1. Introduction

Electromagnetic waves trapped on metal-dielectric interfaces and coupled to propagating free electron oscillations in the metals, known as surface plasmon polaritons (SPPs), are considered as the most promising way for realization of highly integrated optical circuits because they can considerably overcome the classical diffraction limit of light and manipulate light on the subwavelength scale [1–3]. A number of devices based on SPPs, such as Mach-Zehnder interferometers [2,4], all-optical switches [5-6], modulators [7], beam manipulator [8-9], sensors [10-11], polarization analyzer [12], optical amplifier [13], transducer [14], optical buffers [15], Bragg reflectors [16-17], waveguides [18], and mirrors [19] have been numerically simulated and experimentally demonstrated. With the development of artificial fabrication, these devices may be fabricated and applied in future all-optical communications and integrated optical circuits. As an important plasmonic waveguide, the metal-dielectric-metal (MIM) structure has strong confinement of light with an acceptable length for SPP propagation [17]. The MIM waveguide is promising for the design of compact all-optical devices owing to a relatively easy fabrication [20-21]. Based on the MIM waveguide, recently, some simple wavelength-selective structures have been proposed and investigated, such as plasmonic tooth-shaped filters [22], plasmonic filters with rectangular, ring, and disk-shaped resonators [23–26]. As key factors in these devices, optical resonators will be crucial constituent elements of plasmonic wavelength-selective structures because of their symmetry, simplicity, and ease of fabrication [27]. Plasmonic wavelength demultiplexers (WDMs), which can filter specific wavelengths in different channels, will play very important role in the future all-optical communication systems [28]. Noual et al. designed a plasmonic dual-channel WDM based on Y-bent MIM waveguides [29]. However, the device needs a distance of at least 1.5 µm between the cavities and the input port, and is limited to dropping only two operating wavelengths. A plasmonic demultiplexing on the basis of the metallic grating in three-dimensional free space has been proposed [30]. But, its dimension might be unsuitable for the integration and miniaturization of devices owing to the periodic array and 3D conformation. To promote the miniaturization of plasmonic devices, a plasmonic triple-wavelength demultiplexer based on nano-capillary resonators (F-P cavities) has been proposed and investigated [31]. However, the bandwidth is broad due to the weak resonant effect. To overcome the shortcoming, quite recently, an improved compact WDM structure based on arrayed slot cavities (F-P cavities) is proposed [32]. But, the transmission efficiency of each channel is still too low. So there arises an essential problem: how to reinforce the transmission efficiency of plasmonic WDMs based on F-P cavities?

In this paper, a new kind of nanoplasmonic WDM structure based on channel drop filters in MIM waveguide with reflection nanocavities is proposed and numerically investigated. The finite-difference time-domain (FDTD) simulation demonstrates the transmission efficiency of the drop waveguide can be sufficiently reinforced by setting a proper distance between the drop and reflection cavities. The result is accurately analyzed by the temporal coupled-mode theory. A nanoscale plasmonic triple-wavelength demultiplexer with high drop efficiencies is designed as an example to perform the excellent demultiplexing function.

2. Model and principles

Figure 1 shows a three-port plasmonic channel drop filter structure which consists of bus and drop waveguides as well as two rectangular nanocavities in the metallic claddings. The insulators in metallic slits and cavities are set as air. The metal is assumed as silver, whose relative permittivity can be described by the Drude model: ε_m(ω)=ε_∞-ω_p ²/[ω(ω+iγ)] [22]. Here ε_∞ is the dielectric constant at the infinite frequency, γ and ω_p represent the electron collision and bulk plasma frequencies, respectively. ω is the angular frequency of incident light in vacuum. These parameters for silver can be set as ε_∞=3.7, ω_p=9.1 eV, and γ=0.018 eV [16]. TM-polarized wave is emitted from P and propagates to Q. P_P and P_Q stand for incident and drop power flows, respectively. The drop transmission is defined as T_d=P_Q/P_P [26]. In the plasmonic structure, the nanocavities possess symmetric with respect to the reference planes. The amplitudes of the incoming and outgoing waves in the waveguides are depicted by S_+i, S^’_+i and S_-i, S^’_-i (i=1, 2, 3). The temporal evolution of the amplitudes of Cavity a and b as well as the incoming and outgoing waves can be derived from [33-34] and described as,

