Abstract

It has been recently demonstrated that a metallic surface with periodic grooves can support a laterally-confined surface wave called spoof plasmon polaritons (SSPPs). Here we propose a SSPPs waveguide drilled with L-shaped grooves which can support SSPPs efficiently. Dispersion relations based on the modal expansion method (MEM) are derived and discussed. Under the deep subwavelength condition, a concise formula for the dispersion relations is obtained. Our results show that the dispersion relations are sensitive to the transversal depths. The L-shaped groove is equivalent to a deeper rectangular groove, but more compact than the straight one. As an example of the applications, the rainbow-trapping effect is realized by changing the transversal depths of the L-shaped grooves.

© 2016 Optical Society of America

1. Introduction

Surface plasmon polaritons (SPPs) are electromagnetic excitations existing at the interface between two media whose real part of permittivity have opposite signs, for example, a metal and a dielectric. The SPPs fields decay exponentially in the two media and can be excited and propagate in deep subwavelength scales [1]. Thus they are promising to overcome the diffraction limit of light and have potential applications in biosensing, optoelectronics, materials science and nanophotonics [2–5]. Plasmonic devices with different applications have been proposed, such as couplers [6], all-optical switches [7], slow light waveguides [8] and modulators [9].

Recently, considerable efforts have been devoted to develop analogical SPPs in low frequency ranges. However, in microwave and terahertz frequency bands, the metal behaves like a perfect electric conduct (PEC), whose real part of permittivity is negative infinity. In this case, the flat metal/dielectric interface cannot confine SPPs. If some corrugations are drilled into or placed on the interface, another surface waves can exist, which are called spoof SPPs (SSPPs) since they have similar dispersion relations and field distributions with SPPs [10–16]. Unlike SPPs, the properties of SSPPs are primarily controlled by the parameters of corrugations, which provides more possibilities to design plasmonic devices.

A metallic block drilled with periodic rectangular grooves is a common SSPPs waveguide and has been investigated extensively in experiment and theory for its geometric simplicity [11,17–19]. It has been demonstrated that propagation constants of SSPPs increase with groove depths [20,21] and the field-confinement of SSPPs is enhanced remarkably. If we lengthen a rectangular groove in transversal direction, the groove will become an L-shaped one and the geometry of the waveguide will be more compact. In this work, we investigate waveguides drilled with L-shaped grooves. The waveguide we proposed here look like that discussed in [14], which are obtained by attaching periodic L-shaped metallic particles to a metallic surface. Here, we focus on the influence of transversal grooves to dispersion relations. At first, we derive a formula for the dispersion relation of SSPPs based on the modal expansion method (MEM). A concise expression is also obtained under the deep subwavelength condition. Then, we explore properties of SSPPs supported by this kind of waveguides. We found that their dispersion relations are sensitive to transversal depths of the L-shaped grooves. Finally, applications of the waveguides in rainbow-trapping are shown. Dispersion relations for waveguides with multi-transversal-grooves are also discussed.

2. Theoretical method

The proposed waveguide with periodically placed L-shaped grooves is shown in Fig. 1(a). The L-shaped groove consists of two rectangular grooves that have the same width, i. e., a. One of the rectangular grooves is longitudinal and the other is transversal. Their depths are denoted by hl and ht, respectively, x0 depicts the location of the L-shaped groove and d is the periodicity. Figure 1(b) illustrates a three-dimensional perspective of a unit cell of the waveguide. We discuss the two-dimensional structure in this work, that is, the waveguide is infinite in y direction. In the following, we will deduce the dispersion relation for the waveguide using MEM. For simplicity, the metal is approximated as the PEC.

 figure: Fig. 1

Fig. 1 (a) Front view of the waveguide drilled with periodic L-shaped grooves. (b) Three-dimensional perspective of a unit cell of the waveguide. (c) Schematic for the partition of the waveguide unit.

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Taking advantage of the periodic structure, we only need to consider one unit cell of the waveguide. It is convenient to divide the unit cell into four regions, denoted as I, II, III and IV, respectively. The partition is shown in Fig. 1(c). The electromagnetic fields of SSPPs propagating along the waveguide satisfy H=y^Hyand E=x^Ex+z^Ez. According to the MEM, the electromagnetic fields in each region can be expanded to a set of complete solutions for the Maxwell equations. In the region I (z ≥ 0), the magnetic field component Hy can be expressed as

HyI(x,z)=n=+An(1)eqn(1)zeiβnx,
where βn=β+2πn/d (|β|π/d) and qn(1)=βn2k02 (k0 is the wave vector of light in free space) are wave vector components in x and z directions, respectively. n is an integer and represents diffraction order. The electric field component Ex can be obtained directly through Ex(x,z)=1ik0Hyz, which is written as

ExI(x,z)=n=+qn(1)ik0An(1)eqn(1)zeiβnx.

