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

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,



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>



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)



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



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



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



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..



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)

3D



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

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



na

na≠

a

Defects in Periodical Structures

1D 2D

• 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)

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



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)



na

na≠

a

Defects in Periodical Structures

1D 2D

• Point-like defect

• Phase slip - linear defect

(periodicity interruption)



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− =



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



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



Phase Slip – Localized Mode, 1D

Nature (London) 390, 143 (1997)

Dielectric contrast - 12:1

λ

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

g g

ω ω

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



Two Phase Slips, 2D

two phase slips

localized mode

xy0 0

low contrast of dielectric constant



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



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∆ =



localized mode

Magic Crystal: Localized Mode

two phase slips

low contrast of dielectric constantxy

0 0

Q ?Im

ωω

= =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

= + + =

= ∆ + ∆ + ∆

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



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



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⇒ =



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-



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-



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

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= +



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 ≈



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

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)



strongly localized photonic mode in 2d periodic …cophen04/talks/apalkov.pdfstrongly localized...

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