March 15, 2002 / Vol. 27, No. 6 / OPTICS LETTERS 391

Intensity and polarization self-pulsations in vertical-cavitysurface-emitting lasers

Alessandro Scirè, Josep Mulet, Claudio R. Mirasso,* and Maxi San Miguel

Instituto Mediterráneo de Estudios Avanzados, Consejo Superior de Investigaciones Científicas –Universitat de les Illes Balears,Campus UIB, E-07071 Palma de Mallorca, Spain

Received August 13, 2001

We implement a dynamic model that describes the polarization behavior in vertical-cavity surface-emittinglasers that contain an absorbing region surrounding the active zone. We find four regions of qualitativelydifferent behavior: stable linearly polarized operation, intensity pulsations of a linearly polarized component,pulsations of both total-intensity and polarization components, and polarization self-pulsation with constanttotal intensity. We characterize the four regions by computing the polarization-resolved optical and powerspectra. The predicted behavior agrees with recent experimental results. © 2002 Optical Society of America

OCIS codes: 250.7260, 260.5430.

Edge-emitting lasers (EELs) with a narrow-stripe ge-ometry may exhibit self-pulsation (SP) owing to thepresence of a saturable-absorption region surround-ing the active zone.1 – 3 This phenomenon has beenimplemented in clock extraction systems,4 in synchro-nization schemes for encoded communication,5 and asa method to reduce the noise of laser diodes in op-tical disk systems.6 Vertical-cavity surface-emittinglasers (VCSELs) are important, compact light sources7

for applications, e.g., in optical interconnects and opti-cal data storage, owing to their low threshold current,single-longitudinal-mode operation, and easy integra-bility in two-dimensional arrays. As in an EEL, aVCSEL may contain an unpumped region surroundingthe central gain zone, and therefore SP conditions couldbe expected. Recently SP were experimentally demon-strated in small active region VCSELs.8,9 Because thepolarization in VCSELs is not so well stabilized10 as inEELs, it will be interesting to investigate the polar-ization properties of self-pulsating VCSELs. In thisLetter we implement a model that is suitable for de-scribing SP in VCSELs operating in the fundamentaltransverse mode, within the framework of the spin-f lipmodel.10,11 The absorber region is modeled as an ad-ditional carrier reservoir, as was done in Yamada’smodel.2 The complete set of equations reads as

�F6 � 1�2�1 1 ia� �D1 1 D2 6 d1 6 d2 2 1�F6

2 �ea 1 iep�F7 , (1)

�D1,2 � g1, 2�m1, 2 2 D1,2 2 a1, 2�D1, 2 1 d1, 2� jF1j2

2 a1, 2�D1, 2 2 d1, 2� jF2j2 1 c12,21D2,1� , (2)

�d1,2 � 2 gs1, 2d1, 2 2 g1,2�a1, 2�D1,2 1 d1, 2� jF1j2

2 a1, 2�D1, 2 2 d1, 2� jF2j2 2 c12,21d2,1� , (3)

where the subscripts 1 and 2 stand for the pump andabsorbing regions, respectively. F6 are the slowly

0146-9592/02/060391-03$15.00/0

varying complex amplitudes of the two circularlypolarized components of the electric f ield. D1,2 arethe total carrier inversion between the conduction andvalence bands relative to the transparency carrierdensity, and d1, 2 are the carrier inversions with oppo-site spin orientations. The equations are written in adimensionless form12 such that that time is measuredin units of k21, where k is the cavity decay rate. Theeffective injection currents, with respect to trans-parency, are m1, 2. Because the current is suppliedonly in the central region, m1 . 0 and m2 , 0. Carrierdiffusion is introduced in the rate equations throughthe coupling terms c12D2 and c21D1. Linewidthenhancement factor a describes the phase–amplitudecoupling in semiconductor lasers. The parametersg1,2 stand for the total carrier decay rates, and gs1,2are the effective spin-f lip rates. The phase and theamplitude anisotropies10 are ep and ea, respectively.The differential gains in the active and in the absorberregions differ because they have different carrierdensities. This effect is taken into account throughthe parameters aj . By appropriate rescaling of thevariables we can assume that a1 � 1, whereas a2represents the ratio of the effective gain coeff icient inthe absorbing region to that in the pumped zone. Theeffective gain coeff icient is defined1 – 3 as the productof differential gain times the fraction of optical powerwithin each region.

