Photonic CrystalsPhotonic Crystals and Their and Their Various ApplicationsVarious Applications

M Naci Inci

Faculty of Engineering & Natural SciencesSabanci University, Istanbul, Turkey

What is a Photonic Crystal?

A periodic structure of dielectric medium on A periodic structure of dielectric medium on a wavelength scale is called a wavelength scale is called Photonic Photonic CrystalCrystal (PC)or (PC)or Photonic Bangap Photonic Bangap MaterialMaterial (PBG)(PBG)

Light λ

* Periodic along one direction, and extends infinity along other directions.

* Periodic along two axis, and extends infinity along the third axis.

* Periodic along three axes (x, y, z) and can be obtained by filling (eg. sphere, bar) a unit cell of any three dimensional lattice and duplicating through space

Perfect Photonic Band Gap Materials

PERFECT INFINITY (not a slab with a finite height)

The idea of The idea of PhotonicPhotonic Crystals was first introduced byCrystals was first introduced byYablonovitchYablonovitch11 and Johnand John22

1. E Yablonovitch, Phys Rev Lett 58 2059 (1987)2. S John, Phys Rev Lett 58 2486 (1987)

The idea of a The idea of a PhotonicPhotonic Crystal is based on drawingCrystal is based on drawinganalogies between light and electrons:analogies between light and electrons:

Because both have a waveBecause both have a wave--like nature and like nature and can therefore be diffracted !can therefore be diffracted !

Consider Electrons and the Electronic Bandgap first

The electronic bandgap of an insulator arises fromthe diffractive interaction of the electron wavefunction

with the atomic lattice, resulting in destructive interference at certain wavelengths

What about Photons (ie, Light) andOrigin of the Photonic Bandgap ?

Material’s refractive index (or dielectric constant ε) describes the interaction of light with matter !

Interaction of l i g h t with m a t t e r

Setting up a periodic refractive index (like a periodicpotential of an atomic lattice) can result in a similar `band theory’ for photons where

certain frequencies cannot propagate

In other words, the photonic equivalent of an insulator !

However,

Wavelength of Visible Light: λ ∼ 400 nm-700 nm

Wavelength of Electron: λ ∼ 0.1 nm

Electrons and photons are not on the sameElectrons and photons are not on the samewavelength scalewavelength scale

To see the diffractive effects, To see the diffractive effects, we must make large artificial `atoms’ we must make large artificial `atoms’ on the same scale as the wavelengthon the same scale as the wavelength

Schrodinger’s equation governs electrons, but

Maxwell’s equations describe the behavior of light.

With photons, one cannot safely neglect polarizationthe way one can with electrons

Computer models for doing theComputer models for doing thecalculations for semiconductors cannot be calculations for semiconductors cannot be

used for photons !used for photons !

The electromagnetic properties of the photonic crystals are completely determined by the solutions of the

macroscopic Maxwell’s equations

0)( ≡⋅∇ rHrrr

ω(Transversality condition)

)()()(

12

rHc

rHr

rrrrrr

rωω

ωε

=

×∇×∇

(Bloch-Floquet theorem))()(,

rHerHk

rki rrrrr

rr

ωω⋅=

Periodicity of ε(r) is described by the Bravias lattice associated with the crystal

)(rr

ε is the spatially periodic dielectric function that describes the crystal

: Bloch vector (crystal momentum)kr

)()(,

rHerHk

rki rrrrr

rr

ωω⋅=

ΓX

M

)(,

rHk

rrr

ωdenotes the lattice periodic part of the Bloch function, i.e.,

)()(,,

rHRrHkk

rrrrrrr

ωω=+

)()(,

rHerHk

rki rrrrr

rr

ωω⋅=

for all lattice vectors Rr

Restricting the Bloch vector to the first BZ, corresponds to a back-folding of the dispersion relation in the infinitely extended k-space into the 1st BZ by means of translations through reciprocal lattice vectors

This introduces a discrete band index n∈N such that the band structure is described by the set

[ ]{ } NnBZkHk stnkn ∈∈ ,1,),(

,

rrrr

ωω

associated with

)()()(

1 2

rHc

rHr

rrrrrr

rωω

ωε

=

×∇×∇

rGi

BZG

kGGkk

euerHst

rr

r

rrrrr

rrr⋅

∈ =+∑ ∑=

1 2,1

)(,,,

)(λ

λλω

2,1,0, ==⋅ λλ qe q

rrr

)(,

kG

ur

rλ

(transverse unit vectors)

(expansion coefficients - to be determined...)

