Download Dissipative Propagating Spatial Solitons

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
Spatial Soliton
Propagating Spatial Soliton Milestones
Bjorkholm,
Duree,
Mitchell,
White light
Soliton
(1996)
Kerr Solitons
Photorefractive
in Sat. Media
(1993)
Barthelemy,
(1974)
Torruellas,
1D Kerr Soliton
2D Quad. Soliton
(1985)
(1995)
1960
1970
1980
1990
Christodoulides,
Picciante,
NLC (2000)
Silberberg,
Discrete Array
(1997)
2000
SOA
Discrete (1988) Segev,
Chiao
Christodoulides, (2002)
Zakharov,
Photorefractive
Incoherent (1997)
&
Soliton solutions
Solitons
Talanov
(1971)
(1992)
Prediction
Sukhorukov,
Akhmediev
of Kerr
Prediction of Quadratic
Cubic-quintic CLGE
(1964)
Solitons (1975)
(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
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 Soliton Size
Param.
Power
Quadratic
QPM LiNbO3
2
20 x 5 m
100 W
Photorefractive
SBN
1
15 x 5
10W
Kerr
AlGaAs (Eg/2)
1
20 x 4 m
100’s W
Non-Hamiltonian (Dissipative Systems)
Gain balances loss + nonlinearity balances diffraction
Dissipative (SOAs)
AlGaAs
0
15 m
No trade-offs in optical beam properties!!
10’s mWs
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
"Classical " Kerr : n  n2 I
c
phase velocity
Vp 
n0  n2 I
Vp'
Vp < V p'
Self-focusing
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:
1. Robust balance between diffraction and a nonlinear beam narrowing process
2. Stationary solution to a nonlinear wave equation
3. Stable against perturbations
Observed and Studied Experimentally to Date in:
1. Kerr and saturating Kerr media
4. Liquid crystals
2. Photorefractive media
5. Gain media
3. Quadratically nonlinear media
Semiconductor optical
amplifiers
Diffraction in 1D Homogeneous System
Homogeneous in
Diffracting Dimension

E (r )  E ( y, z ) f ( x) exp[ i{t  z}]
f(x)
y
- Insert into wave equation
- Assume slow change over
an optical wavelength

2
 2i E ( y, z ) 
E ( y, z )  0
2
z
y
1D Nonlinear Wave Equation
2 E 
1 2
c 2 t 2
E  0
2
t
L
NL
{
P

P
}
2
depends on nonlinear
mechanism
0 E
(1)
E  exp[ i{t  kz}]
Slowly varying phase and
amplitude approximation
(1st order perturbation theory)

| E | 0
z
2 E 
n02
c
2
2
NL

E




P
0
2

2
 2ik E 
E   2 0 P NL
z
y 2
diffraction
Spatial soliton
nonlinearity
1D Scalar Kerr Solitons
n  n2 I  0

n0 1
y
z
E (r ) 
sec h{ } exp[ i
]
n2 ka
a
2ka2

Psolitona  cons tan t  1 free parameter
x
Output
y
Low Power
(2+1)D Kerr solitons
are unstable
Input
High Power
1D Scalar Kerr Solitons
n  n2 I  0

