## Abstract

We theoretically study the plasmonic modes in metal-multi-insulator-metal (MMIM) waveguides. Two types of symmetric MMIM structures consisting of three insulators are investigated thoroughly. The effective refractive index, energy confinement, propagation length, and figure of merit are given in terms of geometric parameters. Due to the step index modulation, these properties of MMIM structures differ from the metal-insulator-metal (MIM) structure. Compared with the corresponding MIM structures, MMIM structures can possess either better energy confinement or larger propagation length, which depends on the geometric parameters and the index distribution. Propagation length of up to 10^{3} µm and a figure of merit of up to 10^{4} are observed for MMIM structure with core thickness of several hundred nanometers.

© 2012 OSA

## 1. Introduction

Plasmonics, which is photonics based on surface plasmon polaritons (SPPs), has been investigated for over a decade as a promising candidate for merging photonics and electronics at nanoscale [1, 2]. SPPs, which are supported by metal surfaces, are electromagnetic waves coupled to collective oscillations of electron plasma in the metal. By coupling to SPPs, light at visible and near infrared frequencies can be routed and manipulated at nanoscale. A lot of plasmonic systems have been demonstrated and investigated to meet the requirement of both fast signal processing and miniaturization of devices. These include bends [3–6], splitters [4–6], Bragg reflectors [7–9], nanocavities [10–14], interferometers [5, 15], ring resonators [5, 9], etc. In such systems, plasmonic waveguides play an essential role for guiding and confining SPPs [16].

Various types of metallic nanostructures have been proposed for guiding SPPs [16]. Generally, plasmonic waveguides that are compatible with planar technology and suitable for on-chip integration, can be modeled into three basic types: insulator-metal (IM) [4, 9, 17, 18], insulator-metal-insulator (IMI) [8, 19–22], and metal-insulator-metal (MIM) [6, 7, 23–25]. In practice, both IMI and MIM waveguides are referred to symmetric heterostructures, where the SPPs that associated with the two metal surfaces interact with each other. Because of the geometric symmetry, the field profiles of the maintained modes resulting from the coupling effects are of either bilateral or central symmetry [26, 27].

There is a fundamental trade-off between mode confinement and propagation length. Typically, under visible or near infrared excitation, the SPPs supported by IM waveguides penetrate ~10 nm in the metals and ~10^{2} nm in the dielectrics with propagation length on the order of 10 ~10^{2} µm [17]. IMI waveguides can support the long-ranging plasmonic modes, which penetrate several microns in the dielectrics with propagation length of 10^{2} ~10^{4} µm [22]. SPPs can be squeezed in a sub-100 nm insulator core of MIM waveguide, but the propagation length is limited to several microns [10, 24]. As also theoretically discussed in [28], only MIM waveguide can provide true sub-diffraction modal confinement. The high propagation losses suffered by MIM type waveguides can be either alleviated by carefully designing of the structures [29, 30], or compensated by gain materials [31].

The performances of MIM waveguides can be enhanced by introducing step index modulation to the insulator cores [29, 30, 32–37], in which case the MIM structures change into metal-multi-insulator-metal (MMIM) structures. These structures bring more flexibility to the design of MIM type waveguides [29, 30, 32, 33], and have been proposed for efficient coupling between dielectric guides and MIM waveguides [25, 29], designing of CMOS-compatible plasmonic waveguides [30, 34, 35], nanocavities and nanolasing [36, 37]. MMIM and MIM structures are similar in some features (for instance, both of them can squeeze SPPs in deep sub-wavelength insulator cores [10, 35]); however, due to the modulation effect of the step index insulator core of MMIM structure, they differ on the field profiles, energy distributions and dispersion properties.

Previous works have concentrated on the functionalities of various MMIM-based plasmonic devices, and also have provided the properties of the operating modes [29, 30, 32–37]. The characteristics of the operating modes in the devices can be well described by the properties of the corresponding plasmonic modes in planar MMIM waveguides. In these works, the discussions for planar MMIM structures are limited by the geometric parameter ranges, and not systematic or generic enough. Thus a thorough investigation is needed to provide a complete and generalized understanding of the plasmonic modes supported by MMIM structures.

In this paper, we first derive the characteristic equation of the guided modes in a general MMIM structure. Then we focus on the plasmonic modes in two types of symmetric MMIM structures. For simplicity, we only consider MMIM with three insulator layers, which are involved in most of the MMIM-based devices (e.g [32–37].). Certainly our study can be easily extended to MMIM structures with more dielectric layers. Depending on the geometry and index distribution, MMIM structures are found to exhibit either better energy confinement or larger propagation length than the MIM structures with the same core thickness.

