High accuracy determination of optical waveguide geometry using a single spectral measurement of a dispersive device

Abstract

A method for determining a geometry of an optical waveguide, the method comprising: with a ring resonator comprising a predetermined cavity length, analyzing multiple consecutive resonance wavelengths at an operating temperature; performing a constrained numerical optimization process with a regression objective function of the optical waveguide's spectral features to extract dimensions of the optical waveguide, wherein the process minimizes a total mean squared error (MSE) between predicted and measured resonance wavelengths over a selected wavelength range.

Description

HIGH ACCURACY DETERMINATION OF OPTICAL WAVEGUIDE GEOMETRY USING A SINGLE SPECTRAL MEASUREMENT OF A DISPERSIVE DEVICEFIELD

[0001] Aspects of the disclosure relate to methods for manufacturing semiconductor devices.BACKGROUND

[0002] Silicon photonics has seen tremendous growth in recent decades, becoming a foundational platform for integrated photonics with applications ranging from optical communications to quantum computing and Al accelerators. This field is rapidly scaling towards large-volume production of increasingly complex circuits. The high refractive index of silicon facilitates strong light confinement and thus high integration density. On the other hand, it also makes the devices extremely sensitive to fabrication imperfections. Fabrication uncertainties leading to width and thickness variations have been widely recognized and addressed in research over the past decade. Even with state-of-the-art fabrication technology, waveguide in-plane dimensions often deviate from nominal values. The thickness of the silicon on insulator (SOI) wafers also vary undesirably across a wafer and from wafer to wafer. The intrinsic high sensitivity of the optical spectral behavior to these dimensional changes limits the scalability of the circuits and thus the fabrication yields. These dimensional variations often exhibit specific spatial distributions related to the local environment and the device’s location on a wafer. Obtaining accurate knowledge of the waveguide cross-section can inform the design process, open the opportunity of taking compensatory measures to mitigate these issues and enhance overall performance and yield.

[0003] Measuring the dimensions directly seems the obvious choice. Physical metrology tools such as ellipsometry can provide film thickness with typical resolutions in the range of 0.1 to 1 nanometer, but it requires test sites much larger than device sizes and so it is usually done on wafers prior to fabrication. In-plane measurements usually rely on scanning electron microscopy (SEM), which has a precision of a few nanometers. It requires the wafer to be taken out of the fabricationline at the relevant stage and loaded into a vacuum chamber. It is time consuming, and cannot be done post-fabrication or on historical samples.

[0004] The most commonly used structure in silicon photonics is the silicon wire waveguide. Optical properties such as the effective index and group index are directly determined by the waveguide geometry. Be correlating these parameters with measured spectral features in dispersive devices, it has been shown that geometrical parameters can be extracted with higher resolution than direct physical measurements [1-8], There are two main categories of devices used for this purpose: ring resonators and unbalanced Mach-Zehnder interferometers (MZIs). An earlier study introduced a method to retrieve changes in fluidic refractive indices and surface-adsorbed molecular fdm thickness by monitoring resonance shifts in resonator sensors [1], This required extracting the waveguide dimensions first in order to achieve reliable and quantitative index and film thickness results. This foundational work set the stage for later developments. In one study, the performance of silicon photonic integrated circuits predicted by considering layout-dependent correlated manufacturing variability, using a small-radius ring resonator to extract waveguide dimensions [3],

[0005] More recent advancements by others have explored another direction, introducing methods that utilize multiple Mach-Zehnder Interferometers (MZIs) to extract waveguide dimensions [5], Combining specially designed devices to aid the identification of the interference orders, and employing circuit models to fit the measured intensity spectra, these methods have achieved enhanced accuracy of dimension extraction. Applying their method to evaluate wafer level variations, it also provided insights into process control monitoring, crucial for ensuring consistent production quality across wafers [6], These studies highlight the progress in addressing fabrication uncertainties, but they also reveal limitations in current methodologies.

