Transcript Document

Negative Index Materials:
New Frontiers in Optics
C. M. Soukoulis
Ames Lab. and Physics Dept. Iowa State University
and
IESL-FORTH & Materials Dept. - Heraklion, Crete
Left-Handed Materials
History:
• Permittivity e, permeability m and index of refraction n negative
• Reversal of Snell’s Law, perfect focusing, flat lenses, etc.
• Impedance match z=√e/m and n =-1
• l >> a in LHM, while l  a in PBG
Both PBG and LHM exhibit properties not found in naturally materials
Vision:
• Understanding the physics and the exotic properties of LHMs
• Perfect Lens. Near-field optical microscopy, nano-lithography
• Wireless and optical communications. RF sensing.
• Antenna and microwave device miniaturization
________________________________
 Breakthroughs and new concepts in materials processing at nanoscale
 Search for new materials that exhibit m < 0 at THz or optical regime
Computational Methods
 Plane wave expansion method (PWE)
R. Moussa, S. Foteinopoulou & M. Kafesaki
 Transfer matrix method (TMM)
Th. Koschny & P. Markos
 Finite-difference-time-domain-method (FDTD)
M. Agio, M. Kafesaki, R. Moussa, & S. Foteinopoulou
 Effective medium theories
E. N. Economou, Th. Koschny, M. Kafesaki
The LHM effort is in close collaboration with experiments (ISU, Crete,
Bilkent, UCSD, Boeing) Karlsruhe
http://cmpweb.ameslab.gov/personnel/soukoulis
http://gate.iesl.forth.gr/~cond-mat/photonics
Collaborators
 Negative refraction in photonic crystals:
E. N. Economou (Crete, Greece)
S. Foteinopoulou and R. Moussa (Ames, USA)
E.Ozbay (Bilkent, Turkey)
 Left-handed Materials
P. Markos (Ames & Slovakia), T. Koschny (Crete & Ames)
E. N. Economou, M. Kafesaki, & N. Katsarakis (Crete, Greece)
G. Konstandinidis, R. Penciu and T. Gundogdu (Crete, Greece)
E. Ozbay (Bilkent, Turkey)
Lei Zhang J. Zhou & G. Tuttle (Ames, USA)
D. R. Smith (UCSD, USA)
M. Wegener (Karlsruhe)
Boeing’s group (Seattle, USA)
Outline of Talk
 Historical review of left-handed materials
 Results of the transfer matrix method
 Determination of the effective meff and eeff
Negative imaginary parts in e(w) and m(w)
 Periodic effective medium theory. Im e(w) and Im m(w) > 0
 Electric and Magnetic Response of SRRs and LHMs
Effective wp of the LHM is much lower than wp of the wires.
There are “phony” LH peaks when wp < wm. It’s difficult to find a LH peak!




Negative n and FDTD results in PBGs (ENE & SF)
Experiments on negative refraction and superlenses (Ozbay)
Ongoing and future work
Concluding Remarks
Veselago
We are interested in how waves propagate through various
media, so we consider solutions to the wave equation.
 E
2
 E  em
2
(-,+)
e,m space
m
t
2
n   em
(+,+)
e
k  w 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
  
E, H , k
)
is a left set of
(
(
Energy flux in plane waves
• Energy flux (Poynting vector):
– Conventional (right-handed) medium
– Left-handed medium
“Reversal” of Snell’s Law
PIM
RHM
PIM
RHM
PIM
RHM
NIM
LHM
2
1
(1)
2
1
(2)
(1)

k
S
(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
Evanescent wave refocusing: Perfect lensing
J. B. Pendry
Frequency dispersion of LH medium
• Energy density in the dispersive medium
W 
 (ew)
w
E 
2
 (mw )
w
H
2
• Energy density W must be positive and this requires
 (ew)
w
0;
 (mw )
w
0
• LH medium is always dispersive
• According to the Kramers-Kronig relations –
it is always dissipative
Resonances
Medium response
P˜  n p˜ 
P˜ 
( nq / m ) E˜
w 2p e 0 E˜
2
w  w  i w
2
0
2
w 2p e 0 E˜
w 02  w 2  i w

w 02  w 2  i w
  e 0 E˜
Drude-Lorentz forms for e and m
e
e0
 1   1
w 2ep
m
w  w  i w
m0
2
2
0
 1
w 2mp
w 2  w 20  i  w
Resonant response
10
q
E
p
5
q
p
E
0
-5
0
0.5
1
1.5
w/w
2
0
2.5
3
E
p
q
Where are material resonances?
Most electric resonances are THz or higher.
For many metals, wp 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
d
e(w )  1 
w 2p
w
w2
1
c
e
k
2
0
1.5
Drude Model
w
w/wp
e
-1
-2
wp
1
-3
0.5
E
-
-
-4
-5
-
0
1
wp 
2
2
w/wp
2 c
d2
3
Gap
0
k
2
Obtaining electric response (Cut wires)
e(w )  1 

h
w 2p
w w
2
w 
2
0
10
2
5
1.5
c
m
k
w/wp
e
w
0
wp
Drude-Lorentz
-5
E
1
Gap
0.5
-
-
-10
-
1
0
w0 
c
h
w/wp
1
ln( h /  )
2
0