In the region II (hl + az < 0), the electromagnetic fields are bounded in the groove. The y component of magnetic field and x component of electric field are expressed as

HyII(x,z)=m=0ψm(2)(x)[Am(2)eiqm(2)z+Bm(2)eiqm(2)(z+hla)]
and
ExII(x,z)=m=0qm(2)k0ψm(2)(x)[Am(2)eiqm(2)zBm(2)eiqm(2)(z+hla)],
respectively. qm(2) are wave vectors in z direction and m is a natural number associated with the mode number. ψm(2) corresponds to the mth eigen-function along x direction within the groove, which can be written as
ψm(2)(x)={0,if0<x<x01γmacos[mπa(xx0)],ifx0xx0+a0,ifx0+a<x<d.
γm are constants that are equal to 1 when m = 0 and 1/2 when m ≠ 0, resulting from the orthogonality of ψm(2), i. e. 0ddxψm(2)ψm'(2)=δmm'. Thus qm(2) can be obtained through qm(2)=k02(mπ/a)2. Similarly, the electromagnetic fields in the region III (hlz < hl + a) are expressed as
HyIII(x,z)=m=0ψm(3)(x)[Am(3)eiqm(3)(z+hla)+Bm(3)eiqm(3)(z+hl)],
ExIII(x,z)=m=0qm(3)ik0ψm(3)(x)[Am(3)eiqm(3)(z+hla)Bm(3)eiqm(3)(z+hl)],
where ψm(3) are written as
ψm(3)(x)={0,if0<x<x01γm(ht+a)cos[mπht+a(xx0)],ifx0xx0+a+ht0,ifx0+a+ht<x<d.
ψm(3) also satisfy the orthogonality and qm(3)=k02[mπ/(ht+a)]2. In the region IV (z <hl), both the magnetic field and electric field are zero. Thus HyIV(x,z)=ExIV(x,z)=0. According to the boundary condition of electromagnetic fields, Ex is continuous across the whole interfaces between the four regions, while Hy is only continuous at the interfaces between the same media. By imposing the continuity conditions at the three interfaces z = 0, hl + a and hl, we obtain five equations, which can be reduced to two as follows,
(m'=0Wm'm+gm)[Am(2)Bm(2)eiqm(2)(hla)]=fm[Am(2)eiqm(2)(hla)Bm(2)],
fm[Am(2)Bm(2)eiqm(2)(hla)]=(gm+m'=0Tmm'(23)lm'Tm'm(32))[Am(2)eiqm(2)(hla)Bm(2)],
where Wm'm=1dn=+ik0qn(1)Sm'n+Smn with Smn±=0ddxψm(2)e±iβnx. Tmm'(23) and Tm'm(32) are coupling integral between ψm(2) and ψm(3), given by Tmm'(23)=Tm'm(32)=0ddxψm(2)ψm'(3). fm=k0qm(2)isin[qm(2)(hla)], gm=k0qm(2)itan[qm(2)(hla)] and lm=k0qm(3)itan(qm(3)a)are parameters induced. The electromagnetic field of SSPPs can be obtained by solving the continuity equations. The condition making continuity equations solvable yields the dispersion relation of the waveguide. Thus the dispersion relation formula can be written as
|[W]+[g][f][f][g]+[T(23)][l][T(32)]|=0.
All the parameters with brackets represent matrixes. The elements of the matrix W, T(23) and T(32) are given by Wmm', Tmm'(23) and Tm'm(32), respectively. The g, f and l are three diagonal matrixes, which are defined as gmm'=gmδmm', fmm'=fmδmm' and lmm'=lmδmm', respectively. Equation (11) can be simplified and solved analytically under the deep subwavelength condition. First, when λ0>>ht+a, only the zero-order expansion modes (m = 0) in regions II and III can exist. Then, if λ0>>d, the high-order diffraction modes in region I can be neglected. In this case, n = 0. When the two conditions above are satisfied simultaneously, Eq. (11) is simplified to
β2k02k0=ad(aht+a)tan[k0(hla)]+tan(k0a)(aht+a)tan[k0(hla)]tan(k0a).
When ht = 0, Eq. (12) is reduced to
β2k02k0=adtan(k0hl),
which is the dispersion relation of the waveguide with rectangular grooves whose width and depth are a and hl, respectively [11]. Equation (12) reveals the dependence of the dispersion relation of SSPPs on geometric parameters of the waveguide. It shows that the dispersion relation of SSPPs has nothing to do with x0.