[0006] Despite advancements, current techniques have several limitations. Previous work using ring resonators employed small bend radius (12 urn) and therefore large free-spectral range (FSR) to help identifying the interference order. Since waveguide bends cause a deviation in the modal effective index neff, this introduced a certain level of error. Also, a linear approximation was usedto characterize the dispersion, further introducing inaccuracies. Errors in width (W)and height (H )were reported as W=0.85 nm and H=0.55 nm, respectively. MZI- based methods require multiple devices to determine the interference order, which is a crucial parameter to uniquely determine the waveguide dimensions. It is necessary to assume that these devices share the same width and thickness variations. This condition is recognized as often not satisfied ([4], [5]). The method relies on fitting the full spectral profiles, making it susceptible to intensity fluctuations and grating coupler variations. The authors report an error of less than 0.1 nm, from a process that involves more than 10 fitting parameters.SUMMARY

[0007] In one of its aspects, a method for determining a geometry of an optical waveguide, the method comprising: a ring resonator comprising a predetermined cavity length, analyzing multiple consecutive resonance wavelengths at an operating temperature; and performing a constrained numerical optimization process with a regression objective function of the optical waveguide’s spectral features to extract dimensions of the optical waveguide, wherein the process minimizes a total mean squared error (MSE) between predicted and measured resonance wavelengths over a selected wavelength range.

[0008] In another aspect, a computer-implemented method for determining a geometry of an optical waveguide, the method comprising: acquiring multiple consecutive resonance wavelength measurements of a ring resonator comprising a predetermined cavity length, wherein the measurements are performed at a predetermined temperature; with a processor, executing instructions stored in a non-transitory computer readable medium to predict resonance wavelength measurements of the ring resonator comprising the predetermined cavity length, wherein the measurements are performed at the predetermined temperature; with the processor, executing instructions to perform a constrained numerical optimization process with a regression objective function of the measured resonance wavelength to extract dimensions of the optical waveguide, wherein the processminimizes a total mean squared error (MSE) between the predicted resonance wavelengths and the measured resonance wavelengths over a selected wavelength range.

[0009] In another aspect, a system for determining a geometry of an optical waveguide, the system comprising: a ring resonator comprising a predetermined cavity length; a sub-system for acquiring multiple consecutive resonance wavelength measurements of the ring resonator, wherein the measurements are performed at a predetermined temperature; and a processor, execute instructions stored in a non-transitory computer readable medium to predict resonance wavelength measurements of the ring resonator comprising the predetermined cavity length, wherein the measurements are performed at the predetermined temperature; execute instructions to perform a constrained numerical optimization process with a regression objective function of the measured resonance wavelength to extract dimensions of the optical waveguide, wherein the process minimizes a total mean squared error (MSE) between the predicted resonance wavelengths and the measured resonance wavelengths over a selected wavelength range.

[0010] In another aspect, a method described herein determines the dimensions of optical waveguide geometry, in particular its cross-section. The method leverages the unique dependency of the dispersion behavior in spectral features of dispersive devices, in relation with the waveguide cross-section (width W and height H) and operating temperature (T). Using a ring resonator with a long cavity length, we analyze multiple consecutive resonance wavelengths at a known temperature. The waveguide dimensions can be extracted through a constrained numerical optimization process with a regression objective. This process aims at minimizing the total mean squared error (MSE) between predicted and measured resonance wavelengths over a broad wavelength range.

[0011] In another aspect, a method described herein addresses significant challenges in silicon photonics, where fabrication-induced variations in waveguide dimensions is inevitable, affecting device performance and yield. Due to the highrefractive index of silicon, a dimensional change of 1 nm leads to a spectral wavelength shift of approximately 1 nm, the later is comparable to the channel spacing in a wavelength division multiplexer (WDM). There is a strong demand in assessing the fabrication outcome accurately and efficiently, to enable building fabrication- aware circuit models or taking compensatory measures in the design stage.

[0012] Using spectra measured at different temperatures under metrological conditions (with better than 5 mK accuracy), dimensional changes on the order of 0.02 nm caused by a 10 °C temperature change are resolved. On the other hand, for the same spectrum but assuming a temperature inaccuracy of 0.1 °C, the associated dimensional deviation is predicted to be below 0.03 nm. This level of temperature control is commonly available in photonic testing facilities. Using the method described herein, the outcome is the waveguide dimension averaged over the resonator cavity length. On the other hand, this is the quantity of interest for photonic device design and characterization.

