Abstract

Resonant electromagnetic modes are analyzed inside a dielectric cavity of equilateral triangular cross section and refractive index n, surrounded by a uniform medium of refractive index n. The field confinement is determined only under the requirements needed to maintain total internal reflection of the internal electromagnetic fields, matched to external evanescent waves. Two-dimensional electromagnetics is considered, with no dependence on the coordinate perpendicular to the cross section, giving independent TE and TM polarizations. Generally, the mode spectrum becomes sparse and the minimum mode frequency increases rapidly as the index ratio N=nn approaches 2. For specified quantum numbers and N, the TM modes are lower in frequency than the TE modes. Quality factors are estimated by supposing that evanescent boundary waves leak cavity energy at the triangle vertices; diffractive effects are not included. At an index ratio that is large compared with a mode’s cutoff ratio, this method predicts greater field confinement for TE polarization and higher quality factors than for TM polarization.

© 2006 Optical Society of America

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  1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, "Whispering gallery modes in microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
    [CrossRef]
  2. H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
    [CrossRef]
  3. Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
    [CrossRef]
  4. A. W. Poon, F. Courvoisier, and R. K. Chang, "Multimode resonances in square-shaped optical microcavities," Opt. Lett. 26, 632 (2001).
    [CrossRef]
  5. C. Y. Fong and A. W. Poon, "Mode field patterns and preferential mode coupling in planar waveguide-coupled square microcavities," Opt. Express 11, 2897-2904 (2003).
    [CrossRef] [PubMed]
  6. H.-J. MoonK. An, and J.-H. Lee, "Single spatial mode selection in a layered square microcavity laser," Appl. Phys. Lett. 82, 2963-2965 (2003).
    [CrossRef]
  7. U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
    [CrossRef]
  8. I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
    [CrossRef]
  9. M. G. Lamé, Leçons sur la Théorie Mathématique de L'elasticité des Corps Solides, (Bachelier, 1852), Chap. 57.
  10. Y. Z. Huang, "Eigenmode confinement in semiconductor microcavity lasers with an equilateral triangle resonator," Proc. SPIE 3899, 239-246 (1999).
    [CrossRef]
  11. W. H. GuoY. Z. Huang, and Q. M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Padé approximation," IEEE Photonics Technol. Lett. 12, 813-815 (2000).
    [CrossRef]
  12. Y. Z. Huang, W. H. Guo, and Q. M. Wang, "Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator," IEEE J. Quantum Electron. 37, 100-107 (2001).
    [CrossRef]
  13. S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
    [CrossRef]
  14. G. M. Wysin, "Resonant mode lifetimes due to boundary wave emission in equilateral triangular dielectric cavities," J. Opt. A Pure Appl. Opt. 7, 502 (2005).
    [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics (Wiley, 1975) 2nd ed., Chap. 8, pp. 342-346.
  16. M. Brack and R. K. Bhaduri, Semiclassical Physics, Addison-Wesley Frontiers in Physics Series (Addison Wesley, 1997).
  17. Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
    [CrossRef]
  18. J. Wiersig, "Hexagonal dielectric resonators and microcrystal lasers," Phys. Rev. A 67, 023807 (2003).
    [CrossRef]

2005

G. M. Wysin, "Resonant mode lifetimes due to boundary wave emission in equilateral triangular dielectric cavities," J. Opt. A Pure Appl. Opt. 7, 502 (2005).
[CrossRef]

2004

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

2003

C. Y. Fong and A. W. Poon, "Mode field patterns and preferential mode coupling in planar waveguide-coupled square microcavities," Opt. Express 11, 2897-2904 (2003).
[CrossRef] [PubMed]

H.-J. MoonK. An, and J.-H. Lee, "Single spatial mode selection in a layered square microcavity laser," Appl. Phys. Lett. 82, 2963-2965 (2003).
[CrossRef]

J. Wiersig, "Hexagonal dielectric resonators and microcrystal lasers," Phys. Rev. A 67, 023807 (2003).
[CrossRef]

2001

Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, "Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator," IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

A. W. Poon, F. Courvoisier, and R. K. Chang, "Multimode resonances in square-shaped optical microcavities," Opt. Lett. 26, 632 (2001).
[CrossRef]

2000

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

W. H. GuoY. Z. Huang, and Q. M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Padé approximation," IEEE Photonics Technol. Lett. 12, 813-815 (2000).
[CrossRef]

1999

Y. Z. Huang, "Eigenmode confinement in semiconductor microcavity lasers with an equilateral triangle resonator," Proc. SPIE 3899, 239-246 (1999).
[CrossRef]

1998

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

1992

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

Abraham, M.

