Abstract

The microgear is a microdisk resonator surrounded by a circular grating. The circular grating increases the modal selectivity of the cavity. In this paper, a two-dimensional (2-D) analytical method that describes the whispering-gallery mode (WGM) in a microgear resonator is presented. The model based on coupled mode theory needs the normalization of the WGM. In the past, some normalizations have been proposed; they all assume the mode to be bound, which is not true as WGMs are leaky modes. A new normalization with no approximation is presented. The numerical integration of the model is compared to 2-D finite-difference time-domain computations. It shows accurate computations of the resonant wavelength with lower numerical complexity. The method can be generalized to any deformation of the microdisk.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
    [CrossRef]
  2. R. E. Slusher, A. F. J. 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]
  3. A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
    [CrossRef]
  4. T. Baba and D. Sano, "Low-threshold lasing and Purcell effect in microdisk lasers at room temperature," IEEE J. Sel. Top. Quantum Electron. 9, 1340-1346 (2003).
    [CrossRef]
  5. T. Baba, "Photonic crystals and microdisk cavities based on GaInAsP-InP system," IEEE J. Sel. Top. Quantum Electron. 3, 808-830 (1997).
    [CrossRef]
  6. M. Fujita, K. Inoshita, and T. Baba, "Room temperature continuous wave lasing characteristics, of GaInAsP/InP microdisk injection laser," Electron. Lett. 34, 278-279 (1998).
    [CrossRef]
  7. M. Fujita, R. Ushigome, and T. Baba, "Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 µA," Electron. Lett. 36, 790-791 (2000).
    [CrossRef]
  8. D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
    [CrossRef]
  9. L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319-321 (2003).
    [CrossRef]
  10. M. Fujita and T. Baba, "Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity," IEEE J. Sel. Top. Quantum Electron. 37, 1253-1258 (2001).
    [CrossRef]
  11. M. Fujita and T. Baba, "Microgear laser," Appl. Phys. Lett. 80, 2051-2053 (2002).
    [CrossRef]
  12. A. Sakai and T. Baba, "FDTD simulation of photonic devices and circuits based on circular and fan-shaped microdisks," J. Lightwave Technol. 17, 1493-1499 (1999).
    [CrossRef]
  13. B. E. Little, J. P. Laine, and S. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6 (1997).
    [CrossRef] [PubMed]
  14. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
    [CrossRef]
  15. D. Rowland and J. Love, "Evanescent wave coupling of whispering gallery modes of a dielectric cylinder," IEE Proc. Optoelectron. 140, 177-188 (1993).
    [CrossRef]
  16. R. P. Wang and M.-M. Dumitrescu, "Theory of optical modes in semiconductor microdisk lasers," J. Appl. Phys. 81, 3391-3397 (1997).
    [CrossRef]
  17. A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. 13, 233-252 (1977).
    [CrossRef]
  18. D. Marcuse, Light Transmission Optics, Computer Science and Engineering Series (Van Nostrand Reinhold Electrical, 1989).
  19. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  20. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables), Vol. 55 of Applied Mathematics Series (National Bureau of Standards, 1972).
  21. B. E. Little and S. T. Chu, "Estimating surface-roughness loss and output coupling in microdisk resonators," Opt. Lett. 21, 1390-1392 (1996).
    [CrossRef] [PubMed]
  22. J. Zhang and D. Grischkowsky, "Whispering-gallery-mode cavity for terahertz pulses," J. Opt. Soc. Am. B 20, 1894-1904 (2003).
    [CrossRef]
  23. A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, "Analytical study of the microdisk's coupling with a waveguide based on the perturbation theory," J. Lightwave Technol. 22, 827-832 (2004).
    [CrossRef]
  24. D. Marcuse, Theory of Dieclectric Waveguides (Academic, 1991).
  25. A. Taflove, Computational Electrodynamics, the Finite-Difference-Time-Domain-Method (Aretch House, 1995).
  26. Rsoft, "FullWAVE," (2001), http://www.rsoftdesign.com.
  27. J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  28. K. Nozaki, A. Nakagawa, D. Sano, and T. Baba, "Ultralow threshold and single-mode lasing in microgear lasers and its fusion with quasi-periodic photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
    [CrossRef]
  29. K. Chang, V. Shah, and T. Tamir, "Scattering and guiding of waves by dielectric gratings with arbitrary profiles," J. Opt. Soc. Am. 70, 804-813 (1980).
    [CrossRef]

