In the Terahertz (THz) domain, we investigate both numerically and experimentally the directional emission of whispering gallery mode resonators that are perturbed by a small scatterer in the vicinity of the resonators rim. We determine quality factor degradation, the modal structure and the emission direction for various geometries. We find that scatterers do allow for directional emission without destroying the resonator’s quality factor. This finding allows for new geometries and outcoupling scenarios for active whispering gallery mode structures such as quantum cascade lasers and passive resonators such as evanescent sensors. The experimental results agree well with finite difference time domain simulations.
© 2013 OSA
Whispering gallery mode (WGM) resonators are frequently used as laser cavities in quantum cascade lasers (QCLs) [1–3]. This is particularly the case for quantum cascade lasers operating in the THz range (frequencies from 100 GHz to 10 THz, wavelengths between 3 mm and 30 μm), where they have proven to be amongst the most efficient sources of radiation. Besides high efficiency, high directionality of the emitted radiation is a prerequisite for most experiments. Unperturbed WGM resonators are fully symmetric and therefore do not allow for efficient directional free-space coupling. Several different schemes have been developed to induce directional emission. Smoothly deforming the boundary can change the optical mode from a regular to a wave-chaotic behavior. This allows for high quality (Q) factors and directional emission from modes localized on stable periodic lightray-orbit [1,4] or by the short term dynamics of the chaotic light ray-dynamics inside the resonator which channels a direct emission port [5–7]. Efficient coupling of such emitted waves into waveguides has also been studied [8,9]. Recently a different approach was utilized, where the surface of a spherical resonator was strongly perturbed by a small cut  or outside scatterer  and directional emission was observed.
In this paper, we present another way to achieve directional emission: we study directional emission from a WGM resonator with a finite scatterer within the resonator both experimentally and theoretically in the THz frequency range. Theoretical calculations initiated by Wiersig et al. [11–14] already described efficient outcoupling of a high Q mode via interaction with a low Q mode without ruining the Q factor. Here, however, we address the mode directly by perturbing the mode with the scatterer. We will show that the perturbation may be weak enough to maintain a considerably high Q factor while the power is coupled out directionally. Besides directional emission at high Q factors, finite scatterers may further assist to improve the threshold behaviour of WGM-based lasers : they can be used to perturb non-lasing modes which suppresses spontaneous emission. Symmetrically spaced perturbations on the rim or above the disk can act as gratings and efficiently couple light out of the resonator [2, 16] and also provide a novel way to generate optical angular momentum beams . Scatterers therefore present an excellent alternative to improve both directionality and lasing performance of WGM-based lasers, in particular, quantum cascade lasers.
In passive WGM resonators that are frequently used for sensing applications, a finite scatterer may also be implemented. Several detection schemes employ the evanescent field surrounding a WGM resonator for sensing. Gas, fluids, or particles surrounding the resonator influence the evanescent field by either shifting the effective refractive index or by absorption [18–20]. Consequently, this alters the modal structure of the resonator, resulting in mode shifts and altered quality (Q) factors. Instead of addressing the evanescent field, the scatterer can be designed as a hole, which can be flooded with the species under test. Since the species is introduced right into the resonator, the overlap of the WGM mode with the species under test can be engineered by the position of the hole, improving the sensitivity of the device.
For our studies, we examine waveguide-coupled, passive resonators at THz frequencies. The THz frequency range offers unique advantages for studying WGM resonators: the wavelength is roughly a factor of 1000 larger than in the visible domain. Much larger resonators and feature sizes have to be used compared to the optical domain, simplifying the experimental setup. Optical and physical effects remain the same due to the scalability of electromagnetism. It is therefore much easier to characterize the near and far field . However, materials are less advanced than in the optical domain, resulting in losses and therefore much lower quality factors. We developed a THz system based on n-i-pn-i-p superlattice photomixers [22, 23] that allows for characterizing WGM resonators between 60 GHz and several hundreds of GHz .
