Transverse-mode dynamics in vertical-cavity surface-emitting lasers with optical feedback

Transverse-mode dynamics in vertical-cavity surface-emitting lasers with optical feedback M. S. Torre,1C. Masoller,2and Paul Mandel3 1Instituto de Fı ´sica ‘‘Arroyo Seco,’’ UNCPBA Pinto 399 (7000) Tandil, Argentina 2Instituto de Fı ´sica, Facultad de Ciencias, Universidad de la Repu ´blica, Igua 4225, Montevideo 11400, Uruguay 3Universite ´Libre de Bruxelles, Optique Nonline ´aire The ´orique, Campus Plaine Code Postale 231, B-1050 Bruxelles, Belgium 共Received 22 March 2002, revised manuscript received 21 June 2002; published 26 November 2002兲 We study the transverse-mode dynamics of vertical-cavity surface-emitting lasers with weak optical feedback. We use a model that takes into account the spatial dependence of the transverse modes and of two carrier density profiles, associated with confined carriers in the quantum well region of the laser and unconfined carriers in the barrier region. Optical feedback is included as in the Lang-Kobayashi model. We find that for adequate parameter values antiphase dynamics occurs. As the injection current varies, the antiphase dynamics is destroyed through a sequence of periodic mixed states leading to in-phase dynamics. In these mixed states there are time intervals in which the modes are in phase, followed by time intervals in which they are in antiphase. We study the origin of the antiphase dynamics, assessing the role of the different spatial profiles. We show that the competition between the different profiles leads to the observed antiphase behavior. DOI: 10.1103/PhysRevA.66.053817 PACS number共s兲: 42.60.Mi, 42.55.Px, 42.65.Sf, 05.45.Pq I. INTRODUCTION The vertical-cavity surface-emitting laser 共VCSEL兲is a type of semiconductor laser that is emerging as a key element for high-speed information processing systems and optical communication networks 关1兴. The advantages of a VCSEL over a conventional, edge-emitting semiconductor laser are single-longitudinal-mode operation, dense packing capability, low threshold current, high modulation bandwidth, narrow circular beam profile, and simple and efficient coupling to an optical fiber. Near threshold, VCSELs typically emit linearly polarized light in the fundamental transverse mode. However, it is often observed that the polarization state selected at threshold becomes unstable as the injection current is increased, and a switch to the orthogonal polarization state occurs 共see, e.g., Ref. 关2兴and references therein兲. For high-power operation, high-order transverse modes are excited and the VCSEL usually emits multiple transverse modes. The complex polarization and transverse-mode behavior of VCSELs are considered drawbacks from the viewpoint of most applications, and have attracted broad interest, both theoretically and experimentally 关3–20兴. It is well known that optical feedback from an external reflector has important effects on the dynamics of VCSELs 关21–27兴. The effect of optical feedback depends on the amount of power fed back into the laser cavity, and on the round trip time of the field in the external cavity, which determines the feedback phase. Controlled optical feedback might stabilize the laser, reducing its linewidth, but uncontrolled feedback 共unavoidable in many applications兲might destabilize the laser, inducing chaotic intensity fluctuations and a broad linewidth. One particularly complex behavior is known as low-frequency fluctuations 共LFFs兲, and is characterized by abrupt random intensity dropouts followed by gradual, deterministic recoveries. In addition, in multimode lasers optical feedback might induce a variety of complex regimes. Several authors have studied theoretically the dynamics of multiple-longitudinalmode conventional 共edge-emitting兲semiconductor lasers with optical feedback. Sukow et al. 关28兴studied the effect of feedback based on an extension of the single-mode LangKobayashi 共LK兲model 关29兴, which incorporates additional optical modes that are coupled through the carrier inversion and through selfand cross-saturation coefficients. It was found that the statistics of the intensity fluctuations in the LFF regime on a picosecond time scale is essentially independent of the number of optical modes involved in the laser emission. Using a similar model, but with a parabolic gain profile, Rogister et al. 