We use Eq. (12) to analyze the properties of SSPPs sustained by the waveguides drilled with L-shaped grooves. Firstly, we discuss the applicability of Eq. (12). We calculate the dispersion relations for the waveguides with a geometry of d = 50 μm, a = 0.2d, x0 = 0.1d and ht = 0. The calculated results are shown in Fig. 2. The black line is light line that represents the dispersion relation of light in free space. The red, blue and green curves correspond to analytical results for the waveguides with hl = d, 2d and 3d, respectively. The solid, dash and dash-dot-dot curves denote the fundamental, the 1st and 2nd order SSPPs modes. It is easy to observe that the SSPPs mode with an order of m (m = 0 for the fundamental mode) appears as hl > md, which is consistent with the previous work [20,21]. The symbols in Fig. 2 represent simulated results based on finite integration method. We can see that the difference between the analytical and simulated results are obvious when hl = d, resulting from the deep subwavelength condition. As hl increases to 2d, the two results nearly agree with each other. The deviation for the higher order modes are always larger than the lower order ones since their eigen-frequencies are higher.

 figure: Fig. 2

Fig. 2 Dispersion relations of SSPPs supported by waveguides with different hl. d = 50 μm, a = 0.2d, ht = 0 and x0 = 0.1d. The curves and symbols represent the analytical and simulated results, respectively. The solid, dash and dash-dot-dot curves correspond to the fundamental, the 1st and 2nd order modes, respectively.

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According to the results shown in Fig. 2, we set the longitudinal depth hl = 2d. Then we increase the transversal depth ht to see how the dispersion relations change. Figure 3(a) shows the dispersion relations of SSPPs as ht changes from 0 to 0.6d. The solid, dash and dash-dot- dot curves denote the fundamental, the 1st and 2nd order modes, respectively. The inset is an enlarged view for the dispersion curves of the 2nd order modes. It can be seen that all the dispersion curves move down as ht increases. When ht = 0, the 2nd order modes do not exist. They appear as soon as ht > 0. For the waveguide with ht = 0.6d, the Ex distributions of the electrical field of SSPPs at the asymptotical frequencies (correspond to β=π/d) are also given. Figures 3(b)-3(d) represent the fundamental, the 1st and 2nd order mode, respectively. It indicates that the 2nd order mode also has high field-confinement even though its dispersion curves are very close to the light line. The phenomena above state that the dispersion relations of SSPPs are very dependent on the transversal depths of the L-shaped grooves. As we have known, the dispersion relations decide directively the field-confinement of SSPPs. Therefore, we can control the SSPPs by modulating the transversal grooves. Figure 3 also suggests that adding a transversal groove to a straight rectangular groove is just like increasing of its depth. In other words, if we equate an L-shaped groove to a straight one, the equivalent depth is larger than its longitudinal part. Now we set the depth of the equivalent groove as hequ. Correspondingly, Eq. (11) can be written as

β2k02k0=adtan(k0hequ).
Therefore,

 figure: Fig. 3

Fig. 3 (a) Evolution of dispersion relations with the transversal depth ht. (b) - (d) Ex distributions of the SSPPs supported by the waveguide with ht = 0.6d at β=π/d. (b) The fundamental mode. (c) The 1st order mode. (d) The 2nd order mode. d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.

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tan(k0hequ)=(aht+a)tan[k0(hla)]+tan(k0a)(aht+a)tan[k0(hla)]tan(k0a).

It tells us that hequ increases with ht. For the waveguide with straight rectangular grooves whose width and depth are a and hl + ht, the dispersion formula is expressed as

β2k02k0=adtan[k0(hl+ht)].
It is easy to draw that tan[k0(hl+ht)]tan(k0hequ)0. It indicates that the equivalent depth hequ of an L-shaped groove is always smaller than its total length hl + ht. Figures 4 shows evolution of hequ as a function of ht at 0.55 THz, 1.8 THz and 2.95 THz, which is within the frequency band of the fundamental, the 1st and 2nd order modes, respectively. We can see that hequ increases with increasing of ht and it is below the diagonal curve, which represents hl + ht. The results agree with the analytical expressions. For the 1st and 2nd order modes, the difference between hequ and hl + ht is remarkable, while it is very small for the fundamental mode. It means that the fundamental mode is much more sensitive to the transversal depth. A part of the reason is that the deviation of the analytical solutions for the higher order modes are larger than that for the fundamental mode. Therefore, it is reasonable to equate an L-shaped groove to a straight rectangular groove. However, the waveguide with L-shaped grooves is more compact than that with the straight grooves. Thus the former is superior to the latter in application.

 figure: Fig. 4

Fig. 4 Evolution of the equivalent length hequ as fuctions of ht at f = 0.55 THz, 1.8 THz and 2.95 THz.

