Abstract

The choice of the etching depth for semiconductor microcavities is a compromise between a high Q factor and a difficult technique in a practical fabricating process. In this paper, the influences of the etching depth on mode Q factors for mid-infrared quantum cascade microcylinder and microsquare lasers around 4.8 and 7.8μm are simulated by three-dimensional (3D) finite-difference time-domain (FDTD) techniques. For the microcylinder and the microsquare resonators, the mode Q factors of the whispering-gallery modes (WGMs) increase exponentially and linearly with the increase in the etching depth, respectively. Furthermore, the mode Q factors of some higher order transverse WGMs may be larger than that of the fundamental transverse WGM in 3D microsquares. Based on the field distribution of the vertical multilayer slab waveguide and the mode Q factors versus the etching depth, the necessary etching depth is chosen at the position where the field amplitude is 1% of the peak value of the slab waveguide. In addition, the influences of sidewall roughness on the mode Q factors are simulated for microsquare resonators by 2D FDTD simulation.

© 2009 Optical Society of America

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  1. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
    [CrossRef] [PubMed]
  2. J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
    [CrossRef]
  3. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
    [CrossRef] [PubMed]
  4. S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
    [CrossRef]
  5. S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
    [CrossRef]
  6. R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
    [CrossRef]
  7. J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
    [CrossRef]
  8. G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
    [CrossRef]
  9. K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
    [CrossRef]
  10. V. Tamosiunas, Z. Kancleris, and M. Tamosiuniene, “Simulation of modes in terahertz quantum cascade microring and microdisk lasers,” Acta Phys. Pol. A 113, 917-920 (2008).
  11. Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
    [CrossRef]
  12. Y. Z. Huang and Y. D. Yang, “Mode coupling and vertical radiation loss for whispering-gallery modes in 3-D microcavities,” J. Lightwave Technol. 26, 1411-1416 (2008).
    [CrossRef]
  13. B. J. Li and P. L. Liu, “Numerical analysis of the whispering gallery modes by the finite-difference time-domain method,” IEEE J. Quantum Electron. 32, 1583-1587 (1996).
    [CrossRef]
  14. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  15. W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
    [CrossRef]
  16. Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
    [CrossRef]
  17. W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
    [CrossRef]
  18. M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
    [CrossRef]
  19. Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison of Q-factors between TE and TM modes in 3-D microsquares by FDTD simulation,” IEEE Photon. Technol. Lett. 19, 1981-1983 (2007).
    [CrossRef]
  20. L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
    [CrossRef]
  21. B. J. Li and P. L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 33, 791-795 (1997).
    [CrossRef]
  22. Q. Chen, Y. Z. Huang, and L. J. Yu, “Analysis of mode characteristics for deformed square resonators by FDTD technique,” IEEE J. Quantum Electron. 42, 59-63 (2006).
    [CrossRef]
  23. F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
    [CrossRef]

2008

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

V. Tamosiunas, Z. Kancleris, and M. Tamosiuniene, “Simulation of modes in terahertz quantum cascade microring and microdisk lasers,” Acta Phys. Pol. A 113, 917-920 (2008).

F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
[CrossRef]

Y. Z. Huang and Y. D. Yang, “Mode coupling and vertical radiation loss for whispering-gallery modes in 3-D microcavities,” J. Lightwave Technol. 26, 1411-1416 (2008).
[CrossRef]

2007

Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
[CrossRef]

Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison of Q-factors between TE and TM modes in 3-D microsquares by FDTD simulation,” IEEE Photon. Technol. Lett. 19, 1981-1983 (2007).
[CrossRef]

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

2006

Q. Chen, Y. Z. Huang, and L. J. Yu, “Analysis of mode characteristics for deformed square resonators by FDTD technique,” IEEE J. Quantum Electron. 42, 59-63 (2006).
[CrossRef]

2005

Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
[CrossRef]

2004

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

2003

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

2002

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

2001

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

2000

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

1999

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

1998

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

1997

B. J. Li and P. L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 33, 791-795 (1997).
[CrossRef]

1996

B. J. Li and P. L. Liu, “Numerical analysis of the whispering gallery modes by the finite-difference time-domain method,” IEEE J. Quantum Electron. 32, 1583-1587 (1996).
[CrossRef]

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

1994

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Andrews, A. M.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

Audet, R.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Belkin, M. A.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Benz, A.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

Bour, D.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Capasso, F.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Chen, Q.

Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
[CrossRef]

Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison of Q-factors between TE and TM modes in 3-D microsquares by FDTD simulation,” IEEE Photon. Technol. Lett. 19, 1981-1983 (2007).
[CrossRef]

Q. Chen, Y. Z. Huang, and L. J. Yu, “Analysis of mode characteristics for deformed square resonators by FDTD technique,” IEEE J. Quantum Electron. 42, 59-63 (2006).
[CrossRef]

Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
[CrossRef]

Cho, A. Y.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Colombelli, R.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Corzine, S.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Faist, J.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Fan, J. A.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Fasching, G.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

Forchel, A.

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

Gianordoli, S.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

Gmachl, C.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

Gmachl, C. F.

F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
[CrossRef]

Gornik, E.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

Guo, W. H.

Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
[CrossRef]

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hentschel, M.

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Höflerf, G.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Höfling, S.

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

Hofmann, H.

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

Huang, Y. Z.

Y. Z. Huang and Y. D. Yang, “Mode coupling and vertical radiation loss for whispering-gallery modes in 3-D microcavities,” J. Lightwave Technol. 26, 1411-1416 (2008).
[CrossRef]

Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison of Q-factors between TE and TM modes in 3-D microsquares by FDTD simulation,” IEEE Photon. Technol. Lett. 19, 1981-1983 (2007).
[CrossRef]

Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
[CrossRef]

Q. Chen, Y. Z. Huang, and L. J. Yu, “Analysis of mode characteristics for deformed square resonators by FDTD technique,” IEEE J. Quantum Electron. 42, 59-63 (2006).
[CrossRef]

Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
[CrossRef]

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Hvozdara, L.

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

Kaiser, W.

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

Kancleris, Z.

V. Tamosiunas, Z. Kancleris, and M. Tamosiuniene, “Simulation of modes in terahertz quantum cascade microring and microdisk lasers,” Acta Phys. Pol. A 113, 917-920 (2008).

Lee, B. G.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Li, B. J.

B. J. Li and P. L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 33, 791-795 (1997).
[CrossRef]

B. J. Li and P. L. Liu, “Numerical analysis of the whispering gallery modes by the finite-difference time-domain method,” IEEE J. Quantum Electron. 32, 1583-1587 (1996).
[CrossRef]

Li, L.

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

Li, W. J.

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

Lin, K.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Liu, F. Q.

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

Liu, H. E.

F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
[CrossRef]

Liu, J. Q.

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

Liu, P. L.

B. J. Li and P. L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 33, 791-795 (1997).
[CrossRef]

B. J. Li and P. L. Liu, “Numerical analysis of the whispering gallery modes by the finite-difference time-domain method,” IEEE J. Quantum Electron. 32, 1583-1587 (1996).
[CrossRef]

Lu, Q. Y.

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Narimanov, E. E.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Nockel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Painter, O.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Richter, K.

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Roch, T.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

Schrenk, W.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

Semmel, J.

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

Sergent, A. M.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Shao, Y.

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

Sirtori, C.

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
[CrossRef]

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Srinivasan, K.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Stone, D.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

Strasser, G.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

Strasser, L.

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

Striccoli, M.

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Tamosiunas, V.

V. Tamosiunas, Z. Kancleris, and M. Tamosiuniene, “Simulation of modes in terahertz quantum cascade microring and microdisk lasers,” Acta Phys. Pol. A 113, 917-920 (2008).

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

Tamosiuniene, M.

V. Tamosiunas, Z. Kancleris, and M. Tamosiuniene, “Simulation of modes in terahertz quantum cascade microring and microdisk lasers,” Acta Phys. Pol. A 113, 917-920 (2008).

Tennant, D. M.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Toor, F.

F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
[CrossRef]

Troccoli, M.

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

Unterrainer, K.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

Wang, Z. G.

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

Yang, Y. D.

Y. Z. Huang and Y. D. Yang, “Mode coupling and vertical radiation loss for whispering-gallery modes in 3-D microcavities,” J. Lightwave Technol. 26, 1411-1416 (2008).
[CrossRef]

Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison of Q-factors between TE and TM modes in 3-D microsquares by FDTD simulation,” IEEE Photon. Technol. Lett. 19, 1981-1983 (2007).
[CrossRef]

Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
[CrossRef]

Yu, L. J.

Q. Chen, Y. Z. Huang, and L. J. Yu, “Analysis of mode characteristics for deformed square resonators by FDTD technique,” IEEE J. Quantum Electron. 42, 59-63 (2006).
[CrossRef]

Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
[CrossRef]

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Zhu, J. T.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Zobl, R.

