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Abstract
We describe an in-situ technique to characterize the material refractive indices and waveguide geometry for photonic integrated circuits over hundreds of nanometers of optical bandwidth. By combining white light spectroscopy with unbalanced Mach-Zehnder interferometers, we can simultaneously and accurately extract the core thickness, core width, core refractive index, and cladding refractive index. This information is important for the technological maturation of photonic integrated circuit foundry fabrication. Capturing the inter-wafer and intra-wafer variation of these parameters is necessary to predict the yield of photonic components and for overall process quality control. Refractive indices are found with a 1-σ error of between 0.1% and 0.5%, and geometric parameters are found with an error of between 3 nm and 7 nm. Our analysis and validation are implemented and verified using the same waveguide layers as are used in the standard photonic wafer build, without any external techniques such as ellipsometry or microscopy.
1. Introduction
The accurate extraction of photonic parameters for specific photonic integrated circuit platforms has become increasingly important as these platforms mature. Geometrical variations of the core layer thickness or linewidth offset of just a few nanometers can dramatically impact the performance of photonic components. In addition, more platforms are expanding to offer waveguide cores made from deposited materials such as silicon nitride (SiN), alumina (Al$_2$O$_3$), or titanium oxide (TiO$_2$) that can operate at wavelengths from the visible to the mid-wave infrared. The index and dispersion of these materials, which depend on the wavelength and specific deposition process, are critical input parameters for all photonic components built on that platform. Even the exact index of the cladding, often deposited silicon dioxide (SiO$_2$), can vary enough from process to process to impact compact models of foundry process development kit (PDK) components.
Though different techniques exist to measure geometric properties of single-mode waveguides, such as scanning-electron microscopy and atomic-force microscopy, these methods require the removal of wafers from the fabrication line and are often destructive. Similarly, the material refractive indices and dispersion can be measured with ellipsometry, near-field coupling [1,2], prism coupling [3], evanescent-field imaging [4] or Fourier imaging [5] but these ex-situ techniques require specific structures or optics to couple light into and out of the top of the wafers and often are incompatible with the multilayer architectures used in foundry processes today.
Recently, microring resonators [6–8] and unbalanced Mach-Zehnder interferometers (MZIs) [9,10] have successfully been implemented for in-situ geometric (core width and thickness) parameter extraction from silicon waveguides in the optical C-band. Here, we show that with the use of unbalanced MZIs of at least two different widths and measured using the TE$_{\mathrm {00}}$ and TM$_{\mathrm {00}}$ modes, the core and cladding refractive index can be extracted in addition to the core geometry. Accurate knowledge of these four parameters (core linewidth offset, thickness, refractive index, and cladding refractive index) are necessary for process quality control and to predict component performance. In addition, by using white-light spectroscopy [11] to measure the MZIs, the refractive indices and dispersion can be extracted across the entire operation range of the waveguides: demonstrated here from 1000 nm to 1600 nm.
2. Photonic analysis
To constrain the space of parameters, we assume that the waveguide core is rectangular with the same cladding above and below. While these assumptions may not perfectly match the physical fabrication conditions, we will show that the four extracted parameters (width offset from nominal ($w_{\mathrm {ofst}}$), thickness ($t$), core index ($n_{\textrm {core}}$), and cladding index ($n_{\textrm {clad}}$)) are nevertheless capable of accurately predicting the photonic characteristics. We also assume that the waveguide linewidth offset is independent of the width. Our experience with foundry-based fabrication is that this assumption holds for widths significantly larger than the minimum feature size, and for offsets smaller than the minimum feature size.
