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Abstract
There is little literature characterizing the temperature-dependent thermo-optic coefficient (TOC) for low pressure chemical vapor deposition (LPCVD) silicon nitride or plasma enhanced chemical vapor deposition (PECVD) silicon dioxide at temperatures above 300 K. In this study, we characterize these material TOC’s from approximately 300-460 K, yielding values of (2.51 ± 0.08) · 10−5K−1 for silicon nitride and (5.67 ± 0.53) · 10−6K−1 for silicon oxide at room temperature (300 K). We use a simplified experimental setup and apply an analytical technique to account for thermal expansion during the extraction process. We also show that the waveguide geometry and method used to determine the resonant wavelength have a substantial impact on the precision of our results, a fact which can be used to improve the precision of numerous ring resonator index sensing experiments.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The capacity for changes in temperature to alter the fundamental properties of materials is a phenomenon which impacts the behavior of countless physical systems. In the context of integrated photonics, temperature-induced changes to the refractive index of optical materials is one such example of notable importance. Generally referred to as the thermo-optic effect, the change of refractive index with temperature is often quite weak, and as such is usually characterized as a linear response using a single value of $\frac {\partial n}{\partial T}$, the thermo-optic coefficient (TOC). However, there are situations in which considering how a material’s TOC changes with temperature is critical to obtaining accurate results [1–4].
In the past decade, silicon nitride has continued to serve as one of the most promising platforms for integrated photonics [5]. A multitude of major advancements in silicon photonics in recent years have taken advantage of the low loss, wide wavelength range, and CMOS compatibility offered by silicon nitride [6–8]. As such, numerous other works have extracted information regarding the thermo-optic response of silicon nitride and related materials in silicon photonics, manufactured using a variety of processes, over a wide temperature range [9–15]. Due to the weak nature of the thermo-optic effect, these works generally extract the TOC by observing changes in the behavior of some interferometric structure to changes in temperature. Generally, resonators are used, such as optical microcavities [13], microdisk resonators [12], or ring/racetrack resonators [9,11,14]; additionally, non-resonant geometries such as Mach–Zehnder interferometers have also been used [15]. Notably, these methods characterize the TOC in fabricated integrated photonic devices. An alternative approach that has been used for other materials is to characterize bulk samples or thin films on a wafer prior to fabrication using interferometric or ellipsometry-based methods [16,17]. While often very precise, the use of these methods may introduce accuracy issues due to potential material differences between these source materials and the material properties following fabrication. Additionally, these methods require access to the original material or wafers used prior to fabrication, which may not be available to researchers who utilize external foundries for fabrication. Meanwhile, techniques that leverage simple integrated cavities such as ring resonators offer the capability for designers and researchers to perform TOC measurements on materials in the same context as the other functional devices being designed, using only optical equipment standard for the characterization of integrated photonic devices.
Despite the variety of manufacturing processes and temperature ranges covered by these studies, there is no study, to the authors’ knowledge, characterizing the temperature-dependent TOC of LPCVD silicon nitride films at temperatures above 300 K. This temperature range is particularly important for many thermally-actuated integrated photonic devices which operate at room temperature. Additionally, as material properties can vary widely between films manufactured using different techniques, there is a need for TOC data in this temperature range obtained on LPCVD silicon nitride, due to the popularity of LPCVD for manufacturing silicon nitride devices [5]. There are two existing studies on silicon nitride with temperature dependent results in this temperature range [12,13], both of which provide results only on sputtered silicon nitride. Zanatta [13] characterized sputtered amorphous silicon nitride, while Hryciw [12] characterized sputtered erbium-doped silicon nitride; the numerous differences between these materials yielded substantially different results. Such a large difference underscores the need for LPCVD films to be characterized, as their properties can not easily be inferred from this existing literature.
