1. Introduction

Slow light is light with a low group velocity, conventionally slower than one-hundredth of the velocity of light in free space [1]. Recently, there has been growing interest in achieving and utilizing slow light due to its various applications. Optical pulse trains can be delayed by a few bits when they are transmitted through slow light waveguides, allowing for optical buffers [2]. Compression of energy density accompanied with slow light can be used to enhance nonlinear properties of materials with a small feature size [3]. In addition, the large group index of slow light usually leads to the large effective refractive index, which is a very promising characteristic in optical switching [4].

The extreme case of slow light is trapping, which indicates that the group velocity of light vanishes. This phenomenon is also referred to as pinning, stopping, or freezing of light. Methods to realize trapping of light in waveguides can be classified into two groups: one is to use photonic crystal waveguides with resonators and the other is to invoke metamaterial waveguides. The former is associated with multiple reflections of light inside resonators artificially employed in the photonic crystal waveguide, and it is known to have strong points from view points of Q factor and implementation [5–10]. The photon pinning in an optical resonator with the ultrahigh-Q factor of ~5 × 10⁹ was achieved by means of local index tuning in the photonic crystal waveguide [5]. The dynamic release of trapped light was experimentally demonstrated by using adiabatic frequency shift [6].

Research on the usage of metamaterial waveguides for trapping light has been initiated by trailblazing work done by Tsakmakidis et al [11]. The key principle is based on the fact that the direction of energy is directly opposite to that of the phase in a negative index material. By adopting positive- and negative-index materials for the core and the cladding, one can obtain a balance point where the optical power flow in the cladding is exactly compensated by that in the core, resulting in trapping of light. In addition, it was reported that trapping is related to the mode degeneracy [12]. Two distinct modes with different propagation properties can be supported simultaneously in metamaterial waveguides i.e., one has the parallel phase and group velocities, whereas the other exhibits an anti-parallel relation. If these modes degenerate into one mode, its group velocity vanishes. Fu and Gan reported various configurations for trapping of light in the THz [13, 14] and telecom [15] regimes. Kim devised a thin cladding composed of a negative index material for temporary trapping of light [16].

Although considerable research has been devoted to the realization of the negative refractive index materials in the optical frequency regime [17–19], it still remains difficult to implement them. Meanwhile, there has been much interest in the plasmonic waveguide, which consists of only the metal and dielectric and does not require any metamaterial [20–29]. It turned out that the plasmonic waveguide can support a propagating mode with the anti-parallel phase and group velocities [20–22]. The properties of guided modes in the planar plasmonic waveguides were investigated in detail and the existence of solutions with negative group velocities was observed [20]. Dionne and her coworkers also examined the modal dispersion of the metal-insulator-metal (MIM) and insulator-metal-insulator (IMI) waveguides with the relative electric permittivity of the metal that is empirically obtained [21, 23]. In particular, they provided an experimental demonstration for the negative group velocity in the MIM waveguide [24]. However, none of these studies have reported whether light can be trapped inside the plasmonic waveguide. Although Davoyan and his coworkers recently observed that the mode degeneracy can occur in the MIM waveguide under a certain condition, they did not offer physical origin of such degeneracy and the exact condition for its existence [30].

In this study, we seek answers to the following two questions: 1) Is it possible for light to be trapped in the plasmonic waveguide? 2) If possible, what is the condition for that? Full case studies reveal that trapping of light can occur in the plasmonic waveguides under certain conditions. The physical interpretation on the origin of these conditions is provided. In addition, the conditions for the mode to be supported in the plasmonic waveguides are given in detail, which can be used for various design processes of the plasmonic waveguides.

This paper is organized as follows. The basic terms, definitions, conventions are presented in Section 2. Then, the existence of modes and condition for the mode degeneracy in the MIM waveguide are provided in Section 3. The discussion branches into two classes: the anti-symmetric mode in the MIM waveguide (Subsection 3.1) and the symmetric mode in the MIM waveguide (Subsection 3.2). The results of discussion for the anti-symmetric and symmetric modes in the MIM waveguide are provided in Table 2 and Table 3, respectively. The next section offers the case study for the IMI waveguide. Similarly, the discussion branches into two classes: the symmetric mode in the IMI waveguide (Subsection 4.1 and Table 4) and the anti-symmetric mode in the IMI waveguide (Subsection 4.2 and Table 5). Finally, the conclusion is presented.

2. Propagation constant, phase and group velocities in the plasmonic waveguide

Let λ, ω, and k ₀ denote the wavelength, the angular frequency, and the wavenumber in free space, respectively. The relative electric permittivity of the metal and dielectric are denoted by ε_m and ε_d, respectively. The relative magnetic permeability is assumed to be unity in all material. The dependence of ε_m on ω is governed by the Drude model:

where ω_P is the bulk plasma frequency. In this paper, we only consider the lossless case i.e., γ = 0 and ε_m is purely real. In addition, we focus on the case that ε_m is negative i.e., ε_m = -∣ε_m∣ (ω<ω_p). Novel metals such as gold or silver satisfy ε_m = -∣ε_m∣ in the visible and infrared regimes. For convenience, let us define the ratio between ε_m and ε_d as follows:

Note that ρ>0 and ρ decreases as ω increases. By combining Eqs. (1) and (2), we know that ρ>1 for ω<ω_SP and ρ<1 for ω>ω_SP, where ${\omega }_{SP}={\omega }_P/\sqrt{1+{\epsilon }_d}$ is the surface plasma frequency. The bulk plasma wavelength λ_P and wavenumber β_P are defined by 2πc ₀/ω_p and ω_P/c ₀, respectively, where c ₀ denotes the speed of light in free space.

