Abstract

We present comprehensive case studies on trapping of light in plasmonic waveguides, including the metal-insulator-metal (MIM) and insulator-metal-insulator (IMI) waveguides. Due to the geometrical symmetry, the guided modes are classified into the anti-symmetric and symmetric modes. For the lossless case, where the relative electric permittivity of metal (εm) and dielectric (εd) are purely real, we define ρ as ρ = -εm/εd. It is shown that trapping of light occurs in the following cases: the anti-symmetric mode in the MIM waveguide with 1 < ρ < 1.28, the symmetric mode in the MIM waveguide with ρ ≪ 1, and the symmetric mode in the IMI waveguide with ρ < 1. The physical interpretation reveals that these conditions are closely connected with the field distributions in the core and the cladding. Various mode properties such as the number of supported modes and the core width for the mode cut off are also presented.

©2010 Optical Society of America

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 εm and εd, respectively. The relative magnetic permeability is assumed to be unity in all material. The dependence of εm on ω is governed by the Drude model:

εm=1ωp2ω(ω+iγ),

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

ρ=εmεd=εmεd.

Note that ρ>0 and ρ decreases as ω increases. By combining Eqs. (1) and (2), we know that ρ>1 for ω<ωSP and ρ<1 for ω>ωSP, where ωSP=ωP/1+εd is the surface plasma frequency. The bulk plasma wavelength λP and wavenumber βP are defined by 2πc 0/ωp and ωP/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 neff is defined as β/k 0. neff of the surface plasmon polariton (SPP) propagating along the single interface between the metal with εm and the dielectric with εd, called the single interface SPP, is given by nSPP=εmεd/(εm+εd).

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 neff > √εd is referred to as the plasmonic mode and that with neff < √εd 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 (/ )-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 = βr + i), where βr and βi denote the real and imaginary parts of β, respectively [22]. Then the wave function has a factor of exp[j(βrZ-ω)]exp(-βiz). The dispersion relation is plotted as ω versus βr and the propagation length is defined as 1/(2/βi). 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 + i), where ωr and ωi denote the real and imaginary parts of ω, respectively [22]. The wave function has a factor of exp[j(βz - ωrt)]exp(-ωi). From this point of view, the propagation length is not defined, Instead, the life time τ = 1/ωi 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.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the MIM waveguide

Download Full Size | PPT Slide | PDF

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]

κmεm+κdεdcoth(κda)=0(theantisymmetricplasmonicmode),
κmεm+κdεdcot(κda)=0(theantisymmetricphotonicmode),
κmεm+κdεdtanh(κda)=0(thesymmetricplasmonicmode),
κmεm+κdεdtan(κda)=0(thesymmetricphotonicmode).

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

κm2+β2=εmk02,
κd2+β2=εdk02,
κd2+β2=εdk02.

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

W=aκm,
U=aκd,
J=akd,
V=ak0εdεm.

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 ωP.

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=U2+V2 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, U2+V2 asymptotes to U for U → ∞ and the limiting value of U2+V2 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=U2+V2 (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=U2+V2, 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=U2+V2. In other words, for the case ρ>1 i.e., ω < ωSP, 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

ξ=ρεdεm=εmεd(εdεm).

It should be pointed out that putting β = k 0 εd 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=U2+V2, 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.

 figure: Fig. 2.

Fig. 2. Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. (a) The case ρ > 1. (εd,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. (εd,ρ) = (1.0,0.8) and ak 0 = 0.8ξ. The anti-symmetric plasmonic mode exists in the MIM waveguide when ak 0 < ξ.

Download Full Size | PPT Slide | PDF

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=J2+V2 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 neff 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 0εd]. This is because β is purely real and thus β 2 should be positive (See Eqs. (9) and (12).).

 figure: Fig. 3.

Fig. 3. Graphical method for the anti-symmetric photonic mode in the MIM waveguide. (a) The case ρ > V. (εd,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. (εd,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.

Download Full Size | PPT Slide | PDF

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=J2+V2 (the blue solid line) for ρ > V. The bold lines illustrate lines in the domain of definition (J ∈ [0, ak 0εd]), whereas thin lines denote lines outside the domain of definition. In the case ρ > V, if ρJ cot J at J = ak 0εd (the point A in Fig. 3(a)) is smaller than W=J2+V2 at J = ak 0εd (the point B in Fig. 3(a)) i.e., ρ√εdcot(ak 0εd) < √-εm, it is guaranteed that the anti-symmetric photonic mode exists i.e., ak 0 > ζ, where

ζ=1εdcot1εmρεd=1εdcot1εdεm.

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 0εd) > √-εm 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 = ρc, where ρc is a solution satisfying

ρc=tanρc(1+ρc).

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

Tables Icon

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.

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 |neff| 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 |neff|, rather than neff, is used here, owing to the fact that in some cases a negative-valued neff 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 |neff|=√εd. The dash-dotted horizontal lines in Figs. 4(b) and 4(c) correspond to the lines of |neff| = nspp . In Fig. 4(a), it is seen that |neff| 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, |neff| 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].

 figure: Fig. 4.

Fig. 4. (a)-(c) Dependence of the effective refractive index neff 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 Pnorm. The definition of Pnorm is given in Appendix A. (a) and (d) show the results for the case ρ < 1 with (εd,ρ) = (1.0,0.8). (b) and (e) depict the results for the case 1 < ρ < 1.28 with (εd, ρ) = (1.0,1.1). (c) and (f) illustrate the results for the case ρ > 1.28 with (εd, 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 Pnorm = 0 (See Fig. 4(e)) i.e., trapping of light.

