Fabry-Perot (FP) cavities confine light between a pair of reflectors in the form of a standing wave [1]. Due to its sharp spectral responses and capability for field enhancement, the FP cavity plays a central role in a vast range of applications, e.g., wavemeters [2], gravity-wave detection [3], label-free sensing [4], semiconductor lasers [5], linewidth narrowing [6], quantum-information processing [7], frequency comb generation [8,9], and high-capacity communications [10,11]. Compared to traveling-wave cavities, FP cavities have some unique applications due to the standing-wave property, such as cavity-enhanced optical trapping [12] and atom cooling [13]. In addition, bent waveguides are not required in FP cavities, leading to a smaller footprint and a higher integration density. Silicon-on-insulator (SOI) has become a mainstream platform for photonic integration, thanks to its scalability, high yield, and CMOS compatibility [14]. A variety of FP cavities on SOI have been investigated in recent research [8–11,15–22], but their Q factors are all quite limited. The photonic-crystal (PhC) nanocavity can offer high Q factors by trapping the resonant mode in a photonic bandgap [23]. However, the nanocavity mode in PhCs cannot be depicted by FP mechanisms when the cavity length only involves few PhC lattices [24]. Longer FP cavities formed by PhC waveguides have been experimentally characterized, but their Q factors cannot exceed one million [24,25]. To date, the ultrahigh-Q (>106) silicon-integrated FP cavity has not yet been demonstrated.

Unlike traveling-wave cavities, such as microring resonators [26,27] and microdisk resonators [28,29], whose intrinsic Q factors are predominantly governed by propagation loss, FP cavities also suffer from excess losses of reflectors. In order to improve the Q factor of FP cavities, it is essential to reduce the reflection loss. The distributed Bragg reflector (DBR) is the most commonly used on-chip reflector. The DBR relies on grating structures to form the constructive interference of multiple reflected waves. Typically, DBRs can be realized by introducing sidewall corrugations [8–10,16–18], embedded nanoholes [15], and deep trenches [19,20,30]. However, the requirement for broadband reflection necessitates the use of high-contrast gratings with strong out-of-plane diffraction losses [31]. For a silicon-integrated DBR, such diffraction losses can be as high as 5% [19,30] unless the trench depth is scaled up to several micrometers [20]. Therefore, the Q factor of most DBR-based FP cavities is restricted to the order of 104 [10,19,20]. In Refs. [8,9], silicon-nitride-on-insulator (SNOI) with a lower index contrast is exploited to alleviate diffraction losses and lift the Q factor to ≈4.8×105 in a narrow bandwidth, but a higher Q factor (>106) has remained unattainable. As an alternative, on-chip reflectors can also be built by connecting the output ports of a 3-dB coupler to a looped waveguide [11]. The performance of loop structures is, however, limited by the scattering loss in Y-junctions [32] or multimode interference couplers [33]. The scattering loss can be inhibited by using directional couplers (DCs) [34]. Nevertheless, the coupling ratio of DCs is usually susceptible to fabrication defects and typically has a limited optical bandwidth, making it difficult to attain near-unity reflectance over a broad bandwidth. The reported Q factor of loop-based FP cavities is limited to ≈2.3×104 [11]. Other types of on-chip reflectors, such as coated metallic mirrors [21] and inverse-designed structures [22], all suffer from the intrinsically high reflection loss caused by ohmic absorption or stringent feature sizes; hence, the measured Q factors cannot exceed ≈2.0×103. The ultralow-loss on-chip reflector for ultrahigh-Q (>106) FP cavities has remained an unsolved open problem.

In this work, we present the first demonstration on million-Q silicon-integrated FP cavities. Inspired by free-space optics, a nearly lossless retroreflector is developed to address the reflection loss issue. By employing the TE1 mode with zero central field intensity, the scattering loss is minimized at the corner region of retroreflectors. Losses from other mechanisms are also reduced to an ultralow level. An ultrahigh intrinsic Q factor (≈3.4×106) is experimentally achieved. The maximum loaded Q factor is measured to be ≈2.1×106.

