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Abstract
Using the transformation cavity, a gradient index cavity designed by transformation optics, we propose a hybrid resonator system to extract unidirectional narrow-beam emission from high-Q whispering gallery modes by embedding a transformation cavity inside a deformed uniform index cavity that exhibits unidirectional narrow-beam emission. For effective mode coupling between the transformation cavity and enclosing cavity, the embedded transformation cavity is designed to have bidirectional evanescent emission, which enables most of the emission from the transformation cavity to be laterally incident on the rim of the enclosing deformed cavity. Consequently, ultrahigh-Q resonances of this system can provide a sharp free-space light output, which is difficult to achieve by embedding a homogeneous disk cavity instead of the transformation cavity.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical dielectric cavities with rotational symmetry can support very long-lived resonant modes, so called whispering gallery modes (WGMs), by total internal reflection (TIR) along the boundary of the cavity. In particular, the WGMs formed in microcavities having small mode volumes have attracted considerable attention for their potential of diverse photonics applications such as highly sensitive sensors and low-threshold micro lasers [1–8]. However, owing to the rotational symmetry of the cavities, the isotropic emissions of WGMs are a drawback in applications that require directional light sources. To overcome this issue, many studies over the past two decades have focused on achieving directional emission while maintaining a high Q-factor, and two basic approaches have been developed thus far. The first well-known approach is to break the rotational symmetry by deforming the cavity boundary [9–12]. The resulting asymmetrically deformed cavity can be designed to have unidirectional emission by incorporating a ray dynamics effect. The other approach is to introduce defects into the mode field region of the cavity [13–17]. In such cavities, light is scattered by the defects and collimated to exhibit unidirectional emission. These two methods provide a general way to achieve unidirectional emission, but they commonly suffer from degradation of the Q-factor.
Recently, Kim et al. [18] proposed another kind of approach that utilizes the theory of transformation optics [19,20] to control wave propagation in microcavities. Because of the use of optical conformal mapping, these cavities have a gradient refractive index profile and can support high-Q conformal WGMs (cWGMs). This type of cavity is referred to as a transformation cavity (TC) and the spatial variation of the refractive index along the boundary of the TC plays a decisive role in the emission mechanism. The tunneling emissions of cWGMs emerge from a free-space spot off the cavity boundary where the refractive index is lowest and propagate along the tangential direction. These cWGM characteristics can be utilized to achieve directional emission with a high Q-factor by appropriately designing the refractive index distribution of the TC. For example, a limaçon-shaped TC of which the emission emerges from the one free-space spot can exhibit bidirectional emission, and a triangular-shaped TC in which the directions of two dominant emission beams are oriented to be nearly parallel can have unidirectional emission [18,21]. However, a TC design with a narrow-divergence unidirectional emission has not been explored yet.
Fig. 1. (a) Refractive indices of circle-shaped transformation cavity (TC) and (b) face cavity. (c) Far-field patterns of the circle-shaped TC with Re[$kr$] $\approx 15.17$ ($r=0.9$) and (d) face cavity with Re[$kR_0$] $\approx 50.22$ ($R_0=3$). The far-fields are normalized to 1 and the inset of each case shows the external near-field intensity pattern (inside field intensities are deleted).
In this paper, we propose a hybrid dielectric cavity that simultaneously achieves a high Q-factor and unidirectional emission with a narrow divergence angle by steering the emission of evanescent waves of the cWGMs. The key idea of our approach is to couple the evanescent waves emitted from a TC with a resonance of deformed cavity that exhibits highly unidirectional emission. This system consists of an embedded core TC and a deformed homogeneous index cavity which encloses the TC. In this coupled system, the TC is designed to have bidirectional evanescent emission that effectively couple with enclosing deformed cavity. The evanescent waves, which are confined along the boundary of the enclosing deformed cavity, eventually refract out according to the emission characteristics of the enclosing deformed cavity. Consequently, the Q-factor of the resonances formed in this system, which is inherited from the core TC, can be extremely high and simultaneously, the unidirectional emission with a narrow divergence angle can be achieved by virtue of a properly designed enclosing deformed cavity.
2. Results and discussion
As reported in Refs. [18,21,22], the cWGMs in the TCs with a dipole-like refractive index profile have a emission characteristic in which one-spot emission occurs near the boundary where the refractive index is lowest, thereby forming a bidirectional emission. The circle-shaped TC, the simplest case exhibiting bidirectional emission, can be obtained by using the following Möbius transform
which maps the unit disk domain in virtual space (complex $\eta =u+iv$ plane) to a shrunk disk with a shifted center at $\beta \delta$ in physical space (complex $\zeta =x+iy$ plane) where $\beta$ is a positive shrinking parameter with a value less than 1. The corresponding refractive index profile of the circle-shaped TC is given by:Fig. 2. (a) Refractive index profile of TC-embedded hybrid cavity system (THCS). (b) Near-field pattern of a resonance in the designed THCS having Re[kr] $\approx 15.14$ ($r=0.9$) in logarithmic scale. (c), (d) External near-field intensity pattern and far-field pattern of the mode shown in (b), respectively.
