Joint Advanced Students School 2004 Saint Petersburg. … · 2004-11-24 · TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich Photonic crystals:

TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich

Photonic crystals: theory and applications

Alexander PetrovTechnische Universität Hamburg-Harburg

Joint Advanced Students School 2004Saint Petersburg.


ACKNOWLEDGEMENTS

• Steven G. Johnson for some diagrams from his photonic crystal tutorial and for using his MIT-MPB PW software

• CST Darmstadt for supplying us with their MWS Software

INTRODUCTORY BOOKS

• K. Sakoda, Optical Properties of Photonic Crystals, Springer 2001Joannopoulos

• S.G. Johnson, J.D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice, Kluwer 2002

• J.D. Joannopulos et al., Photonic Crystals, Princeton Univ. Press 1995

COWORKERS

G. BöttgerK. Preusser-MellertM. Schmidt

SUPERVISOR

Prof. M. Eich


Theory of infinite PC structure

Beam propagation in PC

PC as omnidirectional mirror

2D PC slab structure

Manufacturing

Possible applications


Photonic crystal is a periodical dielectric materia lEXAMPLES OF PHOTONIC CRYSTALS

[Meyer et al., IPHT, Jena] [Liguda, Eich et al., TU-Hamburg] [Lin et al., Sandia, New Mexico]

[Joannopoulos et al., „Photonic Crystals , Molding the Flow of Light“ (1995) ]

Lattice constant a ~ λ


Maxwell’s equations rewritten to an eigenvalue probl em

MAXWELL’S EQUATIONS WAVE EQUATIONS

∂∂=×∇

∂∂−=×∇

=⋅∇

=⋅∇

t

trErtrH

t

trHtrE

trH

trEr

),()(),(

),(),(

0),(

0)},()({

0

0

r

r

rr

r

r

r

r

r

r

r

r

r

r

εε

µ

ε( )

( ) { }t

trE

ctrE

r

t

trH

ctrH

r

2

2

2

2

2

2

),(1),(

1

),(1),(

1

∂∂−=×∇×∇

∂∂−=

×∇×∇

r

r

r

r

r

r

r

r

r

r

ε

ε

)exp()(

)exp()(

tirHH

tirEE

ωω

−=

−=r

rr

r

rr

)()(

)()(

2

2

2

2

rHc

rHL

rEc

rEL

H

E

r

r

r

r

r

r

r

r

ω

ω

=

=EIGENVALUE PROBLEM


Spatial periodicity allows the use of Fourier expan sionAN APPROACH TO SOLVE THE EIGENVALUE PROBLEM

periodicity

( ) { } )()(1

)(2

2

rErEr

rEc

LE

ωε

=×∇×∇≡ )) rRr (=+( εε

332211 aaaR mmm ++=

rkkk erurErE ⋅== i

nn )()()(

)) ruRru kk (=+( nn

)312(),231(),123()(,)(

)(2

321

=×⋅×

= ijkkj

i aaa

aab

π332211 bbbK lll ++=

∑ ⋅+=K

kk rKkKErE })(exp{)()( inn ∑ ⋅=K

rKKr

)exp()()(

1ie

ε

eigenvalueproblem

Bloch theorem

reciprocal coordinate system

Fourier expansion


Periodical dielectric function couples the spatial harmonics of electromagnetic fieldEXAMPLE OF 1D PHOTONIC CRYSTAL

])cos(1[0 rKMr

r

+= εεa

Kπ2=

,02 =+∆ EkE εr

wave equation

,)exp(∑∞−∞=

⋅==n

nnzz rkiVeEeEr

r

rr

r

Knkkn

rrr

+= 0

,0}{)2/()()( 11222 =++−= +− nnnnn VVkMVkkVq

r

+∞<<∞− n

Setting the determinant of the coefficient matrix to zero leads to the dispersion relation:

Knkkkkn

rrr

+≡ )()( 0

P.Russell Appl.Phys.B 39

Floquet-Bloch Wave: ac

kω=


Local band gap appears at the anti-crossing pointDISPERSION RELATION IN 1D PC

ω

ky

kx

2ω

kx

ky3ω

0kr

1−kr

Kr

−

0kr

kx

1ω2ω3ω

ky1ω

0kr

a

π2

a

π2−


Omni-directional band gap in 3D PC structureBAND DIAGRAM

d

Photonic Bandgap (PBG)

