Strongly Localized Photonic Mode in 2D Periodic …cophen04/Talks/Apalkov.pdfStrongly Localized...

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

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( ) ( ) ( ) ( )ψ ψ ψ∂

− + =∂ V x( )

a localized state

πλ =

Length scale of the potential, aLocalization: a~λ

- de Broglie wave length,

r V r r E r( ) ( ) ( ) ( )ψ ψ ψ−∆ + =

Ø Electrons

E r r E r E rc c

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)

02ω ε

2ω δε

0ε δε>

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

Im 2 | ( ) | ( )

π δτ ′′

= = −

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 −

0 02 2 2,

rc c c

ω ωωπ δε δ ε ε

ω ωε

photon density of states

230 1~ ~

ωλ ω ω

g E E1/ 2( ) ~

g 2( ) ~ω ω

• localization criterion can not be satisfied

electrons

photons

localized states,

~ constl4~l λ

lλ ≤

( )lλ

(Rayleigh law)

Localization Criterion for Photons

Ø Photons

~ ( )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)

Photon Localization: Photonic Band Gap Materials

Photonic crystal – periodic modulation of dielectric constant

a – lattice constantR

ωno emission,if lies in the gap.ω

pseudogap, strong localization

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)

a/πa/π−

1D gap

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

• Disorder-induced localized mode

gLocalized modes

• Point-defect induced localized in-gap mode acts as a high-Q resonator

Strongly localized photon modes

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/

229 citations in PR..

Photonic Crystals

3D2D1D

Natural assembly of colloidal microspheres:- opals - inverted opals - structural defects destroy the bandgap

specially designed

specially designed specially

designed

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

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)

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)

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

M. Meier, et. al., Appl. Phys. Lett. 74, 7 (1999)

g Localized modesDefect-induced localized in-gap mode acts as a high-Q resonator

Defects in Periodical Structures

• Point-like defect

• Phase slip - linear defect

(periodicity interruption)

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)

InGaAs

spontaneous emission spectrum

laser line

515 nm

gDefect-induced localized state inside the gap acts as a high-Q resonator

High dielectric contrast 12:1λ

Two-Dimensional Photonic Band-Gap Defect Mode Laser

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)

Defects in Periodical Structures

• Point-like defect

• Phase slip - linear defect

(periodicity interruption)

Linear Defect-Phase Slip

1D casea

phase slip (stacking fault)

x x a 1( ) ~ cos(2 )δε π φ+ x x a 2( ) ~ cos(2 )δε π φ+

d a2 1 2φ φ π− =

Phase Slip – Localized State, 1D

x x a0 1( ) cos(2 )δε π φ= ∆ +

x x a0 2( ) cos(2 )δε π φ= ∆ +

E x x E x E xc c

02 2( ) ( ) ( ) ( )ω ωδε ε−∆ − =

Solution – localized mode:

x i x i xE x e e e| |( ) ( )γ σ σβ µ− −= +

a/σ π=

σσ−

c3 / 20 0 0ε σ−Ω = ∆gap

σω ω ωε

Ω = − = −

1 cos2 2

φ φ−Ω = ± Ω

с0 2 11/ 20

| |1 sin2 2

φ φγεΩ −

spatial extension of localized mode,

σσ−

0Ωgap

Phase Slip – Localized State, 1D

localized mode, d = 0.5 a

0 2−Ω

1 2φ φ−

d = 0.5 a

Frequency (energy) of localized mode,

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

| |1 2 sin2

с φ φγ

− −=

Phase Slip – Localized Mode, 1D

Nature (London) 390, 143 (1997)

Dielectric contrast - 12:1

Transmission

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

V.M. Apalkov and M.E. Raikh, Phys. Rev. Lett. 90, 253901 (2003)

Two Phase Slips, 2D

two phase slips

localized mode

low contrast of dielectric constant

Localized Mode

σconstω =

E0 const=

localized mode

delocalized modes

no bandgap

| | | |0( , ) cos( )cos( )x yE x y e e x yγ γ σ σ− −=

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-factor:

delocalized modes

localized mode

10∆11∆

11∝ ∆

surface of equal frequency

Fourier harmonics of :( , )x yε

11 0∆ =

localized mode

Magic Crystal: Localized Mode

two phase slips

low contrast of dielectric constantxy

= =Q-factor:

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)

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

= + + =

= ∆ + ∆ + ∆

x1 ( )δε

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

dx dy x y e e/ 2 / 2

/ 2 / 2

( , ) σ σδε− −

∆ = ∫ ∫a a

dx dy x y e/ 2 / 2

/ 2 / 2

( , ) σδε− −

∆ = ∫ ∫

n mn 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

E x y x y E x y E x yc c

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, 0 , 0

( , ) ( , ) cos(2 ) cos(2 )ω ωδε σ σ> >

= = ∆∑ ∑

( )psx yU x y U x d x y d y( )( , ) sign( ), sign( )= + +

nU x U x( )

( ) ( )>

= ∑ psm

mU y U y( )

( ) ( )>

= ∑ ps pspert n m

n mU x y U x y( ) ( )

( , ) ( , )>

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 = =

for magic crystals

pspertH U x y( )

1,1ˆ ( , ) 0= →

pertpert pert

E H EE H H H E

0(2) 10 0 0 0 0

ˆ| |ˆ ˆ ˆ| ( ) |

κ κκ κ

−= − =−∑

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

ps psm n p p p p m n

pertm n m n p p p p

U E E UH 1 1

( ) ( ), , , ,

, , , , 0 ,

ˆκ κ

→−∑ ∑

Higher-order corrections:

Q-factor of Quasilocalized Mode: Higher-Order Corrections

(2)0 0κ⇒ =

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

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

211/ 20 0 0

( )2 4.315

•. 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

RF J q Rq 12 ( )π=

nmq n ma

2 2π= +

Q factor of the Quasilocalized Mode-

εδε

δεε

−+ ⋅

~ 10 δεε

36 00.4 10 ε

δε ⋅

cR Ra−

Q factor of the Quasilocalized Mode-

Weakly disordered media Photonic crystal without bandgap

Q=50 Q=106

hole, 1ε =1 1.5ε ≈

D R20≈

Magic Crystal and Localized Mode: Example

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

ϕδε ϕ θ ϕ∞

∆ = −∫ ∫ i pp

pR R R A e0 0( ) ϕϕ = + ∑

nmiR q ipnmnm p

J q R dR A eq R

2cos2 1 0

πϕ ϕϕπδε

∆ = −

∑ ∫

nmnm p p nm

J q RR A J q Rq R

2 1 00 0

( )2 ( ) 0πδε

∆ = − =

nmq n ma

2 2π= +

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

( ) sin 2 3 0ε ε σ∆ ∝ − =∫

R12 2πσ =

R R2 10.86≈

2 8.5ε ≈

1 12ε ≈

hole, 1ε =1 1.5ε ≈

D R20≈

Magic Crystal and Localized Mode: Example

Weakly disordered media Photonic crystal without bandgap

Q=50 Q=106

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

Linear Defect - Phase Slip: 1D

phase slip (stacking fault)

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− =

aπσ =σ−

c3 / 20 0 0ε σ−Ω = ∆gap

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( ) cos( )xE x e xγ σ−∝

(or d = 1.5 a)

Strongly Localized Photonic Mode in 2D Periodic …cophen04/Talks/Apalkov.pdfStrongly Localized...

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