## Abstract

Measurement uncertainties in the techniques used to characterize loss in photonic waveguides becomes a significant issue as waveguide loss is reduced through improved fabrication technology. Typical loss measurement techniques involve environmentally unknown parameters such as facet reflectivity or varying coupling efficiencies, which directly contribute to the uncertainty of the measurement. We present a loss measurement technique, which takes advantage of the differential loss between multiple paths in an arrayed waveguide structure, in which we are able to gather statistics on propagation loss from several waveguides in a single measurement. This arrayed waveguide structure is characterized using a swept-wavelength interferometer, enabling the analysis of the arrayed waveguide transmission as a function of group delay between waveguides. Loss extraction is only dependent on the differential path length between arrayed waveguides and is therefore extracted independently from on and off-chip coupling efficiencies, which proves to be an accurate and reliable method of loss characterization. This method is applied to characterize the loss of the silicon photonic platform at Sandia Labs with an uncertainty of less than 0.06 dB/cm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Reducing propagation loss in photonic waveguides is critical for optical interconnects and many other integrated photonic applications. Reducing the uncertainty in loss measurements is becoming a more challenging and necessary step to take in order to accurately characterize photonic waveguides as propagation loss is reduced due to improved fabrication techniques. Waveguide loss characterization has been performed in the past using several different methods. By using the cut-back method [1], propagation loss is calculated by measuring several waveguides of different lengths and fitting a line to loss as a function of waveguide length. The slope of the fitted line then is the loss per unit length. The difference in coupling efficiency on and off chip varies for each individual waveguide however, which contributes uncertainty to the measurement. A modification to the cut-back method has recently been explored [2] which involves light being coupled into a single waveguide and routed to an arbitrary number of waveguides using cascaded 3 dB splitters. Transmitted optical power is then measured in each waveguide by using grating couplers and a wavelength appropriate camera. This method allows the measurement of loss independent of the input fiber to waveguide coupling efficiency, however because each waveguide is connected to a different grating coupler, some uncertainty can be attributed to the light emission through each output. Waveguide loss has also been measured through the analysis of the quality factor (*Q)* of microring resonators [3], which is inversely proportional to the waveguide loss. This method does not require knowledge of the on and off chip coupling efficiency; however, for low loss (high-*Q*) resonators small amounts of sidewall roughness can cause sufficient scattering to couple the resonant clockwise mode to the counterclockwise mode. This can significantly alter the spectrum of the ring resonator [4-5], making the extraction of *Q* non-trivial. More sophisticated models have been developed to extract loss from forward and backward propagating resonant modes [6] but rely on knowledge of the ring radius which may not be known accurately due to process variations due to manufacturing [7–15], which will affect the accuracy of the extracted loss. Fringe depth in Fabry-Perot interferometers has also been used as a propagation loss metric [16–18]. This method also is independent of insertion loss, but does require knowledge of the facet reflectivity, which is difficult to accurately measure. Measurement accuracy using this method has been demonstrated to be improved by using a broadband optical source and optical spectrum analyzer rather than a narrow linewidth laser and photodiode [17]. Similarly, propagation loss of photonic waveguides can be found through analysis of fringe depth in unbalanced Mach-Zehnder interferometers [19]. This method relies on measuring the extinction ratio of the through and cross ports for both inputs of the interferometer, imposing a requirement of four individual measurements. Loss has also been investigated through analyzing light which scatters out of the waveguide at different spatial locations along the length of the waveguide [20], however this method may be difficult to apply to low loss waveguides where the scattering due to side-wall roughness is minimal due to improved fabrication technology.

