Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
Optical Beam Instability and CoherentSpatial Soliton Experiments
George Stegeman, School of Optics/CREOL, University of Central Florida
1D Kerr Systems Joachim Maier & Patrick LaycockHomogeneous Waveguides Stewart Aitchison’s Group (Un. Toronto)Discrete Kerr Arrays Yaron Silberberg’s Group (Weizmann)
Demetri Christodoulides Group (CREOL)
1D Quadratic Systems Robert Iwanow & Roland Schiek*
Homogeneous QPM Waveguides Wolfgang Sohler’s Group (Un. Paderborn)Discrete Quadratic Arrays Falk Lederer’s Group
2D Quadratic Systems Ladislav Jankovic, Sergey Polyakov, Hongki KimHomogeneous QPM KTP (PPKTP) Lluis Torner’s Group (Un. Barcelona)
Moti Katz (Soreq)
2D Semiconductor Amplifiers Erdem Ultanir & Stanley ChenChris Lange’s Group (Frederich Schiller Un.)Falk Lederer’s Group (Frederich Schiller Un.)
* Technical University of Munich
Interplay Between Self-Focusing and Diffraction:Spatial Solitons & Modulational Instability
Self-focusing (NLO)Diffraction
+
Narrow Beams
Solitons
Plane Waves (Very Wide Beams)
Modulational Instability (Filaments)
Spatial Solitons
Spatial Solitons (2+1)DSpatial Solitons (1+1)D
Soliton Properties:1. Robust balance between diffraction and a nonlinear beam narrowing process2. Stationary solution to a nonlinear wave equation3. Stable against perturbations
Experimentally:1. Must be “stationary” over multiple diffraction lengths2. Must be stable against perturbations3. Must evolve into a stationary soliton for non-solitonic excitation conditions
(1+1)D - in a slab waveguide- diffraction in one D
(2+1)D - in a bulk material- diffraction in 2D
Material Nonlinear Mechanisms
Discussed here
KerrPNL = ε0χ(3)|E|2E þ ∆n = n2I
QuadraticPNL = ε0χ(2){E(ω) E(ω) + E*(ω)E(2ω)}
ω + ω = 2ω ω = 2ω - ω
(Semiconductor) Gain MediumPNL ∝ f(N, α, π, E)N – carrier density (complex dynamics)α - lossπ - electron pumping rate (determines gain)
Not Discussed hereSaturating KerrPNL = ε0{χ(3)|E|2E + χ(5)|E|4E + …}þ ∆n = ∆nsat I/[I + Isat]
Photorefractive∆n = - ½ n3reffEDC - ½ n3seffE2
DC
Reorientational (liquid crystals)∆n = (n2 - nz )(sin2θ[|E|2] – sin2θ0[0])
Spatial Soliton Systems
ALL spatial soliton generating equations are CW in timeBUT, most spatial soliton experiments use PULSED lasers!!