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

Acta Phys. Pol. A

V. Tamosiunas, Z. Kancleris, and M. Tamosiuniene, “Simulation of modes in terahertz quantum cascade microring and microdisk lasers,” Acta Phys. Pol. A 113, 917-920 (2008).

Appl. Phys. Lett.

J. Faist, C. Gmachl, M. Striccoli, C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Quantum cascade disk lasers,” Appl. Phys. Lett. 69, 2456-2458 (1996).
[CrossRef]

S. Gianordoli, L. Strasser, W. Schrenk, K. Unterrainer, and E. Gornik, “GaAs/AlGaAs-based microcylinder lasers emitting at 10 μm,” Appl. Phys. Lett. 75, 1045-1047 (1999).
[CrossRef]

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. T. Zhu, and G. Höflerf, “Single-mode laser action in quantum cascade lasers with spiral-shaped chaotic resonators,” Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

J. Semmel, W. Kaiser, H. Hofmann, S. Höfling, and A. Forchel, “Single mode emitting ridge waveguide quantum cascade lasers coupled to an active ring resonator filter,” Appl. Phys. Lett. 93, 211106 (2008).
[CrossRef]

K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” Appl. Phys. Lett. 84, 4164-4166 (2004).
[CrossRef]

F. Toor, D. L. Sivco, H. E. Liu, and C. F. Gmachl, “Effect of waveguide sidewall roughness on the threshold current density and slope efficiency of quantum cascade lasers,” Appl. Phys. Lett. 93, 031104 (2008).
[CrossRef]

Chin. Phys. Lett.

L. Li, F. Q. Liu, Y. Shao, J. Q. Liu, and Z. G. Wang, “Low-threshold high-temperature operation of λ~7.4 μm quantum cascade lasers,” Chin. Phys. Lett. 24, 1577-1579 (2007).
[CrossRef]

IEEE J. Quantum Electron.

B. J. Li and P. L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 33, 791-795 (1997).
[CrossRef]

Q. Chen, Y. Z. Huang, and L. J. Yu, “Analysis of mode characteristics for deformed square resonators by FDTD technique,” IEEE J. Quantum Electron. 42, 59-63 (2006).
[CrossRef]

G. Fasching, V. Tamosiunas, A. Benz, A. M. Andrews, K. Unterrainer, R. Zobl, T. Roch, W. Schrenk, and G. Strasser, “Subwavelength microdisk and microring terahertz quantum cascade lasers,” IEEE J. Quantum Electron. 43, 687-696 (2007).
[CrossRef]

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength (λ=10 μm) quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458-464 (2000).
[CrossRef]

B. J. Li and P. L. Liu, “Numerical analysis of the whispering gallery modes by the finite-difference time-domain method,” IEEE J. Quantum Electron. 32, 1583-1587 (1996).
[CrossRef]

W. H. Guo, Y. Z. Huang, Q. Y. Lu, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Q. Chen, Y. Z. Huang, W. H. Guo, and L. J. Yu, “Analysis of modes in a freestanding microsquare resonator by 3D finite-difference time-domain simulation,” IEEE J. Quantum Electron. 41, 997-1001 (2005).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett.

W. H. Guo, W. J. Li, and Y. Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

IEEE Photon. Technol. Lett.

Y. D. Yang, Y. Z. Huang, and Q. Chen, “Comparison of Q-factors between TE and TM modes in 3-D microsquares by FDTD simulation,” IEEE Photon. Technol. Lett. 19, 1981-1983 (2007).
[CrossRef]

J. Lightwave Technol.

Phys. Rev. A

Y. D. Yang, Y. Z. Huang, and Q. Chen, “High-Q TM whispering-gallery modes in three-dimensional microcylinders,” Phys. Rev. A 75, 013817 (2007).
[CrossRef]

Phys. Rev. E

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Science

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556-1564 (1998).
[CrossRef] [PubMed]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553-556 (1994).
[CrossRef] [PubMed]

Other

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (9)

Fig. 1
Fig. 1

Cross sections of (a) microcylinder and (b) microsquare resonators with the refractive index and thickness of each layer of ( n i , d i , i = 1 , 2 , 3 ) . The refractive index of the substrate is n 3 and the etching depth is D = d 1 + d 2 + d 3 . The z = 0 plane is in the center of the active region.

Fig. 2
Fig. 2

Mode Q factor and frequency versus the etching depth are plotted as the solid and open circles, respectively, for (a) TM 22 , 1 WGM in the LW microcylinder and (b) TM 18 , 1 WGM in the SW microcylinder.

