## Transverse mode dynamics of VCSELs through space-time domain simulation

Optics Express, Vol. 5, Issue 3, pp. 55-62 (1999)

http://dx.doi.org/10.1364/OE.5.000055

Acrobat PDF (194 KB)

### Abstract

An approximation to the Maxwell-Semiconductor Bloch equations is used to model transverse mode dynamics of vertical-cavity surface-emitting lasers (VCSELs). The time-evolution of the spatial profiles of the laser field and carrier density is solved by a finite-difference algorithm. The algorithm is fairly general; it can handle devices of any shape, which are either gain or index guided or both. Also there is no a priori assumption about the type or number of modes. The physical modeling includes the nonlinear carrier dependence of the optical gain and refractive index and dispersion effects on the gain and the refractive index are also included.

© Optical Society of America

## 1. Introduction

1. W. W. Chow, S. W. Koch, and M. Sargent, *Semiconductor Laser Physics*, (Springer, Heidelberg, Berlin, 1994). [CrossRef]

2. J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. , **9**, 737 (1997). [CrossRef]

1. W. W. Chow, S. W. Koch, and M. Sargent, *Semiconductor Laser Physics*, (Springer, Heidelberg, Berlin, 1994). [CrossRef]

6. C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective Bloch-equations for semiconductor lasers and ampliers,” IEEE J. Quantum Electron. **33**, 1543 (1997). [CrossRef]

7. T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, “Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A **58**, 3279 (1998). [CrossRef]

8. C. Z. Ning, J. V. Moloney, and R. A. Indik, “A first-principles fully space-time resolved model of a semiconductor laser,” Quantum Semiclass. Opt. , **9**, 681(1997). [CrossRef]

9. A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, “Dynamic Instabilities in MFA-MOPA Semiconductor Lasers,” IEEE J. Quantum Electron. **34**, 166, (1998). [CrossRef]

11. C. Z. Ning, R. A. Indik, and J. V. Moloney, “A self-consistent approach to thermal e ects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B **12**, 1993–2004, 1995. [CrossRef]

15. P. M. Goorjian and C. Z. Ning, “Transverse Mode Dynamics of VCSELs through Space-Time Simulation,” http://science.nas.nasa.gov/egoorjian/Pub/pub.html

## 2. Governing Equations

6. C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective Bloch-equations for semiconductor lasers and ampliers,” IEEE J. Quantum Electron. **33**, 1543 (1997). [CrossRef]

7. T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, “Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A **58**, 3279 (1998). [CrossRef]

11. C. Z. Ning, R. A. Indik, and J. V. Moloney, “A self-consistent approach to thermal e ects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B **12**, 1993–2004, 1995. [CrossRef]

*E*is the complex laser field envelope amplitude,

*N*is the total carrier density

*P*

_{0}and

*P*

_{1}are the effective material polarization functions that have been constructed from microscopic theory [6

6. C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective Bloch-equations for semiconductor lasers and ampliers,” IEEE J. Quantum Electron. **33**, 1543 (1997). [CrossRef]

*δn*(

*x, y*) is the guiding index profile,

*∊*

_{0}is the permittivity of free space, i is the complex number √-1,

*ω*

_{c}is the optical carrier wave frequency in radians per seconds,

*n*

_{b}is the background index of refraction,

*n*

_{g}is the group index of refraction,

*∊*

_{b}=

*ω*

_{c}

*n*

_{b}

*/c*is the optical wavenumber in the cavity with a background index of refraction

*n*

_{b}

*, k*is the cavity loss,

*D*

_{N}is the carrier diffusion coefficient, γ

_{n}is the nonradiative decay constant or carrier loss rate due to spontaneous and nonradiative processes,

*η*is the quantum efficiency, e is the electron charge,

*ħ*=

*h*/2

*π*, where h is Plank’s constant, Γ is the confinement factor, and

*L*is the cavity length.

_{X0}(

*N*), the effective background susceptibility, with real and imaginary parts

_{X0,r}(

*N*) and

_{X0,i}(

*N*) respectively; Γ

_{1}(

*N*), the gain bandwidth,

*ω*

_{1}(

*N*), the detuning, and

*A*

_{1}(

*N*), the strength of the Lorentzian oscillator. The theoretical basis for the Maxwell-Effective Bloch Equations and their derivation from the semiconductor Bloch equations and Maxwell’s equations is given in Reference 6. Also, the derivation of the five density dependent coefficients, which model the optical sus-ceptibility,

_{X}(

*ω,N*), is given in Reference 6.

