1. Introduction

Sub-wavelength confinement and guiding of electromagnetic signals in metallic nanostructures has become a major focus of research in modern nano-optics [1,2]. Strong confinement of electromagnetic energy provided by a variety of metallic nanostructures (plasmonic waveguides) will offer high integration of nano-optical circuits and devices without increasing crosstalk and interference. It was also suggested that the most promising options for the design of efficient plasmonic interconnects might be nano-scale grooves on a metal surface [3–10] and nano-slots in thin metal films [11–15]. Among other advantages provided by metallic groove and slot waveguides [16], one of the most interesting is the possibility of high (nearly 100%) transmission through sharp bends [7,11,17,18].

However, the analysis conducted for transmission through sharp bends in metallic slot waveguides has provided only very preliminary results for specific slot dimensions and fixed wavelength, and did not explicitly consider the fundamental slot mode [11]. Other attempts to investigate numerically the transmission through sharp bends in metal gap guiding structures were predominantly related to metal heterowaveguides [18] and two-dimensional (2D) metal-dielectric-metal (MDM) waveguides [19,20].

It is important to note that the fundamental mode of the slot waveguide [12–15] is physically different from any mode of an MDM waveguide, and as such it has no analogy in the 2D case. In fact, there are several slot waveguide modes (including the fundamental mode) that are formed by four coupled corner modes (wedge plasmons) propagating along the four corners of the rectangular slot [15]. Transmission of such modes through a sharp bend is expected to be different to that of the gap plasmons in an MDM waveguide [19,20].

Therefore, the aim of this paper is to use 3D finite-difference time-domain (FDTD) simulations to analyze numerically the transmission properties of a 90 degree bend in a symmetric slot waveguide with the fundamental mode, where the silver film with the rectangular slot is surrounded by a uniform dielectric (vacuum). We will investigate the bend transmission as a function of the excitation (vacuum) wavelength, width of the slot and thickness of the silver film. Physical effects related to mode transformation at the bend will also be investigated and explained.

2. Structure and computational methods

The analyzed structure is presented in Fig. 1a-b . A dipole (current source) S in the middle of the slot at the distance 1200 nm from the bend oscillates harmonically in the z-direction to excite the fundamental slot mode [12–15] in the first arm of the bend (Fig. 1b). Other modes are either not supported by the slot or not excited by the considered source. The energy flow transmitted through the bend P_Bend is determined by integration of the z-component of the Poynting vector over the cross-section A at the distance of 200 nm from the bend (Fig. 1b). The relative transmissivity T_r = P_Bend/P_str is then determined, where P_str is the energy flow in a straight slot waveguide (i.e. with no bend) with the same parameters at the distance of 1400 nm + w from the generating dipole. Defined in this manner, the relative transmission is normalized to exclude the effect of dissipative losses in the metal, and only the losses directly relating to the bend are taken into account. Dissipative losses can then be easily taken into account by considering the overall distance the mode travels in both the arms of the bend.

 

Fig. 1 (a) The considered slot-waveguide with a 90° bend and a gap width w in a silver film of thickness h, surrounded by vacuum. (b) Top-down view of the bend with generating dipole (current source) S.

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In our FDTD calculations [21], we employ a non-uniform mesh with a grid size of 5 nm in vacuum and the metal, and approaching 1 nm at any of the metal-vacuum interfaces. Perfectly matched layer (absorbing) boundary conditions are used at least 100 nm away from the walls of the slot (Fig. 1b), and 100 nm above and below the interfaces of the silver film. The wavelength-dependant dielectric permittivity of silver is taken from Ref. 22.

