1. Introduction

Large-scale nanophotonic integration [1] – driven by signal-processing [2] and lab-on-a-chip [3] technologies – will require novel and efficient light-routing schemes between heterogeneous optical elements of vastly different dimensions, materials, and layouts with nanoscale footprints. Dielectric waveguides are prevalent, enabling low-loss optical transmission over long distances, while providing a pathway of interlacing optics and electronics at high data rates [4], or for performing photonic biosensing [3]. However, the dimensions of these waveguides are limited by diffraction [5] and cannot confine optical modes below a few hundred nanometers for wavelengths around λ₀ = 1.55 μm. This limitation imposes a fundamental restriction on the local electric field intensities which can be achieved for a given power. Next generation all-optical signal processing and single-molecule detection technologies, which rely on nonlinear optics [6] or plasmonics-enhanced sensing [7], will require large local field intensities while maintaining low input powers, all within an integrated photonic platform.

Plasmonic nanostructures can address the above limitations [8], enabling light confinement well below the diffraction limit [9]. Metal-dielectric-metal (MDM) plasmonic slot waveguides are particularly interesting [10], since their optical modes penetrate weakly into the metal, with most of the field inside the thin dielectric gap. Recent experimental demonstrations in such structures, for example, have shown light confinement down to tens of nanometers [11]), leading to giant nonlinear optical responses after few-wavelength propagation [12,13].

Plasmonic waveguides are inherently unsuitable for long-distance propagation due to their large metallic losses. Ideally, dielectric waveguides should be used for guidance, and plasmonic waveguides for enhancing light-matter interactions – with efficient energy transfer between the two. However, achieving high coupling efficiencies is challenging, since the physical dimensions and modal areas of typical photonic and plasmonic waveguides often differ by several orders of magnitude. Various coupling schemes have addressed this problem. According to 2D calculations, end-fire coupling schemes enable coupling efficiencies of up to 70%, and can be further improved up to 90% using a multi-section taper [14], but their performance drops significantly for large dielectric wavguides or narrow plasmonic waveguides as a result of the modal mismatch. 3D calculations of plasmonic adiabatic tapering [15,16] and directional coupling [17,18] predict efficiencies in the range of 40–90%, depending on plasmonic gap size, but both schemes require modal transition regions with micro-scale footprints.

These coupling schemes are conceptually straightforward and well understood. An alternative, and very compact coupling scheme was proposed by Lau et al. [19], and is conceptually less straightforward. In this orthogonal coupling scheme, the dielectric waveguide and the MDM plasmonic waveguide are orthogonal in orientation, as shown in the Fig. 1 schematic. They show that in this orthogonal scheme, light incident from a dielectric waveguide efficiently enters a sub-wavelength MDM plasmonic waveguide (Fig. 1) with calculated coupling efficiencies of up to 70% in 3D (using silver as the metal) [19], despite the waveguides being orthogonal to each other, and having considerably different lateral dimensions (dielectric/plasmonic waveguide width aspect ratio: ∼ 0.1). It is subsequently experimentally shown that efficient coupling (∼ 50%) is maintained over a broad bandwidth [20]. A similar orthogonal scheme consisting exclusively of plasmonic structures showed coupling efficiencies of nearly 100% in 3D structures [21,22]. For the hybrid dielectric-plasmonic configuration in Fig. 1, it thus remains an open question whether even higher efficiencies can be achieved.

 

Fig. 1 (a) Orthogonal coupler schematic and parameters. Light from the dielectric waveguide of refractive index n_in and width w_in in a background cladding with refractive index n_c (here: n_c = 1) couples into the orthogonal MDM waveguide. The dielectric in the MDM waveguide has refractive index n_out and width w_out. The propagation constants of the modes in the input dielectric waveguide and output MDM waveguide are labelled as β_in and β_out, respectively. Inset: mode profiles (Poynting vector) for the input- (blue) and output- (red) waveguides. (b) Example orthogonal coupling simulation. Magnetic field amplitude |H| in a coupler with parameters n_in = 3.5, w_in = 400 nm, n_out = 1.0 and w_out = 50 nm.

