Dispersion control of silicon nanophotonic waveguides using sub-wavelength grating metamaterials in near-and mid-IR wavelengths

: Controlling the group velocity dispersion of silicon nanophotonic waveguides has been recognized as a key ingredient to enhance the development of various on-chip optical applications. However, the strong wavelength dependence of the dispersion in waveguides implemented on the high index contrast silicon-on-insulator (SOI) platform substantially hinders their wideband operation, which in turn, limits their deployment. In this work, we exploit the potential of non-resonant sub-wavelength grating (SWG) nanostructures to perform a flexible and wideband control of dispersion in SOI waveguides. In particular, we demonstrated that the overall dispersion of the SWG-engineered metamaterial waveguides can be tailored across the transparency window of the SOI platform, keeping easy-to-handle single-etch step manufacturing. The SWG silicon waveguides overcladded by silicon nitride exhibit significant reduction of wavelength dependence of dispersion, yet providing intriguing and customizable synthesis of various attractive dispersion profiles. These include large normal up to low anomalous operation regimes, both of which could make a great promise for plethora of emerging applications in silicon photonics.


Introduction
The silicon-on-insulator (SOI) has been established as a promising waveguide technology to realize key integrated nanophotonic components in various on-chip optical applications. The SOI offers inherent benefits such as nano-scale miniaturization, ultra-dense integration, as well as leveraging manufacturing processes that exist in microelectronics facilities [1,2].
Sub-wavelength gratings (SWG), i.e. nanophotonic segmented structures with dimensions on a scale below the wavelength of light, have enabled a profound technological concept of local refractive index engineering [21]. Since, the first demonstrations of an optical waveguide with a SWG metamaterial core by Cheben et al. [22][23][24], metamaterial SWG waveguides have attracted a strong research interest, because of their great potential for customizing light propagation properties in planar waveguides [21]. More recently, applications include, but not limited to, efficient optical coupling interfaces [25,26], biosensing devices [27,28], optical filters [29,30], broadband directional [31] and multimode interference couplers [32], or suspended Si waveguides for extended mid-infrared (mid-IR) operation [33]. Expanding the spirit of refractive index engineering into a broader sense, SWG metamaterials are poised to locally synthesize different group indices, thereby enabling additional flexibility to control the overall waveguide dispersion. Recently, several broadband metamaterial engineered photonic devices have been reported [26,31,32,34], showing albeit an indirect route of dispersion engineering. From a fundamental point of view, it is practically more attractive to explore dispersion properties of such metamaterial waveguides, since they hold the promise of forming basic building blocks for complex nanophotonic components [35][36][37][38]. So far, the SWG metamaterials have only been partially examined, merely to assist the dispersion controlling in silicon waveguide, exploiting the well-known mode transition effect [36,37]. However, the intrinsic dispersion characteristics of conventional strip SWG waveguides remained unexplored.
In this work, we exploit the potential of SWG metamaterials for the development of flexible and wideband dispersion control in the silicon waveguides overcladded by silicon nitride layer. In particular, we showed that the overall dispersion of SWG-based metamaterial waveguides can be a useful design strategy to substantially relax the typical strong wavelength dependence of conventional silicon waveguides. This way, the dispersion of the metamaterial-controlled silicon waveguide can be tuned in a wide spectral range, yet keeping single-etch step fabrication. The advantageous utilization of SWG nano-structuration enables profound design capability for synthesizing various attractive dispersion profiles, ranging from large normal up to low anomalous operation regimes. Those dispersion characteristics of silicon waveguides engineered by SWG segmentation could be of great interest for broad application scenarios in silicon photonics.

