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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 13 — Jul. 1, 2013
  • pp: 15553–15567
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Simulation of spectral stabilization of high-power broad-area edge emitting semiconductor lasers

Carlo Holly, Stefan Hengesbach, Martin Traub, and Dieter Hoffmann  »View Author Affiliations


Optics Express, Vol. 21, Issue 13, pp. 15553-15567 (2013)
http://dx.doi.org/10.1364/OE.21.015553


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Abstract

The simulation of spectral stabilization of broad-area edge-emitting semiconductor diode lasers is presented in this paper. In the reported model light-, temperature- and charge carrier-distributions are solved iteratively in frequency domain for transverse slices along the semiconductor heterostructure using wide-angle finite-difference beam propagation. Depending on the operating current the laser characteristics are evaluated numerically, including near- and far-field patterns of the astigmatic laser beam, optical output power and the emission spectra, with central wavelength and spectral width. The focus of the model lies on the prediction of influences on the spectrum and power characteristics by frequency selective feedback from external optical resonators. Results for the free running and the spectrally stabilized diode are presented.

© 2013 OSA

1. Introduction

High-Power Diode Lasers (HPDL) operate at high power levels under high electrical to optical conversion efficiencies. Due to their reliability, compactness, economical wafer production and a comparatively long lifetime the devices are widely spread in various fields of industrial and research applications. They are used as pump sources for solid-state lasers, fiber lasers, fiber amplifiers and for a variety of applications in medical technology, telecommunication and direct manufacturing processes, such as marking, cutting or welding of metals and polymers [1

1. F. Bachmann, P. Loosen, and R. Poprawe, eds., High Power Diode Lasers - Technology and Applications (Springer, 2007).

,2

2. M. Traub, M. Bock, H.-D. Hoffmann, and M. Bartram, “Novel high peak current pulsed diode laser sources for direct material processing,” in Proc. SPIE6456, (2007).

]. Direct material processing requires a brightness from 105 up to 108 W cm−2 sr−1 [1

1. F. Bachmann, P. Loosen, and R. Poprawe, eds., High Power Diode Lasers - Technology and Applications (Springer, 2007).

]. Optical power levels of 10 W from a single emitter with electro to optical conversion efficiencies up to 65% are achievable [3

3. G. Erbert, “Progress in high brilliance lasers,” IEEE Photonics Society Summer Topical Meeting Series (2012).

].

The beam quality strongly depends on thermal- and carrier-induced nonlinearities which lead to instable operation of the device and a filamented optical field in the lateral direction due to self-focusing and self-defocusing effects [1

1. F. Bachmann, P. Loosen, and R. Poprawe, eds., High Power Diode Lasers - Technology and Applications (Springer, 2007).

,4

4. G. Erbert, A. Bärwolff, J. Sebastian, and J. Tomm, “High-Power Broad-Area Diode Lasers and Laser Bars,” in High-Power Diode Lasers, R. Diehl, ed. (Topics Appl. Phys. 78, 173–223, 2000).

8

8. P. Crump, S. Böldicke, C. M. Schultz, H. Ekhteraei, H. Wenzel, and G. Erbert, “Experimental and theoretical analysis of the dominant lateral waveguiding mechanism in 975 nm high power broad area diode lasers,” Semicond. Sci. Technol. 27(4), 045001 (2012). [CrossRef]

]. Broad-Area Diode Lasers (BA-LDs) offer poor beam quality in the lateral direction due to lateral multi-mode operation and beam filamentation (see Fig. 1
Fig. 1 Principle of BA-LD epitaxial grow, band gap energy profile, refractive index profile, vertical mode intensity, lateral mode intensity, carrier density, lateral and vertical far-field profiles and axis (lateral, vertical, longitudinal) definition.
).

Incoherent beam combining for pumping of solid-state lasers requires a spectral stability with linewidth < 1 nm [9

9. P. Crump, C. M. Schultz, A. Pietrzak, S. Knigge, O. Brox, A. Maaßdorf, F. Bugge, H. Wenzel, and G. Erbert, “975-nm high-power broad area diode lasers optimized for narrow spectral linewidth applications,” in Proc. SPIE7583, (2010).

], a great bandwidth of emission wavelengths and nearly diffraction limited radiation at high power levels. The spectral width is reduced by frequency-selective external optical feedback, internal distributed feedback (DFB) gratings or distributed Bragg reflector (DBR). The drift of the central wavelength with operation current decreases by suppression of the longitudinal multi-mode operation of the diode laser. Besides, the power characteristics of the spectrally stabilized semiconductor laser are influenced by the external feedback [10

10. P. Crump, S. Hengesbach, U. Witte, H.-D. Hoffmann, G. Erbert, and G. Tränkle, “High-Power Diode Lasers Optimized for Low-Loss Smile-Insensitive External Spectral Stabilization,” IEEE Photon. Technol. Lett. 24(8), 703–705 (2012). [CrossRef]

]. A trade-off between electro-optical efficiency and stabilization-bandwidth is made when designing the external optical system. The characteristics of the spectral stabilization for state-of-the-art concepts are determined by experiment [10

10. P. Crump, S. Hengesbach, U. Witte, H.-D. Hoffmann, G. Erbert, and G. Tränkle, “High-Power Diode Lasers Optimized for Low-Loss Smile-Insensitive External Spectral Stabilization,” IEEE Photon. Technol. Lett. 24(8), 703–705 (2012). [CrossRef]

].

In this paper a multi-frequency 2D cross section laser model with solvers for the transverse electrical, thermal and optical fields with attached external resonators for transverse and longitudinal mode selection is presented. The frequency domain model is implemented in a MATLAB software package for numerical simulation of HPDLs together with micro-optics, such as Volume Bragg Gratings (VBG) and aspherical collimation lenses (Fast-Axis Collimation, FAC) and lens arrays (Slow-Axis Collimation, SAC). Problem specific solvers with reduced complexity predict the effects of external feedback on the emission spectrum, optical output power, near- and far-field distributions of the astigmatic laser radiation and beam caustics. All characteristics are determined dependent on the operation current. Internal device properties as such as local temperature, carrier density and the photon density are iteratively calculated to account for the interaction between the semiconductor material and the light field. Goal of the numerical analysis is to support the design process of external feedback systems, and thus lead to shorter design cycles and reduced development costs.

