## Spatio-spectral dynamics and spontaneous ultrafast optical switching in VCSEL arrays

Optics Express, Vol. 2, Issue 10, pp. 424-430 (1998)

http://dx.doi.org/10.1364/OE.2.000424

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### Abstract

Microscopic simulations on the basis of semiconductor Maxwell-Bloch equations show that in the short-time spatio-temporal dynamics of large aspect vertical cavity surface emitting lasers (VC-SEL) and coupled VCSEL-arrays microscopic and macroscopic effects are intrinsically coupled. The combination of microscopic spatial and spectral dynamics of the carrier distribution functions and the nonlinear polarization of the active semiconductor medium reveal spatio-spectral hole-burning effects as the origin of ultra-fast mode-switching effects. In coupled VCSEL-arrays the simulations predict the emergence of spontaneous ultra-fast spatial switching.

© Optical Society of America

^{2–52. O. Buccafusca, J. L. A. Chilla, J. J. Rocca, C. Wilmsen, S. Feld, and R. Leibenguth, “Ultrahigh frequency oscillations and multimode dynamics in vertical cavity surface emitting lasers,” Appl. Phys. Lett. 67, 185–187 (1995). [CrossRef] }, we numerically model the interrelations of the spatial and spectral distributions of the ultra-high frequency dynamics of VCSELs and phase-coupled VCSEL-arrays. To account for the microscopic processes which act in concert with the macroscopic spatio-temporal interactions we will base our investigation of large-aspect-ratio and coupled VCSELs on the semiconductor laser model derived in

^{66. O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description,” Phys. Rev. A 54, 3347–3359 (1996). [CrossRef] [PubMed] }and applied to the description of broad-area lasers

^{77. O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics,” Phys. Rev. A 54, 3360–3368 (1996). [CrossRef] [PubMed] }. In addition, the polarization of light emitted from VCSELs is generally highly sensitive to small anisotropies in the crystal structure, strain or optical anisotropies in the mirrors

^{8–128. C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. V. 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] }. In practice, however, the polarization of the emitted light is frequently stable under usual cw operating conditions

^{99. F. Koyama, K. Morito, and K. Iga, “Intensity noise and polarization stability of GaAlAs-GaAs surface emitting lasrs,” IEEE J. Quantum Electron. QE-27, 1410–1416 (1991). [CrossRef] }. Indeed, our analysis of the mutual influence of multiple anisotropies (gain-, loss- and frequency-anisotropy) and quantum fluctuations on the emission properties of VCSELs

^{1313. H. F. Hofmann and O. Hess, “Quantum Noise and Polarization Fluctuations in Vertical Cavity Surface Emitting Lasers,” Phys. Rev. A 56, 868–876 (1997). [CrossRef] }allows us here to resort to the assumption of a single stable polarization direction.

*f*

^{e,h}(

*k*, x,

*t*) of electrons (e) and holes (h) and the interband polarization

*p*

_{nl}(

*k*, x,

*t*)

*microscopic*spatial (x = (

*x*,

*y*)) and spectral (

*k*) dynamics of the Wigner distributions with the

*macroscopic*spatio-temporal dynamics

*V*is the volume, and the microscopic and macroscopic generation rates

^{1515. O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).}, where the relevant material properties and parameters employed in the simulation are detailed in Ref.

^{1414. O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis,” Prog. Quantum Electron. 20, 85–179 (1996). [CrossRef] }. Fig. 1 displays snapshots of the spatial distribution of the intensity and density of a typical large-aspect ratio VCSEL with a transverse width

*w*= 30

*μ*m. The time interval between successive snapshots is 3 picoseconds.

*N*(

*x*) determines the induced waveguiding properties. Additionally, the transport of charge carriers leads to a spatial redistribution of the carriers. Due to the difference in characteristic time scales

^{1414. O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis,” Prog. Quantum Electron. 20, 85–179 (1996). [CrossRef] }, however, it occurs on a much slower time scale than the spatial hole burning. As Fig. 1 shows, after only 3 picoseconds the intracavity intensity distribution has fundamentally changed in space by 5–10

*μ*m. After spatial integration, the resulting signal corresponds to the high-frequency spectra observed experimentally.

*δf*

^{e}(

*k*,

*x*,

*y*

_{0},

*t*) =

*f*

^{e}(

*k*,

*x*,

*y*

_{0},

*t*) -

*k*,

*x*,

*y*

_{0},

*t*) along a spatial cut at

*y*

_{0}= 0 during a characteristic time-interval of 100 ps.

