## Ultrafast nonlinear dynamics of whispering-gallery mode micro-cavity lasers

Optics Express, Vol. 14, Issue 7, pp. 2744-2752 (2006)

http://dx.doi.org/10.1364/OE.14.002744

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### Abstract

We explore the ultrafast spatio-temporal dynamics of whispering-gallery micro-cavity lasers. To model the dynamics of the nonlinear whispering-gallery modes we develop a three-dimensional Finite-Difference Time-Domain modelling framework based on the spin and therefore optical polarisation resolved Maxwell-Bloch equations. The numerical algorithm brings together a real value form of the optical Bloch equations with the curl part of Maxwell’s equations. The Hamiltonian of the two-level system contains either linear or circular polarised transitions. In cylindrical micro-cavity lasers the coherent, nonlinear emission process leads to ultrafast fan-like rotational phase dynamics of the degenerate whispering-gallery modes. This rotation is shown to be arrested in gear-shaped micro-cavity lasers followed by an over-damped relaxation oscillation.

© 2006 Optical Society of America

## 1. Introduction

1. K. J. Vahala, “Optical microcavities,” Nature **424**, 939 (2003). [CrossRef]

2. A. F. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometer radius disk laser,” Electron. Lett. **29**, 1666–1667 (1993). [CrossRef]

3. M. Fujita and T. Baba, “Microgear Laser,” Appl. Phys. Lett. **80**, 2051–2053 (2002). [CrossRef]

*Q*) [4

4. M. S. Skolnick, T. A. Fisher, and D. M. Whittaker “Strong coupling phenomena in quantum microcavity structures,” Semicond. Sci. Technol. **13**, 645–669 (1998). [CrossRef]

5. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity *Q*, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express **14**, 1094–1105 (2006). [CrossRef] [PubMed]

6. W. Zakowicz, “Whispering-Gallery-Mode Resonances: A New Way to Accelerate Charged Particles,” Phys. Rev. Lett. **95**, 114801 (2005). [CrossRef] [PubMed]

## 2. Time-domain full vector Maxwell-Bloch modelling framework

*P*by the material relation

*and the dipole density*

**d***n*

_{a}. The dipole moment was set to a reasonable value which should model a quantum dot like material [8

8. P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy, and L. F. Lester, “Transition dipole moment of InAs/InGaAs quantum dots from experiments on ultralow-threshold laser diodes,” Appl. Phys. Lett. **77**, 262–264 (2000). [CrossRef]

**type transition with a complex dipole moment**

*σ***∈ ℂ**

*d*^{3}. In order not to disregard this vital polarisation information and to investigate the associated dynamics we therefore generally have to assume complex-valued electromagnetic fields

*,*

**E***,*

**D***∈ ℂ*

**H**^{3}. The microscopic polarisation

*P*and the population difference

*N*on the other hand are real values (

*P*,

*N*∈ ℝ). We derive the equations of motion of

*P*and

*N*on the basis of a dipole Hamiltonian of a quantum dot like two-level system with an energy separation of Δ

*E*=

**ħω**

_{0}and the occupation probability of the upper level (

*ρ*

_{bb}, with

*N*= 1-2

*ρ*

_{bb}; for details, see Ref. [7]). In addition, we include a phenomenological polarisation damping term (decay constant

*γ*

_{p}) to model a Lorentzian line shape resonance [9

9. P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong Dephasing Time in InGaAs Quantum Dots,” Phys. Rev. Lett. **87**, 157401 (2001). [CrossRef] [PubMed]

^{2}=

*P*and population difference

*N*then read:

*N*

_{0}with a summarising decay constant

*γ*

_{nr}. Because of the interest in the ultrashort, transitional character of the system dynamics and our focus on a quantum dot like active material, carrier diffusion can safely be disregarded [10

10. E. Gehrig, O. Hess, C. Ribbat, R. L. Sellin, and D. Bimberg, “Dynamic filamentation and beam quality of quantum-dot lasers,” Appl. Phys. Lett. **84**, 1650 (2004). [CrossRef]

*ρ*

_{bb, trans}= 0.5).

## 3. Cold-cavity modes and quality factors

^{3}. The time step was chosen according to the Courant criterion. As initial condition, each computational run is seeded with a divergence-free electromagnetic pulse of small amplitude modelling a spontaneously emitted contribution of polarisation.

12. K. P. Huy, A. Morand, D. Amans, and P. Benech, “Analytical study of the whispering-gallery mode in two-dimensional microgear cavity using coupled-mode theory,” J. Opt. Soc. Am. B **22**, 1793–1803 (2005). [CrossRef]

2. A. F. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometer radius disk laser,” Electron. Lett. **29**, 1666–1667 (1993). [CrossRef]

*μ*m and its thickness is 180nm. Embedded in the middle of the disc is a thin layer (60nm thick) of gain material. The Lorentzian resonance is tuned to match the resonance frequency of a cavity eigen-mode (in this microdisc ≈ 1.550nm).