 

Fig. 1 Schematic diagram of plasmonic filter based on the resonant tunneling effect of the cavity (i.e., Cavity b) near a bus waveguide with a side-coupled reflection cavity (i.e., Cavity a).

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Here n_eff denotes the effective refractive index (ERI) of SPP mode. n_eff is related with the wavelength λ and the width w_t of bus waveguide. Their dispersion relation is governed by [22],

Here r=[1/(2Q ₃)]/[j(ω/ω_oa-1)+1/(2Q ₃)+1/Q_oa]. From Eq. (12), we can see that the drop efficiency depends on the phase φ. The transmission of the drop filter is investigated by the FDTD method [35]. In FDTD simulations, the spatial and temporal steps are respectively set as Δx=Δy=5 nm and Δt=Δy/2c [4]. The metal slit width w_t is set as 50 nm. The length d ₁ and width w ₁ of Cavity b are 200 and 50 nm, respectively. The coupling distances g ₁ and t are set to be 15 nm. Due to the resonant tunneling effect, the drop transmission possesses a transmitted peak at 712 nm which corresponds to the resonance wavelength of Cavity b [26]. It is well known that the resonance wavelength of the rectangular nanocavity can be tuned by adjusting the length and width of the cavity as well as the coupling distance between the waveguide and cavity [24,29]. To effectively reflect incident signal in the bus waveguide, we set the geometric parameters of Cavity a as d ₂=205 nm, w ₂=50 nm, and g ₂=15 nm. Thus, the transmitted dip is low enough and also locates at 712 nm (i.e., resonance wavelength of Cavity a). Figure 2(a) shows the evolution of the transmission spectrum with the distance D. It is found that there exists a period of ~250 nm for the transmission response at 712 nm. Meanwhile, the maximum transmission efficiency locates at the distance of ~125 nm. These results can be explained by the temporal coupled-mode theory. From Eq. (12), we find that T_d as a function of φ has the same value at the phase difference Δφ=nπ (n is an integer). The distance difference of adjacent periods is ΔD=λ/(2n_eff). n_eff is about 1.41 at the wavelength of 712 nm as shown in Fig. 2(b). Thus, ΔD is equal to 252 nm which is in good agreement with the FDTD result. When the phase term satisfies φ=(2m+1)π/2 (m=0, 1, 2…) under the condition of ω=ω_oa=ω_ob, the drop transmission possesses the highest value which is

 

Fig. 2 (a) Transmission evolution of the channel drop filter with D from 0 to 600 nm. (b) ERI versus the wavelength with w_t = 50 nm. (c) Reflection R_b, transmission T_b, and transmission T_d with (solid curves) and without (dashed curves) the side-coupled reflection cavity in the bus waveguide. D is 125 nm. The inset depicts the drop transmission obtained by the FDTD simulation and theoretical equation. The parameters are estimated as Q ₁=Q ₃=40, Q ₂=20, and Q_oa=Q_ob=280. =π/2. (d) D for the highest drop efficiency at different resonance wavelengths.