## 2. Theory

Figure 1(a)
illustrates an MMIM structure with three insulator slabs. The thicknesses of the three insulator slabs are denoted as *d*_{1}, *d*_{2}, and *d*_{3}, respectively. And the spatial distribution of dielectric constant is given by

*x*= 0,

*a*

_{1},

*a*

_{2}, and

*a*

_{3}, respectively. All the materials are assumed to be nonmagnetic.

The guided modes are assumed to propagate along *z* direction. We focus on the guided plasmonic modes here. Since SPPs are intrinsically of transverse magnetic polarization, the electromagnetic fields are written as

*β*,

*ω*,

*μ*

_{0}, and

*c*are the propagation constant, the angular frequency, the permeability of free space and the speed of light in vacuum, respectively. The fields of $E$ and $H$ can be obtained by solving the vector wave equation under constraint of tangential field continuity. Uniqueness of the solutions is guaranteed by the Helmholtz theorem.

The transcendental characteristic equation is given by

*N*=

_{pq}*ε*/

_{p}U_{q}*ε*,

_{q}U_{p}*T*= tanh(

_{p}*d*), and

_{p}U_{p}*U*=

_{p}*k*

_{0}(

*n*

_{eff}

^{2}−

*ε*)

_{p}^{1/2}, where

*k*

_{0}is the vacuum wavevector,

*n*

_{eff}=

*β*/

*k*

_{0}is the effective refractive index, and

*p*,

*q*= 0, 1, 2, 3, 4. The complex value of

*n*

_{eff}(hence

*β*) can be obtained by solving Eq. (3). Once

*n*

_{eff}is known, it is straightforward to derive the field components of the corresponding mode. The field components and the calculating method are given in the appendix section.

Then we only discuss two types of symmetric MMIM structures, namely MHLHM [Fig. 1(b)] and MLHLM [Fig. 1(c)], where we denote the insulator slabs of high and low refractive indices by the letters H and L, respectively. In an MHLHM (or MLHLM), the first and the third insulator slabs share the same index that is greater (or less) than the index of the second insulator slab. And both MHLHM and MLHLM here are symmetric structures (*d*_{1} = *d*_{3}). For symmetric structures, the eigenmodes are either (*E _{z}*-)anti-symmetric or (

*E*-)symmetric. Following the terminologies used in [23], we still call the plasmonic modes as bound modes.

_{z}We discuss the dependence of the modes' properties on the geometry, with fixing the operating wavelength and corresponding material parameters. We make the following assumptions for the physical parameters, unless otherwise specified: (1) the operating wavelength *λ*_{0} is 1550 nm; (2) both the metals are silver with dielectric constant given by *ε*_{Ag} = −126 + 2.9i [38]; (3) the indices of H and L slabs are given by *n*_{H} = 3.48 (silicon) and *n*_{L} = 1.44 (silica), respectively. Here we ignore the loss from the dielectrics, and only take the loss generated by the metal layers into account.

## 3. Cutoff properties and critical points of the modes

Both MHLHM and MLHLM waveguides can support anti-symmetric bound (*a _{b}*) modes and symmetric bound (

*s*) modes as well as conventional modes. As in the MIM waveguide, the

_{b}*a*modes exhibit no cutoff of the geometries, but the

_{b}*s*modes and the conventional modes cut off for structures with thin insulator cores. The cutoff properties of MHLHM and MLHLM are shown in Figs. 2(a) and 2(d), respectively. In the regions 1, 2, 3, the number of supported modes is one, two, three, respectively. The

_{b}*s*modes cut off at the blue lines and the conventional TM

_{b}_{1}modes cut off at the red lines. As the insulators become thicker, more conventional modes will emerge (not shown).

Here we examine the real part of the effective refractive index of the bound modes in terms of the geometric parameters (*d*_{1} and *d*_{2}). The bound modes are formed by the interaction of the individual SPPs associated with the two metal surfaces, whose effective refractive index is written as *n*_{sp} = [*ε*_{1}*ε*_{Ag} / (*ε*_{1} + *ε*_{Ag})]^{1/2}. For the M-H and M-L interfaces, the effective refractive indices of the individual SPPs are given by *n*_{spH} = 3.66 + 0.00452i and *n*_{spL} = 1.45 + 0.000282i, respectively. Figures 2(b) and 2(e) show Re(*n*_{eff}) as a function of *d*_{1} with *d*_{2} fixed at 0, 50, 100, 200 nm, for the MHLHM and MLHLM structures respectively. Figures 2(c) and 2(f) show Re(*n*_{eff}) as a function of *d*_{2} with *d*_{1} fixed at 0, 50, 100, 200 nm, for the MHLHM and MLHLM structures respectively. Note that when either *d*_{1} or *d*_{2} is equal to zero, the symmetric MMIM structures reduce to MIM structures.

