nondiffractive light in photonic crystals · 2017-05-10 · 3 relevant harmonics transverse...
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Nondiffractive light in photonic crystals
• Nondiffractive light in linear Photonic Crystals (PCs)
• Nondiffractive pulses in PCs;
• Nondiffractive nonlinear resonators
• Subdiffractive solitons in nonlinear PCs;
• Nondiffractive resonators of PCs;
Group of “Dinàmica i òptica no lineal” , FEN, UPC
Group of “Nondiffractive light” : K.Staliunas, R.Herrero, C.Serrat, C.Cojocaru, J.Trull.
• Nondiffractive light in nonlinear PCs
• Nondiffractive beams in PCs;
Photonic Crystals
• Photonic bands:
• Modification of diffraction k
w
Photonic band I
Photonic bandgap
Photonic band II
• Modification of dispersion:
•diffraction management in fiber arrays (1D PCs)
•diffraction management in BECs (1D lattices)
•diffraction management, numerics and experiments in 2D PCs
Configurations:
n
z
n
z
Step Sinusoidal1D,2D,3D:
conductors and insulators of light
slow light
Elimination of diffraction !!But also
Nondiffractive light in linear Photonic Crystals
220||,)()( ||
⊥⊥
∞
∞−
−−⊥ −== ∫ ⊥ kkkkdeekarA zikxikr
ckkkk 0
022
||ω==+= ⊥
tierArE 0)()( ωrr =
),( ||kkk ⊥=r
rkiekarrr
−)(
z
x
Beam with carrier frequency ω0 propagating in z direction :
⊥k
IIk
Variation of the interference pattern in z
Relative phase variations during the propagation in z
Diffraction
)(rA rProfile in x,y
Different k|| for each plane wave
Diffraction
Diffraction management
⊥k
IIk
⊥k
IIk
),( ||0 kkk ⊥=r
⊥k
IIk1qr
2qr
( ) ( ) ( )rHc
rHr
rrr
21
=
×∇×∇ ωε
H.Kosaka e.a. 1999, J.Witzens e.a. 2002, D.N.Chigrin e.a. 2003, R.Iliew e.a. 2004,…
⊥k
IIk
Constant ω surf. First brillouin zonesFirst branch Second branch
•Localization of non-diffractive regimes
kII
( )( ) ( ) 0,,22 20
220 =∆+∂∂+∂∂ zxAkzxnxzik
)( 000 )()()( tzkiti erAerEr ωω −− == rrrΕ02
2
2
22 =
∂∂−∇tc
n ΕΕ
•Si considerem que :0=∂∂tE 02
20
22 =+∇ E
cnE ω
•Si considerem que :02
2
<<∂∂zA 02 0
2 =∂∂+∇ ⊥ zAikA
•Sols considerant una variable transversa i afegint ∆n:
Slowly Varying Envelope Approximation
•Equació d’ones per l’ona electro-magnètica:
( )( ) ( ) 0,,22 20
220 =∆+∂∂+∂∂ zxAkzxnxzik
⊥k
IIk
Paraxial
• Analytical expressions• Generalizations:
• Linear
Allows:
• Nonlinear (e.g. to BECs);
Paraxial approx. and sinusoidal ∆n
( ))cos()cos(2)( 21 rqrqmrn rrrrr +=∆
( ) ∑=ml
rkiml
mleAzxA,
,,,rr
),(, IIIIml mqklqkk ++= ⊥⊥
r
-1.5 -1 -0.5 0.5 1
-1-0.75-0.5
-0.25
0.250.5IIK
⊥K
• Bloch modes
Sinusoidal
harmonics
Inscription of the index modulation
( ))cos()cos(2)( 21 rqrqmrn rrrrr +=∆ •Photorefractive materials
2D
3D
•Acusto-optic materials
Inscribing usinglight beams
Inscribing usingultrasound waves
Dynamical photonic crystals Dynamical diffraction management
.
.
.
