The Physics of Metamaterials - åœ‹å®¶ç†è«–ç§‘å­¸ç ”ç©¶ä¸­å¿ƒ(南區)

The Physics of Metamaterials
Pi-Gang Luan
Wave Physics/Engineering Lab
Department of Optics and Photonics
National Central University

Contents
t
• Definition and Examples of Metamaterials
• Constitutive i Relations and Effective
parameters
• Isotropic Media
• Anisotropic Media
• Inhomogeneous Media
• Acoustic Metamaterials
• Cloaking and Transformation Optics

What are Metamaterials ?
• Metamaterials are artificial materials
engineered to have properties that may
not be found din nature.
• They usually gain their properties from
structure rather than composition, using
small inhomogeneities to create effective
macroscopic behavior.

Examples
• Thin wire array (negative ε medium)
• Split-ring resonator (SRR) array
(negative μ medium)
• Wire-SRR array (negative n medium)
• Metal-dielectric multilayer structure
(indefinite medium)
• Conducting Helix/Gamada ()
array (chiral medium)
• Fishnet structure (negative refraction
medium)
• Helmholtz resonator/side hole array
(negative modulus medium)

Constitutive relations
and effective parameters
of metamaterials

Effective Permittivity and Permeability
Constitutive relations: D
E,
B
H
0
,
0 0
D
E P P electric dipole density
Np
B H M M nsity Nm
0
, magnetic dipole de
N
number density
1. In a metal, the "free electrons" are driven by the applied
E field, moving long distances.
When PD , we have DE<
0 , which means

Left-Handed Media
D. R. Smith et. al., Physics Today, 17, May (2000).
Phys. Rev. Lett. 84, 4184 (2000) ; Science, 292, 77 (2001)

The Building Blocks of LHM
Electric Dipoles
(high inductance + dilute carrier density
low plasma frequency)
+
Magnetic Dipoles
(LC resonance of ring current
resonance of magnetization)
( ) 1
F
2
2
p
( ) 1
2 2
2
0
(Plasmon-like dispersion)
(Polariton-like dispersion)

Effective ‘Material Parameters’
Ampere & Faraday’slaws
Averaged B field
a
Averaged H field
a
a
Effective permeability
IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)

Dynamical Equation of Polarization P (I)
2
dI
d LI
2
The current equation
L RI V El
RI
IV
dt
dt 2
2
dv
d
m eff
v
2
can be identified as meff
bv qE
bv qEv
dt
dt 2
2 2
l
2
meff
v
LI
if meff
v LI or N
r l , N is the charge concentration.
q
2 2
2
N
r L
2
2
Using the relation
I r Nqv m q
, we find
eff
0
a
0
0l a
I
0
l a
ln
r r
L Hda HldR dR
I I I I 2R 2
r
m
eff
2 2 2
Nr 0 q a 0
e 2
a
2
ln
Nr ln
r 2 r
J. B. Pendry, Phys. Rev. Lett.76, 4773 (1996)
l

Dynamical Equation of Polarization P (II)
Effective concerntration of charge : Neff
Polarization:
P
N qr, where r vdt
eff
t
0
r
2
a
2 2
dv
Nr 0q a
From meff
bv
qE, where meff
ln
dt
2 r
2
d d 2
Equation of motion P
P
2
p0E
dt dt
Plasma frequency: p
N
m
eff
eff
q
2
0
,
2
N
Dissipation coefficient:
Permittivity:
1
2
D P
p
1
0 E 0
E
i
J. B. Pendry, Phys. Rev. Lett.76, 4773 (1996)
b
m
eff

Effective Plasma vs. Real Plasma

Dynamical Equation of Magnetization M (I)
Faraday's law+effect of depolarization field
d
(1 ) dI q
F L RI ,
dt C dt
F : filling fraction of the ring g( (SRR) in one unit cell.
Magenetic fields inside/outsid e the ring: H / H, Applied field:
H
I
Hin
H (from Ampere's law),
l
FH (1 ) (flux
averag in
F H H0 ing)
2 2 2
2 2 r 0I / l r a
rB 0Fa H 0 , L 0
0F
I l l
2
rI I LI
2
M F H 0
H M,
LI 0a M
2
2
a l l a
0
IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)
IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)
J. Appl. Phys. 100, 024915 (2006)
in
0
B
0
a

Dynamical Equation of Magnetization M (II)
Effective inductance of a SRR is (1 F)
L
The term FL is caused by the dipolarization fied
l
dI q 2 dH0
(1 F)
L RI F
0a
dt C dt
2
2 dM R 2 0a
(1 F)
0a
0a M Mdt
dt L LC
2 dH0
F0a
dt
Dynamical equation of M :
d
M 2
d
H
M 0
M dt
dt
F dt
R
2 1
where , 0
L LC
IEEE Trans Microwave Theory Tech 47 2075 (1999)
IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)
J. Appl. Phys. 100, 024915 (2006)
a

