1. Introduction

As the field of photonics continues to advance, increasingly efficient waveguide structures are being developed for various applications. Examples include mode converters [1], waveguide tapers [2], waveguide crossings [3, 4], end-facet couplers [5, 6], bends of various arc lengths and radii [7], etc. Insertion loss is a key metric in the development and implementation of any photonic component, but is often difficult to measure accurately for efficient devices with a modified version of the cut-back technique [8] serving as the most common approach. In this scheme, insertion loss is measured for a set of waveguides with a varying number of test components placed along their lengths. The results are then compared, with the ultimate goal being to use the insertion loss plotted versus number of components to extract the propagation loss of a single structure. Accurately implementing this method requires the optical input and output paths of each device to be identical, with the tolerable variation in loss proportional to the magnitude of the components excess loss. This places an ever-increasing burden on design, fabrication, and alignment tolerance as the efficiency of the component is optimized and requires a prohibitively large number of test structures for the most efficient devices.

Several other measurement techniques have been used to measure propagation loss within waveguides. These include the Fabry-Perot method [9], liquid prism method [10, 11], and analysis of the transmission functions of ring and disk resonators [12–14]. Extension to the interrogation of discrete components in high index contrast waveguides has been limited, however, due to a variety of challenges. The liquid prism method, for example, requires the use of a liquid with higher refractive index than the core of the waveguide leaving the approach poorly suited to high index contrast waveguides. The Fabry-Perot method is limited primarily by the challenge of designing and fabricating high reflectivity, broadband reflective sections integrated along the length of the waveguide. This design and fabrication process requires a level of effort similar to or potentially greater than the device itself with the loss of distributed reflectors of the Bragg variety introducing the very error we hope to eliminate back into the system.

The microphotonic resonator has been extensively studied in recent history proving an indispensable asset in a variety of application spaces ranging from tunable and passive filters [15,16] to optical sources [17–19], modulators [20], and nonlinear optical cavities [21]. The transfer function of the microphotonic resonator is well-characterized and uniquely dependent on the loss within the optical cavity, allowing for direct measurement of this intrinsic loss without the need to calibrate, measure, or otherwise account for fiber-to-waveguide coupling losses or, for that matter, any loss mechanism external to the resonator [15, 22]. Ultimately, this provides a loss measurement technique with extremely low susceptibility to error and a high degree of confidence in the accuracy of the result. While this type of resonator has often been used to measure propagation loss in waveguides, the cylindrical symmetry of the ring and disk resonators typically employed leaves these devices poorly suited to the task of measuring excess loss in structures that may not share their geometric requirements. A combination of the approach presented here and the cutback method was demonstrated [30]; however, improper fitting of the resonator transmission function led to variability in the measured round-trip insertion loss of over 5 dB.

We describe the first complete analysis of the racetrack loss platform (RLP) [31]; a racetrack resonator modified to enable measurement of insertion loss for efficient, compact photonic components. Accurately accounted for and discussed are the error mechanisms present in the application of this measurement technique, the impact of bus-to-resonator coupling and contra-directional coupling on the resonator transmission function, and the ramifications of intracavity losses including bend loss and scattering loss from the straight-to-bent waveguide transitions inherent in racetrack resonator geometries on the estimate of the test component insertion loss. Application of this technique to waveguide width tapers yielded a confidence interval of 0.007 dB; a level of precision that would be difficult, if not impossible, to obtain using the alternative methods previously discussed.

In this scheme, a standard racetrack resonator (Fig. 1(a)) is altered with the straight length opposite the coupling region replaced by two mirror image versions of the structure to be tested (Fig. 1(b)). So long as the structure is symmetric with terminations well suited to couple to the remainder of the racetrack resonator, there are effectively no limitations on the nature of the structure tested thus enabling a flexible platform particularly well suited to the analysis of rectilinear structures. Presented in Fig. 2 are a few examples of devices that could be measured and are difficult to otherwise analyze.

 

Fig. 1 a) Schematic of a racetrack resonator and b) segregation of the RLP by section.

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Fig. 2 Racetrack resonator with example test structures: a) lateral taper, b) vertical transition, c) tap coupler, and d) waveguide crossing.