The paper is structured as follows: First, we measure the influence of the waveguide on the modal structure and near field of the resonator by a near field probe. Second, we experimentally determine the influence of the position of a hole acting as scatterer on the Q factor of the disk. Third, we will explain the numerical algorithm that was used to theoretically determine the radiation pattern of a disk with hole. Last, we will show experimental results on the outcoupling performance and compare the results to theory.
2. Experimental setup
We use a n-i-pn-i-p superlattice photomixer  as THz source that has been developed in house. The measurement setup is illustrated in Fig. 1(a) and (b). The source is mounted on a silicon lens for pre-collimation and efficient out-coupling of the THz beam. The beam is then collimated with a parabolic mirror (PM1). A second parabolic mirror (PM2) focuses the THz beam on a feed horn (FH) with a rectangular Teflon waveguide (dimensions 1 mm x 1.5 mm) mounted inside the horn. The feed horn is used to increase the coupling efficiency. The Teflon waveguide delivers the THz power to the WGM resonator. A Golay cell detector (D2) is used to monitor the transmission through the waveguide. The WGM resonator is situated in the center of a rotation stage. The Golay cell can be mounted on the rotation stage to scan the far field emission from the resonator. A horizontal cylindrical (HL) lens collects THz power emitted perpendicular to the detection plane. A second, vertical cylindrical lens (DL) is mounted in front of the Golay cell to collect power within a 10°–15° angle to improve the signal to noise ratio. The scan range was limited by the Teflon waveguide and the size of the Golay cell and the lenses to about 140°. Alternatively, the Golay cell detector can also be attached to a probe waveguide that touches the resonator in order to scan the near field. In order to reduce the necessary integration times of the Golay cell novel field-effect transistors could also be used [24, 25].
The frequency resolution (120 MHz) of the system is sufficient to measure Q factors up to 1200 in the frequency range from 500 − 350 GHz. Due to the lack of ultra-low-loss materials at THz frequencies, this resolution is sufficient since the ideal loss-limited Q factors are in the range of a few 1000. As an example, the imaginary part of the dielectric constant of polyethylene and Teflon  is in the range of ε″ > 1.8 × 10−3, resulting in an absorption-limited Q factor of < 1100, even if no radiative losses are present. For the investigated disks and frequency ranges, however, the measured Q factor was mostly smaller than 1200, limited by radiative losses.
In a first step, we investigate the perturbation of the coupler on the modal structure of the resonator without scatterer. An unperturbed circular resonator should show a position-independent modal field strength. We resolved the local field strength by a near field tapered probe waveguide that weakly couples to the disk. In Fig. 2(a) we show the transmitted power through the feed waveguide and in Fig. 2(b) the outcoupled power through the probe while scanning both frequency across a resonance and the position of the probe. As expected, the transmitted power through the feed waveguide is high, when little power is coupled out through the probe waveguide. However, we also see a strong, periodic probe position dependence of the power in both sub-figures with a five fold symmetry. This number is much too small to be the modal number (∼ 50) for a resonator with a diameter of 25 mm at a wavelength of 2.24 mm. Already the coupling to the feed waveguide perturbs the modal structure of the resonator considerably. The orbit of the mode is not circular any more, there exist positions where the mode is close to the surface, allowing for enhanced outcoupling, and further away, showing local minima in Fig. 2(b).
As a next step, we investigated the influence of the hole position on the quality factor of the resonances with disks with a diameter of d = 10.5 mm at wavelengths between 2.3 mm and 0.91 mm (0.13–0.33 THz). For holes very far away from the circumference, the outermost radial modes that are excited by the waveguide, are not affected by the hole as their mode volume does not overlap with it. The Q factors are similar to that of a resonator without hole and the modal structure remains the same. When the hole is drilled closer to the circumference, the mode position shifts because the mode gets perturbed by the hole (Fig. 3(b)). At frequencies below 200 GHz, the radiative losses of the unperturbed resonator are larger than the losses introduced by the hole for all investigated hole positions. Therefore, the Q factor is not very sensitive to the hole position. Above 200 GHz, the overall Q-factor decreases as shown in Fig. 3(a). The effect becomes more pronounced if the boundary thickness b (see Fig. 1(c)) becomes comparable to or smaller than the wavelength. The loss-limited quality factor can be estimated using an empiric fit of the formFig. 3 shows the extrapolated values for Qloss. The disks with b = 0.38 mm and the disk with the slit are already loss-limited within the measurement range.
