Transcript Document

The Strange Properties
of
Left-handed Materials
C. M. Soukoulis
Ames Lab. and Physics Dept. Iowa State University
and
Research Center of Crete, FORTH - Heraklion, Crete
Outline of Talk
• Historical review left-handed materials
• Results of the transfer matrix method
• Determination of the effective refractive
index
• Negative n and FDTD results in PBGs (ENE & SF)
• New left-handed structures
• Experiments on negative refraction and
superlenses (Ekmel Ozbay, Bilkent)
• Applications/Closing Remarks
Peter Markos, E. N. Economou & S. Foteinopoulou
Rabia Moussa, Lei Zhang & Gary Tuttle (ISU)
M. Kafesaki & T. Koschny (Crete)
What is an Electromagnetic Metamaterial?
A composite or structured
material that exhibits
properties not found in
naturally occurring materials
or compounds.
Left-handed materials have
electromagnetic properties that
are distinct from any known
material, and hence are
examples of metamaterials.
Veselago
We are interested in how waves propagate through various
media, so we consider solutions to the wave equation.
E
2
 E  em 2
t
2
(-,+)
e,m space
m
n   em
(+,+)
e
k   em
(-,-)
(+,-)
Sov. Phys. Usp. 10, 509 (1968)
Left-Handed Waves
• If e  0, m  0 then
vectors:
• If e  0, m  0 then
vectors:
  
E, H , k


is a right set of


is a left set of
  
E, H , k
Energy flux in plane waves
• Energy flux (Pointing vector):
– Conventional (right-handed) medium
– Left-handed medium
Frequency dispersion of LH medium
• Energy density in the dispersive medium
 e 2  m  2
W
E 
H


• Energy density W must be positive and this requires
 e
0;

 m 
0

• LH medium is always dispersive
• According to the Kramers-Kronig relations –
it is always dissipative
“Reversal” of Snell’s Law
PIM
RHM
PIM
RHM
PIM
RHM
NIM
LHM
2
1
(1)
2
1
(2)

k
S
(1)
(2)

k
S
Focusing in a Left-Handed Medium
RH
RH
RH
RH
LH RH
n=1
n=1.3
n=1
n=1
n=-1 n=1
Left-handed
Right-handed
n=1
n=1
n=-1
n=1,52
n=1
n=1
Source
Source
M. Kafesaki
Objections to the left-handed
ideas
Parallel momentum is not conserved
S1
S2
A
Causality is violated
Fermat’s Principle
Superlensing is not possible
O΄
B
Μ
Ο
 ndl
minimum (?)
Reply to the objections
• Photonic crystals have practically zero absorption
• Momentum conservation is not violated
• Fermat’s principle is OK
 ndlextremum
• Causality is not violated
• Superlensing possible but limited to a cutoff kc or 1/L
Materials with e < 0 and m<0

k

 g opposite to


S  u g


S opposite to k
  1
dn n
d
n  0

p 

c 
g 
, p 
k0

n
p 
em
c2
Photonic Crystals

k

 g opposite to


S  u g


S opposite to k
  1
dn n
, n 

ck

 ,  0
n  0  n   n ,

,   0

p 

c
g 
, p 
k0

n
k e
m
2
2 
S 
E

H

8 



d
u 

p 
k

Super lenses
2  c 2k||2  k2  if k||  /c  k is imaginary
e
ik r
~e
 k r
Wave components with decay, i.e. are lost , then Dmax 
l
If n < 0, phase changes sign
k
k ||
e
ik r
~e
k r
thus
k
if
k  imaginary
k| |   / c
ARE NOT LOST !!!
Resonant response
10
q
E
p
5
q
p
E
0
-5
0
0.5
1
1.5
/
2
0
2.5
3
E
p
q
Where are material resonances?
Most electric resonances are THz or higher.
For many metals, p occurs in the UV
Magnetic systems typically have resonances through the
GHz (FMR, AFR; e.g., Fe, permalloy, YIG)
Some magnetic systems have resonances up to THz
frequencies (e.g., MnF2, FeF2)
Metals such as Ag and Au have regions where e<0,
relatively low loss
Negative materials
e<0 at optical wavelengths leads to important new
optical phenomena.
m<0 is possible in many resonant magnetic systems.
What about e<0 and m<0?
Unfortunately, electric and magnetic resonances do not
overlap in existing materials.
This restriction doesn’t exist for artificial materials!
Obtaining electric response
 2p
e( )  1 2


1
c
e
k
2
0
1.5
Drude Model
/p
e
-1
-2
1
-3
0.5
E
-
-
-4
-5
-
-
0
1
/p
2
3
Gap
0
k
2c 2
1
  2
 2
d ln(d / r) d Le 0
2
p
Obtaining electric response (Cut wires)