For the *a _{b}* mode in both MHLHM and MLHLM, with fixing H layer thickness

*d*

_{H}, Re(

*n*

_{eff}) decreases monotonously with increasing L layer thickness

*d*

_{L}[solid lines in Figs. 2(c) and 2(e), respectively]. Whereas with fixing

*d*

_{L}, Re(

*n*

_{eff}) of the

*a*mode can either decrease or increase with increasing

_{b}*d*

_{H}, and approaches a limit value when

*n*

_{H}

*d*

_{H}>>

*λ*

_{0}, as shown in Figs. 2(b) and 2(f), respectively. The monotonicity depends on the value of L layer thickness. For MHLHM, when

*d*

_{L}is less (or greater) than a critical value

*d*[shown as point

_{C}*C*in Fig. 2(c)], Re(

*n*

_{eff}) decreases (or increases) with

*d*

_{H}; when

*d*

_{L}is equal to

*d*, Re(

_{C}*n*

_{eff}) is independent on

*d*

_{H}, and is exactly equal to the limit value Re(

*n*

_{spH}) [the horizontal orange line in Fig. 2(b)]. For MLHLM, when

*d*

_{L}is less (or greater) than a critical value

*d*

_{C}_{1}[shown as point

*C*

_{1}in Fig. 2(e)], Re(

*n*

_{eff}) decreases (or increases) with

*d*

_{H}; when

*d*

_{L}is equal to

*d*

_{C}_{1}, Re(

*n*

_{eff}) is independent on

*d*

_{H}, and is equal to the limit value

*n*

_{H}[the horizontal orange line in Fig. 2(f)].

For the *s _{b}* mode in MHLHM, Re(

*n*

_{eff}) increases when either

*d*

_{H}or

*d*

_{L}increases [dashed lines in Figs. 2(b) and 2(c)]. For the

*s*mode in MLHLM, with fixing

_{b}*d*

_{L}value, Re(

*n*

_{eff}) increases with increasing

*d*

_{H}[dashed lines in Fig. 2(f)]. Whereas with fixing

*d*

_{H}, Re(

*n*

_{eff}) can either increase or decrease with increasing

*d*

_{L}[dashed lines in Fig. 2(e)]. The monotonicity of Re(

*n*

_{eff}) as a function of

*d*

_{L}for the

*s*mode in MLHLM depends on the H layer thickness

_{b}*d*

_{H}. When

*d*

_{H}is less (or greater) than a critical value

*d*

_{C}_{2}[shown as point

*C*

_{2}in Fig. 2(f)], Re(

*n*

_{eff}) increases (or decreases) with increasing

*d*

_{L}. When

*d*

_{H}is equal to

*d*

_{C}_{2}, Re(

*n*

_{eff}) remains a constant with

*d*

_{L}, and is equal to the limit value Re(

*n*

_{spL}) [the dashed orange line in Fig. 2(e)].

The existence of the critical points *C*, *C*_{1}, and *C*_{2} can be explained by the monotonic and asymptotic properties of Re(*n*_{eff}). For the *a _{b}* mode in MHLHM (or MLHLM), since Re(

*n*

_{eff}) always approaches the limit value Re(

*n*

_{spH}) (or

*n*

_{H}) monotonously as

*d*

_{H}increases to infinity, Re(

*n*

_{eff}) remains unchanged if the condition Re(

*n*

_{eff}) = Re(

*n*

_{spH}) (or Re(

*n*

_{eff}) =

*n*

_{H}) is satisfied at

*d*

_{H}= 0 (M-L-M structure). For the

*s*mode in MLHLM, since Re(

_{b}*n*

_{eff}) always approaches the limit value Re(

*n*

_{spL}) monotonously as

*d*

_{L}increases to infinity, Re(

*n*

_{eff}) remains a constant if the condition Re(

*n*

_{eff}) = Re(

*n*

_{spL}) is satisfied at

*d*

_{L}= 0 (M-H-M structure).

