1. Introduction

Whispering gallery mode (WGM) describing the curvilinear propagation of waves around a concave surface is a classic yet vibrant field of research. Since Lord Rayleigh solved the puzzle of sound enhancement in the St Paul’s cathedral dome in the early 20th century [1], this field-confinement paradigm has been introduced to study the wave mechanics such as light waves [2], elastic waves, and electron waves [3]. In particular, optical WGM cavity has become a cornerstone photonic device in investigating the light-matter interactions for micro- and nano-optics [2,4–8], benefiting from their remarkable ability to confine light to a small volume for a long time, with potential applications ranging from optical sensors, lasers, and nonlinear optics [7–14]. The conventional dielectric WGM cavity consists of a homogeneous high-index dielectric core and a low-index cladding, which as an open-system supports quasinormal modes characterized by complex eigenfrequencies and finite photon lifetimes. The photon lifetime of dielectric WGM cavities is mainly limited by the intrinsic radiation loss due to the light tunneling effect in the curved boundary, especially in low-order modes [2,15,16].

In the past years, researchers have developed various methods to tailor the radiation loss and achieve high quality (Q) factor in microcavities of small mode volumes, which are on demand for cavity quantum electrodynamics [2,16,17]. For instance, by meticulously tuning the structural parameters of high-index dielectric particles, such as spherical or cylindrical shapes, bound states in the continuum with infinite Q factor can be obtained, resulting from an interference mechanism [18–22]. Surface plasmonic resonance is also employed to strongly confine the light fields along with the depression of the radiation in the metal-coated dielectric cavity [23–27]. The zero-index metamaterials are leveraged to realize geometry symmetry-free resonant states of high-Q factors [28–31]. Recently, the conformal mapping method has been utilized to engineer the radiation fields and the Q-factors of WGM cavities, which are of high structural flexibility [32–34]. The radiation loss of broadband WGMs in the ideal optical black hole (OBH) cavity can be completely inhibited by an infinite wide potential barrier [32,35–37].

In this work, we systematically study the WGM fields in a group of generalized OBH cavities with cladding index distributions of $\frac {n_0}{(r/R)^ \alpha }$, as shown in Fig. 1(a). Analytical results show that these generalized OBH cavities ($\alpha \geq 1$) still support WGMs of infinite Q factor. We find that with the increase of $\alpha$, the near-field confinement of OBH cavities is enhanced. Moreover, we reveal that anomalous resonance with external field enhancement beyond the core emerges in high radial-order WGMs (Fig. 1(c)), which can be attributed to the intriguing above-barrier resonance mechanism and potential bending effect. These external resonant WGMs may find applications in light coupling and optical sensing.

 

Fig. 1. (a) Refractive index profiles of two-dimensional generalized optical black hole (OBH) cavities. The cladding index of OBH cavities follows the rule of $\frac {n_0}{(r/R)^ \alpha }~(r>R)$. The zero-index cavity and homogeneous cavity can be regarded as special cases of generalized OBH cavity with refractive index modulation factor $\alpha$ equals to $\infty$ and $0$, respectively. Plots of WGM field patterns of (b) zero-index cavity with $\alpha =\infty$, (c) OBH cavity with $\alpha =2$, and (d) homogeneous cavity with $\alpha =0$. In particular, anomalous WGM with external field enhancement is shown in (c). The white circle is the boundary of the homogeneous dielectric core inside.

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2. Results and discussions

Let us consider the generalized OBH cavities with refractive index distributions in cylindrical coordinates as follows:

It is found that Eq. (3) can have the real eigen-value under certain conditions, which is completely different from the conventional homogeneous cavity with complex eigenfrequency. Thereafter, we have proved that the OBH cavities are a group of radiationless WGM cavities [32]. Based on Eqs. (2–3), the analytical WGM fields in generalized OBH cavities are obtained as shown in Fig. 2(a-d). Here, the structural parameters of generalized OBH cavities are set as follows: $R$=3 cm, $n_1$=4, $n_0$=2, if not specified otherwise. The analytical results of WGM fields agree well with the numerical simulations, as shown in Fig. 2(e-h). The numerical simulation results are computed by finite-element simulation software (COMSOL), and the mesh size of the simulation models is set to be smaller than 1/6 light wavelength. For the conventional homogeneous cavity, a sizeable portion of the field’s tail extends beyond the core, which usually indicates high radiation loss. As $\alpha$ increases, such tails are gradually restrained towards the core and reach the minimum at $\alpha = \infty$ (Fig. 2(d)), suggesting the strongest field confinement in the zero-index cavity. In general, with the increase of $\alpha$, the field confinement of the OBH cavity is enhanced.