10
2
5
1.5
/p
e
 2p
e( )  1 2
   20
0
Drude-Lorentz
m
1
0.5
-
-
-10
-
-
k
Gap
-5
E
c
0
1
/p
2
3
0
k
Obtaining magnetic response
To obtain a magnetic response from conductors, we need to
induce solenoidal currents with a time-varying magnetic field
Introducing
A
A metal
metal ring
diska
gap
into the
is
is also
weakly
ring
creates a
weakly
diamagnetic
resonance
to
diamagnetic
enhance the
response
+
-
-
+
H
Obtaining magnetic response
F 2p
m( )  1 2
   20

3
c
m
k
2
2
1.5
/mp
m
1
0
1
Gap
-1
0.5
-2
-3
0
1
/mp
2
0
k
Metamaterials Resonance Properties
 2p
e    1 2

J. B. Pendry
 2p
m   1 2
   02
First Left-Handed Test Structure
UCSD, PRL 84, 4184 (2000)
Transmitted Power (dBm)
Transmission Measurements
Wires alone
Split rings alone
m>0
e<0
m<0
e<0
m>0
e<0
e<0
Wires alone
4.5
5.0
5.5
6.0
Frequency (GHz)
6.5
7.0
UCSD, PRL 84, 4184 (2000)
Best LH peak observed in left-handed materials
Transmission (dB)
0
SRR
Wire
CMM
-10
-20
-30
-40
-50
3
4
5
6
7
Frequency (GHz)
t
r1
d
r2
Bilkent, ISU & FORTH
w
Single SRR Parameters:
r1 = 2.5 mm
r2 = 3.6 mm
d = w = 0.2 mm
t = 0.9 mm
A 2-D Isotropic Structure
UCSD, APL 78, 489 (2001)
Measurement of Refractive Index
UCSD, Science 292, 77 2001
Measurement of Refractive Index
UCSD, Science 292, 77 2001
Measurement of Refractive Index
UCSD, Science 292, 77 2001
Boeing free space measurements for negative refraction
n
PRL 90, 107401 (2003) & APL 82, 2535 (2003)
Transfer matrix is able to find:
• Transmission (p--->p, p--->s,…) p polarization
• Reflection (p--->p, p--->s,…) s polarization
• Both amplitude and phase
• Absorption
Some technical details:
• Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24
• Length of the sample: up to 300 unit cells
• Periodic boundaries in the transverse direction
• Can treat 2d and 3d systems
• Can treat oblique angles
• Weak point: Technique requires uniform discretization
Structure of the unit cell
EM wave propagates in the z -direction
Periodic boundary conditions
are used in transverse directions
Polarization: p wave: E parallel to y
s wave: E parallel to x
For the p wave, the resonance frequency
interval exists, where with Re meff <0, Re
eeff<0 and Re np <0.
For the s wave, the refraction index ns = 1.
Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm
Typical permittivity of the metallic components: emetal = (-3+5.88 i) x 105
Generic LH related Metamaterials
Typical LHM behavior
m
e
p
p
m
 a/c
 a/c
m
m
 a/c
Resonance and anti-resonance
m
LHM Design used by UCSD, Bilkent and ISU
LHM
SRR
Closed LHM
T
Substrate
GaAs
eb=12.3
f (GHz)
30 GHz FORTH structure with 600 x 500 x 500 mm3
T and R of a Metamaterial
exp(ikd)
ts 
1  1
cosnkd 
z  sinnkd

2
z 
z
d
rs   ts exp(ikd)i(z  1 / z)sin(nkd ) / 2
UCSD and ISU, PRB, 65, 195103 (2002)
m
e
n  me
 2ep
e    1 2
   2e0  ie 0
 2mp
m   1 2
   2m0  im0
Inversion of S-parameters
1
2 m
1  1
2
2 
n  cos  1 r  t  
kd
2t 
kd

e

ik
z, n
te
ik
 ik
re
d
UCSD and ISU, PRB, 65, 195103 (2002)
1  r  t 2
z 
1 r2  t 2
2
n
e
z
m  nz
Effective permittivity e and permeability m of wires and SRRs

e    1

2
p
2
UCSD and ISU, PRB, 65, 195103 (2002)
 m2
m   1 2
   20  i
Effective permittivity e and permeability m of LHM
UCSD and ISU, PRB, 65, 195103 (2002)
Effective refractive index n of LHM
UCSD and ISU, PRB, 65, 195103 (2002)
New designs for left-handed materials
eb=4.4
Bilkent and ISU, APL 81, 120 (2002)
Bilkent & FORTH
Photonic Crystals with negative refraction.
Triangular lattice of rods with
e=12.96 and radius r,
r/a=0.35 in air. H (TE)
polarization.
Same structure as in Notomi,
PRB 62,10696 (2000)
PRL 90, 107402 (2003)
CASE 1
CASE 2
Photonic Crystals with negative refraction.
g
g
Equal Frequency Surfaces (EFS)
Schematics for Refraction at the PC interface
EFS plot of frequency a/l = 0.58
Experimental verification of
negative refraction
a

