1. Introduction

Advances in fabrication technology have put forward the potential of controlling light at the subwavelength scale in order to boost the performance and flexibility of integrated optic components. To this end, silicon subwavelength grating waveguides (SWGs) are a prominent example of subwavelength engineering that led to substantial performance improvements in practical applications [1]. SWGs are one-dimensional periodic waveguides with feature size that are much smaller than their operating wavelengths, such that the relation $\lambda >>2 n_{\textit {eff}} \Lambda$ approximately holds [2]. As such, they support the diffraction-less propagation of a Bloch mode, whose properties can be controlled through not only careful design of their cross-sections but also by controlling the period $\Lambda$ and the duty-cycle $DC = L_{Si}/\Lambda$.

The added degrees of freedom of SWG provide more flexibility in their design; in the deep subwavelength limit, light sees the grating as an homogeneous material with a weighted refractive index, thus opening the possibility of refractive index engineering [3]. Index engineering can be used to multiple ends. A few key examples include on-chip control of group delay [4,5], bandwidth increase of contra-directional couplers [6] or of multimode interferometers [7], on-chip polarization management [7–9], polarization-independent and efficient fiber-to-chip edge couplers [10,11] and engineering of the bandwidth of Bragg filters [12].

In addition to index engineering, SWGs are also interesting to control light-matter interaction with the waveguide top cladding, which can be air or an analyte for sensing [13,14], a material with a negative thermo-optic coefficient $TOC$ for thermal compensation [15] or even a potential rare-earth ions host for amplification [16]. The use of exotic cladding materials is therefore a promising emergent topic with potential to add another degree of freedom in designing SWGs. Among the materials that are amenable for heterogeneous integration with SWG, chalcogenide glasses (ChG) present a set of unique and attractive properties for photonics, including low-loss over a large transparency window, photosensitivity, high non-linearity and a high refractive index [17]. Chalcogenides are also excellent materials for the important field of Brillouin photonics as their low density allow for acoustic waveguiding in addition to optical waveguiding [18]. The development of hybrid chalcogenide-silicon waveguides is a promising approach to harness interesting chalcogenide properties on the silicon platform [19,20].

In this work, we study the combination of silicon SWGs with a high-index chalcogenide glass top cladding. The effect of using a high-index material is first explored through analytical methods. A novel As$_{20}$S$_{80}$-on-Si SWG design is demonstrated using the finite-difference time domain (FDTD) method. Next, microring resonators (MR) based on the designed waveguide are fabricated. The passive optical properties and thermal response of the fabricated MR are measured at telecommunication wavelengths. The MR exhibit intrinsic quality factor of $Q = 5.6\times 10^4$ and athermal behavior with a slightly negative wavelength shift of $\Delta \lambda _r/ \Delta T = -1.54$ pm/K. We also demonstrate that the use of a chalcogenide soft glass can eliminate the formation of air gaps inside the cladding.

2. SWG with high-index claddings

2.1 1D analytical model

In contrast with conventional 2D waveguides (e.g. strips or ribs), SWGs support the propagation of a Bloch mode with properties that are affected by the periodic nature of the waveguide permittivity. Therefore, 2D simulation methods like finite difference eigenmode (FDE) cannot fully capture the intricacy of SWGs optical properties. Rigorous 3D simulation methods like finite-difference time domain (FDTD) can be used to accurately predict the waveguides behavior, but they are computationally expensive and time consuming. Insight into the Bloch mode behavior and initial design guidelines can be obtained by realizing that the peculiarities of the SWGs are akin to the propagation of a plane wave inside a periodic dielectric stack that is semi-infinite in the transversal directions, hereafter referred to as a Bragg stack (BS). The basic features of Bloch mode propagation can then be modelled using an analytical 1D approach. A BS with alternating material permittivities $\epsilon _1 = n_1^2$ and $\epsilon _2 = n_2^2$ and period $\Lambda$ is schematically shown in the inset (i) of Fig. 1(a). At normal incidence, the dispersion relation of a BS is given as [21,22]

 

