## Generalized full-vector multi-mode matching analysis of whispering gallery microcavities |

Optics Express, Vol. 22, Issue 11, pp. 13507-13514 (2014)

http://dx.doi.org/10.1364/OE.22.013507

Acrobat PDF (1169 KB)

### Abstract

We outline a full-vectorial three-dimensional multi-mode matching technique in a cylindrical coordinate system that addresses the mutual coupling among multiple modes co-propagating in a perturbed whispering gallery mode microcavity. In addition to its superior accuracy in respect to our previously implemented single-mode matching technique, this current technique is suitable for modelling waveguide-to-cavity coupling where the influence of multi-mode coupling is non-negligible. Using this methodology, a robust scheme for hybrid integration of a microcavity onto a silicon-on-insulator platform is proposed.

© 2014 Optical Society of America

## 1. Introduction

1. D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature **421**, 925–928 (2003). [CrossRef] [PubMed]

8. M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. **99**, 073701 (2011). [CrossRef]

^{6}[9

9. M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express **13**, 1515–1530 (2005). [CrossRef] [PubMed]

^{8}, such as a silica microtoroid reported in [1

1. D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature **421**, 925–928 (2003). [CrossRef] [PubMed]

10. M. Hossein-Zadeh and K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express **15**, 166–175 (2007). [CrossRef] [PubMed]

11. D. Rowland and J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J **140**, 177–188 (1993). [CrossRef]

15. A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express **21**, 14169–14180 (2013). [CrossRef] [PubMed]

16. 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). [CrossRef]

20. M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. **15**, 083006 (2013). [CrossRef]

21. X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express **21**, 22012–22022 (2013). [CrossRef] [PubMed]

## 2. Theoretical formulations

*ϕ*) may support multiple transverse modes of the same azimuthal order

*M*distinguished by resonance wavelengths. Consequently, continuous wave (CW) laser light delivered to the cavity excites a specific mode if its optical wavelength coincides with the resonance wavelength of that mode. The corresponding electric field distribution of an

*l*

^{th}-order transverse mode can be expressed as

**E**(

**r**) =

*Aẽ̂*(

_{M,l}*ρ*,

*z*)

*e*

^{jm̃lϕ}. The term

*m̃*=

_{l}*M*+

*jm̃*is a complex number whose real part is an integer

_{i,l}*M*representing the azimuthal order of the WGM and whose imaginary part

*m̃*characterizes the intrinsic quality factor of the mode according to

_{i,l}*Q*=

*M*/2

*m̃*. The unit vector

_{i,l}*ẽ̂*(

_{M,l}*ρ*,

*z*) is the normalized

*ϕ*-independent mode field distribution such that the squared amplitude |

*A*|

^{2}represents the total energy of the light stored in the cavity (in units of Joules). For convenience, we have chosen a cylindrical coordinate system (

*ρ*,

*ϕ*,

*z*) as defined in [21

21. X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express **21**, 22012–22022 (2013). [CrossRef] [PubMed]

*m̃*≪ 1 and, with a good approximation, the discrete set of normalized

_{i,l}*M*

^{th}-order azimuthal whispering gallery modes

*ẽ̂*(

_{M,l}*ρ*,

*z*)

*e*

^{jm̃lϕ}span an orthogonal basis in Hilbert space. In a cylindrical coordinate system [22

22. H. A. Haus, *Electromagnetic Noise and Quantum Optical Measurements* (Springer, 2000). [CrossRef]

*ε*(

*ρ*,

*z*,

*ϕ*) =

*ε*(

*ρ*,

*z*)) and Eq. (1) can then be simplified to

*δ*is the Kronecker delta and bra-ket notation is used to define the inner product in the functional space spanned by the WGM’s as 〈

_{lk}*a*|

*b*〉 =

*π*∬

*εa*

^{*}

*·bρdρdz*where

*a*

^{*}is the complex conjugate of

*a*. For convenience, we also drop the subscript

*M*by limiting our discussion to the same azimuthal order

*M*. Also note that in the case of a large cavity, where the field is tightly focused into a small spot of mean radius