The critical values are determined by the material parameters, and therefore wavelength-dependent. For the MIM structures (here M-L-M and M-H-M), Re(*n*_{eff}) of the *a _{b}* (or

*s*) modes decreases (or increases) with increasing insulator thicknesses and increases with increasing insulator indices. The effective refractive indices of the individual SPPs,

_{b}*n*

_{spH}and

*n*

_{spL}, also increase with increasing insulator indices,

*n*

_{H}and

*n*

_{L}. As a result, when we decrease

*n*

_{H}or increase

*n*

_{L}, all the critical values

*d*,

_{C}*d*

_{C}_{1}, and

*d*

_{C}_{2}increase, so that the conditions of the critical points can be still satisfied (Fig. 3 ).

As shown in Fig. 3, the critical values *d _{C}*,

*d*

_{C}_{1}, and

*d*

_{C}_{2}increase sharply as

*n*

_{L}and

*n*

_{H}approach each other. The critical value

*d*exists in all MHLHM structures, and

_{C}*d*

_{C}_{2}exists in all MLHLM structures. When

*n*

_{L}and

*n*

_{H}are eventually equal, and the MMIM structures become MIM structures, the critical values

*d*and

_{C}*d*

_{C}_{2}go to infinity. We note that these observations are consistent with the properties of the modes in the MIM structures [the black lines in Figs. 2(b) and 2(e)]. For the bound modes in MIM, Re(

*n*

_{eff}) approaches Re(

*n*

_{sp}) with increasing insulator thickness. But Re(

*n*

_{eff}) is always greater (or less) than Re(

*n*

_{sp}) for the

*a*(or

_{b}*s*) mode. Thus for both the bound modes in MIM, the relation Re(

_{b}*n*

_{eff}) = Re(

*n*

_{sp}) can be satisfied only if the insulator becomes sufficiently thick, so that the two individual SPPs decouple. So, mathematically, the critical values

*d*and

_{C}*d*

_{C}_{2}for MIM structures are infinity.

The critical value *d _{C}*

_{1}only exists in MLHLM structures with

*n*

_{L}and

*n*

_{H}not very close to each other. We consider an MLHLM structure with

*d*

_{H}= 0 (M-L-M structure). For the

*a*mode of the M-L-M structure, Re(

_{b}*n*

_{eff}) is always greater than Re(

*n*

_{spL}). So, when Re(

*n*

_{spL}) ≥

*n*

_{H}, the relation Re(

*n*

_{eff}) =

*n*

_{H}is never satisfied. For the case shown in Fig. 3(a), as

*n*

_{L}approaches

*n*

_{H}, which is fixed at 3.48, Re(

*n*

_{spL}) increases. When

*n*

_{L}=

*n*

_{Lc}= 3.32 (the red dashed line), Re(

*n*

_{spL}) of the M-L-M structure is equal to

*n*

_{H}, and hence the critical point

*C*

_{1}is no longer supported. Through similar analysis, for MLHLM with

*n*

_{L}fixed at 1.44 [Fig. 3(b)], when

*n*

_{H}≤ Re(

*n*

_{spL}) = 1.452, the condition Re(

*n*

_{eff}) =

*n*

_{H}is never satisfied.

*d*

_{C}_{1}exists only in the region

*n*

_{H}> Re(

*n*

_{spL}) = 1.452. Note that the asymptote

*n*

_{H}= 1.452 is very close to the line

*n*

_{H}=

*n*

_{L}= 1.44, so we do not plot it in Fig. 3(b). When

*n*

_{L}=

*n*

_{Lc}= 3.32 with

*n*

_{H}fixed at 3.48 [Fig. 3(a)] or

*n*

_{H}= 1.452 with

*n*

_{L}fixed at 1.44 [Fig. 3(b)], the condition Re(

*n*

_{eff}) =

*n*

_{H}can be satisfied only if the two individual SPPs decouple, so that Re(

*n*

_{eff}) = Re(

*n*

_{spL}). So, the

*d*

_{C}_{1}value is mathematically infinity at the asymptotes.

