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Abstract
An accurate model for the silicon refractive index including its temperature and wavelength dependence is critically important for many disciplines of science and technology. Currently, such a model for temperatures above 22°C in the optical communication bands is not available. The temperature dependence in the spectral response of integrated echelle grating filters made in silicon-on-insulator is solely determined by the optical properties of the slab waveguide, making it largely immune to dimensional uncertainties. This feature renders the echelle filters a reliable tool to evaluate the thermo-optic properties of silicon. Here we investigate the temperature dependence of silicon echelle filters for the wavelength range of both O and C bands, measured between 22°C to 80°C. We show that if a constant thermo-optic coefficient of silicon is assumed for each band, as is common in the literature, the predictions show an underestimate of up to 10% in the temperature-induced channel wavelength shift. We propose and assess a model of silicon refractive index that encompasses both the wavelength and temperature dependence of its thermo-optic coefficients. We start from literature data for bulk silicon and further refine the model using the echelle filter measurement results. This model is validated through accurate predictions of device channel wavelengths and their temperature dependence, including the quadratic term, over a wide wavelength and temperature range. This work also demonstrates a new high-precision method for characterizing the optical properties of a variety of materials.
1. Introduction
A temperature induced change in the refractive index, or the thermo-optic effect, is present in all optical materials [1–5]. This effect is particularly strong in silicon, with a thermo-optic coefficient ηSi in the range of 1.9 × 10−4/K [6], compared to silicon dioxide which has a thermo-optic coefficient ηoxide in the range of 1 × 10−5/K [1,2,7]. As such, silicon photonic components have a response strongly dependent on temperature [5]. Silicon photonics is widely considered as a key technology for next-generation communications systems and data interconnects [8]. It has also found ever expanding applications in many other fields, ranging from spectroscopy [9], biological and chemical sensing [10], metrology [11], astrophotonic instrumentation [12], to quantum computing [13]. The strong thermo-optic effect in silicon has been used for example in thermo-optic switches, modulators and sensor interrogators. On the other hand, in wavelength sensitive applications such as demultiplexers and spectrometers, the inevitable temperature fluctuations in the local environment cause undesirable performance variations. This requires either active temperature stabilization or on-chip compensation for the practical deployment of such circuits [4,9,14,15]. Regardless of the scenario, accurate predictions of the thermo-optic behavior is critical for the design and operation of silicon photonic components in all areas of science and technology. Such prediction relies on a good model of the silicon refractive index and its temperature dependence.
There is a wide body of literature on the temperature dependence of silicon photonic devices made in silicon-on-insulator (SOI) that are based on wire/ridge waveguides, such as ring resonators [4,16,17], Mach-Zehnder interferometers [14,18,19] and arrayed waveguide gratings [20]. The thermo-optic effect is primarily observed as a temperature induced wavelength shift Δλ/ΔT in their spectral response. These devices have been used to study the thermal tuning sensitivity [21] or even to extract the effective thermo-optic coefficients of SOI waveguides [22], with seemingly adequate agreement between the experiments and theoretical analysis that rely on various forms of approximations. The value of ηSi is usually taken as a constant for the wavelength range of interest [16,17,19]. Although it has been known that ηSi varies with both temperature and wavelength [6,23], these second order variations are rarely taken into account in modeling. For a high index contrast waveguide in SOI, the effective index, including its temperature dependence, is highly sensitive to dimensional fluctuations in the waveguide cross-section associated with the current fabrication technology [19,24]. Therefore small discrepancies between the predicted and measured channel or resonance wavelength shifts have usually been attributed to waveguide dimensional uncertainties. As a result, the potential inadequacy of the thermo-optic model has remained unrecognized.
In this work we investigate the thermo-optic properties of silicon using echelle grating filters (EGFs), in which the optical interference takes place in the free-propagating slab waveguide region. For EGFs operating in the transverse-electric (TE) polarization, the temperature-induced channel wavelength shift Δλ/ΔT has only a very weak dependence on the slab thickness variation which is the only source of possible dimensional uncertainties. The strong light confinement in silicon and the low ηoxide make the contributions from the cladding layers very small. These properties make such EGFs an excellent choice to characterize the thermo-optic properties of silicon. We show that the commonly employed modeling approach using a constant ηSi underestimates the magnitude of Δλ/ΔT by up to 10%. This underestimation can lead to errors in predicting channel wavelengths comparable to a 50 GHz channel spacing for optical DWDM communications over a temperature range of 40°C. To overcome these limitations, we first develop an empirical model for the refractive index of silicon nSi(λ,T) that incorporates both the temperature and wavelength dependence in ηSi based on published data for bulk silicon, measured at a fixed wavelength or a fixed temperature respectively. We demonstrate the validity of this model by comparing the modeled Δλ/ΔT for both the linear and second order temperature terms with that measured from 22 to 80 °C, on a collection of EGFs designed with different interference order and device size in the O- and C-band wavelength range. The agreement between the model and experiment is improved to within 3%. We then propose a refined model based on our optical measurement results, which gives excellent agreement for the full wavelength and temperature ranges investigated.
