Triple-wavelength filter based on the nanoplasmonic metal-insulator-metal waveguides

In this paper, we present a proposal for compact photonic wavelength filtering and 3-dB wavelength splitting device based on a nanoplasmonic metal-insulator-metal structure. The operating performance of the device has been accurately simulated using the temporal coupled-mode theory. We use a numerical simulation method of eigenmode expansion propagation in the overall design process. We show that the transmission efficiency of the drop filter can be significantly enhanced by applying specific optimization of nanotube waveguide. The proposed structure has potential applications in highly efficient, ultra-compact integrated circuits as well as in optical communication systems at the nanoscale size.


Introduction
Recently, surface plasmon polaritons (SPPs) have proved the considerable potential to confine light in a nanoscale beyond the so-called diffraction limit that was unable to overcome by silicon-based photonic waveguides (Barnes et al. 2003;Gramotnev and Bozhevolnyi 2010;Maier 2006). SPPs are regarded as a promising method for constructing highly integrated photonic circuits (Ozbay 2012;Hsieh et al. 2015;Haffner 2015). For instance, long-range SPPs can be realized in a plasmonic hybrid silicon

Model and design principle
The conceptual diagram of the proposed three-wavelength band filter is shown in Fig. 1. The device consists of a SiO 2 bus waveguide, a symmetric double nano-stub in the vertical direction of the propagation waves, and three SiO 2 rectangular nanotube waveguides in the horizontal direction. These nanoscale waveguides are designated to selectively couple with each wavelength band for realizing the functionality of three drop channels. The purpose of the symmetric double stub is to make the possibility of wavelength band blocking for three bands of 1310 nm, 1430 nm, and 1550 nm. The working principle of the stub waveguide is based on the principle of the temporal coupled-mode theory. Whereas plasmonic rectangular nanotube waveguides play the role of nanoscale isolated Fabry-Perot cavities, also following the principle of temporal coupled-mode theory (Lu et al. 2010;Kristensen et al. 2017), to selectively couple wavelength bands depending on the appropriate cavity length. In this design, we labeled three nano-cavities as Cavity1, Cavity2, Cavity3. These three cavities correspond to Port1, Port2, Port3 for the dropping channels of 1310 nm, 1430 nm, and 1550 nm, as seen in Fig. 1, respectively. The proposed structure c MIM waveguides in which the insulating material in metallic slits and cavities is silica (SiO 2 ). The metallic layer in this device is silver, whose relative permittivity could be calculated by using the Drude model (Lin and Huang 2008;Johnson and Christy 1972) as follows: where ∞ is the dielectric constant at the infinite frequency of silver, p is the bulk plasma frequency, and are the angular frequency of incident light in the vacuum and the electron oscillation damping constant, respectively. These parameters have following values ∞ = 3.7 , p = 9.1 eV, and = 0.018 eV (Park et al. 2008;Søndergaard et al. 2008). Silver is selected in this device due to its very small imaginary part in the wavelength ranges of the telecom windows, thus, losses caused by the skin depth absorption are low enabling a long-range propagation of the SPP waves into the device. The refractive index of the fused silica glass material is described by the Sellmeier equation (Ghosh et al. 1994), showing slowly varying values in the telecom wavelength range from 1.44 to 1.46 at the room temperature of 300 °K (Tan and Arndt 2000). In this design, we make use of silica as an insulator instead of air because its high refractive index allows the MIM waveguide to confine lightwave in a nanoscale size easier than air.
As we all know that, following the surface plasmon polariton theory, the s-polarized wave (TE-polarized wave) cannot exist in the mechanism of the SPP mode. Whereas, only the p-polarized wave (TM-polarized wave) can exist and be guided in the mechanism of the SPP mode. Therefore, in our simulation study, the light of the TM-polarized wave is lead from the input port and propagates to three desired output ports for wavelength bands of 1310 nm, 1430 nm, and 1550 nm.
