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Longitudinal mode competition in (Al,In)GaN laser diodes at λ = 445nm and 515 nm with mode competition frequencies from 10 MHz to 150 MHz is observed. Up to two dozen lasing modes oscillate with the lasing mode rolling from the short wavelength edge to the long wavelength edge of the gain profile. The experimental results can be described very well with a set of multi–mode rate equations including self–, symmetric and asymmetric cross gain saturation. By tuning essential parameters of the gain saturation terms, mode competition disappears and single mode operation as well as mode clustering is found. This proves that the mechanisms of gain saturation have not only a profound impact on the complex temporal–spectral behavior but also explains mode clustering in (Al,In)GaN laser diodes, both in pulsed and continuous wave (cw) operation as a natural nonlinear effect without the necessity to add noise.
© 2014 Optical Society of America
1. Introduction
The development of the blue and the green (Al,In)GaN laser diode (LD) experienced great progress in the last years, pushing this technology toward longer wavelengths [1–3], higher output power [4,5], and higher wall plug efficiency [6,7]. These LDs enable not only new types of consumer electronics [8,9], high–value car headlights [10], and have the potential to become the light source for solid–state–lighting [11,12]. As compact stand–alone or original equipment manufacturer (OEM) laser sources they serve a multitude of applications, e.g. in spectroscopy, bio–photonics, and quantum optics. Wavelength stabilization on a single longitudinal mode and tuning in an external cavity [13] is a prerequisite for many of those applications. On the other end of the spectrum of laser dynamics is the generation of ultrafast pulses. High peak–power picosecond [14, 15] and sub–picosecond [16, 17] pulses have been generated in multi–segment (Al,In)GaN LDs. The performance of these devices can be strongly influenced by mode competition and mode clustering. However, the role of gain saturation as a physical cause of these effects has not yet been investigated for wide–bandgap LDs.
A standard Fabry–Pérot type LD without external nor internal feedback normally lases on multiple longitudinal modes. Characteristic for (Al,In)GaN LDs is the formation of few strong lasing modes with a spectral spacing of several longitudinal modes. This has been termed mode clustering and was explained by fluctuations of quantum well thickness or composition [18], fluctuations of the optical gain [19], sub–cavities caused by cracks traversing the waveguide [20], and e–h plasma oscillations [21]. In this work, we show that gain saturation naturally explains the formation of several lasing modes with an enhanced mode spacing, resulting in mode clustering.
Even in cw operation the longitudinal mode comb is not static but exhibits mode competition if asymmetric gain saturation is dominant. The theoretical framework based on four–wave mixing was developed originally to explain longitudinal mode interaction observed in infrared laser diodes [22–26]. For violet (Al,In)GaN LDs mode competition has been observed and was explained by symmetric and asymmetric gain saturation [27]. The time scale of mode competition dynamics is in the MHz range, given by the minute differences in optical gain between individual longitudinal modes, causing one mode to rise while depleting gain from neighboring modes. The gain nonlinearities do not solely act as perturbation for the stabilization of a LD on a single longitudinal mode by distributed feedback or in an external cavity, they also interfere with ultrafast pulse generation by self–Q–switching or mode locking.
In this paper we characterize the temporal–spectral dynamics of blue and green (Al,In)GaN LDs. We employ a streak camera in single–shot mode to measure the rich behavior caused by the interaction of many longitudinal modes. A set of multi–mode rate equations including self–gain saturation and symmetric and asymmetric cross–gain saturation induced by neighboring longitudinal modes is applied, in order to model mode competition. The model was originally developed for GaAs based infrared laser diodes [28]. We use the terminology as described in [29]. The simulations reproduce not only the mode competition quantitatively, but also explain mode clustering in (Al,In)GaN laser diodes naturally as being caused by gain saturation.
2. Experiment
Figure 1 shows spectrochronograms, i.e. the time–and wavelength–resolved laser intensity, of a green LD [Figs. 1(a) and (e)] and a blue LD [Fig. 1(i)]. In this section we describe the experimental results (left pane), the discussion of the simulations (right pane) follows next.
Fig. 1: Experiments (left pane) and simulations (right pane) of the mode competition dynamics. In each case an experimental and simulated spectrochronogram is shown in combination with a time–integrated spectrum illustrating the longitudinal mode comb. (a)–(d): green LD showing mode clustering with indications of mode competition at I = 1.2Ith. (e)–(h): green LD with mode competition at I = 3Ith. (i)–(l): mode competition in a blue LD at I = 4Ith.