In this paper we demonstrate a novel loss measurement technique by using a fiber swept-wavelength interferometer (SWI) to measure an arrayed waveguide structure, depicted in Fig. 1. This arrayed waveguide structure is equivalent to an arrayed waveguide grating (AWG) with a single input and output. In fact, this method is very similar to the cut-back method in the sense that the total loss of several waveguides of various lengths is measured. As in [3,16–19], this method allows for waveguide loss to be measured independently of insertion loss, however it only requires a single measurement which offers the capability of quickly and accurately extracting waveguide loss. The main advantage to measuring waveguides due to this technique lies in the fact that multiple waveguides can be measured simultaneously, enabling the ability to perform statistical analysis on an arbitrary number of waveguides from a single measurement. Additionally, because the input and output to every arrayed waveguide is common, we are able to eliminate any measurement uncertainty associated with on and off chip coupling variation, increasing the accuracy and reliability of using an arrayed waveguide structure. This method relies on measuring the interference pattern of an AWG using a SWI along with a tunable laser source (TLS) in the domain of optical frequency. By taking the fast Fourier transform (FFT) of this interference pattern, we can differentiate the transmission as a function of the group delay of light propagating through the SWI and analyze the transmission in the reciprocal domain of the optical frequency, which translates to the time domain [21]. This provides a method of simultaneously measuring the transmission of each individual waveguide contributing to the overall measured intensity. This measurement takes less than one second to perform data acquisition and processing, and is primarily limited by the laser sweep speed. Given the path length difference of the waveguides within the arrayed waveguide structure we can then calculate propagation loss independent of insertion loss with an uncertainty of less than 0.1 dB/cm.

Additionally, we are able to extract group index of the fundamental mode in the waveguide of interest using this method. Extraction of group index can also be extracted through using the interferometric devices [3, 16–19] by measuring the free-spectral range (FSR) of the device. Because we are able to analyze transmission through the arrayed waveguide structure as a function of time using this method, we can easily find the propagation delay between transmission in each waveguide. This method allows us to gather multiple group delay data points with a single measurement, which provides accurate group index measurement due to the capability to statistically analyze group delay from a single measurement. From the measured group delay and path length difference between waveguides, we can extract the group index of the fundamental mode of our waveguides with an uncertainty of less than 0.03%.

## 2. Arrayed waveguide grating design and fabrication

AWG’s typically are used as wavelength-division multiplexing (WDM) components, making up a system with *N* input channels, *M* arrayed waveguides of different lengths, *P* output channels, and two free propagation regions (FPR) between the arrayed waveguides and inputs/outputs. Input light enters through one of several of the *N* input channels into an FPR, typically a star coupler, and is coupled to all *M* arrayed waveguides. The path length differences in the arrayed waveguides yield different accumulated phase for each of the *M* waveguides. Therefore, when light recombines in the second FPR, which is also a star coupler, the resultant interference pattern results in a focusing of the light onto the plane of output waveguides. Which waveguide this light is focused on will depend on the difference in accumulated phase, or optical wavelength, in adjacent arrayed waveguides. For the purpose of waveguide loss characterization, we are interested in the phase and amplitude transmission of all arrayed waveguides, which can be extracted from transmission through a single input and output channel of the AWG. For this reason, the AWG used in this study is fabricated with only a single input and output channel $(N=P=1)$. Because there is only one channel at the input and output of the AWG, channel spacing and other parameters constituent to the AWG dynamics are ill-defined and not further discussed. The main parameter of interest for this study is the difference in physical path length between adjacent arrayed waveguides. This length is chosen to be sufficiently long to introduce a large group delay between adjacent waveguides to provide adequate resolution during the SWI measurement. At the same time, consideration of the expected propagation loss is taken into account to allow sufficient signal to noise of the longest waveguide. The AWG’s used in this study are based on those in [22], which consist of two star couplers joined by 15 individual waveguides of equal width designed to occupy a 4 × 4 mm^{2} area. Another AWG design is used which consists of identical star couplers joined by 35 waveguides and occupy a space of 8 mm $\times $ 8 mm. The star couplers each consist of silicon tapers which serve as the input and output channels to the AWG structure and are designed so that roughly 80% of light entering the FPR couples to the arrayed waveguides which results in ~1 dB of insertion loss. These silicon tapers couple light to a silicon slab FPR, where the input light fans out to the output waveguides. The output waveguides are arranged on an arc, centered on the center input waveguide. Light entering the FPR from the center input waveguide will reach each output waveguide with the same optical phase. This is analogous to the collimation of light by a lens in free space systems. The arrayed waveguides are located at the opposite side of where the tapers couple to the FPR, as seen in Fig. 1(b). The arrayed waveguide width is held constant for a single AWG. AWG’s of various waveguide widths ranging from 400 nm to 3000 nm are fabricated in order to characterize waveguide loss in Sandia’s silicon photonic platform. Both fully etched (strip) and partially etched (ridge) waveguides were studied. Each waveguide in the array is designed with 32 identical 180° bends resulting in a serpentine geometry to realize designs with smaller footprints. Some of the arrayed waveguides studied here are sufficiently wide to become multimode in the presence of scattering due to sidewall roughness. This is undesirable because we only wish to analyze the loss of the fundamental mode, and the higher order modes obscure this information. To avoid mode conversion from the fundamental to higher order modes, each waveguide transitions to a single mode waveguide before each bent section of waveguide. In all cases, this bent waveguide is a 400 nm wide fully etched strip waveguide, regardless of arrayed waveguide width or type studied (fully or partially etched waveguide) to ensure that the arrayed waveguides remain single mode throughout the entire structure. The bent section of the waveguide is tapered to the arrayed waveguides being studied. Importantly, each of the arrayed waveguides is identical, except for regions of straight waveguide of the desired width and type which is being measured. From one waveguide to the next, these lengths of straight waveguide are increased to add additional optical path length. In the following measurements, we extract differential transmission amplitude and group delay information between waveguides, which can be attributed to this additional straight waveguide of the width and type of interest. Using FDTD simulations, the loss from each bend is estimated to be less than 0.03 dB, which results in a loss of less than 1 dB in for the 3.2 180° bend in each waveguide. These devices are fabricated using the passive silicon photonic flow at the Sandia Microsystems and Engineering Sciences Applications (MESA) facilities. Fabrication consists of an 80 nm partial silicon etch for the ribs, a full etch resulting in silicon thickness of 230 nm for the embedded strips, and waveguide cladding of 2.4 μm high density plasma (HDP) oxide. Devices are fabricated on silicon-on-insulator (SOI) wafers with a buried oxide thickness of 3 μm.

