1. Introduction

Bound states in the continuum (BICs) are completely lossless states, that can exist in the continuum of radiation states of a given system, while being completely decoupled from these radiation states. BICs have first been described in 1929 by von Neumann and Wigner [1] and have since been discussed in a large variety of contexts, for example in electronics, electromagnetics and acoustics – an extensive review on the topic can be found in [2]. The photonics research community has also been exploring BICs [3], as the properties of such lossless states are interesting to achieve resonators with extremely high quality (Q) factors. High-Q BIC-based photonic resonances in photonic crystal slabs have recently been reported with Q-factors exceeding a value of 10⁶ [4], as well as in dielectric metasurfaces, enabling experimental demonstrations in photonic biosensing, spectroscopy and beam shaping [5,6].

In photonic integrated circuits (PICs), BICs have first been experimentally demonstrated unintentionally when investigating low-loss silicon-on-insulator (SOI) thin-ridge waveguides [7]. The authors observed that shallowly etched SOI ridge waveguides, like the structure depicted in Fig. 1(a), can exhibit a high loss for TM-like waveguide modes. The loss occurs as the TM-like waveguide mode exists within the continuum of radiating TE-like radiation modes in the partially etched lateral waveguide cladding and is leaking power into phase-matched radiation modes. However, for certain ‘magic’ widths the waveguides showed low loss transmission, which was caused by the radiation modes interfering destructively and prohibiting the lateral leakage [8,9]. Hence, the shallowly etched ridge waveguide was a single-resonance parametric BIC. Similar devices have been reported in microstrip cavities coupled to slab waveguides [10–12], or low refractive index loadings on high refractive index slabs [13,14], which have recently been explored for BIC waveguiding schemes, in particular for etchless guiding in thin film lithium niobate on insulator [15,16].

 

Fig. 1. a) Leaky mode ridge waveguide. The TM-like guided mode is leaking into TE-like radiation modes of the partially etched slab. b) Asymmetrically excited ridge resonator. The incident TE-like mode in the partially etched slab is exciting a phase-matched TM-like resonator mode and is being reflected.

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In our previous work [17], we showed that a single ridge waveguide can also act as a resonant wavelength filter when the TM-like polarized leaky resonator mode is excited from one side through a phase matched TE slab mode, as depicted in Fig. 1(b). We termed this structure ridge resonators, which are resonators supporting single mode parametric BICs based on ridge waveguides that have deliberately been detuned to form a leaky mode resonator. Coupled systems of ridge resonators and similar BIC-based resonator structures can be engineered to exhibit flat-top filter responses in reflection, making them interesting for telecommunications applications [18,19]. However, the potential for high-Q resonances of the BIC-based ridge resonators has yet to be shown, with the resonators in our previous work being limited to a Q-factor of approximately 500. This could be due to a variety of factors not considered in our earlier models that are potentially limiting the Q-factor of a ridge resonator. In particular, the impact of waveguide losses and the influence of the excitation beam on the spectral response of the resonators has so far not been investigated.

In this contribution, we numerically investigate the influence of the width of the Gaussian shaped excitation beam on the reflectivity and the spectral response of ridge resonators. From the wavelength response of the resonators, we determine an effective Q-factor Q_eff through the bandwidth of the reflection spectrum. We show that for the resonators presented in our previous work this effective Q-factor is limited to ∼790 when exciting with a Gaussian beam width larger than 800 µm. We also evaluate the impact of non-leakage losses such as scattering loss due to waveguide roughness on the resonators spectral response and derive a closed-form expression to estimate the reflectivity of ridge resonators. For simulated waveguide losses of 3 dB/cm we find that effective Q-factors of up to 10⁵ can be achieved. Our investigation of the combined effects of limited excitation beam widths and radiation losses on the ridge resonator finds that the non-leakage waveguide losses are currently not limiting the highest achievable effective Q-factors, but rather the practical limitation on the achievable excitation beam widths for the resonators.

2. Influence of the excitation beam width on the ridge resonator response

Structures supporting BICs are commonly considered to extend into infinity in at least one dimension, as BICs are forbidden in compact structures [2]. In this context, our previous work [17] estimated the spectral response of a ridge resonator by assuming an excitation of the ridge resonator via a single spectral component out of the continuum of slab modes – a delta function in the angular spectrum – which effectively describes a spatially infinitely wide excitation beam, or a planewave. When phase-matched to the leaky TM-like mode in the ridge, the excitation mode and resonator mode are coupled, resulting in a strong resonant field build-up in the ridge and a perfect of the incident beam. However, this approximation is not an accurate resemblance of the experimental demonstration in [17], where the ridge resonator was excited by a beam with a Gaussian intensity profile of finite width. To understand the interaction of ridge resonators with finite wide beams better, we investigate in Section 2.1 the influence of the excitation beam width on the reflectivity and Section 2.2 the impact of the beam width on the achievable effective Q-factor.

