Abstract

We present a numerically improved multipole formulation for the calculation of resonances of multiple disks located at arbitrary positions in a 2-d plane, and suitable for the accurate computation of the resonances of large numbers of disks and of high-wavenumber eigenstates. Using a simple reformulation of the field expansions and boundary conditions, we are able to transform the multipole formalism into a linear eigenvalue problem, for which fast and accurate methods are available. Observing that the motion of the eigenvalues in the complex plane is analytic with respect to a two parameter family, we present a numerical algorithm to compute a range of multiple-disk resonances and field distributions using only two diagonalizations. This method can be applied to photonic molecules, photonic crystals, photonic crystal fibers, and random lasers.

© 2009 Optical Society of America

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References

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  1. B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
    [CrossRef]
  2. T. Carmon, T. Kippenberg, L. Yang, H. Rokhsari, S. Spillane, and K. Vahala, "Feedback control of ultra-high-Q microcavities: application to micro-Raman lasers and microparametric oscillators," Opt. Express 13, 3558-3566 (2005).
    [CrossRef] [PubMed]
  3. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. D¨ohler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, "Coupled whispering gallery mode resonators in the Terahertz frequency range," Opt. Express 16, 7336-7343 (2008).
    [CrossRef] [PubMed]
  4. D. S. Wiersma and A. Lagendijk, "Light diffusion with gain and random lasers," Phys. Rev. E 54, 4256-4265 (1996).
    [CrossRef]
  5. H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
    [CrossRef] [PubMed]
  6. H. E. T¨ureci, L. Ge, S. Rotter, and A. D. Stone, "Strong interactions in multimode random lasers," Science 320, 643-646 (2008).
    [CrossRef] [PubMed]
  7. E. B. Becker, G. F. Carey, and J. T. Oden, Finite Elements (Pentice-Hall, Englewood Cliffs, N.J., 1981).
  8. Q1. J. B. Davies, "Finite element analysis of waveguides and cavities - a review." IEEE T. Magn. 29, 1578 (1993).
    [CrossRef]
  9. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Molding the Flow of Light (Princeton University Press, Pinceton, 2008).
  10. M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
    [CrossRef]
  11. J. Wiersig, "Boundary element method for resonances in dielectric microcavities," J. Opt. Soc. Am. A 5, 53-60 (2003). physics/0206018.
  12. S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004).
    [CrossRef]
  13. A. B. Movchan, N. V. Movchan, and C. G. Poulton, Asymptotics of dilute and densely packed composites (Imperial College Press, London, 2002).
    [CrossRef]
  14. H. E. T¨ureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, "Modes of wave-chaotic dielectric resonators," Prog. Opt. 47, 75-137 (2005). physics/0308016.
    [CrossRef]
  15. W. T. Perrins, D. R. McKenzie, and R. C. McPhedran, "Transport Properties of Regular Arrays of Cylinders," P. Roy. Soc. Lond. A Mat. 369, 207-225 (1979). URL http://www.jstor.org/stable/2398611.
    [CrossRef]
  16. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, "Multipole method for microstructured optical fibers. II. Implementation and results," J. Opt. Soc. B 19, 2331- 2340 (2002). URL http://josab.osa.org/abstract.cfm?URI=josab-19-10-2331.
    [CrossRef]
  17. A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
    [CrossRef]
  18. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999).
  19. Q2. H. E. Tureci and H. G. L. Schwefel, "An efficient Fredholm method for the calculation of highly excited states of billiards," J. Phys. A 40, 13,869-13,882 (2007).
    [CrossRef]

2008

2007

Q2. H. E. Tureci and H. G. L. Schwefel, "An efficient Fredholm method for the calculation of highly excited states of billiards," J. Phys. A 40, 13,869-13,882 (2007).
[CrossRef]

2006

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

2005

2004

2003

J. Wiersig, "Boundary element method for resonances in dielectric microcavities," J. Opt. Soc. Am. A 5, 53-60 (2003). physics/0206018.