In this work, we track the temperature-induced shift of the resonant wavelengths of an integrated waveguide ring resonator (i.e., all pass filter) to determine the temperature-dependent TOC of not just the silicon nitride core of the waveguide, but also the silicon dioxide cladding. To estimate these parameters, we measure the change in resonant wavelength versus temperature for both the TE and TM modes, in a manner similar to some previous studies [9,11]. However, we apply a modified approach to extract the TOC, one which accounts for the coefficient of thermal expansion (CTE) analytically, and which only requires two-dimensional simulations of the modal profile. Our method has numerous other differences in its experimental method and extraction process which enable precise TOC measurements to be performed using equipment available to most integrated photonics researchers.
In performing experiments and data processing for this study, we encountered several challenges introduced by nonidealities in the behavior of our device, including a relatively poor quality factor for our ring resonator as well as the presence of strong parasitic resonances in our data. To overcome these challenges, we employed numerous techniques to optimize the data processing and extraction in order to achieve acceptable precision in the final results. These optimizations, as well as the error analysis we use to evaluate them, have relevance not only to ring-resonator based TOC extraction experiments, but to numerous other applications involving ring resonator refractive index sensors [18].
2. Materials and methods
2.1 Device design, fabrication, and experimental setup
The devices used in this study were fabricated on a silicon substrate with 4.5 µm of thermal oxide. Silicon nitride was deposited using LPCVD and etched using an anisotropic reactive ion etching (RIE) process. The top oxide cladding of the waveguide was then deposited using PECVD. The oxide cladding, waveguides, buried oxide, and handle silicon were all etched using RIE to create edge couplers. This process, which removes only the top 250 µm of silicon, is optimized to provide smooth edge facets suitable for low-loss coupling. The handle silicon was partially removed with deep RIE to allow sufficient clearance for lensed fiber.
We designed our device to use a waveguide geometry of 600 nm width and 400 nm height to ensure single mode operation in both the TE and TM modes (Fig. 1(c)). Due to the sidewall angle introduced by reactive ion etching (RIE) of the waveguide, the actual geometry of the waveguides was measured to be a trapezoid with a 635 nm base width and 535 nm top width. The ring resonators used in this study to track the resonant wavelength shift with respect to temperature were designed with a 60 µm diameter and a 100 nm gap in the bus-ring coupling region (Fig. 1(a)-b). An add-drop resonator was fabricated, but only the through port transmission was used to determine the resonant wavelength shifts in this study. In the fabricated device, the transmission port resonances had a quality factor of approximately 750-850, while the finesse was approximately 2.5-2.8 for the TM mode, and 3.0-3.3 for the TE mode. Simulation results along with our experimental data suggest that the transmission loss was on the order of -0.5 dB, while the loss of each coupler (encountered twice per round trip) was approximately -1.8 dB for the TE mode, and -2.2 dB for the TM mode. To facilitate edge coupling, the waveguides were tapered from 600 nm to 325 nm over a length of 100 µm for all inputs and outputs of the chip. A minimum bend radius of 50 µm was used for all waveguides to minimize bending loss. Five copies of the same ring, each on a different chip, were characterized in this study.
Fig. 1. Device and experimental design. a) SEM image of an add-drop ring resonator on one of the measured devices. b) Close-up SEM image of the top ring coupling region. c) Diagram (not-to-scale) showing the cross section of the ring coupling region, indicating material thicknesses, waveguide dimensions, and coupling gap width. d) Diagram of the experimental setup. FPC: fiber polarization controller; PMF: polarization maintaining fiber; DUT: device under test; TC: thermocouple; PC: personal computer.
