1

Wave-material Interaction in Electromagnetics:

Effects of homogenization of mixtures and classification of complex media responses

Ari SihvolaAalto University

Department of Radio Science and Engineering

Finland

IAS Program HKUST Institute of Advanced Study6 November 2012

Topics to be discussed…

• Basics of electromagnetics• Wave interaction with materials• Homogenization principles and mixing

formulas• Complex material responses:

anisotropy, magnetoelectric coupling, bianisotropy

• Metamaterials concepts

2

Introduction: principles of wave-material interaction, basic dispersion models

Maxwell equations(time-harmonic waves; exp(j t))

0

jj

BD

DJHBE

more connections?

3

Constitutive relations:

or more complex?

HE

BD

F

HBED

Dielectric models for dispersion

4

Debye model

j1

)(''j)(')(

s

Debye modelj1

)( s

10-2 10-1 100 101 1020

10

20

30

40

50

60

70

80

90

'''

5

Lorentz model

j)(

220

2p

Lorentz model

10-1 100 101-0.5

0

0.5

1

1.5

2

2.5

3

3.5

/ 0

j)(

220

2p

6

Drude model

j)(

2

2p

Drude model

10-1 100 101-10

-5

0

5

10

/ p

j)(

2

2p

7

Kramers-Kronig relations

'd'

)'('PV2)(''

'd'

)'('''PV2)('

''j');(1)(

022

022

Homogenization principles and mixing formulas for structurally non-uniform

media

8

Maxwell Garnett mixing formula

)(23

eiei

eieeeff f

f e

eff

i

9

2i

10i 100i

Maxwell Garnett

ei

ei

eeff

eeff

22f

Bruggeman (symmetric)

022

)1(effi

effi

effe

effe ff

10

Maxwell Garnett

Bruggeman

Maxwell Garnett Bruggeman

10j2i

100j2i

11

MG and inverse MG

Raisin pudding Swiss cheese

1

MG and inverse-MG as bounds

Hashin–Shtrikman bounds

12

Bounds in the complex plane

116j2

e

i

13

Aligned ellipsoids

)()1( eieei

eeeff,x

x Nff e

eff

depolarization factor !

i

Randomly oriented ellipsoids

zyxj j

j

zyxj j

NNf

Nf

,, eie

eie

,, eie

ei

eeeff

)()(

3

)(3

e

i

eff(isotropic!)

14

zNtN

axis ratio = at /az

Maxwell Garnett

spheres needles disks

15

Lord Rayleigh mixing formula

Plasmonic mixtures?

predictions differ strongly• Maxwell Garnett: a singularity• Rayleigh: two singularities• Bruggeman: complex-valued effective

permittivity

0'j

16

Singularity/complexity map

Plasmonic inclusions: volume fraction 30%, regular lattice- computational comparison (Comsol Multiphysics)

17

Anisotropic, bi-anisotropic, and extreme-parameter

materials

material constitutive

relations

D = E

B = H

18

chiral

PEMC

Perspectives into complexelectromagnetic materials/surfaces

• Classification in terms of– sign– magnitude– magnetoelectric cross-coupling– anisotropy– boundary/volume relation

19

Anisotropy from geometry

Constitutive relations: bi-anisotropic media

HE

BD

20

ISOTROPIC MEDIA, (2)

ANISOTROPIC MEDIA, (18)

BI-ISOTROPIC MEDIA

, , , (4)

BIANISOTROPIC MEDIA

, , , (36)

Bianisotropy: the word

21

Bianisotropy

Bianisotropic constitutive relations

HEBHED

jj TT

arbitary,0)j(j :Reciprocalreal,)j(j :Lossless

TTTT

*TTTT*

22

Classification of bi-anisotropic materials

Symmetric part:6 parameters

(RECIPROCAL)

Dielectric crystal

Magnetic medium

Chiral medium

Cr2O3

Anti-symmetric part3 parameters

(NON-RECIPROCAL)

Magneto-plasma

Biased ferrite

Omega medium

Moving medium

A. Sihvola, I.V. Lindell (2008), Perfect electromagnetic conductor medium, Ann. der Physik, 17(9-10), 787-802

Optical activity from geometry

23

E pe pm

H pe pm

24

Chiral slab

0

Chiral (handed) media

chirality parameter (Pasteur parameter)

HE

BD

jj

25

Tellegen (NRBI) material

Constitutive relations: bi-isotropic magnetoelectric media

chirality parameter (Pasteur)

non-reciprocity parameter (Tellegen)

= - j

= + j

HE

BD

26

HANDEDNESS IN MATTER:

geometry and structure

Hegstrom & Kondepundi

Scientific American, Jan. 1990

27

Giant frog shell

Marlinspike

Australian trumpet shell

Busycon perversum

Classification of bi-anisotropic materials

Symmetric part:6 parameters

(RECIPROCAL)

Dielectric crystal

Magnetic medium

Chiral medium

Cr2O3

Anti-symmetric part3 parameters

(NON-RECIPROCAL)

Magneto-plasma

Biased ferrite

Omega medium

Moving medium

28

Chirality dyadic (symmetric)

+ +

y

z

00000000

j

y

xz

0000000

j

Omega medium

29

RECIPROCITY ?