We can gain an insight of the critical points from the properties of MIM type waveguide. For the *a _{b}* modes supported in the MMIM structures, as the H layer thickness increases, the mean refractive index of the insulator core increases, which leads to increasing of Re(

*n*

_{eff}), while the total thickness of the insulator core increases as well, which brings decreasing of Re(

*n*

_{eff}). And the two reverse mechanisms can balance at a particular L layer thickness (

*d*for MHLHM or

_{C}*d*

_{C}_{1}for MLHLM). For the

*s*modes in MLHLM, when the L layer thickness increases, the mean refractive index of insulator core decreases, which causes decreasing of Re(

_{b}*n*

_{eff}), while the total thickness of the insulators increases, which causes increasing of Re(

*n*

_{eff}). The two reverse mechanisms can balance at a particular H layer thickness,

*d*

_{C}_{2}. However, the same balance cannot be found by the

*s*modes of MHLHM. Because the energy of the bound modes in MHLHM are always bound at the metal surfaces, and affected much less by the middle insulator thickness,

_{b}*d*

_{L}, than the H layer thickness

*d*

_{H}.

We note that the limit values of the bound modes in MHLHM and MLHLM are attributed to the corresponding energy confinement behaviors, which are discussed in the next section.

## 4. Field profiles and energy confinement behaviors

The electric field component *E _{z}* is directly relevant to drive the plasma oscillations. Typical

*E*profiles of the bound modes are shown in Figs. 4(a) -4(c) for the MHLHM and MLHLM structures. The field profiles of an MIM structure are also plotted in Fig. 4(a) (black lines) for comparison.

_{z}Unlike the MIM structure, as a result of the step refractive index modulation, Re(*E _{z}*) of the bound modes in MMIM have inflection points at the interfaces of H and L layers. For the

*a*modes in both MHLHM and MLHLM, the slope of Re(

_{b}*E*) can be either positive or negative within the H layers. Especially for MLHLM, the slope is determined by the L layer thickness. As shown in Fig. 4(c), when

_{z}*d*

_{L}is less (or greater) than

*d*

_{C}_{1}(4.77 nm), the slope in the H layer is negative (or positive). When

*d*

_{L}=

*d*

_{C}_{1}, Re(

*E*) is zero in the H layer, and moreover the profiles of

_{z}*E*and

_{x}*H*are flat in the H layer (not shown). Namely the

_{y}*a*mode exhibits transverse electromagnetic behavior in the H layer. Meanwhile, the

_{b}*E*field in the metal and L layers coincides exactly with that of the M-L-M structure with core thickness equal to 2

_{z}*d*

_{C}_{1}. Consequently, when

*d*

_{L}=

*d*

_{C}_{1}, all fractions of the field of the

*a*mode experience the same effective index

_{b}*n*

_{H}. Hence, Re(

*n*

_{eff}) of the

*a*mode is equal to

_{b}*n*

_{H}at the critical point

*C*

_{1}.

Then we examine the energy confinement behaviors of the bound modes. The energy density is defined by [39]

*W*is discontinuous at the boundaries as a result of the discontinuity of the normal electric field. For both the

*a*and

_{b}*s*modes in MHLHM, and the

_{b}*a*mode in MLHLM, we find that the energies in each layers of MMIM vary significantly with the H layer thickness,

_{b}*d*

_{H}, however slowly with the L layer thickness,

*d*

_{L}. In contrast, for the

*s*mode in MLHLM, the energies in each layers vary significantly with

_{b}*d*

_{L}, but slowly with

*d*

_{H}. The energy proportions of each layers of MHLHM and MLHLM with respect to the sensitive geometric parameters are shown in Figs. 5(a) -5(c).

For the bound modes in MHLHM [Fig. 5(a)], the energy is dominant in the L layer when *d*_{H} is small. As *d*_{H} increases, the coupling strength of the two individual SPPs decreases and more energy is confined to the metal surfaces, so the energy proportion decreases in the middle insulator (L), while increases in the H layers. When *d*_{H} is sufficiently large, the two individual SPPs tend to decouple, and the energy proportion in the L layer approaches zero. Hence, when *d*_{H} is sufficiently large, the limit value of Re(*n*_{eff}) as a function of *d*_{L} is equal to the effective refractive index of the individual SPP that associated with the M-H interface, Re(*n*_{spH}) [Fig. 2(b)].

The energy confinement behaviors differ greatly between the *a _{b}* and

*s*modes in MLHLM. For the

_{b}*a*mode in MLHLM [Fig. 5(b)], when

_{b}*d*

_{H}is small, the energy is dominant in the L layers. As

*d*

_{H}increases, the energy proportion decreases in the L layers, while increases in the H layer. When

*d*

_{H}is sufficiently large, the energy proportion in the L and metal layers approaches zero. Thus the energy is dominant in the middle layer and the

*a*mode is less affected by the metal claddings. Hence, when

_{b}*d*

_{H}is sufficiently large, the limit value of Re(

*n*

_{eff}) of the

*a*mode as a function of

_{b}*d*

_{L}is equal to the refractive index of the H layer,

*n*

_{H}[Fig. 2(f)].