3. Results and discussion

Figure 2 presents the plots of the relative bend transmissivity T_r for the gap widths (a) w = 50 nm and (b) w = 100 nm, and film thicknesses h = 50, 100 and 200 nm. For comparison, we also plot the transmissivity through a 90 degree bend of an MDM waveguide of the same width (Fig. 2). In particular, it can be seen that the slot waveguide displays similar transmission behavior to that of the MDM waveguide – notably, an increase in the bend transmission with increasing vacuum wavelength λ_vac. The reduced bend transmission of the slot compared to the MDM waveguide can be explained by increased bend losses in the slot waveguides. In an MDM waveguide, bend radiation losses are suppressed by the surrounding metal, whereas in a slot waveguide such screening is only partial and a guided mode can leak from the bend into bulk waves or surface plasmons at the metal film interfaces. This can also explain that the bend transmissivity increases with increasing thickness of the metal film and/or decreasing width of the gap (Fig. 2). In this case, a larger fraction of the mode energy is concentrated towards the middle of the slot, making the slot look more like the 2D MDM waveguide.

 

Fig. 2 Relative transmissivity T_r through a 90° bend in (a) w = 50 nm and (b) w = 100 nm silver slot waveguide at various film thicknesses h = 50 nm (●), 100 nm (○) and 200 nm (∆). The relative transmissivity through a 90° bend in an MDM waveguide of the same gap width (×) is presented for comparison.

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Dissipative losses in the metal may result in further losses of the mode energy, leading to further reduction of the guided energy in the second arm of the bend. These additional losses can be easily taken into account by considering the overall length the mode travels in both arms of the bend. At the same time, for the considered wavelengths and slot parameters, the dissipative losses are typically negligible within the considered propagation distances (Fig. 1b), especially for larger (e.g., telecommunication) wavelengths. Therefore, the relative transmissivity T_r through the bend is approximately equal to the absolute transmissivity that also takes into account dissipative losses.

To better understand the mechanism of mode transmission through the bend, Fig. 3 shows the z-component of the Poynting vector (averaged over one period of the wave) at plane A (Fig. 1b) for a w = h = 100 nm slot at λ_vac = 0.5, 0.75, 1 and 2 µm. At shorter wavelengths, the field in the outgoing arm of the bend is mostly confined to just the outer (with respect to the bend) wall of the slot (Fig. 3a,b). It is proposed that there are two related mechanisms contributing to the observed mode asymmetry in the second arm of the bend. Firstly, at smaller wavelengths, the fundamental mode tends to be mostly localized near the four corners of the gap (i.e., the four wedge plasmons forming this slot mode are only weakly coupled). As a result, the wedge plasmons at the inner corners of the bend experience significant radiative losses, as they are less effectively screened by the other side of the gap. On the contrary, the wedge plasmons at the outer corners of the bend are more effectively screened by the metal and their radiative losses are significantly suppressed. As a result, predominantly wedge plasmons at the outer corners survive the transmission through the bend, leading to the mode asymmetry in the second arm (Fig. 3a,b). As λ_vac increases, coupling between the wedge plasmons forming the fundamental mode increases and the modal field tends to be more localized inside the gap (Fig. 3d). This means increased efficiency of screening (by the outer portions of the metal around the bend) of the wedge plasmons at all four corners, leading to lower radiative losses including from the inner corners of the bend. As a result, the mode in the second arm becomes more symmetric (Fig. 3c,d), and the bend transmissivities at telecommunication wavelengths ~1.5 µm in all the considered slots exceed 0.85 (Fig. 2).

 

Fig. 3 The distribution of the z-component of the Poynting vector (averaged over one period of the wave) at the plane A (z is normal to A; Fig. 1b) of a w = h = 100 nm slot at different excitation wavelengths λ_vac = (a) 0.5 μm, (b) 0.75 μm, (c) 1.0 μm, and (d) 2.0 µm. The right wall of the slot (x > 0) corresponds to the outer wall of the bend (Fig. 1b). Scale is in microns. Red (blue) regions denote regions of maximum (minimum) intensity.