2004

2003

J. Zhang and D. Grischkowsky, "Whispering-gallery-mode cavity for terahertz pulses," J. Opt. Soc. Am. B 20, 1894-1904 (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 photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
[CrossRef]

T. Baba and D. Sano, "Low-threshold lasing and Purcell effect in microdisk lasers at room temperature," IEEE J. Sel. Top. Quantum Electron. 9, 1340-1346 (2003).
[CrossRef]

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319-321 (2003).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
[CrossRef]

2002

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

2001

M. Fujita and T. Baba, "Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity," IEEE J. Sel. Top. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

2000

M. Fujita, R. Ushigome, and T. Baba, "Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 µA," Electron. Lett. 36, 790-791 (2000).
[CrossRef]

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

1999

1998

M. Fujita, K. Inoshita, and T. Baba, "Room temperature continuous wave lasing characteristics, of GaInAsP/InP microdisk injection laser," Electron. Lett. 34, 278-279 (1998).
[CrossRef]

1997

T. Baba, "Photonic crystals and microdisk cavities based on GaInAsP-InP system," IEEE J. Sel. Top. Quantum Electron. 3, 808-830 (1997).
[CrossRef]

R. P. Wang and M.-M. Dumitrescu, "Theory of optical modes in semiconductor microdisk lasers," J. Appl. Phys. 81, 3391-3397 (1997).
[CrossRef]

B. E. Little, J. P. Laine, and S. Chu, "Surface-roughness-induced contradirectional coupling in ring and disk resonators," Opt. Lett. 22, 4-6 (1997).
[CrossRef] [PubMed]

1996

1994

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

1993

D. Rowland and J. Love, "Evanescent wave coupling of whispering gallery modes of a dielectric cylinder," IEE Proc. Optoelectron. 140, 177-188 (1993).
[CrossRef]

R. E. Slusher, A. F. J. 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]

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

1992

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

1980

1977

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. 13, 233-252 (1977).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables), Vol. 55 of Applied Mathematics Series (National Bureau of Standards, 1972).

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 photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
[CrossRef]

T. Baba and D. Sano, "Low-threshold lasing and Purcell effect in microdisk lasers at room temperature," IEEE J. Sel. Top. Quantum Electron. 9, 1340-1346 (2003).
[CrossRef]

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

M. Fujita and T. Baba, "Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity," IEEE J. Sel. Top. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

M. Fujita, R. Ushigome, and T. Baba, "Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 µA," Electron. Lett. 36, 790-791 (2000).
[CrossRef]

A. Sakai and T. Baba, "FDTD simulation of photonic devices and circuits based on circular and fan-shaped microdisks," J. Lightwave Technol. 17, 1493-1499 (1999).
[CrossRef]

M. Fujita, K. Inoshita, and T. Baba, "Room temperature continuous wave lasing characteristics, of GaInAsP/InP microdisk injection laser," Electron. Lett. 34, 278-279 (1998).
[CrossRef]

T. Baba, "Photonic crystals and microdisk cavities based on GaInAsP-InP system," IEEE J. Sel. Top. Quantum Electron. 3, 808-830 (1997).
[CrossRef]

Benech, P.

Benson, T. M.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
[CrossRef]

Berenger, J.-P.

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

Boriskina, S. V.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
[CrossRef]

Chang, K.

Chu, S.

Chu, S. T.

Desieres, Y.

Dumitrescu, M.-M.

R. P. Wang and M.-M. Dumitrescu, "Theory of optical modes in semiconductor microdisk lasers," J. Appl. Phys. 81, 3391-3397 (1997).
[CrossRef]

Fujita, M.

