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We have investigated the optical bistability behavior based on an electromagnetically induced reflection (EIR) effect in a compound metallic grating consisting of subwavelength slits and Kerr nonlinear nanocavities embedded in a metallic film. The theoretical and simulation results show that a narrow peak in the broad reflection dip possesses a red-shift with increasing the refractive index of coupled nanocavities. Importantly, we have obtained an obvious optical bistability with threshold intensity about ten times lower than that of metallic grating coated by nonlinear material. The results indicate that our structure may find excellent applications for nonlinear plasmonic devices, especially optical switches and modulators.
©2013 Optical Society of America
1. Introduction
Plasmonics in nanostructures are widely regarded as an exciting and promising technology for efficient manipulation of photons and realization of highly-integrated optical components [1,2]. So far, a large number of plasmonic devices have been proposed and investigated, such as waveguides [3–5], filters [6], couplers [7], mirrors [8], Bragg reflectors [9], sensors [10], transducer [11], optical buffers [12], modulators [13], solar cells [14], logic gates [15], and optical switches [16,17]. Optical bistability has attracted much attention owing to its widely potential applications in optical devices such as optical logical gates, switches, transistors, memories, and so on [18]. Recently, plasmonic nanostructures were found to be capable of paving another pathway to realize strong nonlinear optical effects and minimize optical components, attributing to the significant field enhancement and light manipulation on deeply subwavelength scale [19]. For example, optical bistability behavior was found in the periodically nanostructured metal films [20]. The specific optical nonlinear property has also been found in one-dimensional metallic slit arrays [17, 19, 21]. For instance, Min et al. investigated the excellent optical bistability and switching behaviors in the subwavelength metallic grating coated by Kerr nonlinear materials [17, 21]. As one of important factors for the nonlinear effect, the threshold intensity can be improved by using nonlinear material with larger third-order nonlinear susceptibility [17, 22]. Due to the limitation of optical materials, it is significant to explore other effective accesses to the decrease of threshold intensity.
As a fascinating phenomenon, electromagnetically induced transparency (EIT) occurs in the atomic systems due to the quantum interference generated by driving the atom with an external laser [23]. The EIT effect has promising applications in enhanced nonlinear optical processes, optical switching, and optical storage due to the strong dispersion [24]. However, chip-scale applications of the atomic EIT are unsuitable owing to the rigorous conditions [24]. Fortunately, optical behaviors analogous to EIT were found in the optical resonator systems [25, 26]. The EIT-like effect is also observed in plasmonic structures [24, 27–29]. Recently, the EIT-like effect was utilized to investigate optical bistability in plasmonic waveguide-coupled resonators [30] and coupled quantum-well nanostructure [31]. However, the investigation of EIT-like effect is still rare for the nonlinear optical enhancement in plasmonic systems such as metallic grating.
In this paper, we have investigated the optical bistability effect an EIT-like response in a novel plasmonic configuration, which consists of a metallic grating with subwavelength slits and nanocavities embedded in a metallic film. It is found that the narrow reflection peak is determined by the coupling strength and detuning between metallic slits and nanocavities. The theoretical results are verified by numerical simulations. The induced reflection peak exhibits a red-shift with the increase of the refractive index in coupled nanocavities. By inserting the Kerr nonlinear material into the nanocavities, an obvious optical bistability are obtained in the compound metallic grating. Especially, the threshold intensity of the bistability is remarkable compared with that of metallic grating coated by nonlinear material.