The blue and green LDs under investigation are R&D devices grown on GaN substrate, with lasing wavelengths around 455 nm and 515 nm, respectively. We stress that the mode competition behavior is generic, as we observe it for many (Al,In)GaN LDs in the violet to green spectral range and from different sources.
The spectrochronograms were taken with a streak camera in combination with a spectrometer with a 600 lines per mm grating. All measurements were taken in single shot mode. It is possible to trigger on the rather regular mode competition oscillations, however only in single shot spectrochronograms the full complexity of the mode dynamics can be observed. For the single shot measurements, rectangular 800 ns electric pulses are applied on the LDs, which corresponds to a quasi cw operation since relaxation oscillations [30] and heating of the electron reservoir [31] appear on a much faster time scale (∼ ns). Heating of the waveguide on a microsecond time scale is observed during the pulse as small red–shift of the longitudinal modes. This effect is most pronounced in the green LD at the highest current [Fig. 1(e)] but does not interfere noticeably with mode competition.
Figure 1 (a), (e) and (i) show the spectrochronograms of a green LD with 600μm resonator length emitting around 513.5nm at currents I = 1.2Ith and I = 3Ith, and of a blue LD with 600μm resonator length emitting around 446.5nm at I = 4Ith. Longitudinal mode spacing is 0.071 nm and 0.043 nm in the green and blue LD, respectively. The individual longitudinal modes can be clearly seen in the spectrochronograms and in the coresponding time–integrated spectra in Figs. 1(b), (f), and (j). Close above threshold only a few longitudinal modes start to lase. In the green sample three modes separated by a couple of modes (approximately 0.5 nm) are dominant at 1.2Ith [see Figs. 1(a) and (b)]. This is known as mode clustering and has been observed and discussed in (Al,In)GaN LDs before [21, 30, 32]. The modulation frequency of a single mode increases with current, and more longitudinal modes (up to two dozens) contribute to lasing and thus to mode competition [see Figs. 1(e) and (i)].
The individual longitudinal modes oscillate in the MHz range, which is fundamentally different from well–known relaxation oscillations in GaN–based LDs. Contrary, these relaxation oscillations typically oscillate in the low GHz range, dependent on the differential gain and current, and are damped [30, 33]. Inhomogeneous pumping of quantum wells that introduces a slow absorber below threshold does not change the oscillation frequency remarkably [34]. By placing an absorber section in the cavity the relaxation oscillations can be stabilized and the longitudinal modes pulsate simultaneously in the GHz range [35]. However, the oscillations in the MHz region, we report on in this work, show the effect that the lasing mode shifts from the shorter–wavelength mode to the longer one. This deterministic dynamical behavior, which is distinct from mode partition noise [36, 37], has been observed previously in infrared LDs [38, 39] and was documented once in a 405 nm (Al,In)GaN LD [27]. Whereas the single mode shows a 100 % modulation, the intensity integrated over all modes does not return to zero. Still, even in cw operation, the mode competition results in considerable intensity noise of the laser output, in particular when combined with wavelength–selective optical elements, e.g. the combination of a waveplate and polarizer.
The regular pattern of mode competition dynamics seen in the experiments indicate a coupling mechanism between longitudinal modes. The modes are rolling from short to long wavelengths at a pace of approximately 50nm/μs. At the long wavelength edge of the spectrum the mode intensity ceases and lasing resumes at the short wavelength edge of the spectrum. This implies that the intensity of all modes oscillate with the same frequency, which is an extreme case of frequency clustering. The latter term arises in antiphase dynamics of the longitudinal modes of InAs quantum dot lasers [40]. The temporal–spectral dynamics, we observe, is caused by spectral hole burning, or more specific by symmetric and asymmetric cross–saturation terms coupling the optical gain of longitudinal modes [23]. This approach was applied before to explain mode competition in violet (Al,In)GaN LDs, but at that time the experimental data allowed only to follow the dynamics of three neighboring longidutinal modes [27]. Now, the spectrochronograms taken with the streak camera allow a quantitative analysis of the mode competition dynamics and also to establish the connection to mode clustering.
