## Simulation of spectral stabilization of high-power broad-area edge emitting semiconductor lasers |

Optics Express, Vol. 21, Issue 13, pp. 15553-15567 (2013)

http://dx.doi.org/10.1364/OE.21.015553

Acrobat PDF (3923 KB)

### 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

^{5}up to 10

^{8}W cm

^{−2}sr

^{−1}[1]. Optical power levels of 10 W from a single emitter with electro to optical conversion efficiencies up to 65% are achievable [3].

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]

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]

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]

## 2. Semiconductor laser model

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. 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. 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]

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. 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]

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]

*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. 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. 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]

*g*the gain,

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).

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

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

### 2.1 Semiconductor material

21. J. R. Botha and A. W. R. Leitch, “Temperature Dependence of the Photoluminescence Properties and Band Gap Energy of In_{x}Ga_{1–x}As/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]

### 2.2 Optical model

*p*,

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]

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]

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

### 2.3 Thermal model

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]

*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

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

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]

## 3. Simulation results

_{0.15}Ga

_{0.85}As 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]

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]

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]

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]

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]

*I*

_{th}= 1.3 A) the spectrum is dominated by spontaneous emission, see Fig. 9 (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.

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]

## 4. Conclusion and future work

## Acknowledgment

## References and links

1. | F. Bachmann, P. Loosen, and R. Poprawe, eds., |

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

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 |

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. |

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. |

7. | W. W. Chow and H. Amano, “Analysis of lateral mode behavior in broad-area InGaN quantum well lasers,” IEEE J. Quantum Electron. |

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. |

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 |

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. |

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. |

12. | J. R. Marciante and G. P. Agrawal, “Nonlinear Mechanisms of Filamentation in Broad-Area Semiconductor Lasers,” IEEE J. Quantum Electron. |

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. LEOS |

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. |

15. | D. Voelz, |

16. | K. J. Ebeling, |

17. | J. R. Marciante and G. P. Agrawal, “Controlling Filamentation in Broad-Area Semiconductor Lasers and Amplifiers,” Appl. Phys. Lett. |

18. | S. W. Koch and W. W. Chow, |

19. | J. Buus, “The Effective Index Method and Its Application to Semiconductor Lasers,” IEEE J. Quantum Electron. |

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. |

21. | J. R. Botha and A. W. R. Leitch, “Temperature Dependence of the Photoluminescence Properties and Band Gap Energy of In |

22. | B. Witzigmann, A. Witzig, and W. Fichtner, “A Multidimensional Laser Simulator for Edge-Emitters Including Quantum Carrier Capture,” IEEE Trans. Electron. Dev. |

23. | J. Ohtsubo, |

24. | C. L. Xu and W. P. Huang, “Finite-Difference Beam Propagation Method for Guide-Wave Optics,” Progress In Electromagnetics Research, PIER |

25. | K. Kawano and T. Kitoh, |

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. |

27. | K. Q. Le and P. Bienstman, “Wide-angle beam propagation method without using slowly varying envelope approximation,” J. Opt. Soc. Am. |

28. | K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun. |

29. | W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron. |

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. |

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. |

**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

- F. Bachmann, P. Loosen, and R. Poprawe, eds., High Power Diode Lasers - Technology and Applications (Springer, 2007).
- 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).
- G. Erbert, “Progress in high brilliance lasers,” IEEE Photonics Society Summer Topical Meeting Series (2012).
- 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).
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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).
- 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]
- 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]
- 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]
- D. Voelz, Computational Fourier Optics (SPIE Press, 2011).
- K. J. Ebeling, Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors (Springer, 1993).
- 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]
- S. W. Koch and W. W. Chow, Semiconductor-Laser Fundamentals (Springer, 1999).
- J. Buus, “The Effective Index Method and Its Application to Semiconductor Lasers,” IEEE J. Quantum Electron.18(7), 1083–1089 (1982). [CrossRef]
- 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]
- 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]
- 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]
- J. Ohtsubo, Semiconductor Lasers – Stability, Instability and Chaos, 2nd Edition (Springer, 2008).
- C. L. Xu and W. P. Huang, “Finite-Difference Beam Propagation Method for Guide-Wave Optics,” Progress In Electromagnetics Research, PIER11, 1–49 (1995).
- K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwell's Equations and the Schrödinger Equation (John Wiley & Sons, Inc.,2001).
- 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]
- 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]
- K. Q. Le, “Complex Padé approximant operators for wide-angle beam propagation,” Opt. Commun.282(7), 1252–1254 (2009). [CrossRef]
- W. Nakwaski, “Static thermal properties of broad-contact double- heterostructure laser diodes,” Opt. Quantum Electron.15(6), 513–527 (1983). [CrossRef]
- 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]
- 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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