Abstract

In this paper, we developed an efficient method for searching the resonant eigenfrequency of dielectric optical microcavities by the boundary element method. By transforming the boundary integral equation to a general eigenvalue problem for arbitrary, symmetric, and multi-domain shaped optical microcavities, we analyzed the regular motion of the eigenvalues against the frequency. The new strategy can predict multiple resonances, increase the speed of convergence, and avoid non-physical spurious solutions. These advantages greatly reduce the computation time in the search process of the resonances. Moreover, this method is not only valuable for dielectric microcavities, but is also suitable for other photonic systems with dissipations, whose resonant eigenfrequencies are complex numbers.

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  1. K. Vahala, “Optical Microavities,” Nature 424, 839–845 (2004).
    [CrossRef]
  2. J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
    [CrossRef]
  3. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
    [CrossRef]
  4. C.-L. Zou, Y. Yang, Y.-F. Xiao, C.-H. Dong, Z.-F. Han, and G.-C. Guo, “Accurately calculating high quality factor of whispering-gallery modes with boundary element method,” J. Opt. Soc. Am. B 26, 2050–2053 (2009).
    [CrossRef]
  5. S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
    [CrossRef]
  6. J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
    [CrossRef] [PubMed]
  7. C.-L. Zou, F.-W. Sun, C.-H. Dong, X.-W. Wu, J.-M. Cui, Y. Yang, G.-C. Guo, and Z.-F. Han, “Mechanism of unidirectional emission of ultrahigh Q whispering gallery mode in microcavities,” http://arxiv.org/abs/0908.3531 .
  8. H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: theory,” Opt. Express 12, 3791–3805 (2004).
    [CrossRef] [PubMed]
  9. E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing,” Opt. Express 15, 10231–10246 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A: Math. Theor. 40, 13869–13882 (2007).
    [CrossRef]
  13. H. G. L. Schwefel and C. G. Poulton, “An improved method for calculating resonances of multiple dielectric disks arbitrarily positioned in the plane,” Opt. Express 17, 13178–13186 (2009).
    [CrossRef] [PubMed]
  14. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
    [CrossRef]
  15. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
    [CrossRef] [PubMed]
  16. A. Bäcker, “Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems,” in Lecture Notes in Physics Vol. 618, S. Graffi and M. Degli Esposti, eds. (Spinger, 2003), pp. 91–144.
    [CrossRef]
  17. J. H. Graf, “Über die Addition und Subtraction der Argumente bei Bessel’schen Functionen nebst einer Anwendung,” Math. Ann. 43, 136–144 (1893).
    [CrossRef]
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Applied Mathematics Series 55) , (National Bureau of Standards1966), Chapter 9, pp. 360, Eq. (9.1.16).
  19. H. E. Türeci, “Wave Chaos in Dielectric Resonators: Asymptotic and Numerical Approaches,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2003).
  20. J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
    [CrossRef]

2011 (1)

2009 (2)

2008 (3)

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

2007 (2)

H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A: Math. Theor. 40, 13869–13882 (2007).
[CrossRef]

E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing,” Opt. Express 15, 10231–10246 (2007).
[CrossRef] [PubMed]

2006 (1)

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

2005 (1)

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

2004 (3)

H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: theory,” Opt. Express 12, 3791–3805 (2004).
[CrossRef] [PubMed]

K. Vahala, “Optical Microavities,” Nature 424, 839–845 (2004).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

2003 (2)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

A. Bäcker, “Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems,” in Lecture Notes in Physics Vol. 618, S. Graffi and M. Degli Esposti, eds. (Spinger, 2003), pp. 91–144.
[CrossRef]

1997 (1)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

1966 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Applied Mathematics Series 55) , (National Bureau of Standards1966), Chapter 9, pp. 360, Eq. (9.1.16).

1893 (1)

J. H. Graf, “Über die Addition und Subtraction der Argumente bei Bessel’schen Functionen nebst einer Anwendung,” Math. Ann. 43, 136–144 (1893).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Applied Mathematics Series 55) , (National Bureau of Standards1966), Chapter 9, pp. 360, Eq. (9.1.16).

Bäcker, A.

