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# Fourier transform spectrometer on silicon with thermo-optic non-linearity and dispersion correction

## Abstract

Miniaturized integrated spectrometers will have unprecedented impact on applications ranging from unmanned aerial vehicles to mobile phones, and silicon photonics promises to deliver compact, cost-effective devices. Mirroring its ubiquitous free-space counterpart, a silicon photonics-based Fourier transform spectrometer (Si-FTS) can bring broadband operation and fine resolution to the chip scale. Here we present the modeling and experimental demonstration of a thermally tuned Si-FTS accounting for dispersion, thermo-optic non-linearity, and thermal expansion. We show how these effects modify the relation between the spectrum and interferogram of a light source and we develop a quantitative correction procedure through calibration with a tunable laser. We retrieve a broadband spectrum (7 THz around 193.4 THz with 0.38-THz resolution consuming 2.5 W per heater) and demonstrate the Si-FTS resilience to fabrication variations—a major advantage for large-scale manufacturing. Providing design flexibility and robustness, the Si-FTS is poised to become a fundamental building block for on-chip spectroscopy.

## Introduction

In recent years, significant efforts have been directed toward the realization of miniaturized optical spectrometers for in situ spectral analysis in numerous areas of science and technology1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17. The widespread use of optical spectroscopy from remote sensing4,7 and planetary sciences13 to medical research18 and pharmaceutical processes19 strongly relies on the large absorption and/or reflection cross-sections of many compounds in the near-infrared and mid-infrared range.

Silicon-based devices operate in such range and the substantial progress in integrated silicon photonics design and fabrication can therefore be leveraged to develop miniaturized spectrometers for mobile platforms. Typical silicon-on-insulator (SOI) waveguides can operate in the 1.1–4 μm wavelength range, limited by the silicon band edge at short wavelengths and by the oxide absorption at long wavelengths. The incorporation of additional complementary metal-oxide-superconductor (CMOS)-compatible materials to the mainstream fabrication process, including silicon nitride (SiN)4,20 for short wavelengths and germanium-on-silicon (Ge-on-Si)4,21,22 for long wavelengths, promises to significantly extend this window of operation. Moreover, the possibility of monolithic integration of spectrometers and photodetectors is a valuable advantage of photonic integration, promising high signal-to-noise ratio and increased sensitivity. Adding heterogeneously integrated light sources23,24,25, all the optical components required for a fully functional spectrometer can be realized in a single chip. Finally, the access to multi-project-wafer services through silicon photonics foundries provides a cost-effective path to developing robust high-performance devices4,26,27,28.

A large variety of integrated photonic spectrometer designs has been recently investigated. These include dispersive devices such as arrayed waveguide gratings4,5 and cavity-enhanced spectrometers6, spatial heterodyne spectrometers (SHS)7,8,9,10,11 based on arrays of interferometers, and stationary wave-integrated Fourier transform spectrometers (SWIFTS)12,13,14. In the latter, a spatial—rather than temporal—interferogram is formed by the standing wave pattern generated by the interference of two counter-propagating beams inside a waveguide. Designs based on the traditional Fourier transform spectrometer (FTS)29, in which a temporal interferogram is measured varying the optical path between the arms of an interferometer, have also been investigated using micro-electro-mechanical systems15,16 and lithium-niobate planar photonic circuits17.

Investigations of the traditional FTS design in the silicon photonics platform have been, however, surprisingly scarce30. Whereas FT-based spectrometers such as SHS are suitable for high-resolution, narrow-band applications, the traditional FTS is a promising candidate to address moderate resolution, broadband applications. Considering the requirements for a silicon photonics-based FTS (Si-FTS), some challenges can be identified. First, the optical path difference between the arms of the interferometer is achieved tuning the refractive index rather than changing the physical length, thus an index tuning mechanism capable of large index changes must be used for high spectral resolution. Fortunately, the thermo-optic effect can deliver large index changes of more than 10−2 for temperature differences around 100 K. For large temperatures, however, thermo-optic non-linearity and thermal expansion of the waveguide become important. Second, silicon waveguides are highly dispersive. As a consequence, the thermo-optic effect will also present strong dispersion. This translates to each optical frequency effectively experiencing a different change in optical path.

In this article, we demonstrate the implementation of a Si-FTS on the SOI platform with integrated microheaters. We show that the issues related to thermo-optic non-linearity, thermal expansion, and dispersion can be properly understood and incorporated in a simple manner. We derive a FT relation between the power spectral density (PSD) and the interferogram with modified optical frequency and arm delay accounting for these effects and we demonstrate a calibration procedure using a tunable laser source. Further, we demonstrate the retrieval of a 7-THz-wide light source around 193.4 THz with spectral resolution of 0.38 THz (12.7 cm−1, 3.05 nm) using a 1 mm2 device with total electric power dissipation around 2.5 W per heater. The Si-FTS shows intrinsic resilience to fabrication variations that allows scalability of its resolution and power consumption performance, enabling robust and versatile portable spectrometers.