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

An, K.

H.-J. MoonK. An, and J.-H. Lee, "Single spatial mode selection in a layered square microcavity laser," Appl. Phys. Lett. 82, 2963-2965 (2003).
[CrossRef]

Bhaduri, R. K.

M. Brack and R. K. Bhaduri, Semiclassical Physics, Addison-Wesley Frontiers in Physics Series (Addison Wesley, 1997).

Brack, M.

M. Brack and R. K. Bhaduri, Semiclassical Physics, Addison-Wesley Frontiers in Physics Series (Addison Wesley, 1997).

Braun, I.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

Cartwright, A. N.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Chang, H. C.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Chang, R. K.

Chen, X. H.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

Courvoisier, F.

Dey, S.

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

Fong, C. Y.

Guo, W. H.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, "Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator," IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
[CrossRef]

W. H. GuoY. Z. Huang, and Q. M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Padé approximation," IEEE Photonics Technol. Lett. 12, 813-815 (2000).
[CrossRef]

Haetty, J.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Huang, Y. Z.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, "Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator," IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
[CrossRef]

W. H. GuoY. Z. Huang, and Q. M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Padé approximation," IEEE Photonics Technol. Lett. 12, 813-815 (2000).
[CrossRef]

Y. Z. Huang, "Eigenmode confinement in semiconductor microcavity lasers with an equilateral triangle resonator," Proc. SPIE 3899, 239-246 (1999).
[CrossRef]

Ihlein, G.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975) 2nd ed., Chap. 8, pp. 342-346.

Kioseoglou, G.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Krauß, O.

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Laeri, F.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Lamé, M. G.

M. G. Lamé, Leçons sur la Théorie Mathématique de L'elasticité des Corps Solides, (Bachelier, 1852), Chap. 57.

Lee, E. H.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Lee, J.-H.

H.-J. MoonK. An, and J.-H. Lee, "Single spatial mode selection in a layered square microcavity laser," Appl. Phys. Lett. 82, 2963-2965 (2003).
[CrossRef]

Lei, H.-B.

Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
[CrossRef]

Levi, A. F. J.

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

Limburg, B.

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Logan, R. A.

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

Lu, Q. Y.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

Luo, H.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Luo, Y.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

McCall, S. L.

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

Mittra, R.

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

Moon, H.-J.

H.-J. MoonK. An, and J.-H. Lee, "Single spatial mode selection in a layered square microcavity laser," Appl. Phys. Lett. 82, 2963-2965 (2003).
[CrossRef]

Na, M. H.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Nöckel, J. U.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

Pearton, S. J.

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

Petrou, A.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Poon, A. W.

Schulz-Ekloff, G.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

Schüth, F.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Slusher, R. E.

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

Vietze, U.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Wang, J.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

Wang, Q. M.

Y. Z. Huang, W. H. Guo, and Q. M. Wang, "Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator," IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

W. H. GuoY. Z. Huang, and Q. M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Padé approximation," IEEE Photonics Technol. Lett. 12, 813-815 (2000).
[CrossRef]

Weiß, Ö.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

Wiersig, J.

J. Wiersig, "Hexagonal dielectric resonators and microcrystal lasers," Phys. Rev. A 67, 023807 (2003).
[CrossRef]

Wöhrle, D.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

Wysin, G. M.

G. M. Wysin, "Resonant mode lifetimes due to boundary wave emission in equilateral triangular dielectric cavities," J. Opt. A Pure Appl. Opt. 7, 502 (2005).
[CrossRef]

Xuan, Y.

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

Yu, L. J.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

Yu, L.-J.

Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
[CrossRef]

Appl. Phys. B: Lasers Opt.

