High-performance electro-optical switch using an anisotropic graphene-based one-dimensional photonic crystal

Shahab Tavana; Shahab Tavana; Shahram Bahadori-Haghighi; Shahram Bahadori-Haghighi; Mohammad Hossein Sheikhi; Mohammad Hossein Sheikhi

doi:10.1364/OE.448607

1. Introduction

Optical switches and modulators are fundamental building blocks of high-speed photonic networks that are used to modify the characteristics of an optical carrier such as amplitude and phase. In the past decades, various optical switches and modulators based on the electro-absorption [1], electro-optic [2], thermo-optic [3] and magneto-optic [4] effects have already been proposed and investigated. In the previous researches, a lot of efforts have been made to reduce the device footprint and the required switching voltage. In this regard, novel CMOS compatible photonic structures have been explored to increase light-matter interaction so that the size and consequently the total volume of the optical circuits are minimized [5,6]. Photonic Crystal (PC) is a novel type of optical structure that can control the flow of electromagnetic radiation to realize different kinds of optical devices [7]. Different optical devices like switches [8], isolators [9] and biosensors [10] based on PCs have already been proposed. One-dimensional (1D) PC is the simplest version of the PC family with low propagation loss and high optical precision to be integrated with existing photonic devices. Besides, the planar construction of 1D PC can be realized by the chemical vapor deposition (CVD) technique [11]. 1D PCs are also known as Bragg mirrors because the refractive indices of their constituents are only periodic in one direction which can block the propagation of certain wavelengths within a range known as photonic bandgap (PBG) [12,13]. Therefore, such a gap is useful to trap or slow down light in order to enhance light-matter interaction. Another approach to realize effective and compact optical switches is to apply new materials like graphene with great tunable characteristics.

Graphene is a two-dimensional atomic crystal which has renowned for its unique properties. It is a zero bandgap material which allows electrons to move easily with lower energy. Graphene is an allotrope of carbon atoms formed in a one-atom thick sheet and arranged in a hexagonal lattice. Very high carrier mobility is one of the most significant characteristics of graphene [14], so that electrons behave like massless Dirac Fermions. Additionally, graphene has a strict linear dispersion curve near its Dirac point so that Fermi level and as a result optical absorption can be effectively tuned by electrostatic gating. One of the most remarkable features is its absorption coefficient that is almost stable at πα=2.3% in the visible and infrared regimes. Furthermore, in contrast to metals, graphene has negligible absorption loss in the THz and far-IR frequency ranges [15].

In recent years, various graphene-based optical devices such as sensors [16], solar cells [17], polarizers [18], modulators and switches [19], absorbers [20] have already been proposed. The tuning capability of graphene is useful for manipulating the PC bandgap without changing the geometrical parameters. Furthermore, the absorption of graphene could be enhanced by PC structures which can be used to design absorbers in the visible and near-infrared spectral ranges [21]. Such an enhancement can be achieved by introducing defect or disorder into the PC structure so that the periodicity is broken and a high-quality micro cavity can be produced. These defect modes appear as very narrow peaks in the transmission spectra of the structures [22]. A low-threshold optical bistability based on Fano resonance in a 1D PC was theoretically examined by Peng et al. [23]. Monfared et al. [24] experimentally produced an electro-optical switch based on 1D graphene plasmonic PCs. The switiching operation is controlled by a gate voltage of 1 V that leads to a significant change in the relative permittivity of bilayer graphene. Liu et al. [25] proposed an electro-optic modulator based on PC nanobeam, combined with graphene/Al₂O₃ multilayer stacks (GAMS), which can considerably regulate the absorption peak. Apart from PC structures, hyperbolic metamaterials and metasurfaces have already been utilized to achieve a switch or modulator. For example, a multifunctional modulator based on a periodic hyperbolic metamaterial was designed by Ma et al. [26]. In 2019, Bahadori-Haghighi et al. [27] demonstrated optical modulators using double-layer graphene-based metasurfaces. According to the results, a high modulation depth of 95% was realized by applying an external voltage of 4.95 V.