[0013] In another aspect, the methods and systems described herein only require a single spectral measurement. It is immune to intensity fluctuations and variations in grating couplers used to access resonator. The compact device size allows for easy distribution across a wafer for process monitoring that can be carrier out in-line, postfabrication, or on historical samples. The methods and systems are adaptable to different material platforms. It is also possible to adapt the method to other dispersive devices for assessing different waveguide geometries. The methods and systems provide unprecedented resolution that is more than 10 time better than SEM, and at least 5 times better than other optical waveguide-based methods reported. All these features make it a valuable tool in modem foundry metrology. It was observed that the dimensional changes as a function of temperature were consistent with the expected thermal expansion. The methods and systems described herein may open new possibilities in nano metrology and device characterization.

[0014] The methods and systems described herein comprise a regression model to extract the waveguide cross-section using ring resonator spectra. This approach can unambiguously identify ring interference order of over 1000, allowing for large bend radii that minimize bend-induced errors. Connected via grating couplers, the deviceenables in-line process monitoring both pre- and post cladding deposition, providing enriched fabrication data. It can also be applied to any sample with an embedded resonator. These methods and system offer more than ten times better resolution than SEM and at least five times the accuracy of other optical waveguide-based methods. Since the resonators are compact, they can be easily distributed across a wafer. Such optical characterization enables better understanding how fabrication deviations impact photonic integrated circuits (PIC) performance, building fabrication-aware circuit models, and informing compensatory design measures. The methods also complement the prefab models by providing local wafer thickness information.

[0015] In another aspect, the methods described herein determine the waveguide cross-section by leveraging the unique spectral feature dispersion behavior of dispersive devices in relation to its waveguide width, height, and temperature. There is no free fitting parameter. The methods make use of the wavelength information only, significantly reduces the impact of intensity noise. By including a large number of resonances, the impact of spectral distorting from various sources such as reflections and grating coupler variations are minimized. The demonstrated dimensional resolution surpasses all previously reported results.BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Several exemplary embodiments of the present disclosure will now be described, by way of example only, with reference to the appended drawings in which:

[0017] Figure la shows a layout of a ring resonator, with the inset showing the waveguide cross-section cut at the orange dashed line;

[0018] Figure lb shows a measured transmission spectrum, in which resonances from order mo to (mo -N) are used for analysis (with ascending wavelength);

[0019] Figure 1c shows predicted free-spectral -range (FSR) of the resonator RR5 at 30 °C, showing a clear wavelength dependence;

[0020] Figure 2 shows a flow chart outlining example steps for the determining waveguide cross-section dimensions;

[0021] Figure 3a shows optimization residue, presented as RMSD= sqrt(MSE) for the thermometer probe RR5, measured in the water bath at 30 C;

[0022] Figure 3b shows measured and predicted FSR over the wavelength range used for the optimization;

[0023] Figure 3c shows deviation between the measured and predicted resonance wavelength over the same range, showing a very small deviation of < 5 pm for approximately 50 resonances; and

[0024] Figure 4 shows predicted variation in the waveguide width and height due to temperature deviations from the true value.DESCRIPTION

[0025] The following detailed description refers to the accompanying drawings. Wherever possible, the same reference numbers are used in the drawings and the following description to refer to the same or similar elements. While embodiments of the disclosure may be described, modifications, adaptations, and other implementations are possible. For example, substitutions, additions, or modifications may be made to the elements illustrated in the drawings, and the methods described herein may be modified by substituting, reordering, or adding stages to the disclosed methods. Accordingly, the following detailed description does not limit the disclosure. Instead, the proper scope of the disclosure is defined by the appended claims.

[0026] Moreover, it should be appreciated that the particular implementations shown and described herein are illustrative of the invention and are not intended to otherwise limit the scope of the present invention in any way. Indeed, for the sake of brevity, certain sub-components of the individual operating components, conventional data networking, application development and other functional aspects of the systems may not be described in detail herein. Furthermore, the connecting lines shown in the various figures contained herein are intended to represent exemplary functional relationships and / or physical couplings between the various elements. It should be noted that many alternative or additional functional relationships or physical connections may be present in a practical system.

[0027] In Figure la, there is shown a system for determining a geometry of an optical waveguide, designated generally by the numeral 10. The system 10 comprises a long cavity ring resonator 12 and an optical waveguide 14 comprising a predetermined cavity length. The geometry of the optical waveguide 14 is determinedby a computing device 16 which analyzes multiple consecutive resonance wavelengths at an operating temperature; and performs a constrained numerical optimization process with a regression objective function of the optical waveguide’s spectral features to extract dimensions of the optical waveguide, as will be described in more detail below.

