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Optics Communications 212 (2002) 353–357 www.elsevier.com/locate/optcom

Filamentation in broad area quantum dot semiconductor lasers C. Sailliot, V. Voignier *, G. Huyet Physics Department, National University of Ireland, University College, Cork, Ireland Received 22 January 2002; received in revised form 23 July 2002; accepted 21 August 2002

Abstract The properties of high power broad area quantum dots lasers are analysed numerically. Although these devices have a low a-factor, transverse modes instabilities may also occur. This instability is associated with the excess of carriers at the edge of the pumped area. It is therefore possible to prevent its appearance with a smooth current proﬁle. Ó 2002 Elsevier Science B.V. All rights reserved.

One of the most demanding applications for high power semiconductor lasers is the pumping of single ﬁber ampliﬁers, e.g., Erbium Doped Fiber Ampliﬁers and Raman Ampliﬁers [1]. For these applications, it is important to couple the laser eﬃciently to a single mode ﬁber, and therefore the spatial coherence of the beam is a major limiting factor. The onset of ﬁlamentation limits the brightness of these lasers and decreases the coupling eﬃciency. Recent fabrication of high power semiconductor lasers with quantum dot gain regions present several interesting features which may improve device performance. The symmetric Lorentzian-like shape of the gain spectrum of quantum dot materials should produce a laser with a very weak, if not zero, phase–amplitude coupling (i.e., zero a-factor). In this case, one expects an

*

Corresponding author. Tel.: +353-214-90-2381; fax: +353214-427-6949. E-mail address: [email protected] (V. Voignier).

absence of ﬁlamentation and also lower sensitivity to optical feedback. In this paper, we analyse the transverse structure of high power quantum dot semiconductor lasers. Our analysis is based on the Maxwell–Bloch equations describing a two-level broad area laser. We ﬁrst discuss the validity of this model, then we demonstrate that the top-hat like transverse pump proﬁle, common in stripe geometry lasers, should lead to the appearance of an instability similar to ﬁlamentation. Subsequently, we present a way of stabilizing the transverse dynamics of the laser by tailoring the pump proﬁle. We show that the threshold of this instability depends on the shape of the chosen proﬁle. Broad area semiconductor lasers are commonly described using semi-classical models where the evolution of the electric ﬁeld is described by the Maxwell equations while the gain properties are derived from quantum mechanics. The level of sophistications of these models may vary. For instance some authors calculate the gain and index

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properties of the medium from many-body theory while others may take a more phenomenological approach where the phase–amplitude coupling or a-factor is added to the classical two-level laser model [2]. In the case of quantum dot semiconductor lasers, the gain spectrum is symmetric and one therefore expects a zero a-factor [3]. From this viewpoint a quantum dot semiconductor laser is very similar to a gas laser and should be described by the Maxwell– Bloch equations for the electric ﬁeld, polarisation and population inversion which read [4] 1h a i 1 ot E ¼ 1 id i r2 E P ; sE 2 sE i 1h ot P ¼ NE þ ð1 þ idÞP ; sP 1 1 ðE P þ EP Þ þ N J ðxÞ ; ot N ¼ sN 2 where E, P and N represent the complex amplitude of the electric ﬁeld, polarisation and the population inversion. J ðxÞ is the pump parameter which is proportional to the current density. sE , sP and sN are the characteristic decay time of the electric ﬁeld, polarisation and carrier density, respectively. d is the detuning. Finally a is the diﬀraction parameter. This model considers only a single longitudinal mode, consequently it cannot describe instabilities arising from multi-mode dynamics. This approximation can be justiﬁed if the laser is in the limit of strong inhomogeneous broadening, i.e., diﬀerent longitudinal modes interact with diﬀerent atoms. As a result, the interaction between the diﬀerent longitudinal modes can be neglected and the total intensity behaves like a single mode. However, the frequency diﬀerence between consecutive transverse modes is much lower than the laser free spectral range and the inhomogeneous broadening does not alter the transverse mode structure. If the laser is homogeneously broadened, then the dynamics of a single mode follows the equations described above. We will also assume d ¼ 0 since the longitudinal mode spacing is much lower than the bandwidth of the material. Previous studies of the Maxwell–Bloch equations have demonstrated that for positive detun-