Fundamental design considerations
In this work, the silicon nitride (Si 3 N 4 ) is utilized as an upper cladding material (superstrate). The appropriate choice of the cladding is a critical design consideration, particularly relevant in case of waveguide dispersion engineering. This choice solely dictates the overall wavelength dependence of the nanophotonic guiding system [8]. More specifically, for conventional waveguides built on inherently high index contrast Si platform [with a crosssection shown in Fig. 1(a)], the overall waveguide dispersion typically exhibits strong wavelength-dependent behavior, as shown in Fig. 1(b). Here, a typical single-mode strip Si waveguide of 220-nm in thickness and 500-nm in width is considered, with different upper cladding materials (air, silicon dioxide (SiO 2 ), and Si 3 N 4 ). In the case of air or SiO 2 claddings, the large index difference between the waveguide core and the respective cladding is responsible for the comparatively higher waveguide dispersion, which hinders the development of wideband optical applications. The Si 3 N 4 as a cladding material still enables index contrast that is sufficiently high to allow for waveguide dispersion engineering, while significantly mitigates the wavelength dependence of dispersion, i.e. the wavelengthdependent profile of waveguide dispersion becomes virtually flat compared to the typically used claddings (air or SiO 2 ). Albeit, beyond the scope of this work, it is worth highlighting that the advantageous combination of Si and Si 3 N 4 could pave a way for extending on-chip functionalities in photonic applications. More specifically: (i) Unlike Si, Si 3 N 4 , has a large bandgap, thereby does not suffer from the undesired nonlinear losses caused by two-photon absorption across the near-infrared (near-IR) spectral region [5]; (ii) Si has no natural secondorder nonlinearities, thereby suffers from the inherent lack of high-speed electro-optic effect [3,5], while the utilization of Si 3 N 4 as a cladding on top of Si waveguides facilitates the development of strain-induced nonlinearities for ultra-fast optical modulation [39,40]; (iii) The cubic nonlinear effects are typically available in both materials [3,5], and both are comparatively larger than that of pure silica [8]; (iv) Both Si and Si 3 N 4 materials are wellestablished for a wide range of passive and active photonic devices, indeed, with a mature CMOS-compatible fabrication [1][2][3]5].
The two-dimensional (2-D) schematic of vertical geometry of the sub-wavelength grating (SWG) metamaterial waveguide is shown in Fig. 1(c). The guiding structure comprises composite waveguide core of thickness H c and width W c . This is formed by the non-resonant nanophotonic structure, here implemented as a longitudinally-invariant periodic array (along the z-direction) of fully etched rectangular trenches of length L t filled by the lower refractive index material separated by unetched silicon (Si) blocks, with a length L b . The lengthwise period of SWG array, defined as Λ = L b + L t , is smaller than half of the effective wavelength in a metamaterial waveguide, thereby mitigating losses and wavelength resonances by frustrating the diffraction [21]. The duty cycle of the SWG structure is denoted asDC = L b /Λ. Our investigations were performed for the transverse electrical (TE) polarization and operation in a wavelength range from 1.3 µm to 3.7 µm. To completely exploit the transparency window of SOI, waveguide dispersion designs have been carried out on various silicon thicknesses. The 220-nm-and 300-nm-thick SOI wafers are typically used by academia and industry for the development of silicon photonics devices [41], while 400-nmand 500-nm-thick Si waveguides are desirable for emerging applications in the mid-IR wavelengths [42].

Simulation strategy
In order to estimate the dispersion trends in SWG metamaterial waveguides, at a first stage, it is necessary to calculate the effective index of waveguides. In the SWG waveguide, the electromagnetic field propagates in the form of Bloch-Floquet modes, which are theoretically lossless [21]. The propagation of Bloch-Floquet modes is restricted by two factors. First, the effective index n BF of the fundamental mode has to meet the following condition: n SiO2 or n Si3N4 < n BF < λ/(2Λ). The left-sided limit (n SiO2 or n Si3N4 < n BF ) assures that the optical field confinement is sufficiently high to guide a mode inside the metamaterial waveguide core, while the upper limit (n BF < λ/(2Λ)) avoids the formation of Bragg reflection zones. To calculate the effective index of the fundamental TE-polarized Bloch-Floquet mode, different simulation strategies were considered: (1) Effective Medium Theory and 3-D homogeneous equivalent waveguide simulations: For most practical cases of interest, the longitudinally periodic structure operates in the SWG regime, i.e. the operation regime, in which the SWG period (Λ) is small enough compared to the designed wavelength (λ) [21]. Here, the inherent behavior of the SWG waveguide may be treated as an optically equivalent waveguide of metamaterial refractive index. This refractive index depends on the grating geometry, polarization, constituent materials, and wavelength. This metamaterial refractive index is estimated by means of Effective Medium Theory (EMT) [43]. The optically equivalent waveguide is described as the conventional strip waveguide of homogeneous metamaterial core that has the same transversal cross-section (thickness and width). Then, the Bloch-Floquet effective index of the SWG waveguide is approximated by the effective index of the fundamental mode in the homogeneous equivalent waveguide. Simulations were performed by full-vectorial three-dimensional (3-D) Finite Difference Eigenmode (FDE) solver [44]. The Perfect Matched Layers (PMLs) were used as boundary conditions. The uniform mesh grid size of 15 nm was used in simulations.
