Polarization-Controlled Cavity Input-Output Relations

Cavity input-output relations (CIORs) describe a universal formalism relating each of the far-field amplitudes outside the cavity to the internal cavity fields. Conventionally, they are derived based on a weak-scattering approximation. In this context, the amplitude of the off-resonant field remains nearly unaffected by the cavity, with the high coupling efficiency into cavity modes being attributed to destructive interference between the transmitted (or reflected) field and the output field from the cavity. In this Letter, we show that, in the strong-scattering regime, the off-resonant field approaches to zero, but more than 90% coupling efficiency can still be achieved due to the classical Purcell effect. As a result, the CIORs turn out to be essentially different than in the weak-scattering regime. With this fact, we propose that the CIOR can be tailored by controlling the scattering strength. This is experimentally demonstrated in a whispering gallery resonator-waveguide coupled system, by the transmission spectra exhibiting either bandstop or bandpass-type behavior according to the polarization of the input light field.

The coupling of light from a waveguide to a microresonator can be physically treated as the scattering of a traveling wave by discrete localized states in the resonator [1,2]. Traditionally, the scattering is assumed to be weak and the coupling is characterized by almost 100% transmission for off-resonant light. Interestingly, in this weak-scattering regime, a near-unity coupling efficiency can be attained if the intrinsic loss of the resonator is equal to the coupling loss induced by the waveguide; this is termed critical coupling [3][4][5]. Intrinsically, critical coupling can be regarded as a consequence of the cavity input-out relation (CIOR) in the weak-scattering regime, i.e., the perfect destructive interference between the direct transmission through the waveguide and the outcoming field from the cavity mode. Critical coupling can be considered as an example of coherent perfect absorption, which was developed in recent years [6,7]. It has been shown that multiple critical couplings could exist within a coupled system [8,9]; however, the rigorous condition required to achieve critical coupling cannot always be satisfied. For instance,in nonlinear optics experiments [10,11], it is challenging to realize critical coupling for two different wavebands simultaneously.
Aside from trapping light in the cavity by creating perfect destructive interference at the coupling point, coupling of light into the cavity modes may be achieved by another mechanism -Purcell-enhanced Rayleigh scattering [12,13]. This method is limited by the small scattering cross-section of the point-like scatterers, thus it is unlikely to achieve a high coupling efficiency from the input field [14,15]. In fact, the Purcell effect does not necessarily have to be explained in terms of the optical density of states, but rather can be described as the constructive interference of waves [13,16,17]. Therefore, the Purcell effect is not restricted to purely the coupling between Rayleigh scatterers or single quantum emitters and a cavity. In this Letter, we show that the resonator-waveguide coupled system may be in the strong-scattering regime; the optical field may be strongly scattered by the resonator and the off-resonant light transmission can drop to zero, i.e., it cannot pass through the coupling junction directly. In this strong-scattering regime, Purcellenhanced coupling between the waveguide and the cavity modes for the resonant light is present. As a result, light can be coupled into the cavity modes with near unity efficiency with a bandpass-type transmission spectrum (Note that we observed this effect before, but the mechanism was not explicitly presented [18]). It implies the corresponding CIOR is essentially different to the conventional formalism, which is only valid for the weakscattering regime. Based on this fact, we propose and realize a tunable CIOR in a resonator-waveguide coupled system. This is achieved by properly designing the geometry of the resonator-waveguide coupled system and ensuring that the coupling can be switched between the weak-scattering regime and the strong-scattering regime by simply controlling the inner degree of light, i.e., the light's polarization.
We consider a silica whispering gallery (WG) resonator coupled with an air-clad, single-mode, tapered optical fiber, as illustrated in Fig. 1. This system has been studied extensively for more than two decades for a variety of applications [5,[19][20][21]. We can treat the WG resonator as a system with two parts: (I) the section of the resonator in the coupling region (indicated by the dashed rectangle) and (II) the rest of the resonator [4]. In the weak-scattering regime, only one cavity mode is considered and the coupling region can be modeled as a two-port beam-splitter, i.e., in the absence of the rest of the resonator the system acts like a directional coupler having so genannten cavity-free guided modes. In contrast, in the strong-scattering regime (i.e., |t 0 | ≈ 0), the light can be partially coupled from the fiber mode, a in , not only into the guided modes, c j , but also into a continuum of radiation modes, b l , where j and l are the mode order numbers, see Fig. 1. Strictly speaking, the guided modes should be termed as quasi-modes because they also have nonzero radiation loss.