## 3. Arrayed waveguide grating transmission model

To model the transmission of the AWG, we first consider the star coupler on the input side of the AWG, which can be seen in Fig. 1(b). Light entering the center waveguide reaches the FPR of the star coupler and begins to expand. We treat the propagation in this region as a Gaussian beam propagating along the *z*-direction, and assume the beam waist is located at the end of the input waveguide. We also assume that the output waveguides are located far from the beam waist, allowing us to treat the phase front as circular. The output waveguides are positioned along a circular curve centered on the input waveguide, and therefore we can assume that the optical phase is uniform across the entrance of each output waveguide. We therefore consider only the amplitude of transmission from the star coupler to the *m ^{th}* arrayed waveguide, which is a Gaussian distribution given by Eq. (1).

*T*is the intensity of the light entering the star coupler,

_{0}*δr*is the spacing between output waveguides, and

*ω(z)*is the beam width as the light propagates along the

*z*-direction, given by Eq. (2), where

*ω*is the beam waist at the input waveguide.

_{0}From these two equations, we see that knowledge of the star coupler transmission can be used to extract the effective index of the FPR (*n _{FPR}*). The effective index is directly related to the material index, as well as the width of the input waveguide and the thickness of the FPR. All three of these parameters can potentially vary with the fabrication process [7–15] and therefore, measuring the effective index provides valuable information for monitoring and correcting process variations. However, because all of these terms will vary with process variations the absolute value of the effective index has significant uncertainty. Additional information, such as physical measurement of the fabricated dimensions would be required to reduce this uncertainty.

Following the star coupler, light propagates through the *M* arrayed waveguides which have a length of ${L}_{0}+m\times \Delta L$. In practice, each waveguide consists of a series of bends and transitions designed to create a compact device. This causes the effective index and waveguide loss to vary throughout the waveguide. However, we design the device such that each of the waveguides has the same number of bends and transitions. The additional length added to each waveguide, $\Delta L$, occurs completely in the waveguide type of interest and not at all in the bend or transition sections. This allows the extra phase and loss associated with the bends and transitions to be pulled out as a global term and can be ignored in the final analysis. The phase and loss term of interest is then given by Eq. (3), where ${t}_{m}{}^{wg}$ is the transmission of the *m ^{th}* arrayed waveguide, ${n}_{eff}(\nu )$ is the effective group index of the fundamental TE mode of the arrayed waveguides, and $\alpha (\nu )$ is the loss of the fundamental mode in units of inverse length.