1220
# Soliton Param.
100’s W20 x 4 µmAlGaAs (Eg/2)1D Kerr10 KW20 x 20 µmPPKTP2D Quadratic100 W20 x 5 µmQPM LiNbO31D Quadratic
10’s mWs15 µmAlGaAsDissipative (SOAs)
PowerSoliton SizeMaterialSoliton Type
I(t)
t Diffracting background
Soliton-formingbeam component
Temporal Pulse – complicates situationSpatial Output
Generic Experimental Layouts
Laser
Beam ShapingBeam Characterization
Sample
Output BeamCharacterization
WavelengthPeak powerPulse widthBandwidth
Elliptical (100:1) or CircularBeam dimensionsM2 (gaussian quality factor)Peak power & Pulse widthPolarization at sampleFlat phase at sample interfaceFrequency spectrum
NLO mechanisms# Diffraction lengthsOptical quality
Beam shape & dimensionsBeam energy distributionBeam frequency spectrumBeam pulse widthBeam transmission (losses)
Nonlinear Wave Equation: Kerr Nonlinearity
Slowly varying phase andamplitude approximation(1st order perturbation theory)
EEc
EEz
ik 2)3(2
22 ||32 χω
−=∇+∂∂
− ⊥
diffraction nonlinearity
NLPEcn
E 022
2
202 µωω −=+∇}]{exp[ kztiE −∝ ω
3ε0χ(3)|E|2E
Stationary Plane Wave Solution
02 =∇⊥E 0|| 0 =∂∂ Ez
Plane Wave Stationary
..21 )(||
02
0,20 cceeEE kztizEnik E += −− ω∆n = n2,E|E|2
Simplest Case: “Plane Waves” in 1D Slab Waveguides
(1+1)D - in a slab waveguide- diffraction in x-dimension
?Slab Waveguidex
y
z
χ(3) þ ∆n = n2,E|E|2 [ ] )(||0
2020)cos()(1 kztizEnikz eeexEE −−+= ωγκκδ
perturbation“plane wave” solution to nonlinear wave equation
þ
Period (Λ) = 2π/κ δ << 1 perturbation amplitude
γ= exponential gain coefficient
Modulational Instability in χ(3) Slab Waveguides
EEc
Ex
Ez
ik 2)3(2
2
2
2||32 χω
−=∂∂
+∂∂
−Insert trial solution into NLWE:
Assume E0 satisfies linear WE
Assume δ <<1
−=
kEnk
k 2||2
2
22
020
22 κκ
γ
For γ real
E
E
nnkEthresholdat
nnkEpeakat
,2020
22
0
,2020
22
0
4|:|
2|:|
κ
κ
=
=40 60 80 100 120 140
2
3
4
5
6
MI G
ain,
cm
-1
Period, µm
- - - - 75 kW 50 kW
60 100 140
----- 75 KW50 KW
6
4
2
γ(c
m-1
)
Period (µm)
-1000 -500 0 500 10000
100
2 kW
3 kW
9 kW
28 kW
55 kW
Position, µm
Nor
m. I
nten
sity
, %
0
100
0
100
0
100
0
100
28 KW
55 KW
1000-1000Position (µm)
0
y [001]
x [110]z
y
n
Al.24Ga.76AsAl.18Ga.82AsAl.24Ga.76As
GaAs substrate
λ = 1.55 µm2 cm
2-3 m
m
n2 ≈2x10-13 cm2/W
(1+1)D Kerr MI Instability: AlGaAs Below Half Bandgap
2 kW
3 kW
9 kW
28 kW
Nor
m. I
nten
sity
, %
1000
100
0
100
0
100
Position (µm)0 1000-1000
Nor
mal
ized
Inte
nsity
2 KW
3 KW
9 KW
Fourier Analysis of Intensity Pattern
0.00 0.01 0.02 0.03 0.04 0.05 0.060
1
2
0 5000
100I n p u t
Inte
nsity
, %
Position, µm
F.T.
of
Fie
ld
Frequency, 1/ µm
2nd harmonic
3rd harmonic? Harmonics growth saturation
Fourier spectrum of small scale noise on profile
Noise Generated χ(3) MI: Period Versus Power
0 10 20 30 40 50 600
50
100
150
200
250
300
350
400P
erio
d, µ
m
Peak Power, kW
Experiment --- Theory
Power (KW)0 20 40 60
Perio
d ( µ
m)
0
100
200
300
400• Experiment
Theory
Nonlinear Wave Equation: Kerr Nonlinearity
EEc
Ex
Ez
ik 2)3(2
2
2
2||32 χω
−=∂∂
+∂∂
−Slowly varying phase andamplitude approximation(1st order perturbation theory) diffraction nonlinearity
NLPEcn
E 022
2
202 µωω −=+∇}]{exp[ kztiE −∝ ω
3ε0χ(3)|E|2E
0|| 0 =∂∂ Ez
Nonlinear EigenmodeSpatial soliton
Stationary NLS Solution
(1+1)D Scalar Kerr Solitons
]2
exp[}{sec1
)( 200000,2
0
wknz
iwy
hwknn
nrE
vacvacE−=∆n = n2,E|E0|2
Low Power
High Power
Input
x
y
Output
Low Power
High Power
Input
x
y
Output
1 parameter family Power x Width = Constant
Kerr Solitons in AlGaAs Waveguides
∼1990
Connection Between MI and Spatial Solitons
Λ−
Λ=
002
22
0,2000
2
22 2
||22
nkEnk
nkE
ππγ Peak γ200
2,2
22
02||
knnE
EΛ=
πMI
|E0|
2w0
Same intensity
18]2[22
20 ≈=
Λ πw
Spatial Soliton Peak field 200
20,2
20
1||wnkn
EE
=
Bandgap core semiconductor: λgap = 736nm
Al0.24 Ga0.76As
Al0.24Ga0.76AsAl0.18Ga0.82As
1.5µ
m1.
