MURI review Purdue MAY 2007, analysis, study

Analysis

1. Simulation and Understanding
of Metamaterials
Th. Koschny, J. Zhou, C. M. Soukoulis
Ames Laboratory and Department of Physics,
Iowa State University.
Th. Koschny, MURI NIMs Review May 2007, Purdue

2. Outline
1.Retrieval
2.Breaking of Scaling
3.Cut-wire pairs
4.Diamagnetic response of SRR
5.Anisotropic & Chiral metamaterials

3. Homogeneous Effective Medium Retrieval
z, n
d
€
e
ik
€
te
ik
€
re
−ik
PRB, 65, 195104 (2002),
Opt. Exp. 11, 649 (2003).

4. Effective medium: Periodicity Artifacts
Resonance/Anti-resonance “coupling”
“cut-off” deformationsnegative imaginary part
PRE, 68, 065602(R) (2003),
PRL 95, 203901 (2005).
Curves are for our 200THz SRR,
315nm x 330nm x 185nm unit cell
2 2 2
()|| ||2||()()Q E H Hnzωε μ ω ωω′′ ′′ ′′′= + =
Energy loss is positive for causal branch Im(n) > 0 Re(z) > 0
ν

5. Periodic Effective medium description
PRB 71, 245105 (2005),
PRE 71, 036617 (2005).
Dashed lines: Underlying physical resonances
Solid lines: Effective response due to periodicity
anti-resonance pseudo-resonance
“cut-off” at Brillouin zone edge
intermediate
band gap
“cut-off” & shift
generic SRR
anti-
pseudo-
resonance

6. Outline
1.Retrieval
2.Breaking of Scaling
3.Cut-wire pairs
4.Diamagnetic response of SRR
5.Anisotropic & Chiral metamaterials

7. Breaking of Scaling
Metals are near-perfect conductors,
the effective LC-resonator


depends on geometry only
Going to THz frequencies
Idea: geometric scaling
Scale:
lenghtSlengthtimeStime× ∧ ×
Such that speed of light invariant and0S→
0rel
A
d
Cεε≈
2
0 0
0
8
log 2
R R
R
l r
L
π
μ μ
⎛ ⎞
≈≈ −⎜ ⎟
⎝ ⎠
densely stacked rings sparse rings
1 1
m
CSLS
SLC
ω∝∧∝⇒ = ∝
linear
scaling

8. PRL 95, 223902 (2005),
Opt. Lett. 31, 1259-1261 (2006).
Upper frequency limit of the SRRs?
55 nm
Theory:
Experiment:

9. 2
2
1
()
2
1
2
e e ee
e
E nVmv
LI
=
=
21
2
=
m m
ELI
e
e
I
v
Sen
=
22
1
~
e
e
e
mV
L
neSa
=
m
La∝ Ca∝
Why saturation of ω
m?
Key point: Kinetic energy of the electrons becomes
comparable to magnetic energy in small scale structures
1
m
m
LC
=ω
1/
m
a∝ω
(a: unit cell size)
V: wire effective volume
S: wire effective cross-section
n
e: e
-
number density
Charge-carriers have non-zero mass !!
2
1 1
( ) .
m
m e
LLC aconst
ω= ∝
+ +

10. Effective permeability
Can be obtained by effective medium retrieval procedure from transmission & reflection
or
directly via the magnetic moment of the SRR
1 1
1 , , 1
2 ()
metal
M rj
M dVji D
H V
μ ω
εω
⎛ ⎞×
=+ = =− −⎜ ⎟
⎝ ⎠
∫

11. Limits of simple LC picture
“magnetic”
modes
circular
current
(anti-symmetric)
“electric”
modes
linear
current
(symmetric)
Magnetic
coupling
or
Electric
coupling
Electric
coupling
current density (arrows) & charge density (color)
~/2λ ~3/2λ ~5/2λ
2~λ×2~/2λ× 2~2λ×

12. Outline
1.Retrieval
2.Breaking of Scaling
3.Cut-wire pairs
4.Diamagnetic response of SRR
5.Anisotropic & Chiral metamaterials

