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 103 µm and a figure of merit of up to 104 are observed for MMIM structure with core thickness of several hundred nanometers.
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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 .
Various types of metallic nanostructures have been proposed for guiding SPPs . 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 ~102 nm in the dielectrics with propagation length on the order of 10 ~102 µm . IMI waveguides can support the long-ranging plasmonic modes, which penetrate several microns in the dielectrics with propagation length of 102 ~104 µm . 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 , 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 .
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.
Figure 1(a) illustrates an MMIM structure with three insulator slabs. The thicknesses of the three insulator slabs are denoted as d1, d2, and d3, respectively. And the spatial distribution of dielectric constant is given by
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
The transcendental characteristic equation is given byEq. (3). Once neff 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 (d1 = d3). For symmetric structures, the eigenmodes are either (Ez-)anti-symmetric or (Ez-)symmetric. Following the terminologies used in , we still call the plasmonic modes as bound modes.
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 λ0 is 1550 nm; (2) both the metals are silver with dielectric constant given by εAg = −126 + 2.9i ; (3) the indices of H and L slabs are given by nH = 3.48 (silicon) and nL = 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 (ab) modes and symmetric bound (sb) modes as well as conventional modes. As in the MIM waveguide, the ab modes exhibit no cutoff of the geometries, but the sb 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 sb modes cut off at the blue lines and the conventional TM1 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 (d1 and d2). 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 nsp = [ε1εAg / (ε1 + εAg)]1/2. For the M-H and M-L interfaces, the effective refractive indices of the individual SPPs are given by nspH = 3.66 + 0.00452i and nspL = 1.45 + 0.000282i, respectively. Figures 2(b) and 2(e) show Re(neff) as a function of d1 with d2 fixed at 0, 50, 100, 200 nm, for the MHLHM and MLHLM structures respectively. Figures 2(c) and 2(f) show Re(neff) as a function of d2 with d1 fixed at 0, 50, 100, 200 nm, for the MHLHM and MLHLM structures respectively. Note that when either d1 or d2 is equal to zero, the symmetric MMIM structures reduce to MIM structures.
For the ab mode in both MHLHM and MLHLM, with fixing H layer thickness dH, Re(neff) decreases monotonously with increasing L layer thickness dL [solid lines in Figs. 2(c) and 2(e), respectively]. Whereas with fixing dL, Re(neff) of the ab mode can either decrease or increase with increasing dH, and approaches a limit value when nHdH >> λ0, as shown in Figs. 2(b) and 2(f), respectively. The monotonicity depends on the value of L layer thickness. For MHLHM, when dL is less (or greater) than a critical value dC [shown as point C in Fig. 2(c)], Re(neff) decreases (or increases) with dH; when dL is equal to dC, Re(neff) is independent on dH, and is exactly equal to the limit value Re(nspH) [the horizontal orange line in Fig. 2(b)]. For MLHLM, when dL is less (or greater) than a critical value dC1 [shown as point C1 in Fig. 2(e)], Re(neff) decreases (or increases) with dH; when dL is equal to dC1, Re(neff) is independent on dH, and is equal to the limit value nH [the horizontal orange line in Fig. 2(f)].
For the sb mode in MHLHM, Re(neff) increases when either dH or dL increases [dashed lines in Figs. 2(b) and 2(c)]. For the sb mode in MLHLM, with fixing dL value, Re(neff) increases with increasing dH [dashed lines in Fig. 2(f)]. Whereas with fixing dH, Re(neff) can either increase or decrease with increasing dL [dashed lines in Fig. 2(e)]. The monotonicity of Re(neff) as a function of dL for the sb mode in MLHLM depends on the H layer thickness dH. When dH is less (or greater) than a critical value dC2 [shown as point C2 in Fig. 2(f)], Re(neff) increases (or decreases) with increasing dL. When dH is equal to dC2, Re(neff) remains a constant with dL, and is equal to the limit value Re(nspL) [the dashed orange line in Fig. 2(e)].
The existence of the critical points C, C1, and C2 can be explained by the monotonic and asymptotic properties of Re(neff). For the ab mode in MHLHM (or MLHLM), since Re(neff) always approaches the limit value Re(nspH) (or nH) monotonously as dH increases to infinity, Re(neff) remains unchanged if the condition Re(neff) = Re(nspH) (or Re(neff) = nH) is satisfied at dH = 0 (M-L-M structure). For the sb mode in MLHLM, since Re(neff) always approaches the limit value Re(nspL) monotonously as dL increases to infinity, Re(neff) remains a constant if the condition Re(neff) = Re(nspL) is satisfied at dL = 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(neff) of the ab (or sb) modes decreases (or increases) with increasing insulator thicknesses and increases with increasing insulator indices. The effective refractive indices of the individual SPPs, nspH and nspL, also increase with increasing insulator indices, nH and nL. As a result, when we decrease nH or increase nL, all the critical values dC, dC1, and dC2 increase, so that the conditions of the critical points can be still satisfied (Fig. 3 ).
