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Polarization switching in VCSELs induced by opticalmi ect ion

Krassimir Panajotov" , Boris Ryvkin2, Eugeny Avrutin3, Ignace Gatare1'4, Marc Triginer' , JordiBuesa1 , Irma Veretennicoff1, Hugo Thienpont' , Marc Sciamanna4

1 Department of Applied Physics and Photonics (TW-TONA),Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium.

2 A.F.Ioffe Physico-Technical Institute, St. Peterburg 1 9402 1, Russia

3 Department ofElectronics, University ofYork, York YOlO 5DD, United Kingdom4Suplec, LMOPS CNRS UMR-7132 (Laboratoire Matriaux Optiques, Photonique et Systmes), 2

Rue Edouard Belin, F-57070 Metz (France).5 Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria.

ABSTRACT

In this paper we summarize our recent theoretical and experimental results concerning optical injection in VCSELs. First,we develop phenomenological rate equation model for the normalized photon densities of the two fundamental VCSELmodes with orthogonal linear polarization when injecting external light polarized along the direction ofpolarization of thenonlazing mode. We then perform a simple analytical analysis of optically induced polarization switching in VCSELs,which takes into account both the current-dependent linear dichroism and the nonlinear dichroism. We also presentdetailed investigation of polarization dynamics induced by optical injection using an alternative rate equation model forVCSELs which takes into account the microscopic spin-flip processes in the active medium. In particular we predict thattwo injection locked solutions may coexist: the first one exhibits the polarization ofthe master laser, while the second onecorresponds to an elliptically polarized injection locking. Continuation techniques allow us to find the bifurcation scenarioleading to such an elliptically locked state. We also report on our first experimental results on polarization switching andinjection locking for the case of an orthogonal optical injection in VCSELs and demonstrate rich dynamical behavior. Thepolarization switching is accompanied by a cascade of bifurcations to timeperiodic and possibly chaotic dynamics. Ourexperimental results show evidence of a period doubling route to chaos.

Keywords: VCSELs, optical injection, polarization switching, optical bistability, chaos, period-doubling, wave mixing.

1. INTRODUCTION

Recently, a new type ofsemiconductor lasers reached the maturity ofmass production, namely the Vertical-Cavity surface-emitting Lasers (VCSELs). These lasers emit light perpendicular to the surface of the quantum wells, compensating forthe decreased confinement factor by very high reflectivity ofthe laser mirrors - specally designed multilayer DBR. Due tothe surface emission and the cylindrical symmetry of the laser cavity polarization direction is not well fixed in VCSELs.Often they undergo polarization switching (PS) between two linear orthogonally polarized (LP) fundamental transversemodes when changing the injection current'9 (for a review see, e.g. 10). This specific VCSELs behavior, which contrastthe polarization behavior ofedge emitting lasers, is due to the fact that VCSELs support two linear orthogonally polarized(LP) modes with slightly different frequencies and net gains even when lasing in the fundamental transverse mode. Thiskind of switching in a bistable manner can be referred to as electro-optical polarization bistability.1113 Due to thesmall dichroizm between the two LP VCSEL modes polarization bistability can also appear in a normally polarizationstable VCSEL when it is subject to weak external perturbation, such as optical feedback or optical injection. In the case

K. Panajotov: Tel: +32 (0)2 629 35 67 Fax: +32 (0) 629 34 50 E-Mail: kpanajotovtona.alna.ac.be

Polarization switching in VCSELs induced by opticalinjection

Krassimir Panajotov" , Boris Ryvkin2, Eugeny Avrutin3, Ignace Gatare1'4, Marc Triginer' , JordiBuesa1 , Irma Veretennicoff1, Hugo Thienpont' , Marc Sciamanna4

1 Department of Applied Physics and Photonics (TW-TONA),Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium.

2 A.F.Ioffe Physico-Technical Institute, St. Peterburg 1 9402 1 , Russia3 Department of Electronics, University of York, York YOlO 5DD, United Kingdom

4Suplec, LMOPS CNRS UMR-7132 (Laboratoire Matriaux Optiques, Photonique et Systmes), 2Rue Edouard Belin, F-57070 Metz (France).

5 Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., 1 784 Sofia, Bulgaria.

ABSTRACT

In this paper we summarize our recent theoretical and experimental results concerning optical injection in VCSELs. First,we develop phenomenological rate equation model for the normalized photon densities of the two fundamental VCSELmodes with orthogonal linear polarization when injecting external light polarized along the direction ofpolarization of thenonlazing mode. We then perform a simple analytical analysis of optically induced polarization switching in VCSELs,which takes into account both the current-dependent linear dichroism and the nonlinear dichroism. We also presentdetailed investigation of polarization dynamics induced by optical injection using an alternative rate equation model forVCSELs which takes into account the microscopic spin-flip processes in the active medium. In particular we predict thattwo injection locked solutions may coexist: the first one exhibits the polarization ofthe master laser, while the second onecorresponds to an elliptically polarized injection locking. Continuation techniques allow us to find the bifurcation scenarioleading to such an elliptically locked state. We also report on our first experimental results on polarization switching andinjection locking for the case ofan orthogonal optical injection in VCSELs and demonstrate rich dynamical behavior. Thepolarization switching is accompanied by a cascade of bifurcations to timeperiodic and possibly chaotic dynamics. Ourexperimental results show evidence of a period doubling route to chaos.

Keywords: VCSELs, optical injection, polarization switching, optical bistability, chaos, period-doubling, wave mixing.

1. INTRODUCTION

Recently, a new type ofsemiconductor lasers reached the maturity ofmass production, namely the Vertical-Cavity surface-emitting Lasers (VCSELs). These lasers emit light perpendicular to the surface of the quantum wells, compensating forthe decreased confinement factor by very high reflectivity ofthe laser mirrors - specally designed multilayer DBR. Due tothe surface emission and the cylindrical symmetry of the laser cavity polarization direction is not well fixed in VCSELs.Often they undergo polarization switching (PS) between two linear orthogonally polarized (LP) fundamental transversemodes when changing the injection current'9 (for a review see, e.g.'°) . This specific VCSELs behavior, which contrastthe polarization behavior ofedge emitting lasers, is due to the fact that VCSELs support two linear orthogonally polarized(LP) modes with slightly different frequencies and net gains even when lasing in the fundamental transverse mode. Thiskind of switching in a bistable manner can be referred to as electro-optical polarization bistability. 11—13 Due to thesmall dichroizm between the two LP VCSEL modes polarization bistability can also appear in a normally polarizationstable VCSEL when it is subject to weak external perturbation, such as optical feedback or optical injection. In the case

K. Panajotov: Tel: +32 (0)2 629 35 67 Fax: +32 (0) 629 34 50 E-Mail: [email protected]

ICONO 2005: Nonlinear Space-Time Dynamics, edited by Nikolai Rosanov, Stephano Trillo,Proc. of SPIE Vol. 6255, 625501, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.676480

Proc. of SPIE Vol. 6255 625501-1

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of optical feedback the result is the so-called channelled light-versus-current rv14 whereby multiple polarizationswitching events occur at equal intervals ofthe injection current.15' 16 Polarization bistability of another, all-optical, typearises when external light with an appropriate polarization and frequency is injected in a VCSEL that is subjected to aninjection current outside the region of electro-optical bistability.'7 Polarization switching is achieved experimentallythrough injection locking, where both the wavelength and the polarization ofthe VCSEL are locked to the injected opticalsignal.'7 Recently, it was also found experimentally that optical injection can stabilize the polarization of VCSELsoperating in the polarization mode hopping 18 Polarization switching induced by optical injection when theVCSEL is biased in its region ofelectro-optical bistability has been experimentally demonstrated by Kawaguchi et al..3' 19In,19 as well as in a number ofother theoretical 1 1, 12, 20 the nonlinear effects give rise to PS and bistability whilethe linear dichroism between the two LP modes is considered to be constant. However, it has been experimentally observedthat current dependent linear dichroism plays an important role for polarization switching in 8 21,22 This ideahas been incorporated in a rate equation model to describe quantitatively the current induced polarization switching in