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3. Application of waveguides with L-shaped grooves

Rainbow-trapping effect is an interesting and meaningful slow-light phenomenon. It refers to that a light wave at a special frequency slows down gradually and stops eventually when it propagates on a slow-wave structure. The waves with different frequencies stop at different places [22–24]. Designing a slow-wave structure that can realize the rainbow-trapping has been a research hotspot. This is because the slow light with a remarkably reduced group velocity offers the possibility for time-domain processing of optical signals, which have potential applications in optical memory [25], information storage [26] and all-optical buffers [27]. For a waveguide corrugated with periodic rectangular grooves, increasing the groove depth can bring down its dispersion curve [20,21]. Meanwhile, the group velocity of SSPPs, given by vg=dω/dβ (ω is the angular frequency), decreases quickly. The waveguides with changing groove depths have been used to realize the rainbow-trapping successfully [28–30].

We demonstrate that the waveguides with L-shaped grooves also possess the rainbow-trapping effect. According to Eq. (11), the group velocity can be written as

vg=c{P2+1+aωdPP2+1acaht+asec2(ωac)[1+tan2(ωhlcωac)]+hlacsec2(ωhlcωac)[(aht+a)2+tan2(ωac)][aht+atan(k0hlωac))tan(k0a)]2}1
with

P=ad(aht+a)tan[k0(hla)]+tan(k0a)(aht+a)tan[k0(hla)]tan(k0a).

We still set d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d. The evolution of the group velocity with a changing transversal depth ht at 0.65 THz is calculated by Eq. (17). For comparison, we also caculate that for a waveguide with straight grooves with the depth of hl + ht. The blue and purple curves in Fig. 5(a) denote the two results, respectively. We can see that there has a critical ht for each waveguide, where the group velocity approaches to zero. Accordingly, we design two slow-wave structures. Both of them are 31d long. The first one is a straight groove array grating whose longitudinal width is 3.6d, as shown in Fig. 5(b). Its groove depths increase from 2d to 2.6d linearly with a step of 0.02d. The second structure is an L-shaped groove array grating with a width of 3d, which is illustrated in Fig. 5(c). Its longitudinal groove depths are fixed at 2d. The transversal depths are gradually changing from 0 to 0.6d with the same step. A plane wave at 0.65 THz propagating along x direction serves as the excitation source. Figures 5(b) and 5(c) depict simulated amplitude distributions of the electric field on the two structures in an xz plane. The corresponding one-dimensional |E| distributions along x direction 1 μm above the structure surfaces are illustrated in Fig. 5(f), which shares the same legend with Fig. 5(a). Obviously, in both cases, the SSPPs slow down and form an energy concentration at a special location. Both the group velocities and electric distributions indicate that there is no big difference between the two slow-wave structures except that the locations where the light is trapped have a little change. However, the structure with L-shaped grooves saves about 20% space.

 figure: Fig. 5

Fig. 5 (a) Changing of c/vg with ht at f = 0.65 THz, 0.62 THz and 0.6 THz. The purple curve denotes waveguides with straight grooves whose depths equate to hl + ht. The blue, green and red curves correspond to waveguides with L-shaped grooves whose longitudinal and transversal depths are hl and ht, respectively. (b) Simulated field distribution on the slow-wave structure with straight grooves at f = 0.65 THz. (c) – (e) Simulated field distributions on the slow-wave structure with L-shaped grooves at f = 0.65 THz, 0.62 THz and 0.6 THz. (f) One-dimensional |E| distributions along x direction 1 μm above the waveguide surfaces. It shares the same legend with (a). d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.

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In order to exhibit the rainbow-trapping effect better, two other excitation frequencies, 0.62 THz and 0.6 THz, are considered. At the two observed frequencies, the changing of the group velocity with transversal depths is shown in Fig. 5(a), represented by the green and red curves, respectively. The propagation of SSPPs on the L-shaped groove array grating structure at 0.62 THz and 0.6 THz is shown in Figs. 5(d) and 5(e), respectively. The green and red curves in Fig. 5(f) denote their one-dimensional |E| distributions along x direction. Figures 5(c)-5(f) demonstrate the rainbow-trapping effect on the slow-wave structure with L-shaped grooves perfectly. It is worth to note that all the concentration positions have deviation with the expectation. That is because we have used the deep subwavelength condition as an approximation in our derivation.