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

M. Fujita and T. Baba, "Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity," IEEE J. Sel. Top. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

M. Fujita, R. Ushigome, and T. Baba, "Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 µA," Electron. Lett. 36, 790-791 (2000).
[CrossRef]

M. Fujita, K. Inoshita, and T. Baba, "Room temperature continuous wave lasing characteristics, of GaInAsP/InP microdisk injection laser," Electron. Lett. 34, 278-279 (1998).
[CrossRef]

Glass, J. L.

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

Grischkowsky, D.

Han, I. Y.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

Hu, E.

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319-321 (2003).
[CrossRef]

Hwang, J. K.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

Inoshita, K.

M. Fujita, K. Inoshita, and T. Baba, "Room temperature continuous wave lasing characteristics, of GaInAsP/InP microdisk injection laser," Electron. Lett. 34, 278-279 (1998).
[CrossRef]

Jang, D.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

Kim, C.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

Laine, J. P.

Lee, Y. H.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

Levi, A. F.

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

R. E. Slusher, A. F. J. 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]

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

Little, B. E.

Logan, R. A.

R. E. Slusher, A. F. J. 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]

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

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

Love, J.

D. Rowland and J. Love, "Evanescent wave coupling of whispering gallery modes of a dielectric cylinder," IEE Proc. Optoelectron. 140, 177-188 (1993).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Marcuse, D.

D. Marcuse, Theory of Dieclectric Waveguides (Academic, 1991).

D. Marcuse, Light Transmission Optics, Computer Science and Engineering Series (Van Nostrand Reinhold Electrical, 1989).

McCall, S. L.

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

R. E. Slusher, A. F. J. 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]

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

Mohideen, U.

R. E. Slusher, A. F. J. 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 photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
[CrossRef]

Nakamura, M.

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. 13, 233-252 (1977).
[CrossRef]

Nosich, A. I.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
[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 photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
[CrossRef]

Pearton, S. J.

R. E. Slusher, A. F. J. 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]

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

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

Phan-Huy, K.

Rowland, D.

D. Rowland and J. Love, "Evanescent wave coupling of whispering gallery modes of a dielectric cylinder," IEE Proc. Optoelectron. 140, 177-188 (1993).
[CrossRef]

Rsoft,

Rsoft, "FullWAVE," (2001), http://www.rsoftdesign.com.

Sakai, A.

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 photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
[CrossRef]

T. Baba and D. Sano, "Low-threshold lasing and Purcell effect in microdisk lasers at room temperature," IEEE J. Sel. Top. Quantum Electron. 9, 1340-1346 (2003).
[CrossRef]

Sewell, P.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
[CrossRef]

Shah, V.

Slusher, R. E.

R. E. Slusher, A. F. J. 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]

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

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

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Song, D. S.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables), Vol. 55 of Applied Mathematics Series (National Bureau of Standards, 1972).

Taflove, A.

A. Taflove, Computational Electrodynamics, the Finite-Difference-Time-Domain-Method (Aretch House, 1995).

Tamir, T.

Ushigome, R.

M. Fujita, R. Ushigome, and T. Baba, "Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 µA," Electron. Lett. 36, 790-791 (2000).
[CrossRef]

Wang, R. P.

R. P. Wang and M.-M. Dumitrescu, "Theory of optical modes in semiconductor microdisk lasers," J. Appl. Phys. 81, 3391-3397 (1997).
[CrossRef]

Yariv, A.

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. 13, 233-252 (1977).
[CrossRef]

Zhang, J.

Zhang, L.

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319-321 (2003).
[CrossRef]

Appl. Phys. Lett.

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

R. E. Slusher, A. F. J. 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]

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, "Directional light coupling from microdisk lasers," Appl. Phys. Lett. 62, 561-563 (1993).
[CrossRef]

L. Zhang and E. Hu, "Lasing from InGaAs quantum dots in an injection microdisk," Appl. Phys. Lett. 82, 319-321 (2003).
[CrossRef]

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

Electron. Lett.

M. Fujita, K. Inoshita, and T. Baba, "Room temperature continuous wave lasing characteristics, of GaInAsP/InP microdisk injection laser," Electron. Lett. 34, 278-279 (1998).
[CrossRef]

M. Fujita, R. Ushigome, and T. Baba, "Continuous wave lasing in GaInAsP microdisk injection laser with threshold current of 40 µA," Electron. Lett. 36, 790-791 (2000).
[CrossRef]

IEE Proc. Optoelectron.

