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Semiconductor Optical Amplifier Semiconductor Optical Amplifier Parameter Extraction using a Wideband Parameter Extraction using a Wideband SteadySteady--State Numerical Model and the State Numerical Model and the
LevenbergLevenberg--Marquardt MethodMarquardt MethodMichael J. Connelly
Optical Communications Research GroupDepartment of Electronic and Computer Engineering
University of Limerick, LimerickIreland
Supported by Science Foundation Ireland
EU 5th Framework IST project
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OutlineOutline1. Introduction
2. Bulk InGaAsP/InP SOA
3. Steady-state Numerical Model
4. Parameter Extraction
5. Simulations and Experiment
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IntroductionIntroductionSOA technology shows great promise for use as basic amplifiers and as functional devices/subsystems in optical communication networks.
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Analytical or numerical models are required to aid device design and predict operational characteristics.
SOAs can be used to amplify signals at different wavelengths – need a wideband SOA model.
SOA models require accurate values for parameters such as material loss and recombination coefficients –need parameter extraction using model predictions and experimental results.
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Bulk InGaAsP/InP SOA Bulk InGaAsP/InP SOA
C. Deguet et al., “Homogeneous buried ridge stripe semiconductor optical amplifier with near polarization independence,” in Proc. Eur. Conf. Optical Communications, 1999. – Corning.
3 dBCoupling losses
5 x 10-5R
0.45Γ
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SOA steadySOA steady--state numerical modelstate numerical modelThe model is based on a set of coupled differential equations that describe the interaction between the internal variables of the amplifier:
mmm ggg ′′−′=
( ) [ ])(1)(224
23
221
23
2
νννντπ vc
g
hhe
hhem ff
hE
mmmm
ncg −−⎥
⎦
⎤⎢⎣
⎡+
=′h
( ) [ ])(1)(224
23
221
23
2
νννντπ cv
g
hhe
hhem ff
hE
mmmm
ncg −−⎥
⎦
⎤⎢⎣
⎡+
=′′h
•carrier density n•signal and ASE photon rates.
Use a wideband model for the material gain
M.J. Connelly, “Wideband Semiconductor Optical Amplifier Steady-State Numerical Model” IEEE J. Quantum Electron.,2001.
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0
2
4
6
8
10
1400 1450 1500 1550 1600
g′m
gm
Wavelength (nm)
g′m
, gm
(104 m
-1)
Typical InGaAsP bulk semiconductor and spectra
mg′ mg
Additive spontaneous emission spectrum
Materialgain
Ignoring band-tail
Cut-offwavelength
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TravellingTravelling--wave equationswave equations
[ ] ±±
⎟⎠⎞
⎜⎝⎛ −Γ+−±= sigssigm
sig EngjdzdE
ανβ ),(21 Signal
ASE is described in terms of photon rates. Nj
+ and Nj- are defined as the travelling-wave ASE photon
rates (TE or TM) in a frequency spacing ∆νM about frequency νj, corresponding to a cavity resonance.
[ ] ),(),( nRNngzd
dNjspjsjm
j ναν ±−Γ±= ±±
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Rsp(νj,n) represents the spontaneous noise coupled into Nj
+ or Nj− per unit length.
Mjmjsp ngnR ννν ∆′Γ= ),(),(
Mν∆ is an integer multiple of the longitudinal mode spacing.
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The amplifier is split into a number of sections. The signal fields and spontaneous emission photon rates are estimated at the section interfaces. The carrier density is estimated at the centre of each section.
i-th longitudinal section
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Carrier density rate equationCarrier density rate equationThe carrier density n obeys the rate equation
[ ])(
)()(),(2
)()(),()()( 22
zQ
zNzNzg
zEzEzgA
nReVI
dtzdn
jjjjm
sigsigsigm
=⎭⎬⎫
++
⎩⎨⎧
⎥⎦⎤
⎢⎣⎡ +Γ−−=
∑ −+
−+
ν
ν
20)( nBnAnR nrad +=
nB0=τRadiative carrier recombination lifetime
Carrier recombination0Ks =αLoss coefficient
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The algorithm updates the carrier density in the amplifier so Q(i)
0
Numerical Numerical AlgorithmAlgorithm
Initial W(i) = 0.1
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Parameter ExtractionParameter ExtractionThe values of the recombination coefficients and material loss can vary from device to device.
