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 impedanceZ, 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 thicknessd. 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]:

(2x2+2z2+kx2+kz2)Hy(x,z)=0,
where kxand kzare x- and z- components of the wave vector, respectively.Hy(x,z)stands for the spatial distribution of the magnetic field, and can be written as Hy(x,z)=u(x)h(z) by the variable in which u(x) and h(z) represent the longitudinal distribution along the center axis of the waveguide(z=0)and the transversal distribution perpendicular to propagation direction, respectively. First, we focus on the field distribution u(x) along the propagation direction, obeying the propagation equation of the wave in an infinite homogeneous one-dimensional system, which is given by:
(kx2+2x2)u(x)=0.
Considering the current as the magnetic source for the TM mode, we find the corresponding Green’s function satisfies:
Zkx(kx2+2x2)G(x,x')=δ(xx'),
where Z is the impedance of the waveguide. Borrowing concepts from transmission line theory [37], the impedance Z has been derived and given approximately by the first-order Taylor expansion as [39]: Z=2kspωεdksp2εdk02tanh(ksp2εdk02d2)kspdωεd. One can easily confirm that G is given by:

 figure: Fig. 1

Fig. 1 Schematic illustrations of (a) infinite MIM waveguide (b) infinite metallic waveguide, and (c) semi-infinite MIM waveguide.

Download Full Size | PPT Slide | PDF

G(x,x')=eikx|xx'|2iZ.

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 TM0 in the waveguide we focus in this paper, when dielectric thickness is small enough (dλsp), the SPP propagation wave vector kspis determined by the dispersion relation [43]:

tanh(ksp2εdk02d2)=(εdksp2εmμmk02εmksp2εdμdk02),
where k0=2π/λ0 is the wave vector in vacuum,εd(μd)andεm(μm)are the relative permittivity (permeability) of dielectric and metal, respectively. With the help of Taylor expansion, it is obtained as:

kspk0εd2εdεdεmk0εdd.

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 impedanceZ, 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 bed', which is arbitrary but has a strong relation to the structure when particular geometry is concerned. Thus we get:

ksp'=k0εm,
Z'=ksp'd'ωεm.
It should be noticed that for these waveguides, including MIM waveguide and metallic waveguide, the impedance in the SPP waveguide is dependent on size. For application of the metallic waveguide, a semi-infinite MIM waveguide is shown in Fig. 1(c), which is composed of an end-terminated MIM waveguide and an infinite metallic block media. As we see from the schematic, metallic media acts as obstacle occurring at the end of the semi-infinite MIM waveguide. The leakage caused by the block metallic material is represented by a semi-infinite metallic waveguide we assumed. In this way, such stub-like structure can be decomposed into a truncated MIM waveguide and a semi-infinite metallic waveguide. By lettingd'=d, the inverse of Green’s function is:
[g(MM)]1=[iZZcot(kspL')Zsin(kspL')Zsin(kspL')iZ'Zcot(kspL')],
whereZ'=ksp'dωεm, and L'=L+δmetal, representing the correction of the finite length.

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 lengthd, (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 Dj 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:

g(MM)(I(MM)+A(MM))=G(MM)inD=[Dj],
where G(MM) is a matrix whose elements are constructed out of the Green’s function of the truncated unit Gj(MM), with I(MM) being the unit matrix, and A(MM) the corresponding surface response operator. Therefore, g(MM) can be obtained by inverting[g(MM)]1, which is built from a juxtaposition of the matrix[gj(MM)]1for each constituent j. The latter matrix [gj(MM)]1 is given by the equation:
[gj(MM)]1=Δj(MM)[Gj(MM)]1,
with:
Δj(MM)=Ij(MM)+Aj(MM),
and:
Aj(x'',x')=Vj(x)Gj(x,x')dx,
where Vj is the cleavage operator, with x,x'Dj and x,x''Mj. For the j-th short-circuit with length Lj [Fig. 2(b)], bounded by two free surfaces at x=0 and x=Lj such that electric field E=0 at these boundaries, in this case, the cleavage operator is:
Vj(x)=δ(x)n+δ(xLj)n,
where n represents the forward direction along the waveguide, thus the inverse of the surface Green’s function is given by:
[gj(MM)]1=(Zjcot(kspLj)Zjsin(kspLj)Zjsin(kspLj)Zjcot(kspLj)).
For a semi-infinite waveguide [Fig. 2(a)], electric field E vanishing on its extremityx=0, the cleavage operator is:
Vj(x)=δ(x)n,
and the inverse of the surface Green’s function is [gj(MM)]1=iZjoriZj'. For the stub structure with lengthLk, 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:
[gk(MM)]1=(Zkcot(kspLk')Zksin(kspLk')Zksin(kspLk')Zkcot(kspLk')),
where Lk'=Lk+δmetal. Here the penetration depth of the electromagnetic field associated with the SPP mode into the metal medium is written asδmetal=1k0|RE(εm)+εdRE(εm)2|12.

 figure: Fig. 2

Fig. 2 Schematic of the type of MIM waveguide networks system. (a) Input and output waveguides (b) short-circuit waveguide (c) open-circuit structure (d) unconnected side-coupled cavity.