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The first example (Fig. 2(a)) is simply that of a waveguide taper [2]. A waveguide taper provides efficient modal transformation between dissimilar waveguide geometries over some propagation distance thus mitigating loss in the transition process. These structures serve vital roles in a variety of applications such as end-facet coupling between single-mode fiber and HIC waveguides [5, 6], transitioning from multi-mode to single-mode waveguides [15], and adiabatic coupling [1], to name a few. An analogous example shown in Fig. 2(b) is that of a vertical transition which provides a similar modal transformation between vertically disjoint waveguides [24]. Next we have the optical tap coupler (Fig. 2(c)) that serves to couple some portion of the input signal into an adjacent waveguide. This can serve diagnostic purposes when a small portion of the signal is coupled, be used to form high extinction ratio interferometric structures [25], or enable coherent detection schemes [26] when designed as 3 dB couplers. In the final example, (Fig. 2(d)), we show that waveguide crossings may also be analyzed using this structure with some care taken in the design of the device.

Presented here is a description of the theory required to extract the insertion loss of a given test structure using transmission data from one test and one reference device. This theory is then applied to lateral waveguide tapers of varying length and profile followed by an analysis of the error present in the technique as applied to this set of devices. In total, 152 resonances from twelve resonators were processed in addition to transmission data from eight waveguides used to estimate propagation loss in the passive devices via the aforementioned cutback approach providing a sufficiently large sample to enable statistically significant estimation of the insertion loss of each taper along with the expected error of each result. Then, using data from reference resonators and straight waveguides of varying length, the estimated device insertion loss can be calibrated to a high degree of accuracy with the expected estimation error detailed.

2. Coupled mode theory

We will use the temporal coupled mode theory(CMT) as developed by Haus [25] and extended by Gorodetsky [26], then Borselli [19] to treat the resonators beginning with the case of negligible coupling between clockwise and counterclockwise traveling modal fields. Doing so will provide a link between the transmission function of the resonator and its insertion loss with extraction of the portion of the loss attributable to the test structure developed later. In the standard two-port add/drop filter configuration (Fig. 1), the equations for the clockwise traveling modal field within the resonator and the transmitted intensity are given by Eqs. 1 and 2, respectively.

The presence of sidewall roughness and the straight-to-bent waveguide transitions introduce scattering loss in addition to coupling the clockwise and counterclockwise traveling modal fields in the resonator via reflection of incident fields. This coupling breaks the degeneracy of the modal fields and introduces an interdependence of their time rate of change of energy amplitudes as they are no longer orthogonal. The result is a splitting of the resonance transmission function (Fig. 3) and a reduction of the accuracy of Eq. (2). A precise model for the transmission function is imperative as the estimated insertion loss is directly derived from the transmission function fitting process. What follows is a mathematical treatment of the doublet resonance and a method for extraction of the resonator insertion loss from the derived transmission function.

 

Fig. 3 Example transmission functions corresponding to several resonance types. Shown are the experimental data (black lines) and theoretical fits to the transmission functions (red lines) along with the theoretical transmission functions of the separable lower (green line) and higher (blue line) frequency eigenmodes.

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The aforementioned degeneracy lifting yields energy amplitude rate equations for the clockwise (E^CW) and counterclockwise (E^CCW) traveling modal fields that are no longer separable [26]. We can rewrite the fields in terms of the standing wave modes originating from interference of the clockwise and counterclockwise traveling modal fields. These fields (E^c and E^s) are identified by their orthogonal azimuthal dependencies with the transformation between eigensystems given by Eq. (3).

The functional form of the energy amplitudes of the standing wave modes with loss rates ${\gamma }_{\alpha }^c$ and ${\gamma }_{\alpha }^s$, respectively, is identical to the original formulation for the singlet mode E^CW with the addition of the backscattering rate γ_β and is given by Eq. (4) with the transmitted intensity now given by Eq. (5).

For an in-depth derivation of the above relations, the interested reader is refered to Jones [31].

3. The racetrack loss platform

In employing the racetrack resonator to measure the excess loss of discrete components, the straight section of the resonator opposite the coupling region is exchanged for some structure under test(SUT) (Fig. 1(b)). Utilization of a resonant structure eliminates external error sources leaving only internal error sources to be accounted for with the aggregate loss α_CW given by the product of the coefficients associated with each of the devices intrinsic loss mechanisms.

In principal, a unique coefficient could be associated with each defect along the length of the waveguide; however, we restrict the analysis here to three loss coefficients to describe loss introduced by the structure under test (α_SUT), the straight-to-bent waveguide transitions (α_STB), and all other loss mechanisms within the structure including absorption, scattering, and bend loss (α_AOL). Separating the losses of the four sections of the resonator (Fig. 1(b)) allows us to write the aggregate loss as follows.

In estimating the loss of the structure-under-test, all other loss components effectively introduce error in the measurement. This error may be reduced via calibration of the loss coefficient by a reference structure. Assuming the structures are effectively identical (less the structure-under-test), the loss coefficient of the reference structure may be written as follows.

