1. Introduction

Three-layer (3L) insulator-metal-insulator (IMI) and metal-insulator-metal (MIM) plasmonic waveguides have been an object of study for a number of decades [1–16]. At the beginning of the 2000s, with the discovery of metamaterials [17] that manifest negative effective values for both magnetic permeability and dielectric permittivity [18], the 3L waveguides attracted considerable attention for their guided plasmonic modes also exhibit negative effective index [1–3], and can be used as the building blocks of photonic metamaterials. For example, it has been shown that metamaterials based on 3L waveguides manifest exotic properties such as negative refraction lensing [4], negative refraction in a stack of bilayer plasmonic waveguides [5], and nanofocusing [6–8]. The structures in most of the previously mentioned studies, however, are considered symmetric (i.e., where the top and bottom layers of the 3L configuration are composed of the same material). At the same time, studies for which more general cases of asymmetric waveguides are considered (i.e., when all of the layers are composed of different materials) have been undertaken less frequently [9–13], and either frequencies that are associated only with the low-loss real part of a wavevector, or lossless materials are generally covered. However, it has been demonstrated that the high-loss and negative slope regions of the dispersion are of great importance for the design of metamaterials that exhibit negative effective index [14]. Considering this, and the lack of detailed studies regarding lossy metals, a thorough investigation of the plasmonic response in 3L asymmetric waveguides at a broad spectral range is presented in our previous work [13]. It has been demonstrated [13] that: 1) for the thin middle layer of thickness t, the modal dispersion is characterized by a double-peak behavior, and 2) the low-loss dispersion constituents of the positive and negative index modes are separated by a spectral zone called the quasi-bandgap, where the propagation of any plasmonic modes is suppressed. The introduction of the quasi-bandgap phenomenon into the theory of plasmonic waveguides has also raised some new questions regarding the understanding of plasmon behavior. For example, for the topical case of tapered MIM plasmonic waveguides [19], an interesting yet previously uncovered question has emerged regarding the single metal-insulator interface mode transformation in an asymmetric configuration at quasi-bandgap frequencies for decreasing t. Further, a fundamental question is whether the negative index mode always exists in 3L waveguides even when t is large.

In this work, we aim to find answers to the previously mentioned questions and to study the details of the dispersion evolution of the positive and negative index modes in 3L asymmetric waveguides. Besides, an attempt is made to approach the theory of 3L plasmonic waveguides in a way that is more generalized and simple, whereby a complicated analytical analysis is avoided and a simple but effective numerical solver is applied. We demonstrate that the obtained results are consistent with those of previous theoretical works, and importantly, complement them through the provision of insight into the transformation of the waveguiding modes as function of both t and the excitation frequency. In this work, we elucidate the way that the plasmonic dispersion of 3L waveguides can be represented in terms of implicit or explicit solutions depending on the thickness value t, and demonstrate that the negative index mode always exists in 3L waveguides, however becoming lossy as t increases.

2. Dispersion equations

Before proceeding to the study of the modes that are supported by multilayer plasmonic waveguides, it is instructive to overview different forms of the dispersion equations, as they can be applied for particular cases. The dispersion equation for the 3L waveguides, for example, can be significantly simplified compared with those considered so far; therefore, the simplified dispersion relations employed in this study allow for an illustrative explanation of the modal evolution that is demonstrated regarding the changing of t at wide spectral range.

2.1 Three-layer waveguides

Starting from the case where the 3L waveguide comprises two effectively separated interfaces for a sufficiently thick middle layer of the thickness t, the plasmonic response of the 3L waveguide can be described by a well-known explicit dispersion of its decoupled single interfaces [for example, like that at the thick part of the tapered waveguide in Fig. 1(a)], as follows [16]:

 

Fig. 1 (a) Typical plasmonic modes supported by the IMI 3L waveguide at different middle layer thicknesses, like the example of the tapered 3L waveguide. (b) Schematic demonstration of how the asymmetry in MIM Ag/ε₂/Au geometry can be replaced by the asymmetry in MIIM Ag/ε₂/ε₃/Ag geometry waveguides.

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For the opposite case, where t is quite thin and the condition k_p² >> k₀²ε_j is satisfied [13] [for example, like that near the tip of the tapered waveguide in Fig. 1(a)], the plasmonic response can be described by the explicit dispersion equations, as follows:

Regarding the IMI waveguide, Eq. (2) and Eq. (3) correspond to the modes with even and odd symmetry, respectively, and this correspondence is vice versa with respect to the MIM waveguides [13].