2. Semiconductor laser model

The diode laser is supposed to run under constant pumping current in Continuous-Wave (CW) operation. The laser simulator is based on the 2.5D laser models reported in [5

5. Z. Dai, R. Michalzik, P. Unger, and K. J. Ebeling, “Numerical Simulation of Broad-Area High-Power Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 33(12), 2240–2254 (1997). [CrossRef]

,11

11. J. J. Lim, S. Sujecki, L. Lang, Z. Zhang, D. Paboeuf, G. Pauliat, G. Lucas-Leclin, P. Georges, R. C. I. MacKenzie, P. Bream, S. Bull, K.-H. Hasler, B. Sumpf, H. Wenzel, G. Erbert, B. Thestrup, P. M. Petersen, N. Michel, M. Krakowski, and E. C. Larkins, “Design and Simulation of Next-Generation High-Power, High-Brightness Laser Diodes,” IEEE J. Quantum Electron. 15, 993–1008 (2009).

14

14. S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, “Nonlinear Properties of Tapered Laser Cavities,” IEEE J. Sel. Top. Quantum Electron. 9(3), 823–834 (2003). [CrossRef]

]. Relevant physical properties, solvers and their interaction are illustrated in Fig. 2
Fig. 2 Flow-diagram illustrating the relevant physical properties, solvers and their interaction.
. As demonstrated, the iteration scheme and slice coupling algorithm are successfully applied for micro cavity laser simulations. The optical field propagation and the updates of the thermal and carrier density distribution are performed for sequences of 2D transverse slices along the cavity. This results in a full 3D output of the calculated properties. The propagation of the optical field between electro-thermal slices is performed with the Wide-Angle Finite-Differences Beam Propagation Method (WA-FD-BPM). Following [11

11. J. J. Lim, S. Sujecki, L. Lang, Z. Zhang, D. Paboeuf, G. Pauliat, G. Lucas-Leclin, P. Georges, R. C. I. MacKenzie, P. Bream, S. Bull, K.-H. Hasler, B. Sumpf, H. Wenzel, G. Erbert, B. Thestrup, P. M. Petersen, N. Michel, M. Krakowski, and E. C. Larkins, “Design and Simulation of Next-Generation High-Power, High-Brightness Laser Diodes,” IEEE J. Quantum Electron. 15, 993–1008 (2009).

14

14. S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, “Nonlinear Properties of Tapered Laser Cavities,” IEEE J. Sel. Top. Quantum Electron. 9(3), 823–834 (2003). [CrossRef]

], we use a Fox-Li iteration scheme to find resonator modes and steady-state output characteristics of the micro cavity laser.

To simulate the coupling between the diode and the external optical resonator, the optical elements (HPDL, VBG, FAC, SAC) are divided into domains. Each domain has specific size, resolution, boundary conditions and different propagation algorithms (WA-FD-BPM, scalar diffraction theory or Fourier-optics [15

15. D. Voelz, Computational Fourier Optics (SPIE Press, 2011).

]) for the optical field. A domain can additionally couple material properties and temperature distributions dynamically to the optical simulation. Changes of the spatial sizes from several nanometers (Quantum Wells, QWs) up to tens of millimeters (in external resonators) are supported. Previously defined domains are combined to a variety of complex optical systems.

During one resonator round-trip the electric field is once propagated forwards through the domains, while partial reflections at the domain boundaries according to the boundary conditions are stored. These values are added to the backwards propagating field at the corresponding position, see Fig. 3
Fig. 3 Schematic representation of the coupling of domains.
. After propagating through all domains in positive direction the procedure is repeated in the opposite way for the backward propagation. The iteration algorithm for the coupled domains is an Fox-Li iteration scheme with partial reflections at the domain interfaces. Inside each domain the Fox-Li iteration algorithm is applied and at each round-trip the fields of both neighboring domains take influence as boundary conditions.

The propagation throughout the domains is repeated until the convergence criteria is reached. If the solution does not converge at all, the procedure is stopped after a distinct number of resonator round-trips. As a convergence criteria the relative change of the optical output power and of the emission spectrum per round-trip are taken. For the simulation of HPDL it is not constructive to take the formation of a stable electromagnetic field profile at the facet or in the radiated far-field as a convergence criteria since the profiles undergo significant changes from one round-trip to another [12

12. J. R. Marciante and G. P. Agrawal, “Nonlinear Mechanisms of Filamentation in Broad-Area Semiconductor Lasers,” IEEE J. Quantum Electron. 32(4), 590–596 (1996). [CrossRef]

].

The final output characteristics are determined by averaging the values of the electromagnetic field over a certain number of round-trips. The incoherent integration of light field intensities for several round-trips leads to a quasi-time-averaged intensity detection. This procedure is equivalent to an experimental intensity measurement.

The electrical model and the optical model are coupled through the laser rate equations [16

16. K. J. Ebeling, Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors (Springer, 1993).

]. At each electro-thermal slice (the optical propagation distance between two electro-thermal slices is ΔzET) the carrier density N is computed with Eq. (1) and material gain by stimulated and spontaneous emission and the carrier induced refractive index change are updated. According to the 2D unipolar electrical model the motion equation for the carriers inside the active region is [5

5. Z. Dai, R. Michalzik, P. Unger, and K. J. Ebeling, “Numerical Simulation of Broad-Area High-Power Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 33(12), 2240–2254 (1997). [CrossRef]

,12

12. J. R. Marciante and G. P. Agrawal, “Nonlinear Mechanisms of Filamentation in Broad-Area Semiconductor Lasers,” IEEE J. Quantum Electron. 32(4), 590–596 (1996). [CrossRef]

,17

17. J. R. Marciante and G. P. Agrawal, “Controlling Filamentation in Broad-Area Semiconductor Lasers and Amplifiers,” Appl. Phys. Lett. 69(5), 593–595 (1996). [CrossRef]

,18

18. S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).

]
dNdt=D2N+ηiJqdBN21τnrNΓc0nacg(ω,N,T)S,
(1)
where S is the photon density, B is the spontaneous emission coefficient, Γ the optical confinement factor, c0 the vacuum speed of light, nac the refractive index of the active area, g the gain, ω the optical frequency, T the temperature, D is the carrier diffusion coefficient, ηi the internal quantum efficiency (defined in [18

18. S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).

]), J the current density, the elementary charge q, the QW thickness d and the nonradiative carrier lifetime τnr. Since a quasi-steady-state solution of Eq. (1) is used for the laser model, fluctuations in time are neglected. Furthermore, longitudinal heat and carrier flow is neglected in the model, which is valid for structures that are slowly varying in the longitudinal direction [11

11. J. J. Lim, S. Sujecki, L. Lang, Z. Zhang, D. Paboeuf, G. Pauliat, G. Lucas-Leclin, P. Georges, R. C. I. MacKenzie, P. Bream, S. Bull, K.-H. Hasler, B. Sumpf, H. Wenzel, G. Erbert, B. Thestrup, P. M. Petersen, N. Michel, M. Krakowski, and E. C. Larkins, “Design and Simulation of Next-Generation High-Power, High-Brightness Laser Diodes,” IEEE J. Quantum Electron. 15, 993–1008 (2009).