4. Waveguides with multi-transversal-grooves

In this section, we add more transversal grooves to a straight groove and see how the dispersion relation changes. Figure 6(a) indicates the dispersion curves for the waveguides with zero, one, two and three transversal grooves, as shown in the Figs. 6(b)-6(e), respectively. All the transversal grooves are 0.6d deep and they are always distributed evenly along the longitudinal direction. The other waveguide parameters are set as before. All the results are obtained by simulation for uniformity. It can be seen that the dispersion curves drop quickly as the number of the transversal grooves increases and the pace of decline gets slow. It once again indicates that adding transversal grooves to a rectangular groove is equivalent to increasing of its depth. It also solves the problem that the transversal depth of an L-shaped groove is limited within a waveguide unit.

 figure: Fig. 6

Fig. 6 (a) Dispersion relations for different waveguides. (b)-(e) Waveguide units with zero, one, two and three transversal grooves, respectively. (f) Simulated field distribution on the slow-wave structure at 0.535 THz. (g) Simulated field distribution on the slow-wave structure at 0.515 THz. d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.

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Now we design another slow-wave structure with multiple transversal grooves, which is shown in Figs. 6(f) and 6(g). The longitudinal grooves are fixed at 2d. First, the depths of the transversal grooves at the bottom increase from 0 to 0.6d linearly with a step of 0.04d and keep unchanged for the subsequent ones. Then the middle transversal grooves appear. Their depths increase gradually with the same step until 0.6d is reached. We simulate the propagation of SSPPs on this structure at 0.535 THz and 0.515 THz, at which the waves cannot be trapped on the stucture exhibited in Fig. 5(c). Figures 6(f) and 6(g) show the electric distributions. Obviously, these two waves are trapped effectively.

5. Conclusion

In conclusion, we derived a rigorous dispersion formula for the waveguides drilled with periodic L-shaped grooves at the PEC approximate. We further simplified the formula using the deep subwavelength approximation and utilized the concise form to describe the dispersion relations within an accepted deviation. We demonstrated that the dispersion curves move down quickly if we add one or more transversal grooves to a straight rectangular groove. The phenomenon is just like increasing its depth. It is an effective way to enhance the field-confinement of SSPPs. We designed two slow-wave waveguides and the rainbow-trapping is realized successfully. We believe that all the properties and applications of the waveguides with straight grooves can be realized on the waveguides with L-shaped grooves. The structures will be more compact. In addition, the L-shaped groove can be regarded as a group of two rectangular grooves. Each of them owns a groove depth and a width, hence the waveguides with L-shaped grooves are more flexible in the applications. Our work is potential in the design of plasmonic devices.

Funding

National Basic Research Program of China (2013CBA01702); National Natural Science Foundation of China (NSFC) (61575055, 61405056, 10974039, 61307072, 61308017, 61377016).

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References

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  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  2. J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
    [Crossref] [PubMed]
  3. M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
    [Crossref] [PubMed]
  4. V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
    [Crossref] [PubMed]
  5. A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
    [Crossref] [PubMed]
  6. T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
    [Crossref]
  7. H. Lu, X. Liu, L. Wang, Y. Gong, and D. Mao, “Ultrafast all-optical switching in nanoplasmonic waveguide with Kerr nonlinear resonator,” Opt. Express 19(4), 2910–2915 (2011).
    [Crossref] [PubMed]
  8. G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
    [Crossref]
  9. M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
    [Crossref]
  10. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
    [Crossref] [PubMed]
  11. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
    [Crossref]
  12. A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
    [Crossref] [PubMed]
  13. D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
    [Crossref] [PubMed]
  14. D. Martin-Cano, O. Quevedo-Teruel, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Waveguided spoof surface plasmons with deep-subwavelength lateral confinement,” Opt. Lett. 36(23), 4635–4637 (2011).
    [Crossref] [PubMed]
  15. J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).
  16. L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
    [Crossref] [PubMed]
  17. L. Shen, X. Chen, and T.-J. Yang, “Terahertz surface plasmon polaritons on periodically corrugated metal surfaces,” Opt. Express 16(5), 3326–3333 (2008).
    [Crossref] [PubMed]
  18. X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
    [Crossref]
  19. S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
    [Crossref]
  20. T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
    [Crossref]
  21. X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
    [Crossref] [PubMed]
  22. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
    [Crossref] [PubMed]
  23. X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
    [Crossref]
  24. H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
    [Crossref] [PubMed]
  25. M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
    [Crossref] [PubMed]
  26. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
    [Crossref] [PubMed]
  27. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
    [Crossref]
  28. 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. 100(25), 256803 (2008).
    [Crossref] [PubMed]
  29. Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
    [Crossref] [PubMed]
  30. Y. Yang, X. Shen, P. Zhao, H. C. Zhang, and T. J. Cui, “Trapping surface plasmon polaritons on ultrathin corrugated metallic strips in microwave frequencies,” Opt. Express 23(6), 7031–7037 (2015).
    [Crossref] [PubMed]

