Erdem Ultanir, Demetri Christodoulides & George I. StegemanSchool of Optics/CREOL, University of Central Florida

Falk Lederer and Christopher LangeFrederich Schiller University Jena

Dissipative Spatial Solitons and Their Applications in Active Semiconductor Optical Amplifiers

Spatial Solitons 1D

DiffractedBeam

Spatial Soliton

waveguide

1960 1970 1980 1990 2000

Chiao&

Talanov Predictionof Kerr (1964)

Zakharov,Soliton solutions

(1971)

Bjorkholm,Kerr Solitonsin Sat. Media

(1974)

Segev, Photorefractive

Solitons (1992)

Sukhorukov,Prediction of Quadratic

Solitons (1975)

Torruellas,2D Quad. Soliton

(1995)

Barthelemy,1D Kerr Soliton

(1985)

Mitchell,White light Soliton (1996)

Silberberg, Discrete Array(1997)

Duree,Photorefractive

(1993)

Christodoulides,Incoherent (1997)

Christodoulides,Discrete (1988)

AkhmedievCubic-quintic CLGE

(1995)

SOA(2002)

Picciante,NLC (2000)

Propagating Spatial Soliton Milestones

• Kerr Solitons, Χ3 effects, integrable system, elastic interactions• Hamiltonian systems (conservative), inelastic interaction, one (or few parameters)• Discrete Hamiltonian systems (includes Kerr)• Dissipative solitons, zero parameter systems

1D Spatial Solitons in Homogeneous Media

A spatial soliton is a shape invariant self guided beam of lightor a self-induced waveguide

Soliton Type Material # Soliton

Param.

Soliton Size Power

Quadratic QPM LiNbO3 2 20 x 5 m 100 W

Photorefractive SBN 1 15 x 5 10W

Kerr AlGaAs (Eg/2) 1 20 x 4 m 100’s W

Dissipative (SOAs) AlGaAs 0 15 m 10’s mWs

Hamiltonian Systems

Nonlinearity balances diffraction

Non-Hamiltonian (Dissipative Systems)

Gain balances loss + nonlinearity balances diffraction

No trade-offs in optical beam properties!!

(1+1)D - in a slab waveguide - diffraction in one D

(2+1)D - in a bulk material - diffraction in 2D

Bulk medium

Diffracting beam Spatial Solitons (2+1)DSpatial Solitons (1+1)D

Planar (slab) waveguide

A spatial soliton is a shape invariant self guided beam of lightor a self-induced waveguide

Vp' Self-focusing

Vp < Vp'

Spatial Solitons in Homogeneous Media

Inn

cVp

20 phase velocityInnKerrClassical 2:""

(1+1)D - in a slab waveguide - diffraction in one D

(2+1)D - in a bulk material - diffraction in 2D

Bulk medium

Diffracting beam Spatial Solitons (2+1)DSpatial Solitons (1+1)D

Planar (slab) waveguide

Soliton Properties:1. Robust balance between diffraction and a nonlinear beam narrowing process2. Stationary solution to a nonlinear wave equation3. Stable against perturbations

Observed and Studied Experimentally to Date in:1. Kerr and saturating Kerr media 4. Liquid crystals2. Photorefractive media 5. Gain media3. Quadratically nonlinear media Semiconductor optical

amplifiers

Spatial Solitons in Homogeneous Media

Homogeneous in Diffracting Dimension

Diffraction in 1D Homogeneous System

0),(),(22

2

zyEy

zyEz

i

}]{exp[)(),()( ztixfzyErE

- Insert into wave equation- Assume slow change over an optical wavelength

f(x)

y

NLPEc

nE 0

222

202

NLPEy

Ez

ik 02

2

22

1D Nonlinear Wave Equation

}]{exp[ kztiE

}{1

2

2

02

2

22 NLL PP

tE

tcE

E)1(0 depends on nonlinear

mechanism

Slowly varying phase andamplitude approximation(1st order perturbation theory)

diffraction nonlinearity

0||

Ez

Spatial soliton

1D Scalar Kerr Solitons

]2

exp[}{sec1

)(02

2

02

ka

zi

a

yh

kan

nrEInn

Low Power

High Power

Input

Output

x

y

(2+1)D Kerr solitonsare unstable

tconsaPsoliton tan 1 free parameter

1D Scalar Kerr Solitons

]2

exp[}{sec1

)(02

2

02

ka

zi

a

yh

kan

nrEInn

0

60

40

20

-20

-40

y (m

icro

ns)