Download Full Size | PPT Slide | PDF

Suppose an incident wave U(x)=eikspx 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(D)=U(D)U(M)[G(MM)]1G(MD)+U(M)[G(MM)]1g(MM)[G(MM)]1G(MD).
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 lengthLj. Within the interface spaceM=(0,Lj), Gj(MM) is: a 2 × 2 square matrix with the formGj(MM)=12iZj(1eikspLjeikspLj1), and the row-vector isUj(M)=(1eikspLj), the column-vectorGj(MD)=12iZj(eikspxeiksp(Ljx)). In this case, we further investigate the Eq. (18) and obtain the distribution of an arbitrarily transmit domain Dj with finite length dj asu(Dj)=2iZin[10]g(MM)Gj1(MM)Gj(MDj), where Zinis the impedance of the input waveguide. Finally, transversal magnetic-field distribution can be expressed in a generalized form as:
u(x)=2iZinsin(αjdj){[g(0,x1)g(0,x2)][sin[αj(djx)]sin(αjx)]},x|x1x2|=dj,
where Zin is the impedance of the input waveguide, g(0,x1) and g(0,x2) representing the response of interface (x1)and (x2)respectively. Here, a simple example is given to illustrate the essence of field distribution. Consider the propagation field of 1D material between interfaces “0” and “1” with finite distanced. The discrete expression of the propagation field is:
u01(x)=ua(x)+ub(x)+uc(x)+ud(x),(0<x<d),
with:
ua(x)=U(0)G01(0,0)g(0,0)G011(0,0)G01(0,x),
ub(x)=U(0)G01(0,0)g(0,0)G011(0,1)G01(1,x),
uc(x)=U(0)G01(0,0)g(0,1)G011(1,1)G01(1,x),
ud(x)=U(0)G01(0,0)g(0,1)G011(1,0)G01(0,x).
We can find a clear physical structure based on Eq. (20), which is the superposition of optical field magnitude of four diverse propagation processes during the 1D material (0→1). Among them, ua(x)and ub(x) denote the time harmonic wave field transmitting through the first interface (0), which is indicated byg(0,0) . While uc(x) and ud(x) describe the propagation process occurring reflection at the second interface (1), which is indicated byg(0,1). By a further observation, oscillation occurs just after wave propagating through interface (0) in the process of ua(x), without reaching at the other interface yet. While, ub(x)represents the wave oscillation with a reflection at the second interface (1). In addition, the propagation process described by uc(x) is a reflecting oscillation at the second interface (1). Finally ud(x) shows the propagation process that optical field is reflected back and forth twice, the first occurs at interface (1) while the second at the interface (0).

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 TM0 plasmonic mode, by using the boundary connection condition, the general formula of magnetic-field spatial distributions in the MIM waveguide is given by

HIy(x,z)=u(x)ekzm(z+d2)(z<d2),
for domain I,
HIIy(x,z)=u(x)(2ekzdd2ekzdd+1cosh(kzdz))(d2<z<d2),
for domain II, and:
HIIIy(x,z)=u(x)ekzm(zd2)(z>d2),
for domain III, with kz(m,d)=ksp2ε(m,d)μ(m,d)k02. Particularly, for metallic waveguides shown in Fig. 1(b, c), the magnetic-field spatial distributions in the propagating range (d2<z<d2) is:

H[metallicWG]y(x,z)=u(x)(2ekzmd2ekzmd+1cosh(kzmz)).

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:

T(1n)=|2iZ1g(1,n)|2,
R1=|12iZ1g(1,1)|2.
For the multi-input case, based on the superposition theorem of the field, the transmittance Tof the port nand the reflectance Rof the incident port k can be rewritten as:
T1,2,3n=|2iZ1A1eiφ1g(1,n)+2iZ2A2eiφ2g(2,n)+2iZ3A3eiφ3g(3,n)+|2,
Rk=|12iZkAkg(k,k)+2iZ1A1eiφ1g(1,k)+2iZ2A2eiφ2g(2,k)+|2,
whereAj,φj,(j=1,2,3)represent the amplitude and initial phase of the wave launched in portj, and Zj is the impedance of port j.

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 [g(MM)]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' for metallic waveguides. A general case is that one backbone MIM waveguide is coupled with jopen circuit stubs. As the schematic illustrated in Fig. 3(a), first, we label the j open circuit stubs by an array of bivariate functions{1(w1,d1),2(w2,d2),,j(wj,dj)}, wherej refers to thejthT-junction on the backbone SPP waveguide and the binary variable (wj,dj) with i=1,2,,j denotes the width and length of the stubs respectively. Δ(a,a+1) is the distance between ath and (a+1)th label of the bus waveguide, where a=1,2,,j1.

 figure: Fig. 3

Fig. 3 Schematic diagrams of three typical constructions. (a) Arbitrary open-circuit stubs model (b) arbitrary side-couped cavities model (c) short circuit junctions’ networks.

Download Full Size | PPT Slide | PDF

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

gn1={iZ0+φright(1)+φs(1),n=1;φleft(j)+φs(j)+iZ0,n=j;φleft(n)+φright(n)+φs(n),n=2,3,,j1.
gn'1=iZn'+φs(n),n=1,2,3,j.
In whichφs(n)=Zncot(kspdj'),Zn=kspwnωεd,Zn'=kspwnωεm,dj'=dj+δmetal. Andφleft(n)=Z0cot(kspΔ(n1,n)),φright(n)=Z0cot(kspΔ(n,n+1)), whereZ0=kspw0ωεd. It is noticed that δmetalis the penetration depth of the field into the metal mentioned in above section. w0 is the width of the bus waveguide, and wn denotes the width of nthstub. Then we discuss the non-diagonal matrix elements. For two neighbor junctions on the bus waveguide, we get:
g(p,q)1=Z0sin(kspΔ(p,q)),if|pq|=1.
For each stub, we present the correlation between n and n'as:
g(n,n')1=Znsin[ksp(dj+δmetal)].
Notice that it is unnecessary to consider the interactions between T-junctions as if the distance between stubs is not too small.

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.{C1(w1,d1),C2(w2,d2),,Cj(wj,dj)}, is utilized to characterize the geometry of the proposed structure. To deal with such structure, the Green’s function should be a (3j×3j) matrix. Additional geometry parameter sn is the gap distance between the nth cavity and the bus waveguide. Because of the appearance of the gap which constitutes a lossy metallic junction, the interface {C1,C2,,Cj} occurs in such structure. {1,2,,j} represents the position of each cavity on the backbone waveguide, while {C1',C2',,Cj'}is the semi-infinite metallic waveguide we supposed, which is equivalent to {1',2',,j'} 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:
    gn1={iZ0+φright(1)+φgap(1),n=1;φleft(j)+φgap(j)+iZ0,n=j;φleft(n)+φright(n)+φgap(n),n=2,3,,j1.

