of modulated quantum well laser diodes · 2018. 11. 13. · laser diodes. these include higher...

VLSI DESIGN1998, Vol. 8, Nos. (1-4), pp. 355-360Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1998 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach SciencePublishers imprint.

Printed in India.

Rate Equation Modelling of NonlinearDynamics in Directly Modulated Multiple

Quantum Well Laser Diodes

STEPHEN BENNETT a’ *, CHRISTOPHER M. SNOWDEN b and STAVROS IEZEKIEL b

Depabrtmen of Electronic and Electrical Engineering, University College London, London, WCIE 7JE, UK;Department of Electronic and Electrical Engineering, University of Leeds, Leeds, LS2 9JT, UK

A theoretical (using rate equations) and experimental study of the nonlinear dynamicsof a distributed feedback multiple quantum well laser diode is presented. The analysis isperformed under direct modulation. Period doubling and period tripling are identifiedin both the measurements and simulations. Period doubling is found over a wide rangeof modulation frequencies in the laser. Computational results using rate equations showgood agreement with the experimental results.

Keywords: MQW laser diodes, rate equations, nonlinear dynamics, period doubling, periodtripling

1. INTRODUCTION

Multiple Quantum Well (MQW) laser diodes arewidely used in many applications because of theirsuperior performance characteristics over bulklaser diodes. These include higher modulationbandwidths and lower threshold currents, both ofwhich are desirable qualities for high-speedoptical fibre links. Directly modulated laserdiodes are operated under conditions that canlead to a wide variety of nonlinear behaviour,including period doubling [1] and chaos [2]. It is

therefore vital to have an accurate model ofnonlinear behaviour that is numerically efficient,and for the most part rate equations are used bymost workers. However, in contrast to a largevolume of work using rate equations for non-linear modelling of bulk structures, little suchwork exists for MQW lasers. This paper presentsa detailed theoretical and experimental study ofthe nonlinear dynamics of an InGaAsP/InGaAsMQW 3,/4 shifted Distributed Feedback (DFB)laser diode with 16 quantum wells [3] under directmodulation.

* Corresponding author.

355

356 S. BENNETT et al.

2. MQW LASER DIODE MODEL

To account for the carrier dynamics in quantumwell lasers, a rate equation model based on thatproposed by Nagarajan et al. [4] is used in ournumerical simulations. The rate equations for thecarrier density in the quantum wells (N) and thebarriers (NB) and the photon density in the opticalcavity (S) can be written as,

FqNdNB_FqI NB (1)dt eV Tc Te

dN NB N[

dt FqTc 7"n 7"evgG(N, S) S (2)

IL I, where PRF is the incident RF input power,FL is the reflection coefficient of the laser and Z0 isthe characteristic impedance. The quantum wellshave one conduction subband so that the relation-ship between the carrier density and the junctionvoltage Vj can be approximated [5],

NkT m2cTrh Lzln[l+exp( eVj-Eph)kT (6)

where me is the electron effective mass, Lz is thequantum well width and Eph is the photon energy.The MQW material gain G (N, S) is modelled by [6]

Go NG(N, S)=ln-- (7)No

dS [rvgG(N’ S) S + (3)

where Fq is the fraction of the MQW region filledby the quantum wells, NB/7-c is the loss rate ofcarriers from the barriers to the quantum wells andN/7"e is the loss rate of carriers from the quantumwells to the barriers. 7-p is the photon lifetime, V isthe active region volume, F is the mode confine-ment factor, and Vg is the group velocity. Thedependence of the carrier lifetime (-n) on N ismodelled as,

1A + BN + CN2 (4)

7"n

where the terms BN and CN2 model bimolecularand Auger recombination respectively. The outputpower per facet P is related to S via,

where R is the facet reflectivity ( 0.32) and L isthe cavity length. The injection current (I) can beexpressed I Ioc + IRv sin (27rft), where Ioc is thebias current, IRV is modulation current amplitudeand f is the modulation frequency. Using anisolator between the microwave source and thelaser to minimise impedance mismatch problemsenables IRF to be approximated, IRF v/2PRF/Zo[

where e is the nonlinear gain coefficient, No is thetransparent carrier density and Go is a constantdependant on well structure.