(2) Effective Index Method and 2-D Bloch-Floquet simulations: The 3-D problem was firstly transformed to the equivalent 2-D problem with the help of the Effective Index Method [38]. Here, the geometrical variations in the transversal cross-section of the waveguide were taken into the account, which in turn, yielded changes in the effective mode index. The calculated effective index is sub-sequently used for the final 2-D longitudinally periodic waveguide and the complex Bloch-Floquet modal analysis was carried out. This was performed by semi-rigorous 2-D Fourier-type simulation tool based on the Eigenmode Expansion Method (F-EEM) [45], optimized for analyzing the periodic waveguides. (

3) The rigorous full-vectorial 3-D Finite Difference Time Domain (FDTD) simulations:
The FDTD Solutions tool [44] was also used for further study. Here, the real 3-D structure of the SWG waveguide was considered. Taking the benefit of the structure periodicity (the SWG waveguide is periodic along the propagation direction), the 3-D structure was simulated along one period, where Bloch boundary conditions were applied. This simulation strategy follows bandstructure calculations used in photonic crystals [46]. For the rest, the PMLs were used as boundary conditions. The simulation grid size was: Δx/Δy/Δz = 10 nm/10 nm/10 nm. For Bloch-Floquet mode excitation, randomly distributed dipoles were used as an input source. From the recorded time signals, the resonant Bloch-Floquet modes were determined with the help of Fourier Transform (FT). It is worth nothing that in afore-described approaches, the intrinsic dispersion of constituent materials (Si, SiO 2 , and Si 3 N 4 ) was taken into the account via Sellmeier equations [47,48]. The dispersion characteristics of SWG metamaterial waveguides were calculated using the formula for dispersion parameter D, expressed in [ps/(km·nm)]: (-λ/c)·(d 2 n BF /λ 2 ). (ii) For SWG metamaterial waveguides with a larger cross-sectional area [see Figs. 2(b), 2(c) and 2(d)], the semi-rigorous 2-D F-EEM + EIM calculations are in very good agreement with the rigorous, although computationally demanding, 3-D FDTD calculations. Indeed, both simulation strategies advantageously take into the consideration the natural resonant/non-resonant behavior of SWG structures. This way, the semi-rigorous 2-D F-EEM approach appears to be a feasible design strategy to estimate the overall trends of dispersion, especially in terms of simulation complexity and computational time.

Results and discussion
The potential of the SWG nano-structuration to control the overall dispersion of Si waveguides is shown in Figs. 3(a) -3(c), respectively. Firstly, different thicknesses of Si waveguides were considered, with a typical transversal aspect ratio of ~1:2. This results in waveguide cross-sections of H c = 300 nm by W c = 600 nm; H c = 400 nm by W c = 800 nm; and H c = 500 nm by W c = 1000 nm, respectively. For these cross-sectional dimensions, both conventional strip Si and metamaterial waveguides are designed to preserve single-mode operation [13,41]. This is generally a superior propagation scenario in order to avoid potential modal cross-talk. However, in practical dispersion-engineered waveguides [5,8], this preferential single-mode condition can be considerably relaxed in order to afford substantial enhancement of the dispersion bandwidth. The longitudinal dimensions of the sub-wavelength grating are: Λ = 200 nm, which is small enough to suppress diffraction over the spectral range under investigation, and different lengths of Si segments, i.e. duty cycles of DC = 25%; DC = 50%; and DC = 75%. For shorter wavelengths of the SOI transparency window, the fundamental mode propagating through the waveguide is typically well-confined inside the composite core of the metamaterial waveguide, thereby waveguide dispersion profiles are mostly governed by the longitudinal geometry of the sub-wavelength nanostructure. Generally, by increasing the duty cycle, i.e. increasing the amount of Si within the SWG period, which in turn, enhances the vertical index contrast between the metamaterial waveguide core and Si 3 N 4 cladding, the modal confinement of the SWG metamaterial waveguide increases as well. More importantly, at shorter wavelengths, the waveguide operates on the edge of the Bragg reflection zone (n Bragg of ~3.25, for Λ = 200 nm at 1.3 μm), thus characteristics of the effective mode index are inherently more affected by the geometry of the sub-wavelength grating.