We start our discussion by looking at the linearlypolarized (LP) steady-state solutions of Eqs. (1)–(3).The electrical fields are expressed as F6�t� �Q exp�i�v6t 6 w��, and the time derivatives of thecarrier densities are set to zero. We find two LPsolutions: w � 0 along the x axis and w � p�2 alongthe orthogonal direction. As we observed that ourresults are only weakly dependent on ea, we in therest of the Letter we assume for simplicity that ea � 0.In this limit, v6 � 2ep for an x-LP solution v6 � epfor a y-LP solution. The total carrier inversions arein turn given by

D̄1, 2 �m1, 2�1 1 a2, 1A� 1 m2, 1c12, 21�1 1 A� �1 1 a2A� 2 c12c21

. (4)

© 2002 Optical Society of America

392 OPTICS LETTERS / Vol. 27, No. 6 / March 15, 2002

The difference in carrier inversion is d̄1, 2 � 0 in theLP states. Total intensity A � 2Q2 satisfies the gain-clamping condition D̄1 6 d̄1 1 D̄2 6 d̄2 � 1, leading toa quadratic equation that reads as

a2A2 1 �a2�1 2 m1� 1 �1 2 m2��A 1 1

2 m1�1 1 c21� 2 m2�1 1 c12� 2 c12c21 � 0 . (5)

For our choice of parameters, Eq. (5) has only one posi-tive root. From a linear stability analysis10 of the twoLP solutions, the linearized system of Eqs. (1)–(3) de-couples into two subsets, one �4 3 4� that accountsfor the stability of the total intensity and one �4 3 4�that describes the polarization stability. Because ofthe arbitrariness in a global phase,11 the actual di-mension of the total intensity subset is three. As thecarrier decay rates g1 and g2 are typically o�1023�, asimplified analysis can be performed. Applying theRouth–Hurwitz criterion13 and neglecting o�g1g2�, weobtain the boundary stability conditions14

g12m1 1 g2

2a2m2 1 �a2g2 2 g1�

3 ��g2c21 1 g1c12�D̄1 2 g1c12� � 0 , (6)

V2 � 1�2�g1D̄1 1 a2g2D̄2�A . 0 . (7)

When Eq. (6) is fulfilled, the total intensity loses itsstability through a Hopf bifurcation, giving rise to pul-sations at frequency V. As was already mentioned,the number of components of the polarization subsetis four, and an analytical study is quite involved.Alternatively, we numerically f ind the conditionsfor the stability boundaries of the polarization in aphase diagram of m1�mth relative to birefringencegp � epk, where mth is the laser threshold. Fourdifferent regions can be distinguished in Fig. 1: inregion A, only the y-LP solution is stable; in region Bwe observe total intensity pulsations in a stable linearpolarization. In this case the time traces are similarto those of SP in EELs if we disregard polarizationdegrees of freedom. The most interesting regionsare C, with coupled pulsations of polarization andtotal intensity, and D, in which polarization pulsa-tions with almost constant total intensity appear.In region C, both total intensity and polarizationare unstable. Time traces of the two polarizationcomponents (Fig. 2A) show amplitude modulationwith the shape of strong pulses. We also analyzedthe modif ication of the stability diagram shown inFig. 1 relative to variations of the device and physicalparameters. In particular, we have found that thenumerical line that delimits regions B and C occursfor larger birefringence values when the spin-f lip rateis increased. To better characterize the dynamics inregions C and D we compute the power spectrum ofthe total intensity �PS�v� � �jP̃ �v�j2�� and the opticalspectrum �OSx,y �v� � �jF̃x, y�v�j2�� of both linear com-ponents, with P �t� � jFxj

2 1 jFy j2. Because the effect

of spontaneous-emission noise has been demonstratedto be important in self-pulsating lasers,1,15 we add

Langevin noise sources to the field equations.16 ThePS (Fig. 2B) shows a fundamental frequency [cor-responding to relation (7) near the bifurcation] andseveral harmonics. In addition to amplitude modu-lation, the two linear components undergo strongphase modulation. The polarization-resolved OSs(Fig. 2C) show, in both polarizations, an asymmetricdouble-peaked profile that is typical of a gain-switchedsemiconductor laser.17 The ratio between the widthof the total OS (Fig. 2C), which represents the averagephase sweep during each pulse, and the fundamentalpeak position is 40, i.e., of the same order as the ex-perimentally reported value.8 When current level m1(at constant gp) is increased continually from region Cto region D, the fundamental frequency increases untilthe locus [Eq. (6)] is crossed, giving rise to a regimein which the total intensity is stable whereas thepolarization is unstable. We observe that this transi-tion is not sharp, but the optical pulses are smoothlyreshaped into a chaotic waveform when they crossthe horizontal line in Fig. 1. Increasing the injection

Fig. 1. Phase diagram (m1�mth relative to gp � epk) ofthe stability of the linearly polarized solutions. The fourregions are A, y-LP stable; B, pulsations in one polar-ization; C, coupled total intensity and polarization pul-sations; and D, polarization pulsations. The parametersare m2 � 232, g1 � 1.09 3 1023, g2 � 1.13 3 1023, a � 3,c12 � 2.84 3 1022, c21 � 1.91, gs1 � gs2 � 0.25, a2 � 8.7,and k � 390 ns21.

Fig. 2. A, Time traces x-LP (dotted curve) and y-LP (solidcurve) in region C of Fig. 1; m1�mth � 2 and gp � 20 ns21.B, PS of the total intensity. C, Polarization-resolved OSsfor x-LP (dotted curve) and y-LP (solid curve).

