## Abstract

Lately, the integration of two-dimensional materials into semiconductor devices has allowed the modification of their effective index by simply applying a modest voltage (between 0 and 3 volts). In this work, we present a device composed of two evanescently coupled silicon microring resonators where both rings have a graphene layer on top. This design is aimed to produce frequency combs with transmission characteristics controlled upon voltage application to the graphene layer. We numerically analyze the device response as a function of the incident wavelength and applied voltage. The results showed a low input intensity (0.6 GW/cm^{2}) needed and a rapid response time (0.1 *μ*s), in comparison to devices controlled by heat injection.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Two-dimensional materials are well known for their electronic and mechanical properties [1,2] leading to a vast number of applications. Among them, graphene is the most popular and studied specimen due to several features, e.g. its peculiar band structure [3–12]. The conduction and valence bands, touch at particular points named Dirac points. Around them, the density of states of carriers is low and consequently, graphene’s Fermi energy can be significantly modified by applying a relatively low voltage. The possibility of tuning the Fermi level allows the variation of graphene’s refractive index. Another extraordinary property is that the optical absorption of graphene does not depend on the wavelength in a wide range of photon energies, including the traditional optical communications bands extending from mid to far infrared [13,14].

By applying an oxide layer on top of a waveguide and a graphene sheet above, a MOS (metal-oxide-semiconductor) capacitor is built. In this way, the effective index of the complete waveguide can be varied by applying voltage to the graphene layer. This effect has been experimentally confirmed [15] and can be used to modify the optical properties of several applications [16,17], among which, we can find frequency combs.

An optical frequency comb is an optical signal whose spectrum consists of equidistant spectral lines, i.e., on equally spaced optical frequency components. These objects have application in diverse fields as optical clocks [18], precision spectroscopy [19–22], ultraviolet and infrared spectroscopy [23–25], precision distance measurement [26], etc.

A usual method used to produce optical combs consists of a continuous wave laser coupled to a resonator which operates in the nonlinear regime. Usually, the resonator is fabricated with a third order nonlinear material which allows degenerated four-wave mixing (FWM). The frequency conversion mechanism that takes place inside the resonator’s material, lies on the intensity dependence of the refractive index. In this process, two pump photons are annihilated and a new pair of photons is created: one of an up-shifted frequency called signal, and another of a downshifted frequency called idler. Provided the momentum and energy conservation, the new born waves are equidistant in frequency from the pump wave. When the optical cavity decay rates are surpassed by the scattering rate into the signal and idler modes, the parametric process occurs, yielding to symmetric sidebands that grow in intensity with increasing pump power and the frequency comb is produced [27–30].

If the signal and idler frequencies coincide with optical microresonator modes, the parametric process is enhanced, resulting in efficient sideband generation.

The power threshold for the initiation of parametric oscillation scales with the inverse of the quality factor Q squared. From this relation, it is possible to see the advantages of using a high quality factor. Furthermore, the required power to start the parametric oscillation can be strongly reduced [30–39]. Linear and nonlinear ring resonators were extensively analyzed in different applications [16,40–42].

When building a device destined to produce frequency combs, many parameters are fixed after the fabrication process, among which we can find the resonance frequency and the coupling strength to the bus waveguide. In the case of having only one resonator, the only parameter that can be varied after building the device, is the resonance frequency. This is achieved by manipulating the resonator effective index, using different techniques [43–45]. In the case of two coupled resonators, when the effective index is changed, not only the the resonance frequency is modified, but also the overall structure coupling to the bus waveguide as it is explained, for instance, in reference [46]. This coupling strength syntonization allows a more precise manipulation of the parameters that play a role in the comb building process [47].

We propose the usage of a graphene layer to change the effective index of two coupled microrring resonators in order to control the properties of the generated frequency comb. Our design aims for the advantages of rapid response time and low power consumption in comparison with the traditional method of heat transfer [48].

In the first part of this work, we analyze the effective index in the SOI-graphene structure. Secondly, we verify that the FWM is present and it is significant in comparison to other non-linear processes. After that, we continue by describing the frequency comb generation device and how the manipulation of the extinction futures is achieved.