To track the resonant wavelength shift of each ring resonator with respect to temperature, we heated the entire chip using a hot plate while measuring the transmission spectrum of the ring’s through port, using the setup shown in Fig. 1(d). Most other TOC studies sweep the temperature by allowing the sample to reach a steady state temperature at desired temperature levels, accomplished using either temperature-controlled stages [12–14], cryostats [11,16,17], or integrated heaters [15]. Instead of allowing the sample to reach a steady state temperature for every desired sample temperature, we used a passive approach in which the hot plate was preheated to the maximum desired temperature, then turned down and allowed to cool at a rate governed by the hot plate’s thermal mass. Due to the large thermal mass of the hot plate, the time for the hot plate to cool from 460 to 300 K was on average approximately 1.5-2 hours. During the cooling process, the temperature of the chip was monitored and recorded at 1 second intervals using a Type-K thermocouple, while transmission spectra were collected repeatedly. Transmission spectra were acquired at a frequency governed by the cooling rate of the setup, such that approximately one spectrum was acquired for every 5 K change in the sample’s temperature. After the experiment, the exact timing of spectrum collection was cross-referenced to the temperature record, allowing for accurate temperature measurements to be assigned to each collected spectrum. This temperature sweep was repeated for both the TE and TM modes on each ring resonator.
In order to utilize this method, care must be taken that the temperature measurements and device behavior are not unexpectedly impacted by it. Multiple measures were taken to ensure that the temperature reported by the thermocouple accurately reflected the temperature of the ring resonator on the chip. These measures are explained in detail in Supplement 1 Section S1, where we show that the temperature error induced by our technique is likely well under 1K, and negligible compared to other errors.
Lensed tapered fibers were used to couple light in and out of the chip’s edge couplers for spectrum measurements during the temperature sweep. A tunable laser and optical power meter (Agilent 8164B) was used to obtain these transmission spectra over a range of approximately 1460-1560 nm at 10 pm resolution. The fibers used were stripped of the polymer coating up to 1 cm from the tip of each fiber so radiative heat from the chip would not discolor and damage the coating of the fiber. In order to maintain acceptable insertion losses throughout the experiment, the tapered fibers must be continuously re-aligned with the edge couplers as the entire setup cools down and shifts due to thermal expansion. This continuous re-alignment was done simply by manually adjusting the 3-axis stages holding each fiber to maximize the transmission for each collected spectrum. This manual method introduces some variability in the insertion loss for each spectrum, but does not impact the resonant wavelength of the resonator, as the spectrum of the ring resonator is independent of coupling efficiency. This independence relies on the assumption that the power in the waveguide is low enough not to induce nonlinear effects. As such, the constant realignment required to use this experimental technique has no impact on the results, as it only serves to ensure that the output signal remains at detectable levels by the photodetector throughout the experiment. Using these techniques, the spectra collected for each device over the duration of experimentation for this study were extremely consistent. When the polarization and device temperature were kept constant, we did not observe any shift or instability of the resonant wavelengths for each device, even over periods of several months. This stability is necessary in order to conclude that the observed resonant wavelength shifts are due to temperature and not some other uncontrolled variable.
2.2 TOC extraction method
The most common extraction method in literature to obtain the TOC from the resonant wavelength shift of ring resonators involves simulating the resonance wavelength shift of each mode in response to perturbations of each material index. Obtaining these values allows for a simple linear system to be solved, directly yielding the material TOC’s [9,11]. However, this approach requires simulations to be performed for the entire ring resonator device, simulations which must be done fully in three dimensions for waveguides with a sidewall angle, which can be estimated using SEM images for our device. Due to the large size of the ring (60 µm) compared to the finest dimensions of the waveguide geometry (50 nm sidewall angle), such simulations require a meshing resolution not feasible on typical hardware.