Size determines the Response
• SRR () 列 例
裂 r
( LC 路 )
烈 ( 路 )
兩
4πr 流
( 度 ) 數
兩 裂 0
數
若
2r 數 a (
)
2πa 數 a 6
a

Indefinite Medium (I)
1D dispersion relation
Long wavelength limit
Define
If

Indefinite Medium (II)
A. Tangential field
a a
P P P , E E E
a
a
ad
am
D//
0
E// P//
Dd
D
a a
ad
am
d
mE// // E//
a a
ad
am
//
d
m
a
a
d
m
// d m // d m
From Faraday’s law
B. Normal field
m
From “div D = 0”
V Vd Vm Ed ad Em am
E
, D
Dd
D
a a a a a
E
ad Ed am Em 1 ad 1 am
1
D a D a D a a
d m
d m
m

Isotropic Media

Negative Refraction in Left-Handed Medium
(ε

Snell’sLaw
v g
v g
v g
v g
k 1x
k 2x
or n 1 sin
1
n
2 sin
2
c c

Veselago’sLeftHandedSlabLens
s Lens
0
0
n
0
The phase
increasing
outside can be
cancelled by the
phase decreasing
inside the slab
V. G. Veselago, Sov. Phys. Usp. 10, , 509 (1968)

Pendry’s idea of Perfect Lens
n 1
1
k vector (phase velocity)
Poynting vector (energy flow)
s
f1
f2
Veselago lens of 1
+ propagating p gwaves
+ evancesent waves
Phase and Amplitude Compensation
J. B. Pendry, “Negative Refraction Makes a
Perfect Lens”, ” Phys. Rev. Lett. 80, 3966 (2000)

Mechanism of Subwavelength Focusing

“Perfect” is unphysical
“Super” is achievable
Source
Virtual images
d
| z |
0
No solution can exist
in this blank region
d
| z |
0

“Imperfect” Superlens for
Subwavelength Imaging
(superlensing effect)
Case 1: x
z
0
1, d
2,
2 / k 0.3,
10 1.0 0.001i 0001i
Case 2: x
z0
1, d
2,
2 / k 2,
1.00 0.000001i

Uncertainty Principle vs.
Subwavelength Focusing
This decaying behavior can be easily explained by the
uncertainty principle. According to this principle, we
must have the relation x k 1, here x
represents
x
the width of the image, and kk
x
represents
s
the fluctuation
of k x
. A subwavelength image is mainly formed by summing
over the Fourier components of those | k | / c terms.
x
Si k 2 2 2 / 2
x
kz
c th t
k z
's. This lea
Since = / , these components must thave imaginary
i
ds to the decaying profile of the field strength.

References for understanding the
‘Subtleties’ of the ‘Perfect Lens’
• Negative Refraction Makes a Perfect Lens
J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)
• Left-Handed Materials Do Not Make a Perfect Lens
N. Garcia et al., Phys. Rev. Lett. 88, 207403 (2002)
• Perfect lenses made with left-handed materials: Alice’s mirror?
Daniel Maystre and Stefan Enoch, J. Opt. Soc. Am. A, 21, 122 (2004)
• Universal Features of the Time Evolution of Evanescent Modes in a
Left-Handed Perfect Lens
G. Gomez-Santos Santos, Phys. Rev. Lett. 90, 077401 (2003)
• Analysis on the imaging properties of a left-handed material slab
Pi-Gang Luan, Hung-Da Chien, Chii-Chang Chen, Chi-Shung Tang,
arXiv:physics/0311122;
Proper boundary conditions for the imaging problem of a negativerefraction
slab lens, WSEAS transactions on electronics, Vol. 1, 236
(2004)

Is the working wavelength much longer
than the lattice constant?
Typical value: λ/a = 5~7

Simulation:
SGI Altix
parallel
l
computer
using 32-48
processors
and require
about
20 h of
processor
time
FDTD simulations of SRR-Wire
based metamaterial slab lens
λ
Appl. Phys. Lett. 91, 154102 (2007)

Anisotropic Media

Negative Refraction in Calcite (Yau et.al. )
http://arxiv.org/abs/cond-mat/0312125

Indefinite media and Hyperbolic Dispersion
Hyperbolic dispersion i relation
If

Hyperlens
Far-field imaging beyond the diffraction limit
k r
k
2 2 2
2
| r
|
c
0, 0
r
OPTICS EXPRESS 14, 8247 (2006)

Effective medium: layered structures
m
d
1
,
2 2
1 1 1
r m d
SCIENCE 315, 1686 (2007)

