## Quick root searching method for resonances of dielectric optical microcavities with the boundary element method |

Optics Express, Vol. 19, Issue 17, pp. 15669-15678 (2011)

http://dx.doi.org/10.1364/OE.19.015669

Acrobat PDF (3323 KB)

### 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.

© 2011 OSA

## 1. Introduction

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]

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]

10. L.-M. Zhou, C.-L. Zou, Z.-F. Han, G.-C. Guo, and F.-W. Sun, “Negative Goos-Hänchen shift on a concave dielectric interface,” Opt. Lett. **36**, 624–626 (2011). [CrossRef] [PubMed]

*kR*), where the finite quality factor of the resonance is defined as

*Q*= −Re(

*kR*)/2Im(

*kR*), which directly corresponds to the photon lifetime

*τ*=

*Q/kc*in the cavity, where

*c*is the speed of light in vacuum. This means that searching for the resonances take places in the two-dimensional complex plane. Newton’s method is usually used in order to search for the minimum of the determinant of discretized boundary integration matrix [3

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

*kR*

_{0}) is very important as

*kR*only converges to the nearest root. In addition, there appear non-physical, so called spurious solutions, which can not be distinguished from high-Q modes with the Newton’s method [3

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

11. 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]

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]

*kR*. Thirdly, we use this analytic behavior of the eigenvalues to predict multiple resonant positions near the starting point, and thus reduce the convergence time significantly. Fourthly, spurious solutions can be easily avoided during the search process. This improved BEM root search strategy can be further extended for various applications, such as resonances of photonic molecules, propagation modes in photonic crystal fibers [8

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]

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]

## 2. Brief Introduction to the Boundary Element Method

*n*is the refractive index of cavity,

*k*= 2

*π*

*/wavelength*is the wavenumber, and Ψ is the wavefunction, which can be either the electric or magnetic component of the electromagnetic field. Here, we use the dimensionless wave number

*kR*for convenience, where

*R*is the size of the photonic structure. The corresponding Green’s function is where

*filled with a homogeneous dielectric, and bounded by a smooth boundary Γ*

_{j}*, the boundary integral equations (BIE) can be written as [3*

_{j}**5**, 53–60 (2003). [CrossRef]

*B*(

*s*′

*,s*) = −2

*G*(

*s, s*′

*,kR*),

*C*(

*s*′

*,s*) = 2

*∂*(

_{v}G*s, s*′

*,kR*) −

*δ*(

*s − s*′), and

*ϕ*(

*s*) =

*∂*

_{v}*ψ*(

*s*) with

*∂*being the normal derivative. By dividing the boundary into small elements, the entire set of BIEs can be written in the matrix form [3

_{v}**5**, 53–60 (2003). [CrossRef]

*B*and

_{j}*C*represent the integral operators in the region Ω

_{j}*. If the number of elements is*

_{j}*l*, then

*M*is a 2

*l*× 2

*l*matrix,

*ϕ*and

*ψ*are

*l*-component vectors.

*kR*to satisfy

*M*(

*kR*) in the two dimensional complex plane. There are several local minima in this figure made visible through the equipotential lines, corresponding to possible eigen-frequency of modes. The locations of the minima are not regularly distributed in the complex plane. Since they are not easily predictable, the root search in the two dimensional space is complex and usually time consuming. A standard numerical procedure for extracting the zeros of this determinant is the Newton’s method [3

**5**, 53–60 (2003). [CrossRef]

*kR*

_{0}and step size Δ, it compares det[

*M*(

*kR*

_{0})], det[

*M*(

*kR*

_{0}+ Δ)], and det[

*M*(

*kR*

_{0}+ Δ

*i*)], then approximately estimates the values of

*kR*to satisfy det[

*M*(

*kR*)] = 0. The step is repeated until |

*kR − kR*

_{0}| ≤

*δ*, where

*δ*is the tolerance precision of

*kR*. The drawbacks of this method are obvious: (1) The root strongly depends on the starting point

*kR*

_{0}. When there are two modes with frequencies close to each other, it is difficult to distinguish and find both of them. (2) Only one root can be found with one given

*kR*

_{0}. (3) It lacks the capability to exclude spurious solutions. For example, as shown with the red circle in Fig. 1(a), several minima near the real axis Im(

*kR*) = 0 are spurious solutions to the interior Dirichlet problem. They are very hard to be distinguished from high-Q modes which also have Im(

*kR*) ≈ 0.