It is assumed that the wavefunction has a factor of exp(-jωt). The guided modes propagate along Z -direction with the propagation constant β i.e., the Z -dependence of the wavefunction is given by exp(jβZ). β is also referred to as the longitudinal wavenumber. The effective refractive index of a guided mode n_eff is defined as β/k ₀. n_eff of the surface plasmon polariton (SPP) propagating along the single interface between the metal with ε_m and the dielectric with ε_d, called the single interface SPP, is given by $n_{SPP}=\sqrt{{\epsilon }_m{\epsilon }_d/\left({\epsilon }_m+{\epsilon }_d\right)}.$

The plasmonic waveguides dealt with in this paper are the two-dimensional planar slab waveguides. Depending on configurations, they can be categorized into two classes. One consists of the dielectric core and the metal cladding, called the MIM waveguides [21]. The other is composed of the metal core and the dielectric cladding, referred to as the IMI waveguide [23]. The plasmonic waveguide symmetric across the line along the center of the geometry has the transverse magnetic field with either symmetric or anti-symmetric distribution. In this paper, the former is called the symmetric mode and the latter the anti-symmetric mode. For convenience the following acronyms are adopted: MIMa, MIMs, IMIa, and IMIs for the anti-symmetric mode in the MIM waveguide, the symmetric mode in the MIM waveguide, the anti-symmetric mode in the IMI waveguide, and the symmetric mode in the IMI waveguide, respectively. In addition, we adopt a convention that the mode with n_eff > √ε_d is referred to as the plasmonic mode and that with n_eff < √ε_d the photonic mode [24].

It is noteworthy that the characteristic equation in a waveguide is a function of β ² rather than β. Consequently, there can be two solutions for β. For example, if β = 1.2k ₀ is the solution, then -1.2k ₀ is also the solution to the characteristic equation. In other words, a solution β always has its twin -β and they are located symmetrically with respect to the origin. After solving a characteristic equation, we have to choose either β or -β as a solution. Consideration on this duality is significant in defining signs of the phase and group velocities, which are given by ω/β and (dβ/ dω)^-1, respectively. The direction of the phase velocity is associated with the sign of the real part of β. The phase propagates toward +Z (-T) direction if the real part of β has a positive (negative) value. Since causality in electromagnetics requires that the energy flow outward from a source, it is necessary to choose β in such a way that the sign of group velocity, which corresponds to the slope in the ω - β dispersion relation, is positive. Detail discussions on the aforementioned argument can also be found in Refs. [22, 31]. Throughout this paper, the positive mode describes a solution with parallel phase and group velocities (both phase and group velocities are positive), whereas the negative mode denotes a solution with anti-parallel phase and group velocities (the negative phase velocity and the positive group velocity).

In this paper, the effect of the material loss is not considered. When considering the material loss, we usually employ the concept of the spatial loss, which means that the characteristic equation is solved under assumption that ω is real and β is complex (β = β_r + iβ_i), where β_r and β_i denote the real and imaginary parts of β, respectively [22]. Then the wave function has a factor of exp[j(β_rZ-ω)]exp(-β_iz). The dispersion relation is plotted as ω versus β_r and the propagation length is defined as 1/(2/β_i). However, dealing with the trapped light, which does not propagate, the concept of the temporal loss is more suitable than the spatial loss. The temporal loss means that we calculate the characteristic equation under assumption that β is real and ω is complex (ω = ωr + iω_i), where ω_r and ω_i denote the real and imaginary parts of ω, respectively [22]. The wave function has a factor of exp[j(βz - ω_rt)]exp(-ω_i). From this point of view, the propagation length is not defined, Instead, the life time τ = 1/ω_i is used to describe the extent to which the wave function decays as time progresses. From the view point of the temporal loss, the trapping of light discussed below is still feasible. The detailed discussion on the effect of the material loss on the trapping of light is under work and will be reported in a next paper.

3. Metal-insulator-metal waveguide

In this section, it is shown that a certain condition exists that the MIM waveguide has two anti-symmetric modes simultaneously and that the optical power flow vanishes at the degenerate point of them. In addition, the symmetric mode also has the mode degeneracy and thus the zero optical power flow takes place in the symmetric mode.

 

Fig. 1. Schematic diagram of the MIM waveguide

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Figure 1 shows the schematic diagram of the MIM waveguide. Four modes are illustrated: the anti-symmetric plasmonic mode, the anti-symmetric photonic mode, the symmetric plasmonic mode, and the symmetric photonic mode. We will derive conditions that each mode is supported in the MIM plasmonic mode. From the boundary condition that tangential electric and magnetic fields are continuous across the interface between the core and the cladding, the characteristic equations are given by [27]

κ_m and κd denote the transverse wavenumber of the plasmonic mode in the metal and dielectric, respectively, and k_d that of the photonic mode in the dielectric. From the momentum conservation relation, we obtain

Here, it is assumed that Κ_m, Κ_d, k ₀, and β are purely real and positive. For convenience, we introduce auxiliary definitions as follows [12, 20]:

In addition, instead of the actual core width 2a, the reduced core width ak ₀ will be used. This is because what matters in the existence of a mode is the ratio between the core width and the wavelength. In this section, the wavelength λ ₀ is set to be 400 nm and it is assumed that various values of ρ originate from its corresponding bulk plasma frequency ω_P.

3.1 Anti-symmetric mode

Let us first consider the condition that the anti-symmetric plasmonic mode can propagate in the MIM waveguide, by using Eqs. (3), (7), and (8) with help of Eqs. (10), (11), and (13). Invoking the graphical method, we want to find the condition that two curves of W = ρU coth U and $W=\sqrt{U^2+V^2}$ have an intersection in the (U,W) space. Considering the asymptotic behavior of coth function, we know that ρU coth U asymptotes to ρU for U → ∞. The limiting value of ρU cothU for U → 0 is given by ρ. Meanwhile, $\sqrt{U^2+V^2}$ asymptotes to U for U → ∞ and the limiting value of $\sqrt{U^2+V^2}$ for U → 0 is simply V.