5µm
4.0µm 8.0µm
41 guides
4.8mm≅2.5 coupling length
Bandgap core semiconductor: λgap = 736nm
Al0.24 Ga0.76As
Al0.24Ga0.76AsAl0.18Ga0.82As
1.5µ
m1.
5µm
4.0µm 8.0µm
41 guides
4.8mm≅2.5 coupling length
AlGaAs Waveguide Arrays
n2 = 1.5x10-13cm2/W @ 1550 nm
Diffraction in Waveguide Arrays
( ) 011 =+++− −+ nnnn aaca
dzda
i β
Coupled mode equation:
Light is guided by individual channels
Neighboring channels coupled by evanescent tails of fields
Light spreads (diffracts) through array by this coupling c
an is field at n-th channel center
ß is propagation constant of single channel
En(x) is the channel waveguide field.
En(x)
an
Diffraction Via Nearest Neighbor Coupling
Channel intensity distribution depends on:1. Field amplitudes in neighboring channels2. Relative phase between channels3. Phase change during coupling process (usually π/2)
Discrete Solitons in Kerr Waveguide Arrays
Eisenberg et al., Phys. Rev. Lett., 83, 2716 (1998)
Moderately localizedsolitons
Single channel input
Strongly localizedsolitons
Single channel output
High Power
Beam Collapse in Waveguide Arrays
-60 -40 -20 0 20 40 60
10-7
10-6
Position [um]
Inp
ut
Pow
er
[W]
0-40 -20 20 40Position (µm)
Inpu
t Pow
er (a
.u.)
1.0
10.0
“Slice” of outputpower distribution
12201
Soliton Param.
100’s W20 x 4 µmAlGaAs (Eg/2)1D Kerr10 KW20 x 20 µmPPKTP2D Quadratic100 W20 x 5 µmQPM LiNbO31D Quadratic
10’s mWs15 µmAlGaAsDissipative (SOAs)mWsPhotorefractive
PowerSoliton SizeMaterialSoliton Type
χ(2): Type I Second Harmonic Generation
PNL = ε0χ(2){E(ω) E(ω) + E*(ω)E(2ω)} Γ ∝ χ(2)
2ω - ω = ω
Up-conversion
Down-conversion
ω + ω = 2ωω + ω = 2ω½ a1 exp[i(ωt – k1z)]+cc ½ a2 exp[i(2ωt – k2z)] + cc
Wavevector (momentum) conservation: 2k1 = k2
Wavevector mismatch: ∆k = 2k1 -k2 Phase-mismatch: ∆kL = (2k1 -k2)L
Diffraction Nonlinear Coupling
0)()()(21)(
0)()(21)(
2*112
2
11
2122
2
22
=Γ−∂∂
−∂∂
=Γ−∂∂
−∂∂
∆
∆−
kzi
kzi
ezazazayk
zaz
i
ezazayk
zaz
i
Characteristic Processes and Lengthsin Second Harmonic Generation
z
Parametric Gain Length
|)0 ,E(|)2( ==
zdcnL
effpg ωω
Coherence LengthkLc ∆= /π
I(2ω)
∆k = 0
On Phase-Match1. Energy exchange betweenfundamental and harmonic2. π/2 phase difference
z
I(2ω)
∆k2 > ∆k1
Off Phase-Match1. Perioidic energy exchange2. Rotating phase difference
χ(2)-Induced Beam Dynamics:1D Beam Narrowing Due to Wave Mixing
(1) e.g. ∆k=0 → exp[±i∆kz] = 1(2) ignore diffraction → ∂ /∂y=0(3) writing ∂a/∂dz as ∆a/∆z