Figures 1(a) and 1(b) depict the schematic configuration of the proposed FP cavity, which consists of an asymmetric directional coupler (ADC) in the middle, a pair of reflectors at each end, and adiabatically tapered waveguides as connectors. The cavity is fed by an accessing waveguide through weak side coupling with a π/2 phase shift in the ADC. The tapered waveguide introduces a round-trip group delay, thereby building up a standing-wave resonance in the cavity. The dropped carrier experiences another π/2 phase shift, resulting in the total π phase shift. At the output port, the destructive interference will dramatically diminish the field intensity for resonant wavelengths, leading to the all-pass filtering of light [21,22]. The on-resonance carriers are reflected back into the input port.

In such a design, loss mechanisms can be categorized into four classes: the propagation loss produced by sidewall roughness, the intermodal conversion in tapered waveguides, the coupling loss in the ADC, and the reflection loss at reflectors, as illustrated in Fig. 1(c). As previously stated, the primary concern is to reduce the reflection loss. Here, we introduce the on-chip retroreflector to realize ultralow-loss reflection, as illustrated in Fig. 1(d). The structure is formed by two total-internal-reflection (TIR) mirrors at an angle of 90°. In a wide waveguide, the guided mode can be approximated as a cluster of weakly divergent rays [35]. Through the TIR effect, each ray will consecutively bounce off at A and A′ [36]. Moreover, outer rays strike the mirror earlier than interior rays yet travel a longer transverse distance, resulting in identical light-path lengths for all launched rays. Thus, the input mode can be reflected while preserving its initial phase profile perfectly. The archetype of this structure, well known as the roof prism [37], has served as a fundamental element in numerous free-space optical systems. The retroreflection mechanism is ineffective at the corner point B, where mirrors A and A′ are joined together, as illustrated in Fig. 1(d). Such corner scattering is the prime cause of reflection losses for on-chip retroreflectors. The TE1 mode is employed to tackle this issue. The maximum field intensity of the TE0 mode occurs at the central position, as shown in the left panel of Fig. 1(e). In contrast, the central field intensity of the TE1 mode is zero, as shown in the right panel of Fig. 1(e). Consequently, the corner scattering can be naturally prevented by choosing the TE1 mode to carry resonances. Figure 1(f) shows the calculated light propagation profiles when TE0 and TE1 modes are injected. For TE1, the corner scattering is significantly mitigated, since all the launched rays reside only on two lateral sides. In the following sections, we will analyze and optimize all the essential components in the proposed FP cavity to achieve an ultrahigh Q factor.

The design is implemented on the SOI platform consisting of a silicon core with Hwg=220nm and a silicon-dioxide buffer with Hbuff=3μm. The upper cladding is a silicon-dioxide layer with a thickness of Hclad=1μm. The central wavelength is chosen as λ=1.55μm. The reference data for refractive indices can be found in Ref. [38]. We calculate the TE1 effective index (neff,TE1) with varying widths (Wwg) by using the finite-difference frequency-domain (FDFD) method [39], as shown in Fig. 2(a). It can be observed that neff,TE1 converges to the effective index of the slab mode (nslab≈2.845) when Wwg is large enough. Under the ray-optic approximation, the TIR angle (θTIR) can be obtained by using the equation below, θTIR=arcsinnSiO2nslab,(1)where nSiO2 denotes the refractive index of silicon dioxide. Figure 2(b) shows the calculated dispersion curve of θTIR. It can be found that θTIR is almost invariant over a 100-nm wavelength range. Thus, the TIR condition (i.e., θTIR<45°) can be fulfilled for mirrors A and A′ regardless of working wavelengths.