Input Power (watts)

500250 750

Output Intensity

tconsaPsoliton tan 1 free parameter

Semiconductor Optical Amplifiers

J

JBottom Electrode

Top Electrode

Output Light

Input Light

Multi-functional Elements for Optics

1. Used as optical amplifiers, with feedback as lasers2. Used as nonlinear optical devices (mW power levels)

- Demultiplexers- All-optical switchers- Wavelength shifters- All-optical logic gates- ….

Freely Propagating Solitons In Gain Systems

• Found also in Erbium-doped fibers, laser cavities

Hamiltonian diffraction+nonlinearity is balanced

Dissipative diffraction+nonlinearity+gain+loss is balanced

•Requires intensity dependent Gain & Loss•Strong attractors

gain gain

loss lossloss

x

z

Self-trapped beams have been observed in SOAs over limited distancesG. Khitrova et al., Phys. Rev. Lett. 70, 920 (1993)

Freely Propagating Solitons In Gain Systems

•Requires intensity dependent Gain & Loss•Strong attractors

Intensity

Saturable gain

Saturable loss

loss

gain

gain gain

loss lossloss

x

z

Diffraction

'))1'('(2

'''

'' NihN

ixxz

Nonlinearindex change

Gain Cladding absorptionand scattering losses

Semiconductor Optical Amplifier Modeling

G. P. Agrawal, J. Appl. Phys. 56, 3100 (1984)Optical field (’) evolution (along z’)

,

kLDD 'BNB tr'

CNC tr2'

2trNaL

)2()1( Lp

)(|| 2 aws trqdNJ kLxx 'zLz '

gain

loss N'1

2/1220 ]/1[ difLzww 2w0

h = Henry factor- change index with N

trNNN /' N – carrier density Ntr – transparency carrier density

Diffusion

Semiconductor Optical Amplifier Modeling

Carrier density equation

232''' |'|'''''' NCNBNND xx

CurrentPumping

Nonradiative RecombinationPhonons Generated

SpontaneousRecomb.

AugerRecomb.

Field absorption

,

trNNN /' N – carrier density Ntr – transparency carrier density

Valence band

Conduction band

Optical Beam

Diffusion

Semiconductor Optical Amplifier Modeling

Carrier density equation

232''' |'|'''''' NCNBNND xx

CurrentPumping

Nonradiative RecombinationPhonons Generated

SpontaneousRecomb.

AugerRecomb.

Field absorption

,

trNNN /' N – carrier density Ntr – transparency carrier density

Complex Ginzburg-Landau Equation

- For small diffusion ( below) and B=C=0, equations simplify to

20

2 2

1 ( 1)( 1)

2 1 | |i i ih ih

z x

- Expanding denominator near the bifurcation point

Complex Ginzburg-Landau Equation

-Solutions in NLO have been investigated systematically by Nail Akhmediev, Soto-Crespo and colleagues since 1995

β, filtering parameter h, linewidth enhancement factor 2bko/aπ, pump parameter α, linear loss coefficient

Potential For Solitary Wave Solution

20

2 2

1 ( 1)( 1)

2 1 | |i i ih ih

z x

|Ψo |

δG

Supercriticalbifurcation

0|| 2

Go

-Defining “small signal” gain relative to transparency point including loss as

)exp(ikzo

1oG

- Nonlinear Dynamics: plane wave field solutions have implications for soliton stability

G

oo ||&,0||

Solutions

Intensity

Saturable gain

Saturable loss

loss

gain

Semiconductor Optical Amplifiers

J

JBottom Electrode

Top Electrode

Output Light

Input Light

The SOA shown above does not support stable plane waves because “noise”

experienceslarger gain

Need to manipulate relative saturable gain and absorption!!