Figure 1(a) shows the refractive index profile of a circle-shaped TC with the parameters $\beta =0.9$, $\delta =0.05$, and $n_0=2.7$ and radius $r=0.9$. Figure 1(c) shows the external near-field and far-field intensity patterns of a cWGM formed in this circle-shaped TC. The depicted mode exhibits a bidirectional emission, which emerges at the small spot near the lowest index position at the boundary. In this work, the resonances are restricted to TM polarization (out-of-plane electric field component) and calculated by finite element method using COMSOL Multiphysics ver. 5.2a.
To turn this bidirectional emission from the TC into unidirectional emission with a narrow beam divergence, we use a deformed cavity that exhibits unidirectional emission. As an example of our approach, we use the following "face cavity" as the enclosing deformed cavity [11]. The boundary of the face cavity is given as
Fig. 3. Changes in (a) Re[kr] and (b) Q-factor of a resonance in logarithmic scale formed in the THCS depending on the distance per wavelength (see the inset of (a)). Each horizontal red line indicates the Re[kr] and Q-factor of the corresponding resonance of an identical non-embedded circle-shaped TC in which the refractive index of the outer region is set to 1.45. (i) ~ (iii) The near-field (left) and far-field patterns (right) of resonant modes are shown for the selected points between 0.5$\lambda$ and 4.5$\lambda$. (iv) The near-field and far-field pattern which exhibit bent bidirectional emission are shown for the selected point near the center of the enclosing face cavity.
Fig. 4. (a), (b) Re[$kr_1$] value and Q-factor of homogeneous disk-embedded cavity system (HDCS) as a function of distance per wavelength. The red solid line indicates the Re[$kr_1$] and Q-factor of the corresponding resonance for an identical non-embedded homogeneous disk, in which the refractive index of the outer region is set to 1.45. (c) The near-field and far-field patterns of a resonance in the HDCS.
By embedding the aforementioned TC in the face cavity, we make a hybrid cavity in which the emitted evanescent waves of the core TC are coupled with the resonance of enclosing face cavity, as shown in Fig. 2(a). Among the many possible positions between core TC and enclosing cavity, first we placed the centers of both cavities on the $x$-axis and matched the maximum intensity position of the enclosing cavity resonance with the center of emission spot where bidirectional emission of the cWGM formed in the core TC emerges. In this coupled system, which we call the TC-embedded hybrid cavity system (THCS), the bidirectional evanescent waves emitted from the circle-shaped core TC are immediately confined along the boundary of the enclosing face cavity, as shown in Fig. 2(b). And then, through dynamical tunneling mechanism of the face cavity, the evanescent waves refract out at its upper and lower boundary, which leads to unidirectional emission. Thus, the resulting far-field emission has high degree unidirectionality with a full-width-at-half-maximum beam divergence angle less than $\pm 5^{\circ }$ and about 75% of the total emission can be collected within $\pm 30^{\circ }$, as shown in Fig. 2(d). In particular, as the cWGM of the core TC is scarcely disturbed by any scatterer or deformation, the Q-factor of the THCS shown in Fig. 2 is extremely high which is approximately $2.3\times 10^{12}$ at Re[$kr$] $\approx 15.14$ while the Q-factor of the stand-alone face cavity resonance is around $1.02\times 10^{4}$ at Re[$kR_0$] $\approx 50.22$.
Next, we investigated the characteristics of THCS by changing the relative position of the core TC. Figure 3 shows the variations of the wavenumber and Q-factor for a specific resonance when the distance $d$ between the left-side boundaries of the two cavities is parametrically swept between approximately 0.1 $\lambda$ and 6.5 $\lambda$, where $\lambda$ is the wavelength of the resonance in the enclosing cavity ($\lambda =\frac {1}{1.45}\frac {2\pi }{k}$). In the range of $d \gtrsim 0.5\lambda$, the Re[kr] value hardly changes, which indicates that the system can be considered in a weak coupling regime and the Re[kr] value of the system is nearly the same as Re[kr] value of the corresponding resonance of an identical non-embedded circle-shaped TC in which the refractive index of the outer region is set to 1.45 (red solid line in Fig. 3(a)). However, there is a noticeable point that the Q-factor sensitively vary depending on the distance $d$, which is in contrast to findings reported previously [23]. In the range of $0.5\lambda \lesssim d \lesssim 3\lambda$, the Q-factor is higher than that of the corresponding resonance of an identical non-embedded circle-shaped TC, in which the refractive index of the outer regions is set to 1.45 (red solid line in Fig. 3(b)). This is because some of the confined evanescent waves feed back into the core TC. On the other hand, near the $d=3.5\lambda$, one can notice that field intensity in the face cavity region is strongly localized to a simple polygonal-shaped pattern, thus the confined fields are quickly refracted out through this lossy channel. Therefore, the Q-factor drastically drops and becomes even lower than that of the corresponding resonance of the identical non-embedded core TC. Beyond a distance of approximately 5$\lambda$, the Q-factor converges to the corresponding resonance of an identical non-embedded circle-shaped TC, as most of the waves emitted from the core TC do not satisfy TIR but directly refract out at the boundary of the enclosing cavity. Otherwise, when $d \lesssim 0.5\lambda$, the Re[kr] value of the resonance changes appreciably and its Q-factor is spoiled [24].
