#### Abstract

We propose and simulate a third-order 3D electro-optically highly tunable compact add-drop filter based on nonlinear microring resonators. The used tuning mechanism relies on enhanced Kerr nonlinearity in a graphene layer integrated on top of a two-photonabsorption-free and low loss silicon-rich nitride core material at telecommunication wavelengths. An ultrahigh tuning efficiency (150 pm/V) over a tuning range of 1.3 nm, ensuring ultralow voltage consumption, was achieved in this work. We used titanium oxide and silicon oxide as the upper-cladding and under-cladding materials, respectively, around the silicon-rich nitride core material, to come up with a polarization-insensitive, and the thermally resilient third-orderadd-drop optical filter in the L band (1565 nm–1625 nm) with a full wave at a half maximum bandwidth of 50 GHz (linewidth of 0.4 nm) around 1570 nm, a high-free spectral range of 18.5 nm, a quality factor of 2580, an extinction ratio of 60 dB, a finesse of 19, and a thermal stability of 0.3 pm/K. A three-dimensional multiphysics approach was used to simulate the propagation of transverse electric and transverse magnetic polarized waves through the filter, combining the electromagnetic features with thermo-optic and stress-optical effects. The contribution of this work to the existing literature is that the designed filter proposes a new and highly tunable material system compatible with the complementary metal-oxide-semiconductor fabrication technology while combining high tunability, polarization insensitivity, and high thermal stability features for an ultracompact and energy-efficienton-chip integrated photonic tunable filter for dense wavelength division multiplexing systems in the less occupied *L* band.

#### 1. Introduction

On-chip silicon photonic integrated devices for the next-generation technologies have attracted great interest where small footprints for ultradense integration, low-power consumption, high-speed data transmission, high efficiency, and ultrafast tuning are highly desired [1, 2]. In that line, ring resonator-based devices such as add-drop filters [3], delay lines [4], all-optical switches [5], high-speed modulators [6], and optical frequency comb generators [7]. have shown their capability in simultaneously addressing challenges such as low drive power, compactness, and a complementary metal-oxide-semiconductor (CMOS)-compatibility in photonic integrated circuits. In the past, particularly for optical filters, different material systems including Si [8], SiN [9], InP/InGaAsP [10], AlN [11], GaAs [12], polymers [13], and chalcogenides [14] have been used to fabricate ring resonator-based devices. The tuning mechanisms investigated in these materials include the thermo-optic effect [15], the electro-absorption effect [16], the electro-optomechanical effect [17], and the electro-optic (EO) effect [1, 18]. Unfortunately, many of the previously mentioned materials suffer from one or many drawbacks and limitations such as low tuning speed and efficiency, temperature and polarization sensitivity, aging, stability, scalability, and large footprint.

For illustration, silicon is an excellent CMOS-compatible material for passive and active ultracompact optical components, but it is challenging to achieve athermal, polarization-insensitive, highly efficient tunable devices using pure silicon material in the near-infrared (IR) at telecommunication wavelengths. The most widely used approach to realize tunable integrated silicon photonic systems is based on the thermal effects in silicon due to its high thermo-optic coefficient (1.8 × 10^{−4}/K). The tuning efficiency for this method can reach 48.4 mW/nm while dynamic tuning measurement shows that the 10%–90% rise and 90%–10% fall time of the filter are 12.63 and 6.31 *μ*s, respectively, which is quite slow [19]. Additionally, it adds complexity to the device by integrating microheaters [20]. This results in high voltage consumption, which impedes the aim of realizing a compact, fast, and energy-efficient optical device for on-chip photonics.

On top of that, other shortcomings such as high bending losses and polarization rotation due to the strong high index contrast for small ring radius, ring resonators in a Si-SiO_{2} material system have pushed the researchers in current photonic technologies to integrate onto the silicon chip low-index contrast III-V compound materials [21] and other materials such as silicon nitride derivatives, glass, and polymers [22], often at the expense of large footprint, CMOS-incompatibility, and the high fabrication cost. Especially, the monolithic integration of III-V compound materials on silicon is impeded by high-production costs and large mismatches in lattice constants and thermal expansion coefficients [23].

To address the thermal dependence, athermal core materials such as Ta_{2}O_{5} materials [24] or top cladding materials with negative thermo-optic coefficients such as titanium dioxide [25, 26], sol-gel materials [27], chalcogenides [14], and polymethyl-methacrylate (PMMA) [28] have previously been used to offset the positive coefficient of the core and substrate materials (Table 1).