In this paper, we propose a high-performance optical switch using an anisotropic graphene (AG)-based 1D PC. In section 2, in order to extract the optical properties of the defective 1D PC, the structure is analyzed by exploiting the finite-difference time-domain (FDTD) and the transfer matrix methods (TMM). In our proposed optical switch a Fabry-Perot cavity is realized by introducing two AG sheets across a dielectric layer as a defect into the 1D PC. A very effective electro-optical switch in the NIR is obtained due to strong field localization inside the defect. The operation of the switch is investigated in section 3 where an extinction ratio of as high as 14.26 dB is obtained. To the best of our knowledge, this is the highest extinction ratio ever reported for a polarization-insensitive free-space switch. Other switching characteristics such as insertion loss, switching voltage, speed and power consumption are also calculated. Finally, the influence of oblique incident angle and temperature on the operation of optical switch are studied.

2. Proposed optical switch and theoretical model

The schematic configuration of the proposed optical switch is shown in Fig. 1. The structure is situated as (AB)^NGCG(AB)^N, where the layers are parallel to the x-y plane and are illuminated by an electromagnetic plane wave propagating along the z-axis with an angle of incidence θ, and a wave vector of k₀ =2π/λ. Here, G is the monolayer sheet of AG with the thickness of 0.34 nm, A, B and C are isotropic dielectric materials with the thicknesses of d_A, d_B and d_C, respectively. The relative permittivity of the layers are respectively ɛ_A, ɛ_B and ɛ_C and the lattice constant of the structure is a = d_A+d_B.

Fig. 1. Perspective view of the optical switch with applied voltage.

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There are various methods for simulating the optical properties of 1D PC structures, such as plane wave expansion (PWE) [28], TMM [29,30], FDTD [31,32], finite element method (FEM) [33] and Green function method (GF) [34]. In this work, FDTD and TMM are applied to study the optical properties of 1D PC. Semi-analytical methods like the TMM are very useful due to their accurate results and relatively small computational volume. TMM in optics is used to analyze the propagation of electromagnetic waves in a homogeneous medium. It is based on Maxwell's equations and the simple continuity conditions for electric fields along the boundaries of the layers. In other words, if the field is known at the beginning of a specified layer, the field at the end of the layer can be obtained from a simple matrix operation.

Now, the optical response of the proposed 1D PC is investigated using TMM. First of all, it should be remembered that the electric fields in anisotropic materials are related by a 3×3 tensor, graphene is an optically uni-axial anisotropic material due to its 2D nature, and its relative anisotropic permittivity in the x-y plane can be described as [35]:

(1)$${\varepsilon _G} = \left( {\begin{array}{ccc} {{\varepsilon_{G,t}}}&0&0\\ 0&{{\varepsilon_{G,t}}}&0\\ 0&0&{{\varepsilon_{G, \bot }}} \end{array}} \right).$$

Since the normal electric field cannot excite any currents in the graphene sheet, the normal component of graphene permittivity is ɛ_G,$\bot$₌ 1. The dielectric function of graphene can be determined as [36,37]:

(2)$${\varepsilon _{G,t}} = 1 + j\frac{{\sigma (\omega )}}{{{\varepsilon _0}\omega {d_G}}}.$$

where ω is the angular frequency, ε₀ is the vacuum permittivity. Surface conductivity (σ) of graphene is described by the Kubo formula which takes into account the interband and intraband transitions according to the following relation:

(3)$$\sigma (\omega )= {\sigma _{{\mathop{\rm int}} ra}} + {\sigma _{{\mathop{\rm int}} er}}$$

The intraband and interband conductivities are determined as [35]:

(4)$${\sigma _{\textrm{intra}}} ={-} j\frac{{{e^2}{k_B}T}}{{\pi {\hbar ^2}(\omega - j\Gamma )}}\left( {\frac{{{\mu_c}}}{{{k_B}T}} + 2\ln ({e^{ - \frac{{{\mu_c}}}{{{k_B}T}}}} + 1)} \right),$$

(5)$${\sigma _{\textrm{inter}}} ={-} j\frac{{{e^2}}}{{4\pi \hbar }}\ln \left( {\frac{{2{\mu_c} - (\omega - j\Gamma )\hbar }}{{2{\mu_c} + (\omega - j\Gamma )\hbar }}} \right),$$

where µ_c is the chemical potential, k_B is the Boltzmann’s constant, $\hbar$ is the reduced Planck constant, T is the temperature (in Kelvin), e is the charge of the electron, n is the carrier density and Γ is the phenomenological scattering rate.