Stabilizing the Background

Contact Pads

solution

SOA SA SOA SASOA SASOA SASOASA

Intensity

Saturable gain

Saturable loss

loss

gain

Saturable gain

Saturable loss

loss

gain

Intensity

Effect of Controlling Saturable Absorption Versus Gain

0

0.05

0.10

Inte

nsity

(m

W/μ

m2 )

Ga

in c

m-1 stable

unstable

(a)

0 5 10 15Intensity MW/cm2

0 0.05 0.1 0.15

Intensity (mW/μm2)

Parameters for Bulk GaAs; D=33 cm2/s, C=10-30 cm6/s, B=1.4x10-10 cm3/sh=3, τ=5x10-9s, a=1.5x10-16cm2, α=5cm-1

Stabilizing Background (Plane Waves)

Unstablebackground

Amplifier saturation

Noncontact regionsaturation

Soliton Bifurcation Diagram

Stable Solitons (finite beams)

gain pumping current

ALL soliton properties(width + peak power)determined by current

ZERO parameter solitons

10-100 mW power levels

Stable Solitons

Subcritical branch

Peak f

ield

lev

el

Small signal gain (cm-1)

current (A)mm

W/

Unstable

Pumping Current (Amps)

Stationary Solutions

)exp()( zix

Stable Solitons (finite beams)

gain pumping current

ALL soliton properties(width + peak power)determined by current

ZERO parameter solitons

10-100 mW power levels

(10

)

Diffraction length

Perpendicular axis (cm)

Inte

nsit

y

Diffraction lengthPerpendicular axis (cm)

Stationary Solutions

)exp()( zix

SOA Sample

SQW InGaAs 950nm grown in Jena

300μm

11μm 9μm

Device fabrication SiN deposition Etching & Au coating

SOA Sample

Cu sheets

TE coolerInsulator

Al mount

Au wires

I Current source

GaAs Contact Layer

InGaAs QW

Al0.2Ga0.8As Waveguide 500nm

Al0.2Ga0.8As Waveguide 500nm

GaAs Buffer

Al0..36Ga0..64As Cladding 1000nm

AlxGa1-xAs x=0.2..0.36 100nm

AlxGa1-xAs x=0.2..0.36 100nm

Single Quantum Well Sample

SQW InGaAs

QW modeling

str

str

NN

NNNNf ln)(

Average system equation

21

221121

)()(),(

ww

wNfwNfNNf

Carrier densities in gain, absorption sections

])1)(,([2 21 ihNNfi

xxz(1)

0)(2

13

12

11 NfCNBNDN xx(2)

0)(2

23

22

22 NfCNBNDN xx(3)

Parameters; D. J. Bossert, Photon. T. Lett. 8, 322 (1996)

SQW Modelling

Propagating Solitons

Current pumping small signal gain Soliton peak intensity and widthZero Optical Parameter System

-

Steady state intensity and phase distribution

Ph

ase

rad

ian

s

Position μmIn

ten

sity

au

20μm

Position μm

Phase

rad

iansIn

tensi

ty

Gaussian beam excitation

(10

)

Diffraction lengthPerpendicular axis (cm)

Inte

nsit

y

Diffraction length Perpendicular axis (cm)

Ti sapphire (CW)910-970nm

CylindricalTelescope

1cmX800μmPatterned SOAat 21.5 oC

100A max, Pulsed Diode Driver

500ns/500Hz

CCDcamera

BS

λ/2

OSA

40x 20x

I

Input Output (@965nm, I=0)

15.2μm FWHM 60.7μm FWHM~4 diffraction lengths

~1 μm

(0mW-200mW)

Sampledefects

Output (@950nm, I=4A)

15.5μm FWHM

Ti sapphire (CW)910-970nm

CylindricalTelescope

1cmX800μmPatterned SOAat 21.5 oC

100A max, Pulsed Diode Driver

500ns/500Hz

CCDcamera

BS

λ/2

OSA

40x 20x

I

Input

15.2μm FWHM ~4 diffraction lengths

~1 μm

(0mW-200mW)

Output Profile vs Intensity ChangePosi

tion μ

mPosi

tion μ

m

Input Power (mW)

BPMSimulations

(10cm)