Download Full Size | PPT Slide | PDF

Figure 4(b) illustrates |neff| 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 |neff| 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 hc, 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.

 figure: Fig. 5.

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

Download Full Size | PPT Slide | PDF

Tables Icon

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

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 |neff| for ρ>1.28 . The functional behavior of |neff| for this case is well established [20, 21, 27]. When ak 0 is sufficiently large, |neff| asymptotes to nspp, as expected. As ak 0 is reduced, |neff| 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, |neff| 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 Pnorm (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 εm and that in the dielectric is positive owing to the positive εd. 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|εm| and 1/εd. Therefore if ρ < 1 i.e., |εm| < εd, 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.

 figure: Fig. 6.

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

Download Full Size | PPT Slide | PDF

So far, we have focused on the geometrical dispersion i.e., the dependence of |neff| 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 εd and the angular frequency ω. The zero power flow can be obtained from another approaches, i.e., varying either εd or ω. Figures 5(a) and 5(b) illustrate |neff| and Pnorm as a function of εd. Here, ak 0 is chosen in such a way that Pnorm vanishes for εd =1. The anti-symmetric plasmonic mode is supported for 1<εd<1.09 and the sign of Pnorm is positive. It is observed that, as εd grows to 1.09 , |neff| diverges to infinite and the anti-symmetric plasmonic mode is cut off. As εd is decreased, |neff| of the anti-symmetric plasmonic mode is also decreased. However, the anti-symmetric photonic mode exists for 1<εd<1.17 with the negative Pnorm. |neff| of the anti-symmetric photonic mode is inversely proportional with εd. It is noteworthy that the mode degeneracy takes place for εd = 1. Figure 5(b) makes sure that the normalized optical power flow vanishes at εd =1. Meanwhile, it appears that Pnorm of the anti-symmetric plasmonic mode could vanish near εd = 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), Ineff diverges at this point.

This would lead to the diverging of κd and κm (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 ωP and the abscissa is normalized with the bulk plasma wavenumber βp . 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 = ωεd/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 εm 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 / 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 UρU as U → ∞ , whereas √U 2 + V 2 + V 2V as U → 0 and √U 2 + V 2 → as Uω. 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.

 figure: Fig. 7.

Fig. 7. Graphical method for the symmetric plasmonic mode in the MIM waveguide. (a) The case ρ>1. (εd,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 . (εd ,p) = (1.0,0.3) and ak 0 = 0.3 . No symmetric plasmonic mode exists regardless of ak 0.

Download Full Size | PPT Slide | PDF

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 0εd] 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 ∈ ( - π/2,), where n is an integer. The fundamental symmetric photonic mode comes from J ∈ (π / 2,π). Hence it is required that ak 0εd>π/2, resulting in the condition ak 0>σ with

σ=π2εd.

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

χ=1εdtan1(εdεm).

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 √εd/(-εm) is positive, χ lies in (π/2√εd), π/ √εd). Thus χ is always larger than σ . Therefore the condition of ak 0>χ is sufficient for the existence of the symmetric photonic mode.

 figure: Fig. 8.

Fig. 8. Graphical method for the symmetric photonic mode in the MIM waveguide. (a) The case with usual ρ. (εd,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. (εd,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.

Download Full Size | PPT Slide | PDF

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.

Tables Icon

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

Now let us consider the dependence of neff and Pnorm 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 (|neff| = √εd). No symmetric plasmonic mode is observed, originating from the fact that a single interface between the metal with εm and the dielectric with εd 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 Pnorm 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 -εm > εd 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 Pnorm (See Figs. 9(c) and 9(f).).

 figure: Fig. 9.

Fig. 9. (a)-(c) Dependence of the effective refractive index neff 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 Pnorm. (a) and (d) show the results for the case ρ ≪ 1 with (εd,ρ) = (1.0,0.01) . (b) and (e) depict the results for the case ρ<1 with (εd,ρ) = (1.0,0.2) . (c) and (f) illustrate the results for the case p>1 with (εd,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.

Download Full Size | PPT Slide | PDF

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.

 figure: Fig. 10.

Fig. 10. Schematic diagram of the IMI waveguide

Download Full Size | PPT Slide | PDF

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]

κdεd+κmεmtanh(κma)=0(thesymmetricplasmonicmode),
κdεd+κmεmcoth(κma)=0(theantisymmetricplasmonicmode).

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.

 figure: Fig. 11.

Fig. 11. Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (εd,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 . (εd,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.

Download Full Size | PPT Slide | PDF

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 |neff| = √εd and |neff| → ∞, 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, |neff| 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.

Tables Icon

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

 figure: Fig. 12.

Fig. 12. (a)-(b) Dependence of the effective refractive index |neff| 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 Pnorm . (a) and (c) show the results for the case ρ < 1 with (εd,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 (εd, ρ) = (1.0,2.0).

Download Full Size | PPT Slide | PDF

Figures 12(a)-(d) show the effect of ak 0 on |neff| and Pnorm 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 |neff| 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 |neff| and Pnorm 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 |neff| asymptotes to √εd. 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 Pnorm → 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.

 figure: Fig. 13.

Fig. 13. Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (εd,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 . (εd, ρ) = (1.0,0.5) and ak 0 = 2.0 . No anti-symmetric mode is guided in the IMI waveguide.