For the rest of t (not satisfying the previously mentioned conditions), the following general implicit form of the dispersion must be considered [Fig. 1(a)] [13]:

In summary, Eq. (4) is a general dispersion equation for a rigorous investigation of the plasmonic response in 3L waveguides, while Eq. (1) to Eq. (3) is a comprehensive set of Eq. (4) approximations.

Further, in agreement with the terminology introduced in [2,3,13,20], the following definitions for positive/negative index modes have been adopted: plasmons that exhibit a positive effective index are characterized by a parallel energy flow and phase velocity so that Re{k_p}>0, Im{k_p}>0; whereas plasmons with a negative effective index are characterized by an oppositely directed phase velocity and energy flow with Re{k_p}< 0, Im{k_p}> 0 (it is assumed that Im{k_p} is always positive).

2.2 Four-layer waveguides

The utilization of two different metals in a particular case of an asymmetric MIM waveguide causes the following major shortcoming: the theoretically expected spectral location of the guided modes typically overlaps with the spectral band that is associated with the interband electronic transitions in one of the metallic layers. To obtain access to the full scope of the supported modes at a broad spectral range, it is therefore convenient to replace an asymmetry that has been introduced by two different metals with an asymmetry that has been introduced via two different dielectrics in a four-layer (4L) MIIM waveguide, whereby identical top and bottom metallic layers are employed [Fig. 1(b)]. Although the dispersion properties of the MIIM and MIM waveguides are very similar, the MIIM structure is more feasible in terms of the electrodynamic response of the metallic layers, and is consequently considered in this work instead of the MIM case; furthermore, another advantage of the MIIM geometry is the possibility of controlling the quasi-bandgap width and position by properly selecting the material for the two insulator layers [13], which is in contrast to the limited choice of plasmonic materials for the metallic layers that exists.

Assuming that the electromagnetic wave propagates along the insulator-metal (IM) interfaces in a 4L waveguide in the form of E(z)e^ißx, whereby the field components of the wave equation for the TM modes $E_x=-i\frac{1}{\omega {\epsilon }_0\epsilon }\frac{\partial H_y}{\partial z}$ and $E_z=-\frac{k_p}{\omega {\epsilon }_0\epsilon }{H}_y$, and the boundary conditions at the interfaces for H_y and E_x are used, a compact implicit plasmonic dispersion relation is derived for the general case of the asymmetric 4L plasmonic waveguide, as follows:

3. Computational technique

Generally used computational techniques for the solving complex implicit dispersion equations like those of Eq. (4) and Eq. (5) are either quite complicated, requiring careful programming (for example, the reflection-pole [21] and argument-principal [22] methods), or they are quite sensitive to the initial parameters (for example, the Newton [23] and iterative [24] techniques). Additionally, it is often mentioned that the standard root searching algorithms that are available from the commercial software (e.g., MATLAB, Mathematica, Mathcad) can be sensitive to the initial guess, and it is therefore difficult to apply them to implicit equations.

An optimized root searching algorithm that is rather simple yet insensitive to the problem regarding the selection of a correct initial guess was therefore employed. The main idea of this technique is the use of the standard root searching function (e.g., MATLAB’s fsolve) along with a very fine randomized grid of the initial guesses of the complex values that cover the whole k₀-k_p space. This leads to a convergence of the root searching function to one of the solutions in vicinity of each initial guess point. Thus, it becomes possible to find a complete set of dispersion solutions in the k₀–k_p space through the employment of a randomized grid of the initial guesses and the frequencies k₀. The proposed method can be successfully applied to solve any implicit equations or systems of equations in complex space. Furthermore, a considerable computational power is not required for this algorithm, and it can even be implemented for conventional multicore desktop computers, whereby the computational time can be drastically decreased by parallelized calculations (e.g., the use of MATLAB’s built-in parfor function). The computational speed of approximately 600 iterations (dispersion solutions) per second is achieved with the use of a commercial quad-core processor (Intel Core i5 in this case). Approximately 10⁴ iterations to 10⁵ iterations (equivalent of 15 seconds to 180 seconds of computational time) are generally needed for the production of smooth dispersion curves.

