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Abstract
We propose and numerically demonstrate that a reconfigurable nanocavity can be created in a graphene-loaded Si photonic crystal waveguide. The cavity formation is caused by the local mode-gap modulation induced by electrostatic gate-tuning of graphene. Although most recent graphene photonic devices are based on a change in the imaginary part of the refractive index, here we make use of a change in the real part of the refractive index for gated graphene. We clarify that nanocavities can be formed in two different cases, red-shifted and blue-shifted tunings. These novel formation mechanisms enable us to create and annihilate a nanocavity in a reconfigurable way by varying the gate voltage, which is promising for novel control in photonic processing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well recognized that the optical absorption of graphene highly depends on its Fermi level which can be controlled by electrostatic gating [1,2]. This is a direct result of Pauli blocking on the inter-band transitions and has been applied to various graphene-based optical modulators [3,4]. On the other hand, the refractive index of graphene can be also changed by electrostatic gating, which is apparent from Kramers-Kronig relation, but there are very few reports of applications, such as for phase modulators [5,6]. The purpose of the present study is to fully employ the real part of the refractive-index tunability of graphene to create and annihilate a nanocavity in a photonic crystal (PhC) line defect waveguide.
It has been known that a local modulation of a two-dimensional PhC line defect waveguide can form a nanocavity which has a strong light confinement and ultrahigh-Q [7–11]. The strong light confinement is achieved by modulating the mode gap which is the bandgap in the projected band structure of the PhC waveguide. The structural modulation, such as a slight shift in the position or radius of air holes, results in the mode gap shifting to a lower frequency. Therefore, the area surrounded by the modulated mode gap becomes a cavity. This type of a cavity is called a modulated mode gap cavity [12]. Instead of the structural modulation, the refractive index modulation can be also used for the cavity formation. In our previous report, the cavity formation can be achieved by a very small local refractive index modulation (Δn/n∼0.1%) in a PhC line defect waveguide [13]. In the present paper, we use the refractive index modulation of graphene to locally modulate the mode-gap in PhC line defect waveguides. Here we assume to employ silicon PhC which has almost no absorption near the telecom band at wavelengths around 1550 nm, and to place a graphene sheet on its top surface. Normally, the effective refractive index modulation induced by the gated graphene sheet is intrinsically small because of its very thin thickness. However, in the case of a graphene-loaded PhC waveguide, we will show that the cavity formation is indeed possible by the gating of graphene. Since high-speed modulation of the Fermi level is possible for a certain type of gated graphene [14–17], it can be expected to form a cavity on demand at a high speed with our method. In addition, as will be described later, in the present method, it is possible to form a cavity with both blue- and red-shifted tuning of the refractive index, making it possible to take full advantage of the characteristics of the refractive index change of graphene.
Usually, the cavity formation of a mode-gap type in a PhC line defect waveguide requires fabrication accuracy to shift holes in several nm [7–11]. The present cavity formation using the refractive index modulation of graphene does not require high fabrication accuracy such as the pattern modulation of several nm. In previous studies, several ways to realize the nanocavity formation by the index modulation have been proposed and demonstrated, such as by selective oxidization [18] and photosensitivity of chalcogenides glass [19]. However, these cavity formations are post-tuning methods based on static irreversible control of the refractive index, and thus they do not allow us to dynamically control the cavity formation in a reconfigurable way. In contrast, our cavity formation method enables dynamic and reconfigurable nanocavity formation because it is based on reversible electrostatic modulation of the refractive index, which leads to dynamic creation and annihilation of a nanocavity by simple electrical control. In addition, our method requires only the simple process of graphene loading onto well-established Si photonic crystal waveguides, which is suitable for integrating in Si photonics circuitries.
2. Proposed method of cavity formation
Here we show the cavity formation mechanism by the modulation of a mode gap in a PhC waveguide using the tunable refractive index of graphene.
As is well known, the refractive index of graphene depends on the Fermi level. In order to account for the optical response of atomically-thin graphene, we assume graphene as a boundary surface having a sheet conductivity in our simulations, because the ac conductivity is directly related to the refractive index. The theoretical 2D sheet conductivity of graphene as a function of the Fermi energy is derived from Kubo formula [20].
Fig. 1. The complex refractive index of graphene as a function of the Fermi level. The black and red lines show the real and imaginary part of the refractive index. The blue line shows the normalized sheet conductivity representing the absorption loss in graphene.