Polarization independence can in principle be achieved by carefully balancing material-dependent and geometry-dependent effects. Therefore, to reduce the polarization dependence due to structural birefringence, Si_{3}N_{4}, SiON as alternative materials to silicon are reported in the literature [26, 29]. The methods mentioned previously are preferred to the use of polarization diversity schemes or additional compensating functions (polarization rotator and polarization splitter-rotator) [30] that come with extra complexity and additional components.

This work combined the use of a highly electro-optically tunable hybrid graphene-coated core material (Si_{3}N_{4}) which exhibits low loss at telecommunication wavelengths and low structural birefringence [31] with proper optimization of the cross-section for polarization-insensitive waveguide. We determined a critical radius for ring resonators to minimize bending losses and polarization rotation. To counter temperature dependence, a top cladding material with a negative thermo-optic coefficient (TiO_{2}) was used. To enhance the tuning efficiency, a 2D layer of graphene for its unprecedented high Kerr coefficient, high mechanical strength, high flexibility, high optical transparency, high carrier mobility [32], and high responsivity respecting CMOS-compatible processes [33] will be deposited on the top of the core material to realize a tunable, polarization insensitive, and temperature-independent filter while retaining the silicon platform for its low-fabrication cost. As a result, we use the photonic confinement and long-range waveguiding properties of silicon-rich nitride while also leveraging graphene’s high optical nonlinearities for electro-optical tuning. Recently, measurements have confirmed that graphene exhibits a strong Kerr nonlinearity [34].

Graphene on a silicon dioxide material system was reported in [35] and is found in a material system with SiO_{2} and Si_{3}N_{4} to realize experimentally a tunable delay line [36]. The proposed electrodes for this device are made of titanium nitride (TiN), which has optical properties that are very similar to those of gold, yet, unlike the latter, TiN is CMOS-compatible [23].

The theoretical background of the research work and the methodology used for the proposed model are given in Section 2. Discussion and results of the simulated model are reported in Section 3. Finally, Section 4 provides the final conclusions of the work.

#### 2. Materials and Methods

The materials and methods are as follows:

##### 2.1. Materials

The transfer functions at the drop and through ports were determined to evaluate the response of the filter in the less occupied *L* band (1565 nm–1625 nm). The coupled mode theory was applied directly on a third-order filter model in Figure 1 and takes into account the amplitude decay rate due to intrinsic loss *γ*_{i}, and the decay rates due to coupling from outer microrings to the input and output waveguides, *γ*_{0}, and *γ*_{3}, related to the ring-to-bus energy coupling coefficients by and . The following differential equations describe the mode coupling in the filter [38]:

**(a)**

**(b)**

The ring-to-ring energy coupling coefficients *μ*_{k} can be related to the corresponding field coupling coefficients *κ*_{k} through

For ring-to-bus waveguides energy coupling coefficients.

Equations (2) and (3) are then later used to determine the coupling gap distances between the microrings and between the microrings and the bus waveguides using the Bahadori model [15].

However, (1) becomes incapable of describing the interactions between light and matter when nonlinear effects are involved. If the nonlinearities are considered, they will modify the refractive index of the cavity and shifts the central frequency to . Therefore, the nonlinear perturbation theory is combined with (1) by considering the following corrections and as perturbations to the model in (1).

Due to the strong Kerr nonlinearity in graphene and silicon rich nitride [39], it is assumed that the Kerr nonlinearity is the main source of the refractive index change , given bythe nonlinear Kerr parameter is given bywhere *n*_{2} is the Kerr coefficient of graphene. *U* is the power derived from the applied bias voltage on the electrodes.

A change in effective refractive induces a resonant frequency shift computed using perturbation theory aswhere is the volume of the ring resonator and *E* is the electric field intensity confined in the ring.

We assume that the ultrafast Kerr effect, relatively low in pure silicon material, is the most dominant phenomenon in graphene [33], while other nonlinearities do not substantially contribute. is reported to suppress the slow two-photon absorption (TPA) nonlinearity for wavelengths greater than 1.2 *μ*m, which is a source of nonlinear losses at high power [40]. exhibits a Kerr nonlinearity *n*_{2} of 2.8 × 10^{−7} m^{2}W^{−1}, approximately ten times higher than silicon material [41]. This effect changes the refractive index and thereby shifts the resonant wavelength [38].

Equation (1) is therefore modified to consider the nonlinearities in microring resonators as follows:

The solutions to (7) gives transfer functions of the filter considering a harmonic input wave excitation and harmonic solutions for the wave amplitudes of the form and .