Ja JbEaEb

dVdV ab?

ba JEJE

isotropic chiral slab

Optical activity

Jb

Ja

dVdV abba JEJE

:reciprocal mediumPasteur

30

Faraday rotation

dVdV abba JEJE

:reciprocal-non smaMagnetopla

Jb

Ja

Igj:typermittivi cAnisotropi

symm

B0

Bi-isotropic media

chirality parameter (Pasteur)

non-reciprocity parameter (Tellegen)

= - j

= + j

HE

BD

jj

31

refractive index & propagation factorin a bi-isotropic medium

2n

Bi-isotropy:instead of DPS, DNG,…

positive-definite, negative-definite,

non-definite materials

(cf. anisotropy)

32

FW, FWBW,

BW

BW, FW

FW, BW

material matrix

positive definite

jj

negative definite

non-definite

(indefinite)

non-definite

(indefinite)

NIM handedness

• non-reciprocal, non-chiral• (Tellegen)

• reciprocal, chiral• (Pasteur)

• non-reciprocal, chiral• (general bi-isotropic)

• reciprocal, non-chiral• (”ordinary” isotropic)

0,0

0,00,0

0,0

NIM

(no anisotropy)

33

znk0-je

rr

22

n

nn

nihility! chiral0,0,0

:EXAMPLEn

plane wave propagation point of view: bi-isotropic media

AXION-only: PEMC material

EHBD MM ,

HE

BD

Function

?

34

Magnetoelectric relations:

H1EB&)HE(DM

qMq

qM

Mq

HE

HE

BD

/111

EH&BD MM

PEMC material

PEC:

000

BE

000

DH

PMC:

0M0M BDEH

35

Reflection from PEMC boundary:

rrr

iii

PS

PS

EEE

EEE

0EH tantan MBoundary condition:

Another definition of the PEMC parameter: angle

PMC2/PEC0

cotM

36

i

i

i

i

r

r

MMMM

M

P

S

P

S2

2

2P

S

EE

2cos2sin2sin2cos

EE

1221

11

EE

Reflected components as functions of incident components

Normal incidence

incident

reflected

rotation angle of the electric field polarization:

2 arctan M

37

Materialization of the PEMC surface

PEC

Gyrotropically uniaxial layer

RPEMC

is

ip

sssp

psppis

ip

rs

rp

EE

RRRR

EE

EE

R

Gyrotropically uniaxial layer:waveguiding (large axial permittivity and

permavility)non-reciprocal (e.g. ferrite)

Performance of the PEMC materialization (1)

t

z

t

z

101001000

38

Performance of the PEMC materialization (2)

t

z

t

z ,

101001000

Extreme-parameter materials

39

DPSENG

MNGDNG

MNZ media

EN

Z m

edia

log

log

”ordinary materials”

EV

L m

edia

MVL media

40

nlog

log

”ordinary materials”ZIM

IZM

IIM

ZZM

MNZ media

EN

Z m

edia

log

log

”ordinary materials”

PEC

ZIM

EV

L m

edia

MVL mediaIZM IIM

ZZM?

41

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

)}(Re{ 01 rka

66 10,10

PEC

1,1060.0 0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

Scattering by a sphere: the first Mie coefficient

1,)j1.01( c 1, cc PEC

Scattering efficiency of spheresmall sphere, ka= 0.1

resonant sphere, ka = 2

42

Philosophy of metamaterials

Metamaterials?

• Qualitatively new properties, not present in the constituent components

• Caused by the structure and interactions• More is different—quantity becomes

quality• Emergence, or at least supervenience• On the other hand: More is less...

(reductionism)

43

1 + 1 > 2 or 1 + 1 < 2 ?emergence or reductionism ?

A. Sihvola (2003): Electromagnetic emergence in metamaterials, in Advances in Electomagnetics of Complex Media and Metamaterials, (S. Zouhdi et al., eds), NATO Science Series, 89, 1-17.

Metamaterial terminology

• DNG (double negative media)• NIM (negative index media)• Backward-wave media• NPV (negative-phase-velocity

media)• Veselago media• Left-handed media

44

What’s in a name? That which we call a rose by any other word would smell as sweet.

Vector directions

< 0:

(E,H,k) left-handed

(E,B,k) right-handed

HBEkBE j

45

E

H

Sk

E

H

Sk

”RHM” ”LHM”

Handedness can be associated with

• vector triplet of electric field, magnetic field, wave vector

• polarization of the propagating wave

• geometrical structure of the medium

46

Circular / Elliptical polarization

k

E(t)

Handedness of circular polarization

temporally RCP, spatially LCP !

http://www.tkk.fi/Yksikot/Sahkomagnetiikka/kurssit/animaatiot/ymppolar.html

47

Handedness in matter: geometry and structure:

chirality

References• A. Sihvola: Electromagnetic mixing formulas and applications, IEE

Publishing, London, 1999• I.V. Lindell, A.H. Sihvola: Realization of the PEMC boundary. IEEE Trans.

Ant. Propagation, 53(9), pp. 3012-3018, September 2005• I.V. Lindell, A.H. Sihvola: Electromagnetic boundary and its realization with

anisotropic metamaterial, Physical Review E, 79, 026604, 2009• A. Sihvola, I.V. Lindell: Perfect electromagnetic conductor medium. Annalen

der Physik (Berlin), 17(9–10), pp. 787-802, 2008• A. Sihvola: Electromagnetic emergence in metamaterials. In (S. Zouhdi, A.

Sihvola, M. Arsalane, editors). NATO Science Series: II, 89, pp. 1-17, Kluwer, Dordrecht, 2003

• A. Sihvola: Metamaterials in electromagnetics. Metamaterials, 1, pp. 2–11, 2007

• A. Sihvola: Metamaterials: a personal view. Radioengineering, 18(2), Part I, pp. 90-94, June 2009

• A. Sihvola, I.V. Lindell, H.Wallén, P. Ylä-Oijala: Material realizations of Perfect Electric Conductor objects. ACES Journal, 25(12), 1007-1016, 2010