*R*and proper normalization is adopted, the orthonormality condition Eq. (2) can be approximated by

21. X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express **21**, 22012–22022 (2013). [CrossRef] [PubMed]

*e*

^{−m̃i,lϕ}is extracted while rendering

*ẽ̂*(

_{l}*ρ*,

*z*) lossless. In this sense, the orthogonality condition for

*ẽ̂*(

_{l}*ρ*,

*z*) will continue to hold. The integral additionally extends to the full space while, in practice, one numerically calculates the modal field with a finite computation window and hence resolves the integral within that window. Such an approximation reduces accuracy due to the neglect of the field radiating to infinity; however, for cases in which the material and mode mismatch losses dominate over the radiation loss, this provides sufficient accuracy for high-Q cavities where the radiation field is small.

*A⃗*becomes a scalar and Eq. (8) reverts to its simplified form as seen in the single-mode propagation equation (i.e. Eq. (11) of [21

**21**, 22012–22022 (2013). [CrossRef] [PubMed]

## 3. Application and discussion

_{2}layer of 220 nm and 2

*μ*m respective thicknesses. The refractive indices of silica (1.44462) and silicon (3.48206) are taken from the literature [23

23. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. **55**, 1205–1208 (1965). [CrossRef]

*μ*m and a minor radius of 5

*μ*m.

^{th}fundamental quasi-TE toroid mode was launched at

*ϕ*= −0.34 rad, wherein the waveguide was 3 wavelengths away from the cavity. The launched transverse electrical field distributions are displayed in the two left insets of Fig. 1b. Beyond that point, the coupling between the cavity and the waveguide falls to a negligible level (with relative Q degradation below 10

^{−10}) and so it is neglected. As shown in the main plot of Fig. 1b, the amplitude of the launched mode diminishes while that of the second lowest-order mode reaches its maximum when the light propagates towards the center of the coupling region. It is also evident from the two right insets of Fig. 1b that the cavity mode launched as the input gradually evolves to a strongly hybridized cavity-taper mode when the light is approaching

*ϕ*= 0 rad. The hybrid mode eventually relaxes to the unperturbed cavity mode at the exit of the cavity-taper coupling region and most of the light energy is coupled back to the originally launched mode in a similar manner as that of a directional coupler. By computing the total energy loss of the launched mode subtracted by the contribution from the intrinsic Q, we obtain a coupling Q-factor of 10

^{5}. The coupling Q calculated here takes the loss from the launched mode to both the tapered waveguide and higher-order cavity modes into account. To precisely estimate the energy delivered to the waveguide, one may simply compute the overlap factor between the output field at

*ϕ*= 0.34 rad and the mode of the tapered waveguide. To characterize the convergence rate of our algorithm, we plotted the reciprocal of the quality factor as a function of azimuthal angle steps and the number of modes included (given by the blue triangles in Figs. 1c and 1d). The expectation value of this quantity at an infinitesimal azimuthal step or infinite mode numbers (i.e. blue dashed line) was extracted through the Richardson extrapolation procedure [25] and the relative error (i.e. red cross markers) was estimated with the extrapolated value as a reference. The black line in Fig. 1c depicts a convergence rate of

*O*(Δ

*ϕ*

^{1.0}), while that of Fig. 1d yields a convergence rate of about

*O*(

*N*

^{0.8}).