The Q factors did not degrade drastically due to the perturbation of the hole. This demonstrates that finite scatterers can be used for resonators and laser cavities. In the following, we discuss the theoretical framework for determining the outcoupling performance by the hole.
3. Theoretical modeling
We now turn to our theoretical approach to model the experimental data. Such modeling is desirable mainly for two reasons. On the one hand, the model gives insight into the physical mechanism for the directional emission, not least since it provides access to system properties which may be difficult to obtain in the experiment. On the other hand, in order to design the emission pattern of WGM resonators towards specific applications, an accurate theoretical modeling is desirable in order to perform numerical parameter studies. For stationary resonance problems a number of numerical methods exist, such as scattering methods , boundary integral methods [28–30] and multipole methods . These methods however do not allow for studying the effects of the coupled waveguide.
The light propagation is thus simulated by the Finite-Difference Time-Domain (FDTD) Method [32–34]. The geometry of waveguide and resonator as well as the material properties are chosen as in the experiment, without fine-tuning of any of the parameters. We however restrict the simulation to a two dimensional rectangular grid in the resonator plane, with grid constant Δx = Δy = 0.02 mm. This corresponds to about λ/90 or λ/110 for the two wavelengths in the mm range we consider below. In two dimensions, Maxwell’s equations for the six components of E⃗ and H⃗ separate into two disjunct sets of three equations each. As in the experiment, we study the propagation of the EM field in TEz polarization, such that Hz, Ex and Ey need to be considered. We drive the Hz component using a soft source placed at one edge of the waveguide, with a transverse mode profile obtained by solving the corresponding Helmholtz equation. Unwanted reflections from the boundaries of the simulation area are suppressed by Berenger type perfectly matched absorbing boundaries .
To predict the emission pattern, we proceed in three steps. First, we excite the system with a temporally short and spectrally broad pulse and monitor the transmission at the waveguide edge opposite to the source. By relating this result to a reference calculation without the resonator, we obtain the modal spectrum of the resonator. We then choose a single resonator mode by matching the numerically obtained mode spectrum to the experimentally observed one.
In the second step, we excite the system with a monochromatic continuous-wave field at the resonance frequency of the mode identified in the first step, and evolve the system into its stationary state (modulo the oscillations at the incident field frequency). This evolution can be monitored, e.g., via the time-dependent power flux through the waveguide downstream the resonator. The steady state, where the resonator is fully loaded, is reached on time scales short enough to neglect the initial cavity build-up time.
In the third step, we calculate the emission pattern of the resonator. A principle problem arises from restraining the computation time to a moderate level. The FDTD simulations therefore are constrained to a small region around the resonator (∼ tens of mm) which are small compared to the experimental detector distance (∼ 10 cm). Thus it is impractical to include the detector itself into the simulation. But while the detector exclusively monitors the far field radiation component of the emission, the vicinity of the resonator is strongly influenced by near field contributions. Thus, a transformation of the FDTD results into the far field is required [32–34]. To this end, we calculate the amplitude and the phase of the electric and magnetic fields on a circle around the resonator center, with radius larger than the resonator radius by a variable offset ρ. At each FDTD grid point r⃗ on the circle, we fit the FDTD time evolution over few cycles of the incident field to the function a sin(ωt + ϕ). Here, ω is the frequency of the incident field; the amplitude a > 0 and the phase ϕ are the fit parameters. From the field configuration, we evaluate the time averaged Poynting vector S⃗(r⃗) at each point r⃗ on the circle. Next, we determine the intersection point R⃗(r⃗) of a ray starting at r⃗ in the direction S⃗ with the detection sphere of radius R around the resonator center. As a result, the point r⃗ contributes with magnitude |S⃗| to the detection signal at detector position R⃗(r⃗). By incoherently summing up the contribution of all points on the circle around the resonator we obtain a prediction for the emission pattern as observed by the detector.