].

Since the index-guided vertical waveguide mode is stable during operation, the focus lies on the analysis of the BA-LDs lateral mode dynamics. Therefore, the reduction of the full spatial 3D model to two spatial dimensions (lateral and longitudinal direction of the device) by the Effective Index Method (EIM) is applied in particular cases [6

6. R. J. Lang, A. G. Larsson, and J. G. Cody, “Lateral Modes of Broad Area Semiconductor Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 27(3), 312–320 (1991). [CrossRef]

,19

19. J. Buus, “The Effective Index Method and Its Application to Semiconductor Lasers,” IEEE J. Quantum Electron. 18(7), 1083–1089 (1982). [CrossRef]

]. The analytical separation of the vertical and lateral fields lead to the 1.5D laser model. If the EIM is applied the effective wave number keff, given in Eq. (2), is
keff(ω,N,T)=k0(neff+Γδn(ω,N,T)+(1Γ)nKerrI)i(Γg(ω,N,T)(1Γ)αint),
(2)
with the optical confinement factor Γ, the light field intensity I, the effective refractive index neff [19

19. J. Buus, “The Effective Index Method and Its Application to Semiconductor Lasers,” IEEE J. Quantum Electron. 18(7), 1083–1089 (1982). [CrossRef]

], the carrier induced refractive index change δn, the Kerr coefficient nKerr accounting for intensity induced refractive index changes and the internal resonator loss coefficient αint.

2.1 Semiconductor material

The presented semiconductor model is based on the free-carrier theory. We follow the approach of [18

18. S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).

] to derive the complex susceptibility of a group III-V direct band gap semiconductor with respect to temperature, frequency of the interacting light field and the carrier density. For the presented semiconductor laser model we simplify the band structure to a two-band model consisting of one parabolic conduction band and one parabolic valence band with the averaged effective-mass of the heavy-, light- and split-off bands. The material properties shown in Fig. 4
Fig. 4 Computed semiconductor properties: amplitude gain (top left), spontaneous emission rate (top right), refractive index change (bottom left, the arrow indicates the direction of increasing carrier density) over wavelength for carrier densities N = 1.5, 2, 2.5, 3, 3.5 1018 cm−3 at T = 293.15, 343.15, 393.15 K. Gain over carrier density for different temperatures (bottom right).
and used in the simulations are calculated based on the parameters listed in Table 1

Table 1. Semiconductor material parameters for In0.15Ga0.85As at T = 300 K

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.

Based on [18

18. S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).

] the susceptibility given in Eq. (3) for a QW structure is
χ(ω,N,T)=nac21mr|μ0|2πε03γd0dεfe(ε,N,T)+fh(ε,N,T)1(1+(ε+εg(T)ωγ)2)(1+εεg(T))2×(i+ε+εg(T)ωγ),
(3)
with the energy of the electron state ε, the reduced electron-hole mass mr, the vacuum permittivity ε0, the Planck constant , the dephasing coefficient γ, the band gap energy εg, given in Eq. (4), and the Fermi-Dirac distributions fα(ε,N,T) for electrons and holes, respectively. The temperature dependence of the band gap energy is modeled via
εg(T)=εg,0αInGaAsT2T+βInGaAs,
(4)
where we introduce the material parameters αInGaAs=5.7104eVK-1 and βInGaAs=249K which are distinguished empirically by fitting the band gap temperature dependence to experimental measurements as reported in [21

21. J. R. Botha and A. W. R. Leitch, “Temperature Dependence of the Photoluminescence Properties and Band Gap Energy of InxGa1–xAs/GaAs Quantum Wells,” J. Electron. Mater. 29(12), 1362–1371 (2000). [CrossRef]

]. Following [18

18. S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).

], in Eq. (3), the dependence of the dipole element on the momentum near the band gap is approximated with the matrix dipole element μ0 at the band gap. This approximation is suitable for InGaAs/AlGaAs systems [22

22. B. Witzigmann, A. Witzig, and W. Fichtner, “A Multidimensional Laser Simulator for Edge-Emitters Including Quantum Carrier Capture,” IEEE Trans. Electron. Dev. 47(10), 1926–1934 (2000). [CrossRef]

]. The coefficient for the optical gain by stimulated emission, Eq. (5), yields
g(ω,N,T)=ω2nacc0(χ).
(5)
The optical gain increases with increasing carrier density and the gain maximum shifts to higher wavelengths with higher temperature (see Fig. 4, top left). The spontaneous emission coefficient is given by
sr(ω,N,T)=Bd(π2mr2qkBTmemh)3/20dεfe(ε,N,T)fh(ε,N,T)1+(ε+εgωγ)2,
(6)
with the Boltzmann constant kB, the electron mass me and the hole mass mh. The absolute refractive index change is given in Eq. (7)

δn(ω,N,T)=12nac((χ)+1nac2).
(7)

The dependence of the refractive index change on the carrier density, temperature and wavelength is plotted in Fig. 4 (bottom left). Local fluctuations of the carrier density inside a BA-LD device from 151018cm3 typically occur at high-power operation. These variations lead to changes in the refractive index of the host medium, compare Fig. 4 (bottom, left).

The nonradiative carrier lifetime, band gap energy offset and the carrier transport efficiency are device specific parameters, since they dependent on the specific epitaxial grow of the heterostructure. In a calibration procedure the nonradiative carrier lifetime and the carrier transport efficiency are determined by iteratively matching the calculated threshold current and slope efficiency to the experimental data. Additionally, the band gap energy offset due to induced strain is determined by comparing simulated emission spectra with measurements.

