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.
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Theory of infinite PC structure
Beam propagation in PC
PC as omnidirectional mirror
2D PC slab structure
Manufacturing
Possible applications
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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=
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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−
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Omni-directional band gap in 3D PC structureBAND DIAGRAM
d
Photonic Bandgap (PBG)
[Toader et al., Science 292, 2001 ]
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Theory of infinite PC structure
Beam propagation in PC
PC as omnidirectional mirror
2D PC slab structure
Manufacturing
Possible applications
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Ultra-refractive phenomena can be demonstrated
EXPERIMENT
H.Kosaka et al. Phys.Rev.B 62
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Theory of infinite PC structure
Beam propagation in PC
PC as omnidirectional mirror
2D PC slab structure
Manufacturing
Possible applications
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
PC is an omni directional reflector at PBG frequenc iesOMNIDIRECTIONAL MIRROR, CAVITY AND WAVEGUIDE
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Theory of infinite PC structure
Beam propagation in PC
PC as omnidirectional mirror
2D PC slab structure
Manufacturing
Possible applications
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Some applications don't nee a complete 3D PBG2D PC RESONATOR IN THE SLAB WAVEGUIDE
Q=1076
CAV
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Modes under light line can have local band gap1D PC SLAB BAND DIAGRAM
ω
k
local band gap
First Brillouin zone
light conekr
a
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Theory of infinite PC structure
Beam propagation in PC
PC as omnidirectional mirror
2D PC slab structure
Manufacturing
Possible applications
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Different approaches are developed for 3D PC EXAMPLES OF 3D PCs
[ Miklyaev et al.,APL. 82]
holography
[ Vlasov et al.,Nature 414]
layer by layerlithography
These structures have to be inverted
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
[Kawashima et. al., J.Quant.Electr. 38]
autocloning
[S. R. Kennedy et al., NanoLetters 2 ]
glancing angle deposition (GLAD)
Different approaches are developed for 3D PC EXAMPLES OF 3D PCs
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Theory of infinite PC structure
Beam propagation in PC
PC as omnidirectional mirror
2D PC slab structure
Manufacturing
Possible applications
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
• 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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
Negative refraction is a new area of refractive opt ics SUPERLENSE
Luo et. al. Phys. Rev.B 65
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
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)
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
TECHNISCHE UNIVERSITÄT HAMBURG-HARBURG Materials in Electrical Engineering and Optics, Eich
DD
d