 

Fig. 2. Analytical and numerical results of WGM fields for generalized OBH cavities. The analytical WGM field patterns of generalized OBH cavities with (a) $\alpha =0$, conventional homogeneous cavity; (b) $\alpha =1$, the fundamental OBH cavity; (c) $\alpha =2$, OBH cavity; (d) $\alpha =\infty$, zero-index cavity. (e-h) Plots of numerical simulation results with cavity structural parameters corresponding to (a-d). The white circles in (a-h) are the boundaries of the homogeneous core. The structural parameters of generalized OBH cavities are as follows: $n_1=4, n_0=2$, the radius of homogeneous core $R=3$ cm. The calculated resonant frequency of (a, e)$\sim$2.3894 GHz, (b, f) $\sim$2.4338 GHz, (c, g) $\sim$2.4648 GHz, (d, h) $\sim$2.5386 GHz.

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To quantitatively characterize the field confinement effect, we extract the skin depth ($\delta$) of WGM fields in the cladding region and trace their dependence on $\alpha$ and azimuthal mode orders $m$, as illustrated in Fig. 3(a). The skin depth is defined as the distance between the cavity boundary and the position with $1/e$ field-amplitude maximum [32]. It is apparent that the larger the $\alpha$, the smaller the skin depth of WGM fields can be obtained. Under the condition of $\alpha$ = 10, its skin depth is close to the ultimate value in the zero-index cladding cavity. The structural design of an OBH cavity provides a trade-off between the field confinement ability and the realistic refractive index parameter. In addition, it is observed that the skin depth reduction effect is significant only when the azimuthal mode number is smaller than 10. Note that although the OBH cavity can achieve an infinite radiating Q factor, the ultimate Q factor of the OBH cavity is still limited by the finite material absorption loss. Besides, it is found that material absorption loss can have a negligible impact on the WGM field distribution.

 

Fig. 3. (a) Comparison of skin depth of evanescent WGM field in typical generalized OBH cavities relating to various azimuthal mode numbers. (b) The effective potential of various generalized OBH cavities. WGM fields are of azimuthal mode number 4, radial order 1. The horizontal dashed line indicates the normalized eigen-energy of WGMs. The blue stars mark respectively the bottom ($k_B^2$) and top ($k_T^2$) of the potential well in the homogeneous cavity. The pink-filling curve is the radial distribution of WGM fields.

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The effective potential function can help clarify the field confinement mechanism in OBH cavities. The effective potential is derived based on the analogy between the radial scalar wave equation of optics and the Schrödinger equation [32,40]: ${V_\mathrm {eff}} = {k_0^2}(1 - {n(r)^2}) + ({m^2} - 1/4)/{r^2}$, where the first and second terms describe the attracting potential and the centrifugal potential, respectively. Figure 3(b) shows that the energy of the bottom of the potential well ($k_B^2$) is only slightly changed with the index modulation $\alpha$ due to the minor change of resonant frequency. On the contrary, the potential barrier region is significantly modified with $\alpha$, attributed to the external index modulation effect. Note that under the aforementioned structural parameters, the potential barrier width is infinite for all the OBH cavities ($\alpha >1$), indicating the bound states with zero radiation loss and infinite radiating Q factor. Since the energy of the top of the potential well ($k_T^2$) is increased with $\alpha$, the field confinement around the periphery is thus enhanced. Moreover, the radial position of $k_T$ is shifted away from the cavity boundary with the variations of $\alpha$, except for the zero-index cavity, as shown in Fig. 3(b). This intriguing potential curve can lead to unconventional external resonant fields for above-barrier resonant states as discussed later.

So far, we have focused on the WGM fields of fundamental radial mode for generalized OBH cavities, of which the electric fields are mostly concentrated in the high-index core region. Recently, there has been increasing research interest in the external resonant modes, featuring the peak of field intensity in the low-index cladding region [41,42]. These unconventional WGMs show promising applications in optical sensors for enhanced light-matter interactions in the environment [24,37,43,44]. However, these external resonant modes along with high radial order in homogeneous cavities are quasinormal modes suffering from considerable optical loss [24,38,45]. Here we show that the external resonant modes featuring bound states emerge in high-radial modes of OBH cavities.