D. Rowland and J. Love, "Evanescent wave coupling of whispering gallery modes of a dielectric cylinder," IEE Proc. Optoelectron. 140, 177-188 (1993).
[CrossRef]

IEEE J. Quantum Electron.

A. Yariv and M. Nakamura, "Periodic structures for integrated optics," IEEE J. Quantum Electron. 13, 233-252 (1977).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. Fujita and T. Baba, "Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity," IEEE J. Sel. Top. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

T. Baba and D. Sano, "Low-threshold lasing and Purcell effect in microdisk lasers at room temperature," IEEE J. Sel. Top. Quantum Electron. 9, 1340-1346 (2003).
[CrossRef]

T. Baba, "Photonic crystals and microdisk cavities based on GaInAsP-InP system," IEEE J. Sel. Top. Quantum Electron. 3, 808-830 (1997).
[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 photonic crystals," IEEE J. Sel. Top. Quantum Electron. 9, 1355-1360 (2003).
[CrossRef]

IEEE Photonics Technol. Lett.

D. S. Song, J. K. Hwang, C. Kim, I. Y. Han, D. Jang, and Y. H. Lee, "InGaAs microdisk lasers on AlxOy," IEEE Photonics Technol. Lett. 12, 954-956 (2000).
[CrossRef]

J. Appl. Phys.

R. P. Wang and M.-M. Dumitrescu, "Theory of optical modes in semiconductor microdisk lasers," J. Appl. Phys. 81, 3391-3397 (1997).
[CrossRef]

J. Comput. Phys.

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

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, "Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators," Opt. Quantum Electron. 35, 545-559 (2003).
[CrossRef]

Other

D. Marcuse, Light Transmission Optics, Computer Science and Engineering Series (Van Nostrand Reinhold Electrical, 1989).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (with Formulas, Graphs, and Mathematical Tables), Vol. 55 of Applied Mathematics Series (National Bureau of Standards, 1972).

D. Marcuse, Theory of Dieclectric Waveguides (Academic, 1991).

A. Taflove, Computational Electrodynamics, the Finite-Difference-Time-Domain-Method (Aretch House, 1995).

Rsoft, "FullWAVE," (2001), http://www.rsoftdesign.com.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

(a) Structure of a two-dimensional (2-D) InP disk of effective refractive index n 1 = 2.05 for TE and n 1 = 2.63 for TM surrounded by air. (b) WGM modes with azimuthal order m = 8 and m = 15 . (c) Structure of a 2-D InP microgear of effective refractive index n 1 = 2.05 for TE and n 1 = 2.63 for TM surrounded by air. (d) Even and odd WGMs with azimuthal order m = 8 .

Fig. 2
Fig. 2

(a) Forced oscillation scheme. (b) Spectral response for forced oscillation. The real part of the E z field is plotted for TE polarization at two different excitation wavelengths. (c) Free-oscillation scheme. (d) Free-oscillation time evolution. (e) Adapted-excitation scheme. (f) Adapted-excitation time evolution.

Fig. 3
Fig. 3

Adapted excitation followed by free-oscillation regime. The amplitude time dependence is shown.

Fig. 4
Fig. 4

Adapted excitation followed by free-oscillation regime. The absolute value of the electric field for the TE polarization is shown.

Fig. 5
Fig. 5

Comparison between CMT and 2-D full-vector FDTD[25] calculations for TM and TE WGM m = 6 , a = 1 μ m ; effective refractive index n 1 = 2.05 for TE and n 1 = 2.63 for TM surrounded by air. The resonant wavelength is shown versus the grating depth. The discontinuities are due to the Cartesian mesh.

Fig. 6
Fig. 6

The unperturbed structure is now a disk surrounded by a ring of mean dielectric permittivity. The modulation of the dielectric permittivity is centered on n r 2 instead of n 1 2 .

