656 OPTICS LETTERS / Vol. 21, No. 9 / May 1, 1996

Locking characteristics of Fabry–Perot semiconductorlaser oscillators with side-mode injection

Pierluigi Debernardi

Cespa–Consiglio Nazionale Delle Ricerche, c/o Dipartimento di Elettronica, Politecnico di Torino,Corso Duca degli Abruzzi, 24, 10129, Torino, Italy

Received February 5, 1996

A model to evaluate frequency locking by means of a power injection near a side mode of the free-runningfrequency is presented. Semiconductor optical nonlinearities govern the locking problem; they are representedby nonlinear coefficients that take into account both carrier f luctuations and all relevant fast phenomena.Field equations are derived from coupled-mode theory, accounting for longitudinal variations by means ofappropriate mean values. Examples of results show both the single locked frequency-stable operation regionand the case when the injected signal and the free-running field coexist. 1996 Optical Society of America

Fabry–Perot (FP) laser oscillators are key componentsin optical communication systems. Recently they havebeen proposed as frequency converters,1 – 3 exploitingfour-wave mixing effects in semiconductor materials.In these devices, if injection locking of the free-runningmode is not adopted it is useful to investigate theparameter range where the signal to be converteddoes not lock the pump, to have correct behavior ofthe converter. On the other hand, it is interesting tostudy injection locking on a side-mode resonance, forexample, to shift the operating wavelength.

The problem of injection locking is usually treatedwith reference to the free lasing mode, the nearest tomaximum modal gain.4,5 Here we investigate the lock-ing properties of a FP semiconductor laser oscillatorwith a locking signal around a side-mode resonance(see Fig. 1). Such a mode cannot lase by itself becauseof the smaller gain, but it can do so under appropriateconditions of the injected locking signal.

The model adopted here is described in Refs. 3 and6, to which we refer for more details. Brief ly, it isbased on coupled-mode theory to compute the f ieldsversus time and space into the FP resonator; then oneeliminates the longitudinal dependences by averagingthe fields, which proves to be a good approximation.3

From Eqs. (1)–(8) of Ref. 3 one can obtain the sim-plified system for the carrier density difference n ­N 2 Nth, the locking signal c1, and the field ampli-tude c2 near the free-running frequency (ci are definedso that jcij

2 are the optical powers):dndt

­ 2n 1 DNi 2 sn 1 DNsd sP1 1 P2d 1 ngP1,

dc1

dt­ c1sGn 2 H 2 sP1 2 m̃P2d 1 Kcin,

dc2

dt­ c2fGn 2 mP1 2 ssP2 2 P20dg , (1)

where K is the coupling coefficient between the in-jected locking signal cin and the average of the intra-cavity field c1,

K ­ tsvgys2Ld∑≥p

R1R2 2 1¥ ≥q

R1yR2 1 1¥

3

≥R1 log

pR1R2

¥21

∏1/2

, (2)

0146-9592/96/090656-03$10.00/0

L is the device length, and R1 and R2 are the facetref lectivities. Other notation is defined in Table 1.In Eq. (1) the conjugate signal is neglected with respectto the other two, because at high beat frequencies thefour-wave mixing process is quite small.

The key parameters in this two-mode competitionproblem are the parameters s, m̃, and m. They ac-count for four-wave mixing and are closely related tothe nonlinear coeff icients S, M , and M̃ of Refs. 3 and6 that represent the optical response of the semicon-ductor to a multicarrier optical f ield. They are com-puted by means of a density matrix formalism andinclude on an equal footing population pulsation andfast relaxation effects such as carrier heating, spec-tral hole burning, and non-Markovian intraband relax-ation.6 The model is then suitable up to the terahertzregion of the beat frequency between free-running andinjected f ields. As already discussed in Ref. 3, theycan be written as

s ­ gDNseh,

m̃ ­ gDNsfeh 1 3/2 ms2Vdg ,

m ­ gDNsfeh 1 3/2 msVdg . (3)

The quantity m represents the intermodulation nonlin-ear effects and can be approximated, as is commonlydone in the literature (see, e.g., Refs. 2 and 7), by a

Fig. 1. Frequency scheme: the vertical lines below theabscissa represent the possible oscillating modes withoutinjection locking, V0 is the free spectral range, and pdefines the number of resonances away from the lasingfrequency sp . 0 if v1 . v2d.

1996 Optical Society of America

May 1, 1996 / Vol. 21, No. 9 / OPTICS LETTERS 657

Table 1. Notation Used

Symbol Meaning

t tyts: normalized timets sdRydN d21: effective lifetimeR Nyts0 1 BN2 1 CN3: recombination rateDNi fJysedactd 2 RsNthdgtsJ Current densitydact Active layer thicknessDNs Nth 2 N0N0 Transparency carrier densityNth Threshold carrier densityPi jcij

2yPs: optical powers normalized to PsPs Saturation powerng Dgya: carrier density induced by DgDg Gain difference between modes 2 and 1a dgydN : differential gainG gs1 1 jada Linewidth enhancement factorg 0.5 vgGatsG Optical confinement factorvg Group velocityH 0.5 vgGtsDgP20 DNiyDNs: normalized free-running power

multipole expression:

msVd ­1 1 ja

1 1 jVts1 et

1 1 jat

1 1 jVtt1 eh

1 1 jah

1 1 jVth

,

(4)where V ­ v2 2 v1 (see Fig. 1). We compute theparameters et, eh, at, ah, tt, and th by fitting the V

dependence of M and M̃ at threshold. The first termin Eq. (4) accounts for carrier f luctuations, and thesecond and third represent hot carrier and spectral-hole effects, respectively.