The reflectance is then calculated for the retroreflector as its width (Wref) is varied, as shown in Fig. 2(c). The simulation is carried out by using the finite-difference time-domain (FDTD) method [39]. For comparison, both TE0 and TE1 are considered in the simulation. The TE0 and TE1 reflectance (Rref,TE0, Rref,TE1) are extracted by applying mode decomposition on the reflected wave [39]. For TE0 and TE1, the calculated reflectance rises as Wref increases, suggesting that the lateral divergence of light is mitigated, and the ray-optic approximation is valid. However, for TE0, the maximum reflectance can only reach Rref,TE0≈0.946 due to the strong corner scattering. In contrast, for TE1, the near-unity reflectance (Rref,TE1≈0.993) can be attained when the width is optimized to be Wref=20μm. Here, the large modal area is also advantageous for the reduction of propagation losses, since the field intensity is diminished at sidewalls [40]. For a narrower width, e.g., Wref=7.5μm, the reflectance is still quite high (Rref,TE1≈0.975), making the proposed reflector well suited for optical filters with smaller footprints and relatively lower Q factors. Figure 2(d) shows the calculated light propagation profile with Wref=20μm. From the profile, high reflectance and negligible corner scattering can be observed. A slanted waveguide sidewall may arise because of the difficulty in the precise control of dry-etching processes. The inclination of sidewalls in retroreflectors will introduce excess losses [41]. To address this concern, we also calculate the reflectance as a function of the sidewall tilt angle (θtilt), as shown in Fig. 2(e). High reflectance of Rref,TE1>0.99 can still be accomplished, even with a considerably large sidewall tilt angle, e.g., θtilt=5°. Such perpendicularity can be ensured by most silicon photonic foundries. The reflection loss (RLTEi=−10log10Rref,TEi) is then calculated to further evaluate the performance, as shown in Fig. 2(f). The calculation results show ultralow reflection losses of RLTE1≈0.01−0.07dB for TE1 over a 100-nm bandwidth. Around the central wavelength, the reflection loss is reduced by >0.2dB compared to the TE0 case. In addition, RL is also slightly affected by the side scattering [see the arrow in Fig. 2(d)], and its strength is tied to the (anti-)node position of lateral standing waves. The ripples in RL spectra are opposite between TE0 and TE1, since their symmetries are inversed and the positions of nodes and antinodes are exchanged.

The ADC is leveraged to achieve the side coupling between TE0 and TE1. The cross-talk-free TE0-to-TE1 conversion relies on the perfect matching between neff,TE0 of the accessing waveguide and neff,TE1 of the bus waveguide [42]. Figure 3(a) shows the calculated neff,TE0 and neff,TE1 with varying widths. The combination of accessing-waveguide and bus-waveguide widths (Wacc, Wbus) can be then determined according to the phase-matching condition. The coupling loss is calculated for the ADC with various structural parameters, as shown in Figs. 3(b) and 3(c). Here, the coupling loss is defined as CL=10log10(TBAR+TCRO,TE1), where TBAR denotes the remnant transmittance in the accessing waveguide, and TCRO,TE1 denotes the TE1 coupling ratio. In the simulation, the gap width is set as Wgap=300nm to ensure a large feature size and easy gap filling. The coupling loss in an ADC is usually sensitive to the perturbation caused by strong evanescent coupling, since the transmission of higher-order modes is more vulnerable. Therefore, we tend to choose a wide width (Wacc=700nm) to depress the evanescent field intensity and a large bending radius (racc=100μm) to ensure the gradual approaching of accessing and bus waveguides, thereby establishing an adiabatic coupling process [26]. According to Fig. 3(a), the bus-waveguide width is then determined as Wbus=1.428μm. The optimal coupling loss is as low as CL<0.01dB over a 100-nm wavelength range. Figure 3(d) shows the calculated TCRO,TE1 with varying coupling lengths (Lc). Figure 3(e) shows the corresponding TCRO,TE1 spectra. To enable high-Q resonances, the coupling length is chosen as Lc=25μm, with a weak coupling ratio of TCRO,TE1≈0.01. Figure 3(f) shows the calculated light propagation profile for the optimized ADC. An efficient TE0-to-TE1 conversion and a weak coupling strength can be clearly observed. In total, four TE modes (i.e., TE0−3) are supported by the bus waveguide with Wbus=1.428μm. To verify the mode selectivity, intermodal cross talks (XTTEi=10log10TCRO,TEi) and extinction ratios (ERTEi=10log10TCRO,TE1/TCRO,TEi) are calculated for the optimized ADC [see Figs. 3(g)–3(i)]. Around the central wavelength, low cross talks (XTTEi<−50dB) and high extinction ratios (ERTEi>30dB) can be attained, so that only resonances carried by the TE1 mode will be excited in the FP cavity. In addition, high performance can be maintained despite fabrication defects, as detailed in Appendix A.