n0 1
y
z
E (r ) 
sec h{ } exp[ i
]
n2 ka
a
2ka2

Psolitona  cons tan t  1 free parameter
Output Intensity
y (microns)
60
40
20
0
-20
Input Power (watts)
-40
250
500
750
Semiconductor Optical Amplifiers
J Top Electrode
Input Light
Output Light
J
Bottom Electrode
Multi-functional Elements for Optics
1. Used as optical amplifiers, with feedback as lasers
2. Used as nonlinear optical devices (mW power levels)
- Demultiplexers
- All-optical switchers
- Wavelength shifters
- All-optical logic gates
- ….
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
gain
z
loss
loss
x
loss
•Requires intensity dependent
Gain & Loss
•Strong attractors
gain
Freely Propagating Solitons In Gain Systems
Saturable gain
loss
Intensity
Saturable loss
gain
gain
z
loss
loss
x
loss
•Requires intensity dependent
Gain & Loss
•Strong attractors
Semiconductor Optical Amplifier Modeling
Optical field (’) evolution (along z’)
G. P. Agrawal, J. Appl. Phys. 56, 3100 (1984)
i '
   x ' x '  (ihN '( N '1)   ) '
2
'
z'
Diffraction
Nonlinear
index change
Gain
Cladding absorption
and scattering losses
w  w0 [1  z 2 / L2dif ]1 / 2
N – carrier density Ntr – transparency carrier density
h = Henry factor
- change index with N
,
D'  kLD
B'  N tr B
C '  N tr2 C
L  aN tr
loss gain
N ' N / N tr
2w0
1
N'
  (1  ) p (2 L) z '  zL
x'  x kL
2   J qdN tr
|  s | 2  w (a )
Semiconductor Optical Amplifier Modeling
Carrier density equation
N ' N / N tr
N – carrier density Ntr – transparency carrier density
D' N x' ' x '    N ' B' N ' 2 C ' N '3  |  ' | 2
Diffusion
,
Current
Pumping
Spontaneous Auger
Recomb.
Recomb.
Nonradiative Recombination
Phonons Generated
Field absorption
Optical Beam


Conduction band


Valence band

Semiconductor Optical Amplifier Modeling
Carrier density equation
N ' N / N tr
N – carrier density Ntr – transparency carrier density
D' N x' ' x '    N ' B' N ' 2 C ' N '3  |  ' | 2
Diffusion
Current
Pumping
Spontaneous Auger
Recomb.
Recomb.
Nonradiative Recombination
Phonons Generated
,

Field absorption
Complex Ginzburg-Landau Equation
- For small diffusion ( below) and B=C=0, equations simplify to
2

  1
    ( 0  1)
 i   i  2  
( ih  1)  ih    
2
z
2
 x
1 |  |

- 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
Potential For Solitary Wave Solution
2

  1
    ( 0  1)
 i   i  2  
(

ih

1)

ih
   
2
z
2
 x
1 |  |

β, filtering parameter h, linewidth enhancement factor 2bko/a
π, pump parameter
α, linear loss coefficient
- Nonlinear Dynamics: plane wave field solutions have implications for soliton stability
   o exp(ikz)
-Defining “small signal” gain relative
to transparency point including loss  as
G   o  1  
|Ψo |
Supercritical
bifurcation
Solutions
G
|  o | 0, & |  o |

 |  o |2
0
G
δG
Semiconductor Optical Amplifiers
J Top Electrode
Input Light
Output Light
J
gain
Bottom Electrode
Saturable gain
loss
Intensity
Saturable loss
The SOA shown above does not support
stable plane waves because “noise” experiences
larger gain
Need to manipulate relative saturable gain and absorption!!
Stabilizing the Background
gain
Contact Pads
Saturable gain
loss
Intensity
Saturable loss
SOA
SA
solution
SOASA
SOA SA SOA SA SOA SA
gain
Effect of Controlling Saturable Absorption Versus Gain
Saturable gain
loss
Intensity
Saturable loss
Stabilizing Background (Plane Waves)
Gain cm-1
stable
Noncontact region
saturation
unstable
0
5
0.05
10
0.1
Intensity (mW/μm2)
Amplifier
saturation
(a)
0.10
0.05
Unstable
background
0
15
0.15
Intensity MW/cm2
Intensity (mW/μm2)
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
Stable Solitons (finite beams)
   ( x) exp( iz )
gain  pumping current
ALL soliton properties
(width + peak power)
determined by current
Peak field level
Stationary Solutions
mW / m
Soliton Bifurcation Diagram
current
(A) (Amps)
Pumping
Current
Stable Solitons
Unstable
Subcritical branch
ZERO parameter solitons
10-100 mW power levels
Small signal gain (cm-1)
Stable Solitons (finite beams)
gain  pumping current
ALL soliton properties
(width + peak power)
determined by current
ZERO parameter solitons
10-100 mW power levels
Intensity
   ( x) exp( iz )
(10 )
Stationary Solutions
SOA Sample
SQW InGaAs 950nm grown in Jena
Device fabrication
Etching & Au coating
SiN deposition
300μm
11μm
9μm
SOA Sample
Au wires
Cu sheets
TE
cooler
Insulator
Al mount
I Current source
Single Quantum Well Sample
SQW InGaAs
GaAs
AlxGa1-xAs
Contact Layer
x=0.2..0.36
Al0..36Ga0..64As Cladding
Al0.2Ga0.8As
Waveguide
InGaAs
QW
Al0.2Ga0.8As
Waveguide
AlxGa1-xAs
x=0.2..0.36
GaAs
Buffer
100nm
1000nm
500nm
500nm
100nm
SQW Modelling
QW modeling
 N N tr  N s
f ( N )  ln 
 N tr  N s