*m′*remains constant between

_{l}*ϕ*

_{0}and

*ϕ*

_{0}+

*δϕ*. A more generalized form which takes the continuous variation of

*m′*into account should be By assuming

_{l}*m′*is a well behaved function between

_{l}*ϕ*

_{0}and

*ϕ*

_{0}+

*δϕ*, we can expand it as a power series around

*ϕ*

_{0}:

*δϕ*where the second power series term dominates the error,

*m′*(

_{l}*ϕ*) =

*m′*(

_{l}*ϕ*

_{0}) +

*O*(

*δϕ*). The 1/

*Q*term characterizes the overall loss contributions from both

*m′*, defined as the imaginary part of

_{i,l}*m′*, and the mode mismatch loss

_{l}*m*. The accuracy of the latter term is determined by the transverse grid spacing, the number of modes involved in the computation, and the relative insensitivity to

_{kl}*δϕ*for small azimuthal steps. Consequently, the adequately small azimuthal steps will procure a convergence rate for 1/

*Q*as a function of

*δϕ*determined by the convergence of

*m′*, which is of

_{i,l}*O*(

*δϕ*). At larger angles, this will be influenced by higher-order power series terms and the mode mismatch loss will deviate from the convergence rate of

*O*(

*δϕ*).

*O*(1). In the particular simulation of our structure, the convergence rate is

*O*(

*N*

^{0.8}). The least square fit excludes the

*N*= 1 point, as it has larger relative error. This is caused by the odd parity at

*N*= 1 and sampling at the center of the structure, whereas the rest of the simulations have even parity and are sampled symmetrically about the center.

*Q*) is equal to the intrinsic Q-factor of the cavity (assumed to be a practical value of 2×10

_{c}^{8}, as specified by the dot-dashed line of Figs. 2a and 2b), we gradually displaced the tapered waveguide horizontally away from the toroid. As displayed in Fig. 2a, a

*Q*on the order of 10

_{c}^{5}is computed when the waveguide touches the cavity surface and 10

^{10}is computed at a gap size of 2.5 wavelengths. For a waveguide sitting in the equatorial plane, a gap size of 0.75

*μ*m is also determined to be desirable in establishing critical coupling. The dependence of the Q-factor as well as the coupling parameter K on the gap is akin to the data in [26

26. S. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. **91**, 043902 (2003). [CrossRef] [PubMed]

*μ*m-diameter microsphere and 1.35

*μ*m-diameter fiber at

*λ*=1550 nm) for the low-Q/accuracy-inhibiting regime in Fig. 2c are in good agreement with the experimental results of [26

26. S. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. **91**, 043902 (2003). [CrossRef] [PubMed]

^{8}to 10

^{7}. Therefore, highly precise alignment is required in order to integrate a toroid onto an SOI platform with this scheme.

*θ*in respect to the equatorial plane, as illustrated in the inset of Fig. 2b. The Q-factor as a function of

*θ*is shown in the main plot as solid circle markers, wherein a Q-factor of 3 × 10

^{5}is calculated at

*θ*= 0. It is also worth mentioning that the

*θ*= 0 position (corresponding to the top surface of the SOI waveguide being aligned with the equator) is different from that in the previous case (i.e. the midplane of the silicon layer being aligned with the equator). At a larger

*θ*, the local field intensity of the cavity mode is smaller. As a result, coupling between the cavity and the waveguide is weaker and a larger coupling Q-factor is observed. The right axis of Fig. 2b represents the magnitude of the electric field squared |

*E*|

^{2}on the surface of the cavity at different

*θ*. It is important to recognize that the Q-factor function inversely resembles the |

*E*|

^{2}curve. For a straight SOI waveguide physically touching the toroid surface and that is placed below the equator, a waveguide location at

*θ*= 65° is favourable in achieving critical coupling. In contrast to the previous case, a waveguide height fluctuation of 1

*μm*yields a drop of coupling Q from 10

^{8}to 10

^{7}. Control of such misalignment tolerance can be easily achieved with conventional technology.

