## Abstract

Novel plasmonic power splitters constructed from a rectangular ring resonator with direct-connected input and output waveguides are presented and numerically investigated. An analytical model and systematic approach for obtaining the appropriate design parameters are developed by designing an equivalent lumped circuit model for the transmission lines and applying it to plasmonic waveguides. This approach can dramatically reduce simulation times required for determining the desired locations of the output waveguides. Three examples are shown, the 1 × 3, 1 × 4, and 1 × 5 equal-power splitters, with the design method being easily extended to any number of output ports.

© 2013 OSA

## 1. Introduction

Plasmonic waveguides have potential use in enabling energy-efficient and ultrahigh-density photonic integrated circuits (PICs) and for integrating optical components into microelectronic integrated circuits (ICs) at the nanoscale level owing to the strong field confinement of the surface plasmon polaritons (SPPs) propagating along the interface between a metal and a dielectric [1, 2]. Among various plasmonic waveguides, the metal–insulator–metal (MIM) structure allows optical modes to be highly confined within the sub-wavelength insulator layer and to propagate in a sharp bend with low additional loss [3, 4]. Additionally, they are able to be easily manufactured using existing nanofabrication techniques [5, 6].

Many compact all-optical devices based on MIM plasmonic waveguides have been proposed and investigated, including filters [7, 8], Bragg reflectors [9, 10], and power splitters [4, 11–14]. Equal-power splitters are essential components to multi-way PIC systems, which distribute the input power equally to several output ports. Several schemes have been proposed, including branching-type power splitters which connect multiple two-branch structures in tandem [4, 11, 12], multimode interference (MMI) power splitters [13] and slot-cavity based power splitters [14]. The size of a cascaded coupler expands as the number of output ports increase, giving rise to a large insertion loss [12]. The relatively low transmission efficiency is observed in a MMI power splitter because of the reflections between the single mode and multimode waveguides [13]. Furthermore, although a plasmonic splitter based on a slot cavity has a compact size, a rather large transmission loss is obtained due to indirect-connected input and output waveguides [14].

In this paper, we present a new equal-power splitter consisting of a rectangular ring resonator with direct-connected input and output waveguides. We analytically establish the equivalent lumped network of the transmission lines for this structure by using microwave engineering approaches and systematically determine the locations of the output waveguides by sequentially obtaining 1:1 voltage ratios between the adjacent output waveguides. To validate the predictions from our model, plasmonic power splitters with various number of output ports are numerically demonstrated by using a two-dimensional (2D) finite difference time domain (FDTD) method.

## 2. Design of plasmonic power splitters

Figure 1
schematically shows the configuration of the proposed N-way power splitter where an input and N output waveguides are directly connected to a rectangular ring resonator with length L and width W. The input signal at port 1 splits among output ports 2, 3, …, and N + 1. The output ports are located symmetrically on either side of the resonator with respect to the input port 1 to obtain identical output powers. The widths of the ring, the input waveguide and the output waveguide are all equal to w. A two-layer antireflection (AR) resonator structure is inserted between the input waveguide and the ring structure to diminish the reflection at port 1 [15]. The widths of these two layers are W_{r} and w, respectively. The corresponding lengths are L_{r1} and L_{r2}, respectively.

Here, a transverse electromagnetic (TEM) transmission line (TL) model is employed to describe the mode propagation of the MIM waveguide [16, 17]. The total equivalent circuit of this structure is illustrated in Fig. 2
. The ring resonator is viewed as a closed-loop transmission line. Each waveguide segment of a ring resonator with length L_{m} is represented by an equivalent T-lumped circuit model with lumped parameters of Z_{a,m} and Z_{b,m}, for m = 1, 2, …, and N + 1 [18]. These parameters are expressed as follows:

_{r}is the propagation constant of the line, and Z

_{o}is the characteristic impedance of the transmission line, calculated by${Z}_{o}=\frac{{\beta}_{r}w}{\omega {\epsilon}_{in}}$ with insulator permittivity ε

_{in}and frequency of incidence ω [16, 17].

It is known that the transmitted power at the m-th port can be obtained by calculating the scattering parameter S_{m,1} which is proportional to the propagation voltage V_{m} on the transmission line of the m-th port as all the output ports are terminated in matched loads Z_{o}. Thus, to obtain equal power at all the output ports, the amplitude of the voltage ratio at any two output ports should be 1.