4. Typical plasmonic response in asymmetric 3L and 4L waveguides

In this section, the most general properties of the plasmonic response in the IMI and MIIM waveguides are demonstrated for the introduction of their main dispersion features, and for the setting of a context for the further study of plasmonic mode evolution.

4.1 IMI waveguides

To comprehensively demonstrate the plasmonic response in a general asymmetric IMI waveguide, the introduced calculation technique is applied to solve the general dispersion of Eq. (4) for q = –1, 0, and 1 (i.e., considering only those solutions with low losses). The family of the resulting dispersion curves is presented in Figs. 2(a) and 2(b) for the IMI waveguide with ε₁ = 2.25, where ε₂(k₀) is the dielectric permittivity of silver (Drude model with parameters as in [25]), ε₃ = 3.9, and t = 12 nm of the silver film. The obtained typical plasmonic response of the waveguide consists of the following three modes: the short range surface plasmon (SRSP) mode, denoted as 0^SRSP (q = 0), the long range surface plasmon (LRSP) mode [26], denoted as 1^LRSP (q = 1), and the negative-index mode, denoted as –1^NEG (q = –1). In general, the dispersion of the 0^SRSP and –1^NEG modes in the asymmetric waveguides is characterized by a double-peak behavior [13]. The notation for the frequencies that are associated with the peaks is adopted from [13] as $k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_1\right|}$ and $k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_3\right|}$, signifying the frequencies where |ε₂| = |ε₁| and |ε₂| = |ε₃|, respectively [see Figs. 2(a) and 2(b)]. As is demonstrated in [13], the quasi-bandgap spectral region between $k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_1\right|}$ and $k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_3\right|}$ corresponds to the frequencies where all of the propagating modes are suppressed [i.e., experience large losses; blue shaded zone in Figs. 2(a) and 2(b)]. At the same time, the LRSP mode is characterized by a cutoff momentum k^cutoff [26] [Fig. 2(a)], below which the LRSP mode becomes leaky [12,13,26,27]. In the presented example with a silver metallic layer, the conditions $k^{cutoff}>k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_1\right|}$ and $k^{cutoff}>k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_3\right|}$ are satisfied for t < 14 nm, such that the LRSP mode’s dispersion is located outside the quasi-bandgap. It should be noted that, although the LRSP and SRSP modes are both a positive index modes, they exhibit opposite field symmetry.

 

Fig. 2 (a) Real and (b) imaginary parts of the typical plasmonic dispersion of the IMI waveguide, demonstrating the SRSP, LRSP, and negative index modes. Dispersion is calculated for ε₁ = 2.25, ε₃ = 3.9, where ε₂(k₀) is the dielectric permittivity of the silver layer of the thickness t = 12 nm. (c) Real and (d) imaginary parts of the typical plasmonic dispersion of the MIIM waveguide, demonstrating the positive index and negative index modes. Dispersion is calculated for ε₁(k₀) = ε₄(k₀), where ε₁(k₀) is the dielectric permittivities of silver, ε₂ = 2.25, ε₃ = 3.9, and t₂ = t₃ = 6 nm.

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4.2 MIIM waveguides

Although it is not possible to simplify the 4L waveguide dispersion [Eq. (5)], its solutions exhibit the corresponding behavior that is strikingly similar to that of the 3L case, as can be seen in Fig. 2. It is therefore convenient to keep the same notations for the dispersion solutions while the mode evolution is described.

The typical plasmonic response of the asymmetric MIIM waveguide that is obtained from Eq. (5) when ε₁(k₀) = ε₄(k₀) is the dielectric permittivity of silver, ε₂ = 2.25, ε₃ = 3.9, and t₂ = t₃ = 6 nm of the dielectric films, as is demonstrated in Figs. 2(c) and 2(d). The response consists of a positive index mode (denoted as 0^SRSP) and a negative index mode (denoted as –1^NEG). According to an IMI waveguide analogy, the quasi-bandgap is located between the low-loss dispersion regions of the supported modes [blue shaded region in Figs. 2(c) and 2(d)] [13]. It should be noted that for the results in Fig. 2, the position and width of the quasi-bandgap in the 4L waveguide coincide with those in the 3L waveguide due to an identical pair of IM interfaces that is utilized in both cases. Adopted from [13], the low and high frequency limits of the quasi-bandgap zone for the 4L waveguide are denoted as $k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_1\right|}$ and $k_0^{\left|{\epsilon }_3\right|=\left|{\epsilon }_4\right|}$ [ε₂ < ε₃; Figs. 2(c) and 2(d)].

