Empirical model for the temperature dependence of silicon refractive index from O to C band based on waveguide measurements

Publisher’s version / Version de l'éditeur:

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.

Questions? Contact the NRC Publications Archive team at

PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

Optics Express, 27, 19, pp. 27229-27242, 2019

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE.

https://nrc-publications.canada.ca/eng/copyright

NRC Publications Archive Record / Notice des Archives des publications du CNRC :

https://nrc-publications.canada.ca/eng/view/object/?id=37356f5b-4d9a-4dd3-ab2b-bfb41577e2ce

https://publications-cnrc.canada.ca/fra/voir/objet/?id=37356f5b-4d9a-4dd3-ab2b-bfb41577e2ce

For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.

https://doi.org/10.1364/OE.27.027229

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

Empirical model for the temperature dependence of silicon refractive

index from O to C band based on waveguide measurements

Xu, Dan-Xia; Delâge, André; Verly, Pierre; Janz, Siegfried; Wang, Shurui;

Vachon, Martin; Ma, Penghui; Lapointe, Jean; Melati, Daniele; Cheben,

Pavel; Schmid, Jens H.

Empirical model for the temperature

dependence of silicon refractive index from O to

C band based on waveguide measurements

D

-X

X

,

A

D

, P

V

, S

J

,

S

W

, M

V

, P

H

M

, J

L

,

D

M

,

P

C

,

J

H. S

Advanced Electronics and Photonics Research Center, National Research Council Canada, Ottawa, Ontario K1A 0R6, Canada

*_{Danxia.Xu@nrc-cnrc.gc.ca}

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 η_Si in the range of 1.9 × 10−4_{/K [6], compared to silicon dioxide which has a thermo-optic}

coefficient ηoxidein 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.

#369061 https://doi.org/10.1364/OE.27.027229

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 ∆λ/∆T 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 ηSiis usually taken as a constant for the wavelength range

of interest [16,17,19]. Although it has been known that ηSivaries 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 ∆λ/∆T 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 ηoxidemake 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 ∆λ/∆T 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 ηSibased 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 ∆λ/∆T 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 section3. In section4we 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:

mλ = neffL, (1)

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]:

mλk= neffΛ(sinθ + sinϕk). (2a) Here θ and ϕκare respectively the incoming and outgoing angle of the light incident on the grating

with respect to its normal, λkthe free-space wavelength of the beam diffracted to the kthchannel at angle ϕκ(k = 0, 1, . . . N), Λ the grating pitch, neffthe 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 σκwhich is the wavelength in the slab for the kthchannel:

σk= Λ(sinθ + sinϕ_m k) =_n λk eff(λk)

. (2b)

Once a grating device is designed and fabricated, its geometrical properties are fixed such that θ and ϕκ are defined directions that intercept the I/O waveguides, and therefore σκ is a constant

for each given output channel independent of the temperature. Any variation in neff due to temperature and other changes is reflected as a shift in the free-space wavelength λkreceived in

the kth_{output channel.}

Fig. 1. (a) Schematic of an echelle grating filter; (b) Optical image of a fabricated device.

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 2ndorder

polynomial:

neff(λk, T) = nref(T) + (λk− λref)α(T) + (λk− λref)2β(T). (3) Here λrefis a reference wavelength chosen for the fit (λref= 1.30 µm for O band and λref= 1.55 µm for C band), nref(T) the corresponding effective index, α and β the fitting coefficients for temperature T. Following Eq. (2b) and (3), the output wavelength at temperature T for the kth

channel can be represented as:

λ_{k= λref} − α 2β − 1 2βσk  − v t  α 2β− 1 2βσk 2 −nref− λref σk β . (4)

As the temperature changes, the grating pitch Λ also changes slightly due to thermal expansion. However, this effect can be neglected for silicon waveguides since the thermal expansion coefficient is nearly two orders of magnitude smaller than the thermo-optic coefficient [5]. The

temperature-induced wavelength shift can be approximated as follows: ∆λk ∆T ≈ λk(T) ng(T) " ∂neff ∂T + ∂2neff ∂T2 ∆T + (λk− λref) ∂2neff ∂T ∂λ # . (5)

When the second order derivatives are neglected, the above becomes the familiar expression that is widely employed in the literature:

dλk dT ≈ λk(T) ng(λk,T) ∂neff(λk, T) ∂T . (6)

Here ng is the slab waveguide group index for the fundamental mode. Since neff and ng are solely determined by the property of the slab section where the phase delay takes place, but independent of other device parameters such as the diffraction order and delay length, it is evident from Eq. (6) that ∆λ/∆T is also independent of the particulars of these device parameters. As will be shown in section3, this prediction is well supported by the experimental data. However, Eq. (6) is inadequate in accurately predicting the shifts ∆λ with temperature. Detailed reasons are explained in section4.1. It is necessary to use Eq. (4) to accurately describe the device temperature dependence, as will be shown in section4.3.

