## Abstract

To increase the brightness of broad area laser diodes, it is necessary to tailor the optical properties of their waveguide region. For this purpose, there is the need for simulation tools which can predict the optical properties of the complete device and thus of the outcoupled light. In the present publication, we show a numerical method to calculate typical intensity distributions of the multimode beam inside a high-power semiconductor laser. The model considers effects of mode competition and the influence of the gain medium on the optical field. Simulation results show a good agreement with near and far field measurements of the analyzed broad area laser diodes.

© 2013 OSA

## 1. Introduction

Broad area semiconductor lasers are one of the most important light sources in industrial applications. A high beam quality and output power of those devices is the main goal of research in this field to provide options for a higher scale of integration, reduce costs, and open up new fields of application which are presently dedicated to expensive disk lasers. Recent publications show possibilities to tailor the optical waveguide such as the integration of optical microstructures or apertures to attenuate higher order modes [1,2]. To enable a proper design of such structured waveguides, it is necessary to analyze the multimode behavior of the waveguide. In the present publication, we show a simulation technique which considers the nonlinear interactions with the high gain amplifying semiconductor material such as spatial hole-burning and self-focusing. Further this simulation provides a high calculation speed which enables the integration into design algorithms and optimization. First, we show an iterative method to calculate the photon density inside the broad area laser waveguide and how to evaluate the propagation of the outcoupled light outside the laser. We demonstrate a fast numerical technique to solve the steady state rate equations for each iteration step. Finally, we illustrate simulation results of the current dependency of beam quality and beam divergence and compare the numerical results with experimental data.

## 2. Eigenvalue equation and multi-mode solution

The transverse modes of the laser are defined by diffraction and interaction with amplitude or phase modifying structures or optical elements inside the laser resonator. In case of a stationary operation of the laser, the complex amplitude of one mode at a certain position is identically reproduced except for a scalar factor. Mathematically, this can be expressed by the eigenvalue equation

Where $\widehat{Z}$ denotes the round-trip operator, ${U}_{i}\left(x,y,z\right)$ expresses the scalar complex amplitude of the $i$^{th}transverse mode and ${\lambda}_{i}$ its eigenvalue. With spatial integration of Eq. (1), we find the round-trip lossof the mode In the stationary operation of a laser, modes are amplified inside the gain medium which compensates the modal loss. In general, the spatial distribution of the gain of the active medium depends on the distribution of the laser light inside the resonator. In this case, the round-trip operator becomes nonlinear and modes of different orders are usually coupled. Therefore, common methods to determine the solution of Eq. (1) fail because they need the operator to be linear [3] or differences in diffraction losses are necessary to obtain a convergence like in case of the Fox-Li iteration [4].To calculate the physical properties of the outcoupled light of a multimode laser, it is not necessary to know the exact complex amplitude of every single mode which is amplified. For most applications, it is sufficient to determine the propagation of the outcoupled beam as a superposition of modes through optical elements or free space. This approach neglects the time dependent spatio-temporal fluctuations in a sub-nanosecond timescale. However a time-averaged propagation of the optical field is of higher interest for the most laser applications. The following iterative procedure shows how it is possible to calculate the intensity distribution of the laser beam as such an incoherent superposition of the intensity of amplified modes inside and outside the resonator.

We consider a starting field

## 3. Steady state rate equations for semiconductor broad area lasers

To calculate the spatial distribution of the gain and the complex refractive index in the waveguide, it is necessary to know the density of excited carriers $N\left(x,{y}_{QW},z\right)=N\left(x,z\right)=N$ in the quantum well $\left(y={y}_{QW}\right)$. The rate equation

In the steady state operation of the laser, the spatial distribution of the density of excited carriers (from now on shortly called carrier density) will not change because the excitation of carriers will be compensated by recombination and optical gain, which means that the rate in Eq. (10) should be zero for a certain equilibrium carrier density ${N}_{0}\left(x,z\right)$. To estimate the spatial change of the carrier density by diffusion one can calculate the carrier lifetime from $\tau ={N}_{0}/d{N}_{I}\ll 0.5ns$ for the equilibrium carrier density. Following this and [5] the resulting diffusion length is assumed by

Carrier diffusion effects have an impact on fast spatio-temporal fluctuations, hole-burning in the direction of propagation and the longitudinal modes. However, as the mode intensity modulations in slow-axis direction have spatial dimensions on a much larger length scale, this estimation indicates that the carrier diffusion has a minor impact on the propagation of transverse slow-axis modes. Thus, we set $d{N}_{diff}=0$ for the following considerations. Setting $dN/dt=0$ allows us to solve Eq. (10) for a certain optical power density and injection current density by a numerical iteration which finds the zero crossings of the curves in Fig. 1:

From this steady state equilibrium carrier density ${N}_{0}\left(x,z\right)$, we obtain the spatial distribution of the optical gain $g\left(x,z\right)$ from the gain spectra which are calculated with a numerical implementation of the $k\cdot p$ method [7]. The effective refractive index ${n}_{eff}\left(x,z\right)$ is calculated by a Kramers-Kronig transformation [8]. The input data of these numerical calculations were provided by OSRAM OS for the specific semiconductor material system of the AlGaAs lasers considered here.

