## Abstract

Metal-insulator-metal (MIM) surface plasmon polaritons (SPPs) waveguides with side-coupled resonators have been widely studied through various approaches. However, few methods are both physically transparent and complete. Here, an analytical approach, which is based on the Green’s function method, is developed in order to investigate electromagnetic wave transmission across SPPs MIM waveguide networks. The proposed method is applied in order to model different MIM-waveguide geometries with weakly-coupled side stubs, comparing to the geometries with strongly-coupled stubs. The weak coupling between the backbone and stubs is taken into account by the electromagnetic field leakage at metal-insulator interface. Analytical expressions for transmittance in cases of single stub and cavity are obtained straightforwardly. Our method shows excellent computational efficiency in contrast with solving Maxwell equations numerically.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Surface plasmon polaritons (SPPs) waveguides with coupled resonators have been widely studied in recent years because of its potential application in highly integrated optical circuits. For example, a designed plasmonic waveguide system with double resonant nano-disks shows a powerful electromagnetically induced transparency (EIT) like effect [1]. Other optical effects such as wavelength splitter [2], total absorption [3], slow light effect [4], and plasmon-induced transparency [5,6] have been realized in plasmonic waveguide with nanoscale side-coupled resonators. A diverse geometry of resonators side-coupled to metal-insulator-metal (MIM) SPPs waveguide have been reported, such as ring resonator [6,7], serial stubs [8,9], rectangular cavity [10,11], as well as irregular cavities [12]. Among them, rectangular side-coupled resonators have the advantage of taking up less physical space. Simplicity in geometry makes rectangular resonators suffer less challenge of fabrications. Recently, various potential applications have been exhibited by MIM SPPs networks with rectangular resonators, including optical switching [13,14], wavelength filter [15–17], optical pressure sensor [18], microfluidic sensor [19], and all-optical logic gates [20].

Methods of treating plasmonic metallic MIM waveguides are either numerical simulation or analytical algorithm. The numerical tools, like Finite Element Method (FEM) and Finite Difference Time Domain (FDTD), are widely used to deal with various SPPs networks. However, generally, they not only lack physical transparency but also are consuming in time and memory. Besides, the use of Cartesian-grid in FDTD may produce artificial surface plasmon around the metallic structures for subwavelength cells [21], resulting in obscures of investigation. In contrast to simulation method, analytical approaches are efficient and rapid in predicting the behavior of light modes in the waveguides [22]. Conventional analysis methods, mainly containing Coupled-Mode Theory (CMT), and Transfer Matrix Method (TMM), have been successfully applied to model various guided-wave devices, such as optical directional couplers [23], nanodisks coupled waveguides [1,24], Kretschmann prism [25], channel fiber [26], etc. But the usage of TMM is limited to 1D model up to now. In prior studies [1,27], however, perfect conductor condition is adopted at the terminal of stubs, which failed to reflect the real leakage of the electromagnetic field at metal-insulator interface. Further, it is attributed to the zero value of the penetration depth of the perfect conductor that the unconnected cavity structure is always troublesome. As an engineering approach, Transmission Line (TL) theory has enjoyed large popularity in modeling varieties of plasmonic devices, including wavelength demultiplexers [28–30], optical filters [31–33], all-optical switching [34,35], sensor [20], resonator [13,36], etc. Importantly, the analogy of waveguide impedance is the ratio of voltage and travelling current,$Z=V/I$, based on the circuit approach [37]. However, the TL theory fails to demonstrate field distribution of arbitrary configuration. Therefore, we find that the analysis approach of electromagnetic wave propagation remains to be improved.

In this paper, we propose a consistent analytical approach based on Green’s function method to model the MIM SPPs waveguides networks containing rectangle side-coupled antennas. In our method, propagating behaviors of waves within the considered system are determined by two key physical quantities, wave vector $k$ and effective impedance$Z$, which are depended on material parameters and magnitude of structure. We map out the complete theoretical framework of our method in details. Algorithms for constructing Green’s function matrix in the cases of arbitrary stubs, cavities and junction networks respectively are also developed. Thereby, we derive analytical expressions for the transmission spectra of the waveguide coupled to single stub and single cavity respectively. Magnetic-field distribution for the two specifications at given frequency are demonstrated both analytically and numerically. It shows that the analytical model agree with the FDTD method very well, which validates the feasibility of the theoretical analysis. Moreover, a multi-stub antenna structure is studied to show the high efficiency of the analytical model on the premise of high accuracy. Our work is expected to give deep insight into the optical properties of the MIM waveguide networks and can provide useful guidance for the design of integrated optical circuits.