To address this issue, we use a slightly different approach which requires less intensive simulation, at the cost of an additional analytical step. This approach has similarities to methods previously used in the literature for ring resonator thermometers and gas index sensors [19,20]. Our method considers the relationship between the material TOC’s to the change in effective index with respect to temperature, instead of directly relating the TOC’s to the resonant wavelength shift. In order to determine the two unknown material TOC’s, we require two independent equations; as in previous works, these independent equations can be obtained simply by considering the relationship for both the TE and TM modes [9,11]. These relationships can be summarized using the linear system
As a system with multiple materials, the expansion of the ring can not be exactly described by a single CTE, and will be impacted by the strain induced by CTE mismatches between materials. However, as our device consists of a 525 µm Si substrate with only 7.5 µm of silicon oxide and just 400 nm of silicon nitride, the impact of the CTE of these thin surface films in changing the ring’s size are negligible compared to the CTE of the substrate. As such, we treat the bulk Si CTE as the dominant factor in determining the ring’s thermal expansion, and use a commonly-referenced result for the temperature-dependent CTE of Si for $\alpha (T)$ in our calculations [24]. Note that in other contexts, such as determining the TOC of silicon waveguides, the effect of the CTE can be considered negligible compared to the TOC, as the TOC of silicon is over an order of magnitude greater than that of silicon nitride [17,19,23]. In the context of this work, the CTE has a non-negligible impact on the extracted TOC due to the relatively low TOC’s of silicon nitride and silicon oxide. In this study, we only consider the impact of thermal expansion on the physical length of the ring. Though the waveguide geometry also changes slightly due to the CTE, these changes are extremely minor and have no significant impact on our results (see Supplement 1 Section S3.2 for more details).
The use of Eq. (2) requires accurate determination of the effective and group indices for each mode. Though these values can be determined from the spectra directly [9], low-noise results with good precision can be achieved through simple mode simulations. Additionally, the index sensitivity terms in the matrix in Eq. (1) must be simulated, and the same simulation can be used to determine the effective and group indices. We performed these simulations using eigenmode analysis in Lumerical MODE. These simulations used room-temperature refractive index and waveguide thickness data obtained from ellipsometry measurements performed on the silicon nitride and silicon dioxide films of the wafer prior to fabrication. We also used SEM images of the waveguides after etching to determine the precise waveguide geometry of the fabricated devices for the simulations. The simulations accounted for the bending radius (30 µm) of the ring, which impacts the mode profile and thus the effective and group indices. These two-dimensional simulations can quickly be performed at a high mesh resolution, regardless of the waveguide geometry. The results of the simulations needed for Eq. (1) are summarized in Fig. 2. We perform further analysis to verify the accuracy of our simulation results in Supplement 1 Section S4.
Fig. 2. Energy density profile for the a) TE and b) TM modes obtained using Lumerical MODE. c) Simulated effective index of the TE and TM modes. d) Sensitivity of the simulated effective index to the core and cladding index for the TE and TM modes.
2.3 Data processing
In order to apply the extraction method discussed in the previous section, we must obtain accurate measurements of the change in resonant wavelength of the ring with respect to temperature (Fig. 3(a)). To maximize the precision of our final result, we track 14 different resonant wavelengths for both the TE and TM modes on each ring. To begin, we approximately identify the resonant wavelengths in each spectrum by locating all local minima with a prominence greater than a certain threshold. Due to the relatively poor quality factor of our rings, the width of each resonance is quite wide, and this initial local minimum of the data is not highly accurate. We further refine the accuracy of the detected resonant wavelength for each mode by transforming the spectrum’s vertical axis to linear transmission, then performing a Lorentzian fit of the resonance shape (Fig. 3(b)-c). Performing such fitting to localize the precise resonant wavelength has been performed in the past to detect small changes of the resonant wavelength in other ring-resonator index sensing experiments [20,25]. For each resonance, we perform this Lorentzian fitting on data within a limited wavelength range (60% of the free spectral range) centered around the rough resonance location.