## 2. SOI-graphene effective index

To model graphene, we follow the approach schemed in [49]. The graphene sheet is modeled as an extremely thin layer with a conductivity *σ*(*ω*, *μ _{c}*, Γ,

*T*), being

*ω*the angular frequency,

*μ*the chemical potential, Γ the scattering rate (assumed to be independent of energy [49]) and

_{c}*T*the temperature. Starting from the Kubo formula [50] and considering no magnetic field, the intraband (

*σ*) and interband (

_{intra}*σ*) contributions to the conductivity are identified:

_{inter}The intraband term results:

*e*is the electron charge,

*k*is the Boltzmann constant and

_{B}*ħ*=

*h*/2

*π*is the reduced Planck constant. Γ can be calculated as Γ =

*e*v

_{F}

^{2}/(

*μ μ*) being v

_{c}_{F}the Fermi velocity in graphene and

*μ*the electron mobility. A typical value is 1/2Γ = 5×10

^{−13}s [51].

When analyzing the energies involved in the conductivity formula we find that thermal energy is two orders of magnitude lower than photon energy and chemical potential (*k _{B}T* ≪ |

*μ*|,

_{c}*ħω*). In this case, the interband conductivity can be approximated as [52]:

Provided the conductivity, the dielectric constant (*∊ _{g}*) can be calculated as:

*∊*

_{0}is the vacuum dielectric constant. A graphene monolayer is only one carbon atom in diameter. Given

*∊*, the real (n

_{g}_{g}) and imaginary (k

_{g}) parts of graphene’s refractive index can be determined using the relation ${\text{n}}_{\text{g}}+{\text{ik}}_{\text{g}}=\sqrt{{\u220a}_{\text{g}}}$.

By applying a suitable voltage *V _{g}* between the graphene layer and the Si core, the chemical potential

*μ*can be modified and, as a consequence, the refractive index is also varied [53]:

_{c}*V*

_{0}= 0.8 V is the offset from zero caused by natural doping and

*η*= 9×10

^{16}1/(Vm

^{2}) [54]. Fig. 2(a) exhibits n

_{g}and k

_{g}calculations as a function of

*V*and

_{g}*μ*. It can be observed that with a modest excursion of 1 V, n

_{c}_{g}can be varied in an order of 3.

This variation of refractive index can be profited to modify the effective index in a semiconductor waveguide. To explore this effect, we have performed finite-element simulations using an ad-hoc software that solves the Helmholtz Eq. and including the electrical conductivity of graphene as a function of the chemical potential, i.e. the applied voltage, and the incident wavelength. The incident wavelength was 1550 nm for the complete set of calculations. Fig. 1 shows the simulation geometry. The dimensions of the waveguide were chosen in order to obtain sufficient evanescent wave so the coupling between adjacent waveguides was possible. The silicon (Si) core was 200 nm width and 500 nm high and it was surrounded by silica. The green layer in Fig. 1 represents a doped Si region used to apply the voltage between the Si-core and the graphene layer. With this transversal area the waveguide results monomode. On top of each microrring resonator an Al_{3}O_{2} layer is deposited with a graphene sheet above. Finally, an air layer covered the waveguide. Scattering boundary conditions were used in the geometry limits. The employed refractive indexes were n_{Si} = 3.47 for silicon, n_{SiO2}=1.44 for silica and n_{Al2O3}=1.74 for Al_{2}O_{3}, for a wavelength of 1550 nm. In the case of graphene, its refractive index was included as a function of applied voltage as it was explained in the previous section. An example of the field distribution in the TM_{0} is shown in Fig. 2(b). The results for the variation of the effective index in the TM_{0} mode are exhibited in Fig. 2(c). It can be seen that with a small voltage variation of 1 V the effective index can be varied in 10^{−3}. These results agree with experimental data [15].

## 3. Non-linear context

Several processes occur in the non-linear medium at the same time. To quantify them we have simulated the beam propagation through the non-linear medium using the non-linear Schrödinger Eq. for the Pump, Stokes, and anti-Stokes signals inside the SOI waveguide [55,56]. We also took into account spontaneous Raman which was modeled adding a term to the Stokes and anti-Stokes modes Eqs., that depends on the pump intensity [57]. These coupled Eqs. were solved using the split-step Fourier method [28]. Since light completes several round trips before leaving the rings it becomes necessary to calculate the effective length, *L _{eff}* thorough which the beam travels. Following Ref. [58],

*L*= 1/(2

_{eff}*Im*(

*k⃗*)), where

*k⃗*is the complex propagation wavevector and |

*k⃗*| = 2

*πn*/

_{eff}*λ*. For our waveguide results

*L*≃1mm. For these simulations, there was no Stokes signal stimulated.

_{eff}On the other hand, the simulations show, as it is expected in Si, losses are very high due to free carriers generated by two photon absorption. We neglect these losses since they can be avoided by removing carriers with a p-n junction in reverse bias as it is the common procedure [59]. The total linear loss calculated was 0.13 GW/cm^{2}.