The ratio of these loss coefficients then gives α_TOT/α_REF = α_SUT/α_AOL3 removing all error factors save the propagation loss of a single straight section of the resonator. Full calibration of the loss measurement requires accurate estimation of the loss of this straight section and may be accomplished via typical loss measurement techniques [9–14]. We take a moment here to distinguish between error sources and noise. As used herein, an error source is some component where the term noise is used to refer to the intrinsic, random variation of all measured parameters. The calibration procedure presented eliminates error sources while noise in each of the measurement and curve fitting processes and fabrication variation in creating the reference and test structures may not be eliminated and ultimately contribute to the precision with which the presented technique may be used. A comparison of the contribution of error factors and intrinsic noise is presented in Sec. 5 to give an indication of the minimum measurable insertion loss and relative importance of the calibration procedures discussed here.

4. Device geometry

The devices tested were fabricated at Sandia National Laboratories‘ MESA facility using a modified CMOS process developed specifically for silicon photonics with waveguides defined via optical lithography and subsequent dry etching. Thermal oxidation was used to reduce side-wall roughness followed by deposition of a thick oxide cladding to passivate and protect the devices. The resulting silicon-on-insulator (SOI) waveguides had a 230 nm thick silicon device layer with three micron silicon dioxide lower and upper cladding layers. All bus waveguides shared a fixed width of 400 nm with end facets tapered to 220 nm to match the optical mode of the tapered lensed fiber used to couple to the devices. This same width was used for the reference portions of the racetrack resonators with a fixed bend radius of 10 μm used to produce reproducible bend loss across the devices tested. Reproducibility was further enhanced via use of a bend in each bus waveguide with a fixed radius of 10 μm. The purpose of this modification was to eliminate dependence of the coupling coefficient on the length of the straight section of the resonator as the modified coupling region behaves approximately as a point coupling.

To demonstrate the proposed technique, waveguide tapers of varying length and profile were analyzed (Fig. 4.). These tapers started at a width of 400 nm to match the passive portion of the structure and ended at a width of 250nm as might be used to couple to a lensed tapered optical fiber. Structures with parabolic, exponential, and linear taper functions were tested along with structures designed to have a linear effective index taper at a wavelength of 1.55 μm. Reference devices with tapers omitted were additionally tested to enable calibration of the measured loss. All devices were laterally coupled to bus waveguides with a 320 nm gap between the bus and resonator waveguides. Data was acquired for taper lengths of 2.5, 5, and 10 μm in an attempt to provide some insight into the length dependence of the excess loss of each taper profile.

5. Experimental method and results

All data presented in this section was acquired using an Agilent 81600B tunable laser source and 81635A dual optical power sensor with a Thorlabs FPC560 polarization controller used to maintain a TE polarized input field. Wavelength scans were acquired over the 1.495 to 1.630 μm range with a 1.1 pm step size. OZ Optics tapered and lensed fiber with a two micrometer spot size was used to couple to the on-chip waveguides that were themselves tapered to optimize coupling efficiency to the lensed fiber. A schematic of this test setup is presented in Fig. 5.

 

Fig. 4 Schematic of lateral tapers examined and analogous RLP test structures along with their taper functions in terms of the initial waveguide width (w₂), final width (w₁), and their difference (Δω = w₂ − w₁).

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Fig. 5 Schematic of the optical test setup.

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It is important to note that, while the transmission function of the resonator is not affected by external loss mechanisms, additive noise sources such as laser RIN and detector noise still contribute to the signal in the typical way. It is also important to leave both the coupling and loss coefficient of the forward and backward traveling modes as free parameters in the fitting process. The presented fitting technique is often applied (particularly in high Q devices) under the assumption that loss is the primary contributor to the width of the lineshape; rigorous use of the presented formalism will avoid error in measurement of the loss coefficient due to variation in coupling behavior as a function of wavelength or device geometry. Finally, in ultra-high-Q devices, it may be preferable to use alternative techniques such as cavity ring down spectroscopy [32] to produce highly accurate Q-factor data where sufficiently stable and/or narrow linewidth sources are unavailable.

The loss and coupling coefficients of each resonance were estimated via least squares optimization [27] of the fit to the transmission function (Eq. (5)) via constrained nonlinear optimization [28]. Loss coefficient data for each resonator was then fit to an exponentially decaying function of the form α̂(λ) = α̂₀ − c₁ · exp ((λ − 1.495μm)/c₂). The validity of this estimation of the wavelength dependence of the loss coefficient was verified via application of the Anderson-Darling test to each set of residuals [29]. The null hypothesis that the residuals follow a normal distribution was accepted for all data sets at the 95% confidence level indicating that α̂(λ) is a reasonable approximation to the wavelength dependence of the loss coefficient for each of the analyzed devices. In total, 152 resonances from twelve test devices were analyzed with a mean residual in the estimated device insertion loss of 1.1 × 10⁻⁵dB and standard deviation of 0.0072 dB. This resulted in estimation of the excess loss of each taper to within 0.0070 dB at a confidence level of 95%.