3. Simulation
To simulate the longitudinal mode dynamics, we use a system of multi–mode rate equations including self–and cross–saturation terms, as described in [41, 42] for GaAs–based LDs and adapted in [27] for (Al,In)GaN LDs. These rate equations result from nonlinear effects of the charge carrier density, susceptibility and polarization [29]. The temporal evolution of the photons of the p–th mode, Sp, is described by
where g̃p is the total gain of the p–th mode, g̃th the threshold gain, βp the spontaneous emission coefficient of the p–th mode, and τr the radiative carrier life time. The rate equation for the charge carriers N has an injection term with current I, injection efficiency ηinj, and the elementary charge qe. τs is the charge carrier lifetime, and the last term represents the stimulated recombination as a sum over all longitudinal modes, which couples the charge carriers to the photons. It is now important, that the total modal gain g̃p saturates at high photon density via different effects. The gain is described by the linear gain Ap for the p–th mode of wavelength λp and quadratic saturation terms. B describes self saturation and Dpq and Hpq symmetric and asymmetric cross saturation, respectively, of the q–th mode on the p–th mode. The terms are given by [28, 42]: In this set of equations, a is the linear gain coefficient, Γ the confinement factor, V the active region volume, Ntr the transparency density, b the gain curvature coefficient that describes the fitted parabola around peak gain, and λ0 the center wavelength. Additionally, in the self saturation coefficient B the vacuum speed of light c0, the vacuum permittivity ε0, the refractive index ngr, the saturation charge carrier number NS, the intraband relaxation time τin, and the transition dipole moment Rcv are introduced, as well as the transparency charge carrier number Ntr, and the antiguiding factor αag in the asymmetric cross saturation term.This model is discussed by Yamada et al. in detail in [43]. There, the underlying mechanism for gain saturation is explained by a reduction of the refractive index and antiguiding effecting the total gain. According to Agrawal, however, the nonlinear gain asymmetry is specifically related to the slope of the gain profile at the frequencies of various longitudinal modes [23]. While we will show that the set of differential equations describes the physical phenomena accurately, we leave the physical interpretation of the parameters such as τin and αag open.
In the mathematical description mode competition is caused by the cross saturation terms [Eqs. (7) and (8)]. Dpq is basically a Cauchy distribution with a full width at half maximum (FWHM):
The FWHMDpq is a quantity describing how many neighboring modes are influenced by symmetric gain saturation. Since the wavelength is given by the LD, the intraband relaxation time τin is the only parameter tuning this quantity. Furthermore, τin influences the amplitude of the Cauchy distribution, as a pre–factor [see Eq. (7)]. The asymmetric gain saturation Hpq is independent of τin, but is proportional to the antiguiding factor, which only enters here. The strong dependence of the mode competition frequency from αag was already shown in simulations by Schmidtke et al.. [27]. Therefore, τin and αag are important parameters, which tune the symmetric and asymmetric gain saturation independently and as a consequence the dynamical behavior of the LD.We solve the multi–mode rate equations numerically and find a set of parameters, which describe the experimental results best. As criteria for a good agreement of simulation and experiment served a comparison of the spectral–dynamic behavior, i.e. the amount of longitudinal modes contributing to mode competition should be similar for each current, and the mode competition frequency vs. current plots should coincide. The most relevant parameters for a blue and green LD are listed in Tab. 1. The photon density–independent term in Eq. (3), Ap, is determined by the linear gain a, which is directly related to dg/dN measured via relaxation oscillations (described in [33, 44]), and by the gain curvature b, which was determined by means of Hakki–Paoli spectroscopy at a current slightly below threshold. Both, the differential gain and the gain curvature decrease from the blue to the green LD due to a smaller wavefunction overlap and a larger inhomogeneous broadening [45]. Furthermore, the transparency density and the internal losses are deduced by Hakki-Paoli gain spectroscopy. Carrier life time τs can be determined by electro–luminescence decay measurements. For the green LD a value of τs = 2.5ns is obtained, whereas the blue LD has a significant longer decay time due to better crystal quality and a smaller Schockley–Read–Hall parameter A [44]. The injection efficiency is assumed to be 1 in all devices. Experimental results on similar devices indicate values above 0.9. The wavelength–dependent modal group refractive index was taken from the work by Scheibenzuber [46]. Transition dipole moment , antiguiding factor αag and intraband relaxation time τin are used as simulation parameters and discussed in Sec. 4.
Table 1:. Most relevant parameters for the simulation of the mode dynamics in blue and green LDs.