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While previous orthogonal dielectric-plasmonic experiments used a silicon waveguide input and an air gap in the orthogonal MDM waveguide, associated calculations showed a comparably high coupling efficiency when using a high-index medium in the sub-wavelength gap [19]. These results are particularly advantageous for nonlinear optics and sensing, which, respectively, rely on highly nonlinear polymers [13] and aqueous solutions [3]. Here, we consider this coupling scheme in greater detail, both numerically and analytically, and with a variety of refractive indices of both the input guide and the output MDM using gold as the metal [13]. To capture the essential physics while minimizing computation time, here we concentrate on 2D slabs, comparing with full 3D case for selected structures of broad interest. We show that coupling efficiencies as high as 78% can be achieved in 2D calculations (and as high as 54% in 3D calculations), and give simple physical arguments that explain and predict under what conditions high coupling efficiencies occur, without the need for full calculations. The insight obtained explains why including high-index dielectrics in the plasmonic gap, under the appropriate conditions, still provides good performance. It also allows us to propose a novel angled geometry that is compatible with both nanoscale photonic integrated circuits and miscroscale optical fiber endfaces. The angled geometry has with a higher coupling efficiency than equivalent orthogonal and end-fire configurations (up to 86% and 61% in 2D and 3D, respectively, with 30–50 nm plasmonic gap widths).

2. Theory and simulations

Since sub-wavelength plasmonic slot waveguides only support propagating TM modes [11], we take the fundamental TM mode of the dielectric slab waveguide as input. The waveguides are assumed to be infinite in y, which allows us to gain physical insight into the coupling mechanism over a large parameter space with relatively fast calculation times. Realistic structures, formed by 3D waveguides of finite height, can then be approximated using suitably adjusted effective indices [23]. Strictly speaking, therefore, the refractive indices used below should be interpreted as effective indices. With reference to Fig. 1(a), we define a coordinate system such that the z-axis is parallel to the dielectric slab waveguide, and the x-axis is parallel to the plasmonic slot waveguide. The magnetic fields are directed in the y-direction. The dielectric slab waveguide has a cladding refractive index n_c and a slab index n_in. The propagation constants of the TM modes of slab waveguides satisfy the transcendental equation

The corresponding transcendental equation for a plasmonic MDM waveguide has a similar form. Defining n_out to be the refractive index of the dielectric gap and n_m to be the complex valued refractive index of the metal leads to [11]

We employ a two-dimensional (2D) finite element method (COMSOL) to simulate the orthogonal dielectric-plasmonic coupler. All results shown here are calculated for λ₀ = 1.55 μm. We consider 20 nm ≤ w_out ≤ 200 nm and 200 nm ≤ w_in ≤ 850 nm, with an interval spacing of 10 nm. Since the plasmonic slot waveguide is lossy, the transmission efficiency depends on the location within the waveguide where the power is detected. Due to the complex coupling mechanism at the orthogonal junction, the numerical detector should be placed far enough from the junction to avoid measuring non-propagating contributions, but close enough so that propagation losses can be neglected. The transmission efficiency is defined as the ratio between the total upwards-directed power (i.e., the integral of the x-component of the Poynting vector in the numerical detector, see Fig. 1(b)), and the input power of the dielectric waveguide.

Here, we measure the transmitted power at a distance of 600 nm from the edge of the dielectric slab waveguide (dashed line in Fig. 1(b)), consistent with the approach of Lau et al. [19]. The propagation constants of the dielectric and plasmonic waveguides are calculated separately, and used for defining the input and output modes in COMSOL. The fundamental TM mode of the dielectric waveguide is defined via the input port, using a domain-backed slit condition with a perfectly matched layer to prevent reflections of higher order modes [24]. The output (plasmonic) boundary is an open port. To remove additional artefacts, we add a perfectly matched layer on all remaining boundaries. We first verify and confirm that coupling efficiencies do not depend on the length of either waveguide (provided that both waveguides support several wavelengths of propagation), or the size of the domain. Our results thus represent the intrinsic coupling efficiency of the dielectric-plasmonic junction, isolated from any extrinsic effects (e.g., Fabry Perot resonances).

We perform mesh size converge tests to maximize precision and minimize computation time. We use a triangular mesh size smaller than λ₀/20 = 77.5 nm in the dielectric slab waveguide and λ₀/50 ≈ 30 nm in the plasmonic gap; the difference in transmission efficiency is < 0.5% compared to that obtained when using a 15.5 nm maximum mesh size in the entire domain.