For the *s _{b}* mode in MLHLM [Fig. 5(c)], when

*d*

_{L}is small, the energy is dominant in the H layer. As

*d*

_{L}increases, more energy is confined to the metal surfaces. The energy proportion decreases in the H layer, and eventually approaches zero. Hence, when

*d*

_{L}is sufficiently large, the limit value of Re(

*n*

_{eff}) as a function of

*d*

_{L}is equal to the effective refractive index of the individual SPP that associated with the M-L interface, Re(

*n*

_{spL}) [Fig. 2(e)].

We define an effective mode width to measure the energy confinement. The effective mode width is given by [39]

where*W*

_{max}is the maximum energy density in the core (0 <

*x*<

*a*

_{3}). Figures 6(a) -6(d) show

*A*

_{eff}of the

*a*and

_{b}*s*modes in MHLHM and MLHLM.

_{b}The maximum energy density in the core can appear at different positions. The regions 1, 2, …, 5 bounded by the gray lines in Figs. 6(a)-6(c) are divided according to the positions of *W*_{max} of the *a _{b}* and

*s*modes, and correspond to the typical energy density profiles shown in Figs. 6(e)-6(i), respectively.

_{b}For MIM structure, the energy confinement is improved when the core refractive index increases. However, for MHLHM, the bound modes can present better energy confinement than the M-H-M structure with the same core thickness (2*d*_{1} + *d*_{2}), albeit the mean refractive index of the core of MHLHM is less than *n*_{H}. For example, for the *a _{b}* mode of MHLHM with

*d*

_{1}=

*d*

_{2}= 120 nm,

*A*

_{eff}is equal to 171 nm, which is ~0.1

*λ*

_{0}and less than 80% of

*A*

_{eff}of the

*a*mode in the M-H-M structure with core thickness equal to 360 nm. The

_{b}*s*mode presents slightly better confinement than the

_{b}*a*mode in the same MHLHM structure. For the

_{b}*s*mode of MHLHM with

_{b}*d*

_{1}=

*d*

_{2}= 120 nm,

*A*

_{eff}is equal to 169 nm. In these cases, the bound modes are confined to the metal surfaces (

*x*= 0

^{+}and

*a*

_{3}

^{−}).

Light can also be enhanced and confined in the middle insulator (L) of MHLHM by utilizing the *a _{b}* mode [Fig. 6(e), and region 1 in Fig. 6(a)]. Similar phenomenon has been found in a type of slot waveguide [40], which contains a low-index slot embedded between two high-index slabs. The mode in the slot waveguide is formed by the interaction between the fundamental eigenmodes of the individual slabs and the mechanism is based on total internal reflection [40]. The MHLHM structure can be considered as the slot waveguide sandwiched between the two metal claddings. However the field confinement in the L layer of MHLHM is due to the interaction between the two individual SPPs. Moreover, for practical applications, as a result of the screening of the metal layers, the integration density of MHLHM-based devices can be significantly improved compared with the slot waveguide.

Compared with MHLHM, the bound modes in MLHLM present relatively low energy confinement, which is due to the smaller mean refractive index of core as well as the energy distribution. The effective mode width *A*_{eff} of the *a _{b}* mode of MLHLM can even be greater than that of the M-L-M structure with the same core thickness [bottom right region bounded by the white lines in Fig. 6(c)], albeit the mean refractive index of MLHLM is greater than

*n*

_{L}. For the

*a*mode in MLHLM, high energy confinement is only found at small

_{b}*d*

_{L}, in which case the energy is highly confined in the low-index layers between metal and H layers. For the

*s*mode in MLHLM, high energy confinement is observed at small

_{b}*d*

_{L}as well as the region near the cutoff line [Fig. 6(d)].

We would like to compare the MLHLM structure with a type of hybrid plasmonic waveguide [39, 41] consisting of low-index material sandwiched between metal and high-index material. The hybrid plasmonic waveguide can be considered as a half of the MLHLM structure and can support the so called hybrid mode [39]. Like the *a _{b}* mode, the hybrid mode can exhibit strong confinement of energy within the low index region. Meanwhile, the energy distribution of the hybrid mode in the low index region is analogous to the

*a*mode in MLHLM. We can consider that the

_{b}*a*mode in MLHLM is formed by the interaction between the two hybrid modes associated with the two halves of the MLHLM structure.