2016 (1)

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

2015 (2)

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Y. Yang, X. Shen, P. Zhao, H. C. Zhang, and T. J. Cui, “Trapping surface plasmon polaritons on ultrathin corrugated metallic strips in microwave frequencies,” Opt. Express 23(6), 7031–7037 (2015).
[Crossref] [PubMed]

2014 (1)

M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
[Crossref]

2013 (3)

X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
[Crossref] [PubMed]

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

2012 (3)

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

2011 (3)

2010 (3)

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref] [PubMed]

2009 (3)

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref] [PubMed]

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

2008 (4)

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

L. Shen, X. Chen, and T.-J. Yang, “Terahertz surface plasmon polaritons on periodically corrugated metal surfaces,” Opt. Express 16(5), 3326–3333 (2008).
[Crossref] [PubMed]

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

2007 (1)

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

2005 (2)

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22(5), 1062–1074 (2005).
[Crossref]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

2004 (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

2001 (1)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Atwater, H. A.

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

Bartoli, F. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Brongersma, M. L.

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Chen, X.

Cheng, X. C.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Cui, T. J.

Y. Yang, X. Shen, P. Zhao, H. C. Zhang, and T. J. Cui, “Trapping surface plasmon polaritons on ultrathin corrugated metallic strips in microwave frequencies,” Opt. Express 23(6), 7031–7037 (2015).
[Crossref] [PubMed]

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Ding, Y. J.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Du, C. L.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Feng, Y.

Fernandez-Dominguez, A. I.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref] [PubMed]

Fernández-Domínguez, A. I.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref] [PubMed]

Fernández-García, R.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Fu, Q. H.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Fu, Z.

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Gan, D. C.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Gan, Q.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Gao, X.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Gao, Y.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Garcia-Vidal, F. J.

D. Martin-Cano, O. Quevedo-Teruel, E. Moreno, L. Martin-Moreno, and F. J. Garcia-Vidal, “Waveguided spoof surface plasmons with deep-subwavelength lateral confinement,” Opt. Lett. 36(23), 4635–4637 (2011).
[Crossref] [PubMed]

D. Martin-Cano, M. L. Nesterov, A. I. Fernandez-Dominguez, F. J. Garcia-Vidal, L. Martin-Moreno, and E. Moreno, “Domino plasmons for subwavelength terahertz circuitry,” Opt. Express 18(2), 754–764 (2010).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

García-Vidal, F. J.

Giannini, V.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Gong, Y.

Hall, W. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Hess, O.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Hu, H.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Huang, J. X.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Ji, D.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Jiang, T.

X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
[Crossref] [PubMed]

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Kee, C.-S.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Khoe, G. D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Khurgin, J. B.

Kim, J.-E.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Kim, K.-J.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Kim, S.-H.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Leijtens, X. J.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Li, L.M.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Liu, C.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

Liu, J.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Liu, K.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Liu, S.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Liu, X.

Liu, X. M.

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

Lu, H.

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

H. Lu, X. Liu, L. Wang, Y. Gong, and D. Mao, “Ultrafast all-optical switching in nanoplasmonic waveguide with Kerr nonlinear resonator,” Opt. Express 19(4), 2910–2915 (2011).
[Crossref] [PubMed]

Luo, C. R.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Luo, W.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Luo, X. G.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Lyandres, O.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Ma, H. F.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Maier, S. A.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Mao, D.

Martin-Cano, D.

Martin-Moreno, L.

Martín-Moreno, L.

A. I. Fernández-Domínguez, E. Moreno, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz wedge plasmon polaritons,” Opt. Lett. 34(13), 2063–2065 (2009).
[Crossref] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

Moreno, E.

Nesterov, M. L.

Oei, Y. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Oh, S. S.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Park, H. Y.

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Pendry, J. B.

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

Polman, A.

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

Quevedo-Teruel, O.

Ran, L. X.

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Roschuk, T.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Shalaev, V. M.

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Shen, L.

Shen, L. F.

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Shen, X.

Shen, X. P.

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Shi, J. H

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

Shih, M. H.

M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
[Crossref]

Smalbrugge, B.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Smit, M. K.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

Song, K.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Sonnefraud, Y.

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Tian, L.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Tomlinson, L. A.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Van Duyne, R. P.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Vezenov, D.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Wagner, K.

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Wang, C. T.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Wang, G. X.

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

Wang, L.

Wood, J. J.

J. J. Wood, L. A. Tomlinson, O. Hess, S. A. Maier, and A. I. Fernandez-Dominguez, “Spoof plasmon polaritons in slanted geometries,” Opt. Lett. 85(7), 075441 (2012).