*u*(

*k*,x,

*t*) and the interband polarization

*p*

_{nl}(

*k*,x,

*t*). The spectral “refill” of such a hole which occurs on ultra-short time scales below 100 femtoseconds is, in turn, governed by the scattering processes with their characteristic time scales

*τ*

_{e,h}. At the same time, transport of charge carriers leads to a spatial redistribution of the various spectral properties. Light emitted from a VCSEL with multiple transverse modes thus carries both, the characteristic spatial and spectral properties of the active semiconductor medium. As a consequence, the linewidth of a VCSEL displays spatial and temporal variations. Next to this spatio-spectral dynamics of the nonlinear gain inside the VCSEL, the optical properties of the VCSEL cavity - i.e. the waveguiding and nonlinear induced refractive index structure - vary dynamically and resemble in their spectral and spatial dependence the microscopic properties. As in the electrically pumped VCSEL electrons and holes are perpetually resupplied via carrier injection, the combined nonlinear dynamic variations of gain and effective refractive index structure result in ultrafast dynamical variations of the transverse optical modes in space and spectrum. Clearly, this interplay would not be possible is small VCSELs and not be included in theories which only consider either the macroscopic spatial dynamics or are based on the assumption of spatially homogeneous spectra.

*s*between the lasers, the array will be strongly or weakly coupled through the evanescent optical waves and charge carriers which may diffuse within and between the lasers.

*w*= 30

*μ*m to

*w*= 5

*μ*m leads to a suppression of higher transverse modes. In spite of identical pumping, however, the lasers show spontaneous ultra-fast spatial optical switching in the free-running condition displayed in animation 4. It is the combination of local spectral holeburning with the diffractive interaction of the evanescent optical fields together and the diffusive transport of charge carriers which leads to a dynamical interplay of gain, self-focusing effects, and dispersion. As a consequence, the VCSELs individually display ultrafast spatially homogeneous pulsations. The array configuration allows coherent intracavity field-coupling between adjacent lasers and thus leads to phase-coupled oscillations of the array, where adjacent lasers oscillate in anti-phase with respect to their neighbor.

## Acknowledgment

## References

1. | C. J. Chang-Hasnain, “Vertical cavity surface-emitting laser arrays,” in |

2. | O. Buccafusca, J. L. A. Chilla, J. J. Rocca, C. Wilmsen, S. Feld, and R. Leibenguth, “Ultrahigh frequency oscillations and multimode dynamics in vertical cavity surface emitting lasers,” Appl. Phys. Lett. |

3. | D. G. H. Nugent, R. G. S. Plumb, M. A. Fischer, and D. A. O. Davies, “Self-pulsations in vertical-cavity surface emitting lasers,” Electron. Lett. |

4. | J. E. Epler, S. Gehrsitz, K. H. Gulden, M. Moser, H. C. Sigg, and H. W. Lehmann, “Mode behavior and high resolution spectra of circularly-symmetric GaAs/AlGaAs air-post vertical cavity surface emitting lasers,” Appl. Phys. Lett. |

5. | I. Hörsch, R. Kusche, O. Marti, B. Weigl, and K. J. Ebeling, “Spectrally resolved mode imaging of vertical cavity semiconductor lasers by scanning near-field optical microscopy,” Appl. Phys. Lett. |

6. | O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description,” Phys. Rev. A |

7. | O. Hess and T. Kuhn, “Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics,” Phys. Rev. A |

8. | C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. V. Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, Polarization, and Transverse mode Characteristics of Vertical Cavity Surface Emitting Lasers,” IEEE J. Quantum Electron. |

9. | F. Koyama, K. Morito, and K. Iga, “Intensity noise and polarization stability of GaAlAs-GaAs surface emitting lasrs,” IEEE J. Quantum Electron. QE- |

10. | D. Vakhshoori, “Symmetry considerations in vertical-cavity surface-emitting lasers: Prediction of removal of polarization isotropy on (001) substrates,” Appl. Phys. Lett. |

11. | K. D. Choquette, J. P. Schneider, K. L. Lear, and R. E. Leibenguth, “Gain-dependent polarization properties of vertical-cavity lasers,” IEEE J. Sel. Top. Quantum Electron. |