2.1 Influence on the reflectivity of the ridge resonator

In this section, we investigate the influence of the beam width on the ridge resonator reflectivity by emulating the interaction of a finite wide Gaussian beam with a ridge resonator. The resonator under investigation is a SOI ridge (220 nm thick silicon thin film, n_Si = 3.478; 3 µm silica buffer layer, n_Oxide = 1.444) with air (n_air = 1) as an upper cladding (Fig. 1(a)). The SOI ridge is defined by a partial etch of the material on either side of the ridge to a thickness of 150 nm, in line with Multi Project Wafer (MPW) foundry fabrication parameters [20,21]. In coherence with our previous work [17], the model is defined with a ridge width of 650 nm. Such a resonator shows a leaky mode resonance at a wavelength of 1.55 µm when excited with a TE-like plane wave incident at the resonator under the phase-matching angle of 47.2° [17]. We consider the width of the Gaussian beam as FWHM of the Intensity distribution, sampled on a cross section perpendicular to the beam principal axis at the half-height of the partially etched silicon slab (75 nm). To calculate the reflectivity of the resonator with a Gaussian beam, we decomposed the incident beams into their plane-wave spectral components through Fourier analysis and employed a Film Mode Matching (FMM) method as described in [17] to simulate the response of the ridge resonator to each angular spectral component. The overall response of the resonator to the Gaussian beam was calculated by superimposing the responses of all angular spectral components.

The angular spectral composition of the Gaussian beams with different FWHM beam widths selected to excite the ridge resonator are shown Fig. 2(a), which also shows the angular reflection spectrum of the ridge resonator under plane wave excitation as a black dashed-dotted line. While the excitation beam spectra follow a Gaussian distribution, the reflection spectrum of the resonator under plane wave excitation describes a Lorentzian distribution. One can observe the trend that an increasing beam width results in a narrower angular spectrum of the excitation beam, which would approach a delta-like spike for infinitely wide excitation beams (i.e. plane waves). One can also see that excitation beams with a FWHM beam width smaller than 200 µm show an angular spectrum that is wider than the resonator angular reflection spectrum, wider excitation beams on the other hand show reflection spectra narrower than the plane-wave reference spectrum.

 

Fig. 2. Ridge resonator excited by beams with Gaussian intensity profiles of different width. a) The angular spectra of Gaussian beams with different FWHM in the partially etched slab. The reflectivity of the ridge resonator under plane wave excitation is shown for reference (black dashed-dotted line). b, c) Top view of the ridge resonator, when excited with different beam widths, showing the square magnitude of the electric field at half height of the unetched slab (110 nm). The resonators are centered at the y = 0 position in the respective plot. Field strength in the ridge has been scaled to the same range for comparability. d) Reflectivity as a function of beam width.

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Figure 2(b) and (c) show the square magnitude of the electric field sampled at half-height (110 nm) of the silicon slab, with the ridge resonator centered in the z-direction of the respective field maps. The excitation beam is launched from the lower left corner of the respective field plot under the phase matching angle. These plots qualitatively illustrate the influence of the excitation beam width on the reflectivity of a ridge resonator. For a beam with a narrow spatial FWHM of 100 µm, as shown in Fig. 2(b)), the incident beam is only partially reflected by the ridge resonator, whereas the remainder of the incident power is transmitted through the resonator. When exciting the same resonator with a Gaussian beam with a spatial FWHM of 1600 µm (Fig. 2(c)), nearly full reflection of the incident beam can be observed. When comparing Fig. 2(b) and c), for both cases an intensity build-up can be observed in the ridge resonator, however, the intensity in the ridge is much higher for excitation with the wide Gaussian beam. This result is consistent with the findings of investigations of similar systems, showing an increasing intensity build-up for increasing excitation beam width [10]. For power-normalized Gaussian excitation beams one can estimate an enhancement in the E_x-polarized field magnitude, sampled at the center of the resonator (y = 0) and the half-height of the unetched silicon slab (110 nm), of approximately 1.67 between the two excitation cases in Fig. 2(b) and (c). From Fig. 2(b) and (c) we can observe that the reflected beams seem to maintain a near Gaussian shape, while the transmitted beams have a non-Gaussian intensity distribution.

