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Abstract
Key for optical microresonator engineering, the total intrinsic loss is easily determined by spectroscopy; however, quantitatively separating absorption and radiative losses is challenging, and there is not a general and robust method. Here, we propose and experimentally demonstrate a general all-optical characterization technique for separating the loss mechanisms with high confidence using only linear spectroscopic measurements and an optically measured resonator thermal time constant. We report the absorption, radiation, and coupling losses for ten whispering-gallery modes of three different radial orders on a Si microdisk. Although the total dissipation rates show order-of-magnitude differences, the small absorptive losses are successfully distinguished from the overwhelming radiation losses and show similar values for all the modes as expected for the bulk material absorption.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical microresonators can combine an ultra-high quality factor with a small mode volume. The highly concentrated energy generates heat via the absorption loss of the resonator. The resulting strong temperature variations largely distort the optical response via changing the refractive index and the size of the resonator, inducing a strong thermo-optical effect [1]. While the thermo-optical effect can be used for resonance wavelength tuning [2–4], thermal sensing [5–7], thermal locking [8,9], and thermal imaging [10–12], the heat from the light absorbed within the cavity generates undesired thermal-instability [13,14] and nonlinear dynamics [15–17], and can strongly limit sensitivity and bandwidth of nanophotonic sensors [18]. A technique for experimentally distinguishing absorption losses from radiation losses with high confidence would help guide the design and fabrication strategies for optical microresonators, including lowering absorptive loss rates to reduce the thermo-optical nonlinearities, as well as quantifying and harnessing these nonlinearities.
The thermo-optical effect itself has been shown to be useful for the analysis of the loss mechanisms in optical microresonators, for example, by measuring the nonlinear absorption rate [19,20], estimating cavity temperature changes based on simulation [15], fitting the loss as a function of microdisk radius [21] or wavelength-dependent absorption coefficient [22], and analyzing the resonance doublet [23,24]. However, the previously proposed methods are not generally applicable since they demand either strong material optical nonlinearities, high quality factors, or multiple devices to sweep parameter space. One example is the method presented in Ref. [19] which calibrates absorption losses and scattering losses using accurately measured nonlinear absorption losses. Although requiring only limited knowledge of input power, materials and geometry information, it can only be applied to limited cases where optical modes are of ultra-high quality factors and strong nonlinear absorption is experimentally evident. Finite element method (FEM) simulations unchecked by experiment may not always accurately capture thermal transport effects such as interfacial thermal impedances, radiative and gas-phase heat transport, and variations in material thermal conductivity. The previous studies so far only focused on a single optical mode or modes with the same radial number and similar quality factors.
Here, we propose a general dissipation analysis method that separates the absorption loss from other sources by calculating the amount of optical power that is being converted into heat to account for the measured resonance shift induced by the thermo-optical effect. The robustness of the method is guaranteed by a novel procedure for characterizing the thermal time constant in the optical microresonator. The procedure is based on capturing the resonator response to optical intensity modulation across the range of frequencies covering the thermal response timescales. Conceptually, the known thermal time constant simplifies the calculation of the heat generation from light, reducing it from the hard question of thermal impedance modeling to a simpler calculation of the heat capacity of the resonator structure. Therefore, we can obtain the heat dissipation with high confidence, which makes the method more robust and generally applicable. Based on this method, we quantify losses separately for a total of ten whispering gallery modes (WGMs) of three different radial orders with drastically distinct quality factors. The small absorption is found to be nearly identical across devices with order-of-magnitude differences in the total dissipation rate, consistent with possible bulk material absorption. Radiation is found to be the dominant source of loss, and comparing radiation losses across the ten measured modes shows that first- and second-radial-order modes radiate energy mainly via surface scattering, while the third-radial-order one additionally exhibits significant leakage. Using only the response to a single optical stimulus, the simple techniques could be applied to quantitatively separate absorptive and non-absorptive losses in a variety of photonic cavities subject to the ubiquitous self-heating and thermo-optical tuning. Quantitative characterization of both the thermal time constants and the self-heating are important for broadband sensing [25], optical thermometry [26,27] and in other photonic cavity applications using the thermal effect [2–12].