Average system equation
f ( N1 ) w1  f ( N 2 ) w2
f ( N1 , N 2 ) 
w1  w2
Carrier densities in gain, absorption sections
(1)
(2)
(3)
i
 z   xx  [ f ( N1, N 2 )(1  ih )   ]
2
2
2
3
DN1xx    BN 1  CN1  f ( N1 )  0
DN 2 xx  BN 2  CN 2  f ( N 2 )  0
2
3
Parameters; D. J. Bossert, Photon. T. Lett. 8, 322 (1996)
2
Propagating Solitons
Intensity au
Intensity (10 )
Intensity
μm
Positionμm
Position
Current pumping  small signal gain  Soliton peak intensity and width
Zero Optical Parameter System
Phase radians
20μm
Phase radians
Steady state intensity
and phase distribution
Gaussian beam excitation
100A max,
Pulsed Diode Driver
500ns/500Hz
(0mW-200mW)
Ti sapphire (CW)
910-970nm
I
CCD
camera
BS
λ/2
Cylindrical
Telescope
40x
OSA
Input
20x
1cmX800μm
Patterned SOA
at 21.5 oC
Output (@965nm, I=0)
~1 μm
15.2μm FWHM
~4 diffraction
lengths
60.7μm FWHM
Sample
defects
100A max,
Pulsed Diode Driver
500ns/500Hz
(0mW-200mW)
Ti sapphire (CW)
910-970nm
I
CCD
camera
BS
λ/2
Cylindrical
Telescope
40x
OSA
20x
1cmX800μm
Patterned SOA
at 21.5 oC
Output (@950nm, I=4A)
Input
~1 μm
15.2μm FWHM
~4 diffraction
lengths
15.5μm FWHM
Experiment
current (A)
Current (amps)
BPM
Simulations
(10cm)
Peak field level
Position μm
Position μm
Output Profile vs Intensity Change
Stable Solitons
Stable solitons
Unstable
X
Subcritical branch
Subcritical
branch
Small signal gain (cm-1)
Input Power (mW)
Output Profile vs Current Change
Position μm
Experiment
Position μm
BPM
simulations
Peak field level
current (A)
Current (amps)
Stable Solitons
Stable
solitons
Unstable
Subcritical
branch
Subcritical
branch
Small signal gain (cm-1)
Current (A)
Soliton Properties
Output FWHM ( μm)
I=4A
(b)
Experiment
Too few soliton
Diffraction
(c)
dominated Solitons
(d)
periods
Solitons
Position μm
Input beam waist FWHM (μm)
Solitons are zero parameter
946nm, 15.9μm
941nm, 39.3μm
Intensity (au)
Soliton Properties
Position (μm)
Solid line g=104cm-1, h=3;
dashed dotted line g=60cm-1, h=3;
dashed line g=60cm-1, h=1
Periodically Patterned Semiconductor Optical Amplifier
J Periodic Electrode
Input Light
Output Light
J
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
Intensity W/cm2 (104)
Bottom Electrode
Dif
frac
t
io n
len
gth
Perp
is
lar ax
u
c
i
d
en
(cm)
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
Experimental Setup
1. 2 Orthogonally polarized Beams
2. 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
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 GinzburgLandau equation
Conclusions
|ψ1|
|ψ2|
Distance (au)
Collisions Between Coherent Solitons
n1
n2 > n1
light bent (drawn) into region
of higher refractive index
Solitons in phase
Solitons out of phase
Other phase angles  Energy Exchange
Collisions Between Coherent Solitons
 - relative phase between solitons
K - Kerr Nonlinearities S - Saturating Nonlinearities