After determining the precise resonant wavelength for each resonance from each polarization in every collected spectrum, we cross-reference each spectrum to its corresponding measured device temperature. This yields a series of resonant wavelength versus temperature curves for that device (Fig. 4(a)). To obtain the temperature-dependent derivative of the resonant wavelength, as is required by Eq. (2), we first perform a polynomial fit of these curves. As these curves are nearly linear with only a slightly discernible curve, a quadratic fit yields an excellent fit (Fig. 4(b)). With such a close fit, examining the residuals of the fit yields information regarding the precision of the resonance wavelength measurement performed in the previous step. Figure 4(c) shows that the standard deviation of the residuals is substantially reduced for both modes when Lorentzian fitting is used instead of the local minimum of the data. In the case of our TM spectra, which had the worst quality factor and substantial parasitic resonances, the Lorentzian fitting improves this measure of the precision of our resonance wavelength determination by approximately 4-5 times from around 0.1 nm to 0.02 nm (see Supplement 1 Section S3.3 for more details and further analysis regarding the use of this metric). We believe that this improvement is due to the noise-suppressing nature of the fit, along with the ability for the fitting to account for the overall shape of the resonance, not just the very bottom of the resonance (Fig. 3(c)). After taking the derivative of the polynomial fit, we use Eq. (2) to compute $\frac {\partial n_\mathrm {eff}}{\partial T}$ for each resonance and mode. Figure 4(d) shows the result of calculating $\frac {\partial n_\mathrm {eff}}{\partial T}$ with and without accounting for thermal expansion using the CTE term in Eq. (2). Neglecting the CTE term yields a result which overestimates the effective index derivative by around $5\cdot 10^{-6}$ RIU/K, a discrepancy which also significantly alters the extracted TOC.
Once $\frac {\partial n_\mathrm {eff}}{\partial T}$ is determined for each mode, it can simply be used with Eq. (1) to determine the TOC for each of the tracked resonances. As the resonant wavelengths of the TE and TM modes do not exactly align, we perform this extraction on pairs of $\frac {\partial n_\mathrm {eff}}{\partial T}\Big |_\mathrm {TE}$ and $\frac {\partial n_\mathrm {eff}}{\partial T}\Big |_\mathrm {TM}$ results corresponding to resonance pairs which are nearest in the original spectra, as indicated by the numbered brackets in Fig. 4(a). Once paired, we solve the system simply by inverting the matrix. It is only once we achieve this final TOC result that data from different resonances can be averaged together to obtain better precision. This is because nearly all the parameters in the extraction (including individual entries in the matrix in Eq. (1)) are wavelength-dependent, meaning that the exact values used in Eqs. (1) and (2) are slightly different for every resonance.
Fig. 3. (a) Spectra measured at a variety of temperatures on the TE mode of one device. The spectra have been normalized such that the mean transmission of each spectrum is 0 dB. (b) Original data and Lorentzian fit of a selected TM mode. (c) Close-up of (b) near the resonance wavelength. The local minima of the raw data and the Lorentzian fits are marked to highlight the discrepancy between the resonance wavelength when identified using each method.
Fig. 4. a) Tracked location of 14 different resonances versus temperature for both the TE and TM modes on one of the five tested devices. Brackets and numbers indicate the 14 separate pairs of TE and TM modes used for simultaneous extraction of the material TOCs for that device. b) Close-up of a pair of TE/TM resonances from a) (pair 5) alongside quadratic fits for each mode. c) Histogram comparing the standard deviation of the quadratic fit residuals for each mode using the two different methods of determining the resonance wavelength as shown in Fig. 3(c); "Lor." = Lorentzian. A lower value indicates a better fit to the quadratic, in turn suggesting a more precise estimate of the resonant wavelength. d) Extracted $\frac {\partial n_\mathrm {eff}}{\partial T}$ for the 14 TE resonances shown in (a), showing the result with and without using the CTE term in Eq. (2).
3. Results
To generate final results, we perform the extraction described in the previous section on each pair of tracked TE/TM resonances independently, then average all the results from one device together to generate a mean result for each of the 5 tested devices. Note that this averaging step yields the average TOC over a bandwidth of 1460-1550 nm, as we did not observe any statistically significant wavelength dependence of the TOC in our results. The extracted temperature-dependent TOC’s for each of the five devices, as well as the mean across the five devices, are compared to the results from relevant literature in Fig. 5.