Fig. 1. Propagation of the Bloch mode in a dielectric stack. (a) Bandstructure simulated using the analytical model of Eq. (1) (solid line) and using a metamaterial approximation (dashed line) with a fixed index contrast $\Delta n = 0.5$. Inset (i): 1D geometry of the BS. Inset (ii): Close-up view near the bandgap showing the deviation between the model of Eq. (1) and the metamaterial approximation (see text). (b) Map of the Bloch mode normalized propagation constant for varying refractive index contrast between the stacks. The vertical dashed line indicates the geometry for which the bandstructure is shown in (a) and the black solid lines delimit the extent of the bandgap. (c) Band-edge location in an infinite crystal with different minimum periods. The minimum feature size for $DC = 0.5$ is half the period $\Lambda /2$. The red shaded regions highlight commonly used spectral bands in telecommunications. (d) Engineering the effective index by tuning the material index and the waveguide duty-cycle $DC$. The normalized wavelength is fixed at $\lambda / \Lambda = 7.75$.

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The effect of high-index cladding is first explored using Eq. (1) by setting $n_2$ as the cladding material refractive index with a constant $n_1 = 2.5$ to represent the index of the silicon period, close to the effective index of a 500 nm wide silicon nanowire at 1550 nm. The effect of index contrast $\Delta n = |n_1 - n_2|$ on the propagation properties is summarized in Fig. 1(a) through (d). The bandstructure near the first order bandgap of a BS with $\Delta n = 0.5$ and $DC = 0.5$ is shown in Fig. 1(a). The result obtained from Eq. (1) is compared with a metamaterial approximation $n_{meta}^2 = \Lambda ^{-1}(l_1 \epsilon _1 + l_2 \epsilon _2)$. As expected, the BS propagation constant $K$ tends toward that using the metamaterial in the deep-subwavelength $\lambda >> \Lambda$ but deviates significantly when approaching the bandgap.

The main effect of reducing the index contrast is the narrowing of the bandgap, as seen in Fig. 1(b), which is expected as the BS approaches a uniform medium in the limit $\Delta n \rightarrow 0$. This narrowing is accompanied by a shift of the band-edges $\lambda _{BE}$, the blue- and red-side wavelength boundaries of the bandgap, which effectively limits the operation in the deep-subwavelength regime at a given wavelength. This is important considering that the minimum feature size of electron-beam lithography and UV photolithography are about 60 nm and 150 nm, respectively. The fabrication minimum feature size then puts a hard limit on how far away from the bandgap the waveguide can be designed to operate. It is well known that silicon SWG do not really operate in the deep-subwavelength regime, where they could be treated using an effective medium theory, but are effectively operating in the transition region [1]. In Fig. 1(c), the position of the red-side band-edge $\lambda _{BE}$ is plotted for various $\Lambda$ and refractive index contrast $\Delta n$. With $DC = 0.5$, the minimum feature size required is half the period $\Lambda / 2$. It is clear that one must be careful when using high-index materials with SWGs as the band-edge creeps rapidly towards common operating wavelengths in the O-band or C-band, for example. We also note that proximity to the band-edge is to be carefully considered, as even if the light is not diffracted in the bandgap, multiple scattering, slow light and Bloch mode reshaping can drastically increase the propagation losses [5,25]. The extent to which the effective index of a BS operating in the SWG regime ($\lambda / \Lambda = 7.75$) can be engineering is shown in Fig. 1(d). A large span of effective index can be achieved by tuning the SWG duty-cycle and using cladding materials with different refractive index.

The analytical model of Eq. (1) is then used to study the effect of high-index cladding on the optical field distribution inside the BS. We use the dimensionless confinement factor $\Gamma _i$, describing the interaction between the Bloch mode and material $i$ such as [26,27]

The normalized wavelength dependence of $\Gamma$ is presented using an analogue to the bandstructure in Fig. 2(a), with the different operation regions identified for clarity. In the deep-subwavelength limit, the confinement factor is directly proportional to the material index and to the duty-cycle, indicating a uniform distribution of the field over the unit cell or, in other words, that the periodicity does not affect the mode distribution. The effect of periodicity becomes significant in the transition region, where $\Gamma _{Si}$ increases until it diverges at the band-edge. This is associated with a reshaping of the Bloch mode and increased(decreased) interaction with the high(low)-index material. This behavior of enhanced light-matter interaction near the band-edge of photonic crystals is responsible for a variety of interesting effects, including gain enhancement for lasers [28] or the optical Borrmman effect [29]. Fig. 2(b) shows the normalized electric field intensity inside along the propagation axis of one unit cell for various parameters, highlighting the mode redistribution into the high index period due to both the wavelength dependence (left panel) and the index contrast (right panel). The sharp increase of $\Gamma _{Si}$ and associated decrease of $\Gamma _{clad}$ in the transition region have consequences in applications where the SWG is used to enhance interaction with the cladding material, such as in thermal compensation [15]. The effect of thermally induced perturbations are important for integrated optics in general and relevant in the context of ChG functionalization [30–32]. Using the confinement factors, the effective thermo-optic coefficient $TOC_{\textit {eff}} = d n_{\textit {eff}} / d T$ can be calculated to a first-order approximation as [32]