Thus, through and drop ports transfer functions of the filter can be expressed, respectively, as

The complex centered frequency around the resonant frequency is written as , and is the expression of the denominator in the through-port transfer function. In the following paragraph, we assume that all ring resonators have the same resonant frequency as they have the same dimensions and are fabricated in the same material system.

The coupling coefficients for this case of a finite series of microrings must be apodized to mitigate wave reflections that cause large ripples in the passband. Closed formulas can be found for Butterworth filters with a flat-top response in the passband [42, 43]. Their values are given in Table 2.

To increase the tuning range of the filter and to suppress interference with neighboring DWDM channels, the second microring radius is modified to expand the free spectral range (Figures 1 and 3). The resulting device exhibited resonance where the individual resonances of all microrings in the device match [44].

The following step consists of investigating the athermal behavior of the proposed device. The resonant wavelength shift of a microring resonator due to temperature change is given bywhere and are the group indexes of the waveguide and the coefficient of thermal expansion (CTE) of the substrate, respectively. From (9), the athermal condition is achieved when becomes zero, thus has to be negative.

We can reduce the temperature-dependent wavelength shift (TDWS) by adjusting the thickness of under-cladding and upper-cladding materials around the core:

and are the under-cladding and upper-cladding confinement factors, respectively.

The zero TDWS condition of the waveguide can then be written as

The objective of this study is to use cladding materials with thermo-optic coefficients which have opposite signs in order to come up with an effective TOC that will counterbalance thermo-expansion effects in the device.

In order to realize simultaneously an athermal and polarization-insensitive device, the authors put extra effort into the engineering of the cross-section of the waveguide. When waveguides exhibit different index contrast in vertical and lateral dimensions, polarization independence may be achieved by modifying the height-width ratio of the waveguide or by simply adapting the waveguide width when the index contrast varies in the lateral direction [44].

The geometrical parameters (Figure 2) are optimized to make which is the zero birefringence condition. In addition, the cross-section dimensions should only allow single-mode operation for both orthogonal polarizations. Furthermore, the energy efficiencies should be such as and, apart from increasing losses, the induced stress enhances the polarization sensitivity and index shifts in the waveguide with ring radius reduction. Therefore, a critical radius was computed to minimize such effects.

As mentioned earlier in Section 1, the predominant nonlinearity in at telecommunication wavelengths is the Kerr effect. The TPA is absent due to its high band gap. The thermo-optic effects are intended to be suppressed by the application of the athermal condition (12). The ability to efficiently tune using the electro-optic (EO) effect is enhanced by depositing a layer of graphene on top of the core to get a material platform cumulating low loss, high nonlinearity, and highly efficient tuning without affecting its optical performance [45].

It is important to put attention to the optical characteristics of the graphene material. The Kubo model describes graphene by defining its surface conductivity and the complex permittivity , from which the refractive index (*n*) and the extinction coefficient(k) can be calculated [46].where is the angular frequency, is the scattering rate, is the chemical potential, is the temperature, is the charge of the electron, eV is the reduced Planck constant, and , the Boltzmann constant.

The permittivity of graphene can be deducted from (14) usingwhere is the vacuum permittivity, is the real part of the permittivity [47], and is the layer thickness.

Then, from (14), the complex refractive index is expressed by

The energy tuning range is determined by considering the surface current on the graphene layer using COMSOL Multiphysics software. When the tuning voltage is applied on TiN electrodes with an equivalent energy close to the the Dirac point (transition point between intraband and interband regions), interband absorption occurs as the electrons are excited by the incident photons. As a result, the graphene layers exhibit high absorption and behave as a lossy medium. When energy is increased up to a value greater than half the photon energy of the incident light, all the electron states in graphene would be filled. As a result, no interband transition is allowed due to Pauli blocking. In this condition, the graphene layers turn to be transparent [48]. Between the two extremes exist a step change region, (0.2 eV < *μ*_{c} < 0.6 eV), suitable for tuning since there is a significant stepping up in the permittivity profile of the graphene layer (Figure 4). This will change the effective index and subsequently shifts the resonant wavelength of the device [46]. When there is no applied voltage , the graphene layers behave as a transparent thin sheet.

