Multilongitudinal-mode dynamics in a semiconductor laser subject to optical injection

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1 Rochester Institute of Technology RIT Scholar Works Articles 1998 Multilongitudinal-mode dynamics in a semiconductor laser subject to optical injection J.K. White Jerome Moloney Athanasios Gavrielides Follow this and additional works at: Recommended Citation IEEE Journal of Quantum Electronics 34N8 (1998) This Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorized administrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.edu.

2 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 8, AUGUST Multilongitudinal-Mode Dynamics in a Semiconductor Laser Subject to Optical Injection John K. White, J. V. Moloney, A. Gavrielides, V. Kovanis, A. Hohl, and R. Kalmus Abstract Multilongitudinal-mode dynamics in a semiconductor laser subject to optical injection are investigated both experimentally and numerically. We found that there are parameter regimes for which the slave laser hops into an adjacent longitudinal mode as we vary the detuning of the optical frequencies between the master and slave laser. A traveling wave model is used to numerically investigate the mode hopping. Good qualitative agreement is found between the numerical computations and the experimental observations. Index Terms Laser modes, laser stability, modeling, nonlinear differential equations, nonlinear optics, optical communications, optoelectronic devices, semiconductor lasers. I. INTRODUCTION SEMICONDUCTOR lasers are widely used as coherent light sources in technological applications such as optical communications. For high-speed and reliable transmission of data, it is essential that the semiconductor laser exhibits stable single-mode operation with a narrow linewidth and may be modulated at high frequencies with little chirping. It was found that injection locking of a slave laser to a master laser may enhance the spectral stability of a laser significantly. The injection-locked slave laser exhibits stable single-mode operation for a specific locking band region [1], its linewidth may be reduced significantly [2], the chirping is minimized in modulated semiconductor lasers [3], and also squeezing can be realized [4]. To employ this locking technique successfully it is essential to know the stability boundaries for a laser subject to optical injection and to understand the underlying dynamical properties beyond these boundaries. The injection-lock band was investigated by many authors experimentally, numerically, and analytically [1], [5] [11] and was found to be asymmetric due to the coupling of the phase and amplitude as expressed by the linewidth enhancement factor. Unstable behavior is especially found for positive detunings, meaning the master laser is detuned to the shorter wavelength with respect to the slave laser. In this regime, multimode operation and leakage of energy into sidemodes has been observed in [8] and [12]. Manuscript received December 22, This work was supported by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under Grant AFOSR DEF and Grant AFOSR The work of A. Hohl was supported by the National Research Council. J. K. White and J. V. Moloney are with the Arizona Center for Mathematical Sciences, University of Arizona, Tucson, AZ USA. A. Gavrielides, V. Kovanis, A. Hohl, and R. Kalmus are with the Nonlinear Optics Group, Phillips Laboratory PL/LIDD, Kirtland AFB, NM USA. Publisher Item Identifier S (98) Mode hopping has been observed in solitary semiconductor lasers as the pump current or the temperature of the laser are varied [13] [19] or due to external feedback from a reflector [20]. The physical cause of mode hopping has been widely attributed to nonlinear gain and it has been shown that including the nonlinear gain into numerical computations is sufficient to induce mode hopping, yet the origin of the nonlinearity is not entirely understood. Some authors attribute the nonlinearity in the gain to spectral hole burning and longitudinal mode beating that modulate the refractive index [13], [14], [16] [18], while others attribute the nonlinear gain to dynamic carrier heating effects [21]. A natural way to account for multimode behavior is to include the longitudinal dependence into the semiconductor laser rate equations. Models that reduce the laser equations to a system of coupled ordinary differential equations (ODE s) [22] are limited to studying the energy in a single mode, usually the lasing mode. Even multimode decompositions [23] where modes are represented by coupled ODE s fail to capture the gain dispersion accurately. Furthermore, all ODE models assume a uniform carrier density. For lasers where one end is antireflection (AR) coated there arise strong longitudinal dependencies in the carriers. Our model addresses all of these issues in a simple self-consistent manor by resolving the longitudinal structure of the forward field, backward field, and carrier density. The paper is organized as follows. We first report on the experimental arrangement and observations on mode hopping due to external optical injection. The description of the model follows and the numerically computed optical spectra are discussed. We conclude with a summarizing section. II. EXPERIMENTAL APPARATUS AND OBSERVATIONS In the experiment two commercially available GaAlAs Fabry Perot lasers (Sharp LT015, double heterojunction diodes in a V-channeled substrate inner stripe (VSIS) structure) were used in a master slave configuration. Both lasers were lasing at 835 nm and during the experiment, both were operated from a low-noise current supply (ILX Lightwave LDC-3722), and the temperature was stabilized to better than 0.01 K. The optical spectrum was monitored with a scanning Fabry Perot interferometer (Newport SR-240C) which had a free spectral range of 2 THz. Isolators of 50-dB attenuation were used to insure that no light was injected into the master laser and no back reflections from the Fabry Perot were injected into either laser. The slave laser was pumped at above threshold, where and and /98$ IEEE