The paper is organized as follows. The theory and modeling approaches are discussed in section 2. We then describe the EGF design, fabrication and the temperature dependence measurements in section 3. In section 4 we describe the procedure of establishing the refractive index model nSi(λ,T), compare the measured data with two different modeling approaches, and demonstrate the significant improvement by using the nSi(λ,T) model that we have developed.
2. Theory and modeling approaches
In interferometric wavelength filters, the center wavelength in free-space λ obeys the following equation:
where L is the physical delay length (such as the circumference of a ring resonator or the difference between the adjacent waveguide arms of an arrayed waveguide grating), m is the integer interference order, and neff the effective index of the waveguide where the phase delay takes place. In an EGF, Eq. (1) takes on the following form [25]: Here θ and φκ are respectively the incoming and outgoing angle of the light incident on the grating with respect to its normal, ${\lambda _k}$ the free-space wavelength of the beam diffracted to the kth channel at angle φκ (k = 0, 1, … N), ${\Lambda }$ the grating pitch, neff the effective index of the fundamental mode of the slab waveguide as shown in the schematic of Fig. 1(a), and N the total number of channels. Here we define a parameter σκ which is the wavelength in the slab for the kth channel:The slab effective index neff(λ,T) varies with wavelength due to the material dispersions of the core and cladding materials as well as the geometric confinement. It also varies with temperature due to the thermo-optic effects in the waveguide core and cladding materials. Within a limited wavelength range, the wavelength dispersion of neff(λ,T) can be well represented by a 2nd order polynomial:
3. Device design, fabrication and optical measurements
We designed and fabricated echelle grating filters in both the O and C-bands, with 4 devices for each band, all having 10 output channels with 400 GHz spacing. The channel positions were chosen to cover the laser wavelength range available for measurements. The main design parameters are summarized in Table 1. A narrow channel passband is desirable for high precision determination of the center wavelength. In theory, the most effective design approach is to increase the Rowland circle radius. In practice, however, the slab thickness fluctuations in this high index waveguide platform may deteriorate the device performance in large devices [26]. The parameters chosen represent this compromise. Other design parameters, such as the input and output waveguide widths and grating mounting angles, have been optimized to reduce the channel passband. The device layouts were generated using an in-house software based on Huygens-Kirchhoff diffraction theory in 2D. The devices were fabricated on silicon-on-insulator (SOI) substrates with 220 nm thick silicon device layer and 2 µm thick buried oxide. The patterns were defined using electron beam lithography, and then transferred by inductively coupled plasma dry etching. Edge couplers taking advantage of subwavelength structuring are used to assist light coupling from lensed fibers [27]. The chips are covered with 2 µm thick silicon dioxide deposited by plasma-enhanced-chemical-vapor-deposition. The optical image of a device is shown in Fig. 1(b).
Table 1. Design parameters for the echelle grating filters used in the measurements. R: Rowland circle radius; m: diffraction order; FSR: free spectral range.