We compared the emission patterns evaluated for different offsets ρ from the resonator edge, and found that for small offsets, the results strongly depend on ρ. However, already with offsets ρ of few wavelengths of the incident light field, the radiation pattern becomes largely independent of the offset, indicating the suppression of near field components. It should be noted, however, that neither the simulation range nor the detector distance in the experiment are large enough to neglect the finite size of the resonator.
4. Comparison of theoretical and experimental results
We investigated two types of resonators for comparison with our theoretical framework. First we consider a polyethylene WGM resonator with R = 5.2 mm, rh = 0.51 mm, b = 0.41 mm and αh = 90°. The relative permittivity is εWGM = 2.56 (nPE = 1.6) for the resonator and εh = 1 (air) inside the hole. This resonator is referred to as small resonator in the following. Its resonance frequencies and Q factors are illustrated in Fig. 3. The far field emission was characterized for angles between 160° and 290° at a resonance frequency of 170 GHz (λ = 1.76 mm).
The dimensions of our second resonator are R = 12.5 mm, rh = 1.25 mm and b = 1 mm. In the following, this resonator is referred to as large resonator. The larger size was chosen to have less radiative losses of the unperturbed resonator and to simplify probing the near field with a probe waveguide. The large resonator was studied at αh = 47° and αh = 315°.
Figure 4(a) shows a FDTD snapshot indicating the considered geometry. Figure 4(b) depicts the result of the Poynting vector analysis for the small resonator. The Poynting vector structure clearly resembles the interference pattern visible in the field configuration. However, the radiation pattern cannot be read off directly from the Poynting vector structure, as the projection of the pattern onto the detection sphere has not yet been applied. The experimental data are compared to the prediction for the far-field pattern from the Poynting analysis in Fig. 4(c). The comparison between experiment and theory in (c) shows excellent agreement. The theoretical curve was obtained for a detector size of 10°, and a detector distance of 10 cm, consistent with the experiment. Apart from an overall scaling, no further free parameter had to be adjusted. The scaling was chosen such that the largest relative emission from the theoretical and the experimental data coincide. The main lobes as well as the intensity minima observed in the experiment are clearly reproduced in the theoretical analysis. The main difference arises at 270°, where the theoretical data exhibits a reduction in the emission intensity. This most likely is due to the waveguide, which is situated at this emission angle in the theoretical calculation, such that an angular range around 270° had to be excluded from the Poynting vector analysis. In the experiment, the waveguide was bent, such that it did not interfere with the emitted radiation at this angle, see Fig 1. The Poynting vector analysis and the comparison between experiment and theory for the large resonator with hole at αh = 315° are shown in Fig. 5. Again, a detector size of 10°, and a detector distance of 10 cm were chosen. Good qualitative matching between experiment and theory is achieved, even though the quantitative agreement is not as good as for the small resonator. The main difference is the width of the second lobe around 130° − 180°, which is wider in the experimental data than in the theoretical prediction. One possible origin for this could be slightly incorrect values for the index of refraction or the distance to the coupling waveguide to the resonator which could not be measured accurately, altering the perturbation of the resonator. Furthermore, the disk center is centered in the rotation stage. There may be some imaging error from radiation emitted at the disk boundary (being 12.5 mm off-axis) to the detector despite the 10° collimation angle (see Fig. 1). This effect is much weaker for the small resonator, where theory and experiment agreed excellently. Finally, the results for the large resonator with hole at αh = 47° are shown in Fig. 5. The first main lobe at around 120° predicted in the theoretical analysis is missing in the experimental data, but experiment and theory agree well for the second lobe around 200°. From the Poynting vector analysis, it can be concluded that the peak around 120° originates from the coupling region of the resonator.