To extract the four parameters at a given wavelength, four different effective index ($n_{\mathrm {eff}}$) measurements are made: both the TE$_{\mathrm {00}}$ and TM$_{\mathrm {00}}$ modes are measured in waveguides of widths $w_1$ and $w_2$ ($n_{\mathrm {eff}}^{\mathrm {TEw_1}} (\lambda )$, $n_{\mathrm {eff}}^{\mathrm {TMw_1}}(\lambda )$, $n_{\mathrm {eff}}^{\mathrm {TEw_2}}(\lambda )$, and $n_{\mathrm {eff}}^{\mathrm {TMw_2}}(\lambda )$). This requires two different unbalanced MZIs (for waveguides with widths $w_1$ and $w_2$) though in practice we use a set of three MZIs for each width and mode [12] to eliminate ambiguity in the interference order of a high-order MZI. Finding the maxima and minima in the unbalanced MZI transmission spectrum (discussed in more detail in Section 3) combined with the interference formulae for transmission maxima:
and minima: give the waveguide effective indices and dispersion for each mode and width. A second-order polynomial fit to $n_{\mathrm {eff}}(\lambda )$ yields a best fit value for $n_{\mathrm {eff}}$ as well as the fit error at any wavelength within the MZI bandwidth.To convert these measured effective index values to the four parameters of interest, a mode-solver is used to calculate the effective indices $n_{\mathrm {eff}(0)}^{\mathrm {TEw_1}}$, $n_{\mathrm {eff}(0)}^{\mathrm {TMw_1}}$, $n_{\mathrm {eff}(0)}^{\mathrm {TEw_2}}$, and $n_{\mathrm {eff}(0)}^{\mathrm {TMw_2}}$ for initial-guess parameters $n_{\mathrm {core}(0)}$, $n_{\mathrm {clad}(0)}$, $t_\mathrm {(0)}$, and $w_{\mathrm {ofst}\mathrm {(0)}}$. For clarity, measured values are shown in brown, calculated values are shown in blue, and the parameters to be solved for are shown in red. Then, if $\Delta n_{\mathrm {core}}/n_{\mathrm {core}},\Delta n_{\mathrm {clad}}/n_{\mathrm {clad}}, \Delta t/t, \Delta w/w$ are each $<<1$, higher-order (nonlinear) dependencies between the effective indices and the photonic and geometric parameters can be dropped and each $n_{\mathrm {eff}}$ can be written as:
To solve for the set of parameters, these equations can be rewritten in vector-matrix form:
Or, equivalently, as:
Fit errors in the measured $n_{\mathrm {eff}}(\lambda )$ values are propagated through this solution to give errors in the final parameter set using a similar process.
3. Experimental results
We fabricate silicon nitride (SiN) waveguides and edge-coupled Mach-Zehnder interferometers using a customized process at AIM Photonics [13]. Briefly, a nominally 220 nm thick SiN waveguide layer on a 5 $\mu$m thick thermal oxide bottom cladding is patterned using 193 nm immersion photolithography with 300-mm wafers [14]. A 5 $\mu$m thick top oxide is then deposited onto the fully-etched waveguides and can be removed for sensing trenches[15], though all waveguides discussed here are fully clad in SiO2.
Shallow-angle MZIs terminated with custom Y-splitters are designed for waveguides of widths of 0.8 $\mu$m, 1.2 $\mu$m, and 1.5 $\mu$m for operation in the Y/J-bands (1000 nm - 1260 nm), O-band (1260 nm - 1370 nm), and S/C/L-bands (1460 nm - 1625 nm), respectively. Compared to MZIs that use 90-degree bends, our 22.5-degree MZIs have less overall bend length, thereby reducing potential errors in measuring $n_{\mathrm {eff}}$. In addition, shallow-angle MZIs occupy less overall space. Three MZIs are used for each width and mode[12] with $\Delta L$ varying based on the MZI order. Each MZI targets constructive interference at either 1064 nm (0.8 $\mu$m wide, 175 $\mu$m minimum bend radius), 1310 nm (1.2 $\mu$m wide, 200 $\mu$m minimum bend radius), or 1550 nm (1.5 $\mu$m wide, 225 $\mu$m minimum bend radius). The fabricated set of 0.8 $\mu$m wide MZIs is show in Fig. 1. The order of the two lowest-order MZIs (7 and 8 at 1064 nm, 7 and 8 at 1310 nm, and 8 and 9 at 1550 nm) are chosen to insure a single unambiguous peak at the target wavelength based on uncertainty in the material indices and geometry. The $\Delta L$ of the highest order MZI is exactly 4 times larger than that of the lowest order one to provide more measurement precision of $n_{\mathrm {eff}}$. The Y-splitter is the same for the TE$_{\mathrm {00}}$ and TM$_{\mathrm {00}}$ modes, but optimized for a specific wavelength band. Both sides of each MZI are terminated in an edge coupler that is also optimized for the design wavelength band.
Fig. 1. Microscope image of six 0.80 $\mu$m wide MZIs used in this analysis. The expected interference order at the design wavelength (1064 nm) is shown for each MZI
Broadband interference spectra are obtained using white-light spectroscopy, as described previously [11]. Briefly, a broadband blackbody light source (Thorlabs SLS201L) is polarized and focused into a polarization-maintaining (PM) single-mode optical fiber (Thorlabs PM980-XP). Polarization-maintaining lensed fibers (Oz Optics TPMJ) are used to couple light into and out of the MZIs via edge couplers. Output light from the sample is then focused into a 0.5 m focal length spectrometer (Princeton Instruments SP2500) and detected with a liquid-nitrogen cooled InGaAs detector (Princeton Instruments Plyon-IR).