A typical magnetic field map of the orthogonal coupler is shown in Fig. 1(b) for n_in = 3.5 (silicon), n_out = n_c = 1 (air), n_m = 0.52406 + 10.742i (gold [25]), w_in = 400 nm, w_out = 50 nm. It shows that a substantial fraction of the field is transmitted into the plasmonic waveguide, despite its deep sub-wavelength width. For these parameters, the coupling efficiency is approximately η = 78%. To gain further insight, we calculate the efficiency for different w_in and w_out while varying refractive indices independently. Figure 2 is a colour map of the coupling efficiency versus w_in and w_out. In Fig. 2(a) we keep n_out = 1 constant while varying n_in, whereas in Fig. 2(b) we keep n_in = 3.5 constant and vary n_out. The example in Fig. 1 is close to optimal for this choice of refractive indices. We now explain the underlying trends using two simple arguments.

 

Fig. 2 Coupling efficiency versus waveguide width and gap width under different configurations. (a) n_out = 1.0 is kept constant, and n_in is increased from 1.5 to 3.5 as labelled. (b) n_in = 3.5 is kept constant, and n_out increased from 1.5 to 3.5 as labelled. The red and magenta dashed lines correspond to Eq. (5) using two different methods for quantifying modal confinement (see main text for details), blue dashed curves correspond to Eq. (7).

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3. Lower- and upper- bounds for orthogonal coupling

Optimization of the coupling between waveguides typically relies on laborious full numerical approaches [26], or semi-analytical methods which calculate appropriate scattering matrices at the junction [27]. For orthogonal junctions, such methods have been applied to hollow metal waveguides [27,28], dielectric waveguides [29,30], and plasmonic waveguides [21]. Increasing the coupling efficiency typically relies on adjusting the geometry at the junction to improve impedance-matching [31] – which is difficult to achieve for the nano-junctions considered. Here, we develop a simple model, which utilizes only modal confinement in the dielectric waveguide, and constructive interference in the plasmonic waveguide, to predict parameter ranges with high coupling efficiencies. This approach immediately suggests avenues for improvement, and provides useful upper- and lower- geometric bounds and practical guidelines for device prototyping.

Optical modes propagating in dielectric slab waveguides have energy in both the slab and the cladding. When the thickness of the slab is much larger than the wavelength, the energy is mostly in the slab. Decreasing the waveguide width confines light into a smaller region, but below a certain point a substantial amount of energy leaks into the cladding. Figure 1(b) suggest that only energy in the slab of the input waveguide couples to the plasmonic waveguide. With little energy in the dielectric core of the waveguide, we expect the transmission efficiency to drop.

A quantitative measure of the modal confinement width is the effective width, w_eff [32], which for the input symmetric TM mode of a dielectric waveguide with effective index n_eff = β_in/k₀ is given by [32]

Figure 3(a) (red) shows w_eff versus w_in for a typical case. The effective width has a unique minimum at $w_{\text{in}}^{\text{min}}$, represented by the vertical red line. For lower values of w_in the energy is pushed into the cladding and the coupling efficiency is expected to drop. A necessary condition for high coupling efficiency is, therefore,

 

Fig. 3 Input waveguide width conditions required for efficient coupling. (a) w_eff (red), and MFD (magenta) versus w_in for refractive indices n_in = 3.5 and n_c = 1.0. Red and magenta dashed lines respectively indicate the $w_{\text{in}}^{\text{min}}$ obtained from the minimum w_eff and MFD. The red shaded region shows regions where w_eff increases due to energy leakage into the air cladding. (b) Flat-phase point source model (concentric circles) used to explain efficient coupling into the MDM waveguide. The green region extends over λ_eff/2 = π/β_out and contributes to in-phase excitation. Sources in the red region interfere destructively with the sources in the green region, lowering the coupling efficiency.

