1. Introduction

Guiding and confining light at the subwavelength scale has become an ever-increasing demand for future high-density photonic integrated circuits [1]. Waveguides and devices based on the surface plasmon polariton (SPP) have been proposed and demonstrated by leveraging the propagation of SPP at the metal/dielectric surface as a novel guiding mechanism [2]. In the past studies of SPP devices, many efforts have been made to balance the tradeoff between the mode confinement and the inevitable propagation loss due to the Ohmic loss in the metal [3]. Recently, dielectric-loaded SPP (DLSPP) waveguides employing a dielectric ridge directly on a metallic surface have been proposed and demonstrated [4]. Compared with those previously proposed long-range SPP waveguides [5], metal slot waveguides [6] and channel SPP waveguides [7], DLSPP waveguides could effectively provide relatively tight confinement of light while maintaining relatively long propagation distance [8]. Many DLSPP waveguide-based devices such as the directional couplers [9], splitters [10] and resonators [11] have also been demonstrated.

Conventional DLSPP waveguides utilize low-index materials, such as polymethylmethacrylate (i.e. PMMA, n~1.5), as the material of the ridge. It enables the relatively low-loss transmission but sets a limit on the downscaling of the waveguide structure. Typically, in order to ensure moderate propagation distance under the single-mode condition, the width and height of the dielectric ridge are as large as ~600 nm for waveguides at 1.55 μm wavelength [12]. This, along with the near-micron-large-mode-size [8], presents challenges for true subwavelength integrations. Introducing a high-index dielectric ridge could enhance the field confinement [13], but the associated increase in the transmission loss is often unacceptable [13, 14]. One widely studied method for loss reduction in plasmonic waveguides is employing a low-index buffer layer between the metal and high-index region [14–17], and the associated hybrid plasmonic mode could enable low-loss transmission with sub-wavelength field confinement. Similarly, introducing air gaps in the vicinity of metals have been found to be useful for lowering the loss in the long-range SPP waveguides [18]. On the other hand, substructures consisting of air in the core have been proven to enhance the field confinement in photonic crystal fibers [19] and dielectric slot waveguides [20].

In this paper, we propose a novel DLSPP structure consisting of a high-index dielectric ridge that has an air nanohole at the interface of the metal and the ridge. The size of the ridge can be greatly reduced due to its higher index. The air nanohole on the other hand helps to reduce the transmission loss. Further field enhancement could also be achieved in the nanohole [15, 17, 21]. Under optimized parameters, a subwavelength mode area could be realized while the propagation loss is also reduced. This relatively simple structure could be compatible with the conventional lithography fabrication process. Similar to the fabrication process of the well-studied air-filled slot waveguides [22], electron-beam lithography and etching, followed by the chemical-vapor deposition process of silicon, could be used.

2. Geometry and modal properties of the proposed DLSPP waveguides

For comparison with the proposed structure, a DLSPP waveguide consisting of a high-index dielectric ridge with a width of w and height of h sitting directly on top of a semi-infinite metallic substrate is shown in Fig. 1(a) . The materials considered are silver (Ag) for the substrate, silicon (Si) for the ridge, and silica (SiO₂) for the cladding, respectively. Our proposed DLSPP waveguide is shown in Fig. 1(b), where a nanohole filled with air is incorporated at the bottom center of the high-index ridge. Its width and height are w_a and h_a. In our simulations, the wavelength is set at λ = 1550 nm. The permittivities of SiO₂, Si and Ag are ε_c = 2.25, ε_d = 12.25 and ε_m = −129 + 3.3i [23], respectively. The width and height of the dielectric ridge are w = h = 200nm, and they are chosen to ensure that the corresponding conventional DLSPP waveguide supports a single TM SPP mode [13]. The modal properties are investigated by means of the finite-element method (FEM) using COMSOL^TM. The eigenmode solver is used with the scattering boundary condition. Convergence tests are done to ensure that the numerical boundaries and meshing do not interfere with the solutions.

