## Abstract

Athermal design of integrated photonic devices can reduce the need for active temperature stabilization and consequently the energy required to operate photonic integrated circuits. For silicon photonic filters such as AWGs which employ wire or ridge waveguides, temperature insensitivity can be achieved using cladding materials with negative thermo-optic coefficients. On the other hand, in echelle grating filters the inteference takes place in the slab free-propagation region, and therefore the modal overlap with the cladding is small, rendering this method ineffective. In this work we present an approach to design an athermal echelle grating filter exploiting a temperature-synchronized Mach-Zehnder interferometer as input. This reduces the spectral shift over a temperature range of 20 K to less than ±45 pm compared to the 1.6 nm shift for the same echelle grating with a conventional waveguide input. Furthermore, the proposed design relies exclusively on a standard fabrication process for silicon-on-insulator photonic devices and exhibits a good tolerance to fabrication uncertainties.

## 1. Introduction

The temperature-dependent behavior of integrated photonic devices is a fundamental issue that affects almost every aspect of circuit and device design and every technological platform. This is particularly true for silicon-on-insulator (SOI) and other semiconductor-based waveguides that exhibit thermo-optic coefficients almost ten times larger than that in silica. This results in large wavelength shifts of the order of several tens of picometers per degree in the optical communication bands and causes severe distortions of the device responses [1,2]. In order to avoid or limit the use of energy-hungry active temperature control, several solutions have been proposed in the literature to mitigate the effect of temperature fluctuations by designing athermal devices. For wavelength filtering and routing, arrayed waveguide grating (AWG), lattice filter and echelle grating are the main device options, each with its advantages and shortcomings [3–6]. For AWGs and lattice filters in which the phase accumulation takes place in wire or ridge waveguides, athermal behavior may be achieved by using a material with a negative thermo-optic coefficient as the waveguide top cladding to counterbalance the positive coefficient of the core and substrates. Exploited materials include resins [7], sol-gel materials [8], polymers [9, 10], and amorphous titanium dioxide [11]. In echelle gratings where the phase accumalation is in the slab free-propagation region, a similair method has been applied to devices based on III–V semiconductors [12]. However, this method is ineffective in echelles based on a submicron silicon platform particularly for devices operating in the transverse electric (TE) polarization since the modal overlap with the cladding material is very small. Additionally, the use of these materials requires a substantial change to the standard SOI photonics fabrication processes, which may hinder their adoption outside of research scenarios. On the other hand, it is possible to utilize the dimensional dependece of the waveguide temperature sensitivity to engineer the overall device thermo-optic behavior [13,14], and an athermal lattice filter has been reported exploiting this technique [15].

In this work we propose a way to design a SOI athermal echelle grating filter operating in the TE polarization. We achieved the desired goal without using materials with negative thermo-optic coefficient and relying exclusively on a standard fabrication process for SOI waveguides. Inspired by the concept of wavelength-synchronized interferometers for flat-top transmission [16,17], we exploit a temperature-synchronized (T-synced) Mach-Zehnder interferometer (MZI) as input to the echelle to match and compensate its focal position shift with temperature, thereby achieving a large reduction in the temperature dependence of the overall transmission spectrum. Unlike for the case of wavelength-synchronized interferometers, the proposed MZI is designed to have a very broad free-spectral-range (FSR) to obtain a T-synced input that has a neglegible variation with wavelength of the generated field profiles. This modified input is similar to that proposed by Kamei et al. [18] but there it was only used to compensate for the small second-order temperature dependence of silica-based AWGs in combination with resin-filled grooves. To the best of our knowledge, this is the first proposal of an athermal echelle grating in TE for the high-index-contrast SOI platform. The designed device shows a residual zero-mean wavelength fluctuation of the output spectrum and a maximum shift of less than 45 pm over a temperature range of 20 K, compared to a 1.6-nm shift for the same grating with a conventional waveguide input. The increase of the channel insertion loss due to the use of the T-synced input is limited to about 0.5 dB. Moreover, the proposed design shows a good tolerance to fabrication uncertainties, with limited degradation of the performance for waveguide width uncertainty of ±10nm.