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 Table1. 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.

Sample R (µm) m FSR (nm) Sample R (µm) m FSR (nm)

R250A(O) 250 25 40.73 R250A(C) 250 25 47.76

R250B(O) 250 46 22.13 R250B(C) 250 37 32.27

R400A(O) 400 25 40.73 R400A(C) 400 25 47.76

R400B(O) 400 46 22.13 R400B(C) 400 37 32.27

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 SiO2surface 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 λkvalues 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, ∆λk/∆T 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 λkon temperature.

The same procedure was applied to all the channels for all the devices and the data for the linear fits are shown in Fig.4for 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 section4.3. Clearly, ∆λk/∆T increases with wavelength but

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.

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 section2.

Fig. 4. Wavelength shift ∆λk/∆T extracted assuming a linear dependence on temperature

for all the devices as listed in Table1. The channel wavelengths at 25°C are used for the x-axis. Symbols: experiments; solid lines: modeled values using Eq. (6) assuming η_Si= 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 ηSito 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 λkvaries with

T, calculating neff(λk,T) and ng(λk,T) requires a-priori knowledge of λk(T) which is obviously lacking. It is common practice to compute the required parameters using a fixed λk(generally the value for room temperature) to estimate ∆λk/∆T. 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 η} oxide

has a negligible impact on the overall ∆λk/∆T predictions (below 0.2%) due to the high modal

confinement in these Si planar waveguides, and the fact that ηoxideis one order of magnitude

smaller than ηSi. We therefore use ηoxide≈1 × 10−5/K [2] for all analysis. With these assumptions,

we compute ∆λk/∆T for a slab thickness of 220 nm, and the results are shown Fig.4together

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. Figure4(b) displays the calculated ∆λk/∆T 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, ∆H = ±10 nm gives variations in ∆λk/∆T 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 nSimeasured 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 Table2.

n2₀(λ, T0) −1 = Õ3 1 Si(T0)λ2 λ2− λ_i2(T0) . (7)

For the index of SiO2 at the reference temperature, we use a 2nd order polynomial fit

(noxide(λ) = 1.459−0.007λ−0.0017λ2) to the data reported by Palik [30].

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].

S1(T0) S2(T0) S3(T0) λ1(T0) λ2(T0) λ3(T0)

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 Table3are therefore modified from the

original values in [36] where T was the variable. ∂n

∂T(λ, T) = C1(λ) + C2(T − T0) + C3(T − T0)

2_{. (8)}

It has been shown experimentally that ηSi varies slowly with both λ and T [23], and so it is

reasonable to assume that this expression can be applied to the full wavelength range covering O to C band, provided a term C1(λ) is introduced to reflect the changes of ηSiwith wavelength.

The expression for C1(λ) is obtained by fitting the dispersion data at 22°C [31]. By so doing, we

obtain the following composite expression that gives the comprehensive silicon refractive index: nSi(λ, T) = n0(λ, T0) + C1(λ)(T − T0) + C2(T − T0)2/2 + C3(T − T0)3/3, (9a)

C1(λ) = R0+ R1λ + R2λ2. (9b)

The wavelength unit is in micron and the temperature can obviously be in either Kelvin or Celsius. The refractive index at 22°C, n0(λ,T0), is defined in Eq. (7). Table3lists the other

fitting coefficients denoted as ‘bulk’. The values for ‘echelle’ are derived based on our optical measurement results, which will be presented in section4.3. The predicted refractive indices at room temperature are listed in Table4and compared with several other published sources. Our obtained index values at 22°C agree with data reported in [31] to ∼2 × 10−3_{for all available}

wavelengths. The difference between model ‘bulk’ and ‘echelle’ is negligible at room temperature. Predicted thermo-optic coefficients of silicon at selected temperatures based on this work are also presented in Table4.

Table 3. Coefficients for the silicon refractive index model as described in Eq. 9.

R0 R1 R2 C1(1523 nm) C2 C3

Bulk 3.1275 × 10−4 _{−1.3462 × 10}−4 _{3.3819 × 10}−5 _{1.862 × 10}−4 _{2.591 × 10}−7 _{−1.490 × 10}−10

Echelle 3.0923 × 10−4 _{−1.2930 × 10}−4

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 Table3, we first examined if it would be sufficient to calculate ∆λk/∆T using Eq. (6) with a

Table 4. The modeled silicon refractive index and its thermo-optic coefficient compared with other published sources.