3 1470 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 8, AUGUST 1998 (a) (b) (c) Fig. 1. Experimentally obtained optical spectra for a fixed injection strength and for decreasing positive detuning 1. For large positive detuning 1= 2.75 GHz (a) the energy is completely in the fundamental mode, but as the detuning is decreased 1=1.65 GHz (b) most of the energy is transferred into the adjacent longitudinal mode and for even smaller detuning 1=0.87 GHz (c) the energy returns to the fundamental mode. denote the pump current and threshold current of the slave, respectively. The strength of injection signal was controlled by a set of two polarizers and was estimated assuming a coupling loss of 30%. We have observed mode hopping for several parameter regimes as we increase the injection strength while keeping the detuning fixed, or for a fixed injection strength while varying the detuning. The amount of energy transfer from one mode into the other may be different depending on the exact parameter values. Fig. 1 shows the optical spectrum of the slave laser when approximately a power ratio injected power/total slave power of 3 10 was injected into the slave laser and the positive detuning was decreased. In Fig. 1(a), the master laser was detuned from the slave laser by 2.75 GHz, where and are the optical frequency of the master and slave, respectively. We see that the main mode of the slave laser had moderately strong relaxation oscillation sidebands due to the injection. This indicates that the laser is in a Hopf regime which is accurately predicted by the single-mode theory of a laser subject to optical injection [12]. As the detuning was decreased to 1.65 GHz, the laser hopped partially into the next adjacent longitudinal mode. For further decrease of the detuning to 0.87 GHz the laser emission returned to the first mode. However, stronger relaxation oscillation sidebands are present in the optical spectrum. From the dynamical point of view we recognize that the strength of the relaxation oscillation sidebands of the main longitudinal mode increased as we decreased the detuning. This is a well-known fact and is again predicted with the single-mode semiconductor laser rate equations [12]. Naturally, mode hopping cannot be predicted with such a single-mode model. We also recognize in Fig. 1(b) that the strength of relaxation oscillation sidebands of the adjacent mode is much weaker than that of the main mode. III. MODEL EQUATIONS Devising a multimode model with all the relevant physics that is computationally feasible presents many challenges. The first of these is how to account for the dependence of the amplitudes of the longitudinal modes. In some instances the laser geometry and boundary conditions are simple enough that is is possible to explicitly solve for the dependence of each mode [23]. For those instances, using a set of coupled single-mode equations is quite accurate and almost trivial to implement. Otherwise it is preferable to use a model which does not rely on knowing the dependence a priori. A second challenge is how to couple the energy among all the different modes. One method that is used frequently with coupled-mode models is to include higher order nonlinearities from the optical susceptibility in the field equations [24]. Finally, the most novel challenge addressed here is the inclusion of a realistic gain dispersion. The full semiconductor Bloch equations [26], [27] have a natural frequency dependence of the gain through the time dependence of the polarization. Since the polarization time scales are so much faster than the field and carrier time scales, the polarization is often adiabatically eliminated. While models have been used which include the actual semiconductor polarization [25], they are computationally expensive and impractical for most theoretical use. Efforts to replace the actual polarization with a simpler approximation are underway [28]. Here we derive a model that addresses the major challenges without modifying too much the usual semiconductor laser rate equations [29]. Our starting point is Maxwell s equations with the usual slowly varying approximation where is the forward field, is the backward field, is the forward polarization, is the backward polarization, is the carrier density, is the background index of refraction, is the speed of light, is the lateral confinement factor, is nonradiative cavity losses, is the current pumping, is the unit electron charge, is the quantum-well thickness, (1)