2.2 Optical model

The heterostructure of the BA-LD (see Fig. 1, refractive index profile) forms a slab waveguide for the electromagnetic field. The presented Beam Propagation Method (BPM) is valid for waveguide structures with small refractive index changes along the propagation direction. In BA-LDs the Transverse Electric (TE) mode is the dominant one [23

23. J. Ohtsubo, Semiconductor Lasers – Stability, Instability and Chaos, 2nd Edition (Springer, 2008).

]. Separation of the stationary part of the transverse electric field into an envelope function and a fast oscillating phase term leads to Eq. (8),
ET(x,t)=p=1NωψTp(x)e-iωpt=p=1Nωϕp(x,y,z)e±iωpc0(nrefxx+nrefyy+nrefzz)e-iωpt,
(8)
where ET is the total transverse electric field, Nω the frequency space resolution, ψTp the transverse electric field of the stationary solution and ωp, the optical frequency for frequency number p, ϕp is the envelope of the electric field and nrefα are the reference refractive indexes for the three directions, respectively. The reference refractive indexes are chosen close to the effective refractive indexes of the mode propagation to match the fast oscillating part so that the envelope varies slowly. The bounds of the wavelength space are λl=2πc0/ωNωλ2πc0/ω1=λh. For small refractive index changes in the propagation direction the extended Helmholtz equation [24

24. C. L. Xu and W. P. Huang, “Finite-Difference Beam Propagation Method for Guide-Wave Optics,” Progress In Electromagnetics Research, PIER 11, 1–49 (1995).

,25

25. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations and the Schrödinger Equation (John Wiley & Sons, Inc.,2001).

] for stationary TE modes is given by Eq. (9)
z2ψTp+T2ψTp+kp2ψTp=T(TψTp1n2T(n2ψTp)),
(9)
where T is the Nabla operator for the transverse directions. A generalized envelope function with two degrees of freedom is introduced in [26

26. K. Q. Le and P. Bienstman, “Fast three-dimensional generalized rectangular wide-angle beam propagation method using complex Jacobi iteration,” J. Opt. Soc. Am. 26(7), 1469–1472 (2009). [CrossRef]

]. In our approach we allow the phase term to depend on both transverse directions. We put Eq. (8) in Eq. (9) and get Eq. (10) with the generalized envelope operator Gp defined for each frequency as a 2x2 matrix operator
p=1Nωz2ϕp+2iωpc0nrefzzϕp+Gpϕp=0.
(10)
The boundary conditions (the reflectivities of the front and back facet) are applied at the domain borders and can optionally be frequency dependent. We apply the Crank-Nichelson scheme to Eq. (10) and get Eq. (11),
(1inrefωpΔz2c0(1+c02Gp(nrefωp)21))Apϕp|k+1=(1+inrefωpΔz2c0(1+c02Gp(nrefωp)21))Bpϕp|k,
(11)
with the longitudinal step size Δz [27

27. K. Q. Le and P. Bienstman, “Wide-angle beam propagation method without using slowly varying envelope approximation,” J. Opt. Soc. Am. 26(2), 353–356 (2009). [CrossRef]

]. The Eq. (11) is solved independently for each frequency to determine the field distribution ϕp|k+1 (at longitudinal grid position k+1) for a given field ϕp|k (at longitudinal grid position k). Following [27

27. K. Q. Le and P. Bienstman, “Wide-angle beam propagation method without using slowly varying envelope approximation,” J. Opt. Soc. Am. 26(2), 353–356 (2009). [CrossRef]

,28

28. K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282(7), 1252–1254 (2009). [CrossRef]

], the square root operator is expressed by a rational fraction. The application of the standard Padé approximation does not give correct damping to evanescent modes. This drawback is circumvented by application of the modified Padé approximation [28

28. K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282(7), 1252–1254 (2009). [CrossRef]

].

Application of the finite-differences (FD) scheme for the discretization of the transverse derivatives converts the operators Ap and Bp from Eq. (11) into banded matrices (field values on the transverse discrete grid are stored in form of a row vector). Consequently, Eq. (11) offers an expression for a step wise propagation of the transverse electric field by solving a linear sparse matrix problem.

There are several applications presented in literature [24

24. C. L. Xu and W. P. Huang, “Finite-Difference Beam Propagation Method for Guide-Wave Optics,” Progress In Electromagnetics Research, PIER 11, 1–49 (1995).

27

27. K. Q. Le and P. Bienstman, “Wide-angle beam propagation method without using slowly varying envelope approximation,” J. Opt. Soc. Am. 26(2), 353–356 (2009). [CrossRef]

] and [28

28. K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282(7), 1252–1254 (2009). [CrossRef]

] where the FD-BPM is successfully applied to compute light field propagations. The right hand side of Eq. (11) is evaluated by direct matrix multiplication, which results in a row vector. The equation is an inhomogeneous sparse linear system, which is either solved directly or iteratively by applying the biconjugate gradient stabilized method (BICGSTAB). For BICGSTAB an incomplete lower-upper (LU) decomposition of matrix Ap is performed to improve the stability of the algorithm and reduce iteration steps. Direct solving of the system becomes inefficient for great transverse grid sizes (NxNy>1000), therefore BICGSTAB is used for the 2.5D simulations.

Furthermore, the contribution of the spontaneous emission to the light field is accounted by Eq. (12) with β¯, the spontaneous emission contribution factor,
ϕ'p|k+1=ϕp|k+1+2ωpΔzβ¯sr(ωp,N,T)ε0c0nref
(12)
by adding the frequency, carrier density and temperature dependent spontaneous recombination term for each field in the frequency space at each propagation step.

2.3 Thermal model

The stationary temperature distribution inside the device is calculated with Eq. (13), the thermal diffusion equation [29

29. W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron. 15(6), 513–527 (1983). [CrossRef]

,30

30. J. Mukherjee and J. G. McInerney, “Electrothermal Analysis of CW High-Power Broad-Area Laser Diodes: A Comparison Between 2-D and 3-D Modeling,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1180–1187 (2007). [CrossRef]

]
(λT(x)T(x))+g(x)=0.
(13)
Here λT is the thermal conductivity distribution and g is the local heat source distribution. The finite element method (FEM) is applied to solve the thermal diffusion equation on a triangular mesh grid (mesh resolution and thermal domain size are listed in Table 3). The temperature is calculated for 2D transverse slices with adiabatic boundaries in lateral direction yT|y=±LT,y/2=0, convection at the n-side of the device λTxT|x=-LT,x/2=h0(TT0) [29

29. W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron. 15(6), 513–527 (1983). [CrossRef]

] and convection at the p-side λTxT|x=LT,x/2=hhs(TThs).

The conductivities and layer thicknesses used for the simulations are listed in Table 2

Table 2. Material parameters of epitaxial layers

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, the thermal boundary conditions and the heat sink and ambient temperature are listed in Table 3

Table 3. Simulation parameters

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.