K :  = 0
S :  = 0
600
600
500
500
400
400
30
100
300
300
20
200
200
50
10
100
100
0
0
-500
0
0
0
-500
0
500
500
K, S :  = 
K, S :  = 0
600
100
500
80
400
100
60
30
40
20
20
10
300
200
50
0
-500
0
-500
0
0
500
K, S :  = 3/2
100
0
0
500
K, S :  = /2
600
600
500
500
400
400
40
40
300
30
200
20
100
10
0
-500
0
0
500
300
30
200
20
100
10
0
-500
0
0
500
Non-local nonlinearity
Phase radians
Intensity au
Phase radians
A
Output
channels
B C
D
Intensity
Soliton Interactions
20μm
Position μm
Position
μm
100μm
100μm
4.09
-
-
Position μm
8.77
Δn (x10-4)
Possibilities
• Gates
• Beam scanners
• Modulation of one output
with optical input
• etc,…
Gain cm-1
-0.58
Non-local nonlinearity
Soliton Interactions
Phase radians
Intensity
Intensity au
Phase radians
A
Output
channels
B C
D
20μm
Position μm
Position
μm
loss
loss
100μm
100μm
4.09
-
loss
-
x
Position μm
8.77
Δn (x10-4)
gain
gain
z
Gain cm-1
-0.58
Dissipative Local Interactions
Parallel excitation
Beam scanner
Output 1 Output 2
22μm
15.3μm
Input 1
Input 2
*exp(jΦ(t))
Propagation length cm
Propagation Distance
3
1.5
0
-200
0
Position
μm
Position μm
200
Dissipative Non-Local Interactions I
Simulation
Input2
Input1
*exp(jΦ(t))
Input2
Input1
51μm
*exp(jΦ(t))
51μm
Position μm
-100
-50
0
50
100
0
π
2π
3π
0
π
2π
3π
output1
output1
output2
output2
50μm
50μm
Position μm
-100
4π
Experiment
-50
0
50
100
Phase difference
4π
Dissipative Non-Local Interactions II
Output 2
Output 1
66μm
3
2
Phase diff = π
2
1
1
0
0
-100 -50 0
50 100 -100 -50 0
Position μm
00
-10
-10
-20
-20
propagation length cm
3
Phase diff = 0
-1-1
Gain
Gaincm
cm
Input 1
Input 2
*exp(jΦ(t))
Propagation distance (cm)
70μm
Gain cm-1
Center sees different waveguide
50 100
Position μm
1cm
8mm
2
1
0
6mm
4mm
2mm
200
0
-200
position m
µm
Position
Position μm
Dissipative Non-Local Interactions II
Output 2
Output 1
Simulation
66μm
Position μm
-100
70μm
Input 1
Input 2
*exp(jΦ(t))
Center sees different waveguide
-50
0
50
100
π
2π
3π
4π
Experiment
5π
π
2π
3π
5π
-100
2
1
1
0
0
-100 -50
0
50 100
-100 -50
Positionμm
0
-20
-20
50 100
Positionμm
Position μm
Gain cm-1
Gain cm-1
2
-10
-10
propagation length cm
Propagation distance (cm)
Phase
diff = 0 Phase diff = π00
3
3
2
1
0
-50
0
50
100
200
0
-200
position µm
Position μm
4π
Phase difference
Dissipative Non-Local Interactions III
Input2
*exp(jΦ(t))
Input1
56μm
Position μm
-100
-50
0
50
Simulation
100
0
output2
46μm
2π
3π
-100
Position μm
output1
π
-50
0
50
Experiment
100
0
π
2π
Phase difference
3π
Modulational Instability
Self-focusing Nonlinearity
Low intensity plane wave
 diffraction dominates
Plane wave
noise fluctuation
High intensity plane wave
 self-focusing dominates
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
Modulational Instability in Kerr Slab Waveguides
2
6
2
----- 75 KW
50 KW
2k0 n2 | E0 |2 


