1 Introduction

Optoelectronics which is a branch of photonics is the technology of light detection and emission in electronic devices (Sun et al. 2016). These devices have a broad range of applications such as in sensing, solar cells, fast signal processing, and optical and electronic converters (Bao and Loh 2012), to name a few. With the ever-growing advancements in technology, the demand for fast and compact optical devices has increased greatly in the past decades. Therefore, optoelectronic devices that are capable of addressing these demands are highly sought after. Among these are fully optical signal processing devices whose main body is made of logic circuits (Rahmani and Mehdizadeh 2018; Saghaei et al. 2017; Sharifi et al. 2016; Wabnitz and Eggleton 2015). These devices incorporate logic gates (Cheraghi et al. 2018; Chhipa et al. 2021; Jiang et al. 2015; Liu and Ouyang 2008; Maleki et al. 2020; Sani et al. 2020; Vali-Nasab et al. 2019), combinational logic circuits, and full-adders to realize high-speed signal processing. To this end, Photonic Crystals (PhCs) have become an alternative approach in the design of all-optical devices (Alipour-Banaei et al. 2015b). Due to their photonic bandgap, PhCs allow for selective and tunable emission paths in waveguides (Yablonovitch 1993). In recent years, an extensive amount of research has been done on the material and design selection of PhCs to achieve application-specific solutions.

Research on material selection such as GaAs, InP, and Si on SiO2 has gained a lot of attention in integrated photonic circuits (Calizo et al. 2007; Danaie and Kaatuzian 2011; Roelkens et al. 2005). Further research on SiO2 as the substrate has shown advantageous functionalities such as very low loss and low thermal sensitivity within the near-infrared (Ebnali-Heidari et al. 2014, 2012; Song and Williams 1993). Further, the role of structural design in photonic crystal ring resonators (PCRR) in realizing compact optical devices was investigated (Alipour-Banaei et al. 2015a; Mehdizadeh et al. 2016; Rakhshani and Mansouri-Birjandi 2014). These devices can be spectrally tailored based on their rod’s refractive index, the structures’ lattice constant, and the design parameters of the rod such as radius, and height (Farmani et al. 2020; Mansouri-Birjandi et al. 2016; Tavousi et al. 2017). Furthermore, the spectrally selective resonances in PCRR devices can be tuned to create either a blue or red shift based on the desired application (Naghizade and Saghaei 2021). The use of PCRRs in PhCs as optical filters have been investigated for various applications such as optical filters (Alipour-Banaei et al. 2014; Guo et al. 2019; Naghizade and Saghaei 2020a; Rakhshani and Mansouri-Birjandi 2013), logic gates (Andalib and Granpayeh 2009; Hussein et al. 2018; Younis et al. 2014), Optical encoders-decoders (Haddadan et al. 2020; Maleki et al. 2019; Moniem 2016; Naghizade and Khoshsima 2018; Naghizade and Saghaei 2020b), comparators (Fakouri-Farid and Andalib 2018; Jile 2020; Seraj et al. 2020), adders and subtractors (Alipour-Banaei and Seif-Dargahi 2017; Hosseinzadeh Sani et al. 2020; Moradi 2019), and switches (Ghadrdan and Mansouri-Birjandi 2017; Jalali Azizpour et al. 2019; Radhouene et al. 2018; Rajasekar et al. 2019).

