Negative refraction in photonic crystals
description
Transcript of Negative refraction in photonic crystals
Negative refraction in photonic crystals
Mike KaliteevskiDurham University
Outine•Photonic Crystals: Introduction
•Negative refraction in left-handed material•Non-diffracting beams •Electromagnetic wiggler
Bragg reflector
r
n1 n2
t
21
21
nn
nnr
21
12
nn
nt
Bragg reflector
n1r n1
d2
n2
tndrndt
2
02
0
2exp
2exp
0
22
02
0
4122
nd
ndnd2
02 4n
d
Bragg reflector
r
n1 n2 n1n2
d2d2
1
01 4n
d
Periodic sequence of the pairs of quarterwave layers is the Bragg reflector. The waves, reflected from different boundaries experience positive interference (enforce each other).
Bragg reflector
BRBRBR ir /)(exp
0,8 1,0 1,20,0
0,2
0,4
0,6
0,8
1,0
-
Arg
(r)
R
Energy, eV
210
21
nnn
nn
Bloch theorem. Dispersion relations
)exp()()( iKzzuzE KK
H
EiKD
H
ET D )exp(ˆ )(
0ˆ)exp(ˆdet )( IiKDT D
KDTT DD cos2ˆˆ )(22
)(11
)sin()sin(2
1)cos()cos()cos( 022011
1
2
2
1022011 kdnkdn
n
n
n
nkdnkdnKD
0
0
0
0 Densityof modes
Densityof modes
k
Im(k)
Im(k)
k /D
/D0
0
1
1 0
0Reflectivity
Reflectivity
BR = c/(n1d1+n2d2)
BRBR
Formation of the photonic band gap in periodic structures
Probability of spontaneous emission
22
22EuedlW
Probability of spontaneous emission
L
LEEnergy 22/
22
22EuedlW
/L )2/(2 LE
Microcavity
Microcavity
L
nRR
n2
n1
Electric field
Magnetic field
0
1
/0
R
Probability of spontaneous emission
L
L
2D Photonic crystal
1D photonic crystal
2D photonic crystal
2D photonic crystal
Dispersion relations in 2D photonic crystal
k
)exp()()( rkirvrH
kk
)()( arvrvkk
)()( arr
Plane waves method
a
)()()(
12
2
rHc
rHr
rGiGr G
exp)()(
1
)exp()()( rkirvrHkk
)(
1
)(
1
arr
)()( arvrvkk
rGkiGkHrHG
k
exp),()(
Bloch theorem
Wave equation
G
Lattice vector
Reciprocal lattice vector
Plane waves method
G
)()()(
12
2
rHc
rHr
Wave equation
Reciprocal lattice vector
),()',(')''(2
2
'
GkHc
GkHGkGkGGG
k
2D photonic crystals
H E
0.0
0.1
0.2
0.3
0.4
0.5Fr
eque
ncy,
c/d
K KM
TE
K M
0.0
0.1
0.2
0.3
k
k
Freq
uenc
y, c
/d
K KM
K M
TM
Disperison relations
H
E
Complete PBG
Transmissiom of light
d=50m
TT
TT
1d=60m
d=70m
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
1
1
0
0
0
d=80m1
0
f, THzExperiment Modelling
PC spectral filter
0.4 0.8 1.2 1.6 2.0 2.4
T
1
0
f, THz
D
G
a
a
Defects in photonic crystals
0,0
0,1
0,2
0,3
0,4
0,5
Fre
quen
cy, c
/d
m = 1
m = 1
m = 2
m = 2
m = 0
m = 3
Photonic crystal waveguide
PC Waveguide
OE_15_12982
3D Photonic crystals
Transmission of light and bandstructure in opals and inverse opals.