)(En∆
)(Pn∆
Diffraction elimination
00
2||
2
=∂∂
=⊥⊥ kkk
Zero diffraction:
Non-diffractive curve
Most homogeneous Bloch mode (A)
2202 ⊥= qmkf
202 ⊥= qkqQ IIII
- modulation depth- geometry
Normalization
Analytical curve for f<<1
:1<<f
( ) +−−≈ 322 181 IIQfd
( ) ...164 54 +− IIQf
...3421 343202
++−≈=
ffQdII
0.0 0.4 0.8 1.2 1.6 2.00.0
0.4
0.8
1.2
1.6
2.0
f
f=0.04
f=0.12
f=0.28
f=0.41
f=0.52
f=0.67
f=0.90
f=1.20
f=1.40
1
1
0-1
K
Q
K
K||
K
3 relevant harmonics
Transverse dispersion relation: Power series expansion
Numerics
Homogeneous material (diffractive)
Photonic crystal(non-diffractive)
2D
Input
Output
Input
Output
Homogeneous material (diffractive)
Photonic crystal(non-diffractive)
Numerics 3D
Input
Outputs
InputDiffractive
outputNon-diffractive
output
Linear effect
Subdiffraction. Analytic
( ) ∑∞
=⊥⊥ =
0n
nnII KdKK n
IIn
n KKnd ⊥∂∂⋅= !1
( ) ( ) 0,444 =∂∂+∂∂ ZXAXdZi B
( ) ...132 524 +−≈ IIQfd
...3404
2+≈ −
=fd
d
( )204
4 8)( ⊥≈∆ qkzdzx π
02 →d
Bloch mode evolution:
0=⊥k
Width evolution:
n even
4)( zazx =∆Example: Beam width=10µm , λ=10µm
Rayleigh length 30µm
Nondiffractive propagation length2m
•Homogeneous media
•Non-diffractive PC λ
π2
00
Wz =
00
2||
2
=∂∂
=⊥⊥ kkk
Subdiffraction. NumericsLarge propagation distance
The smallest non-diffractive structures
Technical applications
Smallest possible structures
-1.0 0.0 1.0-1.0
0.0
1.0
k
∆k∆kk
Non-diffractive propagationlength
Effective diffractionlower than a given value
Plateau width∆K||=0.05
||kL ∆= π
⊥∆=∆ kxmin 2
2D PC3D PC Diferent configurations: n-gons
∑−
=⊥⊥ ++=∆
1
0|| )cos()cos(4)(
n
llyqxqzqmrn ϕβαr
ϕl does not effect the dispersion curve
More symetric, less distorted propagated patterns ?
n
kz isolines
ky
Different PC structures
Q||
f kx
3D Nondiffractive Photonic Crystals
)/2sen()/2cos(nlnl
πβπα
==
Quasicrystal, bessel beam
Experiments with PCs
Nd:YAG 8 ns 300 mJ1064 nm
Spatial filtering and delay
Optical latticeFormation (2D)
Photorefractive crystal20 mm SBN:CeO2
CCD
532 nmProbe beam
- - -+ + + + - - + + -
Light interferences
Charges
Electric field
Refractive index variations
Λ~µm
ϕ=π/2
)(En∆
Light beams
•Photorefractive materials
Wavelength = 532 nmInitial width = 12.6 µm
Propagation length = 4.25 mm
∆n modulation amplitude= 5 10-4
Spatial periodicity: Λ⊥ =7.6µm , Λ||= 483 µm
•Simulations with experimental parameters
•Beam:
•Photonic crystal:
Homogeneous material
Photonic crystal
•Microscopy, photolitography, ...
•Fibers transporting patterns•Wave-guides in electronic circuits•....