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]

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]

## 3. Root Search Strategy

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]

*M*′(

*kR*) =

*M*(

*kR*) –

*N*(

*kR*) and

*u*= {

*ϕ,*

*ψ*}

*is a 2*

^{T}*l*-component vector of boundary values. Solving for the eigenvalues

*λ*, the solutions to Eq. (2) require

*λ*(

*kR*) = 1 + 0i, known as the quantization condition [11

11. 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]

*M*(

*kR*)] is calculated as a single valued function of

*kR*, and only one resonance can be predicted. However, solving Eq. (6) we can find 2

*l*complex eigenvalues for each

*kR*, and extract more information about the BIE. This extra information can be used to find the resonances more efficiently. By carefully choosing the form of the GE equation, we will see a regular behavior of the eigenvalues

*λ*with respect to a change of

*kR*. Figure 1(b) shows such a motion of

*λ*with respect to a change in

*kR*in the complex plane for the general non-symmetric GE method, which is described in detail in the following. A change in Re[

*kR*] induces a rotation of the eigenvalues, while a change of Im[

*kR*] induces a radial motion. The eigenvalues can be traced by an overlap integral with the corresponding eigenvectors, similar to the description in the improved multipole method and scattering quantization approach [11

11. 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]

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]

*λ*with respect to

*kR*, then the resonance with

*λ*= 1 becomes predictable. This would allow us to derive an algorithm for the calculation of the resonant states. In addition, we can also distinguish the physical and non-physical solutions through their dynamics in

*λ*.

*N*from Eq. (6) for dielectric microcavities, including the general, non-symmetric, and the symmetric shape respectively.

### 3.1. General, Non-Symmetric Condition

*ϕ*= ∑

*exp{−i*

_{m}ϕ_{m}*mφ*},

*ψ*= ∑

*exp{−i*

_{m}ψ_{m}*mφ*}. By using Graf’s addition theorems [17

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]

**47**, 75–137 (2005). [CrossRef]

*m*(

*α*– tanh

*α*),

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]

^{−2}

^{Φ}≈ 0 for

*m*<

*nkR*, Eq. (10) can be simplified as In the matrix, only terms tan

*β*, tanh

*α*and e

^{−2}

*vary with*

^{θ}*kR*. Therefore, for any matrix

*N*, the eigenvalues (

*λ*) of the generalized linear eigenvalues equation Eq. (6) should be a function of

*kR*, which can be expressed by these terms. Among these terms, the change of e

^{−2}

*with respect to the change of*

^{θ}*kR*is much faster than others. So, we can expect that the dynamics of

*λ*is predominantly determined by the e

^{−2}

*. If we can express the*

^{θ}*λ*by a simple linear function of e

^{−2}

*, the motion of*

^{θ}*λ*can be very regular and predictable. After several tries, we found that when the eigenvalue can be written as where

*I*is the unit matrix. From Eq. (13), the trace of

*λ*against

*kR*is a circle centered at

*ξ*, with radius |

*ρ*|, and the speed of motion with respect to

*kR*is

*κ*, which is consistent with a regular motion of

*λ*in Fig. 1(b).

### 3.2. Symmetric Condition

*x*-axis symmetric, we can reduce the BIEs to only one half of the boundary. The corresponding size of the matrix will be reduced to only 1/4 of the non-symmetric condition. As a result, the computation complexity is reduced. Therefore, the boundary element matrix

*M*(

*kR*) can be reduced to where

*M*

_{1}and

*M*

_{2}are

*l*×

*l*matrices, and

*ũ*is a

*l*-component vector of boundary values.

*s*= 1 (−1) is the symmetric condition corresponding to even (odd) symmetry.