Design concepts for the Mach-Zehnder interferometer and the interface with the echelle filter are presented in sections 2.1 and 2.2, respectively. Section 3 shows the results of the overall device simulations, the analysis of the performance and sensitivity to uncertainties.

## 2. Principle and design of the device

In an echelle grating filter, the incoming and outgoing angle of the light beams *ϑ* and *φ _{k}* are governed by the following relation [19]:

*λ*the wavelength of the k-th output channel, and

_{k}*n*

_{eff}(

*λ*) the corresponding effective index of the slab waveguide. Since

_{k}*n*

_{eff}varies with temperature T (i.e. thermal dispersion), the diffraction angle

*φ*for a given wavelength shifts with T causing a shift of the focal position on the echelle image plane and hence of the transmission in the output waveguides. For an optical field distribution at the entrance of the slab that is almost insensitive to T and

_{k}*λ*(e.g. when a conventional waveguide input is used) the output focal position (diffracted image) is geometrically determined by the diffraction angle, the radius of the Rowland circle and the grating position.

The proposed temperature-insensitive (athermal) echelle grating filter is schematically shown in Fig. 1. The temperature synchronized Mach-Zehnder interferometer (MZI) placed at the entrance to the slab area is made of two 3-dB directional couplers (marked as first and interface couplers in the figure) and two unbalanced branches with lengths L_{1} and L_{2} and widths w_{1} and w_{2}, respectively. Different waveguide widths allow varying the effective thermo-optic coefficients of the two waveguide branches, thus precisely controlling the dependence of the MZI behaviour on temperature [13, 15]. As depicted in Fig. 1, the coupled waveguides of the interface 3-dB coupler form the input to the echelle filter and, driven by the temperature-dependent phase delays in the MZI, generate a field distribution at the interface that shifts with temperature. The shift of field peak position is designed to match the thermal dispersion of the echelle grating over a selected temperature range ΔT and thus ensures that the field reflected by the grating at a given wavelength is imaged on the same output waveguide for all temperatures. To achieve a temperature-independent behavior for all the wavelength channels of the echelle filter, the MZI is designed to have a large FSR, minimizing the roll-off of its transfer function over the full operational bandwidth and hence ensuring that the field generated at the entrance of the slab area is the same for all the channels.

As a proof-of-concept, we consider a standard SOI technology with a 220-nm silicon core thickness and silicon dioxide upper and lower claddings. The echelle grating is designed to have four channels with 800 GHz spacing centered around *λ*_{0} = 1550 nm and 1-dB bandwidth of about 140 GHz. The grating is in a Rowland circle configuration with a radius of 250 μm, a diffraction order of 20, and a mounting angle of 45 degree. The choice of a small Rowland circle radius helps reducing the impact of the silicon thickness inhomogeneity as pointed out in [3,6]. As will be discussed in later sections, small dimensions are also advantageous for the compensation of the thermo-optic effects. The angular dimension of the grating is 15 degree in order to capture the entire field of the T-synced input. The reflectance of the grating facets is assumed to be 100% in the simulations. High reflectance can be achieved in practice exploiting Bragg reflectors [6]. At the Rowland circle waveguides have a width of 2 μm and are then tapered to 500 nm to ensure single-mode propagation. With the employed 220-nm silicon core with oxide cladding, the temperature-induced angular shift of the echelle grating is 0.0076 deg/K (see Eq. (1)). The corresponding spacial shift is approximately 950 nm over a 20 K temperature range.

