1d photonic crystal struttura a bande. ……. d m0m0 m0m0 m0m0 m0m0 m0m0 mnmn
TRANSCRIPT
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1D Photonic CrystalStruttura a bande
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NN
N
N
N
NN
o
tt
r
t
r
tM
tt
rt
r
tM
1
1
1
1
*
*
*
*
*
*
…….
d
…….
M0 M0 M0 M0 M0
MN
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Sistema con N periodiRiprendiamo
RR
RR
N
NN 2
2
1
IMM NNN
100
11
Re
tse
22
sin
sin
N
N
11
Re
tse
22
sinh
sinh
N
N
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Stop band99.99% riflessione
Leaky modes
11
Re
t
11
Re
t
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Propagazione attraverso un mezzo omogeneo seguita da una slab dielettrica
tt
rt
r
t
jn
nnj
n
nn
jn
nnj
n
nn
dnj
n
nnj
n
nn
jn
nnj
n
nn
M iii
1
1
exp2
exp2
exp2
exp2
2;
exp2
exp2
exp2
exp2
*
*
*
11
121
1
12
11
121
1
12
22
122
2
12
22
212
2
12
d2d1
)(2
U
)(2
U
)(1
U
)(1
U
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Trasmissione dell’elemento singolo
)(2
U
)(2
U
)(1
U
)(1
U
d2d1/2 d1/2
tt
rt
r
t
jn
nnj
n
nn
jn
nnj
n
nn
jn
nnj
n
nn
jn
nnj
n
nn
j
jM
1
1
2/exp2
2/exp2
2/exp2
2/exp2
exp2
exp2
exp2
exp2
2/exp0
02/exp
*
*
*
11
121
1
12
11
121
1
12
22
122
2
12
22
212
2
12
1
1
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Trasmissione dell’elemento singolo
1212
212
1212
212
122
1222
1212
1*
cos4
cos4
1Re
2/expexpexp4
2/exp1
nn
nn
nn
nn
t
jjnnjnnnn
j
t
)(2
U
)(2
U
)(1
U
)(1
U
d2d1/2 d1/2
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Sistema con N periodi
RR
RR
N
NN 2
2
1
sin
sin NN
t
1Recos
1212
212
1212
212 cos
4cos
4
1Re
nn
nn
nn
nn
t
iii dn 2
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Trasmissione dell’elemento singolo
1212
212
1212
212 cos
4cos
4
1Re
nn
nn
nn
nn
t
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
Re(
1/t)
B
B
B
iii
dndn
c
dnc
21
2211 )(2
2
5.3
5.1
2
1
21
n
n
dd
)(2
U
)(2
U
)(1
U
)(1
U
d2d1/2 d1/2
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Trasmissione dell’elemento singolo
212 tripround
N
triproundB
RMax
2@
10
5.3
5.1
2
1
21
N
n
n
dd
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
Re(1
/t)
b
N
2
d2d1/2 d1/2
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Trasmissione dell’elemento singolo
1212
212
1212
212 cos
4cos
4
1Re
nn
nn
nn
nn
t
B
B
iii
dndn
c
dnc
21
2211 )(2
2
5.2;5.3
5.1;5.1
22
11
21
nn
nn
dd
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
Re(
1/t)
b
d2d1/2 d1/2
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5.3
5.1
2
1
n
n21 dd
2211 dndn
2211 dndn 5.35.1 21 nn
5.3
0.1
2
1
n
n
Dependence on ni Dependence on di
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Width of the stop band
14
2/cos
42/
1Re
0
12
212
12
212
12
nn
nn
nn
nn
t B
B
B
2
12
212
212
12
212
212
212
12
212
12
212
12
212
12
212
12
212
2
2cos
42cos
4
41
2cos
4
41
21cos
4
nn
nnnn
nn
nnnnnn
nn
nn
nn
nn
nn
nn
nn
nn
nn
nn
B
B
B
B
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Width of the stop band
12
12
12
12
212
2122
212
212
212
212
212
arcsin4
arcsin4
2
4sin2
2
2cos1
2
2cos
nn
nn
nn
nn
nn
nn
nn
nn
nn
nnnn
B
B
B
B
B
5.3
5.1
2
1
n
n53.0
B
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10N
50N
Dependence on N
.