Transmission characteristics of the stub nano waveguide are analyzed by applying the temporal coupled-mode theory combining the plasmon waves passing through and returning from the stub with the phase matching. In that case, the transmission expressions of the symmetric single stub (T s ) and double stub (T d ) waveguides can be expressed as (Matsuzaki et al. 2008): where gp is the plasmon propagation wavelength in the MIM waveguide, L is the length of the single stub structure.
Optical characterizations were carried out using EME simulation with the grid sizes in a two-dimensional space as Δx = Δz = 5 nm applied for the total area of the proposed device. In this design, we choose all waveguides that have the same width, w. Figure 2 shows the transmission curves of the single stub waveguide at the output port as the functions of the single stub length L for some values of the width w at the operation wavelength of 1550 nm. The waveguide widths for these studies are w = 25 nm, 50 nm, 100 nm, and 200 nm. Here, we placed the stub waveguides at a distance of 1.9 µm from the input. As can be seen from Fig. 2, the wider stub width has higher efficiency. In addition, when the stub width increases the stub length for obtaining the highest resonance also increases. In order to realize the integrated circuit at the nanoscale in this design, we chose the width w = 50 nm.
In Fig. 3, transmission spectra characteristics are illustrated as a function of the stub length L in the range from 0 to 0.4 µm for three wavelengths of 1310 nm (blue), 1430 nm (red), and 1550 nm (green) for the single stub (a) and the double stub (b) at the width w = 50 nm. Both sides of the double stub waveguide have the same length L = 160 nm. The resonant peaks for these three wavelengths are 140 nm, 160 nm, and 170 nm for single stub and double stub cases, respectively. Moreover, the maximum transmissions occur at a length of about 330 nm with a loss of 1.5 dB. However, the full width at half maximum (FWHM) of the double stub's resonant dips is more significant than that of the single stub for each corresponding wavelength. Furthermore, the depth of the resonant peaks corresponding to the minimum transmissions in the double stub case is significantly larger than that of the single stub. Simulation results are in good agreement with analytic characteristics depicted by Eq.
(2) and Eq. (3). Insets in Fig. 3b illustrate the distribution of electromagnetic fields by mean of the eigenmode propagation method for three wavelength bands. It is seen that the double stub plays the role of a three bands holder, which helps enhance the efficiency of the dropping wavelengths in nano-cavities.
In this section, we will design the dropping channels by using nanotubes, which operate like FP cavities for selectively coupling at proper wavelength bands. First, we assume that the TM-polarized wave is injected into the device from the input bus waveguide with the incident power as P I . We denote O 1 , O 2 , and O 3 for the drop ports with outgoing powers as P O 1 , P O 2 , and P O 3 , respectively. Transmission characteristics of the drop channels can be defined as T d 1 = . For the plasmonic structure in our proposed MIM waveguides, the temporal evolution of amplitudes of incoming and outgoing waves are described by the interference between incident and reflected waves. These waves depend on the quality factors of the three nanotube cavities determined by the innate loss of propagating light over cavities. The accumulative phase between two reference planes satisfying the phase-matching condition is expressed by the following equation: where n eff stands for the effective refractive index of the SPP mode propagating into the bus waveguide which depends on the wavelength and the width w of the bus waveguide; k 0 is the wave number of incident light in the vacuum; and spp represents the propagation constant of the SPP mode in the MIM waveguide. Equation (4) presents the phase-matching condition of the dispersion relation in the plasmonic waveguide which is governed by (Lin and Huang 2008): where m and d are the dielectric constants of the silver cladding and silica insulating layers. Three maximum transmissions of the dropping ports are expressed in the final forms owing to the mutual effects between cavities as follows (Lu et al. 2011b): where Q 1 ,Q 2 and Q 3 are the quality factors of Cavity1, Cavity2, and Cavity3, respectively. Also, Q d 1 ,Q d 2 , and Q d 3 are correspondingly to decayed quality factors due to reducing of power into the dropping waveguides, r 1 = 1∕ 1 + 2Q d1 ∕Q 1 , r 2 = 1∕ 1 + 2Q d2 ∕Q 2 , and r 3 = 1∕ 1 + 2Q d3 ∕Q 3 are the reduced quality factors from the bus waveguides due to the Ohmic absorption of the propagation waves. Figure 4 shows the normalized transmission amplitude of the magnetic field | | | H y | | | (TM-polarized SPP mode) that was simulated by utilizing the 2D-EME simulation method at the output Port2 in the absence of the Cavity1 and Cavity3 as well as the double stub waveguides. The inset shows the electromagnetic field distribution of the TM mode (λ = 1430 nm) in the nanoscale size of 50 nm × 210 nm at the output Port2.