As shown in Fig. 3, the critical values dC, dC1, and dC2 increase sharply as nL and nH approach each other. The critical value dC exists in all MHLHM structures, and dC2 exists in all MLHLM structures. When nL and nH are eventually equal, and the MMIM structures become MIM structures, the critical values dC and dC2 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(neff) approaches Re(nsp) with increasing insulator thickness. But Re(neff) is always greater (or less) than Re(nsp) for the ab (or sb) mode. Thus for both the bound modes in MIM, the relation Re(neff) = Re(nsp) can be satisfied only if the insulator becomes sufficiently thick, so that the two individual SPPs decouple. So, mathematically, the critical values dC and dC2 for MIM structures are infinity.
The critical value dC1 only exists in MLHLM structures with nL and nH not very close to each other. We consider an MLHLM structure with dH = 0 (M-L-M structure). For the ab mode of the M-L-M structure, Re(neff) is always greater than Re(nspL). So, when Re(nspL) ≥ nH, the relation Re(neff) = nH is never satisfied. For the case shown in Fig. 3(a), as nL approaches nH, which is fixed at 3.48, Re(nspL) increases. When nL = nLc = 3.32 (the red dashed line), Re(nspL) of the M-L-M structure is equal to nH, and hence the critical point C1 is no longer supported. Through similar analysis, for MLHLM with nL fixed at 1.44 [Fig. 3(b)], when nH ≤ Re(nspL) = 1.452, the condition Re(neff) = nH is never satisfied. dC1 exists only in the region nH > Re(nspL) = 1.452. Note that the asymptote nH = 1.452 is very close to the line nH = nL = 1.44, so we do not plot it in Fig. 3(b). When nL = nLc = 3.32 with nH fixed at 3.48 [Fig. 3(a)] or nH = 1.452 with nL fixed at 1.44 [Fig. 3(b)], the condition Re(neff) = nH can be satisfied only if the two individual SPPs decouple, so that Re(neff) = Re(nspL). So, the dC1 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 ab 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(neff), while the total thickness of the insulator core increases as well, which brings decreasing of Re(neff). And the two reverse mechanisms can balance at a particular L layer thickness (dC for MHLHM or dC1 for MLHLM). For the sb modes in MLHLM, when the L layer thickness increases, the mean refractive index of insulator core decreases, which causes decreasing of Re(neff), while the total thickness of the insulators increases, which causes increasing of Re(neff). The two reverse mechanisms can balance at a particular H layer thickness, dC2. However, the same balance cannot be found by the sb 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, dL, than the H layer thickness dH.
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 Ez is directly relevant to drive the plasma oscillations. Typical Ez 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.
Unlike the MIM structure, as a result of the step refractive index modulation, Re(Ez) of the bound modes in MMIM have inflection points at the interfaces of H and L layers. For the ab modes in both MHLHM and MLHLM, the slope of Re(Ez) 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 dL is less (or greater) than dC1 (4.77 nm), the slope in the H layer is negative (or positive). When dL = dC1, Re(Ez) is zero in the H layer, and moreover the profiles of Ex and Hy are flat in the H layer (not shown). Namely the ab mode exhibits transverse electromagnetic behavior in the H layer. Meanwhile, the Ez field in the metal and L layers coincides exactly with that of the M-L-M structure with core thickness equal to 2dC1. Consequently, when dL = dC1, all fractions of the field of the ab mode experience the same effective index nH. Hence, Re(neff) of the ab mode is equal to nH at the critical point C1.
Then we examine the energy confinement behaviors of the bound modes. The energy density is defined by Figs. 5(a) -5(c).
For the bound modes in MHLHM [Fig. 5(a)], the energy is dominant in the L layer when dH is small. As dH 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 dH is sufficiently large, the two individual SPPs tend to decouple, and the energy proportion in the L layer approaches zero. Hence, when dH is sufficiently large, the limit value of Re(neff) as a function of dL is equal to the effective refractive index of the individual SPP that associated with the M-H interface, Re(nspH) [Fig. 2(b)].
The energy confinement behaviors differ greatly between the ab and sb modes in MLHLM. For the ab mode in MLHLM [Fig. 5(b)], when dH is small, the energy is dominant in the L layers. As dH increases, the energy proportion decreases in the L layers, while increases in the H layer. When dH 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 ab mode is less affected by the metal claddings. Hence, when dH is sufficiently large, the limit value of Re(neff) of the ab mode as a function of dL is equal to the refractive index of the H layer, nH [Fig. 2(f)].
For the sb mode in MLHLM [Fig. 5(c)], when dL is small, the energy is dominant in the H layer. As dL increases, more energy is confined to the metal surfaces. The energy proportion decreases in the H layer, and eventually approaches zero. Hence, when dL is sufficiently large, the limit value of Re(neff) as a function of dL is equal to the effective refractive index of the individual SPP that associated with the M-L interface, Re(nspL) [Fig. 2(e)].
We define an effective mode width to measure the energy confinement. The effective mode width is given by Figures 6(a) -6(d) show Aeff of the ab and sb modes in MHLHM and MLHLM.
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 Wmax of the ab and sb modes, and correspond to the typical energy density profiles shown in Figs. 6(e)-6(i), respectively.