In order to calibrate these results and estimate the expected contribution of each error source, two further data sets were taken. First, transmission data was taken for reference racetrack resonators identical to the above structures with the tapers removed. The result was then fit to an exponentially decaying function as previously discussed with the residuals passing the Anderson-Darling test. This data displayed a 95% confidence interval of 0.0063 dB for the 38 resonances analyzed. The estimated value of the loss coefficients for these devices suggests that omission of calibration using the reference structure data leads to an error in measurement between 0.024 and 0.040 dB over the 1.50 to 1.60 μm wavelength range.

The second set of data taken was transmission data from a set of waveguides with varying lengths. In total, data was taken for eight structures with lengths ranging from seven to 59 mm in excess of the additional length required to route the signal. Care was taken to ensure this additional infrastructure remained identical for all waveguides. At each wavelength, a linear fit was applied to the data and used to estimate the propagation loss of the 400 nm wide waveguide (Fig. 6). The data indicate a linear increase in loss from 3.5 to 4.1 dB over the 1.50 to 1.60 μm wavelength range. Fitting of the result to a line produced an estimated 95% confidence interval of 0.20 dB with the same value calculated for a second order polynomial fit to the data. The loss coefficient of the structure under test may then be modified by the following calibration factor to account for the loss of this straight section as a function of wavelength(λ) and taper length(TL) both assumed to have units of micrometers.

Shown in Fig. 7 is reference racetrack filter insertion loss data for 2.5, 5, and 10 μm taper lengths calculated as the square of the loss coefficient converted to dB scale (IL = 10log₁₀(α²)). The lack of statistically significant variation in loss between devices of differing taper lengths suggests that the portion of the insertion loss attributable to propagation through the straight sections (Fig. 1(b), sections 1 and 3) is within the confidence interval of the measurement. From the cutback loss data, this loss for the devices with 2.5, 5, and 10 μm taper lengths is estimated at 0.0019, 0.0038, and 0.0077 dB, respectively, at a wavelength of 1.55 μm while the confidence interval of the fit to all data is 0.0063 dB. This leaves a difference in loss between the devices with the longest and shortest taper lengths of 0.0058 dB; a value that lies within the confidence interval of the measurement.

 

Fig. 6 Transmission loss data from devices with varying straight section lengths (a) and propagation loss estimated from this data (b).

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Fig. 7 Insertion loss of reference RLP structures (Fig. 1(a)).

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As previously discussed (Sec. 3), the insertion loss of a modified racetrack resonator is composed of the loss attributable to the SUT (Fig. 1(b), section 3) and that of the unperturbed structure (Fig. 1(b), section 1,2, and 4). Proposed in Sec. 3 were three methods of estimating the insertion loss of the SUT that we return to now with a discussion of the devices required to produce and the error associated with each estimate. For clarity, the expression for the total resonator loss is given in Eq. (13) with the error of the estimator ${\widehat{\mathit{IL}}}_{{\mathit{SUT}}_n}$ of the SUT loss IL_SUT for the n^th calibration process defined as in Eq. (14).

Adding a reference resonator sans SUT enables the first calibration by which IL_SUT is estimated by the total loss minus the loss of this reference structure (i.e. ${\widehat{\mathit{IL}}}_{{\mathit{SUT}}_1}\approx {\mathit{IL}}_{\mathit{TOT}}-{\mathit{IL}}_{\mathit{REF}}$). Doing so improves the quality of the estimator with the inherent error bias reduced to −IL_AOL3 (Eq. (16)).

Here, the cutback method was used to provide a reliable means of estimating the propagation loss in the straight waveguide sections while additionally serving as a comparative method by which the relative accuracy of the resonant and cutback techniques could be compared. Using a properly designed structure, the propagation loss estimate could be obtained with a single ring resonator [12] thus yielding a minimum of three resonators required to estimate IL_SUT with optimal precision. It should be noted that calibration of the racetrack loss platform benefits, to some degree, from economies of scale in that structures of similar geometries may share calibration devices; therefore, the number of calibration devices required per test device may be substantially reduced in studies of the parametric variational variety.