2. Measurement of whispering gallery modes
The optical microresonator under investigation is a silicon microdisk which supports WGMs. As shown in Fig. 1, the microdisk is a part of a photonic AFM probe that has been demonstrated in various measurements, presenting outstanding sensitivity and bandwidth [25,28]. The microdisk, supported by a silica post underneath at the center, is of 10 $\mathrm{\mu}$m diameter and 260 nm thickness. A mechanical doubly clamped cantilever probe of 150 nm width surrounds the microdisk with a 200 nm gap. The cantilever is sufficiently narrow and far enough from the disk, so that it causes only a small perturbation to the optical modes. Conducting these experiments on an optomechanical probe instead of a bare microdisk cavity not only demonstrates the use of the technique with a practical optical-microresonator-based sensor, but also enables an additional, independent experimental verification for the thermal time constant. The mechanical cantilever is used to apply frequency modulation to the disk via the optomechanical coupling, providing an alternative way to extract the thermal time constant, to validate the one obtained via the more universally applicable electro-optic intensity modulation approach developed in this work.
Fig. 1. Schematic of the measurement setup. A 10 $\mathrm{\mu}$m silicon microdisk serves as the optical microresonator under research. The injected laser light passes path 1 or 2 (labeled by the dashed frames) in the experiment of the thermal time constant and dissipation analysis, respectively. The laser light is coupled to the microdisk via an on-chip waveguide to excite the WGMs. The transmitted light is collected by a photodetector. The blue and red microdisk overlay schematically illustrated a WGM.
A continuous-wavelength tunable laser working from 1510 nm to 1570 nm is used as the light source for measuring the WGMs. The laser light coupled into a single-mode fiber passes through an optical isolator (OI), a variable attenuator (VA), a polarization controller (PC1), and is connected via one of the two paths shown in Fig. 1. For characterizing the thermal time constant, the light is connected to an electro-optic modulator (EOM) and another polarization controller (PC2), eventually coupled to the microdisk via an on-chip waveguide. For swept-wavelength WGM spectra measurement, the light is directly coupled to the microdisk via path 2. The transmitted light is collected by a photodetector (PD) and recorded by an oscilloscope (OSC) and a lock-in amplifier. The device is measured in air at room temperature.
Figures 2(a) and 2(b) show the WGM resonance dips in the transmission spectrum of the device measured with low and high input powers, respectively. The light passes path 2, and directly couples to the microdisk. While the disk supports both transverse electric (TE) and magnetic (TM) WGMs, the input polarization was adjusted and the integrated waveguide separation was optimized for coupling to TMn,m modes of lowest radial orders n = 1, 2 and 3 [Fig. 2(a), inset], identified by their free spectral range (spacing between modes) which are ${\approx} $ 14.7 nm, 15.3 nm, and 16.4 nm in the detected frequency range. Note, in the vertical direction of the thin microdisk, we can only measure the lowest order modes (only one node in the vertical direction) as shown in the inset of Fig. 2(a). The modes exhibit distinct quality factors, ranging from ≈ 8.7${\times} {10^3}$ to ≈ 111.1${\times} {10^3}$, as discussed later. A single transmission dip is shown in detail in Fig. 2(c). With increasing light intensity, the optical response exhibits nonlinear behavior, shown in Figs. 2(b), 2(d). Changing the direction of the laser wavelength scan makes the optical modes show hysteresis, shown in Fig. 2(d). No mechanical motion excited by the optical forces is observed and the data in Fig. 2 do not depend on the presence of the mechanical probe in any way. Mechanical excitation by the optical forces is negligible because of the low quality factor of the mechanical resonance due to the ambient air damping [18].
Fig. 2. Wavelength scan of the TM modes. Linear and thermo-optically nonlinear transmission spectra are shown in (a) and (b). The colored numbers label the mode families of distinct radial orders. Their radial-cross-sectional mode-shapes, i.e. the distribution of electric field intensity, are shown in the inset. One optical resonance is shown in detail in (c) and (d), for low and high input power, respectively. The hysteresis in (d) is obtained by changing the direction of the laser wavelength scan, as labeled by the black arrows. The solid (dashed) line labels the stable (unstable) state of the hysteresis. It is fitted based on that the wavelength shift is proportional to the energy absorption in the microdisk, as we will discuss later.