Regarding the comb stability, in [60,61] it is explained why the case of anomalous dispersion is much more suitable than normal dispersion. The FWM process is highly affected by the effects of self-phase and cross-phase modulation (SPM and XPM). These processes refer to the nonlinear phase modulation of a beam, caused by its own intensity (SPM) or by other modes intensity (XPM), via the Kerr effect. In order to overcome these phenomena, the waveguide anomalous dispersion can be used. We performed a last set of finite-element simulations to verify that this effect is possible in our waveguides. In this case, we used the geometry of Fig. 1 and implemented the Sellmeier Eqs. to provide the model with an input for the dispersion of Si and SiO_{2} [62]. Once the effective index was obtained as a function of the incident wave frequency, the group velocity dispersion (*GVD*) was calculated through [27],

*GVD*is extracted, the dispersion parameter

*D*can be derived through

*D*= −2

*πc*/(

*λ*

^{2}).

*GVD*. The results are shown in Fig. 3 for a particular chemical potential of

*μ*= 0.8 eV. It can be seen that anomalous dispersion (

_{c}*D*> 0) is achieved for a certain range of wavelengths including

*λ*≃ 1550 nm, which is our selected wavelength.

In recent publication [58], the effective waveguide kerr coefficient *n*_{2eff} is calculated which results at least three times major than that of silicon. We have followed their procedure to calculate this coefficient for our waveguide, finding *n*_{2eff} = 9.75 × 10^{−17} m^{2}/W, which is one order of magnitude higher than that of silicon.

To evaluate these nonlinear effects we have calculated characteristic lengths of each process. For SPM we have *L _{SPM}* = 1/(

*γP*), where

_{norm}*γ*=

*n*

_{2}

*/*

_{eff}ω*c*and

*P*is the incident power normalized by the waveguide area. In our case, the selected value of

_{norm}*P*is justified with in the comb formation theory and will be explained later. We used the value

_{norm}*P*= 1.74

_{norm}*x*103 W/cm

^{2}. The dispersion length has been calculated as

*L*=

_{D}*τ*

^{2}2

*πc*/(

*Dλ*

^{2}), where

*τ*is typical time related to the pump. Since our model uses continuous incident wave, we have chosen

*τ*= 2

*π*/

*ω*= 5×10

^{−15}

*s*. The obtained values for both lengths were

*L*=247 m and

_{SPM}*L*=0.4 m, indicating that the effect of anomalous dispersion is stronger than the self-phase modulation one. The major conclusion is that both lengths result orders of magnitude larger than the waveguide effective length

_{D}*L*≃1 mm, meaning that there will not be a big distortion in our mode profile.

_{eff}## 4. Frequency comb generation device

The proposed analysis consists of two microrings evanescently coupled to a bus waveguide as shown in Fig. 4. Both rings possess a straight coupling section of 0.4 *μ*m length in order to facilitate the passage of the incident wave to the rings. We worked with two options for the radio of the circular parts: 5 *μ*m or 30 *μ*m. These radius were chosen from a trade-off between minimizing the device size and having enough optical path to achieve an effective index change of the order of 10^{−3}, the one achievable with graphene, produces a significant phase shift. Given the high contrast between the refractive index of Si and SiO_{2}, we chose a 200 nm width core in order to have some evanescent field outside the core. This condition was necessary for the coupling between the microring resonators. The width of the total microring resonators, including the silica cladding, was 4 *μ*m. Each ring has a 10 nm thick Al_{2}O_{3} layer on top of the core with a graphene sheet above. With this thickness a MOS capacitor is formed between the graphene layer and the Si core. The resulting waveguide is monomode since the incident frequency is lower than the cutoff frequency of the first TM mode.

Since the two cavities are evanescently coupled, the individual cavity modes hybridize and, as a result, a symmetric and an anti-symmetric supermodes exist. According to the coupled-mode theory, these supermodes eigenfrequencies are given by [63]:

*ω*is the average resonant frequency (between the two cavities), Δ

_{avg}*ω*is the difference between the individual cavity resonances (cavity detuning), and

*K*is the inter-ring temporal coupling rate. In the case of degenerated cavities (Δ

_{ω}*ω*= 0), the supermode resonances are separated by 2

*K*.

_{ω}We performed 2D finite-elements simulations with the aim of investigating the response of our system and comparing it with the predictions of Eq. (7). We solved the Helmholtz Eq. in our domain of interest with an ad-hoc software using the beam envelopes approach, first order elements and scattering boundary conditions in every geometry limit. For the first set of simulations the refractive index of both rings was kept equal.