Wavelength (µm) Refractive index nSi Thermo-optic coefficient ηSi(×10−4)

Ghosh at 20°C [6]

B. J. Frey / This work

22°C [31] Ghosh at 20°C B. J. Frey at22°C This work at22°

1.30 3.5016 3.5062a _{1.926 1.94}a _1.949 1.55 3.4757 3.4788b _{1.807 1.854}b _1.854 Temperature (°C) 22 50 75 100 ηSi(×10−4) Using nSiBulk(λ,T) 1.949 2.020 2.082 2.142 (λ=1.30 µm) Using nSiEchelle(λ,T) 1.983 2.054 2.116 2.176 ηSi(×10−4) Using nSiBulk(λ,T) 1.854 1.925 1.987 2.047 (λ=1.55 µm) Using nSiEchelle(λ,T) 1.901 1.972 2.034 2.094 a_{Measured value as reported;}b_{extracted using a linear fit.}

∼6% over a 50°C range, in obvious contradiction with our experiments which show that ∆λk/∆T

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 α(T), β(T) and nref(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 λ}

kfor

all channels of sample R400B(O) are compared with measured values in Fig.5which 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 ∆λk/∆T, calculated using a linear or a quadratic fit. For a linear fit,

one should be mindful that the value of ∆λk/∆T depends on the extraction temperature range.

Here we present the results for the linear fit first, as shown in Fig.6with the λkat 25°C on the

x-axis. These are compared with the same experimental as well as the modeling results using a constant ηSias already shown in Figs.4(a) and4(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

∆λk/∆T values. The blue and green lines link the two sets of data to guide the eye, which shows

the predicted ∆λk/∆T 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 ∆λk/∆T (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 η_Sifor 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 section2, the temperature dependence of λkhas 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 D2is 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 R0and R1is sufficient to improve

the agreement with the experiment. The modified coefficients are listed in Table3denoted as ‘Echelle’, and the corresponding calculated results for ∆λk/∆T 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 2ndorder

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 n_Si(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 ∆λk/∆T 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 ηSidependence 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 ηSifor a particular wavelength band. Since ∆λk/∆T is sensitive to

waveguide cross-sectional dimensions for devices based on wire/ridge waveguide, thus far the most common configuration, discrepancies in ∆λk/∆T 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 ∆λk/∆T 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 ηSiunderestimate ∆λk/∆T 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 ∆λk/∆T, 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.

References

1. Y. Inoue, A. Kaneko, F. Hanawa, H. Takahashi, K. Hattori, and S. Sumida, “Athermal silica-based arrayed-waveguide grating multiplexer,”Electron. Lett.33_{(23), 1945–1947 (1997).}

2. A. Arbabi and L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of Si3N4and

SiOxusing microring resonances,”Opt. Lett.38(19), 3878–3881 (2013).

3. D. Melati, A. Waqas, A. Alippi, and A. Melloni, “Wavelength and composition dependence of the thermo-optic coefficient for InGaAsP-based integrated waveguides,”J. Appl. Phys.120_{(21), 213102 (2016).}

4. D.-X. Xu, M. Vachon, A. Densmore, R. Ma, S. Janz, A. Delâge, J. Lapointe, P. Cheben, J. H. Schmid, E. Post, S. Messaoudène, and J.-M. Fédéli, “Real-time cancellation of temperature induced resonance shifts in SOI wire waveguide ring resonator label-free biosensor arrays,”Opt. Express18_{(22), 22867–79 (2010).}

5. J.-M. Lee, “Athermal Silicon Photonics,” in Silicon Photonics III. Topics in Applied Physics, D. J. Lockwood and L. Pavesi, eds. (Springer, 2016), pp. 83–98.

6. G. Ghosh, Handbook of Optical Constants of Solids: Handbook of Thermo-Optic Coefficients of Optical Materials

with Applications(Academic, 1998).

7. G.-D. Kim, H.-S. Lee, C.-H. Park, S.-S. Lee, B. T. Lim, H. K. Bae, and W.-G. Lee, “Silicon photonic temperature sensor employing a ring resonator manufactured using a standard CMOS process,”Opt. Express18_{(21), 22215–22221}

(2010).