关30兴showed that in the presence of noise and in the LFF regime two qualitatively different behaviors on the picosecond time scale are possible: the longitudinal modes can emit pulses in phase or oscillate out of phase, depending on the operating parameters. Viktorov and Mandel 关31兴studied a multimode extension of the LK model that takes into account the longitudinal carrier grating associated with a Fabry-Perot configuration and predicted the possibility of antiphase dynamics. In that model, the steady state is destabilized either by a simple Hopf bifurcation leading to in-phase dynamics of the longitudinal modes, or by a degenerate Hopf bifurcation leading to antiphase dynamics 关32兴. Antiphase dynamics is an example of collective behavior in a system of globally coupled oscillators 关33兴. In lasers it results from the phase coherence of time-dependent modal intensities 关34,35兴. In the simplest cases, it is characterized by the fact that the total intensity, which is the direct sum of the modal intensities for rate equation models, has many properties of the single-mode intensity, while modal intensities display a more complex behavior. Several studies of the transverse-mode behavior of VCSELs have been based on a model originally proposed by Valle, Sarma, and Shore 关3,4兴. The model includes spatial profiles for the transverse optical modes and for the carrier density in the quantum well 共QW兲active region of the VCSEL. It also includes carrier diffusion. The model applies to weakly index-guided VCSELs, where the transverse modal profiles and modal frequencies are determined by the built-in refractive index distribution, thus allowing a descripPHYSICAL REVIEW A 66, 053817 共2002兲 1050-2947/2002/66共5兲/053817共9兲/$20.00 ©2002 The American Physical Society66 053817-1 tion in terms of modal amplitudes. For a cylindrical VCSEL, the appropriate transverse modes are the linearly polarized LPm,nmodes 关36兴. They have the property LPm,n(r, ␪ )⫽ ␾ m,n(r)cos(m ␪ ) where (r, ␪ ) are the polar coordinates of the plane transverse to the propagation direction. Several authors 关3,22,26,37,38兴have simplified the numerical simulations by assuming that the azimuthal dependence of the modes with m⬎0 can be neglected, i.e., LPm,n(r, ␪ )⯝ ␾ m,n(r), when these modes are degenerate. However, the resulting approximate modes are not cavity modes, except if m⫽0. In this paper we use a model for VCSELs that is an extension of the model proposed by Valle et al. 关3兴. However, in order to simplify the calculations while keeping the model as complete as possible, we assume that only the first three azimuthally symmetric modes LP0,1 ,LP 0,2 , and LP0,3 can be excited. Carrier transport effects are included by considering two carrier densities, one for the carriers in the QW region 共where the carriers are in two-dimensional quantum states兲, and one for the carriers in the barrier region 共where the carriers are in three-dimensional quantum states兲. The exchange of carriers between these two reservoirs 共carrier capture into the QWs and escape out of the QWs兲is characterized by small but finite capture and escape times. Our approach is the same as in the phenomenological standard rate equations for QW lasers 关39–41兴. External optical feedback is included as in the LK model, by considering a single reflection in the external cavity. We show that a weak optical feedback may induce antiphase dynamics of the transverse modes, and we study how the antiphase dynamics is destabilized as the injection current or the diffusion coefficient varies. We find that the antiphase dynamics is destroyed through a sequence of periodic mixed states. In these states, time intervals in which the modes are in phase alternate with time intervals in which they are in antiphase. We study the origin of the antiphase behavior by considering equal and different spatial profiles for the transverse modes. We show that it is the competition between the different profiles that leads to the observed antiphase behavior. The effects of