    And

    gCn1=φgap(n)+φcavity(n),
    gCn'1=iZn'+φcavity(n),

    withn=1,2,3,,j. The lossy metallic waveguide gap is denoted asφgap(n)=Zn'cot(kspsn), whereZn'kspwnωεm. Andφcavity(n)=Zncot(kspdn'), whereZn=kspwnωεd, dn'=dn+δmetal. Andφleft(n)=Z0cot(kspΔ(n1,n)),φright(n)=Z0cot(kspΔ(n,n+1)), where Z0=kspw0ωεd.

  • (2) Non-diagonal matrix elements:

    For two neighbor junctions on the bus waveguide, we obtain:

    g(p,q)1=Z0sin(kspΔ(p,q)),if|pq|1.

    For each gap, we present the correlation betweennand Cn as:

    g(n,Cn)1=Znsin(kspsn).

    For each cavity, we have the correlation between Cn and Cn' as:

    g(Cn,Cn')1=Znsin(kspdn').

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 lengthLi and widthwi, 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 [g(MM)]1 concerning short-circuit junction is a  (pq×pq) matrix. Junction dots are labeled by a pair of numbers(p,q),{p,q=1,2,3,j}, 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[2(p2)+2(q2)]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:

g(1,1)1=iZL1+iZw1ZL1cot(kspΔw1)Zw1cot(kspΔL1),
g(1,q)1=iZL1+iZwqZL1cot(kspΔwq)Zwqcot(kspΔL1),
g(p,q)1=iZLp+iZwqZLpcot(kspΔwq)Zwqcot(kspΔLp),
g(p,1)1=iZLp+iZw1ZLpcot(kspΔw1)Zw1cot(kspΔLp),
wherei=1.ZLj=kspLj/ωεd, and Zwj=kspwj/ωεd,(j=1,2,3,p,q) represent the impendence of the horizontal and vertical waveguides respectively. For the junctions with one semi-infinite waveguide term, i.e. the junctions on the side ways, the inverse of the surface Green’s function can be described as following style:
g(a,1)1=iZLaZw1cot(kspΔLa1)Zw1cot(kspΔLa+1)ZLacot(kspΔw1),
g(1,b)1=iZwbZL1cot(kspΔwb1)ZL1cot(kspΔwb+1)Zwbcot(kspΔL1),
g(c,q)1=iZLcZwqcot(kspΔLc1)Zwqcot(kspΔLc+1)ZLccot(kspΔwq),
g(p,d)1=iZwdZLpcot(kspΔLd1)ZLpcot(kspΔLd+1)Zwdcot(kspΔLp),
witha,b=2,3,p1, andc,d=2,3,,q1. For the inside junctions, i.e. junctions connected to four short-circuits. The inverse green function respecting its self-energy is written as
g(m,n)1=Zwncot(kspΔLm1)Zwncot(kspΔLm+1)ZLmcot(kspΔwn1)ZLmcot(kspΔwn+1),
with m=2,3,,p1, and n=2,3,,q1.It should be noticed that each junction has four parts of response items as a result of four arms composing one junction in geometry. For arbitrary two different junctions named(r1,s1)and(r2,s2), (r1,r2=1,2...,p, ands1,s2=1,2,q), their interaction is embodied by the non-diagonal elements, which are expressed as:

g[(r1,s1),(r2,s2)]1={ZL(r1)sin(kspΔL[min(s1,s2)]),ifr1=r2,and|s1s2|=1;Zw(s1)sin(kspΔL[min(r1,r2)]),ifs1=s2,and|r1r2|=1;0,else.

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, thusεd=1. The metal is chosen by silver. Its permittivity can be expressed by the well-known Drude model [40]:

εm(ω)=εωp2ω2iωγ.
Here ε=3.7, the bulk plasma frequency and the collision frequency are respectively set to ωp=1.38×1016Hz and γ=2.73×1013Hz [41]. As the simplest example schematically shown in Fig. 4(a), a single side-coupled stub structure is analyzed by the proposed model. The widths of the backbone waveguide and the side-coupled stub are denoted by h andw, respectively. The length of stub and the backbone waveguide are denoted by d andL, respectively. By applying Eq. (29), the transmissionTstub at the waveguide out port is obtained by:
Tstub=|2Z'+iZcot(ksp(d+δmetal))iZZ'cot(ksp(d+δmetal))e2ikspL|2.
The accuracy of the above expression for single stub case where L=400nm, d=300nm, h=w=50nm is shown in Fig. 4(b), where the transmission coefficient is plotted versus the free space wavelength (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 λ=1820nm is shown in Fig. 4(c). Comparing with the insert figure calculated by numerical method, we see close agreement between them.

 figure: Fig. 4

Fig. 4 (a) Schematic of single stub structure. (b) Transmission spectra for single side-coupled waveguide structure withL=400nm,d=300nm andh=w=50nm. (c) Magnetic field at the transmitted-dip wavelengthλ=1820nm . The inset shows the FDTD simulation result for comparison.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 (a) Schematic of single cavity structure. (b) Magnetic field distribution atλ=1412nmwiths=30nm. Inset image: The magnetic-field distribution by the FDTD simulation. (c) Transmission spectrums for single cavity with various gap distancess=10nm,20nm,30nm, respectively. Simulation data (red dashed), conventional model without length modification (black dashed), and the proposed method results (blue solid) are demonstrated.

Download Full Size | PPT Slide | PDF

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 distances. With Eq. (29) we get the transmission Tcavity as:

Tcavity=|4iZsin(α)sin2(β)(Z+iZ'cot(β))iZcot(α)cot(β)Z'cot(α)i(2Z2+(Z')2)[sin(α+2β)sin(α)]3ZZ'[cos(α+2β)cos(α)]e2ikspL|2,
whereα=ksp(d+δmetal),β=ksp's. In Fig. 5(b), the magnetic-field distributions are shown, and the simulation (insert figure) is compared with the result from our theory. Both of them are of perfect agreement with each other. To measure the effect of the length modification, we study the transmission as a function of the gap distancesin Fig. 5(c). The red dashed curves show the simulation data of the transmittance, while the blue block lines result from our improved analytical model. For comparison, the black dash lines show the transmission properties without length modification. It is obvious that the analytical results using the proposed method agree well with the FDTD results, particularly at the transmission-dip wavelength where the proposed length modification takes great effect.