In this study, we employ the large change in the refractive index of graphene shown in Fig. 1 for modulating the mode gap in a PhC waveguide to create a nanocavity. As we mentioned, modulated mode-gap cavities can be formed by a local modulation of the effective refractive index of the PhC waveguide. Here we expect that graphene can produce a sufficiently large index change when loaded on the PhC waveguide. We assume that a graphene sheet is loaded on the top surface of a PhC line-defect waveguide and the wavelength of the incident light is around 1550 nm. The shift direction of the mode gap is determined by the magnitude of the graphene refractive index with respect to the refractive index of surrounding air. If the refractive index of graphene is larger than that of air, the mode gap shifts to the low frequency side (red shift). If the refractive index of graphene is smaller than the refractive index of air, the mode gap shifts to the higher frequency side (blue shift). Hereafter, a shift to the low frequency side is referred to as a red shift, and a shift to the high frequency side is referred to as a blue shift.
Let us examine how we can create a nanocavity for both red- and blue-shifted cases. As shown in the center of Fig. 2, there is no cavity when there is no modulation. This happens when EF is 0.48 eV and the effective refractive index of graphene is close to unity. The mode-gap edge does not change, and thus there is no cavity. When EF < 0.48 eV, the mode-gap edge is red-shifted in the graphene-loaded region, and thus a cavity mode can be created in the red-shifted region if the modulation is strong enough. On the other hand, when EF > 0.48 eV, the mode-gap edge is blue-shifted and it becomes also possible to form a cavity by loading graphene at the barrier regions as shown in Fig. 2. This is because the refractive index of graphene can be made smaller than that of air. In this case, since the mode gap moves to the high frequency side in the region where graphene is loaded, it is expected that light is strongly confined in the region without graphene.
Fig. 2. Local modulation of the mode gap induced by the refractive-index modulation of graphene. The light green area illustrates the frequency range of the mode gap along the PhC line-defect waveguide. (Left) Red-shift tuning. A nanocavity mode is created in the center region where the mode-gap edge is shifted lower. (Center) No cavity is formed. (Right) Blue-shift tuning. A nanocavity is created in the center region. The mode-gap edge of the surrounding barrier regions is shifted higher.
3. Cavity formation by graphene’s tunable refractive index
3.1 Settings of numerical simulations
Next, we investigate whether our proposed idea is possible or not by numerical simulations. We calculate the cavity characteristics and electric field distribution by numerically solving Maxwell equations using the finite-element method (COMSOL Multiphysics). The cavity resonant wavelengths and the Q factors were determined from complex eigen-frequency solution. Figure 3 shows the simulated model of a graphene-loaded Si PhC waveguide. The structure is based on a W1 line defect waveguide in a triangular air-hole lattice PhC slab. The refractive index of Si is 3.46 (hereafter we refer it as n), and the rest area is air (the refractive index is unity). The lattice constant a, hole radius r and slab thickness t are 400 nm, 100 nm, and 215 nm, respectively. The waveguide length is 70 × a, that is, 28 µm. There is no pre-fabricated cavity before loading graphene. The width of the line defect is 0.98 × √3 × a. Because of the infinitesimal small thickness of graphene, we configure the graphene on the Si PhC waveguide as an embedded conductive interface between two dielectrics [26]. Specifically, the top surface of the Si PhC shown as the green area in Fig. 3 is assumed as the 2D boundary plane having the sheet conductivity of graphene using Eqs. (1) and (2) when T = 300 K, τ = 100 fs. Note that the effective isotropic refractive index shown in Fig. 1 is just used for intuitive understanding, and we employ the sheet conductivity model in our all simulations, although the isotropic refractive-index model for graphene was shown to account reasonably well for the experiments [27].
Fig. 3. Schematic of a graphene-loaded PhC waveguide. Graphene is loaded on the surface of the PhC waveguide in the green-shaded region.
3.2 Nanocavity formation based on the red-shift modulation
First, we investigate the case when EF is between 0.30 eV ∼ 0.48 eV, corresponding to the red-shifted modulation. The value of the graphene sheet conductivity does not change significantly until 0.30 eV. So, we start calculations from 0.30 eV. The length of loaded graphene is 10 × a along the waveguide direction. Figures 4(a)–4(c) show the simulated electric field distribution Ey of a Si PhC waveguide without graphene and with graphene (EF = 0.40 eV and 0.48 eV). In the Si PhC waveguide without graphene, the electric field spreads over the entire waveguide, because the mode gap is the same at any position of the waveguide. On the other hand, in the Si PhC waveguide loaded with graphene (EF = 0.40 eV), the light intensity becomes maximum in a graphene-loaded region where the local index modulation is largest. This is exactly what we expected, and clearly shows that a nanocavity mode is created in the graphene-loaded region.