The optical testing was performed using two tunable lasers, one covering the wavelength range of 1260 nm to 1350 nm, while the other from 1470 nm to 1580 nm. The scan step size was 1 pm. The waveguide input facet was coupled to a polarization maintaining lensed fiber with its axis aligned to the TE polarization. For each output channel, the light was collected and focused onto a photodetector using a 20× microscope objective lens. An example of the filter output spectra is shown in Fig. 2(a) (sample R250A(O)), including the output from the designed (m = 25) and the adjacent (m = 24) diffraction order. A copper block was used as the sample stage, and its temperature was adjusted using a Peltier temperature controller. The block temperature was measured using an embedded thermistor. The sample was in direct contact with the copper block while its top surface was exposed to the lab ambient. Optical measurements were carried out at temperatures from 25 °C to 80°C in 5 °C intervals under equilibrium conditions. A one-dimensional steady state heat flow model of the SOI wafer, including thermal radiation to the room temperature ambient, indicates that the waveguide temperature is within 0.2° C of the copper block over this range [28]. In order to confirm this model prediction, calibrated thermistors were attached to the SiO2 surface of the sample and the copper block. The sample surface temperature reading was slightly lower but the difference was within 0.7°C for stage temperatures up to 60°C and increased to 1.8°C for a stage temperature of 80°C. This measurement can serve only as an indication of the maximum temperature difference between the copper block and the silicon waveguide, since the thermistor will always be cooler than the device because of the high thermal conductivity of the thermistor wires combined with the lower thermal conductivity between the thermistor and the silicon layer. Based on both the heat flow model and the measurements we are confident that the silicon waveguide temperature is within 1°C of the copper block temperature over the full temperature range.
Fig. 2. (a) Measured transmission spectra of device R250A(O). System setup loss and waveguide coupling loss are included. a) Transmission spectra at room temperature; (b) Transmission spectra as a function of temperature for one of the channels.
The spectra of a single channel at different temperatures are shown in Fig. 2(b). We take the center of the 3 dB passband as the channel center wavelengths λk. The scatter in λk is within ± 0.05 nm largely caused by small ripples in the passband. The λk values are plotted as a function of temperature for two output channels as shown in Fig. 3(a). We observe that λk increases almost linearly with T. The slope of the linear fit can be used to characterize the device temperature dependence in a small wavelength window. In this case, Δλk/ΔT is about 70.9 pm/K for λc ∼ 1300 nm and 72.2 pm/K for λc ∼ 1320 nm. The residues of the linear and quadratic fits are shown in Fig. 3(b) and 3(c), respectively. Although the residues show similar scatter, the flat distribution for the quadratic fit indicates that a second order relation better represents the dependence of λk on temperature.
Fig. 3. (a) Central wavelength as a function of temperature for two output channels of sample R250(O). The residue, i.e. the difference between the predicted and measured channel wavelength as a function of temperature, for (b) the linear fit and (c) the quadratic fit.
The same procedure was applied to all the channels for all the devices and the data for the linear fits are shown in Fig. 4 for both the O and C bands. The scatter of the data relative to their linear regression is within ± 0.7% for the O band and ± 1.5% for the C band. Analysis for the second order fits will be presented in section 4.3. Clearly, Δλk/ΔT increases with wavelength but it is independent of the particulars of the EGF design parameters such as diffraction order or Rowland circle radius. This is expected from theory as discussed in section 2.
Fig. 4. Wavelength shift Δλk/ΔT extracted assuming a linear dependence on temperature for all the devices as listed in Table 1. The channel wavelengths at 25°C are used for the x-axis. Symbols: experiments; solid lines: modeled values using Eq. (6) assuming ηSi = 1.93 × 10−4/K for the O band and ηSi = 1.85 × 10−4/K for the C band. (a) Data for the 1300 nm wavelength range. The experimental scatter is within ± 0.7%. The discrepancy between the experimental and the modeled values is ∼ 6%. (b) Data for the 1550 nm wavelength range. The experimental scatter is within ± 1.5%. The discrepancy between the experimental and the modeled values is ∼ 10%.
4. Comparison of measured data with different models
In this section, we first apply the expressions with a constant ηSi to model the filter temperature dependence, and discuss the deficiencies of this approach. We then present an empirical model for the silicon refractive index that takes into account the wavelength and temperature dependence of ηSi. Thermo-optic behavior of the devices predicted using this model is then compared with experiments, showing significantly improved agreement.
