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

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References

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  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]
  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 NewWay to Accelerate Charged Particles," Phys. Rev. Lett. 95, 114801 (2005).
    [CrossRef] [PubMed]
  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. 77, 262-264 (2000).
    [CrossRef]
  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]
  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]
  11. A. Taflove and S. C. Hagness, "Computational Electrodynamics: the FDTD method" 2nd ed. (Artech House, Boston, London, 2000) 2006
  12. 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]
  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. 91355- 1360 (2003).
    [CrossRef]
  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.

2006 (1)

2005 (2)

2004 (1)

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]

2003 (2)

K. J. Vahala, "Optical microcavities," Nature 424, 939 (2003).
[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]

2002 (1)

M. Fujita and T. Baba, "Microgear Laser," Appl. Phys. Lett. 80,2051-2053 (2002).
[CrossRef]

2001 (1)

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]

2000 (1)

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]

1999 (1)

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]

1998 (1)

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]

1993 (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]

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]

Amans, D.

Baba, T.

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]

M. Fujita and T. Baba, "Microgear Laser," Appl. Phys. Lett. 80,2051-2053 (2002).
[CrossRef]

Benech, P.

Bimberg, D.

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]

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]

Borri, P.

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]

Borselli, M.

Eliseev, P. G.

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]

Fisher, T. A.

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]

Fujita, M.

M. Fujita and T. Baba, "Microgear Laser," Appl. Phys. Lett. 80,2051-2053 (2002).
[CrossRef]

Gehrig, E.

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]

Hess, O.

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]

Huy, K. P.

Krishna, S.

Langbein, W.

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]

Lester, L. F.

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]

Levi, A. F.

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]

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]

Li, H.

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]

Liu, G. T.

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]

Logan, R. A.

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]

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]

Malloy, K. J.

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]

McCall, S. L.

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]

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]

Mohideen, U.

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]

Morand, A.

Nakagawa, A.

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]

Newell, T. C.

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]

Nozaki, K.

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]

Ouyang, D.

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]

Painter, O.

Pearton, S. J.

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]

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]

Pelton, M.

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]

Ribbat, C.

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]

Sano, D.

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]

Schneider, S.

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]

Sellin, R. L.

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]

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]

Skolnick, M. S.

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]

Slusher, R. E.

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]

Srinivasan, K.

Stintz, A.

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]

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]

Vahala, K. J.

K. J. Vahala, "Optical microcavities," Nature 424, 939 (2003).
[CrossRef]

Whittaker, D. M.

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]

Woggon, U.

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]

Yamamoto, Y.

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]

Zakowicz, W.

W. Zakowicz, "Whispering-Gallery-Mode Resonances: A NewWay to Accelerate Charged Particles," Phys. Rev. Lett. 95, 114801 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (4)

M. Fujita and T. Baba, "Microgear Laser," Appl. Phys. Lett. 80,2051-2053 (2002).
[CrossRef]

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]

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]

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]

Electron. Lett. (1)

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]

IEEE J. Select. Top. Quantum Electron. (1)

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]

J. Opt. Soc. Am. B (1)

Nature (1)

K. J. Vahala, "Optical microcavities," Nature 424, 939 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (1)

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]

Phys. Rev. Lett. (2)

W. Zakowicz, "Whispering-Gallery-Mode Resonances: A NewWay to Accelerate Charged Particles," Phys. Rev. Lett. 95, 114801 (2005).
[CrossRef] [PubMed]

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]

Semicond. Sci. Technol. (1)

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]

Other (3)

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).

A. Taflove and S. C. Hagness, "Computational Electrodynamics: the FDTD method" 2nd ed. (Artech House, Boston, London, 2000) 2006

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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Figures (6)

Fig. 1.
Fig. 1.

Discrete, spatially resolved Fourier transforms of the Cartesian electric and magnetic field components of the cold-cavity HEM510 WGM. The contour lines show the logarithmic decay of the field.

Fig. 2.
Fig. 2.

Geometry of the cylindric resonator cavities used in the simulations. The 3D view on the whole dielectric cavity structure, including pedestal and the active region (yellow), is shown on the left. The right shows the corrugation parameters in the gear-like cavity.

Fig. 3.
Fig. 3.

The logarithmic resonance intensity (I) spectrum of the incompatible and compatible (insets from left to right) HEM510 modes in the microgear. The Lorentzian line shape of the dipole resonance is included (dashed curve).

Fig. 4.
Fig. 4.

Snapshot of the spatial variation in the inversion probability of the dipoles in the microdisc cavity and the evolution of the averaged inversion. The end of the red line marks the time of the snapshot. Animations can be found at [16].

Fig. 5.
Fig. 5.

Snapshot of the spatial variation in the inversion probability of the dipoles in the microgear cavity and the evolution of the averaged inversion. The end of the red line marks the time of the snapshot. Animations can be found at [16].

Fig. 6.
Fig. 6.

Ultrafast dynamic behaviour of the HEM510 lasing mode in a disc (a) and gear (b). Shown is the azimuthal part of the electric field in the disc plane (x-y). The contour lines are even spaced on a logarithmic scale to enhance the visibility of the symmetric properties of the field pattern. The corresponding animations can be found at [16].

Tables (1)

Tables Icon

Table 1. The table summarises the two-level material parameters of the laser simulations; density (n a); dipole moment ( d ); transition frequency (ω 0); polarisation damping constant (γ p); non-radiative decay constant (γ nr); thermal equilibrium (ρ bb,0) and initial occupation probability of the upper level (ρ bb(t = 0)); pump strength (Λ).

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

curlE= t H, curlH= t D.
D= ε r E+2 n a dP
t 2 P=2 γ p t P+ ω 0 2 P= ω 0 2 Ω N|d·E|
t N=2Λ γ nr (N N 0 )4 Ω ω 0 2 |d·E| t P

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