Abstract

The characteristics of whispering-gallery modes (WGMs) in 3-D cylindrical, square, and triangular microcavities with vertical optical confinement of semiconductors are numerically investigated by the finite-difference time-domain (FDTD) technique. For a microcylinder with a vertical refractive index 3.17/3.4/3.17 and a center layer thickness 0.2 $\mu{\hbox {m}}$, $Q$-factors of transverse electric (${\hbox {TE}}$) WGMs around wavelength 1550 nm are smaller than ${\hbox {10}} ^{3}$, as the radius ${R} < 4~\mu{\hbox {m}}$ and reach the orders of ${\hbox {10}} ^{4}$ and ${\hbox {10}} ^{6}$ as $R = 5~{\hbox {and }}~6 ~\mu{\hbox {m}}$, respectively. However, the ${\hbox {Q}}$-factor of transverse magnetic (${\hbox {TM}}$) WGMs at wavelength 1.659 $\mu {\hbox {m}}$ reaches 7.5$\,\times {\hbox {10}} ^{5}$ as $R = 1~\mu{\hbox {m}}$. The mode coupling between the WGMs and vertical radiation modes in the cladding layer results in vertical radiation loss for the WGMs. In the microcylinder, the mode wavelength of ${\hbox {TM}}$ WGM is larger than the cutoff wavelength of the vertical radiation mode with the same mode numbers, so ${\hbox {TM}}$ WGMs cannot couple with the vertical radiation mode and have high $Q$-factor. In contrast, ${\hbox {TE}}$ WGMs can couple with the corresponding vertical radiation mode in the 3-D microcylinder as $R < 5~\mu{\hbox {m}}$. However, the mode wavelength of the ${\hbox {TE}}$ WGM approaches (is larger than) the cutoff wavelength of the corresponding radiation modes at $R = 5~\mu{\hbox {m}}$ (6 $\mu{\hbox {m}}$), so ${\hbox {TE}}$ WGMs have high ${\hbox {Q}}$-factors in such microcylinders too. The results show that a critical lateral size is required for obtaining high $Q$-factor ${\hbox {TE}}$ WGMs in the 3-D microcylinder. For 3-D square and triangular microcavities, we also find that the ${\hbox {Q}}$-factor of ${\hbox {TM}}$ WGM is larger than that of ${\hbox {TE}}$ WGM.

© 2008 IEEE

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  1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
  2. A. F. J. Levi, S. L. McCall, S. J. Pearton, R. A. Logan, "Room temperature operation of submicrometer radius disk laser," Electron. Lett. 29, 1666-1667 (1993).
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  5. M. Fujita, R. Ushigome, T. Baba, "Large spontaneous emission factor of 0.1 in a microdisk injection laser," IEEE Photon. Technol. Lett. 13, 403-405 (2001).
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  8. G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, N. M. Johnson, "Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars," Appl. Phys. Lett. 83, 1710-1712 (2003).
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  10. Y. D. Yang, Y. Z. Huang, Q. Chen, "Comparison of $Q$-Factors between ${\hbox {TE}}$ and ${\hbox {TM}}$ modes in 3-D microsquares by FDTD simulation," IEEE Photon. Technol. Lett. 19, 1831-1833 (2007).
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  12. M. Hentschel, K. Richter, "Quantum chaos in optical systems: The annular billiard," Phys. Rev. E 66, 056207 (2002).
  13. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  14. B. J. Li, 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).
  15. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
  16. W. H. Guo, W. J. Li, Y. Z. Huang, "Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation," IEEE Microw. Wireless Compon. Lett. 11, 223-225 (2001).
  17. Y. Z. Huang, Q. Chen, W. H. Guo, Q. Y. Lu, L. J. Yu, "Mode characteristics for equilateral triangle optical resonators," IEEE J. Sel. Quantum Electron. 12, 59-65 (2006).
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  19. W. H. Guo, Y. Z. Huang, Q. Y. Lu, L. J. Yu, "Modes in square resonators," IEEE J. Quantum Electron. 39, 1563-1566 (2003).
  20. Y. D. Yang, Y. Z. Huang, "Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators," Phys. Rev. A 76, 023822 (2007).

2007 (4)

Y. D. Yang, Y. Z. Huang, Q. Chen, "High-$Q~{\hbox {TM}}$ whispering-gallery modes in three-dimensional microcylinders," Phys. Rev. A 75, 013817 (2007).

Y. D. Yang, Y. Z. Huang, Q. Chen, "Comparison of $Q$-Factors between ${\hbox {TE}}$ and ${\hbox {TM}}$ modes in 3-D microsquares by FDTD simulation," IEEE Photon. Technol. Lett. 19, 1831-1833 (2007).

Q. Chen, Y. H. Hu, Y. Z. Huang, Y. Du, Z. C. Fan, "Equilateral-triangle-resonator injection lasers with directional emission," IEEE J. Quantum Electron. 43, 440-444 (2007).

Y. D. Yang, Y. Z. Huang, "Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators," Phys. Rev. A 76, 023822 (2007).

2006 (1)

Y. Z. Huang, Q. Chen, W. H. Guo, Q. Y. Lu, L. J. Yu, "Mode characteristics for equilateral triangle optical resonators," IEEE J. Sel. Quantum Electron. 12, 59-65 (2006).

2003 (2)

W. H. Guo, Y. Z. Huang, Q. Y. Lu, L. J. Yu, "Modes in square resonators," IEEE J. Quantum Electron. 39, 1563-1566 (2003).