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schüth, U. Vietze, Ö. Weiß, and D. Wöhrle, "Hexagonal microlasers based on organic dyes in nanoporous crystals," Appl. Phys. B: Lasers Opt. 70, 335-343 (2000).
[CrossRef]

Appl. Phys. Lett.

H.-J. MoonK. An, and J.-H. Lee, "Single spatial mode selection in a layered square microcavity laser," Appl. Phys. Lett. 82, 2963-2965 (2003).
[CrossRef]

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

IEEE J. Quantum Electron.

Y. Z. Huang, W. H. Guo, and Q. M. Wang, "Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator," IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

Y. Z. Huang, W. H. Guo, L.-J. Yu, and H.-B. Lei, "Analysis of semiconductor microlasers with an equilateral triangle resonator by rate equations," IEEE J. Quantum Electron. 37, 1259-1264 (2001).
[CrossRef]

IEEE Microw. Guid. Wave Lett.

S. Dey and R. Mittra, "Efficient computation of resonant frequencies and quality factors via a combination of the finite-difference time-domain technique and the Padé approximation," IEEE Microw. Guid. Wave Lett. 8, 415-417 (1998).
[CrossRef]

IEEE Photonics Technol. Lett.

Q. Y. Lu, X. H. Chen, W. H. Guo, L. J. Yu, Y. Z. Huang, J. Wang, and Y. Luo, "Mode characteristics of semiconductor equilateral triangle microcavities with side length of 5-20 μm," IEEE Photonics Technol. Lett. 16, 359-361 (2004).
[CrossRef]

W. H. GuoY. Z. Huang, and Q. M. Wang, "Resonant frequencies and quality factors for optical equilateral triangle resonators calculated by FDTD technique and the Padé approximation," IEEE Photonics Technol. Lett. 12, 813-815 (2000).
[CrossRef]

J. Opt. A Pure Appl. Opt.

G. M. Wysin, "Resonant mode lifetimes due to boundary wave emission in equilateral triangular dielectric cavities," J. Opt. A Pure Appl. Opt. 7, 502 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

H. C. Chang, G. Kioseoglou, E. H. Lee, J. Haetty, M. H. Na, Y. Xuan, H. Luo, A. Petrou, and A. N. Cartwright, "Lasing modes in equilateral-triangular laser cavities," Phys. Rev. A 62, 013816 (2000).
[CrossRef]

J. Wiersig, "Hexagonal dielectric resonators and microcrystal lasers," Phys. Rev. A 67, 023807 (2003).
[CrossRef]

Phys. Rev. Lett.

U. Vietze, O. Krauß, F. Laeri, G. Ihlein, F. Schüth, B. Limburg, and M. Abraham, "Zeolite-dye microlasers," Phys. Rev. Lett. 81, 4628-4631 (1998).
[CrossRef]

Proc. SPIE

Y. Z. Huang, "Eigenmode confinement in semiconductor microcavity lasers with an equilateral triangle resonator," Proc. SPIE 3899, 239-246 (1999).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics (Wiley, 1975) 2nd ed., Chap. 8, pp. 342-346.

M. Brack and R. K. Bhaduri, Semiclassical Physics, Addison-Wesley Frontiers in Physics Series (Addison Wesley, 1997).

M. G. Lamé, Leçons sur la Théorie Mathématique de L'elasticité des Corps Solides, (Bachelier, 1852), Chap. 57.

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Figures (15)

Fig. 1
Fig. 1

x y coordinates, with the origin at the triangle center, and boundaries b 0 , b 1 , and b 2 . Rays represent the six plane waves propagating within the cavity, sketched for the value α = 70 ° . The plane-wave incident angles here determine the reflection amplitudes and phases used in Table 2.

Fig. 2
Fig. 2

Dependence of the TIR Fresnel reflection phase shifts on incident angle, for TM [Eq. (38)] and TE [Eq. (40)] polarization at the indicated refractive index ratios N, assuming unit magnetic permeabilities.

Fig. 3
Fig. 3

Lowest-mode wave vectors for TM polarization at index ratio N = 3.2 plotted versus quantum index m = n 3 n 5 , for different values of the other index n = n 6 + 2 [Eq. (34)] indicated next to the curves, changing by unit increments. The solid circles indicate allowed modes; the solid lines connect those having equal values of n. The dotted lines locate the limits of stable TIR as expected from using the DBC solution for the ETR, Eqs. (45, 46), explained in the text.