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 (2d1 + d2), albeit the mean refractive index of the core of MHLHM is less than nH. For example, for the ab mode of MHLHM with d1 = d2 = 120 nm, Aeff is equal to 171 nm, which is ~0.1λ0 and less than 80% of Aeff of the ab mode in the M-H-M structure with core thickness equal to 360 nm. The sb mode presents slightly better confinement than the ab mode in the same MHLHM structure. For the sb mode of MHLHM with d1 = d2 = 120 nm, Aeff is equal to 169 nm. In these cases, the bound modes are confined to the metal surfaces (x = 0+ and a3−).
Light can also be enhanced and confined in the middle insulator (L) of MHLHM by utilizing the ab mode [Fig. 6(e), and region 1 in Fig. 6(a)]. Similar phenomenon has been found in a type of slot waveguide , 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 . 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 Aeff of the ab 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 nL. For the ab mode in MLHLM, high energy confinement is only found at small dL, in which case the energy is highly confined in the low-index layers between metal and H layers. For the sb mode in MLHLM, high energy confinement is observed at small dL 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 . Like the ab 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 ab mode in MLHLM. We can consider that the ab mode in MLHLM is formed by the interaction between the two hybrid modes associated with the two halves of the MLHLM structure.
5. Propagation length and figure of merit
The propagation length Lp is defined as the 1 / e decay length of the energy, and is given by Lp = 1 / 2Im(β). Lp of the ab and sb modes supported by the MHLHM and MLHLM structures is shown in Fig. 7 , in terms of d1 and d2. For the bound modes of MIM structure, Lp increases with increasing insulator thickness and decreasing insulator index. For MMIM, Lp is additionally affected by distribution of core index. In order to make clear the effect of the contrasted core index, we compare Lp 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 2d1 + d2.
Lp of the ab mode of MHLHM is shown in Fig. 7(a). At the left side of the white line, Lp of the ab mode of MHLHM exceeds that of the M-H-M structure. In contrast, at the right side, Lp of the ab 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.
For the ab mode of MLHLM, as shown in Fig. 7(c), at the left side of the white line, Lp of the ab mode of MLHLM is comparable with but less than that of the M-L-M structure. When d1 is large enough, the ab mode is less affected by the metal claddings. At the right side, Lp of the ab mode of MLHLM is greater than that of the M-L-M structure and increases exponentially with increasing d1 or d2. As an example, when (d1, d2) = (300, 250) nm, Lp is as high as 4719 µm, which is over 20 times of Lp of the ab mode of the M-L-M structure.
High losses are observed for the sb modes. For both MHLHM and MLHLM, the propagation length of the sb modes increases with the insulator thickness. Within our parameter range, the maxima of Lp of the sb 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. Lp of the sb mode never exceeds that of the ab mode that supported by the same structure. As d1 or d2 increase, Lp of the ab and sb modes approach each other for the MHLHM structure, however keep apart from each other for the MLHLM structure.
The real part of the effective index of the ab mode is greater than that of the sb mode (Fig. 2), so the phase velocity of the ab mode is less than that of the sb mode. Whereas high loss of the sb mode compared with the ab mode indicates that the group velocity of the ab mode is greater than that of the sb mode.
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 = Lp / Aeff). Figures 8(a) -8(d) show FOM of the ab and sb modes in the MHLHM and MLHLM structures in terms of d1 and d2.
For the ab mode in MHLHM [Fig. 8(a)], largely, FOM decreases as the H layer thickness increases. For the ab mode in MLHLM [Fig. 8(c)], FOM increases exponentially as core thickness increases. Extremely high FOM (~104) can be achieved by the ab mode of MLHLM with core thickness of several hundred nanometers. For example, when (d1, d2) = (300, 250) nm, FOM is as high as 2.16 × 104, which is 66 times of FOM of the ab mode in the M-L-M structure.
For the sb mode in MHLHM [Fig. 8(b)], FOM increases with increasing core thickness. For the sb 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 d1 value, FOM increases and reaches a maximum then decreases as d2 increases. The maximum also increases as d1 increases. And when d1 is large enough (greater than ~300 nm), the maximum appears at d2 = 0, hence the sb mode of the M-L-M structure with core thickness equal to 2d1 works better than that of MLHLM with geometric parameters near the cutoff line. It is noted that FOM of the sb modes in both MHLHM and MLHLM is less than that of the ab modes supported by the same structures.
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 ab mode in MHLHM (or MLHLM), Re(neff) can be independent on the H layer thickness at a critical L layer thickness dC (or dC1). And for the sb mode in MLHLM, Re(neff) can be independent on the L layer thickness at a critical H layer thickness dC2. The existence conditions of the critical values are given according to the monotonic and asymptotic properties of Re(neff). The critical values dC and dC2 exist for all the material parameters nL and nH. In contrast, the critical value dC1 exists only when nH 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 ab mode of the MLHLM structure, since the field is less affected by the metal claddings when the dL 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 ab modes is higher than that of the sb modes supported by the same structures; and the figure of merit of the MLHLM structure is higher than that of the MHLHM structure. The ab 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 ab mode in the MLHLM structure is of the most applicability.
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:
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
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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