## Abstract

(Al,In)GaN–based laser diodes with ridge widths broader than a few micrometer tend to show filamentation effects in the lateral direction. By time–resolved scanning near–field optical microscopy, we find different kinds of filaments depending on ridge width and lateral position. We investigate these effects systematically and compare them to the results of corresponding simulations, which are based on a simple rate equation model including the lateral dimension. By this comparison we find a consistent and reasonable set of material parameters that can describe the laser diode. Furthermore, we discuss several reasons for filamentation dynamics like ridge asymmetry or spatial hole–burning, as well as critical temperatures that induce filamentation.

© 2008 Optical Society of America

## 1. Introduction

Today there is a wide range of applications for semiconductor quantum well lasers. Depending on the application, specific requirements have to be met. (Al,In)GaN-based laser diodes (LDs) with a peak wavelength of *λ*=405 nm are used e.g. for data storage. Therefore, one needs constant optical output power, good beam stability and a homogeneous Gaussian beam profile in order to be able to focus the beam well. Heating limits the power density that can be used to drive the laser diode. To obtain more output power for applications like laser tv or computer to plate printing, one increases the ridge width. Greater ridge widths lead to the formation of higher order lateral modes, and thus a lateral modulation of the light intensity. The interplay of refractive index influencing effects as heating, growth inhomogeneities, spatial hole–burning, and carrier diffusion leads to filamentation in the lateral direction and to complex variations of the beam profile during the laser pulses.

We provide a comprehensive study of filaments in near–UV (≈405nm) LDs. In particular, we find that temperature is the driving force behind the dynamical behavior of the observed filaments. The refractive index is modified by ridge temperature and the corresponding lateral temperature gradient caused by the operational current. Also, the refraction index profile is significantly influenced by asymmetric sample conditions. Growth impurities, defects or additional monolayers may contribute to an effective asymmetry of the refraction index profile as well as lateral variations of the heat dissipation.

To approach these effects in a simulation, we employ a rate–equation based model to simulate the refraction index modifications and their impact on the lateral intensity distribution. We compare the results to a large, systematic set of measured data from laser diodes with different ridge widths and as a function of operational current. This leads to a better understanding of filamentation processes and to a set of simulation parameters which give good estimations for several material parameters.

We introduce our samples and measurements in section 2, followed by an explanation of the mechanisms causing filamentation and a description of our simulation model in section 3. The latter includes a discussion of our simulation parameters. After that, we compare the experimental data to the simulated results and distinguish different kinds of filaments (section 4). Heating and its influence on filament appearance is discussed extensively in section 5.

## 2. Observed filamentation effects

To investigate filamentation effects we examined different (Al,In)GaN based laser diodes with an emission wavelength of *λ*=405 nm fabricated by MOVPE at *OSRAM OS*. The samples had a standard epitaxial structure [1, 2, 3] with ridge waveguides of 1.5, 2, 2.5, 5, 8 and 10 *µ*m ridge width, a resonator length of 600 *µ*m, and cleaved, uncoated facets with a reflectivity of *R*≈17%. All samples were grown on the same SiC wafer, which allows a comparison of the threshold currents. For all experiments the laser diodes were electrically driven in pulsed mode with a pulse length of about 2 *µ*s. The current was varied between threshold current *I _{th}* and 1.6

*I*. We use a scanning near-field optical microscope (SNOM) to measure the emitted optical intensity [4, 5, 6]. A fiber tip with an apex of less than 100 nm was used to scan in a distance of about 20 nm over the facette, collecting the emitted light at every position. By this method, we achieve a lateral and transversal resolution better than 100 nm, limited mainly by the tip apex.

_{th}Figure 1 shows a time averaged SNOM scan of the facette of a sample laser diode of ridge width *w*=8 *µ*m. While the ridge emits homogeneously in the fundamental mode in transversal direction, we observe a strong variation of the emission along the lateral position, which is a consequence of filamentation. To analyze this lateral distribution it is sufficient to reduce the scanned area to a narrow stripe along the center of the ridge as shown in Fig. 1(a). This helps to reduce measuring time, which is important to measure current series while avoiding aging effects on the laser diode during a set of consecutive scans. Fig. 1(b) shows the optical intensity distribution of this stripe. In all cases, the measured intensity distribution can be reconstructed with high precision by a superposition of single Gaussian shaped filaments.

