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

Z^Ui(x,y,z)=λiUi(x,y,z),
Where Z^ denotes the round-trip operator, Ui(x,y,z) expresses the scalar complex amplitude of the ith transverse mode and λi its eigenvalue. With spatial integration of Eq. (1), we find the round-trip loss
li=1λi2.
of 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

U0(x,y,z0)=i=0n(ξiUi(x,y,z0))+Urest
as a composition of a finite number of modes Ui of the round-trip operator Z^, where ξi denotes the composition factor and Urest the proportion of the starting field, which lies outside the resonators eigenspace or belongs to modes which will not be amplified. In the calculation of the jth iteration, we apply the round-trip operator to the field Uj10(x,y,z0) and find
Uj0(x,y,z0)=i=0n([k=0j1ξi,kλi]Ui(x,y,z0))+Z^...Z^j1timesUrest.
If Z^ is the operator of the laser in the stationary state, the intensity distribution of the incoherent superposition of all modes will remain constant over large time scales. The components Urest of the starting distribution which lie outside the eigenspace of the operator Z^ will lose their energy because they are not stationary by definition. If the round-trip loss of a certain mode is higher than the gain, but the mode was included in the starting distribution, its energy proportions will decrease to zero exponentially during the iterative procedure. In terms of Eq. (4), this means for a sufficiently large number of initialization iterations j0:
k=j0mξi,kλi1forUiisexistingmodek=j0mξi,kλi0forUiisbelowthresholdZ^...Z^mtimesUrest0fornon-stationarypartsofU0.
Following this, we define an average steady state gain coefficient for the ith mode as follows:
gi=1λi2.
During the iteration, the squared magnitudes of the composition factors ξi,j2 will be different from the gain factors gi To provide a convergence of the iteration, it is therefore necessary to adjust the round-trip operator so that gain saturation is considered. Conservation of energy provides that the optical power of the outcoupled beam Pout, which is included in the superposition of the intensity of all modes, is given by the injected pumping power minus the round-trip losses. For the sum of the intensity distribution of the modes averaged over a sufficiently large number of iterations navg, we find,
1navg+1j=j0j0+navgiξi,jλiUi(x,y,zOC)2dxdyPout
at the position of outcoupling zOC. Besides for the total output power Pout, this is also valid for one particular mode because the average gain overlap and round-trip loss define the ratio of power of different modes Pi which leads to
1navg+1j=j0j0+navgξi,jλiUi(x,y,zOC)2dxdyPi.
Only modes which contain energy have to be considered in the outcoupled multimode beam. To calculate the intensity distribution at a certain position inside or outside the resonator, we can reduce the average incoherent superposition of the fields to this set of relevant modes Ω as follows:
1navg+1j=j0j0+navgiξi,jλiUi(x,y,z)2=i1navg+1[j=j0j0+navgξi,jλi2]=1ifiΩ,0otherwiseUi(x,y,z)2=iΩUi(x,y,z)2.
As mentioned before, the round-trip operator depends on the intensity distribution of the modes inside the resonator in case of a real laser. For weak changes of the round-trip operator, we consider this as a perturbation and apply a first order correction. Thus, changes of the round-trip operator only influence the eigenvalues λi but not the normalized complex amplitude Ui(x,y,z) of the modes. Further, the gain overlap for one particular mode depends on the included power in the other modes. This results in a coupling of modes with distinct orders which can be expressed by a change of their gain factors gi. Hence, the coupling also only influences the energy proportions between the coupling modes but not their amplitude shapes. In different iteration steps the power ratio between different modes will oscillate. Depending on the lifetimes of excited states in the gain medium and the photon lifetime, this oscillation can be observed in the time domain as a consequence of temporal mode competition but usually it will be transient shortly after the laser is switched on. Regardless whether there is a time-dependent oscillation, for most applications an averaged intensity distribution is of more interest than effects in a nanosecond scale. In this case, for a sufficiently high number of iterations navg, Eq. (9) remains valid for the intensity distribution averaged over a time which is sufficiently large compared to lifetimes of photons or excited states.