Various devices incorporating PCRR have been proposed among which are rod ring resonators (Daghooghi et al. 2018; Foroughifar et al. 2021) that find applications in the design of switches and filters. PhC-based all-optical encoders and decoders were also proposed using linear and non-linear functionalities of PhCs (Ghadrdan and Mansouri-Birjandi 2013; Neisy et al. 2018). In these proposed systems, one disadvantage is the tunability of the suggested designs. Nonlinearity in PCRRs which is based on the optical Kerr effect sets a threshold for the magnitude of the incident light. As long as this magnitude is higher than the aforementioned threshold the Kerr effect takes place. However, when the magnitude of the incident light is higher than the maximum allowed value, it will harm the device (Carr et al. 2004). Daghooghi et al. proposed another optical decoder based on the slow light effect using a nonlinear ring resonator (Daghooghi et al. 2018). In order to increase the light intensity, it is necessary to reduce the speed of light inside the structure. The intensity increase leads to the appearance of nonlinear phenomena such as the optical Kerr effect, which changes the total refractive index. Finally, resonance occurs at the desired wavelength. Its maximum delay is about 6 ps which is one of the drawbacks of this structure. Rostamizadeh et al. (Rostamizadeh et al. 2020) presented an optical decoder based on nonlinear PCRRs. The rise and fall times are 1.5 ps and 0.7 ps, respectively. Mehdizadeh et al. (2017) proposed an optical 2-to-4 binary decoder based on PhC cavities consisting of an enabling port. Despite all advantages of their study, the structure’s maximum switching frequency was 20 GHz. The main shortcomings of decoders designed in previous works include low data rate, relatively large area, high input power for the appearance of nonlinear phenomena, and lack of adjustability. To overcome such issues, in this paper, we present an optoelectronic 2-to-4 binary decoder using graphene nanoshells (GNSs) coated on SiO2 rods in photonic crystal ring resonators. Compared to similar works, the proposed design benefits from using linear rods. Therefore, an increase of the input power to re-direct the light emission to the preferred output is not needed. The findings revealed that the decoder has the maximum rise and fall times of 0.8 ps and 0.3 ps, respectively, due to its small area of 200 µm2.

In previous works, design parameters such as the radius of the rod and/or the refractive index of the structures were among the factors impacting the resonance frequency of the PhCs. In this work, however, the resonance frequency of the proposed structure depends on the applied voltage to graphene. In addition, the proposed structure in this work benefits from a compact and tunable design that also has a low time response.

The structure of this work is as follows: In Sect. 2, we discuss the GNS characteristics and mathematical formulations, PhC ring resonators, and optoelectronic decoders and their optical behavior. Section 3, covers the design and functionality discussions of the optoelectronic devices and the findings of this work. In Sect. 4, we overview the summary of the findings of this work and present the conclusions.

1.1 Design, model, and methods

1.1.1 Mathematical background

As mentioned earlier in the Introduction, highly tunable, fast, and compact all-optical optoelectronic devices are very desirable. For this reason, in the past decades, graphene which has high electro-optic tunability, optical transmission efficiency, and zero band gap has gained a lot of attention as a 2D material that is ideal for use in fabrications of optoelectronic and plasmonic devices. Enhanced graphene-light interactions are possible due to the strong plasmonic response within a broad range of the spectrum from infrared to terahertz which also results in strong absorption properties. However, in the other parts of the spectrum, such as near-infrared, visible, and UV, a monolayer of graphene has very low absorption in the visible to NIR range and also within the UV region which is due to the absence of the plasmonic response within these regions (Nair et al. 2008).

Tailoring the Fermi energy levels (Ef) of graphene is possible through various methods. One approach is to bias the gate voltage in a substrate-graphene structure set-up. Through this method, one can tune and change the optical characteristics of graphene, the central resonance frequency of the optoelectronic device of interest, and the quality factor of the system. This method which is in line with the graphene sheet electron flexibility allows for the design and modeling of integrated optoelectronic devices. To this end, various structures have been proposed. Among these are the integration of graphene on insulator substrates made of silicon and silicon dioxide. Here we propose the GNS having a thickness of d which is deposited around the silicon dioxide SiO2 rod on the SiO2 substrate. In this proposed structure, the charge carrier density (\({n}_{\mathrm{c}}\)) could be set by applying the gate voltage (Vg) between the substrate and the GNS. It can be formulated based on Eq. (1) (Shi et al. 2016; Tho et al. 2018).