Finally, the light enters the output star coupler. At this point light enters a slab region from each of the arrayed waveguides. Each waveguide enters the slab along the radius of a circle centered on the center waveguide on the opposite side of the slab region. For this analysis, we are only concerned with the light exiting from this center waveguide. We can assume that the phase accumulated by light traveling from each of the arrayed waveguides to this center waveguide is the same, and therefore ignore it as a global phase term. Additionally, each of the waveguides is directed towards this center waveguide. The spacing between arrayed waveguides in the star coupler, *δr*, is sufficiently small such that the angle between all arrayed waveguides satisfy the small angle approximation, so we can therefore assume, to first order, that the transmission from each arrayed waveguide to the center output waveguide will be the same. For these reasons, we ignore the output star coupler completely, treating the amplitude and phase transmission as a global term.

Combining Eq. (1) and Eq. (3) we are left with Eq. (4). This equation provides the basis for extracting the propagation loss and group index of the waveguide of interest. The real part, Eq. (4b), and the imaginary part, Eq. (4c), of the exponent will be used to fit measured data from the SWI, explained in the next section.

## 4. Swept source interferometer

Measurement of the AWG is accomplished using a swept source interferometer as depicted in Fig. 2. The clock Mach Zehnder interferometer (MZI) is imbalanced, with extra optical fiber in one arm. As the TLS is swept, the output from this MZI will be a series of peaks and valleys at regular intervals as the wavelength is swept in time. Hence, the clock MZI of the SWI is used as a clock to trigger sampling in the lower MZI. This is necessary because the TLS source may not tune optical frequency linearly. The clock signal is calibrated using a hydrogen cyanide wavelength reference, and found to provide triggering at a frequency spacing of 1.406 MHz (11.278 fm). Polarization paddles are placed in the clock MZI in order to maximize interference fringe depth in the clock. Similarly, in the measurement MZI, polarization paddles are placed in the upper arm and after the device under test (DUT) in the lower arm to maximize fringe depth.

The loss studied here is the transverse electric (TE) waveguide mode, so a polarization paddle is placed before the DUT. TE polarization is achieved by taking the spectrum of a micro-disk modulator optimized to have high quality factor TE cavity modes. Balanced photodiodes are used to optimize the signal-to-noise ratio (SNR) at the outputs of the SWI. The AWG is placed in one arm of the measurement MZI as the DUT. Light passes through each of the arrayed waveguides and then interferes with the reference arm, of length ${L}_{ref}$, of the measurement MZI at the output. The electric field at one output of the measurement MZI is given by Eq. (5).

Equation (5) has $M+1$terms, and the intensity by $|E(\nu ){|}^{2}$, Eq. (6), has ${(M+1)}^{2}$ terms. Of these ${\left(M+1\right)}^{2}$ terms, ${M}^{2}$ terms are simply interference between light traveling within the AWG and give the expected transmission spectrum of the AWG, which is the first summation of Eq. (6). A single term will be related to the transmission intensity of the reference arm of the MZI $({I}_{ref})$, and lastly $2M$ of these terms will correspond to the interference of the *M* waveguides with the reference arm of the MZI, which is the second summation of Eq. (6), and it is these terms that we are interested in. We will see that as long as the optical path length of the reference arm of the measurement MZI is much longer than the longest optical path length of the AWG, the terms that come from the self-interference terms will not interfere with our analysis.