Depending on whether ρ>1 or ρ<1, the discussion for the existence of the antisymmetric plasmonic mode can be classified into two cases. Figure 2(a) depicts the aforementioned two curves in the first quadrant of U and W for the case of ρ > 1. It is seen that the asymptotes of W = ρU coth U (the red dash-dotted line), which is given by W = ρU (the black dashed line), grows faster than that of $W=\sqrt{U^2+V^2}$ (the blue solid line), which is given by W = U (the green dotted line). Therefore, if the W -intersect of W = ρUcoth U, which is ρ (the point A in Fig. 2(a)), is smaller than that of $W=\sqrt{U^2+V^2}$, which is given by V (the point B in Fig. 2(a)), then it is sure that there exists an intersection between W = ρUcothU and $W=\sqrt{U^2+V^2}$. In other words, for the case ρ>1 i.e., ω < ω_SP, it is guaranteed that the anti-symmetric plasmonic modes can propagate in the MIM waveguide, provided that ρ < V i.e., ak ₀ > ξ, where

It should be pointed out that putting β = k ₀ ε_d also results in Eq. (14). In the same manner, we can induce the condition that the anti-symmetric plasmonic mode exists for the case ρ < 1. Figure 2(b) shows the two curves for ρ < 1. Let us consider the asymptotic behaviors of two curves. For ρ< 1, if the W -intersect of W = ρU cothU, ρ (the point A in Fig. 2(b)), is larger than that of $W=\sqrt{U^2+V^2}$, V (the point B in Fig. 2(b)), then the anti-symmetric plasmonic mode exists. This relation can be expressed in a form of ak ₀<ξ.

 

Fig. 2. Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. (a) The case ρ > 1. (ε_d,ρ) = (1.0,4.0) and ak ₀ = 1.2ξ. The definition of ξ is given in Eq. (14). It is observed that, if ak ₀ > ξ, then the anti-symmetric plasmonic mode is supported in the MIM waveguide. (b) The case ρ < 1. (ε_d,ρ) = (1.0,0.8) and ak ₀ = 0.8ξ. The anti-symmetric plasmonic mode exists in the MIM waveguide when ak ₀ < ξ.

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Next, we look for the condition that the anti-symmetric photonic mode can propagates in the MIM waveguide by employing Eqs. (4), (7), and (9) with help of Eqs. (10), (12), and (13). Two curves of W = pJ cot J and $W=\sqrt{-J^2+V^2}$ will be examined. Owing to the periodic property of the cot function, more than one anti-symmetric photonic mode can be supported, provided that V is sufficiently large. Here, we focus on the fundamental anti-symmetric mode whose n_eff is the largest one. Contrary to the domain of definition of U discussed above, U ∈ [0,∞], J has the domain of definition of J ∈ [0, ak ₀√ε_d]. This is because β is purely real and thus β ₂ should be positive (See Eqs. (9) and (12).).

 

Fig. 3. Graphical method for the anti-symmetric photonic mode in the MIM waveguide. (a) The case ρ > V. (ε_d,ρ) = (1.0,2.0), resulting in ζ < ξ. The definition of ζ is given in Eq. (15). ak ₀ = (ζ + ξ)/2 . It is shown that, if ζ < ak ₀ < ξ, then the anti-symmetric photonic mode is supported in the MIM waveguide. (b) The case ρ < V. (ε_d,ρ) = (1.0,1.1), leading to ζ > ξ . ak ₀ = (ζ + ξ)/2. The anti-symmetric photonic mode exists in the MIM waveguide when ξ < ak ₀ < ζ.

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Depending on whether ρ > V or ρ < V, our discussion branches off. In Fig. 3(a), we show W = ρJ cot J (the red dash-dotted line) and $W=\sqrt{-J^2+V^2}$ (the blue solid line) for ρ > V. The bold lines illustrate lines in the domain of definition (J ∈ [0, ak ₀√ε_d]), whereas thin lines denote lines outside the domain of definition. In the case ρ > V, if ρJ cot J at J = ak ₀√ε_d (the point A in Fig. 3(a)) is smaller than $W=\sqrt{-J^2+V^2}$ at J = ak ₀√ε_d (the point B in Fig. 3(a)) i.e., ρ√ε_dcot(ak ₀√ε_d) < √-ε_m, it is guaranteed that the anti-symmetric photonic mode exists i.e., ak ₀ > ζ, where

It should be mentioned that Eq. (15) can also be derived by putting β = 0 into Eqs. (4), (7), and (9). This property can be found in Table 1 in Ref. [27]. In the same manner, we can obtain the condition that the anti-symmetric photonic mode can propagate in the case of ρ > V : ρ√cot(ak ₀√ε_d) > √-ε_m i.e., ak ₀ (see Fig. 3(b)). Let us remind that ρ > V and ρ < V conditions correspond to ak ₀ < E, and ak ₀ > ξ, respectively. Hence, depending on whether ξ > ζ or ξ < ζ , the condition for the existence of the anti-symmetric photonic mode can be classified into two cases. Meanwhile, the difference between ξ and ζ i.e., ξ - ζ , is a monotonic increasing function of ρ and becomes zero when ρ = ρ_c, where ρ_c is a solution satisfying

ρ_c is calculated as about 1.28. Using this property, we know that ξ > ζ and ξ < ζ correspond to ρ > ρ_c and ρ < ρ_c, respectively. Therefore, the condition for the existence of the anti-symmetric photonic mode is given by ξ < ak ₀ < ζ for ρ < ρ_c and ζ < ak ₀ < ξ for ρ > ρ_c. Table 1 summarizes aforementioned property.

Table 1. Conditions for existence of the anti-symmetric mode in the MIM waveguide. ξ and ζ are defined in Eqs. (14) and (15), respectively.

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So far, we have derived the range of the reduced core width ak ₀ where anti-symmetric plasmonic and photonic modes exist for given range of ρ. Now we are led to the discussion on the mode degeneracy and the zero optical flow of anti-symmetric modes. Let us examine the dependence of the absolute value of the effective refractive index |n_eff| on the reduced core width ak ₀ for each case of ρ<1 , 1 < ρ < 1.28 , and ρ > 1.28 , shown in Figs. 4(a)–(c), respectively. Note that |n_eff|, rather than n_eff, is used here, owing to the fact that in some cases a negative-valued n_eff is chosen. The vertical dash-dotted and dashed lines in Figs. 4(a)–(c) denote the lines of ak ₀ = ξ, and ak ₀ = ζ, respectively. The dotted horizontal lines in Figs. 4(a)–(c) indicate the lines of |n_eff|=√ε_d. The dash-dotted horizontal lines in Figs. 4(b) and 4(c) correspond to the lines of |n_eff| = n_spp . In Fig. 4(a), it is seen that |n_eff| is decreased monotonically as ak ₀ is increased. The anti-symmetric modes is plasmonic for ak ₀ < ξ. As ak ₀ grows and gets larger than ξ, the anti-symmetric mode transits from the plasmonic mode into the photonic mode. As ak ₀ goes from ξ to ζ, |n_eff| decreases monotonically and the anti-symmetric mode vanishes at ak ₀=ζ. This mode cut off is in good agreement with the results of previous studies. It has been reported that this property can be used to offer the lateral confinement in the MIM waveguide [32] and to implement a barrier acting as a mirror, resulting in a plasmonic resonator in the nanoscale [27].