We measure three interference spectra for each waveguide width (0.8, 1.2, and 1.5 $\mu$m wide) and mode (TE$_{\mathrm {00}}$ and TM$_{\mathrm {00}}$). For each set the two lower order spectra are used to coarsely identify $n_{\mathrm {eff}}$ at the design wavelength, and the highest order spectrum is used to precisely measure $n_{\mathrm {eff}}$ using a custom algorithm to identify interference maxima and minima. An example set of spectra is shown in Fig. 2(a) for a 0.8 $\mu$m wide waveguide excited with light polarized in the sample plane (horizontally) to excite the TE$_{\mathrm {00}}$ mode. The effective index is then fit with a second-order polynomial giving a best-fit effective index and error at each wavelength, as shown for the example 0.8 $\mu$m wide waveguide data for the TE$_{\mathrm {00}}$ mode in Fig. 2(b).
Fig. 2. (a): Example interference spectra obtained from the three different orders of MZIs with 0.8 $\mu$m wide waveguides designed for an interference maxima at 1064 nm for the TE$_{\mathrm {00}}$ mode. The interference order is shown in blue for the high-order MZI. Also shown are the maxima and minima found from a custom search algorithm. (b): The resulting effective index and polynomial fit from the high-order MZI spectrum.
To obtain the four photonic parameters of interest, the procedure described in Section 2 is implemented separately for pairs of waveguide widths $w_1$ and $w_2$ = 0.8 $\mu$m wide and 1.2 $\mu$m wide, $w_1$ and $w_2$ = 1.2 $\mu$m wide and 1.5 $\mu$m wide, and $w_1$ and $w_2$ = 0.8 $\mu$m wide and 1.5 $\mu$m wide. Initially, we choose wavelengths (1250 nm and 1300 nm) that are red enough to prevent significant degradation of the TE$_{00}$ MZI spectra due to the presence of the TE$_{10}$ mode in the wider waveguide, and blue enough to prevent substrate or bend loss of the TM$_{00}$ mode in the narrower waveguide. Comsol Multiphysics (Electromagnetic Waves) is used to calculate the effective indices and their partial derivatives. The initial guess and analysis results are shown in Table 1.
Table 1. Extracted photonic parameters from the measured MZIs at wavelengths of 1250 nm and 1300 nm. The errors shown represent 1-$\sigma$ (68%) confidence levels.
Note that since the waveguide widths examined here are all significantly larger than the foundry minimum feature size (100 nm) and the extracted width offset is less than this minimum feature size, the extracted width offset should be independent of the wavelength and the choice of width pairs. The thickness should also be independent of the wavelength and the waveguide widths. Table 1 indicates that the extracted parameters do generally agree within 1-$\sigma$, and always within 2-$\sigma$. However, note that the error is significantly larger for $w_1$, $w_2$ = 1.2 $\mu$m, 1.5 $\mu$m pairs. The $w_1$, $w_2$ = 0.8 $\mu$m, 1.5 $\mu$m and $w_1$, $w_2$ = 0.8 $\mu$m, 1.2 $\mu$m width pairs have significantly smaller error, with the error generally lowest for the $w_1$, $w_2$ = 0.8 $\mu$m, 1.5 $\mu$m width pair. This suggests that small measurement and fitting errors play a larger role as the waveguide widths approach degeneracy. This is noteworthy since it suggests that the possible presence of the TE$_{10}$ mode in the widest waveguide does not appear to add significant analysis uncertainty. Also note that though material dispersion does not require agreement of the refractive indices at different wavelengths, the extracted indices also generally agree within 1-$\sigma$, and always within 2-$\sigma$.
4. Validation
To further validate this analysis and extend it to other wavelengths, we can compare the predicted performance of a test photonic component modeled with the extracted parameters found in Section 3 against the measured performance. To do this, first we set the waveguide width offset and thickness to a single weighted mean found from an analysis across the wavelength range of each width pair: 1050 nm to 1350 nm for $w_1$, $w_2$ = 0.8 $\mu$m, 1.2 $\mu$m; 1200 nm to 1500m for $w_1$, $w_2$ = 1.2$\mu$m, 1.5$\mu$m; and 1200 nm to 1350 nm for $w_1$, $w_2$ = 0.8 $\mu$m, 1.5$\mu$m. This results in $t$ = 213$\pm$4 nm and $w_{\textrm {ofst}}$ = -22$\pm$6 nm.