It is known that the fraction of the spontaneousemission coupled into the lasing mode (/3 factor)plays a large part in determining the nonlineardynamics of laser diodes [2,7-9] and it hasrecently been found that bulk DFB laser diodescan exhibit a reduced /3 factor [2] compared totheir FP counterparts (8 10-7 compared to 10-in typical FP lasers). The value of used in oursimulations was therefore measured using the sametechnique described in [2]. Other parameters whichwere not material or dimensional in nature were

5

o

-0

-15

15

10 t 0.7mW. 1.9mW

0 5 10 15 20

Frequency (GHz)

FIGURE Measured (solid lines) and simulated (points)small-signal modulation frequency response of the laser withincreasing output power.

MODELLING OF MQW LASER DIODES 357

extracted from the small-signal measurementsshown in Figure 1. The parameter values used inthe simulations were: V 5.1 10-17m3 I 0.22,Fq 0.66, -p 1.3 ps, fl 10-6, Go 141107m-1,N0=2.411024m-3,e=3.24 10-23m3,A= 108s -1 B 10 -16 m3s -1 C 3 10-41 m6s -1

-c 20 ps, and 7"e 191 ps. Simulations were per-formed by solving (1)-(3) using a fourth orderRunge-Kutta scheme.

3. NONLINEAR, DYNAMIC ANALYSIS

lO

8

E 6

o

PD

6 10 14 18

Moduletion Frequency (GHz)

3.1. Results

So as to study the nonlinear dynamics resultingfrom direct modulation of the laser alone, opticalfeedback into the laser was minimised by the use ofan optical isolator. The experimental setup used to

analyse the nonlinear dynamics of the lasers isshown in Figure 2. The incident RF input power tothe laser (PRy) was monitored through a direc-tional coupler and the problem of laser mismatchto 509t was minimised through the use of amicrowave isolator.For a range of output power levels * (Pout) and

modulation frequencies (f), PRy was sweptbetween -5 and 22 dBm and the regions wherenonlinear behaviour occurred recorded. Theseregions are shown in Figure 3 along with thesimulated results. There were both regions ofperiod doubling and period tripling. Period

tripling occurred when the laser was biased with

MicrowaveCurrent Spectrum

IlV’t L Source Analyser

Isolator" /Laser Isolator Detector

FIGURE 2 Experimental setup used to investigate the non-linear dynamics of the laser.

10

8

4

0

2

PD

6 10 14 18

Modulofion Frequency (GHz)

FIGURE 3 Regions of where nonlinear behaviour wererecorded for (a) the measured laser, (b) the simulated laser.PD denotes the regions where period doubling were recordedand T the region where period tripling was observed.

Pout levels of between 0.7 mW and 3.6 mW. In theregion where period tripling occurred, as PRF was

increased, the route to period tripling was alwaysvia period doubling. Figure 4 shows the measuredand simulated output frequency spectrum of thelaser under conditions of single period, perioddoubling and period tripling behaviour.

There were features of the period doublingwhich were common to both simulated andmeasured lasers. Namely the upper and lowermodulation frequency limits of the period dou-bling regions seemed to follow one and two times

* Corresponding to a range of bias currents.


the relaxation frequency of the laser (fR) respec-tively. For a fixed ratio of modulation frequency torelaxation frequency the value of PRy required toproduce period doubling increased almost linearlywith Pout. These features are illustrated in Figure 5where the calculated values of PIF required forperiod doubling to occur in the laser are displayedversus Pout and the modulation frequency normal-ised to f. The measured and simulated values of

PF required for period doubling to occur in thelaser versus Pout are displayed in Figure 6. Whenobtaining the data displayed in Figure 6 themodulation frequency was adjusted at each Pout tokeep the ratio of modulation frequency to relaxa-tion frequency constant, which for Figure 6 was1.5. It is expected that this relationship is due tothe corresponding increase in relaxation damping

Measured Simula[ed

0.2 0.4 0.6 0.8 1.0 1.20.2 0.4 0.6 0.8 1.0

Frequency / Modulation Frequency

P1

P2

P3

FIGURE 4 Measured and simulated frequency spectrum ofthe laser under conditions of single period (P 1), period doubling(P2) and period tripling (P3) behaviour.