 

Fig. 4. (a)-(c) Dependence of the effective refractive index n_eff as a function of the reduced core width ak ₀ for the anti-symmetric mode in the MIM waveguide. (d)-(f) Effect of ak ₀ on the normalized optical power flow P_norm. The definition of P_norm is given in Appendix A. (a) and (d) show the results for the case ρ < 1 with (ε_d,ρ) = (1.0,0.8). (b) and (e) depict the results for the case 1 < ρ < 1.28 with (ε_d, ρ) = (1.0,1.1). (c) and (f) illustrate the results for the case ρ > 1.28 with (ε_d, ρ) (1.0,2.0). It is seen that, in the case 1 < ρ < 1.28 the mode degeneracy occurs at a certain point ak ₀ < ξ. This corresponds to the P_norm = 0 (See Fig. 4(e)) i.e., trapping of light.

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Figure 4(b) illustrates |n_eff| as a function of ak ₀ for 1 < ρ < 1.28. Since ak ₀ > ξ, allows the plasmonic mode and ξ < ak ₀ < ζ supports the photonic mode, it can be concluded that the plasmonic and photonic modes exist simultaneously when ξ < ak ₀ < ζ . It is also noteworthy that |n_eff| of the plasmonic mode is decreased as ak ₀ is reduced in the domain ak ₀ > ξ , whereas that of the photonic mode is increased with the decrease of ak ₀ in the domain ξ <ak ₀ < ζ. These two modes come closer to each other in the domain ak ₀ < ξ . It is therefore guaranteed that two modes degenerate in one mode at a specific ak ₀ smaller than ξ . This is one of the most important findings in this paper. It will be shown below that, at this degenerate point, the optical power flow vanishes. Although this kind of curve has been reported in [30], neither the origin nor exact condition was provided.

With respect to the aforementioned observation, it appears that the result in Fig. 2(a) and summary in Table 1 should be supplemented. The region ρ > 1 in 2Fig. 2(a) is classified into two regions: one is the region ρ > 1.28 and the other is the region 1 < ρ < 1.28 . The former results in ξ <ζ , whereas the latter gives rise to ξ < ζ. The U - W relationship shown in Fig. 2(a) is for the case ρ = 4 > 1.28. Thus we show in Fig. 5(a) the supplementary result for 1 < ρ = 1 < 1.28. Note that there are two intersections between two curves W = ρU cothU and W = √U ² + V ². Since these curves are too close to each other, it is hard to distinguish one from the other. In Fig. 5(b), the dependence of ρU coth U -√U ² + V ² as a function of U for various values of ak ₀ can be seen. The zeros, i.e., U -intersections of this curve, correspond to the intersections of W = ρU cothU and W = √U ² + V ² . If ak ₀ = ζ, there is only one zero. Two zeros are observed when ak ₀ = ξ. As ak ₀ is decreased from ξ, these zeros come close to each other. At a certain value, denoted by h_c, the curve ρU coth U - √U ² + V ² has the degenerated zero. This is in good agreement with the result shown in Fig. 4(b). In Table 2, the supplemental result from Table 1 is provided.

 

Fig. 5. (a) Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. The case 1 < ρ < 1.28. (ε_d, ρ) = (1.0,1.1). ak ₀ = 0.98ξ. Two solutions are observed. (b) The graph of ρU cothU - √U ² + V ² for (ε_d,ρ) = (1.0,1.1) . The arrow shows the tendency of decreasing ak ₀ i.e., ak ₀ = ζ, ξ, 0.98ξ, 0.938ξ(= h_c), and 0.91ξ.

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Table 2. Supplemented conditions for existence of the anti-symmetric mode in the MIM waveguide

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Now let us come back to the discussion on the geometric dispersions in Fig. 4. Figure 4(c) shows the effect of ak ₀ on |n_eff| for ρ>1.28 . The functional behavior of |n_eff| for this case is well established [20, 21, 27]. When ak ₀ is sufficiently large, |n_eff| asymptotes to n_spp, as expected. As ak ₀ is reduced, |n_eff| is decreased monotonically. When ak ₀ gets small and crosses ξ , the mode transits from the plasmonic mode to the photonic mode. As ak ₀ reaches ζ, |n_eff| asymptotes to zero and the mode is cut off. Neither the simultaneous presence of two plasmonic and photonic modes nor the degeneracy is observed in this case.

Here, a question arises: what happens at the degenerate point? To answer this question, we calculated the normalized optical power flow P_norm (See Appendix A). Figures 4(d)–(f) show the normalized optical power flow of the anti-symmetric mode for the cases of ρ<1, 1<ρ<1.28 , and ρ > 1.28 , respectively. It is seen in Fig. 4(d) that the anti-symmetric plasmonic and photonic modes for ρ < 1 exhibit a negative normalized optical power flow, indicating that this solution corresponds to the negative mode. Let us remind that the sign of the power flow in the metal is negative due to the negative ε_m and that in the dielectric is positive owing to the positive ε_d. Whether the normalized total optical power flow is positive or negative depends on how much energy is guided through the metal and the dielectric. If the portion of energy guided in the metal cladding is larger than that in the dielectric core, the overall optical power flow becomes negative and vice versa. Note also that the skin depth of the surface mode propagating along the interface between a metal and a dielectric is inversely proportional to the relative electric permittivity i.e., the skin depths into the metal and the dielectric scale with 1|ε_m| and 1/ε_d. Therefore if ρ < 1 i.e., |ε_m| < ε_d, then it is expected that more energy is guided through the metal region. The negative power flow of the anti-symmetric mode in the MIM waveguide can be ascribed to this property [24]. Unlike the anti-symmetric mode in the case ρ < 1, the anti-symmetric mode in the case ρ > 1.28 shows always positive values (Fig. 4(f)). This is because the portion of energy guided through the dielectric core is more than that through the metal cladding. In other words, the anti-symmetric modes for the case ρ > 1.28 are always the positive modes.