The nonlinearity in the transmission dip for such a small micrometer-scale cavity is due to the thermo-optical effect [1]. The temperature variation $\mathrm{\Delta }T$ in the cavity as a function of transmitted light intensity (${\propto} $ input optical power ${P_{in}}$) tunes the resonance wavelength ${\lambda _0}$ by $\Delta \lambda $ by changing the refractive index ${n_0}$ and the size of resonator :
3. Total input optical power to the microdisk
We first need to quantify the total input power to the microdisk. To calculate the input optical power, we characterize the coupling loss between the optical fiber and the on-chip waveguide for both the input and the output. The input and output efficiencies are characterized experimentally by measuring the input and corresponding output powers for light traveling in both directions through the waveguide (i.e., from “terminal 1” to “terminal 2” and vice-versa), as illustrated in Fig. 3(a). The input and output power coupling ratios from the fiber to the on-chip waveguide at the two terminals are ${C_1}$ and ${C_2}$, respectively, which also include the on-chip waveguide losses, if any, to the location of the microdisk. The transmission ratio of the waveguide at the microdisk junction is C. At wavelengths away from cavity resonance, the light does not enter the microdisk, resulting in unity transmission $C = 1$, while at the optical resonance C depends on the wavelength.
Fig. 3. Characterization of the input/output loss of the waveguide. (a) Schematic of the microdisk-waveguide system. (b) Measured transmission signal from each end of the waveguide. The different wavelength shift is due to different coupling losses at each end.
Accordingly, at wavelengths far from the resonance we have
where $P_{in}^i$ and $P_{out}^i$ represent the input and output power from “terminal i”, labeled in Fig. 3(a).At the resonance wavelength with near-optimal waveguide-disk coupling [30], the transmission intensity is nearly zero and $C \approx 0$, i.e. no light passes the junction reaching the output. Based on that, the corresponding resonance shifts $\mathrm{\Delta }\lambda $ due to thermo-optical effect are proportional to the power input to the microdisk at the waveguide-microdisk junction, $P_{in}^i{C_i}$. Therefore, we have:
The measured input/output powers, $P_{in}^1 \approx 3.583\; \textrm{mW}$, $P_{out}^2 \approx 0.014\; \textrm{mW}$, $P_{in}^2 \approx 3.586\; \textrm{mW}$, $P_{out}^1 \approx 0.013\; \textrm{mW}$ give ${C_1} \approx 0.0897$ and ${C_2} \approx 0.0425$. In subsequent experiments, terminal 1 is used as the light input, therefore, the transmission intensity at the waveguide-microdisk junction is $\frac{{{I_{bkg}}}}{{{C_2}}}$, where ${I_{bkg}}$ is the measured transmission intensity at wavelength away from the resonance. The corresponding input power at the waveguide-microdisk junction is written as:
where $\mathrm{\rho }$ is the gain of the photodetector which is measured to be ${\approx} 1.44 \times {10^{ - 5}}\;\textrm{W}/\textrm{V}$.4. Thermal time constant of the optical microresonator
The origin of the thermo-optical effect is the temperature increase due to the optical energy absorption by the microdisk. As shown in Eq. (2), we can characterize the temperature change of the microdisk by measuring the resonance wavelength tuning of the WGMs. In order to further convert the temperature change into the generated heat from the absorption loss, we need an accurate thermal impedance modeling of the system, including the microdisk, silica post, substrate and surroundings. However, modeling the thermal impedance of an optical microresonator is not easy, especially for those of complex geometry and anchors. Errors in thermal impedance modeling will introduce uncertainty in the later dissipation analysis. On the other hand, a known thermal time constant can directly link the harder-to-model thermal impedance to the resonator’s heat capacity, which is well defined by the disk size and the known specific heat capacity of cavity materials, such as Si. Therefore, a direct experimental measurement of the thermal time constant is necessary to obtain the thermal impedance with high confidence. The thermal impedance can then be used to convert the thermo-optically observed temperature change to the absolute absorbed optical power, allowing us to quantify the optical absorption.
To measure the thermal time constant of optical microresonators, the conventional pump-probe method [31,32] requires two lasers separately tuned on different optical resonances and therefore a more complicated experimental setup. Here we propose and demonstrate a novel characterization method that largely simplifies the setup and shows improvement in the characterization accuracy by using large-amplitude optical modulation. The method only needs one laser beam, and therefore it does not require modification of the optical measurement setup except for simply adding an electro-optical modulator for amplitude-modulation as shown in Fig. 1. The simplifications of setup and ability to stably apply strong optical modulation are beneficial to the characterization accuracy, evident by comparing our fitting results shown in Fig. 4 and the one shown in Ref. [32].