Due to the structural symmetry, i.e., L_{1} = L_{N + 1}, L_{2} = L_{N}…, V_{2} = V_{N + 1}, V_{3} = V_{N}…. Then, the analysis of the lumped circuit can be simplified, as shown in Fig. 3(a)
and 3(b) which are the simplified circuit models looking from the port 1 with odd N and even N, respectively, without including the AR structure. Let M = N/2 for even N and M = (N + 1)/2 for odd N. We define VR(m) as the ratio of the voltage V_{m + 1} to V_{m} which is

_{eq,m + 1}is the equivalent impedance as seen from the (m + 1)-th port, calculated by${Z}_{eq,m+1}={Z}_{o}\left|\right|\left[{Z}_{a,m+1}+{Z}_{b,m+1}\left|\right|({Z}_{a,m+1}+{Z}_{eq,m+2})\right]$. Two vertical lines “||” represent the total impedance of two impedances in parallel. ${Z}_{eq,M+1}=2{Z}_{o}$ for odd N, and ${Z}_{eq,M+1}={Z}_{o}\left|\right|\left({Z}_{a,M+1}+2{Z}_{b,M+1}\right)$ for even N.

From Eq. (3), VR(m) is independent of the line lengths of L_{1}, L_{2}, …, and L_{m}. Accordingly, step by step, we can achieve all amplitudes of VR equal to 1. First, we start by finding the appropriate line length of L_{M} to achieve the amplitude of VR(M) equal to 1, which is only a function of L_{M}. In the case that N is even, we can arbitrarily choose the line length of L_{M + 1} to decide the value of Z_{eq,M + 1}. Next, we determine the line length of L_{M-1} such that the amplitude of VR(M-1) is 1, which becomes a function of the single variable L_{M-1} as L_{M} is selected. Successively, we repeat the previous procedure recursively to find the other line lengths except for L_{1}.

The reflected power at port 1 can be realized by the scattering parameter S_{11} expressed as

_{in}is the equivalent input impedance of this whole structure, which is $\left[{Z}_{a,1}+{Z}_{b,1}\left|\right|({Z}_{a,1}+{Z}_{eq,2})\right]/2$.

Since all the line lengths except L_{1} are known, L_{1} becomes the only variable of S_{11}. To diminish the reflected power at port 1, we can select the line length of L_{1} to have the minimal amplitude of S_{11}. However, sometimes the minimal amplitude of S_{11} is unacceptable, and then the AR structure is applied to effectively mitigate the reflected power without changing all the values of VRs. In the following section, we numerically explore several designs to illustrate the above-mentioned design concepts.

## 3. Numerical results

To confirm our design analysis, here we use the example of the Ag-air-Ag waveguide with the dielectric constant of silver described by the five-term Drude-Lorentz model [19]:

_{∞}= 1.0 is the relative permittivity in the infinity frequency, ω

_{p}= 1.258 × 10

^{16}rad/sec is the bulk plasma frequency, and γ = 7.295 × 10

^{13}rad/sec is a damping constant. ω

_{n}, γ

_{n}and Δε

_{n}are the oscillator resonant frequencies,

^{the damping factors}and weighting factors associated with the Lorentzian peaks, respectively. All the parameters of this Drude - Lorentz model can be found in Ref [19].

The commercial software 2D FDTD simulator (Fullwave, RSOFT Design Inc.) is utilized to calculate the field propagation behavior and the performance of the proposed structure with w of 50 nm. The incidence is the fundamental TM mode of this MIM waveguide at the wavelength λ_{0} of 1550 nm. A 50 nm perfectly matched layer (PML) boundary with reflectivity of 10^{−8} is applied. The grid sizes in the transverse direction, x, and transmission direction, z, are Δx = Δz = 5 nm. As the grid sizes are smaller than 5 nm, the transmission varies within ± 2%.

#### 3.1 Odd N

First, take an example of N = 3. In this case, voltage ratio VR(2) is a function of the line length L_{2}. Figure 4(a)
shows the amplitude of VR(2) for varying line length L_{2}. As we can see, the amplitude of VR(2) oscillates with a period of 550 nm, corresponding to an optical length of a half λ_{0}. Furthermore, the amplitude of oscillation gradually decreases because of the complex propagation constant. The amplitude of VR(2) equals 1 when L_{2} equals 0, 515 nm, 588 nm, etc. Let L_{2} be 515nm, then the amplitude variation of S_{11} on L_{1} is illustrated in Fig. 4(b). A periodic oscillation between 0 and 1 with increasing L_{1} is observed and the local minima are obtained at L_{1} = 260 + 550 × *l* nm, with *l* = 0, 1, 2…. As shown in Fig. 4(b), the minimum value is approximately 0.14, which is unsatisfactory for a power splitter. Let L and W be 295 and 1030 nm, respectively, corresponding to L_{1} of 810 nm and L_{2} of 515 nm. Then, the reflection at λ_{0} is mitigated as W_{r}, L_{r1} and L_{r2} are 100, 30 and 155 nm, respectively.