Xu, T.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Yang, T.-J.

Yang, Y.

Zeng, X.

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Zhang, H. C.

Zhang, X. F.

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Zhao, J.

X. Liu, Y. Feng, B. Zhu, J. Zhao, and T. Jiang, “High-order modes of spoof surface plasmonic wave transmission on thin metal film structure,” Opt. Express 21(25), 31155–31165 (2013).
[Crossref] [PubMed]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

Zhao, P.

Zhao, X. P.

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

Zhao, Y. H.

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

Zhou, K.

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

Zhu, B.

Appl. Phys. Lett. (4)

T. Xu, Y. H. Zhao, D. C. Gan, C. T. Wang, C. L. Du, and X. G. Luo, “Directional excitation of surface plasmons with subwavelength slits,” Appl. Phys. Lett. 92(10), 101501 (2008).
[Crossref]

G. X. Wang, H. Lu, and X. M. Liu, “Trapping of surface plasmon waves in graded grating waveguide system,” Appl. Phys. Lett. 101(1), 013111 (2012).
[Crossref]

X. Gao, J. H Shi, X. P. Shen, H. F. Ma, L.M. Li, and T. J. Cui, “Ultrathin dual-band surface plasmonic polariton waveguide and frequency splitter in microwave frequencies,” Appl. Phys. Lett. 102(15), 151912 (2013).
[Crossref]

X. P. Zhao, W. Luo, J. X. Huang, Q. H. Fu, K. Song, X. C. Cheng, and C. R. Luo, “Trapped rainbow effect in visible light left-handed heterostructures,” Appl. Phys. Lett. 95(7), 071111 (2009).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, “Surfaces with holes in them: new plasmonic metamaterials,” J. Opt. A, Pure Appl. Opt. 7(2), S97–S101 (2005).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Mater. (2)

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008).
[Crossref] [PubMed]

A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11(3), 174–177 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

M. H. Shih, “Plasmonics: Small and fast plasmonic modulator,” Nat. Photonics 8(3), 171–172 (2014).
[Crossref]

Nature (3)

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432(7014), 206–209 (2004).
[Crossref] [PubMed]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, “‘Trapped rainbow’ storage of light in metamaterials,” Nature 450(7168), 397–401 (2007).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (3)

Phys. Rev. B (1)

S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015).
[Crossref]

Phys. Rev. Lett. (1)

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. 100(25), 256803 (2008).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

Q. Gan, Y. Gao, K. Wagner, D. Vezenov, Y. J. Ding, and F. J. Bartoli, “Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings,” Proc. Natl. Acad. Sci. U.S.A. 108(13), 5169–5173 (2011).
[Crossref] [PubMed]

Prog. Electromagn. Res. M (1)

T. Jiang, L. F. Shen, X. F. Zhang, and L. X. Ran, “High-order modes of spoof surface plasmon polaritons on periodically corrugated metal surfaces,” Prog. Electromagn. Res. M 8, 91–102 (2009).
[Crossref]

Sci. Rep. (2)

L. Tian, J. Liu, K. Zhou, Y. Gao, and S. Liu, “Investigation of mechanism: spoof SPPs on periodically textured metal surface with pyramidal grooves,” Sci. Rep. 6, 32008 (2016).
[Crossref] [PubMed]

H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow trapping in hyperbolic metamaterial waveguide,” Sci. Rep. 3, 1249 (2013).
[Crossref] [PubMed]

Science (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

M. L. Brongersma and V. M. Shalaev, “Applied physics. The case for plasmonics,” Science 328(5977), 440–441 (2010).
[Crossref] [PubMed]

Small (1)

V. Giannini, A. I. Fernández-Domínguez, Y. Sonnefraud, T. Roschuk, R. Fernández-García, and S. A. Maier, “Controlling light localization and light-matter interactions with nanoplasmonics,” Small 6(22), 2498–2507 (2010).
[Crossref] [PubMed]

Other (1)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Cited By

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Figures (6)

