Abstract

Optimal cross-sectional shapes of whispering-gallery ring resonators with prescribed emission wavelength and resonance mode are generated using topology optimization based on the finite element method. The two critical performance indices, the quality factor (Q factor) and mode volume of a resonator, are treated as the objective functions in the optimization. In our numerical study, characteristics of optimal configurations are identified and analyzed. Since the Q factor and mode volume have a tradeoff relationship, i.e., an increasing Q factor increases mode volume, a Pareto-optimal set of solutions can be identified under certain device specifications. These configurations achieve better performances than existing shapes in producing both a high Q factor and low mode volume.

© 2012 IEEE

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  1. S. McCall, A. Levi, R. Slusher, S. Pearton, R. Logan, "Whispering-gallery mode microdisk lasers," Appl. Phys. Lett. 60, 289-291 (1992).
  2. K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
  3. A. B. Matsko, V. S. Ilchenko, "Optical resonators with whispering-gallery modes—Part I: Basics," IEEE J. Sel. Topics Quantum Electron. 12, 3-14 (2006).
  4. V. S. Ilchenko, A. B. Matsko, "Optical resonators with whispering-gallery modes—Part II: Applications," IEEE J. Sel. Topics Quantum Electron. 12, 15-32 (2006).
  5. A. Oraevsky, "Whispering-gallery waves," Quantum Electron. 32, 377-400 (2002).
  6. M. Oxborrow, "Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators," IEEE Trans. Microw. Theory Tech. 55, 1209-1218 (2007).
  7. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. 71, 013817-1-013817-10 (2005).
  8. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, "Cavity q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots," Opt. Exp. 14, 1094-1105 (2006).
  9. M. P. Bendsøe, O. Sigmund, Topology Optimization: Theory, Methods, and Applications (Springer-Verlag, 2003).
  10. J. Jensen, O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).
  11. W. R. Frei, D. A. Tortorelli, H. T. Johnson, "Topology optimization of a photonic crystal waveguide termination to maximize directional emission," Appl. Phys. Lett. 86, 111114-1-111114-3 (2005).
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  13. J. Andkjær, S. Nishiwaki, T. Nomura, O. Sigmund, "Topology optimization of grating couplers for the efficient excitation of surface plasmons," J. Opt. Soc. Amer. B 27, 1828-1832 (2010).
  14. A. R. Diaz, O. Sigmund, "A topology optimization method for design of negative permeability metamaterials," Struct. Multidisc. Optim. 41, 163-177 (2010).
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  24. G. Allaire, Conception Optimale De Structures (Springer-Verlag, 2007).
  25. J. Nocedal, S. Wright, Numerical Optimization (Springer-Verlag, 2006).

2011

J. Andkjær, O. Sigmund, "Topology optimized low-contrast all-dielectric optical cloak," Appl. Phys. Lett. 98, 021112-1-021112-3 (2011).

2010

A. Takezawa, S. Nishiwaki, M. Kitamura, "Shape and topology optimization based on the phasefield method and sensitivity analysis," J. Comput. Phys. 229, 2697-2718 (2010).

J. Andkjær, S. Nishiwaki, T. Nomura, O. Sigmund, "Topology optimization of grating couplers for the efficient excitation of surface plasmons," J. Opt. Soc. Amer. B 27, 1828-1832 (2010).

A. R. Diaz, O. Sigmund, "A topology optimization method for design of negative permeability metamaterials," Struct. Multidisc. Optim. 41, 163-177 (2010).

2007

W. R. Frei, D. A. Tortorelli, H. T. Johnson, "Geometry projection method for optimizing photonic nanostructures," Opt. Lett. 32, 77-79 (2007).

M. Oxborrow, "Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators," IEEE Trans. Microw. Theory Tech. 55, 1209-1218 (2007).

2006

K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, "Cavity q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots," Opt. Exp. 14, 1094-1105 (2006).

A. B. Matsko, V. S. Ilchenko, "Optical resonators with whispering-gallery modes—Part I: Basics," IEEE J. Sel. Topics Quantum Electron. 12, 3-14 (2006).

V. S. Ilchenko, A. B. Matsko, "Optical resonators with whispering-gallery modes—Part II: Applications," IEEE J. Sel. Topics Quantum Electron. 12, 15-32 (2006).