Using a fast photomultiplier, we achieve a temporal resolution of 5 ns, so that in addition to the lateral behavior, the temporal evolution of the laser pulses is detected, too. Figures 2(a) to (f) show such time-resolved measurements. The horizontal axes are time axes, showing the laser pulses from the beginning to their end after about 2 *µ*s. The vertical axes are equivalent to the lateral cross sections from Fig. 1. Parts (a) to (f) show laser pulses from samples with different ridge widths *w* of 1.5, 2, 2.5, 5, 8 and 10 *µ*m in ascending order. Each figure shows the ridge margins as they were obtained from these specific laser diodes by analyzing the homogeneous emission profile below lasing threshold. One can see in (a,b,c) that the narrow-ridge laser diodes emit laterally in the fundamental mode and do not show significant temporal dynamics, yet not over the full ridge width due to a slight constriction. Each of these intensity profiles may be seen as one single filament. As there are no other filaments, there is no interaction between adjacent filaments, which is the reason for this stable characteristics. The slight constriction at the beginning of the pulse which can be seen in (a) and (c) is not a decrease of the width of the filaments, but simply a loss of intensity due to some turn-on effects as initial heating. In other ridge waveguide LDs with ridge width up to 7*µ*m we observed clearly defined relaxation oscillations with frequencies between 0.15GHz at 1.02*I _{th}* and 2.4GHz at 1.81

*I*with a streak camera in a different set of experiments. These relaxation oscillations can be clearly distinguished from the much slower intensity spike. The spike, which is acompanied by an impedance variation, is typical for InGaN LDs at the leading edge of a longer driving pulse. A likely explanation is a fast initial temperature rise of the p-side due to resistive heating.

_{th}Generally, the full width at half maximum (FWHM) of single filaments is about 1 *µ*m to 1.5 *µ*m, depending on the kind of filament. This is smaller than the 1.9 *µ*m filament width calculated for a similar (Al,In)GaN laser diode in [7, 8]. When the ridge width of the laser diode exceeds two or more filament widths, a critical barrier is passed. As a consequence, dynamic effects between multiple filaments occur. Figure 2(d) shows first filamentation dynamics effects at a ridge width of *w*=5 *µ*m. Two filaments coexist and vary in width, position, and intensity due to their interplay and their dependence on the refraction index and carrier density, as it will be explained in detail in the following sections. These variations are observable at any operational current above lasing threshold. Compared to the samples with narrow ridge widths, the light emitting fraction of the facette is significantly smaller. This comes from the intensity minima between adjacent filaments. In Fig. 2(e), we observe an altering number of filaments during the pulse. Single filaments show up or disappear again. Effects like these are most likely caused by heating of the sample. This heating increases the local refractive index and the slopes of the lateral heat distribution, which affect significantly the refraction index profile. Due to an increased number of filaments, the effective exploitation of the facette width is further decreased. Finally, Fig. 2(f) the intensity profile of a sample with *w*=10 *µ*m is depicted. Again, one can see the different filaments’ dynamic behavior and the rather large area of the facette where no or hardly any light is emitted.

## 3. Simulation

Filamentation can be explained as a consequence of various effects inside the laser diode and their interplay. To understand the associated mechanisms we introduce a simulation model. It is based on an existing model [9, 10] and extended to consider a refractive index rise due to current induced heating of the laser diode and random effects like growth defects or impurities, active region thickness fluctuations in lateral direction or lateral differences in the heat dissipation. Heating is included in the model by a simple heat dissipation model. The random effects are not analyzed individually. Instead, a tilt is added to the lateral refraction index distribution to approximate the overall effect on the laser diode.