It is not possible to measure these coefficients directly.
Use the above numerical model, measurements of signal gain and spontaneous emission spectrum and a variant of the Levenberg-Marquardt method to obtain confident estimates of the SOA parameters.
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LevenbergLevenberg--Marquardt MethodMarquardt MethodThe SOA model is non-linear.
Need to define a χ2 merit function and determine best-fit parameters by its mimimisation.
The minimisation must proceed iteratively from an initial guess of the model parameters.
The procedure is then repeated until χ2 stops decreasing.
Then determine error estimates of the fitted parameters.
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The Levenberg-Marquardt method varies smoothly between the extremes of the inverse-Hessian method and the steepest descent method.
The latter is used far from the minimum, switching continuously to the former as the minimum is approached.
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The SOA parameters we wish to extract can be written as a 3-element vector
Merit function ( ) ( ) ( )∑ ∑=
=
=
−−
+−=bN
i
jj
jjjj
b
PPjj
GGN 1
2,expt
01
2iiexpt,
21
0
11χ
In the parameter extraction algorithm, the following terms are used:
( )T00 BAK nrad=a
1. Mean square difference between experimental and predicted SOA gain vs. bias current characteristic.
2. Mean square difference between experimental and predicted SOA ASE spectrum at a particular operating condition.
1 2
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( )∑∑== ∂
∂∂∂
−+
∂∂
∂∂
=1
011,
11 j
jj l
j
k
jN
i l
i
k
i
blk
o
b
aP
aP
jjaG
aG
NX
( ) ( ) ( )k
jj
jjjj
N
i k
i
bk a
PPP
jjaGG
Ny
b
∂∂
−−
+∂∂= ∑∑
==
1
0
,expt011
iiexpt,1G-1
X, a 3x3 square matrix with elements
3-element vector y with elements
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Parameter Parameter extraction extraction algorithmalgorithm
γ is initialised to 0.01
Good convergenceto a unique set of parameters for a wide range of initial guesses of a
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Experimental ResultsExperimental Results
The extracted SOA parameters (with ∆G = 1 dB) are K0 = 5300 m-1,Anrad = 6.5 x 108 s-1 and B0 = 3.2 x 10-16 m3s-1.
1530 1531 1532
Detail
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Confidence Limits Confidence Limits Calculate covariance matrix (X-1 with γ = 0) to obtainstandard errors in the fitted parameters.
Normalised mean
and standard deviation
%14error Standard ≈
0
0.5
1.0
1.5
B0K0 Anrad
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Internal SOA DistributionsInternal SOA Distributions
Unsaturated SOA
1 0 1 3
1 0 1 4
1 0 1 5
1 0 1 6
1 0 1 7
1 0 1 8
0 2 0 0 4 0 0 6 0 0 8 0 01
2
3
4
C arrie r d en s ity
S ign a l
A S E (-) A S E (+ )
D is tan ce f ro m in p u t (µ m )
Phot
on ra
te (s
-1)
Car
rier d
ensi
ty (1
024 m
-3)
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Internal SOA DistributionsInternal SOA Distributions
Typical carrier density, ASE and signal photon rates spatial distributions for a saturated SOA
1 0 1 3
1 0 1 4
1 0 1 5
1 0 1 6
1 0 1 7
1 0 1 8
0 2 0 0 4 0 0 6 0 0 8 0 01
2
3
4
C arrie r d en s ity
S ign a lA S E (-)
A S E (+ )
D is tan ce f ro m in p u t (µ m )
Phot
on ra
te (s
-1)
Car
rier d
ensi
ty (1
024 m
-3)
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AmpSoft AmpSoft
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ConclusionConclusionWe have developed a numerical model, to enable accurate prediction of SOA steady-state characteristics.
The parameter extraction algorithm can be used to determine material parameters and their confidence limits.
The models have been incorporated into SOA simulation software AmpSoft developed at the University of Limerick.
Thank you.