8. D. Thomson, A. Zilkie, J. E. Bowers, T. Komljenovic, G. T. Reed, L. Vivien, D. Marris-Morini, E. Cassan, L. Virot, J.-M. Fédéli, J.-M. Hartmann, J. H. Schmid, D.-X. Xu, F. Boeuf, P. O’Brien, G. Z. Mashanovich, and M. Nedeljkovic, “Roadmap on silicon photonics,”J. Opt.18_{(7), 073003 (2016).}

9. A. Herrero-Bermello, A. V. Velasco, H. Podmore, P. Cheben, J. H. Schmid, S. Janz, M. L. Calvo, D.-X. Xu, A. Scott, and P. Corredera, “Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers,”Opt. Lett.42_{(11), 2239–2242 (2017).}

10. D. X. Xu, A. Densmore, A. Delâge, P. Waldron, R. McKinnon, S. Janz, J. Lapointe, G. Lopinski, T. Mischki, E. Post, P. Cheben, and J. H. Schmid, “Folded cavity SOI microring sensors for high sensitivity and real time measurement of biomolecular binding,”Opt. Express16_{(19), 15137–15148 (2008).}

11. N. N. Klimov, S. Mittal, M. Berger, and Z. Ahmed, “On-chip silicon waveguide Bragg grating photonic temperature sensor,”Opt. Lett.40_{(17), 3934–3936 (2015).}

12. S. G. Leon-Saval, J. Bland-Hawthorn, and S. Ellis, “Astrophotonics: a promising arena for silicon photonics,” in

13. X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,”Nat. Photonics12_{(9), 534–539 (2018).}

14. M. C. M. M. Souza, A. Grieco, N. C. Frateschi, and Y. Fainman, “Fourier transform spectrometer on silicon with thermo-optic non-linearity and dispersion correction,”Nat. Commun.9_{(1), 665 (2018).}

15. D. Melati, P. G. Verly, A. Delâge, P. Cheben, J. H. Schmid, S. Janz, and D.-X. Xu, “Athermal echelle grating filter in silicon-on-insulator using a temperature-synchronized input,”Opt. Express26_{(22), 28651–28659 (2018).}

16. J.-M. Lee, D.-J. Kim, H. Ahn, S.-H. Park, and G. Kim, “Temperature Dependence of Silicon Nanophotonic Ring Resonator With a Polymeric Overlayer,”J. Lightwave Technol.25_{(8), 2236–2243 (2007).}

17. W. N. Ye, J. Michel, and L. C. Kimerling, “Athermal High-Index-Contrast Waveguide Design,”IEEE Photonics Technol. Lett.20_{(11), 885–887 (2008).}

18. M. Uenuma and T. Motooka, “Temperature-independent silicon waveguide optical filter,”Opt. Lett.34_{(5), 599–601}

(2009).

19. S. Dwivedi, H. D’heer, and W. Bogaerts, “Maximizing Fabrication and Thermal Tolerances of All-Silicon FIR Wavelength Filters,”IEEE Photonics Technol. Lett.27_{(8), 871–874 (2015).}

20. X. Wang, S. Xiao, W. Zheng, F. Wang, Y. Li, Y. Hao, X. Jiang, M. Wang, and J. Yang, “Athermal silicon arrayed waveguide grating with polymer-filled slot structure,”Opt. Commun.282_{(14), 2841–2844 (2009).}

21. T. Baehr-Jones, M. Hochberg, C. Walker, D. Koshinz, E. Chan, W. Krug, and A. Scherer, “Analysis of the Tuning Sensitivity of Silicon-on-Insulator Optical Ring Resonators,”J. Lightwave Technol.23_{(12), 4215–4221 (2005).}

22. S. Dwivedi, A. Ruocco, M. Vanslembrouck, T. Spuesens, P. Bienstman, P. Dumon, T. Van Vaerenbergh, and W. Bogaerts, “Experimental Extraction of Effective Refractive Index and Thermo-Optic Coefficients of Silicon-on-Insulator Waveguides Using Interferometers,”J. Lightwave Technol.33_{(21), 4471–4477 (2015).}

23. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,”J. Phys. Chem. Ref. Data9_{(3), 561–658 (1980).}

24. D.-X. Xu, J. H. Schmid, G. T. Reed, G. Z. Mashanovich, D. J. Thomson, M. Nedeljkovic, X. Chen, V. Thourhout, S. Keyvaninia, and S. K. Selvaraja, “Silicon Photonic Integration Platform-Have We Found the Sweet Spot?”IEEE J. Sel. Top. Quantum Electron.20_{(4), 189–205 (2014).}

25. P. Cheben, A. Delâge, S. Janz, and D.-X. Xu, “Echelle and Arrayed Waveguide Gratings for WDM and Spectral Analysis,” in Advances in Information Optics and Photonics (SPIE, 2008), pp. 599–632.