Fig. 3
Fig. 3

Intensity spectra obtained by 3D FDTD simulation are plotted as solid curves for the even TM modes in (a) the LW microsquare with D = 9.0 μ m and (b) the SW microsquare with D = 7.0 μ m . The dashed curves are results of 2D FDTD with the effective index n eff = 3.295 and 3.312, respectively, for the LW and SW microsquares.

Fig. 4
Fig. 4

Distributions of the electric field E z for (a) TM 10 , 12 , (b) TM 8 , 14 , (c) TM 6 , 16 , and (d) TM 10 , 14 at the x - y plane of z = 0 in the LW microsquare with D = 9.0 μ m .

Fig. 5
Fig. 5

Mode Q factors and frequencies versus the etching depth are plotted as the solid and open symbols, respectively, for (a) TM 10 , 12 and TM 10 , 14 in the LW microsquare and (b) TM 8 , 10 in the SW microsquare.

Fig. 6
Fig. 6

z-directional distribution of (a) the H r for TM 22 , 1 in the LW microcylinder and (b) normalized effective amplitude for TM 10 , 14 in the LW microsquare are plotted as the dotted and dashed curves as D = 7.0 and 8.0 μ m , with the etching plane marked by the dotted and dashed straight lines, respectively. The solid curve is the z-directional distribution of H i ( i = x , y ) for TM mode at wavelength of 7.8 μ m obtained from the corresponding four-layer slab waveguide.

Fig. 7
Fig. 7

Field distributions of magnetic field H i ( i = x , y ) at wavelengths of 4.8 and 7.8 μ m are plotted as the dashed and solid curves, respectively, for the TM mode in the four-layer slab waveguides of the SW and LW structures.

Fig. 8
Fig. 8

(a) TM mode field distributions of magnetic field H i ( i = x , y ) at a wavelength of 7.4 μ m and the profile of refractive index are plotted for the seven-layer slab waveguide. (b) The mode Q factor and frequency versus etching depth are plotted as the solid and open symbols, respectively, for TM 11 , 1 WGM in the microcylinder with R = 5 μ m . The circles and triangles correspond to the emission region thicknesses of 2.2 and 1.8 μ m , respectively.

Fig. 9
Fig. 9

Mode Q factor and frequency versus (a) the perturbation amplitude in LW microsquare with period roughness at sidewalls and (b) the root-mean-square height in LW microsquare with random roughness at sidewalls are plotted as the solid and open symbols, respectively. The squares and triangles correspond to the TM 5 , 7 and TM 3 , 7 modes, respectively. The sidelength of microsquare is 10 μ m .

Tables (1)

Tables Icon

Table 1 Structure Parameters

Equations (20)

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

E r ( r , ϕ , z , t ) = i e r ( r , z , t ) exp ( i υ ϕ ) ,
E ϕ ( r , ϕ , z , t ) = e ϕ ( r , z , t ) exp ( i υ ϕ ) ,
E z ( r , ϕ , z , t ) = i e z ( r , z , t ) exp ( i υ ϕ ) ,
H r ( r , ϕ , z , t ) = h r ( r , z , t ) exp ( i υ ϕ ) ,
H ϕ ( r , ϕ , z , t ) = i h ϕ ( r , z , t ) exp ( i υ ϕ ) ,
H z ( r , ϕ , z , t ) = h z ( r , z , t ) exp ( i υ ϕ ) ,
e r t = 1 ε [ ν r h z h ϕ z ] ,
e ϕ t = 1 ε [ h r z h z r ] ,
e z t = 1 ε [ 1 r ( r h ϕ ) r ν r h r ] ,
h r t = 1 μ 0 [ ν r e z + e ϕ z ] ,
h ϕ t = 1 μ 0 [ e r z e z r ] ,
h z t = 1 μ 0 [ 1 r ( r e ϕ ) r + ν r e r ] ,
H x t = 1 μ 0 ( E y z E z y ) ,
H y t = 1 μ 0 ( E z x E x z ) ,
H z t = 1 μ 0 ( E x y E y x ) ,
E x t = 1 ε ( H z y H y z ) ,
E y t = 1 ε ( H x z H z x ) ,
E z t = 1 ε ( H y x H x y ) .
P ( x 0 , y 0 , z 0 , t ) = exp [ ( t t 0 ) 2 / t ω 2 ] cos ( 2 π f t )
J m ( n eff k R ) H m ( 2 ) ( k R ) = η J m ( n eff k R ) H m ( 2 ) ( k R ) ,

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