We then repeat the parameter extraction process described in Section 2 but now for only the core and cladding index, for a number of individual wavelengths at a single waveguide width ($w_i$, $i$=1,2,3):
This results in the refractive index values for the SiN core and the SiO$_{2}$ cladding shown in Fig. 3, for a number of wavelengths in the Y-band through the C-band. Solving for the refractive index dispersion in this way insures a more physically consistent wavelength dependence that is based on a fixed (wavelength-independent) waveguide geometry. The values at a given wavelength found using different waveguide widths generally agree within error. Also shown for reference are previously reported refractive indices of LPCVD SiN (Tyndall et al. [11] and Luke et al. [16]) and of thermal SiO$_2$ (Lee et al. [17] and Masi et al. [18]). The difference between the referenced index values and those found here highlights the importance of this parameter extraction for a specific foundry process. For example, an error in a SiN refractive index of 0.03 would result in an effective index error of approximately 0.02 at 1064 nm, or a change in a peak wavelength of an MZI of 13 nm: enough to render the device unusable as a filter.
Fig. 3. The core and cladding refractive indices vs. wavelength found from the measured effective indices and the $w_{\mathrm {ofst}}$ and $t$ described in Section 4. Table 1.
Next, these refractive indices and core geometry parameters are used with a mode-solver to predict the wavelength at which the TE$_{\mathrm {10}}$ and TM$_{\mathrm {00}}$ modes are degenerate for a given (nominal) waveguide width. As previously described[11], these mode-crossings are easily observed in simple straight waveguides, and are very sensitive to the photonic waveguide parameters. The transmission spectra are measured using the same white-light spectroscopy set-up that is used to measured the MZIs. Fig. 4(a) shows these measured mode-crossings, appearing as narrow minima in the transmission spectra. Fig. 4(a) also shows the wavelengths of the mode-crossings calculated using the extracted parameters. The agreement is excellent, with an average wavelength error (shown in Fig. 4(b)) of 7 nm, which is comparable to the fitting accuracy of the measured mode-crossing minima. Also note that these mode crossings were measured at a different location on the wafer and predicted wavelength error could result from intra-wafer parameter variation. For comparison, the mode-crossing wavelengths are also calculated using the initial guess parameters and shown in Fig. 4(b). The average error is 55 nm, almost 10$\times$ larger than that using the parameters extracted from our analysis.
Fig. 4. (a): A comparison between the calculated mode-crossing wavelengths (shown with the stars) from the extracted fit parameters shown in Table 1 and Fig. 3 and the measured mode-crossing spectra. (b): The residual between the measured mode-crossing wavelengths and the calculated wavelength for both the extracted fit parameters and the initial guess parameters.
5. Conclusions and future work
We have described a novel method that uses in-situ measurements of unbalanced MZIs and an electromagnetic mode-solver to extract the core thickness, width offset from nominal, refractive index, and cladding refractive index of foundry-processed waveguides. This four-parameter extraction is important for dielectric material cores whose optical properties can vary significantly from one foundry’s process to the next, or even across a single wafer. Ignoring the less accurate width pair of 1.2 $\mu$m and 1.5 $\mu$m, the extracted refractive indices are found here with errors of between 0.1% and 0.5%, and the geometric offsets are found with errors between 3 nm and 7 nm. The analysis has been validated by comparing calculated to measured mode-crossing wavelengths, showing an average accuracy of 0.6% in the calculated wavelength.
Ongoing work is focused on further decreasing the errors in the values of the extracted parameters by examining the assumptions of the matrix analysis and by examining experimental sources of uncertainty. Neither the inclusion of higher-order terms in the analysis matrix nor the inclusion of additional waveguide widths have so far led to reduced parameter errors. Experimental and design issues such as instrumentation noise, MZI wavelength range, MZI geometry, and waveguide widths are also under investigation. For example, the use of Euler bends[19] in the MZIs could reduce higher-order mode generation and lead to cleaner interference spectra. We also continue to examine the group index in addition to the effective index to further reduce measurement uncertainty, or to perform all measurements in a set of single-width MZIs. So far, larger measurement uncertainty of the group index combined with lower sensitivity to the waveguide width have shown much higher error in the extracted parameters using the group index compared with the effective index.
We are also examining adaptions of the technique to narrow wavelength ranges to enable the use of tunable lasers with grating couplers. Such a modification would improve in-situ wafer-scale probing and analysis for process control and quality assurance, both across single wafers (intra-wafer) and from build to build (inter-wafer). Such methods will enable silicon photonics foundries with dielectric-core waveguides to improve reliability and standardize metrology.
Funding
Office of Naval Research (WU 1X06); U.S. Naval Research Laboratory (WU 1X06).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the authors upon request.
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