The propagation loss is mainly induced by the sidewall roughness. The volume-current method is used to model this stochastic process by making sidewall nonuniformity as polarized current sources equivalent. The detailed description of this method can be found in Ref. [43]. In Fig. 4(a), we present the calculated propagation losses (αTE1) with varying Wwg under different mean deviations of roughness (σ). Here, the correlation length of roughness is assumed to be 50 nm [43]. The propagation loss can be reduced to an extremely low level (αTE1<0.01dB/cm) when the waveguide is quite wide (e.g., Wwg>10μm). In the tapered waveguide, the taper width varies linearly from Wbus to Wref; hence, the taper length (Ltp) must be carefully designed to avoid the intermodal conversion. Figure 4(b) shows the calculated TE1 transmission efficiencies (Ttp,TE1) with varying Ltp by using the eigenmode expansion (EME) method. The conversion from TE1 to other modes can be fully eliminated (i.e., Ttp,TE1≈1) when the taper length is long enough (Ltp>1mm). The calculated light propagation profile for the tapered waveguide with Ltp=1.5mm is shown in Fig. 4(c). Other types of taper trajectories can also be adopted to define the waveguide shape and reduce the device footprint [44]. In this work, it is unnecessary to further shorten the taper length, since the long round-trip length is a desirable attribute in the realization of ultrahigh Q factors. In a slow-varying taper, light transmission can be described by the following equations [45]: E=E0Ttp,TE110−α˜TE1Ltp/20eik0n˜eff,TE1Ltp,(2)n˜eff,TE1=∫L=0Ltpneff,TE1dL/Ltp,(3)α˜TE1=∫L=0LtpαtpdL/Ltp,(4)where E0 and E denote the magnitudes of electric field at the beginning and end of the tapered waveguide, k0 denotes the vacuum wavenumber, n˜eff,TE1 denotes the mean effective index, and α˜TE1 denotes the mean propagation loss. Figure 4(d) shows the calculated neff,TE1 map in terms of Wwg and wavelengths. The n˜eff,TE1 spectrum can be then obtained according to Eq. (3), as shown in Fig. 4(e). In addition, we also calculate the corresponding mean group indices (n˜g,TE1=n˜eff,TE1−λ·∂n˜eff,TE1/∂λ), as shown in Fig. 4(f). By using Eq. (4), the mean propagation loss is calculated to be α˜TE1≈0.15dB/cm with σ=5nm at the central wavelength of 1.55 μm. Further calculations show that α˜TE1 is nearly wavelength-independent, which is due to the large modal area. Both n˜eff,TE1 and α˜TE1 are independent with Ltp because of the linear taper trajectory. By using Eqs. (3) and (4), the tapered waveguide can be substituted by an equivalent straight waveguide to simplify the modeling of FP cavities.

Eventually, all the essential components in the FP cavity have been optimized. The optimal structural parameters are listed as follows: Wref=20μm, Wacc=700nm, Wbus=1.428μm, Wgap=300nm, Lc=25μm, Ltp>1mm, L1=50μm, and L2=10μm. Here, L1 and L2 refer to the lengths of straight sections in between retroreflectors and tapers. By using the transient sample mode time-domain (TSMTD) method [39], the transmittance (TFP) and reflectance (RFP) spectra are obtained for the optimized FP cavity. Here, each component is described by a transfer matrix, which is readily available in Lumerical cosimulation suites [39]. In the simulation, the sidewall roughness is set as σ=5nm, as an example. Figure 4(g) shows the calculated TFP and RFP spectra for the FP cavity with Ltp=1.5mm. At resonant wavelengths, the summation of TFP and RFP is nonunity, since a portion of energy is dissipated by intracavity losses. Figure 4(h) is the zoom-in view of the TFP spectrum in the vicinity of a single resonance. The 3-dB bandwidth is calculated to be δλ≈1.0pm with the corresponding loaded Q factor of Qload≈1.5×106. A high extinction ratio of ERres≈3.2dB can be attained at the resonant wavelength. The calculated Qload rises as Ltp increases [see Fig. 4(i)], since the adiabatic taper constitutes the majority of the round-trip length. To determine the intrinsic Q factor, we also calculate the Ltp−Qload curve with a reduced coupling length (Lc=5μm) and a near-zero coupling ratio (TCRO,TE1≈0.001). The intrinsic quality factor is estimated to be Qi≈4.1×106 when the taper length is extended to Ltp>5cm.