The resulting transmittance at the end of the bus waveguide is shown in Fig. 5, as a function of the wavelength *λ*. The symmetric and antisymmetric modes could be identified as described by Eq. (7). The out-of-plane field in the inter-ring coupling zone is also presented to clearly visualize the parity of the supermodes. All the presented results belong to the first transversal electric mode TE_{0}.

In a second set of simulations we varied the relative refractive index of the core of the two rings. We define *ε* as the maximum change that can be produced in n_{eff} by applying voltage to the graphene layer. We have stated before its value being *ε* = 10^{−3} which can be achieved by changing the applied voltage in 1 V. In order to modify the relative effective index we left the first ring with a fixed index of n_{1}=n_{Si} + *ε*/2, and varied the index of the second ring as n_{2} = n_{1} + Δn with −*ε*/2 ≤ Δ*n* ≤ *ε*/2, which means to vary the applied voltage between −0.5 V and 0.5 V. In this way, positive and negative detuning between the two microring resonators was possible preserving the total excursion (1 V).

The calculated transmittance as a function of incident wave frequency is shown in Fig. 6, for rings of 5 *μ*m radius, and in Fig. 7 for rings of 30 *μ*m radius. In both cases the behavior predicted by (7) is observed. The supermodes intensity depends on *K _{ω}* [63] and, since

*K*is a function of the wavelength and refractive indexes [64], the intensity finally depends on these two parameters and also does the transmittance.

_{ω}*K*also depends on the radius of the rings [64] and this is the reason why the shift achieved in the case of the smallest radius is minor than the one obtained for the 30

_{ω}*μ*m radius case.

As a result, the transmittance of the proposed device can be significantly varied by applying a modest voltage. This tunable transmittance can be profited as an envelope for the frequency comb components that allows to diminish, as well as magnify specific spectral lines.

To exemplify this point, we have performed numeric simulations. The comb lines, can be modeled by a system of coupled Eqs. [60]:

*A*|

_{μ}^{2}is the instantaneous number of photons in the mode

*μ*. Each mode

*μ*has a frequency

*ω*=

_{μ}*ω*

_{0}+

*D*

_{1}

*μ*+ 0.5

*D*

_{2}

*μ*

^{2}, where

*D*

_{1}corresponds to the free spectral range (FSR) of the resonator and

*D*

_{2}to the difference between two neighboring FSRs at the center frequency

*ω*

_{0}. In Eq. (8),

*A*is the amplitude of the mode

_{μ}*μ*,

*t*is the time, Δ

*ω*is the modal bandwidth,

_{μ}*F*is the external pumping factor which is normalized in a way such that |

*F*|

^{2}represents the total number of photons that are coupled into the cavity and

*g*

_{0}is the FWM reference gain.

*ω*is the modal frequency and

_{μ}*ω*is the pump frequency. The second term in Eq. (8) describes the external pumping and it adds up only for

_{p}*μ*= 0 and the last term describes the frequency mixture that occurs in the FWM process.

We solved Eq. (8), that is stated for a single resonator, using the methodology proposed in [65] and a split-step Fourier algorithm. To model the structure, i.e. two coupled resonators, we then split each obtained mode following Eq. (7) and, after that, we multiplied the result by the coupled resonators efficiency. The calculus were carried out considering Δ*ω _{μ}* = Δ

*ω*

_{0}which is a fair assumption as can be observed in Fig. 7. Δ

*ω*

_{0}was extracted from the finite-element simulations being Δ

*ω*

_{0}= 45 GHz. The pump frequency was chosen to be

*ω*= 1232 rad/s and

_{p}*g*

_{0}= 3MHz. As it is explained in [60] there is a power threshold for the initiation of the nonlinear effects, that can be translated into a threshold number of photons ${\left|{A}_{0}^{\mathit{th}}\right|}^{2}=0.5\mathrm{\Delta}{\omega}_{0}/{g}_{0}$. We chose to express the parameter

*F*in units of this quantity. Once we have the threshold number of photons we can calculate the power threshold and normalize it by the waveguide area resulting 4.31 × 10

^{−5}GW/cm

^{2}. For the simulations we used

*F*= 3|

*A*

_{0}|

^{2}. The selected total time for the simulations was 0.1

*μ*s.

Results are shown in Fig. 8. It can be observed that the same line can be suppressed or amplified depending on the value of Δ*n*, i.e. on the applied voltage. The insets zoom in the evolution of the central modes. We can observe that for Δ*n* = 0.5 × 10^{−3} the first mode has a slower intensity than the second mode. When Δ*n* = −0.5 × 10^{−3} the second mode has higher intensity than the first one. This shows that the relation between the intensities of these two modes is inverted when we change Δ*n*, that is to say when we change the applied voltage.