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 dandL, respectively. Figure 6(b) gives the transmission spectra for such serial stubs structure withL=d=400nmand h=w=50nm 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.

 figure: Fig. 6

Fig. 6 (a) Schematic of MIM waveguide with four serial stubs structure. (b)Transmission spectra for serial stubs structure with L=d=400nmand h=w=50nm.

Download Full Size | PPT Slide | PDF

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).

References

1. G. Lai, R. Liang, Y. Zhang, Z. Bian, L. Yi, G. Zhan, and R. Zhao, “Double plasmonic nanodisks design for electromagnetically induced transparency and slow light,” Opt. Express 23(5), 6554–6561 (2015). [CrossRef]   [PubMed]  

2. K. Wen, Y. Hu, L. Chen, J. Zhou, L. Lei, and Z. Guo, “Design of an Optical Power and Wavelength Splitter Based on Subwavelength Waveguides,” J. Lightwave Technol. 32(17), 3020–3026 (2014). [CrossRef]  

3. A. Sellier, T. V. Teperik, and A. de Lustrac, “Resonant circuit model for efficient metamaterial absorber,” Opt. Express 21(Suppl 6), A997–A1006 (2013). [CrossRef]   [PubMed]  

4. C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016). [CrossRef]  

5. Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016). [CrossRef]   [PubMed]  

6. M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016). [CrossRef]  

7. S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014). [CrossRef]  

8. Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express 16(21), 16314–16325 (2008). [CrossRef]   [PubMed]  

9. R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014). [CrossRef]  

10. Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015). [CrossRef]  

11. M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012). [CrossRef]  

12. X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017). [CrossRef]  

13. A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015). [CrossRef]  

14. A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017). [CrossRef]  

15. S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016). [CrossRef]  

16. S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017). [CrossRef]  

17. X. Lin and X. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B 26(7), 1263–1268 (2009). [CrossRef]  

18. J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016). [CrossRef]  

19. X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016). [CrossRef]  

20. H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

21. R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

22. I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017). [CrossRef]  

23. L. Y. He, T. J. Wang, Y. P. Gao, C. Cao, and C. Wang, “Discerning electromagnetically induced transparency from Autler-Townes splitting in plasmonic waveguide and coupled resonators system,” Opt. Express 23(18), 23817–23826 (2015). [CrossRef]   [PubMed]  

24. H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012). [CrossRef]  

25. H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010). [CrossRef]  

26. S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014). [CrossRef]  

27. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009). [CrossRef]   [PubMed]  

28. F. Hu, H. Yi, and Z. Zhou, “Wavelength demultiplexing structure based on arrayed plasmonic slot cavities,” Opt. Lett. 36(8), 1500–1502 (2011). [CrossRef]   [PubMed]  

29. H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011). [CrossRef]  

30. G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19(4), 3513–3518 (2011). [CrossRef]   [PubMed]  

31. A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015). [CrossRef]  

32. S. Wu and D. Wu, “Adjusting Spectrum of MIM Optical Filters by Stub Inclination,” in Frontiers in Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTh2A.157.

33. S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016). [CrossRef]  

34. S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016). [CrossRef]  

35. S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016). [CrossRef]  

36. H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016). [CrossRef]  

37. K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, Berlin, 2008).

38. J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004). [CrossRef]  

39. Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013). [CrossRef]  

40. S. A. Maier, Plasmonics: fundamentals and applications (Springer Science & Business Media 2007).

41. J. Tao, X. G. Huang, X. Lin, Q. Zhang, and X. Jin, “A narrow-band subwavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure,” Opt. Express 17(16), 13989–13994 (2009). [CrossRef]   [PubMed]  

42. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010). [CrossRef]   [PubMed]  

References

  • View by:

  1. G. Lai, R. Liang, Y. Zhang, Z. Bian, L. Yi, G. Zhan, and R. Zhao, “Double plasmonic nanodisks design for electromagnetically induced transparency and slow light,” Opt. Express 23(5), 6554–6561 (2015).
    [Crossref] [PubMed]
  2. K. Wen, Y. Hu, L. Chen, J. Zhou, L. Lei, and Z. Guo, “Design of an Optical Power and Wavelength Splitter Based on Subwavelength Waveguides,” J. Lightwave Technol. 32(17), 3020–3026 (2014).
    [Crossref]
  3. A. Sellier, T. V. Teperik, and A. de Lustrac, “Resonant circuit model for efficient metamaterial absorber,” Opt. Express 21(Suppl 6), A997–A1006 (2013).
    [Crossref] [PubMed]
  4. C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
    [Crossref]
  5. Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
    [Crossref] [PubMed]
  6. M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
    [Crossref]
  7. S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
    [Crossref]
  8. Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express 16(21), 16314–16325 (2008).
    [Crossref] [PubMed]
  9. R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
    [Crossref]
  10. Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
    [Crossref]
  11. M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
    [Crossref]
  12. X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
    [Crossref]
  13. A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
    [Crossref]
  14. A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
    [Crossref]
  15. S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
    [Crossref]
  16. S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
    [Crossref]
  17. X. Lin and X. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B 26(7), 1263–1268 (2009).
    [Crossref]
  18. J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
    [Crossref]
  19. X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
    [Crossref]
  20. H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).
  21. R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).
  22. I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
    [Crossref]
  23. L. Y. He, T. J. Wang, Y. P. Gao, C. Cao, and C. Wang, “Discerning electromagnetically induced transparency from Autler-Townes splitting in plasmonic waveguide and coupled resonators system,” Opt. Express 23(18), 23817–23826 (2015).
    [Crossref] [PubMed]
  24. H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
    [Crossref]
  25. H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
    [Crossref]
  26. S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
    [Crossref]
  27. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009).
    [Crossref] [PubMed]
  28. F. Hu, H. Yi, and Z. Zhou, “Wavelength demultiplexing structure based on arrayed plasmonic slot cavities,” Opt. Lett. 36(8), 1500–1502 (2011).
    [Crossref] [PubMed]
  29. H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
    [Crossref]
  30. G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19(4), 3513–3518 (2011).
    [Crossref] [PubMed]
  31. A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
    [Crossref]
  32. S. Wu and D. Wu, “Adjusting Spectrum of MIM Optical Filters by Stub Inclination,” in Frontiers in Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTh2A.157.
  33. S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
    [Crossref]
  34. S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
    [Crossref]
  35. S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
    [Crossref]
  36. H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
    [Crossref]
  37. K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, Berlin, 2008).
  38. J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
    [Crossref]
  39. Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
    [Crossref]
  40. S. A. Maier, Plasmonics: fundamentals and applications (Springer Science & Business Media 2007).
  41. J. Tao, X. G. Huang, X. Lin, Q. Zhang, and X. Jin, “A narrow-band subwavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure,” Opt. Express 17(16), 13989–13994 (2009).
    [Crossref] [PubMed]
  42. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010).
    [Crossref] [PubMed]