)()(,

rHerHk

rki rrrrr

rr

ωω⋅=

)()()(

1 2

rHc

rHr

rrrrrr

rωω

ωε

=

×∇×∇

One Dimensional Photonic Crystals

a x

y

z

unit cell(lattice constant)

For the sake of simplicity, if we assume the structure itself the same with its reciprocal lattice (a 2π/a), we would describe the irreducible Brillouin zone with the half of the unit cell [0, π/a] shown in the figure.

wave vector

One Dimensional One Dimensional Photonic Photonic Crystal Crystal

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

wavevector (ka/2π)

frequency (wa/2πc)

0.30

0.25

0.20

0.15

0.10

0.05

0

(ωa/2πc)

(ka/2π)0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

PHOTONIC BAND GAPPHOTONIC BAND GAP

0.5

0.0

GaAs/Air Multilayer Filmalternating layers of width 0.5a

dielectric band

air band

(0.5,0.15)

(0.5,0.25)

Light λ

λ/4nH

λ/4n L

nH=1.52nL=1.38

(square lattice of dielectric rods, ε = 8.9 r = 0.2a)

a

Two Dimensional Two Dimensional Photonic Photonic CrystalCrystal(square lattice of dielectric rods, ε = 8.9 r

= 0.2a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 3 5 7 9 11 13 15

Γ X Μ Γ

(ωa/2πc)

0.7

0.6

0.5

0.4

0.3

0.2

0.1(ka/2π)

TM PHOTONIC BAND TM PHOTONIC BAND GAPGAP

TE PHOTONIC BAND TE PHOTONIC BAND GAPGAP

TE modesM

0TM modes

Γ X

M

2D Hexagonal lattice structure(ε = 13, r = 0.48a )

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

COMPLETE PHOTONIC BAND GAPCOMPLETE PHOTONIC BAND GAP

TM modesTE modes

Γ M K Γ

(ωa/2πc)

(ka/2π)

Γ

M K

2D Hexagonal

ε = 13, r = 0.48a

(Joannoupoulos’s group / MIT)

(Joannoupoulos’s group / MIT)

f=w/d=0.28c/d=1.414

S Y Lin et al Nature 394 251 (1998)

Complete bandgap(0.46c/a - 0.56c/a)

Face centered tetragonal latticesymmetry

O Toader and S John, Science 292 1133(2001)

What we did ...

Working Principle

* A 2D PBG Structure for Surface Temperature Mapping

* Based on BB Radiation Characteristics of the Target

Intensity

wavelengthλi λj λk

I(λi,T)I(λj,T)

I(λk,T)

O(λi,T)O(λj,T) O(λk,T)

I(λ,Τ) = [2πhc2]/[λ5(ehc/λkT-1)]

I(λ,Τ) → Ο(λ,Τ) = SγΙ(λ,Τ)

* 2D Photonic Crsytal Slab

* Triangular array (ie, 2D Hexagonal)of air holes

* GaAs: Lossless around 1.55-µm

* r/a = 0.3 (a = 0.382 -µm)

* ε = 11.4

* Complete TE band-gap: 0.213-0.280 (c/a)

* Defect radii: 0.51a, 0.54a, 0.57a.

Proposed PBG Structure: Design Parameters

RadiationGuiding EM Modes through the waveguide

Trapping by the corresponding point defects

Obtaining the Scaled Intensities

Obtaining temperature using BBRC and PC transmission response

Relation between measured optical power and resonant wavelengths (output radiation and BBR are linked)λ

λλ

λλ

λλ

λ

λ

λλ

dddP

dddP

TP

ii

i

i

ii

vvivi ∫∫

∆+

∆−

−≡2

2

),(

∫∆+

∆−

=2/

2/

),(),(ii

ii

dTISTP iivi

λλ

λλ

λλγλ

∫∫=

λλ

λλ

γγ

λλ

dTI

dTI

TPTP

j

i

jvj

ivi

),(

),(

),(),(Ratio of optical powers for any

two defects i and j

[ ][ ] j

m

l jl

in

k ik

j

i

m

ljl

jl

vjl

n

kik

ik

vik

Thc

Thc

P

P

δλλ

δλλ

γγ

δλδλ

δ

δλδλδ

∑∑

∑

∑−

−=

)1/exp(/1

)1/exp(/1 Discrete formulisation for numerical analysis

}{

}

iiiik k δλλλλ )2/1(2/ −+∆−≡

jjjjl l δλλλλ )2/1(2/ −+∆−≡

iin λδλ ∆=− )1(

Notation}[ ][ ]