Experiment

Stable Solitons

Subcritical branch

Pea

k fi

eld

lev

el

Small signal gain (cm-1)

current (A)

X

Current (amps)

Subcritical branch

Unstable

Stable solitons

Output Profile vs Current Change

Current (A)

Posi

tion μ

mPosi

tion μ

m

BPMsimulations

Experiment

Stable Solitons

Subcritical branch

Pea

k f

ield

lev

el

Small signal gain (cm-1)

current (A)

Subcritical branch

Current (amps)

Unstable

Stable solitons

Input beam waist FWHM (μm)

Soliton Properties

Solitons are zero parameter

Position μm

(b)

(c)

(d)Solitons

ExperimentO

utp

ut

FW

HM

(μ

m)

I=4A

SolitonsDiffractiondominated

Too few solitonperiods

Soliton Properties

Position (μm)In

tens

ity (

au)

Solid line g=104cm-1, h=3; dashed dotted line g=60cm-1, h=3;

dashed line g=60cm-1, h=1

946nm, 15.9μm

941nm, 39.3μm

Inte

nsity

W/c

m2

(104 )

Diffraction lengthPerpendicular axis (cm)

Inte

nsity

W/c

m2

(104 )

Diffraction lengthPerpendicular axis (cm)

Input Light

J

JBottom Electrode

Periodic Electrode

Output Light

1. Periodic regions of gain and absorption.

2. Absorption region saturates before gain

3. Stable “autosoliton” with gain=loss

4. For given pumping current J, soliton power & width fixed (zero parameter soliton family)

5. Soliton has a strong phase chirp

6. 10-100 mW power levels

Periodically Patterned Semiconductor Optical Amplifier

Do Multi-Component Dissipative Solitons Exist?

- In Kerr (n=n2I systems “Manakov” solitons exist and are stable!- Simplest case is two orthogonal incoherent polarizations

Spatial width invariant for TE/TM = 0.1 10

- AlGaAs at 1.55 m n2 same for both TE and TM, and n2 = n2 - coherence between TE and TM eliminated by passing through different dispersive optics- Manakov solitons have 1. Spatial width independent of polarization ratio 2. No energy exchange between polarizations

1. 2 Orthogonally polarized Beams

2. Different Wavelengths from 2 Different Lasers Mutually Incoherent Beams

Grating to separate beamsat different wavelengths

Experimental Setup

TS – tunable wavelength and power, titanium sapphire laser operated at =943nmLD – laser diode, very limited temperature tunability, operated at =946nm, 40mW power

λTS=943nm

Conclusions

1. There are no completely stable, multi-component dissipative solitons in this case

2. The two beams form quasi-stable solitary waves over cm distances which depend on input power

3. Even though optical beams are incoherent, they do interact for by competing for excited carriers in order to compensate for loss

4. Although the wavelengths are almost identical, the gain, loss etc. coefficients are slightly different!

5. Similar results found by using the quintic complex Ginzburg-Landau equation

Conclusions

Distance (au)

|ψ1|

|ψ2|

Collisions Between Coherent Solitons

light bent (drawn) into regionof higher refractive index

n1 n2 > n1

Solitons in phase

Solitons out of phaseOther phase angles Energy Exchange

-500 0 500

0

100

200

300

400

500

600

0

50

100

K : = 0

Collisions Between Coherent Solitons

-500 0 500

0

20

40

60

80

100

0

50

100

K, S : = 0

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

40

K, S : = /2

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

K, S : =

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

40

K, S : = 3/2

-500 0 500

0

100

200

300

400

500

600

0

10

20

30

S : = 0

- relative phase between solitons K - Kerr Nonlinearities S - Saturating Nonlinearities

Possibilities• Gates• Beam scanners• Modulation of one outputwith optical input• etc,…

A B C D

Outputchannels

Soliton Interactions

Non-local nonlinearity

100μm

-0.58

-

4.09

- 8.77

Gain

cm

-1

Position μm

Δn

(x1

0-4)