Exemplary results of the numerical solutions are displayed on the right pane in Fig. 1. The numerical solutions for the time–dependent photon densities Sp(t) are illustrated as spectrochronograms and time–integrated spectra for comparison with the experimental results at the same wavelength and current. Experiment and simulations are in good agreement and show a similar number of longitudinal modes contributing to mode competition [see Figs. 1(e), (i) and (g), (k)]. At the lowest current displayed in this paper mode clustering is visible in the experiment [see Figs. 1(a) and (b)] with indications of mode competition pulsations. The simulations [see Figs. 1(c) and (d)] rather indicate mode competition behavior with a slight tendency to mode clustering. The peaks of the spectra in Figs. 1(b), (f) and (j) are smeared out in the experiment since the longitudinal modes still slightly shift into the red approximately 300 ns after the pulse beginning due to heating of the waveguide [31]. The smearing out in the temporal axis in the experimental data is caused by the streak camera resolution and intensity fluctuations. In general, the experimental data shows irregularities in the intensity, which are not found in the simulation, since no gain fluctuations or other noise terms are introduced for the sake of clarity.
Figure 2 shows exemplary numerical solutions for the total gain g̃p and photon density Sp of the multi–mode rate equations for the 515nm LD at a current I = 2.5Ith at various points in time. Each dot and square in the plots symbolize one longitudinal mode in the cavity. The zero position of the photon density (blue squares and line) on the right axis is chosen to coincide with threshold gain (black line). At 210 ns [Fig. 2(a)] a main single mode lases together with the two weak neighboring modes. The photon density effects the gain profile through Eq. (3). Through self–saturation and the symmetric cross saturation the gain of the lasing mode and its neighboring modes is decreased. By the asymmetric cross saturation term Hpq the gain of the mode next to the main lasing mode on the long wavelength side of the gain profile is larger. As a consequence, the photon density of this mode increases as visible at 215 ns [Fig. 2(b)]. This mechanism causes rolling of the modes from shorter to longer wavelengths. Furthermore, at 210 ns [Fig. 2(c)] around λ = 513.3nm the gain is slightly larger than threshold, leading to an increase of the photon density and the appearance of a second lasing mode near 513.5nm at t = 215ns. This mode also moves toward longer wavelengths, increasing in intensity, whereas the previous lasing mode vanishes [Fig. 2(d)], since its gain cannot exceed the threshold condition for longer wavelengths. It is thus possible that not only one mode but several ones are in lasing condition at the same time and move through the gain profile. Therefore, the frequency determined by the time for a lasing mode to move through the whole gain profile is smaller than the pulse repetition frequency of a single mode, which we refer to as mode competition frequency here.
Fig. 2: Simulated temporal evolution (5 ns steps) of the total gain (black open dots) and the photon density (blue filled squares) for the longitudinal modes of a green LD at I = 2.5Ith. The black horizontal line corresponds to the modal gain at threshold. The local gain maxima exceeding the threshold condition move from shorter to longer wavelengths with the photon density following.
4. Comparison of simulated and measured mode competition frequency
The spectrochronograms in Fig. 1 show a generic behavior which was verified by measurements for green, blue, and violet LDs of several sources. For further quantitative analysis we compare measured and simulated mode competition frequencies of two blue and two green LDs as function of current (see Fig. 3). The blue LDs are identical in their design and show similar threshold current densities. The green LDs are also identical except for one LD having a 1.5 times larger resonator length than the other and thus a smaller threshold current density. Single–shot spectrochronograms were taken at each current, and the mode competition frequencies were determined from trails of a single longitudinal mode and plotted as circles (blue LD) and squares (green LD). The mode competition frequencies reach from 10 MHz slightly above threshold to 150 MHz at I = 6Ith in the blue devices and from 10 to 60 MHz in the green devices. The frequencies show a sublinear dependence on current. The variation of some data points are also due to the noisy character of mode competition. At lower frequencies the pulsation frequency can be determined easier, since only one mode is above threshold at at time, whereas at higher currents several modes are above threshold and intensity fluctuations lead to a more complex dynamic behavior as visible in Fig. 1(i). We especially observe this behavior in commercially available violet LD, in which the dynamics are faster and mode competition appears in a more chaotic way.
Fig. 3: Measurement (circles and squares) and simulation (dashed and solid lines) of the current dependence of the mode competition frequency of two 445 nm (blue coloring) and two 515 nm (green coloring) LDs. The blue dashed curve shows the simulated mode competition frequency for a blue LD and the green dashed and solid lines for the two green LDs with different cavity lengths.