2000

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

1997

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

1996

D. S. Wiersma and A. Lagendijk, "Light diffusion with gain and random lasers," Phys. Rev. E 54, 4256-4265 (1996).
[CrossRef]

1993

Q1. J. B. Davies, "Finite element analysis of waveguides and cavities - a review." IEEE T. Magn. 29, 1578 (1993).
[CrossRef]

Benson, T. M.

Boriskina, S. V.

Cao, H.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Carmon, T.

Chang, R. P. H.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Chang, S.-H.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Chu, S.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

D¨ohler, G. H.

Davies, J. B.

Q1. J. B. Davies, "Finite element analysis of waveguides and cavities - a review." IEEE T. Magn. 29, 1578 (1993).
[CrossRef]

Foresi, J.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Freude, W.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

Fujii, M.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

Ge, L.

H. E. T¨ureci, L. Ge, S. Rotter, and A. D. Stone, "Strong interactions in multimode random lasers," Science 320, 643-646 (2008).
[CrossRef] [PubMed]

Gossard, A. C.

Hanson, M.

Haus, H.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Ho, S. T.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Kippenberg, T.

Koos, C.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

Lagendijk, A.

D. S. Wiersma and A. Lagendijk, "Light diffusion with gain and random lasers," Phys. Rev. E 54, 4256-4265 (1996).
[CrossRef]

Laine, J.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Leuthold, J.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

Little, B.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Liu, X.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Malzer, S.

Nosich, A. I.

Poulton, C.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

Preu, S.

Rokhsari, H.

Rotter, S.

H. E. T¨ureci, L. Ge, S. Rotter, and A. D. Stone, "Strong interactions in multimode random lasers," Science 320, 643-646 (2008).
[CrossRef] [PubMed]

Schwefel, H. G. L.

Seelig, E. W.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Sewell, P.

Spence, A.

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

Spillane, S.

Stone, A. D.

H. E. T¨ureci, L. Ge, S. Rotter, and A. D. Stone, "Strong interactions in multimode random lasers," Science 320, 643-646 (2008).
[CrossRef] [PubMed]

T¨ureci, H. E.

H. E. T¨ureci, L. Ge, S. Rotter, and A. D. Stone, "Strong interactions in multimode random lasers," Science 320, 643-646 (2008).
[CrossRef] [PubMed]

Tureci, H. E.

Q2. H. E. Tureci and H. G. L. Schwefel, "An efficient Fredholm method for the calculation of highly excited states of billiards," J. Phys. A 40, 13,869-13,882 (2007).
[CrossRef]

Vahala, K.

Wang, L. J.

Wiersig, J.

J. Wiersig, "Boundary element method for resonances in dielectric microcavities," J. Opt. Soc. Am. A 5, 53-60 (2003). physics/0206018.

Wiersma, D. S.

D. S. Wiersma and A. Lagendijk, "Light diffusion with gain and random lasers," Phys. Rev. E 54, 4256-4265 (1996).
[CrossRef]

Xu, J. Y.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Yang, L.

Zhang, D. Z.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Zimmerman, J. D.

IEEE Photon. Technol. Lett.

M. Fujii, C. Koos, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear FDTD analysis and experimental verification of four-wave mixing in InGaAsP-InP racetrack microresonators," IEEE Photon. Technol. Lett. 18, 361-363 (2006).
[CrossRef]

IEEE T. Magn.

Q1. J. B. Davies, "Finite element analysis of waveguides and cavities - a review." IEEE T. Magn. 29, 1578 (1993).
[CrossRef]

J. Comput. Phys.

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

J. Lightwave Technol.

B. Little, S. Chu, H. Haus, J. Foresi, and J. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

Q2. H. E. Tureci and H. G. L. Schwefel, "An efficient Fredholm method for the calculation of highly excited states of billiards," J. Phys. A 40, 13,869-13,882 (2007).
[CrossRef]

Opt. Express

Phys. Rev. E

D. S. Wiersma and A. Lagendijk, "Light diffusion with gain and random lasers," Phys. Rev. E 54, 4256-4265 (1996).
[CrossRef]

Phys. Rev. Lett.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, "Spatial Confinement of Laser Light in Active Random Media," Phys. Rev. Lett. 84, 5584-5587 (2000).
[CrossRef] [PubMed]