The plane wave amplitudes at each side of an interface can be related by transfer matrix M_i (i = A, B, G, C and D for Si, SiO₂, AG, Al₂O₃ and air layers, respectively) which is given by [38]:

(6)$${M_i} = \left( {\begin{array}{cc} {\cos {k_{iz}}{d_i}}&{\frac{{ - i}}{{{p_i}}}\sin {k_{iz}}{d_i}}\\ { - i{p_i}\sin {k_{iz}}{d_i}}&{\textrm{cos }{k_{iz}}{d_i}} \end{array}} \right).$$

where ${p_i} = \frac{{{k_{iz}}}}{{\omega {\mu _0}}}$, ${k_{iz}} = \sqrt {{k_0}^2{\varepsilon _i} - {k_{ix}}^2}$, ${k_{ix}} = {k_0}\sin ({\theta _0})$ for the TE wave and ${p_i} = \frac{{\omega {\varepsilon _0}{\varepsilon _i}}}{{{k_{iz}}}}$, ${k_{G,z}} = \sqrt {{k_0}^2{\varepsilon _{G,t}} - {k_{ix}}^2({\varepsilon _{G,t}}/{\varepsilon _{G, \bot }})}$, ${k_{iz}} = \sqrt {{k_0}^2{\varepsilon _i} - {k_{ix}}^2}$, ${k_{ix}} = {k_0}\sin ({\theta _0})$ for the TM wave. The transfer matrix for N-period PC is obtained by multiplying the mentioned matrices which is expressed according to the following equation:

(7)$$M = {({{M_A}{M_B}} )^N} = \left( {\begin{array}{cc} {{m_{11}}}&{{m_{12}}}\\ {{m_{21}}}&{{m_{22}}} \end{array}} \right).$$

The total transfer matrix of the proposed defective PC could also be extracted in the same way:

(8)$$M = {({{M_A}{M_B}} )^N}{M_G}{M_C}{M_G}{({{M_A}{M_B}} )^N} = \left( {\begin{array}{cc} {{m_{11}}}&{{m_{12}}}\\ {{m_{21}}}&{{m_{22}}} \end{array}} \right).$$

Finally, the reflection and transmission coefficients are calculated by using the components of the total transfer matrix:

(9)$$R = |r{|^2},{\kern 7pt}T = \left\{ \begin{array}{ll} |t{|^2}&\textrm{For TE}\\ \frac{{{p_0}}}{{{p_{N + 1}}}}|t{|^2}&\textrm{For TM} \end{array} \right., A = 1 - T - R.$$

(10)$$r = \left( {\frac{{{m_{11}}{p_0} + {m_{12}}{p_0}{p_{N + 1}} - {m_{21}} - {m_{22}}{p_{N + 1}}}}{{{m_{11}}{p_0} + {m_{12}}{p_0}{p_{N + 1}} + {m_{21}} + {m_{22}}{p_{N + 1}}}}} \right), $$

(11)$$t = \left( {\frac{{2{p_0}}}{{{m_{11}}{p_0} + {m_{12}}{p_0}{p_{N + 1}} + {m_{21}} + {m_{22}}{p_{N + 1}}}}} \right), $$

(12)$${p_{N + 1}} = \left\{ \begin{array}{l} \sqrt {{\varepsilon_0}/{\mu_0}} \cos ({\theta_{N + 1}})\\ \sqrt {{\varepsilon_0}/{\mu_0}} /\cos ({\theta_{N + 1}}) \end{array} \right.,{\kern 7pt}{p_0} = \left\{ \begin{array}{l} \sqrt {{\varepsilon_0}/{\mu_0}} \cos ({\theta_0})\\ \sqrt {{\varepsilon_0}/{\mu_0}} /\cos ({\theta_0}) \end{array} \right.,$$

where ɛ₀ and ɛ_N+1 are the dielectric constants at the input and output planes (i.e. air), respectively, θ₀ and θ_N+1 are also angles of the input and output plane. The dispersion relation of our optical switch is also calculated as follows:

(13)$$\cos ({k_z}a) = \cos ({k_{A,z}}{d_A})\cos ({k_{B,z}}{d_B}) - \frac{1}{2}\left( {\frac{{{P_A}}}{{{P_B}}} + \frac{{{P_B}}}{{{P_A}}}} \right)\sin ({k_{A,z}}{d_A})\sin ({k_{B,z}}{d_B}). $$

Another method that is used to analyze the proposed PC is the FDTD method. The FDTD method is the most accurate tool for electromagnetic analyses of structures, providing reliable solutions to Maxwell's equations and the ability to control complex-valued AG properties. We apply the periodic boundary conditions along the x and y axes and the perfectly matched layers (PML) along the z axis. The mesh size near the AG layers is reduced down to 0.01 nm along the z axis. In the following section, the results of the TMM are presented and discussed where they are compared with those obtained by the FDTD method.

3. Results and discussions

In order to perform optical simulations of the proposed switch we assume normal incidence of light which is the same for TE and TM polarizations according to ${k_{iz}} = {k_0}\sqrt {{\varepsilon _i}}$. Therefore, TM polarization of light at normal incidence is considered to perform calculations. Two lossless materials Si and SiO₂ are employed with relative permittivities of ɛ_A= 11.7, ɛ_B = 2.1 and thicknesses of d_A = 325 nm, d_B = 380 nm, respectively. AG and Al₂O₃ are also placed as the defect layers where Al₂O₃ is sandwiched by AG sheets. Other optical and geometrical parameters are assumed to be ɛ_C = 3.1, d_C = 75 nm, T = 300 K and Γ = 1 THz. The number of periods at each side of the defect layers is N = 7. Our analyses are based on TMM and FDTD methods and it will be shown that the results of the FDTD method agree with those of the TMM which confirm the validity of the numerical calculations presented in this paper. The total sizes of the device in the x and y directions are the same as 10 µm. A unit cell of the perfect 1D PC is required to examine the band structure of the proposed optical switch which is depicted in Fig. 2.

Fig. 2. Real part of the dispersion relation of the Si/SiO₂ 1D PC at normal incidence.

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The thicknesses of Si and SiO₂, their refractive index contrast and the lattice constant that is equivalent to the periodicity are the fundamental characteristics that define the PBG of the structure. As it is shown in Fig. 2, there are two main PBGs. The first bandgap is in the wavelength range between 1.51 µm to 1.87 µm and the second from 2.8 µm to 4.4 µm. Our optical switch operates within the first PBG. The optical switch is realized by creating a multilayer Fabry-Perot cavity at the middle of the PC consisting of an Al₂O₃ layer between two AG sheets. The incident wave at the resonance wavelengths are trapped in the defect region and the energy is stored. Therefore, the defect provides time for the trapped photons to tunnel through the crystal to reach the output, the effect of the resonance inside the cavity and the consequent tunneling with the lowest absorption losses lead to a crack inside the PBG [39].

Assuming that both AG layers have uniform conductivities and chemical potentials of µ_c1=µ_c2= 0.42 eV, the optical switch is normally in the OFF state. This is due to the fact that when there is no applied voltage, the interband transitions are Pauli blocked and the AG layers are transparent with negligible absorption. The transmission and reflection spectra of optical switch in the OFF state are shown in Figs. 3(a) and 3(b).

Fig. 3. (a) Transmittance and (b) Reflectance spectra of the optical switch in OFF state.

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According to Fig. 3(a), it is worth noting that the defect mode appears as sharp transmission peaks at the wavelengths of 1547 nm and 1552.4 nm with amplitudes of 0.91 and 0.96 for TMM and FDTD methods, respectively. Hence, the results from the two methods show a 5.4 nm wavelength difference. It should be noted that the TMM is originally a 1D analytical approach where continuity equations are applied at the interfaces. Therefore, TMM has limited accuracy due to its inherent approximations. In contrast, as you know, FDTD method is a numerical approach that can be used for 3D simulations (like in our paper). Furthermore, in FDTD method, PML boundary conditions are applied to the computational cell and the structure is also broken into small domains determined by the mesh size. Hence, it can generally be concluded that the 3D FDTD method is much more accurate than the 1D TMM and a 5.4 nm difference of the resonance wavelengths between the two methods is expectable and inevitable.

The variations of the transmission spectrum versus the number of periods is shown in Fig. 4(a). As it is illustrated, the spectral width of the transmission peak is reduced and the center wavelength redshifts by increasing the number of periods. Therefore, the degree of defect mode localization is enhanced by increasing the number of periods so that the quality factor of the defect mode can be adjusted. The reduction of the linewidth is due to the fact that the proposed device is actually a Fabry-Perot structure where a defect is surrounded by grating mirrors. When the number of periods increases the reflectivity of the mirrors increases, too which leads to a more confined defect mode and a high quality factor resonance. On the other side, the physical reason behind the redshift of the resonance is that the field profile of the defect mode is more confined to the defect layer so that the concentration of the field within the adjacent layer with lower refractive index is reduced which results in a resonance with lower frequency or longer wavelength.

Fig. 4. (a) Color map of the transmittance versus the number of periods and wavelength, (b) Absorption spectra of the optical switch in OFF state.

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The quality factor is one of the important factors to evaluate the performance of optical switches that is calculated as Q=λ/Δλ where λ is the central wavelength and Δλ is the spectral width of the transmission peak. Therefore, the high quality factors of 6222 and 5000 are obtained from TMM and FDTD methods for N = 7. According to Fig. 4(b), partial absorption of 0.049 for TMM and 0.045 for FDTD methods is observed due to the absorption of two AG sheets.

In order to clarify the previous results, the electric field distribution is presented to exhibit the significant field enhancement at the transmission peak. The electric field distribution of the optical switch at the wavelength of 1552.4 nm is plotted in Fig. 5. It can be seen that high electric field intensity is trapped in the defect region and constantly bounces back and forth between the two mirrors. At this condition, only the wavelengths with constructive interference in the defect pass through the optical switch. Consequently, the interaction between AG and light increases that can fundamentally increase the performance of the optical switch.

Fig. 5. The electric field profile distributions of the optical switch in OFF state at 1552.4 nm.

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Now, the operation of the switch in response to the applied external voltage is investigated. The principle operation of the AG sheets in our proposed switch is schematically shown in Fig. 6. As it is shown in Fig. 6(a), when there is no voltage across the AG sheets and the switch is in the OFF state, the chemical potentials of the two AG layers are equal and a little above half the photon energy of the incident light (E_f > hν/2), which does not allow the interband optical transitions due to Pauli blocking because the conduction band is approximately occupied. Therefore, both AG layers are transparent, simultaneously. However, when an appropriate external voltage is applied across the two AG layers, the electron density inside the left AG increases while that of the right AG sheet decreases. Since the band structure of the AG layers are not symmetric around the chemical potential of 0.42 eV, applying an external voltage does not shift the chemical potentials of the left and right AG layers equally.

Fig. 6. Principle operation of the proposed switch based on energy band diagram of AG. (a) Intraband transition dominates when optical switch is in OFF state. Energy band diagram of the (b) left and (c) right AG sheets in the ON state.

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To clarify this theorem assume that the chemical potential of the left AG changes from 0.42 eV to 0.4 eV. As a result, the changes of carrier concentration in both AG layers are the same as Δn = 1.2×10¹² 1/cm² according to ${\mu _c} = \hbar {\nu _F}\sqrt {\pi n}$[40]. Where ν_F is the Fermi velocity that is 10⁶ m/s, and ħ represents the Planck constant. Therefore, the chemical potential of the right AG must shift to 0.439 eV to support the required change of the carrier concentration. As it is seen in Fig. 6(b), the chemical potential of the left AG goes below the incident photon energy level (i.e. E_f < hν/2) which allows the absorption of photons and interband transitions of electrons so that the switch is set at the ON state. This is while according to Fig. 6(c), the chemical potential of the right AG is still above the photon energy level (i.e. E_f > hν/2) and the right AG remains transparent.

The transmission and reflection spectra of the optical switch in the ON state are shown in Figs. 7(a) and 7(b), respectively. As it was mentioned earlier, when an appropriate voltage is applied to the AG layers, the left AG sheet becomes lossy while the right AG remains transparent. This leads to a significant reduction of the transmission peak as well as broadening of the resonance line width. As it is shown in Fig. 7(a), transmission peaks in the ON state are located at the wavelengths of 1546.5 nm and 1551.8 nm with amplitudes of 0.034 and 0.053 for TMM and FDTD methods, respectively. Hence, there is a blueshift of the resonance compared with the OFF state which could help to boost the extinction ratio that is a critical parameter for evaluating the performance of optical switches. The extinction ratio of the optical switch is calculated from the following equation:

(14)$$ER = 10\log (\frac{{{T_{OFF}}}}{{{T_{ON}}}})$$

where T_OFF and T_ON are the transmission amplitudes of the optical switch in the OFF and ON states. T_OFF and T_ON for TMM (FDTD) at the wavelength of 1547 nm are respectively 0.91 and 0.0211 (0.96 and 0.036 at the wavelength of 1552.4 nm) which lead to the ER of 16.34 dB (14.26 dB). The insertion loss of the optical switch is also defined as L = |10log (T_OFF)|. The insertion loss is calculated to be 0.4 dB for TMM and 0.18 dB for FDTD methods.

Fig. 7. (a) Transmittance and (b) Reflectance spectra of the optical switch in ON state.

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The reflection spectra of the optical switch in the ON state is also shown in Fig. 7(b) where the reflection amplitudes at the resonant wavelength for the TMM and FDTD methods are 0.71 and 0.64, respectively. The corresponding absorption spectra are also shown in Fig. 8(a). As it can be seen, the absorption peaks are respectively increased to 0.25 and 0.3 for TMM and FDTD methods compared to those in the OFF state because the left AG acts as an absorber. The electric field distribution of the optical switch at 1551.8 nm when it is in the ON state is shown in Fig. 8(b). The weak electric field intensity in the defect is obviously seen. Furthermore, the optical switch works as a Bragg reflector and reflects light due to the lack of confinement field inside the defect.

Fig. 8. (a) Absorption spectra of the optical switch in ON state, (b) The electric field profile distributions of optical switch in ON state at 1551.8 nm.

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Figure 9 shows the refractive index of an AG sheet as a function of the chemical potential. The inset Fig. 9 depicts the comparison of transmission peak in the ON and OFF state. According to the diagram, there are two main phenomenon. Firstly, the reduction of the transmission peak in the ON state compare to the OFF state due to increased imaginary part of the refractive index of the left absorber AG. Secondly, as it can be seen, the transmission peak in the ON state experiences a blueshift compared to that of the OFF state. This is due to the fact that real part of the refractive index decreases for the right AG, while that of the left AG rises. Although the Re(neff) of the left AG increases, there is a slight decrease near the µ_c= 0.4 eV that leads to a reduction of the effective total refractive index of two AG layers. The other reason could be due to the distribution of the defect mode profiles in the ON and OFF states. In other words, when the switch is in the OFF state, the graphene sheets are transparent and the defect mode profile is highly confined within the defect region. However, when it turns to the ON state, the graphene sheets are lossy and the Q-factor of the resonance is reduced. Therefore, the mode profile is less confined in the defect layer so that it penetrates into the adjacent layers. However, the refractive index of SiO₂ is closer to that of the defect (Al₂O₃) (compared to the refractive index of Si) and consequently the penetration into the adjacent SiO₂ layer is much higher than that into the Si layer. Therefore, the mode profile tends to move toward the SiO₂ layer with lower refractive index which results in the blueshift of the resonance.

Fig. 9. The refractive index of monolayer AG as a function of the Fermi energy at 1547 nm. The inset shows the wavelength shift in the transmission diagram of optical switch for TMM.

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Now, the effect of incident angle of light and temperature on the proposed optical switch are investigated using TMM. Contour plots of the transmission spectra as functions of the incident angles for TE and TM polarizations in the OFF state are depicted in Figs. 10(a) and 10(b), respectively. It can be seen that the structure is sensitive to the angle of incident light and when it just changes 3°, the resonant wavelength approximately shifts 1 nm to lower wavelengths for both the TE and TM polarizations. The reason of this blueshift is the displacement of the PBG for both polarizations. As it is shown in Fig. 10, when the incident angle increases, the resonant wavelengths decrease for both polarizations. Furthermore, the amplitudes of the resonances decreases because by increasing the incident angle fewer incoming photons penetrate the structure and as a result, an incomplete resonance is formed within the defect region.

Fig. 10. Color map of the transmission versus incident angles and wavelengths for (a) TE and (b) TM polarizations in the OFF state.

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Since graphene is a naturally anisotropic material, the angle of incidence has an impact on its optical properties, too. The dependence of the transmission spectra on the incident angles for both TE and TM polarizations in the ON state is illustrated in Fig. 11. It is clear that the resonant wavelength shifts to lower wavelengths as the incident angle increases for both polarizations. This blueshift in TE polarization is accompanied by a decrease in the transmission peak. In contrast, the amplitude of the defect mode for TM polarization increases when the incident angle rises.

Fig. 11. Color map of the transmission versus incident angles and wavelengths for (a) TE and (b) TM polarizations in the ON state.

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In order to evaluate the effect of temperature on the structure, two important factors must be considered simultaneously. One of the factors is the thermal expansion coefficient (α) which influences the thickness of the crystal layers. Another factor is the thermo-optical coefficient (β) which affects the refractive index of the layers as the temperature varies. The values of the two mentioned coefficients for each layer are given in Table 1. It should be said that the two coefficients for the AG layers are ignored since the thickness of AG is minimal compared to other layers and also according to Eq. (4) and (5), the surface conductivity of graphene is slightly sensitive to temperature variations. The effects of the two coefficients on the thickness and refractive indices of the layers are obtained from the following equations [41–43]:

(15)$$\left\{ \begin{array}{l} {d_i}(T )= {d_i}({1 + \alpha \Delta T} )\\ {n_i}(T )= {n_i} + \beta \Delta T \end{array} \right.$$

where n_i and d_i are respectively the refractive index and thickness of ith layer at room temperature. ΔT is the temperature change relative to room temperature. The variations of the transmission spectrum of the optical switch in the OFF state versus temperature changes are shown in Fig. 12(a). As it is seen, the defect mode shifts toward lower wavelengths while its amplitude remains unchanged when the temperature increases from 100 K to 300 K. According to Fig. 12(a), the sensitivity of the optical switch to temperature in the OFF state is S = Δλ/ΔT = 0.01 nm/K.

Fig. 12. Transmission spectra of the defect mode at three different temperature for (a) OFF state and (b) ON state.

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Table 1. Thermo-optical and expansion coefficients of the applied materials

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The transmission spectra of the optical switch for various temperatures in the ON state are shown in Fig. 12(b). It is clear that likewise the OFF state, the resonant wavelength of defect mode shifts to lower wavelengths when the temperature increases. Moreover, the amplitude of the transmission decreases slightly as temperature rises that is due to the enhanced absorption of the left AG layer. According to the presented results, the temperature sensitivity of our proposed optical switch is relatively low in both OFF and ON states.

One of the key features of the device is switching voltage, which indicates the required voltage to change the chemical potentials of the AG sheets that is obtained from the following equation [40]:

(16)$$V = \frac{{qn}}{{{C_{ox}}}} + \frac{{\Delta {\mu _c}}}{q}$$

where Δn is the change of the carrier concentration, Δµ_c is the total changes of the chemical potentials and C_ox is the oxide capacitance per unit area which is calculated by the simple capacitor model C_ox = ɛ/d = 0.0413 µF/cm². Therefore, according to Eq. (16), the proper voltage for switching is calculated to be as low as 4.68 V for the defect thickness of 75 nm. The variations of the switching voltage versus the total changes of the chemical potential and the thickness of the defect layer is shown in Fig. 13. The thickness of the defect layer plays an essential roles in the switching voltage and speed of the optical switch. It should be noted that the total capacitance of the device and as a result the switching voltage are affected by the thickness and refractive index of the defect layer. As it can be seen in Fig. 13, by increasing the thickness of the defect layer, the required voltage also increases. In order to achieve a low switching voltage and at the same time a reasonable speed or bandwidth, the optimum defect thickness of 75 nm is selected for the proposed structure. On the other side, according to Fig. 13, if the total changes of the chemical potentials increase, the switching voltage increases, too. In this paper, the initial chemical potentials of the two AG sheets are set to 0.42 eV so that by a total change of chemical potential as low as Δµ_c = 0.039 eV, the switching operation is performed.

Fig. 13. Variation of the switching voltage versus the total difference of chemical potential and defect thickness.

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Regarding the bandwidth of the proposed switch, it should be noted that the total capacitance (C_T) of the device is the series combination of the oxide capacitance (C_ox) and quantum capacitor (C_Q) of the left and right AG layers. the quantum capacitance per unit area for a single monolayer AG is expressed as follows [44]:

(17)$${C_Q} = \frac{{2{e^2}{k_B}T}}{{\pi {{(\hbar {\nu _F})}^2}}}\ln \left[ {2(1 + \cosh (\frac{{{\mu_c}}}{{{k_B}T}})} \right]$$

The calculated quantum capacitance per unit area is equal to 9.86 µF/cm² which is much larger than the oxide capacitance (C_Q >> C_ox), so the quantum capacitance can be neglected. According to device size, the total capacitance of the structure is calculated as 41.3fF. The total resistance of the structure consists of the sheet resistances of the left and right AG layers and contact resistances which is obtained according to the following equation [45,46]:

(18)$${R_T} = 2{R_{sheet}}\frac{W}{L} + 2\frac{{{R_c}}}{L}$$

where R_sheet is the graphene sheet resistance, R_C is the metal-graphene contact resistance, L and W are the length and width of the AG layers. Assuming that each AG layer is connected to a metal strip of palladium (Pd), the value of R_C is approximately equal to 100 Ω.µm. The typical values of the graphene sheet resistances of are between 100-300 Ω/Sq [47]. Therefore the total ohmic resistance of the structure becomes 220 Ω for a sheet resistance of 100 Ω/Sq. According to the relation f_3dB = 1/2πRC, the 3 dB bandwidth of the optical switch is obtained as high as 17.5 GHz. According to E = CV²/4, the total energy consumption of the optical switch is also estimated to be as low as 226 fJ/bit. Table 2 presents the comparison of the performance of our proposed optical switch with the previously reported optical switches or modulators. It can be seen that our proposed optical switch is the only polarization-insensitive switch for which the required switching voltage and 3 dB bandwidth are reasonable and even much better than some references. Moreover, the ER of 16.34 dB (14.26 dB) for TMM (FDTD) is the highest ER reported for a polarization-insensitive optical switch.

Table 2. Comparison of the performance of various optical switches and modulators

View Table | View all tables in this article

4. Conclusion

In conclusion, we proposed an electro-optical switch based on a defective one-dimensional photonic crystal. The optical properties of switch are calculated by the TMM and 3D FDTD method. A defect layer consists of a dielectric layer sandwiched between two AG layers so that a narrow band resonance appears in the PBG. When a low voltage of 4.68 V is applied across the AG layers, the switch changes from OFF to ON state so that the transmission peak is reduced and a high ER of 14.26 dB is calculated. According to the calculations, the switching speed is as high as 17.5 GHz and the total energy consumption of the optical switch is 226 fJ/bit. Finally, the effect of the temperature and oblique incident angle on the resonance wavelength has been investigated by the TMM. The high extinction ratio, low switching voltage, high bandwidth and low energy consumption of our proposed switch make it a good candidate for near-infrared optical systems.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Material	α (1/K)	β (1/K)
Si	2.62 × 10⁻⁶	166 × 10⁻⁶
SiO₂	5.5 × 10⁻⁷	1 × 10⁻⁵
Al₂O₃	7.15 × 10⁻⁶	12.6 × 10⁻⁶

Structure type	Parameters
Structure type	Voltage (v)	ƒ_3dB(GHz)	ER(dB)	Polarization sensitivity
Hybrid plasmonic waveguide [48]	2.3	83.91	NA	sensitive
Fano resonance in metasurface [27]	4.95	0.63	12.98	sensitive
Graphene plasmonic coupler [40]	6	60	10.5	sensitive
Lithium niobate photonic-crystal [49]	2.5	17.5	11.5	sensitive
Lithium niobate waveguide [50]	40	0.045	17.39	sensitive
U-shaped micro-ring graphene [51]	1.2	NA	17	sensitive
Plasmonic Bragg microcavity [52]	5	NA	19.8	sensitive
This work	4.68	17.5	16.34	insensitive

Material	α (1/K)	β (1/K)
Si	2.62 × 10⁻⁶	166 × 10⁻⁶
SiO₂	5.5 × 10⁻⁷	1 × 10⁻⁵
Al₂O₃	7.15 × 10⁻⁶	12.6 × 10⁻⁶

Structure type	Parameters
Structure type	Voltage (v)	ƒ_3dB(GHz)	ER(dB)	Polarization sensitivity
Hybrid plasmonic waveguide [48]	2.3	83.91	NA	sensitive
Fano resonance in metasurface [27]	4.95	0.63	12.98	sensitive
Graphene plasmonic coupler [40]	6	60	10.5	sensitive
Lithium niobate photonic-crystal [49]	2.5	17.5	11.5	sensitive
Lithium niobate waveguide [50]	40	0.045	17.39	sensitive
U-shaped micro-ring graphene [51]	1.2	NA	17	sensitive
Plasmonic Bragg microcavity [52]	5	NA	19.8	sensitive
This work	4.68	17.5	16.34	insensitive

High-performance electro-optical switch using an anisotropic graphene-based one-dimensional photonic crystal

Abstract

1. Introduction

2. Proposed optical switch and theoretical model

3. Results and discussions

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (2)

Equations (18)

Optics Express