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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
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Introduction: principles of wave-material interaction, basic dispersion models
Maxwell equations(time-harmonic waves; exp(j t))
0
jj
BD
DJHBE
more connections?
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Constitutive relations:
or more complex?
HE
BD
F
HBED
Dielectric models for dispersion
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Debye model
j1
)(''j)(')(
s
Debye modelj1
)( s
10-2 10-1 100 101 1020
10
20
30
40
50
60
70
80
90
'''
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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
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Drude model
j)(
2
2p
Drude model
10-1 100 101-10
-5
0
5
10
/ p
j)(
2
2p
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Kramers-Kronig relations
'd'
)'('PV2)(''
'd'
)'('''PV2)('
''j');(1)(
022
022
Homogenization principles and mixing formulas for structurally non-uniform
media
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Maxwell Garnett mixing formula
)(23
eiei
eieeeff f
f e
eff
i
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2i
10i 100i
Maxwell Garnett
ei
ei
eeff
eeff
22f
Bruggeman (symmetric)
022
)1(effi
effi
effe
effe ff
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Maxwell Garnett
Bruggeman
Maxwell Garnett Bruggeman
10j2i
100j2i
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MG and inverse MG
Raisin pudding Swiss cheese
1
MG and inverse-MG as bounds
Hashin–Shtrikman bounds
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Bounds in the complex plane
116j2
e
i
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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!)
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zNtN
axis ratio = at /az
Maxwell Garnett
spheres needles disks
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Lord Rayleigh mixing formula
Plasmonic mixtures?
predictions differ strongly• Maxwell Garnett: a singularity• Rayleigh: two singularities• Bruggeman: complex-valued effective
permittivity
0'j
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Singularity/complexity map
Plasmonic inclusions: volume fraction 30%, regular lattice- computational comparison (Comsol Multiphysics)
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Anisotropic, bi-anisotropic, and extreme-parameter
materials
material constitutive
relations
D = E
B = H
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chiral
PEMC
Perspectives into complexelectromagnetic materials/surfaces
• Classification in terms of– sign– magnitude– magnetoelectric cross-coupling– anisotropy– boundary/volume relation
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Anisotropy from geometry
Constitutive relations: bi-anisotropic media
HE
BD
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ISOTROPIC MEDIA, (2)
ANISOTROPIC MEDIA, (18)
BI-ISOTROPIC MEDIA
, , , (4)
BIANISOTROPIC MEDIA
, , , (36)
Bianisotropy: the word
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Bianisotropy
Bianisotropic constitutive relations
HEBHED
jj TT
arbitary,0)j(j :Reciprocalreal,)j(j :Lossless
TTTT
*TTTT*
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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
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E pe pm
H pe pm
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Chiral slab
0
Chiral (handed) media
chirality parameter (Pasteur parameter)
HE
BD
jj
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Tellegen (NRBI) material
Constitutive relations: bi-isotropic magnetoelectric media
chirality parameter (Pasteur)
non-reciprocity parameter (Tellegen)
= - j
= + j
HE
BD
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HANDEDNESS IN MATTER:
geometry and structure
Hegstrom & Kondepundi
Scientific American, Jan. 1990
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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
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Chirality dyadic (symmetric)
+ +
y
z
00000000
j
y
xz
0000000
j
Omega medium
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RECIPROCITY ?
Ja JbEaEb
dVdV ab?
ba JEJE
isotropic chiral slab
Optical activity
Jb
Ja
dVdV abba JEJE
:reciprocal mediumPasteur
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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
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refractive index & propagation factorin a bi-isotropic medium
2n
Bi-isotropy:instead of DPS, DNG,…
positive-definite, negative-definite,
non-definite materials
(cf. anisotropy)
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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)
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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
?
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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
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Reflection from PEMC boundary:
rrr
iii
PS
PS
EEE
EEE
0EH tantan MBoundary condition:
Another definition of the PEMC parameter: angle
PMC2/PEC0
cotM
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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
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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
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Performance of the PEMC materialization (2)
t
z
t
z ,
101001000
Extreme-parameter materials
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DPSENG
MNGDNG
MNZ media
EN
Z m
edia
log
log
”ordinary materials”
EV
L m
edia
MVL media
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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?
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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
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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)
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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
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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
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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
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Circular / Elliptical polarization
k
E(t)
Handedness of circular polarization
temporally RCP, spatially LCP !
http://www.tkk.fi/Yksikot/Sahkomagnetiikka/kurssit/animaatiot/ymppolar.html
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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