Basically, a laser diode can be described by a rate equation model [9, 10, 11, 12]. Equations 1 and 2 represent the rate equations for the carrier density and the photon density, including the lateral direction (x-axis). Note that this model does not consider the spectral distribution of the beam, as all the calculations are based on the assumption of monochromatic photons with *λ*=405 nm.

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-\frac{c}{\stackrel{~}{n}}g\left(N(x,t)\right)\Gamma \sum _{p}{I}_{p}(x,t){S}_{p}\left(t\right)$$

In Eq. (1), the four right–hand terms describe from left to right: carrier injection from the operation current, carrier loss by spontaneous recombination, lateral carrier diffusion, and loss by stimulated emission [9]. The right–hand side in Eq. (2) includes the modal gain, reduced by the internal losses and mirror losses (in brackets) and the additional photons from spontaneous emission coupling into the laser modes. *N*(*x*,*t*) describes the carrier density distribution, *e* is the unit charge and *d* is the active layer’s thickness. The current density distribution *J*(*x*,*t*) is constant within the borders of the ridge and is assumed as zero in adjacent regions. Lateral carrier diffusion is taken into account; *D*=*L*
^{2}
_{d}τ^{-1}
* _{eff}* is the diffusion constant with

*τ*as the effective carrier lifetime for spontaneous (radiative) and nonradiative recombination and

_{eff}*L*being the diffusion length;

_{d}*c*is the vacuum speed of light,

*ñ*describes the group refractive index;

*g*(

*N*(

*x*,

*t*)) is the material gain per length. The confinement factor is called Γ, and

*I*and

_{p}*S*are the (lateral) modal intensity distribution and photon numbers for the

_{p}*p*lateral mode.

^{th}*G*is used for the net modal gain,

_{p}*α*are the internal losses.

_{int}*L*is the resonator length and

*R*is the mirror reflectivity, where we assume the same reflectivity for both mirrors. Finally, the last term describes the contribution of spontaneous emission into the

_{m}*p*lateral mode with average modal carrier density

^{th}*N*. The be correct, one should use the radiative recombination time in the last term of Eq. (2). Because we do not know the ratio between radiative and nonradiative recombination rate, we use the effective recombination time

_{p}*τ*in this term, too. This amounts to a decrease of the spontaneous emission factor

_{eff}*β*compared to its real value. This is eligible because an exact value of the spontaneous emission factor

*β*is not known and this term has no significant impact on the results of the simulation.

For this work, we consider the static case for the carrier dynamics, only. This is justified because we want to describe the dynamics on a time scale of 100 ns to microseconds, which is caused by thermal modifications of the refractive index profile, as we will show in section 5. Carrier and photon number fluctuations happen on a much smaller, sub–nanosecond or nanosecond timescale. As mentioned above, we observed relaxation oscillations with periods of 7ns and slower for a similar InGaN LD. It would be very interesting to observe the carrier and mode dynamics on this fast time scale in analogy to similar work for GaAs/GaAlAs and GaAs/GaInAs LDs [13, 14]. Yet our current experiment addresses a different time scale.

In this quasi–static approximation the mode profile for a certain time during the pulse can be described as static case for a given refractive index profile. Therefore, we set the left–hand sides of Eq. (1) and Eq. (2) to zero and drop the time dependence in all terms to simulate the time independent conditions of a continuous–wave (cw) driven laser diode.

The lateral waveguide is defined by the refractive index and gain profiles. They are expressed by the complex effective permittivity, which is modeled as

with

For the shallow etched ridge, transverse waveguiding is guaranteed inside and outside the ridge. The corresponding effective refractive index *n _{eff}*+Δ

*n*

_{ridge}(

*x*) is calculated by one–dimensional (transverse)waveguide simulations [2], resulting in a value of

*n*=2.475 outside and

_{eff}*n*+Δ

_{eff}*n*=2.505 inside the ridge, respectively. Beside the contribution from the ridge waveguide, Δ

_{step}*n*(

*x*) contains an estimation for structural asymmetry effects Δ

*n*

_{asym}(

*x*) and thermal induced modifications of the refractive index profile Δ

*n*

_{therm}(

*x*), as it is shown in Fig. 3. The asymmetry term is an empirical expression which is necessary to explain the experimental results. The static asymmetric refractive index profile may be caused by growth inhomogeneities or an asymmetric heat conduction due to asymmetric bond pads. Also other parameters like the injected current density

*J*(

*x*), the material gain

*g*or internal losses

*α*may be affected by processing inhomogeneities or fluctuations, which are not considered here, and lead to an effective asymmetry.