13 23

In this paper we summarize our results concerning optical injection in VCSELs. In section 2 we develop phenomeno-logical rate equation model for the normalized photon densities of the LP modes when injecting external light into they—LP mode. We then perform a simple analytical analysis ofoptically induced polarization switching in VCSELs, whichtakes into account both the current-dependent linear dichroism and the nonlinear dichroism. 24in section 3 we use alternative rate equation model for VCSELs based on the well known San Miguel, Feng and Moloney(SFM) 526 and present detailed investigation of polarization dynamics induced by optical injection. Of particularimportance of this model is that it predicts that two injection locked solutions may (co)exist: the first one exhibits thepolarization of the master laser (y), while the second one corresponds to an elliptically polarized injection locking, i.e.the LP modes of the slave laser both lock to the master laser frequency. Continuation techniques allow us to map thebifurcation boundaries in the plane of the frequency detuning versus the injection strength.In section 4 we report on our first experimental results on the polarization switching and injection locking for the caseof an orthogonal optical injection in VCSELs and demonstrate rich dynamical behavior. We show that the injection of aLP light orthogonal to that of the free-running VCSEL leads to PS in an VCSEL based outside the polarization bistableregion. The PS is accompanied by a cascade ofbifurcations to timeperiodic and possibly chaotic dynamics. Experimentalresults show evidence for a period doubling route to chaos.Finally, in sections 5 we conclude.

2. ANALYTICAL MODEL FOR VCSELS UNDER OPTICAL INJECTION: STEADY STATECHARACTERISTICS AND KINETICS OF POLARIZATION SWITCHING IN VCSELS

Considering the case of optical injection into the y—polarized VCSEL fundamental mode, the VCSEL rate equations forthe photon densities ofthe polarization modes S and S3, read.(see e.g.27'28):

FSRS + vGS (1)

f FspRsp+vgGySy+cos(y) (2)

co+vgGyisin(y) (3)

where the net gains are G3 = JT3g — a3 (j = x,y) , IT, g3 and a3 are the optical confinement factors, material gains, andtotal losses for the two modes, respectively, Vg 5 the group velocity of light and a is the linewidth enhancement factor.

<<1 is the fraction of the spontaneous emission (with a total rate of =Nth/tp(Nth)) captured by the modes, whereNth is the carrier density at laser threshold and t5 is the carrier lifetime, S is the photon density of the injected lightinside the laser cavity, i.e. S = (1 — R)St, with R being the VCSEL output mirror reflectivity and S' the incident

of optical feedback the result is the so-called channelled light-versus-current rv14 whereby multiple polarizationswitching events occur at equal intervals ofthe injection rr15' 16 Polarization bistability of another, all-optical, typearises when external light with an appropriate polarization and frequency is injected in a VCSEL that is subjected to aninjection current outside the region of electro-optical ty'7 Polarization switching is achieved experimentallythrough injection locking, where both the wavelength and the polarization ofthe VCSEL are locked to the injected opticalsignal.17 Recently, it was also found experimentally that optical injection can stabilize the polarization of VCSELsoperating in the polarization mode hopping 18 Polarization switching induced by optical injection when theVCSEL is biased in its region ofelectro-optical bistability has been experimentally demonstrated by Kawaguchi et al..3' 19In,19 as well as in a number ofother theoretical 1 1, 12, 20 the nonlinear effects give rise to PS and bistability whilethe linear dichroism between the two LP modes is considered to be constant. However, it has been experimentally observedthat current dependent linear dichroism plays an important role for polarization switching in 8 21,22 This ideahas been incorporated in a rate equation model to describe quantitatively the current induced polarization switching in

13 23

In this paper we summarize our results concerning optical injection in VCSELs. In section 2 we develop phenomeno-logical rate equation model for the normalized photon densities of the LP modes when injecting external light into they—LP mode. We then perform a simple analytical analysis ofoptically induced polarization switching in VCSELs, whichtakes into account both the current-dependent linear dichroism and the nonlinear dichroism. 24in section 3 we use alternative rate equation model for VCSELs based on the well known San Miguel, Feng and Moloney(SFM) 526 and present detailed investigation of polarization dynamics induced by optical injection. Of particularimportance of this model is that it predicts that two injection locked solutions may (co)exist: the first one exhibits thepolarization of the master laser (y), while the second one corresponds to an elliptically polarized injection locking, i.e.the LP modes of the slave laser both lock to the master laser frequency. Continuation techniques allow us to map thebifurcation boundaries in the plane of the frequency detuning versus the injection strength.In section 4 we report on our first experimental results on the polarization switching and injection locking for the caseof an orthogonal optical injection in VCSELs and demonstrate rich dynamical behavior. We show that the injection of aLP light orthogonal to that of the free-running VCSEL leads to PS in an VCSEL based outside the polarization bistableregion. The PS is accompanied by a cascade ofbifurcations to timeperiodic and possibly chaotic dynamics. Experimentalresults show evidence for a period doubling route to chaos.Finally, in sections 5 we conclude.

2. ANALYTICAL MODEL FOR VCSELS UNDER OPTICAL INJECTION: STEADY STATECHARACTERISTICS AND KINETICS OF POLARIZATION SWITCHING IN VCSELS

Considering the case of optical injection into the y—polarized VCSEL fundamental mode, the VCSEL rate equations forthe photon densities ofthe polarization modes S and S, read.(see e.g.27'28):

f!? TSRS + vgGxSx (1)

f!; (2)

(3)

where the net gains are G3 = Jjgj —a (j = x,y) , IT, gj and a are the optical confinement factors, material gains, andtotal losses for the two modes, respectively, Vg 5 the group velocity of light and a is the linewidth enhancement factor.

<<1 is the fraction of the spontaneous emission (with a total rate of =Nth/tp(Nth)) captured by the modes, whereNe,, is the carrier density at laser threshold and ;, is the carrier lifetime, Si,, is the photon density of the injected lightinside the laser cavity, i.e. S = (1 — R)St, with R being the VCSEL output mirror reflectivity and St the incident



photon density at the outer surface of the output mirror. tj. is the roundtrip time in the VCSEL cavity. Finally,

LU) = (OyO—

COin = xO — &OBF —COin (4)

is the frequency detuning, xO and co being the frequencies of the free-running x-mode of the VCSEL and the injectedlight, respectively, and &OBF = uxO (Oy() the frequency splitting between the two polarization modes due to the small

birefringence that can be due to residual stress through the elasto-optic effect or to the electro-optic effect. '?,10 In ouranalysis, we consider the injection phase-locked case and very fast adaptation ofthe phase,i.e.:

y = arcsin [+a2 (+ tL] —arctan(a). (5)

We solve equations (1)- (4) by making use of the conservation law for the photon densities, S(I, t) + S(I, t) =and neglecting any time dependence of the total intensity, i.e. dS(I,t)/dt + dS(I,t)/dt =0 holds. In dimensionlessunits, the reduced equations reads:

= 2Ey3 — [3Es(i) +B(i)]y2 + [Es2(i) +B(i)s(i) +2]y—s(i) + [s(i) —y]J,s, ), (6)

where I =t/tL, tL = , a= VgIl(otSsptL, B(i) : and

Jin,sin, ) = cos = cos {)], (7)dt s5 tL1 /55f aldy\=arcsin /(l+a2)Vsn(+2ydi) —arctan(a),

—. (ex —es)S . _ I— 'th _ S • _ S(I) S5p SE— , 1— , S—;--—, y(i)_—-——, s5__--——, s,,—-——,

L)Sp SW th sw SW SW SW

Here, I is the constant injection current, 'th and 'SW are the injection currents at threshold and at polarization switching;SSW 5 the total photon density at 'SW; iGL(I = Ith) is the linear part ofthe net gain dichroism; atot is the total loss value,and r is the photon lifetime, r = (vgatot)' . S are the photon densities of the two polarization modes, so the totalconstant photon density is S = S + S,. We also introduce, following,'3 the spontaneous photon density in the lasingmode = n5/V0, where n5 ' 1 is the population inversion factor ,27 and V0 is the volume of the laser mode. €sand x are the net gain self-and cross-suppression coefficients which combine the nonlinearities of gain and absorption. 13The parameter b =B(i)/(1 — i) describes the linear change of the gain dichroism with the injection current. VCSELsteady-state characteristics as obtained from Eqns.(6-7) read:

s(i)— • — (b—3E)y2 +by— 1 —fin + /((b—3e)y2 +by— 1 —fin)2 +4(b—E)y2(2Ey2 —by+2+J)(8)1

2(b—E)y

In Fig.1 , we plot the dependencies of the normalized y-polarized photon density y on the normalized injection current Ifor zero frequency detuning &o = 0 and several values of the injected light photon density. The parameters are: S5, -2 x 1 010 cm3 , SSW 2 x 1 0 14 cm3 , b = 20, 1 0, tp/tL 1 50, and a = 3 . For a fixed value of the injection current ofi = 1 the state ofthe system will be determined by the intersection ofthe straight vertical line in Fig.1(left) passing throughi = 1 with the stable lower or upper branches of the curves y(i). Without optical injection, the system is in the lower-branch (x-LP) state (point 1 on the thick solid line in Fig.1(left)). At a small injection photon density (s =5 xthe state of the system changes only slightly (point 1' on the thin solid line in Fig.1). However, with an optical injection ofs = 2 x 1012, the change in the state of the system is dramatic - an abrupt transition from point 1 to point 2 on the upperbranch of the dashed bistable curve takes place. After such a transition, the y-LP (upper-branch) state becomes dominant.

photon density at the outer surface of the output mirror. tL is the roundtrip time in the VCSEL cavity. Finally,

&L) (0y0 in = -0xO &OBF 0jn (4)

is the frequency detuning, WxO and wm being the frequencies of the free-running x-mode of the VCSEL and the injectedlight, respectively, and AO)BF = -oxO C0yO the frequency splitting between the two polarization modes due to the small

birefringence that can be due to residual stress through the elasto-optic effect or to the electro-optic effect. '?,10 In ouranalysis, we consider the injection phase-locked case and very fast adaptation of the phase,i.e.:

y = arcsin[ a2) ) tL]

—arctan(a). (5)

We solve equations (1)- (4) by making use ofthe conservation law for the photon densities, S(I,t) + S(I,t) =and neglecting any time dependence of the total intensity, i.e. dS(I, t)/dt + dS(I, t)/dt =0 holds. In dimensionlessunits, the reduced equations reads:

= 2ey3 — [3Es(i) +B(i)]y2 + [Es2(i) +B(i)s(i) +2]y—s(i) + [s(i) —yJJ,s, ), (6)

where I =t/tL, tL = a = VgIl(otSsptL, B(i) and

fin ,Sin , ) = i? cos [)}, in cos [)] , (7)dt tL

1 rj:;-f ctldy\4y = arcsin/(i +a2)Vsin (M)tL+ —) — arctan(cz),

—. (ex—es)S . _ I— 'th _ S • _ S(I) Sp•

SinE— , 1— , s—E—, y(i)_—--—, s,,—-—-, Sin,sp 'SW 'th )sw )sw )sw )sw

Here, I is the constant injection current, 'th and I are the injection currents at threshold and at polarization switching;Ssw S the total photon density at I; AGL(I = Ith) is the linear part of the net gain dichroism; a101 is the total loss value,and t is the photon lifetime, 'r = (vgatot)' . S are the photon densities of the two polarization modes, so the totalconstant photon density is S = S+ 5,,. We also introduce, following,'3 the spontaneous photon density in the lasingmode S = n/V0t, where n5 1 is the population inversion factor ,27 and V0, is the volume of the laser mode. sand x are the net gain self-and cross-suppression coefficients which combine the nonlinearities of gain and absorption. 13The parameter b =B(i)/(1 — i) describes the linear change of the gain dichroism with the injection current. VCSELsteady-state characteristics as obtained from Eqns.(6-7) read:

s()—— (b3c)y2+by1Jin+((b3e)y2+by1fin)2+4(b)y2(2Ey2by+2+Jin)(8)1 1

2(b—E)y

In Fig.1 , we plot the dependencies of the normalized y-polarized photon density y on the normalized injection current ifor zero frequency detuning i±tü = 0 and several values of the injected light photon density. The parameters are:2 x 1 0 cm3 , 2 x 1 0 cm3 , b = 20, = 1 0, tp/tL = I50, and a = 3 . Fora fixed value of the injection current ofi = 1 the state ofthe system will be determined by the intersection ofthe straight vertical line in Fig.1(left) passing throughi = 1 with the stable lower or upper branches of the curves y(i). Without optical injection, the system is in the lower-branch (x-LP) state (point 1 on the thick solid line in Fig.1(left)). At a small injection photon density (Sin =5 X 1O'),the state of the system changes only slightly (point 1' on the thin solid line in Fig.1). However, with an optical injection ofin = 2 x 10—12, the change in the state of the system is dramatic - an abrupt transition from point 1 to point 2 on the upperbranch of the dashed bistable curve takes place. After such a transition, the y-LP (upper-branch) state becomes dominant.



___Figure 1. (left) Dependence of y-polarized normalized output power y on the normalized injection current i for &o = 0 and different

injection strength: s = 0 - thick solid line,s, = 5 x 1013 solid line, s = 2 x 1012 dashed line, = 10b0 dashed-dotted line,sin = iO7 - dotted line.

Figure 2. (right) (a)Dependence of y-polarized normalized output power y on the normalized injection current i. The thick solid curve

(1) is for the case without optical injection. (a) optical injection strength ofs = 2.5x 10 12 and negative frequency detuning: LOYtL = 0- solid line (2), &OtL —2 x 10_6 dashed line (3), &OtL —3.3 x 106 dashed-dotted line (4), LWtL = —1.5 x i0 - dotted line;(b) optical injection strength ofs = 1 x 10_li and positive frequency detuning: LWtL = 0 - solid line (2), &JYtL = 1 x i0- dashedline (3), LWtL 1 .5 x iO - dashed-dotted line (4).