$${\text{n}}_{{\text{c}}} = \frac{{{\text{V}}_{{\text{g}}} \varepsilon_{0} \varepsilon_{{\text{r}}} }}{{{\text{er}}}} = {\text{V}}_{g} {\text{C}}$$
(1)

where \({\upvarepsilon }_{0}\) and \(\upvarepsilon\) denote the permittivity of air and the relative permittivity of the substrate of choice, respectively. Further, e represents the electron charge and r is the SiO2 rod radius and \(\frac{{\upvarepsilon }_{0}{\upvarepsilon }_{\mathrm{r}}}{\mathrm{er}}=\mathrm{C}\) where C is the gate capacitance. The chemical potential of graphene is shown as (µc) and could be found as follows (Farmani et al. 2017a; Ju et al. 2011):

$$\mu_{c} {\text{ = v}}_{{\text{f}}} { }\sqrt {{\pi n}_{{\text{c}}} } = {\text{ v}}_{{\text{f}}} { }\sqrt {{\pi V}_{g} {\text{C}}}$$
(2)

In Eq. (2), \(\mathrm{\hbar}\) is the reduced Plank's constant, and the Fermi velocity is represented by \({\mathrm{v}}_{\mathrm{f}}\). Moreover, the resonance wavelength of the GNS, λ0, is obtained through the method described in (Yan et al. 2012):

$${\uplambda }_{0} \approx \frac{{2{\pi c}}}{{\text{e}}}\sqrt {\frac{{{\upvarepsilon }_{{{\text{eff}}}} {\upvarepsilon }_{0} {\text{d}}\zeta}}{{{\text{E}}_{{\text{f}}} }}}$$
(3)

In Eq. (3), \({\upvarepsilon }_{\mathrm{eff}}\) is the effective permittivity of the outer medium of the GNS and is given as \({\upvarepsilon }_{{{\text{eff}}}} = \left( {{\upvarepsilon }_{{{\text{SiO}}_{2} }} + {\upvarepsilon }_{0} } \right)/2\), and \(d\) represents the GNS thickness (Farmani et al. 2017b). \(\upzeta =3.1\) which is the fitting parameter which is a dimensionless constant. The conductivity \(\updelta\) of a GNS at T = 3000 k for infrared to THz range could be estimated via the formulation given in Eq. (4) (Casiraghi et al. 2007; Naghizade and Saghaei 2020a).

$${\updelta } \approx \frac{{{\text{ie}}^{2} {\text{E}}_{{\text{f}}} }}{{{\uppi }^{2} \left( {{\upomega } + {\text{i }}\tau^{ - 1} } \right)}}$$
(4)

here τ and ω represent the electron relaxation time and the angular frequency, respectively. Furthermore, the conductivity for N-layer GNS is shown as Nδ. Here N denotes the number of GNS layers. In recent works on PhCs, the finite difference time domain method (FDTD) has been suggested to analyze electromagnetic wave propagation in compact optical devices. FDTD can provide a comprehensive solution to Maxwell’s equations in a small volume of the PhC of interest (Gedney 2011). In this work, we have used the FDTD method to solve Maxwell’s equations and analyze the light propagation in the waveguide of interest. Further, the plane wave expansion (PWE) method is used to calculate the photonic band gap (PBG). In the following equation, the formulation for the optical absorption of graphene is provided

$$A\left( \lambda \right) = \frac{4\pi c}{\lambda }n\left( \lambda \right)k\left( \lambda \right)\int\limits_{V}^{{}} {\left| {E_{l} } \right|^{2} } dV$$
(5)

In Eq. (5), c is the speed of light in a vacuum, V stands for the volume of graphene and El represents the local electric field. Based on Eq. (5), we could observe that the light absorption is directly proportional to the square of local electric field intensity.