26. K. Okamoto, “Progress and technical challenge for planar waveguide devices: silica and silicon waveguides,”Laser Photonics Rev.6_{(1), 14–23 (2012).}

27. P. Cheben, J. H. Schmid, S. Wang, D.-X. Xu, M. Vachon, S. Janz, J. Lapointe, Y. Painchaud, and M.-J. Picard, “Broadband polarization independent nanophotonic coupler for silicon waveguides with ultra-high efficiency,”Opt. Express23_{(17), 22553–22563 (2015).}

28. Y. Cengel and A. Ghajar, Heat and Mass Transfer: Fundamentals and Applications, 5th edition (McGraw-Hill, 2015). 29. R. B. Priti, Y. Xiong, and O. Liboiron-Ladouceur, “Efficiency Improvement of an O-band SOI-MZI Thermo-optic

Matrix Switch,” in IEEE Photon. Conf., Waikoloa, HI, USA (2016), pp. 823–824. 30. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998).

31. B. J. Frey, D. B. Leviton, and T. J. Madison, “Temperature-dependent refractive index of silicon and germanium,” in

SPIE Optomechanical Technologies for Astronomy, E. Atad-Ettedgui, J. Antebi, and D. Lemke, eds. (SPIE, 2006), 6273, pp. 62732J.

32. F. Lukes, “The temperature-dependence of the refractive index of silicon,”J. Phys. Chem. Solids11_{(3–4), 342–344}

(1959).

33. J. Humlíček and J. Šik, “Optical functions of silicon from reflectance and ellipsometry on silicon-on-insulator and homoepitaxial samples,”J. Appl. Phys.118_{(19), 195706 (2015).}

34. B. Andreas, Y. Azuma, G. Bartl, P. Becker, H. Bettin, M. Borys, I. Busch, M. Gray, P. Fuchs, K. Fujii, H. Fujimoto, E. Kessler, M. Krumrey, U. Kuetgens, N. Kuramoto, G. Mana, P. Manson, E. Massa, S. Mizushima, A. Nicolaus, A. Picard, A. Pramann, O. Rienitz, D. Schiel, S. Valkiers, and A. Waseda, “Determination of the Avogadro Constant by Counting the Atoms in a Si 28 Crystal,”Phys. Rev. Lett.106_{(3), 030801 (2011).}

35. D. Franta, A. Dubroka, C. Wang, A. Giglia, J. Vohánka, P. Franta, and I. Ohlídal, “Temperature-dependent dispersion model of float zone crystalline silicon,”Appl. Surf. Sci.421_{, 405–419 (2017).}

36. G. Cocorullo, F. G. Della Corte, and I. Rendina, “Temperature dependence of the thermo-optic coefficient in crystalline silicon between room temperature and 550 K at the wavelength of 1523 nm,”Appl. Phys. Lett.74_(22),

3338–3340 (1999).

37. F. G. Della Corte, M. Esposito Montefusco, L. Moretti, I. Rendina, and G. Cocorullo, “Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models,”J. Appl. Phys.88_{(12), 7115–7119}

(2000).

38. J. A. Mccaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide,”Phys. Rev. B49_{(11), 7408–7417 (1994).}

39. “www.photond.com/products/fimmwave”.

40. Y. Xing, J. Dong, S. Dwivedi, U. Khan, and W. Bogaerts, “Accurate extraction of fabricated geometry using optical measurement,”Photonics Res.6_{(11), 1008–1020 (2018).}

41. Y. Atsumi, D.-X. Xu, A. Delâge, J. H. Schmid, M. Vachon, P. Cheben, S. Janz, N. Nishiyama, and S. Arai, “Simultaneous retrieval of fluidic refractive index and surface adsorbed molecular film thickness using silicon wire waveguide biosensors,”Opt. Express20_{(24), 26969–26977 (2012).}

42. D.-X. Xu, “Polarization Control in Silicon Photonic Waveguide Components Using Cladding Stress Engineering,” in

Silicon Photonics II. Topics in Applied Physics, Vol 119., D. J. Lockwood and L. Pavesi, eds. (Springer, 2011), pp. 31–70.

43. W. N. Ye, D.-X. Xu, S. Janz, P. Cheben, M.-J. Picard, B. Lamontagne, and N. G. Tarr, “Birefringence Control Using Stress Engineering in Silicon-on-Insulator (SOI) Waveguides,”J. Lightwave Technol.23_{(3), 1308–1318 (2005).}