.10log..
trasm
incid
I
IDO
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Field amplitude within the stop bandFinite multilayer: Bragg mirror
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Field amplitude within the stop bandFinite multilayer: Bragg mirror
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Evanescent fielddue to interference
Evanescent wavefunction
Field amplitude within the stop bandInfinite multilayer: 1D Photonic crystal
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Bragg mirrorTunneling out of a barrier
1D PhCEvanescent wave in the barrier
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Infinite Bragg
N
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Origine del band gap
1
k
0
Mezzo uniforme
ck
1
Legge di dispersione
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1
(x) = (x+a)a
k
0 π/a–π/a
[ Lord Rayleigh, “On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure,” Philosophical Magazine 24, 145–159 (1887). ]
Trattiamolo come periodico
bands are “folded”by 2π/a equivalence
ei
a
x
,e i
a
x
cosa
x
, sin
a
x
Legge di dispersione è ripiegata nella FBZ
Origine del band gap
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(x) = (x+a)a
1
0 π/a x = 0
Trattiamolo come periodico
Origine del band gap
Gli stati degeneri abordo zona sono riscrivibili come
xa
AxE
xa
AxE
cos)(
sin)(
2
1
)(
)(
2
1
xE
xE
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(x) = (x+a)a
1 2 1 2 1 2 1 2 1 2 1 2
0 π/a
Aggiungiamo una piccola anisotropia
2 = 1 +
x = 0
Tutti i sistemi 1d hanno gap
Stato con ventre in 1
Stato con ventre in 2
xa
AxE
xa
AxE
cos)(
sin)(
2
1
)(
)(
2
1
xE
xE
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Principio variazionale
Principio variazionale: gli autostati minimizzano il funzionale energia, quindi i modi fotonici di più bassa frequenza
In (MQ) le funzioni d’onda di più bassa energia hanno ampiezza concentrata nelle regioni a potenziale minore. Vale anche in MQ la “legge dei nodi”.
23
23
)()(
Errd
ErdHU f
23
*3 ˆˆ
rd
HrdH
hanno ampiezza concentrata nella regione ad alto dielettrico. Inoltre un dato modo in generale conterrà più nodi rispetto a un modo di minore frequenza.
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band gap
0 π/a
sina
x
cosa
x
(x) = (x+a)a
1 2 1 2 1 2 1 2 1 2 1 2
x = 0
Splitting della degenerazione:state concentrated in higher index (2)
has lower frequencyAggiungiamo una piccola anisotropia
2 = 1 +
Air band
Dielectric band
Origine del band gap
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band gap
0 π/a
Air band
Dielectric band
Valore del mid gap
a
dndnn
n
ck
eff
eff
2211
B
eff
Mmidgap
dndn
c
n
ck
22211
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band gap
0 π/a
Air band
Dielectric band
Stati nel band gap
Ck
Rk
ak
2
2
2
2
Nel band gap onde evanescenti
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Ingegnerizzazione del band gap
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Gap/mid gap: quarter wave stack
1212
212
12
212 cos
4cos
4
1Re
nn
nn
nn
nn
t B
021
12
12
212
2122
212
212
212
12
212
122
12
21212
12
212
12
212
arcsin42
2sin
2118
cos
81
4)(cos
14
)(cos
4
nn
nn
nn
nn
nn
nn
nn
nn
nn
nn
nn
nnnn
nn
nn
nn
nn
B
B
B
B
B
B
B
2
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
Re(
1/t)
B
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Et
Ht
y x TM
Near Brewsterangle
Incidenza obliqua: perdita del band gap
Perdita gap
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Modi Ex (TE)