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(x,yQW,z)=N(x,z)=N in the quantum well (y=yQW). The rate equation

dNdt=R(N)N(a)g(N)p(x,z)dxhνΓdz(b)+dNI(x,z)(c)+dNdiff(d)
defines the temporal change of the spatial distribution of excited carriers. The incoherent recombination is given by part (a) and the stimulated emission is given by (b) with g(N(x,z)) being the optical gain. The optical power density is p(x,z)=ε0U(x,y,z)2dy and Γ is the confinement factor of the waveguide. The excitation of carriers by the pump source due to current injection is defined by part (c). It can be calculated from the current density distribution IQW(x,z) by dividing with the electron charge e which yields dNI(x,z)=IQW(x,z)/e. Lateral carrier diffusion in the active region is expressed by part (d), which is given by dNdiff=DAM(N/x+N/y) where DAM is the ambipolar diffusion coefficient [5]. eThe recombination term (a) is given by
R(N)=A(a)+B(N)N(b)+CN2(c),
where part (a) defines the Shockley–Read–Hall (SRH) recombination, term (b) defines the bimolecular (band to band) recombination and (c) represents the Auger recombination [6].

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 N0(x,z). To estimate the spatial change of the carrier density by diffusion one can calculate the carrier lifetime from τ=N0/dNI0.5ns for the equilibrium carrier density. Following this and [5] the resulting diffusion length is assumed by

LDiffDAMτ20cm2s10.5ns=1μm.

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 dNdiff=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:

 

Fig. 1 Change of carrier density vs. carrier density for different power and current densities for two different injection current densities (2500 and 1500 A/cm2) and 8 optical power densities (0,0.005..0.095 W/µm)

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dN0(x,z)dt0N0(x,z)=N0(p(x,z),I(x,z)).

From this steady state equilibrium carrier density N0(x,z), we obtain the spatial distribution of the optical gain g(x,z) from the gain spectra which are calculated with a numerical implementation of the kp method [7]. The effective refractive index neff(x,z) 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(x,z)for two different injection current densities I(x,z)(2500 and 1500 A/cm2) and 8 optical power densities p(x,z) (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 1012 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 navg, 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 α is multiplied to the optical power density of the auxiliary field of the previous iteration.

pj+1(x,z)=(1α)pj(x,z)+αε0Uj+10(x,y,z)2dy.
Further we introduce a damping factor β to limit the change of the round-trip operator to small changes of the carrier density during one iteration step as follows:
N0(x,z)j+1=(1β)N0(x,z)j+βN0(pj+1(x,z),I(x,z)).
This ensures that the complex amplitudes of the modes of the round-trip operator remain approximately constant over a couple of iterations to provide the validity of the first order correction and to avoid numerical instabilities. For the examined AlGaAs lasers, the values of the damping factors were chosen to be 0.05<α<0.2 and 0.2<β<0.5. Smaller damping factors increase the number of iterations in which the mode oscillation takes place and thus increases the computation time. Higher damping factors will cause an unstable numerical behavior. It should be remarked that the exact values of the damping factors do not change the statistical results of the algorithm. It also has to be annotated, that the field Uj=00(x,y,z0) is set to a band-limited random distribution at the beginning of the algorithm, which means that the resulting shape of the calculated multimode beam may vary for different calculations of the same configuration. However, the average results for the beam characteristics are not influenced. A statistical analysis of different calculations of the same laser configuration show that the standard deviation is below ±3% for each value of injection current.

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 j0 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 M2 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.

 

Fig. 2 Schematic sketch of the algorithm

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To obtain a high computation speed for the optical simulation, we compute a lookup table for N0(x,z), g(x,z) and neff(x,z) over a certain range of optical power densities p(x,z) and injection current densities I(x,z)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.

 

Fig. 3 Near field diameter (top, red) and farfield full angle (bottom, blue) of an AlGaAs broad area laser (100µm stripe width, 4mm resonator length) vs. injection current; Simulation results (triangles) are compared with measurements (squares)

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Fig. 4 Optical output power (bottom, red) and beam parameter product (top, blue) vs. injection current; Simulation results (triangles) are compared with measurements (squares)

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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>j0 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.

 

Fig. 5 Simulated (red) and measured (blue) intensity distributions. Left: on the outcoupling facet (near field), right: and angular distribution of the output beam (far field)

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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.

 

Fig. 6 Distribution of excited carriers and optical power density inside the waveguide and change of the refractive index. The stripe width of the contacted region is 100 µm, the resonator length is 4 mm.

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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).

References and links

1. A. Büttner and U. Zeitner, “Experimental Realization of Monolithic Diffractive Broad-Area Polymeric Waveguide Dye Lasers,” IEEE J. Quantum Electron. 43(7), 545–551 (2007). [CrossRef]  

2. A. Büttner, U. D. Zeitner, and R. Kowarschik, “Design considerations for high-brightness diffractive broad-area lasers,” J. Opt. Soc. Am. B 22, 796 (2005). [CrossRef]  

3. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators (United States Governm. Pr. Office, 1950).