[Toader et al., Science 292, 2001 ]






Manufacturing



Light beam propagates with the group velocity

EXAMPLE OF BIREFRINGENT CRYSTAL

kr

Sr ky

kx

Sr

kr

)(kc kg

r

r ω∇=

Real space Huygens approach

Reciprocal space Wave vector diagram

Mathematical representation


Dispersion relation of PCs can be quite complexEXAMPLE OF 2D PC DISPERSION DIAGRAM

2ra

n=1.54

c

af

41.0,4.0=

c

af

First Brillouin zone


Snell's law can be applied at the PC interface

SCHEMATIC WAVE VECTOR DIAGRAM

H.Kosaka et al. Phys.Rev.B 62

Crystal surface

Incident light

Propagating light

k

ko

Dispersion surface in a vacuum (circle)

Dispersion surface in a PC


Ultra-refractive phenomena can be demonstrated

EXPERIMENT

H.Kosaka et al. Phys.Rev.B 62






Manufacturing



PC is an omni directional reflector at PBG frequenc iesOMNIDIRECTIONAL MIRROR, CAVITY AND WAVEGUIDE


Defect creates a mode inside PBG region

AIR DEFECT MODE FROM REDUCED ROD SIZE

0

0.1

0.2

0.3

0.4

0.5

0.6

Photonic Band Gap

Γ X M Γ

freq

uenc

y (c

/a)

ω0

ω

S. Johnson, tutorial , MIT

Γ X

M

r

k


0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

Line defect allows modes propagating along the defe ct

PBG

BAND DIAGRAM OF A PHOTONIC CRYSTAL WAVEGUIDE






Manufacturing



Some applications don't nee a complete 3D PBG2D PC RESONATOR IN THE SLAB WAVEGUIDE

Q=1076

CAV


kr

Only modes below light line are guided in slab wave guideDISPERSION RELATION OF DIELECTRIC SLAB WAVEGUIDE

guided modes

k

ω kn

c

0

=ωlight line

light coneradiative modes

• radiative modes are lossy to air or cladding - continuum

• guided modes are discreteand confined to slab, evanescent in cladding

kn

c

i

=ω

n1

n0


Modes under light line can have local band gap1D PC SLAB BAND DIAGRAM

ω

k

local band gap

First Brillouin zone

light conekr

a


Cavity in the 2D PC slab has intrinsic vertical los ses

BAND DIAGRAM OF A DEFECT IN 2D SLAB PC

even (TE-like) bands

odd (TM-like) bands

defect mode

0

0.1

0.2

0.3

0.4

0.5

Γ M K Γ

PC PCcav


The useful bandwidth of the PC waveguide is reducedBAND DIAGRAM OF PC SLAB WAVEGUIDE (AIR-BRIDGE)

0 30 60 90 120 150 1800

100

200

300

IDX

(3µm)

(1µm)

(1.5µm) GA

P

CORE

AIR

N23AIR A560 R160 D400 W1 [even]

f [T

Hz]

phase

W1 M1


All the advantages of lithography technology are in favor of 2D PC slab structures

EXAMPLES OF 2D PC STRUCTURES

McNab et. al. Opt.Expr. 11; Akahane et. al. APL 83






Manufacturing



Different approaches are developed for 3D PC EXAMPLES OF 3D PCs

microassembly

[Lin et al.,JOSA B 18]

[ Johnson et al.,APL. 77]

layer by layerlithography



[ Miklyaev et al.,APL. 82]

holography

[ Vlasov et al.,Nature 414]

layer by layerlithography

These structures have to be inverted


[Kawashima et. al., J.Quant.Electr. 38]

autocloning

[S. R. Kennedy et al., NanoLetters 2 ]

glancing angle deposition (GLAD)



waveguide

substrate

resist metal

C RIE (O2 / CF4) wet etch of metal

B IBE (Ar)A E-beam and development

waveguide

substrate substrate

PMMA 300nm

TiO2 0.4µm

Mes.SiO2 1 µm

SiO2 2.3µm

Si-Wafer

NiCr 50nm

2D slab structures are are manufactured using a thr ee step lithography process

TiO2 FABRICATION PROCESS






Manufacturing



• Modification of the density of states (Purcell effect, low threshold laser)