100μm

Ph

ase

rad

ian

s

Position μm

Inte

nsi

ty a

u

20μm

Position μm

Phase

rad

ians

Inte

nsi

ty

A B C D

Outputchannels

Soliton Interactions

Non-local nonlinearity

100μm

-0.58

-

4.09

- 8.77

Gain

cm

-1

Position μm

Δn

(x1

0-4)

100μm

Ph

ase

rad

ian

s

Position μm

Inte

nsi

ty a

u

20μm

Position μm

Phase

rad

ians

Inte

nsi

ty

gain gain

loss lossloss

x

z

Output 2Output 1

Input 2*exp(jΦ(t))

Input 1

22μm

15.3μm

Parallel excitation

Dissipative Local Interactions

Beam scanner

-200 2000

3

1.5

Position μm

Pro

pag

ati

on

len

gth

cm

Position μm

Pro

pagati

on D

ista

nce

0

Input1

output2output1

50μm

51μm

Input2*exp(jΦ(t))

Dissipative Non-Local Interactions I

Input1

output2output1

50μm

51μm

Input2*exp(jΦ(t))

0 π 2π 3π 4π

-100

-50

0

50

100

-100

-50

0

50

100

Po

siti

on

μm

Po

siti

on

μm

Phase difference

0 π 2π 3π 4π

Simulation

Experiment

Dissipative Non-Local Interactions II

Output 1Output 2

Input 1Input 2*exp(jΦ(t))

66μm

70μm

-100 -50 0 50 1000

2

3Phase diff = π

Position μm-100 -50 0 50 1000

2

1

3Phase diff = 0

Position μm

1

Pro

paga

tion

dis

tanc

e (c

m)

0

1

2

prop

agat

ion

leng

th c

m

-2000200

position µm

-20

-10

0

Ga

in

c

m-1

0

-10

-20

Ga

in c

m-1

Position μm

1cm

8mm

6mm

4mm

2mm

Position m

Gai

n cm

-1

Center sees different waveguide

Dissipative Non-Local Interactions II

Output 1Output 2

Input 1Input 2*exp(jΦ(t))

66μm

70μm

0

1

2

prop

agat

ion

leng

th c

m

-2000200

position µm

-20

-10

0

Ga

in

c

m-1

0

-10

-20

Ga

in c

m-1

Position μm-100 -50 0 50 1000

2

3Phase diff = π

Position μm-100 -50 0 50 1000

2

1

3Phase diff = 0

Position μm

1

Pro

pa

ga

tio

n d

ista

nc

e (

cm

)

Center sees different waveguide

π 2π 3π 4π 5π

-100

-50

0

50

100

-100

-50

0

50

100

Po

sit

ion

μmP

os

itio

n μm

Phase difference

π 2π 3π 4π 5π

Simulation

Experiment

0 π 2π 3π

-100

-50

0

50

100

-100

-50

0

50

100

Po

siti

on

μm

Po

siti

on

μm

Phase difference0 π 2π 3π

Input1Input2*exp(jΦ(t))

output2output1

46μm

56μm

Simulation

Experiment

Dissipative Non-Local Interactions III

Modulational Instability

Self-focusing Nonlinearity

Low intensity plane wave diffraction dominates

High intensity plane wave self-focusing dominates

Plane wave noise fluctuation

Noisyplane wave

Low intensity plane wave diffraction dominates beam remains noisy

High intensity plane wave self-focusing dominates periodic noise components amplified

Occurs in (2) and (3) media - should occur in dissipative systems

Modulational Instability in Kerr Slab Waveguides

kEnk

k 2

22

0202

22 2

||22

For (gain coefficient) real

40 60 80 100 120 1402

3

4

5

6

MI

Ga

in,

cm

-1Period, m

- - - - 75 kW 50 kW

60 100 140

----- 75 KW 50 KW

(c

m-1

)Period (m)

6

4

2

Connection to Soliton Power

Same intensity 18]2[22

20

w

),(

),(

)exp()exp()exp()exp(),(

)exp()),((

2202

1101

00

0

zxnNN

zxnNN

xjzbxjzazx

zjzx

Spatial frequency = 2/

Analysis of MI in SOAs

Noise Fluctuations in Optical Fields

Noise Fluctuations in Carrier Density Gain Coefficient

1. Substitute into field and carrier equations

2. Solve for small variables 0->> (x,z) and N(1,2(>>n(1,2).

3. No simple analytical solutions.4. Very messy!

Numerical Solutions

- Actually there are 3 solutions, but only one leads to growth of noise!

