erdem ultanir, demetri christodoulides & george i. stegeman school of optics/creol, university...
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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)