2
2
 k 
 k 
 (cm-1)
2 
2 2 
For  (gain coefficient) real
4
Connection to Soliton Power
2
Same intensity
60
100
140
Period  (m)
[2w0 ]2

2

8

2
1
Analysis of MI in SOAs
Noise Fluctuations in Optical Fields
Spatial frequency = 2/
  ( 0   ( x, z )) exp( jz )
 ( x, z )  a0 exp( z ) exp( jx)  b0 exp( z ) exp(  jx)
Noise Fluctuations in Carrier Density
Gain Coefficient
N1  N10  n1 ( x, z )
N 2  N 20  n2 ( x, z )
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
-1) -1

(cm
Gain (cm )
- Actually there are 3 solutions, but only one leads to growth of noise!
-
-1
p (mm
(mm-1) )
Physical Solution For MI
ain (cm-1)
 (cm-1)
Higher Pumping
 (mm-1)
Beam Propagation Calculations of MI 1
1. Plane wave seeded with weak sine wave modulation
2. Gain is calculated taking the Fourier transform of
simulations after some distance
3. Gain calculated
π = 50, h = 30
Gain (cm-1)
Intensity (au)
=16.91mm-1
=9.51mm-1
-
Position (μm)
Propagation length (cm)
Beam Propagation Calculations of MI 2
Current change
Note the saturation with increasing pumping!
Wavelength tuning
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
x axis um
• Output beam at 950 nm breaks into
3 solitons which have identical
17m fitted beam waists
Injected current (A)
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
Discrete Dissipative Solitons: What Are They?
an
En(x)
Parallel channel waveguides, weakly coupled by evanescent fields
Discrete solitons already found in Kerr, quadratic, photorefractive, liquid crystal media
Discrete Dissipative Solitons: What Are They?
Relative Power in Waveguide
1.2
lin diffracted
FH Soliton
Theory Diffracted
Soliton 117W
0.8
0.4
0.0
-20
-10
0
Waveguide
10
20
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 propagation
after all, it is a periodic system
4. New varieties of solitons exist
e.g. solitons guided by boundary between continuous and discrete media
see poster by Suntsov
5. Large range of angles over which no filamentation occurs at high powers
6. Etc.
Fascinating Properties of Propagation in Arrays
11µm
1.5µm
41 guides
11µm
4.0µm
1.5µm
4.0µm
Al0.24Ga0.76As
Al0.18Ga0.82As
Al0.24Ga0.76As
Al0.18Ga0.82As
Al0.24Ga0.76As
Al0.24Ga0.76As
41 guides
Band 1:
 (1/m)
Band 2:
kkzz
Band 3:
ß
Band 4:
 (units of )
Relative phase between channels


-
 k d=
x kxd
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
Some Numerical Solutions
1.0026
34
(a)
1.0020
|Ψ|peak ( W / cm )
FB1 

1.0
(d)
16
k sol
(c)
ψpeak
/ k0 neff
FB2
0.99
0
60
πo 120
(b)
0
Distance (cms)
(e)
120
π
mo o
60
Distance (cm)
(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
More Complex Solutions
(a)
(c)
|Ψ|2
(e)
|Ψ|2
Position (μm)
(b)
|ΨF|
(d)
|ΨF|
(f)
|ΨF|
f (1/Λ)
Fourier Spectra
Intensity Profiles
|Ψ|2
Exciting?
Sample Preparation is Really Tough!
Just ask Tony Ho (poster)
Semiconductor Amplifier Modeling Parameters
Carrier diffusion coefficient
Nonradiative recombination
time
Confinement factor
Gain parameter
Index parameter
Antiguiding parameter
(linewidth enhancement factor)
Spontaneous recombination
coefficient
Auger coefficient
Internal loss
Wavelength
D
1/Anr
33cm2/s
5ns

a
b
h
0.5
1.5x10-16cm2
150cm-1
3
B
1.4x10-10cm3/s
C
αint

1.0x10-30cm6/s
2.5 cm-1
0.82um
G. P. Agrawal “ Fast-Fourier-transform based beam propagation model for stripe-geometry
semiconductor lasers” J. Appl. Phys. 56, 3100-3108 (1984)