 0
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 w mp
2
m (w )  1 
w w
2
w 
2
m
3
c
m
k
2
2
1.5
1
w/wmp
m
w
0
w mp 1
Gap
-1
wm 
2
0.5
1
-2
LC
-3
0
1
w/wmp
2
0
k
Metamaterials Resonance Properties
e (w )  1 
w 2p
m (w )  1 
w2
J. B. Pendry
w 2p
w 2  w 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
Transfer matrix method to compute scattering amplitudes
continuum Homogeneous Effective Medium inversion
Generic LH related Metamaterials
Typical LHM behavior
wm
e
w
p
w
p
wm
w a /c
w a /c
m
wm
w a /c
Resonance and anti-resonance
wm
PRL 93, 107402(2004)
Closing the gaps of the SRRs,---> magnetic response disappears
FORTH and Ames
PRL 93, 107402(2004)
Electric response of LHMs is the sum of wires and closed SRRs
FORTH and Ames
PRL 93, 107402(2004)
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
Left-Handed Materials
t
SRR Parameters:
r1
d
r2
w
r1=2.5 mm,
r2=3.6 mm,
d=w=0.2 mm
t=0.9 mm
Parameters:
ax=9.3 mm
ay=9mm
az=6.5 mm
Nx=15
Ny=15
Transmission data for open and closed SRRs
Magnetic resonance
disappears for closed SRRs
Bilkent, Crete & Ames
Effective wp of closed SRRs & wires is much lower than wp of the wires.
Bilkent, Crete & Ames
Best LH peak in a left-handed material
Peak at
f=4 GHz
l=75 mm
much larger
than
size of SRR
a=3.6 mm
Losses: -0.3 dB/cm
Bilkent, Crete & Ames
Electric coupling to the magnetic resonance
APL 84, 2943 (2004)
Ames Lab. & Crete
Magnetic response at 100 THz, almost optical frequencies
l
10
S. Linden & M. Wegener, Karlsruhe
Magnetic response at 100 THz, almost optical frequencies
S. Linden & M. Wegener, Karlsruhe
Magnetic response at 100 THz, almost optical frequencies
S. Linden & M. Wegener, Karlsruhe
T and R of a Metamaterial
ts 
exp(  ikd )
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
n
e
e (w )  1 
m (w )  1 
me
w 2ep
w 2  w 2e0  i e 0
w 2mp
w2 w
2
m0
)
 i m 0
Inversion of S-parameters
n
e
1
kd
cos
1
2 m
 1
2
2 
1  (r  t ) 
 2 t 