p (mm-1)

Gain

(cm

-1)

-

(mm-1)

(c

m-1

)

p (mm-1)

Gain

(cm

-1) -

(mm-1)

(c

m-1)

Physical Solution For MI

Higher Pumping

Gai

n (

cm-1)

Propagation length (cm)

=16.91mm-1

=9.51mm-1

-

π = 50, h = 30

1. Plane wave seeded with weak sine wave modulation 2. Gain is calculated taking the Fourier transform of

simulations after some distance3. Gain calculated

Inte

nsity

(au

)

Position (μm)

Beam Propagation Calculations of MI 1

Beam Propagation Calculations of MI 2

Current change

Note the saturation with increasing pumping!

Wavelength tuning

Onset of Modulational Instabilityx

axis

um

Injected current (A)

Output behavior (168 mW input, λ=950nm) Input beam waist 22.75m

Output beam waist at 965nm is 33.89m

• Output beam at 950 nm breaks into 3 solitons which have identical 17m fitted beam waists

Possible Applications

• Beam stabilization in broad area devices• Beam scanners• Low power (mW), fast soliton (ps) interactions• Fast reconfigurable interconnects• Cascadable all-optical logic gates• Multiple functions on a single chip

controlled by electrode geometry

Issues and Questions• Collisions between incoherent solitons (incoherent solitons

sharing the same gain profile are quasi-stable, OK over 10 cms)• Discreteness – coupled channels – anything new and useful?• Modulational instability analysis implies sub-10m in width

solitons

Discrete Dissipative Solitons: What Are They?

Parallel channel waveguides, weakly coupled by evanescent fieldsDiscrete solitons already found in Kerr, quadratic, photorefractive, liquid crystal media

En(x)

an

Discrete Dissipative Solitons: What Are They?

-20 -10 0 10 200.0

0.4

0.8

1.2

Rel

ativ

e P

ower

in W

aveg

uide

Waveguide

lin diffracted FH Soliton Theory Diffracted Soliton 117W

Fascinating Properties of Propagation in Arrays

1. Linear beams can slide across the array

2. No beam spreading occurs in specific directions

3. Multiple bands occur for propagationafter all, it is a periodic system

4. New varieties of solitons existe.g. solitons guided by boundary between continuous and discrete mediasee poster by Suntsov

5. Large range of angles over which no filamentation occurs at high powers

6. Etc.

Fascinating Properties of Propagation in Arrays

(1

/m)

Band 1:

Band 2:

Band 3:

Band 4:

(units of )Relative phase between channels

kxd

kz

ß

kz

- kxd=

Same equations for carrier density and optical field

Introduce an index modulation n(x) = n0+n(x) and (x) to describe the array

Solve for the Block modes of the structure

w2 w1

Discrete SOA Solitons

Some Numerical Solutions

mo

ψpeak

(b)

(c)

(d)

(e)

0 60 120

16

34 |Ψ|peak

πo Distance (cm)

)/( cmW

60 120πo

1.0026

0.99

1.0

1.0020

0

eff

sol

nk

k

0/

FB1

FB2

(a)

Distance (cms)

(b) Stability diagram of discrete dissipative solitons

(a) Discrete solitons in first Fourier Block band

Propagation of solution on (d) stable branch and (e) unstable branch

f (1/Λ)Position (μm)

|Ψ|2(a) (b)

(c) (d)

(e) (f)

|Ψ|2

|Ψ|2 |ΨF|

|ΨF|

|ΨF|F

ourier Spectra

More Complex Solutions

Inte

nsit

y P

rofi

les

Exciting?

Sample Preparation is Really Tough!Just ask Tony Ho (poster)

Semiconductor Amplifier Modeling Parameters

G. P. Agrawal “ Fast-Fourier-transform based beam propagation model for stripe-geometry semiconductor lasers” J. Appl. Phys. 56, 3100-3108 (1984)