1. Introduction

Optical bistability (OB) has attracted much attention since the seminal work of Gibbs in the late 1970s [1] due to its widely potential applications in photonic devices such as optical logical gates, switches, optical regeneration, all-optical transistors, and memories etc [2]. So far, OB has been theoretically predicted and experimentally demonstrated in various optical systems, including Fabry-Perot (F-P) cavities [1], layered periodic dielectric structures [3], nonlinear prism coupler [4], and waveguide-ring resonators [5]. However, in these systems, both relatively strong light power and/or large enough nonlinear optical material are generally needed to achieve a sizeable nonlinear response. To overcome these drawbacks, photonic crystal cavities [6–7] and quantum well structures [8] etc. were proposed to enhance the nonlinear effects so as to reduce the material volume. With the assistance of surface plasmon polaritons (SPPs) to the effects of confining and enhancing the local optical field intensity, OB has also been demonstrated in different metal nanostructures such SPP crystals, metal-dielectric multilayers, and metal gratings etc [9–11].

Recent experimental result shows that SPPs can be confined into two parallel metal films sandwiched SiO₂ nanocavities s (≈ 0.001λ³) with much shorter effective wavelength (several tens of nanometers) than that of the incident wavelength in free space [12]. In terms of this idea, here we introduce an alternative plasmonic nanocavity for realizing the OB effect. The nanocavity is constructed by filling a piece of nonlinear optical medium into the metal gap waveguides (MGWs). Finite-difference time-domain (FDTD) simulations show that this structure can produce obvious OB effect with greatly reduced incident light power in a nanoscale nonlinear optical medium.

2. Design and theory

 

Fig. 1. Scheme of MGW F-P nanocavities for OB effect.

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The proposed structure is schematically shown in Fig. 1. A piece of GaAs layer with thickness w and length L is filled into Ag film-constructed MGWs. Where h_m is the thickness of the metal films, ε ₁, ε ₂, and ε _m denote the dielectric constants of air, GaAs, and the metal films, respectively. The Ag-GaAs-Ag waveguide region forms a plasmonic cavity and the interfaces between air and GaAs layer in the MGWs form two mirrors. Hence the structure can function as a F-P cavity. The physics of OB occurred in the F-P cavities is due to the intrinsic positive feedback resulted from the optical Kerr effect [13].

For an ordinary F-P cavity filled with a nonlinear material, the transmittance T can be written as [14]:

and

respectively. In which A and B are the coefficients related to the length, the absorption of the cavity, and the intensity reflectivity of the mirrors, F is the finesse, I_in and I̅ are the intensity of the incident light and the average optical intensity in the nonlinear medium, respectively, and φ is the phase shift of light passing through the cavity, which is expressed as [15]:

where λ ₀ is the wavelength of the incident light in air, Re(n _eff2) is the real part of effective refractive index n _eff2 of the cavity to SPPs. The n _eff2 can be defined as [16, 17]:

k ₀ is the wave number of light in air, β _spp2 is the propagation constant of SPPs in the cavity, which is related to ε ₂ [17]. The dielectric constant of the nonlinear optical material ε ₂ is read as [9, 15, 18]:

Where ε ⁽⁰⁾ ₂ and χ ⁽³⁾ are the linear dielectric constant and the third-order susceptibility of GaAs, respectively, D_d is the degeneracy factor. The electric field intensity $|\stackrel{\rightharpoonup }{E}{|}^2$ is related to the magnetic field intensity (|H_z|²) through the Maxwell equation $\nabla \times \stackrel{\rightharpoonup }{k}\cdot H_z={\epsilon }_0{\epsilon }_2\partial \stackrel{\rightharpoonup }{E}/\partial t$ ($\stackrel{\rightharpoonup }{k}$ is the unit vector in the z direction), and can be written as [19] |$|\stackrel{\rightharpoonup }{E}{|}^2$ ≈ (β _spp2/ω ₀ ε ₀ ε ₂)²·|H_z|² here (ω ₀ is the wave frequency of light in air). In our FDTD simulations, we will use |H_z|² to denote the optical field intensity of light.

Equations (1) and (2) respectively represent a usual periodic Airy function and a linear relations between the transmission T and the average optical intensity I̅. Geometrically, the OB effect can be regarded as the case where the straight line of transmittance from Eq. (2) has multiple intersections with the periodic Ariy curve from Eq. (1). By changing I_in one can modulate the transmission to produce multiple intersections between the curves and straight line from Eqs. (1) and (2), respectively.

Generally,, n _eff2 of the interface between metals and dielectrics to SPPs is much lager than (ε ₂)^1/2 of the dielectrics [20]. Therefore, the n _eff2 is more sensitive to the change of ε ₂, and so the nanocavity structure provides a feasible way to achieve sufficiently strong nonlinear response in a nonlinear medium with small size and low operating light power [From Eq. (1) and Eqs. (3) – (5)].