Fig. 7
Fig. 7

Comparison between improved perturbation theory and FDTD for WGM m = 6 , a = 1 μ m ; effective refractive index n 1 = 2.05 for TE and n 1 = 2.63 for TM surrounded by air. The resonant wavelength is shown versus the grating depth. The discontinuities are due to the Cartesian mesh.

Fig. 8
Fig. 8

Error between improved perturbation theory and FDTD for WGM m = 6 , a = 1 μ m ; effective refractive index n 1 = 2.05 for TE and n 1 = 2.63 for TM surrounded by air. The error on the resonant wavelength is shown versus the grating depth.

Fig. 9
Fig. 9

Contour C used to demonstrate the modal orthogonality.

Fig. 10
Fig. 10

Dielectric permittivity ϵ r evolution with r and its derivative. As ϵ r is a step function, ϵ r r exhibits Dirac delta functions. Each Dirac delta function is weighted with the corresponding step amplitude. It can be decomposed as the difference of two Dirac delta functions.

Equations (68)

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

Ψ m = F m ( r ) exp [ j ( m θ + ω t ) ] cos ( β z ) ,
2 Ψ m r 2 + 1 r Ψ m r + ( k 0 2 n i 2 m 2 r 2 ) Ψ m = 0 ,
E z = A 0 J m ( k 0 n 1 r ) exp [ j ( m θ + ω t ) ] ,
H θ = 1 j ω μ 0 k 0 n 1 A 0 d J m d u u = k 0 n 1 r exp [ j ( m θ + ω t ) ] ,
E z = [ A 1 H m ( 1 ) ( k 0 n 2 r ) + A 2 H m ( 2 ) ( k 0 n 2 r ) ] exp [ j ( m θ + ω t ) ] ,
H θ = k 0 n 2 j ω μ 0 [ A 1 d H m ( 1 ) d u u = k 0 n 2 r + A 2 d H m ( 2 ) d u u = k 0 n 2 r ] , × exp [ j ( m θ + ω t ) ]
det ( J m ( k 0 n 1 a ) H m ( 2 ) ( k 0 n 2 a ) n 1 d J m d u u = k 0 n 1 a n 2 d H m ( 2 ) d u u = k 0 n 2 a ) = 0 .
Ψ m free = J m ( ω 0 + j α c n 1 r ) exp ( j m θ ) exp [ ( j ω 0 α ) t ] .
lim r + H m ( 2 ) ( ω 0 + j α c n 2 r ) exp [ j ( ω 0 + j α c n 2 r ) ] r 1 2 ,
exp [ j ( ω 0 + j α c n 2 r ) ] r 1 2 + with r + ,
exp ( α t ) + with t ,
G ( t ) = ω p ( ω ) exp ( j ω t ) d ω .
+ P ( ω ) Ψ m forced exp ( j ω t ) d ω = { Ψ m adapted exp ( j ω t + α t ) , t [ , T 0 ] Ψ m free exp ( j ω t α t ) , t [ T 0 , + ] } .
ϵ r = ϵ r 0 + Δ ϵ r ( θ , r ) ,
ϵ r 0 ( r ) = { n 1 2 r < a n 2 2 r > a } ,
Δ ϵ r ( θ , r ) = ( n 2 2 n 1 2 ) Π ( θ ) h ( r ) ,
Π ( θ ) = { 0 for π 4 m < θ [ π m ] < π 4 m 1 for π 4 m < θ [ π m ] < 3 π 4 m } ,
