Strongly Localized Photonic Modein 2D Periodic Structure Without Bandgap
V. M. APALKOV M. E. RAIKHPhysics Department, University of Utah
The work was supported by: the Army Research Office under Grant No. DAAD 19-0010406; the Petroleum Research Fund under Grant No. 37890-AC6NSF
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Disordered Media
Localization of Electrons
P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1958)
“tail” state
diffusion
l
Length scale: - mean free path, l - a step of diffusion motion
Localization: l2 ~ 1πλ
(Ioffe-Regel criterion)
Potential Well x V x x E xx
2
2( ) ( ) ( ) ( )ψ ψ ψ∂
− + =∂ V x( )
E<0
x
a localized state
mE22
πλ =
Length scale of the potential, aLocalization: a~λ
- de Broglie wave length,
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r V r r E r( ) ( ) ( ) ( )ψ ψ ψ−∆ + =
Ø Electrons
E r r E r E rc c
2 2
02 2( ) ( ) ( ) ( )ω ωδε ε−∆ − = Ø Photons
Schroedinger-Maxwell Analogy
• The electron can have a negative energy, can be trapped in deep potentials
• Frequency-dependent potential
• “Energy” is positive,
(unlike plasmons)
c
2
02ω ε
c
2
2ω δε
0ε δε>
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Localization Criteria for Electrons and Photons
Ø ElectronsV r V r r r( ) ( ) ( )δ′ ′= Γ −
• Golden Rule
( )k k kk kE V r E E
g E
2,
Im 2 | ( ) | ( )
~ ( )
π δτ ′′
= = −
Γ
∑
electron density of states
1/ 21~ ~( )
l E Eg Eλ Γ
• localization criterion is satisfied for low enough E
Ø Photonsr r r r( ) ( ) ( )δε δε δ′ ′= Γ −
2
22 22
0 02 2 2,
3
3 20
Im( )
2 ( )
( )~
k k
k k
k kl
rc c c
gc
ω ωωπ δε δ ε ε
ω ωε
′
′
= =
= −
Γ
∑
photon density of states
230 1~ ~
( )l
gε
ωλ ω ω
−
Γ
g E E1/ 2( ) ~
g 2( ) ~ω ω
• localization criterion can not be satisfied
λ
l
electrons
photons
localized states,
~ constl4~l λ
lλ ≤
( )lλ
( )lλ
(Rayleigh law)
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Localization Criterion for Photons
Ø Photons
20
~ ( )glλ
ω ωεΓ
g 2( ) ~ω ω
weak disorder
Rayleighl 4~ λ
geometric ray optics
strong scattering (resonance) free space value
Bragg Resonance – Photonic Crystalpseudogap, strong localization Geometric optics
Rayleigh
Scattering resonances (Mie resonances, Bragg resonances) strongly modify photon density of states
frequency
S. John, Phys. Rev. Lett. 58, 2486 (1987)
λ
l
g
ω
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Photon Localization: Photonic Band Gap Materials
Photonic crystal – periodic modulation of dielectric constant
a – lattice constantR
δε
ωno emission,if lies in the gap.ω
pseudogap, strong localization
g
ω
Two Fundamental Optical Principles:
• Localization of Light
- S. John, Phys. Rev. Lett. 58, 2486 (1987)
• Inhibition of Spontaneous Emission
- E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987)
ω
a/πa/π−
1D gap
k
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Complete bandgap:
• The frequency domain where the propagation of light is completely forbidden
• Requires high contrast of the dielectric constant, > 10:1
• Difficult to achieve
Photonic Band Gap Materials: Complete Bandgap
g
ω
• Disorder-induced localized mode
g
ω ω
gLocalized modes
• Point-defect induced localized in-gap mode acts as a high-Q resonator
Strongly localized photon modes
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Periodic lattice of dielectric spheres Diamond structure – complete photonic bandgapContrast of dielectric constant > 4:1
Complete Bandgap: Computational Demonstration
The MIT Photonic-Bands package: http://ab-initio.mit.edu/mpb/
3D
229 citations in PR..
