Transcript Document

Negative refraction and Left-handed
behavior in Photonic Crystals:
FDTD and Transfer matrix method studies
Peter Markos, S. Foteinopoulou and C. M. Soukoulis
Outline of Talk
•
•
•
•
What are metamaterials?
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 refractions (Bilkent)
• Applications/Closing Remarks
E. N. Economou & S. Foteinopoulou
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.
Electromagnetic Metamaterials
Example: Metamaterials based on repeated
cells…
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
(-,+)
k   em
(-,-)
n   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;

• LH medium is always dispersive
 m 
0

• 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
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)/4p
opposite to k
 Snell’s law with
 g opposite to k n   em
 Flat lenses
 Super lenses
< 0 (NIM)
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


 g opposite to k


S  u g

S

opposite to k
  1

S

opposite to k
dn n
d
n  0




c 
p
g 
, p 
k0

n


m 
k    2
m  2 
p  2 S 
E 
H 
c
8p  


u 

p 
k

Photonic Crystals


 g opposite to k


S  u g
  1
dn n
d
n 
n  0  n   n ,

 ,  0
,   0

p 
c 
g 
, p 
k0

n


ck
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 !!!
Metamaterials Extend 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)
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
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
Structure of the unit cell:
SRR
EM waves propagate
in the z-direction.
Periodic boundary
conditions are used
in the xy-plane
LHM
Left-handed material: array of SRRs and wires
Resonance frequency
as a function of
metallic permittivity
 complex em
 Real em
Dependence of LHM peak on metallic permittivity
The length of the system is 10 unit cells
Dependence of LHM peak on metallic permittivity
PRB 65, 033401 (2002)
Example of Utility of Metamaterial
exp(ikd)
ts 
1  1
cosnkd 
z  sinnkd

2
z 
The transmission coefficient
is an example of a quantity
that can be determined
simply and analytically, if the
bulk material parameters are
known.
z
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
Effective permittivity e and permeability m of wires and SRRs
UCSD and ISU, PRB, 65, 195103 (2002)
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)
Determination of effective parameters from transmission studies
From transmission and reflection data, the index of refraction n was calculated.
Frequency interval with Re n<0 and very small Im n was found.
Pe rm ittivitye, Pe rm e abilitym and Im pe danceZ
Re m < 0
Re e < 0
Im m> 0
Im e < 0 ???
Im Z < 0
Re  > 0
Energy lossesQ are always positive in spite of the fact thatIm e is negative:
Q()= 2/(2p) |H|2 Im (n) Re (Z) > 0
Another 1D left-handed structure:
Both SRR and wires are located on the same side of the dielectric board.
Transmission depends on the orientation of SRR.
Bilkent & ISU APL 2002
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
New designs for left-handed materials
eb=4.4
Bilkent and ISU, APL 81, 120 (2002)
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
Frequency (GHz)
Bilkent & ISU APL 2002
14
ax=6.5 mm
t= 1 mm
Transmission (dB)
0
SRR
Wire
LHM
-10
-20
-30
-40
-50
-60
7
8
9
10
11
12
13
Frequency (GHz)
Bilkent & ISU APL 2002
14
Cut wires: Positive and negative n
Phase and group refractive index
•In both the LHM and PC literature there is still a lot
of confusion regarding the phase refractive index np
and the group refractive index ng. How these
properties relate to “negative refraction” and LH
behavior has not yet been fully examined.
•There is controversy over the “negative refraction”
phenomenon. There has been debate over the allowed
signs (+ /-) for np and ng in the LH system.
DEFINING phase and group refractive index np and ng
In any general case:
The equifrequency surfaces (EFS) (i.e. contours of
constant frequency in 2D k-space) in air and in the PC are
needed to find the refracted wavevector kf (see figure).
vphase=c/|np|
and
vgroup= k  c/|ng|
Where c is the velocity of light
So from k// momentum conservation: |np|=c kf () /.
Remarks
In the PC system vgroup=venergy
so |ng|>1. Indeed this holds !
np <1 in many cases, i.e. the phase velocity is larger than c in many cases.
np can be used in Snell’s formula to determine the angle of the propagating
wavevector. In general this angle is not the propagation angle of the signal. This
angle is the propagation angle of the signal only when dispersion is linear (normal),
i.e. the EFS in the PC is circular (i.e. kf independent of theta).
ng can never be used in a Snell-like formula to determine the signals propagation
angle.
Index of refraction of photonic crystals
– The wavelength is comparable with the period of the
photonic crystal
– An effective medium approximation is not valid
ky
Effective index
kx
Refraction angle
Equifrequency surfaces
Incident angle