Download Full Size | PPT Slide | PDF

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.

Tables Icon

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

In Figs. 14(a) and 14(b), we illustrate |neff| and P3 as a function of ak 0 in the case ρ > 1. If ak 0 is large, |neff| asymptotes to the line |neff| = nspp (Fig. 14(a)). As ak 0 dwindles, |neff| 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 |neff| and Pnorm 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.

 figure: Fig. 14.

Fig. 14. (a) Dependence of the effective refractive index |neff| 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 Pnorm. (εd,ρ) = (1.0,2.0) in common. Regardless of ak 0, a plasmonic positive mode always exists.

Download Full Size | PPT Slide | PDF

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 εd and the dielectric εd i.e., ρ = -εm/εd. 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.

Appendix A

The y -component of the magnetic field of the anti-symmetric plasmonic mode in the MIM waveguide is expressed as

Hy(x,z;t)={A1exp[κm(xa)]exp[j(βzωt)](xa)B1sinh(κdx)exp[j(βzωt)](axa)A1exp[κm(xa)]exp[j(βzωt)](xa)

From the Maxwell’s curl equation, the x - and z - components of the electric field of the anti-symmetric plasmonic mode are given by

Ex(x,z;t)={(βωε0εm)A1exp[κm(xa)]exp[j(βzωt)](xa)(βωε0εd)B1sinh(κdx)exp[j(βzωt)](axa)(βωε0εm)A1exp[κm(xa)]exp[j(βzωt)](xa).
Ez(x,z;t)={(κmjωε0εm)A1exp[κm(xa)]exp[j(βzωt)](xa)(κdjωε0εd)B1cosh(κdx)exp[j(βzωt)](axa)(κmjωε0εm)A1exp[κm(xa)]exp[j(βzωt)](xa)

The relation between the coupling coefficients A 1 and B 1 is obtained from the boundary condition for Eqs. (A1) and (A3) as

A1=B1sinh(κda),
κmεmA1=κdεdB1cosh(κda),

which results in the characteristic equation

κmεm=κdεdcoth(κda).

The z -component of the Poynting vector Pz = ExH * y/2 is given from Eqs. (A1), (A2), and (A4) as

Pz(x,z;t)={(βB12ωε0)(1εm)sinh2(κda)exp[2κm(xa)](xa)(βB12ωε0)(1εd)sinh2(κdx)(axa)(βB12ωε0)(1εm)sinh2(κda)exp[2κm(xa)](xa).

Note that κd and κm are assumed to be purely real here. By taking an integral in Eq. (A7), we get the total power propagating through the metal cladding and the dielectric core as follows:

Pmetal=2a{(βB12ωε0)(1εm)sinh2(κda)exp[2κm(xa)]}dx
=2[(βB12ωε0)(1εm)sinh2(κda)]0exp[2κmx]dx
=(βB12ωε0)(sinh2(κda)εmκm),
Pdielectric=2aa(βB12ωε0)(1εd)sinh2(κdx)dx
=2(βB12ωε0)(1εd)0asinh2(κdx)dx
=2(βB12ωε0)(1εd)[sinh(2κdx)4κdx2]0a
=(βB12ωε0)(1εdκd)[sinh(2κda)2aκd].

The normalized power flow Pnorm = (Pdielectric + Pmetal)/(Pdielectric + Pmetal) for the anti-symmetric plasmonic mode in the MIM waveguide is defined as

Pnorm=sinh(2κda)/2aκdεdκd+sinh2(κda)εmκmsinh(2κda)/2aκdεdκdsinh2(κda)εmκm.

The counterparts of Eqs. (A6), (A10) for the anti-symmetric photonic mode, the symmetric plasmonic and photonic modes in the MIM waveguide, and the anti-symmetric and symmetric plasmonic modes in the IMI waveguide can be derived in the same way by replacing Eq. (A1) as

Hy(x,z;t)={A2exp[κm(xa)]exp[j(βzωt)](xa)B2sin(κdx)exp[j(βzωt)](axa)A2exp[κm(xa)]exp[j(βzωt)](xa),
Hy(x,z;t)={A3exp[κm(xa)]exp[j(βzωt)](xa)B3cosh(κdx)exp[j(βzωt)](axa)A3exp[κm(xa)]exp[j(βzωt)](xa),
Hy(x,z;t)={A4exp[κm(xa)]exp[j(βzωt)](xa)B4cos(κdx)exp[j(βzωt)](axa)A4exp[κm(xa)]exp[j(βzωt)](xa),
Hy(x,z;t)={A5exp[κd(xa)]exp[j(βzωt)](xa)B5sinh(κmx)exp[j(βzωt)](axa)A5exp[κd(xa)]exp[j(βzωt)](xa),
Hy(x,z;t)={A6exp[κd(xa)]exp[j(βzωt)](xa)B6cosh(κmx)exp[j(βzωt)](axa)A6exp[κd(xa)]exp[j(βzωt)](xa),

respectively.

Appendix B

From the characteristic equation for the anti-symmetric plasmonic mode in the MIM waveguide i.e., Eq. (3), we calculate the partial derivative of ak 0 on neff as follows:

neff2εmεm+neff2εdεdcoth(ak0neff2εd)=0.

