Electro-optic modulator side-coupled with a photonic crystal nanobeam loaded graphene/Al<sub>2</sub>O<sub>3</sub> multilayer stack

Hanqing Liu; Peiguo Liu; Li-an Bian; Chenxi Liu; Qihui Zhou

doi:10.1364/OME.8.000761

1. Introduction

Photonic crystal cavities provide powerful means for modifying the interactions between light and matter due to their high quality factor and small mode volume, which have inspired great interests on the fields including quantum information processing, nonlinear optics, optical trappings, and optofluidics [1–3]. Particularly, created by etching a one-dimensional lattice of cylindrical air holes into silicon waveguide, the photonic crystal nanobeam (PCN) can be easily applied to electro-optic (EO) modulators, which show merits regarding device footprints, modulation speeds, power consumptions, and mass-productivity [4,5]. Particularly, the diameter of unit hole and periodic constant length of PCN can be optimized to achieve a ultrahigh quality factor [6,7]. However, the optical dispersion relations, symmetries, and spatial distribution are still hardly tuned once the structural parameters are fixed, which means the resonant characteristic of modulator based on PCN cannot be controllable after fabrication.

Since its discovery in 2004, graphene, a newly emerged two dimensional atomically thin material arranged in honeycomb lattice, processes exceptional electrical and optical properties like ultra-high carrier mobility as well as broadenband tunable optical absorption [8–10]. It is verified in experiments that the surface conductivity of graphene can be tuned effectively under electric/magnetic biasing or chemical doping, which makes it possible to solve the non-tunability of devices and achieve the dynamical wavelength control over mid- and far-infrared region [11,12]. In previous research, Pan et al. designed a hybrid modulator implemented by a compact PCN cavity coupled to a bus waveguide with monolayer graphene on the top surface, which achieve the modulation of quality factor and resonance wavelength through applied voltage [13]. Xu et al. obtained a strong enhancement of light-graphene (LG) interaction by combining graphene with one-dimensional PCN resonator [14]. However, these EO modulators are often limited by narrow tuning range and insensitive response for gate voltage, since the monolayer graphene is too thin to sustaining an intense resonance, which makes it hard to realize a significant modulation for light wave [15,16]. Besides, the surface defects could probably cause electrons to deflect and back scattering, result in weakening the LG interaction [17,18].

To make up the limitation caused by monolayer graphene, in this paper, we introduce the graphene/Al₂O₃ multilayer stack (GAMS) which is formed by depositing alternating graphene layers and thin dielectric layers. It is indicated that the distributing Dirac fermions in monolayer graphene disks into GAMS can drastically increase the plamonic resonance due to the strong Coulomb interaction of adjacent graphene layer [19,20]. Here, we propose a novel EO modulator side-coupled with PCN loaded GAMS for the first time. Benefiting from the enhanced LG interaction in GAMS and the tunable and strong EO effect of the whole device, the high-speed and wide-range modulation of light wave can be easily realized. And we also demonstrate that the modulation depth of our proposed device is much better than that of previous studied EO modulators using monolayer graphene.

2. Theoretical analysis and device design

To prove the feasibility of the proposed EO modulator, we firstly analyze the electrical tunability of the permittivity of GAMS in theory. Generally, the monolayer graphene is electrically modeled either as a 2D infinitesimally thin conductive layer by the complex surface conductivity $σ_{g}$ , or as a 3D actual medium by the permittivity $ε_{g}$ [21]. The relationship is $ε_{g} = 1 + j σ_{g} / ω ε_{0} h_{g}$ , where $ω$ is the angular frequency of light wave, $ε_{0}$ is the permittivity of air, $h_{g}$ (~1nm) is the thickness of monolayer graphene. In terms of conductivity, $σ_{g}$ can be calculated as $σ_{g} = σ_{intra} (ω) + σ_{inter} (ω)$ , which contains two contributed portions [22–24]: $σ_{intra} (ω)$ represents absorption due to intraband electron photon scattering, while $σ_{inter} (ω)$ is caused by interband electron transition process. Their expressions are given by [25]

σ_{intra} (ω) = j \frac{q^{2}}{π ℏ (ℏ ω + j Γ_{c})} [μ_{c} + 2 k_{B} T \ln (e^{- μ_{c} / k_{B} T} + 1)],

σ_{inter} (ω) = j \frac{q^{2}}{4 π ℏ} \ln [\frac{2 | μ_{c} | - (ℏ ω + j Γ_{c})}{2 | μ_{c} | + (ℏ ω + j Γ_{c})}],

where

q

is the charge of electron,

ℏ

is the reduced planck constant,

k_{B}

is the Boltzman constant, and

μ_{c}

is the chemical potential of graphene.

Γ_{c}

represents the damping constant which can be defined as

Γ_{c} = q ℏ ν_{f}^{2} / μ μ_{c}

, where

ν_{f}

(~c/300m/s) is the Fermi velocity and

μ

(~10000cm²/Vs) is the electron mobility [26]. Hence, it is clearly seen from Eq. (1) and Eq. (2) that the conductivity of monolayer graphene is closely in connection with its chemical potential.

As shown in Fig. 1(a), the GAMS can be fabricated by repetitive monolayer graphene transfer and dielectic Al₂O₃ layer deposition with the desirable shapes, such as cylinder or cuboid. In Fig. 1(b), the graphene layers are separated by Al₂O₃ layers with thickness $h_{d}$ (~100nm) and relative permittivity $ε_{d}$ (~4.9). In addition, $h_{d}$ is thick enough to avoid the complex interactions between the adjacent graphene layers (e.g., interlayer transitions). Under this assumption, the characterization of GAMS can be homogenized and utilized through effective medium theory [27]. Considering the fact that graphene layers have negligible thickness, the components of the relative permittivity of GAMS can be expressed as [19,28]

\begin{array}{l} ε_{| |} = ε_{d} + \frac{i σ_{g} (ω)}{ω ε_{0} h_{d}}, \\ ε_{⊥} = ε_{d}, \end{array}

Where

ε_{| |}

and

ε_{⊥}

are the permittivity components parallel and perpendicular to the horizontal plane, respectively. In this case, we can obtain that the GAMS is obviously a uniaxial anisotropic material. On the basis of Eq. (1) to (3),

ε_{||}

can be directly controlled by varying the chemical potential of graphene while

ε_{⊥}

is equivalent to the permittivity of Al₂O₃ all the way. Furthermore, the relationship between the chemical potential of graphene and applied voltage can be denoted by the formula [29]

μ_{c} = ℏ ν_{F} \sqrt{π (n_{S} + \frac{C | V |}{q})},

Where

ν_{F}

(~1 × 10⁶m/s) is the Fermi velocity for graphene and

C

(~24mF/m²) is the effective capacitance per unit area,

V_{g}

is the biased voltage,

n_{S}

is the intrinsic carrier concentration. It is verified from Eq. (4) that improving the gate voltage could result in an enlargement of

μ_{c}

. Thereby, it can be convincingly modified from the formula derivations above that the permittivity of GAMS can be conveniently modulated through the applied voltage, makes GAMS as a promising candidate for the applications in EO modulators.

Fig. 1 (a) Fabrication process of graphene/Al₂O₃ multilayer stack (GAMS); (b) Side view.

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The novel EO modulator are designed as schematically visualized in Fig. 2(a), which is mainly composed by bus waveguide, PCN, GAMS, and SiO₂ substrate. The waveguide is very close to the PCN with a gap and the GAMS is embedded into the central air hole of PCN. Figure 2(b) shows the top view of PCN, which consists of finite periodic cylindrical air holes in a silicon-on-insulator strip waveguide. On both side of PCN, the radius of each hole is identically defined as $r$ and the distance between adjacent holes is defined as $d$ . On the defect cavity, the radius of holes firstly decrease and then increase, which presents a mirror symmetrical distribution. For the front half part, the radius of holes reduces from $r_{1}$ to $r_{6}$ while the distance between adjacent holes reduces from $d_{1}$ to $d_{5}$ . The defect cavity is inherently integrated with the feeding waveguide, and it is simple to control the coupling rate of the cavity to the feeding waveguide. Thus the defect cavity provides an effective mechanism for realizing a single resonance within a large spectral window of high propagation [30,31]. In addition, Fig. 2(c) shows the cross-section view of our proposed device, where the thickness of substrate is defined as $h_{1}$ and both the heights of bus waveguide and PCN are defined as $h_{2}$ . The widths of bus waveguide and PCN are defined as $w_{1}$ and $w_{2}$ , where the width of gap between them is defined as $w_{3}$ . In addition, the radius $r_{g}$ of GAMS is designed a bit smaller than that of air holes to keep the integrate periodicity of defect cavity in PCN. To achieve a relatively high quality factor as well as a obvious coupling phenomenon, the geometric dimension of our proposed EO modulator listed above are optimized as follows: $d = 407.25 nm$ , $r = 113.13 nm$ , $w_{1} = 180.22 nm$ , $w_{2} = 301.58 nm$ , $w_{3} = 167.21 nm$ , $r_{g} = 67.88 nm$ , $h_{1} = 200.00 nm$ , $h_{2} = 220.00 nm$ , $r_{1} = 0.98 r$ , $r_{2} = 0.92 r$ , $r_{3} = 0.88 r$ , $r_{4} = 0.84 r$ , $r_{5} = 0.80 r$ , $r_{6} = 0.76 r$ , $d_{1} = 0.98 d$ , $d_{2} = 0.92 d$ $d_{3} = 0.88 d$ , $d_{4} = 0.84 d$ , $d_{5} = 0.80 d$ .

Fig. 2 (a) Schematic illustration of the proposed EO modulator side-coupled with PCN loaded GAMS; (b) Top view of PCN; (c) Cross-section view of the proposed EO modulator.

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It should be noted that a patch of n-doped silicon slab with thickness $h_{3}$ is covered on the top surface of GAMS, where the electrodes are mounted on the slab and SiO₂ substrate, respectively. In this case, the chemical potential of GAMS can be efficiently tuned through the applied electric field. Furthermore, Table 1 shows the relationship between the thickness $h_{g}$ of dielectric Al₂O₃ layer and the layers number of graphene in GAMS. The total height of GAMS is defined equal to the value of $h_{2}$ , thus $h_{g}$ decreases as the layers number of graphene goes up. Figure 3(a) and (b) show the calculation results about the conductivity $σ_{g}$ and permittivity $ε_{| |}$ of GAMS as the function of chemical potential at the wavelength of 1650nm, respectively. As the chemical potential increases, the real part of $σ_{g}$ gradually decreases while the imaginary part of $σ_{g}$ firstly increases and then reduces; on the contrary, the real part of $ε_{| |}$ firstly increases and then decreases with $μ_{c}$ , while the imaginary part of $ε_{| |}$ reduces rapidly when the $μ_{c}$ is in the region $0.4 eV < μ_{c} < 0.6 eV$ but changes slowly in the other regions. Particularly, the permittivity components $ε_{| |}$ of GAMS parallel to the horizontal plane is connected with $h_{g}$ according to Eq. (3), thus the modulation range of imaginary part in Fig. 3(b) is enhanced three times as the layers number of graphene in GAMS increases from three to seven.

Table 1. Designed structural parameters of GAMS

View Table

Fig. 3 In the case of chemical potential varies from 0.1eV to 1.1eV: (a) The real part and imaginary part of the conductivity of monolyer graphene; (b) The real part and imaginary part of the permittivity $ε_{| |}$ of GAMS with different layers number (the wavelength is fixed at 1650nm).

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In previous studies, the GAMS is only fabricated as the form of array on the substrate to analyze its optical and electric properties [19]. However, embedding the GAMS into a specific optical device is still difficult to achieve in experiment. Here we give the method of fabricating our proposed device in theory: the first step is coating dielectric layer and monolayer graphene in turn on the silicon substrate to obtain the unpatterned multilayer structure; the second step is patterning multilayer structure into the desirable shape with height $h_{2}$ and radius $r_{g}$ , and then building the bus waveguide and PCN on the substrate; the third and last step is adding electrodes on GAMS to realize the electrical tunablity of the whole device.

3. Simulation results

Before discussing the simulation results, we present a general theoretical description of our proposed EO modulator to prove its feasibility, and the whole system can be established via the transfer matrix method as shown in Fig. 4. Provided that the quality factor of PCN cavity is not too low, then the system can be described by the coupled mode theory with the following equations [14]:

- i ω A = - i ω_{0} A - A τ_{1} - A τ_{2} - A τ_{r} - A τ_{a} + \sqrt{2 τ_{1}} S_{1 +},

S_{1 -} = - S_{1 +} + \sqrt{2 τ_{1}} A

S_{2 -} = \sqrt{2 τ_{2}} A,

where

A

denotes the overall amplitude of the PCN.

S_{1 +}

,

S_{1 -}

and

S_{2 -}

represent the input, reflection, and output field amplitude, respectively.

τ_{1}

and

τ_{2}

are the coupling rate to the input port and output port, respectively.

ω_{0}

is the resonant frequency.

τ_{r}

and

τ_{a}

denote the radiation loss rate and absorption rate of the PCN cavity. Thus we can obtain the transmission of our proposed device by solving the equations above

T (ω) = {| \frac{S_{2 -}}{S_{1 +}} |}^{2} = \frac{4 τ_{1} τ_{2}}{{(ω - ω_{0})}^{2} + {(τ_{1} + τ_{2} + τ_{a} + τ_{r})}^{2}},

and the expression of absorption can be described as

A (ω) = 1 - T (ω) - R (ω) = \frac{4 τ_{1} τ_{a}}{{(ω - ω_{0})}^{2} + {(τ_{1} + τ_{2} + τ_{a} + τ_{r})}^{2}} .

From Eq. (8) and (9) we can see that the performance of transmission and absorption of our proposed device can be effectively tuned by varying the absorption rate of PCN cavity, which is closely connected with the LG interaction in GAMS. In the ideal situation, the radiation loss rate is low enough to be ignored, and then we discuss the transmission and absorption for different values of

τ_{1} / τ_{a}

and

τ_{2} / τ_{a}

with the premise of

ω = ω_{0}

. As shown in Fig. 5(a), the transmission gradually increases as

τ_{1} / τ_{a}

and

τ_{2} / τ_{a}

rise, but it couldn’t become too high since the value of

τ_{1}

and

τ_{2}

are both at a low level when the incident light wave is satisfied with the resonant frequency. On the other hand, as shown in Fig. 5(b), the absorption up to more than 90% is available when

τ_{2} < < τ_{a}

and

τ_{1} = τ_{a}

, which is called nearly critical coupling conditions [32]. Thus from the theoretical analysis above, the property of our proposed EO modulator can be electrically tuned through changing the absorption effect of PCN. Furthermore, it is verified in experiments that to make the effective medium theory of Eq. (3) agree well with the transfer matrix method and to guarantee an effective tunability of PCN, the layers number of graphene in GAMS need to be larger than three [33].

Fig. 4 The model of PCN cavity as a single-mode optical resonator coupled with bus waveguide.

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Fig. 5 (a) Transmission and (b) absorption of our proposed modulator in the case of different value of $τ_{1} / τ_{a}$ and $τ_{2} / τ_{a}$ .

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As GAMS only interacts with the tangential electric field, the absorption of GAMS is polarization-sensitive. In this paper, only the fundamental mode of TE-like polarization is considered [34]. Assume that the graphene conductivity $σ_{g}$ is in the case of $T = 300 K$ with $Γ = 1.3 meV$ . Using the rigorous finite element method implemented in the Multi-physics commercial software COMSOL, the transmission and absorption spectra of the proposed device as shown in Fig. 6 are calculated over the wavelength from 1644nm to 1667nm, under the circumstance of controlling the GAMS in different chemical potentials. For the transmission spectra as shown in Fig. 6(a), when $μ_{c} = 0.8 eV$ , the propagation loss is relatively weak within pass band and more than 80% incident light can transmits through the bus waveguide, while a resonant wavelength $λ^{*}$ appears at 1658.6nm regarded as a complete stop band. It should be noted that $λ^{*}$ shows a distinct blue shift with the decline of $μ_{c}$ , reduces to 1654.1nm when $μ_{c} = 0.4 eV$ . In order to evaluate the modulating effect about transmission, we introduce the concept of modulation depth as follows

T_{m o d} = \frac{λ_{m a x}^{*} - λ_{m i n}^{*}}{μ_{c m a x} - μ_{c m i n}},

where

λ_{m a x}^{*} - λ_{m i n}^{*}

represents the modulating range of resonant wavelength,

μ_{c m a x} - μ_{c m i n}

is the variation of chemical potential corresponding to

λ^{*}

. Thus the value of

T_{m o d}

for our proposed device is about 11.25nm/eV, which has a great improvement compared with the simulation results of the EO modulators based on monolayer graphene in previous studies [13]. For the absorption spectra as shown in Fig. 6(b), as expected, the wavelength of absorbing peak shows an obvious blue shift with the varying of

μ_{c}

. The decrease of

μ_{c}

results in the increase of absorbing peak value and the maximal absorption value 51.27% appears at 1654.2nm when

μ_{c} = 0.4 eV

. Similar to

T_{m o d}

, we can also determine the definition for modulation depth of absorption as

A_{m o d}

, which is about 12.50nm/eV. Moreover, to better understand the physical mechanism in Fig. 3 as well as the LG interaction in the side-coupled PCN, we plot the power flow distributions of the designed EO modulator when

μ_{c} = 0.4 eV

in Fig. 7. At about 1665.0nm, as shown in Fig. 7(a), most of the incident light wave could pass through the bus waveguide and export from the output port although a little energy loss in the side PCN. However, at about 1654.1nm, it can be obtained from Fig. 7(b) that the light wave is extremely centralized in the defect cavity of PCN and dissipative rapidly in propagation process, making the resonant stop band engender. As a result, through the analysis about power flow distribution in the on/off states, we can investigate GAMS is suitable to be used in EO modulators owe to its good performance on LG absorption.

Fig. 6 (a) Transmission and (b) absorption spectra of the proposed EO modulator based on GAMS in the case of different chemical potentials.

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Fig. 7 Power flow distribution of the proposed EO modulator based on GAMS when $μ_{c} = 0.4 eV$ at (a) 1665.0nm and (b) 1654.1nm.

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Then we further introduce the perturbation theory to explain the reason for the blue shift phenomenon of transmission as shown in Fig. 6. Assume that $E_{0}$ , $H_{0}$ , and $λ_{0}$ represent E-field intensity, H-field intensity, and proper wavelength of PCN before perturbation, while $E$ , $H$ , and $λ$ represent corresponding to values after perturbation. Since the perturbation caused by the change of relative permittivity of GAMS is tiny enough, we can consider that the field intensity distribution in PCN is invariable consistently before and after perturbation. In this case, the relationship between $λ$ and $λ_{0}$ can be written as

\frac{λ_{0} - λ}{λ_{0}} \approx \frac{∭_{V} (Δ ε {| E_{0} |}^{2} + Δ μ {| H_{0} |}^{2}) d V}{∭_{V} (ε {| E_{0} |}^{2} + μ {| H_{0} |}^{2}) d V},

where

ε

and

μ

is the relative permittivity and permeability of the whole cavity,

Δ ε

and

Δ μ

is the variation on account of perturbation,

V

is the volume of air holes in PCN. The resonant wavelength of incident light wave almost coincides with the proper wavelength of PCN mode of an isolated cavity [35]. Thus, it can be verified from Eq. (11) that the transmission of incident wave can be modulated distinctly by controlling the permittivity of GAMS with applied voltage. On the one hand, the real part of

ε_{| |}

drops gradually as the chemical potential

μ_{c}

of GAMS rises in the region of

μ_{c} > 0.4 eV

in Fig. 3(b), leading to the proper wavelength of PCN goes up at the same time, and the blue shift of resonant wavelength in Fig. 6 occurs accordingly since it almost coincides with the proper wavelength of PCN. On the other hand, the imaginary part of

ε_{| |}

also declines gradually as

μ_{c}

raises in Fig. 3(b), thus we can obtain that the value of absorption peak in Fig. 6(b) decreases as

μ_{c}

raises due to the absorption ability of GAMS is positively related to the imaginary part of

ε_{| |}

[36,37]. The layers number of graphene in GAMS is also a significant factor for the modulation capacity of our designed device. Figure 8 shows the relationship between the value of absorption peak and the chemical potential of GAMS. As we can see, the absorption effect is enlarged a lot when the layers number of graphene in GAMS increases from three to seven, especially in the range of

0.4 eV < μ_{c} < 0.6 eV

, which highly accords with the variation of imaginary part in Fig. 3(b). Moreover, as plotted in the illustration of Fig. 4, the modulation depth

A_{m o d}

is about 12.50nm/eV when the layers number of graphene in GAMS is three, while it can reach to 15.93nm/eV as layers number increases to seven.

Fig. 8 The relationship between the value of absorption peak of our proposed EO modulator and the chemical potential of GAMS (different color represents different layers number of graphene in GAMS); the illustration is the variation trend of modulation depth $A_{m o d}$ with the layers number increasing from three to seven.

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Equation (4) presents the relationship between the chemical potential of monolayer graphene and the gate voltage, where the intrinsic carrier concentration $n_{S}$ is fixed at 1.17 × 10¹⁷m⁻³. However, in the graphene multilayer structure, $n_{S}$ is changed by applying a voltage $V_{g}$ as [29]

n_{S} = \frac{ε_{0} ε_{d}}{q h_{d}} (V_{g} + V_{0}),

Where

ε_{0}

(~8.854 × 10⁻¹²F/m) denotes the permittivity of the vacuum and

ε_{d}

(~4.9) is the relative permittivity of Al₂O₃ layer.

V_{0}

denotes the offset voltage caused by doping. According to Eq. (4) and (12), it should be noticed that the layers number of graphene is connected with the gate voltage for the chemical potential tuning, since the thickness of Al₂O₃ dielectric reduces as the layer number of graphene goes up as shown in Table 1. Assume that the graphene in GAMS is non doped (

V_{0} = 0

), we can calculate the required voltage on the electrodes as the function of

μ_{c}

in the case of different value of

h_{d}

. As shown in Fig. 9, to realize the modulation of chemical potential from 0.4eV to 0.8eV, the required voltage decrease from 37.1V to 18.2V when the layers number increases from three (corresponding to

h_{d} = 54.25 nm

) to seven (corresponding to

h_{d} =26 .63nm

). As a result, compared with other silicon based EO modulators before [38-39], our proposed device need to apply a higher gate voltage in order to achieve an effective modulation, which could lower down its feasibility. To overcome this limitation, a effective method is using the doped graphene in GAMS which can markedly improve the value of

V_{0}

.

Fig. 9 The relationship between chemical potential of graphene and the required voltage applied on the GAMS in the case of different thickness of Al₂O₃ layer.

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4. Further discussion

Figure 10(a) shows the EO modulator side-coupled with a two-defect-cavity PCN, where two GAMSs are embedded into the center air holes of two defect cavities, respectively. The structural parameters of this device are completely equal to the homologous values as defined in Fig. 2. The radius of GAMS 1 and GAMS 2 are consistent to ensure the identical absorbing ability for the incident light wave. Define that the chemical potential of GAMS 1 and GAMS 2 as $μ_{c 1}$ and $μ_{c 2}$ , respectively. We firstly simulate the transmission and absorption spectra in the case of tuning $μ_{c 1}$ and $μ_{c 2}$ synchronously, and the variation trend of results are resemble with Fig. 6. Furthermore, by applying different bias voltage on GAMS 1 and GAMS 2, an interesting modulation phenomenon is occurred. For the transmission spectra as shown in Fig. 11(a), in the case of $μ_{c 1} = 0.4 eV$ and $μ_{c 2} = 0.8 eV$ , there are two resonant dips over the operating band from 1644nm to 1667nm, one is at 1654.7nm and the other is at 1661.9nm. Then fixing $μ_{c 1}$ still at 0.4eV and declining $μ_{c 2}$ begin with 0.8eV, we can find dip 1 keeps constant while dip 2 exhibits a blue shift, and finally these two dips finally roll into one resonant band when $μ_{c 1} = μ_{c 2}$ . The mechanism of blue shift here can also be explained by the perturbation theory as discussed above. On the other hand, for the absorption spectra as shown in Fig. 11(b), absorption peak 1 remains unchanged while absorption peak 2 displays a blue shift as $μ_{c 2}$ decreases. It should be noted that peak 1 and peak 2 gradually merge into one peak and the value of total absorption can attain 60%. Thus owe to the superimposed absorption of GAMSs, $A_{\mod}$ could reach to 14.31nm/eV which is enhanced compared with the single-defect-cavity modulator. We also plot the power flow distribution in the case of $μ_{c 1} = 0.4 eV$ and $μ_{c 2} = 0.8 eV$ as shown in Fig. 10(b) and (c): at about 1654.7nm, the power of incident light wave mainly concentrates on defect cavity 1, which means peak 1 is resulted from the LG absorption in GAMS 1; at about 1661.9nm, the power mainly concentrates on defect cavity 2, which means peak 2 arises from the LG absorption in GAMS 2. Additionally similar variation trend of transmission and absorption can be obtained if fixing $μ_{c 2}$ and varying $μ_{c 1}$ . Thereby, under the circumstance of adding different gate voltages on GAMS 1 and GAMS 2, propagation characteristics of incident light wave can be tuned validly, at this point it is of great practical interest for using the proposed two-defect-cavity EO modulator as a tunable dual-channel sensor.

Fig. 10 (a) Schematic illustration of the proposed EO modulator side-coupled with two-defect-cavity PCN loaded GAMSs; In the case of $μ_{c 1} = 0.4 eV$ and $μ_{c 2} = 0.8 eV$ : Power flow distribution at (b) 1654.7nm and (c) 1661.9nm.

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Fig. 11 (a) Transmission and (b) absorption spectra of the proposed EO modulator side-coupled with a double-defect-cavity PCN in the case of different chemical potentials.

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5. Conclusion

In summary, we have studied the electro-optic (EO) modulator which is side-coupled with photonic crystal nanobeam (PCN) loaded graphene/Al₂O₃ multilayer stack (GAMS). The transmission and absorption characteristics can be controlled over the operating wavelength band from 1644nm to 1667nm by electrically tuning the chemical potential of the graphene, which can provide a efficient method to achieve light modulation at telecommunication band. According to the simulation results through the multi-physics software COMSOL, the modulation depth of transmission and absorption can reach to 11.25nm/eV and 12.50nm/eV, respectively. Moreover, we also design a EO modulator side-coupled with double-defect-cavity PCN loaded two GAMSs. Both the resonant band as well as the absorption peak could split into two parts if adding different voltages on the different GAMS. All of these results perform excellent characteristics which are promising for the further researches about high-density integration of the photonic circuit, optical bistable devices, and sensing applications.

Funding

National Natural Science Foundation of China (61372029).

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Total height 220nm	Thickness of monolayer graphene (h_g) 1nm
Layers number	Thickness of dielectric layer (h_d)
3	54.25nm
4	43.20nm
5	35.83nm
6	30.57nm
7	26.63nm

Electro-optic modulator side-coupled with a photonic crystal nanobeam loaded graphene/Al₂O₃ multilayer stack

Abstract

1. Introduction

2. Theoretical analysis and device design

3. Simulation results

4. Further discussion

5. Conclusion

Funding

References and links

Cited By

Figures (11)

Tables (1)

Equations (12)

Optical Materials Express