)()1/exp(/1

)1/exp(/1Tf

Thc

Thc

P

P

jl l

k ik

j

im

lvjl

n

kvik

≡−

−−

∑∑

∑

∑δλλ

δλλ

γγ

δ

δAnalytical Solution {

0)(lim'

=→

TfTT

Obtain temperature, T’}RXS ∏Θ=Constant scaling factors {

TM modes

TE modes

TE Photonic Band Gap

TM modes

TE modes

Our Photonic Crystal Structure(ε = 11.4, r = 0.3a)

Γ Μ Κ Γ

0.4

0.3

0.2

0.1

0

(ωa/2πc)

(ka/2π)

TE Photonic Band Gap

Γ

M K

(A line defect is introduced in the periodic array)

TE modes

TETE PhotonicPhotonic Band GapBand Gap

waveguide modes

(ka/2π)

(ωa/2πc)

0 0.1 0.2 0.3 0.4 0.5

0.30

0.20

0.10

0

0.25

0.15

0.05

Line defect is introduced in the periodic array...

(ε = 11.4, r = 0.3a)

0 0.1 0.2 0.3 0.4 0.5

0.30

0.15

0.20

0.25

(ωa/2πc)

(ka/2π)

waveguide modes

λ1

λ2

λ3

… and point defects are introduced.

TE modes

RadiationGuiding EM Modes through the waveguide

Trapping by the corresponding point defects

Obtaining the Scaled Intensities

Obtaining temperature using BBRC and PC transmission response

Surface plot for the intensity of Hycomponent of EM field for the defect radius of 0.51a, corresponding to the resonant wavelength of 1.73 µm.

Amplitude of the Hy component of the field in a.u. for the defect, 0.51a. The each colored line indicates one slice passing through the isolated point defect.

Defect Size: r = 0.51a

localisation

waveguide region

point-defect region

Surface plot for the intensity of Hycomponent of EM field for the defect radius of 0.54a, corresponding to the resonant wavelength of 1.699 µm.

Amplitude of the Hy component of the field in a.u. for the defect, 0.54a. The each colored line indicates one slice passing through the isolated point defect.

localisation

waveguide region

point-defect region

Defect Size: r = 0.54a

Surface plot for the intensity of Hycomponent of EM field for the defect radius of 0.57a, corresponding to the resonant wavelength of 1.658 µm.

Amplitude of the Hy component of the field in a.u. for the defecct, 0.57a. The each colored line indicates one slice passing through the isolated point defect.

localisation

waveguide region

point-defect region

Defect size: r = 57a

λ (µm)

r(a)

Coup.(Γ)

Inp. Rad.(W/cm2.µm)

Out. Rad.(W/cm2.µm).S

1.730 0.51 0.95 28.500 27.0751.699 0.54 1.00 28.775 28.7751.658 0.57 0.77 29.040 22.3611.584 0.60 0.49 29.310 14.362

Calculated input and output radiations and coupling ratios, corresponding to the resonant wavelengths in the first column, are illustrated. Second column gives the corresponding defect radii, while the third column indicates the relative power coupling for these defects, which is calculated by the FDTD method, in terms of a constant Γ. In the fourth and fifth columns are given the input radiation in BB radiation form and output radiation, which is emitted from the crystal structure, respectively. Note that the output radiation has a constant scaling factor S, which our method for temperature reading makes use of. Blue colour (last row) indicates the fourth defect, which has a different profile than the others.

SOME OTHER PC APPLICATIONSSOME OTHER PC APPLICATIONS

ultra-small multiwavelength light source

ultra-small wavelength DEMUX circuit

Ultra-small optical integrated circuits by 3D Photonic Crystals

(S Noda, Kyoto University, Japan)

Photonic Bandgap Lasers on InP Substrates

* A small defect (ring with no holes) is introduced in pattern.

* Light is trapped in the ring defect and generates laser oscillations.

* World’s smallest laser.

Ring defect

PBG Laser

(Yokohama National University/Baba Research Lab)

2D PBG Laser

H Park et al, APL79 3032(2001)

Photonic Crystal Waveguide

(A Mekis,et al Phys. Rev Lett. 77 (1996)

High Density Multi-layer PBG Interconnects

(University of Delaware)

Crosstalk ReductionUsing Photonic Crystal Resonators

(Johnson et al, Optics Letters, Dec 1998)

Tapped Delay Line FilterRF Signal Modulated on Optical Carrier

(MIT Lincoln Laboratory)

J C Knight and P St Russell, Science 296 276 (2002)

New Ways to Guide Light