We compare the measured mode competition frequencies with those determined by the numerical solutions of the multi–mode rate equations (1) and (2). The most relevant parameters for the mode competition simulation are listed in Tab. 1. The transition dipole moment is smaller for the green than the blue device and smaller than the violet device in [19] due to a reduced wavefunction overlap in the longer wavelength devices. In detail, the optical matrix parameter Ep in bulk GaAs, Ep = 28.8eV, is about twice as large as in bulk GaN and bulk InGaN, Ep = 14.0eV and Ep = 14.6eV [47]. With consideration of the reduced wavefunction overlap in InGaN quantum wells due to the quantum confined Stark effect, Ep is further reduced. The transition dipole moment Rcv2, used in [41, 42], is related to transition matrix element |MT|2 as follows: [48]
where ω is the photon angular frequency and m0 the electron mass. |MT| can be expressed with |M| via a basis transformation as described in A10.1 in [48]. |M|2 is the momentum matrix element and is calculated from Ep via the relation . Rcv is thus estimated to be between 1 · 10−29 Cm−1 and 5 · 10−29 Cm−1 for green and blue (Al,In)GaN LDs, whereas the values applied in the simulations for GaAs–based LDs are larger [49]. B and Dpq are proportional to Rcv2.The antiguiding factor is αag = 9 in all simulated devices and is essential for mode competition since it is the only parameter in Hpq, which does not appear in any other gain terms. The used value is thus two times larger than measured values [50], but necessary to generate mode competition with a similar amount of contributing modes.
The intraband relaxation time τin is important for tuning the width of Dpq, as Eq. (9) shows. We use τin as a simulation parameter. Measurements and simulations of the gain in violet In-GaN/GaN quantum wells suggest τin ≈ 25fs [51]. As discussed in Sec. 3 it is open, which physical processes cause cross–gain saturation in (Al,In)GaN LDs. This may affect how the quantities τin, Rcv and αag enter Eqs. (1)–(8). With the chosen parameters, the measurements are described very accurately. The number of modes contributing to mode competition and the mode competition frequency are in good agreement at various currents for the investigated devices.
5. Mode clustering
The phenomenon, that more than one mode, separated by a few longitudinal modes, start to lase and lead to mode clustering is well–known in InGaN LDs [30, 32, 52]. It has been proposed, that quantum–well thickness or compositional fluctuations promote mode clustering around threshold [18]. Optical gain fluctuations [19], and e–h plasma oscillations in the tilted quantum well [21] were also cited as possible reasons for mode clustering.
In this paper, we report that numerical solutions of the multi–mode rate equations show mode clustering naturally, without the use of gain fluctuations or e–h plasma oscillations. The gain saturation terms play a crucial role in this description of mode clustering.
Figure 4(a) shows the simulated spectrochronogram with the same set of parameters for a green LD as listed in Tab. 1. After approximately 100 ns stable mode competition occurs with modes covering a wavelength range of over 0.5 nm. Figure 5(a) shows the gain and the gain saturation terms at 240 ns. Next to the lasing mode, the photon density of a new mode on the long–wavelength side starts to build up. Due to the asymmetric gain term ∑q HpqSq the gain of the main lasing mode decreases. By this the lasing mode hops from shorter to longer wavelengths caused by asymmetric gain cross-saturation. Furthermore, it is visible that the total gain is close to threshold at λ = 513.1nm, preparing the next lasing mode to move through the gain spectrum. The multi–mode spectrum is shown in Fig. 4(b).
Fig. 4: Transition from mode competition to mode clustering. Simulated spectrochronograms and spectra for a green LD at I = 1.5Ith. (a), (b) τin = 113fs, αag = 9: mode competition, (c), (d) τin = 113fs, αag = 3: single–mode operation, (e), (f) τin = 225fs, αag = 3: two stable modes (mode clustering).
Fig. 5: Normalized photon number (upper plots), total gain and gain terms (lower plots) at I = 1.5Ith for a green LD. The dots represent a single longitudinal mode. Besides the total gain (black, axis to the left), the self–saturation term BSp (magenta), the cross saturation terms ∑q DpqSq, ∑q HpqSq (blue and red) and the total gain saturation as a sum of these contributions (green) are plotted as a change in the gain Δg (axis to the right). Parameters in (a) – (c) are the same as in Fig. 4 and show their relevance for mode competition (a), single–mode operation (b), and mode clustering (c).
