IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 5, MAY 1996 839 Instabilities and Nonlinear L- I Characteristics in Complex-Coupled DFB Lasers with Antiphase Gain and Index Gratings Bjorn Jonsson, Arthur James Lowery, Senior Member, IEEE, Henning Olesen, and Bjame Tromborg, Member, IEEE Abstruct- Complex-coupled DFB lasers can be designed to to give “antiphase” coupling, chirp can be reduced [4], [6] provide large bandwidths and low chirp by adjusting the strength and the modulation bandwidth can be improved [ 151, [ 161. and phase of their index and gain gratings. In this paper we Schatz 1171 found that conventional index-coupled DFB develop a self-consistent method of calculating the coupling co- efficient of complex-coupled DFB lasers with a corrugated active lasers lasing at the short-wavelength side of the Bragg wave- region. Basic geometrical and structural parameters are used as length have the potential of becoming unstable when biased inputs to the model. We show that antiphase coupling implies above some critical output power. He also showed that these lasing on the short-wavelength side of the Bragg wavelength instabilities cannot occur when the laser is operated on the which in turn leads to instabilities and/or nonlinearities in the long-wavelength side and that the instabilities become increas- light-current characteristic. The critical output power for onset of instabilities is typically in the order of a few milliwatts. Time- ingly pronounced as the lasing mode is pushed away from domain simulations are used to assess the potential effects of the Bragg wavelength toward the short-wavelength side, for these instabilities on optical communicationsystems. We find that example by changing the magnitude of a discrete phase shift. A transitions from a state of equal output powers from the facets simple threshold analysis was given in [ 171 to estimate the crit- to a state of dramatically different output powers occur within ical output power above which instabilities occur. Instabilities a few nanoseconds of turn-on. The origin of these instabilities is explained using a simple physical model and possible ways of in conventional index-coupled DFB lasers were also studied increasing the critical power for instabilities are discussed. For extensively by Olesen et al. [18]-[20]. We have previously example, we clearly show that the critical power for instability found [2 11 that complex-coupled DFB lasers with antiphase increases when the carrier lifetime is decreased. coupling show a qualitatively similar type of instability to that reported in [17]. Lowery and Novak [ll] as well as Zhang and Carroll [ 161 have recently performed above-threshold 1. INTRODUCTION calculations of the dynamic properties of complex-coupled ISTRIBUTED feedback (DFB) semiconductor lasers lasers, but unstable behavior was not addressed. D have a built-in periodic longitudinal variation of the In this paper, a detailed analysis of DFB lasers with a com- refractive index andor the net optical gain. This modulation plex coupling coefficient is presented. Above-threshold sta- produces a coupling between the forward- and backward- bility is studied using two different computer simulation pro- traveling optical waves, providing the feedback mechanism grams: a semianalytic model based on the traveling-wave ap- for lasing with a wavelength determined by the period of proach [22], and a large-signal analysis with the transmission- the modulation. It is well-known that index-coupled DFB line laser model (TLLM) [ 111. Previously, these models have lasers with antireflection coated facets have two degenerate been used to simulate index-coupled [19], [20], [23] and modes, unless the waveguide contains a perturbation, such complex-coupled lasers [6], [11], [15], 1211, 1241, and the com- as a phase-shift or a taper. As was shown by Kogelnik and bined features of the two models yield detailed information Shank [l], DFB lasers with pure gain coupling prefer a single about the laser’s behavior. mode of oscillation. A very high side mode suppression We show that complex-coupled DFB lasers optimized for ratio (SMSR) can be achieved in gain-coupled or complex- high bandwidth and low chirp, using antiphase coupling, will coupled DFB lasers, and this is one of the main reasons for oscillate on the short wavelength side of the Bragg wavelength. the current interest in these devices [2]-[14]. Furthermore, Due to this fact, such lasers will be susceptible to instabilities numerical studies of complex-coupled lasers have shown that and nonlinear behavior 1191, [21]. This is unfortunate as we by adjusting the ratio and phase of the gain and index gratings shall show that for some grating geometries the laser may become unstable at a