4.1 Commonly used approximation with a constant silicon thermo-optic coefficient
In the current literature, Eq. 6 is widely used to predict the temperature dependence of a filter. Several approximations are used to apply this equation in practice. First, since λk varies with T, calculating neff(λk,T) and ng(λk,T) requires a-priori knowledge of ${\lambda _k}$(T) which is obviously lacking. It is common practice to compute the required parameters using a fixed ${\lambda _k}$ (generally the value for room temperature) to estimate Δλk/ΔT. Second, only a constant value of thermo-optic coefficient is used for each of the waveguide materials. The most commonly used values are: ηSi ≈1.93 × 10−4/K for the O band [29], ηSi ≈1.85 × 10−4/K for the C band [7,17,19], and ηoxide ≈(0.8–1.5) × 10−5/K for the silicon dioxide cladding layers [1,2,7]. The precise value of ηoxide has a negligible impact on the overall Δλk/ΔT predictions (below 0.2%) due to the high modal confinement in these Si planar waveguides, and the fact that ηoxide is one order of magnitude smaller than ηSi. We therefore use ηoxide ≈1 × 10−5/K [2] for all analysis. With these assumptions, we compute Δλk/ΔT for a slab thickness of 220 nm, and the results are shown Fig. 4 together with the experimental values. We observe that the modeling results clearly underestimate the wavelength shifts, by ∼ 6% in the O band and ∼ 10% in the C band.
The first possibility to examine is the potential deviation in the assumed slab silicon thickness H, the only geometrical parameter in an EGF that has an appreciable effect on the temperature dependence of neff and ng. Figure 4(b) displays the calculated Δλk/ΔT in the C band for two additional slab thicknesses of 210 nm and 230 nm. The results for the O band are similar. As can be observed, ΔH = ±10 nm gives variations in Δλk/ΔT by only ∼ 0.5%, far smaller than the observed discrepancies. The actual device layer thickness has been measured using ellipsometry and the discrepancies from the nominal thickness of 220 nm are smaller than ± 2 nm. Alternative explanations are required.
4.2 Silicon refractive index model encompassing the λ and T dependence in ηSi
To understand these observed discrepancies, we re-examine the available silicon refractive index data (nSi) and the thermo-optic coefficients. For room temperature, there are a few commonly cited sources for nSi measured using bulk material [6,23,30]. A more recent comprehensive data set was provided by B. J. Frey et al. where the absolute refractive index of bulk silicon prisms was measured from cryogenic temperatures to 22°C across a wide wavelength range [31], and the index data was presented as a Sellmeier expression with wavelength and temperature dependent coefficients. The uncertainties for the index measurements were cited as within ± 1 × 10−4, however the authors emphasized that discrepancies on the order of ± 5 × 10−3 from different batches of raw material would not be surprising. We compared the data between the Sellmeier model reported by Frey et al. (at 22°C), Ghosh (at 20°C) [6], and the discrete data reported by Palik [30]. We found the discrepancies to be within ± 5 × 10−3, consistent with the range anticipated by Frey. The slight differences in the dispersion lead to differences of ≤ 0.5% in the slab group index ng. Therefore the particular choice of the Si index model at room temperature is of negligible consequence. In this work, we adopt the Sellmeier expression for the silicon index n0(λ,T0) at the reference temperature of T0=295 K (22°C) reported in [31], as given in Eq. 7 where the wavelength is in micron and the temperature is in Kelvin. The corresponding Sellmeier coefficients are cited in Table 2.
Table 2. Sellmeier coefficients for the refractive index of silicon at the reference temperature of T0=295 K (22 °C) as in Frey et al. [31].
We now consider the wavelength and temperature dependence of the Si thermo-optic coefficient. Although silicon is one of the most studied materials, there is only limited literature data on ηSi(λ,T) for the near infrared optical communication wavelength bands and the relevant temperature range (centered around room temperature). Li et al. proposed a Sellmeier-like model for nSi (λ,T) [23] where the temperature coefficient was based on a single set of available data by Lukes et al. published in 1959 [32]. The most comprehensive data by Frey et al. cover from cryogenic temperatures to 22°C [31]. However their Sellmeier coefficients cannot be extended to higher temperatures since the model shows a divergent pole at 46°C. Efforts to replace the international kilogram prototype using silicon spheres has sparked renewed interest in improving the knowledge of the silicon optical constants [33–35]. These efforts, however, focused on wavelength ranges with electronic transitions.
Among the available data for ηSi [6,23,30–32,36–38], we adopt the following two sets since they provide the best agreement with our experiment. For the wavelength dependence, we use the values in Frey et al. [31] measured at 22°C. For the temperature dependence, we use the values published by Cocorullo et al. [36] measured from room temperature to 550 K at a fixed wavelength of 1523 nm, given as a 2nd order polynomial. The formula is re-written in Eq. (8) with (T-T0) as the variable. The coefficients cited in Table 3 are therefore modified from the original values in [36] where T was the variable.
Table 3. Coefficients for the silicon refractive index model as described in Eq. 9.
