Transcript Outline

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
Outline
1.
2.
3.
4.
5.
Retrieval
Breaking of Scaling
Cut-wire pairs
Diamagnetic response of SRR
Anisotropic & Chiral metamaterials
Homogeneous Effective Medium Retrieval
e
ik
z, n
 ik
re
d
PRB, 65, 195104 (2002),
Opt. Exp. 11, 649 (2003).
te
ik
Effective medium: Periodicity Artifacts
Resonance/Anti-resonance “coupling”
“cut-off” deformations
negative imaginary part
Energy loss is positive for causal branch Im(n) > 0 Re(z) > 0
ν
Q()    | E |2  | H |2  2 | H |2 n() z()
Curves are for our 200THz SRR,
315nm x 330nm x 185nm unit cell
PRE, 68, 065602(R) (2003),
PRL 95, 203901 (2005).
Periodic Effective medium description
anti-resonance
pseudo-resonance
antipseudoresonance
“cut-off” & shift
“cut-off” at Brillouin zone edge
intermediate
band gap
generic SRR
PRB 71, 245105 (2005),
PRE 71, 036617 (2005).
Dashed lines: Underlying physical resonances
Solid lines: Effective response due to periodicity
Outline
1.
2.
3.
4.
5.
Retrieval
Breaking of Scaling
Cut-wire pairs
Diamagnetic response of SRR
Anisotropic & Chiral metamaterials
Breaking of Scaling
Going to THz frequencies
Idea: geometric scaling
linear
scaling
Metals are near-perfect conductors,
the effective LC-resonator
C   0  rel
A
d
0
 R2
l


8R
 L  0 R  log
 2
r0


densely stacked rings
sparse rings
depends on geometry only
lenght S  length  time S  time
Scale: Such
that speed of light invariant and S  0
C  S  L  S  m 
1
1

S
LC
Upper frequency limit of the SRRs?
55 nm
Theory: PRL 95, 223902 (2005),
Experiment: Opt. Lett. 31, 1259-1261 (2006).
Why saturation of ωm?
m 
1
LmC
1
Em  Lm I 2
2
Lm  a
m  1/ a
Ca
(a: unit cell size)
Charge-carriers have non-zero mass !!
Key point: Kinetic energy of the electrons becomes
comparable to magnetic energy in small scale structures
1
Ee  (neV )me ve2
2
1
2
 Le I
2
V: wire effective volume
S: wire effective cross-section
ne: e- number density
I
ve 
S e ne
me V
1
Le 
~
2
2
ne e S
a
1
1
m 

( Lm  Le )C
a 2  const.
Effective permeability
Can be obtained by effective medium retrieval procedure from transmission & reflection
or
directly via the magnetic moment of the SRR
M
1
  1 , M 
H
V
r j
 2 dV ,


1
j   i 1 
D

(

)
metal


Limits of simple LC picture
~ /2
~ 3 / 2
~ 5 / 2
“magnetic”
modes
Magnetic
coupling
or
circular
current
Electric
coupling
(anti-symmetric)
“electric”
modes
Electric
coupling
linear
current
(symmetric)
2 ~  / 2
2 ~ 
2  ~ 2
current density (arrows) & charge density (color)
Outline
1.
2.
3.
4.
5.
Retrieval
Breaking of Scaling
Cut-wire pairs
Diamagnetic response of SRR
Anisotropic & Chiral metamaterials
Electric resonance
Electric mode
of coupled electric resonances
Magnetic mode
of coupled electric resonances
Periodic Short-wire Pair arrays
With periodicity:
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).
6
Permeability
(a)
Imaginary
2
2
0
-2
0
0
Permittivity
Refractive Index
4
4
Real
-2
14
15
16
17
18
Frequency (GHz)
(b)
-2
-4
-6
APL 88, 221103 (2006)
14
L
L
b
b
2
ay 2
ay
(a)
(a)
Ce
Ce
1
1
ax
ax
Cm
C
Lm Lmm
Lm L m
Cm
Cm
(c)
magnetic(c)
resonance
2
2
C
C
(b)
(b)
Cm
Cm
15.0
Ce
Ce
14.5
a
14.0
13.5
13.0
fm
b
12.5
12.0
1.01
fe
1.02
1.03
ay/l
1.04
18
The cross-over of the
magnetic and electric
resonance frequencies
is difficult to achieve!
Lm  Cm  Ce 
 e 
Lm  Cm 



