Semiconductor Optical Amplifier Parameter Extraction using a Wideband Steady-State State Numerical Model and the Levenberg-Marquardt Method

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Connelly Optical Communications Research Group Department of Electronic and Computer

1 Semiconductor Optical Amplifier Parameter Extraction using a Wideband Steady-State State Numerical Model and the Levenberg-Marquardt Method Michael J. Connelly Optical Communications Research Group Department of Electronic and Computer Engineering University of Limerick, Limerick Ireland Supported by Science Foundation Ireland EU 5 th Framework IST proect NUSOD-04

2 Outline. Introduction. Bulk InGaAsP/InP SOA 3. Steady-state Numerical Model 4. Parameter Extraction 5. Simulations and Experiment NUSOD-04

3 Introduction SOA technology shows great promise for use as basic amplifiers and as functional devices/subsystems in optical communication networks. NUSOD-04 3

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 4

Bulk InGaAsP/InP SOA Γ R Coupling losses 0.45 5 x 0-5 3 db C. Deguet et al.

5 Bulk InGaAsP/InP SOA Γ R Coupling losses x db C. Deguet et al., Homogeneous buried ridge stripe semiconductor optical amplifier with near polarization independence, in Proc. Eur. Conf. Optical Communications, 999. Corning. NUSOD-04 5

6 SOA steady-state state numerical model The model is based on a set of coupled differential equations that describe the interaction between the internal variables of the amplifier: carrier density n signal and ASE photon rates. Use a wideband model for the material gain g m g m = = 4 4 c 3 π c 3 π n n τν τν h h m m ( m + m ) e e m m hh hh ( m + m ) e e hh hh ( ν ) g = g g NUSOD-04 M.J. Connelly, Wideband Semiconductor Optical Amplifier Steady-State Numerical 6 Model IEEE J. Quantum Electron., E ν h E ν h g g f v f c ( ν ) m [ f ( ν )] v [ f ( ν )] c m m

7 0 Additive spontaneous emission spectrum g m, g m (0 4 m - ) g m g m Material gain Wavelength (nm) Ignoring band-tail Cut-off wavelength g gm Typical InGaAsP bulk semiconductor m and spectra NUSOD-04 7

8 Travelling-wave equations de dz ± sig Signal [ ] ± = ± β + Γgm( ν sig, n) α s Esig ASE is described in terms of photon rates. N + and N - are defined as the travelling-wave ASE photon rates (TE or TM) in a frequency spacing ν M about frequency ν, corresponding to a cavity resonance. dn dz ± = ± Γ [ ] ± g ( ν, n) α N ± R ( ν, n) m s sp NUSOD-04 8

9 R sp (ν,n) represents the spontaneous noise coupled into N + or N per unit length. R sp ( ν, n) = Γg ( ν, n) ν m M ν M is an integer multiple of the longitudinal mode spacing. NUSOD-04 9

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 0

11 Carrier density rate equation The carrier density n obeys the rate equation dn( z) I Γ R( n) g (, z) + = m ν sig Esig( z) + Esig( z) dt ev A + = Q( z) g m ( ν, z) [ + N ( z) + N ( z) ] Carrier recombination R( n) = Anrad n + B0n Loss coefficient α s = K 0 Radiative carrier recombination lifetime τ = B 0 n NUSOD-04

12 Numerical Algorithm The algorithm updates the carrier density in the amplifier so Q(i) 0 Initial W(i) = 0. NUSOD-04

13 Parameter Extraction The 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 3

14 Levenberg-Marquardt Method The SOA model is non-linear. Need to define a χ 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 χ stops decreasing. Then determine error estimates of the fitted parameters. NUSOD-04 4

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 5

16 The SOA parameters we wish to extract can be written as a 3-element vector a = K A N b = Merit function χ = ( Gexpt,i Gi ) + ( Pexpt, P ) Nb i= ( ) T 0 nrad B 0 In the parameter extraction algorithm, the following terms are used: ( ) 0 =. Mean square difference between experimental and predicted SOA gain vs. bias current characteristic.. Mean square difference between experimental and predicted SOA ASE spectrum at a particular operating condition. 0 NUSOD-04 6

17 NUSOD-04 7 ( ) = = + = 0, l k N i l i k i b l k o b a P a P a G a G N X ( ) ( ) ( ) k N i k i b k a P P P a G G N y b + = = = 0 expt, 0 i expt,i - G X, a 3x3 square matrix with elements 3-element vector y with elements

18 Parameter extraction algorithm γ is initialised to 0.0 Good convergence to a unique set of parameters for a wide range of initial guesses of a NUSOD-04 8

19 Experimental Results Detail The extracted SOA parameters (with G = db) are K 0 = 5300 m -, A nrad =6.5 x 0 8 s - and B 0 = 3. x 0-6 m 3 s -. NUSOD-04 9

20 Confidence Limits Calculate covariance matrix (X - with γ = 0) to obtain standard errors in the fitted parameters..5 Standard error 4% Normalised mean and standard deviation.0 K 0 A nrad B NUSOD-04 0

21 Internal SOA Distributions Photon rate (s - ) A S E (-) A S E (+ ) S ign al C arrier d en sity 3 Carrier density (0 4 m -3 ) Distance from input (µ m) Unsaturated SOA NUSOD-04

22 Photon rate (s - ) Internal SOA Distributions A S E (-) S ignal A S E (+ ) C arrier d en sity 4 3 Carrier density (0 4 m -3 ) Distance from input (µ m) Typical carrier density, ASE and signal photon rates spatial distributions for a saturated SOA NUSOD-04

23 AmpSoft NUSOD-04 3

24 Conclusion We 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. NUSOD-04 4

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