1. Introduction

Graphene is a two dimensional single layer of carbon atoms arranged in an hexagonal (honeycomb) lattice featuring an energy vs momentum dispersion diagram where the conduction and valence bands meet at single points (Dirac points) [1–3]. In the vicinity of a Dirac point, the band dispersion is linear and electrons behave as fermions with zero mass, propagating at a speed of around 10⁶ m s^–1 and featuring mobility values of up to 10⁶ cm²V^–1s^–1. The density of states of carriers near the Dirac point is low, and as a consequence, its Fermi energy can be tuned significantly with relatively low electrical energy (applied voltage) [1–4]. This tuning modifies the refractive index of graphene and, if this material is incorporated into integrated dielectric waveguide structures, it also affects the effective index and the absorption of the propagated modes opening new possibilities of obtaining tunable components in different regions of the electromagnetic spectrum. Devices exploiting this effect have been reported, for instance, in the terahertz [3], photonic [4] and microwave [5] regions of the electromagnetic spectrum.

A particularly active area of research during the last years is related to the design of tunable integrated photonic components where different contributions have theoretically and experimentally reported a variety of functionalities including electroabsorption modulation in straight waveguides [5–8] and resonant structures [9], channel switching [10], and electrorefractive modulation [11]. Most of these are based either on a straight waveguide configuration or in ring cavities. In this paper we propose the use of interleaved graphene sections on top of a silicon waveguide to implement tunable distributed Bragg gratings. Some properties of graphene relevant to its tunable conductivity and dielectric constant are briefly reviewed in section 2. The proposed silicon graphene waveguide Bragg grating (SGWBG) structure is presented in section 3. We begin describing the configuration of the tunable silicon graphene upon which it is based and then illustrate by numerical means the variation of the effective indexes of the transversal magnetic (TM) and electric (TEM) fundamental modes by suitable change of the chemical potential. The principle of the SGWBG is then introduced, which is based on interleaving sections of graphene on top of a silicon waveguide. In section 4 we explore the behavior of the proposed device including its wavelength tunability, bandwidth and also the possibility of side-lobe reduction by apodization of the chemical potential. Section 5 concludes the paper with a summary and some relevant future directions of research

2. Graphene conductivity and dielectric constant

Graphene has noteworthy optical properties due to its conical band structure that allow both intra-band and inter-band transitions [1–3]. Both types of transitions contribute to the material conductivity [12].

A typical value for the collision rate is 1/Γ = 5x10⁻¹³ sec. The dielectric constant of graphene follows from (1)-(4):

3 Graphene silicon waveguide and Bragg grating

Graphene can be incorporated into silicon to implement graphene silicon waveguides (GSWs). For the implementation of the Bragg grating we consider the layout shown in Fig. 1 that consists in placing a monolayer graphene sheet on top of a silicon bus waveguide, separated from it by a thin Al₂O₃ layer.

 

Fig. 1 Deep silicon waveguide with a layer of graphene placed on top of it.

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The incorporation of the graphene layer modifies the propagation characteristics (field profile, losses and effective index) of the guided modes and these can be, in turn, controlled and reconfigured changing the chemical potential by means of applying a suitable voltage. Furthermore, these properties are wavelength dependent so a complete description requires the use of numerical and or mode solving techniques. In Fig. 2 we show for λ = 1550 nm, the effective index and the losses (in insets) vs the chemical potential for the transversal magnetic (TM) fundamental mode (a similar behavior is obtained for the transversal electric TE mode). These results have been numerically calculated using a Finite Difference (FD) based commercial Field Designer mode solver, from PhoeniX Software B.V.

 

Fig. 2 Effective index and losses (inset) for the TM fundamental mode of a deep GSW versus the chemical potential (T = 300°K and $1/\Gamma =5\text{x}{10}^{-13}$sec).

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Referring to Fig. 2, the value of effective index of the TM mode when no graphene is placed on top of the waveguide ($n_{eff}^o=1.8595$) corresponds to the case of μ_c = 0 (with no losses). Note that this same value is obtained for μ_c = μ_co close to 0.52 eV. Note as well that increasing the value from 0.52 to 0.82 eV results in a tunable change of the effective index $\Delta n_{eff}^{}\left({\mu }_c\right)=\left|n_{eff}\left({\mu }_c\right)-n_{eff}^o\right|$ in the range of 0 to 0.3%. This effect can be exploited to implement a tunable Bragg grating by interleaving graphene sections on top of the silicon waveguide as shown in Fig. 3.

 

Fig. 3 Layout of the proposed silicon graphene Bragg grating.

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One of the main advantages brought by the use of graphene is that its band structure close to the Dirac points leads to an extremely high carrier mobility, enabling high-speed reconfiguration of its chemical potential with extremely low energy consumption. For instance it has been anticipated that modulation speeds of up to 500 GHz with tens of femto joule energy consumption can be achieved [6], [13].

4 Results, discussion and applications

The proposed device can be fabricated in three steps of which the first two are similar to those reported in [6]. In a first step a thick Si layer can be employed to connect the Si bus waveguide and the ground electrode followed by uniform deposition of a Al2O3 spacer on the surface of the waveguide by atom layer deposition. In the second step a graphene sheet grown by chemical vapour deposition can be mechanically transferred onto alternate sections of the silicon waveguide and the active electrode extended by means of a platinum film deposited on top of the graphene layer as shown in Fig. 1. The final step involves the patterning the graphene sections. This can be achieved by different methods [14]: direct mechanical cleavage, scanning probe lithography, chemical etching or plasma etching. The latter can be carried out in a standard CMOS compatible SOI process.