Fig. 4
Fig. 4

Lowest-mode wave vectors for TM polarization at index ratio N = 8.0 plotted as described in Fig. 3. Note also that the fundamental mode here has lower ( m , n ) and k a than that for N = 3.2 .

Fig. 5
Fig. 5

Fundamental TM modes of oscillation at N = 3.2 , with ( m , n ) = ( 3 , 5 ) , where the pixel intensity is proportional to Re { ψ } 1 2 , which enhances the definition of the nodal curves. The black nodal curves separate alternating regions of positive and negative Re { ψ } . Two degenerate wave functions are displayed. In (a) the phase is 0 = 0 ; in (b) the phase is θ 0 = π 2 , where the mode amplitude is A 0 = e i θ 0 .

Fig. 6
Fig. 6

First excited TM modes of oscillation at N = 3.2 , with ( m , n ) = ( 4 , 6 ) , as described in Fig. 5. Two degenerate wave functions are displayed. In (a) the phase is θ 0 = 0 ; in (b) the phase is θ 0 = π 2 , where the mode amplitude is A 0 = e i θ 0 .

Fig. 7
Fig. 7

TM mode wave vectors as functions of the index ratio N, for modes indicated by quantum indexes ( m , n ) .

Fig. 8
Fig. 8

Lowest-mode wave vectors for TE polarization at index ratio N = 3.2 , plotted as described in Fig. 3. The fundamental mode here has lower ( m , n ) and k a than that for TM polarization. On the other hand, at fixed ( m , n ) , the mode wave vectors here are higher than those for TM polarization.

Fig. 9
Fig. 9

Fundamental TE modes of oscillation at N = 3.2 , with ( m , n ) = ( 2 , 4 ) , as described in Fig. 5. Two degenerate wave functions are displayed. In (a) the phase is θ 0 = 0 ; in (b) the phase is θ 0 = π 2 , where the mode amplitude is A 0 = e i θ 0 .

Fig. 10
Fig. 10

Lowest-mode wave vectors for TE polarization at index ratio N = 2.5 , plotted as described in Fig. 3.

Fig. 11
Fig. 11

TE mode wave vectors (solid curves) as functions of the index ratio N for modes indicated by quantum indexes ( m , n ) . The dotted lines show the DBC wave vectors, terminating at the cutoffs given by Eq. (56).

Fig. 12
Fig. 12

Fundamental (a) TE and (b) TM modes of oscillation at N = 8.0 , with ( m , n ) = ( 1 , 3 ) , as described in Fig. 5, both for phase θ = 0 . (The choice θ = π 2 would instead produce a vertical nodal line.)

Fig. 13
Fig. 13

Mode quality factors for (a) TE and (b) TM modes calculated from boundary wave-emission; see (54, 55), as functions of the index ratio N, with modes labeled by ( m , n ) .

Fig. 14
Fig. 14

Comparison of TE wave functions at (a) N = 2.5 , only slightly above cutoff, and very weakly bound with small Q and (b) N = 8.0 , substantially above the cutoff, with much higher Q, and a field amplitude concentrated away from the boundaries.

Fig. 15
Fig. 15

Comparison of some mode (a) lifetimes and (b) quality factors, as functions of the index ratio N, from the present theory, using dielectric boundary conditions (solid curves), and from the simpler theory using Dirichlet boundary conditions (dashed curves). The modes are labeled by ( m , n ) .

