Optical Beam Instability and Coherent Spatial Soliton Experiments George Stegeman, School of Optics/CREOL, University of Central Fl 1D Kerr Systems Joachim Maier & Patrick Laycock Homogeneous 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 Kim Homogeneous QPM KTP (PPKTP)Lluis Torner’s Group (Un. Barcelona) Moti Katz (Soreq) 2D Semiconductor Amplifiers Erdem Ultanir & Stanley Chen Chris Lange’s Group (Frederich Schiller Un. Falk Lederer’s Group (Frederich Schiller Un * Technical University of Munich
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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)
Elliptical (100:1) or CircularBeam dimensionsM2 (gaussian quality factor)Peak power & Pulse widthPolarization at sampleFlat phase at sample interfaceFrequency spectrum
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
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
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