 

Fig. 2. Confinement factor of the Bloch mode in a dielectric stack simulated using the model of Eq. (1). (a) Confinement factor in the silicon (red solid line) and in the glass (solid blue line) as a function of normalized wavelength $\lambda /\Lambda$ with a fixed $\Delta n = 0.5$. The vertical dashed lines serve to approximately delimit the operation regions (SWG, transition and band-edge). The bandgap is indicated as the dark grey area. Inset: Confinement factor ratio $\Gamma _{clad}/\Gamma _{Si}$ that tends towards $n_2/n_1$ in the SWG limit (dashed line). (b) Normalized electric field intensity along one period (top) for various normalized wavelengths $\lambda /\Lambda = [4.85, 5, 6, 10]$ and fixed $\Delta n = 0.5$ and (bottom) for various index contrast $\Delta n = [0.7, 0.5, 0.3, 0.1]$ and fixed $\lambda /\Lambda = 7.75$.

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In this work, we consider the case where $n_1$ is silicon and has a thermo-optic coefficient of $dn_{Si/1}/dT = 1.8\times 10^{-4}$ RIU/K [33] while the material $n_2$ is As$_{20}$S$_{80}$, with $n_2 = n_{As_{20}S_{80}} = 2.15$ and $dn_{ChG/2}/dT = -5\times 10^{-5}$ [31].

2.2 FDTD simulations

Following the basic design guidelines provided by the analytical simulations of the previous section, we simulated an SWG with a 800 nm As$_{20}$S$_{80}$ cladding, which is expected to provide thermal compensation while allowing to operate sufficiently far away from the band-edge to avoid detrimental effects. The periodicity in the FDTD simulation was achieved using Bloch boundary conditions along the propagation axis and the simulation time window was 2000 fs (time step of 0.02 fs). The simulation window is 16 $\mathrm{\mu}$m wide and 8 $\mathrm{\mu}$m high and the meshing is set to 10 nm in all direcitons around the silicon block. The simulated waveguide parameters are $w = 450$ nm, $t = 220$ nm and $DC = 0.5$. Three periods are considered: $\Lambda = [200, 250, 300]$ nm.

The effective index and group index of the heterogeneous SWG are shown in Fig. 3(a) and (b), respectively. We observe the absence of bandgap related distortion down to at least $\lambda = 1400$ nm for $\Lambda = 200$ nm, so that the SWG is expected to operate in the transition region. The SWG with $\Lambda = 250$ nm shows a moderate increase of $n_g$ at shorter wavelength, indicating operation in the transition region that approaches the bandgap while the SWG with $\Lambda = 300$ nm has its bandgap just short of $\lambda = 1500$ nm. The position of the bandgap is represented as a divergence of the effective and group indices in the plots. The 3D distribution of the electric field intensity at $\lambda = 1550$ nm for $\Lambda = 250$ nm is shown in Fig. 3(c). The relatively narrow waveguide geometry and reduced effective refractive index results in a delocalization of the mode outside of the silicon block in the transversal directions. The right panel of Fig. 3(c) shows a top view where the concentration of $|E|^2$ inside the silicon is clearly visible, indicating that the SWG does not operate in the deep-subwavelength region. The confinement factors are $\Gamma _{Si} = 0.28$, $\Gamma _{clad} = 0.89$ and $\Gamma _{SiO_2} = 0.06$. The thermo-optic coefficient of the waveguide is predicted to be $\frac {d n_{\textit {eff}}}{d T} = 6.3 \times 10^{-6}$ using the material thermo-optic coefficient ($TOC$) reported in Ref. [31]. This corresponds to a resonant wavelength shift of $\Delta \lambda _r/ \Delta T = 6.2$ pm/K in a microring resonator with $R = 100\,\mathrm{\mu}$m. We note that the large ratio of $\Gamma _{clad} / \Gamma _{Si} > 3$ in TE polarization is a good example of the SWG advantage in increasing light-matter interaction with top cladding materials.