This leads us to the study at hand wherein the insertion loss of lateral tapers in SOI waveguides with varying geometry, as discussed in Sec. 4, was estimated. Presented in Fig. 8 is insertion loss estimates of single tapers with 2.5, 5, and 10 μm lengths, respectively. Shown are the device insertion loss data (blue asterisks) and a decaying exponential fit to the device insertion loss (blue line, ${\widehat{\mathit{IL}}}_{{\mathit{SUT}}_0}$) along with the first (red dashed line, ${\widehat{\mathit{IL}}}_{{\mathit{SUT}}_1}$) and second (black line, ${\widehat{\mathit{IL}}}_{{\mathit{SUT}}_2}$) calibrations as discussed above. From this data, the confidence interval associated with the decaying exponential fit to IL_TOT for all devices was calculated at 0.014dB corresponding to the value $\mathit{err}\left({\widehat{\mathit{IL}}}_{\mathit{TOT}}\right)$. We then estimate the bias reduction associated with the first and second calibrations as the step-wise difference in successive calibration steps. Derivation of the confidence intervals of the taper insertion loss estimates for all devices and for each individual device then yields the values shown in Tbl. 1.

 

Fig. 8 Taper insertion loss data for a) 2.5, b) 5, and c) 10 μm long tapers of varying lateral profile. Shown are the uncalibrated insertion loss estimates (circles) along with fits to this data (solid lines), and calibrated device insertion loss estimates (dashed lines).

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Table 1. Compiled insertion loss (IL) and confidence interval (CI) data for the twelve tapers analyzed along with the central value $\left({{\widehat{\mathit{IL}}}_2|}_{\lambda =1.55\mu m}\right)$ and range ( $\mathrm{\Delta }{\widehat{\mathit{IL}}}_2$) of the estimated insertion loss across the 1.50 to 1.60μm wavelength range. Subscripts of the insertion loss estimates ${\widehat{\mathit{IL}}}_n$ correspond to the n^th calibration process (Eqs. 15 to 17).

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We may conclude, then, that the insertion loss of the racetrack resonators analyzed here is dominated by the bending loss in the bent waveguide sections and the straight-to-bent waveguide transitions with the straight waveguide sections minimally contributing to the sum total. The results of Tbl. 1 further indicate that a confidence interval of less than 0.05 dB may be obtained in the SOI system presented here for tapers lengths up to 50 μm long at a wavelength of 1.55 μm yielding high precision with only a first order calibration. Inclusion of the secondary calibration yields the highest accuracy with a 0.0070 dB confidence interval that is effectively independent of taper length while the confidence interval of the un-calibrated result at 0.037 dB remained substantially lower than that of the cutback method estimated here to be approximately 0.20 dB.

6. Conclusion

Presented herein is a method by which the excess loss of a discrete photonic component may be estimated with a high degree of accuracy while requiring minimal on-chip real estate. The use of a resonant device eliminates all error sources external to the resonator thus mitigating challenges associated with alignment error and reducing sensitivity to fabrication variations. Of further benefit is the extreme sensitivity of the resonator transmission function to intracavity losses enabling highly accurate estimation of the loss coefficient associated with each resonance when appropriate curve fitting methods are employed. This behavior enabled an insertion loss measurement to within 0.0070 dB for lateral width tapers in SOI waveguides with taper lengths from 2.5 to 10 μm over the 1.50 to 1.63 μm wavelength range using a reference resonator, test resonator, and propagation loss data obtained via the cutback method.

The racetrack resonator loss platform(RLP) represents a versatile measurement technique capable of providing fairly high accuracy using a single device and extremely high accuracy using three resonant devices; all of which are impervious to extracavity loss mechanisms. This technique was applied to lateral width tapers in SOI waveguides starting at 400 nm and tapering to 250 nm. Several taper functions were examined including linear, parabolic and exponential functions with a fourth taper designed to exhibit linearity in the effective index(n_eff) at a wavelength of 1.55 μm. Analysis of devices across the 1.50 to 1.63 μm wavelength range with taper lengths of 2.5, 5, and 10 μm yielded estimated confidence intervals less than 0.037 dB as-measured with appropriate calibration reducing this value to 0.007 dB overall.

The future of this measurement technique is bright with the potential to enable accurate estimation of the insertion loss of a variety of efficient, compact photonic structures. Examples posited here include tapers in the geometric profile of the waveguide, vertical transitions in multi-layer optical interconnects, optical tap couplers, and waveguide crossings. The intent of this technology is to serve as a fundamental enabler of rapid design, characterization, and optimization cycles for efficient photonic devices and it is our hope that this work aids in the development of a variety of disruptive advances in device technology.