## Results

### Experimental device

The device consists of a standard Mach-Zehnder Interferometer (MZI) integrated with metal microheaters fabricated in full compatibility with standard silicon photonics foundry processes (Fig. 1). The external light is butt-coupled into and out of the chip using inverse tapers and adiabatically transitions to the highly confined quasi-TE mode of the access strip waveguide before splitting in the two arms of the interferometer and subsequently recombining into the output waveguide through broadband y-branch couplers (Fig. 1e)31. The output light is coupled out of chip directly into a photodetector. Each arm of the MZI consists of a spiral (Fig. 1d) with total length of 30.407 mm and is covered by independently actuated nichrome microheaters. The propagation losses of the waveguides are estimated to be around 2 dB cm−1. The total device footprint is 1 mm2.

### Spectrometer modeling

In this section, we present the main results that allow to define a modified FT relation between the varying optical power at the output of the MZI, I, and the PSD of the incoming light, PSD (ν). A detailed discussion is provided in Supplementary Note 2.

The operation of the Si-FTS includes a simple data acquisition step consisting of measuring the output power as a function of the phase difference Δϕ between the two arms of the MZI. The Δϕ-dependent term is given by

$$I({\mathrm{\Delta }}\phi ) = {\int}_{ - \infty }^{ + \infty } {\kern 1pt} T(\nu ){\kern 1pt} {\mathrm {PSD}}(\nu )e^{j{\mathrm{\Delta }}\phi (\nu )}{\kern 1pt} \mathrm {d}\nu,$$
(1)

where ν is the optical frequency and T(ν) is the transfer function of the MZI—ideally 1. The phase difference is

$${\mathrm{\Delta }}\phi (\nu ) = \frac{{2\pi \nu }}{c}\left[ {n_{{\mathrm{eff}},{\mathrm{1}}}(\nu )L_1 - n_{{\mathrm{eff}},{\mathrm{2}}}(\nu )L_2} \right],$$
(2)

where c is the speed of light, neff,i and L i are the effective index and the total length of arm i.

The discussion is facilitated by first considering the response of an idealized device. In this case T(ν) = 1, the two arms are identical with length L, the effective indices are identical and dispersionless, neff,i(ν) ≡ neff, and the effective index change due to temperature change ΔT depends only on a linear thermo-optic coefficient (TOC) ∂ T n, such that Δneff = ∂ T nΔT. We use a contracted notation for partial derivatives, $$\frac{{\partial n_{{\mathrm{eff}}}}}{{\partial x}} \equiv \partial _xn$$. The time delay between the arms of the MZI is defined as $$\tau = \frac{L}{c}\partial _Tn{\mathrm{\Delta }}T$$ and the phase difference is simply

$${\mathrm{\Delta }}\phi (\nu ) = 2\pi \nu \tau .$$
(3)

The phase difference in the form 2π × frequency × delay establishes a direct FT relation between I(τ) and PSD(ν), with the conjugate variables ν and τ,

$$I(\tau ) = {\int}_{ - \infty }^{ + \infty } {\kern 1pt} {\mathrm {PSD}}(\nu )e^{j2\pi \nu \tau }{\mathrm {d}}\nu = {\cal{F}}\left[ {{\mathrm {PSD}}(\nu )} \right],$$
(4)

where $${\cal F}$$[] denotes the Fourier transform. Thus, PSD(ν) can be directly obtained from the inverse FT (IFT) of the interferogram,

$${\mathrm {PSD}}(\nu ) = {\int}_{ - \infty }^{ + \infty } {\kern 1pt} I(\tau )e^{ - j2\pi \nu \tau }{\kern 1pt} {\mathrm {d}}\tau = {\cal{F}}^{ - 1}[I(\tau )].$$
(5)

In practice, the Si-FTS with thermal tuning includes other effects that must be taken into account. First, the strong mode dispersion of silicon waveguides causes significant frequency dependence on the effective index. Second, a large temperature excursion is required to achieve large phase imbalances and the non-linearity of the thermo-optic response must be considered. The large temperature excursion also induces changes in the arm length (ΔL) due to thermal expansion. Finally, chip-scale variability20,26 and fabrication imperfections often introduce small differences between the two arms of the MZI, identical by design. Such variations may affect the arm length (δL) as well as the effective index (δn(ν)). Assuming the heater on top of arm 1 (H1) is actuated and including the deviations from the designed parameters in arm 2, the effective indices and arm lengths are

$$\begin{array}{*{20}{l}} {n_{{\mathrm{eff}},{\mathrm{1}}}(\nu ,{\mathrm{\Delta }}T)} \hfill & = \hfill & {n_{{\mathrm{eff}}}(\nu ) + \Delta n_{{\mathrm{eff}}}(\nu ,{\mathrm{\Delta }}T)}, \hfill \\ {n_{{\mathrm{eff}},{\mathrm{2}}}(\nu )} \hfill & = \hfill & {n_{{\mathrm{eff}}}(\nu ) + \delta n(\nu )}, \hfill \\ {L_1({\mathrm{\Delta }}T)} \hfill & = \hfill & {L + {\mathrm{\Delta }}L({\mathrm{\Delta }}T)}, \hfill \\ {L_2} \hfill & = \hfill & {L + \delta L} \hfill \end{array}.$$
(6)