The behavior in the case of 1<ρ<1.28 is noteworthy (Fig. 4(e)). When the single-interface SPPs are considered, it is expected that the total power flow is positive since the skin depth to the dielectric is longer than that to the metal (ρ>1). However, the anti-symmetric mode in the MIM waveguide originates from the coupling between two single-interface SPPs. Note that the anti-symmetric mode has a node line along the center of the dielectric core (x = 0), indicating that a certain portion of energy should be pumped out from the dielectric core and guided in the metal. Consequently, there exists chance that the energy guided in the metal cladding is the same as that in the dielectric core. If the reduced core width ak ₀ is large enough, then the normalized optical power flow is positive due to the fact that more energy resides in the dielectric core. As ak ₀ is decreased, the portion of the energy in the metal is increased, resulting in the decrease of the normalized optical power flow. If ak ₀ < ζ, then the anti-symmetric photonic mode whose power flow is negative emerges and two anti-symmetric modes are supported simultaneously. As ak ₀ continues to decrease and reaches a certain point, two modes degenerate into one mode whose normalized optical power flow vanishes, as can be shown in Fig. 4(e). In other words, the condition 1<ρ<1.28 guarantees that there exists a certain value of the reduced core width at which the normalized optical power flow vanishes. The rigorous verification of this property is presented in Appendix B.

 

Fig. 6. Dependence of (a) |n_eff| and (b) P_norm on ε_d. (ε_d,ak ₀) = (-1.1,0.712) (c) ω - β dispersion relation. (ε_d,2a,ω_P) = (1.0,90nm,6.83 × 10¹⁵). The abscissa is normalized with the bulk plasma wavenumber β_P , while the ordinate is normalized with the bulk plasma frequency ω_P. (d) The dependence of P_norm on ω. The parameters are the same as those in (c).

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So far, we have focused on the geometrical dispersion i.e., the dependence of |n_eff| on ak ₀. It was shown that, when 1<ρ<1.28, a specific core width exists where the normalized optical power flow vanishes for given values of the dielectric material ε_d and the angular frequency ω. The zero power flow can be obtained from another approaches, i.e., varying either ε_d or ω. Figures 5(a) and 5(b) illustrate |n_eff| and P_norm as a function of ε_d. Here, ak ₀ is chosen in such a way that P_norm vanishes for ε_d =1. The anti-symmetric plasmonic mode is supported for 1<ε_d<1.09 and the sign of P_norm is positive. It is observed that, as ε_d grows to 1.09 , |n_eff| diverges to infinite and the anti-symmetric plasmonic mode is cut off. As ε_d is decreased, |n_eff| of the anti-symmetric plasmonic mode is also decreased. However, the anti-symmetric photonic mode exists for 1<ε_d<1.17 with the negative P_norm. |n_eff| of the anti-symmetric photonic mode is inversely proportional with ε_d. It is noteworthy that the mode degeneracy takes place for ε_d = 1. Figure 5(b) makes sure that the normalized optical power flow vanishes at ε_d =1. Meanwhile, it appears that Pnorm of the anti-symmetric plasmonic mode could vanish near ε_d = 1.1, which has not been predicted from our analysis. This point seems not to be associated with the mode degeneracy. As can be seen in Fig. 5(a), In_eff diverges at this point.

This would lead to the diverging of κ_d and κ_m (See Eqs. (7) and (8).). Unfortunately, it is not certain how the normalized optical power flow is affected by the diverging. It is thought that more study is required for this phenomenon.

Figures 5(c) and 5(d) show β and Pnorm as a function of ω according to the convention that ω is set along the ordinate. In Fig. 6(c), the ordinate is normalized with the bulk plasma frequency ω_P and the abscissa is normalized with the bulk plasma wavenumber β_p . The horizontal dash-dotted line indicates the surface plasma frequency ω = ωSP i.e., ρ = 1. The dashed straight line shows the light line β = ω√ε_d/c ₀. The dotted curve lying at the right side of the light line corresponds to the dispersion relation of the single-interface SPP. Note that ε_m is also a function of ω, which is given by Eq. (1). The solid curve denotes the dispersion relation of the anti-symmetric mode in the MIM waveguide. In Fig. 6(c), it is seen that a certain range of ω has two solutions of β. As ω decreases, two branches come close and the mode degeneracy takes place at a certain value of ω. It is remarkable that the group velocity, defined by dω/dβ becomes zero at the degenerate point. To best of our knowledge, the zero-group velocity in the plasmonic waveguide, which does not include any metamaterials for the core nor the cladding, has not been reported. Figure 6(d) confirms that the normalized optical power flow vanishes at the degenerate point.

Before continuing to examine other cases in the plasmonic waveguide, let us make a brief summary for the anti-symmetric mode in the MIM waveguide. It was shown that the condition of 1<ρ<1.28 guarantees that two anti-symmetric modes can be supported simultaneously for ξ < ak ₀ < ζ. These two modes degenerate into one mode at a certain reduced core width of ak ₀ < ξ, where the normalized optical power flow vanishes. The dispersion relation shows that the group velocity becomes zero at the degenerate point.

3.2 Symmetric mode

In this section, by employing a similar method used above for the case of the anti-symmetric mode, we investigate the condition for the existence of the symmetric plasmonic and photonic mode. It will be shown that the symmetric mode also has the mode degeneracy where the optical power flux vanishes.

The characteristic equation of the symmetric plasmonic mode is obtained from Eqs. (5), (7), and (8) with help of Eqs. (10), (11), and (13). Figures 7(a) and 7(b) illustrate two curves of W = -ρUtanhU and W = √U ² + V ² for those two cases of ρ<1 and ρ>1 , respectively. Note that ρUtanhU → 0 as U → 0 and ρU tanh U → ρU as U → ∞ , whereas √U ² + V ² + V ² → V as U → 0 and √U ² + V ² → as U → ω. Due to the behavior of the tanh function, ρU tanh U is always smaller than ρU. In addition, The change in the reduced core width ak ₀ affects only V, which is the W - intersect of the W = √U ² + V ² curve. Consequently, if ρ, which is the slope of the asymptote line of ρU tanh U, is smaller than one, there is no chance that two curves (W = ρUtanhU and W = √U ² + V ²) have an intersection (Fig. 7(b)). However, if ρ>1, then it is guaranteed that the symmetric plasmonic mode can always be supported (Fig. 7(a)). Table 3 summarizes the aforementioned discussion.