 

Fig. 3. (a) and (b) Effective index and group index simulated using 3D-FDTD for an As$_{20}$S$_{80}$ clad SWG with $w = 450$ nm, $DC = 0.5$ and different periods(b) Electric field intensity distribution of the Bloch mode for $\Lambda = 250$ nm in the (left) $xy$ plane in the glass, (center) $xy$ plane in the Si and (right) $zx$ plane at mid-height in the waveguide for 3 periods.

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3. Fabrication

Next, the waveguide simulated using FDTD in the last section was used to design and fabricate microring resonators. MRs were chosen instead of straight waveguides because they provide measurement of the group index, propagation loss and thermo-optic coefficient in a very compact footprint. In addition, estimating the waveguide loss from the MRs response instead of using the cutback methods alleviates the error arising from variations in the fiber-to-chip coupling when depositing the cladding on individual chips. The silicon chips were processed using electron-beam lithography by Applied NanoTools through the SiEPICfab consortium [34] on 220 nm thick silicon-on-insulator wafers with 2 $\mathrm{\mu}$m thick buried oxide (BOx) layer. The glassy thin-films were deposited in-house using the same method as described in Ref. [31], including the 120 s annealing step at $T = 150\,^{\circ }$C. The measured deposited cladding thickness is $890$ nm. A schematic of the heterogeneous SWG is provided in Fig. 4(a). Scanning electron microscope (SEM) images were taken before the glass deposition. A top view SEM image of an MR is shown in Fig. 4(b) with the inset showing a close-up of the coupling region between the ring and the bus waveguide. The routing strip waveguides (500 nm $\times$ 220 nm) were converted into SWGs using linear geometrical tapers of length $L_{taper} = 100\,\mathrm{\mu}$m, similar to those used in [5]. The fabricated device had a radius $R = 100\,\mathrm{\mu}$m, a width $w=450$ nm, a period $\Lambda = 250$ nm and a duty-cycle $DC = 0.5$.

 

Fig. 4. Silicon SWG waveguides and microresonators with high-index ChG cladding. (a) Schematic drawing of an SWG waveguide with period $\Lambda$, width $w$. Inset (i): Top view of a single unit cell. Inset (ii): Geometry of a single silicon block. (b) Top view SEM image of a fabricated SWG microresonator of radius $R$ with a close-up view of the coupling as inset (right).

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A common difficulty when fabricating oxide-clad silicon devices with tiny features like SWGs is the formation of air voids that are not filled by the oxide [35]. These voids can lead to significant performance deviation from the intended design. The use of materials that can be reflowed at low temperature, like chalcogenides, can suppress these gaps and lead to a conformal structure where the cladding material completely covers the silicon. This effect in the As$_{20}$S$_{80}$ cladded SWGs was investigated using focused ion beam milling (FIB) to image the waveguide along its propagation axis in a sample before annealing (as-deposited) and in a sample after thermal annealing. The as-deposited sample had systematic voids, as visible in Fig. 5(a), where the air appears as darker patterns. Thermal annealing resulted in the reflow of the glass and a complete filling between the silicon blocks, as visible in Fig. 5(b). We note that this aggressive reflow at moderate temperature ($T = 150^{\circ }$C) is possible due to the quasi-polymeric behavior of the sulfur-rich chalcogenide composition [36]. A similar effect was reported in silicon slot waveguides covered with As$_2$S$_3$ for nonlinear photonics [20].

 

Fig. 5. SEM images of fabricated heterogeneous SWG waveguides with FIB cut along the propagation axis before (a) and after (b) thermal annealing. The insets show close-up views of the glass between the silicon pillars, showing the air gap before annealing and the filling of the gap by the glass after annealing.

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4. Optical characterization

4.1 Passive response

The optical response of an SWG MR with $g = 300$ nm was measured using an optical vector analyzer (OVA5000 from LUNA). The chip stage was mounted on a Peltier module with a temperature controller (TEC) to stabilize the operation at $T = 20\,^{\circ }$C. The results are summarized in Fig. 6(a) through (d).