2017 (4)

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

2016 (11)

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

2015 (5)

G. Lai, R. Liang, Y. Zhang, Z. Bian, L. Yi, G. Zhan, and R. Zhao, “Double plasmonic nanodisks design for electromagnetically induced transparency and slow light,” Opt. Express 23(5), 6554–6561 (2015).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

L. Y. He, T. J. Wang, Y. P. Gao, C. Cao, and C. Wang, “Discerning electromagnetically induced transparency from Autler-Townes splitting in plasmonic waveguide and coupled resonators system,” Opt. Express 23(18), 23817–23826 (2015).
[Crossref] [PubMed]

2014 (4)

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

K. Wen, Y. Hu, L. Chen, J. Zhou, L. Lei, and Z. Guo, “Design of an Optical Power and Wavelength Splitter Based on Subwavelength Waveguides,” J. Lightwave Technol. 32(17), 3020–3026 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

2013 (2)

A. Sellier, T. V. Teperik, and A. de Lustrac, “Resonant circuit model for efficient metamaterial absorber,” Opt. Express 21(Suppl 6), A997–A1006 (2013).
[Crossref] [PubMed]

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

2012 (2)

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

2011 (3)

2010 (2)

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (1)

2006 (1)

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

2004 (1)

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Abouti, O. E.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Agrawal, G. P.

Akjouj, A.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Bayati, M. S.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Bian, Z.

Bonyadi Ram, S.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Cao, C.

Cao, G.

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

Chen, L.

Chen, X.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Chen, Z.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

Cheng, B. H.

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Cherbi, L.

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

de Lustrac, A.

Djafarirouhani, B.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Djafari-Rouhani, B.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Dobrzynski, L.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Duan, L.

Ebadi, S. M.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

El Boudouti, E. H.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Elbialy, S.

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Elboudouti, E.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Fang, G.

Fikri, R.

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

Fukui, M.

Gao, Y. P.

Gong, Y.

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Guo, Z.

Gwon, H. R.

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

Haddouche, I.

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

Haraguchi, M.

Hattori, H. T.

He, J.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

He, L. Y.

He, Z.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Helmy, A. S.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

Hu, F.

Hu, Y.

Huang, X.

Huang, X. G.

Jamili, M.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Jin, X.

Kaatuzian, H.

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

Lai, G.

Lan, Y. C.

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Lang, P.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Lee, S. H.

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

Lei, L.

Li, B.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Li, C.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Li, G.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Li, H.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Li, X.

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

Liang, R.

Lin, X.

Liu, J.

Liu, S.

Liu, X.

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19(4), 3513–3518 (2011).
[Crossref] [PubMed]

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Lu, H.

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19(4), 3513–3518 (2011).
[Crossref] [PubMed]

Lu, W.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Mansouri-Birjandi, M. A.

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

Mao, D.

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19(4), 3513–3518 (2011).
[Crossref] [PubMed]

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Matsuzaki, Y.

Nakagaki, M.

Noual, A.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Okamoto, T.

Ouyang, M.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Pannipitiya, A.

Paul, S.

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

Pennec, Y.

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

Poursajadi, S. M.

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

Premaratne, M.

Rakhshani, M. R.

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

Ray, M.

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

Rukhlenko, I. D.

Salim, M.

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

Samra, A.

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Sellier, A.

Song, J.

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

Su, R.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Su, X.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Swillam, M. A.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

Taheri, A. N.

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

Tang, B.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Tao, J.

Teperik, T. V.

Vasseur, J.

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Vigneron, J. P.

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

Wang, C.

Wang, G.

Wang, L.

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Wang, T. J.

Wang, Y.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Wang, Z.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

Wen, K.

Wu, H. H.

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Wu, J.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Xu, H.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Yang, H.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

Yi, H.

Yi, L.

Yousif, B.

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Zafar, R.

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

Zhan, G.

Zhan, S.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

Zhang, J. X. J.

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

Zhang, Q.

Zhang, R.

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

Zhang, X.

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

Zhang, Y.

Zhao, H.

Zhao, R.

Zhao, W.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Zhao, Z.

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Zheng, M.

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Zhou, J.

Zhou, X.

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Zhou, Z.

Zhu, Q.