4 WHITE et al.: MULTILONGITUDINAL-MODE DYNAMICS IN A SEMICONDUCTOR LASER 1471 TABLE I PARAMETERS USED IN THE NUMERICAL COMPUTATIONS Fig. 2. Schematic of the system for optical injection showing the boundaries and the forward and backward propagating fields. is the current stripe width, is the cavity length, is the nonradiative recombination time, is the free-space permitivity, is the material permitivity, and is Planck s constant We define the polarization in the frequency domain as We assume that has a linear gain and refractive index and a parabolic gain profile where is the linear gain coefficient, is the linewidth enhancement factor, is the bandwidth, and is the lasing frequency. A parabolic approximation is accurate only when close to the peak gain. This is justified since the bandwidth is huge (10 THz) compared to longitudinal mode spacing (100 GHz). If it is necessary to resolve modes far from peak gain then a higher order polynomial expansion in may be necessary to properly fit the true gain dispersion. Now make use of the dispersion relation to transform the gain dispersion from the frequency domain to the longitudinal wavevector domain. This is so that when we take the inverse Fourier transform, the gain dispersion is in terms of spatial derivatives which are easier to handle. The final equations are (2) (3) (4) where is the detuning of the injection frequency from the free-running laser frequency, is the amplitude of the injected laser (this is as defined in Section II), and is the coupling coefficient (which here is the transmission coefficient of the output facet). We numerically solve the equations along the characteristics To handle the gain dispersion, we split the problem into two operators (8) (9) (10) Next we discretize, solving the part explicitly and the part implicitly. After rearranging terms (and replacing with where or )we find the discretized evolution equation broken into two distinct operators (11) Notice that if then the discretized equations reduce to the formally integrated equations. To integrate forward in time, we first solve for, then evaluate and at that value. Then we apply the exponential operator and finally the gain dispersion operator. Finally the boundary conditions are (see Fig. 2) (5) (6) (7) IV. NUMERICAL OBSERVATIONS In our numerical computations we used typical parameters for the GaAlAs laser used in the experiment (see Table I). The microscopic approach requires detailed knowledge of the laser gain medium (i.e., relative material compensation, strain (compressive or tensile), well thickness, band structure, etc.). As these lasers are commercial devices, this information was not available to us and we used the material parameters from [31]. The laser geometry was determined by empirically

5 1472 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 8, AUGUST 1998 (a) phenomena persists. Had the system needed noise to reproduce the mode hop then it might be argued that the fundamental mode becomes saturated and an adjacent mode lases out of noise. Instead we must assume that the mode hop is a dynamical instability and as such defies a simple explanation. In a future publication we hope to present a dynamical explanation of the phenomena. (b) (c) Fig. 3. Numerically obtained optical spectra for a fixed injection strength and for decreasing positive detuning 1. For large positive detuning 1= 2.75 GHz (a) the energy is completely in the fundamental mode, but as the detuning is decreased 1=1.65 GHz (b) all of the energy is transferred into the adjacent longitudinal mode (note the different scales) and for even smaller detuning 1=0.87 GHz (c) the energy returns to the fundamental mode. fitting the mode spacing and laser threshold. Further, there exists a current-pumping-dependent mode hop. This was used to determine the relative sign of the numerically computed longitudinal modes. Finally, the current pumping was chosen to be 0.76 above threshold, as defined in Section II. Numerically we observe a mode hop that is qualitatively similar to the experimental results. Fig. 3 shows the results of numerical simulations for the same detuning that was used in the experiment. Initially for a detuning of 2.75 GHz (a) the energy is concentrated in the free running mode. The mode shows relaxation oscillation sidebands at about 5 GHz indicating that the laser underwent a Hopf bifurcation. As the detuning is decreased to 1.65 GHz (b) all of the energy is transfered to the higher adjacent longitudinal mode. Here the mode is narrow corresponding to a stable laser output. No relaxation oscillation sidebands are present. As the detuning is further decreased to 0.85 GHz the energy is again found in the free running mode. The relaxation oscillation sidebands are stronger than for Fig. 3(a), indicating that the laser is further into the Hopf regime. It must be noted that this mode hop is a deterministic effect. The results presented here were obtained without spontaneous emission noise in the model. When noise is included the V. SUMMARY AND FUTURE WORK We have experimentally observed mode hopping in a semiconductor laser subject to optical injection. Mode hopping is an energy transfer from the free-running laser mode to an adjacent longitudinal mode at higher energy. Previous numerical work on the dynamics of a semiconductor laser subject to optical injection has solely used single-mode models that correctly describe the dynamical behavior when the laser is single mode but fail to capture the multilongitudinal-mode dynamics. In this paper we have presented a simple multilongitudinalmode model that includes gain dispersion. This model predicts quite accurately mode hopping in our system, the parameter regime where this instability is seen experimentally, and it also correctly describes the dynamical behavior of each mode in accordance with the single-mode model. Multimode instabilities are not due to spontaneous emission noise. Instead multimode instabilities modify the underlying dynamics of the semiconductor laser. Work to control and eliminate chaos in semiconductor lasers depends on accurately understanding the underlying dynamics. Without the inclusion of multimode dynamics, control schemes are at best valid for a limited range of parameters. With the existence of multimode instabilities shown experimentally and the presentation of a simple multimode model that captures the dynamics accurately, it is our future hope to better understand how multilongitudinal dynamics modify the well-studied single-mode dynamics of chaotic semiconductor lasers. 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