In order to separate three wavelengths bands into the desired output ports, we need to design Cavity1, Cavity2, and Cavity3 properly so that Cavity1 is highly resonant with the wavelength of 1310 nm but is not resonant with the other wavelengths, and vice versa. Similarly, Cavity2 and Cavity 3 are designed for the resonant wavelengths of 1430 nm and 1550 nm, respectively. Note that no analytical solution exists to describe the resonant effects of propagation waves in plasmonic waveguides. Thus, this paper will apply numerical simulation methods for investigation. As we all know both EME and FDTD (finite-difference time-domain) simulations are rigorous solutions for simulating plasmonic circuits that have been efficiently proved in many of the previous related works (Søndergaard et al. 2008;Dolatabady and Granpayeh 2017;Geng et al. 2016b;Yu et al. 2017;He 2013;Kulchin et al. 2014). However, the FDTD method is more time-consuming to simulate and characterize the circuit than the EME method. Therefore, in this paper, we utilize the EME method to simulate and optimize design processes. Nonetheless, the obtained results are accurate. In the simulation, we set the initial baffle thickness between rectangular cavities and the corresponding dropping waveguides as t 1 = t 2 = t 3 = t = 10 nm, the gap between cavities and the bus waveguide is also initially set the same value of g as zero nm. We design selectively to realize highly efficient transmissions at the nanocavities via numerical simulation. The process is carried out by searching for the highest transmission efficiency at an accumulative phase condition of Cavity1 1 = (2m + 1) ∕2 for the wavelength of 1310 nm by changing both the distance D 1 and the length L 1 of Cavity1. Here D 1 denotes the distance from Cavity1 to the input port which was initially set as 455 nm. m is an integer that we expect to be a minimum value, capably. Similar steps are also carefully performed in cases of Cavity2 and Cavity3. Finally, we found that the lengths of Cavity1, 2, and 3 are 400 nm, 640 nm, and 575 nm, respectively. The reference distances D 1 , D 2 , D 3 from the corresponding cavities numbers to the input port are found out as 455 nm, 1305 nm, and 860 nm for obtaining the highest transmission efficiencies. Figure 5 presents the transmission efficiencies of the proposed drop filter at three desired output ports for three desired wavelength bands of 1310 nm, 1430 nm, and 1550 nm by using numerical simulation in two cases, with and without the contribution of the double stub waveguide. Herein, the normalized powers (to the input power) for the Port1, Port2, and Port3 are denoted as P1, P2, and P3 with the contribution of the double stub waveguide, and vice versa without the contribution of the double stub waveguide as P'1, P'2, and P'3, respectively. Simulated results show that the transmission efficiencies of the SPP modes at the desired output ports are significantly enhanced for all three wavelength bands when introducing the double stub waveguides. Based on the working principle of the filter device, we can design a 3-dB three-wavelength band splitter by mirroring the output ports over the straight bus waveguide. The detailed characterization of these devices will be presented in the next section.

Device performance and characterization
First, we use numerical simulations to evaluate the functions of the three-wavelength band filter and the three-wavelength band 3-dB splitter as well as characterize the optical performances of the proposed device by mean of EME method. Figures 6a-c and 7a-c show magnetic field distributions, , of plasmonic modes at the individual wavelength bands for the wavelength filter and the wavelength splitter, respectively. The 2D-plots of the field distribution for the modes at three wavelength-bands clearly demonstrate functions of filter and 3 dB-splitter of the proposed device with high efficiencies with negligible crosstalk between optical fields at each output ports. In addition, 2D-plots corresponding to Fig. 6d-f also exhibit the confinement of light at Cavity1, 2, 3 for wavelengths of 1200 nm, 1650 nm and 1800 nm, respectively.