The derivative of ak 0 on neff is obtained as follows:

neffk0εmκm+neffk0εdκdcoth(ak0neff2εd)κdεdk0csch2(ak0neff2εd)(d(ak0)dneffκdk0+ak02neffκd)=0,

which results in

d(ak0)dneff=neffεdk03κd2[sinh(2κda)/2aκdεdκd+sinh2(κda)εmκm].

Comparison between Eqs. (A10) to (B3) reveals that the sign of d(ak 0)/dneff is the same as that of Pnorm. Therefore the normalized power flux vanishes at the mode degenerate point, where d(ak 0)/dneff = 0. In addition, it can be shown that positive is the normalized power flux of the upper branch of the anti-symmetric plasmonic mode for 1< ρ < 1.28, where neff is decreased with the decrease of ak 0 (d(ak 0)/d neff > 0), whereas negative is that of the lower branch owing to the fact that d(ak 0)/dneff, < 0 (See Fig. 4(a).). The counterparts of d(ak 0)/d neff for the anti-symmetric photonic mode, the symmetric plasmonic and photonic modes in the MIM waveguide, and the anti-symmetric and symmetric plasmonic modes in the IMI waveguide can be derived in the similar way.

Acknowledgment

This work was supported by the National Research Foundation and the Ministry of Education, Science and Technology of Korea through the Creative Research Initiatives Program (Active Plasmonics Application Systems).

References and links

1. T. F. Krauss, “Why do we need slow light?,” Nature Photon. 2, 448–450 (2008). [CrossRef]  

2. L. Thevenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008). [CrossRef]  

3. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002). [CrossRef]  

4. D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008). [CrossRef]   [PubMed]  

5. M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16, 18657–18666 (2008). [CrossRef]  

6. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009). [CrossRef]   [PubMed]  

7. T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D 40, 2659–2665 (2007). [CrossRef]  

8. T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007). [CrossRef]   [PubMed]  

9. D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007). [CrossRef]   [PubMed]  

10. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008). [CrossRef]  

11. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007). [CrossRef]   [PubMed]  

12. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006). [CrossRef]  

13. Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008). [CrossRef]  

14. Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008). [CrossRef]  

15. Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009). [CrossRef]   [PubMed]  

16. K. Y. Kim, “Tunneling-induced temporary light trapping in negative-index-clad slab waveguide,“ Jpn. J. Appl. Phys. 47, 4843–4845 (2008). [CrossRef]  

17. V. M. Shalaev, “Optical negative-index metamaterials,“ Nature Photon. 1, 41–48 (2007). [CrossRef]  

18. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008). [CrossRef]   [PubMed]  

19. J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008). [CrossRef]   [PubMed]  

20. B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991). [CrossRef]  

21. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006). [CrossRef]  

22. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008). [CrossRef]  

23. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005). [CrossRef]  

24. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007). [CrossRef]   [PubMed]  

25. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006). [CrossRef]   [PubMed]  

26. J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,“ Opt. Express 16, 413–425 (2008). [CrossRef]   [PubMed]  

27. J. Park, H. Kim, I.-M. Lee, S. Kim, J. Jung, and B. Lee, “Resonant tunneling of surface plasmon polariton in the plasmonic nano-cavity,“ Opt. Express 16, 16903–16915 (2008). [CrossRef]   [PubMed]  

28. J. Park and B. Lee, “An approximate formula of the effective refractive index of the metal-insulator-metal surface plasmon polariton waveguide in the infrared region,“ Jpn. J. Appl. Phys. 47, 8449–8451 (2008). [CrossRef]  

29. E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,“ Opt. Express 17, 2465–2469 (2009). [CrossRef]   [PubMed]  

30. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,“ Opt. Express 16, 21209–21214 (2008). [CrossRef]   [PubMed]  

31. E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).

32. F. Kusunoki, T. Yotsuya, and J. Takahara, “Confinement and guiding of two-dimensional optical waves by low-refractive-index cores,“ Opt. Express 14, 5651–5656 (2006). [CrossRef]   [PubMed]  

33. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986). [CrossRef]  

34. J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,“ Optics and Photonics News 6, 54–59 (2004). [CrossRef]  

35. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,“ Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]  

36. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,“ Phys. Rev. Lett. 93, 137404 (2004). [CrossRef]   [PubMed]  