An array of FP cavities with varying Ltp was fabricated at Applied Nanotools Inc. [46] by applying electron beam lithography (EBL) and reactive ion etching (RIE) on a commercial SOI wafer, as shown in Figs. 5(a) and 5(b). An oxide upper cladding was deposited through plasma-enhanced chemical vapor deposition (PECVD). Titanium–tungsten heaters were patterned on the top of FP cavities to offer thermo-optical tunability. Efficient edge coupling with lensed fibers is enabled by leveraging inverse tapers [47]. On-chip polarizers were monolithically integrated with inverse tapers to ensure single-polarization operations [48]. Several straight waveguides were fabricated in close proximity for normalization purposes.

Figure 5(c) illustrates the measurement setup. The fabricated devices were characterized by scanning the wavelength of a narrow-linewidth tunable laser (Keysight 8164B) while reading the output power from a synchronized high-detectivity power meter (Keysight 81636B). The polarization state was adjusted by a polarization controller (PC). The chip was mounted on a thermo-electric cooler (TEC) to stabilize the ambient temperature. We used an optical isolator to filter out the reflected light, since the performance of an FP cavity can be fully depicted by its TFP spectrum. Figure 5(d) shows the measured TFP spectra around λ≈1.55μm with varying Ltp. In Figs. 5(e) and 5(f), we show the TFP spectra in the vicinity of a single resonance with Ltp=1.5mm and 4.0 mm. By applying Lorentz fitting, loaded Q factors are characterized to be Qload≈1.1×106 and 1.9×106, respectively. The measured line shape is virtually symmetric even if the launched laser power is up to −5dBm, indicating a cold-cavity state and suppressed optothermal responses. This phenomenon originates in the large modal area of multimode waveguides. Numerous applications can benefit from the optothermal stability, such as linewidth narrowing [6] and frequency stabilization [49]. The measured Ltp−Qload curve is shown in Fig. 5(g). Here, the measurement error is estimated by the standard deviation of Qload obtained from several neighboring resonances. In Appendix B, we also measured the TFP spectra at different wavelength bands. Due to the dispersion of TCRO,TE1, the measured Qload is higher at shorter wavelengths. In the experiment, we observe a maximum loaded Q factor of Qload≈2.1×106 around λ≈1.51μm. It is impossible to obtain the intrinsic Q factor by directly measuring the 3-dB bandwidth of resonances, since the cavity length is constrained by the die size. Here, we propose to use the cavity-scattering theorem to resolve Qi [21]. To achieve this, it is necessary to extract several hidden parameters (i.e., Rref,TE1, α˜TE1, and TCRO,TE1) from the measured TFP spectra by using the following equations: Qload=k0LFPn˜g,TE1(Rref,TE1a)−1+TCRO,TE1−1,(5)ERres=10log10[1+Rref,TE1a1−Rref,TE1a·1−Rref,TE1a(1−TCRO,TE1)1+Rref,TE1a(1−TCRO,TE1)]2,(6)FSR=λ22LFPn˜g,TE1,(7)where FSR denotes the free spectral range, LFP denotes the cavity length (≈L1+2L2+2Ltp), n˜g,TE1 denotes the mean group index throughout the cavity, and a denotes the attenuation over LFP. Three unknown variants (i.e., LFP−n˜g,TE1 product, Rref,TE1−a product, and TCRO,TE1) can be derived from Eqs. (5)–(7) given the measured Qload,ERres, and FSR (see Appendix B). The propagation loss dominates the attenuation process; hence, the Rref,TE1−a product can be formulated as a log-scale linear regression on LFP, 10log10Rref,TE1a=−α˜TE1LFP/10+10log10Rref,TE1.(8)