Science

H. E. T¨ureci, L. Ge, S. Rotter, and A. D. Stone, "Strong interactions in multimode random lasers," Science 320, 643-646 (2008).
[CrossRef] [PubMed]

Other

E. B. Becker, G. F. Carey, and J. T. Oden, Finite Elements (Pentice-Hall, Englewood Cliffs, N.J., 1981).

A. B. Movchan, N. V. Movchan, and C. G. Poulton, Asymptotics of dilute and densely packed composites (Imperial College Press, London, 2002).
[CrossRef]

H. E. T¨ureci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, "Modes of wave-chaotic dielectric resonators," Prog. Opt. 47, 75-137 (2005). physics/0308016.
[CrossRef]

W. T. Perrins, D. R. McKenzie, and R. C. McPhedran, "Transport Properties of Regular Arrays of Cylinders," P. Roy. Soc. Lond. A Mat. 369, 207-225 (1979). URL http://www.jstor.org/stable/2398611.
[CrossRef]

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, "Multipole method for microstructured optical fibers. II. Implementation and results," J. Opt. Soc. B 19, 2331- 2340 (2002). URL http://josab.osa.org/abstract.cfm?URI=josab-19-10-2331.
[CrossRef]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: Molding the Flow of Light (Princeton University Press, Pinceton, 2008).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of disk geometry.

Fig. 2.
Fig. 2.

a) Distribution of the eigenvalues for two coupled disks at kR=20.0, d=0.1, and n=1.5 b) Overlap of some representative eigenfunctions (defined by the scalar product), with respect to kR.

Fig. 3.
Fig. 3.

a) Several representative eigenvalues traced in the complex plane for changes of Re [kR] (circular motion) and Im[kR] (radial motion). b) Angular speed (change of ϕ with respect to kR) of eigenvalues, with respect to the angular position ϕ in the complex plane. c) Log of the radial speed of the eigenvalues with respect to Im[kR]. In all the calculations we used (kR=[20, …,22], d=0.1, n=1.5, N trunc=33).

Fig. 4.
Fig. 4.

(left) false color representation of the a wavefunction with m=28 at kR=21.60 (top) and one iteration later at kR=24.74-0.04i (bottom). On the right the trace of the eigenvalue, in red, through one rotation in the complex plane. Also shown are the eigenvalues of the final diagonalization in black.

Fig. 5.
Fig. 5.

a) False color representation of the absolute value squared wavefunction of 16 disks on a randomized rectangular grid, for the state with kR=4.502758-0.07872i,n=1.5,N trunc=10. b)Wavefunction of a high frequency whispering gallery mode in three coupled disks at kR=40.52586-0.0005924i,n=1.5,d=0.1,R=1,N trunc=60. c) Wavefunction in a randomized grid of 1024 dielectric disks of kR~1,n=1.5,N trunc=2. d) Traces of the eigenvalues of Eq. (9) in the 16 disk randomized on a rectangular grid. Six representative eigenvalues are traced with the highest overlap criteria. The ‘radial’ motion of the eigenvalues is generated by tracing the eigenvalues through a change in Im[kR], kR=[3.69-0.08i, …,3.69-0.78i], the ‘rotational’ motion by a change in the real component kR=[3.69-0.08i, …,5.30-0.08i],n=1.5,N trunc=8. Contrary to Fig. 3 the motion is not any longer completely regular, however the general behavior is good enough for a general prediction.

Equations (11)

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

(2+n2(r)(kR)2)ψ=0
ψ(ri,θi)=Σm=[amiHm(1)(nikri)+bmiHm(2)(nikri)] exp (imθi)
ψ(ri,θi)=Σm=[cmiHm(1)(kri)+dmiJm(kri)] exp (imθi) .
dmi=Σn=ΣjiXmnijcnj,
Xmnij=(1)mnHnm(1)(kdij)ei(nm)θij.
B1a+B2b=B3c+B4d
B1a+B2b=B3c+B4d,
(1P(kR))u=0 .
(B1(B3+B4X)B'1(B3+B4X))(ac)=(B2bB2c).
(B1(B3+B4X)B1(B3+B4X))(ac)=(B20B20)(ac).
M(kR)u=λ N (kR)u,

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