1. Dissipative Spatial Solitons and Their Applications
in Active Semiconductor Optical Amplifiers
Erdem Ultanir, Demetri Christodoulides & George I. Stegeman
School of Optics/CREOL, University of Central Florida
Falk Lederer and Christopher Lange
Frederich Schiller University Jena
Spatial Solitons 1D
Diffracted
Beam
waveguide
Spatial Soliton

2. Prediction of Quadratic
Mitchell,
White light
Soliton
(1996)
Propagating Spatial Soliton Milestones
Picciante,
NLC (2000)
Bjorkholm,
Kerr Solitons
in Sat. Media
(1974)
Duree,
Photorefractive
(1993)
Silberberg,
Discrete Array
(1997)
Barthelemy,
1D Kerr Soliton
(1985)
Torruellas,
2D Quad. Soliton
(1995)
1960
1970
1980
1990
2000
Christodoulides,
Discrete (1988)
SOA
(2002)
Segev,
Photorefractive
Solitons
(1992)
Chiao
&
Talanov
Prediction
of Kerr
(1964)
Zakharov,
Soliton solutions
(1971)
Christodoulides,
Incoherent (1997)
Sukhorukov,
Prediction of Quadratic
Solitons (1975)
Akhmediev
Cubic-quintic CLGE
(1995)
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

3. 1D Spatial Solitons in Homogeneous Media
A spatial soliton is a shape invariant self guided beam of light
or a self-induced waveguide
Hamiltonian Systems
Nonlinearity balances diffraction
Soliton Type
Material
# Soliton
Param.
Soliton Size
Power
Quadratic
QPM LiNbO3 2
20 x 5 m
100 W
Photorefractive
SBN 1
15 x 5
10W
Kerr
AlGaAs (Eg/2)
20 x 4 m
100’s W
Non-Hamiltonian (Dissipative Systems)
Gain balances loss + nonlinearity balances diffraction
Dissipative (SOAs)
AlGaAs
15 m
10’s mWs
No trade-offs in optical beam properties!!

4. Spatial Solitons in Homogeneous Media
Spatial Solitons (1+1)D
Diffracting beam
Spatial Solitons (2+1)D
Planar (slab) waveguide
Bulk medium
(1+1)D - in a slab waveguide
- diffraction in one D
(2+1)D - in a bulk material
- diffraction in 2D
A spatial soliton is a shape invariant self guided beam of light
or a self-induced waveguide
phase velocity
Vp'
Self-focusing
Vp < Vp'
n2 > 0

5. Spatial Solitons in Homogeneous Media
Spatial Solitons (1+1)D
Diffracting beam
Spatial Solitons (2+1)D
Planar (slab) waveguide
Bulk medium
(1+1)D - in a slab waveguide
- diffraction in one D
(2+1)D - in a bulk material
- diffraction in 2D
Soliton Properties:
Robust balance between diffraction and a nonlinear beam narrowing process
Stationary solution to a nonlinear wave equation
Stable against perturbations
Observed and Studied Experimentally to Date in:
Kerr and saturating Kerr media Liquid crystals
Photorefractive media Gain media
Quadratically nonlinear media Semiconductor optical amplifiers
n2 > 0

6. Diffracting Dimension
Diffraction in 1D Homogeneous System
Homogeneous in
Diffracting Dimension
f(x) y
Insert into wave equation
Assume slow change over
an optical wavelength

7. 1D Nonlinear Wave Equation
depends on nonlinear
mechanism
Slowly varying phase and
amplitude approximation
(1st order perturbation theory)
nonlinearity
diffraction
Spatial soliton

8.  1 free parameter 1D Scalar Kerr Solitons x y (2+1)D Kerr solitons
are unstable

9.  1 free parameter 1D Scalar Kerr Solitons Output Intensity60 40 20
-20
-40
y (microns)
Input Power (watts)
500
250
750
Output Intensity

10. J J Semiconductor Optical Amplifiers Top Electrode Bottom Electrode
Input Light
Output Light J
Bottom Electrode
Multi-functional Elements for Optics
1. Used as optical amplifiers, with feedback as lasers
Used as nonlinear optical devices (mW power levels)
Demultiplexers
All-optical switchers
Wavelength shifters
All-optical logic gates ….