The non-Gaussian field distribution of the beams transmitted through BIC-based resonators have been described previously with application in optical differentiation and integration operations [22,23]. To quantify the reflectivity of the ridge resonators for different excitation beam widths, we sampled the complex field by using line detectors perpendicular to the incident and reflected beams. The reflectivity R is then determined as the ratio between incident and reflected power. Figure 2(d) shows the quantitative analysis of the reflectivity as a function of the excitation beam width. For increasing beam width, the reflectivity asymptotically approaches the theoretical upper limit of R = 1. The increasing reflectivity of the resonator with increasing excitation beam width is due to the spectral composition of the excitation beams. For increasing excitation beam width, the respective angular spectrum becomes increasingly narrow (see Fig. 2(a)). Hence, for a very wide beam most power is carried in the central spectral components of the beam, which is phase-matched to the resonator mode and thus causes a high resonant energy build-up in the resonator and a strong reflection. For a narrow excitation beam, on the other hand, the excitation beam spectrum is broader than the angular reflection spectrum of the resonator. While the central spectral component is still phase matched to the resonator mode and experiences full reflection off the resonator, the outer spectral components of the excitation beam – which in the case of a spatially narrow beam still carry a significant portion of the incident power – only interact weakly with the resonator, resulting in a lower overall reflectivity of the beam.

From the requirement of a certain excitation beam width to excite the resonance in a ridge resonator, we can derive a condition for the length of the so that it can fully interact with the incident beam. A Gaussian intensity distribution would decay to 1% of its peak intensity for a beam width of [24,25]

To estimate the resonator length required for interaction with this beam, we project the beam width w on the resonator structure under the phase matching angle α_PM, and get

Hence, from the perspective of a Gaussian beams launched with phase matching angles of ∼47° we can estimate resonator lengths of ∼3.8 * FWHM to be sufficient.

While this experiment strictly considers the excitation of the ridge resonators with Gaussian beams of different width “on resonance”, i.e. a situation where the central spectral component of the excitation beam is phase-matched to the resonator mode, considerations on similar structure excited with Gaussian beams that are detuned in angle or wavelength have been made in [10] and for excitation under plane-wave conditions in our previous work [17].

In this part of our investigation, we have shown that the ridge resonator reflectivity at the resonant wavelength strongly dependents on the width of the excitation beam. In order to achieve a high reflectivity, the angular spectrum of the excitation beam needs to be narrow compared to the plane-wave angular reflection spectrum of the ridge resonator (black dashed-dotted line in Fig. 2(a)).

2.2 Influence on the Q-factor of the resonator

In Section 2.1, we found that the reflectivity of the ridge resonator at the resonant wavelength depends on the width of the Gaussian excitation beam, as different spectral components in the excitation beam are differently phase-matched to the TM-like mode of the resonator and hence experience different reflectivities. This effect likely also plays a role at non-resonant wavelengths, therefore influencing the wavelength-spectral response of the resonator, in turn influencing the FWHM of the wavelength response and the achievable effective Q-factor of a ridge resonator.

To investigate how the beam width of the excitation beam influences the effective Q-factor of a ridge resonator, we used the same method that was employed for calculating the peak reflectivity in Section 2.1. The reflectivity of a 650 nm wide ridge resonator was calculated for a range of excitation wavelengths from 1.545 to 1.555 µm, with 1.55 µm as the resonant wavelength and a wavelength resolution of 0.2 nm. From the reflection spectra we then determined the effective Q-factor for a given excitation beam.

Figure 3 shows the reflection spectra of the ridge resonator when excited by Gaussian beams of FWHM width of 100, 200, 400, 800 and 1600 µm as well as for plane wave excitation. For increasing beam width, an increase in peak reflectivity can be observed. Simultaneously, the reflectivity away from the resonant wavelength decreases for increasing excitation beam width. Both effects affect the resonant bandwidth, hence the effective Q-factor for a given excitation beam width as shown in the inset in Fig. 3. The inset in Fig. 3 also shows that for increasing excitation beam width, the effective Q-factor of the resonator increases and asymptotically approaches the effective Q-factor that is achieved under plane wave excitation, indicated by the horizontal dashed-dotted line. For beam widths > 800 µm we calculate a effective Q-factor of the resonator that is – within the resolution of our numerical experiment – indistinguishable from the Q-factor determined for plane wave excitation.