[0028] In the spectral features of dispersive devices, there is a unique signature in its dispersion behavior dependent on the waveguide cross-section and its operating temperature. The seemingly subtle variations can be accurately discerned from spectroscopic measurements that offer high precision. Using a set of N measured consecutive resonances at a known temperature, as shown in Figure lb, the optical waveguide 14 dimensions [W, H] can be extracted.

[0029] The unique aspect of the present disclosure is treating the determination of waveguide cross-section dimensions as a constrained numerical optimization problem with a regression objective function. The algorithm aims to minimize the difference between predicted and measured resonance wavelengths directly, for multiple resonances over a broad wavelength range. The methods and systems of the present disclosure avoids the use of measured free spectral range in conventional methods.

[0030] Figure 2 shows a flow chart 100 outlining example steps for the determining waveguide cross-section dimensions.

[0031] In step 102, the computing device 16 computes a matrix of waveguide modal effective index neff as a function of its width, height, wavelength and temperature. This process uses a comprehensive silicon refractive index model established in

[0011] , which accounts for the complex interdependencies among these parameters. This model provides accurate index dispersion relations neff(W,T enabling precise predictions of resonance wavelengths for all orders without relying on FSR for parameter extraction as in prior art.

[0032] In step 104, the computing device 16 determines the measured resonance wavelengths A™easwithin a selected range. The first resonance is assigned as having interference order mo (see Fig. 1), and the consecutive N resonances are used for analysis, using a peak-finder algorithm. In one example, the resonator cavity length L is assumed to be known (e.g. L= 950 pm).

[0033]

[0034] In step 106, following the resonator equation, the computing device 16 estimates the resonator interference order as an initial guess mO(initial) using the waveguide nominal dimensions [Wo, Ho].

[0035]

[0036] In step 108, within a selected range of potential actual interference orders (mo = mo(initial)±order_scan), the computing device 16 performs the following optimization by adjusting the waveguide dimensions [W, H]:

[0037] minimize

[0038] In step 110, resonance wavelength prediction for ^edlctisachieved by satisfying the resonator equation (1) through a regression process executed by the computing device 16. Since there are multiple interdependent variables in the equation, the available information is insufficient to solve the problem analytically.

[0039]

[0040] In step 112, the computing device 16 evaluates the error MSE from step 108 over the selected range of interference orders. It is important to note that due to the cyclic nature of resonator spectra, multiple mo values can produce a suitable XRwith the corresponding [W, H] that are plausible within the known bounds ofdimension variations. However, it may be demonstrated that when the error MSE is evaluated over many resonances covering a wide wavelength range, the dispersion behavior locks in the choice of mo. In other words, there is a particular mo value that leads to a global minimum in MSE, as shown in Figure 3a. The mo that gives the global minimum is considered the correct actual order, with the corresponding waveguide dimensions [Woptimized, Hoptimized].

[0041] As described in the previous section, finding the correct interference order m is a key challenge. Other methods use devices of low and higher order (m<150) to locate the proper value for m. This requires several assumptions, specifically that the two devices share the same width and thickness deviations, that are recognized as often inaccurate. The methods and systems described herein overcome this limitation, allowing the order to be uniquely determined, even for higher order devices (m~ 1500 in the current devices). This high order is critical in the significant enhancement in the resolution of dimensional extraction.

[0042] Shown below are the analysis results performed on a number of chips, in which a thermometer probe RR5 was measured from 20°C to 80°C under metrological conditions in a water bath. Other chips were measured in a photonics lab environment. In Table I for the RR5 results, it can be seen that an increase in W and a reduction in H with temperature, resolving dimensional changes on the order of 20 pm. These changes are consistent in magnitude with the expected thermal expansion. In current literature, the effects of thermal expansion on device behaviour are ignored, since the problem becomes too complex.

[0043] The method described herein provides a very high precision beyond currently available alternative methods, so the validation of the accuracy requires further investigation. On the other hand, the method described herein could open a new possibility in nano metrology, providing knowledge of device properties with unprecedented precision.

[0044] Table I. Parameters extracted from thermometer probe RR5 measured in a temperature-controlled water bath at the Metrology Research Center.