Heat is generated by nonradiative recombination of carriers, absorption of light in the waveguide and cladding layers and Joule heating gel(x)=ρ(x)J2(x) due to the transport of carriers [29

29. W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron. 15(6), 513–527 (1983). [CrossRef]

,30

30. J. Mukherjee and J. G. McInerney, “Electrothermal Analysis of CW High-Power Broad-Area Laser Diodes: A Comparison Between 2-D and 3-D Modeling,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1180–1187 (2007). [CrossRef]

]. The heat sources due to nonradiative recombination and reabsorption of spontaneous emission are distributed throughout the active region, the heat sources due to absorption of light are distributed among the waveguide (proportional to the light intensity, equally if EIM is used) and the Joule heating is distributed according to the electric resistivity inside the injection stripe.

The maximum temperature occurs inside the active region (see Fig. 5
Fig. 5 Computed transverse intra-cavity temperature profiles. Left: temperature profile over vertical (x) and lateral (y) direction for I = 6 A, right: lateral temperature distributions inside active region (x = 0 µm) for injection currents of I = 1 A to 10 A (the arrow indicates the direction of increasing injection current).
and [5

5. Z. Dai, R. Michalzik, P. Unger, and K. J. Ebeling, “Numerical Simulation of Broad-Area High-Power Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 33(12), 2240–2254 (1997). [CrossRef]

]), since greatest optical intensity and nonradiative recombination processes are located there. Small ripples in the temperature profile due to the non-uniform light field distribution are observed along the lateral direction (compare Fig. 5, right). Nevertheless, along the lateral device direction the temperature shows a smooth bell-shaped distribution which increases with the current.

To reproduce the particular dependence of the temperature on the current the thermal boundaries and the heat sink convection term are matched to the specific experimental setup.

3. Simulation results

The device modeled and tested is a Extreme LOw Divergence (ELOD) single emitter with three In0.15Ga0.85As quantum wells and asymmetric waveguide as presented in [31

31. P. Crump, A. Pietrzak, F. Bugge, H. Wenzel, G. Erbert, and G. Tränkle, “975 nm high power diode lasers with high efficiency and narrow vertical far field enabled by low index quantum barriers,” Appl. Phys. Lett. 96(13), 131110 (2010). [CrossRef]

]. Several simulation runs with different current levels are performed with the input parameters listed in Table 3 to create output power and spectral power density depending on the point of operation.

The astigmatism of the BA-LD radiation is based on the different guiding mechanisms in both transverse directions. Since refractive index steps in lateral direction are missing, the guiding of the light arises caused by a channel of high gain, which is created inside the injection stripe width by induced carriers. Therefore, the light is gain-guided in the lateral direction.

High optical intensities result in a depressed local gain (depressed carrier density) caused by the stimulated recombination of electron and hole pairs, while regions with low light intensities show maxima in the carrier density (inversion is not reduced by stimulated recombination and additional carriers are injected due to the pumping current). The minima of the carrier density distribution thus fall together with the maxima of the optical intensity and vice versa. This spatial modulation in charge carriers leads to a local perturbation of the refractive index and gain which forces the light field to split up into filaments [6

6. R. J. Lang, A. G. Larsson, and J. G. Cody, “Lateral Modes of Broad Area Semiconductor Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 27(3), 312–320 (1991). [CrossRef]

,7

7. W. W. Chow and H. Amano, “Analysis of lateral mode behavior in broad-area InGaN quantum well lasers,” IEEE J. Quantum Electron. 37(2), 265–273 (2001). [CrossRef]

,14

14. S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, “Nonlinear Properties of Tapered Laser Cavities,” IEEE J. Sel. Top. Quantum Electron. 9(3), 823–834 (2003). [CrossRef]

,23

23. J. Ohtsubo, Semiconductor Lasers – Stability, Instability and Chaos, 2nd Edition (Springer, 2008).

] (see Fig. 6
Fig. 6 Plots of the computed envelope of the electromagnetic intensity and the carrier density for a lateral (y direction), –longitudinal (z direction) slice inside the active region. The intensity of the forward (left) and backward (center) travelling waves and the carrier density (right) are displayed over the lateral (y) and longitudinal (z) direction for a) I = 10 A and b) I = 2 A.
, Fig. 7
Fig. 7 Computed transverse profile of the optical intensity in the waveguide (combination of false color and height plot, left). The calculated vertical light intensity is displayed together with the refractive index profile in vertical direction (right). The QWs are positioned at x = 0 µm.
and Fig. 8
Fig. 8 Computed optical intensity profiles: Averaged (from round-trip 20 to 100) near-field profiles over lateral direction at the diodes facet (left) and averaged (from round-trip 90 to 100) far-field profiles over lateral angle (right) for several injection currents.
). In vertical direction the waveguide (refractive index steps between the epitaxial layers) supports the formation of a stable eigenmode (compare [31

31. P. Crump, A. Pietrzak, F. Bugge, H. Wenzel, G. Erbert, and G. Tränkle, “975 nm high power diode lasers with high efficiency and narrow vertical far field enabled by low index quantum barriers,” Appl. Phys. Lett. 96(13), 131110 (2010). [CrossRef]

] and Fig. 7, right), which results in a gaussian like profile in the vertical far-field and a higher beam quality in this direction. The calculated, (incoherent) averaged optical intensity shows irregular peaks in lateral direction inside the injection stripe width, see Fig. 7 and Fig. 8.

The strength of the filaments increases with current (compare [6

6. R. J. Lang, A. G. Larsson, and J. G. Cody, “Lateral Modes of Broad Area Semiconductor Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 27(3), 312–320 (1991). [CrossRef]

]) and the spikes consequently appear in the far-field, see [9

9. P. Crump, C. M. Schultz, A. Pietrzak, S. Knigge, O. Brox, A. Maaßdorf, F. Bugge, H. Wenzel, and G. Erbert, “975-nm high-power broad area diode lasers optimized for narrow spectral linewidth applications,” in Proc. SPIE7583, (2010).

] and Fig. 8 (left and right). High spatial frequencies in the near-field result in large far-field divergence angles and consequently a reduced brightness. A lateral far-field divergence angle of approximately 5.5° at 6 A injection current (see Fig. 8) matches the experimental value of about 6.5° as measured in [9

9. P. Crump, C. M. Schultz, A. Pietrzak, S. Knigge, O. Brox, A. Maaßdorf, F. Bugge, H. Wenzel, and G. Erbert, “975-nm high-power broad area diode lasers optimized for narrow spectral linewidth applications,” in Proc. SPIE7583, (2010).

]. Due to the chaotic light propagation under the influence of nonlinearities, the light field changes its form on a small time-scale of several picoseconds (and equivalent after each round-trip).