_{x}*ϕ*= ∑

*cos(*

_{m}ϕ_{m}*mφ*),

*ψ*= ∑

*cos(*

_{m}ψ_{m}*mφ*) for

*s*= 1, and

_{x}*ϕ*= ∑

*sin(*

_{m}ϕ_{m}*mφ*),

*ψ*= ∑

*sin(*

_{m}ψ_{m}*mφ*) for

*s*= −1. Similar to the general case, we can rewrite the boundary integrals in a matrix form

_{x}*χ*= (

*s*− 1)

_{x}*π*/4. For integer

*m*, the Bessel and Hankel functions satisfy

*J*

_{−}

*(*

_{m}*z*) = (−1)

*(*

^{m}J_{m}*z*) and

*ξ*,

*ρ*and

*κ*by repeatedly solving the eigenvalues three times at a starting point

*kR*

_{0}, where

*λ*

_{0}=

*λ*(

*kR*

_{0}) and

*λ*

_{±}=

*λ*(

*kR*

_{0}± Δ). Therefore, we can predict the motion of eigenvalues as

## 4. Application

*kR*

_{0}, and then solve for the parameters {

*ξ,ρ,κ*}. For a 2

*l*× 2

*l*matrix, there are 2

*l*eigenvalues that can be obtained. Thus, we can get 2

*l*sets of parameters {

*ξ*}, with

_{j},ρ_{j},κ_{j}*j*= 1,...,2

*l*. We then can predict 2

*l*possible resonant frequencies as As a first example, we considered a circular silica microdisk, with refractive index

*n*= 1.45. Figure 2(a) shows the non-symmetric GE method for a number of eigenvalues

*λ*(red circles) with

*kR*= {99.5 – 0.4i,...,100.5 – 0.4i}, and shows the motion with respect to

*kR*. Through solving Eq. (17)–(19), at the given

*kR*

_{0}= 99.5 – 0.4i, we can find the parameters {

*ξ,ρ,κ*} for all

*λ*, and then predict the motion of them by Eq. (13), shown in the blue lines. The predictions are very consistent to the actual motion of eigenvalues.

*kR*

_{0}= 100.0 – 0.4i, we can predict the resonances by Eq. (21), shown with blue crosses in Fig. 2(b). Comparing to the exact analytical solution to circular cavities [20

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]

*kR*

_{0}(black solid dot). However, for some eigenvalues far away from the start point

*kR*

_{0}, the predicted roots do not agree well with the exact roots. The reason for this, as analyzed above in Eq. (13), is that the regular motion function is just an approximation, which only works in a small area around

*kR*

_{0}. When

*|kR*–

*kR*

_{0}| is large, the approximation can lead to large errors. We can define a working distance

*L*= |

*kR*

*–*

_{p}*kR*

_{0}|, where the area with radius

*L*around the starting point is within the predictable region. From Fig. 2(b), we can estimate

*L*≈ 1.0, resulting in a valid prediction of about 100 resonances.

*L*≈ 1.0 in Fig. 2(d).

*ϕ*e

_{m}*} and {*

^{imφ}*ψ*e

_{m}*} are orthogonal, the field in a deformed cavity can be expanded in a superposition of multiple components of different*

^{imφ}*m*. According to Eq. (13), we can approximately express

*λ*(

*kR*) ≈

*ξ*+ ∑

_{m}*ρ*e

_{m}^{kR×κm}. For small deformations, the dynamics of lambda is predominantly determined by a single or only a few components. And for different

*m*close to each other, the corresponding

*κ*are nearly the same. Therefore, the approximate function of

_{m}*λ*, Eq. (13) is still adequate to describe the regular eigenvalue motion for deformed cavities. When increasing the deformation, more components of m get involved and give rise to much more complicated dynamics and the motion of

*λ*is less regular, thus the work distance

*L*is reduced. Our numerical results indicated that the regular rotation and radial motion of

*λ*is maintained even for squares and full chaotic cavities. As an example, we use GE method for two coupled stadium cavities with aspect ratio 2:1 [5

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]

*δ*= 10

^{−6}, albeit with a reduced working distance of about

*L*< 0.1, resulting in a prediction of 20 resonances. One of the resonances in the coupled cavities is plotted in Fig. 3(b).

## 5. Discussion

*L*varies for different cavity boundary shapes, we must know the value of

*L*before applying the GE method to search roots. A simple way to estimate the

*L*is to solve the GE at two separate start points

*kR*and

_{a}*kR*, and obtain two arrays of predicted roots {

_{b}*kR*

_{a}_{1},

*kR*

_{a}_{2},...,

*kR*

_{a}_{2}

*} and {*

_{l}*kR*

_{b}_{1},

*kR*

_{b}_{2},...,

*kR*

_{b}_{2}

*}. Since the predictable region are circles of radius*

_{l}*L*with center at starting points, we can estimate the unknown

*L*from the overlap of the two arrays. Plot them in a graph and find the correspondence between the predicted roots of the two different start points, similar to Fig. 2(b). The distance between the two starting points should be smaller than

*L*, for example, we can take |

*kR*–

_{b}*kR*| = 0.05. Through this method, we can estimate the

_{a}*L*quickly by solve GE only twice without knowing the exact roots.