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Another parameter that characterizes energy confinement is the mode field diameter (MFD), which will be useful for analyzing 3D waveguides, and which we introduce here for completeness. For a 1D waveguide, we adapt the standard definition of the MFD [33] as:

To determine an upper bound, we consider the MDM waveguide to be excited by point sources positioned along its length – which in this case corresponds to the end of the dielectric waveguide (see the concept schematic in Fig. 3(b)). Since bound modes of dielectric slab waveguides have flat phase fronts, all point sources have the same phase. When the point sources all lie within half an effective wavelength of the MDM waveguide (λ_eff/2 = π/β_out), we expect their contributions to the MDM mode to add in phase (Fig. 3(b), green), leading to efficient coupling. In contrast, when the point sources extend over more than λ_eff/2 (Fig. 3(b), red), then we expect some destructive interference, reducing the energy flow into the MDM waveguide. Therefore we can estimate the upper bound to the thickness of the dielectric slab waveguide

A systematic numerical investigation involving multiple refractive index combinations of the slab and the gap show that conditions Eq. (5) and Eq. (7) act as excellent lower and upper bounds for all parameters. As we show in Fig. 2, adjusting the refractive indices n_in and n_out correspondingly shift the lower and upper bounds, allowing for a visualisation of the bounds in action.

In particular, Fig. 2 demonstrates that for a significant range of parameters, the lower and upper bounds gives rise to a region in which both conditions are simultaneously met, allowing for efficient coupling. Note also that the borders of the highly efficient regions are well aligned with these lower and upper bounds. Furthermore, in Fig. 2(a), when the lower bound is relaxed, the intersection point of the two bounds shifts downwards. We see that the highly efficient region also shifts downwards, reinforcing our confidence in our arguments. Finally, when the the two conditions cannot be simultaneously met (e.g. n_out = 1.0, n_in = 1.5 or n_in = 3.5, n_out = 2.5 or 3.5), the efficiency drops dramatically. Ignoring material dispersion, these results also imply that strong coupling is maintained over a broad bandwidth, which is consistent with the findings of Lin et al. [20]. Note that the fields from the dielectric waveguide contribute a non-trivial point-like sources to the MDM input, which are affected by the various structural factors (e.g., reflections from the gold wall, the waveguide-to-MDM junction, and other interfaces). Therefore, the transition from high- to low- coupling by changing material- and geometric- parameters shows a gradual decrease, rather than an abrupt jump. Our model accurately describes what parameter windows contain the highest coupling efficiencies, although good performance can also occur outside those windows.

4. Angled coupling

Having established that the conditions of Eq. (5) and Eq. (7) capture key elements of the underlying physics, we consider whether it is possible to improve the coupling efficiency further. To see how this could be achieved we note that since the sources act as a phased array, the coupling efficiency is expected to be largest when the direction of the plasmonic waveguide aligns with the direction in which the wave is emitted. This alignment can be achieved if the plasmonic waveguide is angled with respect to the dielectric waveguide [34], see the schematic of Fig. 4(a). Since the interface is now sloped, the point sources do not radiate exactly in-phase. We choose ϑ such that the phases of point sources are consistent with the propagation constant in the plasmonic waveguide.

 

Fig. 4 (a) Angled coupled schematic and parameters, where ϑ is the angle between the input waveguide and the plasmonic waveguide. Also shown is the magnetic field for n_in = 3.5 and n_out = 1.0 (b) Coupling efficiency versus ϑ and w_in, for fixed n_in = 3.5, n_out = 1.0, and w_out = 50 nm. The blue line represents the result of Eq. (10). (c) Normalized coupling efficiencies for w_out = 50 nm, n_in = 3.5, n_out = 1.0, and w_in as labelled. For the purpose of comparison, the blue, red, yellow and purple curves have all been normalized. The actual peak coupling efficiencies are 0.82, 0.62, 0.40, and 0.26, respectively

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Neglecting the edge effects of the gold corner, the difference in phase across the two edges of the dielectric slab is

To further verify our simple model, we consider what happens when increasing w_in (Fig. 4(c)). Even though for orthogonal coupling Eq. (7) ceases to be valid, and the coupling efficiency drops, the angled coupling satisfying Eq. (10) is not subject to this limitation, since all sources contribute with optimal phase. Figure 4(c) shows the normalized coupling efficiency for different values of ϑ and w_in, with the original peak values given in the caption. The figure exemplifies the transition between these two regimes: for smaller w_in = 500 nm, where Eq. (7) holds, the coupling efficiency is relatively insensitive to waveguide angle (blue line); for larger values of w_in, the efficiency is highest at the angle given by Eq. (10) (purple line). For the largest w_in, we observe a clear maximum at the phase-matching condition (Eq. (10)).