_{int}For the calculation of the thermal contribution we use a simple two-dimensional heat conduction model with the assumption that the injected heat is deposited in the region of the *pn*–junction below the ridge waveguide. Sources are resistive heating, predominantly in the p–side with its lower conductivity; heating due to nonradiative recombination of carriers outside the active region due to carrier overflow; heating due to nonradiative recombination in the active region, and reabsorption. We neglect the lateral variation of heat influx due to the filaments. Before including this effect one should use a model on a much higher level of sophistication, e.g. a fully self–consistent 2d simulation of current, carrier, and photon density and temperature [15, 16].

The heat conduction model with parameters from Ref. [17] gives us the time resolved waveguide temperature as well as the corresponding lateral temperature profile. The calculated rise in temperature with time is in good agreement with time resolved waveguide temperature measurements by analyzing the temperature induced spectral shift of single longitudinal modes during the driving pulse [18, 19]. The lateral temperature profile can then be translated into a thermal induced refractive index modulation Δ*n*
_{therm}(*x*) with a thermal induced change of the refractive index of *dn*/*dT*=1.5·10^{-4} K^{-1} [20, 21]. The increasing temperature and therefore increasing thermally induced refractive index change is the driving force for the observed filament dynamics.

The last two terms in Eq. (3) describe carrier induced changes of the real and the imaginary part of *ε _{eff}*, where

*n*

_{1}and

*n*

_{2}are the refractive indices of the active and passive layers, respectively. A higher carrier concentration

*N*(

*x*) leads thereby to a higher material gain

*g*(

*N*)=

*aN*+

*b*, but a lower real part of the refractive index. The antiguiding factor $\alpha =-2k\left(\frac{\frac{\mathrm{dn}}{\mathrm{dN}}}{a}\right)$ describes the ratio of these two effects with

*k*=2

*πλ*

^{-1}beeing the wave number. The parameters

*a*and

*b*are extracted from many body simulations of gain spectra [1] and according to [22] a value of about

*α*=4 is assumed in this model to derive the value of $\frac{\mathrm{dn}}{\mathrm{dN}}$ from the differential gain parameter

*a*.

We introduce the electric field of the *p ^{th}* lateral mode

*E*(

_{p}*x*), which is given by the one–dimensional wave equation:

The eigenvalue
${\beta}_{{z}_{p}}$
in this equation is the propagation constant in z–direction of the p* ^{th}* mode. It defines the modal gain
${G}_{p}\left(t\right)=2\mathrm{Im}\left({\beta}_{{z}_{p}}\right)$
. Now, the modal intensity distribution is obtained from

where the ridge width *w* is used as a scaling factor. All the parameters mentioned above are summarized and commented in table 1.

Figure 3(a) depicts the essential starting conditions for the simulation model, consisting of the waveguide refractive index profile given by Eq. (4) and the carrier density distribution, affected by diffusion (Eq. (1). Equations (1) and (2) are then solved self–consistently. In this loop the influence of the carrier density distribution on the refractive index is calculated via Eq. (3). The normalized modal intensity distribution is calculated using Eq. (5) and Eq. (6). The solution of the rate Eq. (1) and (2) then leads to the photon densities of the different lateral modes. The overall intensity distribution

will in turn influence the carrier distribution via stimulated emission. Typically, stable conditions are established after about three iterations of the self–consistency loop. Figure 3(b) shows the results after five iterations of the loop. One can see clearly a modulation of the refractive index compared to the initial shape in (a). The lobes at the edges are mainly a consequence of carrier diffusion and are responsible for the formation of *edge channels* [6], which will be explained in section 4.