_{b}## 5. Propagation length and figure of merit

The propagation length *L _{p}* is defined as the 1 /

*e*decay length of the energy, and is given by

*L*= 1 / 2Im(

_{p}*β*).

*L*of the

_{p}*a*and

_{b}*s*modes supported by the MHLHM and MLHLM structures is shown in Fig. 7 , in terms of

_{b}*d*

_{1}and

*d*

_{2}. For the bound modes of MIM structure,

*L*increases with increasing insulator thickness and decreasing insulator index. For MMIM,

_{p}*L*is additionally affected by distribution of core index. In order to make clear the effect of the contrasted core index, we compare

_{p}*L*of MHLHM and MLHLM with that of the M-H-M and M-L-M structures, which is calculated from Eq. (3) and not given here for simplicity. In this section, the insulator thicknesses of both M-H-M and M-L-M are assumed to be 2

_{p}*d*

_{1}+

*d*

_{2}.

*L _{p}* of the

*a*mode of MHLHM is shown in Fig. 7(a). At the left side of the white line,

_{b}*L*of the

_{p}*a*mode of MHLHM exceeds that of the M-H-M structure. In contrast, at the right side,

_{b}*L*of the

_{p}*a*mode of MHLHM is less than that of the M-H-M structure, albeit the mean insulator index of MHLHM is less than the M-H-M structure.

_{b}For the *a _{b}* mode of MLHLM, as shown in Fig. 7(c), at the left side of the white line,

*L*of the

_{p}*a*mode of MLHLM is comparable with but less than that of the M-L-M structure. When

_{b}*d*

_{1}is large enough, the

*a*mode is less affected by the metal claddings. At the right side,

_{b}*L*of the

_{p}*a*mode of MLHLM is greater than that of the M-L-M structure and increases exponentially with increasing

_{b}*d*

_{1}or

*d*

_{2}. As an example, when (

*d*

_{1},

*d*

_{2}) = (300, 250) nm,

*L*is as high as 4719 µm, which is over 20 times of

_{p}*L*of the

_{p}*a*mode of the M-L-M structure.

_{b}High losses are observed for the *s _{b}* modes. For both MHLHM and MLHLM, the propagation length of the

*s*modes increases with the insulator thickness. Within our parameter range, the maxima of

_{b}*L*of the

_{p}*s*mode of MHLHM and MLHLM are 17 µm and 182 µm, respectively, and they are found at the upper right corners of Figs. 7(b) and 7(d), respectively.

_{b}*L*of the

_{p}*s*mode never exceeds that of the

_{b}*a*mode that supported by the same structure. As

_{b}*d*

_{1}or

*d*

_{2}increase,

*L*of the

_{p}*a*and

_{b}*s*modes approach each other for the MHLHM structure, however keep apart from each other for the MLHLM structure.

_{b}The real part of the effective index of the *a _{b}* mode is greater than that of the

*s*mode (Fig. 2), so the phase velocity of the

_{b}*a*mode is less than that of the

_{b}*s*mode. Whereas high loss of the

_{b}*s*mode compared with the

_{b}*a*mode indicates that the group velocity of the

_{b}*a*mode is greater than that of the

_{b}*s*mode.

_{b}As shown in Figs. 6 and 7, there is a trade-off between mode confinement and propagation length. We define a figure of merit (FOM) to give a comprehensive evaluation of the MMIM structures. The FOM is defined as the ratio of propagation length to effective mode width (FOM = *L _{p}* /

*A*

_{eff}). Figures 8(a) -8(d) show FOM of the

*a*and

_{b}*s*modes in the MHLHM and MLHLM structures in terms of

_{b}*d*

_{1}and

*d*

_{2}.

For the *a _{b}* mode in MHLHM [Fig. 8(a)], largely, FOM decreases as the H layer thickness increases. For the

*a*mode in MLHLM [Fig. 8(c)], FOM increases exponentially as core thickness increases. Extremely high FOM (~10

_{b}^{4}) can be achieved by the

*a*mode of MLHLM with core thickness of several hundred nanometers. For example, when (

_{b}*d*

_{1},

*d*

_{2}) = (300, 250) nm, FOM is as high as 2.16 × 10

^{4}, which is 66 times of FOM of the

*a*mode in the M-L-M structure.