Fig. 4. Cavity formation using the red-shift modulation. Graphene is loaded on the central region. (a, b, c) Electric field distribution Ey of Si PhC waveguide (a) without graphene, (b) with graphene (EF = 0.40 eV), and (c) with graphene (EF = 0.48 eV). The green area shows the graphene. (d) Calculated cavity wavelength as a function of the Fermi level. The shaded area shows the mode gap. (e) Calculated Q and effective mode volume as a function of the Fermi level. Blank squares indicate that the simulated values do not show enough convergence.
We examine the range of the cavity formation with varying EF, and discuss the performance of the created cavity. Figures 4(d) and 4(e) shows the cavity-mode wavelength, the quality factor Q of the cavity mode and the effective mode volume Veff as a function of EF. There are resonant cavity modes even out of the mode gap below 0.38 eV. These modes will be mixed with PhC waveguide modes and thus they are not useful for cavity applications. On the other hand, at EF between 0.40 eV and 0.48 eV, the resonant cavity wavelength is within the mode gap. As already shown in Fig. 4(b), we have confirmed a nanocavity formation when the Fermi level is 0.40 eV. The cavity mode wavelength is 1539.00 nm which is located inside the mode gap of the Si PhC line-defect waveguide without graphene. Veff and Q are 4.64 (λ/n)3 and 2400, respectively. In the range of EF between 0.40 and 0.42 eV, there exist similar nanocavity modes. The graphene-loaded Si PhC waveguide with this EF range shows the wavelength-scale light confinement within the mode gap. Thus, these results clearly demonstrate that a nanocavity mode can be formed by the local mode-gap modulation induced by the gate control of graphene.
As seen in Fig. 4(e), when EF > 0.42 eV (shown by squares and dotted lines), Veff becomes significantly increased. We confirmed that the calculated Veff does not converge in this region and depends on the length and size of Si PhC waveguide assumed in the simulations. We regard that in this Fermi level range, we cannot prove the cavity formation with the present data. Although we may be able to confirm the cavity formation even in this range if we can enlarge the computation size, Veff should be even larger than the values of blank squares in Fig. 4(e). Therefore, this range is not appealing for our purpose. Finally, when EF = 0.48 eV, the electric field distribution is essentially the same as that in the Si PhC waveguide without graphene, which means the electric field is spread over the entire waveguide. This is consistent with the result in Fig. 2, where the refractive index of graphene becomes unity at EF = 0.48 eV.
Consequently, the red-shift modulation enables a nanocavity formation in a Si PhC waveguide when EF is between 0.40 eV to 0.42 eV. It is worth to note that the gate control of the real part of the refractive index plays a key role in this cavity formation, which contrasts with the previous studies of the gate-controlled graphene photonic devices where the imaginary part variation is employed. In addition, it is expected that one can reversibly create and annihilate a nanocavity by simply changing the electrostatic voltage.
3.3 Nanocavity formation based on the blue-shift modulation
In this section, we investigate a nanocavity formation using the blue shift of the mode gap for EF ≥ 0.48 eV. As we pointed out before, another way of cavity formation becomes possible for the blue-shifted modulation. We employ a graphene-loaded area as a confinement barrier, and a nanocavity will be formed in the central region where no graphene is loaded. We use the same parameter as the previous section, and the length of the central area where graphene is not loaded is 10 × a along the waveguide direction.
Figures 5(a) and 5(b) show the simulated electric field distribution Ey of a Si PhC waveguide loaded with graphene when EF = 0.48 eV (a) and 0.80 eV (b). In Fig. 5(a), there is no cavity, as expected, and the field distribution is the same as that without graphene. In contrast, when EF = 0.80 eV, Fig. 5(b) shows clear light confinement around the central unloaded region. The estimated Q, Veff, and the mode wavelength are 1.8 × 105, 3.63 (λ/n)3, and 1538.34 nm, respectively. The calculated resonant cavity wavelength is located inside the graphene-modulated mode gap (the gap edge is 1537.90 nm). Hence, this result clearly demonstrates the nanocavity formation by the modulated-mode gap cavity mechanism.