optical feedback on the dynamics of VCSELs were previously studied by Law and Agrawal 关23,24兴, based on a model similar to ours but that takes into account several reflections in the external cavity and does not consider carrier capture and escape. In-phase and antiphase regimes were found in that model but that aspect of the dynamics was not the topic of these papers. Here we focus on studying in detail the in-phase and antiphase behavior. This paper is organized as follows. The model is described in Sec. II. Analytical results for the steady state are presented in Sec. III. Results of numerical simulations that show distinct dynamical regimes of the transverse modes are presented in Sec. IV. Finally, Sec. V contains a summary and the conclusions. II. THE MODEL We consider a cylindrically symmetric structure, whose active region 共consisting of several quantum wells兲is modeled as a single effective quantum well of radius aand thickness dQW . Barrier regions of thickness dblimit the QW region. Two highly reflecting mirrors separated by a distance L along the longitudinal zaxis define the laser cavity. The injected current is azimuthally uniform over the transverse area and varies stepwise: j(r)⫽jofor r⬍aand j(r)⫽0 otherwise. The emission behavior is determined by the built-in index guiding introduced by the transverse refractive index step in the surrounding region. The core 共cladding兲refractive index is taken to be ncore (nclad), i.e., the transverse refractive index profile is n(r)⫽ncore for r⬍aand n(r)⫽nclad for r⬎a. For this geometry the appropriate transverse modes are the linearly polarized LPmn modes 关36兴, for which the transverse variation of the field is given by ␺ mn共r, ␪ 兲⫽Jm共umnr/a兲 Jm共umn兲cosm ␪ for r⬍a, ␺ mn共r, ␪ 兲⫽Km共wmnr/a兲 Km共wmn兲cosm ␪ for r⬎a,共1兲 where Jmand Kmare Bessel functions of the first and second kinds, respectively, umn⫽a关(ncorekmn)2⫺ ␤ 2兴1/2,wmn ⫽a关 ␤ 2⫺(ncladkmn)2兴1/2, ␤ L⫽q ␲ ,qis an integer, and the wave vector kmn is obtained from eigenvalue equations. To simplify the calculations we consider that only three modes, having azimuthal symmetry, are excited in the range of parameters considered in this paper: ␺ 1共r兲⬅ ␺ 01共r, ␪ 兲⫽LP01 , ␺ 2共r兲⬅ ␺ 02共r, ␪ 兲⫽LP02 , and ␺ 3共r, ␪ 兲⬅ ␺ 03共r兲⫽LP03 .共2兲 The mode profiles are normalized such that 兰 0 ⬁ 兩 ␺ i 兩 2(r)rdr ⫽1. Since the mode profiles are exponentially small outside the active region, this normalization hardly differs from the physical normalization 兰 0 a 兩 ␺ i 兩 2(r)rdr⫽1. The equations for the slowly varying complex amplitude of the ith mode, ei(t), the density of carriers confined in the QW region, nw(r,t), and the density of 共unconfined兲carriers in the barrier region, nb(r,t), are 关3,29,38兴 dei dt ⫽1⫹j ␣ 2 冉 gi⫺1 ␶ pi 冊 ei共t兲⫹kiei共t⫺ ␶ 兲exp共⫺j ␻ i ␶ 兲, 共3兲 ⳵ nb ⳵ t⫽j共r兲 edb ⫺nb ␶ cap ⫹VQW Vb nw ␶ esc ⫺nb ␶ n ⫹Db 1 r ⳵ ⳵ r 冉 r ⳵ nb ⳵ r 冊 , 共4兲 ⳵ nw ⳵ t⫽Vb VQW nb ␶ cap ⫺nw ␶ esc ⫺nw ␶ n ⫺go共nw⫺nt兲兺 兩 ei 兩 2 兩 ␺ i 兩 2 ⫹Dw 1 r ⳵ ⳵ r 冉 r ⳵ nw ⳵ r 冊 .共5兲 In these equations the modal amplitude eiis normalized such that 兩 ei 兩 2 兩 ␺ i 兩 2is the photon density in the ith mode. The carrier variables are averaged along the longitudinal axis. TORRE, MASOLLER, AND MANDEL PHYSICAL REVIEW A 66, 053817 共2002兲 053817-2 Therefore, nw(r,t)关nb(r,t)兴represents the average carrier density in the transverse plane in the QW 共barrier兲region. If the QW region consists of several QWs, interwell carrier transport effects are not considered, and nwrepresents the average carrier density in the QWs. The first term in the right-hand side of Eq. 共3兲accounts for optical gain, losses, and phase-amplitude coupling. Here, ␣ is the linewidth enhancement factor and giis the modal gain, gi共t兲⫽ 冕 0 ⬁go⌫i共nw⫺nt兲 兩 ␺ i 兩 2rdr,共6兲 where gois the gain coefficient, ⌫iis the confinement factor for the ith mode, and ntis the transparency carrier density. ␶ pi is the photon lifetime for the ith mode. The second term in the right-hand side of Eq. 