Abstract

In this paper, we proposed a useful algorithm to optimize the shape of asymmetric resonant cavities (ARCs). At first, we studied the repeatability and accuracy of the ray simulation on one kind of ARCs (half-quadrupole-half-circle shaped cavity), and confirmed the robustness of the directionality under shape errors. Based on these, we proposed a hill-climbing algorithm to optimize the ARCs for unidirectional emission. Different evaluation functions of directionality were tested and we suggested using the function of energy contained in a certain angle for highly collimated and unidirectional emission. By this method, we optimized the ARCs to obtain about 0.46 of the total radiated energy in divergence angle of 40° in the far field. This optimization method is very powerful for the shape engineering of ARCs and could be applied in future studies of ARCs with specific emission properties.

© 2013 IEEE

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  1. H. Schwefel, H. Tureci, A. D. Stone, R. Chang, Optical Microcavities ( World Scientific, 2004) pp. 415-496.
  2. Y.-F. Xiao, C.-L. Zou, C.-H. Dong, Y. Li, Q. Gong, Z.-F. Han, "Asymmetric resonant cavities and their applications in optics and photonics: A Review," Front. Optoelectron. China 3 , 109-124 (2010 ).
  3. G. Kozyreff, J. L. Dominguez-Juarez, J. Martorell, "Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes," Laser Photon. Rev. 5, 737 -749 (2011).
  4. C.-H. Dong, Y. Yang, Y.-L. Shen, C.-L. Zou, F.-W. Sun, H. Ming, G.-C. Guo, Z.-F. Han, "Observation of microlaser with Er-doped phosphate glass coated microsphere pumped by 780 nm," Opt. Commun 283, 5117-5120 (2010).
  5. F. Vollmer, S. Arnold, " Whispering-gallery-mode biosensing: Label-free detection down to single molecules," Nature Methods 5, 591-596 (2008).
  6. Y. S. Park, A. K. Cook, H. Wang, "Cavity QED with diamond nanocrystals and silica microspheres," Nano Lett. 6, 2075-2079 (2006).
  7. Y.-S. Park, H. Wang, " Resolved-sideband and cryogenic cooling of an optomechanical resonator," Nature Phys. 5, 489-493 (2009).
  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).
  9. J. Wiersig, M. Hentschel, "Combining directional light output and ultralow loss in deformed microdisks," Phys. Rev. Lett. 100, 033901 (2008).
  10. F. Albert, C. Hopfmann, A. Eberspächer, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, S. Reitzenstein, " Directional whispering gallery mode emission from Limaçon-shaped electrically pumped quantum dot micropillar lasers," Appl. Phys. Lett. 101, 021116 (2012).
  11. Q. Song, W. Fang, B. Liu, S. T. Ho, G. S. Solomon, H. Cao, "Chaotic microcavity laser with high quality factor and unidirectional output," Phys. Rev. A 80, 041807 (2009).
  12. M. S. Kurdoglyan, S.-Y. Lee, S. Rim, C.-M. Kim, "Unidirectional lasing from a microcavity with a rounded isosceles triangle shape," Opt. Lett. 29, 2758-2760 (2004).
  13. M. Hentschel, Q. J. Wang, C. Yan, F. Capasso, T. Edamura, H. Kan, "Emission properties of electrically pumped triangular shaped microlasers," Opt. Express 18, 16437-16442 (2010 ).
  14. X. Wu, H. Li, L. Liu, L. Xu, "Unidirectional single-frequency lasing from a ring-spiral coupled microcavity laser ," Appl. Phys. Lett. 93, 081105 (2008).
  15. Q. J. Wang, C. Yan, N. Yu, J. Unterhimminghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, F. Capasso, " Whispering-gallery mode resonators for highly unidirectional laser action," Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
  16. C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, G.-C. Guo, "High Q and unidirectional emission whispering gallery modes: Principles and design," IEEE J. Sel. Top. Quant. 19, 9000406 (2013).
  17. X.-F. Jiang, Y.-F. Xiao, C.-L. Zou, L. He, C.-H. Dong, B.-B. Li, Y. Li, F.-W. Sun, L. Yang, Q. Gong, " Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities," Adv. Mater. 24, OP260-OP264 (2012).
  18. J. Wiersig, "Boundary element method for resonances in dielectric microcavities ," J. Opt. A: Pure Appl. Opt. 5, 53-60 (2003).
  19. C.-L. Zou, H. G. L. Schwefel, F.-W. Sun, Z.-F. Han, G.-C. Guo, " Quick root searching method for resonances of dielectric optical microcavities with the boundary element method ," Opt. Express 19, 15669-15678 (2011).
  20. C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, G.-C. Guo, "Accurately calculating high quality factor of whispering-gallery modes with boundary element method," J. Opt. Soc. Amer. B 26, 2050-2053 (2009).
  21. H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-massoud, J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Amer. B 21, 923-934 (2004).
  22. M. Hentschel, H. Schomerus, "Fresnel laws at curved dielectric interfaces of microresonators," Phys. Rev. E 65, 045603 (2002).
  23. Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, G. S. Solomon, "Directional laser emission from a wavelength-scale chaotic microcavity," Phys. Rev. Lett. 105, 103902 (2010).
  24. B. Redding, L. Ge, Q. Song, J. Wiersig, G. Solomon, H. Cao, "Local chirality of optical resonances in ultrasmall resonators," Phys. Rev. Lett. 108, 253902 (2012).
  25. J. Nocedal, S. J. Wright, Numerical Optimization (Springer-verlag, 2006).