 

Fig. 3. Reflection spectra and effective Q-factor (inset) for a 650 nm wide ridge resonator when excited with Gaussian beam of different widths. For reference, we included plane wave excitation in the respective plots as black dashed-dotted line.

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The increase of the effective Q-factor with increasing beam width is caused by two changes to the wavelength-spectral response of the resonator: i) the increase in peak reflectivity, and ii) the decrease in the off-resonance reflectivity for wider beams, both influencing the spectral width of the resonator response. The observed increase in peak reflectivity with increasing beam width agrees with the results shown in Section 2.1 and have already been discussed. Like the increasing peak reflectivity, the decreasing off-resonance reflectivity that can be observed for wider excitation beams is due to the angular spectral composition of the excitation beams. As discussed earlier, wide Gaussian beams have an angular spectral composition that resembles a delta-like function with respect to the angular reflection spectrum of the resonator under investigation. Hence, the spectral response of the resonator closely resembles the ideal reflection spectrum under plane wave excitation. For the wide angular spectrum of a spatially narrow beam, on the other hand, parts of the excitation spectrum are phase-matched to the resonator mode at off-resonant wavelength and hence reflected. This results in a wider reflectivity spectral bandwidth and a decrease in the effective Q-factor.

In this part of our investigation, we have shown the influence of the excitation beam width on the effective Q-factor of a given ridge resonator. We found that if the angular spectral composition of the excitation beam is wider than the plane wave angular reflection spectrum of the resonator, the effective Q-factor of the resonator is lowered. This effect is a combination of reduced peak reflectivity and increased off-peak reflectivity for spatially narrow beams.

3. Impact of non-leakage loss on the resonator response

In Section 2, we found that the effective Q-factor of the ridge resonator can be limited by the width of the excitation beam. As losses are a limiting parameter for the Q-factor of other types of integrated optical resonators (e.g. ring resonators), it is expected that (non-leakage) losses, such as scattering losses and absorption losses, will also pose a limit on the effective Q-factor that can be achieved in a ridge resonator. To investigate the impact of the losses on the resonators we first provide some background information (Section 3.1), before considering the impact of losses on ridge resonators when using plane wave excitation (Section 3.2).

3.1 Background

In this section we provide the background information for the investigation of the waveguide losses on the resonator performance. Scattering and absorption losses generally impose a limit on the Q-factor of resonant photonic microcavities [26]. However, in [17] we considered the ridge resonator as an ideal structure, where the loss in the resonator was exclusively attributed to the lateral leakage from the leaky TM-like mode of the ridge resonator into its phase-matched leakage channel. This is also reflected in the closed-form expression used to describe the reflectivity of a ridge resonator derived in [17]. Based on a temporal coupled mode theory [27], the analytically derived expression attributes all the loss in the ridge resonator structure to coupling into the leakage channel, thereby ignoring non-leakage losses in the resonator. By extending the analytically derived expression in [17] with an additional term to account for non-leakage losses in the ridge resonator, the reflection coefficient can be expressed as:

Equation (4) indicates that the influence of non-leakage losses in the resonator only become significant once they become non-negligible with respect to the intrinsic leakage loss of the resonator mode. Based on this consideration one can hypothesize that for ridge resonators which are strongly detuned from the BIC – i.e. resonators with a leaky resonator mode that is strongly coupled to a radiation mode – the resonator can accept considerable non-leakage losses while still largely performing as designed. For resonances close to the BIC on the other hand, for which leakage loss of the resonator mode approaches 0, the non-leakage loss in the ridge becomes the dominating factor in Equation (4), thus limiting the reflectivity R and with it the Q-factor of the resonator.

3.2 Ridge resonators with non-leakage loss under plane wave excitation

To evaluate the impact of non-leakage loss on the wavelength response of a ridge resonator, we calculated the resonator wavelength-spectral response using the film mode matching technique. The additional loss in the ridge resonator was emulated through a material with a complex refractive index in the ridge region as shown in the inset of Fig. 4(a). The material was defined so that the real part of its refractive index is identical to the surrounding silicon slab material, the imaginary part of the refractive index can be tuned to emulate different levels of non-leakage losses. We first investigate the impact of non-leakage loss on a resonator with relatively low effective Q-factor. For this purpose, different levels of non-leakage losses were imposed on a 650 nm wide ridge resonator. We further investigate the influence of different levels of non-leakage loss on the effective Q-factor achievable under plane wave excitation. As described in [26], the Q-factors for resonators with and without additional non-leakage loss can be determined via their respective complex effective refractive indices. The total Q-factor is then calculated as the reciprocal sum of the individual Qs.