Table 4. The modeled silicon refractive index and its thermo-optic coefficient compared with other published sources.
4.3 Thermo-optic behavior predicted using the new silicon refractive index model
Using the new Si index model nSi(λ,T) expressed in Eq. (9) and using the coefficients for ‘bulk’ in Table 3, we first examined if it would be sufficient to calculate Δλk/ΔT using Eq. (6) with a constant λk (at room temperature). However, this leads to a Δλk/ΔT that increases with T by ∼6% over a 50°C range, in obvious contradiction with our experiments which show that Δλk/ΔT is nearly independent of T with a 2nd order residue term smaller by two orders of magnitude. It becomes clear that it is necessary to find the EGF channel wavelength λk as a function of T, and then analyze the data using either a linear or second order fit. The modeling procedure is as follows. We first numerically calculate the slab effective index neff(λ) for a range of T using the new silicon index model. Since the slab is only a one dimensional system, a rigorous analytical mode solver was used [39]. Within a limited wavelength range (1260 m −1350 nm and 1470 nm - 1580 nm respectively, corresponding to the available scan range of our lasers), neff(λ) can be fitted very well to a 2nd order polynomial with residues of < 1 × 10−5. A set of effective dispersion coefficients $\alpha $(T), $\beta $(T) and ${n_{ref}}$(T) is then obtained for the expansion of Eq. (3). The wavelength for the kth channel at T can be obtained using Eq. (4). Examples of predicted λk for all channels of sample R400B(O) are compared with measured values in Fig. 5 which show very good agreement. The largest wavelength discrepancy δ occurs for channel 9 and 10 (0.25 nm at 30°C and 0.18 nm at 80°C), while for channels 1–7 the maximum δ is 0.045 nm. The averaged δ of all 10 channels is less than 0.08 nm across the temperature range.
Fig. 5. Measured (blue diamond) and predicted (red square) channel wavelength (all 10 channels for diffraction order 46) as a function of temperature for sample R400B(O).
Comparing the predicted and measured channel wavelength is the most direct validation of the proposed refractive index model. Nonetheless it is still informative to represent the device thermo-optic property using Δλk/ΔT, calculated using a linear or a quadratic fit. For a linear fit, one should be mindful that the value of Δλk/ΔT depends on the extraction temperature range. Here we present the results for the linear fit first, as shown in Fig. 6 with the λk at 25°C on the x-axis. These are compared with the same experimental as well as the modeling results using a constant ηSi as already shown in Figs. 4(a) and 4(b). The agreement between the predictions and the measurements is now < 3% for both the O and C bands, a significant improvement compared to the results using a constant ηSi. The blue and green symbols here indicate the computed Δλk/ΔT values. The blue and green lines link the two sets of data to guide the eye, which shows the predicted Δλk/ΔT increases linearly with λ over the entire investigated wavelength span, in agreement with experiments. The remaining differences are, however, distinct and larger than the experimental scatter.
Fig. 6. Comparison of the measured and calculated Δλk/ΔT (using a linear fit) as a function of wavelength for both the O and C bands using the proposed models for the silicon refractive index and a silicon thickness of 220 nm. Results using a constant ηSi for each wavelength band, as already presented in Fig. 4, are also included for comparison. The calculations were performed for the wavelength range of 1260–1370 nm and 1470–1580 nm respectively, as indicated by the blue and green symbols. The solid blue and green lines are for guiding the eye.
As already discussed in section 2, the temperature dependence of λk has a small but appreciable 2nd order term. This is particularly important when devices undergo large temperature excursions [11,14]. Using the same experimental and modeling data, we analyzed the relations as λk = D0 + D1(T − T0) + D2(T − T0)2/2. The results for D1 and D2 are shown in Fig. 7. Although the experimental data show larger scatter compared to the linear fit, it can be observed that the discrepancy to the predicted values is mainly in the linear coefficient D1, both in terms of the dispersion slope and the absolute value. The modeled second order term D2 is consistent with the measurements in the order of magnitude as well as in the weak dependence on wavelength. Inspecting Eq. (9), we observe that adjusting the coefficients R0 and R1 is sufficient to improve the agreement with the experiment. The modified coefficients are listed in Table 3 denoted as ‘Echelle’, and the corresponding calculated results for Δλk/ΔT are shown as green symbols in Fig. 7. This single model provides excellent agreement with experiments for both O and C bands (see also Fig. 6), indicating that it may be applied to the wavelength range in between, and it can account for the quadratic dependence on temperature of the EGF channel wavelengths.