Since the presented model is derived in frequency domain the spectral power density is determined by evaluating the power of each partial electric field ϕp with frequency ωp. Below the lasing threshold (Ith = 1.3 A) the spectrum is dominated by spontaneous emission, see Fig. 9
Fig. 9 Spectral power density and output power over current – simulation and experiment. The computed center of mass wavelength (solid line), 95% power inclusion (dashed line) and the output power (dashed line) are displayed (left). The triangular markers indicate the experimental measurements. Simulated (solid line) and experimental (dotted line, performed with a spectral resolution of 50 pm) spectral power densities (normalized) over wavelength for currents I = 2, 4, 6, 8, 10 A (right). The spectrum and power is determined by averaging over 50 round-trips.
(left). The spectrum is comparatively broad and the central wavelength is higher below threshold current than above. Above threshold the spectrum contracts around a central wavelength. The energy pumped into the semiconductor by carrier injection is distributed to the light field according to the gain profile and the local spectral power density. As demonstrated, the temperature increases with increasing injection current. Consequently, see Eq. (4), the band gap energy decreases with increasing temperature, resulting in a shift of the gain maximum to higher wavelengths (compare Fig. 4 gain over wavelength for different temperatures). Since nonlinear effects are included and the temperature distribution is calculated in the model, spectral broadening and the central wavelength shift are successfully reproduced by the numerical results, see Fig. 9.

An external optical system including a reflective VBG (with a thickness of 3.5 mm and peak reflectivity of 7.5% at 976 nm) as frequency selective filter in an inverse self-imaging setup (reported in [10

10. P. Crump, S. Hengesbach, U. Witte, H.-D. Hoffmann, G. Erbert, and G. Tränkle, “High-Power Diode Lasers Optimized for Low-Loss Smile-Insensitive External Spectral Stabilization,” IEEE Photon. Technol. Lett. 24(8), 703–705 (2012). [CrossRef]

]) is presented to demonstrate the calculation of a spectrally stabilized diode laser. The angular selectivity of the VBG is about 2.4° and the spectral selectivity is 0.2 nm (the selectivity is determined from the distance between the first minima of the filter profile).

The simulation includes 9 domains (the HPDL, a FAC, SAC, VBG and the free space between the components), see Fig. 10
Fig. 10 Inverse self-imaging external optical system with FAC, SAC and VBG. False color plot of optical intensity with 2nd momentum of intensity for the forward (top) and backward (bottom) propagating field.
. The refraction at the acylindrical surface of the FAC is calculated with the scalar diffraction integral (evaluated for curved surfaces). For the SAC the thin lens approximation is applied, thus the field is multiplied by a quadratic phase term. Propagation in homogeneous media is performed by Fourier-optical methods with the impulse response function or transfer function dependent on the sampling conditions [15

15. D. Voelz, Computational Fourier Optics (SPIE Press, 2011).

]. To model the angular and frequency dependent diffraction at the VBG the coupled wave theory is applied for finite field distributions.

For an external reflectivity of 6% the spectral width (at 90% power inclusion) decreases to 0.2 nm in the injection current interval from 5 to 8 A, see Fig. 11
Fig. 11 Calculated (top row) and measured (bottom row with spectral resolution of 50 pm) spectral stabilization with external reflectivity of 6%. Spectral power density (normalized for each injection current) over wavelength for several injection currents (left, top and bottom), and spectral map in false colors (right top and bottom) with 90% power inclusion (dashed line) and central wavelength (dash-dotted line).
. The HPDL is spectrally stabilized in this interval. The spectrum shows a maximum for injection currents above threshold at the stabilization wavelength of 976 nm and side lobes at wavelengths corresponding to the spectrum of the free running diode laser, compare Fig. 11.

4. Conclusion and future work

A 2.5D multi-frequency model for high-power edge-emitting semiconductor lasers is developed to compute the spatial distributions of the carrier density, temperature and optical field inside the micro cavity and the light propagation in the external optical system. For transverse slices along the longitudinal direction of the device the optical field is propagated (with WA-FD-BPM) and dependent on the carrier density and temperature the refractive index- and gain-profile is determined. The software provides interfaces to connect models of complex frequency- and angle-selective optical elements, as such as VBGs to the diode laser simulation domain.

The simulation results reproduce experimental measurements for the output power, the astigmatic beam profile and the spectral power density (including central wavelength and spectral width) dependent on the current for both the spectrally stabilized and for the free running diode laser. The effect of wavelength locking over a distinct injection current range by external feedback is reproduced. Good qualitative and quantitative agreement between simulation results and experimental measurements of the tested HPDL is obeyed.

Purpose of the analysis is the identification of influences from external feedback on the beam characteristics, the transverse laser modes, output power and the emission spectrum (including the stabilization range over wavelength and current) to derive guidelines for design modifications of edge-emitting semiconductor lasers and external optical systems for frequency and angular selective feedback. The introduced model is used to design future concepts for increased brightness of HPDL based on spectral and transverse mode selection with external resonators.

Acknowledgment

This work was supported in part by the German Federal Ministry of Education and Research Program SpektraLAS under Contract 13N9729 (ILT).

References and links

1.

F. Bachmann, P. Loosen, and R. Poprawe, eds., High Power Diode Lasers - Technology and Applications (Springer, 2007).

2.

M. Traub, M. Bock, H.-D. Hoffmann, and M. Bartram, “Novel high peak current pulsed diode laser sources for direct material processing,” in Proc. SPIE6456, (2007).

3.

G. Erbert, “Progress in high brilliance lasers,” IEEE Photonics Society Summer Topical Meeting Series (2012).

4.

G. Erbert, A. Bärwolff, J. Sebastian, and J. Tomm, “High-Power Broad-Area Diode Lasers and Laser Bars,” in High-Power Diode Lasers, R. Diehl, ed. (Topics Appl. Phys. 78, 173–223, 2000).

5.

Z. Dai, R. Michalzik, P. Unger, and K. J. Ebeling, “Numerical Simulation of Broad-Area High-Power Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron. 33(12), 2240–2254 (1997). [CrossRef]

6.

R. J. Lang, A. G. Larsson, and J. G. Cody, “Lateral Modes of Broad Area Semiconductor Lasers: Theory and Experiment,” IEEE J. Quantum Electron. 27(3), 312–320 (1991). [CrossRef]

7.

W. W. Chow and H. Amano, “Analysis of lateral mode behavior in broad-area InGaN quantum well lasers,” IEEE J. Quantum Electron. 37(2), 265–273 (2001). [CrossRef]

8.