 

Fig. 6. Passive optical characterization of an As$_{20}$S$_{80}$ clad SWG microring resonator with $\Lambda = 250$ nm, $DC = 0.5$, $w = 450$ nm and $g = 300$ nm. (a) Normalized transmittance in the C-band. (b) Single resonance with $Q_i=5.6\times 10^4$ near 1530 nm (identified as the shaded region in (a)). (c) Quality factors measurement over the C+L band. (d) Measured (red markers) and simulated (blue line) group index over the C+L band.

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The transmittance is shown in Fig. 6(a) while a single resonance is identified and zoomed on in (b). From the resonances, the group index can be calculated and is presented in Fig. 6(d) alongside simulated values using 3D-FDTD. We note that the linear increase of group index with frequency suggests the absence of band-edge proximity effect and operation away from the bandgap, as expected from the FDTD simulations. The difference between the simulated and measured values of group index could arise from a variety of fabrication errors that are non-trivial to separate. We also mention that the dispersion of As$_{20}$S$_{80}$ used in the simulations was obtained using a prism-coupling method with a certain degree of uncertainty, which could explain the difference [31]. Next, the waveguiding loss is assessed through the MR quality factor $Q = \lambda / FWHM$, calculated using a Lorentzian fit on each resonance, as in Fig. 6(b). The wavelength dependence of the loaded and intrinsic $Q$ is shown in 6(c). The intrinsic quality-factor $Q_i$ and the associated propagation loss $\alpha$ are calculated using the formulas [37,38]

4.2 Thermal response

Finally, the thermal response of the heterogeneous SWG was investigated by increasing the TEC temperature in steps of $3\,^{\circ }$C and monitoring the MR resonant wavelength shift. The results are summarized in Fig. 7(a) through (d), where (a) and (b) are examples of measurement of two resonances: one at shorter wavelength with a nearly temperature independent behavior and one at longer wavelength with a strong negative dependence, respectively.

 

Fig. 7. Thermal measurements of a As$_{20}$S$_{80}$ clad SWG with $\Lambda = 250$ nm, $DC = 0.5$ and $w = 450$ nm. (a,b) Normalized transmittance of a single resonance with a low (a) and high (b) thermo-optic coefficient. (c) Temeperature dependent resonance shift coefficient $\Delta \lambda _r / \Delta T$ measured over the C+L band. Linear fits used to extract $\Delta \lambda _r / \Delta T$ from the resonances shown in blue (a) and red (b).

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The resonance shift $\Delta \lambda _r$ is extracted for each measured trace and a linear fit is used to extract the resonance shift coefficient $\Delta \lambda _r / \Delta T$. Two examples of this process are shown in Fig. 7(d) while the full wavelength dependent data is shown in Fig. 7(c). $\Delta \lambda _r / \Delta T$ decreases at longer wavelength as the confinement factor with the negative $TOC$ As$_{20}$S$_{80}$ increases. Thermal compensation between the silicon positive $TOC$ and ChG negative $TOC$ is nearly perfect at shorter wavelengths. The minimum value measured is $\Delta \lambda _r / \Delta T = -1.54$ pm/K, indicating that the hybrid MR is virtually athermal at this wavelength. The value of $\Delta \lambda _r / \Delta T$ is slightly lower than the one predicted in Sect.2. using FDTD simulation. This discrepancy could arise from difference between the simulated and fabricated SWG, similar to the effect on group index noted in sect.4.1. We also note that the deposited cladding was slightly thicker than in the simulation (by 90 nm) thus slightly increasing the overlap with the chalcogenide. We note that the athermal wavelength could be shifted, resulting in a zero crossing of $\Delta \lambda _r/ \Delta T$, by slightly adjusting the geometry of the waveguide. Increasing the feature size of the silicon will reduce $\Gamma _{clad}$ and shift the curve in Fig. 7(c) upwards. It is also interesting to calculate the effective thermo-optic coefficient of waveguide $TOC_{\textit {eff}} = d n_{\textit {eff}} / d T$, which can be obtained from measurements using the expression [40,41]

5. Conclusion

In summary, we demonstrated a novel heterogeneous silicon subwavelength grating waveguide with a high-index chalcogenide glass cladding that exhibits athermal operation and propagation loss of $8.6$ dB/cm. Further work will focus on using the athermal SWG in concrete applications and improve the loss performance. This work demonstrates the potential of heterogeneous SWGs to engineer light-matter interaction for some of the most critical challenges facing silicon photonics.