2005

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. 71, 013817-1-013817-10 (2005).

W. R. Frei, D. A. Tortorelli, H. T. Johnson, "Topology optimization of a photonic crystal waveguide termination to maximize directional emission," Appl. Phys. Lett. 86, 111114-1-111114-3 (2005).

2004

J. Jensen, O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).

2003

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).

2002

A. Oraevsky, "Whispering-gallery waves," Quantum Electron. 32, 377-400 (2002).

1992

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

1985

M. Koshiba, K. Hayata, M. Suzuki, "Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides," IEEE Trans. Microw. Theory Tech. 33, 227-233 (1985).

1946

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Appl. Phys. Lett.

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

J. Jensen, O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).

W. R. Frei, D. A. Tortorelli, H. T. Johnson, "Topology optimization of a photonic crystal waveguide termination to maximize directional emission," Appl. Phys. Lett. 86, 111114-1-111114-3 (2005).

J. Andkjær, O. Sigmund, "Topology optimized low-contrast all-dielectric optical cloak," Appl. Phys. Lett. 98, 021112-1-021112-3 (2011).

IEEE J. Sel. Topics Quantum Electron.

V. S. Ilchenko, A. B. Matsko, "Optical resonators with whispering-gallery modes—Part II: Applications," IEEE J. Sel. Topics Quantum Electron. 12, 15-32 (2006).

IEEE J. Sel. Topics Quantum Electron.

A. B. Matsko, V. S. Ilchenko, "Optical resonators with whispering-gallery modes—Part I: Basics," IEEE J. Sel. Topics Quantum Electron. 12, 3-14 (2006).

IEEE Trans. Microw. Theory Tech.

M. Oxborrow, "Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators," IEEE Trans. Microw. Theory Tech. 55, 1209-1218 (2007).

M. Koshiba, K. Hayata, M. Suzuki, "Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides," IEEE Trans. Microw. Theory Tech. 33, 227-233 (1985).

J. Comput. Phys.

A. Takezawa, S. Nishiwaki, M. Kitamura, "Shape and topology optimization based on the phasefield method and sensitivity analysis," J. Comput. Phys. 229, 2697-2718 (2010).

J. Opt. Soc. Amer. B

J. Andkjær, S. Nishiwaki, T. Nomura, O. Sigmund, "Topology optimization of grating couplers for the efficient excitation of surface plasmons," J. Opt. Soc. Amer. B 27, 1828-1832 (2010).

Nature

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).

Opt. Lett.

W. R. Frei, D. A. Tortorelli, H. T. Johnson, "Geometry projection method for optimizing photonic nanostructures," Opt. Lett. 32, 77-79 (2007).

Opt. Exp.

K. Srinivasan, M. Borselli, O. Painter, A. Stintz, S. Krishna, "Cavity q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots," Opt. Exp. 14, 1094-1105 (2006).

Phys. Rev.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. 71, 013817-1-013817-10 (2005).

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Quantum Electron.

A. Oraevsky, "Whispering-gallery waves," Quantum Electron. 32, 377-400 (2002).

Struct. Multidisc. Optim.

A. R. Diaz, O. Sigmund, "A topology optimization method for design of negative permeability metamaterials," Struct. Multidisc. Optim. 41, 163-177 (2010).

Other

K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms (Wiley, 2009).

M. P. Bendsøe, O. Sigmund, Topology Optimization: Theory, Methods, and Applications (Springer-Verlag, 2003).

D. W. Prather, S. Shi, A. Sharkawy, J. Murakowski, G. J. Schneider, Photonic Crystals, Theory, Applications and Fabrication (Wiley, 2009).

S. H. Gould, Variational Methods for Eigenvalue Problems: An Introduction to the Methods of Rayleigh, Ritz, Weinstein, and Aronszajn (Dover, 1995).

G. Allaire, Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation (Oxford Univ. Press, 2007).

E. J. Haug, K. K. Choi, V. Komkov, Design Sensitivity Analysis of Structural Systems (Academic, 1986).

G. Allaire, Conception Optimale De Structures (Springer-Verlag, 2007).

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

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