Inhomogeneous Media

M. Notomi, Phys. Rev. B 62, 10 696 (2000).

Negative Refraction by PC

Snell’sLaw—The Generalized Form
Square Lattice Triangular Lattice

Constant Frequency Curve
Square Lattice v.s. Triangular Lattice

Superlens and Subwavelength Imaging

Negative Refraction
or Evanescent Wave
Coupling ?
Pi-Gang Luan and Kao-Der Chang,
“Superlensing effect without obvious
negative refraction ”,
J. Nanophotonics, 1, 013518 (2007)
Evanescent wave can be transmitted
from source to image via evanescent
wave coupling effect (using a stack of
dielectric gratings)

Is PhC Superlens Flat Lens ? (I)
J. Phys.: Condens. Matter 23 (2011) 035301

Is PhC Superlens Flat Lens ? (II)
J. Phys.: Condens. Matter 23 (2011) 035301

Is PhC Superlens Flat Lens ? (III)
J. Phys.: Condens. Matter 23 (2011) 035301

Wave Propagation in Periodic
Structures — Electric Filters and
Crystal Lattices
“Waves always behave in a similar
way, whether they are longitudinal
or transverse, elastic or electric.
Scientists of the last (19th) century
always kept this idea in mind.
”
--- L. Brillouin

Acoustic Metamaterials

Locally Resonant Sonic Material
Negative Dynamic Mass leads to
Polariton-like band gap
i t
i t
Fe
0
0
F F
a
a e
m
M , m m, Displacement: x,
y
box
ball
elastic constant: K,
K
Mx F 2 K( x y), my 2 K ( y
x
)
2
F f
0
m 2K
meff
M 1 , f ,
2 2
0
a 0
M m
Ping Sheng et. al., Science 289, 1734 (2000)
g g , , ( )
Physica B: Condensed Matter 394, 256 (2007)

Pl Polariton-like lik gap

Ultrasonic metamaterials
with negative modulus
Xiang Zhang et al
Xiang Zhang et. al.,
Nature Materials, 5, 452 (2006)

1D HR array metamaterial
v
1 pp
sh
p , v
u
t B t A
'
M du p p' S u,
dp BS u
dt dt V
du
BS
M u x
pS (Driven Oscillation)
dt V
For monochromatic wave
p
u
i
p
sh
Sp i
p
v
B A BS
iM
Beff
iV
2
1
1
F
0
Beff
(
) B
1
2 2
0
i
v
p '

Transformation Optics
and Invisibility Cloak

Principle of Invisibility cloak (I)
料
裡 不 不 連
不 不
不
不
裡 不
John B. Pendry 2006 年 Science 了
Maxwell 理 論 理 論 連
滑 Maxwell
不 數 率 —
(anisotropic)
(inhomogeneous) 更 精 說 量 度
(tensor densities)

Principle of Invisibility cloak (II)
兩
了 ()
(representation) 了
了
來
(transformed medium) 不
( 不 不 ) (embedded)
了 了

Principle of Invisibility cloak (III)
Duke University 了
()
了 易 參 數
理 論 參 數 了
參 數 參 數 (reduced
parameters)
裂 (split-ring resonator, SRR)
狀 列 參 數 () 理
參 數 不
不
理 論 了 力
了 料 更

Principle of Invisibility cloak (IV)

Invisibility Cloak utilizes Anisotropic
and Inhomogeneous Metamaterials
Science 312, 1777 (2006) Science 312, 1780 (2006)
PRE 74, 036621 (2006) New J. Phys. 8, 247 (2006)
Phys. World, Sep. 30 (2006)
Science 313, 1399 (2006)

Experimental Realization
Ideal theoretical Parameters
Parameters for real experiment
2 2
r
a r
r
r
,
r a
, 1,
b
r r
r
z
a
r b
a
2
b r
a
z
z
b
a
r Science 314, 977 (2006)

Shuang Zhang, Dentcho A. Genov, Cheng Sun, and Xiang Zhang
Phys. Rev. Lett. 100, 123002 (2008)
Cloaking of Matter Waves
A cloaking of matter wave can be realized at given energy by
designing i the potential and effective mass of the matter waves
in the cloaking region

Aharonov-Bohm effect

Cloaking Matters under
Aharonov-Bohm effect
D. H. Lin & P. G. Luan, PRA 79, 051605(R) (2009)

Conclusion
• Most proposed metamaterials consist of periodically arranged
resonators or resonator pairs.
• A single resonator couples almost only to its nearest neighbors.
• If the spatial dispersion of the effective medium can be neglected,
then its local properties are not influenced under arbitrary bending.
Examples: hyperlens, invisibility cloak,…
• Photonic crystals (PhC) and metamaterials (MtM) are both periodic
structures, however, spatial dispersion effect cannot be neglected in
PhC. This might be the main difference between PhC and MtM.
• In order to solve the more and more challenging problems
encountered din metamaterial t research, in the future we need dto
develop a different education/discipline system to train up more
“Sciengineers”.

Thank you for your attention!