To verify that it is difficult to achieve unidirectional emission by embedding a homogeneous cavity instead of TC, we compared the THCS with a homogeneous disk-embedded cavity system (HDCS). Figure 4 shows the calculation results for an HDCS using a homogeneous disk with $n=2.7$ and radius $r_1=1$. Similar to the THCS case, when $d \lesssim 0.5\lambda$, the Re[$kr_1$] value changes significantly and the Q-factor is spoiled. However, beyond a distance of approximately 0.5$\lambda$, the HDCS resonance shows slightly fluctuating Q-factor, which contrasts with the THCS case. Moreover, the HDCS has difficulty in realizing unidirectional emission, as most of evanescent waves of core cavity are not confined along the enclosing cavity but directly refract out at the boundary of the enclosing cavity. Figure 4(c) shows the near-field and far-field pattern of the resonance of the HDCS. To some extent, the far-field exhibits an unidirectional emission as some of the waves emitted from the homogeneous disk are confined along the boundary of enclosing cavity. However, there are lots of emissions beside in $\pm 30^{\circ }$, and only approximately 30% of the total emission is collected within $\pm 30^{\circ }$. To quantitatively compare the unidirectionality of HDCS with THCS, we measure the unidirectionality by defining a factor U as
We also investigated the origin of the differences between the THCS and HDCS by using Husimi function. In general, the Husimi function is used to represent the resonance of a cavity in the phase space, and defined at the boundary of the cavity by overlapping of a Gaussian wave packet with the resonant mode of the cavity, which implies the intensities of the incident and emerging waves at the cavity boundary [25,26]. In this case, to see how much of the emission from the core cavity is laterally incident on the rim of the enclosing cavity, we considered the resonances of non-embedded circle-shaped TC and homogeneous disk in which the refractive index of outer region is set to 1.45, and calculated the Husimi function of evanescent waves of these two cavities at the virtual boundary of the same face cavity in THCS and HDCS, respectively. Figures 6(a) and 6(b) show the near-field patterns of the resonances of the circle-shaped TC and homogeneous disk which correspond to the red solid lines in Figs. 3 and 4, respectively and the white dashed-lines indicate the virtual boundary where the Husimi function is calculated with $d \approx 2.5\lambda$. Figures 6(c) and 6(d) show the calculated Husimi functions at the virtual boundary for each case where the yellow solid lines in the Husimi plots correspond to the critical angle of original face cavity which is placed in the air. In the case of the circle-shaped TC, most of the intensities are above the critical line. By contrast, in the case of the homogeneous disk, only small part of intensities are above the critical line, which means that the HDCS is barely coupled with the enclosing cavity, whereas the THCS can be effectively coupled with the enclosing cavity.
Fig. 6. (a), (b) Logarithmic-scaled near-field patterns of the resonances of an circle-shaped TC and a homogeneous disk, respectively, in which the refractive index of the outer region is set to 1.45. The white dashed lines indicate face cavity shaped virtual boundaries where the Husimi function is calculated. (c), (d) Husimi plots at the virtual boundary obtained by overlapping a Gaussian wave packet with the emitted waves of the cWGM of an circle-shaped TC and the WGM of a homogeneous disk, respectively. The yellow solid lines correspond to the critical angle of original face cavity placed in the air.
3. Conclusions
In summary, we proposed a TC-embedded hybrid cavity system (THCS) to achieve narrow beam divergence unidirectional light output with an ultrahigh Q-factor. The proposed hybrid cavity consist of bidirectional TC as core cavity and deformed homogeneous cavity as an enclosing cavity. This THCS can exhibit simultaneously a high-Q factor by the core TC and a highly unidirectional emission by the enclosing cavity, which is difficult to achieve by a deformed homogeneous cavity or using scatterer. We expect this system to be utilized in a diverse range of photonics applications by changing the enclosing cavity appropriately, and low-threshold micro lasers by making the core cavity as an active medium and the enclosing cavity as a passive medium.
Funding
National Research Foundation of Korea (2020R1A2C3007327, 2020R1A4A1019518).
Acknowledgments
We would like to thank Dr. J.-W. Ryu for the helpful discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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