A. Bäcker, “Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems,” in Lecture Notes in Physics Vol. 618, S. Graffi and M. Degli Esposti, eds. (Spinger, 2003), pp. 91–144.
[CrossRef]

Cheng, H.

Collier, B.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

Crutchfield, W.

Doery, M.

Dong, C.-H.

Ge, L.

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

Graf, J. H.

J. H. Graf, “Über die Addition und Subtraction der Argumente bei Bessel’schen Functionen nebst einer Anwendung,” Math. Ann. 43, 136–144 (1893).
[CrossRef]

Greengard, L.

Guo, G.-C.

Han, Z.-F.

Hassani, A.

Hentschel, M.

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

Jacquod, P.

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

Kabashin, A.

Kim, C.-M.

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Kurdoglyan, M. S.

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Lacroix, S.

Lee, S. -Y.

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

Lee, S.-Y.

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Nöckel, J. U.

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

Park, Y.-J.

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

Pone, E.

Poulton, C. G.

Rim, S.

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Rin, S.

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

Rotter, S.

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

Ryu, J.-W.

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

Schwefel, H. G. L.

H. G. L. Schwefel and C. G. Poulton, “An improved method for calculating resonances of multiple dielectric disks arbitrarily positioned in the plane,” Opt. Express 17, 13178–13186 (2009).
[CrossRef] [PubMed]

H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A: Math. Theor. 40, 13869–13882 (2007).
[CrossRef]

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

Skorobogatiy, M.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Applied Mathematics Series 55) , (National Bureau of Standards1966), Chapter 9, pp. 360, Eq. (9.1.16).

Stone, A. D.

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

Sun, F.-W.

Türeci, H. E.

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A: Math. Theor. 40, 13869–13882 (2007).
[CrossRef]

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

Vahala, K.

K. Vahala, “Optical Microavities,” Nature 424, 839–845 (2004).
[CrossRef]

Wiersig, J.

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

Xiao, Y.-F.

Yang, Y.

Zhou, L.-M.

Zou, C.-L.

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).
[CrossRef]

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

J. Phys. A: Math. Theor. (1)

H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A: Math. Theor. 40, 13869–13882 (2007).
[CrossRef]

Math. Ann. (1)

J. H. Graf, “Über die Addition und Subtraction der Argumente bei Bessel’schen Functionen nebst einer Anwendung,” Math. Ann. 43, 136–144 (1893).
[CrossRef]

Nature (2)

K. Vahala, “Optical Microavities,” Nature 424, 839–845 (2004).
[CrossRef]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Lett. A (1)

J.-W. Ryu, S. Rin, Y.-J. Park, C.-M. Kim, and S. -Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531–3536 (2008).
[CrossRef]

Phys. Rev. A (2)

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

Prog. Opt. (1)

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

Science (1)

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

Other (4)

A. Bäcker, “Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems,” in Lecture Notes in Physics Vol. 618, S. Graffi and M. Degli Esposti, eds. (Spinger, 2003), pp. 91–144.
[CrossRef]

C.-L. Zou, F.-W. Sun, C.-H. Dong, X.-W. Wu, J.-M. Cui, Y. Yang, G.-C. Guo, and Z.-F. Han, “Mechanism of unidirectional emission of ultrahigh Q whispering gallery mode in microcavities,” http://arxiv.org/abs/0908.3531 .

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Applied Mathematics Series 55) , (National Bureau of Standards1966), Chapter 9, pp. 360, Eq. (9.1.16).

H. E. Türeci, “Wave Chaos in Dielectric Resonators: Asymptotic and Numerical Approaches,” Ph.D. thesis (Yale University, New Haven, Connecticut, 2003).

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Figures (4)

Fig. 1
Fig. 1

(a) The distribution of the determinant det[M(kR)] for a circular cavity (n = 1.45). Red and white circles indicate two minima, corresponding to a spurious and physical resonance, respectively. (b) The motion of the generalized eigenvalues λ for the same cavity, with different real parts kR = 14.5+{0.0, 0.02,...,0.50} (blue circles), and different imagine parts kR = 14.5 – {0.0i, 0.02i,...,0.70i} (red circles).