The expressions for neff(ν), Δneff(ν, ΔT), δn(ν), and ΔLT) are presented in Supplementary Note 2 and contain high-order terms in ΔT and/or Δν, with Δν = ν − ν0.

The phase difference of the Si-FTS can be written in the form 2π × frequency × delay, similar to Eq. 3 but with modified frequency and delay terms. Substituting Eq. 6 in Eq. 2 yields Δϕ with contributions in ΔνiΔTj for i from 0 to 4 and for j from 0 to 5. The simplification of the resulting expression depends on the dispersion and thermo-optical properties of the waveguide as well as on the maximum temperature excursion and on the bandwidth of the light source. Using the properties of our 250-by-550 nm2 SOI waveguides (Supplementary Table 1) with temperature excursion of <100 K and optical bandwidth of a few tens of terahertz, we show that Δϕ can be simplified to

$${\mathrm{\Delta }}\phi \approx \varphi (\nu ) + 2\pi u{\cal T}.$$
(7)

The first term φ(ν) depends only on ν (Supplementary Eq. 19) and therefore does not contribute to the kernel of the FT. It contributes to shifting and distorting the interferogram, but it has no influence on the retrieved PSD as will be shown. Thus, it is not discussed in detail here.

The second term, $$2\pi u{\cal T}$$ is similar to Eq. 3, but with modified optical frequency and time delay

$$\begin{array}{*{20}{l}} u \hfill & = \hfill & {{\mathrm{\Delta }}\nu (1 + \xi _1) + \nu _0} \hfill \\ {\cal T} \hfill & = \hfill & {\tau + \gamma _2\tau ^2.} \hfill \end{array}$$
(8)

u represents the broadening of the original optical frequency ν around ν0 by a factor 1 + ξ1. In our case, ξ1 is dominated by ∂ν,Tn. $${\cal T}$$ represents a correction of the original optical delay τ that includes the non-linear contribution γ2τ2, with γ2 dominated by $$\partial _{T^2}n$$. τ in turn is related to ΔT by τ = η1ΔT, with η1 dominated by ∂ T n. The expressions for ξ1, γ2, and η1 are presented in Supplementary Note 2.

Substituting Eq. 7 in Eq. 1 and changing the integration variable from ν to u, the interferogram and the PSD are finally related through a FT in the modified conjugate variables u and $${\cal T}$$, denoted by $${\cal F}$$[],

$$I( {\cal T}) = \frac{1}{{1 + \xi _1}}{\cal{F}}\left[ {e^{j\varphi (u)}T(u){\kern 1pt} {\mathrm {PSD}}(u)} \right].$$
(9)

Neglecting the constant term (1 + ξ1)−1 multiplying the FT, the PSD is then retrieved from the absolute value of the IFT of the interferogram, normalized by the MZI transfer function T(u),

$${\mathrm {PSD}}(u) = \frac{{\left| {{\cal{F}}^{ - 1}{\kern 1pt} [I({\cal T})]} \right|}}{{T(u)}}.$$
(10)

Finally, the frequency axis must be transformed back to the original optical frequency ν,

$${\mathrm {PSD}}(u)\mathop{\longrightarrow}\limits^{{\nu = {\textstyle{{u - \nu _0} \over {1 + \xi _1}}} + \nu _0}}{\mathrm {PSD}}(\nu ).$$
(11)

### Spectrometer calibration with tunable laser

As in free space, the Si-FTS must be calibrated to provide good absolute frequency accuracy. In addition, parameters ξ1, γ2, and T(ν) should also be ideally determined in a calibration step such that the transformations uν, $$\tau \to {\cal T}$$, and the re-normalization of Eq. 10 can be properly performed in later use. A calibration process realized with a narrow linewidth tunable laser source allows to address all these requirements.

