Cristalli fotonici

Y(r,t) Magnetic vector field H(r,t)

Wave function

Schrodinger equation Maxwell equations

  

2 2

p 1

 

+ Y = Y     =

V ( r ) ( r ) E ( r )

  H ( r ) H ( r )

 



E E 2

 

2 m ( r ) c

 

− ( )

ic

=  

 

E ( r ) H r

 



 

( r )

Periodicity of the lattice potential Periodicity of the dielectric constant

(

( )

)  

=

= +

+

V r

r V (

( r

r R

R ) )

Bloch theorem for wave function Bloch theorem for harmonic modes

( )

( ) = 

Y =  H ( r ) u ( r ) exp ik r

( r ) u ( r ) exp ik r k k

k k

Bloch theorem for wave function Bloch theorem for harmonic modes

( )

( ) = 

Y =  H ( r ) u ( r ) exp ik r

( r ) u ( r ) exp ik r k k

k k

The values for the wave vector k The values for the wave vector k

lie in the Brillouin zone in lie in the Brillouin zone in

reciprocal space. reciprocal space.

Quantum Mechanics in a Electromagnetism in a Periodic

Periodic Potential (Crystal) Dielectric (Photonic Crystal)

Periodicity of the lattice potential Periodicity of the dielectric constant

(

( )

)  

=

= +

+

V r

r V (

( r

r R

R ) )

Dispersion Dispersion

diagram of diagram of

electron energy photon energy

 (k)

E = E(k) =

1D Photonic Crystals For wavelength in band gap

 1

 2 a For wavelength not in band gap

 =  

1 2 1 2

1D Photonic Crystals: DBR

layer numbers spectrum sharpness reflectance

within stop band

8 periods

6 periods

4 periods

1,0 2 periods

0,8

Reflectance 0,6

0,4

0,2

0,0

0,4 0,6 0,8 1,0 1,2 1,4

energy [eV]

1D Photonic Crystals: DBR

refractive index stop band width

contrast n =4.5 n =1.7

1 2

n =4.0 n =1.7

1 2

n =3.5 n =1.7

1 2

n =3.0 n =1.7

1,0 1 2

0,8

Reflectance 0,6

0,4

0,2

0,0

0,4 0,6 0,8 1,0 1,2 1,4

energy [eV]

1D Photonic Crystals: DBR

1,0

 1

 0,8

2 Transmittance 0,6

0,4

0,2 Stop band

0,0 frequency (arb. units)

1D Photonic Crystals: Fabry-Perot cavity

Defect state in the gap

1,0

 1 0,8

 2 Transmittance 0,6

 3 0,4

0,2 Stop band

0,0 frequency (arb. units)

Surface Modes on 1D Photonic Crystals

One-dimensional photonic crystals (1DPC) made by a stack of dielectric

bi-layers deposited on a glass substrate

SiO

2

127 nm

SiO

2

137 nm

Ta O

2 5

95 nm glass substrate

Surface Modes on 1D Photonic Crystals

One-dimensional photonic crystals (1DPC) made by a stack of dielectric

bi-layers deposited on a glass substrate

excitation l=532

Calculated Reflectivity map Field intensity at nm

R(q,l) (TE-pol)

1.5D Photonic Crystals: DBR on multilayer

Bragg’s = −2

According to law, almost full reflection occurs when

Λ

Τ

= 2

where is the grating vector of the DBR

DBR

2D FDTD Model

Normalized power

at the monitor

(Reflected BSW)

Λ = 260nm

DBR

1.5D Photonic Crystals: cavity on multilayer

Fabry-Perot cavity mode for Bloch Surface Waves 2D FDTD Model

Normalized

power at the

monitor

(Reflected BSW)

D= 570 nm

Λ = 260 nm

DBR

1.5D Photonic Crystals: patterned multilayer

2D Photonic Crystals Band diagram for a triangular

lattice of air column drilled in a



dielectric medium with =12

(Silicon)

Filling fraction

r/a=0.48

2D Photonic Crystals field

Elecric

TM Elecric field

TE

2D Photonic Crystals: cavities, waveguides

No defects Point defect Line defect

2D Photonic Crystals: cavities, waveguides

No defects Point defect Line defect

resonant cavity waveguide

3D Photonic Crystals

Eli Yablonovitch invented the first successful photonic crystal while

at Bell Communication Research (U.S.A. Patent n°5172267-1992)

Yablonovite Defects in

Yablonovite

3D Photonic Crystals

Eli Yablonovitch invented the first successful photonic crystal while

at Bell Communication Research (U.S.A. Patent n°5172267-1992)

Yablonovite Defects in

Yablonovite

3D Photonic Crystals

E. Yablonovitch et al PRL 67 (91) 3380 Defects in

Yablonovite

11 12 13 14 15 16 17

11 12 13 14 15 16 17 Frequency (GHz)

Frequency (GHz)

3D Photonic Crystals

Spheres arranged in a face centred

cubic FCC structure (bare opal)

Photonic Crystals synthesis: 1D

multilayers grown by Plasma Assisted Chemical Vapor Deposition

substrate

plasma

several gas lines (SiH , C H , NH , H , CO , N )

4 2 2 3 2 2 2

Load-Lock chamber mc

•13.56 MHz Plasma Enhanced CVD amorph. and Si based alloys (SiC , SiN , SiO )

X X X

•2.45 GHz Electron Cyclotron Resonance CVD poly-cryst. Si-based materials, a-SiN , a-SiO

X X

Photonic Crystals synthesis: 1D

a-Si N :H Distributed Bragg Reflectors with stop band

1-x x from the visible to the IR range

high refractive

l/4

index n d =

nm H

3.5

1500 3.0

at low refractive

index l/4

index n d =

2.5 L

refractive 2.0

1.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 Corning glass

x=N/(Si+N)