Photonic Crystals with negative refraction.
Photonic Crystals with negative refraction.
S. Foteinopoulou, E. N. Economou and C. M. Soukoulis
Schematics for Refraction at the PC interface
EFS plot of frequency a/l = 0.58
Schematics for Refraction at the PC interface
EFS plot of frequency a/l = 0.535
Negative refraction and left-handed behavior for a/l = 0.58
Negative refraction but NO left-handed behavior for a/l = 0.535
Superlensing in 2D
Photonic Crystals
Lattice constant=4.794 mm
Dielectric constant=9.73
r/a=0.34, square lattice
Experiment by Ozbay’s group
Negative Refraction in a 2d Photonic Crystal
Band structure, negative refraction and experimental set up
Frequency=13.7 GHz
Negative refraction is achievable
in this frequency range for
certain angles of incidence.
Bilkent & ISU
Superlensing in photonic crystals
Subwavelength Resolution in PC based Superlens
The separation between the two point sources is l/3
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.
Photonic Crystals with negative refraction.
t0=1.5T
T=l/c
Photonic Crystals with negative refraction.
Photonic Crystals with negative refraction.
Photonic Crystals with negative refraction.
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, Appl. Phys. Lett. (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, accepted (2003)
 S. Foteinopoulou and C. M. Soukoulis, submitted Phys. Rev. B (2002)
 P. Markos and C. M. Soukoulis, submitted to Opt. Lett.
 E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, submitted to Nature
 P. Markos and C. M. Soukoulis, submitted to Optics Express
The keen interest to the topic
40
30
20
10
0
1968 1980 1990 1996 1999 2001
• Terminology
•
•
•
•
•
•
Left-Handed Medium (LH)
Metamaterial
Backward Medium (BW)
Double Negative Medium (DNG)
Negative Phase Velocity (NPV)
Materials with Negative Refraction (MNR)
Collaboration between Crete, Greece and Bilkent University, Turkey
Crete
GM
Image Plane
Experimental Setup
Network Analyzer
Scanned Power Distribution at the Image Plane
13.698 GHz
Average Intensity
1.0
Experiment
Theory
0.8
0.6
0.4
0.2
0.0
-5
-4
-3
-2
-1
0
1
Detuning (cm)
2
3
4
5
Dependence of LHM peak on L and Im em
Dependence on the incident angle
Transmission peak does not depend
on the angle of incidence !
This structure has an additional
xz - plane of symmetry
Transition peak strongly depends
on the angle of incidence.
Transmission depends on the orientation of SRR
Transmission properties depend on the orientation of the SRR:
Lower transmission
Narrower resonance interval
Lower resonance frequency
Higher transmission
Broader resonance interval
Higher resonance frequency
Dependence of the LHM T peak on the Im eBoard
In our simulations, we have:
Periodic boundary condition,
therefore no losses due to
scattering into another direction.
Very high Im emetal therefore
very small losses in the metallic
components.
Losses in the dielectric board are crucial for the transmission
properties of the LH structures.
New / Alternate Designs
Superprism Phenomena in Photonic Crystals
Experiment
– H.Kosaka, T.Kawashima et. al. Superprism phenomena in photonic
crystals, Phys. Rev. B 58, 10096 (1998)
Scattering of the photonic crystal
Hexagonal 2D photonic crystal
-M.Natomi, Phys. Rev. B 62, 10696 (2000)
Using an equifrequency surface (EFS) plots
Vanishingly small index modulation
Small index modulation
Photonic crystal as a perfect lens
C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry
Phys. Rev. B, 65, 201104 (2002)
Resolution limit 0.67 l