First, the calibration of the absolute optical frequency, ξ1 and γ2 is achieved measuring the interferogram of the laser source at different laser frequencies (at least three) in the spectral region of interest. Calibrating the absolute optical frequency reduces to determining κ τ that connects the electric power dissipated in the heater with the resulting arm delay, τ = κ τ W. The interferogram of a laser source at frequency ν is, according to Eq. 9,

$$I({\cal T}) = A{\kern 1pt} {\mathrm{cos}}{\kern 1pt} (2\pi u{\cal T} + \varphi _\nu ),$$
(12)

where A and φ ν are constant amplitude and phase. Using this relation and Eq. 8, the interferogram can be written as a function of the electric power and the original laser frequency as

$$I(W,\nu ) = A{\kern 1pt} {\mathrm{cos}}{\kern 1pt} [2\pi K(\nu ){\kern 1pt} W(1 + \gamma _{W}W) + \varphi _\nu ]$$
(13)

with

$$\begin{array}{*{20}{l}} {K(\nu )} \hfill & = \hfill & {\kappa _\tau (1 + \xi _1){\kern 1pt} {\mathrm{\Delta }}\nu + \kappa _\tau \nu _0} \hfill \\ {\gamma _{W}} \hfill & = \hfill & {\kappa _\tau \gamma _2.} \hfill \end{array}$$
(14)

K(ν) and γ W can be determined for each heater (H1 and H2) curve-fitting the experimental interferograms using a cosine with non-linear argument. Following, the linear fit of K(ν) with ν0 = 193.414 THz allows to determine κ τ for each heater and ξ1. Finally, using κ τ , we obtain γ2 = γ W /κ τ .

The calibration results using the laser interferograms are presented in Fig. 2a–e and the extracted parameters are summarized in Table 1. The interferogram for the laser frequency 187.37 THz (1600 nm) with heater H2 actuated and its curve-fit are shown in Fig. 2a–c as an example of the procedure realized for multiple frequencies and both heaters. The decrease in the mean optical power in Fig. 2a and the small features in the envelope of Fig. 2b are caused by slight misalignment and vibration of the input/output fibers as the heater power increases due to the thermal expansion of the chip. In a practical application, these features would not be present as the fiber would be permanently attached to the silicon chip. Combining the interferogram fit results for each heater, K(ν) follows a linear dependence with frequency (Fig. 2d) while γ W has a constant value (Fig. 2e), in agreement with the Eq. 14. The non-linear term γ2 obtained separately for each heater has the same value within the experimental error, (101 ± 1) × 10−3 ps−1, while ξ1 only slightly deviates for each heater, 0.22 ± 0.02 for H1 and 0.24 ± 0.01 for H2, yielding 0.23 ± 0.02.

The current vs voltage (IV) curves depicted in Fig. 2f confirm the thermo-optic origin of the measured non-linearity and allow to understand the difference between κτ,1 and κτ,2. The IV plots show a fairly linear behavior for both heaters, with resistances 15,876 kΩ for H1 and 15,162 kΩ for H2, and indicates that any non-linearity originating from the heaters is small compared to the non-linearity intrinsic to the thermo-optic effect. Moreover, the difference in electric resistance causes a difference in heater efficiency k T that explains the observed discrepancy in the measured κ τ ’s. The heater efficiency is determined such that the temperature change at the waveguide level ΔT is related to the dissipated electric power W by ΔT = k T W, thus κτ,i = η1kT,i. The agreement in γ2 and ξ1 obtained independently for each heater indicates the two arms of the MZI are fairly similar, so that η1 can be considered the same. Using η1 ≈ 1.94 × 10−2ps K−1 obtained from simulations (Supplementary Note 1), the heater efficiencies are estimated at kT,1 ≈ 18.5 K W−1 and kT,2 ≈ 20.6 K W−1.

The interferometer transfer function T(ν) is obtained from the transmission spectrum of the MZI, as depicted in Fig. 2g, h. The optical power at the output of the MZI is

$$I_{{\mathrm{out}}}(\nu ) = I_0(\nu ) + T(\nu ){\kern 1pt} {\mathrm{cos}}{\kern 1pt} [{\mathrm{\Delta }}\phi (\nu )].$$
(15)

T(ν) is the envelope of the transmission oscillations and can be obtained by adjusting the experimental trace. It is recommended that the transmission spectrum be measured at a non-null phase difference Δϕ so that T(ν) can be decoupled from the frequency dependence of the average power I0(ν). In our case, the experimental transmission (black trace) represents the transmission spectrum of the passive device when none of the heaters are actuated. The oscillations have a free-spectral range of 3.96 THz around 193.414 THz (32.7 nm around 1550 nm) originated from slight differences between the two arms incorporated as δL and δn(ν) as discussed in the previous section. The non-flat transmission of Fig. 2h is attributed to the non-ideal response of the power splitter/combiner for frequencies higher than 194 THz. In practice, the limited bandwidth of these components will dictate the bandwidth of the Si-FTS. For the specific y-junction design used here31, operation over >25 THz (200 nm), excess loss lower than 0.3 dB and low reflection is expected.