Fig. 5. Cavity formation using the blue shift modulation. Graphene is loaded on two barrier regions. (a, b) Electric field distribution Ey of a graphene-loaded Si PhC waveguide at (a) EF = 0.48 eV and (b) EF = 0.80 eV (The green area shows graphene. (c) Calculated Q and effective mode volume as a function of the Fermi level. Blank squares indicate that the simulated values do not show enough convergence. (d) Calculated cavity wavelength as a function of the Fermi level.
We examine the range of the cavity formation, and discuss the performance of the cavity. When 0.48 eV < EF ≤ 0.55 eV (shown by squares and dotted lines in Fig. 5(c)), the calculated Veff does not converge and depends on the length and size of Si PhC waveguide assumed in the simulation. With the same reason as in 3.2, this range is not appealing for our purpose. On the other hand, we have confirmed light confinement for a wide range of EF between 0.60 eV and 0.80 eV. We summarize the cavity parameters for various EF values in Figs. 5(c) and 5(d). As shown in the figures, as EF increases from 0.48 eV, Q increases and Veff, decreases. Note that interestingly, the two cavity parameters are simultaneously improved, which shows a stark contrast with the case of the red-shift modulation. The optimum cavity formation is achieved when EF = 0.80 eV.
When the loaded length of graphene for the red shifted modulation and the unloaded length for the blue shifted modulation are compared, the Q is larger by a factor of 102 for the blue shift case. There are two reasons for this difference. First, it is because the electric field distribution is concentrated in the central region where no graphene is present and thus the light field is not strongly absorbed in graphene. Secondly, the cavity formation is achieved at the Fermi level where the absorption loss of graphene itself is small, as shown in Fig. 1. In addition to the advantage in the cavity Q, the blue shift type leads to the smallest Veff 3.63 (λ/n)3
4. Cavity length optimization
Here we investigate the relationship between the mode volume and the length of the central region (which means the graphene-loaded region in the red-shift modulation and the graphene-unloaded region in the blue-shift modulation, and hereafter we call it “cavity length”). In the previous sections, we fixed the cavity length and varied other parameters. Here, we optimize the cavity length to further improve the cavity performance as regards Q and Veff. First, we examine the case for the red-shift modulation. Figure 6(a) shows Veff as a function of the cavity length ranging from 4a to 19a with EF = 0.40 eV and 0.42 eV. This plot shows that Veff is minimized when the cavity length is 12a for EF = 0.40 eV, and 15a for EF = 0.42 eV. Since larger modulation is generally required for making the cavity volume smaller, the observed increase of Veff for shorter cavity length is due to the smaller modulation. In addition, at the long limit of the cavity length, Veff should be proportional to the cavity length. The observed slight increase of Veff for longer cavity length is a signature for this tendency. As a result of these two factors, Veff shows a minimum value at 12a (EF = 0.40 eV) and 15a (EF = 0.42 eV).
Fig. 6. Calculated the dependences of cavity performance on the cavity length about red-shift modulation with EF = 0.40 eV and 0.42 eV. (a) Effective mode volume. (b) Q factor.
Figure 6(b) shows that Q monotonically decreases as the cavity length becomes longer. In this parameter range, the cavity loss is reduced for shorter cavity length because the light field is more extended outside the graphene-loaded region. Since there is a certain absorption loss in the graphene-loaded region, the tight confinement within the central region leads to a reduction of Q. That is, there is some trade-off between Q and Veff in the case of the red shift modulation.
Next, we investigate the cavity length dependence for the blue-shift modulation case. Here, we vary the length of the central unloaded region. Figure 7(a) shows Veff as a function of the cavity length ranging from 8a to 19a with EF = 0.70 eV and 0.80 eV. Similar to the result for the red-shift modulation, Veff becomes minimum when the cavity length is around 11a. In the case of the blue shift modulation, the changes of Veff in Fig. 7(a) are smaller than that for the red shift case in Fig. 6(a) because the refractive index change in the blue shift case is larger than that in the red shift case.
Fig. 7. Calculated the dependences of cavity performance on the cavity length about blue-shift modulation with EF = 0.70 eV and 0.80 eV. (a) Effective mode volume. (b) Q factor.
Although the behavior of Veff in Fig. 7(a) is qualitatively similar to the red-shift modulation (Fig. 6(a)), the behavior of Q shown in Fig. 7(b) is fundamentally different from Fig. 6(b). Q monotonically increases as the cavity length becomes longer. This can be understood as the absorption loss in the central cavity region is smaller in the blue shift modulation than in the red shift case. Thus, the tight confinement is advantageous both for Q and Veff simultaneously, and thus there is no trade off. This final finding is very important for designing graphene-loaded PhC nanocavities, and shows a clear advantage for the blue shift modulation in comparison with the red shift modulation.