30

25

20

35

"I.2 "I.4 "1.6 "1.8 2 2.20.5 (roW)

Normalised ModulationFrequency fa

FIGURE 5 Three dimensional plot displaying the calculatedvalues of PRF required for period doubling to occur in the laserversus Pout and the modulation frequency normalised to J,.

2o

15

10m-O

5

0-o

-5

-10

-150.0 5.50.5 1.0 1.5 2.0 2.5 3.0

Power (roW)

FIGURE 6 Measured (points) and simulated (solid line) RFinput power required to produce period doubling versus outputpower per facet. The modulation frequency to relaxationfrequency ratio is kept constant at 1.5.

2 0.9 0.7 0.5 0.3 0.1

[ Factor (10-5)

15

io.

5 n

0

FIGURE 7 Simulated dependence of nonlinear behaviour on

PRF and /3 in the laser. Simulations were performed with amodulation frequency of 6 GHz and Pout set at 0.6 mW. Theblack areas in the diagram indicate regions of higher orderbifurcations.

with power level which is known to be an

important factor in characterising the nonlinear

dynamics of laser diodes [9].Liu et al. [2] suggest that the promotion ofperiod

tripling in their bulk DFB laser was due to .theunusually low value of/3 measured in their laser. Toinvestigate whether this was the case in our MQWDFB laser we analysed the effect of variations in/3

MODELLING OF MQW LASER DIODES 359

on our simulation results. Figure 7 shows theregions of nonlinear behaviour as/3 and PRF arevaried. It is clear that as/3 is reduced the periodtripling behaviour becomes more prominent. Itwould therefore seem likely that the observed periodtripling in our DFB laser can be attributed to thelow value of/3 measured in this laser compared tothat measured in typical Fabry Perot lasers.

4. CONCLUSION

In summary, we have performed a detailedtheoretical (using rate equations) and experimentalanalysis of the nonlinear dynamics of a MQWDFB laser diode under direct modulation. Weobserved period doubling and period tripling inboth measurements and simulations. Period dou-bling was evident over a wide range of modulationfrequencies. Computational results using rate

equations show good agreement with the experi-mental results.

Acknowledgements

The authors would like to acknowledge Dr. L. DoWestbrook and Dr. I. F. Lealman of British Tele-com Research Laboratories for informative dis-cussions and the supply of sample lasers.

References

[1] Hemery, E., Chusseau, L. and Lourtioz, J. M. "Dynamicbehaviors of semiconductor lasers under strong sinusoidalcurrent modulation: Modeling and experiments at1.3 lam", IEEE J. Quantum Electron., 26, 633-641, April1990.

[2] Liu, H. F. and Ngai, W. F. "Nonlinear dynamics of adirectly modulated 1.55 tm InGaAsP distributed feedbacksemiconductor laser", IEEE J. Quantum Electron., 29,1668-1675, June 1993.

[3] Lealman, I. F., Harlow, W. F. and Perrin, S. D. "Effectsof Zn doping on modulation bandwidth of 1.55gmInGaAs/InGaAsP multiquantum well DFB lasers", Elec-tron. Lett., 29, 1197-1198, June 1993.

[4] Nagarajan, R., Ishikawa, M., Fukushima, T., Geels, R. S.and Bowers, J. E. "High speed quantum-well lasers andcarrier transport effects", IEEE J. Quantum Electron., 28,1990- 2007, Oct. 1992.