In Fig.2(a), we show the dependence ofthe y-polarized photon density y on the injection current i for negative (Fig.2(a))and positive (Fig.2(b)) frequency detunings &JYtL. The thick solid curve denoted by I is calculated for the case ofa solitarylaser (s = 0). The injection (s = I x 1012 in Fig.2(a) - curve 2) shifts the bistable curve to the left (towards smallercurrents) and also shrinks the bistability region, thus facilitating polarization switching. However, introducing a smallnegative detuning brings the bistable curve almost back to the initial position (Fig.2(a) -curve 3; our simulations showedthe same effect even for twice the injection strength). Increasing the negative detuning further (Fig.2(a) -curve 4) shiftsthe bistable curve to the right, towards larger currents, but at the same time the upper branch of the hysteresis curve candisappear, meaning that no CW phase-locked operation is possible in this case. In the opposite case ofa positive detuningshown in Fig.2(b), as the detuning is increased, the hysteresis region first moves to the left and shrinks (Fig.2(b)- curve 2)and then disappears (Fig.2(b)- curve 3). At a still larger detuning, EtWtL =I .5 x iO, the second steady state ceases toexist at injection power above a certain value, and so no transition from the low state can occur for i =1 (Fig.2(b)- curve4). Note that this loss of a stable branch is due to the argument ofthe arcsine in (4) exceeding one and is thus not specificto the dual-polarization laser: even with a single-frequency laser, no stable injection-locked state is known to exist whenboth the detuning and the free-running laser power are large enough.

We now determine the characteristic time of polarization switching induced by optical injection, i.e. the time oftransition from a stationary state near the turning point ofthe bistable loop without optical injection (point I in Fig.1) to astate at the upper branch of the bistable curve with injection (point 2). To this end, we discuss the reaction of the VCSEL

. . . . . . dS(at the state 1) to a step signal of optical injection applied at t = 0. If we use the steady-state (- = 0) solution of theEqn. (5) the phase is a function ofy and the normalised injection J1, can also be expressed as a function J,, (y) . In sucha way, the kinetics of the transition from the state Ydown to the state Yup (Ydown < Y � Yup) are easily determined from thesolution of Eqn.(6) by separation of variables:

t [Yup d — [Yup i dy9

tL Ydown

—Ydown a[—2y3 + (3Ei— bi+b)y2 + (bi2 — bi— —2)y+ i] + (y)n&)' (

with .J y) calculated from (7) using , from the steady-state version of Eqn.(5).

___Figure 1. (left) Dependence of y-polarized normalized output power y on the normalized injection current i for Ao = 0 and different

injection strength: s = 0 - thick solid line,s, = 5 x 1013 solid line, s = 2 x 1012 dashed line, s,, = 10b0 dashed-dotted line,sin = lO - dotted line.

Figure 2. (right) (a)Dependence of y-polarized normalized output powery on the normalized injection current i. The thick solid curve

(1) is for the case without optical injection. (a) optical injection strength ofs, = 2.5x l012 and negative frequency detuning: MYtL = 0- solid line (2), &OIL —2 x l0_6 dashed line (3), &JYCL —3.3 x 106 dashed-dotted line (4), LO)tL = —1.5 x i0 - dotted line;(b) optical injection strength ofs = I x 10h1 and positive frequency detuning: EIWtL = 0 - solid line (2), EWEL = I x l0- dashedline (3), LO)TL I .5 x I 0 - dashed-dotted line (4).

In Fig.2(a), we show the dependence of the y-polarized photon density y on the injection current i for negative (Fig.2(a))and positive (Fig.2(b)) frequency detunings &JYtL. The thick solid curve denoted by 1 is calculated for the case ofa solitarylaser (s, = 0). The injection (s1 = 1 x 1012 in Fig.2(a) - curve 2) shifts the bistable curve to the left (towards smallercurrents) and also shrinks the bistability region, thus facilitating polarization switching. However, introducing a smallnegative detuning brings the bistable curve almost back to the initial position (Fig.2(a) -curve 3; our simulations showedthe same effect even for twice the injection strength). Increasing the negative detuning further (Fig.2(a) -curve 4) shiftsthe bistable curve to the right, towards larger currents, but at the same time the upper branch of the hysteresis curve candisappear, meaning that no CW phase-locked operation is possible in this case. In the opposite case ofa positive detuningshown in Fig.2(b), as the detuning is increased, the hysteresis region first moves to the left and shrinks (Fig.2(b)- curve 2)and then disappears (Fig.2(b)- curve 3). At a still larger detuning, EtWtL =1 .5 x i0, the second steady state ceases toexist at injection power above a certain value, and so no transition from the low state can occur for i =1 (Fig.2(b)- curve4). Note that this loss of a stable branch is due to the argument ofthe arcsine in (4) exceeding one and is thus not specificto the dual-polarization laser: even with a single-frequency laser, no stable injection-locked state is known to exist whenboth the detuning and the free-running laser power are large enough.

We now determine the characteristic time of polarization switching induced by optical injection, i.e. the time oftransition from a stationary state near the turning point of the bistable loop without optical injection (point 1 in Fig.l) to astate at the upper branch ofthe bistable curve with injection (point 2). To this end, we discuss the reaction ofthe VCSEL

. . . . . . dS(at the state 1) to a step signal of optical injection applied at t = 0. If we use the steady-state (- = 0) solution of theEqn. (5) the phase is a function ofy and the normalised injection J1, can also be expressed as a function J,(y).In sucha way, the kinetics of the transition from the state Ydown to the state Yup (Ydown < Y Yup) are easily determined from thesolution of Eqn.(6) by separation of variables:

t [Yup d — [Yup i dy9

Ydown

—'Ydown a{2y3 + (3Ei— bi+b)y2 + (bi2 — bi— i2 —2)y+ i] + (i—y)n)' (

with fin y) calculated from (7) using , from the steady-state version of Eqn.(5).



0 1 2 3 4 5 6 7

tltL X 1061o4

(b)

1.5

sin1 x i07

0.8

>: _____________________________ _____________________________0 1 2 3 4 5 6 .2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

UtL x 106 sini x i0_6

Figure 3. (left) Kinetics of increasing of the photon density of the y LP mode during polarization switching process and for different

photon densities ofthe injected light: s = 2x 1012 solid line (1); s = 5 x 1O12 dashed line (2) and s = 2 x 10_b dotted line(3); (a) and (b) are with (without) the term responsible for the critical slowing down in eqn.(9).

Figure 4. (right) Dependence of the polarization switching time tsw/tL between the states Ydown 0 1 1 y (Fig. 1) and Yup 0.9 Y2(Fig. 1) on the photon density ofthe injected light and for different frequency detunings: (a) LWtL = 0 solid line (1), LWtL = — 1 x iOdashed line (2) and LWtL 3 x iO dot line (3); (b) LWtL = 0 solid line (1), LWtL = —3 x iO dashed line (2) and &OtL = 1 x iOdot line (3).