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

Appl. Phys. B (1)

H. Lu, X. Liu, L. Wang, D. Mao, and Y. Gong, “Nanoplasmonic triple-wavelength demultiplexers in two-dimensional metallic waveguides,” Appl. Phys. B 103(4), 877–881 (2011).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (2)

S. Paul and M. Ray, “Analysis of plasmonic subwavelength waveguide-coupled nanostub and its application in optical switching,” Appl. Phys., A Mater. Sci. Process. 122(1), 21 (2016).
[Crossref]

A. Noual, O. E. Abouti, E. H. El Boudouti, A. Akjouj, Y. Pennec, and B. Djafari-Rouhani, “Plasmonic-induced transparency in a MIM waveguide with two side-coupled cavities,” Appl. Phys., A Mater. Sci. Process. 123(1), 49 (2017).
[Crossref]

EPL (1)

Q. Zhu and Z. Wang, “The Green’s function method for metal-dielectric-metal SPP waveguide network,” EPL 103(1), 17004 (2013).
[Crossref]

IEEE Photonic. Tech. L (4)

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-Efficiency Nanoplasmonic Wavelength Filters Based on MIM Waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

S. M. Ebadi, M. S. Bayati, S. Bonyadi Ram, S. M. Poursajadi, and M. Jamili, “High-efficiency nanoplasmonic wavelength filters based on MIM waveguides,” IEEE Photonic. Tech. L 28(22), 2605–2608 (2016).
[Crossref]

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed form model,” IEEE Photonic. Tech. L 24(6), 497–499 (2012).
[Crossref]

R. Zafar and M. Salim, “Wideband Slow Surface Plasmons in Double Resonator Plasmonic Grating Waveguide,” IEEE Photonic. Tech. L 26(22), 2221–2224 (2014).
[Crossref]

J. Appl. Phys. (1)

S. Paul and M. Ray, “Plasmonic switching and bistability at telecom wavelength using the subwavelength nonlinear cavity coupled to a dielectric waveguide: A theoretical approach,” J. Appl. Phys. 120(20), 203102 (2016).
[Crossref]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

M. R. Rakhshani and M. A. Mansouri-Birjandi, “Dual wavelength demultiplexer based on metal–insulator–metal plasmonic circular ring resonators,” J. Mod. Opt. 63(11), 1078 (2016).
[Crossref]

J. Wu, P. Lang, X. Chen, and R. Zhang, “A novel optical pressure sensor based on surface plasmon polariton resonator,” J. Mod. Opt. 63(3), 219–223 (2016).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. D Appl. Phys. (1)

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Yang, “Slow light based on plasmon-induced transparency in dual-ring resonator-coupled MDM waveguide system,” J. Phys. D Appl. Phys. 47(20), 205101 (2014).
[Crossref]

Mater. Trans. (1)

H. R. Gwon and S. H. Lee, “Spectral and angular responses of surface plasmon resonance based on the Kretschmann prism configuration,” Mater. Trans. 51(6), 1150–1155 (2010).
[Crossref]

Opt. Commun. (4)

I. Haddouche and L. Cherbi, “Comparison of finite element and transfer matrix methods for numerical investigation of surface plasmon waveguides,” Opt. Commun. 382, 132–137 (2017).
[Crossref]

C. Li, R. Su, Y. Wang, and X. Zhang, “Theoretical study of ultra-wideband slow light in dual-stub-coupled plasmonic waveguide,” Opt. Commun. 377, 10–13 (2016).
[Crossref]

X. Li, J. Song, and J. X. J. Zhang, “Design of terahertz metal-dielectric-metal waveguide with microfluidic sensing stub,” Opt. Commun. 361, 130–137 (2016).
[Crossref]

X. Zhou, M. Ouyang, B. Tang, Z. Wang, and J. He, “Transparency windows of the plasmonic nanostructure composed of C-shaped and U-shaped resonators,” Opt. Commun. 384, 65–70 (2017).
[Crossref]

Opt. Express (8)

A. Sellier, T. V. Teperik, and A. de Lustrac, “Resonant circuit model for efficient metamaterial absorber,” Opt. Express 21(Suppl 6), A997–A1006 (2013).
[Crossref] [PubMed]

Y. Matsuzaki, T. Okamoto, M. Haraguchi, M. Fukui, and M. Nakagaki, “Characteristics of gap plasmon waveguide with stub structures,” Opt. Express 16(21), 16314–16325 (2008).
[Crossref] [PubMed]

L. Y. He, T. J. Wang, Y. P. Gao, C. Cao, and C. Wang, “Discerning electromagnetically induced transparency from Autler-Townes splitting in plasmonic waveguide and coupled resonators system,” Opt. Express 23(18), 23817–23826 (2015).
[Crossref] [PubMed]

G. Lai, R. Liang, Y. Zhang, Z. Bian, L. Yi, G. Zhan, and R. Zhao, “Double plasmonic nanodisks design for electromagnetically induced transparency and slow light,” Opt. Express 23(5), 6554–6561 (2015).
[Crossref] [PubMed]

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009).
[Crossref] [PubMed]

G. Wang, H. Lu, X. Liu, D. Mao, and L. Duan, “Tunable multi-channel wavelength demultiplexer based on MIM plasmonic nanodisk resonators at telecommunication regime,” Opt. Express 19(4), 3513–3518 (2011).
[Crossref] [PubMed]

J. Tao, X. G. Huang, X. Lin, Q. Zhang, and X. Jin, “A narrow-band subwavelength plasmonic waveguide filter with asymmetrical multiple-teeth-shaped structure,” Opt. Express 17(16), 13989–13994 (2009).
[Crossref] [PubMed]

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010).
[Crossref] [PubMed]

Opt. Lett. (1)

Opt. Quantum Electron. (3)

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

A. N. Taheri and H. Kaatuzian, “Numerical investigation of a nano-scale electro-plasmonic switch based on metal-insulator-metal stub filter,” Opt. Quantum Electron. 47(2), 159–168 (2015).
[Crossref]

S. Elbialy, B. Yousif, and A. Samra, “Modeling of active plasmonic coupler and filter based on metal-dielectric-metal waveguide,” Opt. Quantum Electron. 49(4), 145 (2017).
[Crossref]

Phys. Rev. A (1)

H. Lu, X. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multi-nanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012).
[Crossref]

Plasmonics (3)

S. Zhan, H. Li, G. Cao, Z. He, B. Li, and H. Xu, “Theoretical analysis of plasmon-induced transparency in ring-resonators coupled channel drop filter systems,” Plasmonics 9(6), 1431–1437 (2014).
[Crossref]

H. H. Wu, B. H. Cheng, and Y. C. Lan, “Coherent-controlled all-optical devices based on plasmonic resonant tunneling waveguides,” Plasmonics 12(6), 1–7 (2016).