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The second explanation of the mode asymmetry is related to the generation of other propagating, non-propagating and leaky slot modes at the bend. These modes interfere with the fundamental transmitted mode resulting in the observed modal asymmetry in the second arm near the bend. For example, at λ_vac = 750 nm, only the fundamental (i.e., the (s_fa_g)_cw) mode with symmetric charge distribution across the film, and anti-symmetric charge distribution across the gap (slot) [15] can exist as a guided wave in the considered slots. However, its interaction with the bend is expected to result in generation of a leaky (s_fs_g)_cw mode [15] with the symmetric charge distribution across the slot in the second arm of the bend. Such a mode has a wave number that is slightly smaller than that in the surrounding vacuum.

The mode asymmetry in this case (Fig. 3b) is caused by the interference of the fundamental mode (having the antisymmetric charge distribution across the slot [15]) with the leaky mode (having the symmetric charge distribution across the slot [15]). Because of their opposite symmetry of the charge distribution across the slot, interference of these two modes produces the observed asymmetry of the field intensity distribution in the second arm near the bend (Fig. 3b). As these two modes have slightly different wave numbers, their interference pattern must be in the form of beats in the second arm of the bend. However, as the leaky (s_fs_g)_cw mode [15] leaks out of the slot in the form of bulk waves and surface film plasmons, the modal asymmetry (i.e., the amplitude of the beat pattern) must decay with increasing distance from the bend.

Indeed, Fig. 4a shows the typical relaxation of the mode asymmetry in the second arm of the bend. In particular, it can be seen that the mode asymmetry occurs in both 2D MDM and 3D slot waveguides (Figs. 4a,b). However, the decay of the mode asymmetry in the case of the MDM waveguide is much faster than that for the slot waveguide. This difference can be explained as follows. There are no leaky modes in the MDM waveguide, and there can only be non-propagating modes that quickly decay into the second arm, resulting in a rapid (within about 200 nm) decay of the mode asymmetry in the second arm (the insert in Fig. 4a). In the slot waveguide, there is a weakly leaking (s_fs_g)_cw mode [15] that can propagate a noticeable distance along the second arm of the bend before it leaks out. The resultant interference between the fundamental and (s_fs_g)_cw leaky mode produces the indicated beat pattern in the second arm of the bend (Fig. 4a). Other leaky and non-propagating slot modes generated at the bend may also contribute to the predicted beat pattern (Fig. 4a). Decreasing slot width should result in more effective leakage of the (s_fs_g)_cw mode from the gap, which means faster decay of the associated beat pattern (Fig. 4b). If the slot parameters are such that the (s_fs_g)_cw mode is a true guided mode, then the beat pattern persists for a long distance in the second arm of the bend until the modes dissipate in the metal.

 

Fig. 4 (a) The dependencies of the z-component of the Poynting vector (averaged over one period of the wave) at the half-height inside the slot near the outer (solid curve) and near the inner (dotted curve) slot walls on distance z from the bend into the second arm (z = w/2 corresponds to the inner corner of the bend). The structural parameters are the same as for Fig. 3 with λ_vac = 750 nm. The inset shows the same dependencies for the 2D MDM waveguide with w = 100 nm at the same wavelength. (b) The same dependencies, but for the slot with w = h = 50 nm.

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As has been discovered previously, special design of the sharp corner of the bend may result in a significant increase of the transmissivity through a sharp bend in a V-groove waveguide [17]. For example, an additional defect in the outer corner of the bend in the form of a triangular pillar [17] increased transmissivity through a sharp bend in a V-groove waveguide from about 0.86 to approximately 1. It is therefore interesting to find out if the same design of the corner of the bend in a slot waveguide (Fig. 5 ) leads to a similar enhancement of transmission through the bend. The defect is assumed to have the size such that its corner is exactly in the middle of the bend (Fig. 5a,b). This was one of the optimal defect dimensions for a V-groove waveguide to ensure nearly 100% transmission through the bend.