The kinetics ofthe polarization switching process, i.e. the dependence ofthey-LP mode photon densityy (yi �y Y2)on the normalized time 1, is shown in Fig.3 for different injected photon densities. The curves are calculated eithertaking into account (Fig.3(a)) or not taking into account (Fig.3(b)) the square-bracketed critical slowing down term inthe denominator of the Eqn.(9). In fig.4, we show the dependence of the total polarization switching time tsw/'tL onthe photon density of the injected light for different frequency detunings and for two ranges of the injection strength2 x 1O_8 2 x iO in Fig.4(a) and 2 x iO —2 x 106 in Fig.4(b). Also shown is the lower limit ofthe switching time(curves denoted by 4), calculated using (1 1). As can be expected, the switching time decreases with and approachesthe theoretical limit. On the other hand, it increases with the (negative) detuning, especially for the case ofweak injection.Still, with cL "-j 1014s normal for VCSELs, switching times of the order of -' 5 are predicted by our results,confirming the possibility ofusing optically triggered polarization switching for all optical flip-flop operation. 29

3. POLARIZATION DYNAMICS IN OPTICALLY INJECTED VCSELS ON THE BASE OF SANMIGUEL - FENG - MOLONEYMODEL

Alternative rate equation model for VCSEL has been proposed by San Miguel, Feng and Moloney (SFM)25 extended tothe case of optical injection in.26 The SFM model adds some features to the classical twomode equations (1)-(3), such asthe inclusion ofmicroscopic spin relaxation processes. In the reference frame ofthe master laser, our model writes:

— = K(1 (10)

= K(1 + i) (DF — idF — ) + i(p _AW)Fy+ + K141, (11), =_[D(1+IFXI2+wI2)] +_jd(*_FF), (12)

(13)

0 1 2 3 4 5 6 7

tltL X 1061o4

(b)

1.5

sn1 x 10

0.8

___________ 0iTHi:iji:i::I:j::Ijijitt::Il:ffl0 1 2 3 4 5 6 .2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

UtL x 106 sini x 10_6

Figure 3. (left) Kinetics of increasing of the photon density of the y LP mode during polarization switching process and for different

photon densities ofthe injected light: se,, 2x 1012 solid line (1); s = 5 x lO_12 dashed line (2) and s = 2 x 1O_10 dotted line(3); (a) and (b) are with (without) the term responsible for the critical slowing down in eqn.(9).

Figure 4. (right) Dependence ofthe polarization switching time tsw/tL between the statesydown 0.11 y (Fig.1) andy = 0.9=Y2(Fig. 1) on the photon density ofthe injected light and for different frequency detunings: (a) LWtL = 0 solid line (1), &JYCL = — 1 x iOdashed line (2) and L(.O'CL 3 x iO dot line (3); (b) MYtL = 0 solid line (I), LWtL = —3 x lO dashed line (2) and &OTL = 1 x iOdot line (3).

The kinetics ofthe polarization switching process, i.e. the dependence ofthey-LP mode photon densityy (yi y Y2)on the normalized time 1, is shown in Fig.3 for different injected photon densities. The curves are calculated eithertaking into account (Fig.3(a)) or not taking into account (Fig.3(b)) the square-bracketed critical slowing down term inthe denominator of the Eqn.(9). In fig.4, we show the dependence of the total polarization switching time tSW/'tj. onthe photon density of the injected light for different frequency detunings and for two ranges of the injection strength2 x 1O_8 2 x iO in Fig.4(a) and 2 x iO —2 x 106 in Fig.4(b). Also shown is the lower limit ofthe switching time(curves denoted by 4), calculated using (1 1). As can be expected, the switching time decreases with Sin and approachesthe theoretical limit. On the other hand, it increases with the (negative) detuning, especially for the case ofweak injection.Still, with cL 1014s normal for VCSELs, switching times of the order of -' S are predicted by our results,confirming the possibility ofusing optically triggered polarization switching for all optical flip-flop operation.29

3. POLARIZATION DYNAMICS IN OPTICALLY INJECTED VCSELS ON THE BASE OF SANMIGUEL - FENG - MOLONEYMODEL

Alternative rate equation model for VCSEL has been proposed by San Miguel, Feng and Moloney (SFM)25 extended tothe case of optical injection in.26 The SFM model adds some features to the classical twomode equations (1)-(3), such asthe inclusion ofmicroscopic spin relaxation processes. In the reference frame ofthe master laser, our model writes:

.. = K(1 + Ia) (DFX + idF — F) — i(p+ Aw)F — YaFx, , (10)

= K( + i) (D — idF — F)+ i(p _W)Fy+ aFy + K141, (1 fl, = ie [D(1+Fx2+Fy2)] (12)

Ysd_yed(IFxI2+II2) _jyeD(*FxF). (13)



: 0.8(jcJ

'T

C

oC(0

coO2Co 8

pinJJto [101%1

Figure 5. Bifurcation diagram ofI and I, as a function of P11/Io.

are the slowly varying x- and y-LP components ofthe electric field. D and d are two carrier inversion numbers.25 Themeaning of the parameters can be found elsewhere.25 Parameters are chosen such that the free-running VCSEL exhibitsa stationary x-LP state, i.e. the injection is LP and orthogonal to the free-running VCSEL polarization: y =30(rad/ns),Ia O.5ns1, Ys 50ns1, a = 3, K = 300ns1, Ye 1ns1, andp = 1.5. i' is the injection rate, E1 is the injectedfield amplitude, and LU) is the detuning: Aco = cn — cnth, where flth = (u+ u3,)/2. o, are the frequencies of the twoVCSEL LP modes: = Fi cqa. 1'inj 5 fixed to = i, which corresponds to the optimal case ofa mode-matched

injected input beam.

Figure 5 shows the bifurcation diagram of IFI2 as a function of the normalized injected power P/Io andfor &o = Yp• Pinj EJ2 and I is the total intensity without injection. As we increase the injection strength, theintensity ofthe normally depressed y-LP mode increases and the VCSEL successively bifurcates to qualitatively differentdynamics; see time-traces in Fig. 6 corresponding to the labels (a)-(h) in Fig. 5. For very small injected power, the twoLP mode intensities exhibit a time-periodic_dynamics (al), (a2) at ' 0.48 GHz, which is much smaller than the relaxation

oscillation (RO) frequency [fRo /iYe (U _ 1)/(2it) 2.77GHz] or the beating frequency between the two LP modes{2y/(2ic) 2 9.5GHz]. The frequency offset between the two fields is given by fj aya/(2ic) 0.24GHz. The wavemixing is linked to a pulsation of the population inversion, which in turn yields an intensity modulation at about 2 f°jj.For a slightly larger injection amplitude the LP mode intensities still exhibit a slow time-periodic dynamics but now theyrelax with faster oscillations at the RO frequency (bi), (b2). The LP mode intensities are anticorrelated at the time-scaleof the slow oscillations and are in phase at the RO time-scale. For a still larger amount of optical injection, the two LPmodes exhibit a stationary behavior (ci), (c2) and are locked to the frequency ofthe master laser defining anew kind ofelliptically polarized injection locking (EPIL) state. The EPIL steady-state undergoes a Hopfbifurcation to time-periodic,inphase dynamics at the RO frequency in the two LP modes (dl), (d2). As we increase the injection strength further thetime-periodic dynamics bifurcates to period-two (el), (e2), period-four (1.1), (f2) etc. until the chaotic regime (gi), (g2)with irregular bursts ofpulses that extend over a large range ofintensity values. Finally, the bifurcation cascade leads to aPS induced by optical injection with a switch-offofthe x-LP mode and an injection locking ofthe y-LP mode (hi), (h2).It is worth noting that bistability is also found in a quite large range ofinjection strength between the y-LP injection lockedsteady-state and either time-periodic, period-doubled or even chaotic regimes, which emerge from the EPIL steady-state.In order to gain insight into the bifurcations on the EPIL steady-state, we have used the package DDE-BIFTOOL3° whichallows to follow branches of steady-states and time-periodic solutions irrespective of their stability. In Fig. 7 we plot inblack (gray) the values ofI (Li,) corresponding to the injection locked solutions as a function ofP1/Io and for differentAco. Stable (unstable) parts of the branches are indicated by solid (dashed) lines. Saddle-node, transcritical and Hopfbifurcations are labeled with squares, stars and diamonds respectively. Bold diamonds emphasize the supercritical Hopfbifurcations, i.e. which modify the stability ofthe steady-states.A particular case is (d) for which &o = = +ala. As we increase the injected power from zero, a stable EPIL

steady-state appears with I > I. For larger P/Io, it undergoes a supercritical Hopf bifurcation to a time-periodic