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Photonic Crystals
3D2D1D
Natural assembly of colloidal microspheres:- opals - inverted opals - structural defects destroy the bandgap
specially designed
specially designed specially
designed
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Synthetic Opals: Thin Layers 3D
Silica (SiO2) microspheres
- sediment by gravity into
close-packed fcc lattice
Nature (London) 414, 289 (2001)Silica opal
Inverted silicon opal
- filled with silicon (Si)
- silica template removed by wet etching
7 layers
Theory – blackExperiment red/blue
Thick layers –structural defects
V.N. Astratov, et. al., PRB 66, 165215 (2002)
contrast ~ 12:1
contrast ~ 1.5:1
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Complete Bandgap: Inverted Opals - 1.5 micrometers
Nature (London) 405, 437 (2000)
calculations
experiment (many structural defects which destroy bandgap)
fcc lattice of air sphere in silicon: contrast ~ 12:1
(theoretical threshold for complete bandgap ~ 8:1)
3D
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Photonic Band Gap Materials: Specially Designed 3D
O. Toader, S. John, Science, 292, 1133 (2001)
S. Fan, et. al., Appl. Phys. Lett. 65, 1466 (1994)
M. E. Povinelli, et. al., Phys. Rev. B 64, 75313 (2001)
A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003)
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Photonic Band Gap Materials
• Two –dimensionally periodic structures of finite height (photonic-crystal slabs)
• Light is confined by a combination of an in-plane photonic band gap and out-of-plane index guiding.
• Advantage – easy to manufacture
complete band gap: contrast ~ 10:1
2D
M. Meier, et. al., Appl. Phys. Lett. 74, 7 (1999)
ω
g Localized modesDefect-induced localized in-gap mode acts as a high-Q resonator
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na
na≠
a
Defects in Periodical Structures
1D 2D
• Point-like defect
• Phase slip - linear defect
(periodicity interruption)
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Point Defect-Induced Localized Mode in 2D
O. Painter1, R. K. Lee1, A. Scherer1, A. Yariv1, J. D. O’Brien2, P. D. Dapkus2, I. Kim2
1California Institute of Technology;2University of Southern California.
Science 284, 1819 (1999).
Hexagonal array of air holes (radius 180 nm)
480
nm
InGaAs
spontaneous emission spectrum
laser line
515 nm
ω
gDefect-induced localized state inside the gap acts as a high-Q resonator
High dielectric contrast 12:1λ
Power
Two-Dimensional Photonic Band-Gap Defect Mode Laser
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2D Photonic Crystal: Weak Contrast of Dielectric Constant
dielectric contrast 1.7 : 1.46 : 1
no complete bandgap
organic substrate air
(solid organic gain media)
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na
na≠
a
Defects in Periodical Structures
1D 2D
• Point-like defect
• Phase slip - linear defect
(periodicity interruption)
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Linear Defect-Phase Slip
1D casea
phase slip (stacking fault)
a d
d < a
x x a 1( ) ~ cos(2 )δε π φ+ x x a 2( ) ~ cos(2 )δε π φ+
d a2 1 2φ φ π− =
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Phase Slip – Localized State, 1D
x x a0 1( ) cos(2 )δε π φ= ∆ +
x x a0 2( ) cos(2 )δε π φ= ∆ +
x 0<
x 0>
E x x E x E xc c
2 2
02 2( ) ( ) ( ) ( )ω ωδε ε−∆ − =
Solution – localized mode:
x i x i xE x e e e| |( ) ( )γ σ σβ µ− −= +
k
ω
a/σ π=
σσ−
c3 / 20 0 0ε σ−Ω = ∆gap
c0
0
σω ω ωε
Ω = − = −
0ω
2 10
1 cos2 2
φ φ−Ω = ± Ω
с0 2 11/ 20
| |1 sin2 2
φ φγεΩ −
=
da
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spatial extension of localized mode,
k
ω
σσ−
0Ωgap
Phase Slip – Localized State, 1D
localized mode, d = 0.5 a
0 2−Ω
1 2φ φ−
Ω
c0
2Ω
0
1γ
d = 0.5 a
d = 0.5 a
Frequency (energy) of localized mode,
da
d/a 10.5( -phase slip)/ 2π
( -phase slip)/ 2π
| |0( ) cos( )xE x e xγ σ−∝
2 10 0
1 cos2 2
φ φω ω
−Ω = − = ± Ω
0 2Ω
1 2 1
0
| |1 2 sin2
с φ φγ
− −=
Ω
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Phase Slip – Localized Mode, 1D