4. A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51(1), 80–89 (1963). [CrossRef]  

5. B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett. 97(26), 262119 (2010). [CrossRef]  

6. L. A. Coldren, S. W. Corzine, and M. Mashanovitch, Diode lasers and photonic integrated circuits, 2nd ed. (Wiley, 2012).

7. J. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Phys. Rev. 97(4), 869–883 (1955). [CrossRef]  

8. D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron. 24(1), 1–30 (1992). [CrossRef]  

9. A. E. Siegman, “How to (Maybe) Measure Laser Beam Quality,” in OSA Trends in Optics and Photonics, Vol. 17 (1998), MQ1.

10. M. D. Feit and J. A. Fleck Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5(3), 633 (1988). [CrossRef]  

References

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  1. A. Büttner and U. Zeitner, “Experimental Realization of Monolithic Diffractive Broad-Area Polymeric Waveguide Dye Lasers,” IEEE J. Quantum Electron.43(7), 545–551 (2007).
    [CrossRef]
  2. A. Büttner, U. D. Zeitner, and R. Kowarschik, “Design considerations for high-brightness diffractive broad-area lasers,” J. Opt. Soc. Am. B22, 796 (2005).
    [CrossRef]
  3. C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators (United States Governm. Pr. Office, 1950).
  4. A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE51(1), 80–89 (1963).
    [CrossRef]
  5. B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
    [CrossRef]
  6. L. A. Coldren, S. W. Corzine, and M. Mashanovitch, Diode lasers and photonic integrated circuits, 2nd ed. (Wiley, 2012).
  7. J. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Phys. Rev.97(4), 869–883 (1955).
    [CrossRef]
  8. D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
    [CrossRef]
  9. A. E. Siegman, “How to (Maybe) Measure Laser Beam Quality,” in OSA Trends in Optics and Photonics, Vol. 17 (1998), MQ1.
  10. M. D. Feit and J. A. Fleck., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B5(3), 633 (1988).
    [CrossRef]

2010

B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
[CrossRef]

2007

A. Büttner and U. Zeitner, “Experimental Realization of Monolithic Diffractive Broad-Area Polymeric Waveguide Dye Lasers,” IEEE J. Quantum Electron.43(7), 545–551 (2007).
[CrossRef]

2005

1992

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
[CrossRef]

1988

1963

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE51(1), 80–89 (1963).
[CrossRef]

1955

J. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Phys. Rev.97(4), 869–883 (1955).
[CrossRef]

Büttner, A.

A. Büttner and U. Zeitner, “Experimental Realization of Monolithic Diffractive Broad-Area Polymeric Waveguide Dye Lasers,” IEEE J. Quantum Electron.43(7), 545–551 (2007).
[CrossRef]

A. Büttner, U. D. Zeitner, and R. Kowarschik, “Design considerations for high-brightness diffractive broad-area lasers,” J. Opt. Soc. Am. B22, 796 (2005).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Fox, A. G.

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE51(1), 80–89 (1963).
[CrossRef]

Hagan, D. J.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
[CrossRef]

Hutchings, D. C.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
[CrossRef]

Kohn, W.

J. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Phys. Rev.97(4), 869–883 (1955).
[CrossRef]

Kowarschik, R.

Li, T.

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE51(1), 80–89 (1963).
[CrossRef]

Luttinger, J.

J. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Phys. Rev.97(4), 869–883 (1955).
[CrossRef]

Ruzicka, B. A.

B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
[CrossRef]

Samassekou, H.

B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
[CrossRef]

Sheik-Bahae, M.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
[CrossRef]

Stryland, E. W.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
[CrossRef]

Werake, L. K.

B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
[CrossRef]

Zeitner, U.

A. Büttner and U. Zeitner, “Experimental Realization of Monolithic Diffractive Broad-Area Polymeric Waveguide Dye Lasers,” IEEE J. Quantum Electron.43(7), 545–551 (2007).
[CrossRef]

Zeitner, U. D.

Zhao, H.

B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
[CrossRef]

Appl. Phys. Lett.