共3兲takes into account the field reflected from the external cavity. We consider a single reflection, and therefore the model is valid for weak and moderate feedback levels. kiis the feedback coefficient of the ith mode: ki⫽(1⫺R2) 冑 R2Rext ␩ c/(R2 ␶ in)关24兴, where R2and Rext are the output and external mirror reflectivities, ␶ in is the solitary laser round-trip time, and ␩ cis the coupling efficiency. In general ␩ ccan be different for different transverse modes, but in this study we take ␩ cto be mode independent. ␻ iis the optical frequency of the ith mode in the absence of feedback, and ␶ is the external-cavity round-trip time. The terms on the right-hand side of Eq. 共4兲correspond, from left to right, to 共i兲the rate at which carriers are injected into the barrier region, 共ii兲the rate at which carriers are captured into the QWs, 共iii兲the rate at which carriers escape out of the QWs, 共iv兲the carrier loss owing to various nonradiative recombination processes, and 共v兲carrier diffusion across the barrier region. The transport effects are included by a capture time ␶ cap , an escape time ␶ esc , and a diffusion coefficient Db. The carrier loss is included by a carrier lifetime ␶ n. Since the variables nband nwrefer to carrier densities, the different sizes of the barrier and QW regions must be taken into account. This is done by the ratio Vb/VQW , where Vb⫽db ␲ a2is the volume of the barrier region, and VQW ⫽dQW ␲ a2is the volume of the QW region. The terms in the right-hand side of Eq. 共5兲correspond, from left to right, to 共i兲the carriers captured into the QWs, 共ii兲the carriers that escape out of the QWs, 共iii兲the nonradiative carrier loss, 共iv兲the carrier loss owing to stimulated recombination, and 共v兲carrier diffusion across the QWs. For simplicity we consider the same nonradiative recombination time for the carriers in the QW region and for the carriers in the barrier region 共the effect of different recombination times was studied in 关38兴兲. III. STEADY-STATE SOLUTIONS The stationary solutions of Eqs. 共3兲–共5兲are ei共t兲⫽ei sexp关i共 ␻ i s⫺ ␻ i兲t兴, nw共r,t兲⫽nw s共r兲,nb共r,t兲⫽nb s共r兲, where ei s, ␻ i s,nw s(r), and nb s(r) satisfy gi s⫽ 冕 0 ⬁go⌫i共nw s⫺nt兲 兩 ␺ i 兩 2rdr⫽1/ ␶ pi⫺2kicos共 ␻ i s ␶ 兲, 共7兲 ␻ i s⫺ ␻ i⫽⫺ ␣ kicos共 ␻ i s ␶ 兲⫺kisin共 ␻ i s ␶ 兲,共8兲 nb s 冉 1 ␶ cap ⫹1 ␶ n 冊 ⫽j共r兲 edb ⫹VQW Vb nw s ␶ esc ⫹Db 1 r ⳵ ⳵ r 冉 r ⳵ nb s ⳵ r 冊 , 共9兲 nw s 冉 1 ␶ esc ⫹1 ␶ n 冊 ⫽Vb VQW nb s ␶ cap ⫺go共nw s⫺nt兲兺 兩 ei s 兩 2 兩 ␺ i 兩 2 ⫹Dw 1 r ⳵ ⳵ r 冉 r ⳵ nw s ⳵ r 冊 .共10兲 Equation 共7兲shows that the stationary values of the modal gains depend on the feedback level but not on the carrier capture and escape times. Equation 共8兲determines the optical frequencies of the transverse modes in the presence of feedback, which are also independent of ␶ cap and ␶ esc . Integrating Eqs. 共9兲and 共10兲between r⫽0 and r⫽⬁gives 共 ␥ cap⫹ ␥ n兲 冕 0 ⬁nb srdr⫽1 edb 冕 0 ⬁j共r兲rdr⫹VQW Vb ␥ esc 冕 0 ⬁nw srdr ⫹Db 冕 0 ⬁1 r ⳵ ⳵ r 冉 r ⳵ nb s ⳵ r 冊 rdr,共11兲 共 ␥ esc⫹ ␥ n兲 冕 0 ⬁nw srdr⫽Vb VQW ␥ cap 冕 0 ⬁nb srdr ⫺兺 兩 ei s 兩 2 冕 0 ⬁go共nw s⫺nt兲 兩 ␺ i 兩 2rdr ⫹Dw 冕 0 ⬁1 r ⳵ ⳵ r 冉 r ⳵ nw s ⳵ r 冊 rdr,共12兲 where ␥ esc⫽1/ ␶ esc , ␥ cap⫽1/ ␶ cap , and ␥ n⫽1/ ␶ n. The numbers of carriers in the barrier and QW regions are Nb共t兲⫽2 ␲ db 冕 0 ⬁nb共r,t兲rdr, Nw共t兲⫽2 ␲ dQW 冕 0 ⬁nQW共r,t兲rdr, and Eqs. 共11兲and 共12兲can be rewritten as 共 ␥ cap⫹ ␥ n兲Nb s⫽2 ␲ e 冕 0 ⬁j共r兲rdr⫹ ␥ escNw s ⫹Db2 ␲ dbr ⳵ nb s ⳵ r 冏 r⫽0 r⫽⬁ ,共13兲 TRANSVERSE-MODE DYNAMICS IN VERTICAL-CAVITY . . . PHYSICAL REVIEW A 66, 053817 共2002兲 053817-3 共 ␥ esc⫹ ␥ n兲Nw s⫽ ␥ capNb s⫺2 ␲ dQW兺 兩 ei s 兩 2gi s/⌫i ⫹Dw2 ␲ dQWr ⳵ nw s ⳵ r 冏 r⫽0 r⫽⬁ ,共14兲 where Nb s⫽2 ␲ db 兰 0 ⬁nb s(r)rdr and Nw s ⫽2 ␲ dQW 兰 0 ⬁nQW s(r)rdr. Now it is clear why it is convenient to define the normalization condition and all integrals between r⫽0 and r⫽⬁.nw(r⫽⬁)⫽nb(r⫽⬁)⫽0 and the diffusion terms vanish. Equations 共13兲and 共14兲can be simplified to 共 ␥ cap⫹ ␥ n兲Nb s⫽J⫹ ␥ escNw s,共15兲 共 ␥ esc⫹ ␥ n兲Nw s⫽ ␥ capNb s⫺ ␥ pIT,共16兲 where J⫽2 ␲ 兰 0 ⬁j(r)rdr/eis the number of injected carriers per unit time, and IT⫽2 ␲ dQW兺 兩 ei s 兩 2is the total number of photons in the QW region. In Eq. 共16兲we have assumed that gi s⬃1/(⌫i ␶ pi)⫽ ␥ p. This approximation is valid for low feedback levels such that 1/ ␶ piⰇki. From Eqs. 共15兲and 共16兲we can eliminate Nb sand obtain ␥ pIT⫽J 1⫹ ␥ n/ ␥ cap ⫺ ␥ nNw s 1⫹ ␥ n/ ␥ cap 冉 1⫹ ␥ n ␥ cap ⫹ ␥ esc ␥ cap 冊 . 