The Si-FTS is validated by recovering the spectrum of the amplified spontaneous emission (ASE) of a C-band erbium-doped fiber amplifier (EDFA). The ASE provides a good test spectrum in the telecom band, suitable for testing with the available equipment in our lab. Also, the broad features of the ASE spectrum are suitable for this demonstration given the limited resolution achieved here (0.38 THz). The reference ASE spectrum, measured with a tabletop optical spectrum analyzer, is shown in Fig. 3a, and its theoretical ideal interferogram for a dispersionless, perfectly balanced Si-FTS is depicted in Fig. 3b. The ASE presents two resolved peaks at 192.6 and 195.9 THz in addition to an unresolved peak at 194 THz, with total bandwidth around 7 THz (56 nm, 233.5 cm−1).

The experimental interferogram is presented in Fig. 3c. The delay axis corresponds to the transformed delay $${\cal T}$$ obtained using the parameters from Table 1. Positive delay corresponds to heater H1, while negative delay corresponds to H2. It spans $${\mathrm{\Delta }}{\cal T} = 2.13$$ ps, corresponding to maximum dissipated powers around 2.6 and 2.5 W in heaters H1 and H2 and maximum temperature excursions around 54 and 46 K at the waveguide level.

A comparison between experimental and ideal interferograms shows some noticeable distinctions. The zero delay (maximum envelope amplitude) is shifted to around 0.172 ps and the interferogram envelope is asymmetric. These effects result from the non-zero contribution of φ(ν) in Eq. 9, which carries the contribution of the difference between arms, δL and δn(ν) (Supplementary Eq. 20). The zero-delay shift is caused by the first-order in ν, while the envelope distortion is caused by high-order terms.

The difference between arms that cause these observable changes in the interferogram correspond to typical variations expected from the silicon photonics process, rather than strong differences due to a non-ideal fabrication process. Considering the difference in arm length δL negligible, the zero delay is centered at

$${\cal T}_0 \approx \frac{L}{c}\frac{{\delta \left( {n_{{\mathrm{eff}}}{\mathrm{|}}_{\nu _0}} \right) + \nu _0\delta (\partial _\nu n)}}{{1 + \xi _1}}.$$
(16)

Using the known values for the other parameters, we estimate $$\delta \left( {n_{{\mathrm{eff}}}{\mathrm{|}}_{\nu _0}} \right) + \nu _0\delta (\partial _\nu n) \approx 2 \times 10^{ - 3}$$. This value agrees with the expected order of magnitude for effective index fluctuations due to chip-scale variations in the silicon device layer thickness20,26. The same is true for the differences in high-order dispersion terms.

The interferogram oscillations are highlighted in the insets of Fig. 3c, where a cosine oscillation at the ASE mean frequency (νs = 193.44 THz) is superposed (gray trace). In regions with a smooth envelope variation the interferogram oscillates at νs (blue-colored zoom-in), while an additional phase is introduced as the envelope varies more abruptly (green-colored zoom-in). The phase change due to envelope variations supports the interest of using a narrow laser source with almost flat envelope to calibrate the thermo-optic non-linearity, as the non-linear fit would be compromised by such phase change.

The PSD obtained from the experimental interferogram and the effects of thermo-optic non-linearity (γ2), dispersion (ξ1), and MZI transfer function (T(ν)) are presented in Fig. 3d–f. The spectra were calculated using the mean value of the calibration parameters summarized in Table 1.

The PSD obtained directly form the as-measured interferogram—with the delay axis corresponding to τ and without performing any correction—is presented in Fig. 3d. The TOC non-linearity distorts, broadens, and shifts the PSD to higher frequencies as the interferogram oscillates faster with increasing delay.

After the optical delay axis of the interferogram is properly transformed to $${\cal T}$$, the resulting PSD becomes very similar to the reference spectrum (Fig. 3e). Both resolved peaks are clearly identified and the unresolved peak is also present around 194 THz. However, since the spectrum has not been re-scaled to the original frequency ν, it is broadened by the factor 1 + ξ1 around ν0. In addition, since it has not been re-normalized by T(ν), the high frequency peak appears attenuated relatively to the low frequency peak.

Failing to re-scale the frequency axis according to Eq. 11 introduces a frequency error u − ν = ξ1Δν that degrades the absolute frequency accuracy. In this demonstration, the effect is modest since the detuning of the peaks with respect to ν0 is small. The peak around 196 THz, for instance, is shifted by 0.5 THz, which is roughly half of its full-width at half maximum. Nonetheless, the effect can be very significant for frequencies further from ν0.

The PSD corrected for the thermo-optic non-linearity, dispersion, and the MZI transfer function reproduces satisfactorily well the reference spectrum (Fig. 3f). This spectrum was obtained performing only the aforementioned corrections, with no additional data processing such as zero-filling or apodization of the interferogram. The experimental spectral resolution of δν = 0.38 THz (δσ = 12.7 cm−1, δλ = 3.05 nm) is comparable to other on-chip spectrometers aimed at broadband operation and it is sufficient for a large range of Raman and infra-red (IR) absorption spectroscopy applications14,16.