2013 (1)

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, G.-C. Guo, "High Q and unidirectional emission whispering gallery modes: Principles and design," IEEE J. Sel. Top. Quant. 19, 9000406 (2013).

2012 (3)

X.-F. Jiang, Y.-F. Xiao, C.-L. Zou, L. He, C.-H. Dong, B.-B. Li, Y. Li, F.-W. Sun, L. Yang, Q. Gong, " Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities," Adv. Mater. 24, OP260-OP264 (2012).

F. Albert, C. Hopfmann, A. Eberspächer, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, S. Reitzenstein, " Directional whispering gallery mode emission from Limaçon-shaped electrically pumped quantum dot micropillar lasers," Appl. Phys. Lett. 101, 021116 (2012).

B. Redding, L. Ge, Q. Song, J. Wiersig, G. Solomon, H. Cao, "Local chirality of optical resonances in ultrasmall resonators," Phys. Rev. Lett. 108, 253902 (2012).

2011 (2)

G. Kozyreff, J. L. Dominguez-Juarez, J. Martorell, "Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes," Laser Photon. Rev. 5, 737 -749 (2011).

C.-L. Zou, H. G. L. Schwefel, F.-W. Sun, Z.-F. Han, G.-C. Guo, " Quick root searching method for resonances of dielectric optical microcavities with the boundary element method ," Opt. Express 19, 15669-15678 (2011).

2010 (3)

C.-H. Dong, Y. Yang, Y.-L. Shen, C.-L. Zou, F.-W. Sun, H. Ming, G.-C. Guo, Z.-F. Han, "Observation of microlaser with Er-doped phosphate glass coated microsphere pumped by 780 nm," Opt. Commun 283, 5117-5120 (2010).

Q. J. Wang, C. Yan, N. Yu, J. Unterhimminghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, F. Capasso, " Whispering-gallery mode resonators for highly unidirectional laser action," Proc. Natl. Acad. Sci. USA 107, 22407 (2010).

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, G. S. Solomon, "Directional laser emission from a wavelength-scale chaotic microcavity," Phys. Rev. Lett. 105, 103902 (2010).

2009 (3)

Q. Song, W. Fang, B. Liu, S. T. Ho, G. S. Solomon, H. Cao, "Chaotic microcavity laser with high quality factor and unidirectional output," Phys. Rev. A 80, 041807 (2009).

Y.-S. Park, H. Wang, " Resolved-sideband and cryogenic cooling of an optomechanical resonator," Nature Phys. 5, 489-493 (2009).

C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, G.-C. Guo, "Accurately calculating high quality factor of whispering-gallery modes with boundary element method," J. Opt. Soc. Amer. B 26, 2050-2053 (2009).

2008 (3)

J. Wiersig, M. Hentschel, "Combining directional light output and ultralow loss in deformed microdisks," Phys. Rev. Lett. 100, 033901 (2008).

X. Wu, H. Li, L. Liu, L. Xu, "Unidirectional single-frequency lasing from a ring-spiral coupled microcavity laser ," Appl. Phys. Lett. 93, 081105 (2008).

F. Vollmer, S. Arnold, " Whispering-gallery-mode biosensing: Label-free detection down to single molecules," Nature Methods 5, 591-596 (2008).

2006 (1)

Y. S. Park, A. K. Cook, H. Wang, "Cavity QED with diamond nanocrystals and silica microspheres," Nano Lett. 6, 2075-2079 (2006).

2004 (2)

M. S. Kurdoglyan, S.-Y. Lee, S. Rim, C.-M. Kim, "Unidirectional lasing from a microcavity with a rounded isosceles triangle shape," Opt. Lett. 29, 2758-2760 (2004).

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-massoud, J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Amer. B 21, 923-934 (2004).

2003 (2)

J. Wiersig, "Boundary element method for resonances in dielectric microcavities ," J. Opt. A: Pure Appl. Opt. 5, 53-60 (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 (1)

M. Hentschel, H. Schomerus, "Fresnel laws at curved dielectric interfaces of microresonators," Phys. Rev. E 65, 045603 (2002).