Lattice constant
a=4.794 mm
Dielectric
constant=9.61
R/a=0.329
Frequency=13.698 GHz
square lattice
E(TM) polarization
Bilkent & ISU
Band structure, negative refraction and experimental set up
Negative refraction is
achievable in this frequency
range for certain angles
of incidence.
Bilkent & ISU
Frequency = 13.7 GHz
l= 21.9 mm
17 layers in the x-direction and 21 layers in the y-direction
Superlensing in photonic crystals
Image Plane
FWHM = 0.21 l
Distance of the source from the PC interface is 0.7 mm (l/30)
Subwavelength Resolution in PC based Superlens
The separation between the two point sources is l/3
Subwavelength Resolution in PC based Superlens
Power distribution along the image plane
The separation between the two point sources is l/3 !
Controversial issues raised for
negative refraction
Among others
1) What are the allowed signs
for the phase index np and group
index ng ?
PIM
NIM
2) Signal front should move
causally from AB to AO to AB’;
i.e. point B reaches B’ in infinite
speed.
Does negative refraction violate causality and
the speed of light limit ?
Valanju et. al., PRL 88, 187401 (2002)
Photonic Crystals with negative refraction.
Photonic
Crystal
vacuum
FDTD simulations were used to study the time evolution of an EM
wave as it hits the interface vacuum/photonic crystal.
Photonic crystal consists of an hexagonal lattice of dielectric rods
with e=12.96. The radius of rods is r=0.35a. a is the lattice constant.
QuickTime™ and a
BMP decompressor
are needed to see this picture.
We use the PC system of case1 to
address the controversial issue raised
Time evolution of negative refraction shows:
The wave is trapped initially at the interface.
Gradually reorganizes itself.
Eventually propagates in negative direction
Causality and speed of
light limit not violated
S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, 107402 (2003)
Photonic Crystals: negative refraction
The EM wave is trapped temporarily at the interface and after a long time,
the wave front moves eventually in the negative direction.
Negative refraction was observed for wavelength of the EM wave
l= 1.64 – 1.75 a (a is the lattice constant of PC)
Conclusions
• Simulated various structures of SRRs & LHMs
• Calculated transmission, reflection and absorption
• Calculated meff and eeff and refraction index (with UCSD)
• Suggested new designs for left-handed materials
• Found negative refraction in photonic crystals
• A transient time is needed for the wave to move along the - direction
• Causality and speed of light is not violated.
• Existence of negative refraction does not guarantee the existence of
negative n and so LH behavior
• Experimental demonstration of negative refraction and superlensing
• Image of two points sources can be resolved by a distance of l/3!!!
$$$ DOE, DARPA, NSF, NATO, EU
Publications:
 P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002)
 P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002)
 D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002)
 M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, APL 81, 120 (2002)
 P. Markos, I. Rousochatzakis and C. M. Soukoulis, Phys. Rev. E 66, 045601 (R) (2002)
 S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, 107402 (2003)
 S. Foteinopoulou and C. M. Soukoulis, Phys. Rev. B 67, 235107 (2003)
 P. Markos and C. M. Soukoulis, Opt. Lett. 28, 846 (2003)
 E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and CMS, Nature 423, 604 (2003)
 P. Markos and C. M. Soukoulis, Optics Express 11, 649 (2003)
 E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and CMS, PRL 91, 207401 (2003)
 T. Koshny, P. Markos, D. R. Smith and C. M. Soukoulis, PR E 68, 065602(R) (2003)
PBGs as Negative Index Materials (NIM)
Veselago :
Materials (if any) with e < 0 and m< 0
 em>0

Propagation
 k, E, H Left Handed (LHM)  S=c(E x H)/4
opposite to k
 Snell’s law with
 g opposite to k n   em
 Flat lenses
 Super lenses
< 0 (NIM)
0.33 mm
w
t»w
t
t=0.5 or 1 mm
w=0.01 mm
0.33 mm
l=9 cm
3 mm
0.33 mm
3 mm
ax
Periodicity:
ax=5 or 6.5
mm
ay=3.63 mm
az=5 mm
Polarization: TM
y
E
x
B
y
z
x
Number of SRR
Nx=20
Ny=25
Nz=25
ax=6.5 mm
t= 0.5 mm
Transmission (dB)
0
-10
-20
-30
SRR
Wire
LHM
-40
-50
-60
7
8
9
10
11
12
13
14
Frequency (GHz)
Bilkent & ISU APL 81, 120 (2002)