Fig. 1
Fig. 1 (a) Front view of the waveguide drilled with periodic L-shaped grooves. (b) Three-dimensional perspective of a unit cell of the waveguide. (c) Schematic for the partition of the waveguide unit.
Fig. 2
Fig. 2 Dispersion relations of SSPPs supported by waveguides with different hl. d = 50 μm, a = 0.2d, ht = 0 and x0 = 0.1d. The curves and symbols represent the analytical and simulated results, respectively. The solid, dash and dash-dot-dot curves correspond to the fundamental, the 1st and 2nd order modes, respectively.
Fig. 3
Fig. 3 (a) Evolution of dispersion relations with the transversal depth ht. (b) - (d) Ex distributions of the SSPPs supported by the waveguide with ht = 0.6d at β=π/d . (b) The fundamental mode. (c) The 1st order mode. (d) The 2nd order mode. d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.
Fig. 4
Fig. 4 Evolution of the equivalent length hequ as fuctions of ht at f = 0.55 THz, 1.8 THz and 2.95 THz.
Fig. 5
Fig. 5 (a) Changing of c/ v g with ht at f = 0.65 THz, 0.62 THz and 0.6 THz. The purple curve denotes waveguides with straight grooves whose depths equate to hl + ht. The blue, green and red curves correspond to waveguides with L-shaped grooves whose longitudinal and transversal depths are hl and ht, respectively. (b) Simulated field distribution on the slow-wave structure with straight grooves at f = 0.65 THz. (c) – (e) Simulated field distributions on the slow-wave structure with L-shaped grooves at f = 0.65 THz, 0.62 THz and 0.6 THz. (f) One-dimensional |E| distributions along x direction 1 μm above the waveguide surfaces. It shares the same legend with (a). d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.
Fig. 6
Fig. 6 (a) Dispersion relations for different waveguides. (b)-(e) Waveguide units with zero, one, two and three transversal grooves, respectively. (f) Simulated field distribution on the slow-wave structure at 0.535 THz. (g) Simulated field distribution on the slow-wave structure at 0.515 THz. d = 50 μm, a = 0.2d, hl = 2d and x0 = 0.1d.

Equations (18)

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

H y I ( x,z )= n= + A n (1) e q n (1) z e i β n x ,
E x I ( x,z )= n= + q n (1) i k 0 A n (1) e q n (1) z e i β n x .
H y II ( x,z )= m=0 ψ m (2) ( x )[ A m (2) e i q m (2) z + B m (2) e i q m (2) (z+ h l a) ]
E x II ( x,z )= m=0 q m (2) k 0 ψ m (2) ( x )[ A m (2) e i q m (2) z B m (2) e i q m (2) (z+ h l a) ] ,
ψ m ( 2 ) (x)={ 0, if 0<x< x 0 1 γ m a cos[ mπ a ( x x 0 ) ], if x 0 x x 0 +a 0, if x 0 +a<x<d .
H y III ( x,z )= m=0 ψ m (3) (x)[ A m (3) e i q m (3) (z+ h l a) + B m (3) e i q m (3) (z+ h l ) ] ,
E x III ( x,z )= m=0 q m (3) i k 0 ψ m (3) (x)[ A m (3) e i q m (3) (z+ h l a) B m (3) e i q m (3) (z+ h l ) ] ,
ψ m ( 3 ) (x)={ 0, if 0<x< x 0 1 γ m ( h t +a) cos[ mπ h t +a ( x x 0 ) ], if x 0 x x 0 +a+ h t 0, if x 0 +a+ h t <x<d .
( m'=0 W m'm + g m )[ A m (2) B m (2) e i q m (2) ( h l a) ]= f m [ A m (2) e i q m (2) ( h l a) B m (2) ],
f m [ A m (2) B m (2) e i q m (2) ( h l a) ]=( g m + m'=0 T mm' (23) l m' T m'm (32) )[ A m (2) e i q m (2) ( h l a) B m (2) ],
| [W]+[g] [f] [f] [g]+[ T (23) ][l][ T (32) ] |=0.
β 2 k 0 2 k 0 = a d ( a h t +a )tan[ k 0 ( h l a) ]+tan( k 0 a ) ( a h t +a )tan[ k 0 ( h l a) ]tan( k 0 a ) .
β 2 k 0 2 k 0 = a d tan( k 0 h l ),
β 2 k 0 2 k 0 = a d tan( k 0 h equ ).
tan( k 0 h equ )= ( a h t +a )tan[ k 0 ( h l a) ]+tan( k 0 a ) ( a h t +a )tan[ k 0 ( h l a) ]tan( k 0 a ) .
β 2 k 0 2 k 0 = a d tan[ k 0 ( h l + h t ) ].
v g =c { P 2 +1 + aω d P P 2 +1 a c a h t +a se c 2 ( ωa c )[ 1+ tan 2 ( ω h l c ωa c ) ]+ h l a c se c 2 ( ω h l c ωa c )[ ( a h t +a ) 2 + tan 2 ( ωa c ) ] [ a h t +a tan( k 0 h l ωa c ) )tan( k 0 a ) ] 2 } 1
P= a d ( a h t +a )tan[ k 0 ( h l a) ]+tan( k 0 a) ( a h t +a )tan[ k 0 ( h l a) ]tan( k 0 a) .

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