11. Freely Propagating Solitons In Gain Systems
Self-trapped beams have been observed in SOAs over limited distances
G. Khitrova et al., Phys. Rev. Lett. 70, 920 (1993)
Hamiltonian diffraction+nonlinearity is balanced
Dissipative diffraction+nonlinearity+gain+loss is balanced
Found also in Erbium-doped fibers, laser cavities
gain
loss x z
Requires intensity dependent
Gain & Loss
Strong attractors

12. Freely Propagating Solitons In Gain Systems
Intensity
Saturable gain
Saturable loss
loss
gain
gain
loss x z
Requires intensity dependent
Gain & Loss
Strong attractors

13. N – carrier density Ntr – transparency carrier density
Semiconductor Optical Amplifier Modeling
Optical field (’) evolution (along z’)
G. P. Agrawal, J. Appl. Phys. 56, 3100 (1984)
Nonlinear
index change
Cladding absorption
and scattering losses
Diffraction
Gain
2w0
N – carrier density Ntr – transparency carrier density
gain
loss N' 1
h = Henry factor
- change index with N ,

14. N – carrier density Ntr – transparency carrier density
Semiconductor Optical Amplifier Modeling
Carrier density equation
N – carrier density Ntr – transparency carrier density
Current
Pumping
Diffusion
Auger
Recomb.
Spontaneous
Recomb.
Field absorption
Nonradiative Recombination
Phonons Generated
Optical Beam ,  
Valence band
Conduction band  

15. N – carrier density Ntr – transparency carrier density
Semiconductor Optical Amplifier Modeling
Carrier density equation
N – carrier density Ntr – transparency carrier density
Current
Pumping
Diffusion
Auger
Recomb.
Spontaneous
Recomb.
Field absorption
Nonradiative Recombination
Phonons Generated , 

16. Complex Ginzburg-Landau Equation
- For small diffusion ( below) and B=C=0, equations simplify to
- 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

17. Potential For Solitary Wave Solution
β, filtering parameter h, linewidth enhancement factor 2bko/a
π, pump parameter α, linear loss coefficient
Defining “small signal” gain relative
to transparency point including loss  as
- Nonlinear Dynamics: plane wave field solutions have implications for soliton stability
|Ψo | δG
Supercritical
bifurcation
Solutions

18. The SOA shown above does not support
Semiconductor Optical Amplifiers J
Bottom Electrode
Top Electrode
Output Light
Input Light
Intensity
Saturable gain
Saturable loss
loss
gain
The SOA shown above does not support
stable plane waves because “noise” experiences
larger gain
Need to manipulate relative saturable gain and absorption!!

19. Stabilizing the Background
Contact Pads
solution
Intensity
Saturable gain
Saturable loss
loss
gain
SOA SA
SOA SA
SOA SA
SOA SA
SOA SA

20. Effect of Controlling Saturable Absorption Versus Gain
Saturable gain
Intensity
Saturable loss
loss

21. Stabilizing Background (Plane Waves)
Amplifier
saturation
Unstable
background
Noncontact region
saturation
Parameters for Bulk GaAs; D=33 cm2/s, C=10-30 cm6/s, B=1.4x10-10 cm3/s
h=3, τ=5x10-9s, a=1.5x10-16cm2, α=5cm-1

22. Stable Solitons (finite beams)
Soliton Bifurcation Diagram
Unstable
Pumping Current (Amps)
Stationary Solutions
gain  pumping current
ALL soliton properties
(width + peak power)
determined by current
ZERO parameter solitons
mW power levels

23. Stable Solitons (finite beams)
(10 )
Diffraction length
Perpendicular axis (cm)
Intensity
Stationary Solutions
gain  pumping current
ALL soliton properties
(width + peak power)
determined by current
ZERO parameter solitons
mW power levels

24. SOA Sample SQW InGaAs 950nm grown in Jena SiN deposition
Etching & Au coating
Device fabrication
300μm
11μm
9μm

25. SOA Sample

26. Single Quantum Well Sample
SQW InGaAs

27. SQW Modelling QW modeling Average system equation
Carrier densities in gain, absorption sections
(1)
(2)
(3)
Parameters; D. J. Bossert, Photon. T. Lett. 8, 322 (1996)

28. Zero Optical Parameter System
Propagating Solitons
Steady state intensity
and phase distribution
20μm
Position μm
Phase radians
Intensity
Gaussian beam excitation
(10 )
Diffraction length
Perpendicular axis (cm)
Intensity
Current pumping  small signal gain  Soliton peak intensity and width
Zero Optical Parameter System -