ing, the laser selects a travelling wave with a transverse k-vector such that its frequency is the atomic frequency [8,11,12]. The wavelength of the travelling wave is therefore selected by the value of the detuning. However, travelling waves can be observed even with a zero detuning as it has been demonstrated that imposing a zero electric ﬁeld as a boundary condition generates transverse travelling wave solutions [5,6] in analogy with spiral waves in nonlinear media [7]. This can be illustrated by ﬁxing the electric ﬁeld to zero with an homogeneous pump proﬁle in a semi-inﬁnite medium. Near the threshold, the electric ﬁeld can be written as Eðx; tÞ ¼ RðxÞ expðiðxt uðxÞÞÞ; qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 RðxÞ ¼ j 1 ðxsP Þ tanhðpxÞ; u0 ðxÞ ¼ q tanhðpxÞ; where the frequency x, the size of the boundary 1=p and the transverse wavenumber q can be directly calculated from the Maxwell–Bloch equations. These solutions correspond to travelling waves selected by the border, whose wavelength 1=q is proportional to the size of the boundary 1=p, with resulting oﬀ-axis emission. We integrated numerically the Maxwell–Bloch equations with a top-hat pump proﬁle (J ðxÞ ¼ 0 for jxj > W =2 and J ðxÞ ¼ J0 for jxj < W =2), with a zero detuning and a high carrier decay rate sN =sE ¼ 1. As expected from the above argument, we observed two counter-propagating travelling waves emitted from the boundaries (see Fig. 1). In our case, the pump proﬁle deﬁnes boundary conditions because it damps the ﬁeld to zero. Since each edge selects a wave which travels toward the center, one could imagine that these would interfere to form a standing wave. However, these solutions are unstable due to the strong nonlinear coupling of the medium [8] and the two travelling waves occupy diﬀerent spatial regions of the transverse section of the laser as shown in Fig. 1. This is similar to the behaviour of a ring cavity where the two counterpropagating travelling waves cannot co-exist due to the strong nonlinear interaction [9,10]. This behaviour has not been observed so far in conventional broad area semiconductor lasers as self-focusing destabilises these solutions. New

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Fig. 1. Near-ﬁeld (left) and far-ﬁeld (right) of the laser with a top-hat pump proﬁle and a large carrier decay rate. Real part of electric ﬁeld (straight line) and intensity (dashed line) are shown in the near-ﬁeld, and intensity is shown in the far-ﬁeld. We observe two counterpropagating travelling waves emmited from the boundary and propagating toward the center in the near-ﬁeld, and a double lobed far-ﬁeld. Here W ¼ 140, L ¼ 512, J0 ¼ 2, a ¼ 1, sE ¼ 1, sP ¼ 0:1, sN ¼ 1.

semiconductor lasers such as quantum dot lasers seem to be very promising devices since the theoretically calculated (and experimentally measured) a-parameter remains very small [3]. As a result no ﬁlamentation instability should occur. However, numerical simulations of the Maxwell–Bloch equations with a long carrier lifetime sN =sE ¼ 200, characteristic of semiconductor lasers, does not show such a behaviour with a top-hat pump proﬁle. A complicated spatio-temporal behaviour is observed instead of the two counter-propagating traveling waves, as shown in Fig. 2. This unstable

Fig. 2. Average far-ﬁeld intensity (left) and near-ﬁeld spacetime diagram for a laser with a small carriers decay rate sN ¼ 1=200. The laser dynamics is unstable. All parameters other than sN are indentical to Fig. 1.

behaviour produces ﬂuctuations of the laser intensity on the order of the relaxation oscillation time scale. The mechanism responsible for the instability is the following. Diﬀraction prevents the ﬁeld intensity from matching exactly at the sharp edges of the top-hat pump proﬁle. As a result, an excess of carriers builds up at the boundary of the proﬁle (Fig. 3). This carrier excess, far from the carrier equilibrium value, destabilising the source of the travelling waves at large values of the carrier life time (sN =sE 1). In order to demonstrate this, we analysed the dynamics for a case with just a single

Fig. 3. The top-hat current proﬁle (dashed-line) induces a large amount of carriers (dotted-line) and a low power (full line) at the edge. Here sE ¼ 1, sP ¼ 0:1, sN ¼ 1.

