NUSOD-04 1

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

NUSOD-04 2

OutlineOutline1. Introduction

2. Bulk InGaAsP/InP SOA

3. Steady-state Numerical Model

4. Parameter Extraction

5. Simulations and Experiment

NUSOD-04 3

IntroductionIntroductionSOA technology shows great promise for use as basic amplifiers and as functional devices/subsystems in optical communication networks.

NUSOD-04 4

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.

NUSOD-04 5

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

NUSOD-04 6

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.

NUSOD-04 7

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

NUSOD-04 8

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±= ±±

NUSOD-04 9

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.

NUSOD-04 10

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

NUSOD-04 11

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

NUSOD-04 12

The algorithm updates the carrier density in the amplifier so Q(i)

0

Numerical Numerical AlgorithmAlgorithm

Initial W(i) = 0.1

NUSOD-04 13

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.

NUSOD-04 14

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.

NUSOD-04 15

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.

NUSOD-04 16

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

NUSOD-04 17

( )∑∑== ∂

∂∂∂

−+

∂∂

∂∂

=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

NUSOD-04 18

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

NUSOD-04 19

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

NUSOD-04 20

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

NUSOD-04 21

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)

NUSOD-04 22

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)

NUSOD-04 23

AmpSoft AmpSoft

NUSOD-04 24

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.