Fig. 5. Comparison between the results from this paper and various results from literature for the temperature dependent TOC of a) silicon nitride and b) silicon dioxide. The spread of grey crosses at each temperature point represents the extracted material TOC at that temperature for the 5 separate devices, averaged across the 14 resonances on that device. The black line represents the average of these device results, with the error bars representing the standard error of the five-device mean (see Supplement 1 Section S3.3 for more details on this error computation).
As the five devices were manufactured on the same wafer in close proximity (<2 cm), the mean across all devices can be interpreted as an estimator of the population mean temperature-dependent TOC for the silicon nitride and silicon dioxide in this region of this wafer. Further discussion regarding this averaging as well as the error reported in the final result can be found in Supplement 1 Section S3.4. Polynomial fits for these total means, valid over the temperature range $300 \text { K} \leq T \leq 460 \text { K}$ investigated in this study can be written as
The provided error is the worst-case standard error of the 5-device mean in this temperature range. The error bars in Fig. 5 accurately show the slight variations in the propagated error across the temperature range. The final relative error is less than $\pm$4% for silicon nitride and $\pm$10% for silicon dioxide. Note that a quadratic fit was used to compute the derivatives of the resonant wavelength shifts, which would usually yield a linearly increasing TOC with temperature for both materials [12]. However, as the CTE of silicon over this range does not follow a linear shape, accounting for the CTE introduces a slight curve into the final result. Additionally, designers using the same waveguide profile (600 nm x 400 nm) on this material platform may find the underlying effective index temperature coefficient useful. Second order polynomial fits for the temperature coefficient of the TE and TM effective indices (valid over the same 300 - 460 K range) can be written as
Table 1 compares the numerical values for the TOC of silicon nitride and silicon dioxide at 300 K from this work (averaged across all devices) to the results from relevant literature.
Table 1. Comparison of silicon nitride and silicon dioxide TOC values at 300 K between this work and relevant literature.
Our silicon nitride result at 300 K has excellent agreement with several other results from the literature at the same temperature [9,11,12]. The slope (second order TOC) of this silicon nitride result also has good agreement overall with two other available results in this temperature range, though the slightly shallower slope of our result is noticeably different from [12]. Notably, the TOC does not appear to change substantially with wavelength for these materials, with only slight deviations between values reported at 880 nm [11] compared to the various results around 1500 nm. While the results in this work and other works are similar, there are still noticeable discrepancies between all available results. This reinforces the impact that the details of the silicon nitride deposition (particularly the temperature, recipe, and post-deposition annealing for LPCVD) have on the final material properties. The nitride used in this study was measured to have a room-temperature refractive index of 1.997 at 1550 nm, indicating that this nitride and the TOC results in this work are likely similar to stoichiometric Si3N4.
The waveguide geometry in this work has a thermal oxide beneath the waveguide, but a PECVD oxide on all other sides. This means that the extracted TOC is the weighted average of the underlying TOC’s for these two different oxide variants, which do not necessarily have the same thermo-optic response. Our silicon dioxide result is significantly lower than the other two values in similar literature, which could arise due to differences in our extraction process or sample preparation. While the two other available references shown in Fig. 5 also have a mix of thermal and PECVD oxide, variations in the exact fabrication parameters of either the thermal or PECVD oxide could contribute to the difference. In particular, differences in the PECVD oxide such as the density, oxygen content, and hydrogen content can vary depending on the deposition conditions as well as other steps in the fabrication process. The fabrication of the chips used in this work also included process steps for metal heaters and routing (not used in this study), which may have impacted the material properties of the oxide cladding. Interestingly, while the first-order TOC term differs substantially, the slope of our result is quite comparable to that observed near room temperature in [11], indicating good agreement of the second-order term.