critical output power of only a few Manuscript received July 18, 1995; revised January 23, 1996. This work was milliwatts. The dynamics of such unstable lasers are simulated supported by the Australian Department of Industry, Science and Technology, and the output powers at the two facets are found to deviate under an International Scicnce & Technology Grant. B. Jonsson, H. Olcsen, and E. Tromborg are with Tclc Danmark Rescarch, from each other after a few nanoseconds. In order to identify DK-2970 H@rsholm,Denmark. laser designs less susceptible to instabilities, we extend our A. J. Lowery is with the Photonics Research Laboratory, Australian Pho- analysis to lasers with various corrugation geometries of the tonics Cooperative Research Centre, Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, Vic. 3052, Australia. active region, and show that the critical output power can be Publisher Item Identifier S 001 8-9197(96)03207-1. increased by altering the modulation depth and duty cycle of 0018-9197/96$05.00 0 1996 IEEE 840 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 5, MAY 1996 where neffis the waveguide effective index, and X is the lasing wavelength. The periodic structure of Fig. 1 can be represented by periodic perturbations of the waveguide effective index, the modal gain and the internal absorption. The standard technique is to expand this perturbation in a Fourier series and to retain only the first-order terms Fig. 1. Grating structure in a complex-coupled DFB laser. the gain grating. We also develop a simple physical model where neE 0. go. and azo denote the average values over of the instability mechanism, and use this model to identify one grating period; An,,, Ag, and Aa, are the peak-to-peak methods of increasing the critical output power. Spontaneous deviations within one grating period; 'p, is a phase offset of the recombination is an important damping mechanism and we corrugation relative to the left end-facet (z = 0). The detailed find that the critical power is inversely proportional to the geometry is represented by the Fourier coefficients A,, A,, carrier lifetime. and AL for the index-, gain-, and loss-gratings, respectively. The paper is organized as follows. In Section I1 the theo- Note that using the explicit forms (3a)-(3c) implies that retical background for the self-consistent calculation of the the three types of gratings cannot have an arbitrary phase complex coupling coefficient as a function of the grating relationship. They must either be in-phase or in antiphase with geometry is presented. Section I11 gives a brief description each other, which will be the case for all practical geometries. of the two numerical methods. In Section IV, results for the It is important to realize that a variation of, for example, dependence of the complex coupling coefficient on the grating the active region parameters affects the refractive index and geometry, below and at threshold, are presented. In Section the absorption as well as the gain. All three of the modal V, operation above threshold is investigated. In particular, parameters (n,~,g. and a,) depend on the material parameters unstable regions are analyzed and the time-domain behavior of all transverse layers within the intensity profile and, in of such lasers is simulated. The origin of these instabilities addition, may depend on the injected carrier density. and possible ways of increasing the critical output power are Inserting (3a)-(3c) in (1) and collecting the slowly- discussed in Section VI. Finally, our results are summarized varying and first-order terms, the perturbation of the complex in Section VII. wavenumber is given as 11. THEORY (4) Complex coupling can be realized by incorporating a pe- riodic modulation of the active region itself, or by having where is the slowly varying part of the wavenumber. We separate grating layers to provide loss or index coupling. have here introduced the complex coupling coefficient K, and A typical multilayer laser structure is outlined in Fig. 1. in the presence of all three perturbations the general expression The longitudinal coordinate is denoted by z. and h is the for K is grating period. The active region provides gain coupling with 7l li some parasitic index coupling. The index grating allows the K = -x A, An,, + - (A,Ag - AL Aai). (5) amount of index coupling to be independently controlled. In 4 the following, we only use the longitudinal dependence z The real part of the coupling coefficient (the index coupling) explicitly to denote variations within one grating period. It is determined completely by the variation in the waveguide is implicitly assumed that all variables are allowed to vary effective index, while the imaginary part (the gain or loss slowly with x