All our simulations considered the first seven lateral modes. In Fig. 3(c), the same simulation as in (b) is depicted, showing the individual intensity distributions of the seven incorporated lateral modes. From the very low intensity of the seventh mode one can see that this number of modes is sufficient enough to ensure accurate simulation results. Here, the consideration of less than six modes would obviously distort the calculated results. These simulations also identify the origin of filamentation as a superposition of lateral modes modified by and interacting via the refractive index profile and the carrier density distribution. However, there is one important restriction in the simulation model: to obtain the lateral intensity distribution, we add the single lateral modal intensities. As these modes are each influenced by different effective refraction indexes *n _{eff}*, they are not coherent. Our simple model ignores this incoherence. A full spatial and spectral model would be necessary to consider this, but there would be a huge rise in

complexity and in computation time without a significant improvement in the results, describing filaments, thermal and asymmetry effects.

## 4. Different types of filaments

The optical intensity profile in Fig. 3(b) shows several filaments, localized mainly at the maxima of the refraction index profile. The filament on the right side is a type of filament which we call *edge channel* [6]. These filaments appear constantly at one or at both edges of the ridge, depending on asymmetry effects. They are guided in the edge lobes of the lateral refractive index profile. Edge channels can also be seen in the experiment, for example, Fig. 2(e) contains an edge channel at *x*≈-3 *µ*m. Due to the structure geometry, the position of the refraction index lobes can not vary much, which is the reason why a filament like this does not change its position during the pulse. Other filaments, which are called *central filaments* in this work, however, do show a significant variation of their lateral position during the pulse, especially during the first microsecond. The reason for this difference is mainly the difference in the overlap of single filaments with adjacent filaments. As it can be seen in Fig. 1(b) and in Fig. 2, edge channels have little overlap with other filaments. This leads to an undisturbed development of the edge channel throughout the pulse, while central filaments interact strongly with each other by modulating the refraction index and the carrier density in the overlap area.

A main goal of the simulations was to find a consistent set of material parameters that describe the lateral mode shape of the laser diodes with different ridge widths. Table 2 lists the parameters that are not exactly known and thus were modified to find the best simulation results and to get a feeling how critical the single parameters are. Note that it was not attempted to reproduce one individual sample, which is in principle possible. The idea was rather to get a consistent set of parameters that describe the typical characteristics of laser diodes with different ridge widths. This parameter set consisting of the optimum values given in table 2 was used for all simulations shown in this paper. Variations within the given range of the internal losses or the structure geometry induced refraction index step have insignificant impact on the simulation result. The listed modifications of the diffusion length, the effective lifetime, the lateral asymmetry and the carrier induced refractive index modulation may lead to a better fitting result for a single measurement but worse agreement with other measurements.

To compare experimental data and simulations, averaged cross–sections of the intensity distribution towards the end of the pulses are regarded. At this time of the pulse, heating of the diode does increase less than in the beginning and thus the diode emits more or less in a stable quasi–cw mode, which is suited best to be compared to the cw simulations. Figure 4 compares the experimental data with their corresponding simulations. (a) shows the data from a 1.5 *µ*m laser diode, (b) compares experiment and simulation in case of a 10 *µ*m sample. In (a), one can see that the shape of the simulated filament corresponds quite well with the measured data while the width of the filament is slightly overestimated. One would have to adjust different simulation parameters to reproduce the exact shape of the individual measurement, but this was not the objective of this work. In fact, the set of parameters was chosen to achieve reasonable simulation results for all ridge widths. When varying the current, the filament shape does not differ significantly between experiment and simulation. With the same set of simulation parameters we obtained the simulated intensity distribution for the 10 *µ*m laser diode shown in Fig. 4(b). The number of different filaments increases with the current in the same way as it was observed in a measurement on a 10 *µ*m sample diode. One can see that the rise of the right–hand edge channel with increasing current was reproduced accurately. Also, we achieved a good qualitative match of the lateral mode shape and even quantitative compliance concerning the threshold current and the slope of the diodes PI–characteristics, as depicted in Fig. 4(c). This gives proof that the parameters as listed above are reasonable.