Fig. 4. Measured amplitude of modulated TM signal. Yellow, blue, and orange lines are amplitude R, in-phase component X, and out-of-phase component Y. The black lines are the fit of Eq. (7)’s amplitude, real and imaginary parts.
We switch the connection to the path 1. The light intensity is modulated by an EOM. The light of high intensity couples to the microdisk and excites the thermo-optically nonlinear WGMs. The output light coupled to the thermal dynamics in the microdisk is demodulated by a lock-in amplifier at the modulation frequency.
Under perturbative modulation of intensity, the amplitude of modulated transmission light is proportional to both the modulation magnitude and temperature variation due to the thermo-optical effect. By considering the thermodynamics in the optical microresonator, we obtain the transmission light coupled to the disk (TM components) as [See Supplement 1]:
where $\tau $ is the thermal time constant, $\omega $ is the modulation frequency, ${c_1}$ and ${c_2}$ are two positive numbers.Figure 4 shows the measured TM after calibration of the propagation delay in the coaxial cable and removing the TE component arising from the birefringent effect of the EOM [See Supplement 1]. When $\omega \gg 1/\tau $, the thermodynamics cannot follow the modulation speed and the thermal effect is negligible, $TM = {c_1}$. When $\omega \ll 1/\tau $, the temperature of the disk adiabatically follows the intensity variation, adding another term ${c_2}$ accounting for the temperature-induced intensity changes, $TM = {c_1} + {c_2}$. We obtain ${c_1} \approx 5.5$ mV from the real part of TM at $\omega \gg 1/\tau $ and ${c_1} + {c_2} \approx 22.8$ mV at $\omega \ll 1/\tau $. Only one adjustable parameter $\tau = ({6.8 \pm 0.3} )$ µs is used to fit the real part of TM based on Eq. (7). The uncertainty is the statistical uncertainty of the fit parameter. All uncertainties reported are one standard deviation unless noted otherwise. The corresponding in-phase, out-of-phase, and amplitude of TM with the fitted parameters are shown as the black lines in Fig. 4, showing excellent agreement.
The thermal time constant can also be obtained by using frequency (phase) modulation, rather than amplitude modulation. However, the depth of frequency modulation using an EOM phase modulator decreases linearly with the modulation rate, making it difficult to obtain pure frequency modulation without being overwhelmed by the residual intensity modulation at low frequencies. Therefore, the intensity modulation is easier to use for the characterization of the thermal time constant of optical microresonators. Besides the proposed all-optical method, the mechanical cantilever evanescently coupled to the microdisk provides an alternative way to measure and corroborate the thermal time constant on this specific device. We drive the cantilever electrostatically to generate frequency modulation on the WGMs optomechanically. By changing the modulation frequency and measuring the amplitude of the modulated transmission signal [18], we obtain $\tau \approx 7.0$ µs, consistent with the one obtained above, verifying the all-optical method presented here.
Deviations between the FEM thermal model and the actual device behavior are particularly likely to arise in three situations: inaccurate knowledge or representation of the actual geometry, materials parameters, and boundary conditions at external boundaries or internal boundaries with sizeable boundary thermal impedances. Inaccurate geometry may arise with fabrication variations which are difficult to verify after fabricating, or when geometry is complex and simplifications are used in the model. For example, in the present case, the truncated-cone shaped oxide post, created by timed hydrofluoric acid undercut, is hidden under a microdisk and requires careful scanning electron microscopy measurement. The post largely defines the effective thermal impedance and up to 80% error can arise from an inaccurate representation of the sidewall slope. Furthermore, while the material parameters of thermal oxide and silicon are accurately known, densities, stiffnesses, and thermal properties of silicon nitride can vary significantly. This also applies to other deposited inorganic materials and organic polymers whose material properties depend significantly on the method of deposition and deposition parameters. Finally, thermal modeling of solid-air interface and heat conduction through air is challenging. For example, even for a very simple waveguide geometry [33] authors chose to consider the upper and lower bounds via different boundary conditions, producing a 50% difference, rather than accurately modeling boundary heat flow. Similar difficulties also arise for internal boundary modeling between very dissimilar materials. These situations occur for many optical microresonators. The described experimentally straightforward optical measurement of the time constant serves as a check for these uncertain factors, necessary for accurate quantitative analysis based on thermal modeling.