12. | A. K. J. van Doorn, M. P. van Exter, and J. P. Woerdman, “Elasto-optic anisotropy and polarization orientation of vertical-cavity surface-emitting semiconductor lasers,” Appl. Phys. Lett. |

13. | H. F. Hofmann and O. Hess, “Quantum Noise and Polarization Fluctuations in Vertical Cavity Surface Emitting Lasers,” Phys. Rev. A |

14. | O. Hess and T. Kuhn, “Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis,” Prog. Quantum Electron. |

15. | O. Hess, |

**OCIS Codes**

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

(140.2010) Lasers and laser optics : Diode laser arrays

(140.5960) Lasers and laser optics : Semiconductor lasers

(250.0250) Optoelectronics : Optoelectronics

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 13, 1998

Revised Manuscript: December 31, 1997

Published: May 11, 1998

**Citation**

Ortwin Hess, "Spatio-spectral dynamics and
spontaneous ultrafast optical switching in VCSEL arrays," Opt. Express **2**, 424-430 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-10-424

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### References

- C. J. Chang-Hasnain, "Vertical cavity surface-emitting laser arrays," in Diode Laser Arrays, D. Botez and D. R. Scrifres, eds., (Cambridge University Press, Cambridge, 1994), pp. 368-413. [CrossRef]
- O. Buccafusca, J. L. A. Chilla, J. J. Rocca, C. Wilmsen, S. Feld, and R. Leibenguth, "Ultrahigh frequency oscillations and multimode dynamics in vertical cavity surface emitting lasers," Appl. Phys. Lett. 67, 185-187 (1995). [CrossRef]
- D. G. H. Nugent, R. G. S. Plumb, M. A. Fischer, and D. A. O. Davies, "Self-pulsations in vertical-cavity surface emitting lasers," Electron. Lett. 31, 43-44 (1995). [CrossRef]
- J. E. Epler, S. Gehrsitz, K. H. Gulden, M. Moser, H. C. Sigg, and H. W. Lehmann, "Mode behavior and high resolution spectra of circularly-symmetric GaAs/AlGaAs air-post vertical cavity surface emitting lasers," Appl. Phys. Lett. 69, 2312-2314 (1996). [CrossRef]
- I. Hoersch, R. Kusche, O. Marti, B. Weigl, and K. J. Ebeling, "Spectrally resolved mode imaging of vertical cavity semiconductor lasers by scanning near-eld optical microscopy," Appl. Phys. Lett. 79, 3831-3833 (1996).
- O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers I: Theoretical Description," Phys. Rev. A 54, 3347-3359 (1996). [CrossRef] [PubMed]
- O. Hess and T. Kuhn, "Maxwell-Bloch equations for spatially inhomogeneous semiconductor lasers II: Spatio-temporal dynamics," Phys. Rev. A 54, 3360-3368 (1996). [CrossRef] [PubMed]
- C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. C. V. 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]
- F. Koyama, K. Morito, and K. Iga, "Intensity noise and polarization stability of GaAlAs-GaAs surface emitting lasrs," IEEE J. Quantum Electron. QE-27, 1410-1416 (1991). [CrossRef]
- D. Vakhshoori, "Symmetry considerations in vertical-cavity surface-emitting lasers: Prediction of removal of polarization isotropy on (001) substrates," Appl. Phys. Lett. 65, 259-261 (1995). [CrossRef]
- K. D. Choquette, J. P. Schneider, K. L. Lear, and R. E. Leibenguth, "Gain-dependent polarization properties of vertical-cavity lasers," IEEE J. Sel. Top. Quantum Electron. 1, 661-666 (1995). [CrossRef]
- A. K. J. van Doorn, M. P. van Exter, and J. P. Woerdman, "Elasto-optic anisotropy and polarization orientation of vertical-cavity surface-emitting semiconductor lasers," Appl. Phys. Lett. 69, 1041-1043 (1996). [CrossRef]
- H. F. Hofmann and O. Hess, "Quantum Noise and Polarization Fluctuations in Vertical Cavity Surface Emitting Lasers," Phys. Rev. A 56, 868-876 (1997). [CrossRef]
- O. Hess and T. Kuhn, "Spatio-Temporal Dynamics of Semiconductor Lasers: Theory, Modeling and Analysis," Prog. Quantum Electron. 20, 85-179 (1996). [CrossRef]
- O. Hess, Spatio-Temporal Dynamics of Semiconductor Lasers (Wissenschaft und Technik Verlag, Berlin, 1993).

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