## Discussion

The ultimate performance of the on-chip Si-FTS is quite promising considering recent advancements in silicon photonics design and fabrication. First, the window of operation for a given device will be dictated by the finite bandwidth of the waveguide optical power couplers/splitters. Such components offering flat optical response over tens of terahertz31,32,33 (hundreds of nanometers) and extremely low excess loss may allow Si-FTS operating over large bandwidths. Second, fine spectral resolution could be achieved using long low-loss silicon waveguides fabricated in tight footprints28,34,35,36 combined with high temperature excursions endured by CMOS-compatible silicon devices37. Finally, the power efficiency can be significantly improved by applying suitable design changes. For instance, using Michelson interferometers38 instead of MZIs can double the optical path in a given footprint, while introducing heat isolating structures can significantly increase heating efficiency39,40.

In addition to high performance, a valuable advantage of the Si-FTS compared to other on-chip spectrometer approaches is its robustness to fabrication variations. Although the interferogram is strongly affected by the difference in effective index between the arms of the MZI (Fig. 3c), as previously discussed, the PSD remains unaffected (Fig. 3f). This result is expected from our model since φ(u) is canceled calculating the PSD by taking the absolute value of the IFT (Eq. 10).

It is worth noticing that the presence of TOC non-linearity and dispersion have an upside in the Si-FTS performance. Ideally, one seeks to lower the resolution × dissipated power product, given by

$$\delta \nu \times W_{{\mathrm{total}}} = \kappa _\tau ^{ - 1}$$

in the case where the two effects are absent (κ τ is assumed to be identical for the two heaters). In the presence of ξ1 and γ2, this product is modified to

$$\delta \nu \times W_{{\mathrm{total}}} = \kappa _\tau ^{ - 1}{\kern 1pt} [(1 + \xi _1)\,(1 + \gamma _2\kappa _\tau W_{{\mathrm{total}}})]^{ - 1}$$

and is effectively reduced, resulting in decreased power dissipation for a given resolution or, conversely, lower δν given a maximum power. In our case, the resolution of 0.38 THz is achieved dissipating a total power of 5.1 W, representing a 35% decrease in the power (6.9 W) that would be required to achieve this resolution if dispersion and thermo-optic non-linearity were absent.

In summary, we demonstrated the realization of a FTS with true time delay in silicon photonics. Considering the non-linearity of the thermo-optic effect as well as thermal expansion and dispersion, we derived simple corrections that effectively account for these effects and allow to use well established FT techniques to obtain accurate spectral responses. We showed how the Si-FTS can be calibrated using a tunable laser source and we demonstrated the successful recovery of a broadband spectrum, resilient to fabrication variations. Our discussion proposes a simple approach to tackle the hurdles of doing FT spectrometry using dispersive integrated platforms with high temperature excursions that can be readily applied to other device geometries and extended silicon photonics platforms such as SiN and Ge-on-Si, paving the way for robust, cost-effective, and versatile FT-based portable spectrometers.

## Methods

### Fabrication

The Si-FTS was fabricated in 15 × 15 mm2 dies from 250-nm-thick SOI wafers with 3 μm of buried oxide. All the fabrication steps correspond to standard CMOS-compatible processes. The waveguides (550 nm wide) and input/output taper regions (150 nm wide at the tip) were patterned using electron-beam (e-beam) lithography with the negative tone resist hydrogen silsequioxane (HSQ). The separation between adjacent waveguides in the spiral sections is 5 μm center-to-center and the bending radius is 10 μm. After patterning, the waveguides are dry etched, the HSQ is striped with a dip in buffered oxide etch solution and the waveguides are covered by 1.5 μm of plasma-enhanced chemical vapor deposition (PECVD)-deposited silicon oxide cladding. After the oxide deposition, the heaters are patterned on top of each arm using PMMA and the metals are deposited via sputtering in a liftoff process. The heaters consist of a serpentine nichrome (NiCr) trail of 6 μm-wide 280 nm-thick sections separated by 4 μm border-to-border and totaling 17.33 mm in length. The NiCr heaters are terminated by 170 × 120 μm2 titanium-gold (Ti:Au, 5 nm:300 nm) pads. A second PECVD silicon oxide layer (500 nm-thick) is deposited on top of the heaters and 100 × 100 μm2 windows are opened on top of the Ti:Au pads to allow electric contact. After fabrication, the devices are diced and the input/output facets are polished to mitigate optical losses.