共17兲 Two limits are interesting to analyze. When the carriers do not escape out of the QW region ( ␥ esc⬃0), we recover the simple connection between the photon number, the injected current, and the carrier number in the QW region, ␥ pIT⫽Jeff⫺ ␥ nNw s, with a modified, ‘‘effective’’ injected current Jeff⫽J(1⫹ ␥ n/ ␥ cap)⫺1, which is slightly lower than the actual injected current 共typically, ␶ nis of the order of nanoseconds, and ␶ cap is of the order of picoseconds; thus ␥ n/ ␥ capⰆ1). The factor (1⫹ ␥ n/ ␥ cap)⫺1represents the loss of carriers 共due to nonradiative processes兲during the capture time. In other words, a finite capture time slightly diminishes the current density effectively injected into the QW region. The other limit corresponds to a large ratio between the capture and the escape times, R⫽ ␶ cap / ␶ esc⫽ ␥ esc / ␥ cap . This leads to a larger, negative contribution of the last term in Eq. 共17兲, and, therefore, to a significant reduction of the total number of photons in the QW region. We will show in the next section that this favors emission in the fundamental transverse mode. A complementary way to understand the effect of carrier capture and escape is by considering the dependence of the threshold current of the fundamental transverse mode on the capture and escape times. The threshold current can be estimated from Eq. 共17兲as Jth⫽ ␥ nNw s 冉 1⫹ ␥ n ␥ cap ⫹ ␥ esc ␥ cap 冊 .共18兲 Clearly, an increase of ␥ esc increases the threshold current, and, therefore, for a fixed injection current the laser operates closer to threshold. IV. DYNAMICAL REGIMES We integrated the model equations with the parameters a⫽6 ␮ m, dQW⫽0.024 ␮ m共three QWs each of thickness 0.08 ␮ m), db⫽1.2 ␮ m, index step⫽0.1, ␣ ⫽3, go ⫽vg ⳵ g/ ⳵ nwith vg⫽0.0715 ␮ m/ns and ⳵ g/ ⳵ n⫽5.95 ⫻10⫺8 ␮ m2,nt⫽1.33⫻106 ␮ m⫺3, ␶ cap⫽5 ps, ␶ esc ⫽25.5 ps, ␶ n⫽1.52 ns, ␶ ⫽1 ns, and Db⫽0.5 ␮ m2/ns. The time integration step is ⌬t⫽10⫺4ps and the space integration step is ⌬r⫽0.02 ␮ m. First, we consider a degenerate situation, in which all modes have the same confinement factor ⌫i⫽0.038, frequency ( ␻ i ␶ ⫽0 rad), losses ( ␶ pi ⫽2.2 ps), and feedback level (ki⫽k). The feedback level, the diffusion coefficient Dw, and the injection current I ⫽jo ␲ a2are the free parameters of our study. We show the existence of an antiphase dynamic regime for weak feedback and adequate parameter values. Next, we study the effect of carrier diffusion and modal profiles on the antiphase regime. Finally, we show that the antiphase regime is also observed in a more realistic situation, in which the modes have different optical frequencies. Without feedback and close to threshold, the single-mode steady state is stable, while for larger injection the transverse multimode steady state is stable. The fundamental LP01 transverse mode has the lowest threshold and it is stable for low current. As the current increases the LP02 mode turns on. FIG. 1. Total and modal intensities as the injection current increases. The diffusion coefficient is Dw⫽0.5 ␮ m2/ns. 共a兲Without feedback. 共b兲The feedback level is k⫽1ns ⫺1. The thick 共thin兲line shows the value of the total intensity 共injection current兲. The modes are represented as LP01 , dashed line; LP02 , dot-dashed line; LP03 , dotted line. TORRE, MASOLLER, AND MANDEL PHYSICAL REVIEW A 66, 053817 共2002兲 053817-4 For even larger injection, the LP03 mode turns on and the three modes coexist. Figure 1共a兲displays the total and modal intensities in the absence of feedback, as the injection current 共thin solid line兲gradually increases. The thick line corresponds to the total power, while the other lines correspond to the modal powers (LP01 dashed line; LP02 dot-dashed line; LP03 dotted line兲. Weak feedback levels modify this picture quantitatively but not qualitatively. Figure 1共b兲corresponds to k⫽1ns ⫺1. Considering the internal round trip ␶ in⫽0.045 ps, the coupling efficiency ␩ c⫽1, and the output-mirror reflectivity R2⫽0.995, this feedback level corresponds to an externalmirror reflectivity of Rext⫽8.06⫻10⫺5, i.e., we work in the very weak feedback regime. Figure 1共b兲shows that for this feedback the modal intensities exhibit oscillations, and we find distinct dynamical regimes with increasing current. To investigate in more detail what happens for different injection currents, we