Without loss of generality, we consider the OBH cavity with $\alpha =2$. From Eq. (3), the specific characteristic equation is obtained:

The calculated resonant spectrum is shown in Fig. 4(a). We find that the conventional inner resonant states lie within the potential well enclosed by the dispersion curves of $k_BR$ (solid black curve) and $k_TR$ (red dashed curve) at the cavity boundary, for example, the WGM field with azimuthal mode number 4 and radial order ($l$) 1 as shown in Fig. 2(b). For these inner resonant states, the electric fields are mainly confined around the cavity boundary and the peak field amplitude decreases monotonically from inside to outside of the cavity [38].

 

Fig. 4. Properties of external resonant modes in OBH cavity. (a) Resonant spectrum for the OBH cavity with $\alpha =2$. The labels of i-iv, represent the WGMs with radial orders of 2, 4, 7, and 10, respectively. The solid black line and the dashed red line represent the dispersion for the bottom and top of the potential well at the boundary position. (b-e) Effective potentials and field distributions of various WGMs corresponding to labels i-iv in (a). The pink-filling curves are the radial field distribution of WGMs, and the mark of the red star indicates the external turning points of potentials. The upper-right insets show a zoomed-in view of the potentials around the cavity boundary. The lower-right insets show standing-wave patterns of WGMs along with the radial cross-section field distributions.

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With the increase of radial order, the eigen-energy of WGMs is increased. Under the regime of above-barrier resonance [45,46], the potential curve around the cavity boundary is bent upward, and the external turning point is also shifted away from the boundary, as shown in Fig. 4(b)-e. The turning point is defined as the position where the eigen-energy equals the potential function. WGM fields are confined within the inner core boundary, where the resonance energy $k_0^2>V_\mathrm {eff}$. As the photon passes inward through the external turning point (red star), it enters a classically restricted zone since $k_0^2 < V_\mathrm {eff}$, and consequently, the WGM fields will behave as the evanescent wave with an imaginary radial propagation wavevector. We can define a virtual boundary (dashed white circle) enclosed by the turning points, around which propagating fields are confined. As a result, for the high-order OBH cavities, the field confinement depends both on the core’s real boundary and the virtual boundary. For the WGM of radial order 2, the electric field distribution of the OBH cavity still follows the conventional WGM (Fig. 4(b)), although its resonance energy is already slightly higher than $k_{T}^2$ at the boundary. As the radial order increases, such as $l$ = 4,7,10 illustrated in Fig. 4(c)-e, the resonance energy is always higher than $V_\mathrm {eff}$ until the radial position of $\sim$1.9R, $\sim$2.7R, and $\sim$3.5R, respectively. Intriguingly, there are enhanced field distributions between the real and virtual boundaries, and their resonance intensity is even stronger than that of the core region. We note that in a homogeneous cavity, no such external virtual boundary exists; when a photon crosses the external turning point, it instead enters a classically allowed zone ($k_0^2 > V_\mathrm {eff}$) and thus WGM fields radiate outward boundlessly.

3. Conclusion

In this work, we systematically explore the WGMs in a group of generalized OBH cavities, featuring bound states without radiating loss. We show that the increase of index-modulation factor $\alpha$ factor contributes to the enhanced near-field confinement of OBH cavities resulting from the increase of potential barrier. Furthermore, we study the external resonant modes of high radial order, exhibiting intriguing field enhancement in the low-index region. These anomalous WGMs are attributed to the potential bending effect and above-barrier resonance. Further study shows that such external WGMs are also immune to boundary deformation of the core. For example, Fig. 5 shows the standing wave field patterns of the face-shaped [32,47] and flowercavity [42], where the nodes of WGMs are still regularly distributed along the virtual boundaries even with the larger deformation factors of 2.5 and 0.2, respectively. Such OBH cavities can be practically and technologically implemented based on effective medium theory [32,48,49]. Our work may provide a new avenue to cavity optics for photonic applications, such as light coupling and optical sensing.

 

Fig. 5. External resonant modes of (a) face-shaped cavity and (b) flowercavity. The deformation factors of face-shape cavity and flower cavity are 2.5 and 0.2, respectively. The white solid and dashed lines respectively, denote the undeformed and actual boundaries of the core.

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