5. Modal evolution in IMI waveguide

5.1 Dispersion evolution of the positive index modes

Through the consideration of the asymmetric IMI waveguide dispersion, started in Section 4.1, it can be shown that for the increasing value of the silver film thickness t, the real part of the SRSP mode’s wavevector gradually descends and eventually transforms into the single interface mode at the ε₃|Ag (ε₃ > ε₁) interface when t → ∞ [Fig. 3(a)]. An unexpected behavior is observed, however, for the imaginary part of the wavevector for frequencies $k_0>k_0^{\left|{\epsilon }_3\right|=\left|{\epsilon }_2\right|}$, where irregular jumps are revealed for an increasing t [Fig. 3(b)], and this is further explained in Section 5.1.1. Moreover, for the growing values of t, the real part of the LRSP mode’s wavevector experiences a rapid increase of its peak value and an abrupt shift of the peak’s spectral position for the silver film thickness between t = 28 nm and t = 29 nm, whereby it is eventually transformed into the single interface mode at the ε₁|Ag (ε₃ > ε₁) interface when t → ∞ [Fig. 3(c)]. Similarly, specific behavior is observed for the imaginary part of the LRSP mode wavevector [Fig. 3(d)], and this is further explained in Section 5.1.2.

 

Fig. 3 Dispersion evolution for (a) real and (b) imaginary parts of the SRSP mode wavevector for the different thicknesses t of the silver layer in the IMI waveguide. Dispersion evolution for the (c) real and (d) imaginary parts of the LRSP mode wavevector for different thicknesses t of the silver layer in the IMI waveguide. The model parameters of the IMI waveguide are the same as those of Fig. 2.

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To understand the non-trivial dispersion evolution for a continuously changing t, it is necessary to recall the dispersion equation Eq. (4) for which the low-loss solutions when |q| ≤ 1 (low order solutions) are considered, along with the periodic lossy solutions when |q| > 1 (high order solutions), and this is further discussed in the following sections.

5.1.1 Dispersion evolution of SRSP mode

To evidently demonstrate the dispersion evolution of the SRSP mode for the continuously changing t, the dispersion for a set of t in vicinity of $k_0^{}=k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_3\right|}$ (where the mode experiences a transition from a low-lossy to a high-lossy regime (Fig. 4)) is successively solved. Thus, Fig. 4(a) shows two solutions of Eq. (4) for the silver film when t = 12 nm: a generally considered SRSP mode [q = 0; see Fig. 2], and for a first order lossy solution “1” (q = 1; not the LRSP solution). For the further growth of t, the imaginary parts of these solutions gradually approach each other [red curves in Figs. 4(a)–4(c)], followed by a mutual interchange of their real and imaginary parts, as is shown in Figs. 4(c) and 4(d). For the further increasing of t, this behavior is continually repeated, involving lossy solutions of the higher orders q = 2, 3, … [Figs. 4(e)–4(h)] until the single interface (i.e., surface plasmon) mode SP₁ is eventually formed at the ε₃|Ag interface (ε₃ > ε₁) when t → ∞ [Fig. 4(i)]. As one can see from Fig. 4, starting from the silver film thickness t = 18 nm, the continuous interaction between the low and high order solutions of Eq. (4) explains the irregular behavior of the wavevector’s imaginary part at a broad frequency range that is shown in Fig. 3(b).

 

Fig. 4 The SRSP mode dispersion evolution (blue curve: real part of wavevector; red: imaginary part of wavevector) for various values of t in vicinity of $k_0^{}=k_0^{\left|{\epsilon }_3\right|=\left|{\epsilon }_2\right|}$, showing active interactions with high order lossy solutions. The model parameters of the IMI waveguide are the same as those of Fig. 2.

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It must be noted that, for the silver film thickness t < 18 nm, the dispersion of the SRSP mode can be well approximated by the explicit dispersion of Eq. (2), while the general dispersion of Eq. (4) must be used for t > 18 nm. Furthermore, dealing only with the low-loss part of the SRSP dispersion when $k_0^{}<k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_3\right|}$ (commonly considered in the literature) and starting from t ≈40 nm, the dispersion can be approximated as that of the single interface plasmonic mode [Eq. (1)] and can be seen in Figs. 3(a) and (b).