 

Fig. 7. Graphical method for the symmetric plasmonic mode in the MIM waveguide. (a) The case ρ>1. (ε_d,ρ) = (1.0,1.8) and ak ₀ = 0.8 . The symmetric plasmonic mode is always supported in the MIM waveguide regardless of ak ₀. (b) The case ρ<1 . (ε_d ,p) = (1.0,0.3) and ak ₀ = 0.3 . No symmetric plasmonic mode exists regardless of ak ₀.

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Next, let us examine the condition for the existence of the symmetric photonic mode in the MIM waveguide. From Eqs. (6), (7), and (9) with help of Eqs. (10), (12), and (13), we obtain two curves of W = -pJtanJ and W = √-j ² + V ² (See Fig. 8(a).). As in the anti-symmetric photonic mode, the domain of definition of J is given by J ∈ [0,ak ₀√ε_d] and we focus on the fundamental symmetric photonic mode. Let us consider the case when p is not so small (Fig. 8(a)). It will be shown that the condition for existence of the symmetric photonic mode gets more complex for ρ ≪ 1(Fig. 8(b)). Note that -ρJtanJ is positive only for J ∈ (nπ - π/2,nπ), where n is an integer. The fundamental symmetric photonic mode comes from J ∈ (π / 2,π). Hence it is required that ak ₀√ε_d>π/2, resulting in the condition ak ₀>σ with

If -ρJ tan J at J = ak ₀√ε_d (the point A in Fig. 8(a)) is smaller than √-J ² + V ² at J = ak ₀√ε_d (the point B in Fig. 8(a)), i.e., -ρ(ak ₀√ε_d) tan (ak ₀√ε_d) < ak ₀√-ε_m , then it is guaranteed that the symmetric photonic mode is supported. This relation reduces to a simple form of ak ₀ > χ, where

It should be pointed out that Eq. (18) can also be obtained with the substitution β = 0 into Eqs. (6), (7), and (9). Here, let us take the range of the tan^-1 function as (π/2,3π,2). Since √ε_d/(-ε_m) is positive, χ lies in (π/2√ε_d), π/ √ε_d). Thus χ is always larger than σ . Therefore the condition of ak ₀>χ is sufficient for the existence of the symmetric photonic mode.

 

Fig. 8. Graphical method for the symmetric photonic mode in the MIM waveguide. (a) The case with usual ρ. (ε_d,ρ = (1.0,2.0) and ak ₀ = 1.2χ. The definition of χ is given in Eq. (18). If W at the point A is smaller than that at the point B, it is guaranteed that the symmetric plasmonic mode is supported in the MIM waveguide. (b) The case ρ≪1. (ε_d,ρ) = (1.0,0.01) . ak ₀ =0.99χ. Two solutions are observed. As ak ₀ decreases, the radius V of the circle W = √-J ² + V ² also decreases and two solutions come closer to each other. It is expected that, at a certain value of ak ₀, two solutions would degenerate into one, resulting in trapped light.

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Now we deal with the case of ρ ≪ 1. It will be shown that the MIM waveguide can support two symmetric photonic modes for a certain combination (not clearly specified) of ρ and ak ₀. Let us recall that -ρJ tan J and √-J ² + V ² are decreasing function of J. Thus it is expected that those curves have more than one intersection under a certain criterion. In Fig. 8(b), it is seen that two curves (-ρJtanJ and √-J ² + V ²) have two intersections, indicating that two symmetric photonic modes are simultaneously supported. The decrease of ak ₀ gives rise to the decrease of the radius (V) of the circle √-J ² + V ² and two intersects come close to each other. Therefore, it is expected that, at a certain value of ak ₀, those two intersections degenerate into single point of contact, giving rise to trapped light. Table 3 summarizes the aforementioned discussion.

Table 3. Conditions for existence of the symmetric mode in the MIM waveguide. χ is defined in Eq. (18).

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Now let us consider the dependence of n_eff and P_norm on ak ₀ for the symmetric mode in the MIM waveguide. Figures 9(a)–(c) show results for the cases of ρ ≪ 1, ρ < 1, and ρ > 1, respectively. The vertical dashed and dash-dotted lines in Figs. 9(a)–(f) denote ak ₀ = σ and ak ₀ = χ, respectively. We first examine the ρ < 1 case (Fig. 9(b)). The horizontal dotted line indicates the refractive index of the cladding (|n_eff| = √ε_d). No symmetric plasmonic mode is observed, originating from the fact that a single interface between the metal with ε_m and the dielectric with ε_d does not support the plasmonic mode when ρ < 1. Note that this property is different from the result of the anti-symmetric plasmonic mode, which can be supported even if ρ < 1 (See Fig. 4(a)). In contrast to the symmetric plasmonic mode, the symmetric photonic mode is supported even for ρ < 1, provided that ak ₀ > χ. Figure 9(e), illustrating P_norm for the case ρ < 1, shows that the optical power flow of the symmetric photonic mode is positive.

Meanwhile, the functional behavior of the symmetric mode for ρ > 1 (Figs. 9(c) and 9(f)) is well established [20, 21, 26–28]. Ordinary plasmonic devices that consist of noble metals such as gold or silver and dielectric materials such as oxides or air satisfy -ε_m > ε_d in the visible and infrared regimes. As can be seen in Fig. 7(a), the symmetric plasmonic mode exists no matter how narrow the core width is. In other words, the symmetric plasmonic mode does not exhibit the mode cut off associated with the narrow core width [27]. It was also reported that the effective refractive index of the symmetric plasmonic mode in the MIM waveguide reveals an asymptotic behavior as the angular frequency is increased to the infrared regime [28]. Figure 9(f) reveals that its normalized optical power flow is always positive. The symmetric mode for ρ > 1 has a photonic mode when ak ₀> χ with a positive P_norm (See Figs. 9(c) and 9(f).).