Z. He, H. Li, S. Zhan, B. Li, Z. Chen, and H. Xu, “π-Network transmission line model for plasmonic waveguides with cavity structures,” Plasmonics 10(6), 1581–1585 (2015).
[Crossref]

Proc. SPIE 6343, Photonics North (1)

R. Fikri and J. P. Vigneron, “Discrete plasmonic nano-waveguide: numerical and theoretical studies,” Proc. SPIE 6343, Photonics North 2006, 63432Q (2006).

RSC Advances (1)

H. Yang, G. Li, X. Su, W. Zhao, Z. Zhao, X. Chen, and W. Lu, “A novel transmission model for plasmon-induced transparency in plasmonic waveguide system with a single resonator,” RSC Advances 6(56), 51480–51484 (2016).
[Crossref]

Sci. Rep. (1)

Z. Chen, H. Li, S. Zhan, B. Li, Z. He, H. Xu, and M. Zheng, “Tunable high quality factor in two multimode plasmonic stubs waveguide,” Sci. Rep. 6(1), 24446 (2016).
[Crossref] [PubMed]

Surf. Sci. Rep. (1)

J. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafarirouhani, and E. Elboudouti, “Photon, electron, magnon, phonon and plasmon mono-mode circuits,” Surf. Sci. Rep. 54(1), 1–156 (2004).
[Crossref]

Other (3)

S. A. Maier, Plasmonics: fundamentals and applications (Springer Science & Business Media 2007).

K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics, 2nd ed. (Springer, Berlin, 2008).

S. Wu and D. Wu, “Adjusting Spectrum of MIM Optical Filters by Stub Inclination,” in Frontiers in Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper JTh2A.157.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic illustrations of (a) infinite MIM waveguide (b) infinite metallic waveguide, and (c) semi-infinite MIM waveguide.
Fig. 2
Fig. 2 Schematic of the type of MIM waveguide networks system. (a) Input and output waveguides (b) short-circuit waveguide (c) open-circuit structure (d) unconnected side-coupled cavity.
Fig. 3
Fig. 3 Schematic diagrams of three typical constructions. (a) Arbitrary open-circuit stubs model (b) arbitrary side-couped cavities model (c) short circuit junctions’ networks.
Fig. 4
Fig. 4 (a) Schematic of single stub structure. (b) Transmission spectra for single side-coupled waveguide structure with L = 400 nm , d = 300 nm and h = w = 50 nm . (c) Magnetic field at the transmitted-dip wavelength λ = 1820 nm . The inset shows the FDTD simulation result for comparison.
Fig. 5
Fig. 5 (a) Schematic of single cavity structure. (b) Magnetic field distribution at λ = 1412 nm with s = 30 nm . Inset image: The magnetic-field distribution by the FDTD simulation. (c) Transmission spectrums for single cavity with various gap distances s = 10 nm , 20 nm , 30 nm , respectively. Simulation data (red dashed), conventional model without length modification (black dashed), and the proposed method results (blue solid) are demonstrated.
Fig. 6
Fig. 6 (a) Schematic of MIM waveguide with four serial stubs structure. (b)Transmission spectra for serial stubs structure with   L = d = 400 nm and h = w = 50 nm .