Key considerations for the design of the complete device are athermal operating temperature range, overall device size, excess loss and tolerance to fabrication uncertainties. The design of the Mach-Zehnder interferometer is presented in Sec. 2.1 while Sec. 2.2 discusses in detail the design of the coupler at the interface with the slab area. The electromagnetic simulations presented in the next sections for the Mach-Zehnder were obtained with a commercial finite difference mode solver and a 3D mode expansion propagation solver [20]. The design and simulation of the echelle grating were performed with a dedicated 2D semi-vectorial in-house tool based on the Huygens-Kirchhoff diffraction theory and the effective index approximation. The software can handle input fields of arbitrary and strongly dispersive shapes as could be possibly generated by some T-synced input configurations.

#### 2.1. Design of the Mach-Zehnder interferometer

During the design of the MZI, we considered for the first 3-dB coupler a width of 500 nm for both waveguides and a gap of 300 nm, obtaining a 3-dB length L* _{f}* = 46 μm. Interconnecting waveguide sections were assumed to have the same length in the two MZI branches and a width of 500 nm, thus not contributing to the overall transfer function. The design was performed by defining for each branch a section of length L

_{1}and L

_{2}and width w

_{1}and w

_{2}, respectively, to simultaneously maximize the FSR and completely switch the Mach-Zehnder output from bar to cross condition (or vice-versa) within the given temperature variation ΔT to match the echelle focal spot shift with temperature.

Assuming a linear frequency dispersion around the central wavelength *λ*_{0}, the FSR of the Mach-Zehnder (in wavelength) is defined as

*λ*

_{0}and T

_{0}are the central wavelength and temperature of the considered ranges, L

*is the length of the waveguide (i=1,2), and*

_{i}*n*

_{g,i}(

*λ*

_{0},

*T*

_{0}) is the group index of the i-th waveguide at

*λ*

_{0}and T

_{0}. The FSR can hence be easily maximized imposing that In order to match the temperature dispersion of the echelle grating, the phase difference between the modes propagating in the two branches of the MZI must change between −

*π*/2 and

*π*/2 when the temperature varies from T

_{0}−ΔT/2 to T

_{0}+ΔT/2. Assuming a linear dependence of the phase effective index on temperature, this condition is equivalent to

*n*

_{eff,i}(

*λ*

_{0},

*T*) the temperature-dependent phase effective index of the i-th waveguide at central wavelength

*λ*

_{0}. Although the same behaviour could be obtained varying the phase difference between

*π*/2 and 3

*π*/2, this would cause the field at the output of the MZI to be anti-symmetric, with detrimental effects for both the propagation in the slab area and the overlap with the output waveguide. This condition is avoided in the design and the analysis of fabrication uncertainty discussed in Sec. 3 ensures it is not reached with linewidth uniformity guaranteed by standard fabrication technologies.

Equations (3) and (4) allow to calculate the lengths L_{1} and L_{2} for any chosen combination of w_{1} and w_{2} in order to simultaneously maximize the FSR and match the shift of the grating focal point over the temperature range ΔT. To this purpose, the effective index of the fundamental TE-like mode of an SOI waveguide was simulated as function of waveguide width, temperature and wavelength. To model silicon properties, we considered at *λ* = 1550 nm and T = 300 K a refractive index of 3.479, a thermo-optic coefficient of 1.87 ×10^{−4} K^{−1} and a spectral dispersion of −0.083 μm^{−1}.The same parameters for silica were chosen as 1.444, 0.85×10^{−5} K^{−1} and −0.012 μm^{−1}, respectively [21]. From these simulations, the dependence of the group index n* _{g}* and the (effective) thermo-optic coefficient

*∂n*

_{eff}/

*∂*T on waveguide width were calculated. Figure 2 shows the obtained results for

*λ*

_{0}= 1550 nm and T

_{0}= 300 K.