13. Electric mode
of coupled electric resonances
Magnetic mode
of coupled electric resonances
Electric resonance

16. Periodic Short-wire Pair arrays
Lagarkov & Sarychev, PRB 53, 6318
(1996);
Panina et al., PRB 66, 155411 (2002);
Shalaev et al., Opt. Lett. 30, 3356 (2005).
Opt. Lett. 31, 3620 (2006),
Opt. Lett. 30, 3198 (2005).
With periodicity:

17. 1213141516
-8
-6
-4
-2
0
2
4


101112131415


1/101/10
Frequency (GHz)
a b
14 15 16 17 18
-6
-4
-2
0
(b)

Frequency (GHz)
-2
0
2
4
6


(a)
APL 88, 221103 (2006)
14 15 16 17 18
-2
0
2
4
Real

Frequency (GHz)
Imaginary
 < 0 and  < 0
1.01 1.02 1.03 1.04
12.0
12.5
13.0
13.5
14.0
14.5
15.0
a

ay/l
f
e
f
m
b
LmLm
Ce
Cm
Ce
Cm
(b)
(c) (d)
L
C
1 1
2 2
LeLe
Ce
Cm
Ce
Cm
(a)
2
1
b
ay
ax
l
magnetic resonance electric resonance
Opt. Lett. 31, 3620 (2006)
The cross-over of the
magnetic and electric
resonance frequencies
is difficult to achieve!
2
1 1
e m m
m e e
L C
L C
ω
ω
⎛⎞ ⎛ ⎞
= + <⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
1
e
ee
LC
ω=
( )
1
m
m m e
LCC
ω=
+
LmLm
Ce
Cm
Ce
Cm
(b)
(c) (d)
L
C
1 1
2 2
LeLe
Ce
Cm
Ce
Cm
(a)
2
1
b
ay
ax
l

18. “Fishnet” structure
Zhang et al., PRL 95, 137404 (2005).
With periodicity:
Opt. Lett. 31, 1800 (2006).
Realization n<0 at 1.5Realization n<0 at 1.5m, Karlsruhe & ISUm, Karlsruhe & ISU

19. Since the first demonstration of an artificial LHM in 2000, there has been rapid
development of metamaterials over a broad range of frequencies.
A Brief History of Left-handed Metamaterials
Iowa State University involved in designing, fabrication and testing
of LHMs from GHz to optical frequencies [4,6,7,10,11,13,14].
Open symbol: µ<0Solid symbol: n<0
n<0 for 1.5 µm
(ISU & Karlsruhe)
Science 312, 892 (2006)
n<0 for 780 nm
(ISU & Karlsruhe)
Opt. Lett. 32, 53 (2007)
µ<0 for 6 THz
(ISU & Crete)
Opt. Lett. 30, 1348 (2005)
n<0 for 4 GHz
(ISU & Bilkent )
Opt. Lett. 29, 2623 (2004)
Science 315, 47 (2007)

20. Outline
1.Retrieval
2.Breaking of Scaling
3.Cut-wire pairs
4.Diamagnetic response of SRR
5.Anisotropic & Chiral metamaterials

21. Magnetic moment around resonance
2
2 2
()1
m
F
i
ω
μω
ωωγω
=+
−+
according to
μ(ω) should return to unity below and above the resonance?

22. Two types of diamagnetic response
0B= 0B=
below resonance
B eliminated from
area of ring metal
above resonance
B eliminated from
all enclosed area
at resonance

23. Diamagnetic & Resonant currents
below resonance
at resonance
(note: scale is 10x larger)
L=10μm
f=300GHz
L=10μm
f=3.2THz
1
() 1 ()
()
metal
j i Dωω ω
εω
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
r ur
we describe metal by Drude model permittivity
then current density is available as:
Skin-depth

24. good
conductor
lossy negative
“dielectric”Im
Re
Metals at THz frequencies
Drude model permittivity qualitatively good description for Au, Ag, Cu up to optical frequencies
Aluminum
Copper
Gold
SilverSkin-depth saturates
at optical frequencies !
Ratio
Skin-depth/structure size
becomes larger !!