[5] Seltzer, C. P., Westbrook, L. D. and Wickes, H. J. (1995)."The ’Gain-Lever’ effect in InGaAsP/InP multiple quan-tum well lasers", IEEE J. Lightwave Technol., 13(2),283-289.

[6] McIlroy, P. W. A., Kurobe, A. and Uematsu. "Analysisand application of theoretical gain curves to the design ofmulti-quantum-well lasers", IEEE J. Quantum Electron.,21, 1958-1963, Dec. 1985.

[7] Lee, C. H., Yoon, T. H. and Shin, S. Y. "Period doublingand chaos in a directly modulated laser diode", Appl.Phys. Lett., 46, 95-97, Jan. 1985.

[8] Hori, Y., Serizawa, H. and Sato, H. "Chaos in a directlymodulated semiconductor laser", J. Opt. Soc. Amer. B., 5,1128-1133, May 1988.

[9] Yoon, T. It., Le, C. H. and Shin, S. Y. "Perturbationanalysis of bistability and period doubling bifurcations indirectly-modulated laser diodes", IEEE J. QuantumElectron., 25, 1993-2000, Sept. 1989.

Authors’ Biographies

Stephen Bennett was born in Blackpool, England,in 1971. He received the B.Eng. (Honours) degreein electronic and electrical engineering from theUniversity of Leeds in 1993. After graduating hejoined the Microwave and Terahertz TechnologyGroup at the University of Leeds where he hasbeen working towards the Ph.D. degree on thehigh frequency nonlinear dynamics of semicon-ductor lasers. Since 1996 he has worked as a

Research Fellow within the Microwave Optoelec-tronics group at University College London. Hisresearch interests include semiconductor laserdynamics, soliton control, optical comb generationand MQW saturable absorption.

Christopher M. Snowden received the B.Sc.,M.Sc. and Ph.D. degrees from the University ofLeeds. After graduating in 1977 he worked as an

Applications Engineer for Mullard, Mitcham. HisPh.D. studies were conducted in association withRacal-MESL and were concerned with the large-signal characterisation of MESFET microwaveoscillators. In 1982 he was appointed Lecturer inthe Department of Electronics at the University ofYork. He joined the Microwave Solid State Groupin the Department of Electrical and ElectronicEngineering at the University of Leeds in 1983. Henow holds the Chair of Microwave Engineering inthe Microwave and Terahertz Technology Re-search Group and is also currently Head of the


Department of Electronic and Electrical Engineer-ing. During 1987 he was a Visiting ResearchAssociate at the California Institute of Technol-ogy. He has been a Consultant to M/A-COM Inc.,Corporate Research and Development since 1989,where he was on sabbatical leave during the period1990-91. During this year he represented M/A-COM as Senior Staff Scientist. He was Chairmanof the 1995 international Microwaves and RFConference. He is a Member of the MIT Electro-magnetics Academy. He is also a Top Scientist atthe International Research Centre for Telecom-munications-Transmission and Radar, Delft Uni-versity of Technology, Netherlands.

Professor Snowden is a Fellow of the IEEE and aFellow of the lEE. He is a Distinguished Lecturer(1996/7) for the IEEE (Electron Devices Society).He is co-Chairman of the MTT-1 Committee and aMember of the 1997 IEEE MTT-S Technical

Program Committee. His main research interestsinclude compound semiconductor device model-ling, microwave, terahertz and optical nonlinearsubsystem design and advanced semiconductordevices. He has written 7 books and over 190 papers.

Stavros lezekiel was born in Coventry, England,in 1966. He received the B.Eng. and Ph.D. degreesin electronic and electrical engineering from theUniversity of Leeds in 1987 and 1991, respectively.From 1991 to 1993, he worked as a ResearchFellow developing hybrid optoelectronic packa-ging systems for M/A-Com Corporate Researchand Development Centre. In 1993, he was

appointed as a Lecturer in high-frequency analo-gue electronics at the University of Leeds. Hisresearch interests include high-speed semiconduc-tor laser diode modelling, optoelectronic packaging,nonlinear systems, and microwave-optoelectronicsubsystem design.

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