1.2 Optoelectronic decoder and its ring resonator

The 2D PhC proposed in this work has a square type of lattice made of Si and SiO2 rods. The Si rod has a radius of 118 nm, a lattice constant of a = 590 nm, and a refractive index of 3.46 at λ = 1550 nm. The SiO2 rod has a 108 nm radius, and a linear refractive index of 1.4. Figure 1 depicts the PBG of the proposed structure. It could be observed that in TM mode, the first PBG is at 0.29 < a∕λ < 0.4. This bans wavelengths within the 1439 nm < λ < 2034 range from propagation in any direction within the PhC. From the results of Fig. 1, it can be seen that the proposed structure is suitable for working at 1550 nm.

Fig. 1
figure 1

Fig. 4. The fundamental PhC’s band structure diagram in TM mode

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Figure 2 shows the suggested PCRR structure. Its working principle is discussed by Chhipa et. al in (Chhipa et al. 2021). The GNS and SiO2 defect rods are depicted in red and green colors, respectively. The defect rods are covered by 10 nm thick GNSs. It has a single input port and a dual output port for A and B. A beam of light with a center frequency of 1550 nm is injected into the input port of the waveguide and dropped to the PCRR. The output signal of the PCRR is dependent on the value of GNS chemical potential (µc). In Figs. 3a, b, the time-domain light propagation inside the PCRR for two values of µc are depicted.

Fig. 2
figure 2

The schematic of the proposed PhC structure in this work

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Fig. 3
figure 3

Light propagation inside the GNS-based PCRR for different values of chemical potential a µc = 0.3 eV and b µc = 0.6 eV

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Figure 3a shows that when the µc is 0.3 eV, the optical beam cannot enter the resonator and continues the propagation path directly to the output port of B. In this case, the measured power is more than 90%. Figure 3b illustrates that when µc is 0.6 eV, the optical beam can propagate inside the resonator and is directed to the output port (i.e., Port A). The measured power, in this case, is more than 95%.

As it could be observed in Fig. 4, the suggested optoelectronic 2-to-4 binary decoder is made of multiple waveguides and GNS-based ring resonators in the proposed PhC structure. As depicted in Fig. 4a, E represents the enable port, X and Y display the input ports, and O0, O1, O2, and O3 are the output ports of the decoder. The total size of this miniaturized device is 850 µm2. The Y input port, W2 waveguide, and R1 resonator with μc1 form the first switch. The X input port, W4 waveguide, and R2 resonator with μc2 make the second switch. The Y input port, W3 waveguide, and R3 resonator with μc3 also form the third switch. For the sake of clarity, the 3D view of the presented decoder is represented in Fig. 4b, the SiO2 substrate is shown in grey and is 2 µm thick. Gate voltages of V1, V2, and V3 are applied to the first, second, and third PCRRs, respectively. This will tune the output states of the decoder by altering the µc values in the ring resonators.

Fig. 4
figure 4

Schematics of the proposed optoelectronic 2-to-4 binary decoder, a top view in XY plane, and b the perspective view

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2 Simulation and results

In this work, we have used the FDTD method to computationally analyze the light propagation inside the suggested decoder consisting of two input ports of X and Y. The PML boundary conditions were assumed for all boundaries. The mesh size is chosen so fine that all changes in the geometric structure can be applied in the calculations. According to the computational principle, we have 22 (2 N, N denotes the number of input ports of the binary decoder) outputs and an optical signal centered at 1550 with an electric field of 1 V/m is injected into every active input port. For all four states of input ports, the computational findings are conferred as follows.

Case #1: In this scenario, input ports X and Y are kept inactive, which means that the applied signal strength is insignificant. We set the µc1 to 0.6 eV. Since µc2 and µc3 are less than 0.6 eV, the optical beam coming from E inside the W1 waveguide will be dropped into W2, and no optical waves travel in the direction of R2 and R3. Thus, O0 will be ON, and the other ports will be OFF in this case (see Fig. 5a). Figure 5b illustrates that the normalized power (the ratio of total output power to input power) at O0 is around 80%; however, the normalized powers at other ports are < 5%. Moreover, for O0, the rise and fall times are around 0.6 ps and 0.1 ps, respectively.