kd


ik
z, n
re
te
ik
 ik
d
UCSD and ISU, PRB, 65, 195103 (2002)
(1  r )  t 
z 
2
(1  r )  t 2
2
e
n
z
2
m  nz
Refractive index n
Im n > 0
Re n > 0
Permittivity e
Permeability m
Im e < 0 ???
Re e > 0
Im m > 0
Re m < 0
Energy Losses Q in a passive medium are always positive in spite of the fact that Im e < 0
Q( w )  e  | E |  m | H |
2
2
Q( w )  2w | H | n (w ) z (w )
2
Q(w) > 0, provided that Im n(w) > 0 and Re z(w) > 0
Band structure, negative refraction and experimental results
f = 13.7 GHz
l= 21.9 mm
Negative refraction is
achievable in this
frequency range.
PRL 91, 207401 (2003)
Nature 423, 604 (2003)
17 layers in the x-direction and 21 layers in the y-direction
Bilkent & Ames
Nature 423, 604 (2003)
2D field snapshot for two incoherent sources viewed
from the image plane
Same as in previous slide but zoomed in
Coupling to surfaces waves can improve focusing
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.
Negative refraction in photonic crystals
QuickTime™ and a
BMP decompressor
are needed to see this picture.
PRL 90, 107402 (2003)
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)
Ongoing and future work
 Improvement of transfer matrix code.
Off-normal incidence and non-uniform discetization.
 Improvement of the retrieval code and understanding of why Ime Imm < 0.
 Effective medium theory. Effects of periodicity. Origin of losses.
 Isotropic 2d and 3d designs of LHMs. Fabrication and testing.
 Propose and fabricate LH structures at 94 GHz, THz and 10.6 mm.
 Superlattices of negative e and negative m. Negative n?
Objectives of the LHM effort
 A better understanding of the physics of left-handed (LH) materials.
 Improvement of the existing tools for modeling and simulating more complicated
structures than can be done today.
 Fabrication of LH-materials, using various approaches, materials and processes.
 Testing the electromagnetic behavior of these materials.
 Identifying several different applications where such materials can make a big
contribution.
Conclusions
• Simulated various structures of SRRs & LHMs.
• Calculated transmission, reflection and absorption.
• Calculated meff and eeff and refraction index. Ime(w) Imm(w) < 0.
• Periodic effective medium theory.
• Suggested new designs for left-handed materials and SRRs.
• A criterion was proposed for finding if a T peak is LH or RH.
• Magnetic response in 100 THz regime! (Experiment).
• Found negative refraction in photonic crystals. Low losses.
• Experimental demonstration of negative refraction and superlensing.
• Image of two points sources can be resolved by a distance of l/3!!!
• Evanescent wave refocusing. Role of surface termination. Surface waves.
$$$ 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)
 N. Katsarakis, T. Koschny, M. Kafesaki, ENE and CMS, APL 84, 2943 (2004)
 T. Koschny, M. Kafesaki, E. N. Economou and CMS, PRL 93, 107402 (2004)
 Lei Zhang, G. Tuttle and CMS, Photonics and Nanostructures (accepted, 2004)
Material Response: Lorentz Oscillators
Driven harmonic oscillator:
F (t )  m Ý
xÝ qE ( t)  kx   xÝ
Harmonic dependence:
 iw t
˜
E ( t)  E e
x (t )  x˜ e
E
q
p
 i wt
2
˜
 m w x  m w 0 x˜  i w x˜  q E˜
2
x˜ 
(q / m ) E˜
w  w  i w
2
0
2
p  q x˜ 
2
(q / m ) E˜
w 02  w 2  i w
Electric response of wires
Electric response of cut wires
Electric and magnetic
response of SRR
Electric response of LHM
E and M response of LHM
1d single-ring SRR: retrieved Re n() via cHEM inversion
for different length of the unit cell: 6x10x9 ... 6x10x14
TMM simulated 1d single-ring SRR:
retrieved Re n() via cHEM inversion
for different resonance frequencies
π/(Nz)
Vacuum case as before
Emulate small SRR gap: we fill the
gap with dielectic, eg.eps=300
TMM simulated 1d single-ring SRR:
retrieved eps() and mu() via cHEM inversion
for different resonance frequencies
Vacuum case as before
Emulate small srr gap:
we fill the gap with
dielectic, eg. eps=300
No negative Im e(w) and Im m(w) are observed !
Photonic Crystals with negative refraction.
ug
ug
Equal Frequency Surfaces (EFS)
Schematics for Refraction at the PC interface
EFS plot of frequency a/l = 0.58
Electric and Magnetic Response of SRRs and LHMs
• Electric and Magnetic Response are independent.
• One can change the magnetic response without changing
the electric response.
• GHz and THz magnetic response in artificial structures!
• The SRR has strong electric response. It’s cut-wire like.
• Effective electric response of LHM is the sum of wire and SRR.
• Effective wp of the LHM is much lower than wp of the wires.
• There are “phony” LH peaks when wp < wm
PRL 93, 107402(2004)
Some Significance, and Unique Properties of LHMs
 LH materials will exhibit a negative index of refraction in 1D, 2D
and 3D
 The change in phase of propagating waves in the LH frequency band
will evolve in time with opposite sign of the change in phase for waves
in a RH band.
 By constructing LHMs, the magnitudes of the effective index n and
surface impedance Z at a chosen frequency can be separated designed
over an appreciable continuous range, with the algebraic sign of n going
from positive to negative values. Match both Z and n of free space with
LHMs, i.e. Z=+1, while n=-1!
 LH materials can reconstitute evanescent wave components in space
(passively, without active components). Thus, the superposition of
reconstituted evanescent components can result in refocusing of “point
sources” below the traditional far-field diffraction limit!
Background and goals
 Left-handed materials (LHM), as well as photonic crystals (PC)
are composite metamaterials whose properties are not determined by
the fundamental physical properties of their constituents but by the
shape and distribution of specific patterns included in them.
 LHM have the unique property of having both the effective
permittivity and the effective permeability negative.
 The aim of the research is the theoretical understanding, analysis,
development, fabrication and testing of LHM, and also the
investigation of their feasibility for applications.
The LHM effort is in close collaboration with experiments (ISU,
Crete, Bilkent, UCSD, Boeing) Karlsruhe
More generally…
The general response of a material is a sum over oscillators
e
1
e0
w
k
fk
2
0,k
 w  iw
2
This implies a low frequency electrical permittivity:
e
e0
1
w
k
Insulator
fk
2
0,k
e
e0
1 

w ( w  i )
Conductor
More generally
The permittivity and permeability must be causal analytic functions,
implying Kramers-Kronig relations hold:
e(w )  1 
1

