Abstract

Starting from the time-harmonic Maxwell's equations in cylindrical coordinates, we derive and solve the finite-difference (FD) eigenvalue equations for determining vector modes of axially symmetric resonator structures such as disks, rings, spheres and toroids. Contrary to the most existing implementations, our FD scheme is readily adapted for both eigenmode and eigenfrequency calculations. An excellent match of the FD solutions with the analytically calculated mode indices of a microsphere resonator provides a numerical confirmation of the mode-solver accuracy. The comparison of the presented FD technique with the finite-element method highlights the relative strengths of both techniques and advances the FD mode-solver as an important tool for cylindrical resonator design.

© 2010 IEEE

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2009 (1)

M. Ferrera, D. Duchesne, L. Razzari, M. Peccianti, R. Morandotti, P. Cheben, S. Janz, D.-X. Xu, B. E. Little, S. Chu, D. J. Moss, "Low power four wave mixing in an integrated, micro-ring resonator with $Q = 1.2$ million," Opt. Exp. 17, 14 098-14 103 (2009).

2008 (3)

R. D. Kekatpure, M. L. Brongersma, "Quantification of free-carrier absorption in silicon nanocrystals with an optical microcavity," Nano Lett. 8, 3787 (2008).

R. D. Kekatpure, M. L. Brongersma, "Fundamental photophysics and optical loss processes in Si-nanocrystal-doped microdisk resonators," Phys. Rev. A 78, 023829 (2008).

J. Xiao, H. Ni, X. Sun, "Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems," Opt. Lett. 33, 1848-1850 (2008).

2007 (5)

S. F. Preble, Q. Xu, M. Lipson, "Changing the colour of light in a silicon resonator," Nature Photon. 1, 293-296 (2007).

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

P. T. Rakich, M. A. Popovic, M. Soljacic, E. P. Ippen, "Trapping, corralling and spectral bonding of optical resonances through optically induced potentials," Nat. Photon. 1, 658-665 (2007) 11.

P. DelH´aye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, T. J. Kippenberg, "Optical frequency comb generation from a monolithic microresonator," Nature 450, 1214-1217 (2007).

K. Srinivasan, O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862-865 (2007).

2006 (2)

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).

T. J. Kippenberg, J. Kalkman, A. Polman, K. J. Vahala, "Demonstration of an erbium-doped microdisk laser on a silicon chip," Phys. Rev. A 74, 051802 (2006).

2005 (2)

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (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. A 71, 013817 (2005).

2004 (2)

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, K. J. Vahala, "Ultralow-threshold erbium-implanted toroidal microlaser on silicon," Appl. Phys. Lett. 84, 1037-1039 (2004).

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).

2002 (2)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-623 (2002).

N.-N. Feng, G.-R. Zhou, C. Xu, W.-P. Huang, "Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers," J. Lightw. Technol. 20, 1976-1980 (2002).

1999 (1)

B. Little, J.-P. Laine, H. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightw. Technol. 17, 704-715 (1999).

1998 (1)

W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, W.-P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightw. Technol. 16, 910-914 (1998).

1997 (3)

B. Little, S. Chu, H. Haus, J. Foresi, J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).

F. Teixeira, W. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).

F. Teixeira, W. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guided Wave Lett. 7, 371-373 (1997).

1994 (1)

W. C. Chew, "Electromagnetic theory on a lattice," J. Appl. Phys. 75, 4843-4850 (1994).

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. (2)

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).

A. Polman, B. Min, J. Kalkman, T. J. Kippenberg, K. J. Vahala, "Ultralow-threshold erbium-implanted toroidal microlaser on silicon," Appl. Phys. Lett. 84, 1037-1039 (2004).

IEEE Microw. Guided Wave Lett. (2)

F. Teixeira, W. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).

F. Teixeira, W. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guided Wave Lett. 7, 371-373 (1997).

IEEE Trans. Microw. Theory Techn. (1)

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

J. Appl. Phys. (1)

W. C. Chew, "Electromagnetic theory on a lattice," J. Appl. Phys. 75, 4843-4850 (1994).

J. Lightw. Technol. (4)

B. Little, J.-P. Laine, H. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightw. Technol. 17, 704-715 (1999).

B. Little, S. Chu, H. Haus, J. Foresi, J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997).

W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, W.-P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightw. Technol. 16, 910-914 (1998).

N.-N. Feng, G.-R. Zhou, C. Xu, W.-P. Huang, "Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers," J. Lightw. Technol. 20, 1976-1980 (2002).

Nano Lett. (1)

R. D. Kekatpure, M. L. Brongersma, "Quantification of free-carrier absorption in silicon nanocrystals with an optical microcavity," Nano Lett. 8, 3787 (2008).

Nat. Photon. (1)

P. T. Rakich, M. A. Popovic, M. Soljacic, E. P. Ippen, "Trapping, corralling and spectral bonding of optical resonances through optically induced potentials," Nat. Photon. 1, 658-665 (2007) 11.

Nature (5)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-623 (2002).

K. Srinivasan, O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862-865 (2007).

V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).

Q. Xu, B. Schmidt, S. Pradhan, M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).

P. DelH´aye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, T. J. Kippenberg, "Optical frequency comb generation from a monolithic microresonator," Nature 450, 1214-1217 (2007).

Nature Photon. (1)

S. F. Preble, Q. Xu, M. Lipson, "Changing the colour of light in a silicon resonator," Nature Photon. 1, 293-296 (2007).

Opt. Exp. (2)

M. Ferrera, D. Duchesne, L. Razzari, M. Peccianti, R. Morandotti, P. Cheben, S. Janz, D.-X. Xu, B. E. Little, S. Chu, D. J. Moss, "Low power four wave mixing in an integrated, micro-ring resonator with $Q = 1.2$ million," Opt. Exp. 17, 14 098-14 103 (2009).

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).

Opt. Lett. (1)

Phys. Rev. A (3)

R. D. Kekatpure, M. L. Brongersma, "Fundamental photophysics and optical loss processes in Si-nanocrystal-doped microdisk resonators," Phys. Rev. A 78, 023829 (2008).

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. A 71, 013817 (2005).

T. J. Kippenberg, J. Kalkman, A. Polman, K. J. Vahala, "Demonstration of an erbium-doped microdisk laser on a silicon chip," Phys. Rev. A 74, 051802 (2006).

Other (5)

M. Popovic, Theory and design of high-index-contrast microphotonic circuits Ph.D. dissertation Massachusetts Institute of TechnologyCambridge (2008).

Photon Design, Fimmwave http://www.photond.com/products/fimmwave.htm.

(2007) Rsoft Design Group, Femsim http://www.rsoftdesign.com.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

(2007) Comsol Multiphysics v. 3.3a http://www.comsol.com.

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