plot in Fig. 2 the time-averaged value of the total and modal intensities, as a function of the injection current. In this figure the injection current Iwas kept constant until the stable regime was reached, and then the average value of the total and modal intensities was calculated. Clearly, with weak feedback the transverse-mode dynamics is such that the total intensity looks single mode, i.e., it increases linearly with the injection current except for the weak nonlinear response close to threshold. To display another facet of this intriguing coherence, we show in Fig. 3 the maximum and minimum values of the total 关Fig. 3共a兲兴 and the modal 关Fig. 3共b兲兴 intensities whose time average is shown in Fig. 2. Figure 3 reveals a complex underlying transverse-mode behavior, which is typical of antiphase dynamics in globally coupled nonlinear oscillators. A. Antiphase dynamics For I⭐1.7 mA the VCSEL is single mode. The intensity of the LP01 mode exhibits undamped relaxation oscillations, whose amplitude decreases for increasing I. Figure 3 indicates that for I⫽1.7 mA there is single-mode steady-state operation involving only the LP01 mode. For Islightly larger, the LP02 mode emerges, destabilizing the steady-state LP01 mode. In the interval 1.7⬍I⬍2.2 mA, the two modes oscillate in phase. As the current is increased, a two-mode steady state is reached for I⫽2.3 mA. As for the case I⫽1.7 mA, increasing Idestabilizes the steady-state operation via the emergence of a new mode. For I⫽2.4 mA the LP03 mode emerges and for I⬎2.4 mA we observe different regimes of three-mode operation. First, for 2.4⬍I⬍2.7 mA there are oscillations of the total intensity, since the maximum and minimum values differ 关Fig. 3共a兲兴. For I⬎2.7 mA the maximum and minimum values of the total intensity are nearly equal, and Fig. 3共b兲shows that two distinct dynamical regimes actually occur. For 2.7⬍I⬍3.7 mA there are oscillations of the modal intensities which nearly compensate in the total intensity, while for I⭓3.7 mA each transverse mode is in steady state. Figures 3共a兲and 3共b兲suggest that in the interval 2.7⬍I ⬍3.7 mA antiphase dynamics occurs, since the modal intensities oscillate while the total intensity remains nearly constant. In order to study the phase relations among the modes in the different dynamical regimes, Fig. 4 shows for increasing values of the injection current the total and modal intenFIG. 2. Total and modal averaged intensities as a function of the injection current. All parameters are as in Fig. 1共b兲. The thick line shows the value of the total intensity. The modal intensities are represented as LP01 , dashed line; LP02 , dot-dashed line; LP03 , dotted line. FIG. 3. Maximum and minimum values of the 共a兲total and 共b兲 modal intensities as a function of the injection current. All parameters are as in Fig. 1共b兲. The modal intensities are represented as LP01 , dashed line; LP02 , dot-dashed line; LP03 , dotted line. TRANSVERSE-MODE DYNAMICS IN VERTICAL-CAVITY . . . PHYSICAL REVIEW A 66, 053817 共2002兲 053817-5 sities. As before, the thick line shows the total power, while the thin lines show the modal powers (LP01 , dashed line; LP02 , dot-dashed line; LP03 , dotted line兲. For low current there is the single-mode periodic regime shown in Fig. 4共a兲. Figure 4共b兲displays a two-mode in-phase regime with very different oscillation amplitudes. Figure 4共c兲corresponds to a two-mode steady-state regime. The emergence of the third mode leads to in-phase oscillations for the other two modes, Fig. 4共d兲. The transition to antiphase oscillations, Fig. 4共f兲,is through a mixed state displayed in Fig. 4共e兲. Note that in Fig. 4共e兲there are time intervals in which the LP01 and LP02 mode pulses are in phase followed by time intervals in which they are in antiphase. As a whole, this regime is periodic. For even larger injections a three-mode steady state is reached 关Fig. 4共g兲兴. Antiphase behavior was also found by Valle 关42兴, in the competition of the LP11 c关with a cos2( ␪ ) dependence intensity profile兴, and the LP11 s关with a sin2( ␪ ) dependence intensity profile兴, when the VCSEL is subjected to injection current modulation. Moreover, previous studies by Law and Agrawal 关23,24兴of