Cono di luce Evanescent waves
)0,,0( yk
)/,,0( ak y
kz
Assenza band gap completo
Struttura a bande per propagazione nel piano
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a
1 2 1 2 1 2 1 2 1 2 1 2
Modi EE Extended-Extended
All’interno di una banda e dentro il cono di luce
Tipologia dei modi
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a
1 2 1 2 1 2 1 2 1 2 1 2
Modi ED Extended-Decay
All’interno di un gap e dentro il cono di luce
Tipologia dei modi
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a
1 2 1 2 1 2 1 2 1 2 1 2
Tipologia dei modi
Modi DE Decay-Extended
All’interno di una banda e oltre il cono di luce
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Cono di luce Evanescent waves
Modi Ex (TM)
LEGENDA
ED=Extended in air, Decay in PhC DE=Decay in air, Extended in PhCEE=Extended in air, Extended in PhC DD=Decay in air, Decay in PhC
Tipologia dei modi
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a
1 2 1 2 1 2 1 2 1 2 1 2
Tipologia dei modi
Modi DD Decay-Decay
Stati di interfaccia
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Nel band gap propagazione proibita: Modo EDTutta l’energia è riflessa
True band gap Omnidiretional mirror
Omnidiretional mirror True band gap
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Bande Bragg mirror
No band gap
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Bande Bragg mirror
1
1sin
sin
n
nc
k
ck
L
Ly
y
Cono di luce
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Bande Bragg mirror
1
1sin
sin
n
nc
k
ck
L
Ly
y
Cono di luce
Omnidirectional mirror
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Angolo di Brewster
Angolo di Brewster
2
12
1
21
22
11
tantan
sin
sin
n
n
n
n
nc
k
nc
k
BB
By
By
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2
tan
tan
tB
iti
tir EE
Angolo di Brewster
Onda TM non è riflessa
1
2tann
nB
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Angolo di Brewster è simmetrico
tantan1
tantantan
2
tan
tan
21
2
12
1
21
BB
B
B
n
n
n
n
n1 n2 n1n2
1B2B
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Angolo di Brewster1
2,1tan
n
nB
n1 n2 n1 n2
Cono di luce1
,1
1sin
nL n1 n2 n1
LB
BL
,1,1
,1,1
Onda esterna TM può propagarsi a Brewster
Onda esterna TM non può propagarsi a Brewster
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1,0 1,5 2,0 2,5 3,0 3,530
35
40
45
50
55
60
65
70
1,0 1,5 2,0 2,5 3,0 3,510152025303540455055606570
n2
n2
Ang
olo
Ang
olo
1
2arctann
nB
1
1arcsin
nL
2
1arctann
nB
2
1arcsin
nL
n1 n2 n1 n2
Air
n2 n2
Air
n1 n1
111
22
21
nn
Confronto angolo Brewster vs angolo limite (n1=1.5)
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1
2arctann
nB
1
1arcsin
nL
1
2
1
1
2
1sin
1cos
1sin
tan
n
n
n
n
nLB
111
22
21
nn
Confronto angolo Brewster vs angolo limite
Se i due angoli coincidono
Quindi
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Specchio Omnidirezionale
gap
LB ,1,1
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Specchio Omnidirezionale
Gap/midgap
111
22
21
nn
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[ Y. Fink et al, Science 282, 1679 (1998) ]
Omnidirectional Mirrors in PracticeQuickTime™ and aGraphics decompressorare needed to see this picture. 11.21.41.61.822.22.42.62.8311.11.21.31.41.51.61.71.81.92
S m a lle r in d e x , n
1
0%10%20%30%40%50%/mid
6 9 1 2 1 50
5 0
1 0 0
0
5 0
1 0 0
0
5 0
1 0 0
0
5 0
1 0 0
0
5 0
1 0 0
normal
450 s
450 p
800 p
800 s
Re
flec
tanc
e (
%)
Wavelength (microns)
Te / polystyrene
Ref
lect
ance
(%
)
contours of omnidirectional gap size