29. 100A max, Pulsed Diode Driver 500ns/500Hz (0mW-200mW) I
Ti sapphire (CW) nm
CCD
camera BS
40x
20x
λ/2
Cylindrical
Telescope
1cmX800μm
Patterned SOA
at 21.5 oC
OSA
Input
Output I=0)
~1 μm
Sample
defects
15.2μm FWHM
~4 diffraction
lengths
60.7μm FWHM

30. 100A max,
Pulsed Diode Driver
500ns/500Hz
(0mW-200mW) I
Ti sapphire (CW) nm
CCD
camera BS
40x
20x
λ/2
Cylindrical
Telescope
1cmX800μm
Patterned SOA
at 21.5 oC
OSA
Input
Output I=4A)
~1 μm
15.2μm FWHM
~4 diffraction
lengths
15.5μm FWHM

31. Output Profile vs Intensity Change
Experiment
Current (amps)
Position μm
Unstable
Stable solitons
BPM
Simulations
(10cm) X
Position μm
Subcritical branch
Input Power (mW)

32. Output Profile vs Current Change
Experiment
Current (amps)
Position μm
Unstable
Stable solitons
BPM
simulations
Position μm
Subcritical branch
Current (A)

33. Solitons are zero parameter
Soliton Properties
I=4A
(b)
Experiment
Too few soliton
periods
Output FWHM ( μ m)
Diffraction
dominated
(c)
Solitons
(d)
Solitons
Position μm
Input beam waist FWHM ( μ m)
Solitons are zero parameter

34. dashed dotted line g=60cm-1, h=3;
Soliton Properties
946nm, 15.9μm
Pictures from 11/05/02
941nm, 39.3μm
Solid line g=104cm-1, h=3;
dashed dotted line g=60cm-1, h=3;
dashed line g=60cm-1, h=1

35. J Periodically Patterned Semiconductor Optical Amplifier
Input Light J
Bottom Electrode
Periodic Electrode
Output Light
Periodic regions of gain and absorption.
2. Absorption region saturates before gain
Stable “autosoliton” with gain=loss
For given pumping current J, soliton power
& width fixed (zero parameter soliton family)
Soliton has a strong phase chirp
mW power levels

36. Do Multi-Component Dissipative Solitons Exist?
In Kerr (n=n2I systems “Manakov” solitons exist and are stable!
Simplest case is two orthogonal incoherent polarizations
- 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 
Spatial width invariant for TE/TM = 0.1  10

37. Experimental Setup 2 Orthogonally polarized Beams
Different Wavelengths from 2 Different Lasers
 Mutually Incoherent Beams
Grating to separate beams
at different wavelengths
TS – tunable wavelength and power, titanium
sapphire laser operated at =943nm
LD – laser diode, very limited temperature
tunability, operated at =946nm, 40mW power

38. λTS=943nm

40. Conclusions
There are no completely stable, multi-component dissipative solitons in this case
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
Although the wavelengths are almost identical, the gain, loss etc. coefficients are slightly different!
Similar results found by using the quintic complex Ginzburg-Landau equation

41. Conclusions

42. Collisions Between Coherent Solitons
n2 > n1
light bent (drawn) into region
of higher refractive index
Solitons in phase
Solitons out of phase
Other phase angles  Energy Exchange

43. Collisions Between Coherent Solitons
 - relative phase between solitons K - Kerr Nonlinearities S - Saturating Nonlinearities
K :  = 0
S :  = 0
K, S :  = 0
K, S :  = 
K, S :  = /2
K, S :  = 3/2

44. Soliton Interactions Δn (x10-4) Non-local nonlinearity Output channels
20μm
Position μm
Phase radians
Intensity A B C D
Output
channels
100μm
-0.58 -
4.09
8.77
Gain cm-1
Position μm
Δn (x10-4)
Possibilities
Gates
Beam scanners
Modulation of one output
with optical input
etc,…

45. Soliton Interactions Δn (x10-4) Non-local nonlinearity Output channels
20μm
Position μm
Phase radians
Intensity A B C D
Output
channels
100μm
-0.58 -
4.09
8.77
Gain cm-1
Position μm
Δn (x10-4)
gain
loss x z

46. Interaction Apparatus
100A max,
Pulsed Diode Driver
500ns/500Hz
2 beams of 50mW power
 20m wide at input
Cylindrical
Telescope I M1
CCD
camera
BS2
PBS
40x
20x
λ/2
1cmX800μm
Patterned SOA
at 21.5 oC Ti
sapph
BS1 M2
Mounted on PZ
for phase scan
OSA
Note: Piezoelectric was moved in steps, with
a pause at each step to gather data