Fig. 5
figure 5

a The electric field distribution along with b the optical decoder’s time response diagram when both X and Y are OFF

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Case #2: In this scenario, X = 1, and Y = 0, by biasing the V3, we set the µc3 to 0.6 eV. Since µc1 and µc2 are less than 0.6 eV, the electromagnetic waves coming from E propagate close to R3 through W1 and drop into W3. Therefore, the O1 output port associated with W3 will be active (O1 = 1), and other output ports remain inactive (O0 = 0, O2 = 0, and O3 = 0). Figure 6a depicts the electric field distribution within the suggested decoder. The time response diagram is represented in Fig. 6b, where the amount of normalized power at port O1 is about 85% while it is less than 3% for other ports. The rise and fall times for O1 are 0.8 ps and 0.3 ps, respectively.

Fig. 6
figure 6

a The electric field distribution as well as b the optical decoder’s time response diagram when X = 1 and Y = 0

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Case #3: IN this scenario, X = 0, and Y = 1, by biasing the V2, µc2 = 0.6 eV. Since µc1 and µc3 are less than 0.6 eV, the light coming from E propagates close to R2 through W1 and drops into W4. Therefore, the O2 output port corresponding to W4 will be active (O2 = 1), and other output ports remain inactive (O0 = 0, O1 = 0, and O3 = 0). Figure 7a demonstrates the electric field distribution inside the decoder structure. The time response diagram is represented in Fig. 7b, where the amount of normalized power at O2 is about 90% and less than 5% for other ports. The rise and fall times of O2 are around 0.7 ps and 0.2 ps, respectively.

Fig. 7
figure 7

a The electric field distribution as well as b the optical decoder’s time response diagram when X = 0 and Y = 1

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Case #4: In this scenario, both input ports X and Y are active, resulting in X = 1, and Y = 1. Since µc1, µc2 and µc3 are less than 0.6 eV, the light propagating from E travels closely to R1, R2, and R3 through W1. Since the bias voltages of the resonators are below a certain threshold, none of the resonators can absorb the incoming light. As a result, the optical waves traveling from the E port reach O3. Thus, O3 will be active (O3 = 1), and other output ports remain inactive (O0 = 0, O1 = 0, and O2 = 0). Figure 8a demonstrates the electric field distribution inside the proposed decoder. Figure 8b shows the time response diagram where the quantity of normalized power at O3 is around 82% and is less than 3% for other ports. The rise and fall times for O3 are 0.5 ps and 0.15 ps, respectively. Table 1 summarizes the simulation results for the presented structure. The results discussed in this work, confirm that the suggested structure can operate as an optoelectronic 2-to-4 binary decoder. The ON/OFF contrast ratio for each of the outputs can easily be determined by dividing the percentage of logic 1 by the percentage of logic 0. These calculated ratio results are presented in Table 2. For the proposed structure, the worst rise and fall times are about 0.8 ps and 0.3 ps. The ON/OFF contrast ratio for the proposed structure is about 13.5 dB. The comparison of the proposed decoder with other works is listed in Table 2.

Fig. 8
figure 8

a The electric field distribution as well as b the optical decoder’s time response diagram when X = 1 and Y = 1

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Table 1 The input and output states of the decoder
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Table 2 The comparison between the proposed decoder and other structures in recently published papers
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3 Conclusion

In this work, we have proposed a new design for a fast and tunable optoelectronic 2-to-4 binary decoder using graphene-coated silica rods in photonic crystal ring resonators. The presented device consisted of three GNS-based ring resonators and several waveguides for input and output ports. Four different states were simulated and discussed for given inputs. Based on the computational findings of this work, it could be observed that the presented decoder is tunable by changing the µc of GNS material of the ring resonator. The maximum rise and fall times were about 0.8 ps and 0.3 ps. The ON/OFF contrast ratio is about 13.5 dB. Compared to previous works published so far, the most important advantages of the presented structure are its lower rise and fall times and tunability. Therefore, it is a proper tool for fast data processing devices.