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]

*L*does not increase by increasing of the precision of BEM. The limited

*L*of the GE method does not come from the computation errors, which are intrinsic to the irregular motion of the eigenvalues.

*kR*) < 20.5 and −1 < Im(

*kR*) < 0 with BEM, we have to solve all determinants in order to produce a figure like Fig. 1(a). In order to find all resonances, we set the step size as Δ = 0.01 to get sufficient resolution, which means that we need to solve Eq. (5) 10,000 times. In contrast, in the GE method, we can set the step size to Δ = 0.2 to find all resonances, thus we only need to solve the GE problem for 3×25 = 75 times. In the numerical calculation, the computation time for solving a determinant is similar to that of solving the GE problem. Therefore, the GE method is more than two orders of magnitude faster than the previous method which searches the minima of det(

*M*). In addition, when two modes have eigenfrequencies near to each other, with a distance smaller than Δ, it is difficult find both of them by the Newton’s method.

*λ*, we can also expect a much faster convergence speed of the root search. In Fig. 4, we compare the Newton’s method and GE method for various cavities. All results show a much better convergence of the GE method. Noting that the starting point of the GE mode is placed 0.05 away from the exact resonance, whereas the starting point for the Newton method is 0.005 away from the exact resonance. The Newton method is limited to the convergence to the root which is closest to the starting point, whereas the GE mode can sustain a starting point further away within the working distance

*L*.

*κ*to exclude the spurious roots. In Fig. 1(a), there are some spurious minima in the determinant distribution near the real axis. However, these spurious solutions are avoided in the GE method, as we can see from Fig. 2(b). There are some eigenvalues with very small

*R*

_{0}, corresponding to resonances very far away from the starting point, a threshold is also needed to exclude them.

## 6. Conclusion

## References and links

1. | K. Vahala, “Optical Microavities,” Nature |

2. | J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature |

3. | J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. |

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 |

5. | S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A |

6. | J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. |

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 |

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 |

10. | L.-M. Zhou, C.-L. Zou, Z.-F. Han, G.-C. Guo, and F.-W. Sun, “Negative Goos-Hänchen shift on a concave dielectric interface,” Opt. Lett. |

11. | H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. |

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. |

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 |

14. | H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A |

15. | H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science |

16. | A. Bäcker, “Numerical aspects of eigenvalue and eigenfunction computations for chaotic quantum systems,” in |

17. | J. H. Graf, “Über die Addition und Subtraction der Argumente bei Bessel’schen Functionen nebst einer Anwendung,” Math. Ann. |

18. | M. Abramowitz and I. A. Stegun, |

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 |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(230.5750) Optical devices : Resonators

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 3, 2011

Revised Manuscript: June 21, 2011

Manuscript Accepted: July 15, 2011

Published: August 1, 2011

**Citation**

Chang-Ling Zou, Harald G. L. Schwefel, Fang-Wen Sun, Zheng-Fu Han, and Guang-Can Guo, "Quick root searching method for resonances of dielectric optical microcavities with the boundary element method," Opt. Express **19**, 15669-15678 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15669

Sort: Year | Journal | Reset

### References

- 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]
- J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003). [CrossRef]
- 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]
- 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]
- J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008). [CrossRef] [PubMed]
- 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 .
- 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]
- 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]
- L.-M. Zhou, C.-L. Zou, Z.-F. Han, G.-C. Guo, and F.-W. Sun, “Negative Goos-Hänchen shift on a concave dielectric interface,” Opt. Lett. 36, 624–626 (2011). [CrossRef] [PubMed]
- 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]
- 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. 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, 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, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008). [CrossRef] [PubMed]
- 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]
- J. H. Graf, “Über die Addition und Subtraction der Argumente bei Bessel’schen Functionen nebst einer Anwendung,” Math. Ann. 43, 136–144 (1893). [CrossRef]
- 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).
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.