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

( 2 x 2 + 2 z 2 + k x 2 + k z 2 ) H y ( x , z ) = 0 ,
( k x 2 + 2 x 2 ) u ( x ) = 0.
Z k x ( k x 2 + 2 x 2 ) G ( x , x ' ) = δ ( x x ' ) ,
G ( x , x ' ) = e i k x | x x ' | 2 i Z .
tanh ( k sp 2 ε d k 0 2 d 2 ) = ( ε d k sp 2 ε m μ m k 0 2 ε m k sp 2 ε d μ d k 0 2 ) ,
k sp k 0 ε d 2 ε d ε d ε m k 0 ε d d .
k sp ' = k 0 ε m ,
Z ' = k sp ' d ' ω ε m .
[ g ( M M ) ] 1 = [ i Z Z cot ( k sp L ' ) Z sin ( k sp L ' ) Z sin ( k sp L ' ) i Z ' Z cot ( k sp L ' ) ] ,
g ( M M ) ( I ( M M ) + A ( M M ) ) = G ( M M ) i n D = [ D j ] ,
[ g j ( M M ) ] 1 = Δ j ( M M ) [ G j ( M M ) ] 1 ,
Δ j ( M M ) = I j ( M M ) + A j ( M M ) ,
A j ( x ' ' , x ' ) = V j ( x ) G j ( x , x ' ) d x ,
V j ( x ) = δ ( x ) n + δ ( x L j ) n ,
[ g j ( M M ) ] 1 = ( Z j cot ( k sp L j ) Z j sin ( k sp L j ) Z j sin ( k sp L j ) Z j cot ( k sp L j ) ) .
V j ( x ) = δ ( x ) n ,
[ g k ( M M ) ] 1 = ( Z k cot ( k sp L k ' ) Z k sin ( k sp L k ' ) Z k sin ( k sp L k ' ) Z k cot ( k sp L k ' ) ) ,
u ( D ) = U ( D ) U ( M ) [ G ( M M ) ] 1 G ( M D ) + U ( M ) [ G ( M M ) ] 1 g ( M M ) [ G ( M M ) ] 1 G ( M D ) .
u ( x ) = 2 i Z in sin ( α j d j ) { [ g ( 0 , x 1 ) g ( 0 , x 2 ) ] [ sin [ α j ( d j x ) ] sin ( α j x ) ] } , x | x 1 x 2 | = d j ,
u 0 1 ( x ) = u a ( x ) + u b ( x ) + u c ( x ) + u d ( x ) , ( 0 < x < d ) ,
u a ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 0 ) G 0 1 1 ( 0 , 0 ) G 0 1 ( 0 , x ) ,
u b ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 0 ) G 0 1 1 ( 0 , 1 ) G 0 1 ( 1 , x ) ,
u c ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 1 ) G 0 1 1 ( 1 , 1 ) G 0 1 ( 1 , x ) ,
u d ( x ) = U ( 0 ) G 0 1 ( 0 , 0 ) g ( 0 , 1 ) G 0 1 1 ( 1 , 0 ) G 0 1 ( 0 , x ) .
H I y ( x , z ) = u ( x ) e k z m ( z + d 2 ) ( z < d 2 ) ,
H II y ( x , z ) = u ( x ) ( 2 e k z d d 2 e k z d d + 1 cos h ( k z d z ) ) ( d 2 < z < d 2 ) ,
H III y ( x , z ) = u ( x ) e k z m ( z d 2 ) ( z > d 2 ) ,
H [ metallicWG ] y ( x , z ) = u ( x ) ( 2 e k z m d 2 e k z m d + 1 cos h ( k z m z ) ) .
T ( 1 n ) = | 2 i Z 1 g ( 1 , n ) | 2 ,
R 1 = | 1 2 i Z 1 g ( 1 , 1 ) | 2 .
T 1 , 2 , 3 n = | 2 i Z 1 A 1 e i φ 1 g ( 1 , n ) + 2 i Z 2 A 2 e i φ 2 g ( 2 , n ) + 2 i Z 3 A 3 e i φ 3 g ( 3 , n ) + | 2 ,
R k = | 1 2 i Z k A k g ( k , k ) + 2 i Z 1 A 1 e i φ 1 g ( 1 , k ) + 2 i Z 2 A 2 e i φ 2 g ( 2 , k ) + | 2 ,
g n 1 = { i Z 0 + φ right ( 1 ) + φ s ( 1 ) , n = 1 ; φ left ( j ) + φ s ( j ) + i Z 0 , n = j ; φ left ( n ) + φ right ( n ) + φ s ( n ) , n = 2 , 3 , , j 1.
g n ' 1 = i Z n ' + φ s ( n ) , n = 1 , 2 , 3 , j .
g ( p , q ) 1 = Z 0 sin ( k s p Δ ( p , q ) ) , if | p q | = 1.
g ( n , n ' ) 1 = Z n sin [ k sp ( d j + δ metal ) ] .
g n 1 = { i Z 0 + φ right ( 1 ) + φ gap ( 1 ) , n = 1 ; φ left ( j ) + φ gap ( j ) + i Z 0 , n = j ; φ left ( n ) + φ right ( n ) + φ gap ( n ) , n = 2 , 3 , , j 1.
g C n 1 = φ gap ( n ) + φ cavity ( n ) ,
g C n ' 1 = i Z n ' + φ cavity ( n ) ,
g ( p , q ) 1 = Z 0 sin ( k sp Δ ( p , q ) ) , if | p q | 1.
g ( n , C n ) 1 = Z n sin ( k sp s n ) .
g ( C n , C n ' ) 1 = Z n sin ( k sp d n ' ) .
g ( 1 , 1 ) 1 = i Z L 1 + i Z w 1 Z L 1 cot ( k sp Δ w 1 ) Z w 1 cot ( k sp Δ L 1 ) ,
g ( 1 , q ) 1 = i Z L 1 + i Z w q Z L 1 cot ( k sp Δ w q ) Z w q cot ( k sp Δ L 1 ) ,
g ( p , q ) 1 = i Z L p + i Z w q Z L p cot ( k sp Δ w q ) Z w q cot ( k sp Δ L p ) ,
g ( p , 1 ) 1 = i Z L p + i Z w 1 Z L p cot ( k sp Δ w 1 ) Z w 1 cot ( k sp Δ L p ) ,
g ( a , 1 ) 1 = i Z L a Z w 1 cot ( k sp Δ L a 1 ) Z w 1 cot ( k sp Δ L a + 1 ) Z L a cot ( k sp Δ w 1 ) ,
g ( 1 , b ) 1 = i Z w b Z L 1 cot ( k sp Δ w b 1 ) Z L 1 cot ( k sp Δ w b + 1 ) Z w b cot ( k sp Δ L 1 ) ,
g ( c , q ) 1 = i Z L c Z w q cot ( k sp Δ L c 1 ) Z w q cot ( k sp Δ L c + 1 ) Z L c cot ( k sp Δ w q ) ,
g ( p , d ) 1 = i Z w d Z L p cot ( k sp Δ L d 1 ) Z L p cot ( k sp Δ L d + 1 ) Z w d cot ( k sp Δ L p ) ,
g ( m , n ) 1 = Z w n cot ( k sp Δ L m 1 ) Z w n cot ( k sp Δ L m + 1 ) Z L m cot ( k sp Δ w n 1 ) Z L m cot ( k sp Δ w n + 1 ) ,
g [ ( r 1 , s 1 ) , ( r 2 , s 2 ) ] 1 = { Z L ( r 1 ) sin ( k sp Δ L [ min ( s 1 , s 2 ) ] ) , if r 1 = r 2 , and | s 1 s 2 | =1; Z w ( s 1 ) sin ( k sp Δ L [ min ( r 1 , r 2 ) ] ) , if s 1 = s 2 , and | r 1 r 2 | =1; 0 , else .
ε m ( ω ) = ε ω p 2 ω 2 i ω γ .
T stub = | 2 Z ' + i Z cot ( k sp ( d + δ metal ) ) i Z Z ' cot ( k sp ( d + δ metal ) ) e 2 i k sp L | 2 .
T cavity = | 4 i Z sin ( α ) sin 2 ( β ) ( Z + i Z ' cot ( β ) ) i Z cot ( α ) cot ( β ) Z ' cot ( α ) i ( 2 Z 2 + ( Z ' ) 2 ) [ sin ( α + 2 β ) sin ( α ) ] 3 Z Z ' [ cos ( α + 2 β ) cos ( α ) ] e 2 i k sp L | 2 ,

Metrics