Photonic microstructures in nature
Negative refraction in left-handed material
Right - hand materials
2
000 Enk
kHE
00
kS
0grv
0 n
•Usual electromagnetic word
Left - hand materials
V.G.Veselago, Electrodinamics of the materials with negative dielectric and magnetic constant (1967)
2
000 Enk
kHE
00
kS
0grv
0 n
•Inversed Doppler effect•Inversed Vavilov – Cherenkov effect•Negative refraction
Refraction
Kτ
Kτ
Kτ
Kτ
kS
Positive refraction
Kτ
Kτ
kS
Negative refraction
Left - hand materials
kS
0grv
Negative refraction
Flat Lense
L
n1 n2
A
D
ALD
Flat lence
n1
n2 =-n1
Superlence ???
L
n1 n2
A
D
ALD
n1
n2 =-n1
Comment: John Michael Williams, Some Problems with Negative Refraction, Phys. Rev. Lett. 87, 249703 (2001) Comment: G. W. 't Hooft, Comment on “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. 87, 249701 (2001) Reply: M. Nieto-Vesperinas and N. Garcia, Nieto-Vesperinas and Garcia Reply:, Phys. Rev. Lett. 91, 099702 (2003)
J. B. Pendry , Negative Refraction Makes a Perfect Lens, Phys. Rev. Lett. 85, 3966 - 3969 (2000)
Автор ввел понятие "суперлинза", ...утверждая, что для этого устройства отсутсвует дифракционный предел. Наверное, наиболее убедительное доказательство ошибочности подобного рода утверждений можно найти в ... [ В.Г.Веселаго, УФН, 173 (7) 790 (2003) ]
With a conventional lens sharpness of the image is always limited by the wavelength of light. An unconventional alternative to a lens, a slab of negative refractive index material, has the power to focus all Fourier components of a 2D image, even those that do not propagate in a radiative manner. Such “superlenses” .....
Realization of left-hand materials
MetamaterialsPhotonic crystals
Negative refraction in photonic crystals
Band 2
fX
1.0
1.2
1.4
1.6
1.8
Band 1
fJ
XJ
f, T
Hz
vgr<0
vgr>02D hexagonal metallic PC, D =200 microns, d = 60 microns
Negative refraction in 2D hexagonal photonic crystals
Band 2
fX
1.0
1.2
1.4
1.6
1.8
Band 1
fJ
XJ
f, T
Hz
PRF
NRF
IFSOURCE
(a)
PRF
NRF
IFSOURCE
(a)
PRF
NRF
IFSOURCE
(b)
Refraction of wave in photonic crystal prism
vgr<0Band 2
fX
1.0
1.2
1.4
1.6
1.8
Band 1
fJ
XJ
f, T
Hz
vgr>0
Refraction of wave in photonic crystal prism
0.5 1.0 1.5 2.0
T1
0
f, THz
PRF
NRF
IFSOURCE
(a)
PRF
NRF
IFSOURCE
(a)
PRF
NRF
IFSOURCE
(b)
Refraction of wave in photonic crystal prism
0.5 1.0 1.5 2.0
T
1
0
f, THz
PRF
NRF
IF
SOURCE
(c)
PRF
NRF
IF
SOURCE
(c)
Band 2
fX
1.0
1.2
1.4
1.6
1.8
Band 1
fJ
XJ
f, T
Hz
PRF
NRF
IFSOURCE
Refraction of wave in photonic crystal prism
1n
Experimental study of negative refraction
Experimental study of negative refraction of THz
using QCL
Experimental study of negative refraction of THz
using QCL
SIGNAL WITHOUT SAMPLE
Negatively refracted beam
Non-diffracting beams
W
W sin
L
l1l2
A
D
0 AnLD /
An
LALD
22
2
sin
sin1
tan
tan
nsin D
Non-diffracting beams
n1 n2 n1
21 nn
11 n02 n
0 AnLD /
An
LALD
22
2
sin
sin1
tan
tan
nsin D
Non-diffracting beams
21 n
11 n02 n
L
l1l2
A
D
L
l1l2
A
D
n1 n2 n1
1.0
1.2
1.4
1.6
f, T
Hz
-0.5 0 XJneff
L
L
L
D0
D0
A
(c)
(b)
(a)
A
16275 m
4000
m
Non-diffracting beams
L
L
L
D0
D0
A
(c)
(b)
(a)
A
16275 m
4000
m =185m
=180m
-2 -1 0 1 2
I n
t e n
s i
t y, a
. u.