Figure 5 shows calculations of the coupling efficiency versus ϑ for w_in = 2 μm for different values of n_in and n_out. We find that the transmission peaks for constant n_in (varying n_out, Fig. 5(a)) and for constant n_out (varying n_in, Fig. 5(b)) are in good agreement with the angled phase matching condition (Eq. (10)) as shown in Fig. 5(c) and the vertical lines in Figs. 5(a)–5(b), validating our model’s assumptions. We note that the coupling efficiency for large-waveguides are higher than those obtained for end-fire coupling [14], which we calculate to be ∼ 7 – 8 % for the materials and dimensions considered in Fig. 5.

 

Fig. 5 Coupling efficiencies for w_in = 2 μm, w_out = 50 nm as a function of ϑ, (a) constant n_in = 3.5 μm and varying n_out, and (b) constant n_out = 1.0 μm while varying n_in. Dashed lines indicate where Eq. (10) is satisfied. (c) The angles of the transmission maxima (circles) agree with the condition of Eq. (10) (solid lines).

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Although increasing the width of the dielectric waveguide results in higher coupling losses (see Fig. 5(a) and Fig. 5(b)) such designs have the remarkable advantage of being applicable to the tips of optical fibers [35, 36] – arguably the most convenient and widely-used optical waveguide. This aligns with the current paradigm of nanophotonic integrated circuit packaging, where light is coupled into optical circuits using angled fibers [37]. For a given waveguide and MDM, the method presented here immediately predicts what angle leads to efficient coupling, in a very simple geometry that does not require any intermediate circuits or gratings. As an example, we consider the case where the fundamental mode for n_in ≈ 2.5, w_in = 2 μm, and n_c = 1 (e.g., a chalcogenide taper, such as As₂S₃ [38]) couples to the MDM mode for n_out ≈ 1.5 and w_in = 50 nm (e.g., a highly nonlinear polymer (MEH-PPV), used for compact efficient plasmonically enhanced four-wave mixing [13]). Figure 6(a) shows an angled waveguide with ϑ = 58° found from Eq. (10), with the magnetic field superimposed. Note that, as expected from Eq. (10), the period of the magnetic field at the end of the input waveguide equals that in the plasmonic waveguide. For this geometry we calculate a coupling efficiency of 21%. As a direct comparison, we find that the angled coupler can be superior to the more conceptually straightforward end-fire coupler, as shown in Fig. 6(b) using equivalent materials and geometries.

 

Fig. 6 Angled coupler (λ₀ = 1.55 μm) with w_in = 2 μm, w_out = 50 nm, n_in = 2.5, n_out = 1.5 at ϑ = 58°, corresponding to the phase-matching condition as shown. Coupling efficiency: 21%. (b) End-fire coupling between the same waveguides results in a lower coupling efficiency of 7%.

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This method can, for example, be used to reduce the footprint of existing devices by incorporating the optical functionality directly on the fiber end-face. Note that a single-mode step-index configuration possesses a comparable effective index, and would function at a similar angle. An equivalent fiber structure should also be polarization-maintaining for optimal performance [34]. Note that tapered dielectric waveguides can provide high coupling efficiencies to plasmonic slots, and are a competitive option for input waveguides with sub-micrometer lateral dimensions [39]. However, there are significant experimental challenges in tapering large fibers to tip dimensions of tens of nanometers. In contrast, an angled end-face is more easily achieved, for example by mechanically cleaving a precisly defined micro-cut via focussed ion beam milling [36].

Finally, note that the phase matching condition, which we have found to adequately predict the region of highest coupling, applies to an MDM waveguide which is external to the coupling region itself (Fig. 6(a)). Elucidating the role of the junction for larger waveguides will require a more detailed analysis beyond the simple model presented here.