## 2. Method

Consider an infinite two-dimensional MIM waveguide, as shown in Fig. 1(a), which is composed of two semi-infinite metallic media and a thin gap insulator layer with thickness$d$. We know that only the TM modes can be excited in the MIM waveguide under the boundary condition for the dielectric-metal interface, and their propagation equation can be expressed as [39]:

Before we employ the Green’s function of the magnetic field, it is necessary to seek the propagation wave vector of the MIM waveguide. For the only existing symmetric mode TM_{0} in the waveguide we focus in this paper, when dielectric thickness is small enough ($d\ll {\lambda}_{\text{sp}}$), the SPP propagation wave vector ${k}_{\text{sp}}$is determined by the dispersion relation [43]:

In order to characterize the leakage of the field which is unavoidable in SPP MIM networks, here we introduce the conception of *metallic waveguides*. This is particularly important when solving the problem containing side-coupled antennas. Since two key physical quantities, wave vector $k$ and effective impedance$Z$, have been obtained in MIM waveguides, we simply consider an infinite metal media, as shown in Fig. 1(b). The thickness of the metallic layer is assumed to be$d\text{'}$, which is arbitrary but has a strong relation to the structure when particular geometry is concerned. Thus we get:

In what follows, we present the basic idea of the Green’s function method in the MIM network waveguide system. Any composite structure can be decomposed into the following categories, as shown in Fig. 2. They are (a) input and output waveguides, (b) short-circuit waveguide with finite length$d$, (c) open-circuit stub like structure waveguide and (d) unconnected cavity side-coupled structure. Input and output waveguides act as exterior ports, which can be taking as semi-infinite waveguide. Let ${D}_{j}$ indicate the domain and $M$ denote all the existing interfaces for each constituent. Based on the interface response theory [38], one can calculate the Green’s function $g(MM)$ of each composited system which is related to the Green’s function $G(MM)$ by:

*j*. The latter matrix ${\left[{g}_{j}(MM)\right]}^{-1}$ is given by the equation:

*j*-th short-circuit with length ${L}_{j}$ [Fig. 2(b)], bounded by two free surfaces at $x=0$ and $x={L}_{j}$ such that electric field $E=0$ at these boundaries, in this case, the cleavage operator is:where $n$ represents the forward direction along the waveguide, thus the inverse of the surface Green’s function is given by:

*E*vanishing on its extremity$x=0$, the cleavage operator is:and the inverse of the surface Green’s function is ${\left[{g}_{j}(MM)\right]}^{-1}=i{Z}_{j}\text{or}i{Z}_{j}^{\text{'}}$. For the stub structure with length${L}_{k}$, as shown in Fig. 2(c), we take penetration depth of metal into account to represent the insulator-metal interface across the propagation direction. Thus we get the inverse of the surface Green’s function:

Suppose an incident wave $U(x)={e}^{i{k}_{\text{sp}}x}$ is launched in one homogeneous piece of composite material, the magnetic-field distribution $u(D)$along the wave propagation direction in the waveguide can be calculated by [38]:

_{$U(M)$}and $G(MM)$ are the row-vector and matrix, respectively, constructed by taking a finite number of discrete interfaces in the $M$ space. $G(MD)$ is a column-vector, whose value is got by calculating two points belonging to the $M$ space and $D$ domain. For example, consider a finite waveguide of length${L}_{j}.$ Within the interface space$M=(0,{L}_{j}),$ ${G}_{j}(MM)$ is: a 2 × 2 square matrix with the form${G}_{j}(MM)=\frac{1}{2i{Z}_{j}}\left(\begin{array}{cc}1& {e}^{i{k}_{\text{sp}}{L}_{j}}\\ {e}^{i{k}_{\text{sp}}{L}_{j}}& 1\end{array}\right),$ and the row-vector is${U}_{j}(M)=\left(\begin{array}{cc}1& {e}^{i{k}_{\text{sp}}{L}_{j}}\end{array}\right),$ the column-vector${G}_{j}(MD)=\frac{1}{2i{Z}_{j}}\left(\begin{array}{c}{e}^{-i{k}_{\text{sp}}x}\\ {e}^{i{k}_{\text{sp}}({L}_{j}-x)}\end{array}\right).$ In this case, we further investigate the Eq. (18) and obtain the distribution of an arbitrarily transmit domain ${D}_{j}$ with finite length ${d}_{j}$ as$u({D}_{j})=2i{Z}_{\text{in}}\left[\begin{array}{cc}1& 0\end{array}\right]g(MM){G}_{j}^{-1}(MM){G}_{j}(M{D}_{j}),$ where ${Z}_{\text{in}}$is the impedance of the input waveguide. Finally, transversal magnetic-field distribution can be expressed in a generalized form as:

As the transversal field distribution can be calculated by Eq. (19), the longitudinal distribution of the magnetic field $h(z)$ should be also considered thus magnetic-field spatial distribution in the waveguide can be obtained. The infinite MIM waveguide structure is divided into three parts denoted by I, II and III, as shown in Fig. 1(a). For the TM_{0} plasmonic mode, by using the boundary connection condition, the general formula of magnetic-field spatial distributions in the MIM waveguide is given by

Suppose the wave has been launched in the port 1 of the SPP network, the transmittance $T$ of the port $n$ and reflectance $R$ can be obtained directly in terms of the green function:

For the multi-input case, based on the superposition theorem of the field, the transmittance $T$of the port $n$and the reflectance $R$of the incident port $k$ can be rewritten as:Thereby we get the two deterministic physical quantities in wave propagation of both MIM and metallic waveguides. In implementing of side-coupled antennas structure, we assume that stubs are terminated by semi-infinite metallic output waveguides. Similarly for cavities, metallic waveguides are supposed within the interconnecting direction with a finite length between the cavity and the backbone waveguide. Armed with these results, we can now set up network analogs for any system including interested-guides, stubs, cavities, etc.-interconnected in arbitrary ways and excited either externally or internally. Since precise numerical values are assigned to all the quantities involved, we can then calculate frequency response, radiated power, and other quantities of concern.

## 3. Main constructions in SPPs networks

A network structure consisted of four X-junctions whose corresponding ${\left[g(MM)\right]}^{-1}$ can be written as a (12 × 12) matrix, has already been discussed in details [39]. However, the analysis on such problem is limited to short-circuit waveguides which is directly connected to each other composing the network structure with several junctions. The previous framework lacked of skills to model the side-coupled antennas structures, such as T-shaped waveguides [9,35] and unconnected cavities [10,36]. In order to mature the Green’s function method into a complete way for various SPPs network compositions, we introduce an explicit implemental algorithm of the Green function to deal with three typical constructions, namely (i) open circuit stubs, (ii) side-coupled cavities, and (iii) short circuit junctions.

#### 3.1 Open-circuit stub model

By assuming metallic waveguides connected to the ends of the stub structure, arbitrarily side-coupled stubs system can be analyzed by the Green’s function method. In this manner, the side stub-like open circuit structure is considered as two short-circuit junction with distinct impedance $Z$ for MIM waveguides and $Z\text{'}$ for metallic waveguides. A general case is that one backbone MIM waveguide is coupled with $j$open circuit stubs. As the schematic illustrated in Fig. 3(a), first, we label the $j$ open circuit stubs by an array of bivariate functions$\left\{1({w}_{1},{d}_{1}),2({w}_{2},{d}_{2}),\dots ,j({w}_{j},{d}_{j})\right\}$, where$j$ refers to the${j}^{\text{th}}$T-junction on the backbone SPP waveguide and the binary variable $({w}_{j},{d}_{j})$ with $i=1,2,\dots ,j$ denotes the width and length of the stubs respectively. ${\Delta}_{(a,a+1)}$ is the distance between ${a}^{\text{th}}$ and ${(a+1)}^{\text{th}}$ label of the bus waveguide, where $a=1,2,\dots ,j-1$.

Then we focus on the Green’s function, which is considered to be a $\left(2j\times 2j\right)$ matrix. Two kinds of interfaces occur in such configuration. They are $\left\{1,2,\dots j\right\}$representing the T-junctions on the backbone waveguide, and $\left\{1\text{'},2\text{'},\dots j\text{'}\right\}$representing the semi-infinite metallic waveguides. The diagonal matrix elements representing self-energy of each labeled stub open circuits can be expressed as:

#### 3.2 Arbitrary cavity model

In this part, we emphasize on the side-coupled cavities. Owning the view of metallic waveguide, we consider the gap between backbone waveguide and cavities to be metallic waveguides with loss. In detail, one backbone waveguide is coupled with $j$ cavities, which is shown in Fig. 3(b). Similar labeling method, i.e.$\left\{{C}_{1}({w}_{1},{d}_{1}),{C}_{2}({w}_{2},{d}_{2}),\dots ,{C}_{j}({w}_{j},{d}_{j})\right\}$, is utilized to characterize the geometry of the proposed structure. To deal with such structure, the Green’s function should be a $\left(3j\times 3j\right)$ matrix. Additional geometry parameter ${s}_{n}$ is the gap distance between the ${n}^{\text{th}}$ cavity and the bus waveguide. Because of the appearance of the gap which constitutes a lossy metallic junction, the interface $\left\{{C}_{1},{C}_{2},\dots ,{C}_{j}\right\}$ occurs in such structure. $\left\{1,2,\dots ,j\right\}$ represents the position of each cavity on the backbone waveguide, while $\left\{{C}_{1}^{\text{'}},{C}_{2}^{\text{'}},\dots ,{C}_{j}^{\text{'}}\right\}$is the semi-infinite metallic waveguide we supposed, which is equivalent to $\left\{1\text{'},2\text{'},\dots ,j\text{'}\right\}$ in the case of general side stubs configuration. Following the matrix elements expression above, we get the Green’s function straightforward.

- (1) Diagonal matrix elements:
And

with$n=1,2,3,\dots ,j.$ The lossy metallic waveguide gap is denoted as${\phi}_{\text{gap}}(n)=-{Z}_{n}^{\text{'}}\mathrm{cot}\left({k}_{\text{sp}}{s}_{n}\right),$ where${Z}_{n}^{\text{'}}\frac{{k}_{\text{sp}}{w}_{n}}{\omega {\epsilon}_{\text{m}}}.$ And${\phi}_{\text{cavity}}(n)=-{Z}_{n}\mathrm{cot}\left({k}_{\text{sp}}{d}_{n}^{\text{'}}\right),$ where${Z}_{n}=\frac{{k}_{\text{sp}}{w}_{n}}{\omega {\epsilon}_{\text{d}}},$ ${d}_{n}^{\text{'}}={d}_{n}+{\delta}_{\text{metal}}.$ And${\phi}_{\text{left}}(n)=-{Z}_{0}\mathrm{cot}\left({k}_{\text{sp}}{\Delta}_{(n-1,n)}\right),$${\phi}_{\text{right}}(n)=-{Z}_{0}\mathrm{cot}\left({k}_{\text{sp}}{\Delta}_{(n,n+1)}\right),$ where ${Z}_{0}=\frac{{k}_{\text{sp}}{w}_{0}}{\omega {\epsilon}_{\text{d}}}.$

- (2) Non-diagonal matrix elements:
For two neighbor junctions on the bus waveguide, we obtain:

$${g}_{(p,q)}^{-1}=\frac{{Z}_{0}}{\mathrm{sin}\left({k}_{\text{sp}}{\Delta}_{(p,q)}\right)},\text{if}\left|p-q\right|-1.$$For each gap, we present the correlation between$n$and ${C}_{n}$ as:

$${g}_{\left(n,{C}_{n}\right)}^{-1}=\frac{{Z}_{n}}{\mathrm{sin}\left({k}_{\text{sp}}{s}_{n}\right)}.$$For each cavity, we have the correlation between ${C}_{n}$ and ${C}_{n\text{'}}$ as:

#### 3.3 Short-circuit junction networks

Such networks contain orthogonally connected MIM-waveguide forming junctions separately at their terminals as shown in Fig. 3(c). Clearly, four short-circuit MIM waveguides with length${L}_{i}$ and width${w}_{i}$, constitute a junction structure. Our networks constructed by short circuits junctions are compatible to any number of arms, junctions and ports. The inverse of Green’s function ${\left[g(MM)\right]}^{-1}$ concerning short-circuit junction is a $\left(pq\times pq\right)$ matrix. Junction dots are labeled by a pair of numbers$\left(p,q\right),\left\{p,q=1,2,3,\dots j\right\}$, and can be divided into three types. The first is the four junction dots at the corner composed with two semi-infinite waveguides and two finite waveguides (*red colored*), second type is the$\left[2(p-2)+2(q-2)\right]$junction dots on the side way with one semi-infinite waveguide and three finite waveguides (*blue colored*), and the last one is the rest inside junctions (*black colored*) with four finite waveguides. For the junctions with two semi-infinite waveguide terms, i.e. the junctions on the corner, their inverse of the surface Green’s function are as follows:

Though, of course, it is necessary to state the limitation of the analytical method here. The distance between side antennas or two paralleled waveguides cannot be smaller than the width of the waveguide. Further, the gap distance between the side cavity antennas cannot be too large. Thus far, we have derived the general expression for each element of the green’s function. In this way, transmission properties of plasmonic waveguide system contained by any combination of junctions, stubs or cavities can be characterized analytically. The theoretical analysis mentioned above may provide a guideline for the accurate adjustment of spectral responses.