There are several considerations for choosing w_{1} and w_{2}. An MZI with w_{1} < 350 nm and w_{2} > 500 nm (or vice-versa) maximizes the difference between the two thermo-optic coefficients and the two group indices, which together contributes in defining the required lengths of the branches through Eqs. (3) and (4). Consequently this would reduce L_{1} and L_{2} and the overall device size. On the other hand, this choice would make the device very sensitive to fabrication uncertainties because of the steep dependence of both n* _{g}* and

*∂n*

_{eff}/

*∂*T on waveguide width for w < 350 nm. In the proposed design we hence chose w

_{1}= 380 nm and w

_{2}= 800 as a viable compromise between robustness and device compactness (dashed lines in Fig. 2). In this case

*∂n*

_{eff,1}/

*∂*T = 1.981×10

^{−4}K

^{−1},

*∂n*

_{eff,2}/

*∂*T = 1.936×10

^{−4}K

^{−1}, n

_{g,1}= 4.426, and n

_{g,2}= 3.909 for T

_{0}= 300 K and

*λ*

_{0}= 1550 nm. For an athermal operating temperature range ΔT = 20 K, the corresponding lengths of the two branches obtained with Eqs. (3) and (4) are L

_{1}= 1707 μm and L

_{2}= 1933 μm. The choice of the temperature range is discussed in Sec. 2.2.

#### 2.2. Design of the coupler-slab interface

The design of the 3-dB directional coupler at the interface with the echelle slab area is particularly important for minimizing the excess loss generated by replacing a simple input waveguide with the proposed T-synced input. It should be noted that the modal mismatch between the coupler and the slab area also generates back-reflections that might be critical for certain applications and should hence be minimized. A schematic of the directional coupler and interface section is shown in Fig. 3(a). Here the design parameters are the waveguide distance d and the waveguide width w* _{c}*. The goal is to match the focal point shift with temperature in the echelle for as large a temperature range as possible, while maximizing the modal overlap between the field generated by the T-synced input (and imaged by the grating to the echelle outputs) and the fundamental mode of the 2-μm-wide output waveguides.

To this purpose, the distance d must be equal to the shift produced by the echelle slab over the selected temperature range ΔT. A small value of d is desirable to both improve the overlap with the output mode and reduce the length of the 3-dB interface coupler. Since the echelle angular shift with temperature is determined by the thermo-optic properties of the slab waveguide according to Eq. (1), a small d can be achieved by reducing the Rowland radius. On the other hand, considerations for example on channel cross-talk, grating truncation loss and fabrication issues limit the minimum radius usable. As a balanced choice, we selected a Rowland radius R = 250 μm that for a temperature range ΔT = 20 K centered at 300 K is expected to generate a focal point shift of 950 nm, which is hence chosen as value for d.

Overlap losses and back-reflections depend on w* _{c}* in two opposite ways. Narrow waveguides are required to expand the field profile in the horizontal direction and improve the in-plane overlap with the fundamental mode of the echelle output. On the other hand, narrow waveguides delocalize the field also in the vertical direction, decreasing the out-of-plane overlap and increasing the back-reflections. For our design, w

*= 255 nm represents a good compromise, giving an overlap loss with the fundamental mode of a 2-μm waveguide at T*

_{c}_{0}= 300 K of about 0.7 dB and back-reflections smaller than −14 dB. The increase of the overlap losses at T = 290 K and T = 300 is smaller than 0.4 dB. Two 35-μm-long linearly tapered sections were used to adiabatically adjust the width from the initial 500 nm to w

*= 255 nm with negligible losses. A total coupler length L*

_{c}*= 36 μm (including tapers) ensures a 3-dB operation at*

_{O}*λ*

_{0}= 1550 nm and T

_{0}= 300 K.