Figure 4 shows the fundamental TM mode profile in a grating region without (Upper) and with (Lower) graphene (μ_c = 0.58 eV) computed with the Field Designer mode solver. For each region we plot the bidimensional normalized field amplitude in the left hand-side and the normalized field amplitude for the X = 0 axis in the right hand-side. The strong waveguide field interaction with the graphene layer is clearly appreciated.

 

Fig. 4 Bidimensional normalized field amplitude (left) and normalized field amplitude for the X = 0 axis (right) in a grating section without (upper) and with (lower) graphene (μ_c = 0.58 eV) cover layer.

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For a grating structure of length L and period $\Lambda$ the N-th order resonance central operating wavelength ${\lambda }_D$and grating bandwidth are given by [15]:

A Bragg grating has been designed with the following parameters: N = 3, Λ = 1250.4 nm L = 1500 μm, μ_co = 0.52 eV, $n_{eff}^o=1.8591$. Figure 5 shows the results for the power transmission (lower part) and reflection (lower part) transfer functions of the device versus the optical wavelength, taking the chemical potential as a parameter. The results were computed by numerical solution of the grating structure using a commercial software package CAMFR [16] that implements a transfer matrix approach based on the Eigenmode Expansion Method (EME). CAMFR is not based on spatial discretisation or finite differences, but rather on frequency-domain eigenmode expansion techniques hence, instead of specifying the fields on a discrete set of grid points in space, the fields are described as a sum of local eigenmodes in each z-invariant layer of the structure.

 

Fig. 5 Transmission (Left) and Reflection (Right) intensity transfer function of a Silicon graphene Bragg grating with N = 3 Λ = 1250.4 nm, L = 1500 μm, μ_co = 0.52 eV, n_eff^o = 1.8591, T = 300°K and$1/\Gamma =5\text{x}{10}^{-13}$sec.

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As it can be observed as the chemical potential is increased the grating central wavelength is shifted to lower values. Note as well that the grating bandwidth and the maximum value of the reflection increase. These features can be explained from Eqs. (7) and (8) and the dependence of the effective index on the chemical potential shown in Fig. 2. Note from Fig. 2 that as μ_c increases then $n_{eff}\left({\mu }_c\right)$ decreases and hence ${\overline{n}}_{eff}\left({\mu }_c\right)$ and ${\lambda }_D\left({\mu }_c\right)$ decrease as well. The non-constant value of the average effective index with μ_c is due to the fact that while the change in the effective index is increased with μ_c the maximum value remains constant. On the other hand $\Delta n_{eff}\left({\mu }_c\right)$increases with μ_c and the quotient $\Delta n_{eff}\left({\mu }_c\right)/{\overline{n}}_{eff}^o\left({\mu }_c\right)$ dominates over the square root factor in Eq. (8). Figure 6 shows the evolution of the central operating wavelength ${\lambda }_D$ and the grating bandwidth obtained by numerical simulation.

 

Fig. 6 Central operating wavelength λ_D and the grating bandwidth evolution versus the chemical potential of a Silicon graphene Bragg grating with Λ = 1250.4 nm, L = 1500 μm, μ_co = 0.52 eV, n_eff^o = 1.8591, T = 300°K and$1/\Gamma =5\text{x}{10}^{-13}$sec.

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The proposed design can also implement apodized Bragg grating configurations which are useful to modify their spectral response in order to reduce the sidelobes at one or both sides of the central wavelength. This can be achieved, for instance, by changing the applied voltage in the propagation region. Figure 7 shows as an example the results for the reflection and transmission functions in a Gaussian apodized (μ_c = 0.62 eV) design where:

 

Fig. 7 Transmission (left) and Reflection (right) intensity transfer function of a Silicon graphene, Gaussian-apodized, Bragg grating with N = 3, Λ = 1250.48 nm, L = 1500 μm, μ_c = 0.62 eV, T = 300°K and $1/\Gamma =5\text{x}{10}^{-13}$sec. taking the Gaussian window coefficient G as a parameter.

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In (9) G is the apodization parameter that controls the steepness of the variation in refractive index as compared to the value in the central position of the grating (z = L/2). As expected, sidelobe suppression is obtained only in the long wavelength region taking the central wavelength of the grating as a reference. This is due to the fact that the average effective index in the grating is not constant [17]. In order to obtain a symmetric spectral behavior, several techniques such as average index pre-compensation can be implemented. These, and other techniques are under current investigation and will be reported in the near future.

The realization of gratings in integrated form has been proposed using a variety of techniques and materials as described in an authoritative review [18]. Instead of being based on exploiting the photosensitivity of the waveguide material as in fiber devices [19], integrated Bragg gratings are usually fabricated by means of physical corrugation or perturbation of the waveguide geometry and on the depth of the etching process. This brings two basic advantages [18]. First, as already mentioned, the fabrication does not rely upon a photosensitive process expanding the variety of materials can be used. Second, it allows the realization of much higher grating coupling strengths enabling the implementation of devices with significant reduced footprint. The device proposed opens the possibility of implementing fast tunable, low energy consumption integrated silicon Bragg gratings which are not based on waveguide corrugation. However, it should also be mentioned that the proposed principle technique could certainly be incorporated into the alternate sections of corrugated silicon Bragg gratings [18], [19] to further enhance the coupling strength providing extra fast tunability. The applications of the proposed device would be similar to those of other integrated silicon Bragg gratings, most of which are covered in detail in [18].

5. Summary and conclusions

We have proposed the use of interleaved graphene sections on top of a silicon waveguide to implement tunable Bragg gratings. The filter central wavelength and bandwidth can be controlled changing the chemical potential of the graphene sections. Apodization techniques have also been presented and discussed.