2. Model and theories
In Fig. 1, we show the schematic of the plasmonic system, which consists of a compound metallic grating with periodic subwavelength dielectric slits and rectangular nanocavities. The dynamic transmission through the simple metallic grating can be attributed to two types of transmission resonances, which are identified as horizontal and vertical surface-plasmon resonances [32]. The first one is due to the coupled surface plasmon polaritons (SPPs) excited on the interfaces of the metallic grating, and the other is a Fabry-Perot (F-P) like waveguide resonance formed in the narrow slits. The slits work as “cavities” with regions II/I and II/III acting as two mirrors. Different from coupled SPPs, the waveguide resonance is mainly dependent on geometrical features of slits, not the period of the grating. The F-P like resonance is considered in the grating structure. Here, the metal is assumed as silver, whose permittivity can be determined by the Drude model: εm(ω) = ε∞-ω2p/(ω2 + jωγ). Here ε∞ is the dielectric constant at the infinite frequency. ωp and γ represent the bulk plasma and electron collision frequencies, respectively. These parameters for silver can be set as ε∞ = 3.7, ωp = 9.1 eV, and γ = 0.018 eV [7, 33, 34]. The regions III and I are assumed as air for simplicity. The permittivity of dielectric material in slits and nanocavities is assumed as 2.25. The p-polarized (magnetic field parallel to the z direction) plane wave is incident normally on the metallic grating. To analyze the optical response, a simple two-oscillator system is described in Fig. 2(a). The lower cavity is represented by oscillator 1, which is driven by the incident light only launched from the left port. The upper cavity is represented by oscillator 2, which is excited only by the coupling between two cavities. By the temporal coupled-mode theory [35, 36], the equations for the temporal evolution of the cavity modes a and b in oscillators 1 and 2 can be respectively described as,
Here ω1 and ω2 represent the resonance frequencies of oscillators 1 and 2. γ11 and γ12 are the decay rates due to the radiative and internal loss in oscillator 1, respectively. γ2 stands for the decay rate due to the internal loss in oscillator 2. κ is the coupling coefficient between the two oscillators. Si, Sr, and St depict the incident, reflection, and transmission waves, respectively. The input and reflection waves satisfy a relationship: Sr = −Si + a. We assume that the decay rates of the two resonators have the relationship: γ12, γ2<<γ11<<ω1. The incident frequency is ω. For the frequency difference ω-ω1<<ω1, the reflection spectrum of the two-oscillator system can be expressed as,Here γ1 = γ11 + γ12 denotes the total decay rate of oscillator 1. δ = ω2-ω1 is the detuning of the resonance frequency of oscillator 2 from oscillator 1. Oscillators 1 and 2 can be regarded as the radiative and dark elements in the EIR system, respectively. In the simplified model, the decay rates are assumed rationally and set as γ11 = 0.09ω1, γ12 = 0.0013ω1, and γ2 = 0.0025ω1 for the analysis of spectral features of induced reflection phenomenon. Figure 2(b) illustrates the evolution of reflection spectrum with the coupling coefficient κ when δ = 0. An obvious reflection peak locates in the middle of a broader dip. Here, we employ a brief terminology “induced reflection” to describe spectral response, which results from the destructive interference between the two optical excitation pathways, namely, the direct excitation of resonant mode in oscillator 1 by the external light and the excitation by coupling with oscillator 2 [29]. The induced peak increases with the coupling coefficient between two oscillators, while the spectral width is broadened. Figure 2(c) denotes that the induced peak possesses a blue-shift with increasing the detuning δ when the resonance frequency ω1 is fixed.
Fig. 1 Schematic of the metallic grating with geometrical parameters: the period of grating P, thickness of metallic film h, slit width a, coupling distance between silts and nanocavities g, as well as length and width of cavities L and w.
Fig. 2 (a) Schematic of the simple two-oscillator model. (b) Dynamic evolution of reflection spectra with the coupling coefficient κ with the frequency detuning δ = 0. (c) Spectral evolution with the detuning δ when κ = 0.033ω1. The results in (b) and (c) are calculated by theoretical modeling.