From Eq. (8), Rref,TE1 and α˜TE1 can be reconstructed by using the intercept and slope of the regression curve. This method is applied at different wavelengths; the reconstruction results are shown in Fig. 5(h). The reflectance is measured to be Rref,TE1≈0.99, with a flattened dispersion over an 80-nm bandwidth. The corresponding reflection loss is RLTE1≈0.05dB, which is in good agreement with the simulation results. The measured propagation loss is α˜TE1≈0.18dB/cm. This loss level is quite close to the experimental results demonstrated in Ref. [50]. The reconstructed TCRO,TE1 spectrum is presented in Appendix B. The intrinsic Q factor is then estimated to be Qi≈3.4×106 by assuming TCRO,TE1→0 and LFP→∞ in Eq. (5). Figure 5(i) shows the measured TFP spectra at different ambient temperatures (t). The temperature sensitivity is measured to be ∂λ/∂t≈60pm/K. The thermo-optical tuning was implemented by applying different electric power (P) with a power source (Keithley 2400), as shown in Fig. 5(j). A high tuning efficiency of ∂λ/∂P≈5.6pm/mW is experimentally realized.

In this work, we report a silicon-integrated FP cavity with an ultrahigh intrinsic Q factor of Qi≈3.4×106. From the measured spectra, we observe a maximum loaded Q factor of Qlaod≈2.1×106. These results represent the highest Q factor ever demonstrated in an FP cavity on SOI. The high performance is enabled by a novel on-chip retroreflector working in the multimode regime. The reflectivity of the proposed retroreflector is supported by the naturally low-loss and broadband TIR effect. Moreover, the TE1 mode is employed to dampen the corner scattering and bring the reflection loss to an ultralow level. The experimental results provide strong evidence that this new type of reflector possesses ultralow reflection losses (RLTE1≈0.05dB) over a broad bandwidth (>100nm). The performance of several on-chip FP cavities is summarized and compared in Table 1. The reflection loss of the proposed retroreflector is comparable to that of the bulk DBR with microscale trenches. Compared to the previous results on silicon, the demonstrated Q factor has been improved by more than 2 orders of magnitude. Usually, such high Q factors can only be accomplished by using bulky silica waveguides with an ultralow index contrast (≈1%) [51]. The loaded Q factor can be further enhanced by decreasing the coupling ratio of ADCs. In addition, it is possible to improve the intrinsic Q factor through the utilization of more advanced fabrication technologies (e.g., the photonic Damascene process [52]) or material platforms with a lower index contrast (e.g., SNOI) to reduce the sidewall roughness. The reflection loss can be mitigated by using the hard mask in the dry-etching process to improve the perpendicularity of sidewalls [53]. The measured finesse is as high as F>100, which can be improved to F>500 by reducing the propagation loss (e.g., α<0.1dB/cm) and coupling ratio (e.g., TCRO,TE1<0.005).Table 1.

Performance Comparison of On-Chip FP Cavities

Our design has the potential to serve as a versatile platform for numerous applications, including, but not limited to, comb lasers and linewidth narrowing. The proposed FP cavity possesses an ultrahigh Q factor and a relatively small free spectral range. These features are desirable for electrically pumped narrow-linewidth dense comb lasers [5]. It is anticipated that a low-threshold low-repetition-rate comb laser could be realized via the heterogeneous integration with thin-filmed III-V gain materials. Another potential application is linewidth narrowing, which is to suppress close-to-carrier noises of lasers by using an ultrahigh-Q cavity isolated in a cryogenic environment [6]. The large modal volume of the proposed FP cavity could help to mitigate optical nonlinearity and optothermal phase noises [40]. Overall, this work demonstrates a novel method for realizing ultralow-loss on-chip reflectors for the use in ultrahigh-Q integrated FP cavities, which enriches the silicon photonic toolbox and paves ways for numerous applications.

Acknowledgment. The authors acknowledge Applied Nanotools Inc. for enabling the device fabrication. H. Xu thanks ITF Research Talent Hub for the financial support.