As an example, Figs. 3(b)–3(d) shows the simulated field profile |E* _{x}*|

^{2}generated at the entrance to the slab area by the interface coupler at temperature 290 K, 300 K and 310 K, respectively. For these simulations, we assumed the fundamental quasi-TE modes being excited at the beginning of the two 500 nm wide waveguides with the same power and a phase difference of −

*π*/2 rad, 0 rad and

*π*/2 rad respectively, as generated by the Mach-Zehnder designed in section 2.1 for the three considered temperatures. As expected, the center of the spot shifts about 950 nm with a temperature variation of 20 K. The field shape also changes with temperature, causing a slight variation of the shape of the echelle pass-bands as described in the next section.

## 3. Athermal echelle simulations and performance

For the evaluation of the entire device we considered the parameters optimized in the previous sections: ΔT = 20 K centered at T_{0} = 300 K; central wavelength *λ*_{0} = 1550 nm; for the first 3-dB coupler, L* _{f}* = 46 μm, waveguide width 500 nm and gap 300 nm; for the MZI branches w

_{1}= 380 nm, w

_{2}= 800 nm, L

_{1}= 1707 μm, and L

_{2}= 1933 μm; for the interface 3-dB coupler d = 950 nm, w

*= 255 nm and length L*

_{c}*= 36 μm; a waveguide width of 2 μm for the echelle output waveguides. The simulation of the entire T-synced input was performed exploiting a transfer matrix approach, describing its transmission as a function of wavelength and temperature as [17]*

_{O}*) while*

_{f}*β*

_{1}and

*β*

_{2}are the propagation constants of the two branches of the Mach-Zehnder (lengths L

_{1}and L

_{2}). The phase delay of the symmetric and anti-symmetric modes of the interface coupler are defined as ${\phi}_{s/a}^{o}={\int}_{0}^{{L}_{O}}{\beta}_{s/a}^{o}(\text{s})\hspace{0.17em}\text{ds}$, where L

*is the length of the output coupler and ${\beta}_{s/a}^{o}$ the propagation constants of the two modes. The integral is required because ${\beta}_{s/a}^{o}$ change along the coupler due to the tapered sections. Propagation losses are omitted. The matrix C is defined as and is needed to convert transfer matrices defined on the fundamental modes of isolated waveguides to matrices defined on the symmetric and anti-symmetric modes of the couplers. All the propagation constants*

_{O}*β*in Eq. (5) were simulated as function of wavelength and temperature as described in Sec. 2. It is worth noting that in this way the simulations take into account the dependence of the transfer function with wavelength and temperature not only for the two MZI branches but also for the two 3-dB couplers, providing an accurate result for the entire input circuit. Matrix M was used to compute the field profile at the entrance to the slab area, and the transfer function of the echelle filter was then computed with our in-house tool using these fields as input.

The simulation results are presented in Fig. 4. As a reference, Fig. 4(a) shows the transmission spectra obtained with a conventional simple 2-μm-wide input waveguide for the four channels of the echelle grating at 290 K, 300 K and 310 K, respectively. At 300 K, the four channels are located at *λ*_{1} = 1540.56 nm, *λ*_{2} = 1546.92 nm, *λ*_{3} = 1553.33 nm, and *λ*_{4} = 1559.80 nm. Channel 3-dB bandwidth is about 250 GHz. The minimum insertion loss is 0.4 dB, the power roll-off is about 0.3 dB and channel cross-talk is smaller than −35 dB. By using the proposed T-synced input, the spectral shift is significantly reduced, as shown in Fig. 4(b). This design modification introduces a slight dependence of the pass-band on temperature due to the variation of the corresponding field distribution generated at the entrance to the slab area (see Figs. 3(b)–3(d)). Channel cross-talk slightly increases only at the edges of the design temperature range.

Figure 4(c) shows the shift of the central wavelength of the channels. For the conventional waveguide input, a linear shift with temperature of approximately 1.6 nm over the 20-K temperature range (or 80 pm/K, see the black solid line) is observed. For the T-synced input case, red diamonds and blue crosses report the residual wavelength shift for channels 1 and 2. Channels 3 and 4 have similar properties and are omitted here for clarity. As can be seen, there is a residual zero-mean fluctuation of the channel spectral position with temperature. This is due to the fact that the “center-of-gravity” of the field distribution generated by the MZI depends on temperature through a squared sine function. The maximum of the average residual fluctuation is about ±45 pm at 295 K and 305 K.