B. A. Ruzicka, L. K. Werake, H. Samassekou, and H. Zhao, “Ambipolar diffusion of photoexcited carriers in bulk GaAs,” Appl. Phys. Lett.97(26), 262119 (2010).
[CrossRef]

IEEE J. Quantum Electron.

A. Büttner and U. Zeitner, “Experimental Realization of Monolithic Diffractive Broad-Area Polymeric Waveguide Dye Lasers,” IEEE J. Quantum Electron.43(7), 545–551 (2007).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Quantum Electron.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. Stryland, “Kramers-Kronig relations in nonlinear optics,” Opt. Quantum Electron.24(1), 1–30 (1992).
[CrossRef]

Phys. Rev.

J. Luttinger and W. Kohn, “Motion of Electrons and Holes in Perturbed Periodic Fields,” Phys. Rev.97(4), 869–883 (1955).
[CrossRef]

Proc. IEEE

A. G. Fox and T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE51(1), 80–89 (1963).
[CrossRef]

Other

A. E. Siegman, “How to (Maybe) Measure Laser Beam Quality,” in OSA Trends in Optics and Photonics, Vol. 17 (1998), MQ1.

C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators (United States Governm. Pr. Office, 1950).

L. A. Coldren, S. W. Corzine, and M. Mashanovitch, Diode lasers and photonic integrated circuits, 2nd ed. (Wiley, 2012).

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Figures (6)

Fig. 1
Fig. 1

Change of carrier density vs. carrier density for different power and current densities for two different injection current densities (2500 and 1500 A/cm2) and 8 optical power densities (0,0.005..0.095 W/µm)

Fig. 2
Fig. 2

Schematic sketch of the algorithm

Fig. 3
Fig. 3

Near field diameter (top, red) and farfield full angle (bottom, blue) of an AlGaAs broad area laser (100µm stripe width, 4mm resonator length) vs. injection current; Simulation results (triangles) are compared with measurements (squares)

Fig. 4
Fig. 4

Optical output power (bottom, red) and beam parameter product (top, blue) vs. injection current; Simulation results (triangles) are compared with measurements (squares)

Fig. 5
Fig. 5

Simulated (red) and measured (blue) intensity distributions. Left: on the outcoupling facet (near field), right: and angular distribution of the output beam (far field)

Fig. 6
Fig. 6

Distribution of excited carriers and optical power density inside the waveguide and change of the refractive index. The stripe width of the contacted region is 100 µm, the resonator length is 4 mm.

Equations (15)

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Z ^ U i ( x,y,z )= λ i U i ( x,y,z ),
l i =1 λ i 2 .
U 0 ( x,y, z 0 )= i=0 n ( ξ i U i ( x,y, z 0 ) ) + U rest
U j 0 ( x,y, z 0 )= i=0 n ( [ k=0 j1 ξ i,k λ i ] U i ( x,y, z 0 ) ) + Z ^ ... Z ^ j1times U rest .
k= j 0 m ξ i,k λ i 1 for U i isexistingmode k= j 0 m ξ i,k λ i 0 for U i is belowthreshold Z ^ ... Z ^ mtimes U rest 0 fornon-stationary parts of U 0 .
g i = 1 λ i 2 .
1 n avg +1 j= j 0 j 0 + n avg i ξ i,j λ i U i ( x,y, z OC ) 2 dxdy P out
1 n avg +1 j= j 0 j 0 + n avg ξ i,j λ i U i ( x,y, z OC ) 2 dxdy P i .
1 n avg +1 j= j 0 j 0 + n avg i ξ i,j λ i U i ( x,y,z ) 2 = i 1 n avg +1 [ j= j 0 j 0 + n avg ξ i,j λ i 2 ] =1ifiΩ,0otherwise U i ( x,y,z ) 2 = iΩ U i ( x,y,z ) 2 .
dN dt = R( N )N (a) g( N ) p( x,z )dx hν Γdz (b) + d N I ( x,z ) (c) +d N diff ( d )
R( N )= A (a) + B(N)N (b) + C N 2 (c) ,
L Diff D AM τ 20c m 2 s 1 0.5ns =1μm.
d N 0 ( x,z ) dt 0 N 0 ( x,z )= N 0 ( p( x,z ),I( x,z ) ).
p j+1 ( x,z )=( 1α ) p j ( x,z )+α ε 0 U j+1 0 ( x,y,z ) 2 dy .
N 0 ( x,z ) j+1 =( 1β ) N 0 ( x,z ) j +β N 0 ( p j+1 ( x,z ),I( x,z ) ).

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