37. E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,“ Opt. Express 16, 45–57 (2008). [CrossRef]   [PubMed]  

References

  • View by:
  • |
  • |
  • |

  1. T. F. Krauss, “Why do we need slow light?,” Nature Photon. 2, 448–450 (2008).
    [Crossref]
  2. L. Thevenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008).
    [Crossref]
  3. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
    [Crossref]
  4. D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
    [Crossref] [PubMed]
  5. M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16, 18657–18666 (2008).
    [Crossref]
  6. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
    [Crossref] [PubMed]
  7. T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D 40, 2659–2665 (2007).
    [Crossref]
  8. T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007).
    [Crossref] [PubMed]
  9. D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
    [Crossref] [PubMed]
  10. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
    [Crossref]
  11. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
    [Crossref] [PubMed]
  12. K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
    [Crossref]
  13. Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
    [Crossref]
  14. Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
    [Crossref]
  15. Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009).
    [Crossref] [PubMed]
  16. K. Y. Kim, “Tunneling-induced temporary light trapping in negative-index-clad slab waveguide,“ Jpn. J. Appl. Phys. 47, 4843–4845 (2008).
    [Crossref]
  17. V. M. Shalaev, “Optical negative-index metamaterials,“ Nature Photon. 1, 41–48 (2007).
    [Crossref]
  18. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
    [Crossref] [PubMed]
  19. J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
    [Crossref] [PubMed]
  20. B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991).
    [Crossref]
  21. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
    [Crossref]
  22. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008).
    [Crossref]
  23. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
    [Crossref]
  24. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007).
    [Crossref] [PubMed]
  25. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006).
    [Crossref] [PubMed]
  26. J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,“ Opt. Express 16, 413–425 (2008).
    [Crossref] [PubMed]
  27. J. Park, H. Kim, I.-M. Lee, S. Kim, J. Jung, and B. Lee, “Resonant tunneling of surface plasmon polariton in the plasmonic nano-cavity,“ Opt. Express 16, 16903–16915 (2008).
    [Crossref] [PubMed]
  28. J. Park and B. Lee, “An approximate formula of the effective refractive index of the metal-insulator-metal surface plasmon polariton waveguide in the infrared region,“ Jpn. J. Appl. Phys. 47, 8449–8451 (2008).
    [Crossref]
  29. E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,“ Opt. Express 17, 2465–2469 (2009).
    [Crossref] [PubMed]
  30. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,“ Opt. Express 16, 21209–21214 (2008).
    [Crossref] [PubMed]
  31. E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).
  32. F. Kusunoki, T. Yotsuya, and J. Takahara, “Confinement and guiding of two-dimensional optical waves by low-refractive-index cores,“ Opt. Express 14, 5651–5656 (2006).
    [Crossref] [PubMed]
  33. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986).
    [Crossref]
  34. J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,“ Optics and Photonics News 6, 54–59 (2004).
    [Crossref]
  35. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,“ Phys. Rev. B 61, 10484–10503 (2000).
    [Crossref]
  36. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,“ Phys. Rev. Lett. 93, 137404 (2004).
    [Crossref] [PubMed]
  37. E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,“ Opt. Express 16, 45–57 (2008).
    [Crossref] [PubMed]

2009 (3)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref] [PubMed]

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009).
[Crossref] [PubMed]

E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,“ Opt. Express 17, 2465–2469 (2009).
[Crossref] [PubMed]

2008 (17)

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,“ Opt. Express 16, 21209–21214 (2008).
[Crossref] [PubMed]

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008).
[Crossref]

J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,“ Opt. Express 16, 413–425 (2008).
[Crossref] [PubMed]

J. Park, H. Kim, I.-M. Lee, S. Kim, J. Jung, and B. Lee, “Resonant tunneling of surface plasmon polariton in the plasmonic nano-cavity,“ Opt. Express 16, 16903–16915 (2008).
[Crossref] [PubMed]

J. Park and B. Lee, “An approximate formula of the effective refractive index of the metal-insulator-metal surface plasmon polariton waveguide in the infrared region,“ Jpn. J. Appl. Phys. 47, 8449–8451 (2008).
[Crossref]

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,“ Opt. Express 16, 45–57 (2008).
[Crossref] [PubMed]

K. Y. Kim, “Tunneling-induced temporary light trapping in negative-index-clad slab waveguide,“ Jpn. J. Appl. Phys. 47, 4843–4845 (2008).
[Crossref]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
[Crossref]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
[Crossref]

T. F. Krauss, “Why do we need slow light?,” Nature Photon. 2, 448–450 (2008).
[Crossref]

L. Thevenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008).
[Crossref]

D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Ultracompact and low-power optical switch based on silicon photonic crystals,” Opt. Lett. 33, 147–149 (2008).
[Crossref] [PubMed]

M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16, 18657–18666 (2008).
[Crossref]

2007 (6)

T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D 40, 2659–2665 (2007).
[Crossref]

T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007).
[Crossref] [PubMed]

D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
[Crossref] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

V. M. Shalaev, “Optical negative-index metamaterials,“ Nature Photon. 1, 41–48 (2007).
[Crossref]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007).
[Crossref] [PubMed]

2006 (4)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

F. Kusunoki, T. Yotsuya, and J. Takahara, “Confinement and guiding of two-dimensional optical waves by low-refractive-index cores,“ Opt. Express 14, 5651–5656 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
[Crossref]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

2005 (1)

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
[Crossref]

2004 (2)

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,“ Phys. Rev. Lett. 93, 137404 (2004).
[Crossref] [PubMed]

J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,“ Optics and Photonics News 6, 54–59 (2004).
[Crossref]

2002 (1)

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[Crossref]

2000 (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,“ Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991).
[Crossref]

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Atwater, H. A.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008).
[Crossref]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007).
[Crossref] [PubMed]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
[Crossref]

Baba, T.

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Bartoli, F. J.

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009).
[Crossref] [PubMed]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
[Crossref]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
[Crossref]

Beggs, D. M.

Berini, P.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,“ Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

Boyd, R. W.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[Crossref]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Davoyan, A. R.

Ding, Y. J.

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009).
[Crossref] [PubMed]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
[Crossref]

Ding, Y. J. J.

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
[Crossref]

Dionne, J. A.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008).
[Crossref]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007).
[Crossref] [PubMed]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
[Crossref]

Feigenbaum, E.

E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,“ Opt. Express 17, 2465–2469 (2009).
[Crossref] [PubMed]

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).

Fu, Z.

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
[Crossref]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
[Crossref]

Gan, Q. Q.

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009).
[Crossref] [PubMed]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
[Crossref]

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
[Crossref]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

Heebner, J. E.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[Crossref]

Hermann, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

Jamois, C.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

Jung, J.