### 2.1 Optical Susceptibility

1. W. W. Chow, S. W. Koch, and M. Sargent, *Semiconductor Laser Physics*, (Springer, Heidelberg, Berlin, 1994). [CrossRef]

*t*) and

*t*), the occupation probabilities for electron and holes respectively, and

*p*

_{k}(

*t*), the interband dipole expectation function with momentum

*ħk*. Next,

*P*(

*t*), the induced polarization is computed [1

*Semiconductor Laser Physics*, (Springer, Heidelberg, Berlin, 1994). [CrossRef]

*p*

_{k}(

*t*). Then the optical susceptibility

_{X}(

*ω;N*) is determined from the Fourier transforms of the electric field,

*E*(

*ω*), and the induced polarization,

*P*(

*ω*) by the equation

*δn*(

*ω;N*), and the optical gain,

*G*(

*ω;N*), are determined by the equation,

_{X}(

*ω,N*) is approximated by a background susceptibility

_{X0}(

*N*) and one Lorentzian oscillator

_{X1}(

*ω;N*).

## 3. Computed Results

_{1}(

*N*), the gain bandwidth, 1/Γ

_{1}(

*N*) was approximately 13.8 femtoseconds initially in the current aperture area and the light dynamics changed on that time scale and γ

_{n}, the nonradiative decay constant, 1/γ

_{n}was 1 nanosecond, and at that time scale the laser field was approximately steady. The light wave length was 0.98 microns, the circular current apertures was 10.0 microns in diameter, the cavity length

*L*was 144 nanometers and the confinement factor was one fourth. The value used for index guiding was -0.05, which was the decrease in the refractive index outside of the active region.

15. P. M. Goorjian and C. Z. Ning, “Transverse Mode Dynamics of VCSELs through Space-Time Simulation,” http://science.nas.nasa.gov/egoorjian/Pub/pub.html

### 3.1 Case 1. Disk without index guiding

### 3.2 Case 2. Disk with index guiding and low pumping current

### 3.3 Case 3. Ring with index guiding and low pumping current

### 3.4 Case 4. Disk with index guiding and high pumping current

### 3.5 Case 5. Ring with index guiding and high pumping current

## 4. Conclusion

## References and links

1. | W. W. Chow, S. W. Koch, and M. Sargent, |

2. | J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, “Effects of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers,” Quantum Semiclass. Opt. , |

3. | P. M. Goorjian and G. P. Agrawal, “Computational Modeling of Ultrashort Optical Pulse Propagation in Nonlinear Optical Materials,” Paper NME31, |

4. | P. M. Goorjian and G. P. Agrawal, “Computational Modeling of Ultrafast Optical Pulse Propagation in Semiconductor Materials,” Paper QThE9, Quantum Optoelectronics, Spring Topical Meeting, OSA, Washington, D. C, Nevada, March 17–21, 1997. |

5. | P. M. Goorjian and G. P. Agrawal, “Maxwell-Bloch Equations Modeling of Ultrashort Optical Pulse Propagation in Semiconductor Materials,” Paper WB2, OSA 1997 Annual Meeting, Washington, D. C, October 12–17, 1997. |

6. | C. Z. Ning, R. A. Indik, and J. V. Moloney, “Effective Bloch-equations for semiconductor lasers and ampliers,” IEEE J. Quantum Electron. |

7. | T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, “Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A |

8. | C. Z. Ning, J. V. Moloney, and R. A. Indik, “A first-principles fully space-time resolved model of a semiconductor laser,” Quantum Semiclass. Opt. , |

9. | A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, “Dynamic Instabilities in MFA-MOPA Semiconductor Lasers,” IEEE J. Quantum Electron. |

10. | C. Z. Ning, S. Bischoff, S. W. Koch, G. K. Harkness, J. V. Moloney, and W. W. Chow “Micro-scopic Modeling of VCSELs: Many-body interaction, plasma heating, and transverse dynamics,” Optical Engineering, April, 1998. |

11. | C. Z. Ning, R. A. Indik, and J. V. Moloney, “A self-consistent approach to thermal e ects in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B |

12. | P. M. Goorjian and C. Z. Ning, “Computational Modeling of Vertical-Cavity Surface-Emitting Lasers,” Paper Thc15, Nonlinear Optics Topical Meeting, Kauai, HI, August 9–14, 1998. |

13. | P. M. Goorjian and C. Z. Ning, “Simulation of Transverse Modes in Vertical-Cavity Surface-Emitting Lasers,” 1998 Annual Meeting of the Optical Society of America, Washington, D. C, October 5–9, 1998. |

14. | P. M. Goorjian and C. Z. Ning, “Transverse Mode Dynamics of VCSELs through Space-Time Simulation,” Paper 3625–45, Integrated Optoelectronic Devices, Photonics West, 1999, (SPIE), San Jose, CA, January 23–29, 1999. |