PV

(w e )
w


e( x )
xw
e( w )  
dx
(w m )
1
Umedium 
1 (ew )
2 w
w
E 
2
1

w


1
1 (mw )
2
PV

H 0
2
e(x )  1
x w
dx
Electric coupling to the magnetic resonance
The second electric coupling hinders the appearance of LH
behavior. Problem in higher dimensions
For higher dimensions  More symmetric structures are required
E
k
H
Hperp at the
magnetic and the
electric resonance,
for normal
incidence
Effective permittivity e(w) and permeability m(w) of wires and SRRs
e (w )  1 
w
2
p
w
2
UCSD and ISU, PRB, 65, 195103 (2002)
wm
2
m (w )  1 
w 2  w 20  i  w
Effective permittivity e(w) and permeability m(w) of LHM
UCSD and ISU, PRB, 65, 195103 (2002)
Effective refractive index n(w) of LHM
UCSD and ISU, PRB, 65, 195103 (2002)
Negative Index Materials:
New Frontiers in Optics
C. M. Soukoulis
Ames Lab. and Physics Dept. Iowa State University
Intermediate summary:
continuum homogeneous effective material (cHEM)
●
cHEM inversion basically works,
we find length-independent(!) effective material behavior
but problems:
Re n(w) seems to be cut-off at Brillouing zone.
 Discrepancy between n(w) and z(w): where is the resonance?
 Resonance/anti-resonance coupling.
 Negative imaginary parts in e(w) or m(w)
 Deformed resonances, i.e. unexpected shallow negative m(w)
 What is all this structure at higher frequencies?

Going to multi-gap structures
(1)
Reason: requirement for higher symmetry, for use in 3D
LH structures
a)
b)
c)
d)
e)
(a) better than (b) (wider SRR dip); (c) better than (d) (stronger dip); (e)
like the conventional SRR but weaker dip (for large separation)
Problem: Increase of wm (wm close to w0 )
Gaps act like capacitors in series: wm2(n gaps) ~ n wm2(1 gap)
Going to multi-gap structures
(2)
Solution: Make the gaps smaller or change the design
Improvements?
Up to a
point


Only the left one
Promising multi-gap structures from 1D study
(a): Detailed study on
progress (in 1D)
a)
c)
b
)
(b): Not studied in
detail yet
(c): Good LH T
3D structures
a)
b
)
c)
Best
combination:
(b)+(c)
Two-sided SRR Structures: No coupling to Electric Field
Two-sided SRRs do not have coupling to electric field
1 .0
0 .8
0 .6
S21
kp a rE p a r
0 .4
kp a rE p e n
kp e n E p a r
1.0
kp e n E p e n
0 .2
0.8
0 .0
2
4
6
8
10
12
14
fre q u e n cy(G H z)
0.6
t = 0.25
S 21
t = 0.25 * 0.5
0.4
t = 0.25 * 0.3
0.2
0.0
2
4
6
8
10
frequency (G H z)
12
14
Electric coupling to the magnetic resonance
The external electric field excites the resonant circular currents,
i.e. the SRR resonance
Reason: system asymmetry
E
E
Hperp at the
magnetic
resonance,
for normal
incidence
Result: Electric resonance close to SRR magnetic resonance
It happens also for the in-plane propagation
APL 84, 2943 (2004)
Analytic model for the electric and magnetic response of SRRs
Analytic model of the electric and magnetic response of LHMs
PRL 93, 107402(2004)
re trie v e d m fo r (a )
30
20
re a l m
m
im a g m
10
0
-1 0
8
4
9
10
F re q u e n c y (G H z)
11
re trie v e d m fo r (d )
3
re a l m
20
im a g m
m
re trie v e d e fo r (d )
30
re a l e
2
e
1
im a g e
10
0
0
8
9
10
F re q u e n c y (G H z)
11
-1 0
8
9
10
F re q u e n c y (G H z)
11
Photonics and Nanostructures (accepted, 2004)
TMM simulated 1d single-ring off-plane LHM:
retrieved Re n() and Im n() via cHEM inversion
Im n()
Re n()
TMM simulated 1d single-ring off-plane LHM:
retrieved e() and m() via cHEM
Model: Effective periodic material (PEM)
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)