Qualitatively, all emission patterns can be interpreted in the following way. On the one hand, there is emission in the direction of the tangent to the resonator at the hole position. This best visible for the small resonator around 270° in Fig. 4, but also around 120° in for the large resonator with hole at αh = 47°, see Fig. 5(a–c). For the large resonator with αh = 315° in Fig. 5(d–f), this contribution is masked by the waveguide and the coupling region. A naive picture for this tangential emission is that a fraction of the light is deflected by the scatterer such that it acquires a larger incidence angle on the resonator surface compared to the unperturbed resonator mode, and therefore is able to scatter out of the resonator. Next to the main lobe in tangential direction, also side lobes with higher scattering angle are visible. These appear as side lobes at 210° and 180° in Fig. 4 for the small resonator. Note that the structure at 180° almost corresponds to a radial emission from the resonator, i.e., almost perpendicular to the energy flow in the resonator mode. The structure around 210° corresponds to about 45° deflection. From the simulation data, it also appears that similar higher deflection angles also occur towards the resonator center, even though they are masked in the emission pattern since they would have to pass both the resonator and the waveguide before they could reach the detector.
Contributions with larger scattering angles originating from positions far away from the scatterer are also visible for the large resonator. The field configurations in Fig. 5(a,d) suggest that these originate from perturbations of the mode by the scatterer. These perturbations lead to coupling to other modes. The total modal orbit observed in the theoretical simulation in Fig. 5(a) therefore approximately resembles a hexagon, while that of Fig. 5(d) resembles a pentagon. This results in out-coupled power at the edges of the modal orbits, also far away from the hole.
The emission patterns for the two hole positions shown in Fig. 5 with respect to the coupling position substantially differ. This is in agreement with our findings in Fig. 2. The waveguide already perturbs the modal structure. This effect interferes with the effect of the hole. Both perturbations have to be taken into account in order to determine the emission pattern.
In conclusion, we demonstrated both experimentally by measurements in the THz domain and theoretically by a finite difference time domain simulation, that a finite scatterer close to the circumference of the whispering gallery mode resonator allows for directional outcoupling without ruining the quality factor. The main emission direction was tangential, originating at the scatterer. However, there exist side lobes with larger emission angles. Influence of other perturbations such as a coupling waveguide have to be taken into account in order to numerically determine the emission direction. With a finite difference time domain method, we achieved good agreement between the experimental data and the theoretical modeling in all investigated cases.
We thank H. Lu and A. C. Gossard at MRL Santa Barbara, CA, USA, for growing the substrate of the THz devices. We further would like to thank G. Leuchs and his devision at the MPL for the creative atmosphere and the financial support. We acknowledge support by the Deutsche Forschungsgemeinschaft and the Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding program Open Access Publishing.
References and links
1. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998) [CrossRef]
2. L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie, “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009) [CrossRef]
3. Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. U.S.A. (2010).
4. S.-K. Kim, S.-H. Kim, G.-H. Kim, H.-G. Park, D.-J. Shin, and Y.-H. Lee, “Highly directional emission from few-micron-size elliptical microdisks,” Appl. Phys. Lett. 84, 861–863 (2004) [CrossRef]
5. H. G. L. Schwefel, N. B. Rex, H. E. Türeci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004) [CrossRef]
7. X.-F. Jiang, Y.-F. Xiao, C.-L. Zou, L. He, C.-H. Dong, B.-B. Li, Y. Li, F.-W. Sun, L. Yang, and Q. Gong, “Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities,” Adv. Mater. 24, OP260–OP264 (2012) [CrossRef]