15. | P. M. Goorjian and C. Z. Ning, “Transverse Mode Dynamics of VCSELs through Space-Time Simulation,” http://science.nas.nasa.gov/egoorjian/Pub/pub.html |

16. | C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. |

17. | Y. Satuby and M. Orenstein, “Small-Signal Modulation of MultitransverseModes Vertical-Cavity Surface-Emitting Lasers,” IEEE Photonics Tech. Letters , |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Focus Issue: Spatial and Polarization Dynamics of Semiconductor Lasers

**History**

Original Manuscript: April 1, 1999

Published: August 2, 1999

**Citation**

Peter Goorjian and C.Z. Ning, "Transverse mode dynamics of VCSELs through space-time domain simulation," Opt. Express **5**, 55-62 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-3-55

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### References

- W. W. Chow, S. W. Koch and M. Sargent, Semiconductor Laser Physics, (Springer, Heidelberg, Berlin, 1994). [CrossRef]
- J. Y. Law, G. H. M. van Tartwijk and G. P. Agrawal, " Effects of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclass. Opt., 9, 737 (1997). [CrossRef]
- P. M. Goorjian and G. P. Agrawal, "Computational Modeling of Ultrashort Optical Pulse Propagation in Nonlinear Optical Materials," Paper NME31, Nonlinear Optics: Materials, Fundamentals and Applications, 11, 1996 OSA Technical Digest Series, Washington, D.C., 1996, 132-133.
- P. M. Goorjian and G. P. Agrawal, "Computational Modeling of Ultrafast Optical Pulse Propagation in Semiconductor Materials," Paper QThE9, Quantum Optoelectronics, Spring Topical Meeting, OSA, Washington, D. C, Nevada, March 17-21, 1997.
- P. M. Goorjian and G. P. Agrawal, "Maxwell-Bloch Equations Modeling of Ultrashort Optical Pulse Propagation in Semiconductor Materials," Paper WB2, OSA 1997 Annual Meeting, Washington, D. C, October 12-17, 1997.
- C. Z. Ning, R. A. Indik and J. V. Moloney, "Effective Bloch-equations for semiconductor lasers and amplifiers," IEEE J. Quantum Electron. 33, 1543 (1997). [CrossRef]
- T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney and C. Z. Ning, "Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface- emitting lasers," Phys. Rev. A 58, 3279 (1998). [CrossRef]
- C. Z. Ning, J. V. Moloney and R. A. Indik, "A first-principles fully space-time resolved model of a semiconductor laser," Quantum Semiclass. Opt., 9, 681(1997). [CrossRef]
- A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert and J. G. McInerney, "Dynamic Instabilities in MFA-MOPA Semiconductor Lasers," IEEE J. Quantum Electron. 34, 166, (1998). [CrossRef]
- C. Z. Ning, S. Bischoff, S. W. Koch, G. K. Harkness, J. V. Moloney and W. W. Chow "Microscopic Modeling of VCSELs: Many-body interaction, plasma heating, and transverse dynamics," Optical Engineering, April, 1998.
- C. Z. Ning, R. A. Indik and J. V. Moloney, "A self-consistent approach to thermal effects in vertical-cavity surface-emitting lasers," J. Opt. Soc. Am. B 12, 1993-2004, 1995. [CrossRef]
- P. M. Goorjian and C. Z. Ning, "Computational Modeling of Vertical-Cavity Surface-Emitting Lasers," Paper Thc15, Nonlinear Optics Topical Meeting, Kauai, HI, August 9-14, 1998.
- P. M. Goorjian and C. Z. Ning, "Simulation of Transverse Modes in Vertical-Cavity Surface- Emitting Lasers," 1998 Annual Meeting of the Optical Society of America, Washington, D. C, October 5-9, 1998.
- P. M. Goorjian and C. Z. Ning, "Transverse Mode Dynamics of VCSELs through Space-Time Simulation," Paper 3625-45, Integrated Optoelectronic Devices, Photonics West, 1999, (SPIE), San Jose, CA, January 23-29, 1999.
- P. M. Goorjian and C. Z. Ning, "Transverse Mode Dynamics of VCSELs through Space-Time Simulation," http://science.nas.nasa.gov/~goorjian/Pub/pub.html
- C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. Von Lehmen, L. T. Florez and N. G. Stoffel, "Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers," IEEE J. Quantum Electron. 27, 1402-1409, (1991). [CrossRef]
- Y. Satuby and M. Orenstein, "Small-Signal Modulation of Multitransverse Modes Vertical-Cavity Surface-Emitting Lasers," IEEE Photonics Tech. Letters, 10, 757-759, (1998). [CrossRef]

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