr<1 or µµµµr<1

εreffz

1real part

1<0

•µµµµr real part

•Imaginarypart

•Drude model •Lorentz model

Both of these models are highly dispersive for0< εr<1 (or 0< µr<1)

fpf0

imaginary part

fmpf0f

0

1 part

Resonance is always present in passive

metamaterials – WHY? )

+

-

+

-- + - +

E

dielectric

- + - +

•URL: http://www.walter-fendt.de/ph14e/resonance.htm© Walter Fendt, September 9, 1998

-+

-+

-

+

-

- +

- + - +

E

- +

- + - +

Can one go around the dispersion-energy limitations?

20

20

)(

2

1)(

2

1HEWWW rr

me ωωµµ

ωωεε

∂∂+

∂∂=+=

)()( ∂∂ ωµωεW<0, ???

The only way is to use active medium (i.e. to introduce active elements) !

.0)(

,0)( >

∂∂>

∂∂

ωωµ

ωωε rr

Active medium should introduce ‘assisting negative restoring force’ !

•URL: http://www.walter-fendt.de/ph14e/resonance.htm© Walter Fendt, September 9, 1998

Fas•One would need a ‘negative spring’ or a‘negative mass’

•Going Active - Non-Foster reactive elements (negative C and negative L)

CX

CX

f

LX

LX

jX

-jX

ll

l

in

inin Z

I

V

I

VZ −=

−==

C-jX

New idea : Introduction of negative capacitance into TL (Hrabar et al , APS 2008)

•Positivecapacitance

•Negativecapacitance

21 CCC +=

−+−+ −=+= CCCCCe

1000 <<⇒−=+= −− εεεεεε e

)x=λ/30

Realistic

)x=λ/30

Z0

Realistic

CCRealistic

C

)x=λ/30

•Experimental RF Active ENZ TL with Three Unit Cells (l = 1 m), (Hrabar et.al 2011)

•Measurement of effective permittivity of ENZ Active TL with Three Unit Cells (Hrabar et.al 2011)

ENZ switched OFF

ENZ switched ON

Potential Engineering ApplicationsVI – Going nano?VI – Going nano?

Graphene-based One-atom-thick Metamaterials

Figures taken from N. Engheta (University of Pennsylvania)presentation slides

Graphene-based Metamaterials

Figures taken from N. Engheta (University of Pennsylvania)presentation slides

What’s next?

• Nanotechnolgy + metamaterials (plasmonics, ,photonics, lasing, gain materials, superluminal materials, graphene, nano-spheres and nano-films)

• Gravity, matter, acoustics + metamaterials (black hole metamaterials, seismic metamaterials, acoustic lenses)

Conclusions

• Metamaterials offer new exciting physical phenomena, which are very often counter-intuitive !

• In author’s opinion, this area is not in its infant phase any more and applications, that

• It is an interdisciplinary field which needs extensive collaboration of different worlds (math, physics, engineering, at least!)

infant phase any more and applications, that make use of new intriguing phenomena are expected to come in coming years!