P. Crump, S. Böldicke, C. M. Schultz, H. Ekhteraei, H. Wenzel, and G. Erbert, “Experimental and theoretical analysis of the dominant lateral waveguiding mechanism in 975 nm high power broad area diode lasers,” Semicond. Sci. Technol. 27(4), 045001 (2012). [CrossRef]

9.

P. Crump, C. M. Schultz, A. Pietrzak, S. Knigge, O. Brox, A. Maaßdorf, F. Bugge, H. Wenzel, and G. Erbert, “975-nm high-power broad area diode lasers optimized for narrow spectral linewidth applications,” in Proc. SPIE7583, (2010).

10.

P. Crump, S. Hengesbach, U. Witte, H.-D. Hoffmann, G. Erbert, and G. Tränkle, “High-Power Diode Lasers Optimized for Low-Loss Smile-Insensitive External Spectral Stabilization,” IEEE Photon. Technol. Lett. 24(8), 703–705 (2012). [CrossRef]

11.

J. J. Lim, S. Sujecki, L. Lang, Z. Zhang, D. Paboeuf, G. Pauliat, G. Lucas-Leclin, P. Georges, R. C. I. MacKenzie, P. Bream, S. Bull, K.-H. Hasler, B. Sumpf, H. Wenzel, G. Erbert, B. Thestrup, P. M. Petersen, N. Michel, M. Krakowski, and E. C. Larkins, “Design and Simulation of Next-Generation High-Power, High-Brightness Laser Diodes,” IEEE J. Quantum Electron. 15, 993–1008 (2009).

12.

J. R. Marciante and G. P. Agrawal, “Nonlinear Mechanisms of Filamentation in Broad-Area Semiconductor Lasers,” IEEE J. Quantum Electron. 32(4), 590–596 (1996). [CrossRef]

13.

J. Wykes, L. Borruel, S. Sujecki, I. Esquivias, P. Sewell, T. M. Benson, E. C. Larkins, P. Moreno, and M. Krakowski, “Hot-Cavity Modelling of High-Power Tapered Laser Diodes using Wide-Angle 3D FD-BPM,” in Proc. LEOS1, 91–92 (2002). [CrossRef]

14.

S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, “Nonlinear Properties of Tapered Laser Cavities,” IEEE J. Sel. Top. Quantum Electron. 9(3), 823–834 (2003). [CrossRef]

15.

D. Voelz, Computational Fourier Optics (SPIE Press, 2011).

16.

K. J. Ebeling, Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors (Springer, 1993).

17.

J. R. Marciante and G. P. Agrawal, “Controlling Filamentation in Broad-Area Semiconductor Lasers and Amplifiers,” Appl. Phys. Lett. 69(5), 593–595 (1996). [CrossRef]

18.

S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).

19.

J. Buus, “The Effective Index Method and Its Application to Semiconductor Lasers,” IEEE J. Quantum Electron. 18(7), 1083–1089 (1982). [CrossRef]

20.

K. S. Chan, H. H. Li, and C. Y. Chan, “Optical Gain of Interdiffused InGaAs-GaAs and AlGaAs-GaAs Quantum Wells,” IEEE J. Quantum Electron. 34(1), 157–165 (1998). [CrossRef]

21.

J. R. Botha and A. W. R. Leitch, “Temperature Dependence of the Photoluminescence Properties and Band Gap Energy of InxGa1–xAs/GaAs Quantum Wells,” J. Electron. Mater. 29(12), 1362–1371 (2000). [CrossRef]

22.

B. Witzigmann, A. Witzig, and W. Fichtner, “A Multidimensional Laser Simulator for Edge-Emitters Including Quantum Carrier Capture,” IEEE Trans. Electron. Dev. 47(10), 1926–1934 (2000). [CrossRef]

23.

J. Ohtsubo, Semiconductor Lasers – Stability, Instability and Chaos, 2nd Edition (Springer, 2008).

24.

C. L. Xu and W. P. Huang, “Finite-Difference Beam Propagation Method for Guide-Wave Optics,” Progress In Electromagnetics Research, PIER 11, 1–49 (1995).

25.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations and the Schrödinger Equation (John Wiley & Sons, Inc.,2001).

26.

K. Q. Le and P. Bienstman, “Fast three-dimensional generalized rectangular wide-angle beam propagation method using complex Jacobi iteration,” J. Opt. Soc. Am. 26(7), 1469–1472 (2009). [CrossRef]

27.

K. Q. Le and P. Bienstman, “Wide-angle beam propagation method without using slowly varying envelope approximation,” J. Opt. Soc. Am. 26(2), 353–356 (2009). [CrossRef]

28.

K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. 282(7), 1252–1254 (2009). [CrossRef]

29.

W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron. 15(6), 513–527 (1983). [CrossRef]

30.

J. Mukherjee and J. G. McInerney, “Electrothermal Analysis of CW High-Power Broad-Area Laser Diodes: A Comparison Between 2-D and 3-D Modeling,” IEEE J. Sel. Top. Quantum Electron. 13(5), 1180–1187 (2007). [CrossRef]

31.

P. Crump, A. Pietrzak, F. Bugge, H. Wenzel, G. Erbert, and G. Tränkle, “975 nm high power diode lasers with high efficiency and narrow vertical far field enabled by low index quantum barriers,” Appl. Phys. Lett. 96(13), 131110 (2010). [CrossRef]

OCIS Codes
(250.0250) Optoelectronics : Optoelectronics
(250.5960) Optoelectronics : Semiconductor lasers

ToC Category:
Optoelectronics

History
Original Manuscript: April 22, 2013
Revised Manuscript: June 13, 2013
Manuscript Accepted: June 13, 2013
Published: June 21, 2013

Citation
Carlo Holly, Stefan Hengesbach, Martin Traub, and Dieter Hoffmann, "Simulation of spectral stabilization of high-power broad-area edge emitting semiconductor lasers," Opt. Express 21, 15553-15567 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15553