4. Discussion
The passive temperature sweep procedure used in this study has numerous advantages. Unlike experiments in which the chip is heated to a specific desired temperature, our setup does not need to maintain a precise temperature in order to obtain acceptable accuracy of the temperature measurement. This allows for the experiment to be performed using simple heating hardware, which is much more accessible than the high-precision heating chambers usually necessary for experiments that require a wide temperature range and decent accuracy. Additionally, as this method does not rely on steady-state conditions, extended settling times are not needed for each data point, allowing arbitrarily many spectra to be recorded without increasing the duration of the experiment. This time advantage allows for many more spectra to be obtained on more devices, a sample size improvement which in turn substantially improves the precision of the final TOC results.
It is important to note that while our extraction method accounts for thermal expansion to accurately extract the effective index shift, this effective index shift is not only determined by thermo-optic effects. Stress-optic effects induced by CTE mismatch between the substrate and the waveguide materials will also impact the material index. It is not possible using our methods to isolate index changes due to material TOCs from those due to strain. However, the TOC results in this paper can be interpreted as a total thermal coefficient for material indices which encompasses index shifts both due to material TOCs and any strain-optic effects that may be present. These figures can be used to calculate the thermal behavior for other devices fabricated using similar film thicknesses on a silicon substrate, as the strain effects would have a similar contribution to the thermal behavior of the device. This combined coefficient allows for thermo-optic and strain-optic effects to be accounted for using a single figure, instead of considering each effect separately, simplifying thermal calculations when designing future devices.
Although increasing the sample size through the use of a more efficient procedure helps improve precision of the mean, it does not improve the precision of individual samples. In Supplement 1 Section S3.2, we perform a thorough error analysis of our experiment and extraction procedure for each individual resonance. One of the main contributors to the error on the extracted TOC for each individual resonance is the ambiguity in the exact wavelength caused by the wide FWHM of the ring resonator used in this work. While improving the quality factor of the resonator is the best method to reduce this resonant wavelength error, it is not always feasible to redesign or manufacture a new resonator with a narrower linewidth. In such cases, our results reinforce the value of performing Lorentzian fitting for a much more precise determination of the resonant wavelength. This improvement is significant both for clean spectra (such as our TE spectra, where implementing the fitting improved precision from around 20 pm to 10 pm) but is even more impactful for spectra containing undesirable parasitic resonances, as was the case for the TM spectra in this study. In both cases, our final precision was on the order of 10-20 pm, around two orders of magnitude better than the FWHM of the resonances (which was around 2 nm). This is a result significant not just to improving the sensitivity of TOC experiments, but to many more general ring resonator index sensors, which also track shifting resonance wavelengths [18–21,25]. Finally, the precision of such effective index shift measurements can be further enhanced by averaging over multiple resonances, as was done in this study. The use of a longer ring to measure and average the results from more resonances has the potential to achieve higher precision in the final result (more details regarding this improvement are discussed in Supplement 1 Section S3.2).
The other major contributor to the error in this paper as discussed in Supplement 1 Sections S3.1-2 does not arise from measurements or processing strategy, but is intrinsic to the equations used for extracting the TOC, namely Eq. (1). Since this linear system must be solved to determine the final TOC values, the conditioning of the matrix of effective index sensitivities plays a key role in the error of that final result. If the TE and TM modes share a very similar energy density profile, the sensitivity of their effective index to each material index will be similar, resulting in row vectors that are not highly independent, and consequently to an ill-conditioned linear system. The extreme example of this is a square waveguide geometry, in which the TE and TM modes will have the exact same response to thermal shifts, and as such measurements of the two modes will not provide linearly independent information with which to extract the two TOC values. For the waveguide design used in this study (approximately 600 nm x 400 nm), the condition number of this matrix ranged from approximately 9-10 over the wavelength range used. As the condition number is highly related to the factor by which error is amplified when solving the linear system, this corresponds to TOC results for each resonance which have substantially worse precision than their corresponding values of $\frac {\partial n_\mathrm {eff}}{\partial T}$. While we are able to achieve acceptable precision of our final result by averaging a large number of samples, in general error from this source can be minimized by performing analysis of the waveguide geometry before the chip is manufactured. Namely, a highly asymmetrical waveguide design with substantially different TE and TM mode profiles can yield row vectors with better orthogonality, reducing the error introduced by the ill-conditioning of the linear system.