5.1.2 Dispersion evolution of LRSP mode

To understand the dispersion evolution of the LRSP mode for the continuously changing t in the same manner as that of the SRSP mode, the dispersion for a set of t in vicinity of $k_0^{}=k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_1\right|}$ is successively solved [Fig. 5]. Again, for the increasing thickness of the silver film t, the first order lossy solution “1” (q = 1) of Eq. (4) must be considered together with the LRSP solution 1^LRSP (also obtained when q = 1). When t increases, the real and imaginary parts of the solution “1” approaches those of the LRSP, as shown in Figs. 5(a)–5(g). At t ≈29 nm, the interchange between the real and imaginary parts of these solutions occurs according to what is shown in Fig. 5(e), while the LRSP solution becomes lossy at the whole spectrum range. For the further increasing of t > 30 nm, solution “1” continues to interchange with solutions of higher orders q = 2, 3, 4, … [Fig. 5(g)], and it is eventually transformed into the single interface mode SP₂ at the ε₁|Ag interface (ε₃ > ε₁) when t → ∞ [Fig. 5(h)].

 

Fig. 5 The LRSP mode dispersion evolution (blue curves: real part of wavevector; red: imaginary part of wavevector) for various values of t in vicinity of $k_0^{}=k_0^{\left|{\epsilon }_1\right|=\left|{\epsilon }_2\right|}$, showing active interactions with high order lossy solutions. The model parameters of the IMI waveguide are the same as those of Fig. 2.

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The transformation into the surface plasmon mode at the ε₁|Ag interface (ε₃ > ε₁) for the increasing t is the most distinctive feature of the LRSP mode in the 3L waveguide. This transformation is characterized by an abrupt shift of its wavevector’s peak value to the higher frequencies k₀ and a considerable amplification of its magnitude. For the silver film, this abrupt change is observed at t ≈28.5 nm [Figs. 5(d) and 5(e)], and clearly explains the dispersion behavior that is previously demonstrated in Figs. 3(c) and 3(d).

It must be noted that a “pure” LRSP mode exists only up to t ≈28.5 nm, and its dispersion must be described by a general dispersion equation for the 3L waveguide [Eq. (4)]. For t > 28.5 nm, the single interface mode starts to form, eventually becoming a surface plasmon mode at the ε₁|Ag interface (ε₃ > ε₁) when t → ∞. However, when the low-loss part of the dispersion ($k_0^{}<k_0^{\left|{\epsilon }_2\right|=\left|{\epsilon }_1\right|}$) is exclusively dealt with, the surface mode approximation [Eq. (1)] becomes valid from t ≈40 nm [Figs. 3(c) and 3(d)].

It is interesting to note that the value of skin depth in silver at considered frequencies (≈2 nm) is much smaller than considered metal film thicknesses (12-100 nm). However, as demonstrated in Figs. 4 and 5, plasmonic response of the IMI waveguide reflects the presence of interaction between modes of both IM interfaces even when they are separated by t ~100 nm. We speculate that this fact is related to the presence of plasmonic resonance in the metal.

5.1.3 Numerical confirmation of LRSP mode and SRSP mode evolutions

Asymmetric tapered adiabatic waveguides, which are characterized by the absence of any considerable scatterings [6,7], are an ideal platform for the verification of the demonstrated results. The small tapering angle of tapered adiabatic waveguides allow to consider the latter as a sequence of 3L waveguides with a continuously decreasing t in the direction of the tip. Therefore, a plasmonic mode, excited on the thick part of such waveguide, is propagating towards its tip and experiences the dispersion evolution described in Section 5.1.1 and Section 5.1.2 (although, in a “reverse” order, from a larger to a smaller t). As has been analytically demonstrated, the SRSP mode can be exclusively excited via the surface plasmon mode at the ε₃|Ag (ε₃ > ε₁) interface, whereas the excitation of the LRSP mode is possible only via the surface plasmon at the opposite interface ε₁|Ag (ε₃ > ε₁). This can be confirmed by a simple numerical simulation using the COMSOL software package (finite element method; frequency domain with dipole excitation). Figure 6(a) shows the simulation results when the surface mode is excited at the ε₃|Ag (ε₃ > ε₁) interface. As expected, nanofocusing is experienced at the tip, which is a direct consequence of the mode coupling to the high index SRSP mode [6, 7]. At the same time, Fig. 6(b) demonstrates the way that the mode which is excited at the ε₁|Ag (ε₃ > ε₁) interface effectively couples into the free space radiative mode via a coupling with the low index LRSP mode, whereby it becomes leaky at a small t [12,13,26,27].