 

Fig. 4. a) Reflection spectra for ridge resonators with different materials extinction coefficients (resulting additional losses) of $\kappa = 0$ (0 dB/cm); $\kappa = {10^{ - 4}}$ (35 dB/cm); $\kappa = {10^{ - 3}}$ (350 dB/cm); $\kappa = {10^{ - 2}}$ (3500 dB/cm) determined through Eq. (4) (solid line) and the mode matching method (crosses) in comparison. Inset shows the schematic of the ridge resonator emulating additional loss in the ridge region. b) Q-factors of resonators with and without additional losses over the resonator width for material losses (resulting additional losses) $\kappa = 0$ (0 dB/cm);

$\kappa = 1.42 \cdot {10^{ - 6}}$ (0.5 dB/cm);$\kappa = 2.48 \cdot {10^{ - 6}}$ (1 dB/cm); $\kappa = 8.52 \cdot {10^{ - 6}}$ (3 dB/cm);

$\kappa = 1.42 \cdot {10^{ - 5}}$ (5 dB/cm); $\kappa = 2.84 \cdot {10^{ - 5}}$ (10 dB/cm).

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Figure 4(a) shows the reflection spectra for ridge resonators afflicted by non-leakage loss obtained through the closed form expression (Eq. (4)) shown as continuous lines and the rigorous FMM method (crosses) for different levels of non-leakage loss. Both results are in good agreement and show the reflectivity of the resonator being reduced across the observed wavelength spectrum for increasing non-leakage loss in the ridge region. The decrease in peak reflectivity results in a reduced effective Q-factor, as it widens the half-maximum bandwidth. The results obtained through the closed form expression show overall a slightly narrower spectral reflectivity. The small discrepancy between rigorous and analytical solution can be attributed to the non-resonant reflection and transmission terms being ignored in Eq. (4). The non-resonant reflection and transmission are due to the step discontinuity at the slab/ridge interface [17] and additional material permittivity discontinuity between lossless and lossy regions. With the analytical solution we can estimate a effective Q-factor of approximately 800 for the resonator without additional loss. Increasing the loss in the ridge region to a value of 35 dB/cm decreases the peak reflectivity to 93% and the effective Q-factor to approximately 630. Further increase of the loss in the ridge to 3.5 × 10² and 3.5 × 10³ reduces the effective Q-factor further to values of 530 and 160, respectively.

Figure 4(b) shows the simulated effective Q-factor as a function of the resonator ridge width – and hence resonator leakage loss – for different non-leakage loss levels. The imaginary effective index contributions of the resonator mode alone (black dashed line) and the non-leakage losses (colored lines) over resonator width are shown as an inset. For resonators without any non-leakage loss and a ridge width close to 722 nm, the resonator approaches the BIC with effective Q-factor approaching infinity. For loss-afflicted resonators, the maximum effective Q-factor can still be observed at the resonator width at which the BIC would occur. However, the maximum Q-factor is reduced for increasing non-leakage losses. For a loss of 3 dB/cm – a typical loss value for waveguides on the SOI platform [28] – the maximum achievable effective Q-factor for the ridge resonator is approximately 10⁵ at a ridge width of 722 nm. When changing the ridge width away from the BIC, the difference in effective Q-factors between the loss-afflicted and ideal resonator structures decreases. The point at which a loss-inflicted resonator starts behaving like the ideal, lossless resonator of identical dimension is dependent on the non-leakage loss in context of the leakage loss (see inset in Fig. 4(b) and Eq. (4)).

In this investigation we have shown the limits that non-leakage waveguide losses in the ridge region impose on the maximum achievable effective Q-factor of the ridge resonators. We have derived a closed form expression for the spectral reflectivity of the loss-afflicted resonators and compared it with the results of the rigorous FMM method. We have further shown that for waveguide losses typical for the SOI platform effective Q-factors as high as 10⁵ can be achieved in ridge resonators.

4. Ridge resonators with non-leakage loss under Gaussian beam excitation

In the previous two sections we showed separately that the excitation beam width as well as non-leakage losses can be limiting factors for the achievable effective Q-factor and the peak reflectivity of ridge resonators. In the following we investigate how the combination of the two effects influence the ridge resonator response, providing a guide on achievable effective Q-factors and reflectivity for ridge resonators of different widths that are afflicted by scattering losses and Gaussian beam excitation.