By a decrease of the antiguiding factor from αag = 9 to αag = 3, which is closer to the experimental αag = 4 [50], the asymmetric gain saturation term is reduced by a factor of three. As a consequence mode competition stops as Fig. 4(c) demonstrates, and the laser operates longitudinally single–mode [Fig. 4(d)]. Figure 5(b) shows that the gain of only mode, specifically the lasing mode, is above gain threshold. The neighboring mode stays below threshold due to the decrease of Hpq.
By increasing τin from 113 fs to 225 fs and keeping αag = 3, a second mode separated by 0.8 nm on the short wavelength side starts to lase as shown in Figs. 4(e) and (f). The increase of τin yields a larger symmetric gain saturation Dpq, which, however, is narrower, visible in Fig. 5(c) and Eq. (9). In total, a much broader gain spectrum is closer at threshold, so that the second lasing mode arises, well separated from the first lasing mode. At 514.4nm a third mode is at the edge to lasing and would get more intense, if τin was further increased or if the current was increased, resulting in a larger charge carrier number and an enhancement of Dpq.
It is remarkable, that the multi–mode rate equations do not only describe mode competition very accurately. But by tuning the essential parameters, single mode operation as well as multi–mode operation with separations of the lasing modes typically found in lasers with mode clustering are generated. Since the dependence of the linear gain Ap on wavelength is described by a quadratic approximation [see gain curvature coefficient b in Eq. (4)] and no gain fluctuation is built into the set of equations, only the photon–assisted gain saturation terms are necessary to generate mode clustering. By applying gain fluctuations as in [19] in the regime of mode clustering, the position of the lasing wavelength can be changed and the turn–on delay is reduced. However, we stress that gain fluctuations are not necessary in this set of equations for mode clustering. The gain saturation terms depend on αag and τin and consequently on threshold carrier density. This explains why LDs with higher threshold exhibit stronger effects of mode clustering than recent LDs with improved linear gain, reduced internal losses and consequently reduced threshold carrier density. We conclude, that gain saturation is the origin for mode clustering in GaN–based LDs.
6. Conclusion
In conclusion, we report on longitudinal mode competition and mode clustering in blue and green LDs. The mode competition frequencies vary from 10 to 150 MHz. The formalism of symmetric and asymmetric gain saturation allows to model the dynamics of mode competition in (Al,In)GaN laser diodes with high accuracy. Equations (1)–(8) together with the appropriate parameter set provides an accurate quantitative description of the dynamic and static spectro–temporal behavior of a lateral single–mode, but longitudinal multi–mode laser diodes in the green to near UV spectral region. The measured mode competition frequencies and the number of contributing modes are truly reproduced by the simulations. For mode competition the asymmetric gain cross saturation Hpq is essential. By tuning αag and τin single mode operation and mode clustering could be induced. The symmetric gain cross saturation Dpq is essential for the number of lasing modes. Since neither gain fluctuations nor plasma oscillations are necessary for generating mode clustering effects, we propose gain saturation to be the main cause of mode clustering in (Al,In)GaN LDs. Still, gain fluctuations my enhance mode clustering and lead to spectral pinning of the strongest longitudinal modes.
The gain saturation mechanism perturb not only LDs in cw operation and wavelength stabilization, but also have an effect on the generation of ultrafast pulses. We point out that the spectro–temporal dynamics can lead in combination with a wavelength selective optical element, e.g. a filter with a sharp edge or a waveplate, to an intensity modulation of the laser beam, as well as in systems with optical feedback [53].
However, an understanding of the underlying microscopic mechanisms is still missing in these GaN–based devices. We do not expect symmetric and asymmetric gain saturation to be necessarily different in a fundamental way from the mechanisms discussed in the context of GaAs– and InP–based LDs. Yet, these mechanisms will be different in the presence of large inhomogeneous broadening caused by indium fluctuations in InGaN quantum wells and by polarization effects, i.e. the quantum confined Stark effect. Rcv is decreased in (Al,In)GaN LDs and is not only very dependent on the design of the active region, but also strongly influences the modal dynamics. To gain an understanding of the complex dynamics, a theoretical study of spectral hole burning by rigorous many–body simulations of (Al,In)GaN LDs on the ultra–fast time scale is necessary.
Acknowledgments
The authors would like to thank Katarzyna Holc, Angela Thränhardt and Agata Bojarska for fruitful discussions. This research has been supported by the German Ministry for Education and Research (BMBF) under contract No. 13N12291.
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