The trends of the overall dispersion in SWG metamaterial waveguides are more pronounced, once we provide a comprehensive comparison of the calculated dispersion profiles in respect to the waveguide thickness, as shown in Fig. 3, for the particular longitudinal geometry of the SWG nanostructure. Generally, for a fixed SWG geometry, it is apparent that the metamaterial waveguides become less dispersive for thicker Si layers, originating from the improved vertical optical confinement. As a result, the increase in the Si waveguide thickness produces a direct vertical upshift of the dispersion profile. The absolute values of the waveguide dispersion were smaller, gradually approaching zero dispersion regions up to the ultra-low anomalous (positive) dispersion regime, as shown in the insets of Figs. 3(b) and 3(c), respectively. In all cases illustrated in Fig. 3, it can be observed that for shorter wavelengths, the SWG metamaterial waveguides exhibited ultra-large normal dispersion characteristics. Since, for thicker Si waveguide, the effective index of the Bloch-Floquet mode proportionally increases, the effect of normal dispersion is even stronger in that case. The absolute values of dispersion are comparatively higher at shorter wavelengths and for larger lengths of Si segments. The combined interplay and collective contribution of: (i) the SWG geometry, operating close to the Bragg reflection zone; (ii) increased vertical confinement of the Si layer; and (iii) strong material dispersion of Si close to the material bandgap, results in the obtained ultra-strong normal dispersion profiles. Meanwhile, the normal dispersion operation regime could be useful for the realization of integrated photonic components for on-chip pulse compression or development of optical delay lines.
On the other hand, with the increase in wavelength, in general, by increasing the wavelength-to-period ratio, in particular, one can observe very sharp increase of the dispersion curves. In this spectral region, the effective mode index (for fixed transversal and longitudinal geometry) becomes more delocalized from the waveguide core and its optical properties are comparatively more affected by the optical properties of Si 3 N 4 segments instead of the Si properties. Far away from the Bragg zone, i.e. the SWG metamaterial waveguide operates in true SWG regime, it can be seen that dispersion profiles are virtually flat over the wide spectral range. The sub-wavelength grating waveguide acts as an optically equivalent homogenous metamaterial, whose propagation properties are driven by the spatial averaging effect. This clearly demonstrates that the SWG nano-structuration of the Si waveguides holds the great promise for controlling the overall dispersion characteristics, particularly in term of their absolute values as well as in term of their spectral profile, thereby substantially alleviating the strong wavelength dependence of the waveguide dispersion. Furthermore, for developing feasible photonic waveguides and complex components with engineered dispersion, waveguide losses play a critical role. This is particularly important for SWG-structured metamaterial waveguides, since propagation properties of such waveguides depend on the waveguide geometry comparatively more than in case of conventional Si waveguides. For dispersion controlling, the modal confinement is of essential interest, since it is directly related to both propagation properties such as waveguide losses and effective mode index, respectively. To estimate the loss evolution in SWG waveguides, the 3-D full-vectorial analysis was performed [44]. In Figs. 4(a) -4(c) is shown the spectral evolution of losses of investigated SWG metamaterial waveguides, considering different cross-sectional and longitudinal geometries. From the performed simulations, it becomes apparent that for smaller duty cycle (larger gap of grating trenches with lower refractive index material, [see Fig. 4(a)]), SWG metamaterial waveguides experience more losses due to the reduced confinement of the optical field, i.e. the optical field is more delocalized from the waveguide core. The modal confinement dramatically diminishes with the increasing wavelength, thereby making the space for losses. This way, the useful transparency window of the guiding structure is substantially limited. Conversely, as shown in Figs. 4(b) and 4(c), for larger duty cycles (smaller gaps of grating trenches), the mode delocalization effect decreases (more high refractive index material within the SWG period), which in turn, extends the transparency window of the guiding structure. As predicted by simulations, while the optical field confinement is optimal, the waveguide losses may be considered to be negligible. This confirms the theoretically lossless