fotonica 3d - studenti di fisica · inverse-opal band gap good agreement between theory (black)...
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Fotonica 3D
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TE gapTM gap
( )zyx HEE ,,
( )zyx EHH ,,
•Interconnessioni nella direzione di E
•Alto contrasto di indice
Ricetta per un band gap completo: caso 2D
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band gap
ω
0 π/a
sinπ
ax
cosπ
ax
ε(x) = ε(x+a)a
ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2
x = 0
Splitting della degenerazione:state concentrated in higher index (ε2)
has lower frequencyAggiungiamo una
piccola anisotropia
ε2 = ε1 + ∆ε
Air band
Dielectric band
Bang gap si apre al bordo della FBZ
Richiamo esempio 1D
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Bang gap si apre al bordo della FBZ
band gap
ω
Γ X
Air band
Dielectric band
M
Se sistema anisotropo
non c’è sovrapposizione
di gap.
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Bang gap si apre al bordo della FBZ
band gap
ω
Γ X
Air band
Dielectric band
M
Se sistema isotropo
c’è sovrapposizione di gap.
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Reticoli cubici
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FBZ
FCC ha FBZ
più simmetrico
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FCC non ha PhC band gap
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fcc (face-centered-cubic)
with two “atoms” per unit cell.
• Same FBZ of fcc
• Closer packing
Diamond PhC with band gap
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Diamante ha PhC band gap
overlapping Si spheres
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Diamante ha PhC band gapa
a43.0
4
3=
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Diamante ha PhC band gapa
a43.0
4
3=
Interconnessioni
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Diamante ha PhC band gap
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•PhC band gap è raro
•Per ogni reticolo c’è un valore di soglia del
contrasto di indice sopra il quale si apre il gap
•Soglia inferiore se sistema ha FBZ poco isotropa
•Il gap cresce con il contrasto
•Esistono valori ottimali (raggio sfere, lunghezza
tubi).
Regole generali per un PhC 3D
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Inverse Opalsfcc solid spheres do not have a gap…
…but fcc spherical holes in Si do have a gap
Infiltration
sub-micron colloidal spheres
Template(synthetic opal)3D
Remove
Template
“Inverted Opal”
complete band gap
~ 10% gap between 8th & 9th bands
small gap, upper bands: sensitive to disorder
[ figs courtesy
D. Norris, UMN ]
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Inverse-Opal Photonic Crystal[ fig courtesy
D. Norris, UMN ]
[ Y. A. Vlasov et al., Nature 414, 289 (2001). ]
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Inverse-Opal Band Gap
good agreement
between theory (black)
& experiment (red/blue)
[ Y. A. Vlasov et al., Nature 414, 289 (2001). ]
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Elementi unitari per un PhC 3D
Molti gradi di libertà
Spot Vein
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First PCBG
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Yablonovite
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Yablonovite
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New diamond-like fcc crystal
(Topology is very
similar to 2000
layer-by-layer crystal)
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New diamond-like fcc crystal
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Layered structure
rod layer
hole layerhole layer
Vertical cut
vein
spot
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Layer-by-Layer Lithography
• Fabrication of 2d patterns in Si or GaAs is very advanced
(think: Pentium IV, 50 million transistors)
So, make 3d structure one layer at a time
…inter-layer alignment techniques are only slightly more exotic
Need a 3d crystal with constant cross-section layers
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A More Realistic Schematic
[ M. Qi, H. Smith, MIT ]
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The Woodpile Crystalan earlier design:
(& currently more popular)
[ K. Ho et al., Solid State Comm. 89, 413 (1994) ] [ H. S. Sözüer et al., J. Mod. Opt. 41, 231 (1994) ]
Up to ~ 17% gap for Si/air
(diamond-like, “bonds”)
[ Figures from S. Y. Lin et al., Nature 394, 251 (1998) ]
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The Woodpile Crystal
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Other diamond-like fcc crystal
There is a gap
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PAD
PCD
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microcavities waveguides (“wires”)
Defects in 3D PhC
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Example
rod layer
hole layerhole layer
Vertical cut
vein
spot
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PhC
Difetto di punto in un taglio verticale
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PhC
Difetto di punto in un taglio orizzontalero
d la
ye
r
rod
la
ye
r
Air defect
rod
la
ye
r
Dielectric defect
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Difetto di aria (monopolo)
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Difetto di dielettrico (dipolo)
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Difetto di linea
Air defectro
d la
yer
ayer
Taglio verticale Taglio orizzontale
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Difetto di linea
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Difetto di linea
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Fotonica su slab
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Cono di
luce
Modi guidati
Slab omogenea
Entro il cono di luce
Oltre il cono di luce
zkkc2
//2
+=ω
2
2
//2
//c
kikck z
ωω −=→<
//2
2
2
// kc
kck z −=→≥ω
ω
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1
2sinn
ni >θ
Rappresentazione modi guidati:
Confinamento 1D della luce
onda evanescente
onda evanescente
Slab
Confinamento 1D
della luce dovuto ad
“index guiding”
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Diagramma a bande 2D
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Diagramma a bande 2D + cono di luce
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Diagramma a bande slab
Confinamento
2D nel PBG e 1D
da index guiding
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Diagramma a bande slab
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Guide d’onda slab
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Guide d’onda slab
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Guide d’onda slab
Più confinamento con riflessione PCB