 

Fig. 5 (a) Bend in the slot waveguide with the addition triangular pillar defect of side d = w/2. (b-c) Top–down view of the slot waveguide with bend and defect (b) and straight slot waveguide of equivalent length (c), showing relative positions of the planes B and A, and the source S.

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Table 1 presents the comparison of the energy flows through the planes A and B in the second and the first arms of the bend, respectively, in the presence and in the absence of the triangular pillar defect (Figs. 5a-c). For comparison, the same energy flows are also calculated through the planes within the same equivalent distance from the generating dipole in the straight slot waveguide (Fig. 5d).

Table 1. Comparative energy flows with and without the corner defect

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Several important aspects can be seen from Table 1. Firstly, energy flow through plane A in the straight waveguide (the last column in Table 1) is always larger than in the presence of the bend with or without the defect. This is expected, because the bend always results in considerable losses due to radiation and reflection of the incident mode, which reduces the energy reaching plane A.

Secondly, energy flow through plane B in the straight waveguide is also always larger than in the presence of the bend with or without the defect. This is also expected, because the bend results in partial reflection of the incident mode back into the first arm of the bend, which reduces the overall energy flow through plane B in the positive x-direction.

Thirdly, for all three wavelengths, introducing the defect into the bend results in increasing the energy flow through the plane A in the second arm of the bend (compare columns 3 and 5 in Table 1). This means increasing transmissivity through the bend, which is similar to V-groove waveguides [17]. However, this increase in transmissivity is rather small and decreases with increasing wavelength (Table 1). Simultaneously, it can be seen that introducing the defect into the bend results in increasing energy flow in the first arm of the bend. Both these effects are caused by the same mechanism – the defect leads to the destructive interference of the reflected (from different parts of the defect) waves in the first arm of the bend very much similar to the bend with the defect in a V-groove waveguide [17]. As a result, the energy flow in the first arm of the bend increases (due to suppression of the reflected wave), as well as the energy flow in the second arm of the bend, causing an increase in the transmissivity (similar to the V-groove waveguide [17]).

The obtained results do not allow to conclude that introduction of the triangular pillar defect into the bend caused increase of radiation bend losses, though this is to an extent still possible. The main conclusion is that the defect in the sharp 90° bend of a slot waveguide performs very similarly to that in the bend in a V-groove waveguide [17], i.e., increased transmissivity through the bend. However, it is important to note that at larger telecommunication wavelengths this increase in transmissivity is typically negligible. This is because at such larger wavelengths reflection losses at the bend are next to zero even in the absence of the defect. The similar finding for 2D MDM waveguides was made in Ref. 20. Therefore, the defect should typically be used only at smaller wavelengths.

4. Conclusions

In conclusion, we have used the 3D FDTD method to determine the transmission properties of a 90 degree bend in a metallic slot waveguide. We have shown that the transmission through the bend is increased as either the width of the slot is decreased, or the film thickness (height of the slot) is increased. We have identified an additional loss mechanism in the slot that is not present in the 2D metal-dielectric-metal waveguides – the generation of additional leaky modes at the bend junction, which propagate along the outgoing arm of the bend. The additional energy losses associated with the generation of these leaky modes (compared to just dissipative losses in the metal) further reduces the effective transmissivity through the bend, especially if the second arm is relatively long (with respect to the distance of relaxation of the beat pattern (Fig. 4)). However, if a bend has small arms (e.g., hundreds of nanometers – Fig. 1b), such as in highly integrated optical circuits, neither leakage, nor dissipative losses are expected to have a noticeable impact on the bend transmissivity. If the main requirement is just high energy transmissivity through a sharp bend on the nano-scale, generation of additional modes at the bend does not seem to be of a major practical importance.

We have also considered the effect of a triangular pillar defect on the transmission through the bend. We have shown that the presence of the defect results in a significant increase in transmission at smaller wavelengths, although negligible improvement at larger (telecom) wavelengths where the transmission is anyway close to 100%.