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The even-parity (16,2) TM resonant mode in the limaçon-shaped TC with = . 0 24 , n 0 = 2.0 and β = 1.0. (a,b) Wave intensity patterns obtained by BEM and FEM, respectively. (c) Far-field intensity distribution which are obtained by BEM (solid black lines) and FEM (dashed red lines). Inset of (c) is a zoom-in plot for the range of 90~270 degree.
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In the case of two-dimensional gradient index cavities designed by the conformal transformation optics, we propose a boundary integral equation method for the calculation of resonant mode functions by employing a fictitious space which is reciprocally equivalent to the physical space. Using the Green’s function of the interior region of the uniform...
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Context 1
... the BEM, we also obtained another resonant mode in a limaçon-shaped TC with = . 0 24 , n 0 = 2.0, and β = 1.0. The complex wave number of the mode is k res R 0 = 11.913 − i 0.107 and the intensity pattern of the resonant mode is shown in Fig. 6(a). This resonant mode has low-Q factor (Q ≈ 56) and the corresponding far-field intensity distribution is shown in Fig. 6(c). Contrary to the bidirectional far-field distribution of the above (14,1) cWGM, this low-Q mode has a unidirectional far-field intensity distribution. From these results, one can note that the Q-factors and ...
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... obtained another resonant mode in a limaçon-shaped TC with = . 0 24 , n 0 = 2.0, and β = 1.0. The complex wave number of the mode is k res R 0 = 11.913 − i 0.107 and the intensity pattern of the resonant mode is shown in Fig. 6(a). This resonant mode has low-Q factor (Q ≈ 56) and the corresponding far-field intensity distribution is shown in Fig. 6(c). Contrary to the bidirectional far-field distribution of the above (14,1) cWGM, this low-Q mode has a unidirectional far-field intensity distribution. From these results, one can note that the Q-factors and emission directionalities of the resonant modes in TCs depend on the scaling factor β as well as the deformation parameter 35 . ...
Context 3
... well as the deformation parameter 35 . We also obtained the resonant mode under the same parameters by FEM and the resultant wave number of the mode is k res R 0 = 11.913 − i 0.108, which agrees well with the BEM result. Also, the intensity pattern of the corresponding mode and the far-field intensity distribution obtained by FEM are shown in Fig. 6(b,c), respectively, which nearly match the results of BEM as well. www.nature.com/scientificreports ...
Citations
... However, special care must be devoted when it comes to the TCs because of their inhomogeneous refractive index. Here, we introduce the reciprocal virtual (RV) space, which is given by inverse-conformal mapping h −1 : z → w ′ = u ′ + iv ′ of the physical space [21,[23][24][25]. In the RV space, because the refractive index inside the cavity is constant (n 0 ), we can rewrite Eq. (7) as ...
Unlike the ideal circular whispering gallery cavities, those without mirror symmetry intrinsically support resonant modes exhibiting chirality which indicates an imbalance between clockwise and counterclockwise wave components. In extreme cases, nearly degenerate pairs of copropagating modes can be found around the chiral exceptional points (EPs) in parameter spaces. The chiral EPs have been studied in various schemes; however, most attention has been focused on the cases with piecewise constant or periodic refractive index profiles. In this Letter, we report the formation of a chiral EP in a gradient-index cavity designed by conformal transformation optics. Here, the mirror symmetry of the cavity is broken solely by its gradient index profile, and the parameter space is constructed with coordinate transformation parameters. We unveil the chirality, nonorthogonality, and complex-square-root topology near the chiral EP, which can be explained by the non-Hermitian model Hamiltonian.
... Since we applied the linear coordinate transformation only to the inside of the cavity, the OV space does not have complete electrodynamic correspondence to the physical space. The virtual space that truly corresponds to the entire physical space is called the reciprocal virtual (RV) space which can be obtained from the physical space by applying the inverse of the coordinate transformation [24,25]. One can simply deduce the permittivity profile of the RV space of the bTC as ...
It was reported that whispering gallery cavities designed by conformal transformation optics can support high-Q resonant modes with emission directionality. Intrinsically, these cavities have gradient index profiles implementing conformal mappings in physical space. In this paper, using the linear coordinate transformation, we propose another design scheme of whispering gallery cavities with (piecewise-) homogeneous, anisotropic index profile. We numerically show that so-designed cavities are also able to support high-Q whispering gallery modes with directional far-field emission patterns. We verify such characteristics by using a phase space representation (called the Poincaré Husimi function) of the intracavity wave function.
... In this work, we focus on TM polarization modes without loss of generality since TE polarization mode can be treated similarly. In the following sections, we numerically investigate the variation of resonant modes as a function of ε and β using the boundary element method 23,24 . ...
Directional light emission from high-Q resonant modes without significant Q-spoiling has been a long standing issue in deformed dielectric cavities. In limaçon-shaped gradient index dielectric cavities recently proposed by exploiting conformal transformation optics, the variation of Q-factors and emission directionality of resonant modes was traced in their system parameter space. For these cavities, their boundary shapes and refractive index profiles are determined in each case by a chosen conformal mapping which is taken as a coordinate transformation. Through the numerical exploration, we found that bidirectionality factors of generic high-Q resonant modes are not directly proportional to their Q-factors. The optimal system parameters for the coexistence of strong bidirectionality and a high Q-factor was obtained for anisotropic whispering gallery modes supported by total internal reflection.
We study the resonant modes in double-layered transformation cavities consisting of inner and outer layer boundaries where the mode intensity is mainly located inside the inner layer. This is an intermediate design between a transformation cavity and a cavity in a wholly transformed space of transformation optics. We demonstrate the crossover between these two extreme cases as the outer layer of the cavity varies and also explore the properties of the resonant modes in the cavity. While the near-field patterns of the resonant modes do not change as the outer layer becomes larger, the Q factors approach those of a cavity in the wholly transformed space of transformation optics, and the far-field patterns are modified.