 

Fig. 9. (a)-(c) Dependence of the effective refractive index n_eff as a function of the reduced core width ak ₀ for the symmetric mode in the MIM waveguide. (d)-(f) Effect of ak ₀ on the normalized optical power flow P_norm. (a) and (d) show the results for the case ρ ≪ 1 with (ε_d,ρ) = (1.0,0.01) . (b) and (e) depict the results for the case ρ<1 with (ε_d,ρ) = (1.0,0.2) . (c) and (f) illustrate the results for the case p>1 with (ε_d,ρ) = (1.0,1.5) . Note that the mode degeneracy occurs at a certain point ak ₀<χ in the case ρ ≪ 1, leading to trapping of light.

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A notable feature is that the symmetric mode can hold two photonic modes simultaneously if ρ ≪ 1 (Fig. 9(a)). This was expected in the analysis in Fig. 8(b). It turns out in Fig. 9(d) that the upper branch of these two photonic modes corresponds to the positive mode, whereas the lower branch the negative mode. At the degeneracy point, the symmetric mode conveys the optical power flow neither forward nor backward. Note that the reduced core width of the degeneracy is around ak ₀ = χ.

4. Insulator-metal-insulator plasmonic waveguide

In this section, we show that a certain condition exists that two symmetric modes can be simultaneously supported in the IMI waveguide. The optical power flow becomes zero at the degenerate point. On the other hand, the anti-symmetric mode has only one solution, and thus the zero optical power flow does not occur.

 

Fig. 10. Schematic diagram of the IMI waveguide

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Figure 10 shows the schematic diagram of the IMI waveguide. There are two kinds of modes: one is the anti-symmetric plasmonic mode and the other is the symmetric plasmonic mode. In contrast to the MIM waveguide, where the photonic mode as well as the plasmonic mode is allowed, the IMI waveguide does not support the photonic mode. This is because the metallic core of the IMI waveguide is opaque. As in the previous section, the wavelength is chosen to be 400 nm. We derive conditions that each mode is guided in the IMI waveguide. From the boundary condition that tangential electric and magnetic fields are continuous across the interface between the core and the cladding, the characteristic equations are given by [23]

4.1 Symmetric mode

Let us first deal with the symmetric mode. Only plasmonic mode is allowed. The characteristic equation of the symmetric plasmonic mode in the IMI waveguide is derived from Eqs. (19), (7), and (8) along with Eqs. (10), (11), and (13). Figures 11(a) and 11(b) depict the graphical method to determine the solution of the characteristic equation. The solution corresponds to the intersection between two curves U = ρ ^-1tanhW (the red dash-dotted lines in Figs. 11(a) and 11(b)) and U = √W ² - V ² (the blue solid lines in Figs. 11(a) and 11(b)). Their asymptotic lines are U = ρ ^-1 W (the black dashed lines) and U = W (the green dotted lines), respectively. The lines W = V are shown as the horizontal dashed lines. For U to be real, W should be larger than or equal to V, i.e., the domain of definition of W is W ∈ [V,∞]. As in the analysis for the MIM waveguide, discussion splits depending on whether ρ > 1 or ρ < 1.

 

Fig. 11. Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (ε_d,ρ) = (1.0,1.8) and ak ₀ = 0.8 . There is one intersection regardless of the change in ak ₀, indicating that one symmetric plasmonic mode is always allowed in this case. (b) The case ρ < 1 . (ε_d,ρ) = (1.0,0.5) and ak ₀ = 0.1 . For ak ₀ below a certain value, there are two intersections. As ak ₀ increases, two intersections come closer to each other, and at the certain value, they degenerate into single intersection.

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In Fig. 11(a), we illustrate the aforementioned two curves and their asymptotic lines for the case ρ > 1. The curve U = ρ ^-1 W tanh W grows from zero with the increase of W and is present to the right to the straight line U = ρ ^-1 W , the slope of which is smaller than one. On the other hand, the curve U = √w ² - V ² asymptotes to the line U = W , whose slope is one. As a result, if ρ > 1 , then two curves U = ρ ^-1 WtanhW and U = √W ² - V ² always have an intersection. Although the change in the core width, which contributes to the change in V, may affect the value of β of the mode, the number of the mode is still one.

Figure 11(b) show the two curves U = ρ ^-1 WtanhW and U =√W ² - V ² for the case ρ < 1. Let us first consider two extreme cases: one is the case where the core width is very small (V → 0) and the other is the case in which the core width is considerably large (V ≫ 1). In the former case, it is expected that there are two intersections at U = 0 and U = W , which corresponds to |n_eff| = √ε_d and |n_eff| → ∞, respectively. In the latter case, where the curve U = √W ² - V ² is moved to the right far away, there is no chance that U = ρ ^-1 WtanhW and U = √W ² - V ² have an intersection (see Fig. 11(b)). Consequently, it can be inferred that, as ak ₀ is increased from zero, |n_eff| of two modes would come closer to each other, and at a certain value of ak ₀, they would degenerate into single mode. Unfortunately, we could not extract the exact formulation to get the value of ak ₀ where the mode degeneracy occurs. Table 4 summarizes the aforementioned properties.

Table 4. Conditions for existence of the symmetric mode in the IMI waveguide

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Fig. 12. (a)-(b) Dependence of the effective refractive index |n_eff| as a function of the reduced core width ak ₀ for the symmetric mode in the IMI waveguide. (c)-(d) Effect of ak ₀ on the normalized optical power flow P_norm . (a) and (c) show the results for the case ρ < 1 with (ε_d,ρ) = (1.0,0.5 ) . It is noteworthy that the mode degeneracy occurs at a certain point in the case ρ<1 . (b) and (d) depict the results for the case ρ>1 with (ε_d, ρ) = (1.0,2.0).