Figure 4(c) shows the design and the field evolution of this 1 × 3 power splitter. The transmitted powers obtained by the FDTD method are −5.41, −5.39 and −5.41 dB with respect to ports 2, 3, and 4. Notice that the powers at ports 2 and 4 are identical owing to structural symmetry. Additionally, the output powers at the ports 2 and 3 are approximately identical. This small deviation is due to the slight inaccuracy of the TL model at λ_{0}. The reflected power is effectively reduced to −48.15 dB. The insertion loss of this device is −0.63 dB, mainly resulting from the transmission loss propagating through the ring and AR resonator structures. Figure 4(d) depicts the wavelength dependence of the powers at the ports 1, 2 and 3, calculated both by the TL model and by the FDTD method. As shown, the two simulated results are in close agreement. Moreover, the simulated output powers at the ports 2 and 3 are very close to each other over the broad wavelength range of 1400 to 1700 nm. On the other hand, the reflection is very wavelength selective with a steep V-shaped spectral curve. The bandwidth for reflection less than −20 dB is obtained over a wavelength range of 1460 to 1630 nm.

Next, we extend the aforementioned design to realize a 1 × 5 power splitter. We first set L_{3} to be 515 nm to achieve the amplitude of VR(3) of 1. Then, we search for the line length of L_{2} to obtain the amplitude of VR(2) equal to 1. Figure 5(a)
shows the amplitude variation of VR(2) with L_{3} of 515 nm on L_{2}. An oscillation with a period of 550 nm is observed, and the amplitude of VR(2) becomes 1 only as L_{2} = 0 nm. Let L_{2} = 0 nm and L_{3} = 515 nm. The corresponding line lengths of L_{1} to acquire the minimal amplitude of S_{11} are 270 + 550 × *l* nm, with *l* = 0, 1, 2…, as illustrated in Fig. 5(b). The reflection can be further minimized as W_{r}, L_{r1} and L_{r2} are 40, 45 and 115 nm, respectively.

Figure 5(c) shows the layout and its FDTD field evolution of the design with L of 1030 nm and W of 305 nm, corresponding to L_{1} of 820 nm, L_{2} of 0 nm and L_{3} of 515 nm. The transmitted powers obtained by the FDTD method are −7.78, −7.62, −7.70, −7.62 and −7.78 dB with respect to output ports 2, 3, 4, 5 and 6. The reflected power is effectively reduced to −69.07 dB. The insertion loss of this device is −0.70 dB. Figure 5(d) depicts the power at the ports 1, 2, 3, and 4 as a function of wavelength. The performance has similar tendencies as those obtained in the aforementioned 1 × 3 power splitter except that the transmission is smaller, roughly less by −2.3 dB. The bandwidth for reflection less than −20 dB is obtained over a wavelength range of 1520 to 1580 nm.

#### 4.2 Even N

Here, we study the design example of a 1 × 4 power splitter. As before, we first arbitrarily choose the line length of L_{3} to acquire the value of Z_{eq,3}. Figure 6(a)
illustrates the amplitude of VR(2) as the line length L_{2} is varied at L_{3} = 250 nm. The amplitude of VR(2) oscillates with period of 550 nm and becomes 1 as L_{2} is 0, 200 nm, 557 nm, …. Let L_{2} be 200 nm, and then the amplitude variation of S_{11} on L_{1} is displayed in Fig. 6(b). It shows the local minima at L_{1} = 239 + 550 × *l* nm, *l* = 0, 1, 2…. Let L and W be 464 and 650 nm, respectively, corresponding to L_{1} of 789 nm, L_{2} of 200 nm and L_{3} of 250 nm. The reflection is minimized as W_{r}, L_{r1} and L_{r2} are 185, 0 and 30 nm, respectively. Figure 6(c) shows the layout and the field evolution of this 1 × 4 power splitter. The transmitted powers are −6.76, −6.51, −6.51, and −6.76 dB with respect to the output ports 2, 3, 4, and 5. The reflected power is reduced to −38.89 dB. The insertion loss of this device is −0.61 dB. Figure 6(d) depicts the power at the ports 1, 2, and 3 as a function of wavelength. The calculated transmission spectra obtained by the FDTD method is shifted to shorter wavelengths by roughly 35 nm compared with those obtained by the TL model. The bandwidth for reflection less than −20 dB is obtained over a wavelength range of 1520 to 1580 nm.

## 4. Conclusion

A new type of plasmonic power splitter is proposed and analyzed. This device consists of a rectangular ring resonator with direct-connected input and output waveguides. The theoretical structure is investigated using an equivalent circuit model and analytical expressions to obtain equal output powers at all the output ports. The appropriate line lengths are attained by finding the solution that results in all VR amplitudes equal to 1. Three illustrative examples with different numbers of output ports are simulated by using the FDTD method to confirm our analytical model. Simulation results substantiate that this approach can effectively acquire the targeted design parameters without lengthy computation time. In addition, this structure can be easily extended to designs with a greater number of output ports. The insertion loss of this structure is primarily attributed to propagation losses in the ring resonator and the AR structure, and the bandwidth is predominantly limited by the wavelength response of the AR structure. Enhanced predictions can be achieved by including equivalent circuits for photonic T-junctions, crossings and 90° bends.

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