Kaminski, N.

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).

Kawasaki, T.

Kim, H.

Kim, K. Y.

K. Y. Kim, “Tunneling-induced temporary light trapping in negative-index-clad slab waveguide,“ Jpn. J. Appl. Phys. 47, 4843–4845 (2008).
[Crossref]

Kim, S.

Kivshar, Y. S.

Klaedtke, A.

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

Kobayashi, T.

J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,“ Optics and Photonics News 6, 54–59 (2004).
[Crossref]

Krauss, T. F.

Kubo, S.

Kuipers, L.

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref] [PubMed]

Kusunoki, F.

Lee, B.

Lee, I.-M.

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007).
[Crossref] [PubMed]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

Liu, Y. M.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Liu, Z. W.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Mori, D.

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991).
[Crossref]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref] [PubMed]

M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16, 18657–18666 (2008).
[Crossref]

O’Faolain, L.

Orenstein, M.

E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,“ Opt. Express 17, 2465–2469 (2009).
[Crossref] [PubMed]

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).

Park, J.

Park, Q. H.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[Crossref]

Polman, A.

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008).
[Crossref]

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,“ Opt. Express 16, 45–57 (2008).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
[Crossref]

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991).
[Crossref]

Sasaki, H.

Shadrivov, I. V.

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,“ Nature Photon. 1, 41–48 (2007).
[Crossref]

Stacy, A. M.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Stockman, M. I.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,“ Phys. Rev. Lett. 93, 137404 (2004).
[Crossref] [PubMed]

Sun, C.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
[Crossref]

Takahara, J.

F. Kusunoki, T. Yotsuya, and J. Takahara, “Confinement and guiding of two-dimensional optical waves by low-refractive-index cores,“ Opt. Express 14, 5651–5656 (2006).
[Crossref] [PubMed]

J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,“ Optics and Photonics News 6, 54–59 (2004).
[Crossref]

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref] [PubMed]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref] [PubMed]

M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16, 18657–18666 (2008).
[Crossref]

Thevenaz, L.

L. Thevenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008).
[Crossref]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

Verhagen, E.

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991).
[Crossref]

Wang, Y.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

White, T. P.

Yao, J.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Yotsuya, T.

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

Zhang, X.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

IEEE J. Sel. Topics Quantum Electron. (1)

Z. Fu, Q. Q. Gan, Y. J. J. Ding, and F. J. Bartoli, “From waveguiding to spatial localization of THz waves within a plasmonic metallic grating,“ IEEE J. Sel. Topics Quantum Electron. 14, 486–490 (2008).
[Crossref]

J. Phys. D (1)

T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D 40, 2659–2665 (2007).
[Crossref]

Jpn. J. Appl. Phys. (2)

K. Y. Kim, “Tunneling-induced temporary light trapping in negative-index-clad slab waveguide,“ Jpn. J. Appl. Phys. 47, 4843–4845 (2008).
[Crossref]

J. Park and B. Lee, “An approximate formula of the effective refractive index of the metal-insulator-metal surface plasmon polariton waveguide in the infrared region,“ Jpn. J. Appl. Phys. 47, 8449–8451 (2008).
[Crossref]

Nano Lett. (1)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,“ Nano Lett. 6, 1928–1932 (2006).
[Crossref] [PubMed]

Nature (2)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,“ Nature 455, 376–U332 (2008).
[Crossref] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “Trapped rainbow’ storage of light in metamaterials,” Nature 450, 397–401 (2007).
[Crossref] [PubMed]

Nature Photon. (4)

V. M. Shalaev, “Optical negative-index metamaterials,“ Nature Photon. 1, 41–48 (2007).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

T. F. Krauss, “Why do we need slow light?,” Nature Photon. 2, 448–450 (2008).
[Crossref]

L. Thevenaz, “Slow and fast light in optical fibres,” Nature Photon. 2, 474–481 (2008).
[Crossref]

Opt. Express (10)

M. Notomi and H. Taniyama, “On-demand ultrahigh-Q cavity formation and photon pinning via dynamic waveguide tuning,” Opt. Express 16, 18657–18666 (2008).
[Crossref]

T. Kawasaki, D. Mori, and T. Baba, “Experimental observation of slow light in photonic crystal coupled waveguides,” Opt. Express 15, 10274–10281 (2007).
[Crossref] [PubMed]

D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
[Crossref] [PubMed]

J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,“ Opt. Express 16, 413–425 (2008).
[Crossref] [PubMed]

J. Park, H. Kim, I.-M. Lee, S. Kim, J. Jung, and B. Lee, “Resonant tunneling of surface plasmon polariton in the plasmonic nano-cavity,“ Opt. Express 16, 16903–16915 (2008).
[Crossref] [PubMed]

F. Kusunoki, T. Yotsuya, and J. Takahara, “Confinement and guiding of two-dimensional optical waves by low-refractive-index cores,“ Opt. Express 14, 5651–5656 (2006).
[Crossref] [PubMed]

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,“ Opt. Express 16, 45–57 (2008).
[Crossref] [PubMed]

E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,“ Opt. Express 17, 2465–2469 (2009).
[Crossref] [PubMed]

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear plasmonic slot waveguides,“ Opt. Express 16, 21209–21214 (2008).
[Crossref] [PubMed]

J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,“ Opt. Express 16, 19001–19017 (2008).
[Crossref]