Figure 1 shows a numerical computation of the change rate of the carrier density for the examined AlGaAs lasers with an operation wavelength of 970nm. The chart shows the rate vs. the area carrier density $N\left(x,z\right)$for two different injection current densities $I\left(x,z\right)$(2500 and 1500 A/cm^{2}) and 8 optical power densities $p\left(x,z\right)$ (0 to 0.095 W/µm). From this illustration one can deduce the range of the equilibrium carrier density at different positions of the waveguide and the resulting rates for non-equilibrium conditions. It shows, that for the most parts of the waveguide the carrier density is around 1.5 to 2 x 10^{12} cm^{−2}. Changes of the carrier density will take place in time scales around 100ps.

Because of the strong dependency of the round-trip operator from the photon density inside the waveguide, it is necessary to consider the change of the round-trip operator during the iterative procedure. Therefore, instead of the static number of average iterations ${n}_{avg}$, we use a moving average to calculate the optical power density from the incoherent superposition of the complex amplitude compared to Eq. (9). Thus, a damping factor $\alpha $ is multiplied to the optical power density of the auxiliary field of the previous iteration.

It further has to be remarked that the optical power density and carrier density from Eqs. (14) and (15) do not converge during the iteration. The changing distributions represent effects of mode competition without considering the exact time dependence of these fluctuations.

After a number of ${j}_{0}$ initialization iterations, we calculate the average of the intensity distribution of the auxiliary field from Eq. (9) for different positions outside the resonator, especially at the outcoupling facet and after propagation in some cm of air to compute the far field. Using this, we calculate the beam parameter product or the M^{2} value of the outcoupled beam with the method of second moments or with 95% power inclusion [9].

Figure 2 shows a schematic chart of one $j$- iteration step.

To obtain a high computation speed for the optical simulation, we compute a lookup table for ${N}_{0}\left(x,z\right)$, $g\left(x,z\right)$ and ${n}_{eff}\left(x,z\right)$ over a certain range of optical power densities $p\left(x,z\right)$ and injection current densities $I\left(x,z\right)$for about 2000 x 2000 samples in order to interpolate the parameters during the iteration.

In the presented implementation of the algorithm, the calculation of the round-trip operator consists basically of the propagation inside the waveguide, which is calculated with the split step Fourier transform beam propagation method [10]. This method considers the optically induced spatial inhomogeneity of the effective refractive index.

## 4. Results and comparison with experimental data

In the calculation and measurements we used AlGaAs broad area lasers with 4 mm resonator length and a rectangular shaped contact region with 100 µm stripe width. The emitting wavelength of the laser is 970 nm. The measurements were performed with short pulses (1 µs) at low repetition rates (100 Hz) to avoid unwanted heating of the device. Figure 3 and Fig. 4 show a comparison of simulated and measured values for different injection currents from 1A to 10A. The simulation results indicate an adequate accordance with the measured values. Nevertheless, measurement results show a slightly higher far field angle which means a worse beam quality. It has to be remarked that even in short pulse operation of the laser, the beam quality is degraded due to thermal induced change of the refractive index which increases the effect of filamentation. This implies that low order modes are affected by higher diffraction losses and higher order modes can be amplified. This increases the intensity in high far-field angles and worsens the efficiency of the laser.

Using this algorithm, it is possible to compute beam profiles inside and outside the laser resonator. It is also possible to calculate the propagation of the light through optical elements or fibers. Therefore, all fields from the $j$- iterations with $j>{j}_{0}$ have to be propagated through these elements and the intensity distribution of the multimode beam can be computed using Eq. (9). Figure 4 shows a comparison between simulated and measured intensity distributions on the outcoupling facet and in the far field.

Here, it can be observed that the highest optical power density is found on the sides of the emitting region. This effect can be explained with the higher effective refractive index on the outside of the contact region because of the lower excited carrier density. This causes an anti-guiding effect which broadens the near field diameter and increases the intensity at the sides of the emitting region. The measured far-field pattern shows higher intensities in high angular positions. First forthcoming investigations show that this might be a result of thermal aberrations. Figure 5 shows the density of excited carriers, the photon density and the effective refractive index of the waveguide at an injection current of 6A.

The intensity distribution inside the waveguide exposes the formation of filaments inside the waveguide as shown in Fig. 6. The reduced carrier density at the positions of the filaments causes a self-focusing effect. Further it shows the rise of the effective refractive index to the sides of the contacted region, which causes the mentioned anti-guiding effect to the optical field.

## 5. Conclusion

We presented a numerical model which approximately simulates the multimode optical field inside highly nonlinear active media laser resonators. We demonstrated a simplified solution to the excited carrier rate equations which allows the integration in the optical simulation with a high computation speed. The model allows to analyze the beam characteristics of the laser in about 5 minutes on a desktop computer. This enables the possibility to optimize waveguide structures to improve the beam quality and efficiency of the laser. In cases of a micro structured waveguide or in case of spatially inhomogeneous material parameters or temperature distributions, a more sophisticated round-trip operator could be used as well. Further, it might be possible to use the optical part of this simulation technique for different types of laser resonators if an appropriate model for the gain medium is used. Moreover other types of optoelectronic devices with semiconductor waveguides such as superluminescent diodes and optical amplifiers could be analyzed with our model of the gain medium.

The presented simulation results show a good agreement with measured data in the pulse operation of high-power broad area semiconductor lasers. To improve the accuracy for cw-operation and consider effects of thermal lensing we will integrate a thermal modal to the calculation, which will be published soon.

## Acknowledgments

We kindly acknowledge Osram Opto Semiconductors GmbH for providing the laser specific simulation parameters. We further acknowledge the German Federal Ministry of Education and Research (BMBF) for supporting our work in the project IMOTHEB (grant number: 13N12312).

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