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 −=∝=