##### 2.2. Methods

Nonlinear coupled mode theory (NCMT) was used to compute energy coefficients needed to determine the spectral responses of a maximally flat-response Butterworth filter in the L band from the spectral requirements for DWDM filters (ITU-T grid) [49]. The nonlinear coupled-mode equations were solved in MATLAB software. The design space exploration of the filter geometry enabled us to relate the spectral and the physical parameters of the filter. A material combination was chosen for the waveguide to ensure low (radiation and bending) losses, CMOS compatibility, compactness, a high tunability efficiency as well as athermal and polarization-insensitive behaviour using a mode analysis solver in COMSOL software. An optimized cross-section, a critical radius, and gap distances between ring resonators and between rings and waveguides were determined at this stage. As the study involves simultaneously electromagnetic wave propagation, heat transfer, and electro-optic tuning, a Multiphysics approach was used to investigate those different aspects of the problem to create conditions making the simulation as close to real life as possible. The finite element, beam envelope propagation method was used to solve the electromagnetic problem in the filter model. Simulations were performed on the ideal material combination chosen to build a highly tunable athermal and polarization-insensitiveadd-drop filter. A flow chart of the methodology is given in Figure 5.

#### 3. Results and Discussion

Given the cross-section dimensions that allow fundamental quasi-TE and quasi-TM modes and athermal behavior with high confinement of the fields in the core of the filter waveguides, this step aims to determine the effective index, the group index as well as a critical bending radius that determines the upper limit for bending losses, stress-induced resonance shift, and birefringence in the filter. To select the critical bending radius, the bent waveguide was replaced with an equivalent straight waveguide with coordinate transformation following conformal mapping [22]. The bending loss was computed from the imaginary part of the effective index for radii in the range of 6 to 37.5 *μ*m at 1570 nm. When the waveguide is bent, the effective index turns complex, and its imaginary part is used to estimate the bending loss. A critical bending radius value of 27.5 *μ*m corresponding to a maximum bending loss of 0.0001 dB/cm was used for the device (Figure 6. The results of this study are found in Table 3. A slight difference in effective indices in TE and TM modes has been tolerated as a tradeoff to realize ring resonator-based devices with different materials that are nearly polarization insensitive. The results have shown that a small birefringence coupled with a selected value of of a trapezoidal cross-section (Figure 2) can minimize the polarization rotation effects on the spectral performances of the filter [50].

Using the serially-coupled ring resonators topology shown in Figure 1, transfer functions of a three-ringresonator-basedadd-drop filter were determined by solving the nonlinear coupled mode equations (7). The design parameters to consider are as follows: a full wave at half maximum (FWHM) bandwidth of 50 GHz and an FSR of 3.7 nm and a resonance wavelength of 1570 nm. We assume that the waveguide cross-section geometry is constrained by the fabrication process, temperature and polarization stability, and an upper limit of bending losses by using COMSOL Multiphysics. The results from (7) were compared to simulation solutions using a finite element method. Figure 7 shows simulation results of flat-top spectrum responses of a third-order Butterworth filter and demonstrates an excellent agreement between the predictions of the nonlinear coupled mode theory (NCMT) and simulation in the frequency domain finite element mode method (FEM) using COMSOL. In this simulation, the graphene response is modeled as a surface electric current through the electrostatics module. However, a loss increment up to 0.8 dB in the passband from 0.51 dB observed during the NCMT simulation is measured in FEM simulation results. This can be explained by different loss contributions from different sections of the filter, especially polarization losses and coupling losses at the interfaces between microrings and between microrings and bus waveguides. The gap distances are related to the field coupling coefficients given in Table 2 where we notice a slight difference in TE and TM modes values, due to the small birefringence in the material system.

The confinement of the field in the straight waveguide depends on the choice of the materials, the waveguide cross-section engineering, together with the operating wavelength range. With a bent waveguide, particular attention was put on the determination of the critical radius to keep the field confined in the center of the core, or equivalently, to reduce the bending losses (Figure 6). Additionally, there is a correlation between the critical radius and the induced stress in the waveguide that can contribute to additional fluctuations of resonant wavelength. In the ring resonators, only wavelengths such that , where is an integer and is the effective index of the microring, will propagate through the filter structure and exit at the drop port. Therefore, a sweep-step study is performed on the *L* band wavelength range to determine wavelengths that satisfy the resonance condition. A value of = 1570 nm was found, and strong field confinement in the filter structure was observed (Figure 8).

To expand the free spectral range (FSR) or equivalently increase the distance between two consecutive resonant wavelengths, a ring resonator with a smaller ring radius is used (Vernier effect). The interstitial resonance amplitudes are lower than 20 dB (7) and thus cannot interfere with other DWDM signals. As a result, the FSR was expanded from 3.7 nm to 18.5 nm, i.e., by a factor of five.