3. Simulation and discussion

In the following, we perform FDTD numerical simulations to demonstrate the OB property of the plasmonic nanostructures, where w=30 nm, L=300 nm, h_m=150 nm, ε ₁=1, D_d=3, and χ ⁽³⁾=6.5×10^-4 esu (9.0757×10^-12 (m/V)²) [15]. ε ⁽⁰⁾ ₂ of the GaAs is from the measured values [21], which is from 13.8425 to 12.2657 in the wavelength range of from 750 nm to 1010 nm [21, 22]. The dielectric function of Ag films is deduced from the fitting of the measured values in the visible and near-infrared range by using the Drude model [21] ε_m=ε _∞-ω ² _p/(ω ²+jγω) with (ε _∞, ω_p, γ) (1, 7.75 eV, 0.08267 eV). The incident light used to excite SPPs is a TM-polarized (the magnetic field is parallel to the z axis) plane wave. It should be noted that as w=30 nm, our calculated results (not shown here) reveal that SPPs excited by the frequencies of the illuminating field above a vacuum wavelength of 623 nm would not be supported in the GaAs region due to the absence of coupling between the SPPs on the surfaces of two Ag films. Therefore, in our case, we choose the excitation wavelength beyond 700 nm so as to achieve the observable plasmonic bistability. In the FDTD simulation, the spatial and temporal steps are set at Δx=4 nm, Δy=2 nm, and Δt=Δy/2c, respectively, and the magnetic field intensity |H_z|² _in (|H_z|² _out) at the center of a plane outside the F-P cavity 8 nm (1 nm) away from the front (back) mirror is used to represent the optical field intensity of the incident light I_in (output light I_out).

 

Fig. 2. Transmission spectra of MGW F-P nanocavities as the |H^z|² _in of the incident light is (a) 1.18 (A/m)², (b) 4.73×10⁵ (A/m)², (c) 1.06×10⁶ (A/m)², respectively.

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Figure 2 shows the calculated transmission spectra of the F-P cavity structure as the incident wavelength is in the range of from 750 nm to 1010 nm. Optical field intensity of the incident light |H_z|² _in is assumed to be (a) 1.18 (A/m)², (b) 4.73×10⁵ (A/m)², (c) 1.06×10⁶ (A/m)², respectively. The transmittance T is defined as |H_z|² _out/|H_z|² _in. From the figure we see that the transmission peaks show a red shift and become to be more asymmetric with the increasing intensity of the incident light, indicating the feasible of OB effect [5, 13]. This can be understood from the following analysis. On the one hand, when the intensity of the incident light is increased, the dielectric constant ε ₂ of nonlinear medium (GaAs) will increase due to the Kerr effect [Eq. (5)]. Correspondingly, the effective refractive index n _eff2 of F-P cavity to SPPs will increase [Eq. (4)], which results in the phase shift ϕ of the light field in the cavity [Eq. (3)]. On the other hand, a higher input light intensity will produce a stronger electric field intensity in the nonlinear medium, which results in the increase of both the dielectric constant ε ₂ of nonlinear material [Eq. (5)] and the effective refractive index n _eff2 of the cavity to SPPs [Eq. (4)]. As a result, the transmission peak (corresponding to a fixed phase ϕ) will shift to longer wavelength [Eq. (3)]. In addition, from Eqs. (1) and (3) we see that the transmittance T shows a nonlinear dependence on the phase ϕ and hence on the wavelength of the incident light. Consequently, more asymmetric shape of transmission peaks appears in transmission spectra at longer wavelengths as the intensity of the incident light is increased (Fig. 2).

 

Fig. 3. Output-input intensity relation of MGW F-P nanocavities as the incident light is with wavelength λ₀=(a) 860 nm, (b) 870 nm, (c) 880 nm, (d) 956 nm, (e) 966 nm, and (f) 976 nm, respectively. Inset, magnified view of the hysteresis loop as the incident light is with λ₀=976 nm.

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Figure 3 displays the output-input intensity relation of the SPP cavity as the intensity of the incident light with wavelength λ₀=(a) 860 nm, (b) 870 nm, (c) 880 nm, (d) 956 nm, (e) 966 nm, and (f) 976 nm is increased and decreased, respectively. The figure shows a series of hysteresis loops, manifesting the occurrence of OB around different incident wavelengths. For instance, as the incident light is with λ₀=976 nm, we can see that the OB occurs as the light intensity is around |H_z|² _in=1.18×10⁶ (A/m)², corresponding to I_in=37.14 kW/cm ² [Inset of Fig. 3]. The occurrence of the hysteresis loops means that the transmission spectra resulted from Eqs. (1) and (2) show two intersections around λ₀=976 nm. On the other hand, Fig. 3 shows that there are two sets of hysteresis loops in the output-input intensity relation of the plasmonic cavity. The lower set corresponds to three weaker transmission peaks while the upper one to three stronger peaks of Fig. 2. And furthermore, only around the asymmetric transmission peaks are the hysteresis loops formed. The stronger the asymmetry of the transmission peaks, the larger the hysteresis loops [13, 23, 24].

Regarding to the OB effect observed in conventional metal-dielectric optical systems, we see that much stronger incident power and/or larger nonlinear optical materials are needed. For instance, a metal-dielectric multilayer structure requires 2.5 GW/cm ² light power and 1.17 µm thick nonlinear materials for obtaining a visible nonlinear response [10], while a metallic grating coated with about 880 nm thick nonlinear materials requires more than 1×10¹⁶ (V ²/m ²) (I_in=1.33 GW/cm ²) light power to achieve the OB [11]. In contrast, in our case, only about I_in=37.14 kW/cm ² light power and 300 nm long nonlinear medium can realize a remarkable plasmonic bistability at λ⁰=976 nm.

4. Conclusion

In conclusion, we have demonstrated an alternative MGW F-P cavity for OB phenomenon by using FDTD simulation. The results show that the present structures can work with much weaker optical power and smaller nonlinear medium comparing to the previous reports. We attribute it to the strong local field enhancement and nanoscale confinement of SPPs to light energy. Our results imply a feasible way for constructing nanoscale optical logical gates, switches, all-optical transistors etc. for high density integration of optical circuits.