To illustrate the effect of the cavity on the light coupling in the strong-scattering regime, we can directly compare the coupling power into the cavity-modified guided modes, P c j , (i.e., in the presence of part II) and the cavity-free guided modes, P w j . Their ratio is defined as the cavity impact factor, G j (ω), such that where κ j = κ 0 j +κ e j , κ 0 j (κ e j ) represents the field amplitude decay rate due to the intrinsic loss (waveguide coupling). τ j is the circulation time for the mode traveling inside the resonator. ω j is the cavity resonant frequency. For the resonant case, G j (ω j ) = 2/π(λ/n j )(Q j /L). Here, n j is the effective refractive index and Q j is the qualify factor of the cavity mode, j. L is the circumference of the resonator. Note that the cavity impact factor, G, is very similar to the well-known Purcell factor, F = 3/4π 2 (λ/n) 3 (Q/V ) [12], which is widely used for point-like emitters or scatters. Essentially, the similarity between G and F stems from the identical underlying physics -wave interference. Therefore, we could treat G as a generalized Purcell factor. It is worth emphasizing that this cavity impact factor is invalid in the weak scattering regime therefore critical coupling cannot be attributed to the Purcell effect. Compared to the guided modes, the presence of the cavity has a much weaker affect on the distribution of the radiation modes [22], thus we could assume the scattering rates into the radiation modes remains the same with or without the existence of part II of the resonator. Therefore, in the strong scattering regime, we can define the channeling efficiency, Γ j (ω), which represents the fraction of power coupled from the waveguide into the cavity-modified mode, j: where γ g k and γ rad stand for the scattering rates into the cavity-free guided mode k and all radiation modes.
In order to compare the weak-scattering and strongscattering regimes, we performed measurements using a silica microsphere coupled to tapered optical fibers of different diameters [18], as presented in Fig. 2. With a thick fiber, the coupling can be classified as weak-scattering: only the resonant fields are absorbed by the resonator and the transmission spectra are bandstop type, see Fig.  2(a). In contrast, with a thin fiber, the coupling enters the strong-scattering regime: only the resonant fields can pass through and the resulting transmission spectra are bandpass type, see Fig. 2(b). The existence of two distinct spectra is due to their distinct coupling regions. To study the coupling region experimentally, we placed a tiny droplet of ultraviolet (UV) adhesive (NOA 81, Thorlabs) onto a small area of the microsphere opposite to the coupling region and cured it using a UV gun, see Fig. 2. The adhesive has no direct influence on the coupling region, but it does act to prevent the circulation of the guided cavity modes; in other words, the WG cavity modes degrade into cavity-free guided modes. However, the radiation modes should remain nearly unmodified. For each case, the transmission spectra are plotted for the input fields with two orthogonal polarizations, i.e., horizontal, H (red), and vertical, V (green), corresponding to the HE x 11 and HE y 11 modes in the fiber. After the adhesive was added all sharp features in the transmitted spectra disappeared and the transmission became constant (almost identical for both H and V), represented by the dashed lines in Fig. 2. The polarization of the guided mode in the tapered fiber was controlled using the method presented in [23]. During experiments, the fiber was in contact with the resonator at all times.
Note that, for the thicker fiber, see Fig. 2(a), the transmissions for resonant and off-resonant frequencies are affected by the UV adhesive. The importance of this behavior must be stressed at this point, bearing in mind that the adhesive does not physically affect the coupling junction. For light in the fiber, the coupling system essentially looks like a directional coupler, i.e., there are no resonances and the cavity simply acts like another waveguide. On removing the adhesive, off-resonant light cannot enter the restored cavity.
Here, it is appropriate to point out that, for a multimode resonator, even in the weak-scattering case, the magnitude of the direct transmission coefficient, t 0 , can be much less than unity (here |t 0 | 2 ≈ 0.2). This is not in conflict with the widely used assumption in this regime, i.e., that the field passes through the coupling region with unity transmission. In fact, the real direct transmission coefficient, t j , is actually modified by all cavity modes. Thus, the CIOR in the vicinity of ω j for weak-scattering is [24,25]: where t j = 1 = t 0 . For a thin fiber, as shown in Fig. 2(b), only resonant light can partially pass through the coupling region, implying that the off-resonant light is completely lost into radiation modes. In this case, |t j | ≈ |t 0 | ≈ 0 (here the measured |t 0 | 2 = 0.02), therefore the expression for the CIOR in the strong-scattering regime reduces to a out = 2κ e j c j .
The observation of non-vanishing (larger than 20% for some modes) transmission of resonant light in the strongscattering case can be explained by Eq. (2) and (4). Even if γ g k γ rad , however, the large cavity impact factor, G j (ω j ), can cause the channelling efficiency Γ j to approach to unity, e.g., G 10 4 for a mode with Q = 10 7 , and L = 400 µm.
When the scattering strength lies between the weak and strong regime, the CIOR is similar to Eq. 3 but with |t j | < 1. In this case, destructive interference-induced trapping of the light in the cavity and Purcell factor enhanced channelling both contribute to the light coupling.
The radically different CIORs for the weak and strong coupling regimes imply that the resonances can be selectively controlled to induce either absorption or transparency of light. However, in general, for a given set of system parameters, e.g., the fiber size used above, it is difficult to utilize both the bandpass and bandstop functions. By noting the CIOR is determined from the scattering strength (or t 0 ) it is feasible to achieve a tunable CIOR in a system with a polarization-dependent scattering strength (or t 0 ).