5. Comparison with 3D calculations

We now compare our model and 2D results with full 3D finite element calculations. We consider the industry standard silicon waveguide geometry (height h = 220 nm; n_in = 3.5; w_in = 300 – 700 nm) coupled to sub-wavelength MDM waveguides (w_out = 20 – 200 nm), keeping n_c = 1 for simplicity. We consider the coupling efficiency for n_out = 1.0, corresponding to recently reported experiments [20], and for n_out = 1.5, which has yet to be experimentally demonstrated and has important implications for nonlinear optics [12, 13]. The results are shown in Fig. 7(a),7(b), with the overall trend showing good agreement with our model. Here, the lower bound was obtained from the w_in possessing the smallest MFD (Eq. (6)). The peak efficiency, while lower than for the 2D case (∼ 54% at w_in = 400 nm, w_out = 30 nm, n_out = 1.0), is comparable to the value of 50.6% experimentally measured in a similar structure [20]. Discrepancies with the previously calculated value of ∼ 70% are likely due to the different materials used (the imaginary part of the gold permittivity is three times larger than that of silver at λ₀ = 1.55 μm [25]), in combination with the strong dependence of results on position of the (numerical) power monitor for such a lossy device. Furthermore, our simple theoretical model predicts additional properties that previous treatments did not: firstly, that the introduction of a higher-index dielectric in the plasmonic gap enables a comparably high coupling efficiency (∼ 54% at w_in = 390 nm, w_out = 50 nm, n_out = 1.5) that lies within our bounds (Fig. 7(b)); secondly, that this coupling efficiency can be further improved by increasing the angle between waveguides. Figure 7(c) compares the coupling efficiency as a function of ϑ for comparable 2D and 3D waveguides, which demonstrates the qualitative similarities between the two (see Fig. 2). The 3D case shows a peak efficiency of 61% – comparable to the experimental state of the art of 1.7 dB, or ∼ 68%, obtained for tapered junctions with similar gap sizes [39] – but without demanding any additional intermediate tapering region. Finally, note that 3D calculations are ∼ 400 times slower than their 2D counterparts, which limits practical calculation times over large parameter spaces. Our model will be particularly useful for providing initial design guidelines for future orthogonal and angled dielectric-plasmonic junctions.

 

Fig. 7 Coupling efficiency versus waveguide width and gap width for 3D waveguides with height 220 nm, n_in = 3.5 and n_c = 1. (a) n_out = 1.0. (b) n_out = 1.5. The magenta dashed lines correspond to Eq. (5) using Eq. (6) to obtain $w_{\text{in}}^{\text{min}}$, blue dashed curves correspond to Eq. (7). (c) Coupling efficiency as a function of ϑ for comparable parameters. n_in = 3.5; n_out = 1.0. 2D: w_in = 400 nm; w_out = 50 nm. 3D: w_in = 400 nm; w_out = 30 nm.

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6. Conclusion

In summary, we presented a simple model to predict regions in parameter space of high coupling efficiency for the orthogonal coupling scheme between dielectric slab waveguides and plasmonic MDM waveguides, as verified by finite element calculations. We find that efficiencies can reach approximately 78% at optical communication wavelengths when considering 2D structures, and 54% in equivalent 3D structures. This, combined with small footprint makes this method highly competitive in comparison to other schemes [14,15]. We propose two simple arguments which give lower and upper bounds to the region of phase space with high efficiencies. These bounds explain the rapid drops in transmission efficiencies as the waveguide parameters are varied. Our idea of the interaction between the dielectric slab waveguide and the MDM waveguide as point-like sources at the interface provides some insight into the mechanism that leads to the high coupling efficiencies of the orthogonal coupler. It also allows us to find a novel angled coupler for which the coupling efficiency is as high as 86% in 2D structures, and 61% in equivalent 3D structures with a plasmonic gap as small as 30 nm, which is comparable to the state of the art [39]. This model also allows us to predict what material combinations and angles produce high coupling efficiencies to MDM modes in large waveguides, such as optical fibers. In spite of this success, this simple model does not predict a value of the coupling efficiency, which requires a more detailed analysis at the junction. Future work will consider the influence of the characteristic impedance [21,22] and plasmonic Purcell factor [40,41], which are likely to play a significant role, and could potentially lead to even higher coupling efficiencies, e.g., via multi-section tapers [14]. These results lead to new and compact designs in hybrid dielectric-plasmonic photonic interconnects.