Nature (London) 390, 143 (1997)
Dielectric contrast - 12:1
λ
Transmission
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Localization of Photons in 2D Crystals with Incomplete Bandgap
• Strong localization of a photon can be achieved in 2D photonic crystals with low contrast of dielectric constant
• A long-living photon mode exists only in 2D photonic crystal with a certain MAGIC GEOMETRY of a unit cell
strongly localized mode
complete bandgap no bandgap
Conventional approach Our result
g g
ω ω
V.M. Apalkov and M.E. Raikh, Phys. Rev. Lett. 90, 253901 (2003)
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Two Phase Slips, 2D
two phase slips
localized mode
xy0 0
low contrast of dielectric constant
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Localized Mode
xk
yk
σ
σconstω =
E0 const=
gap
localized mode
delocalized modes
no bandgap
g
ω
xy0 0
| | | |0( , ) cos( )cos( )x yE x y e e x yγ γ σ σ− −=
E0
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Strongly Localized Mode: Magic Geometry of a Unit Cell
LEAKAGE
3 / 211 1(2 )J Rσ∆ ∝
10 01 1(2 )J Rσ∆ = ∆ ∝
radius of cylinders
сR u3 / 202 3.8σ ≈ ≈
J u1 0( ) 0=
сR a0.43≈
RR0.32
Magic geometry of a unit cell:
aπσ =
11 0∆ = ⇒
Q ?Im
ωω
= =Q-factor:
xk
yk
σ
σ
gap
delocalized modes
localized mode
10∆11∆
01∆
11∝ ∆
surface of equal frequency
Fourier harmonics of :( , )x yε
11 0∆ =
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localized mode
Magic Crystal: Localized Mode
two phase slips
low contrast of dielectric constantxy
0 0
Q ?Im
ωω
= =Q-factor:
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2D Photonic Crystal: Weak Contrast of Dielectric Constant
dielectric contrast 1.7 : 1.46 : 1
no complete bandgap
organic substrate air
(solid organic gain media)
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Two Phase Slips - Quasilocalized Mode
x y x y x yx y x y
1 2 11
10 10 11
( , ) ( ) ( ) ( , )cos(2 ) cos(2 ) cos(2 ) cos(2 )
δε δε δε δεσ σ σ σ
= + + =
= ∆ + ∆ + ∆
σ
σ
σ−
σ−
x1 ( )δε
xk
ykx1 ( )δε
y2 ( )δε
x y11 ( , )δε
- localization in x-direction
- localization in y-directiony2 ( )δε
x y11 ( , )δε - destroys localization a a
i x i y
a a
dx dy x y e e/ 2 / 2
2 211
/ 2 / 2
( , ) σ σδε− −
∆ = ∫ ∫a a
i x
a a
dx dy x y e/ 2 / 2
210
/ 2 / 2
( , ) σδε− −
∆ = ∫ ∫
n mn m
n mn m
n m
x y x n a y m a n x m y
x y x y n x m y
1 1 ,,
1 2 11 ,, 0
( , ) (1,1)
( , ) ( , ) cos(2 ) cos(2 )
( ) ( ) cos(2 ) cos(2 ) cos(2 ) cos(2 )
δε δε σ σ
δε δε σ σ σ σ>
≠
= + + = ∆ =
= + + ∆ + ∆
∑
∑aπσ =
“separable” part
+ phase slip
+ phase slip
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E x y x y E x y E x yc c
2 2
02 2( , ) ( , ) ( , ) ( , )ω ωδε ε−∆ − =
Q-factor of Quasilocalized Mode: Higher-Order Corrections
( ) pspertE x y U x U y E x y U x y E x y E x y( )
1 2( , ) ( ) ( ) ( , ) ( , ) ( , ) ( , )κ−∆ − + − =
n m n mn m n m
x y U x y n x m yс с
2 2
, ,2 2, 0 , 0
( , ) ( , ) cos(2 ) cos(2 )ω ωδε σ σ> >
= = ∆∑ ∑
( )psx yU x y U x d x y d y( )( , ) sign( ), sign( )= + +
psn
nU x U x( )
1 ,00
( ) ( )>
= ∑ psm
mU y U y( )
2 0,0
( ) ( )>
= ∑ ps pspert n m
n mU x y U x y( ) ( )
,, 0
( , ) ( , )>
= ∑
H E x y E x y0 0 0 0ˆ ( , ) ( , )κ=
H E x y0ˆ ( , ) pertH E x yˆ ( , )
pertE H E(1)0 0 0| | 0κ = =
E0
x
0Imκ
for magic crystals
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pspertH U x y( )
1,1ˆ ( , ) 0= →
pertpert pert
E H EE H H H E
2
0(2) 10 0 0 0 0
0
ˆ| |ˆ ˆ ˆ| ( ) |
µ
µ µ
κ κκ κ
−= − =−∑
E0
x
0Imκ( )pertE H E
2(2)0 0 0
ˆIm Im | | µ µµ
κ κ π δ κ κ= = −∑
resonant term for magic crystals
x y x y
x y x y
ps psm n p p p p m n
pertm n m n p p p p
U E E UH 1 1
1 1
( ) ( ), , , ,
, , , , 0 ,
ˆκ κ
→−∑ ∑
Higher-order corrections:
Q-factor of Quasilocalized Mode: Higher-Order Corrections
(2)0 0κ⇒ =
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2 21 2
0 1 20 0 0
Im cR RQ Ca
ω δε δε δεα αω ε ε ε
− − = = − +
12 2→22 2→
n m n m
n m
F Fu J u n n m m
, 1, 1 31 2 2 2
, 00 0 0
27 2 2 5 108 ( )
α α + + −
>
= + ≈ ⋅+ + +∑
cR R a10
δε αε
= +
fine tuning of R (Fano resonance)
•.
