Filamentation in broad area quantum dot semiconductor lasers

Filamentation in broad area quantum dot semiconductor lasers

1 November 2002 Optics Communications 212 (2002) 353–357 www.elsevier.com/locate/optcom Filamentation in broad area quantum dot semiconductor lasers...

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1 November 2002

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

0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 9 5 5 - 7

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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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