=175m
Position, mm
(a)
(b)
(c)
Non-diffracting beams
1.0
1.2
1.4
1.6
f, T
Hz
-0.5 0 XJneff
Negative refraction in 1D photonic crystals
n1 n2
d1 d2
Problem: Veselago lens based on 1D PC Bragg reflector does not work.
Because system is anisotropic: negative effective mass is required for negative refraction, and for 2nd , 4th , etc bands mz<0, but always mx>0
0 0
0
2 2
( , ) [exp( ) exp( )]exp( )
2
( / )
Ry p p p p
p
p
p p
E x z i x R i x iK z
pK K
D
c K
n1 n2
d1 d 2
x
zK
0
1
( , ) ( ) exp( )Brm m m
m
E x z a u z i x
0 0
0
2 2
( , ) [exp( ) exp( )]exp( )
2
( / )
Ry p p p p
p
p
p p
E x z i x R i x iK z
pK K
D
c K
Field of the wave in the structure
Modes in Bragg reflector
0),,( 2 Kf
1 2 1 2
1( ) cos( )cos( ) sin( )sin( ) cos( ) 0
2f d d d d KD
2 2 21( / )n c
2 2 22 ( / )n c
Amplitude of waves
* * *1,1 0,1 ,1
* * *, 0, ,
1 1,1 1 ,
1 100 0,1 1 0 0,
1 1,1 1 ,
1 0
0 1
1 0 0
0 1
1 0
0 0 1
P P
mP M M P M
PP P P P M M
M M
PP P P P M M
aJ J J
aJ J J
RJ J
RJ J
RJ J
*0,1
*0,
0
1
0
M
J
J
*0[ ]p p pn n
p
R J a
0
1
[ ]q q q m m qmm
R a J
-3000 -2500 -2000 -1500 -1000 -500 0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
log 1
0(|a
m|2 )
f (2 )
21E-3
0.01
0.1
1
-3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500
-4
-2
0
2
4
log 1
0(|a
m|2 )
f (2
)
21E-6
1E-5
1E-4
1E-3
0.01
0.1
1
10
High contrast:n1=3.7n2=1
Low contrast:n1=1.4n2=1.8
-40 -20 0 20 40
-3
-2
-1
0
1
2
3
KD
(m -1)
6
-40 -30 -20 -10 0 10 20 30 40
-3
-2
-1
0
1
2
3
KD
(m -1)
5
-30 -20 -10 0 10 20 30
-3
-2
-1
0
1
2
3
KD
(m -1)
4
0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
3.0
45
6
3
2
( eV
)
KD
1-20 -15 -10 -5 0 5 10 15 20
-3
-2
-1
0
1
2
3
KD
(m -1)
3
-5 -4 -3 -2 -1 0 1 2 3 4 5-2
-1
0
1
2
KD
(m -1)
1
-8 -6 -4 -2 0 2 4 6 8
-3
-2
-1
0
1
2
3
KD
(m -1)
2
0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
3.0
( eV
)
KD
-5 -4 -3 -2 -1 0 1 2 3 4 5-2
-1
0
1
2
KD
(m-1) aa
0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
3.0
( eV
)
KD
-30 -20 -10 0 10 20 30
-3
-2
-1
0
1
2
3
KD
(m-1)
bb
Negative refraction
cc
0.0 0.5 1.0 1.5 2.0
-1.0
-0.5
0.0
0.5
1.0
1.5
, <S
z>K
frequency (eV)
photonicband gap
negativerefractionarea
DKS z 0sin~
Normal channelling
Normal channelling
-3000 -2500 -2000 -1500 -1000 -500 0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
log 1
0(|a
m|2 )
f (2 )
21E-3
0.01
0.1
1
Low contrast:n1=1.4n2=1.8
xS z )(cos 21
Electromagnetic wiggler
Electromagnetic wggler
Conclusion:
• One can hardly make Veselago lense based 1D photonic crystal
• But there are some interesting effects like “electromagnetic snake”, normal channeling, etc.