∆a2 ∝ a12∆z → a2 is narrower than a1 along y-axis
e.g. a1 ∝ exp[-y2/w02] → ∆ a2 ∝ exp[-2y2/w0
2] ∆z
∆a1 ∝ a2a1* ∆z → a1 is narrowed along y-axis
e.g. a2 ∝ exp[-2y2/w02] → ∆ a1 ∝ exp[-3y2/w0
2] ∆z
a1
a2
y
0]exp[21 2
122
2
2
2 =∆−Γ−∂∂
−∂∂ kzia
ya
kza
i
0]exp[21
2*12
12
1
1 =∆Γ−∂∂
−∂∂
kziaaya
kza
i
Recipe For Plane Wave Instability & Solitons
1. Find plane wave stationary solutions, i.e. solve nonlinearwave equations in absence of diffraction.
2. Add to plane wave solution: noise with spatial Fouriercomponent κ, amplitude δ<<1 and gain coefficient γ
3. Solve for intensity regimes with γ real and > 0. If they exist,plane wave solutions are unstable over those parameter ranges.
4. If plane wave solutions are unstable at high intensity, nonlineareigenmodes are solitons
1D Plane Wave Eigenmodes and Modulational Instability
- there are 2 unstable stationary eigenmodes, each consists of a fundamental and harmonic wave- fundamental and harmonic fields are either co-directional or counter-directional
- fundamental only at input and +ve phase-mismatch → co-directional dominates
Nonlinear Wave Equations (No Diffraction)
da1/dz = iΓb2a1*exp[-i∆kz] db2/dz = iΓa1
2exp[i∆kz] ∆k = 2k(ω) - k(2ω)
- consider a perturbation with periodicity 2π/κ, gain coefficient γand amplitudes F1 and F2
a1 = [ρ1 + F1cos(κx)exp(γz)]exp[iK1z] b2 = [ρ2 + F2cos(κx)exp(γz)]exp[iK2z]
For γ real, periodic pattern grown exponentially with distance z!
a1 b2
MI: Gain Versus Period
102W ∆kL = 9π56W35W
102W ∆kL = 21π56W35W
MI Evolution: 1D SH Eigenmodes
Fundamental
Harmonic
Prop
agat
ion
Dis
tanc
e (m
m)
50
100
150
150
100
50
Transverse Position (µm)
Transverse Position (µm)0 200 400-200-400
0 200 400-200-400
High Intensity SH Eigenmode
1D case energy trappedbetween peaks in waveguide
1D Modulational Instability
∆kL = 21π
∆kL = 9π
TM0(ω)1550 nm
TE0(2ω)LiNbO3
550-750µm
TM0(ω)1550 nm
TE0(2ω)LiNbO3
550-750µm
5 cms
Power Dependence of Breakup Period: χ(2)
20 40 60 80 10050
100
150
200
250
Experiment
Theory
Peak Power [kW]
∆kL=9π
Perio
d [µ
m]
Solitons are excited by focusing 10’smicron diameter beams at entrance facet
pgdif LLL ≥>>
SolitonFundamental
2w0
L
a1
a2
y
a1
a2
a1
a2
y
Quadratic Solitons
Beam narrowing mechanisms robustly balance diffractionFor the quadratic nonlinearity: quadratic solitons
Consist of both a fundamental and harmonic component, constant with distance, ratio depends on soliton intensity and phase-mismatch!
The fundamental and harmonic are in phase!
