## 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 *h _{l}* and

*h*, respectively,

_{t}*x*

_{0}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.

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 $H=\widehat{y}{H}_{y}$and $E=\widehat{x}{E}_{x}+\widehat{z}{E}_{z}$. 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 *H _{y}* can be expressed as

*k*

_{0}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

*E*can be obtained directly through ${E}_{x}(x,z)=\frac{1}{i{k}_{0}}\frac{\partial {H}_{y}}{\partial z}$, which is written as

_{x}In the region II ($-$*h _{l}* +

*a*≤

*z*< 0), the electromagnetic fields are bounded in the groove. The

*y*component of magnetic field and

*x*component of electric field are expressed as

*z*direction and

*m*is a natural number associated with the mode number. ${\psi}_{m}^{(2)}$ corresponds to the

*m*

^{th}eigen-function along

*x*direction within the groove, which can be written as

*m*= 0 and 1/2 when

*m*≠ 0, resulting from the orthogonality of ${\psi}_{m}^{(2)}$, i. e. $\int}_{0}^{d}dx{\psi}_{m}^{(2)}{\psi}_{m\text{'}}^{(2)}}={\delta}_{mm\text{'$. Thus ${q}_{m}^{(2)}$ can be obtained through ${q}_{m}^{(2)}=\sqrt{{k}_{0}^{2}-{(m\pi /a)}^{2}}$. Similarly, the electromagnetic fields in the region III ($-$

*h*≤

_{l}*z*< $-$

*h*+

_{l}*a*) are expressed as

*z*<$-$

*h*), both the magnetic field and electric field are zero. Thus ${H}_{y}^{IV}(x,z)={E}_{x}^{IV}(x,z)=0$. According to the boundary condition of electromagnetic fields,

_{l}*E*is continuous across the whole interfaces between the four regions, while

_{x}*H*is only continuous at the interfaces between the same media. By imposing the continuity conditions at the three interfaces

_{y}*z*= 0, $-$

*h*+

_{l}*a*and $-$

*h*, we obtain five equations, which can be reduced to two as follows,

_{l}*W*,

*T*

^{(23)}and

*T*

^{(}^{32)}are given by ${W}_{mm\text{'}}$, ${T}_{mm\text{'}}^{(23)}$ and ${T}_{m\text{'}m}^{(32)}$, respectively. The

*g*,

*f*and

*l*are three diagonal matrixes, which are defined as ${g}_{mm\text{'}}={g}_{m}{\delta}_{mm\text{'}}$, ${f}_{mm\text{'}}={f}_{m}{\delta}_{mm\text{'}}$ and ${l}_{mm\text{'}}={l}_{m}{\delta}_{mm\text{'}}$, respectively. Equation (11) can be simplified and solved analytically under the deep subwavelength condition. First, when ${\lambda}_{0}>>{h}_{t}+a$, only the zero-order expansion modes (

*m*= 0) in regions II and III can exist. Then, if ${\lambda}_{0}>>d$, 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

*h*= 0, Eq. (12) is reduced to

_{t}*a*and

*h*, 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

_{l}*x*

_{0}.

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* = 0.2*d*, *x*_{0} = 0.1*d* and *h _{t}* = 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

*h*=

_{l}*d*, 2

*d*and 3

*d*, 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

*h*>

_{l}*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

*h*=

_{l}*d*, resulting from the deep subwavelength condition. As

*h*increases to 2

_{l}*d*, 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.

According to the results shown in Fig. 2, we set the longitudinal depth *h _{l}* = 2

*d*. Then we increase the transversal depth

*h*to see how the dispersion relations change. Figure 3(a) shows the dispersion relations of SSPPs as

_{t}*h*changes from 0 to 0.6

_{t}*d*. 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

*h*increases. When

_{t}*h*= 0, the 2nd order modes do not exist. They appear as soon as

_{t}*h*> 0. For the waveguide with

_{t}*h*= 0.6

_{t}*d*, the

*E*distributions of the electrical field of SSPPs at the asymptotical frequencies (correspond to $\beta \text{=}\pi /d$) 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

_{x}*h*. Correspondingly, Eq. (11) can be written as

_{equ}It tells us that *h _{equ}* increases with

*h*. For the waveguide with straight rectangular grooves whose width and depth are

_{t}*a*and

*h*+

_{l}*h*, the dispersion formula is expressed as

_{t}*h*of an L-shaped groove is always smaller than its total length

_{equ}*h*+

_{l}*h*. Figures 4 shows evolution of

_{t}*h*as a function of

_{equ}*h*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

_{t}*h*increases with increasing of

_{equ}*h*and it is below the diagonal curve, which represents

_{t}*h*+

_{l}*h*. The results agree with the analytical expressions. For the 1st and 2nd order modes, the difference between

_{t}*h*and

_{equ}*h*+

_{l}*h*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.

_{t}## 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 ${v}_{g}=d\omega /d\beta $ ($\omega $ 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

We still set *d* = 50 μm, *a* = 0.2*d*, *h _{l}* = 2

*d*and

*x*

_{0}= 0.1

*d*. The evolution of the group velocity with a changing transversal depth

*h*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

_{t}*h*+

_{l}*h*. The blue and purple curves in Fig. 5(a) denote the two results, respectively. We can see that there has a critical

_{t}*h*for each waveguide, where the group velocity approaches to zero. Accordingly, we design two slow-wave structures. Both of them are 31

_{t}*d*long. The first one is a straight groove array grating whose longitudinal width is 3.6

*d*, as shown in Fig. 5(b). Its groove depths increase from 2

*d*to 2.6

*d*linearly with a step of 0.02

*d*. The second structure is an L-shaped groove array grating with a width of 3

*d*, which is illustrated in Fig. 5(c). Its longitudinal groove depths are fixed at 2

*d*. The transversal depths are gradually changing from 0 to 0.6

*d*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 $\left|E\right|$ 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.

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 $\left|E\right|$ 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 0.6*d* 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.

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 2*d*. First, the depths of the transversal grooves at the bottom increase from 0 to 0.6*d* linearly with a step of 0.04*d* and keep unchanged for the subsequent ones. Then the middle transversal grooves appear. Their depths increase gradually with the same step until 0.6*d* 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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