 

Fig. 1 Geometries of (a) the conventional and (b) the proposed DLSPP waveguides.

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Our results indicate that the proposed DLSPP waveguide can support a fundamental quasi-TM plasmonic mode under a wide range of geometric parameters of the nanohole. As an example, Ey distributions and the cross-sectional curves of Ey of the fundamental plasmonic mode of the conventional and proposed DLSPP waveguides with w_a = 100nm and h_a = 30nm are shown in Fig. 2 and Fig. 3 . For comparison, the field distribution of a pure dielectric structure with a SiO₂ substrate is also drawn in Fig. 2(c). As shown in Fig. 2(a), the 200nm × 200nm Si ridge is capable of effectively confining the field due to its high index. By introducing the air nanohole in the ridge, the tails of the field distribution spread out slightly more, as the low-index nanohole slightly weakens the field confinement capability of the ridge. However, in either case, subwavelength mode confinement can be achieved, because of the smaller, high-index ridge. The effective mode sizes (D_eff), defined as the diameter of the effective mode area (A_eff), of the structures in Fig. 2 (a), (b) and (c) are 0.24, 0.28 and 1.56 μm, respectively. A_eff is calculated using $A_{eff}=({\displaystyle \iint W(r)dA{)}^2}/({\displaystyle \iint W{(r)}^{2}dA})$, where, to accurately account for the energy in the metal region, the electromagnetic energy density W(r) is defined as [24, 25]:

 

Fig. 2 (a)-(c) |Ey| distributions of the fundamental modes of two DLSPP waveguides and quasi-TM mode of the pure dielectric waveguide with the same-size air nanohole (w = h = 200nm, w_a = 100nm, h_a = 30nm). Note that the field distributions are normalized so that the surface integrals of the power flow in the cross section are equal.

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Fig. 3 (a)-(c) Ey distributions along x direction (The profiles are taken at 15 nm above the silver-silica/silver-air/silica-air interface); (d)-(f) Ey distributions along y direction at the center position of the nanohole.

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The dimensions of the nanohole strongly influence the modal properties. Figure 4 shows the electric field distributions at different nanohole heights with w_a fixed at 100nm. It is noted that smaller h_a would cause stronger field enhancement for both horizontal and vertical directions in the nanohole region. At larger h_a’s, the field enhancement is less obvious, and more electric field is distributed in the upper part of the high-index ridge, leading to an increased mode area. D_eff of the structures in Fig. 4 (a)-(d) are 0.27, 0.28, 0.29 and 0.33 μm, respectively. Comparisons of Ey distributions at various nanohole widths in Fig. 5 illustrate a similar trend but the resultant changes are less obvious. Widening the nanohole leads to a slight increase in the spreading of the field distribution. D_eff of the structures in Fig. 5 (a)-(d) are 0.25, 0.27, 0.28 and 0.33 μm, respectively.

 

Fig. 4 (a)-(d) |Ey| distributions of the fundamental novel DLSPP mode at various nanohole heights (w_a = 100nm, w = h = 200nm); (e)-(h) Ey distributions along x direction at the center position of the nanohole; (i)-(l) Ey distributions along y direction at the center position of the nanohole

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Fig. 5 (a)-(d) |Ey| distributions of the fundamental novel DLSPP mode at various nanohole heights (h_a = 100nm, w = h = 200nm); (e)-(h) Ey distributions along x direction at the center position of the nanohole; (i)-(l) Ey distributions along y direction at the center position of the nanohole.