Fig. 2
Fig. 2

The motion of eigenvalues λ of a two-dimensional circular microdisk cavity with n = 1.45 for the non-symmetric condition (a) and symmetric condition (c). The circles are the eigenvalues with kR = 99.5–0.4i,...,100.5–0.4i, and the blue lines correspond to the prediction retrieved at kR0 = 99.5 – 0.4i. (b) and (d) show the predicted roots (blue crosses) at starting point kR0 = 100.0 – 0.4i (black dot) and compare them to the analytical results (red circles), for the non-symmetric and symmetric conditions, respectively.

Fig. 3
Fig. 3

(a) The predicted roots of two coupled stadium cavities (blue crosses) at start point kR0 = 20.025 – 0.07i (black dot), compared to the exact resonant frequencies (calculated via BEM) (red circles). (b) False color representation of the electromagnetic field intensity of a resonance in the two coupled stadium cavities. The refractive index of the cavity is n = 2.0 and the aspect ratio is 2 : 1. The centers of the two cavity are at (3.0, 2.0) and (4.0, 4.1).

Fig. 4
Fig. 4

Comparison of the convergence between the Newton (solid line) and GE (dashed line) method, for a single disk cavity, a single stadium cavities and two coupled stadium cavities. “G” and “S” stand for general, non-symmetric and symmetric condition, respectively.

Equations (21)

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

[ 2 + ( nkR ) 2 ] Ψ = 0 ,
[ 2 + ( nkR ) 2 ] G ( r , r ; kR ) = δ ( r r ) ,
Γ j [ B ( s , s ) ϕ ( s ) + C ( s , s ) ψ ( s ) ] d s = 0 ,
( B 1 C 1 B 2 C 2 B J C J ) ( ϕ ψ ) = M ( ϕ ψ ) = 0 ,
det [ M ( kR ) ] = 0.
[ M ( kR ) + λ N ( kR ) ] u = 0 ,
H 0 ( 1 ) ( k | r r | ) = m H m ( 1 ) ( kr ) J m ( k r ) e i m ( φ φ ) ,
J m ( kR ) H m ( 1 ) ( kR ) J m ( kR ) H m ( 1 ) ( kR ) = 2 i π kR ,
( i π R H m ( 1 ) ( nkR ) J m ( nkR ) i π nkR H m ( 1 ) ( nkR ) J m ( nkR ) i π R H m ( 1 ) ( kR ) J m ( kR ) i π kR H m ( 1 ) ( kR ) J m ( kR ) 2 ) ( ϕ m ψ m ) = M ( ϕ ψ ) = 0.
( i m tan β ( 1 + e 2 θ ) e 2 θ 1 i m tanh α ( 1 + i 2 e 2 Φ ) i 2 e 2 Φ 1 ) ( ϕ m ψ m ) = 0 ,
( i m tan β ( 1 + e 2 θ ) e 2 θ 1 i m tanh α 1 ) ( ϕ m ψ m ) = 0.
N = ( 0 2 I 0 0 ) ,
λ ( kR ) = ( 1 + tanh α tan β ) ( 1 + e 2 θ ) ξ + ρ e kR × κ ,
( M 1 + s x M 2 ) u ˜ = 0 ,
0 = e i [ m φ + χ ] 2 ( i π R H m ( 1 ) ( nkR ) J m ( nkR ) i π nkR H m ( 1 ) ( nkR ) J m ( nkR ) i π R H m ( 1 ) ( kR ) J m ( kR ) i π kR H m ( 1 ) ( kR ) J m ( kR ) e i m φ 2 ) ( ϕ m ψ m ) + e i [ m φ + χ ] 2 ( i π R H m ( 1 ) ( nkR ) J m ( nkR ) i π nkR H m ( 1 ) ( nkR ) J m ( nkR ) i π R H m ( 1 ) ( kR ) J m ( kR ) i π kR H m ( 1 ) ( kR ) J m ( kR ) 2 ) ( ϕ m ψ m )
λ ( kR ) ξ + ρ e kR × κ .
κ = ln ( λ + λ 0 λ 0 λ ) / Δ ,
ρ = ( λ + λ 0 ) ( λ 0 λ ) ( λ + λ 0 ) ( λ 0 λ ) ,
ξ = λ 0 ρ ,
λ ( kR ) ξ + ρ e ( kR k R 0 ) × κ .
k R j = k R 0 + 1 κ j log ( 1 ξ j ρ j ) .

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