5. Dissipation analysis
In the following, we study the thermo-optical nonlinearity of WGMs with different radial numbers and quantitatively separate their absorption loss from other loss mechanisms. Figures 5(a)–5(c) present the normalized transmission signal for modes of radial orders of 1, 2, and 3 (family 1, 2, and 3) where the normalized linear and nonlinear responses are colored in blue and purple, respectively. The input power ${P_{in}}$ at the microdisk-waveguide junction is measured to be (1.$6\; \pm \; 0.1)$ µW and (163.7 $\pm \; 2.7)$ µW for the linear and nonlinear cases, based on Eq. (6). It is clear that the total dissipation rates for different families are highly distinct, evident from their quality factors. The narrow family-1 resonances show as partially resolved doublets arising from the weak coupling and hybridization between the clockwise (CW) and counterclockwise (CCW) propagating WGMs. Here, we define ${\gamma _c}$ as the rate of dissipation induced by the coupling loss from the microdisk into the waveguide, ${\gamma _a}$ is the absorption loss converting optical power into heat and ${\gamma _r}$ is the radiative dissipation rate of the disk. We write the total dissipation rate ${\gamma _t}$ of the microdisk cavity as the sum of these terms:
Using the conventional waveguide-cavity-coupling model [30,34], we write the normalized linear transmission intensity as [19]:
Fig. 5. Thermo-optical nonlinearity for WGMs of distinct radial numbers. Normalized linear (blue) and nonlinear (purple) transmission signals of the modes with the radial orders of 1, 2, 3 are shown in (a), (b) and (c). The black lines are the corresponding fits from Eqs. (9) and (10), respectively. The input power ${P_{in}}$ before the microdisk is measured to be around 1.6 µW and 163.7 µW for the linear and nonlinear cases. (d) Distinguished loss rates of different modes. Yellow, purple, blue and green bars (arranged left to right for each mode) show the total, radiative, coupling, and absorptive loss rates, respectively. The numbers on the top label the corresponding radial family. The modes are grouped by their radial order as in Table 1. The error bars are one standard deviation statistical uncertainties propagated from fit parameters.
Table 1. Separated loss rates for different modes of radial orders 1, 2, and 3. Uncertainties in the last significant digit are shown in parenthesis, which are one standard deviation statistical uncertainties propagated from fit parameters.
To distinguish the absorption loss from radiation loss in the microdisk, we consider the high-power spectrum of the mode. Different from the method shown in Ref. [19], which determines the absorption loss by analyzing strong nonlinear absorption, here, we determine the absorption loss by calculating the amount of power that is being converted into heat to account for the measured resonance shift. The proposed method is generally applicable to photonic resonators subject to thermo-optical nonlinearity, does not rely on material optical nonlinearities, and is capable of analyzing the dissipation mechanisms of optical modes of different radial orders with very different loss rates.
Based on Eq. (2) $\frac{{\Delta \lambda }}{{{\lambda _0}}} \approx \mathrm{\alpha} \Delta T = \mathrm{\alpha} \kappa {Q_a}$, we rewrite the relative resonance frequency shift as:
Expressing power through the normalized measured transmission, $({{P_{in}} - {P_{out}}} )= \frac{{\mathrm{\rho }\;}}{{{C_2}}}({{I_{bkg}} - {I_{out}}} )= \frac{{\mathrm{\rho }\;}}{{{C_2}}}\eta ({1 - I} )$ with a normalization coefficient $\eta $ and using Eqs. (10) and (9), we fit the normalized nonlinear transmission signal shown as the black lines in Fig. 5. The fitting procedure is summarized in three steps. First, we fit the linear resonance under low input power using Eq. (9), shown as the blue colored peaks and their fit in Fig. 5. Next, we shift the wavelength of the linear resonance as a function of the absorption energy in the cavity $\sigma ({{P_{in}} - {P_{out}}} )$ as shown in Eq. (10). Last, by sweeping the value of $\sigma $ we fit the thermo-optically nonlinear resonance based on the least square fit. There is only one fitting parameter $\sigma = \frac{{{\gamma _a}}}{{{\gamma _a} + {\gamma _r}}}$. From $\sigma $ and the known ${\gamma _t}$ and ${\gamma _c}$, we obtain the ${\gamma _a}$ and ${\gamma _r}$. They are shown in Table 1 and the green and purple bars in Fig. 5(d).