[0045] Figure 4 shows the deviation in the predicted waveguide width and height for the same measured spectrum, assuming varied level of temperature inaccuracies. With a temperature measurement uncertainty of 0.1 °C, the potential errors in the dimensions determined by the fitting algorithm are very small ([AW, AH] = [-0.025, 0.004] nm). Even for a temperature error of 0.5 °C, [AW, AH] = [-0.126, 0.02] nm, which is an order of magnitude smaller than the typical uncertainty in SEM measurements. Temperature control to these levels is easily achieved in photonic testing facilities.

[0046] Table II and III present the results from chips fabricated in separate runs, measured in the photonic testing facility. The results are within the expected range and in agreement with available physical metrology data. They show that the method can be applied to resonators of different cavity length (Table II). Devices made in separate runs but with substrates from the same wafer show similar waveguide height but varied widths (Table I vs Table II). For a chip showing varied performance from die to die, the results (Table III) show similar waveguide height, but large change in waveguide width (up to 10 nm), consistent with device performance. Further comparisons with SEM and ellipsometry measurements are on-going.

[0047] Table II. Parameters extracted for resonators with two cavity lengths from several locations on the same chip, measured in a photonics lab environment, with the stage temperature controlled at 25C.

[0048] Table III. Parameters extracted for resonators from a third chip where fabrication variations were observed in other devices. All resonators have a cavity length of 950 microns, measured in a photonics lab environment, with the stage temperature controlled at 25C and 30C respectively.

[0049] This invention offers significant advantages over current technologies. The method requires only one transmission measurement for one (TE) polarization at a known temperature with moderate accuracy of <0.5 °C. This makes it easily achievable in photonic testing laboratories. By relying solely on the dispersion behavior of resonance wavelengths over a wide range, the method is immune to intensity fluctuations and variations in grating couplers or misalignments. Given that multiple resonances are utilized (ranging from 20 to 50 currently), the problem becomes an over-determined system of equations with two unknown variables. This approach yields a deterministic outcome without any free fitting parameters. Using multiple resonances over a broad wavelength range also minimizes the impact of noise and imperfections, such as resonance shape distortion and splitting. As shown in Fig. 2(b), the measured FSR scatters about the predicted values. The above reported results are achieved in the presence of such imperfections. The method can unambiguously identify the interference order, even for large orders (1000 - 1500), allowing the use of large bend radii to minimize neff deviations. The associated long cavity length also increases the resonator quality factor, enabling higher accuracy in determining resonance wavelengths.

[0050] The compact device size (currently 100 pm x 400 pm, can be made smaller) allows for easy distribution across a wafer for process monitoring. The use of grating couplers enables in-line process monitoring before or after upper claddingdeposition, providing enriched fabrication information. Post-process measurements provide the final dimensions. The procedure can also be applied to historical sample, applicable to any wafer with an embedded resonator.

[0051] The method can evaluate a wide range of waveguide width selections without requiring specialized designs, unlike unbalanced MZIs previously reported. It can also be adapted to waveguides with non-negligible sidewall angles, simply requiring a new set of neff values to be calculated.

[0052] Currently only the TE polarization is used, however, the methods described herein can be extended to include TM polarization to extract more parameters, including both geometry and core / cladding indices. This adaptability makes it suitable for other material platforms like SiN, which is prevalent in quantum and nonlinear-optics applications. There is significant interest in this topic [8-10],

[0053] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations of particular inventions. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0054] Accordingly, the above description of example implementations does not define or constrain this disclosure. Other changes, substitutions, and alterations are also possible without departing from the spirit and scope of this disclosure.