: 0.8£ (jcJ T 'T

C

C(coO2

Co 8PinJJto [1O_1%]

Figure 5. Bifurcation diagram °fI and I, as a function of P,1/Io.

are the slowly varying x- and y-LP components ofthe electric fie'd. D and d are two carrier inversion numbers.25 Themeaning of the parameters can be found elsewhere.25 Parameters are chosen such that the free-running VCSEL exhibitsa stationary x-LP state, i.e. the injection is LP and orthogonal to the free-running VCSEL polarization: y =30(rad/ns),Ia O.5ns1, Ys 50ns1, a = 3, K = 300ns1, y 1ns1, andp = 1.5. is the injection rate, is the injectedfield amplitude, and LQ) is the detuning: Aw = — üth, where Wth = (o+ u3,)/2. are the frequencies of the twoVCSEL LP modes: w3, = Fy a?a. K1 5 fixed to = K, which corresponds to the optimal case ofa mode-matched

injected input beam.

Figure 5 shows the bifurcation diagram of IFI2 as a function of the normalized injected power P,/Io andfor &o = P,,1 EJ2 and Jo is the total intensity without injection. As we increase the injection strength, theintensity of the normally depressed y-LP mode increases and the VCSEL successively bifurcates to qualitatively differentdynamics; see time-traces in Fig. 6 corresponding to the labels (a)-(h) in Fig. 5. For very small injected power, the twoLP mode intensities exhibit a time-periodic_dynamics (al), (a2) at ' 0.48 GHz, which is much smaller than the relaxation

oscillation (RO) frequency [fRo /iYe (U _ l)/(2it) 2.77GHz] or the beating frequency between the two LP modes{2y/(2ic) 2 9.5GHzI. The frequency offset between the two fields is given by fjj aya/(2ic) 0.24GHz. The wavemixing is linked to a pulsation of the population inversion, which in turn yields an intensity modulation at about 2 f0j.For a slightly larger injection amplitude the LP mode intensities still exhibit a slow time-periodic dynamics but now theyrelax with faster oscillations at the RO frequency (bi), (b2). The LP mode intensities are anticorrelated at the time-scaleof the slow oscillations and are in phase at the RO time-scale. For a still larger amount of optical injection, the two LPmodes exhibit a stationary behavior (ci), (c2) and are locked to the frequency ofthe master laser defining anew kind ofelliptically polarized injection locking (EPIL) state. The EPIL steady-state undergoes a Hopfbifurcation to time-periodic,inphase dynamics at the RO frequency in the two LP modes (dl), (d2). As we increase the injection strength further thetime-periodic dynamics bifurcates to period-two (el), (e2), period-four (fi), (f2) etc. until the chaotic regime (gi), (g2)with irregular bursts ofpulses that extend over a large range ofintensity values. Finally, the bifurcation cascade leads to aPS induced by optical injection with a switch-offofthe x-LP mode and an injection locking ofthe y-LP mode (hi), (h2).It is worth noting that bistability is also found in a quite large range ofinjection strength between the y-LP injection lockedsteady-state and either time-periodic, period-doubled or even chaotic regimes, which emerge from the EPIL steady-state.In order to gain insight into the bifurcations on the EPIL steady-state, we have used the package DDE-BIFTOOL30 whichallows to follow branches of steady-states and time-periodic solutions irrespective of their stability. In Fig. 7 we plot inblack (gray) the values ofI (4,) corresponding to the injection locked solutions as a function ofP1/Io and for differentAco. Stable (unstable) parts of the branches are indicated by solid (dashed) lines. Saddle-node, transcritical and Hopfbifurcations are labeled with squares, stars and diamonds respectively. Bold diamonds emphasize the supercritical Hopfbifurcations, i.e. which modify the stability ofthe steady-states.

A particular case is (d) for which &o = co = — +ala. As we increase the injected power from zero, a stable EPILsteady-state appears with I > I. For larger P,,j/Io, it undergoes a supercritical Hopf bifurcation to a time-periodic



—x___________________________

—x____________________________

—>'

I3I _______________—>'

x oL. (hi)1

— o.[ (h2)

o 5 . iO i5 20Time (ns)

Figure 6. Time-traces ofI for (al), (a2): P,1/Io =0.03 %, (bi), (b2): P1/Io = 0.16 %, (ci), (c2): P1/Io = 018 %, (dl), (d2):Pinh/Jo = 0.35 %, (el), (e2): P1/Io = 0.47 %, (fi), (f2): P1/Io = 0.51 %, (gi), (g2): P,1/Io =0.54%, (hi), (h2): P,1/Io = 0.55 %.

005P iI

irtj 0

Figure 7. Steady-states and their stability for (a) &i = — i6 (radlns), (b) M = —20 (radlns), (c) &i = —28 (rad/ns), (d) & = —28.5

(rad/ns), (e) &J) = —30 (radlns), (f) zw = —34 (radlns). I (4) are plotted in black (gray) as a function ofP11/Io. Symbols are defi nedin the text.

solution in the two LP modes. For a still larger value ofP1/Io, the only stable steady-state is the y-LP injection lockedsolution which appears from a saddle-node bifurcation. The node corresponds to the branch with the largest I, and I =0.The branches ofEPIL and y-LP injection locked solutions cross at a transcritical bifurcation. For larger negative detunings(e), (f), the EPIL solution suddenly unfolds. A saddle-node bifurcation leads to two branches of solutions with differentIx,y. The branch with the largest (smallest) I (If) values is a stable solution (node). The second branch correspondsto a saddle-type solution. The node undergoes a supercritical Hopf bifurcation to a stable time-periodic solution in the

_x 04

-(c2)I

-x

—>.

Time (ns)

_)( ,0.2\

0002 000

* 042

: \\ !%06 0O80,05P iI

InJ 0

XH\JrY\iY\f\JTJ\l) q—>'

—x —x

ç.45

—>'

-(c2)I

-K —x o.F

—>'

°05101520 O[Time (ns)

-JII

o 5 . 10 15 20Time (ns)

Figure 6. Time-traces ofI for (al), (a2): Pj/Io = 0.03 %, (bi), (b2): P1/Io = 0.16 %, (ci), (c2): Pmj/Io 0.18 %, (dl), (d2):Pini/Jo = 0.35 %, (el), (e2): P1,j/Io = 0.47 %, (fi), (f2): P/Io = 0.51 %, (gi), (g2): P1/Io =0.54%, (hi), (h2): P1,1/Io = 0.55 %.

0.05P iI

irtj 0

Figure 7. Steady-states and their stability for (a) Ao = — i6 (rad/ns), (b) &o = —20 (radlns), (c) M = —28 (rad/ns), (d) Ao = —28.5

(rad/ns), (e) &i) = —30 (radlns), (f) w = —34 (rad/ns). I (4) are plotted in black (gray) as a function ofP1/Io. Symbols are deli nedin the text.

solution in the two LP modes. For a still larger value ofP1/Io, the only stable steady-state is the y-LP injection lockedsolution which appears from a saddle-node bifurcation. The node corresponds to the branch with the largest 4, andI =0.The branches ofEPIL and y-LP injection locked solutions cross at a transcritical bifurcation. For larger negative detunings(e), (f), the EPIL solution suddenly unfolds. A saddle-node bifurcation leads to two branches of solutions with differentIx,y. The branch with the largest (smallest) I (If) values is a stable solution (node). The second branch correspondsto a saddle-type solution. The node undergoes a supercritical Hopf bifurcation to a stable time-periodic solution in the

—x(c1) ___________________________

—>.