h ( r ) = { 1 a h d < r < a 0 otherwise } ,
Δ ϵ r = n = 0 + ( n 2 2 n 1 2 ) h ( r ) b n exp ( j n 2 m θ ) + exp ( j n 2 m θ ) 2 ,
b n = 2 n π sin ( n π 2 ) n N * ,
b 0 = 1 2 ,
Δ E grad ( div E ) = μ 0 2 D t 2 ,
ϵ 0 ϵ r div E = div D E grad ( ϵ 0 ϵ r ) = 0 .
2 E z r 2 + 1 r E z r + 1 r 2 2 E z θ 2 = μ 0 2 D z t 2 ,
D z = ϵ 0 [ ϵ r 0 ( r ) + n = 0 + ( n 2 2 n 1 2 ) h ( r ) b n exp ( j n 2 m θ ) + exp ( j n 2 m θ ) 2 ] E z .
E z = k Z * a k ( t ) Ψ k ( r , θ , t ) = k Z * a k ( t ) F k ( r ) exp ( j k θ ) exp ( j ω t ) ,
k Z * a k ( t ) exp [ j ( k θ + ω t ) ] [ d 2 F k d r 2 + 1 r d F k d r + ( k 0 2 ϵ r 0 k 2 r 2 ) F k ] 1 c 2 ( ϵ r 0 + Δ ϵ r ) k Z * exp [ j ( k θ + ω t ) ] F k ( r ) 2 j ω a k ( t ) t + 1 c 2 Δ ϵ r k Z * a k ( t ) exp [ j ( k θ + ω t ) ] F k ( r ) ( j ω ) 2 .
2 ( ϵ r 0 + Δ ϵ r ) k Z * exp [ j ( k θ ) ] F k ( r ) a k ( t ) t = ( n 2 2 n 1 2 ) h ( r ) 2 k Z * n = 0 j ω b n exp { j ( k ± n 2 m ) θ } F k ( r ) a k ( t ) .
δ k , m = + exp [ j ( k m ) θ ] d θ ,
t ( a m ( t ) a m ( t ) ) = j ω m K ( 2 b 0 b 1 b 1 2 b 0 ) ( a m ( t ) a m ( t ) ) ,
K = ( n 2 2 n 1 2 ) 4 r h ( r ) r F m ( r ) F m * ( r ) d r r ϵ r 0 ( r ) r F m ( r ) F m * ( r ) d r .
e 0 = 2 b 0 b 1 , v 0 = { 1 1 } ,
e 1 = 2 b 0 + b 1 , v 1 = { 1 1 } .
Ψ m 0 = sin ( m θ ) F m ( r ) exp { j ω m [ 1 K ( 2 b 0 b 1 ) ] t } ,
Ψ m 1 = cos ( m θ ) F m ( r ) exp { j ω m [ 1 K ( 2 b 0 + b 1 ) ] t } .
λ 0 = 2 π c ω m [ 1 K ( 2 b 0 b 1 ) ] ,
λ 1 = 2 π c ω m [ 1 K ( 2 b 0 + b 1 ) ] .
grad ( div H ) Δ H = t [ grad ( ϵ 0 ϵ r ) E μ 0 ϵ 0 ϵ r H t ] ,
grad ( ϵ 0 ϵ r ) E = ϵ 0 ( E r r ϵ r θ E θ ϵ r r ) z ,
2 H z r 2 + 1 r H z r + 1 r 2 2 H z θ 2 1 c 2 ϵ r 2 H z t 2 = t [ ϵ 0 ( E r r ϵ r θ E θ ϵ r r ) ] .
E r = 1 j ω ϵ 0 ϵ r 1 r H z θ ,
E θ = 1 j ω ϵ 0 ϵ r H z r ,
2 H z r 2 + 1 r H z r + 1 r 2 2 H z θ 2 1 c 2 ϵ r 2 H z t 2 = [ 1 j ω ϵ r ϵ r θ 1 r 2 t ( H z θ ) + 1 j ω ϵ r t ( H z r ) ϵ r r ] .
H z = k Z * a k ( t ) Ψ k ( r , θ , t ) = k Z * a k ( t ) F k ( r ) exp ( j k θ + j ω t ) ,
λ 0 = 2 π c ω m [ 1 ( 2 b 0 + b 1 ) ( K 1 + K 3 ) + b 1 K 2 ( 2 b 0 + b 1 ) K 3 + b 1 K 2 + K 4 ] 1 ,
λ 1 = 2 π c ω m [ 1 ( 2 b 0 b 1 ) ( K 1 + K 3 ) b 1 K 2 ( 2 b 0 b 1 ) K 3 b 1 K 2 + K 4 ] 1 ,