## 5. Influence of temperature on filaments

The specific behavior of filaments as predicted in the simulations is the result of different effects. A strong influence of the waveguide temperature on the lateral mode profile of broad ridge laser diodes is observed. Figures 5(a) to (d) show the time-resolved lateral intensity distribution of a 10 *µ*m broad diode at currents of *I _{th}*, 1.2

*I*, 1.4

_{th}*I*and 1.6

_{th}*I*. During one laser pulse the temperature as well as the lateral temperature gradient rise inside the laser diode with time. An increased current increases the heating rate. Due to operating the laser diodes in pulsed mode with a pulse duration of 2

_{th}*µ*s and a duty cycle of 0.1% we assume a constant temperature of 300 K at the beginning of each pulse. As the measurements show, filaments appear earlier in the pulse when the current is higher. As an example, two of the marked filaments in Fig. 5(c) will be discussed in detail.

Figure 6 shows the temporal evolution of the optical intensity of the filaments marked as *D* and *E* in Fig. 5. In Fig. 6(a), the intensity of the left edge channel *E* at different driving currents is plotted as solid line. At threshold current and 10% above, the filament does not emerge, as the critical temperature is not reached in the measured time window. Starting with a current of 1.2 *I _{th}*, the edge channel rises. As one can see, the rise happens earlier during the pulse when the current increases. Figure 6 also shows the corresponding calculated time-dependent temperature inside the pumped region as dashed lines. These calculations are based on the model described in section 3 and agree with experimental temperature data from spectral resolved measurements on a similar sample. The absolute temperature values for the sample shown in Fig. 6 may differ on some percent, but the shape of the temperature curves is correct. So they work well enough to show that the rise of the filament is connected with the laser diode reaching a certain critical temperature estimated around 316 K for this particular laser diode and filament. Possible sources of errors in this estimation are the lack of knowledge of absolute temperatures and the difficulties with the definition of the exact moment when the filament emerges due to the background noise of the intensity.

The idea of a critical temperature gets confirmed when Fig. 6(b) is taken into account. Again, the time-dependent intensity and temperature are compared, this time relating to the central filament marked as *D* in Fig. 5. In this case, the filament already rises at threshold current. Like in Fig. 6(a), an increasing current means an earlier rise of the filament. Here, one finds a critical temperature of about 311 K. For this value, there is the same level of uncertainty as for the edge channel, but it is clear that the appearance of the individual filaments shows a distinct temperature dependency.

## 6. Conclusions

We have shown that the occurrence of filaments is based on regular lateral modes, influenced by lateral variations of the refractive index, carrier diffusion effects, spatial hole burning, and heating effects. Filaments are usually guided in local maxima of the refractive index. Depending on the position of this maximum, we distinguish between edge channels and central filaments. An edge channel is significantly narrower and more stable with respect to time, lateral position and to current increase. Central filaments interact with each other, causing strong dynamics in the lateral intensity distribution for the first microsecond of a laser pulse. A separation of the beam into multiple filaments does not occur for narrow ridge widths. Nevertheless, the intensity distribution of narrow ridge laser diodes is constricted to roughly the same width as single filaments of broad ridge samples. This indicates that these effects have the same origin, i.e. filamentation.

We observe for a given waveguide at specific temperatures reproducible changes in the filament patterns. Thus, while the width and formation of the filaments is governed by carrier dynamics, their position is dictated by the ridge asymmetry and thermally induced refractive index profile. The waveguide mode dynamics on a 100ns to several *µ*s time scale is dominated by thermal effects. Both, asymmetry of the ridge waveguide and lateral temperature gradients are important factors concerning the lateral stability of (Al,In)GaN laser diodes.

Furthermore, we have developed a simulation model that reproduces the measured intensity distributions of quasi–cw driven laser diodes accurately. Together with the model we provide a set of material parameters which can be used to describe the lateral dynamics of GaN–based laser diodes.

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