## 4. Validity of method

#### 4.1 Single-side antenna

In order to validate our analytical method and highlight the improvement of the coupling efficiency by the metallic waveguides, we calculate transmittance spectra and the field distribution for (i) single stub [Fig. 4] and (ii) single cavity [Fig. 5], both analytically and numerically. For the sake of simplicity, vacuum is used as the insulating media, thus${\epsilon}_{\text{d}}=1$. The metal is chosen by silver. Its permittivity can be expressed by the well-known Drude model [40]:

*solid*) and is compared with the FDTD result (

*dotted*). For reference, we also show the transmittance spectrum reported in [42] with the same geometric and material parameters. Obviously, our transmission spectrum does a better match with the numerical result, especially at low frequency range. Besides, the magnetic field in the structure at the incident wavelength $\lambda =1820\text{nm}$ is shown in Fig. 4(c). Comparing with the insert figure calculated by numerical method, we see close agreement between them.

Transmission properties of the MIM waveguide side-coupled to single unconnected cavity structure are also studied, shown as Fig. 5(a). The additional geometry parameter is the gap distance$s$. With Eq. (29) we get the transmission ${T}_{\text{cavity}}$ as:

#### 4.2 Multi-stub antenna

To show the efficiency in computation, we further consider a MIM waveguide with four periodical stubs perpendicular to it as shown in Fig. 6(a). The insulator medium of the waveguide is assumed to be air and the metal medium is chosen by silver. The length of the stubs and the interval between two stubs are denoted by $d$and$L$, respectively. Figure 6(b) gives the transmission spectra for such serial stubs structure with$L=d=400\text{nm}$and $h=w=50\text{nm}$ by FDTD (*dotted, red*) and Green function method (*solid, blue*). It shows that the analytical method provides more detailed features in transmission on the premise of high accuracy. Compared to the case of single stub, taking 1.544s, the computing time of the transmission of the four periodical stubs structure is 2.260s. Swept range is from 500nm to 2000nm with step length 1nm. Resulting from the structural complexity, the increase in computing time is acceptable. The only difference is the increase in matrix dimensions, from 4 × 4 (single stub) to 10 × 10 (four stubs). In comparison, it takes 138s (2 minutes, 18 seconds) and 303s (5 minutes, 3 seconds) for Comsol to calculate the transmission of single stub structure and four periodical stubs structure respectively, with the same swept range and step length. Thus it can be seen that simulation tool is far more time-consulting than the method of Green’s function, which is very efficient and rapid to predict the behavior of light modes in waveguides.

## 5. Conclusions

We have developed a complete analytical method based on Green function to calculate the propagation problem in the MIM SPPs waveguide networks containing rectangle side-coupled antennas. Our method provides analytical solutions for arbitrary composition of side-coupled stubs or cavities. In proposed method, phase shift effect caused by metals-insulator interface has been taken into account by utilizing length modification of the stubs or cavities in propagating direction. The leakage of the electromagnetic field at metal-insulator interface described by metallic waveguides reveals the weak coupling between the backbone and antennas structure. Using the proposed theory, we demonstrated the algorithms for constructing Green’s function matrix in the cases of arbitrary stubs, cavities as well as junction networks, which are important in practical. Thus the effectiveness of our method stemming from superposition arithmetic in Green function formalism is estimated. Through the calculation of the transmission spectra and magnetic field distribution in cases of single stub and cavity, it is shown that our method is capable of providing results in high accuracy. The proposed analytical method can significantly reduce the computational time and allow us to design geometrically flexible couplers for plasmonic waveguides. Our work may help to understand the fundamental physics of the MIM waveguide networks and to guide the corresponding applications.

## Funding

National Key R&D Program of China (Grant No. 2017YFA0303400) and National Natural Science Foundation of China (NSFC) (Grant No. 91630313).

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