March 15, 2002 / Vol. 27, No. 6 / OPTICS LETTERS 393

Fig. 3. A, Time traces x-LP (dotted curve) and y-LP (solidcurve) in region D of Fig. 1; m1�mth � 4 and gp � 20 ns21.B, Total-intensity PS. C, Polarization-resolved OSs forx-LP (dotted curve) and y-LP (solid curve). Note that thearbitrary units are different from those in Fig. 2.

current further, well inside region D, causes thesechaotic forms to lead to regular oscillation. There thetime traces (Fig. 3A) show nearly out-of-phase ampli-tude modulation composed of harmonic pulsations inthe two polarization components. Consequently thePS (Fig. 3B) shows a single peak at this frequency.Harmonic phase modulation is also present in both po-larizations, as shown in the polarization-resolved OS(Fig. 3C). In the experiments8 a similar transitionfrom strongly chirped pulses to harmonic pulsationswas also found when the pump was increased. Ourresults for regions C and D are then in agreementwith the experimentally reported PSs and OSs. Thebehavior observed in region C suggests that thephase dynamics is dominated by the phase-amplitudecoupling, whereas the total intensity shows the usualself-pulsation induced by means of saturable absorp-tion as in EELs.6 In region D, however, polarizationinstability leads to time-dependent harmonic solutionsat almost constant total intensity. This result canbe regarded as a sign of mode partition among thetwo polarization components. Our conjecture is alsosupported by the arrangement of the OS in multiplepeaks, a clear sign of the out-of-phase dynamics of thepolarization pulsations.

In conclusion, we have demonstrated that polar-ization self-pulsations in VCSELs are possible undercertain operation conditions. As a result of analyticaland numerical analysis of our model, we have foundregions of stable operation, coupled pulsations of thepolarization and the total intensity, and polariza-tion self-pulsation at nearly constant total intensity.These regions, which appear when the injection cur-

rent is changed, compare successfully with reportedexperimental results. This fact suggests that ourmodel is a suitable framework for the study of self-pulsation in VCSELs in the presence of polarizationeffects.

This study has been funded by the EuropeanCommission through VISTA HP-TRN network, MarieCurie grant IF MCFI-2000-00617, and the SpanishMinistry of Science and Technology under projectsTIC99-0645-C05-02 and BFM2000-1180. A. Scirèe-mail address is scire@imedea.uib.es.

*Permanent address, Departament de Física, Uni-versitat de les Illes Balears, E-07071 Palma de Mal-lorca, Spain.

References

1. C. R. Mirasso, G. H. M. van Tartwijk, E. Hernandez-Garcia, D. Lenstra, S. Lynch, P. Landais, P. Phelan, J.O’Gorman, M. San Miguel, and W. Elsässer, IEEE J.Quantum Electron. 35, 764 (1999).

2. M. Yamada, IEEE J. Quantum Electron. 29, 1330(1993).

3. R. W. Dixon and W. B. Joyce, IEEE J. Quantum Elec-tron. 15, 470 (1979).

4. P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P.Landais, L. Pesquera, and J. Hegarty, IEEE J. Quan-tum Electron. 35, 221 (1999).

5. C. Juang, T. M. Hwang, J. Juang, and L. Wei-Wei,IEEE J. Quantum Electron. 36, 300 (2000).

6. M. Yamada and T. Higashi, IEEE J. Quantum Electron.27, 380 (1991).

7. K. J. Ebeling, in Semiconductor Quantum Optoelectron-ics, A. Miller, M. Ebrahimzadeh, and D. M. Finlaynson,eds. (Institute of Physics, Bristol, U.K., 1999), p. 339.

8. M. B. Willemsen, A. S. van de Nes, M. P. van Exter,J. P. Woerdman, M. Brunner, and R. Hövel, Appl. Phys.Lett. 77, 3514 (2000).

9. J. A. Hudgings, R. J. Stone, C.-H. Chang, S. F. Lim,K. Y. Lau, and C. J. Chang-Hasnain, IEEE J. Sel. Top.Quantum Electron. 5, 512 (1999).

10. M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev.A 52, 1728 (1995).

11. J. M. Regalado, F. Prati, M. San Miguel, and N. B.Abraham, IEEE J. Quantum Electron. 33, 765 (1997).

12. T. W. Carr and T. Erneux, IEEE J. Quantum Electron.37, 1171 (2001).

13. G. A. Korn and T. M. Korn, Mathematical Handbookfor Scientists and Engineers (McGraw Hill, New York,1968).

14. A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel,Proc. SPIE 4646, 293 (2002).

15. G. H. M. van Tartwijk and M. San Miguel, IEEE J.Quantum Electron. 32, 1191 (1996).

16. J. Mulet, C. R. Mirasso, and M. San Miguel, Phys. Rev.A 64, 0238-17 (2001).

17. See, for instance, G. P. Agrawal and N. K. Dutta, LongWavelength Semiconductor Lasers (Van Nostrand, NewYork, 1986), p. 265.