Here, we achieve this goal using a hollow microbub- ble WG resontor, as shown in Fig. 3(a). The hollow microbubble was fabricated using a CO 2 laser focused onto a silica microcapillary [26] and the wall of the microbubble can be as thin as a few hundred nanometers [27]. When a tapered fiber couples to the resonator, the coupling region of the system may demonstrate strong birefringence due to its unique geometry. Specifically, the thin wall of the resonator may act as a curved 2D waveguide that can support two polarized, guided modes, i.e., TE and TM modes with different propagation constants, in addition to radiation modes. Therefore, the coupling coefficients, such as t 0 , are quite sensitive to the polarization of the input field. A calculation of |t 0 | 2 through the coupling region was performed using a finite element method (COMSOL), the results of which are shown in Figs. 3(c) and (e). In contrast to solid microsphere resonators [18], |t 0 | 2 does not monotonically depend on the fiber diameter, d, and the resonator wall thickness, w. The phenomenon is reminiscent of two coupled waveguides, where the optical energy is periodically exchanged between them.
Using the same technique as before (i.e., transmission measurements with a fiber and UV adhesive on the resonator), |t 0 | 2 for the microbubble-fiber coupled system can be measured experimentally. The measured transmittance as a function of fiber diameter is shown in Fig.  3(d) and the measured data correspond well with the simulated results plotted in Fig. 3(c). For some specific sets of parameters, e.g., w = 0.9 µm and d = 0.5 µm (shaded region I), and H polarization at the input, reasonable transmission is observed (|t 0 | 2 = 0.3), whereas for V input polarization it is almost completely lost (|t 0 | 2 ≈ 0). Therefore, unlike for microspheres, polarization-controlled CIORs should be achievable in this system. Certainly, in resonators with different geometries, it is possible to have CIORs with different dependence on the polarization, see the color regions in Fig.  3(e). Figure 3(f) shows how the measured |t 0 | 2 , varies as a function of the input polarization for two different microbubbles with wall thicknesses corresponding to the shaded regions, I and II, in Fig. 3(e).
To demonstrate the feasibility of achieving a polarization-controlled CIOR, the transmission spectra of two microbubble samples are measured and presented in Fig. 4. With the input polarization changing from H to V, the transmission spectrum evolves continuously from a bandstop to a bandpass type for sample A (blue) and the opposite for sample B (red). During this process, the excited cavity modes also switch from TE to TM.
Comparing the results obtained using a microbubble to those for a microsphere resonator (Fig. 2), the highest peaks in the transmission spectra in the strong-scattering regime are much higher, with the measured maximum value exceeding 93% (Mode 1 in Fig. 4). This can be understood by calculating the ratio of the power coupling  Experimental demonstration of polarizationcontrolled CIOR with two microbubble resonators, samples A (blue) and B (red), corresponding to Fig. 3(f). The transmittance is normalized to the transmission of the bare fiber. From top to bottom the input polarization changes from H to V. into the waveguide: The higher transmission can be achieved at the expense of increasing the coupling rates, i.e., decreasing the external Q-factor. For instance, with an intrinsic Q 0 = 10 8 and external Q e = 5 × 10 6 , T (ω j ) = 0.95 × Γ j . Due to the unique geometry of the microbubble resonator, the effective refractive indices of the cavity modes are closer to that of the fiber mode compared to those of a microsphere. Accordingly, the relatively large coupling leads to a reduction in the total Q-factors of those cavity modes, see Fig. 4. Nevertheless, a Q-factor of 3 × 10 6 can be achieved (Mode 2 in Fig. 4) and its transmission is 91%. The observation of the high transmission peaks also demonstrates that the cavity-enhanced channelling efficiency, Γ j , can indeed approach unity. To conclude, we have shown that, when the familiar resonator-waveguide coupled system is in the strongscattering regime, there exists a Purcell factor enhanced channelling mechanism for light coupling, that is similar to the traditional cavity QED modified spontaneous and stimulated emission, as well as Rayleigh scattering of a dipole near an optical cavity. Thus it can be utilized as a complimentary method to achieve high efficiency coupling where critical coupling cannot be accessed, for example, in broadband frequency comb generation [11] and third harmonic generation [10]. In this regime, the CIOR exhibits radically different behavior from the conventional case. On this basis, we have shown that a tunable CIOR is possible. We have experimentally demonstrated this effect in a hollow resonator-waveguide system, which can be switched between the weak-scattering and strongscattering regimes by controlling the input polarization. This counter-intuitive demonstration of a tunable CIOR could have wide impact in designing optical circuits [28] for optical switching [29], tunable filtering [18,30], and integrated polarization elements [31,32]. Furthermore, the polarization-dependent CIOR allows for the preparation of entangled states in a cQED system in a novel way [33] and could be used for cavity-based quantum information processing [34,35].
This work was supported by Okinawa Institute of Science and Technology Graduate University (OIST).