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References

  1. F. Bachmann, P. Loosen, and R. Poprawe, eds., High Power Diode Lasers - Technology and Applications (Springer, 2007).
  2. M. Traub, M. Bock, H.-D. Hoffmann, and M. Bartram, “Novel high peak current pulsed diode laser sources for direct material processing,” in Proc. SPIE6456, (2007).
  3. G. Erbert, “Progress in high brilliance lasers,” IEEE Photonics Society Summer Topical Meeting Series (2012).
  4. G. Erbert, A. Bärwolff, J. Sebastian, and J. Tomm, “High-Power Broad-Area Diode Lasers and Laser Bars,” in High-Power Diode Lasers, R. Diehl, ed. (Topics Appl. Phys. 78, 173–223, 2000).
  5. Z. Dai, R. Michalzik, P. Unger, and K. J. Ebeling, “Numerical Simulation of Broad-Area High-Power Semiconductor Laser Amplifiers,” IEEE J. Quantum Electron.33(12), 2240–2254 (1997). [CrossRef]
  6. R. J. Lang, A. G. Larsson, and J. G. Cody, “Lateral Modes of Broad Area Semiconductor Lasers: Theory and Experiment,” IEEE J. Quantum Electron.27(3), 312–320 (1991). [CrossRef]
  7. W. W. Chow and H. Amano, “Analysis of lateral mode behavior in broad-area InGaN quantum well lasers,” IEEE J. Quantum Electron.37(2), 265–273 (2001). [CrossRef]
  8. P. Crump, S. Böldicke, C. M. Schultz, H. Ekhteraei, H. Wenzel, and G. Erbert, “Experimental and theoretical analysis of the dominant lateral waveguiding mechanism in 975 nm high power broad area diode lasers,” Semicond. Sci. Technol.27(4), 045001 (2012). [CrossRef]
  9. P. Crump, C. M. Schultz, A. Pietrzak, S. Knigge, O. Brox, A. Maaßdorf, F. Bugge, H. Wenzel, and G. Erbert, “975-nm high-power broad area diode lasers optimized for narrow spectral linewidth applications,” in Proc. SPIE7583, (2010).
  10. P. Crump, S. Hengesbach, U. Witte, H.-D. Hoffmann, G. Erbert, and G. Tränkle, “High-Power Diode Lasers Optimized for Low-Loss Smile-Insensitive External Spectral Stabilization,” IEEE Photon. Technol. Lett.24(8), 703–705 (2012). [CrossRef]
  11. J. J. Lim, S. Sujecki, L. Lang, Z. Zhang, D. Paboeuf, G. Pauliat, G. Lucas-Leclin, P. Georges, R. C. I. MacKenzie, P. Bream, S. Bull, K.-H. Hasler, B. Sumpf, H. Wenzel, G. Erbert, B. Thestrup, P. M. Petersen, N. Michel, M. Krakowski, and E. C. Larkins, “Design and Simulation of Next-Generation High-Power, High-Brightness Laser Diodes,” IEEE J. Quantum Electron.15, 993–1008 (2009).
  12. J. R. Marciante and G. P. Agrawal, “Nonlinear Mechanisms of Filamentation in Broad-Area Semiconductor Lasers,” IEEE J. Quantum Electron.32(4), 590–596 (1996). [CrossRef]
  13. J. Wykes, L. Borruel, S. Sujecki, I. Esquivias, P. Sewell, T. M. Benson, E. C. Larkins, P. Moreno, and M. Krakowski, “Hot-Cavity Modelling of High-Power Tapered Laser Diodes using Wide-Angle 3D FD-BPM,” in Proc. LEOS1, 91–92 (2002). [CrossRef]
  14. S. Sujecki, L. Borruel, J. Wykes, P. Moreno, B. Sumpf, P. Sewell, H. Wenzel, T. M. Benson, G. Erbert, I. Esquivias, and E. C. Larkins, “Nonlinear Properties of Tapered Laser Cavities,” IEEE J. Sel. Top. Quantum Electron.9(3), 823–834 (2003). [CrossRef]
  15. D. Voelz, Computational Fourier Optics (SPIE Press, 2011).
  16. K. J. Ebeling, Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors (Springer, 1993).
  17. J. R. Marciante and G. P. Agrawal, “Controlling Filamentation in Broad-Area Semiconductor Lasers and Amplifiers,” Appl. Phys. Lett.69(5), 593–595 (1996). [CrossRef]
  18. S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).
  19. J. Buus, “The Effective Index Method and Its Application to Semiconductor Lasers,” IEEE J. Quantum Electron.18(7), 1083–1089 (1982). [CrossRef]
  20. K. S. Chan, H. H. Li, and C. Y. Chan, “Optical Gain of Interdiffused InGaAs-GaAs and AlGaAs-GaAs Quantum Wells,” IEEE J. Quantum Electron.34(1), 157–165 (1998). [CrossRef]
  21. J. R. Botha and A. W. R. Leitch, “Temperature Dependence of the Photoluminescence Properties and Band Gap Energy of InxGa1–xAs/GaAs Quantum Wells,” J. Electron. Mater.29(12), 1362–1371 (2000). [CrossRef]
  22. B. Witzigmann, A. Witzig, and W. Fichtner, “A Multidimensional Laser Simulator for Edge-Emitters Including Quantum Carrier Capture,” IEEE Trans. Electron. Dev.47(10), 1926–1934 (2000). [CrossRef]
  23. J. Ohtsubo, Semiconductor Lasers – Stability, Instability and Chaos, 2nd Edition (Springer, 2008).
  24. C. L. Xu and W. P. Huang, “Finite-Difference Beam Propagation Method for Guide-Wave Optics,” Progress In Electromagnetics Research, PIER11, 1–49 (1995).
  25. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations and the Schrödinger Equation (John Wiley & Sons, Inc.,2001).
  26. K. Q. Le and P. Bienstman, “Fast three-dimensional generalized rectangular wide-angle beam propagation method using complex Jacobi iteration,” J. Opt. Soc. Am.26(7), 1469–1472 (2009). [CrossRef]
  27. K. Q. Le and P. Bienstman, “Wide-angle beam propagation method without using slowly varying envelope approximation,” J. Opt. Soc. Am.26(2), 353–356 (2009). [CrossRef]
  28. K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun.282(7), 1252–1254 (2009). [CrossRef]
  29. W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron.15(6), 513–527 (1983). [CrossRef]
  30. J. Mukherjee and J. G. McInerney, “Electrothermal Analysis of CW High-Power Broad-Area Laser Diodes: A Comparison Between 2-D and 3-D Modeling,” IEEE J. Sel. Top. Quantum Electron.13(5), 1180–1187 (2007). [CrossRef]
  31. P. Crump, A. Pietrzak, F. Bugge, H. Wenzel, G. Erbert, and G. Tränkle, “975 nm high power diode lasers with high efficiency and narrow vertical far field enabled by low index quantum barriers,” Appl. Phys. Lett.96(13), 131110 (2010). [CrossRef]

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