the dynamics of VCSELs with feedback 共based on a model similar to ours, but that includes several reflections in the external cavity and does not take into account carrier capture and escape兲revealed the existence of in-phase and antiphase behavior. In 关23,24兴the authors considered high reflectivity, the competition of only two transverse modes, and different contact geometries for the injection current. For instance, with a disk-contact geometry, periodic in-phase and antiphase behaviors of the LP01 and LP11 modes were found 共see Figs. 5b and 5c of 关23兴兲 for different values of the feedback parameter. Thus, antiphase dynamics seems to be a robust general feature independent of the details of the model. As discussed in Sec. III, the effect of carrier capture and escape is to modify the current effectively injected into the laser cavity. Our numerical simulations verify that increasing ␶ esc is indeed equivalent to increasing the injection current, and a transition from in-phase to antiphase behavior can be observed. In the following sections we study the influence of carrier diffusion, modal profiles, and different optical frequencies on the antiphase behavior. B. Influence of the carrier diffusion Varying the diffusion coefficient also changes the threshold current, and therefore decreasing Dwhas an effect similar FIG. 4. Dynamic regimes for increasing injection current. 共a兲I ⫽1.7 mA; 共b兲I⫽2.1 mA; 共c兲I⫽2.3 mA; 共d兲I⫽2.55 mA; 共e兲I ⫽2.67 mA; 共f兲I⫽3.3 mA; 共g兲I⫽3.7 mA. All other parameters are as in Fig. 1共b兲. FIG. 5. Dynamic regimes for decreasing carrier diffusion. k ⫽1ns ⫺1,I⫽2.8 mA. 共a兲Dw⫽3.0 ␮ m2/ns; 共b兲Dw ⫽2.0 ␮ m2/ns; 共c兲Dw⫽1.5 ␮ m2/ns; 共d兲Dw⫽0.5 ␮ m2/ns; 共e兲 Dw⫽0.01 ␮ m2/ns. TORRE, MASOLLER, AND MANDEL PHYSICAL REVIEW A 66, 053817 共2002兲 053817-6 to increasing the injection. Figure 5 shows the transversemode dynamics for five decreasing values of the diffusion coefficient. For large diffusion 关Fig. 5共a兲兴 the laser operates on the fundamental transverse mode in a periodic regime. As the diffusion coefficient decreases, we observe a transition to antiphase oscillations involving the LP01 and LP02 modes 关Figs. 5共c兲and 5共d兲兴. As the diffusion coefficient is further decreased, we find a more complex type of antiphase regime: antiphase oscillations involving the LP02 and LP03 modes, while the LP01 mode exhibits small oscillations that are in phase, alternately with the LP02 and the LP03 modes 关Fig. 5共e兲兴. The results obtained for different values of the diffusion coefficient clearly show that in this model it is the population grating due to the Fabry-Perot configuration that induces the multimode behavior, as occurs in other types of homogeneously broadened lasers. C. Transverse profile competition In order to understand the origin of antiphase dynamics in this model, we analyze the competition among the three transverse modes. For that purpose, we shall replace the LP profiles used up to here 关Eq. 共1兲兴 by other profiles. Figure 6 shows the effect of different profiles. Figure 6共a兲 is used as a reference: the transverse profiles are the LP01 , LP02 , and LP03 modes, and an antiphase oscillation is observed between the LP01 and LP02 modes. Next, we assume that modes 2 and 3 have uniform profiles within the active region 关 兩 ␺ i(r) 兩 2⫽cifor r⬍aand 兩 ␺ i(r) 兩 2⫽0 otherwise兴 while the third mode is the fundamental transverse mode LP01 . This case is shown in Fig. 6共b兲, where we observe that the two modes with equal profiles are identical 共dot-dashed line兲, while the third mode 共dashed line兲oscillates such that large 共small兲peaks in the identical modes correspond to small 共large兲peaks in the third mode. Last, we consider the case in which all modes have uniform profiles. In this case, all three modes are identical and oscillate in phase, and the total intensity is exactly three times the intensity of any mode 关Fig. 6共c兲兴. The initial conditions are the same in Figs. 6共a兲,6共b兲, and 6共c兲. The initial conditions are taken all through the paper FIG. 6. Effect of different transverse mode profiles. k ⫽1ns ⫺1,I⫽2.8 mA, Dw⫽0.5 ␮ m2/ns. 共a兲The transverse modes are LP01 ,LP 02 , and LP03 .共b兲Two transverse modes have the same uniform profiles, and the third mode is the LP01 mode. 