As the optical frequency *ν* of the TLS is tuned linearly, the measured intensity out of the interferometer oscillates due to the cosine terms in Eq. (6). By taking the FFT of this intensity as a function of optical frequency, we are given the intensity as a function of time (inverse of frequency). To understand the meaning of the time axis, we can look at the cosine terms in Eq. (6). When taking the FFT, we expect these cosine terms to result in peaks centered at positive and negative values associated with the ‘frequency’ of the cosine term. The instantaneous frequency of the cosine terms in the second summation of Eq. (6) is given by the derivative of the phase of the cosine function with respect to the optical frequency. This is shown in Eq. (7), and from this we can see that the cosine will oscillate with a frequency proportional to the difference in group delay between light traveling in the reference arm and the *m ^{th}* arrayed waveguide of the AWG. Therefore, the time axis in the FFT can be associated with the relative group delay of light traveling through the DUT. Once calibrated, we can use the relative group delay between arms of the AWG to extract the group index of the waveguide type being studied.

By this same reasoning, the terms in the first summation of Eq. (6) will oscillate at a frequency proportional to the relative group delay between individual arms of the AWG. This series of terms will have many overlapping frequencies; for example, the $(m=1,k=2)$ term will have the same relative group delay as the $(m=2,k=3)$ term. For this reason, these terms do not provide meaningful data and we wish to ignore them. By choosing the length of the reference are to be sufficiently long, we can ensure that the terms from the first summation in Eq. (6) will appear at frequencies (group delays) much lower than the terms from the second summation. In the final analysis we will only care about the second summation, shown in Eq. (8), which in the group delay domain, will be a series of peaks with intensity proportional to $\mathrm{exp}[{\xi}_{m}(\nu )]$.

## 5. Parameter extraction

In order to extract the propagation delay and group index of the waveguide type of the AWG, we perform a sweep of the SWI over some optical frequency range. A greater frequency range will provide higher resolution in the group delay domain. For this work, the laser was swept over approximately 1.5 THz, which provides a group delay resolution of approximately 680 fs. The maximum group delay that can be measured is fixed by the frequency step of the clock MZI at approximately 350 ns. Because we are measuring the group delay relative to the reference arm of the measurement MZI, this time is further limited (i.e., zero physical group delay does not show up at zero group delay in the FFT data).

Once we have measured the interferogram produced by the measurement MZI, with the DUT in one arm, we perform the FFT on this data. By performing the Fourier transform of Eq. (6), we find that the intensity is a summation of Dirac-delta functions centered at group delays ${\tau}_{m}$, which is the group delay of the *m ^{th}* arrayed waveguide in the group delay domain, as seen in Eq. (9). For this analysis, we assume that the transmission amplitude and group delay of the waveguides is nearly constant over the bandwidth of the laser sweep. If these values vary significantly over the bandwidth of the sweep, the associated peak will be broadened and may contain additional structure (e.g. side-bands).

Since we are assuming a nearly constant propagation loss and group index, the intensity corresponding to the transmission of each arrayed waveguide in Eq. (9) is found by integrating the experimentally measured power around each of the *M* peaks in the group delay domain. This yields a discrete data set that contains *M* points which correlate to the transmission through each of the *M* arrayed waveguides. Prior to integrating transmission round each peak, experimental data is averaged over 100 measurements in order to reduce any noise present in the measurement. A typical group delay spectrum is seen in Fig. 3. By taking the natural logarithm of this discrete data set we are left with the terms in Eq. (4b) as a function of waveguide number *m*. After fitting to a second-order polynomial to this data set, we are able to accurately extract waveguide loss as the linear coefficient of the fit. The linear term is normalized by the differential waveguide path length ($\Delta L$) to get the loss in units of inverse centimeters, and scaled by $10{\mathrm{log}}_{10}e$ to convert to units of dB/cm. The quadratic coefficient of the fit is related to the material parameters and dimensions of the FPR in the star coupler, as explained in section 3. A typical data set of intensity as a function of waveguide number, along with a quadratic fit in which loss is extracted from, is shown in Fig. 3. The loss uncertainty is extracted from the fit of the quadratic function to experimental data as a function of waveguide number *m*, and is typically on the order of 0.05 dB/cm.

Finally, the relative group delay of each peak is determined and a linear fit is made to the group delay as a function of waveguide number *m*. The slope of this fit is the group delay per waveguide. Dividing this number by the differential waveguide path length ($\text{\Delta}L$) provides the group velocity, and dividing the speed of light by the velocity gives the group index. The uncertainty in this value can be determined from the linear fit and is typically on the order of 0.015%.