• Light guiding around tight corners (ultra compact optics)

• High Q resonators (optical filtering, switching, sensor)

• Refractive optics

• Time delay, dispersion control

• Microwave antenna designs

• Pigments

• PC fibers

MOTIVATION FOR PC RESEARCH

Several applications are discussed in PC community


Small light circuits can be obtained with PC struct uresSCHEMATICAL VIEW OF INTEGRATED DEVICES

Y - splitters

Z - bends

T and X intersections

l 1.55 microns

λ

S. Johnson, tutorial , MIT


Less transmission of comparable channel waveguide

COMPARISON OF PC-BEND VS. BENT CHANNEL WG

PC wg bend channel wg bend

f = 199.45 THzncore = 2.3nsub = 1.14


Novel design proposal for tuning drop cavities (Nod a)

W1-WG SIDE-COUPLING TO L3-CAVITY EIGENSTATE (n=3.4)

L3 A00/A15 Eigen

Q~45000

Akahane et. al. Nature 425


188 189 190 191 192 193 1940.0

0.2

0.4

0.6

0.8

1.0

1550

nm

1580

nm

1560

nm

1570

nm

1590

nm

L3 A

15 E

igen

L3 A

00 E

igen

Dis

pers

ion

edge

L20 A15 S21 AmplitudeDrop 191.3THz (1568nm)

L20 A15 Norm. Def. Amplitude

L20 A00 S21 AmplitudeDrop 192.6THz (1558nm)

W1 L20 reference

N34AIR A420 R120 D250nm L20xL3 A00 / A15

Am

plitu

de

f [THz]

Combination of W1-WG with cavity yields high Q dropW1 WAVEGUIDE SIDE-COUPLING TO L3 CAVITY (DESIGN NODA)

Transm.1500nm

Drop: 1568nmCut-Off: 1587nm

L30xL3a15


PC anisotropy can lead to negative refraction of li ght

WAVEVECTOR DIAGRAM AND BEAM PICTURE

2D square lattice of holes in dielectric

Luo et. al. Phys. Rev.B 65


Negative refraction is a new area of refractive opt ics SUPERLENSE

Luo et. al. Phys. Rev.B 65


Dispersion limits the available bandwidth of the fi berDISPERSION IN OPTICAL FIBERS

initial signals

20

-20

01.2

1.4 1.6

⋅ kmnm

psDc

[ ]mµλ

dispersion coefficient to account for:

dispersion shifted

dispersion flattened

kmnm

psDc ⋅

±≈ 20

kmL 100≈

nm

ps2000±≈Ddispersion broadening

standard


Coupled modes can be used for dispersion compensati onANTI-CROSSING POINT OF TWO COUPLED MODES

nω

nk 1υn

ng kd

dωυ =

2υ

nω

cD

−∆

==21

111)/1(

υυλλυ

d

dD g

c

nω∆c=max2υ

↓↓∆ 1, υλ ↑D


Mode anti-crossing can appear inside PBG region BAND DIAGRAM OF A PC WAVEGUIDE

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

nk

nω

1,41.4,333.0/ War == ε

index guided

gap guided,

2υ

1υvery small group velocity


Quasi-constant dispersion is achieved in PC wavegui desPC WAVEGUIDE, BAND DIAGRAM AND DISPERSION

8.0,90.4,366.0/ War == ε

)50(1 GHzchannel≈∆λmmL 5≈

nk

nωPBG

0.3 0.4 0.5

0.2

0.3

0.4

0.5 mmnm

psD

⋅,

nω0.3968 0.3970 0.3972 0.3974

0

500

1000


Theory of infinite PC structure (band diagram, band gap)

Beam propagation in PC (group velocity direction, Snell's law)

PC as omnidirectional mirror (cavity, waveguides)

2D PC slab structure (light line, losses)

Manufacturing (3D, 2D)

Possible applications (Q-cavities, waveguides, refraction, dispersion)



DD

d

Documents

Joint Advanced Students School 2004 Saint Petersburg. … · 2004-11-24 · TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich Photonic crystals: