Tailoring Optical Bistability In Nonlinear Photonic Crystals Based on Metallic Nanoparticles And Graphene

In this work, we revisit the optical response of a one-dimensional photonic crystal consisting of graphene monolayers and a plasmonic nanocomposite as a defect layer in the structure. By taking advantage of the modiﬁed transfer matrix approach, the analytical solution of the light transmission and ﬁeld distribution of the photonic crystal are evaluated. Besides, by considering one of the layers as a Kerr-nonlinear medium, we delve into optical bistability phenomenon in the model for two diﬀerent cases. Our numerical results reveal that the proposed photonic crystal can enhance the ﬁeld distribution and reduce the optical bistability’s threshold in comparison to the conventional photonic crystals. Furthermore, the optical bistable switch-up and switch-down thresholds of the proposed resonator can be tailored ﬂexibly by plasmon-plasmon interactions in the defect layer. Finally, the electric ﬁeld distribution amelioration and optical bistability by means of graphene layers in the structure are attainable. The inﬂuences of the parameters such as the graphene and the nanocomposite on the performance of OB are analyzed and compared in the two diﬀerent cases. Therefore, present approach can lay the groundwork for designing highly sensitive surface plasmon resonance biosensors and switches where the proposed technique may ﬁnd unprecedented capabilities.

In this work, we revisit the optical response of a one-dimensional photonic crystal consisting of graphene monolayers and a plasmonic nanocomposite as a defect layer in the structure. By taking advantage of the modified transfer matrix approach, the analytical solution of the light transmission and field distribution of the photonic crystal are evaluated. Besides, by considering one of the layers as a Kerr-nonlinear medium, we delve into optical bistability phenomenon in the model for two different cases. Our numerical results reveal that the proposed photonic crystal can enhance the field distribution and reduce the optical bistability's threshold in comparison to the conventional photonic crystals. Furthermore, the optical bistable switch-up and switch-down thresholds of the proposed resonator can be tailored flexibly by plasmon-plasmon interactions in the defect layer. Finally, the electric field distribution amelioration and optical bistability by means of graphene layers in the structure are attainable. The influences of the parameters such as the graphene and the nanocomposite on the performance of OB are analyzed and compared in the two different cases. Therefore, present approach can lay the groundwork for designing highly sensitive surface plasmon resonance biosensors and switches where the proposed technique may find unprecedented capabilities.

II. INTRODUCTION
Photonic crystals (PCs) have been considered as one of the most appealing structures due to innumerable different materials which can be used in designing the photonic crystals such as linear and nonlinear dielectric materials [4], metals [5], graded-index materials [6] and metamaterials [7]. Based on structural arrangement, materials used in the structure and the number of underlying structural layers, photonic crystals demonstrate * Electronic address: tayebe.naseri@gmail.com mesmerizing optical responses. The nonlinear optical responses of the photonic crystal has given the momentum for exploring its applications in designing all-optical devices in optics communication and information arenas.
Nonlinear photonic crystals (NPCs) pave the way to improve and modify nonlinear optical processes. Various nonlinear optical phenomena such as nonlinear frequency conversion [8], second and high harmonic generation [9], four-wave mixing [10] and optical bistability [11,16,17] have been explored in photonic crystals both theoretically and experimentally.
Optical bistability (OB) as the existence of two stable output states of an input light intensity is an important means to control light propagation with another light. This has been studied in various media such as multi-level atomic systems [18], Graphene and plasmonic nanostructures [19], and NPCs [11,12]. The optical bistability and multistability in a defect structure doped with polaritonic materials and three-level nanoparticles have been investigated recently [13][14][15]. It was shown that the threshold of optical bistability can be manipulated by some controllable parameters such as Rabi frequency, line width of upper level, and thickness of defect structure. The OB can be realized by the dynamic shifting of band edges in NPCs. In OB phenomenon, the system shows two stable outputs for a specific input in presence of optical nonlinearity and positive feedback of the structure. Needless to mention that inquiring new optical materials with tunable parameters is the cornerstone for dynamically tunable optical bistable devices.
Graphene as a 2D material consisting of one layer of carbon atoms arrayed in a hexagonal lattice has a broadband and tunable optical response in the IR to visible frequency range [20][21][22]. It further reveals extremely large Kerr nonlinearity due to the linear band structure near the Fermi energy level that has been explored widely [23][24][25]. Fascinatingly, the bistablity behavior in the graphene-based structures can be electrically controlled and tuned just by changing the applied voltage on the graphene.
On the other hand, recent advanced progresses in nanotechnology have shown that metallic nanoparticles have emerged as powerful structures in various optical applications as a result of their superb optical properties [26][27][28]. That is to say, the optical properties of metallic nanoparticles are explained through surface plasmon resonances (SPRs) strongly depending on various parameters such as nanoparticles size, shape, concentration and spatial distribution as well as the properties of the background medium [29,30]. The remarkable optical properties of metallic nanoparticles have been studied extensively and metal-dielectric composites have found a plethora of applications in nanophotonic [31][32][33]. The unique optical characteristics of these composite structures can be achievable as the contribution of the local field of the nanoparticles is tunable which is one of prerequisites of designing nonlinear optical devices.
Combining the nonlinearity of graphene and plasmonic excitation properties of metallic nanocomposites together have come forth to study Kerr nonlinearity and consequently OB phenomena. In this work, we study a graphene monolayer in a 1DPC and investigate the formative impact of graphene on the transmission properties. We stumble on that graphene significantly modifies the characteristics of the optical bistability behavior. A shift in defect modes and consequently OB can be seen via changing the refractive index and the light localization in the nonlinear defect layer.
Creating such a strong light localization prepares the ground for the tunable and low threshold OB with wide bistability hysteresis loop in the strong light matter interaction regime in the PCs. metallic nanocomposite and graphene are a big boon for strengthening the interaction between light and the PC.
In this manuscript, we present a new scheme to enhance the optical response of a one-dimensional photonic crystal that consists of graphene monolayers and a plasmonic nanocomposite as a defect layer in the structure. The numerical results demonstrate that the proposed photonic crystal can enhance the field distribution and reduce the optical bistability threshold in comparison with the common photonic crystals. The nonlinear photonic crystal coupled with graphene sheets and metallic nanocomposite sets the stages for the bistable response of the transmitted light intensity. Realizing a low intensity threshold of OB is a critical challenge in scheming nonlinear devices and decreasing the switching threshold becomes practical with this NPC structure; with this end in view, such this controllable bistable system could find potential applications in optical all-optical switching and sensing.

III. THEORETICAL MODEL AND METHODS
In this study, we consider two different structures, as shown in Fig.1, for Case'I' the proposed structure is (BA) N D(AB) N , where A, B are isotropic dielectric layers with high and low refractive indexes, respectively and N is the number of periods. The interfaces of the layers are parallel to the (x − y) plane, and the z axis is normal to the structure. In Case'II', it is assumed that the linear graphene monolayers G are embedded between adjacent dielectric layers (see Fig. 2). The defect layer D is a nanocomposite with a refractive index distribution function ǫ D based on Maxwell Garnett approximation [34]: where the dielectric function of a plasmonic noble metal is described by the Drude model and given by ǫ d = , and ω p , γ and ǫ ∞ are the plasma frequency of the metal, the collision frequency of free electrons and the high-frequency part of the dielectric function [35]. In this work, gold (Au) is used due to its strong plasmon resonance with tunable resonance wavelength [36]. Also, ǫ h , f are permittivity of host material and metal filling fraction of defect layer, respectively. In this paper, the graphene monolayer is described by its The frequency dependent surface conductivity of graphene σ G is written as the superposition of the intraband σ intra and the interband term σ inter [19], which is expressed in the following forms: where ω, E F , τ , T , e, k B , η 0 , d G , , and k 0 = 2π/λ are the frequency of the incident field, the Fermi energy, the electron-phonon relaxation time, the temperature, the elemental electron charge, the Boltzmann constant, the impedance of air, the thickness of the graphene nanolayer, the reduced Planck constant, and the vacuum wavevector, respectively. It is worth mentioning that only the transverse electric field (T E) is being studied here. So, the electric fields in layers are written as: Here, k x = k 0 sin(θ), with k 0 = ω/c, and k zj = k 0 ǫ j µ j (1 − sin 2 (θ)/ǫ j µ j ). θ is the angle of the incident light. Inside the layers, the electric field is governed by the Helmholtz equation [37]: The subscript j denotes the number of the layers. At the interface between two layers, the boundary conditions are imposed [38][39][40]: µj ω sin(k zj ∆z j ) cos(k zj ∆z j ) (7) Here, l = 1, 2, . . . , N , where N is the total number of the layers in the structure. Consequently, we have the total transfer matrix: The tangential components of the electric and magnetic fields at the incident site z = 0 and the transmitted site z = L are related by the following matrix equation: Finally, the transmission can be obtained as T = tt * where t = 2p/(pM tot. 11 + pM tot. 22 + p 2 M tot. 12 + M tot. 21 ) and p = cos(θ). It is important to note that subscript "j" in equations 7,8 represents the linear layer (A, G, D) with thicknesses of dA, dD, and dG. However, in the nonlinear layer (B), the subscript "j" refers to very thin sublayers with a thickness of dB/N sub. , where N sub. denotes the number of sublayers. It is possible to use the same transfer matrix for both linear and nonlinear regimes by using this strategy (dividing the nonlinear layer into very thin sublayers) [6,41,42].

IV. RESULTS AND DISCUSSION
As mentioned earlier, in both cases (Case'I'and Case'II') the thickness of the layers are fixed to d A = 80nm, d B = 120nm,d D = 20nm, d G = 0.33nm and f = 0.1. It is assumed that the layers are nonmagnetic and their permeabilities are µ A = µ B = µ D = 1.

A. Case 'I'
For the first case, we study nonlinear properties of the structure without graphene sheets, and the optical parameters of the system are chosen as: N = 5, ǫ A = 6.25, ǫ h = 2.89 and ǫ B = 2.25 + χ (3) |E| 2 . Although negative Kerr nonlinearity has been experimentally proven in various composite crystals [43][44][45][46], the majority of naturally occurring materials have a positive Kerr coefficient, so here χ (3) is considered as a small positive, third-order susceptibility coefficient accounting for the nonlinear interactions in the structure and E is the electric field. The optical Kerr effect is a change in the refractive index of a medium caused directly by the electric field of incident light. The importance of this type of nonlinearity in PCs has been demonstrated in the literature when designing several nonlinear devices based on the optical Kerr effect, such as optical diodes [47][48][49], switches, and limiters [50,51].   Fig. 2(a) the transmission spectrum of a defect free structure plotted to draw an analogy with defective structures, it is obvious from this figure that by considering a defect layer in the middle of the structure the lower edge of the band gaps a bit shifted to the lower energies and by changing the electric permittivity of defect layer from host material to the permittivity of nanocomposite the band gap is broadened. The optical responses of the metallic nanocomposite comes from the surface plasmons that are collective oscillations of the conduction electrons in metallic particles, and by embedding the nanocomposite defect in the middle of the structure the light interaction with surface plasmons of metallic nano particles and also plasmonplasmon interactions would enhanced.Therefore, an enormous shift happens in the energy band gap and the energy band gap becomes wider (see Fig. 2(c)). To clarify the connection between the nanocomposite defect layer and the OB threshold intensity, the normalized transmitted light intensity (χ (3) |E t | 2 ) through the structure versus the normalized incident light intensity (χ (3) |E i | 2 ) are illustrated in Fig.3. Here, we consider the frequencies lying close to the low-frequency edges of the defect modes. By introducing a positive Kerr coefficient, the enhancement of the electric field (self-focusing effect) [52] occurs near the low-frequency edge of the defect mode which leads to the optical bistability. In the case of a negative Kerr coefficient, optical bistability can be seen only near the high-frequency edge where the electric field does not get enhanced (self-defocusing effect) [53]. According to this figure, there is a decrease in threshold intensity of OB by introducing the defect layer and this reduction in threshold intensity of OB intensify by embedding the nanocomposite layer, due to plasmonplasmon interactions in the defect layer. Such strong plasmon-plasmon interactions in the system allow an enhancement in optical nonlinearities due to the energy transfer between the plasmons. Moreover, the metallic nanoparticles are able to provide large tunability of the plasmon resonance wavelength which leads to boost the nonlinearity in the proposed structure, and as it is known, the bistable region and bistability threshold is strongly dependent on the nonlinearity of the structure. The ad-vantage of the defective structures over the defect-free structures is clearly demonstrated in Fig. 3 by comparing their OB threshold, so we will now focus on the defective structures in this paper. In Fig.4, we considered the normalized transverse electric field intensities of the proposed defective structure in the frequency of defect mode in two different conditions. As it is seen from the Fig.4a, for the ordinary defect layer the electric field intensity reaches the peak in the middle of the nonlinear layers and the magnitude of these peaks is maximized symmetrically in the vicinity of the defect layer. On the contrary, when the structure includes nanocomposite defect layer, the electric field intensity inside the structure at the frequency of the defect mode for the situations described in Fig.2c, the field intensity grows gradually from the incident side to the transmitted side and this growth intensified after passing through the defect layer (Fig.4b).
To study the dependence of the photonic band gaps (PBGs) on the incidence angle θ and nano particles filling factor f , we plot the photonic band structure of the system as a function of θ and f in figures 5 and 6, respectively. Here, the gaps are represented by the dark areas while the bright areas show the allowed bands for the TE polarization. It is clear that by increasing the incident angle PBGs and the defect modes shift to the higher frequencies, moreover when defect layer contains metallic nanoparticles in large angles the defect mode disappears. Figure 6 depicts how the filling factor of metallic nanoparticles affects defect mode and photonic bandgap, the defect mode sharpens and slightly moves to the lower energies when the filling factor is increased from 0.1 to 0.4, meaning that the Q-factor of the defect modes, which is one of the main parameters in designing Bio-sensors and refractometers, can be improved by fine-tuning with the filling factor. Furthermore, changing the filling factor severely affects the higher edge of the photonic bandgap; however, the lower edge does not exhibit high sensitivity to filling factor variance, making it a strong choice for designing low pass filters.

B. Case 'II'
In the second case, we studied the structure of Case 'I' with embedded linear graphene sheets between the layers (see Fig.1(b)).
The linear transmission of the graphene based structure has been represented in Fig. 7 as a function of frequency in comparison to the first case to investigate the effect of the graphene layers on the frequency of the defect mode when the structure includes nanocomposite. It is obvious that by embedding graphene sheets between the layers, the defect mode slightly moves to the lower energies, since. As shown in Eq. 2, the surface conductivity of graphene includes the imaginary part and the real part, which shows that graphene acts like a very thin metallic layer. The addition of graphene layers to the photonic crystal leads to extra positive phase shift and energy loss because of the real part and the imaginary part of the surface conductivity of graphene.

FIG. 7:
The transmission spectrum of the defective structure at normal incidence for (a) (BA) 5 D(AB) 5 (b) (GBGA) 5 GDG(AGBG) 5 with other parameters unchanged. In both cases defect layer is nanocomposite Figure. 8 shows the effect of linear graphene monolayers on nonlinear response of the system when the structure consists of a nanocomposite defect layer. In this figure the normalized transmitted light intensity (χ (3) |E t | 2 ) through the structure has been plotted versus the normalized incident light intensity (χ (3) |E i | 2 ).
In line with this figure, threshold intensity of OB declined dramatically by inserting graphene sheets between the layers. It should be noted that the OB hysteresis loop shrinks. In the presence of graphene monolayers in the photonic crystal, the nonlinearity of this structure would be enhanced considerably.
To investigate the effect of graphene sheets on the behavior of electric field distribution inside the defective PC we made an analogy between two structures related to the Fig. 7. Both of the structures defect layer consist of metallic nanoparticles and the geometry of the structures are the same except in one of them graphene monolayers are inserted between the layers. As shown in Fig. 9, in both structures, the field intensity is being enhanced from the incident side to the transmitted side and this growth intensified after passing through the nanocomposite layer, but enhancement of the electric field intensity is higher in the graphene-based PC. In conclusion, we have investigated the issue of optical transmission and optical bistability in the one dimensional photonic crystal. In this model, the defect mode is deemed as a metallic nanocomposite including spherical golden nanoparticles. We have shown that tailoring of the photonic crystal in both cases (with and without graphene monolayers) allows to obtain the high optical responses. The numerical results demonstrate that a tremendous shift happens in the energy band gap and consequently the energy band gap becomes squarely wider. That is to say, light intensity enhancement stems from surface plasmon resonances in graphene and plasmonic nanoparticles embedded in the defect layer. This exceptional phenomenon reduces the threshold input intensity in the OB hysteresis loop as well.