3. Numerical simulations
The coupling strength between the oscillators is determined by the coupling distance g in the plasmonic system. The larger coupling distance corresponds to the weaker coupling strength. Here, the period of metallic grating is set as 500 nm. The thickness and slit width of the grating are 250 nm and 100 nm. The length and width of coupled nanocavities are 210 nm and 50 nm, respectively. The coupling distance between nanocavities and slits is assumed as 25 nm. The spectral response is calculated by the FDTD method [37]. In FDTD simulations, perfectly matched layer absorbing boundary conditions are employed at the top and bottom of the computational space, and periodic boundary conditions are used at the left and right boundaries [17]. When high Q (quality)-factor cavities couple with low Q-factor slits through the resonant tunneling effect, a narrow reflection peak exhibits in the center of the broad dip. The reflection spectra with different g are plotted in Fig. 3(a). It is found that the bandwidth of the induced reflection peak around 1046 nm is sharper with increasing g, whereas, the larger g leads to a drop of the reflection peak. There is a trade-off between the bandwidth and peak reflection. These results agree well with the above theoretical modeling. The resonance frequency of nanocavities is determined by the refractive index nd of coupled nanocavities. Figure 3(b) shows the reflection spectra with different nd. The induced reflection peak has a red-shift for increasing nd. The spectral evolution is consistent with the result in Fig. 2(c).Here, the dielectric in nanocavities is changed as a Kerr nonlinear material. The dielectric constant of the Kerr nonlinear material is dependent on the intensity of electric field |E|2: εd = ε0 + χ(3)|E|2. The linear dielectric constant ε0 is set as 2.25. χ(3) is the third-order nonlinear susceptibility and assumed as 1 × 10−18 (m2/V2). These parameters are the same as that of Ref [21]. The intensity-dependent change of the dielectric constant in Kerr-nonlinear cavities will result in the shift of induced reflection peak. The nonlinear optical feature in the plasmonic configuration is investigated by the FDTD method. Figure 3(c) shows that the reflection spectrum possesses a red-shift with the increase of incident intensity. This optical feature may contribute to the important nonlinear behavior: optical bistability effect, which can be calculated by slowly increasing and decreasing the input intensity for a fixed wavelength. The operating wavelength is fixed at 1078 nm where the original spectral value is low enough. Moreover, the resonant spectral peak shifts toward this selected wavelength with increasing the intensity, this feature is similar to that of Refs [21]. With increasing the input intensity, as depicted in Fig. 3(d), the output intensity will rapidly increase and jump to high values at about 1200 V2/μm2. However, when the incident intensity is decreased, the output intensity drops slowly. It is noted that the threshold intensity of this bistability is about ten times lower than that of reported simple nonlinear metallic grating [21], and about six times lower than that of the plasmonic EIT-like waveguide with nanometeric size [30]. The magnetic field distributions in one period of grating for incident intensities of 400 V2/μm2 and 1300 V2/μm2 are shown in Figs. 4(a) and 4(b), respectively. The incident light completely passes through the metallic grating film when the intensity is 400 V2/μm2, which agrees well with the spectral profile in Fig. 3(c). With inputting high intensity, the stronger resonance is established in the rectangular nanocavities, but the original oscillation in slits is weakened and the incident light is reflected due to the destructive interference between the two excitation pathways. The strong resonance in coupled cavities gives rise to the reduction of threshold intensity.
Fig. 3 (a) Reflection spectra with different g for L = 210 nm. (b) Reflection spectra with different refractive index nd of nanocavities for g = 30 nm and L = 210 nm. (c) Reflection spectra with different incident intensities from the metallic grating with Kerr-nonlinear nanocavities. (d) Output intensities versus different incident intensities for λ = 1078 nm. The dashed (solid) line corresponds to increasing (decreasing) intensities. In (c) and (d), g is set as 30 nm.
Fig. 4 Magnetic field distributions |Hz| with incident intensities of (a) 400 V2/μm2 (Media 1) and (b) 1300 V2/μm2 at the wavelength of 1078 nm. The upper dotted lines and white arrows denote the incident plane and direction, respectively.
4. Conclusions
We have demonstrated an optical bistability performance based on the EIR effect in a compound metallic grating system, which consists of subwavelength slits and coupled nanocavities embedded in a metallic film. There exists a trade-off between the reflection spectral width and peak height, which are determined by the coupling strength. The induced reflection peak possesses a red-shift with the increase of the refractive index of nanocavities. The FDTD simulations agree well with the theoretical modeling. An obvious optical bistability loop has been observed by inserting Kerr nonlinear material into the nanocavities. It is noteworthy that the threshold intensity of the bistability is excellent and about ten times lower than that of the nonlinear metallic grating reported in [21], and about six times lower than that based on EIT-like effect in plasmonic waveguide-coupled resonators [30]. The results show that the EIR-assisted plasmonic nonlinear grating would have more potential for the design of compact optical devices, especially optical switches and modulators.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants 10874239, 10604066, and 11204368. Corresponding author (X. Liu). Tel.: + 862988881560; fax: + 862988887603; electronic mail: liuxueming72@yahoo.com and liuxm@opt.ac.cn.
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