Opt. Lett. (1)

Optics and Photonics News (1)

J. Takahara and T. Kobayashi, “Low-dimensional optical waves and nano-optical circuits,“ Optics and Photonics News 6, 54–59 (2004).
[Crossref]

Phys. Rev. B (6)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,“ Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal-films,“ Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,“ Phys. Rev. B 72, 075405 (2005).
[Crossref]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric-constant,“ Phys. Rev. B 44, 13556–13572 (1991).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,“ Phys. Rev. B 73, 035407 (2006).
[Crossref]

K. L. Tsakmakidis, C. Hermann, A. Klaedtke, C. Jamois, and O. Hess, “Surface plasmon polaritons in generalized slab heterostructures with negative permittivity and permeability,“ Phys. Rev. B 73, 085104 (2006).
[Crossref]

Phys. Rev. E (1)

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[Crossref]

Phys. Rev. Lett. (4)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref] [PubMed]

Q. Q. Gan, Z. Fu, Y. J. Ding, and F. J. Bartoli, “Ultrawide-bandwidth slow-light system based on THz plasmonic graded metallic grating structures,“ Phys. Rev. Lett. 101, 169903 (2008).
[Crossref]

Q. Q. Gan, Y. J. Ding, and F. J. Bartoli, ““Rainbow“ trapping and releasing at telecommunication wavelengths,“ Phys. Rev. Lett. 102, 056801 (2009).
[Crossref] [PubMed]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,“ Phys. Rev. Lett. 93, 137404 (2004).
[Crossref] [PubMed]

Science (2)

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,“ Science 316, 430–432 (2007).
[Crossref] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,“ Science 321, 930–930 (2008).
[Crossref] [PubMed]

Other (1)

E. Feigenbaum, N. Kaminski, and M. Orenstein, “Negative group velocity: Is it a backward wave or fast light?,“ arXiv:0807.4915 (2008).

Cited By

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

Alert me when this article is cited.


Figures (14)

Fig. 1.
Fig. 1. Schematic diagram of the MIM waveguide
Fig. 2.
Fig. 2. Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. (a) The case ρ > 1. (εd ,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. (εd ,ρ) = (1.0,0.8) and ak 0 = 0.8ξ. The anti-symmetric plasmonic mode exists in the MIM waveguide when ak 0 < ξ.
Fig. 3.
Fig. 3. Graphical method for the anti-symmetric photonic mode in the MIM waveguide. (a) The case ρ > V. (εd ,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. (εd ,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.
Fig. 4.
Fig. 4. (a)-(c) Dependence of the effective refractive index neff 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 Pnorm . The definition of Pnorm is given in Appendix A. (a) and (d) show the results for the case ρ < 1 with (εd ,ρ) = (1.0,0.8). (b) and (e) depict the results for the case 1 < ρ < 1.28 with (εd , ρ) = (1.0,1.1). (c) and (f) illustrate the results for the case ρ > 1.28 with (εd , 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 Pnorm = 0 (See Fig. 4(e)) i.e., trapping of light.
Fig. 5.
Fig. 5. (a) Graphical method for the anti-symmetric plasmonic mode in the MIM waveguide. The case 1 < ρ < 1.28. (εd , ρ) = (1.0,1.1). ak 0 = 0.98ξ. Two solutions are observed. (b) The graph of ρU cothU - √U 2 + V 2 for (εd ,ρ) = (1.0,1.1) . The arrow shows the tendency of decreasing ak 0 i.e., ak 0 = ζ, ξ, 0.98ξ, 0.938ξ(= hc ), and 0.91ξ.
Fig. 6.
Fig. 6. Dependence of (a) |neff | and (b) Pnorm on εd . (εd ,ak 0) = (-1.1,0.712) (c) ω - β dispersion relation. (εd ,2a,ωP ) = (1.0,90nm,6.83 × 1015). The abscissa is normalized with the bulk plasma wavenumber βP , while the ordinate is normalized with the bulk plasma frequency ωP . (d) The dependence of Pnorm on ω. The parameters are the same as those in (c).
Fig. 7.
Fig. 7. Graphical method for the symmetric plasmonic mode in the MIM waveguide. (a) The case ρ>1. (εd ,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 . (εd ,p) = (1.0,0.3) and ak 0 = 0.3 . No symmetric plasmonic mode exists regardless of ak 0.
Fig. 8.
Fig. 8. Graphical method for the symmetric photonic mode in the MIM waveguide. (a) The case with usual ρ. (εd ,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. (εd ,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.
Fig. 9.
Fig. 9. (a)-(c) Dependence of the effective refractive index neff 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 Pnorm . (a) and (d) show the results for the case ρ ≪ 1 with (εd ,ρ) = (1.0,0.01) . (b) and (e) depict the results for the case ρ<1 with (εd ,ρ) = (1.0,0.2) . (c) and (f) illustrate the results for the case p>1 with (εd ,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.
Fig. 10.
Fig. 10. Schematic diagram of the IMI waveguide
Fig. 11.
Fig. 11. Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (εd ,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 . (εd ,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.
Fig. 12.
Fig. 12. (a)-(b) Dependence of the effective refractive index |neff | 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 Pnorm . (a) and (c) show the results for the case ρ < 1 with (εd ,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 (εd , ρ) = (1.0,2.0).
Fig. 13.
Fig. 13. Graphical method for the anti-symmetric plasmonic mode in the IMI waveguide. (a) The case ρ > 1. (εd ,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 . (εd , ρ) = (1.0,0.5) and ak 0 = 2.0 . No anti-symmetric mode is guided in the IMI waveguide.
Fig. 14.
Fig. 14. (a) Dependence of the effective refractive index |neff | 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 Pnorm . (εd ,ρ) = (1.0,2.0) in common. Regardless of ak 0, a plasmonic positive mode always exists.