K 1 = ( n 2 2 n 1 2 ) 2 c 2 j ω m r h ( r ) r Ψ m Ψ m * d r ,
K 2 = ( n 2 2 n 1 2 ) j ω r m 2 r 3 h ( r ) ϵ r 0 Ψ m Ψ m * d r ,
K 3 = ϵ 0 2 ( n 2 2 n 1 2 ) 1 j ω m ϵ 0 [ 1 n 1 2 Ψ m r r = a h d Ψ m * ( a h d ) a h d 1 n 2 2 Ψ m r r = a Ψ m * ( a ) a ]
K 4 = 2 c 2 j ω m r ϵ r 0 r Ψ m Ψ m * d r ,
E z = [ A 1 H m ( 1 ) ( k 0 n r r ) + A 2 H m ( 2 ) ( k 0 n r r ) ] exp [ j ( m θ + ω t ) ] ,
H θ = 1 j ω μ 0 k 0 n r [ A 1 d H m ( 1 ) d u u = k 0 n r r + A 2 d H m ( 2 ) d u u = k 0 n r r ] × exp [ j ( m θ + ω t ) ]
E z = A 3 H m ( 1 ) ( k 0 n 2 r ) + A 4 H m ( 2 ) ( k 0 n 2 r ) exp [ j ( m θ + ω t ) ] ,
H θ = 1 j ω μ 0 k 0 n 2 [ A 3 d H m ( 1 ) d u u = k 0 n 2 r + A 4 d H m ( 2 ) d u u = k 0 n 2 r ] × exp [ j ( m θ + ω t ) ]
Δ ϵ r new ( r , θ ) ( n 2 2 n 1 2 ) h ( r ) b 1 exp ( j 2 m θ ) + exp ( j 2 m θ ) 2 ( n 2 2 n 1 2 ) h ( r ) [ b 0 + b 1 exp ( j 2 m θ ) + exp ( j 2 m θ ) 2 ] ,
rot H n = j ω ϵ 0 ϵ r E n ,
rot E m = j ω μ 0 H m .
r z ( E m rot H n * H n * rot E m ) d r d z = j ω r z ( ϵ 0 ϵ r E m E n * μ 0 H m H n * ) d r d z .
r z ( E n * rot H m H m rot E n * ) d r d z = j ω r z ( ϵ 0 ϵ r E m E n * μ 0 H m H n * ) d r d z .
r z ( E n * rot H m H m rot E n * ) d r d z = r z div ( H m E n * ) d r d z = j ( n m ) r z 1 r ( H m E n * ) z d r d z + r z div ( r , z ) ( H m E n * ) d r d z ,
r z div ( r , z ) ( H m E n * ) d r d z = C H m E n * d n ,
j ( m n ) r z 1 r ( H n * E m + H m E n * ) z d r d z = 0 .
j ( m n ) r z 1 r ( H n * E m ) z d r d z = 0 .
2 H z r 2 + 1 r H z r + 1 r 2 2 H z θ 2 1 c 2 ϵ r 2 H z t 2 = [ 1 j ω ϵ r ϵ r θ 1 r 2 t ( H z θ ) + 1 j ω ϵ r t ( H z r ) ϵ r r ] ,
1 j ω m ϵ r t ( Ψ m r ) ϵ r r = 2 n 1 2 ϵ r Ψ m r [ δ ( r a + h d ) ] + 2 n 1 2 + ( 1 n 1 2 ) ( 2 b 0 + b 1 ) ϵ r Ψ m r [ δ ( r a + h d ) δ ( r a ) ] + 2 n 2 2 ϵ r Ψ m r [ δ ( r a ) ] .
1 j ω m ϵ r t ( Ψ m r ) ϵ r r = ( 1 n 1 2 ) ( 2 b 0 + b 1 ) ϵ r Ψ m r [ δ ( r a + h d ) δ ( r a ) ] + 2 ( n 2 2 n 1 2 ) ϵ r Ψ m r [ δ ( r a ) ] .
E θ m = 1 j ω m ϵ 0 ( 1 ϵ r Ψ m r ) ,
K 3 = ϵ 0 2 ( n 2 2 n 1 2 ) 1 j ω m ϵ 0 [ 1 n 1 2 Ψ m r r = a h d Ψ m * ( a h d ) a h d 1 n 2 2 Ψ m r r = a Ψ m * ( a ) a ] .

Metrics