Nature Methods (1)

F. Vollmer, S. Arnold, " Whispering-gallery-mode biosensing: Label-free detection down to single molecules," Nature Methods 5, 591-596 (2008).

Adv. Mater. (1)

X.-F. Jiang, Y.-F. Xiao, C.-L. Zou, L. He, C.-H. Dong, B.-B. Li, Y. Li, F.-W. Sun, L. Yang, Q. Gong, " Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities," Adv. Mater. 24, OP260-OP264 (2012).

Appl. Phys. Lett. (3)

X. Wu, H. Li, L. Liu, L. Xu, "Unidirectional single-frequency lasing from a ring-spiral coupled microcavity laser ," Appl. Phys. Lett. 93, 081105 (2008).

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).

F. Albert, C. Hopfmann, A. Eberspächer, F. Arnold, M. Emmerling, C. Schneider, S. Höfling, A. Forchel, M. Kamp, J. Wiersig, S. Reitzenstein, " Directional whispering gallery mode emission from Limaçon-shaped electrically pumped quantum dot micropillar lasers," Appl. Phys. Lett. 101, 021116 (2012).

Front. Optoelectron. China (1)

Y.-F. Xiao, C.-L. Zou, C.-H. Dong, Y. Li, Q. Gong, Z.-F. Han, "Asymmetric resonant cavities and their applications in optics and photonics: A Review," Front. Optoelectron. China 3 , 109-124 (2010 ).

IEEE J. Sel. Top. Quant. (1)

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, G.-C. Guo, "High Q and unidirectional emission whispering gallery modes: Principles and design," IEEE J. Sel. Top. Quant. 19, 9000406 (2013).

J. Opt. A: Pure Appl. Opt. (1)

J. Wiersig, "Boundary element method for resonances in dielectric microcavities ," J. Opt. A: Pure Appl. Opt. 5, 53-60 (2003).

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

C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, G.-C. Guo, "Accurately calculating high quality factor of whispering-gallery modes with boundary element method," J. Opt. Soc. Amer. B 26, 2050-2053 (2009).

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

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-massoud, J. Zyss, "Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers," J. Opt. Soc. Amer. B 21, 923-934 (2004).

Laser Photon. Rev. (1)

G. Kozyreff, J. L. Dominguez-Juarez, J. Martorell, "Nonlinear optics in spheres: from second harmonic scattering to quasi-phase matched generation in whispering gallery modes," Laser Photon. Rev. 5, 737 -749 (2011).

Nano Lett. (1)

Y. S. Park, A. K. Cook, H. Wang, "Cavity QED with diamond nanocrystals and silica microspheres," Nano Lett. 6, 2075-2079 (2006).

Nature Phys. (1)

Y.-S. Park, H. Wang, " Resolved-sideband and cryogenic cooling of an optomechanical resonator," Nature Phys. 5, 489-493 (2009).

Opt. Commun (1)

C.-H. Dong, Y. Yang, Y.-L. Shen, C.-L. Zou, F.-W. Sun, H. Ming, G.-C. Guo, Z.-F. Han, "Observation of microlaser with Er-doped phosphate glass coated microsphere pumped by 780 nm," Opt. Commun 283, 5117-5120 (2010).

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (1)

M. Hentschel, H. Schomerus, "Fresnel laws at curved dielectric interfaces of microresonators," Phys. Rev. E 65, 045603 (2002).

Phys. Rev. A (1)

Q. Song, W. Fang, B. Liu, S. T. Ho, G. S. Solomon, H. Cao, "Chaotic microcavity laser with high quality factor and unidirectional output," Phys. Rev. A 80, 041807 (2009).

Phys. Rev. Lett. (3)

J. Wiersig, M. Hentschel, "Combining directional light output and ultralow loss in deformed microdisks," Phys. Rev. Lett. 100, 033901 (2008).

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, G. S. Solomon, "Directional laser emission from a wavelength-scale chaotic microcavity," Phys. Rev. Lett. 105, 103902 (2010).

B. Redding, L. Ge, Q. Song, J. Wiersig, G. Solomon, H. Cao, "Local chirality of optical resonances in ultrasmall resonators," Phys. Rev. Lett. 108, 253902 (2012).

Proc. Natl. Acad. Sci. USA (1)

Q. J. Wang, C. Yan, N. Yu, J. Unterhimminghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, F. Capasso, " Whispering-gallery mode resonators for highly unidirectional laser action," Proc. Natl. Acad. Sci. USA 107, 22407 (2010).

Other (2)

H. Schwefel, H. Tureci, A. D. Stone, R. Chang, Optical Microcavities ( World Scientific, 2004) pp. 415-496.

J. Nocedal, S. J. Wright, Numerical Optimization (Springer-verlag, 2006).

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