Tables (5)

Tables Icon

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.

Tables Icon

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

Tables Icon

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

Tables Icon

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

Tables Icon

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

Equations (43)

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

ε m = 1 ω p 2 ω ( ω + i γ ) ,
ρ = ε m ε d = ε m ε d .
κ m ε m + κ d ε d coth ( κ d a ) = 0 ( the anti symmetric plasmonic mode ) ,
κ m ε m + κ d ε d cot ( κ d a ) = 0 ( the anti symmetric photonic mode ) ,
κ m ε m + κ d ε d tanh ( κ d a ) = 0 ( the symmetric plasmonic mode ) ,
κ m ε m + κ d ε d tan ( κ d a ) = 0 ( the symmetric photonic mode ) .
κ m 2 + β 2 = ε m k 0 2 ,
κ d 2 + β 2 = ε d k 0 2 ,
κ d 2 + β 2 = ε d k 0 2 .
W = a κ m ,
U = a κ d ,
J = a k d ,
V = a k 0 ε d ε m .
ξ = ρ ε d ε m = ε m ε d ( ε d ε m ) .
ζ = 1 ε d cot 1 ε m ρ ε d = 1 ε d cot 1 ε d ε m .
ρ c = tan ρ c ( 1 + ρ c ) .
σ = π 2 ε d .
χ = 1 ε d tan 1 ( ε d ε m ) .
κ d ε d + κ m ε m tanh ( κ m a ) = 0 ( the symmetric plasmonic mode ) ,
κ d ε d + κ m ε m coth ( κ m a ) = 0 ( the anti symmetric plasmonic mode ) .
H y ( x , z ; t ) = { A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 1 sinh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a )
E x ( x , z ; t ) = { ( β ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ( β ω ε 0 ε d ) B 1 sinh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) ( β ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) .
E z ( x , z ; t ) = { ( κ m j ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ( κ d j ω ε 0 ε d ) B 1 cosh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) ( κ m j ω ε 0 ε m ) A 1 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a )
A 1 = B 1 sinh ( κ d a ) ,
κ m ε m A 1 = κ d ε d B 1 cosh ( κ d a ) ,
κ m ε m = κ d ε d coth ( κ d a ) .
P z ( x , z ; t ) = { ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) exp [ 2 κ m ( x a ) ] ( x a ) ( β B 1 2 ω ε 0 ) ( 1 ε d ) sinh 2 ( κ d x ) ( a x a ) ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) exp [ 2 κ m ( x a ) ] ( x a ) .
P metal = 2 a { ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) exp [ 2 κ m ( x a ) ] } d x
= 2 [ ( β B 1 2 ω ε 0 ) ( 1 ε m ) sinh 2 ( κ d a ) ] 0 exp [ 2 κ m x ] d x
= ( β B 1 2 ω ε 0 ) ( sinh 2 ( κ d a ) ε m κ m ) ,
P dielectric = 2 a a ( β B 1 2 ω ε 0 ) ( 1 ε d ) sinh 2 ( κ d x ) d x
= 2 ( β B 1 2 ω ε 0 ) ( 1 ε d ) 0 a sinh 2 ( κ d x ) d x
= 2 ( β B 1 2 ω ε 0 ) ( 1 ε d ) [ sinh ( 2 κ d x ) 4 κ d x 2 ] 0 a
= ( β B 1 2 ω ε 0 ) ( 1 ε d κ d ) [ sinh ( 2 κ d a ) 2 a κ d ] .
P norm = sinh ( 2 κ d a ) / 2 a κ d ε d κ d + sinh 2 ( κ d a ) ε m κ m sinh ( 2 κ d a ) / 2 a κ d ε d κ d sinh 2 ( κ d a ) ε m κ m .
H y ( x , z ; t ) = { A 2 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 2 sin ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 2 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 3 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 3 cosh ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 3 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 4 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 4 cos ( κ d x ) exp [ j ( β z ω t ) ] ( a x a ) A 4 exp [ κ m ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 5 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 5 sinh ( κ m x ) exp [ j ( β z ω t ) ] ( a x a ) A 5 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
H y ( x , z ; t ) = { A 6 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) B 6 cosh ( κ m x ) exp [ j ( β z ω t ) ] ( a x a ) A 6 exp [ κ d ( x a ) ] exp [ j ( β z ω t ) ] ( x a ) ,
n eff 2 ε m ε m + n eff 2 ε d ε d coth ( a k 0 n eff 2 ε d ) = 0 .
n eff k 0 ε m κ m + n eff k 0 ε d κ d coth ( a k 0 n eff 2 ε d ) κ d ε d k 0 csch 2 ( a k 0 n eff 2 ε d ) ( d ( a k 0 ) d n eff κ d k 0 + a k 0 2 n eff κ d ) = 0 ,
d ( a k 0 ) d n eff = n eff ε d k 0 3 κ d 2 [ sinh ( 2 κ d a ) / 2 a κ d ε d κ d + sinh 2 ( κ d a ) ε m κ m ] .

Metrics