*Q*was estimated (from the FDTD simulation) to be ≈ 800. The neighbouring high

*Q*resonances are at 1.4 and 1.7nm. With the introduction of the gear-like corrugation around the circumference of the disc (see Fig. 2), this resonance is split up into two, as the continuous azimuthal degeneracy of the mode is removed (see also Ref. [3

3. M. Fujita and T. Baba, “Microgear Laser,” Appl. Phys. Lett. **80**, 2051–2053 (2002). [CrossRef]

*Q*resonance (the compatible mode) has a

*Q*≈ 770, almost the same as the degenerate mode in the disc without corrugation. The lower

*Q*resonance features a much lower

*Q*≈ 130 and therefore called the incompatible mode. The higher

*Q*mode (the compatible HEM510) was chosen to coincide with the resonance frequency of the Lorentzian transition (see Fig. 3) of the active medium. Table 1 summarises the material parameters which were used in the microdisc and microgear laser simulations, now solving the full, nonlinear set of Maxwell-Bloch equations including 3 and 4.

## 4. Spatio-temporal dynamics of whispering-gallery mode lasers

13. R. E. Slusher, A. F. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. **63**, 1310–1312 (1993). [CrossRef]

14. M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev. A **59**, 2418–2421 (1999). [CrossRef]

15. K. Nozaki, A. Nakagawa, D. Sano, and T. Baba, “Ultralow Threshold and Single-Mode Lasing in Microgear Lasers and Its Fusion With Quasi-Periodic Photoic Crystals,” IEEE J. Select. Top. Quantum Electron. **9**1355–1360 (2003). [CrossRef]

_{510}WG like nonlinear material cavity “mode”, that allows such regular accumulation and removal of inversion.

_{510}mode is excited in both cavities. It initially displays straight radially extending node lines with the mode itself oscillating in place. Soon after the relaxation oscillation swing, however, the ultrafast dynamics of the electric field in the microdisc laser shows a rotating HEM

_{510}like optical field pattern that resembles the characteristics of a vortex structure. The direction of the rotation depends on the initial conditions of the simulation. Figure 6(a) shows a sequence of six snap-shots that reveal the fan-like rotation of the optical field with the symmetry properties of the cold cavity WGM with the corresponding eigen-frequency. We link this rotation to spatial hole burning effects in the inversion density and an interplay of the different continuously degenerate HEM510 modes. These cause the electric field anti-nodes to start moving to regions with higher inversion that formerly had been occupied by the nodes of the mode. This ultra-fast rotation of the electric field is possible in a cylindrical cavity since all cold cavity modes have an azimuthal degeneracy by an arbitrary angle. It is this rotating electric field which leads to the smeared-out inversion profile as seen in Fig. 4.

_{510}mode is significantly more well-behaved. It is the higher loss of the incompatible cold-cavity mode in the gear geometry that prevents the nodes of the electric field to shift to areas of higher inversion (i.e. higher gain). Most notably, no vortex-like electric field structures such as in the microdisc laser were observed. Indeed, the time series of the axial electric field component in Fig. 6(b) shows the arrested HEM

_{510}mode fixed in place. In the case of the microdisc, the rotating field pattern smears out the inversion profile near the circumference of the disc. Due to the nonlin-earity, the mode profile in transitional phases does not necessarily represent an imprinted image in the inversion. In the microgear laser, the introduction of a corrugation around the discs circumference removes the azimuthal degeneracy and arrests the mode in place. Here, the mode profile in the transitional phase matches the spatial hole burning pattern in the inversion. This shows that the WGM is truly arrested in place.

## 5. Conclusion

## Acknowledgements

## References and links

1. | K. J. Vahala, “Optical microcavities,” Nature |

2. | A. F. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Room temperature operation of submicrometer radius disk laser,” Electron. Lett. |

3. | M. Fujita and T. Baba, “Microgear Laser,” Appl. Phys. Lett. |

4. | M. S. Skolnick, T. A. Fisher, and D. M. Whittaker “Strong coupling phenomena in quantum microcavity structures,” Semicond. Sci. Technol. |

5. | K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity |

6. | W. Zakowicz, “Whispering-Gallery-Mode Resonances: A New Way to Accelerate Charged Particles,” Phys. Rev. Lett. |

7. | A. Klaedtke, J. Hamm, and O. Hess, “Simulation of Active and Nonlinear Photonic Nano-Materials in the Finite-Difference Time-Domain (FDTD) Framework,” Lecture Notes in Physics 642, Computational Material Science – From Basic Principles to Material Properties, 75–101, Springer (2004). |

8. | P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy, and L. F. Lester, “Transition dipole moment of InAs/InGaAs quantum dots from experiments on ultralow-threshold laser diodes,” Appl. Phys. Lett. |

9. | P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong Dephasing Time in InGaAs Quantum Dots,” Phys. Rev. Lett. |

10. | E. Gehrig, O. Hess, C. Ribbat, R. L. Sellin, and D. Bimberg, “Dynamic filamentation and beam quality of quantum-dot lasers,” Appl. Phys. Lett. |