[0055] REFERENCES[1] Yuki Atsumi, Dan-Xia Xu, Andre Delage, Jens H. Schmid, Martin Vachon, Pavel Cheben, Siegfried Janz, Nobuhiko Nishiyama, and Shigehisa Arai,"Simultaneous retrieval of fluidic refractive index and surface adsorbed molecular film thickness using silicon wire waveguide biosensors," Opt. Express 20, 26969-26977 (2012).[2] Chen X, Li Z, Mohamed M, Shang L, Mickelson AR. Parameter extraction from fabricated silicon photonic devices. Appl Opt. 2014 Mar 1;53(7): 1396-405. doi:10.1364 / AO.53.001396. PMID: 24663369.[3] Zeqin Lu, Jaspreet Jhoja, Jackson Klein, Xu Wang, Amy Liu, Jonas Flueckiger, James Pond, and Lukas Chrostowski, "Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability," Opt. Express 25, 9712-9733 (2017).[4] Yufei Xing, Jiaxing Dong, Sarvagya Dwivedi, Umar Khan, and Wim Bogaerts, "Accurate extraction of fabricated geometry using optical measurement," Photon. Res. 6, 1008-1020 (2018).[5] Yufei Xing, Mi Wang, Alfonso Ruocco, Joris Geessels, Umar Khan, and Wim Bogaerts,"Compact silicon photonics circuit to extract multiple parameters for process control monitoring," OSA Continuum 3, 379-390 (2020)[6] Yufei Xing, Jiaxing Dong, Umar Khan, and Wim Bogaerts, “Capturing the effects of spatial process variations in silicon photonic circuits”, ACS Photonics 2023 10 (4), 928-944, DOI: 10.1021 / acsphotonics.2c01194.[7] Wim Bogaerts and Lukas Chrostowski, “Silicon Photonics Circuit Design: Methods, Tools and Challenges”, Laser & Photonics Reviews 2018, 12, 1700237. https: / / doi.org / 10.1002 / lpor.201700237.[8] Zhengxing Zhang, Sally I. El-Henawy, Carlos A. Rios Ocampo, and Duane S. Boning, "Inference of process variations in silicon photonics from characterization measurements," Opt. Express 31, 23651-23661 (2023).[9] Todd H. Stievater, Nathan F. Tyndall, Marcel W. Pruessner, Dmitry A. Kozak, and William S. Rabinovich, "Optical and geometric parameter extraction for photonic integrated circuits," Opt. Express 30, 14453-14460 (2022).

[0010] Jordan N. Butt, Nathan F. Tyndall, Marcel W. Pruessner, Kyle J. Walsh, Benjamin L. Miller, Nicholas M. Fahrenkopf, Ahn O. Antohe, Todd H. Stievater;Optical and geometric parameter extraction across 300-mm photonic integrated circuit wafers. APL Photonics 1 January 2024; 9 (1): 016104. https: / / doi.Org / 10.1063 / 5.0173914

[0011] Dan-Xia Xu, Andre Delage, Pierre Verly, Siegfried Janz, Shurui Wang, Martin Vachon, PengHui Ma, Jean Lapointe, Daniele Melati, Pavel Cheben, and Jens H. Schmid, "Empirical model for the temperature dependence of silicon refractive index from O to C band based on waveguide measurements," Opt. Express 27, 27229-27241 (2019).

Claims

CLAIMS:

1. A method for determining a geometry of an optical waveguide, the method comprising: a ring resonator comprising a predetermined cavity length, analyzing multiple consecutive resonance wavelengths at an operating temperature; with a processor, executing instructions stored in a non-transitory computer readable medium to perform a constrained numerical optimization process with a regression objective function of the optical waveguide’s spectral features to extract dimensions of the optical waveguide, wherein the process minimizes a total mean squared error (MSE) between predicted and measured resonance wavelengths over a selected wavelength range.

2. The method of claim 1, wherein the spectral features of the optical waveguide are dependent on the optical waveguide’s cross-section and the operating temperature.

3. The method of claim 1, wherein the dimensions of the optical waveguide comprise width and height.

4. The method of claim 3, wherein the optimization process comprises the step of: computing a matrix of waveguide modal effective index neff as a function of the width and the height of the optical waveguide, the wavelength and the operating temperature.

5. The method of claim 4, wherein the optimization process uses a comprehensive silicon refractive index model to account for complex interdependencies among the width and the height of the optical waveguide, the wavelength and the operating temperature.

6. The method of claim 5, wherein the comprehensive silicon refractive index model provides accurate index dispersion relations and allows for predictions of the resonance wavelengths for all orders without relying on free-spectral-range (FSR) for extraction of the width and the height of the optical waveguide.

7. The method of claim 6, comprising further steps of determining the measured resonance wavelengths within the selected wavelength range; and assigning a first resonance as having interference order mo and wherein consecutive N resonances are used for analysis.

8. The method of claim 7, comprising a further step of estimating the resonator interference order as an initial guess mo(initial) using the optical waveguide nominal dimensions [Wo, Ho].