Tables (5)

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Table 1 Definitions of the Parameters of the Plane Waves Labeled by Wave Vectors k l , within the Triangular Cavity a

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Table 2 Relations between the Incident and Reflected Wave Amplitudes on the Lower Boundary ( b 0 ) , the Upper Right Boundary ( b 1 ) , and the Upper Left Boundary ( b 2 ) a

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Table 3 Properties of Some of the Lower TM Modes for N = 3.2 , Labeled by ( m , n ) or by Underlined Indexes ( m ̱ , l ̱ ) , Where Compared with Results of Ref. [12], in Parentheses

Tables Icon

Table 4 Properties of Some of the Lower TE Modes for N = 3.2 , Labeled by ( m , n ) or by Underlined Indexes ( m ̱ , l ̱ ) , Where Compared with Results of Ref. [12], in Parentheses

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Table 5 Mode Frequencies and Free-Space Wavelengths in the Range Around 1.3 to 1.6 μ m , for Cavities with N = 3.2 and Edge Lengths a a

Equations (67)

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sin θ c = 1 N , N = n n .
k l = k ( cos α l , sin α l ) .
R = ( 1 2 3 2 3 2 1 2 ) .
ψ = l = 1 6 A l e i k l r ,
θ i , 3 = α 30 ° , θ i , 5 = α 150 ° , θ i , 6 = 90 ° α .
A 6 e i k 6 y ( a 2 3 ) + A 1 e i k 1 y ( a 2 3 ) = F 6 .
ψ refl ψ inc = r ( θ i ) = e i δ ( θ i ) .
r 6 = e i δ 6 = A 1 e i k 1 y a 2 3 A 6 e i k 6 y a 2 3 = A 1 A 6 e i k 6 y a 3 .
A 1 = A 6 e i δ 6 e i k 6 y a 3 , F 6 = A 6 ( 1 + e i δ 6 ) e i k 6 y a 2 3 .
A 2 = A 5 e i δ 5 e i k 5 y a 3 , F 5 = A 5 ( 1 + e i δ 5 ) e i k 5 y a 2 3 .
A 4 = A 3 e i δ 3 e i k 3 y a 3 , F 3 = A 3 ( 1 + e i δ 3 ) e i k 3 y a 2 3 .
Δ l δ l k l y a 3 , l = 3 , 5 , 6 .
A 4 A 3 = A 2 A 1 = A 6 A 5 = e i Δ 3 ,
A 2 A 5 = A 6 A 3 = A 4 A 1 = e i Δ 5 ,
A 1 A 6 = A 5 A 4 = A 3 A 2 = e i Δ 6 .
A 2 A 4 A 6 A 1 A 3 A 5 = e i 3 Δ 3 = e i 3 Δ 5 = e i 3 Δ 6 .
e i 3 ( Δ 3 + Δ 6 ) = 1 , e i 3 ( Δ 5 + Δ 6 ) = 1 .
e i ( Δ 3 + Δ 5 + 2 Δ 6 ) = 1 .
Δ 3 + Δ 6 = 2 π 3 n 3 , Δ 5 + Δ 6 = 2 π 3 n 5 .
Δ 3 + Δ 5 + 2 Δ 6 = 2 π n 6 ,
n 3 + n 5 = 3 n 6 ,
k 3 y = k sin ( α 120 ° ) = k 2 ( sin α + 3 cos α ) ,
k 5 y = k sin ( α + 120 ° ) = k 2 ( sin α 3 cos α ) ,
k 6 y = k sin ( α ) = k sin α .
Δ 3 + Δ 6 = ( δ 3 + δ 6 ) + 1 2 k a ( 3 sin α + cos α ) ,
Δ 5 + Δ 6 = ( δ 5 + δ 6 ) + 1 2 k a ( 3 sin α cos α ) ,