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Fig. 4. Pump proﬁle with linearly varying edges (left) and selected transverse k-vector versus the inverse of the boundary size (right). The limit case 1=p ¼ 0 deﬁnes a top-hat proﬁle, for which the k-vector reaches a saturation value.

edge as depicted in Fig. 3. This was achieved by ﬁxing Eð0Þ ¼ Rð0Þeiuð0Þ with R0 ð0Þ and u00 ð0Þ ¼ 0, with a pump proﬁle deﬁned as (J ðxÞ ¼ 0 for x < W =2 and J ðxÞ ¼ J0 for x > W =2). With such a proﬁle, a single travelling wave is created at the edge (x ¼ W =2) and propagates towards the non-reﬂecting edge (x ¼ 0). For large values of the ratio sN =sE the solution becomes unstable, although pure travelling waves (E ¼ E0 eðixtkxÞ ) with the same wavenumber would be stable in an inﬁnite medium. The source of the instability starts with oscillation of the intensity at the edge of the proﬁle, with a frequency of the same order as the relaxation oscillation. In order to investigate the eﬀect of this instability in a more realistic case, we investigated the spatio-temporal dynamics of the laser with a spatially tailored pump proﬁle as shown on Fig. 4. Here the pump increases linearly at the boundaries and is constant in the centre (Fig. 4(a)). With such a pump proﬁle, the characteristic length for the variation of the transverse intensity distribution (or the near-ﬁeld) is the size of the boundary, which is inversely proportional to the slope of the pump proﬁle edge (p in Fig. 4). Therefore the wavenumber of the selected travelling wave increases with the slope (4b). For low values of the slope, the ﬁeld should follow the pump proﬁle and the excess of carriers at the edge should be reduced. Numerical simulations of the Maxwell–Bloch equations demonstrate that the two counter-propagating travelling waves are stable for low values of the slope. An instability, similar to that observed for the top-hat pump

proﬁle, occurs when the slope is increased. It is worthwhile to note that we observe the same behaviour (travelling solutions and associated instabilities) when the edges of the current proﬁle are smoothed. Interestingly, the threshold of this instability depends on the width (W) of the pump proﬁle as illustrated in Fig. 5. This dependence should therefore be associated with the interaction between the two sources: their coupling tends to destabilise them. This coupling decreases as the distance between the two sources increases and the threshold of the instability consequently increases. We also note that at the onset of the instability, the sources oscillate synchronously and out of phase, during a transient leading to spatio-temporal disorder. When the instability is fully developed, the dynamics are similar to ﬁlamentary behaviour as

Fig. 5. Phase space for the wave sources instability. The threshold for the instablity depends on the width of the proﬁle. For small width, the coupling of the two sources is strong, which lower the threshold.

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described by Fisher et al. [13], although our model does not include an explicit self-focusing term such as the a-factor. The results we describe were obtained for a zero detuning, but they remain valid for a non zero detuning. It is worthwhile to note that the transverse k-vector varies with detuning [5,8]. As a result, the instability threshold decreases for small values of detuning. An amplitude instability can also be observed for large detuning for any pump proﬁle. This instability has already been described [5,8]. In conclusion, our numerical results demonstrate that pump proﬁling will remain extremely important even if semiconductor lasers have a zero a-factor. The current proﬁle should remain smooth in order to limit the excess of carriers at the edges of the current injection.

Acknowledgements We acknowledge John Houlihan and Eamonn OÕNeill for useful discussion.

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