Possible applications:
1 fiber transport 1 bit (light or no light)
1 fiber transport 1 pattern
Diffractive fibers Nondiffractive fibers
•Multimode nondiffractive wave-guides
Shift in x
Lens effect
Bending
Multimode Nondiffractive Fibers and Wave-Guides:
Radi de curvatura = 9 x diàmetre fibra
2202 ⊥= qmkf
202 ⊥= qkqQ IIII
- modulation depth;- geometry;
ck 00 ω=
0.0 0.4 0.8 1.2 1.6 2.00.0
0.4
0.8
1.2
1.6
2.0
f
f=0.04
f=0.12
f=0.28
f=0.41
f=0.52
f=0.67
f=0.90
f=1.20
f=1.40
1
1
0-1
K
Q
K
Non-diffractive pulses
( ) ( ) 0,,,2
102
2
0
=
∆−
∂∂−
∂∂+
∂∂ tzxAkzxni
xki
ztc
δω
-0.6 -0.4 -0.2 0.2 0.4 0.6
0.1
0.2
0.3
0.4
IIK
⊥K
Small variations of the carrierfrequency ω=ω0(1+δω)
•Carrier frequencies interval
04
12
2
2
2
00, =
∂∂
∂∂−
∂∂+
∂∂+−
∂∂ A
XTTi
TViK
Z II α
22
00, 4 ⊥⋅+++= KV
KK IIII δωαδωδω ( )4
1 20,
0,II
II
QK
−=
( ) ( )2
311 0,0,
0
IIII QQV
−⋅−=
( )0,13
IIQ−=α
Small index modulations (f<<1)
Input Output
•Gaussian pulse in x and t Propagation length: 375 µm
Integration of the main equation
( ) ( ) 0,,,2
102
2
0
=
∆−
∂∂−
∂∂+
∂∂ tzxAkzxni
xki
ztc
Propagation length: 0.45 mm
•Gaussian pulse in x and t
Filtrat per treure les oscil·lacions degudes a la modulació de ∆n
InputOutput with PC
Output without PC
( ) constKKII =⊥ δω,
⊥K
δω
⊥K
δω
0 0
-1-1 -10 0
1 1
For different groupvelocities V0
Invariant spacio-temporal shapes
Adding modes of the sameisoline
No dispersive, no diffractivePropagation
22
00, 4 ⊥⋅+++= KV
KK IIII δωαδωδω
Non-diffractive resonators
•Homogeneous spacer
•Free propagationalong distance L
•Non-diffractive cavitySpacer filled with PC
R=0.9
Input beam
L
Gaussian spectrum filteredby the resonator
Broader beam
Output beams
ϕiertkAkA 2
2
0 1)()(
−= ⊥⊥
2
0
2
22
)( ⊥⊥
⊥ −=−≅ dkLkkkϕ
reflection r, transmission t and phase shift ϕ of the cavity
Different ϕ for each transverse component
effective diffraction of the cavity d=λL/2π.
r,t r,tInput beam Output beam
Homogeneous spacerHomogeneous resonators
Gaussian spectrum filtered by the resonator
•Transmitted field
•Scann of the cavity length
400 λ 400.5 λ
Output intensityRings
w0input width
•Rings
•Free spectral range: λ/2+a , a(w0)
•Output width depends on L
Diffraction
Maximum of intensity Waist
Free spectral range
Photonic crystal spacer
Photonic crystal resonators
•No Rings. Only the central peak remains (all transversewaves in phase).
•Free spectral range does not depend on w0.
•Output width does not depends on L.
Propagation in Nonlinear PC’s (1D transvers)
( ) 0,22 2202
2
0 =
−∆+
∂∂+
∂∂ AAckzxn
xzik
W0=5
0.5
1.5
0
0
Output solitonInput beam
Width
Intensity
1
0
1
0
d2 0
W0=3
2
2
xA
∂∂ 2Ac
Solitons
Diffraction Nonlinearity
Solitons: Amplitude-Width relationship
Near the linear non-diffractive curve
( ) 0,22 2202
2
0 =
−∆+
∂∂+
∂∂ AAckzxn
xzik
AACX
dX
dTAi B
+−= 2
4
4
42
2
2 ∂∂
∂∂
∂∂
02 >d
02 <d
Modulation wavenumber
Modulation amplitude
0.5
1
1.5
25 50 750x
2A
a)
25 50 750
0.5
1
1.5
2
2.5
x
2A
b)0.8 1.2 1.4 1.6 1.8
0.9
1.1
1.2
1.3
1.4
( )010 ELog
( )010 xLog
a)
b)
constxE =⋅ 0
constxE =⋅ 30
Nondiffractivecurve
Nonlinear PCs (2D transvers)
( ) 0,22 220
20 =
−∆+∇+
∂∂ AAckzxnz
ik
( )AACddZAi B
244
22 +∇−∇=
∂∂
NLSE colapses in 2D and 3D . No stable solitons appear.