We calculate the effective Q-factors and peak reflectivity based on Eq. (4) for ridge resonators with widths between 650 nm and 790 nm for resonators afflicted by a typical waveguide loss for the SOI platform of 3 dB/cm. The resonators are excited through Gaussian beams with beam widths between 100 µm and 1 cm. The resulting effective Q-factor is shown in as a heatmap and the peak reflectivity on resonance as black dashed contours in Fig. 5.

 

Fig. 5. Map of achievable effective Q-factors (colormap) and reflectivities (black, dashed contours) for ridge resonators afflicted with 3 dB/cm non-leakage losses as a function of the resonator width and excitation beam width. Q-factors exceed 10⁴ are observed at a ridge width of approximately 722 nm, which is where the BIC occurs. For the highest Q resonances the reflectivity is restricted to values below -10 dB of the incident beam power.

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One can see that for all resonator widths, the calculated effective Q-factor that is achievable under Gaussian beam excitation is increasing with beam width. For intrinsically low-Q resonators, the theoretical maximum for the effective Q-factor under plane wave excitation is being approached asymptotically when increasing the beam width to 1 cm, the maximum beam width considered in this experiment. For resonators close to the BIC at approximately 720 nm, the highest effective Q-factors are being calculated to approximately 3.1 × 10⁴ for the considered maximum beam width.

This result is in coherence with the findings in the previous sections. It shows that the excitation beam spectrum must be narrow compared to the ridge resonators angular reflection spectrum to achieve the nominal effective Q-factor for a given resonator. The maximum effective Q-factor for loss inflicted ridge resonators therefore stays behind the predicted effective Q-factor under plane wave excitation of up to 10⁵ (Fig. 4(b)), This suggests that the maximum achievable effective Q-factor for the ridge resonator on the SOI platform is not limited by the scattering losses in the waveguide, but rather by the size of the beam required to excite such a resonance.

The reflectivity of the ridge resonators increases with excitation beam width for any resonator width, this result is consistent with the results presented in Fig. 2(d). While for low-Q_eff resonators almost perfect reflection can be achieved at the resonant wavelength, the reflectivity of the ridge resonator decreases when the resonator width approaches the BIC at a ridge width of approximately 720 nm. This result was implied by the expression in Eq. (4), where at the resonant wavelength the non-leakage loss puts a limitation on the maximum reflectivity even under plane wave excitation conditions. While for strong leakage losses the reflectivity is only weakly influenced by the non-leakage term, it becomes the dominating factor for resonators with an intrinsically low leakage loss.

In this investigation we have shown the limitations that ridge resonators face when simultaneously afflicted by non-leakage losses and finite-wide Gaussian beam excitations. We found that for high-Q_eff resonances the reflectivity of the resonators is greatly reduced. The effective Q-factors of the loss-afflicted resonators even when excited by beams with a width of 1 cm FWHM is lower than theoretical limits under plane wave excitation conditions. This suggests that the resonators Q_eff is limited by the excitation beam widths rather than the loss in the waveguides and higher effective Q-factors could be achievable if the excitation beam width can be increased.

5. Conclusion and outlook

In this work we have investigated the influence of the excitation beam width and non-leakage losses on the effective Q-factor and reflectivity of BIC-based ridge resonators on the SOI platform. We have shown that the spectral composition of the excitation beam is influencing both resonator characteristics and has to be considered when designing such integrated photonic BIC-based resonators.

We also have shown the limiting factor that non-leakage loss has on the performance of ridge resonators. While for intrinsically leaky (i.e. far away from the BIC) resonators additional waveguide loss has barely an influence on the ridge resonators effective Q-factor, the waveguide loss becomes increasingly important the closer the leaky mode resonator is to the actual BIC. As for all integrated photonic resonators, the potential for high-Q_eff BIC-based resonators is therefore strongly tied to the waveguide loss that can be achieved on the respective platform.

When both, finite beam width and non-leakage loss, are considered, it becomes clear that the waveguide scattering loss plays a minor role in restricting the effective Q-factors of the resonators. For resonators close to the BIC it is much more the required excitation beam width that poses a limit for the effective Q-factor of the ridge resonators. In particular, for excitation with a Gaussian beam that is launched through a parabolic reflector, as used in [17], the beam shaping takes up a large part of the integrated photonics devices footprint, and a balance must be struck between size restrictions and resonator performance.