The absorption losses are found to be similar across all the investigated highly-confined modes, favoring the bulk absorption of silicon as the root cause. The uniform absorptive loss rate makes the optical modes of a low quality factor show less thermo-optical nonlinearity for the same input power since the absorption loss is a relatively smaller portion of the total loss. It means that low-quality-factor modes possess a larger power dynamic range before entering nonlinearity where thermal-instability and nonlinear dynamics show up.
As expected, the waveguide coupling rates are found to increase systematically with higher radial and lower azimuthal numbers, as the WGM evanescent tail increases and more strongly overlaps with the waveguide mode. While they also depend on the effective index of the waveguide, the leading trend is the coupling increase with the lowering of the mode effective index inside the microdisk and the corresponding lengthening of the evanescent tail.
The radiation losses are the dominant losses for all modes and are larger for the higher radial order families. As shown in Fig. 6, for first and second radial order families, the radiation loss does not significantly depend on the azimuthal number, showing the loss is mainly from scattering loss, presumably due to surface roughness and point defects. The surface scattering notably increases with the radial number in agreement with theoretical predictions [36]. Additionally, for the modes of the third radial order, the radiation loss increases with decreasing azimuthal number and is linearly correlated with the waveguide coupling loss, suggesting that the radiation leakage plays a significant role in addition to scattering due to imperfections. As the radiation losses exponentially increases with decreasing the disk size [37], we expect the radiation loss remains as the dominant loss mechanism for smaller disks.
Fig. 6. Radiative loss rate plotted vs. waveguide coupling loss rate. The blue diamonds, orange circles and green squares correspond to radial orders 1, 2, and 3, respectively. The black lines are drawn to guide the eye. The error bars are smaller than the size of marks for families 1 and 2.
As a summary, the uniform absorption losses observed across the mode families are consistent with the optical power remaining well confined inside the high index Si of the microdisk, even for the largest radial order modes measured here. On the other hand, for the higher radial order families the reduced mismatch of the mode wavelength to free space, with the resulting evanescent tail increase, leads to higher radiation losses and larger waveguide coupling.
Besides the agreement with theoretical predictions, the measurement results are also consistent with the reported values on similar systems, including the thermal time constant [31,32] and intrinsic absorption loss rates [19,38].
6. Conclusions
We proposed and demonstrated a general all-optical method for measuring microresonator absorption and scattering losses using a single interrogation laser. By quantitatively characterizing the thermal time constant of the system, we obtain a better understanding of thermo-optical dynamics in the system, which enables us to perform a systematic study and disambiguation of the absorption and radiation optical loss mechanisms in the optical cavity modes, including their dependence on mode frequency and radial structure. Using this method, we analyzed optical modes of largely distinct properties, such as an order of magnitude difference in quality factors. For the microdisk resonators studied here, the loss is dominated by scattering due to surface roughness, with additional radiative leakage loss for the high-radial-order mode family. The uniform absorption losses, independent of mode radial number, favor bulk over surface-localized absorption mechanism. As the thermo-optical nonlinearity is ubiquitous in nanophotonic systems [1], this general and robust method provides a systematic solution for dissipation analysis on general optical resonators. Besides, the novel method to characterize the thermal time constant and self-heating can benefit optical thermometry, photonic sensing and other photonic cavity applications.
Funding
Center for Nanoscale Science and Technology (70NANB14H209); University of Maryland (70NANB14H209).
Acknowledgments
We thank Dr. Jeffrey Schwartz, Dr. Biswarup Guha, Dr. J. Alexander Liddle and Dr. Marcelo Davanco for reviewing this paper and giving meaningful suggestions. M.W. and D.J.P. are supported by the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Center for Nanoscale Science and Technology, Award 70NANB14H209, through the University of Maryland.
Disclosures
The authors declare no conflicts of interest.
Supplemental document
See Supplement 1 for supporting content.
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