11. | A. Taflove and S. C. Hagness, “Computational Electrodynamics: the FDTD method” 2nd ed. (Artech House, Boston, London, 2000) |

12. | K. P. Huy, A. Morand, D. Amans, and P. Benech, “Analytical study of the whispering-gallery mode in two-dimensional microgear cavity using coupled-mode theory,” J. Opt. Soc. Am. B |

13. | R. E. Slusher, A. F. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. |

14. | M. Pelton and Y. Yamamoto, “Ultralow threshold laser using a single quantum dot and a microsphere cavity,” Phys. Rev. A |

15. | K. Nozaki, A. Nakagawa, D. Sano, and T. Baba, “Ultralow Threshold and Single-Mode Lasing in Microgear Lasers and Its Fusion With Quasi-Periodic Photoic Crystals,” IEEE J. Select. Top. Quantum Electron. |

16. | A. Klaedtke research white paper, “Nanolasers,” (University of Surrey, Advanced Technology Institute, Theory and Advanced Computation, 2006), http://www.ati.surrey.ac.uk/TAC/research/nanolasers. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(030.4070) Coherence and statistical optics : Modes

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

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 10, 2006

Revised Manuscript: March 27, 2006

Manuscript Accepted: March 27, 2006

Published: April 3, 2006

**Citation**

A. Klaedtke and O. Hess, "Ultrafast nonlinear dynamics of whispering-gallery mode micro-cavity lasers," Opt. Express **14**, 2744-2752 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2744

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### References

- K. J. Vahala, "Optical microcavities," Nature 424, 939 (2003). [CrossRef]
- A. F. Levi, S. L. McCall, S. J. Pearton and R. A. Logan, "Room temperature operation of submicrometer radius disk laser," Electron. Lett. 29,1666-1667 (1993). [CrossRef]
- M. Fujita and T. Baba, "Microgear Laser," Appl. Phys. Lett. 80,2051-2053 (2002). [CrossRef]
- M. S. Skolnick, T. A. Fisher and D. M. Whittaker "Strong coupling phenomena in quantum microcavity structures," Semicond. Sci. Technol. 13,645-669 (1998). [CrossRef]
- K. Srinivasan, M. Borselli, O. Painter, A. Stintz and S. Krishna, "Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots," Opt. Express 14,1094-1105 (2006). [CrossRef] [PubMed]
- W. Zakowicz, "Whispering-Gallery-Mode Resonances: A NewWay to Accelerate Charged Particles," Phys. Rev. Lett. 95, 114801 (2005). [CrossRef] [PubMed]
- A. Klaedtke, J. Hamm and O. Hess, "Simulation of Active and Nonlinear Photonic Nano-Materials in the Finite- Difference Time-Domain (FDTD) Framework," Lecture Notes in Physics 642, Computational Material Science - From Basic Principles to Material Properties, 75-101, Springer (2004).
- P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy and L. F. Lester, "Transition dipole moment of InAs/InGaAs quantum dots from experiments on ultralow-threshold laser diodes," Appl. Phys. Lett. 77, 262-264 (2000). [CrossRef]
- P. Borri,W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang and D. Bimberg, "Ultralong Dephasing Time in InGaAs Quantum Dots," Phys. Rev. Lett. 87, 157401 (2001). [CrossRef] [PubMed]
- E. Gehrig, O. Hess, C. Ribbat, R. L. Sellin and D. Bimberg, "Dynamic filamentation and beam quality of quantum-dot lasers," Appl. Phys. Lett. 84, 1650 (2004). [CrossRef]
- A. Taflove and S. C. Hagness, "Computational Electrodynamics: the FDTD method" 2nd ed. (Artech House, Boston, London, 2000) 2006
- K. P. Huy, A. Morand, D. Amans and P. Benech, "Analytical study of the whispering-gallery mode in twodimensional microgear cavity using coupled-mode theory," J. Opt. Soc. Am. B 22,1793-1803 (2005). [CrossRef]
- R. E. Slusher, A. F. Levi, U. Mohideen, S. L. McCall, S. J. Pearton and R. A. Logan, "Threshold characteristics of semiconductor microdisk lasers," Appl. Phys. Lett. 63, 1310-1312 (1993). [CrossRef]
- M. Pelton and Y. Yamamoto, "Ultralow threshold laser using a single quantum dot and a microsphere cavity," Phys. Rev. A 59, 2418-2421 (1999). [CrossRef]
- K. Nozaki, A. Nakagawa, D. Sano and T. Baba, "Ultralow Threshold and Single-Mode Lasing in Microgear Lasers and Its Fusion With Quasi-Periodic Photoic Crystals," IEEE J. Select. Top. Quantum Electron. 91355- 1360 (2003). [CrossRef]
- A. Klaedtke research white paper, "Nanolasers," (University of Surrey, Advanced Technology Institute, Theory and Advanced Computation, 2006), http://www.ati.surrey.ac.uk/TAC/research/nanolasers.

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