Subdiffractive NLSE
Near d2 0:
01−
5.0−
5.01
5.055.06.0
65.0
0
1−5.0−
5.01YK
zK
XK
different geometries:
n=4 not usefull to stabilize nonlinear solitons
kz isolines
ky
Q||
f kx
Nondiffractive curves
NLSE: ( )AAditA
eff22 −∇=
∂∂
Stable solitons•Analyticaly•numericaly
V(x)
x
ψ (x)
Light beam Bose-Einstein Condensate
Stable solitons in 2D and 3D
Bose-Einstein Condensates
•Optical nonlinearities•Bose-Einstein Cond. (BEC’s)
NLSE: ( )AAditA
eff22 −∇=
∂∂( ) 0,22 22
02
2
0 =
−∆+
∂∂+
∂∂ AAckzxn
xzik
( )AACddTAi B
244
22 +∇−∇=
∂∂
A ψ
Diffraction Effective mass
Non-diffractive Nonlinear resonators
•Nonlinear resonators:
QLdsize ⋅⋅≈≈ λ2/1
Homogeneous material
...)(...)( 2 +∇+=∂∂ rAidtrA
mµλ 1≈ mL µ10≈ 100≈Q
mx µ300 ≈ Taranenko, Weiss, Kuszelewicz , 2000
CV in VCSELs, lasers with saturableabsorber, OPOs, Photorefractives,
gain
intensity
Cavity Solitons
Patterns and dissipativestructures;
Switch on/off in any point
•Memory •Processing
...3421 343202
++−≈=
ffQdII
0.5 0.1 0.15 0.2 0.25 0.3 0.35
0.5
0.6
0.7
0.8
0.9IIQ
f20 40 60 80 100
0.2
0.4
0.6
0.8
1
1.2
Reduction of spatial scale
λ≈size
•Nonlinear photonic crystal resonators:
Modulation of refraction index
AB
C
ABC
Bibliography
•E. Yablonovitch, 1987; S. John, 1987;
•H.S.Eisenberg et al. 2000; R. Morandotti et al . 2001;
•Pertsch et al. 2002; O. Zobay et al. 1999;
•Yu.S.Kivshar et al. 2003; H.Kosaka e.a. 1999,
•J.Witzens e.a. 2002, D.N.Chigrin e.a. 2003,
•R.Iliew e.a. 2004
•K.Staliunas, Mid-band Dissipative Spatial Solitons, Phys. Rev. Letts, 91, 053901(2003)
•K. Staliunas and R. Herrero, Nondiffractive propagation light in photonic crystals,Phys.Rev.E 73, 023601 (2006) ; subm. 2004, arXiv: physics/0501115.
•K. Staliunas, C.Serrat, C.Cojocaru, J.Trull and R. Herrero, Nonspreading Light Pulses in Photonic Crystals , subm. 2005, arXiv: physics/0505153, accepted in PRE
•K.Staliunas, R.Herrero, G.Valcarcel and N.Achmediev, Subdiffractive Solitons in Two-Dimensional Nonlinear Schrödinger Equation 2006, submitted
•K.Staliunas, G.Valcarcel and R.Herrero, Sub-Diffractive Band-Edge Solitons in Bose-Einstein Condensates in Periodic Potencials , 2006, submitted
642 0 ∇⇒=∇=∇(Q||,f) with
2202 ⊥= qmkf
202 ⊥= qkqQ IIII
- modulation depth;- geometry;
ck 00 ω=
0.0 0.4 0.8 1.2 1.6 2.00.0
0.4
0.8
1.2
1.6
2.0
f
f=0.04
f=0.12
f=0.28
f=0.41
f=0.52
f=0.67
f=0.90
f=1.20
f=1.40
1
1
0-1
K
Q
K
•Sub-sub-diffractive?
•Nondiffractive-nondispersive pulses?
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