•.
•.
⇒ QC
3 360 0
max 20 2
1 0.4 10ε εδε δεα
= ≈ ⋅
HIGH-Q mode
J u u1 0 0( ) 0 3.83= =u J uС
uJ
211/ 20 0 0
00
1
( )2 4.315
2
= ≈
•. n m n m n m
n m
F F Fu J u n n m m
2 2 2, 1, 1 , 1 3
2 2 2, 00 0 0
24 3 0.8 1021 ( )
α + + + −
>
+ += ≈ ⋅
+ + +∑
•.
cnm nm c
nm
RF J q Rq 12 ( )π=
nmq n ma
2 2π= +
Q factor of the Quasilocalized Mode-
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c
QR R
a
0
225
0
0.23
6 10
εδε
δεε
−
=
−+ ⋅
Q
3
0
~ 10 δεε
−
36 00.4 10 ε
δε ⋅
cR Ra−
Q factor of the Quasilocalized Mode-
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Weakly disordered media Photonic crystal without bandgap
Q=50 Q=106
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hole, 1ε =1 1.5ε ≈
D R20≈
Magic Crystal and Localized Mode: Example
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Long-living Quasilocalized Mode: Fine Tuning
For R=Rc 11 0⇒ ∆ =Fine tuning of the shape of cylinders
nm n m Q11 0 and 0 (for all , 0)⇒ ∆ = ∆ = > ⇒ = ∞
( )nmirqnm d rdr e R r
2cos
0 0
( )π
ϕδε ϕ θ ϕ∞
∆ = −∫ ∫ i pp
pR R R A e0 0( ) ϕϕ = + ∑
nmiR q ipnmnm p
pnm
J q R dR A eq R
0
2cos2 1 0
00 0
( )22
πϕ ϕϕπδε
π+
∆ = −
∑ ∫
nmnm p p nm
pnm
J q RR A J q Rq R
2 1 00 0
0
( )2 ( ) 0πδε
∆ = − =
∑
nmq n ma
2 2π= +
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Light Localization in 3D Photonic Crystals with Incomplete Bandgap
Face Centered Cubic (FCC) lattice (closed packed)
• introduce three phase slips along the major axes
• separable part of localized mode
• two “diagonal” components, and , destroy localization
• magic crystal:
• specific feature of FCC lattice
• magic crystal: only one condition
• composite particles
110∆ 111∆x y z x y z1 2 3 100 010 001( ) ( ) ( ) cos(2 ) cos(2 ) cos(2 )δε δε δε σ σ σ+ + = ∆ + ∆ + ∆
x y z( , , )δε ⇒
110 0∆ = 111 0∆ =
110 0∆ ≡
111 0∆ =
R1 1,ε
R2 2,ε
( ) ( )R
r dr r r1
1110
( ) sin 2 3 0ε ε σ∆ ∝ − =∫
R12 2πσ =
R R2 10.86≈
2 8.5ε ≈
1 12ε ≈
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hole, 1ε =1 1.5ε ≈
D R20≈
Magic Crystal and Localized Mode: Example
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Weakly disordered media Photonic crystal without bandgap
Q=50 Q=106
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Photon Localization
Disordered medium Photonic crystals (background medium)
• strongly modify the photon density of states
• density of states becomes similar to the electron density of states disorder-induced localized states
• custom-made defects and in-gap localized modes
The main problems:
• realization of strong enough scattering
Ø metallic particles (Mie resonances)
Ø semiconductor particles (GaAs, GaP) with very large refractive index
• observation of photon localization
Ø exponential scaling of transmission coefficient
Ø rounding of the top of the backscattering cone
Ø variance of relative fluctuations
• absorption
⇒
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Linear Defect - Phase Slip: 1D
a
phase slip (stacking fault)
ad
d < a (d > a)
0 1( ) cos(2 )x x aδε π φ= ∆ + 0 2( ) cos(2 )x x aδε π φ= ∆ +d a2 1 2φ φ π− =
k
ω
aπσ =σ−
c3 / 20 0 0ε σ−Ω = ∆gap
0ω
2 10 0
1 cos2 2
φ φω ω
−Ω = − = ± Ω
с0 2 11/ 20
| |1 sin2 2
φ φγεΩ −
=
0 2−Ω
1 2φ φ−
d = 0.5 a
( -phase slip)/ 2π
0 2Ω
| |0( ) cos( )xE x e xγ σ−∝
(or d = 1.5 a)
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