_)( '-,.

0.2\ o5tI-::J

0002 000

\ 042

: "\ !%o oo0.05P iI

101 0



two LP modes. Interestingly, the saddle-node bifurcation on the EPIL steady-state might be located at larger injectionstrength than the saddle-node on the y-LP injection locked solution (f), hence leading to bistability in a quite large rangeof injected powers. Once the EPIL steady-state destabilizes with a Hopf bifurcation, the bistability is between a time-periodic dynamics in the two LP modes and the y-LP injection locking. For smaller negative detunings (a)-(c), a similarunfolding mechanism occurs with a saddle-node bifurcation appearing on the EPIL solution. In (c), a stable branch ofEPIL solution is still possible but, as we decrease the detuning, the stability ofthe EPIL solution is lost as soon as the Hopfbifurcation coalesces with the saddle-node bifurcation. For detuning values beyond this co-dimension two bifurcation,the Hopfbifurcation determining the stability ofthe EPIL steady-state is now on the saddle branch and the EPIL solutionis then completely unstable (a)-(b).

4. POLARIZATION DYNAMICS IN OPTICALLY INJECTED VCSELS - EXPERIMENTALRESULTS

In this section we summarize some of our preliminary experimental results on the nonlinear polarization dynamics of aVCSEL with orthogonal optical injection. Our experimental setup is detailed in Fig. 8. In our experiment, we use anoxide-confined Al- GaAs/GaAs quantum well VCSEL emitting at 845nm. It emits a vertical polarization at thresholdbut switches to the horizontal, lower frequency mode, at about 1 .6 mA (type I PS). The VCSEL switches back to thevertical, higher frequency mode at about 3.5 mA (type II PS). The type I PS is accompanied by a polarization modehopping behavior. The type II PS occurs with hysteresis. Our master laser (ML) is an external cavity diode laser (TEC100 Littrow). A non-polarizing 50/50 beamsplitter (BS) is used to guide the light to the detection branch. A half-waveplate (HWP1) allows detecting the desired polarization (horizontal or vertical). An optical isolator (1501) prevents backreflections from the fiber coupling unit (FCU) or the fiber facets. A second optical isolator (1502) prevents opticalfeedback from the master laser. The polarizer P1 is used to vary the optical injection strength. The half-wave plateHWP2 and the polarizer P2 help to improve the linearity of the polarization of the master laser light and to ensure that thepolarization ofthe injected light is orthogonal to the polarization ofthe free-running VCSEL. A computer (PC2) has beenused to display and record the optical spectrum, being connected to the detector-amplifier of a Fabry-Prot interferometer

SI :ii: __ i1

LcJL L!TI ::i l I i ; I '

JLj1 L11

() UI

I :__

Figure 8. (left)Experimental setup for the investigation of VCSEL dynamics with orthogonal optical injection. Labels are defi ned inthe text;

Figure 9. (right) Polarization-resolved optical spectra of the vertical (grey) and horizontal (black) polarizations, for a frequency detun-ing of 6 GHz and increasing values of the injected power. The vertical line shows the ML frequency. The spectra illustrate a typicalsequence of bifurcations to chaos.

two LP modes. Interestingly, the saddle-node bifurcation on the EPIL steady-state might be located at larger injectionstrength than the saddle-node on the y-LP injection locked solution (f), hence leading to bistability in a quite large rangeof injected powers. Once the EPIL steady-state destabilizes with a Hopf bifurcation, the bistability is between a time-periodic dynamics in the two LP modes and the y-LP injection locking. For smaller negative detunings (a)-(c), a similarunfolding mechanism occurs with a saddle-node bifurcation appearing on the EPIL solution. In (c), a stable branch ofEPIL solution is still possible but, as we decrease the detuning, the stability ofthe EPIL solution is lost as soon as the Hopfbifurcation coalesces with the saddle-node bifurcation. For detuning values beyond this co-dimension two bifurcation,the Hopfbifurcation determining the stability ofthe EPIL steady-state is now on the saddle branch and the EPIL solutionis then completely unstable (a)-(b).

4. POLARIZATION DYNAMICS IN OPTICALLY INJECTED VCSELS - EXPERIMENTALRESULTS

in this section we summarize some of our preliminary experimental results on the nonlinear polarization dynamics of aVCSEL with orthogonal optical injection. Our experimental setup is detailed in Fig. 8. In our experiment, we use anoxide-confined Al- GaAs/GaAs quantum well VCSEL emitting at 845nm. It emits a vertical polarization at thresholdbut switches to the horizontal, lower frequency mode, at about 1 .6 mA (type I PS). The VCSEL switches back to thevertical, higher frequency mode at about 3.5 mA (type II PS). The type I PS is accompanied by a polarization modehopping behavior. The type II PS occurs with hysteresis. Our master laser (ML) is an external cavity diode laser (TEC100 Littrow). A non-polarizing 50/50 beamsplitter (BS) is used to guide the light to the detection branch. A half-waveplate (HWP1) allows detecting the desired polarization (horizontal or vertical). An optical isolator (1501) prevents backreflections from the fiber coupling unit (FCU) or the fiber facets. A second optical isolator (1502) prevents opticalfeedback from the master laser. The polarizer P1 is used to vary the optical injection strength. The half-wave plateHWP2 and the polarizer P2 help to improve the linearity ofthe polarization of the master laser light and to ensure that thepolarization ofthe injected light is orthogonal to the polarization ofthe free-running VCSEL. A computer (PC2) has beenused to display and record the optical spectrum, being connected to the detector-amplifier of a Fabry-Prot interferometer

SI

I:__ [; I1•:5 i •?':1Li L:

::

ULjiILI() UI

Figure 8. (left)Experimental setup for the investigation of VCSEL dynamics with orthogonal optical injection. Labels are defi ned inthe text;

Figure 9. (right) Polarization-resolved optical spectra of the vertical (grey) and horizontal (black) polarizations, for a frequency detun-ing of 6 GHz and increasing values of the injected power. The vertical line shows the ML frequency. The spectra illustrate a typicalsequence of bifurcations to chaos.



(FP) with a free spectral range (FSR) of 30 GHz and a finesse of about 150. To measure large frequency detunings, wehave used an optical spectrum analyzer (OSA) Ando-AQ6317B. The presence ofthe mirror (M), which can be covered,is necessary to have a reference of the position (frequency) of the master laser in the spectrum. The box called arm-I corresponds to a two-positions mobile stage that consists of two photodetectors connected to a power meter (PM2).When placed in the middle of the trajectory of the beams, they can measure the averaged total power of the output lightfrom the VCSEL and the injected power. Time-traces of the polarized intensities are analyzed with the combination of4 GHz photodiode (PD) and digital oscilloscope (OSC) Tektronix CSA 7404 or spectrum analyzer (SA). Time-traces ofthe polarized intensities are analyzed with the combination of 4 GHz photodiode (PD) and digital oscilloscope (OSC)Tektronix CSA 7404 or spectrum analyzer (SA).