In steady-state conditions the system of Eqs. (1)gives2n 1 DNi 2 sn 1 DNsd sP1 1 P2d 1 ngP1 ­ 0,

fGn 2 jx2 2 mP1 2 ssP2 2 P20dg ­ 0,

P1jGn 2 jxin 2 H 2 sP1 2 m̃P2j2 ­ K2Pin , (5)where x ­ tsdv are normalized detunings from free-running FP resonances. The system of Eqs. (5) per-mits the determination of the static working point. Itcan be reduced to a sixth-order polynomial form in n;some of its six roots are not physical (e.g., complex num-bers) and are not considered; a stability analysis of theremainder yields the steady-state solutions. We carryout the stability analysis in a standard way by intro-ducing small deviations around the static point; tryingan exponential solution expsltd, we f ind stable opera-tion if the characteristic equation has all the roots withRehlj , 0.

The procedure outlined above permits the determi-nation of two-signal stable operations. For particularpower and frequency ranges of the v1 signal, single-mode injection-locked operation can be achieved; thefree-running mode disappears, and then one has tosolve a problem simpler than that discussed above.We can obtain such operation by considering only thefirst two of Eqs. (1) with P2 ­ 0 and proceeding as wehave discussed in the determination of steady-state so-lution and stability analysis.

In the numerical simulations a simple planar wave-guide based on lattice matched In12xGaxAsyP12y com-pounds and operating near l ­ 1.55 mm has beenconsidered sdact ­ 0.05 mm, G ­ 0.082d. For carrierrecombinations the following parameter values wereadopted: ts0 ­ 0.3 ns, B ­ 1 3 10210cm3 s21, C ­6 3 10229cm6 s21. The device length is L ­ 250 mm,the facet ref lectivities are R1 ­ R2 ­ 0.3, and wave-guide losses of 10 cm21 are assumed. For this struc-ture the threshold gain is 68 cm21; the correspondingcarrier density is ,3.18 3 1018 cm23, and the thresholdcurrent density 10.9 kAycm2. Around this workingpoint a fit to the results computed by the micro-scopic model of Ref. 6 gives Ps ­ 11.54 mWymm, a ­400 3 10218 cm2, a ­ 1.71, N0 ­ 1.1 3 1018 cm23,at ­ 0.65, et ­ 0.052, tt ­ 3.3 ps, ah ­ 20.08, eh ­0.021, and th ­ 77 fs.

Figure 2 shows the stable locking region (blackareas) and the two-frequency stable operation domain(gray areas) versus current at the p ­ 110 side reso-nance sv1 . v2d. The locking region is wider for lowercurrents and disappears above a certain current valuewhere the free-running mode is too strong to be over-come; this upper current value increases when the lock-ing power is increased. The white areas correspond to

Fig. 2. Stable locking range (black areas) versus normal-ized current for two different power injections displaced110 V0 from the free-running pulsation: The gray areascorrespond to stable two-signal operation; the white onesare unstable regions.

Fig. 3. Output powers for signals at frequencies 1 (curves)and 2 (curves with squares) along the two-signal stabilityborderlines of Fig. 2. Continuous curves, upper boundary;dashed curves, lower borderline; thick solid lines, free-running power.

658 OPTICS LETTERS / Vol. 21, No. 9 / May 1, 1996

Fig. 4. Stable locking range (black areas) versus injectedpower displaced 110 V0 from the free-running pulsation fortwo different currents: The gray areas correspond to stabletwo-signal operation; the white ones are unstable regions.

Fig. 5. As in Fig. 4 but for two different frequencies of thelocking signal and for IyIth ­ 2.

regions where neither a locked regime nor multicarrieroperation is stably achieved; in these regions multiperi-odic or chaotic behavior or both can occur.8

The output powers at frequencies 1 and 2 (see thefrequency scheme in Fig. 1) along the boundary of thetwo-frequency stability region of Fig. 2 are shown inFig. 3. It can be observed that the sum of the twopowers is roughly equal to the free-running power,shown by the thick solid lines. Moreover, along theupper boundary there is a current range where P2is almost zero; in this condition a small variation ofthe detuning out of the gray area leads to a negative(unphysical) value of P2.

Figure 4 is similar to Fig. 2, but it is computed withrespect to injected locking power sPloc ­ jcinj2d for two

different current values. It can be seen that the two-frequency stable operation is always achieved up toa certain locking power; above this threshold power,which increases with the current, all the possibleregimes can occur as a function of the detuning.

The effect of changing the frequency of the injectedsignal is shown in Fig. 5, where two different sideresonances are considered. Because the free spectralrange V0ys2pd ­ 185 GHz, for p ­ 1 the gain differenceis small sDg ­ 0.08 cm21d; for p ­ 30 (correspondingto 5.55 THz of frequency spacing) Dg ­ 71 cm21 is al-ready remarkable (almost 10% of the maximum gain),and it is responsible for the upward shift of the stablelocking operation. As expected, the latter case dis-plays a wider multicarrier stable region, whereas theunstable region is smaller. For negative p values asimilar behavior occurs, at least until the semiconduc-tor gain symmetry holds.

The following conclusions can be drawn: Asym-metic stable operation regions with respect to detun-ing are found. Around the side resonances next to thefree-running mode the unstable region between single-and double-signal stable operation is narrow for neg-ative detunings and in some cases also disappears.This is directly connected to the well-known asym-metry of the locking region in semiconductor lasers4

because of the carrier f luctuation nonlinearity thatdominates at small detunings. Only fast phenomenaare relevant when the two-signal regime occurs, andthen the corresponding stable region is more nearlysymmetric. The results presented show interestingfeatures of the locking properties near a side-moderesonance in a FP laser oscillator; they can be success-fully exploited both to prevent the locking regime infrequency-conversion applications or to achieve it whena frequency shift of the free-operating wavelength isneeded.

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