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To further quantify the field confinement and enhancement in the air nanohole region, the normalized optical power (NOP) and normalized average optical intensity (NAOI) in the nanohole are compared at different widths and heights, as shown in Fig. 6 . The NOP is defined as the ratio of the power inside the nanohole to the total power of the waveguide. It increases at bigger h_a when w_a is small. However, when w_a is relatively large (e.g. 100nm or 150nm), the NOP may decrease at bigger h_a, as a large air hole significantly weakens the field confinement capability of the ridge. The NAOI is given by NOP divided by the nanohole area and normalized to the peak intensity in the dielectric ridge of the corresponding traditional DLSPP waveguide shown in Fig. 2(a). It is shown that when h_a is small (e.g. <10nm), the calculated NAOI exceeds far beyond 1, especially when the nanohole is wider, indicating a several-fold enhancement in the optical intensity over that in the traditional DLSPP. However, as h_a is getting larger (e.g. >30nm), the NAOI at larger w_a drops more rapidly, due to the greater weakening of mode confinement in the nanohole region.

 

Fig. 6 The dependence of (a) the normalized optical power (NOP) and (b) the normalized average optical intensity (NAOI) of the fundamental plasmonic mode of the proposed novel DLSPP waveguide on nanohole sizes.

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N_eff, L_p, the normalized mode area (A_eff /A₀) and figure of merit (FOM) of the SPP mode of our proposed structure with different nanohole sizes are shown in Fig. 7(a)-(c) . A₀ is the diffraction-limited mode area and defined as λ²/4. FOM is defined as the ratio of L_p to D_eff [26]. Figure 7(a) shows that both of the modal effective index and propagation loss decrease monotonically when the nanohole size (width or height) increases. As shown in Fig. 7(b), at relatively smaller nanohole height, such as h_a<50nm, the mode area undergoes only small expansion, yet the propagation length could see a notable increase. For example, when h_a = 10nm, w_a = 150nm, the propagation length can reach ~35 μm, more than 2 times as large as that of the conventional DLSPP mode (~17 μm), while there is merely a less than 30% increase in its mode area. Therefore a shallow and wide nanohole could not only cause strong field enhancement as shown in Fig. 6, but also effectively reduce the transmission loss while maintaining deep-subwavelength confinement as shown in Fig. 7. At larger h_a, the mode area increases rapidly as the width of the hole increases. In such cases, the greatly extended propagation length comes at a price of much larger mode sizes. When h_a = 100nm or 150nm, the propagation distance could exceed 100 μm, but its normalized mode area also approaches 0.5. It is also noted that at rather large nanohole size, the FOM could decrease with increasing h_a (e.g. FOM(w_a = 150nm,h_a = 150nm) < FOM(w_a = 150nm,h_a = 100nm)), as the increase of the mode area outruns the increase of the propagation length,

 

Fig. 7 The dependence of modal properties of fundamental plasmonic mode of the proposed DLSPP waveguide on nanohole widths, where dashed lines correspond to the conventioal DLSPP waveguide: (a) the modal effective index (N_eff) and the propagation length (L_p); (b) the normalized mode area(A_eff /A₀); (c) the figure of merit (FOM).

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We note that there is a special case that deserves further discussions. When w_a = 200nm, i.e. the width of nanohole equals that of the ridge, the air nanohole turns into a nano-gap and the DLSPP structure transforms into a hybrid plasmonic waveguide [15–17]. However, such configuration with a suspended silicon structure over an air gap on top of the metal surface is physically infeasible to implement and the air gap has to be replaced by a low-index dielectric buffer layer. Due to the higher index of the buffer layer, the field confinement effect would be weaker in that case compared to ours.

In fact, for practical applications, etching and further deposition may generate nanoholes with various geometric shapes (e.g. trapezoidal) and positions (e.g. non-concentric). Consequently it could result in different modal behaviors, which may be more complicated than what is studied here and deserve further investigation in our later studies.

3. Conclusions

In this paper, we have proposed and studied a novel DLSPP waveguide by simply incorporating an air nanohole in the dielectric ridge. The existence of a shallow and wide, low-index nanohole could cause strong field enhancement in the nanohole and thus significant changes in the modal behavior of the plasmonic mode. Reductions in the transmission loss could be achieved when the mode area is only slightly increased under such a design. Our proposed structure could provide a relatively simple and effective way to alleviate the loss problem in the plasmonic waveguides, and could be appealing for building ultra-compact plasmonic devices and high-density photonic integrated circuits.