Multilayers deposited by r.f. Plasma Enhanced CVD

(SiH + NH + H gas mixtures)

4 3 2 Photonic Crystals synthesis: 1D

a-Si N :H Distributed Bragg Reflectors with stop band

1-x x from the visible to the IR range

N=7 n =3,4 n =1,74

1,0 H L

N=6 n =2,9 n =1,8

H L

N=6 n =2,2 n =1,8

H L

0,8

Reflectance 0,6

0,4

0,2

0,0

400 500 600 700 800 1400 1600 1800 2000 2200

Wavelength [nm]

Multilayers deposited by r.f. Plasma Enhanced CVD

(SiH + NH + H gas mixtures)

4 3 2 Photonic Crystals synthesis: 1D

a-Si N :H based Fabry-Perot microcavities

1-x x high refractive

l/4

index n d =

H

a-Si N :H

1-x x low refractive

a-Si N :H l/4

3 4 index n d =

L

luminescent

layer

l

d = 50 nm

Substrate:

Corning glass

Multilayers deposited by r.f. Plasma Enhanced CVD Cross sectional

TEM viewgraph

(SiH + NH + H gas mixtures)

4 3 2 l = l =

Microcavities tuned @ 640 nm and 1200 nm

C C

6 periods for each DBR Thickness homogeneity: 3% on 8 cm x 8 cm

Transmittance

Reflectance

1,0

R

T,

0,5

0,0 500 600 700

m

-cavity

units] x 1

[arb. l = 5 nm emitter

intensity 0

x 40 units] 20

1.0 40

[arb. 0.8

intensity

PL 60

0.6 PL angular

0.4 dependence

500 600 700 PL 80

0.2

l [nm] 0.0

Photonic Crystals synthesis: 2D

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates

submicrometer sized

First step: lithography

Resist (PMMA) [~50-200 nm]

Si substrate Photonic Crystals synthesis: 2D

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates

submicrometer sized

First step: lithography (electron-beam)

max resolution ~ 10 nm

Photonic Crystals synthesis: 2D

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates

submicrometer sized 200 nm

First step: lithography (electron-beam) (a-Si) Polytechn. of Torino, Italy

max resolution ~ 10 nm 100 nm

(InP) University of St. Andrews, UK

Photonic Crystals synthesis: 2D

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates 298 nm, 193 nm

submicrometer sized

First step: lithography (deep UV)

max resolution ~ 150 nm mask

Photonic Crystals synthesis: 2D

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates

submicrometer sized

First step: lithography (deep UV)

max resolution ~ 150 nm mask University of Gent, Belgium

Photonic Crystals synthesis: 2D

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates

submicrometer sized

nd

2 step: etching Photonic Crystals synthesis: 2D

Standard RIE

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates

React. gas i.e.

CF , O , SF

submicrometer sized 4 2 6

nd

2 step: etching (Reactive Ion Etching [RIE]) 13.56 MHz

RF power supply:

direct control of bias voltage

Photonic Crystals synthesis: 2D ICP/ECR

Procedures aimed at obtaining cylindric 13.56 MHz

hole arrays on semiconductor substrates

React. gas i.e.

submicrometer sized CF /O , SF / O ,

4 2 6 2

CCl F , CHF ,

2 2 3

nd

2 step: etching (Remote plasma-RIE) HBr,Cl

2 13.56 MHz

Independent control of bias

voltage respect rf power supply

Photonic Crystals synthesis: 2D

CAIBE

Procedures aimed at obtaining cylindric

hole arrays on semiconductor substrates i.e. XeF 2

submicrometer sized

nd

2 step: etching (Ion assisted-RIE)

Photonic Crystals synthesis: 2D

Previous technology yields aspect ratio ~ 1:10

ICP-RIE c-Si InP

University of St. Andrews, UK

a-Si

University of Ghent, Belgium

Photonic Crystals synthesis: 2D

Porous Si technology can reach aspect ratio ~ 1:100

Max-Planck-Institute, Weinberg Germany

Photonic Crystals synthesis: 3D

Procedures aimed at obtaining

FCC structures with 3D PBG

nd

1 step: bare opal fabrication

DIP COATING

SAMPLE

COLLOIDAL

SUSPENSION OF

SiO OR POLYSTYRENE

2

NANOSPHERES

THERMOSTATIC CELL

strong capillary forces at a meniscus between a substrate and a colloidal solution can induce

crystallization of spheres into a 3D array of controllable thickness

Photonic Crystals synthesis: 3D

Procedures aimed at obtaining

FCC structures with 3D PBG

nd

1 step: bare opal fabrication

SOME EXAMPLES

Polystyrene opal Silica opal

Polytechn. of Torino, Italy NEC Res. Institute, NJ, USA

Photonic Crystals synthesis: 3D

Procedures aimed at obtaining

FCC structures with 3D PBG

nd

2 step: opal infiltration

Titania infiltration and template removal (TiEt infiltration and calcination @ 575 °C)

TiO inverse opal

2

Polystyrene opal Polytechn. of Torino, Italy

Photonic Crystals synthesis: 3D

Procedures aimed at obtaining

FCC structures with 3D PBG

nd

2 step: opal infiltration

Si infiltration by CVD and template removal

Silicon inverse opal

SiO opal

2 Ioffe Institute St. Petersbourg, Russia

3D-Photonic Crystals: inverted opals

Si infiltration and template removal: Si deposition by thermal CVD (SiH precursor)

4

and Silica removal by plasma (SF RIE) and wet etching (HF)

6 

Sphere