first ~ω
1/2 then ~o(1)
Drude model parameters from Experimental data:
Johnson & Christy, PRB 6, 4370 (1972);
El-Kady et al., PRB 62, 15299 (2000).
1/2
S
c
l
μωσ
⎛ ⎞
≈
⎜ ⎟
⎝ ⎠
1/2
2
2
1
,
Im
S
l q
q c
μεω⎛ ⎞
≈ =⎜ ⎟
⎝ ⎠
for f < 1THz

25. Diamagnetic response of open and closed SRR ring
dependence on the ring width
L=10μm
f~3THz
L=100nm
f~70THz

26. Outline
1.Retrieval
2.Breaking of Scaling
3.Cut-wire pairs
4.Diamagnetic response of SRR
5.Anisotropic & Chiral metamaterials

27. 200250300350400450500550600650
-10
0
10
20
30
40
50
Re(ε)
Im(ε)
Re(μ)
Im(μ)
Permittivity
,P
ermeability


Frequency (THz)
ω
p
200250300350400450500550600650
-10
-8
-6
-4
-2
0
2
4
Re(ε)
Im(ε)
Re(μ)
Im(μ)
Permittivity
,P
ermeability


Frequency (THz)
ω
p
Short wires: radius=30nm, length=300nm, Drude-model Gold: F=11%
Continuous wires: radius=30nm, Drude-model Gold, (130nm)
2
unit cell: F=16%
Anisotropic Arrays of Continuous or Short Nanowires
500550600650700750800
-1.0
-0.5
0.0
0.5
1.0
1.5
Re(ε)
Im(ε)
Re(μ)
Im(μ)
Permittivity
,P
ermeability


Frequency (THz)
ω
p
wiresE
ur
P
500550600650700750800
0
1
2
3
4
Permittivity
,P
ermeability


Frequency (THz)
Re(ε)
Im(ε)
Re(μ)
Im(μ)
wiresH
uur
P
Beware:
Periodicity
artifacts

28. 1,(1,0.5)++−
anisotropic
negative
refraction
1,(1,1)++−
1,(1,1)−−−
left-handed
negative
refraction
Note that the hyperbolic dispersion supports propagating
modes for arbitrarily high parallel momenta
(which would be evanescent in air).

29. •Bilayer chiral metamaterials
exhibits strong gyrotropy
at optical frequencies.
•Specific rotatory power:
Wavelength  (nm) 660, 980, 1310
Optical activity (°/mm) 600, 670, 2500
Eigenmodes in chiral medium:
right circularly polarized (RCP, +) and
left circularly polarized (LCP, -), whose
wavenumbers and effective indices are:
0
0
(),
/()
kkn
nkkn
χ
χ
±
± ±
= ±
= =±
nχ>
0, 0,k n
− −
< <
If the chirality parameter is very large,
the refractive index for the LCP
eigenmode becomes negative.
00
00
j
j
εχμε
μχμε
=−
= +
DE H
BH E
then
Constitutive relations
V. A. Fedotov, CLEO Europe 2007
50nm Al
50nm dielectric
Chiral Metamaterials: large gyrotropy & negative index

30. Experimental results
2 2
||||
s s
t t
++ −−
Δ= −
arg()arg()
s s
t tδ
++ −−
= −
LCP
RCP
5.25 GHz
A
ν=
6.50 GHz
B
ν=
5.25 GHz
A
ν=
Frequency (GHz)
T
r
a
n
s
m
is
s
io
n

(
d
B
)
6.58 GHz
B
ν=
Frequency (GHz)
Δ

(
d
B
)
Frequency (GHz)
δ

(
d
e
g
r
e
e
)
A.V. Rogacheva, et al., PRL 97, 177401 (2006)
Simulations, J. Dong et al.
D
Electron
Elastic
coupling
Electron
Inductive
Co u pl ing
D_
Svirko-Zheludev-Osipov
Metamaterial (APL 78, 498 (2001))
Circular Dichroism: Experiment & Simulation

31. Thanks for your attention