9. The method of claim 8, comprising a further step of performing the constrained numerical optimization process by adjusting the optical waveguide dimensions [W, H] within a selected range of potential actual interference orders (mo = mo(initial) order_scan) to determine the total mean squared error (MSE).

10. The method of claim 9, comprising a further step of evaluating the MSE over the selected range of interference orders.

11. The method of claim 10, comprising a further step of determining a particular mo value that leads to a global minimum in MSE, whereby the particularmo value is considered a correct actual order having corresponding optimized optical waveguide dimensions [Woptimized, Hoptimized] .

12. A computer-implemented method for determining a geometry of an optical waveguide, the method comprising: acquiring multiple consecutive resonance wavelength measurements of a ring resonator comprising a predetermined cavity length, wherein the measurements are performed at a predetermined temperature; with a processor, executing instructions stored in a non-transitory computer readable medium to predict resonance wavelength measurements of the ring resonator comprising the predetermined cavity length, wherein the measurements are performed at the predetermined temperature; and with the processor, executing instructions to perform a constrained numerical optimization process with a regression objective function of the measured resonance wavelength to extract dimensions of the optical waveguide, wherein the process minimizes a total mean squared error (MSE) between the predicted resonance wavelengths and the measured resonance wavelengths over a selected wavelength range.

13. The method of claim 12, wherein the dimensions of the optical waveguide comprise width and height.

14. The method of claim 13, wherein the optimization process comprises the step of: computing a matrix of waveguide modal effective index neff as a function of the width and the height of the optical waveguide, the wavelength and the operating temperature.

15. The method of claim 14, wherein the optimization process uses a comprehensive silicon refractive index model to account for complexinterdependencies among the width and the height of the optical waveguide, the wavelength and the operating temperature.

16. The method of claim 15, wherein the comprehensive silicon refractive index model provides accurate index dispersion relations and allows for predictions of the resonance wavelengths for all orders without relying on free-spectral-range (FSR) for extraction of the width and the height of the optical waveguide.

17. The method of claim 16, comprising further steps of determining the measured resonance wavelengths within the selected wavelength range; and assigning a first resonance as having interference order mo and wherein consecutive N resonances are used for analysis.

18. The method of claim 17, comprising a further step of estimating the resonator interference order as an initial guess mo(initial) using the optical waveguide nominal dimensions [Wo, Ho].

19. The method of claim 18, comprising a further step of performing the constrained numerical optimization process by adjusting the optical waveguide dimensions [W, H] within a selected range of potential actual interference orders (mo = mo(initial) order_scan) to determine the total mean squared error (MSE).

20. The method of claim 19, comprising a further step of evaluating the MSE over the selected range of interference orders.

21. The method of claim 20, comprising a further step of determining a particular mo value that leads to a global minimum in MSE, whereby the particular mo value is considered a correct actual order having corresponding optimized optical waveguide dimensions [Woptimized, Hoptimized] .

22. A system for determining a geometry of an optical waveguide, the system comprising: a ring resonator comprising a predetermined cavity length; a sub-system for acquiring multiple consecutive resonance wavelength measurements of the ring resonator, wherein the measurements are performed at a predetermined temperature; and a processor, execute instructions stored in a non-transitory computer readable medium to predict resonance wavelength measurements of the ring resonator comprising the predetermined cavity length, wherein the measurements are performed at the predetermined temperature; execute instructions to perform a constrained numerical optimization process with a regression objective function of the measured resonance wavelength to extract dimensions of the optical waveguide, wherein the process minimizes a total mean squared error (MSE) between the predicted resonance wavelengths and the measured resonance wavelengths over a selected wavelength range.

23. The system of claim 22, wherein the dimensions of the optical waveguide comprise width and height.

24. The system of claim 23, wherein the optimization process comprising computing a matrix of waveguide modal effective index neff as a function of the width and the height of the optical waveguide, the wavelength and the operating temperature.

25. The system of claim 24, wherein the optimization process uses a comprehensive silicon refractive index model to account for complex interdependencies among the width and the height of the optical waveguide, the wavelength and the operating temperature.

26. The system of claim 25, wherein the comprehensive silicon refractive index model provides accurate index dispersion relations and allows for predictions of theresonance wavelengths for all orders without relying on free-spectral-range (FSR) for extraction of the width and the height of the optical waveguide.

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