The advantage of this method of measurement can now clearly be seen. For a given structure consisting of *M* arrayed waveguides, we are able to extract *M* loss terms in a single measurement. This allows the possibility of statistically analyzing a set of loss data after performing a single measurement. Therefore, in a single measurement, we are able to calculate an average value for propagation loss, along with the associated uncertainty of this value. As with any measurement, accuracy can be statistically improved through increasing the sample size of the experiment, which in this case is achieved through increasing *M,* the number of arrayed waveguides. Furthermore, accuracy can be improved by applying Eq. (4b) to a data set with higher contrast between data points, which here is the total propagation loss in each waveguide. This can be accomplished by increasing the path length difference between waveguides, $\text{\Delta}L$. Therefore, the tolerance of this method can be further improved by increasing the path length difference between the longest and shortest arrayed waveguide, $(M-1)\times \text{\Delta}L$. As a result of being able to acquire a large amount of data points per measurement, an accurate measurement can be found even in the presence of anomalous data points (which for example may arise from process variation or waveguide defects) through identifying and removing these statistical outliers. The optimal value of $(M-1)\times \text{\Delta}L$ resulting in minimum relative uncertainty will then depend on the propagation loss of the waveguide being studied, and as will be seen in section 6.

Sources of measurement uncertainty may arise from noise in the laser and detectors. For this reason, we use a laser with low source spontaneous emission (SSE) and −145dB/Hz relative intensity noise (RIN), as well as balanced photodetectors with more than 20dB common mode rejection. Furthermore, environmental fluctuations may introduce noise. For these measurements, the temperature of the arrayed waveguide structure is held constant to within 0.01°C with a thermo-electric cooler and PID feedback in order avoid fluctuations in propagation loss due to the thermo-optic effect in silicon.

In order to improve accuracy of this method, uncertainty can be decreased by increasing the path length between the shortest and longest arrayed waveguide, $\left(M-1\right)\times \text{\Delta}L$, in the structure. For sufficiently low loss waveguides, the total propagation loss between the shortest and longest arrayed waveguide can be small. In this case, the linear term of Eq. (4b) as a function of waveguide number *m* negligibly contributes to the polynomial fit of transmission, resulting in higher relative uncertainty in this fit. However, by increasing $\left(M-1\right)\times \text{\Delta}L$ by either increasing the number of arrayed waveguides *M* or the path length difference $\text{\Delta}L$, the overall footprint of the structure will increase, revealing a tradeoff between accuracy and space consumption.

## 6. Silicon photonic waveguide characterization

Using this method of loss characterization, we are able to characterize loss as a function of waveguide width in the silicon photonic platform at Sandia for both fully etched embedded strip waveguides and partially etched rib waveguides at widths ranging from 400 nm to 3000 nm, which is shown in Fig. 4. This experimental data agrees well with the loss model in Eq. (10), which consists of loss due to scattering from sidewall roughness and free carrier absorption.

The first term in Eq. (10) is derived by Payne and Lacey in [23] for a three-layer dielectric slab waveguide with an RMS deviation of *σ* from a smooth surface. $g\left(V\right)$ depends on the waveguide geometry; $V={k}_{0}d\times {\left({n}_{1}^{2}-{n}_{2}^{2}\right)}^{1/2},U=d\times {\left({n}_{1}^{2}{k}_{0}^{2}-{\beta}^{2}\right)}^{1/2},W=d\times {\left({\beta}^{2}-{n}_{2}^{2}{k}_{0}^{2}\right)}^{1/2}$, *β* is the propagation constant of the fundamental waveguide mode, *k _{0}* is the wavenumber in free space,

*d*is the half width of the waveguide, and

*n*and

_{1}*n*are the refractive index of the core and cladding of the waveguide, respectively. ${f}_{e}\left({L}_{c},\gamma \right)$ depends on the statistical distribution of surface roughness which has an exponential autocorrelation function for sidewalls that are manufactured lithographically [24], where $x=W{L}_{c}/d$, $\gamma ={n}_{2}V/\left({n}_{1}W\Delta {}^{1/2}\right)$ and $\text{}\text{}\Delta \text{}\text{}=\left({n}_{1}^{2}-{n}_{2}^{2}\right)/\left(2{n}_{1}^{2}\right)$. In order to apply this model to our data, the effective index method is employed to reduce the waveguides in our study to equivalent three-layer dielectric slabs, which is the same procedure in [25]. The RMS roughness and correlation length are found to be 0.95 nm and 57.48 nm respectively through analysis of SEM images taken from 500 nm wide SOI waveguides.