1 
  1

L
C
e 
e 
 m
2
2
(d)
electric(d)
resonance
Opt. Lett. 31, 3620 (2006)
1
2
Cm
Cm
Le Le
Le Le
Ce Ce
Ce Ce
1
m 
LeCe
e 
1
1
Permittivity,Permeability
l
l
Frequency (GHz)
1
1
15
16
17
Frequency (GHz)
 < 0 and  < 0
4
a
b
2
0
-2
-4
-6
-8
12
1/10
13
14
1/10
15
16 10
11
12
Frequency (GHz)
13
14
15
“Fishnet” structure
With periodicity:
Realization n<0 at 1.5m, Karlsruhe & ISU
Zhang et al., PRL 95, 137404 (2005).
Opt. Lett. 31, 1800 (2006).
A Brief History of Left-handed Metamaterials
Since the first demonstration of an artificial LHM in 2000, there has been rapid
development of metamaterials over a broad range of frequencies.
1000
(11)
(10)
(7)
(8)
100
(9)
(12)
(14)
n<0 for 780 nm
(ISU & Karlsruhe)
1 µm
(13)
Opt. Lett. 32, 53 (2007)
10 µm
10
(6)
200 nm
1
100µm
500 nm
(5)
Wavelength
Magnetic resonance frequency (THz)
100 nm
Science 315, 47 (2007)
n<0 for 1.5 µm
(ISU & Karlsruhe)
Science 312, 892 (2006)
1 mm
0.1
3 mm
0.01
1 cm
(2)
(3)
200 nm
(4)
µ<0 for 6 THz
(ISU & Crete)
Opt. Lett. 30, 1348 (2005)
(1)
10 cm
2000
2001
2002
2003
2004
2005
2006
2007
Year
Solid symbol: n<0
Open symbol: µ<0
n<0 for 4 GHz
(ISU & Bilkent )
Opt. Lett. 29, 2623 (2004)
Iowa State University involved in designing, fabrication and testing
of LHMs from GHz to optical frequencies [4,6,7,10,11,13,14].
Outline
1.
2.
3.
4.
5.
Retrieval
Breaking of Scaling
Cut-wire pairs
Diamagnetic response of SRR
Anisotropic & Chiral metamaterials
Magnetic moment around resonance
according to
F 2
 ( )  1  2
m   2  i
μ(ω) should return to unity below and above the resonance?
Two types of diamagnetic response
below resonance
B eliminated from
area of ring metal
above resonance
B eliminated from
all enclosed area
at resonance
B0
B0
Diamagnetic & Resonant currents
we describe metal by Drude model permittivity
then current density is available as:
L=10μm
f=300GHz
below resonance
Skin-depth


1
j ( )  i 1 
 D( )

(

)
metal


L=10μm
f=3.2THz
at resonance
(note: scale is 10x larger)
Metals at THz frequencies
Drude model permittivity qualitatively good description for Au, Ag, Cu up to optical frequencies
Drude model parameters from Experimental data:
lossy negative
“dielectric”
Im
Re
good
conductor
Johnson & Christy, PRB 6, 4370 (1972);
El-Kady et al., PRB 62, 15299 (2000).
1/ 2
 c 
lS  

  
for f < 1THz
1/ 2
   2 
1
lS 
, q   2 
Im q
 c 
Skin-depth saturates
at optical frequencies !
Silver
Ratio
Skin-depth/structure size
becomes larger !!
Copper
first ~ω1/2 then ~o(1)
Gold
Aluminum
Diamagnetic response of open and closed SRR ring
dependence on the ring width
L=10μm
f~3THz
L=100nm
f~70THz
Outline
1.
2.
3.
4.
5.
Retrieval
Breaking of Scaling
Cut-wire pairs
Diamagnetic response of SRR
Anisotropic & Chiral metamaterials
Anisotropic Arrays of Continuous or Short Nanowires
Continuous wires: radius=30nm, Drude-model Gold, (130nm)2 unit cell: F=16%
4
1.0
E wires
Re()
Im()
Re()
Im()
Permittivity,Permeability
Permittivity,Permeability
1.5
0.5
p
0.0
-0.5
-1.0
500
550
600
650
700
Frequency (THz)
750
800
Re()
Im()
Re()
Im()
3
H wires
2
Beware:
Periodicity
artifacts
1
0
500
550
600
650
700
Frequency (THz)
750
800
Short wires: radius=30nm, length=300nm, Drude-model Gold: F=11%
4
Re()
Im()
Re()
Im()
40
30
20
10
0
p
-10
200 250 300 350 400 450 500 550 600 650
Frequency (THz)
Permittivity,Permeability
Permittivity,Permeability
50
2
0
-2
p
-4
Re()
Im()
Re()
Im()
-6
-8
-10
200 250 300 350 400 450 500 550 600 650
Frequency (THz)
left-handed
negative
refraction
1,(1, 1)
anisotropic
negative
refraction
1,(1, 0.5)
Note that the hyperbolic dispersion supports propagating
modes for arbitrarily high parallel momenta
(which would be evanescent in air).
1,(1, 1)
Chiral Metamaterials: large gyrotropy & negative index
Constitutive relations
1310 nm
60
D   E  j  0 0 H
980 nm
B   H  j  0 0 E
40
660 nm
Transmission (%)
80
20
0.6
0.8
1.0
1.2
1.4
Wavelength (m)
50nm Al
50nm dielectric
1.6
Eigenmodes in chiral medium:
right circularly polarized (RCP, +) and
left circularly polarized (LCP, -), whose
wavenumbers and effective indices are:
• Bilayer chiral metamaterials
exhibits strong gyrotropy
at optical frequencies.
k  k0 (n   ),
n  k / k0  (n   )
If the chirality parameter is very large,
 n
• Specific rotatory power:
Wavelength  (nm)
Optical activity (°/mm)
then
660, 980, 1310
600, 670, 2500
V. A. Fedotov, CLEO Europe 2007
k  0, n  0,
the refractive index for the LCP
eigenmode becomes negative.
Circular Dichroism: Experiment & Simulation
Svirko-Zheludev-Osipov
Metamaterial (APL 78, 498 (2001))
Experimental results
LCP
Simulations, J. Dong et al.
 A  5.25 GHz
 B  6.50 GHz
Transmission (dB)
RCP
 A  5.25 GHz
 B  6.58 GHz
Frequency (GHz)
Δ (dB)
s 2

δ (degree)
 | t |  | t |
s
s
  arg(t
)  arg(t
)
s 2

Frequency (GHz)
A.V. Rogacheva, et al., PRL 97, 177401 (2006)
Frequency (GHz)
Thanks for your attention