propagation of the fundamental Bloch-Floquet modes. Furthermore, it is interesting to follow that for SWG waveguides of fixed duty cycle and of thinner waveguide core, the sharp increase of waveguide losses can be observed. This is particularly caused by the fact that with increasing wavelength, by natural behavior, the effective index of the Bloch-Floquet mode gradually approaches the material index of the superstrate (Si 3 N 4 ). This effect is more pronounced (the rate of this approach) in case of thinner waveguides, because the proper optical confinement is rapidly lost. Therefore, beyond the cut-off wavelengths, the mode is no longer guided inside the metamaterial waveguide core and it will radiate into the upper cladding, yielding significant contribution to losses. In Fig. 5, it is shown the evolution of the dispersion as a function of the wavelength for different transversal and longitudinal geometries of the SWG metamaterial waveguides. Here, various transversal aspect ratios of 1:2; 1:3; and 1:4 are considered, with a fixed waveguide thicknesses of 400-nm [Figs. 5(a) -5(c)] and 500-nm [Figs. 5(d) -5(f)], respectively. The duty cycles were: DC = 25%; DC = 50%; and DC = 75%. Generally, for larger Si waveguide widths and fixed Si waveguide thickness, it can be observed that spectral curves of the overall dispersion are red-shifted, i.e. the peak values of the absolute dispersion are moved towards longer wavelengths. Furthermore, for a fixed longitudinal geometry of the SWG structure, the increase in the transversal aspect ratio of the Si waveguide affords enhancement in the absolute value of the waveguide dispersion. In other words, this pushes the spectral dispersion profile from the normal dispersion regime towards the anomalous regime. In addition, the increase of the transversal aspect ratio yields at the same time the substantial enlargement of the spectral bandwidth of the respective dispersion profile. This way, the cut-off wavelengths are moved towards the longer wavelengths, i.e. hindering the mode radiation towards the superstrate and facilitating the exploitation of the full SOI transparency window. It is interesting to follow that this way, the SWG waveguide affords a useful ultra-wideband dispersion control. It can be seen in Figs. 5(a) -5(c) that the calculated dispersion profiles are virtually flat in wide wavelength range, with an ultra-low or a moderate normal dispersion. This spectral dispersion features would be desirableto implement photonic functionalities for linear broadband dispersion compensation in optical communications, Kerr-type combs generation, or optical pulse shaping. On the other hand, in case of 500-nm-thick Si waveguide and DC = 75% [see Fig. 5(f)], the dispersion reaches the peak of 24 ps/(km·nm) at 2.20 µm, with an anomalous dispersion bandwidth of ~250 nm, for a transversal aspect ratio of 1:2. For an aspect ratio of 1:3, the peak value of dispersion is 156 ps/(km·nm) at 2.75 µm, with an extraordinary 1240-nm-wide dispersion bandwidth. Compared to case of 1:2 aspect ratio, this yields a five-hold enhancement of the anomalous dispersion bandwidth. In case of 1:4 aspect ratio, the peak dispersion is 142 ps/(km·nm) at a wavelength of 3.3 µm and the bandwidth is theoretically larger than 1300-nm, since 3.7-µm in wavelength is the edge of the SOI transparency [42]. These attributes make the SWG metamaterial waveguides particularly attractive choice in nonlinear photonic applications, demanding appropriate controlling of the phase-matched processes across wide spectral bandwidths such as supercontinuum and frequency comb generations or parametric conversions.

Conclusions
In summary, we report on waveguide dispersion designs by exploiting the profound technological concept of the SWG nano-patterning implemented in a dominant silicon-oninsulator platform. Specifically, we demonstrated that the overall dispersion of SWG metamaterial waveguides can be seamlessly engineered, allowing flexible and wideband dispersion control across the transparency window of the silicon-on-insulator platform, while keeping simplicity in the waveguide fabrication. The advantageous utilization of Si and Si 3 N 4 segments interlaced on a scale bellow the wavelength of light alleviates the strong wavelength dependence of conventional Si waveguides. We showed that the SWG structures enable customizable synthesis of different attractive dispersion profiles, ranging from ultra-large normal up to low anomalous operation regimes. The SWG metamaterial waveguides hold immense promise for broad nanophotonic application scenarios, particularly interesting for the development of complex photonic components with extended functionality for optical communications, dispersion compensation, and optical signal processing.