Silicon photonics has been developed rapidly in the past ten years ^[1–4] because of seamless photonic integration with electronics and compatibility with mature CMOS fabrication technology. In silicon photonics, microresonator-based devices have the advantages of a small footprint, high performance, and low power consumption ^[5–7]. However, in contrast to electronic circuits that can function well with a temperature variation of tens of degrees celsius, integrated photonic devices, especially for microresonators, suffer from a high sensitivity to temperature, which has been an obstacle for their practical applications. This is due to the large thermo-optic coefficients (TOCs) of silicon used in integrated platforms. In an on-chip photonic system that may contain both active and passive devices, the local temperature of devices varies both spatially and temporally, which changes the refractive index and thus induces a large wavelength shift in resonator-based devices. At 1550 nm, the temperature-dependent wavelength shift (TDWS) of silicon is 110 pm/K ^[8]. For a high-Q microring resonator, this large wavelength drift may make the device completely unusable. Designing athermal photonic devices becomes an attractive solution to achieve temperature-insensitive integrated devices and systems.

To mitigate the thermal sensitivity of silicon photonic devices, some solutions have been proposed and demonstrated. An athermal technique is to utilize a Mach–Zehnder interferometer with asymmetric arms to fully compensate for the resonance shift ^[9], but this approach requires a large footprint and causes much power consumption. Besides, a heater or thermal feedback control ^[8,10,11] has also been used to stabilize the local temperature and the resonance wavelength, but it increases power consumption. Using negative thermo-optic polymer materials such as polymethyl methacrylate and hyperlinked fluoropolymer developed by Enablence ^[12] as cladding layers can compensate for the large positive TOC of Si. Polymer-cladded waveguides have been utilized to realize athermal filters ^[12–18] and modulators ^[19]. However, polymers show unstable properties with the change of environment conditions like thermal condition, mechanical strength, and moisture retention. A better candidate of negative-TOC materials is titanium dioxide (TiO2), and it has a thermal expansion coefficient larger than the polarizability change due to thermal variations ^[20]. TiO2 shows high stability during the fabrication process, and its negative TOC has been utilized to realize various athermal devices, including microresonators ^[21–23], modulators ^[24], arrayed waveguide gratings ^[25], filters ^[26], and lasers ^[27].

The effective TOC of an athermal waveguide, which consists of a Si core and a compensating material as cladding, can be expressed as Eq. (1). It is weighted by optical confinement factors in different parts of the waveguide. By properly tailoring waveguide structural parameters, the negative-TOC material can fully compensate for the TDWS caused by positive-TOC material at a specific wavelength. However, as wavelength increases, the guided mode always extends to the negative-TOC cladding, and thus, the effective TOC of the waveguide becomes negative. This is why the previously reported athermal waveguides ^[13–27] can only achieve the athermal property at a single wavelength, which means that the zero TDWS can be produced for only one of the resonant peaks of a microresonator. However, for sensing and communication applications, it would be highly desirable to have wideband athermal devices: dneffdT=Γc(λ)dncdT+Γcl(λ)dncldT+Γsub(λ)dnsubdT.(1)In this work, we propose a broadband athermal waveguide, which shows a low effective TOC of ±1×10−6/K from 1400 to 1700 nm. The proposed waveguide has a low-loss and bending-insensitive broadband athermal property when it is applied to ring resonators, and the bending loss of ring resonators with a radius of >30μm is 1 dB/cm.

A broadband athermal waveguide, as shown in Fig. 1, is based on a dual-cladding structure. The lower cladding, made by TiO2, has a negative TOC, which compensates for the positive TOC of the Si core. By using another positive-TOC material, Si3N4, as an upper cladding, we can correct the wavelength dependence of the TOC to obtain a broadband athermal property. The key idea is to properly control the fractional powers of the guided mode in the core, the upper cladding, the lower cladding, and the substrate, Γc(λ), Γu(λ), Γl(λ), and Γsub(λ), which are highly wavelength dependent, as expressed in Eq. (2). This requires a careful control of all the layer thicknesses. Thus, using TiO2 instead of a polymer is advantageous in terms of both precise thickness and CMOS compatibility: dneffdT=Γc(λ)dncdT+Γu(λ)dnudT+Γl(λ)dnldT+Γsub(λ)dnsubdT.(2)Four structural parameters of this proposed waveguide can be tailored, which are waveguide width (W), Si3N4 cladding height (H1), TiO2 cladding height (H2), and Si core height (H3). The material TOCs of Si3N4, TiO2, Si, and SiO2 are 2.45×10−5/K ^[28,29], −1×10−4/K, 1.86×10−4/K, and 1×10−5/K ^[21], respectively. As wavelength increases, the optical confinement factors in both the TiO2 cladding and the Si3N4 cladding become larger. At the beginning, the mode is mainly in the Si core and the TiO2 cladding, and thus, the effective TOC decreases with wavelength. When the wavelength is long enough, the mode confinement factors in the Si3N4 layer and the SiO2 substrate become significantly larger, and the effective TOC turns positive and increases with wavelength. In this way, one can infer from Eq. (2) that the effective TOC experiences a non-monotonic change with wavelength, with two zero-TOC points.