_{2}The second term in Eq. (10) is the change in absorption due to excess hole concentration and is derived from the Soref & Bennett model [26], where ${\text{\Gamma}}_{Si}$ is the modal confinement factor in silicon. The excess hole concentration is derived from the resistivity of the SOI film and found to be approximately 1e15 cm^{−3}. These numbers are applied to the model in Eq. (10) and plotted against waveguide width in Fig. 4 for both fully and partially etched waveguides, along with the independently extracted experimental propagation loss. The agreement between experiment and model indicates that our measurement method is accurate, and can be further improved by performing atomic force microscopy (AFM) measurements on the waveguide sidewalls in order to more accurately extract the roughness statistics. As expected from the model in Eq. (10), the loss decreases as the waveguide becomes wider. This is because the modal overlap with the Si/SiO_{2} interface is reduced as the waveguide widens. As modal overlap with the sidewalls becomes sufficiently small, the free carrier absorption becomes a dominant source of loss. The variation in experimental data can be attributed to process non-uniformities, which can significantly affect the performance of photonic devices [7–15]. The relative uncertainty of these particular measurements is inversely proportional to the measured loss, which can be prominently seen as the relatively high uncertainty in the wider waveguides in Fig. 4. The accuracy for measured propagation loss of these wide waveguides can be improved through increasing the total loss between waveguides by increasing the path length difference between the longest and shortest arrayed waveguides $\left(M-1\right)\times \text{\Delta}L$. In the measurement reported here, $(M-1)\times \text{\Delta}L$ is held constant for every arrayed waveguide structure, and as a consequence we can easily observe the relative measurement uncertainty increase as loss decreases, indicating that an optimal value for $(M-1)\times \text{\Delta}L$ will depend on the actual propagation loss in the studied waveguide.

To further illustrate the advantages of using this method of measurement, the effective group index from the fundamental mode of each waveguide dimension is extracted and seen in Fig. 5 along with the simulated group index. The group index is experimentally extracted from the time delay between the transmission of adjacent arrayed waveguides, which is ${\tau}_{m+1}-{\tau}_{m}={n}_{g}\left(\nu \right)\text{\Delta}L/c$, where the path length difference between adjacent waveguides, $\text{\Delta}L$ is known by design, and *c* is simply the speed of light. The simulated group index is found through an eigenmode solver and plotted against waveguide width along with the independently extracted experimental data. We find that the effective group index follows a decreasing trend as waveguide width increases, which is expected due to the optical mode overlap with the waveguide cladding as the waveguide dimensions decrease. The experimentally extracted values show excellent agreement with simulation results, suggesting that this measurement method can accurately measure group delay to a high degree of accuracy, which can be attributed to the ability to gather group index data from ($M-1$) waveguide simultaneously.

## 7. Conclusion

We have shown a measurement method consisting of arrayed waveguide structures using SWI techniques that can quickly and accurately extract waveguide loss and group delay with a single measurement independent of on and off chip coupling efficiencies. This measurement technique enables measurement of low loss waveguides which previously could not be measured within a sufficient degree of accuracy. By varying the environmental parameters of the arrayed waveguides, it is possible to accurately and quickly investigate both sources of loss and the group index in photonic waveguides using this method due to its accuracy and simplicity. This technique is applied to the silicon photonic process flow at Sandia labs to characterize waveguide loss as a function of waveguide width for both embedded strip and rib silicon waveguides, and can easily be expanded to study various sources of loss in photonic waveguides and improve waveguide fabrication processes.

## Funding

Laboratory Directed Research and Development #100007000.

## Acknowledgments

Sandia National Laboratories is a multi mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

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