There are numerous opportunities for future work in light of the results and analysis performed in this work. Firstly, the experimental procedure used in this study relied on a substantial amount of manual alignment, as thermal expansion in the experimental setup caused the chip to constantly drift out of alignment with the lensed tapered fibers over the course of each temperature sweep. This constant manual re-alignment had to be performed at a fast pace in order to collect spectra at the desired 5 K intervals. Instead of requiring such time-sensitive manual alignment for every collected spectrum, the use of automated alignment or a packaged sample would substantially reduce the amount of human involvement necessary to perform the experiment. Additionally, as the resonant wavelength and linear system conditioning were the main sources of error in this study, future experiments with higher quality factor resonators and highly asymmetric waveguides could yield substantially better precision than what is achieved here. This is a design insight that can be applied to any experiment in which a similar linear system-based method is used to extract multiple index changes simultaneously. While this method has primarily been used previously for the determination of material TOC’s, this same method could be used to characterize index changes in multiple materials simultaneously due to some stimulus other than temperature, for example stress-optic or electro-optic effects. Alternatively, performing two-polarization measurements in situations where only one index is changing or is unknown would provide two independent sample points, thus improving precision for the one unknown index. Finally, given the precision improvements obtained here through the use of Lorentzian fitting, employing Lorentzian fitting on narrow linewidth resonances measured with very high spectral resolution could enable ring resonator sensors with extremely high sensitivity.
5. Conclusion
In summary, we have determined the temperature-dependent TOC’s for both silicon nitride and silicon dioxide for temperatures ranging from 300 K to 460 K. These material results can be used in the design and simulation of advanced thermally actuated integrated photonic devices which operate at and above room temperature. The method we use to perform this characterization is simple and can be utilized for similar characterizations on a wide variety of material platforms. Additionally, we have performed analysis which provides multiple important considerations when designing ring-resonator based index sensing experiments. Firstly, we show that waveguide design has a large impact on the error introduced when solving the linear systems required to simultaneously extract the TOC for two different materials. We also show that performing Lorentzian fitting for precise determination of the resonance wavelength substantially reduces the error in determining the resonance wavelength. This result is particularly significant for resonators with a low quality factor, as it provides the means to detect resonant wavelength shifts that are substantially smaller than the FWHM. Finally, we show that thermal expansion has a substantial impact on the resonant wavelength shift in the measurements of materials with low ($<< 10^{-4} \text {K}^{-1}$) TOC’s, an effect which must be accounted for during the extraction process. Altogether, our analysis as well as our final material results provide insight regarding the design, characterization, and data processing for numerous applications across the field of integrated photonics.
Funding
Cymer; LEED: A Lightwave Energy-Efficient Datacenter funded by the Advanced Research Projects Agency-Energy; Basic Energy Sciences (DE-SC0019273); Quantum Materials for Energy Efficient Neuromorphic Computing-an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE) Office of Science; San Diego Nanotechnology Infrastructure (SDNI) supported by the NSF National Nanotechnology Coordinated Infrastructure (ECCS-2025752); Army Research Office; National Science Foundation (ECCS-180789, ECCS-190184, ECCS-2023730); Office of Naval Research; Defense Advanced Research Projects Agency.
Acknowledgments
Naif Alshamrani and Dhaifallah Almutairi would like to thank King Abdulaziz City for Science and Technology (KACST) for their support during their study.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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