Δ 3 + Δ 5 + 2 Δ 6 = ( δ 3 + δ 5 + 2 δ 6 ) + 3 k a sin α .
Δ 3 Δ 5 = ( δ 3 δ 5 ) + k a cos α = 2 π 3 ( n 3 n 5 ) .
k x a = k a cos α = 2 π 3 ( n 3 n 5 ) ( δ 3 δ 5 ) ,
k y a = k a sin α = 1 3 [ 2 π n 6 ( δ 3 + δ 5 + 2 δ 6 ) ] .
p ( α ) = [ 2 π 3 ( n 3 + n 5 ) ( δ 3 + δ 5 + 2 δ 6 ) ] cos α 3 [ 2 π 3 ( n 3 n 5 ) ( δ 3 δ 5 ) ] sin α = 0 .
k a = 2 π 3 { [ n 3 n 5 3 2 π ( δ 3 δ 5 ) ] 2 + 3 [ n 6 1 2 π ( δ 3 + δ 5 + 2 δ 6 ) ] 2 } 1 2 .
Δ 3 = 1 3 [ 2 π 3 ( 2 n 3 n 5 ) + ( δ 3 + δ 5 δ 6 ) ] ,
Δ 5 = 1 3 [ 2 π 3 ( 2 n 5 n 3 ) + ( δ 3 + δ 5 δ 6 ) ] ,
Δ 6 = 1 3 [ 2 π 3 ( n 3 + n 5 ) ( δ 3 + δ 5 δ 6 ) ] ,
φ ( α ) δ 3 + δ 5 δ 6 .
A 1 = 1 2 A 0 exp { i [ 2 π 9 ( n 3 + n 5 ) + 1 6 φ ] } ,
ψ = A 0 { e i k x x cos [ k y y 2 π 9 ( n 3 + n 5 ) 1 6 φ ] + e i k 3 x x cos [ k 3 y y 2 π 9 ( n 5 2 n 3 ) 1 6 φ ] + e i k 5 x x cos [ k 5 y y 2 π 9 ( n 3 2 n 5 ) 1 6 φ ] } ,
k 3 x = 1 2 k x + 3 2 k y , k 3 y = 3 2 k x 1 2 k y ,
k 5 x = 1 2 k x 3 2 k y , k 5 y = 3 2 k x 1 2 k y .
k x a = 2 π 3 ( n 3 n 5 ) , k y a = 2 π 3 ( n 6 + 2 ) .
n 5 n 3 < 2 n 5 + 3 .
m ( n 3 n 5 ) , n ( n 6 + 2 ) .
k a = 2 π 3 { m 2 + 3 n 2 } 1 2 .
ψ refl ψ inc = e i δ = ϵ μ cos θ i ϵ μ cos θ ϵ μ cos θ i + ϵ μ cos θ        ( TM ) .
cos θ = i γ i ( sin θ i sin θ c ) 2 1 ,
tan δ 2 = μ μ cos 2 θ c cos 2 θ i 1     ( TM ) .
ψ refl ψ inc = e i δ = ϵ μ cos θ i ϵ μ cos θ ϵ μ cos θ i + ϵ μ cos θ       ( TE ) .
tan δ 2 = ϵ ϵ cos 2 θ c cos 2 θ i 1       ( TE ) .
n 5 < n 3 < 2 n 5 + 3 .
n 3 = 2 n 5 , n 5 = 1 , 2 , 3
60 ° < α < 90 ° θ c ,
m > n 3 N 2 1 .
k a = ω a c * < 2 π N 3 m .
k a = ω a c * > 4 π 3 m .
N > N c 3 n 2 m 2 + 1 .
τ = U P , Q = ω τ = 2 π f τ ,
U = h d x d y ϵ E 2 8 π = h d x d y μ H 2 8 π ,
S = c 8 π μ E 2 n sin θ i x ̂ = c 8 π ϵ H 2 n sin θ i x ̂ ,
ψ 2 = A i [ 1 + e i δ ( θ i ) ] e i k x x e k γ y 2 .
P x = h c 2 4 π ω A i 2 cos 2 [ 1 2 δ ( θ i ) ] 1 ( sin θ c sin θ i ) 2 { 1 μ ( TM ) 1 ϵ ( TE ) } . .
P = 3 ( P x , 3 + P x , 5 + P x , 6 ) .
τ c * a = 2 3 k a d x a d y a ψ 2 i = 3 , 5 , 6 A 0 2 cos 2 [ 1 2 δ i ] 1 ( sin θ c sin θ i ) 2 { μ μ ( TM ) ϵ ϵ ( TE ) } .
Q = ω τ = c * k τ = ( k a ) ( τ c * a ) ,
cos 2 δ 2 = ( ϵ ϵ ) 2 cos 2 θ i sin 2 θ i sin 2 θ c + ( ϵ ϵ ) 2 cos 2 θ i .
k y a = 2 π 3 ( m ̱ + 1 ) , m ̱ = 0 , 1 , 2 , .
k x a = 2 π 3 l ̱ 2 θ ̱ , l ̱ = 3 , 4 , 5 , ,

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