Lastly, Fig. 4(d) reports the insertion loss for channels 1 and 2 at their central wavelengths *λ*_{1} = 1540.56 nm and *λ*_{2} = 1546.92 nm, respectively, for both the waveguide and T-synced input cases. Results are similar for channels 3 and 4. With a conventional waveguide input, central channels (2 and 3) have an insertion loss of about 0.4 dB at 300 K that quickly increases to more than 2 dB with temperature variations of ±10 K due to the shift of the pass-band. The use of the T-synced input increases the insertion loss at 300 K for these channels to about 0.9 dB due to the additional overlap losses with the fundamental mode of the 2-μm output waveguide. On the other hand, this insertion loss has a much smaller dependence on temperature and reaches only 1.3 dB at 290 K and 310 K. In this case the additional losses mainly come from a slight residual wavelength dependence of the Mach-Zehnder transfer function and hence of the field generated at the entrance of the slab area. For both types of input designs, the outer channels (1 and 4) have behavior similar to the central channels but with an additional loss of about 0.3 dB due to the power roll-off of the echelle grating.

The careful selection of the waveguide width for the two Mach-Zehnder branches presented in Sec. 2.1 and the balanced design of the interferometer provide a good robustness of the proposed device to fabrication errors. In order to test this sensitivity we repeated the simulations assuming a variation Δw = ± 10 nm of the waveguide width in the MZI, including both 3-dB couplers and the two branches. This range of dimensional changes well represents the expected linewidth uniformity with 193 nm deep-UV lithography [22]. The dimensions of the echelle grating are kept unchanged as variations on this scale on the grating facet positions do not significantly change the filter performance, causing at most a slight increase of the channel cross-talk. Figure 5 reports the simulation results along with the ideal response (Δw = 0, same data shown in Fig. 4(b)) for channel 1 at 290 K, 300 K, and 310 K, respectively. Since the entire MZI design is performed at the central wavelength 1550 nm, the outer channels 1 and 4 are expected to suffer more spectral distortion compared to the central channels 2 and 3. As can be seen, the device exhibits a good tolerance to fabrication uncertainties. The effect of the waveguide width variation on the pass-band shape and central wavelength are negligible at the center of the considered temperature range (300 K). At the edges (290 K and 310 K) the compensation of the thermal shift of the transfer function is still effective, preserving the functionality of the device, with some minor effects on the shape of the pass-band. Similar results were obtained for channel 4 while channel 2 and 3 showed an expected higher tolerance to waveguide width deviations, with negligible effects also at 290 K and 310 K. The same device tolerance is predicted also considering waveguide width deviations between the two branches that are uncorrelated but within the same range.

## 4. Conclusions

We have proposed a design for an athermal echelle grating filter operating in TE polarization that relies exclusively on standard silicon-on-insulator photonics fabrication processes without using any additional material with negative thermo-optic coefficient. The device exploits a Mach-Zehnder interferometer with a very large free spectral range as a temperature-synchronized input of the echelle grating to match the shift of the grating focal point with temperature. In this way, a large reduction of the sensitivity of the overall transfer function with temperature has been achieved, with a residual zero-mean fluctuation of about ±45 pm over a temperature range of 20 K compared to a 1.6-nm shift obtained with a simple waveguide input. The use of the Mach-Zehnder at the input slightly increased the device insertion loss by about 0.5 dB. The device functionality has been numerically demonstrated taking into account also a waveguide width fabrication uncertainty of ±10 nm. The covered temperature range could be increased by further minimizing the radius of the echelle grating and by exploiting different types of interface couplers (e.g. multi-mode interference couplers).

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