_{b}For the *s _{b}* mode in MHLHM [Fig. 8(b)], FOM increases with increasing core thickness. For the

*s*mode in MLHLM [Fig. 8(d)], when the geometric parameters are far away from the cutoff line [region 5 in Fig. 6(d)], FOM increases with increasing core thickness. However, when the geometric parameters are near the cutoff line [region 4 in Fig. 6(d)], generally, with fixing

_{b}*d*

_{1}value, FOM increases and reaches a maximum then decreases as

*d*

_{2}increases. The maximum also increases as

*d*

_{1}increases. And when

*d*

_{1}is large enough (greater than ~300 nm), the maximum appears at

*d*

_{2}= 0, hence the

*s*mode of the M-L-M structure with core thickness equal to 2

_{b}*d*

_{1}works better than that of MLHLM with geometric parameters near the cutoff line. It is noted that FOM of the

*s*modes in both MHLHM and MLHLM is less than that of the

_{b}*a*modes supported by the same structures.

_{b}## 6. Conclusions

In conclusion, the optical properties of the guided modes in MMIM waveguide have been discussed thoroughly. The characteristic equation of the MMIM structure has been derived. With fixing the operating wavelength and the corresponding material parameters, the cutoff property, effective refractive index, energy confinement behavior, propagation length, and figure of merit have been presented with respect to the geometric parameters for two symmetric MMIM structures, MHLHM and MLHLM.

The plasmonic modes supported by MMIM exhibit some unique properties that could not be observed in the MIM structures. In particular, for the *a _{b}* mode in MHLHM (or MLHLM), Re(

*n*

_{eff}) can be independent on the H layer thickness at a critical L layer thickness

*d*(or

_{C}*d*

_{C}_{1}). And for the

*s*mode in MLHLM, Re(

_{b}*n*

_{eff}) can be independent on the L layer thickness at a critical H layer thickness

*d*

_{C}_{2}. The existence conditions of the critical values are given according to the monotonic and asymptotic properties of Re(

*n*

_{eff}). The critical values

*d*and

_{C}*d*

_{C}_{2}exist for all the material parameters

*n*

_{L}and

*n*

_{H}. In contrast, the critical value

*d*

_{C}_{1}exists only when

*n*

_{H}is greater than the effective index of the individual SPPs associated with the M-L interface. These phenomena are explained by the properties of the plasmonic modes in MIM waveguide.

The MHLHM structure shows better energy confinement than the MLHLM structure, yet suffers relatively high loss compared with the MLHLM structure. For the *a _{b}* mode of the MLHLM structure, since the field is less affected by the metal claddings when the

*d*

_{L}is large, long propagation length of up to millimeters can be observed for core thickness of several hundred nanometers. Overall, the figure of merit of the

*a*modes is higher than that of the

_{b}*s*modes supported by the same structures; and the figure of merit of the MLHLM structure is higher than that of the MHLHM structure. The

_{b}*a*mode in the MLHLM structure is also superior to that in the MIM structure in terms of propagation length and the figure of merit. Therefore, the

_{b}*a*mode in the MLHLM structure is of the most applicability.

_{b}These results provide physical insights into the guided plasmonic modes in the MMIM structures, and are useful for flexible designing of MIM type waveguides and MIM-based devices.

## Appendix: field components and calculating method

The nonzero field components are written as follows:

*S*= sinh(

_{pq}*a*),

_{p}U_{q}*C*= cosh(

_{pq}*a*), with

_{p}U_{q}*p*,

*q*= 1, 2, 3.

In this work, the complex transcendental characteristic equation is solved by an iterative algorithm. Taking the right side of Eq. (3) equal to *f*, the iteration formula is written as

*f*= Re{

_{r}*f*},

*f*= Im{

_{i}*f*},

*n*= Re(

_{r}*n*

_{eff}),

*n*= Im(

_{i}*n*

_{eff}), and

*p*= 1, 2, ... is the iteration number. The initial value

*n*

_{i}^{(0)}is set to zero in the iterative procedure. In our work, the residues of Re{

*f*} and Im{

*f*} are controlled to be less than 10

^{−14}and 10

^{−16}, respectively, except for the cases that the modes nearly cut off (when a mode cuts off, Im(

*n*

_{eff}) approaches infinity, so strictly no solution exists at a cutoff point).

## Acknowledgments

This research is supported by the Chinese National Key Basic Research Special Fund (2011CB922003), the National Natural Science Foundation of China (11074132), the Fundamental Research Funds for the Central Universities, Tianjin Natural Science Foundation (09JCYBJC01600), and the 111 Project (B07013).

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