共c兲The three transverse modes have the same uniform profiles. FIG. 7. Dynamic regimes for increasing injection current, when the transverse modes have different optical frequencies. 共a兲I ⫽1.7 mA; 共b兲I⫽2.1 mA; 共c兲I⫽2.3 mA; 共d兲I⫽2.55 mA; 共e兲I ⫽2.67 mA; 共f兲I⫽3.3 mA; 共g兲I⫽3.7 mA. ␭1⫽851.8 nm, ␭2 ⫽852.0 nm, and ␭3⫽852.2 nm. All other parameters are as in Fig. 4. TRANSVERSE-MODE DYNAMICS IN VERTICAL-CAVITY . . . PHYSICAL REVIEW A 66, 053817 共2002兲 053817-7 with the laser off, i.e., the modal amplitudes are at the noise level and the carrier densities are at the transparency value. The different dynamic regimes shown in Figs. 6共a兲,6共b兲, and 6共c兲are thus a consequence of the transverse-mode profiles considered. From these results it is clear that in this model the antiphase dynamics has its origin in the mode coupling and competition 共fourth term兲in Eq. 共5兲. The competition among the transverse modes with different spatial profiles to ‘‘burn holes’’ in the same reservoir of carriers 共i.e., the carriers in the QWs兲leads to the observed antiphase dynamics. Equivalently, one can interpret this mechanism as a transverse carrier grating induced by the different weights 兩 ␺ i 兩 2of the lasing modes. This behavior has also been found recently in the dynamics of the longitudinal modes of an edge-emitting laser with optical feedback 关31,32兴and it is the same mechanism that leads to antiphase dynamics. D. Influence of different optical frequencies and different feedback levels In the preceding subsections, the three transverse modes have the same optical frequencies, and one might question if the antiphase regime found is not a singular property of the degenerate equations. Figure 7 shows results in which the modes have different optical frequencies: ␭1⫽851.8 nm, ␭2⫽852.0 nm, and ␭3⫽852.2 nm, all other parameters being as in Fig. 4. The antiphase regime persists, and this confirms that, given the smallness of the intermode frequency separation with respect to the optical frequency, intermode frequency differences may be treated by a perturbation theory of which Eqs. 共3兲–共5兲are the zero order approximation. The antiphase regime is also robust with respect to slightly different feedback levels. Figure 8 shows results for k1⫽k2⫽1ns ⫺1,k3⫽2ns ⫺1, all other parameters being as in Fig. 4共f兲. Clearly the antiphase regime survives. V. SUMMARY AND CONCLUSIONS The transverse-mode dynamics of an index-guided vertical-cavity surface-emitting laser with weak optical feedback was analyzed using a model that takes into account three transverse modes and two carrier density profiles, associated with confined carriers in the quantum well region of the laser. We found antiphase dynamics and studied the destabilization of this regime when the injection current or the carrier diffusion varies. We found that the antiphase dynamics is destabilized though a sequence of mixed states, in which time intervals in which the mode pulses are in phase alternate with time intervals in which they are in antiphase. We have also shown that in this model the antiphase dynamics is due to the transverse profiles of the optical modes, which compete for the same reservoir of carriers. Antiphase dynamics is usually studied in the frame of modal rate equations 关30–32兴, where the carrier density is expanded either in Fourier series or in modal series. These series have to be truncated and the approximation induced by this truncation is difficult to assess 关43兴. The power of the model studied in this paper is that no such expansion has been introduced and the carrier density dynamical equation is complete, including diffusion. As a result, there is no ambiguity as to the origin of the antiphase dynamics. This is especially clear from the analysis of Sec. IVC, where antiphase dynamics was clearly attributed to the transverse grating of the carriers. The negative aspect of this model is that little can be concluded analytically about the timedependent regimes of the model equations 共3兲–共5兲. Numerical simulations are essential to understand the dynamical regimes. 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PHYSICAL REVIEW A 66, 053817 共2002兲 053817-9