Fig.9 shows experimental optical spectra for different values ofthe injected power and frequency detuning of6 GHz.It illustrates a typical sequence that is observed for a small positive detuning. The free-running VCSEL emits only inhorizontal polarization. A small injected power in the vertical polarization induces first a wave mixing dynamics [Fig. 9(a)]. For a larger injected power the PS is achieved and still accompanied by wave mixing [Fig. 9 (b)J. Increasing theinjection strength further leads to undamping of the relaxation oscillation and side peaks appear on each side of theslave laser peak. This limit cycle dynamics accompanies the wave mixing and occurs when the frequency differencebetween the master laser and the (pushed) slave laser is close to twice the relaxation oscillation frequency. For still largerinjected power the laser enters into a period doubling route to chaos and interestingly, the horizontal polarization, whichis normally depressed after the PS, is now strongly enhanced [Fig. 9 (d), (e)]. The chaotic dynamics disappears for largerinjected power and the laser then again exhibits the combination ofa limit cycle with wave mixing [Fig. 9 (f)J, and finallya much more steady behavior for very large injection strength [Fig. 9 (g)J.

5. CONCLUSIONS

In summary, we have presented a survey on our recent theoretical and experimental results on optical injection in VCSELs.We discuss a simple twomode phenomenological rate equation model for the photon densities ofthe two fundamental VC-SEL modes with orthogonal linear polarization which leads to simple analytical analysis of optically induced polarizationswitching in VCSELs. We have also discussed an alternative rate equation model for VCSELs, namely the SFM modelwhich predicts an elliptically polarized injection locking. The bifurcation scenario leading to such an elliptically lockedstate is also analyzed. Finally, we report experimental results demonstrating polarization switching and injection lockingas well as an evidence of a period doubling route to chaos.

ACKNOWLEDGMENTS

The authors acknowledge the support from the Inter-University Attraction Pole (lAP V/19) "Photon" program of theBelgian government, the Fund for Scientific Research (FWO) - Flanders, Belgium and the the support of COST 288"Nanoscale and ultrafast photonics". This work also benefited from project support from the Geconcertande OnderzoeksAktie (GOA) and the Onderzoeksraad (OZR) ofthe Vrije Universiteit Brussels. We thank Thomas Erneux for fruitful andstimulating discussions on the optical injection problem.

REFERENCESI . K. D. Choquette, D. A. Richie, and R. E. Leibenguth, Appl. Phys. Lett., 64, 2062, 1994.2. K. D. Choquette, R. P. Schneider Jr., K. L. Lear, and R E. Leibenguth, IEEE J. Sel. Top. Quant. Electr., 1, 661, 1995.

3. H.Kawaguchi, I.S.Hidayat, Y.Takahashi, Y. Yamayoshi, Electr. Lett., 31, 109, 1995.4. J.Martin-Regalado, F.Prati, M.SanMiguel, and N.B.Abraham, IEEE J.Quant.Electr., 33, 765, 1997.5. J.Martin-Regalado, J.L.a.Chilla, J.J.Rocca, and P.Brusenbach, Appl. Phys. Lett., 70, 3350(1997).6. M.P.van Exter, M.B.Willemsen, and J.P.Woerdman, Phys. Rev.A, 58, 4191, 1998.

7. K. Panajotov, B. Ryvkin, J. Dackaert, M. Peeters, H. Thienpont, I. Veretennicoff, IEEE Phot. Technol. Lett., 10, 6,(1998).

(FP) with a free spectral range (FSR) of 30 GHz and a finesse of about 150. To measure large frequency detunings, wehave used an optical spectrum analyzer (OSA) Ando-AQ63 1 7B. The presence of the mirror (M), which can be covered,is necessary to have a reference of the position (frequency) of the master laser in the spectrum. The box called arm-I corresponds to a two-positions mobile stage that consists of two photodetectors connected to a power meter (PM2).When placed in the middle of the trajectory of the beams, they can measure the averaged total power of the output lightfrom the VCSEL and the injected power. Time-traces of the polarized intensities are analyzed with the combination of4 GHz photodiode (PD) and digital oscilloscope (OSC) Tektronix CSA 7404 or spectrum analyzer (SA). Time-traces ofthe polarized intensities are analyzed with the combination of 4 GHz photodiode (PD) and digital oscilloscope (USC)Tektronix CSA 7404 or spectrum analyzer (SA).

Fig.9 shows experimental optical spectra for different values of the injected power and frequency detuning of 6 GHz.It illustrates a typical sequence that is observed for a small positive detuning. The free-running VCSEL emits only inhorizontal polarization. A small injected power in the vertical polarization induces first a wave mixing dynamics [Fig. 9(a)] . For a larger injected power the PS is achieved and still accompanied by wave mixing [Fig. 9 (b)]. Increasing theinjection strength further leads to undamping of the relaxation oscillation and side peaks appear on each side of theslave laser peak. This limit cycle dynamics accompanies the wave mixing and occurs when the frequency differencebetween the master laser and the (pushed) slave laser is close to twice the relaxation oscillation frequency. For still largerinjected power the laser enters into a period doubling route to chaos and interestingly, the horizontal polarization, whichis normally depressed after the PS, is now strongly enhanced [Fig. 9 (d), (e)]. The chaotic dynamics disappears for largerinjected power and the laser then again exhibits the combination of a limit cycle with wave mixing [Fig. 9 (f)], and finally

a much more steady behavior for very large injection strength [Fig. 9 (g)].

5. CONCLUSIONS

In summary, we have presented a survey on our recent theoretical and experimental results on optical injection in VCSELs.We discuss a simple twomode phenomenological rate equation model for the photon densities ofthe two fundamental VC-SEL modes with orthogonal linear polarization which leads to simple analytical analysis of optically induced polarizationswitching in VCSELs. We have also discussed an alternative rate equation model for VCSELs, namely the SFM modelwhich predicts an elliptically polarized injection locking. The bifurcation scenario leading to such an elliptically lockedstate is also analyzed. Finally, we report experimental results demonstrating polarization switching and injection lockingas well as an evidence of a period doubling route to chaos.

ACKNOWLEDGMENTS

The authors acknowledge the support from the Inter-University Attraction Pole (lAP V/19) "Photon" program of theBelgian government, the Fund for Scientific Research (FWO) - Flanders, Belgium and the the support of COST 288"Nanoscale and ultrafast photonics". This work also benefited from project support from the Geconcertande UnderzoeksAktie (GOA) and the Underzoeksraad (OZR) ofthe Vrije Universiteit Brussels. We thank Thomas Emeux for fruitful andstimulating discussions on the optical injection problem.

REFERENCES1. K. D. Choquette, D. A. Richie, and R. E. Leibenguth, Appl. Phys. Lett., 64, 2062, 1994.2. K. D. Choquette, R. P. Schneider Jr., K. L. Lear, and R E. Leibenguth, IEEE J. Sel. Top. Quant. Electr., 1, 661, 1995.

3. H.Kawaguchi, l.S.Hidayat, Y.Takahashi, Y. Yamayoshi, Electr. Lett., 31, 109, 1995.4. J.Martin-Regalado, F.Prati, M.SanMiguel, and N.B.Abraham, IEEE J.Quant.Electr., 33, 765, 1997.

5. J.Martin-Regalado, J.L.a.Chilla, J.J.Rocca, and P.Brusenbach, Appl. Phys. Lett., 70, 3350(1997).6. M.P.van Exter, M.B.Willemsen, and J.P.Woerdman, Phys. Rev.A, 58, 4191, 1998.

7. K. Panajotov, B. Ryvkin, J. Dackaert, M. Peeters, H. Thienpont, I. Veretennicoff, IEEE Phot. Technol. Lett., 10, 6,(1998).



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