Because the three-wavelength 3-dB power splitter is designed by using a symmetrical structure which is ideal for the design of three-wavelength filtering device, and therefore, we only consider the characteristics of the proposed device in the role of the wavelength filter. We investigate the performance of this device in the wavelength domain by scanning wavelength from 1200 to 1700 nm via EME simulation. Figure 8 illustrates the transmission properties of three output ports as functions wavelength. As can be seen, the power transmissions at three output ports obtain the highest resonant peaks at the wavelength bands of 1310 nm, 1430 nm, and 1550 nm with the corresponding powers of − 5.37 dB, − 6.19 dB, and − 5.68 dB, respectively. Besides, a 3-dB bandwidth  the output ports Port1, Port2, and Port3 in these cases are as much as 90 nm, 40 nm, 100 nm. The bandwidths of these calculated transmission spectra are relatively broad, and therefore, the proposed filter is suitable for broadband access optical network applications. In order to make narrower bandwidth and increase the wavelength resolution for the wavelength selective filter in the application of wavelength division multiplexing (WDM) technique, we can modify the resonant properties of the isolated FP resonator to Fano-line shape resonator by adjusting the thickness, position of the silver baffles and the distance, and the length of FP resonators (Chen et al. 2011(Chen et al. , 2014. To fabricate the MIM waveguide at the nanoscale size, high-resolution lithography techniques such as focused ion-beam (FIB) lithography, based on heavier particles, can be used for patterning across a surface in order to create nanoscale structures instead of using ultraviolet or X-ray photolithography techniques. Although the FIB technique has high accuracy, it still exists a finite tolerance because these heavier particles have more momentum for etching the patterns. Therefore, it is important to investigate the effect of geometrical fabrication tolerance regarding the width of cavities and the primary bus waveguide for the proposed wavelength filter performance. Figure 9 illustrates the variation of power transmissions at three output ports corresponding to the wavelength bands of 1310 nm (a), 1430 nm (b), 1550 nm (c), and total absorption and total reflection of the wavelength filter (d). With a tolerance of Δw = ± 2 nm, the power transmissions are not smaller than 7 dB, the crosstalk powers are below − 15 dB, the absorptions are not exceeded 1 dB, and reflections are lower than − 10 dB. Finally, we study the effect of insulating materials on the characteristics of the proposed wavelength-selective filter. To do that, we will investigate the device performance with the refractive index change, Δn, of different materials compared to that of the crystalline silica used in the previous section using the Sellmeier model (Ghosh et al. 1994). Figure 10 presents the power transmissions at output ports Port1, Port2, Port3, and output port Port0 of the main bus waveguide corresponding to wavelength bands of 1310 nm (a), 1430 nm (b), and 1550 nm (c). Figure 10d presents the total absorption and power transmission of individual wavelength bands. One can see a strong dependence of the optical characteristics at the output ports with the insulator materials' index difference. However, no clear behavior trend was observed in the fluctuation rather. Both power transmission and absorption are the highest for silica material, about − 5 dB to − 6 dB for power transmission and 1 dB for absorption.

Conclusion
In conclusion, a triple-wavelength filter using MIM waveguides with thin baffles and a double nano-stub was introduced as isolated FP resonators and wavelength band selectors. A compact and highly efficient broadband wavelength filter and a broadband 3-dB wavelength (a) (b) (c) (d) Fig. 10 Optical transmission characteristics as functions of index difference of insulating material at four output ports: a at 1310 nm, b at 1430 nm, c at 1550 nm, and d total absorption and total power at three wavelength bands for three output port. Δn = 0 is corresponding to silica splitter were demonstrated numerically via the strong coupling effect of various isolated FP resonators and enhancement of nano-stub waveguide EME simulation. The proposed structure exhibits high transmission, low absorption, relatively large tolerances, and relatively broadband for optical access network applications at the telecom windows. In addition, the narrower bandwidth and the high wavelength resolution can be obtained via the Fano-line shape resonance mechanism, which may provide a novel method for designing nanoscale and high-resolution wavelength components in optical communications and computing, especially in wavelength division multiplexing and switching systems.