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, N. M. Johnson, "Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars," Appl. Phys. Lett. 83, 1710-1712 (2003).

2002 (2)

K. Djordjev, S.- J. Choi, S.- J. Choi, P. D. Dapkus, "High-${\hbox {Q}}$ vertically coupled InP microdisk resonators," IEEE Photon. Technol. Lett. 14, 331-333 (2002).

M. Hentschel, K. Richter, "Quantum chaos in optical systems: The annular billiard," Phys. Rev. E 66, 056207 (2002).

2001 (3)

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

M. Arzberger, G. Böhm, M.-C. Amann, G. Abstreiter, "Continuous room-temperature operation of electrically pumped quantum dot microcylinder lasers," Appl. Phys. Lett. 79, 1766-1768 (2001).

M. Fujita, R. Ushigome, T. Baba, "Large spontaneous emission factor of 0.1 in a microdisk injection laser," IEEE Photon. Technol. Lett. 13, 403-405 (2001).

1999 (1)

S. M. K. Thiyagarajan, D. A. Cohen, A. F. J. Levi, S. Ryu, R. Li, P. D. Dapkus, "Continuous room-temperature operation of microdisk laser diodes," Electron. Lett. 35, 1252-1253 (1999).

1996 (1)

B. J. Li, 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).

1994 (2)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).

D. Y. Chu, M. K. Chin, W. G. Bi, H. Q. Hou, C. W. Tu, S. T. Ho, "Double-disk structure for output coupling in microdisk lasers," Appl. Phys. Lett. 65, 3167-3169 (1994).

1993 (1)

A. F. J. Levi, S. L. McCall, S. J. Pearton, R. A. Logan, "Room temperature operation of submicrometer radius disk laser," Electron. Lett. 29, 1666-1667 (1993).

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).

Appl. Phys. Lett. (1)

M. Arzberger, G. Böhm, M.-C. Amann, G. Abstreiter, "Continuous room-temperature operation of electrically pumped quantum dot microcylinder lasers," Appl. Phys. Lett. 79, 1766-1768 (2001).

Appl. Phys. Lett. (3)

G. D. Chern, H. E. Tureci, A. D. Stone, R. K. Chang, M. Kneissl, N. M. Johnson, "Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars," Appl. Phys. Lett. 83, 1710-1712 (2003).

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).

D. Y. Chu, M. K. Chin, W. G. Bi, H. Q. Hou, C. W. Tu, S. T. Ho, "Double-disk structure for output coupling in microdisk lasers," Appl. Phys. Lett. 65, 3167-3169 (1994).

Electron. Lett. (1)

S. M. K. Thiyagarajan, D. A. Cohen, A. F. J. Levi, S. Ryu, R. Li, P. D. Dapkus, "Continuous room-temperature operation of microdisk laser diodes," Electron. Lett. 35, 1252-1253 (1999).

Electron. Lett. (1)

A. F. J. Levi, S. L. McCall, S. J. Pearton, R. A. Logan, "Room temperature operation of submicrometer radius disk laser," Electron. Lett. 29, 1666-1667 (1993).

IEEE J. Quantum Electron. (3)

B. J. Li, 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).

Q. Chen, Y. H. Hu, Y. Z. Huang, Y. Du, Z. C. Fan, "Equilateral-triangle-resonator injection lasers with directional emission," IEEE J. Quantum Electron. 43, 440-444 (2007).

W. H. Guo, Y. Z. Huang, Q. Y. Lu, L. J. Yu, "Modes in square resonators," IEEE J. Quantum Electron. 39, 1563-1566 (2003).

IEEE Photon. Technol. Lett. (1)

Y. D. Yang, Y. Z. Huang, Q. Chen, "Comparison of $Q$-Factors between ${\hbox {TE}}$ and ${\hbox {TM}}$ modes in 3-D microsquares by FDTD simulation," IEEE Photon. Technol. Lett. 19, 1831-1833 (2007).

IEEE J. Sel. Quantum Electron. (1)

Y. Z. Huang, Q. Chen, W. H. Guo, Q. Y. Lu, L. J. Yu, "Mode characteristics for equilateral triangle optical resonators," IEEE J. Sel. Quantum Electron. 12, 59-65 (2006).

IEEE Microw. Wireless Compon. Lett. (1)

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

IEEE Photon. Technol. Lett. (2)

M. Fujita, R. Ushigome, T. Baba, "Large spontaneous emission factor of 0.1 in a microdisk injection laser," IEEE Photon. Technol. Lett. 13, 403-405 (2001).

K. Djordjev, S.- J. Choi, S.- J. Choi, P. D. Dapkus, "High-${\hbox {Q}}$ vertically coupled InP microdisk resonators," IEEE Photon. Technol. Lett. 14, 331-333 (2002).

J. Comput. Phys. (1)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).

Phys. Rev. A (1)

Y. D. Yang, Y. Z. Huang, "Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators," Phys. Rev. A 76, 023822 (2007).

Phys. Rev. A (1)

Y. D. Yang, Y. Z. Huang, Q. Chen, "High-$Q~{\hbox {TM}}$ whispering-gallery modes in three-dimensional microcylinders," Phys. Rev. A 75, 013817 (2007).

Phys. Rev. E (1)

M. Hentschel, K. Richter, "Quantum chaos in optical systems: The annular billiard," Phys. Rev. E 66, 056207 (2002).

Other (2)

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

Y. D. Yang, Y. Z. Huang, "High $Q$-factor ${\hbox {TM}}$ modes in three-dimensional semiconductor microresonators," Proc. CLEO/Pacific Rim (2007).

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