Fig. 7. Comparison of measured and calculated (a) D1 and (b) D2 using a 2nd order polynomial fit to the channel wavelengths, as a function of wavelength for both the O and C bands using the new models for the silicon refractive index nSi(λ, T). The calculations were performed for the wavelength range of 1260–1370 nm and 1470–1580 nm respectively, as indicated by the blue and green symbols. The solid green and blue lines are for guiding the eye.
Optical measurements using integrated photonic devices have been shown to extract silicon waveguide dimensions with submicron accuracy [40,41], more precise than physical metrology tools. This is due to the long optical path length available in integrated photonics devices, providing higher resolution than measurement techniques that only make use the very small layer thickness, such as in ellipsometry or Fabry-Perot interferometry. The method we presented here benefits from the same advantage. For the remaining small discrepancy using the ‘bulk’ model, we can make some speculations. Sample temperature is unlikely the cause. For the experiments carried out between 25–80°C, the silicon layer temperature can only be slightly lower than the copper block temperature. Such a discrepancy would lead to even larger experimental Δλk/ΔT values. Another possibility is that the thermo-optic behavior of thin silicon films in SOI is slightly modified from that of bulk silicon, potentially caused by a residual stress in the silicon layer [42,43]. We have attempted to use variable angle ellipsometry to measure the silicon index at different temperatures in SOI samples with varied silicon thickness, which should affect the residue stress levels. However, the complexity in such a multilayer system led to limited accuracies and we were not able to arrive to a conclusion. Optical measurements on EGF devices with different silicon thickness may shed light on this question, and may be the topic of future work. Since the data reported in [36] for the ηSi dependence on temperature used in the ‘bulk’ model has an error estimate of 8%, it is likely that our modified model in fact provides more accurate predictions of the silicon index.
5. Conclusions
Silicon is one of the most important materials for science and technology. During the last 20 years, the quality of silicon wafers, particularly in the form of silicon-on-insulator, has vastly improved. On the other hand, the refractive index of silicon and its temperature and wavelength dependence, properties of critical importance, has not received updated attention. Due to a lack of comprehensive data, modeling the thermo-optic properties of silicon photonic devices has relied on using a constant ηSi for a particular wavelength band. Since Δλk/ΔT is sensitive to waveguide cross-sectional dimensions for devices based on wire/ridge waveguide, thus far the most common configuration, discrepancies in Δλk/ΔT between the predicted and measured values could easily be attributed to fabrication uncertainties. In echelle grating filters, on the other hand, the dispersive property is determined only by the slab waveguide of the free-propagating region. This feature makes Δλk/ΔT in such devices nearly immune to silicon thickness variations. For EGFs with a silicon thickness of 220 nm operating in the TE-polarization, the sensitivity is only 0.02pm/K·nm.
We experimentally characterized a collection of EGFs with 10 channels at 400 GHz spacing using different diffraction order and Rowland circle radius designed to provide variety. We found the modeled results using a constant ηSi underestimate Δλk/ΔT by up to 10%, a discrepancy that cannot be attributed to silicon thickness variations.
We examined the literature data on the temperature coefficients of bulk silicon refractive index. We selected available data sets for the wavelength [31] and temperature [36] dependence of its thermo-optic coefficients, respectively, and proposed a composite model of silicon refractive index that combines the λ and T dependence. We demonstrate that this model can predict the device channel wavelength with good agreement. For a filter in the O band, the maximum δ is only 0.045 nm for the temperature range of 22–80 °C for most channels. Analyzing the temperature dependence as Δλk/ΔT, the agreement between the predictions and measurements is now < 3% for both O and C bands, a significant improvement compared to the results using a constant ηSi. We further proposed a modified silicon index model using the optical measurement data. The predictions now agree with measurements to within the experimental scatter (< ±0.7% for O band and < ±1.5% for C band). Since current literature data has limited precision, we postulate that our modified model in fact provides more accurate predictions of the silicon refractive index for the wavelength and temperature range investigated. We believe the optical properties of many materials can be more accurately characterized using planar waveguide devices such as EGFs, particularly for materials with a high refractive index.
In conclusion, the proposed thermo-optic model for the refractive index of silicon will improve the design and implementation of silicon photonic devices, whether the temperature change is a cause for concerns or a tool for functionalities.
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