 

Fig. 6 (a) The magnitude of electric field E_x and the time averaged energy flow W at k₀ = 1.25 × 10⁷ m⁻¹ for the plasmon, propagating along the ε₃|ε₂(Ag) interface of the tapered IMI waveguide. (b) The magnitude of electric field E_x and the time averaged energy flow W at same frequency k₀ for the plasmon, propagating along the ε₁|ε₂(Ag) interface. Waveguide materials in both cases are the same as those of the model employed in Fig. 2.

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5.2 Dispersion evolution of the negative index mode

Equation (4) is also a governing equation for the negative index mode –1^NEG [q = –1; Fig. 2] in the considered IMI waveguide [13]. Contrary to the LRSP and SRSP modes, the solution for the negative index mode does not experience interchanges with high-loss periodic solutions, and its evolution is therefore significantly simpler, as is presented in Fig. 7. In fact, while the negative index mode always exists in the 3L waveguides, they become lossy as the t increases. For the considered IMI waveguide with silver (see Section 4.1), the thickness of the transition in the lossy regime is approximately 20 nm [Fig. 7].

 

Fig. 7 Typical dispersion evolution of the negative index mode in the IMI waveguide. The model parameters of the IMI waveguide are the same as those of Fig. 2.

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6. Modal dispersion evolution in MIIM waveguides

6.1 Dispersion evolution of the positive index modes

The difference between the modal evolution of the MIIM waveguide and that of the IMI waveguide is mainly due to the absence of the LRSP mode; however, the underlying evolution (i.e., multiple interchanges between the low and high order solutions) remains the same (Fig. 8). The SRSP solution 0^SRSP, for example, experiences the first interchange with the lossy solution “1” at t₂ + t₃ ≈24 nm [Fig. 8(c)]; then, for the further increasing of t₂ + t₃, it survives the interchanges with lossy solutions of higher orders as it gradually transforms into the single interface mode SP₁ [Fig. 8(c)–8(f)]. The lossy solution “1” simultaneously and gradually becomes a low-lossy solution [Figs. 8(c)–8(e)] until it completely transforms into the single interface mode SP₂ for t₂ + t₃ → ∞ [Fig. 8(f)].

 

Fig. 8 Typical dispersion evolution (blue curves: real part of wavevector; red: imaginary part of wavevector) of the positive index modes in the MIIM waveguide for various t values. The model parameters of the MIIM waveguide are the same as those of Fig. 2.

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6.2 Dispersion evolution of the negative index mode

The behavior of the negative index mode –1^NEG [Fig. 2(c) and 2(d)] in the MIIM waveguide is shown in Fig. 9 and is very similar to that of the IMI waveguide. The negative index mode exists at a very broad range of the thicknesses of the dielectric layers, which can be explained by a field confinement that is weaker than that of IMI geometry. It should be noted that, contrary to the IMI case, the mode’s dispersion curve might descend below the light lines; this feature was utilized for the design of metamaterials with effective negative refraction index [5].

 

Fig. 9 Typical dispersion evolution of the negative-index mode in the MIIM waveguide for various t values. The model parameters of the MIIM waveguide are the same as those of Fig. 2.

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7. Conclusion

Employing a novel dispersion solving approach, our results elucidate the modal dispersion regarding any middle layer waveguide thickness at any frequency of interest. It has been demonstrated that the high-order lossy solutions of a general dispersion play a crucial role in the evolution of guided modes. Further, the continuous transition between a generally considered single interface (t → ∞) and the thin film (t → 0) plasmonic modes in the 3L waveguides is revealed and explained in detail. The present study also demonstrates that the negative index mode experiences a transition between low- and high-lossy regimes depending on the thickness of the waveguide. The discussed results is perfectly integrated in the context of previous theoretical works while a complicated analysis has been avoided.