Consequently, although the red shift type and blue shift type cavities are based on the same cavity formation mechanism, their cavity performances are rather different. The red shift modulation device is simple in fabrication and a nanocavity can be realized at a lower Fermi level compared to the blue shift type, but Q is limited less than 104. On the other hand, the blue shift type is a bit structurally complicated and higher Fermi levels are required. But the overall cavity performance is improved against that for the red shift modulation, and a higher Q over 104 can be realized with a mode volume smaller than the red shift type.
5. Conclusion and discussion
In conclusion, we have numerically demonstrated that a nanocavity can be created by the local mode-gap modulation induced with gate-tuning in a graphene-loaded Si photonic crystal waveguide. We have found that the nanocavity formation is possible in two different regimes of the index tuning: red-shifted and blue-shifted tuning cases. In the red-shifted modulation case, the light field is confined in the graphene-loaded region. When the graphene-loaded length is 10a with EF = 0.40 eV, Veff and Q are 4.64 (λ/n)3 and 2400, respectively. The achievable Q values ranges from 103 to 104 for the red-shifted modulation. In the blue-shifted modulation case, the light field is confined in the central graphene-unloaded region. When the graphene-unloaded length is 10a with EF = 0.80 eV, Veff and Q are 3.63 (λ/n)3 and 1.8 × 105, respectively. The achievable Q values range from 105 to 106, which are higher than that for the red-shifted modulation, showing an advantage for the blue-shifted modulation. In addition, these nanocavities can be annihilated in both cases by setting EF = 0.48 eV where the refractive index becomes unity. Thus, these results have demonstrated that one can create and annihilate a nanocavity by simply changing the gate voltage. This is a remarkable feature of the present method and it suggests a possibility for dynamical control to trap and release a light pulse using the high-speed modulation of the Fermi level within the cavity life time. It would be worth noting that our method utilizes a large change in the real part of the refractive index for graphene, which shows a stark contrast to previous graphene-based modulators which rely on the tuning of the imaginary part of the refractive index for graphene.
Finally, we briefly discuss achievable Fermi levels of graphene in theoretical and experimental aspects. The theoretical sheet conductivity of the graphene shown as Eqs. (1) and (2) may hold as far as the Dirac-corn band structure is maintained. It has been reported that graphene keeps its linear band dispersion up to about 1.0 eV [28], which is higher than EF in our simulations. Technically, the Fermi level higher than 0.8 eV has been reported in various methods employing ion-gel and ion-liquid gates [4,29–33]. Thus, we believe that the EF values assumed in our simulations should be experimentally possible.
Ion gating is effective for the static cavity formation and elimination. On the other hand, thin-film oxide gates enable modulation at much faster speed. In a previous report, the modulation frequency of the Fermi level by an oxide gate reached about 30 GHz [14–17]. In other reports, the Fermi level of 0.50 eV was achieved by the oxide gate [34]. As a different approach, we can utilize optical pumping to change the Fermi energy of graphene. It is well known that graphene exhibits saturable absorption with an ultrafast recovery time, ranging from 100 fs to a few ps. This suggests a possibility that we can control the Fermi level of graphene by optical pumping at ultrafast speed. For example, the response time of graphene saturable absorption is reported 2.2 picoseconds using 220 fs light pulse [35].
These modulation methods would enable the high-speed cavity creation and elimination, and pave the way for dynamical control of trap-and-release of a light pulse using the high-speed EF modulation within the cavity photon lifetime. The estimated cavity photon lifetime is about several tens of ps for the blue-shifted modulation, which is a hard but not impossible target. In addition, the blue-shift-type configuration enables us to control the Fermi level of graphene for input-side/output-side barriers individually. Hence, we can selectively open and close a gate to input/output waveguides, by which a stored light pulse can be released dynamically in a desired direction. We believe that various novel ways of optical processing would arise from high-speed modulation of graphene-loaded PhC waveguides.
Funding
Japan Society for the Promotion of Science (KAKENHI JP15H05735).
Acknowledgments
We would like to thank Dr. Hisashi Sumikura, Dr. Masaaki Ono, Dr. Shota Kita, Dr. Kenta Takata, Dr. Yuto Moritake, Masato Tsunekawa, and Masanori Hata for their useful advices and constructive discussions about graphene and the method of numerical calculation.
Disclosures
The authors declare no conflicts of interest.
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