Parameters such as the scattering rate , the graphene layer thickness , and the applied voltage were analyzed. The increase in the applied voltage induces changes in the transmission spectrum. Wavelength resonance shifting can be observed as long as the breakdown voltage is not reached. The increase in the applied voltage beyond this point is associated with fluctuations of the amplitude response (Figure 9). In the simulated device, the breakdown was observed at the value slightly below 10 V. To investigate the tuning efficiency, the Electrostatics physics was coupled with the Electromagnetic Waves Beam Envelope physics and the simulation results are illustrated in Figures 9 and 10. The simulation results revealed tuning efficiencies of ∼150 pm/V and 21 pm/V over a tuning range of 1.3 nm, with and without the graphene layer, respectively, which confirm the enhancement role of the integrated graphene layer.

A particular attention has been placed on the choice of the electrodes. The material (TiN), size, and location were chosen to transfer efficiently the applied voltage to the ring resonators without altering the optical performance of the filter. As a consequence, the center wavelength was shifted with the applied voltage (Figure 9) as a result of the variation of the refractive and group indices. Figure 11 shows a uniform distribution of the voltage in different microrings. Such a distribution is necessary to ensure a synchronous tuning of individual resonances of the different microrings and will avoid differences with the overall resonance wavelength of the filter.

In this work, the optical filter performance was investigated in the presence of temperature changes. In fact, the thermal variations may result in structural deformation of the filter geometry, i.e., the expansion of the rings’ circumference and/or different layers of the filter structure. Thus, the resonant wavelengths may be affected. A temperature gradient of 60 K from the substrate was considered in the study. We assume a situation where the temperature excursion in the filter is consecutive to overheating due to power dissipation in the filter itself or in the surrounding components, or simply due to an external heat source. The temperature propagates through the structure from the bottom of the substrate (Figure 12). The figure shows a quasiuniform profile except in joints and in curvatures of the ring resonators. To study their effects on the spectral properties of the filter, Heat Transfer in Solids physics and Solid Mechanics physics coupled to thermal expansion Multiphysics were added to the electromagnetics simulation in the model. The simulation results for temperature and stress distributions are given in Figures 12 and 13, respectively. We observed that these effects did not distort the quasifundamental TE and TM modes of the filter at resonance but induced a slight shift of the S-parameters (Figure 14). The overall thermal effects over a gradient of 60 K (within a temperature range from 25 to 85°C) are a 19 pm-resonance shift, thereby temperature stability of 0.3 pm/K.

**(a)**

**(b)**

Thermo-optic effects on the filter are not significant due to the athermal condition imposed by the choice of the material system used. However, the mismatch of coefficients of thermal expansions of different layers in the structure of the filter and particularly the expansion in ring resonators (stress-optical effects) explain the slight resonant wavelength detune. Figure 14 shows a slight shift in the center wavelength and a small increase of loss in the passband (0.1 dB). However, these changes do not affect the performance of the filter for DWDM systems.

These results enabled us to state that particular attention should be put on the choice of the material system, the computation of the cross-section dimensions, the size of different layers, and the ring resonator bending radius.

#### 4. Conclusion

This work investigated the athermality, polarization-insensitivity, and tunability of an add-drop filter based on ring resonators for DWDM systems. Using a 3D Multiphysics approach, the main goal was to relate the spectral parameters specified by the ITU-grid for DWDM systems and the material, physical, and optical parameters of an athermal, polarization insensitive, and tunable filter. The results demonstrate an innovative tunable filter exploiting nonlinearities in silicon-rich nitride enhanced by a graphene layer on top of the core material. We found that the electro-optically tunable filter can arbitrarily operate with TM and TE polarized signals in the less occupied L band (1565 nm–1625 nm) with high thermal stability (0.3 pm/K) and without interference with neighbouring DWDM channels on a range of 18.5 nm.

In the future, this research intends to consider all-optical tuning methods and compare the performance with the electro-optical method used in the work for modern DWDM systems.

#### Data Availability

This is to state that data supporting the results will be made available on request through the corresponding author (Filston Rukerandanga, [email protected]).

#### Disclosure

The study is part of the academic requirements for a Ph.D. degree.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors thank the African Union Commission for Education for granting the software facilities to make the simulations and the Kenya Education Network (KENET) for granting the simulation facilities (remote High-Performance Virtual PC).