After that, we will estimate the response time of the system in order to compare it to other methods. In [60], it is established that the difference between the pump frequency and the resonator’s resonance frequency, *σ*, establishes a condition for certain modes to be excited. At power threshold, for the mode *μ* to be stable it is necessary that $\sigma +0.5{\overline{\omega}}_{\mu}<-\sqrt{3}/2\mathrm{\Delta}{\omega}_{0}$ where *ω̄ _{l}* =

*ω*−

_{α}*ω*+

_{β}*ω*−

_{γ}*ω*referencing the frequencies involved in the four-wave mixing process. In our case, this stability condition is achieved for the six first modes,

_{μ}*μ*∈ {−6, ..., 6}, and this was verified in the simulations (not shown). Only the six first spectral lines were activated when the incident power was at threshold.

On the other hand, after 0.1 *μ*s these six components had already appeared so we considered this time as representative for the response time in a single resonator. We established this time as the lower limit for the response time of the coupled resonators device. Furthermore, we need to consider the speed at which the change in the refractive index is produced. In the literature we can find modulators made of Si and graphene that reach modulations speeds ranging from 1 to 30 GHz [1]. With this information we estimate a time response for this effect of the order of 30 ps. We notice that this time is significantly minor than the time needed to stabilize the comb. So, our final estimation of the response time is 0.1 *μ*s. This is a remarkable advantage when we compare this device with others that employ the thermo-optic effect [48] since they have slower response time, i.e. in the order of microseconds [66] and they also need to deal with issues of thermal volumetric expansion.

A word must be said concerning SPM and XPM. The processes can introduce additional detunings Δ*ω _{SPM}* and Δ

*ω*and, as a consequence, they can limit the comb span. To investigate these effects we have calculated the detunings they produce, taking into account the graphene presence by using the kerr index

_{XPM}*n*

_{2eff}calculated as explained in Ref. [58]. Since we are using a continuous incident wave, the SPM frequency shift coefficient can be calculated as 2

*πn*

_{2}

*I*

_{0}

*L*

_{eff}ω_{0}/

*λ*

_{0}where

*I*

_{0}is the incident intensity and all the other parameters were defined before. Evaluating this expression we obtain Δ

*ω*≃ 23 rad/s and Δ

_{SPM}*ω*≃ 88 rad/s. To calculate the XPM contribution we took an extreme case were the six modes present at threshold condition contribute in the same way to XPM and each mode intensity was extracted in relation to the central mode. With the obtained values the new detuning

_{XPM}*σ*was calculated. Finally, the stability condition was once more verified to find that only six modes are allowed at threshold, meaning that SPM and XPM effect do not change the comb structure.

Continuing with the comparison with the method of thermal injection, we can say that voltage can be applied much more locally than heat.

In addition to this, since the envelope of the comb components can be manipulated in-situ, the proposed system also has the property of simplifying, to certain extent, the amendment of unavoidable imperfections in the fabrication process after the device is built.

Another important advantage of the proposed device arises when looking at the consumed power. The comb simulations results show that a power 1.3 × 10^{−4}GW/cm^{2} destined to the FWM process is enough to produce the comb. To achieve this situation, an input intensity of 0.6 GW/cm^{2} is sufficient. This is considered a low input intensity [67] and makes the device suitable for many applications.

We have also estimated the electrical power consumption. Using as starting point reference [68] we have estimated power consumption of 5 mW which results in the same order of magnitude of that needed by the electric heating method [48].

## 5. Conclusion

We have proposed a device for frequency comb generation that allows in-situ tunability of the comb characteristics. The geometry contains two coupled SOI microrings coupled to a bus waveguide. The inter-ring coupling can be modified through low intensity voltage applied to a graphene layer lying on top of one of the microrings. We have numerically analyzed the response of the prototype as a function of the applied voltage and incident wavelength.

In comparison to thermal heating, which is commonly used to manipulate the comb characteristics, this device has the potential of offering lower response time, low input intensity and small footprint.

In addition, the calculated power consumption resulted low which makes it suitable for many applications.

## Funding

Universidad Nacional de Cuyo (C014); CONICET, Comisión Nacional de Energía Atómica (CNEA); Sofrecom Argentina; Universidad de la Empresa (Research project P17T02).

## Acknowledgments

We would like to acknowledge Condensed Matter Theory Group belonging to Bariloche Atomic Center for the provision of the cluster where the calculations were performed.

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