Generally speaking, the TOCs of Si, Si3N4, and TiO2 could vary with temperature and wavelength. For example, Si and Si3N4 are characterized in Refs. ^[30,31]. In practice, these should be measured, and the results would depend on a specific fabrication process in which the films are deposited. After being patterned to be optical waveguides, the stress in different layers may change, and designed parameters may have to be modified again to take the effect of stress into account ^[27].

We calculate the effective TOC of the proposed athermal waveguide using a mode solver twice, with and without a temperature-induced index change added to the original material indices. Note that the material dispersion is taken into account for all the materials. The structural parameters W, H1, H2, and H3 are 400, 250, 170, and 143 nm, respectively. The effective TOC is shown in Fig. 2. We note that the TOC curve of the proposed waveguide has two zero points, at 1450 and 1650 nm. By properly choosing the parameters above, the effective TOC remains within a small range of −1×10−6/K in a wide wavelength band of 300 nm, from 1400 to 1700 nm. Mode fields of the athermal waveguide are shown in Fig. 2, which expand increasingly more into the lower cladding and then the upper cladding as wavelength increases. This confirms our idea above that a broadband athermal waveguide can be achieved by adding a positive-TOC material as the second cladding.

It is important to examine the impact of fabrication errors on the effective TOC. Specific to different fabrication steps, the error ranges are also different. We choose the error ranges of W, H1, H2, and H3 to be ±10%, ±50%, ±5%, and ±1%, respectively. The Si core can have a precisely controlled thickness by oxidization, so we choose a small height variation for H3, around ±1%. The TOC changes caused by the fabrication errors are shown in Fig. 3. One can see that there is still a large bandwidth with athermal properties obtained with an effective TOC within ±1×10−6/K. We find that the effective TOC is quite insensitive to W and H1 variations, especially for H1. Width is typically less well-controlled in device fabrication, depending on lithography, and we see in Fig. 3(a) that the TOC curves are not changed much with a width variation of ±10%. The Si3N4 layer is grown by chemical vapor deposition, and it can be controlled to be ±5%, but here we intentionally increase the variation to show the robustness. Figure 3(b) shows that, as the positive-TOC material, Si3N4, has an increased height from 100 to 200 nm, the effective TOC curve counterintuitively becomes negative. This is because the Si3N4 layer will drag the optical field up, with more power into the negative-TOC material, TiO2. As H1 increases further to 300 nm, the TOC curve has a very limited shift because the fractional power of the mode in the SiO2 substrate does not reduce due to an increased H1, and thus, the mode profile is stabilized.

We note in Fig. 2 that the mode fields of the waveguide increasingly extend to the substrate as wavelength increases. With the same structural parameters, waveguide loss increases rapidly with wavelengths beyond 1860 nm, as shown in Fig. 4(a), but in the bandwidth of interest, with a small TOC, we have low loss due to substrate leakage. In simulations, we consider the Si3N4 film deposited using plasma enhanced chemical vapor deposition, in which N-H bonds may induce an absorption loss of 10 dB/cm. The propagation loss of the guided mode is found to be around 1 dB/cm over the wavelength range of interest. This is mainly due to the N-H absorption loss caused by Si3N4. This low-loss performance proves its compatibility with integrated photonic devices.

The proposed waveguide is the building block of a broadband athermal microresonator. When it is employed in micro-ring resonators, the bending loss of the ring and the absorption loss of silicon nitride should also be taken into consideration. For comparison, we choose three bending radii, 100, 30, and 10 μm, and calculate the TOC and the optical loss of these micro-ring resonators. Figure 4(b) shows that the effective TOCs of the micro-ring resonators with different radii remain almost the same. This result shows the high insensitivity of the proposed broadband athermal resonators to the bending radius. In contrast, the optical loss of the micro-ring resonator increases greatly when the ring radius decreases. Nevertheless, the optical bending loss is still negligible for a ring radius of >30μm over the wavelength range of interest from 1400 to 1700 nm. The relation between the TDWS and the effective TOC is given in Eq. (3) ^[12]. We calculate the TDWS of the broadband athermal ring resonator with a ring radius of 30 μm, as shown in Fig. 5. A variation within ±0.5pm/K in the wavelength range of 1400 to 1700 nm is obtained. Assuming the local temperature change is 30°C, this low TDWS is less than one resonance linewidth of a resonator with a cavity Q-factor smaller than 1×105: 1λrdλrdT(λ)=1ngdneffdT(λ).(3)

We have proposed a broadband athermal waveguide and an associated ring resonator with low loss and a wideband and small TDWS of ±0.5pm/K from 1400 to 1700 nm. This proposed resonator shows nearly unchanged temperature stability with different radii, which greatly broadens the applications of silicon ring resonators in multi-channel optical communications and broadband sensing systems.