Recall: for SHG with ∆k=0, a1 and a2 are π/2 out of phase and a2grows with distance!
d331.55 µm d33d331.55 µm
Beam Dynamics in 1D QPM LiNbO3 Slab Waveguides
uniform periodicity (Λ) ∆k = 2kω - k2ω + 2π/Λ
Distance (mm)10 20 30 40 50
FW
HW
Inte
nsity
Lpg
Rapid energy change and relative phase rotation
L >> Ldif ≥Lpg Fundamental Wave (FW) Only Input
Narrow Beam Inputs
0 250 500 750 1000
4 kW
POSITION, µm
0 250 500 750 1000
7.2 kW
POSITION, µm0 250 500 750 1000
5.2 kW
POSITION, µm
0 250 500 750 10000
25
50
75
100 Input beam
NO
RM
. IN
TEN
SIT
Y, %
POSITION, µm0 250 500 750 1000
2.4 kW
POSITION, µm0 250 500 750 1000
0.75 kW
POSITION, µm
0 250 500 750 10000
25
50
75
100
Nor
m. I
nten
sity
, %
Position µm
10 kW
Solitonw0≅70µm
Onset of MI
Width for single soliton generationIntensity >> single soliton intensity
M.I. period at peak of gain at input intensity
“Noise” on input beam
Input intensity and widthfor single soliton generation
M.I. period at peak of gain at input intensity
Multi-Soliton Generation
QPM Engineered Waveguides
-0.02 µm steps for Λ in fabrication-intermediate values of Λ by averaging over multiple discrete periods
-e.g. 17.605 µm 4 periods 17.60 1 period 17.62averaged periodactual periods (too dense to separate out)
0 10000 20000 30000 40000 5000017.48
17.52
17.56
17.60
17.64
17.68
Perio
d [µ
m]
Position [mm]0 10 20 30 40 50
Fundamental Wave (FW)Only Input
Peak
Inte
nsity
Distance (mm)10 20 30 40 50
FW
SH
Rapid energy changeand relative phase rotation
Fields in PhaseFixed Amplitudes
Soliton
0 50 100 15050
100
150
200
FD 210.5C, 0.7π SH 210.5C FD 211C, 0π FD 211.5C, -0.7π INPUT Diffracted
FWH
M [µm
]
Peak Input Power [W]
Diffracted
Input
-300 -200 -100 0 100 200 300
0
20
40
60
80
ϑ=219.5oCPpeak=190 W
POSITION [µm]
INTE
NS
ITY
[%
]
FD (65.6%) SH (34.4%)∆kL ≅0
FW (66.6%)SH (34.4%)
PPLN Waveguide Arrays
xz
y
crystal axes
TM 00 an
width
ω
TM 00 bn
2ω
refractive index diffusion profile
Ψb
cbcb βb
ca caβa
separation § z-cut LiNbO3 substrates
§ 16.8 µm QPM structure
§ λPM = 1557nm @130°C
§ 7 µm Ti stripes in-diffused
§ 101 guides, 51mm long
§ lossFH = 0.2 dB/cm
zx
d
zx
zx
dd
FundamentalHarmonicλ = 1557nm
SecondHarmonicλ = 778.5nm
transverse energy transport
discrete diffraction
no transverse energy transport
χ(2) nonlinearcoupling
Coupled FH – Decoupled SH
Qualitative Results – FH = 1085W
-200 -100 0 100 2000.0
0.5
1.0
Temp = 1520C 1W 400W 1085W
N
orm
aliz
ed p
ower
[au]
Position [µm]
Separation 13.5µmCoupling length 13mm ∆kL = 50π
Summary
1. Spatial solitons are a very rich and diverse field 2. Spatial solitons have been studied in homogeneous and
discrete systems, in waveguides and bulk media3. Many nonlinear mechanisms can be used for solitons4. 0, 1 and 2 parameter families of solitons demonstrated5. Most solitons require temporal pulses to reach soliton threshold
6. Many spurious factors complicate understanding results7. Dissipative solitons exist at 10’s mWs power with
nanosecond response – many applications (not discussed) 8. Interactions are a very rich field with novel applications
Dispersion Relation For Waveguide Arrays
∆θ = 2π
kz
kxdπ-π
β
( ) 011 =+++− −+ nnnn aaca
dzdai β }]{exp[0 dnkzktiEa xzn −−= ω“plane waves”:
}]exp{}[exp{ dikdikck xxz −++= β ]cos[2 dkck xz += β
kz
kxd π-π
β
Input Beam Width
kxd/π
Out
put B
eam
Wid
th
Diffraction in Arrays
“Negative diffraction”
Zero diffraction
)cos(2 22
2dkcdDndiffractio
dkkdD xx
z −=∝=