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Figures 12(a)-(d) show the effect of ak ₀ on |n_eff| and P_norm in the case ρ < 1 and ρ > 1. As explained above, two symmetric modes are simultaneously allowed for the core width under a certain point (See Fig. 12(a).). The difference of |n_eff| between the upper and lower branches is decreased with the increase of ak ₀. At a certain point of ak ₀, these two modes degenerate into single mode. In Fig. 12(c), it is confirmed that the normalized optical power flow vanishes at the degenerate point, resulting in trapping of light. Contrarily, the results in the case ρ > 1 (Figs. 12(b) and 12(d)) show that the dependences of |n_eff| and P_norm on ak ₀ are monotonic. If ak ₀ is sufficiently large, the functional behavior asymptotes to the result of the single interface SPP (See Fig. 12(b).). As ak ₀ is decreased, more power is guided through the dielectric, which makes |n_eff| asymptotes to √ε_d. This is in good agreement with results of previous studies [23, 33, 34]. In the presence of material loss, this mode exhibits a long propagation length compared to the anti-symmetric mode and is referred to as the long range SPP [35]. In Fig. 12(d), it is observed that P_norm → 1 as ak ₀ → 0, which indicates that all power is guided through the dielectric cladding, leading to a disadvantage from a view point of confinement.

4.2 Anti-symmetric mode

The analysis on the anti-symmetric mode in the IMI waveguide is quite simple. In this section, it is shown that the anti-symmetric mode in the IMI waveguide exists only when ρ > 1 and this mode is a positive mode. In the case ρ < 1, there is no anti-symmetric mode regardless of the core width. In addition, the mode degeneracy does not occur for the anti-symmetric mode in the IMI waveguide.

 

Fig. 13. Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (ε_d,ρ) = (1.0,1.8) and ak ₀ = 0.8 . The anti-symmetric plasmonic mode is supported in the IMI waveguide regardless of the core width. (b) The case ρ < 1 . (ε_d, ρ) = (1.0,0.5) and ak ₀ = 2.0 . No anti-symmetric mode is guided in the IMI waveguide.

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As in the analysis for the symmetric mode in the IMI waveguide, the photonic mode is not supported and only the plasmonic mode is considered. The characteristic equation of the anti-symmetric plasmonic mode in the IMI waveguide is obtained from Eqs. (20), (7), and (8) along with Eqs. (10), (11), and (13). In Figs. 13(a) and 13(b), we illustrate two curves of U = ρ ^-1 WcothW (the red dash-dotted lines) and U =√W ² - V ² (the blue solid lines). Their asymptotic lines are given by U = ρ ^-1 W (the black dashed lines) and U = W (the green dotted lines), respectively. The horizontal dashed lines denote lines W = V . The domain of definition of W is given by W ∈ [V,∞].

Let us first consider the case ρ > 1 (Fig. 13(a)). As W is increased, U = √W ² - V ² grows monotonically from zero and asymptotes to the line U = W. In contrast, U = ρ^-1 WcothW increases with the increase of W and it asymptotes to the line U = ρ ^-1 W, the slope of which is smaller than that of U = W. Therefore, two curves U = ρ ^-1 WcothW and U = √W ² - V ² always have an intersection. The variance in ak ₀ may increase or decrease V, which is the W - intersect in the graph. However, the change in V does not affect the number of intersections, i.e., there is only one intersection regardless of the core width.

Next we move to the discussion for the case ρ < 1 (Fig. 13(b)). Note that the curve U = ρ ^-1 WcothW is always located in the upper region of its asymptotic line U = W ^-1 W, while the curve U = √W ² - V ² is under its asymptotic line U = W . Since the slope of U = ρ ^-1 W is larger than that of U = W , there is no chance that U = ρ ^-1 WcothW and U = √W ² - V ² have an intersection. Table 5 summarizes the aforementioned properties.

Table 5. Conditions for existence of the anti-symmetric mode in the IMI waveguide

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In Figs. 14(a) and 14(b), we illustrate |n_eff| and P3 as a function of ak ₀ in the case ρ > 1. If ak ₀ is large, |n_eff| asymptotes to the line |n_eff| = n_spp (Fig. 14(a)). As ak ₀ dwindles, |n_eff| is increased monotonically and cut off is not observed. This coincides well with published results [23, 33, 34]. In particular, it has been reported that the absence of cut off can be used to confine light in the subwavelength scale [36, 37]. The nanofocusing of light in a metallic nano rod was theoretically expected [36], which was experimentally demonstrated by using the asymmetric cladding and adiabatic tapering [37]. The guided modes explored in these reports are associated with the anti-symmetric mode in the IMI waveguide [33, 37]. If the material loss is taken into account, this mode shows a short propagation length compared to the symmetric mode and is called the short range SPP [35]. The dependence of |n_eff| and P_norm on ak ₀ in the case ρ < 1 is not provided since no mode exists in that case. Thus the mode degeneration and trapping of light do not occur for the anti-symmetric mode in the IMI waveguide.

 

Fig. 14. (a) Dependence of the effective refractive index |n_eff| as a function of the reduced core width ak ₀ for the anti-symmetric mode in the IMI waveguide for the case ρ > 1. (b) Effect of ak ₀ on the normalized optical power flow P_norm. (ε_d,ρ) = (1.0,2.0) in common. Regardless of ak ₀, a plasmonic positive mode always exists.

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5. Conclusion

Comprehensive case studies on trapping of light in the plasmonic waveguide are presented. The fundamental principle is simply based on the fact that the direction of the optical power flow in a metal composing a plasmonic waveguide is opposite to that in a dielectric. If amounts of these two optical power flows are the same, then the total optical power flow vanishes. However, the problem on the existence of such a balance point in a plasmonic waveguide is not trivial. The ratio between the optical power flows guided in the metal and the dielectric is governed by two factors: one is a ratio ρ between the relative electric permittivity of the metal ε_d and the dielectric ε_d i.e., ρ = -ε_m/ε_d. The other is a ratio between the core width 2a and the operating wavelength λ ₀ = 2π/k ₀, and we use the reduced core width ak ₀. The rigorous case studies conducted for the lossless two dimensional MIM and IMI waveguides demonstrate that trapping of light occurs in the following cases: 1) the anti-symmetric mode in the MIM waveguide with 1<ρ<1.28 and ak ₀ < ξ (Figs. 4(b) and 4(e)), 2) the symmetric mode in the MIM waveguide with ρ ≪ 1 and ak ₀ < χ (Figs. 9(a) and 9(d)), and 3) the symmetric mode in the IMI waveguide with ρ < 1 and small ak ₀ (Figs. 12(a) and 12(c). The explicit condition on the reduced core width is not given). The definitions of ξ, and χ are presented in Eqs. (14) and (18), respectively. It is believed that the aforementioned results of this study can open a way to implement various optical devices with trapped or slow light.