8. A. A. Danylov, J. Waldman, T. M. Goyette, A. J. Gatesman, R. H. Giles, K. J. Linden, W. R. Neal, W. E. Nixon, M. C. Wanke, and J. L. Reno, “Transformation of the multimode terahertz quantum cascade laser beam into a gaussian, using a hollow dielectric waveguide,” Appl. Opt. 46, 5051–5055 (2007) [CrossRef]
9. M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Angular emission characteristics of quantum cascade spiral microlasers,” Opt. Express 17, 10335–10343 (2009) [CrossRef]
10. Y.-C. Liu, Y.-F. Xiao, X.-F. Jiang, B.-B. Li, Y. Li, and Q. Gong, “Cavity-QED treatment of scattering-induced free-space excitation and collection in high-Q whispering-gallery microcavities,” Phys. Rev. A 85, 013843 (2012) [CrossRef]
11. J. Wiersig and M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. A 73, 031802 (2006) [CrossRef]
12. C. P. Dettmann, G. V. Morozov, M. Sieber, and H. Waalkens, “Unidirectional emission from circular dielectric microresonators with a point scatterer,” Phys. Rev. A 80, 063813 (2009) [CrossRef]
13. J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A 81, 053827 (2010) [CrossRef]
14. R. F. M. Hales, M. Sieber, and H. Waalkens, “Trace formula for a dielectric microdisk with a point scatterer,” J. Phys. A 44, 155305 (2011) [CrossRef]
15. S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176–178 (1999) [CrossRef]
16. D. C. Aveline, L. Baumgartel, B. Ahn, and N. Yu, “Focused ion beam engineered whispering gallery mode resonators with open cavity structure,” Opt. Express 20, 18091–18096 (2012) [CrossRef]
17. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. OBrien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012) [CrossRef]
19. H.-P. Loock, J. A. Barnes, G. Gagliardi, R. Li, R. D. Oleschuk, and H. Wächter, “Absorption detection using optical waveguide cavities,” Can. J. Chemistry 88, 401–410 (2010) [CrossRef]
20. E. N. Shaforost, N. Klein, S. A. Vitusevich, A. Offenhusser, and A. A. Barannik, “Nanoliter liquid characterization by open whispering-gallery mode dielectric resonators at millimeter wave frequencies,” J. Appl. Phys. 104, 074111 (2008) [CrossRef]
21. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, 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]
22. S. Preu, F. H. Renner, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, A. C. Gossard, T. L. J. Wilkinson, and E. R. Brown, “Efficient terahertz emission from ballistic transport enhanced n-i-p-n-i-p superlattice photomixers,” Appl. Phys. Lett. 90, 212115 (2007) [CrossRef]
23. S. Preu, G. H. Döhler, S. Malzer, L. J. Wang, and A. C. Gossard, “Tunable, continuous-wave terahertz photomixer sources and applications,” J. Appl. Phys. 109, 061301 (2011) [CrossRef]
24. S. Preu, S. Kim, R. Verma, P. G. Burke, M. S. Sherwin, and A. C. Gossard, “An improved model for non-resonant terahertz detection in field-effect transistors,” J. Appl. Phys. 111, 024502 (2012) [CrossRef]
26. I. Microtech Instruments, “http://www.mtinstruments.com/thzresources/thzpropertiesofmaterials.pdf,”.
27. H. E. Türeci, H. G. L. Schwefel, P. Jacquod, and A. D. Stone, “Modes of wave-chaotic dielectric resonators,” in “Progress in Optics” 47E. Wolf, ed. (Elsevier Science Bv, Amsterdam), pp. 75–137 (2005) [CrossRef]
28. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A 5, 53–60 (2003) [CrossRef]
29. H. E. Türeci and H. G. L. Schwefel, “An efficient Fredholm method for the calculation of highly excited states of billiards,” J. Phys. A 40, 13869–13882 (2007) [CrossRef]
30. C.-L. Zou, H. G. L. Schwefel, F.-W. Sun, Z.-F. Han, and G.-C. Guo, “Quick root searching method for resonances of dielectric optical microcavities with the boundary element method,” Opt. Express 19, 15669–15678 (2011) [CrossRef]
31. 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]
32. S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, “FDTD microcavity simulations: design and experimental realization of waveguide-coupled single-mode ring and whispering-gallery-mode disk resonators,” J. Lightwave Technol. 15, 2154–2165 (1997) [CrossRef]
33. A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method,3rd ed. (Artech HouseBoston, 2005).
34. S. I. Schmid, K. Xia, and J. Evers, “Pathway interference in a loop array of three coupled microresonators,” Phys. Rev. A 84, 013808 (2011) [CrossRef]
35. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. of Comput. Phys. 114, 185–200 (1994) [CrossRef]