### Electrical and optical measurements

The calibration described in the main text was performed using a tunable laser source (Agilent 81600B) operating between 1460 and 1630 nm with optical power around 0 dBm. The EDFA broadband source (Amonics) delivered around 15 dBm of total optical power in the spectral range between 1528 and 1564 nm and was connected to an in-line fiber polarizer. Light was coupled into and out of the device using polarization maintaining cleaved fibers, oriented to excite the quasi-TE mode of the SOI waveguide. The output fiber was directly connected to an InGaAs powermeter. The coupling fibers were positioned using precise translation stages (Thorlabs NanoMax) driven by piezoelectric controllers. The fibers made physical contact with the sample, but were not permanently attached to it. The sample was mounted on top of a temperature stabilized station (20 °C). The heaters were driven using a sourcemeter (Keithley 2400) allowing for simultaneous precise current drive (up to 2.1 A) and voltage monitoring (up to 210 V). The electric contact with the sample was done using microprobes with 50 × 50 μm2 tips connected to the heater pads. The total parasitic series resistance of the electric apparatus was <4 ohm. All the equipment was connected to a computer and controlled via Matlab routines. The interferograms were measured sweeping the electric current from zero to 2 A using a non-linear vector increasing with the square of the current such that the electric power vector would be linearly spaced. The interval between consecutive measurements was <0.5 s. Notice that a continuous current sweep was not employed only because the instruments used (Keithley 2400) did not support such continuous sweep.

### Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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## References

1. 1.

Wan, N. H. et al. High-resolution optical spectroscopy using multimode interference in a compact tapered fibre. Nat. Commun. 6, 7762 (2015).

2. 2.

Redding, B., Liew, S. F., Bromberg, Y., Sarma, R. & Cao, H. Evanescently coupled multimode spiral spectrometer. Optica 3, 956–962 (2016).

3. 3.

Redding, B., Liew, S. F., Sarma, R. & Cao, H. Compact spectrometer based on a disordered photonic chip. Nat. Photonics 7, 746–751 (2013).

4. 4.

Subramanian, A. Z. et al. Silicon and silicon nitride photonic circuits for spectroscopic sensing on-a-chip. Photonics Res. 3, B47–B59 (2015).

5. 5.

Ma, X., Li, M. & He, J. J. CMOS-compatible integrated spectrometer based on echelle diffraction grating and MSM photodetector array. IEEE Photonics 5, 6600807 (2013).

6. 6.

Kyotoku, B. B. C., Chen, L. & Lipson, M. Sub-nm resolution cavity enhanced micro-spectrometer. Opt. Express 18, 102–107 (2010).

7. 7.

Florjaczyk, M. et al. Development of a slab waveguide spatial heterodyne spectrometer for remote sensing. Proc. SPIE 7594, 75940R (San Francisco, California, United States, 2010).

8. 8.

Nedeljkovic, M. et al. Mid-infrared silicon-on-insulator Fourier-transform spectrometer chip. IEEE Photonics Technol. Lett. 28, 528–531 (2016).

9. 9.

Podmore, H. et al. Demonstration of a compressive-sensing Fourier-transform on-chip spectrometer. Opt. Lett. 42, 1440–1443 (2017).

10. 10.

Akca, B. I. Design of a compact and ultrahigh-resolution Fourier-transform spectrometer. Opt. Express 25, 1487–1494 (2017).

11. 11.

Herrero-Bermello, A. et al. Temperature dependence mitigation in stationary Fourier-transform on-chip spectrometers. Opt. Lett. 42, 2239–2242 (2017).

12. 12.

Coarer, E. L. E. et al. Wavelength-scale stationary-wave integrated Fourier-transform spectrometry. Nat. Photonics 1, 473–478 (2007).

13. 13.

Coutant, O., De Mengin, M. & Le Coarer, E. Fabry-Perot optical fiber strainmeter with an embeddable, low-power interrogation system. Optica 2, 400–404 (2015).

14. 14.

Nie, X., Ryckeboer, E., Roelkens, G. & Baets, R. CMOS-compatible broadband co-propagative stationary Fourier transform spectrometer integrated on a silicon nitride photonics platform. Opt. Express 25, A409–A418 (2017).

15. 15.

Sabry, Y. M., Khalil, D. & Bourouina, T. Monolithic silicon-micromachined free-space optical interferometers on chip. Laser Photonics Rev. 9, 1–24 (2015).

16. 16.

Erfan, M. et al. On-chip micro-electro-mechanical system Fourier transform infrared (MEMS FT-IR) spectrometer-based gas sensing. Appl. Spectrosc. 70, 897–904 (2016).

17. 17.

Li, J., Lu, D.-f. & Qi, Z.-m. Miniature Fourier transform spectrometer based on wavelength dependence of half-wave voltage of a LiNbO3 waveguide interferometer. Opt. Lett. 39, 3923–3926 (2014).

18. 18.

Ferrari, M. & Quaresima, V. A brief review on the history of human functional near-infrared spectroscopy (fNIRS) development and fields of application. Neuroimage 63, 921–935 (2012).

19. 19.