47. Dissipative Local Interactions
Parallel excitation
Beam scanner
Output 2
Output 1
Input 2
*exp(jΦ(t))
Input 1
22μm
15.3μm 3
1.5
Propagation Distance
Propagation length cm
-200
200
Position μm
Position μm

48. Dissipative Non-Local Interactions I
Input2
*exp(jΦ(t))
Input1
51μm
output1
output2
50μm

49. Dissipative Non-Local Interactions II
Output 2
Output 1
66μm
Center sees different waveguide
70μm
1cm
8mm
6mm
4mm
2mm
Position m
Gain cm-1
Input 2
*exp(jΦ(t))
Input 1

50. Dissipative Non-Local Interactions II
Output 2
Output 1
Simulation
Experiment
66μm
Center sees different waveguide
70μm
Input 2
*exp(jΦ(t))
Input 1 -
100 2 3
Phase diff = π
Position μ m 1
Phase diff = 0
Propagation distance (cm)

51. Dissipative Non-Local Interactions III
Input1
Input2
*exp(jΦ(t))
output2
output1
46μm
56μm
Simulation
Experiment

52. Modulational Instability
Self-focusing Nonlinearity
Low intensity plane wave
 diffraction dominates
High intensity plane wave
 self-focusing dominates
Plane wave
noise fluctuation
Noisy
plane 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

53. Modulational Instability in Kerr Slab Waveguides60
100
140 KW
50 KW
(cm-1)
Period  (m) 6 4 2
For  (gain coefficient) real
Connection to Soliton Power
Same intensity

54. Noise Fluctuations in Optical Fields
Analysis of MI in SOAs
Noise Fluctuations in Optical Fields
Spatial frequency = 2/
Noise Fluctuations in Carrier Density
Gain Coefficient
Substitute into field and carrier equations
Solve for small variables 0->> (x,z) and N(1,2(>>n(1,2).
No simple analytical solutions.
Very messy!

55. - Numerical Solutions  (cm-1)  (mm-1)
- Actually there are 3 solutions, but only one leads to growth of noise! -
 (mm-1)
 (cm-1)

56. Physical Solution For MI
Higher Pumping -
 (mm-1)
 (cm-1)

57. Beam Propagation Calculations of MI 1
1. Plane wave seeded with weak sine wave modulation
Gain is calculated taking the Fourier transform of
simulations after some distance
3. Gain calculated
π = 50, h = 30
Gain (cm-1)
Propagation length (cm)
=16.91mm-1
=9.51mm-1 -

58. Beam Propagation Calculations of MI 2
Wavelength tuning
Current change
Note the saturation with increasing pumping!

59. Onset of Modulational Instability
Output behavior
(168 mW input, λ=950nm)
Input beam waist 22.75m
Output beam waist at 965nm is 33.89m
Output beam at 950 nm breaks into
3 solitons which have identical
17m fitted beam waists
x axis um
Injected current (A)

60. 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-10m in width
solitons

61. Discrete Dissipative Solitons: What Are They?
En(x) an
Parallel channel waveguides, weakly coupled by evanescent fields
Discrete solitons already found in Kerr, quadratic, photorefractive, liquid crystal media

62. Discrete Dissipative Solitons: What Are They?

63. Fascinating Properties of Propagation in Arrays
Linear beams can slide across the array
No beam spreading occurs in specific directions
Multiple bands occur for propagation
after all, it is a periodic system
New varieties of solitons exist
e.g. solitons guided by boundary between continuous and discrete media
see poster by Suntsov
Large range of angles over which no filamentation occurs at high powers
Etc.

64. Relative phase between channels
Fascinating Properties of Propagation in Arrays
 (1/m)
Band 1:
Band 2:
Band 3:
Band 4:
 (units of )
Relative phase between channels kz - 
kxd=

65. Discrete SOA Solitons w2 w1
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

66. Some Numerical Solutions
Distance (cms)
(a) Discrete solitons in first Fourier Block band
(b) Stability diagram of discrete dissipative solitons
Propagation of solution on (d) stable branch and (e) unstable branch

67. More Complex Solutions
Fourier Spectra
Intensity Profiles

68. Sample Preparation is Really Tough! Just ask Tony Ho (poster)
Exciting?
Sample Preparation is Really Tough!
Just ask Tony Ho (poster)

69. Semiconductor Amplifier Modeling Parameters
G. P. Agrawal “ Fast-Fourier-transform based beam propagation model for stripe-geometry
semiconductor lasers” J. Appl. Phys. 56, (1984)