## 4. Conclusion

## References and links

1. | D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature |

2. | F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods |

3. | V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters |

4. | T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA |

5. | Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, and X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst |

6. | G. Bahl, X. Fan, and T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. |

7. | V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. |

8. | M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. |

9. | M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express |

10. | M. Hossein-Zadeh and K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express |

11. | D. Rowland and J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J |

12. | M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B |

13. | M. A. C. Shirazi, W. Yu, S. Vincent, and T. Lu, “Cylindrical beam propagation modelling of perturbed whispering-gallery mode microcavities,” Opt. Express |

14. | C.-G. Xu, X. Xiong, C.-L. Zou, X.-F. Ren, and G.-C. Guo, “Efficient coupling between dielectric waveguide modes and exterior plasmon whispering gallery modes,” Opt. Express |

15. | A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, and H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express |

16. | M. Oxborrow, “Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microw. Theory Tech. |

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

18. | Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, and Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A |

19. | I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B |

20. | M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. |

21. | X. Du, S. Vincent, and T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express |

22. | H. A. Haus, |

23. | I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. |

24. | M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, and E. Van Stryland, |

25. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

26. | S. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. |

27. | H. Derudder, F. Olyslager, D. De Zutter, and S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. |

**OCIS Codes**

(040.1880) Detectors : Detection

(230.3990) Optical devices : Micro-optical devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: February 3, 2014

Revised Manuscript: April 29, 2014

Manuscript Accepted: May 15, 2014

Published: May 28, 2014

**Citation**

Xuan Du, Serge Vincent, Mathieu Faucher, Marie-Josée Picard, and Tao Lu, "Generalized full-vector multi-mode matching analysis of whispering gallery microcavities," Opt. Express **22**, 13507-13514 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13507

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### References

- D. Armani, T. Kippenberg, S. Spillane, K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]
- F. Vollmer, S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008). [CrossRef] [PubMed]
- V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Letters 13, 3347–3351 (2013). [CrossRef] [PubMed]
- T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Nat. Acad. Sci. USA 108, 5976–5979 (2011). [CrossRef] [PubMed]
- Y. Sun, J. Liu, G. Frye-Mason, S.-J. Ja, A. K. Thompson, X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009). [CrossRef] [PubMed]
- G. Bahl, X. Fan, T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012). [CrossRef]
- V. Ilchenko, A. Matsko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006). [CrossRef]
- M. Santiago-Cordoba, S. Boriskina, F. Vollmer, M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011). [CrossRef]
- M. Borselli, T. Johnson, O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005). [CrossRef] [PubMed]
- M. Hossein-Zadeh, K. J. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007). [CrossRef] [PubMed]
- D. Rowland, J. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” Optoelectronics, IEEE Proceedings J 140, 177–188 (1993). [CrossRef]
- M. L. Gorodetsky, V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999). [CrossRef]
- M. A. C. Shirazi, W. Yu, S. Vincent, T. Lu, “Cylindrical beam propagation modelling of perturbed whispering-gallery mode microcavities,” Opt. Express 21, 30243–30254 (2013). [CrossRef]
- C.-G. Xu, X. Xiong, C.-L. Zou, X.-F. Ren, G.-C. Guo, “Efficient coupling between dielectric waveguide modes and exterior plasmon whispering gallery modes,” Opt. Express 21, 31253–31262 (2013). [CrossRef]
- A. Kaplan, M. Tomes, T. Carmon, M. Kozlov, O. Cohen, G. Bartal, H. G. L. Schwefel, “Finite element simulation of a perturbed axial-symmetric whispering-gallery mode and its use for intensity enhancement with a nanoparticle coupled to a microtoroid,” Opt. Express 21, 14169–14180 (2013). [CrossRef] [PubMed]
- 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). [CrossRef]
- J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53 (2003). [CrossRef]
- Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012). [CrossRef]
- I. Teraoka, S. Arnold, F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003). [CrossRef]
- M. R. Foreman, F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys. 15, 083006 (2013). [CrossRef]
- X. Du, S. Vincent, T. Lu, “Full-vectorial whispering-gallery-mode cavity analysis,” Opt. Express 21, 22012–22022 (2013). [CrossRef] [PubMed]
- H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000). [CrossRef]
- I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]
- M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. Macdonald, V. Mahajan, E. Van Stryland, Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992).
- S. Spillane, T. J. Kippenberg, O. J. Painter, K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef] [PubMed]
- H. Derudder, F. Olyslager, D. De Zutter, S. Van Den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antennas. Propag. 49, 185–195 (2001). [CrossRef]

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