Reich, G. Near-infrared spectroscopy and imaging: basic principles and pharmaceutical applications. Adv. Drug Deliv. Rev. 57, 1109–1143 (2005).

20. 20.

Rahim, A. et al. Expanding the silicon photonics portfolio with silicon nitride photonic integrated circuits. J. Lightwave Technol. 35, 639–649 (2017).

21. 21.

Malik, A. et al. Ge-on-Si and Ge-on-SOI thermo-optic phase shifters for the mid-infrared. Opt. Express 22, 28479–28488 (2014).

22. 22.

Shen, L. et al. Mid-infrared all-optical modulation in low-loss germanium-on-silicon waveguides. Opt. Lett. 40, 268–271 (2015).

23. 23.

Zhou, Z., Yin, B. & Michel, J. On-chip light sources for silicon photonics. Light Sci. Appl. 4, e358 (2015).

24. 24.

Crosnier, G. et al. Hybrid indium phosphide-on-silicon nanolaser diode. Nat. Photonics 11, 297–300 (2017).

25. 25.

Volet, N. et al. Semiconductor optical amplifiers at 2.0-µm wavelength on silicon. Laser Photonics Rev. 11, 1600165 (2017).

26. 26.

Zortman, Wa, Trotter, D. C. & Watts, M. R. Silicon photonics manufacturing. Opt. Express 18, 23598–23607 (2010).

27. 27.

Hochberg, M. & Baehr-Jones, T. Towards fabless silicon photonics. Nat. Photonics 4, 492–494 (2010).

28. 28.

Chrostowski, L. & Hochberg, M. Silicon Photonics Design (Cambridge University Press, Cambridge, 2015).

29. 29.

Griffiths, P. R. The early days of commercial FT-IR spectrometry: a personal perspective. Appl. Spectrosc. 71, 329–340 (2017).

30. 30.

Dong, B. et al. Nano-silicon-photonic Fourier transform infrared (FTIR) spectrometer-on-a-chip. In Conference on Lasers and Electro-Optics STu4I.1 (Optical Society of America, California, 2015).

31. 31.

Zhang, Y. et al. A compact and low loss Y-junction for submicron silicon waveguide. Opt. Express 21, 1310–1316 (2013).

32. 32.

Yun, H. et al. Broadband 2×2 adiabatic 3 dB coupler using silicon-on-insulator sub-wavelength grating waveguides. Opt. Lett. 41, 3041–3044 (2016).

33. 33.

Wang, Y., Gao, S., Wang, K. & Skafidas, E. Ultra-broadband and low-loss 3 dB optical power splitter based on adiabatic tapered silicon waveguides. Opt. Lett. 41, 2053–2056 (2016).

34. 34.

Dong, P. et al. Low loss shallow-ridge silicon waveguides. Opt. Express 18, 14474–14479 (2010).

35. 35.

Yurtsever, G., Weiss, N., Kalkman, J., van Leeuwen, T. G. & Baets, R. Ultra-compact silicon photonic integrated interferometer for swept-source optical coherence tomography. Opt. Lett. 39, 5228–5231 (2014).

36. 36.

Teng, C. C. et al. Fiber-pigtailed silicon photonic sensors for methane leak detection. In Conference on Lasers and Electro-Optics AM3B.2 (Optical Society of America, California, 2017).

37. 37.

Sedky, S., Witvrouw, A., Bender, H. & Baert, K. Experimental determination of the maximum post-process annealing temperature for standard CMOS wafers. IEEE Trans. Electron Devices 48, 377–385 (2001).

38. 38.

Patel, D. et al. High-speed compact silicon photonic Michelson interferometric modulator. Opt. Express 22, 26788–26802 (2014).

39. 39.

Dong, P. et al. Low power and compact reconfigurable multiplexing devices based on silicon microring resonators. Opt. Express 18, 9852–9858 (2010).

40. 40.

Fang, Q. et al. Ultralow power silicon photonics thermo-optic switch with suspended phase arms. IEEE Photonics Technol. Lett. 23, 525–527 (2011).

## Acknowledgements

This work was performed in part at the San Diego Nanotechnology Infrastructure (SDNI) of UCSD, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the National Science Foundation (grant ECCS-1542148). M.C.M.M.S. and N.C.F. acknowledge financial support from the São Paulo Research Foundation (2014/04748-2 and 2015/20525-6). M.C.M.M.S. acknowledges fruitful discussions with colleagues from Fainman’s group at UCSD and from the Device Research Laboratory at Unicamp.

## Author information

A.G. and Y.F. conceived and supervised the project. M.C.M.M.S. and A.G. developed the mathematical model. N.C.F. co-supervised the project mostly with the mathematical model development. M.C.M.M.S. performed simulations, fabricated the samples, and realized the experiments. All authors contributed to discussing the results and preparing the manuscript.

### Competing interests

The authors declare no competing financial interests.

Correspondence to Yeshaiahu Fainman.