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Here r=1/[1+2Q ₃/Q_oa]. When m=0, D=λ/(4n_eff)=126 nm which is also consistent with the result in Fig. 2(a). Figure 2(c) reveals the drop transmission spectra without and with the side-coupled reflection nanocavity in the bus waveguide. In simulations, we select the distance D as 125 nm. The length from the incident port to Cavity b is about 200 nm. It is found that the drop efficiency is very high and has been improved by more than 50% when compared to the case without the reflection cavity. Meanwhile, the reflection R_b and transmission T_b in the bus waveguide are zero at 712 nm, which illustrates that the incident power only outgoes from the drop waveguide. The drop transmission T_d does not reach unity which is mainly attributed to the intrinsic loss in metal cavities. The inset of Fig. 2(c) shows that T_d obtained by solving Eq. (12) is consistent with the FDTD result. By calculating Eq. (13), T_d,max≈0.77, which is coincident with the result in Fig. 2(a). The deviation of widths of cavities will result in the degradation of drop transmission. Moreover, if t is zero, the drop transmission will decrease intensively because most power is reflected from the bus waveguide.

3. Design and numerical simulations

As mentioned above, the transmission efficiency of the drop filter can be enhanced by setting the phase between the two reference planes as φ=(2m+1)π/2. To realize the ultra-compact WDM, we choose m=0. According to Eqs. (10) and (11), the distance D for the highest drop efficiency at different resonance wavelengths can be obtained as depicted in Fig. 2(d). It is found that the curve is nearly linear. To design WDM structures by employing the highly efficient drop filter, we choose a triple-wavelength demultiplexer with three channel drop filters to investigate the transmission response, as shown in Fig. 3(a) . The results can be extended to other multi-wavelength demultiplexing structures. The widths of waveguide and cavities are fixed as 50 nm. The lengths of Cavity 1, 2, and 3 are set as 280, 240, and 200 nm, respectively. The coupling distances between Cavity 1-3 and waveguides are 15 nm. The corresponding resonance wavelengths of Cavity 1-3 are 928, 820, and 712 nm, respectively. The lengths of Cavity 4-6 are 280, 240, and 205 nm. The coupling distances between Cavity 4-6 and the bus waveguide are 10, 10, and 15 nm, respectively. The resonance wavelengths of Cavity 4-6 are the same as that of Cavity 1-3, respectively. In the bus waveguide, thus, the transmitted powers at the operating wavelengths are effectively reflected by these side-coupled cavities. From Fig. 2(d), the optimal distances D ₁, D ₂, and D ₃ for maximal transmission at 928, 820, and 712 nm are 166, 145, and 125 nm, respectively. The length from Cavity 1 to the incident port is about 200 nm. Figure 3(b) shows the transmission spectra of the three drop waveguides. We find that the transmission efficiency of each channel is up to 70% and two times higher than the result in Ref [31]. Moreover, the efficiencies are improved by more than 50% when compared to the case without side-coupled reflection nanocavities. Figures 3(c)-(d) depict the field distributions of |H_z|² with launching continuous waves at 712, 820, and 928 nm, respectively. The field distributions are in good agreement with the transmission spectra in Fig. 3(b).

 

Fig. 3 (a) Schematic diagram of a plasmonic triple-wavelength demultiplexer. D ₁=166 nm, D ₂=145 nm, and D ₃=125 nm. (b) Transmission spectra of the three channel drop waveguides with (solid curves) and without (dashed curves) the reflection nanocavities. Field distributions of |H_z|² at (c) 712 nm (Media 1), (d) 820 nm (Media 2), and (e) 928 nm (Media 3).

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4. Conclusions

In this paper, we have proposed and numerically investigated a novel kind of nanoscale WDM structure on the basis of channel drop filters in MIM plasmonic structure with side-coupled reflection nanocavities. The FDTD simulation shows that the transmission efficiency of the drop filter can be effectively reinforced by choosing a special distance between the drop and reflection cavities. The result has been exactly analyzed by the temporal coupled-mode theory. In particular, we have designed a nanoscale plasmonic triple-wavelength demultiplexer with drop efficiencies up to 70%, which is twice as high as the result in Ref [31]. and improved by more than 50% when compared to the case without reflection nanocavities [32]. The proposed structure has important potential for the design of ultra-compact and high-efficient WDM systems in highly integrated optical circuits and optical communications.