1. Introduction

The optical waveguide is an important building block which is heavily relied upon to enable photonic applications in data communications [1], biosensing [2], 3D imaging and light detection and ranging systems [3,4], inertial sensing [5], nanoparticle manipulation [6], hardware security [7,8], photonic circuits for classical or quantum information processing [9,10] and more. Attractively, silicon offers a high refractive index alongside its infrared transparency, which promotes small mode dimensions, compact device size, and the opportunity to enhance light-matter interactions. However, the mode dimensions and field enhancement of conventional waveguides, such as the silicon strip waveguide depicted in Fig. 1(a), are generally restricted by the diffraction limit. This limit can be broken by exploiting subdiffraction phenomena that locally enhance the near-field, on the subwavelength scale, through mechanisms that are distinct from interference phenomena, as exemplified in slot waveguides [11], hybrid dielectric-plasmonic waveguides [12], and anisotropy-engineered waveguides [13] excited under the appropriate polarization.

 

Fig. 1. (a) Electric field mode profile of a silicon strip waveguide (350 nm x 220 nm) and (b) silicon V-groove waveguide with h = 14 nm cladded by SiO₂ (fundamental quasi-TE modes). (c) electric field energy density, ${u_e} = \frac{1}{2}\epsilon {|E |^2}$, for the strip waveguide and (d) silicon V-groove waveguide normalized to the same color scale. Values above 1/6^th the max value, ${u_{e,max}}$, are saturated to show detail. (e) Schematic of the evanescent coupling architecture and two prospective approaches: (f) non-adiabatic directional coupling, and (g) adiabatic mode evolution.

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Recently, we have designed and introduced novel silicon based subdiffraction waveguides [14], such as the silicon V-groove waveguide depicted in Fig. 1(b). This waveguide design exploits two boundary conditions of Maxwell’s equations to constrain the eigenmode solution, enhance the electric field within silicon, and achieve ultra-small mode areas, A_n ≈ 10⁻³$\lambda_0^2$, on par with plasmonic nanowires but in an all-dielectric platform. Reductions in mode area correspond to enhancements in the maximum electric field energy density, ${{\boldsymbol u}_{\boldsymbol e}} = \frac{1}{2}\mathrm{\epsilon }{|{\boldsymbol E} |^2}$, as observed in Fig. 1(d) for the silicon V-groove waveguide when compared to the silicon strip waveguide depicted in Fig. 1(c). The silicon V-groove waveguide is amenable to fabrication by wet etching of crystalline (100) silicon, an approach which fosters low surface roughness and precisely controlled critical dimensions [15]. However, to harness the prospective benefits of this ultra-small mode area waveguide in enhancing light-matter interactions, it is necessary to first identify a solution for efficient input/output coupling. While prior works have studied and developed optical couplers for interfacing with slot waveguides [16,17] and plasmonic waveguides [18,19], no such coupler has been introduced for all-dielectric subdiffraction waveguides such as the silicon V-groove or diabolo type waveguides as described in Ref. [14]. The motivation of this work is to develop and investigate an efficient, broadband, and fabrication tolerant optical coupling solution for interfacing with subdiffraction all-dielectric waveguides with ultra-low mode area, using the silicon V-groove waveguide as a prototypical example.

The ultra-small mode area of the silicon V-groove waveguide coincides with a significant mode and phase mismatch relative to conventional silicon strip waveguides. Moreover, the propagation constant of the silicon V-groove waveguide exhibits amplified sensitivity to certain geometric parameters, such as the bridge height, h, of silicon remaining below the groove, where the electric field energy density is strongly enhanced, as visualized in Fig. 1(d),(e) [14]. To overcome these issues and facilitate efficient coupling, we consider two prospective coupling strategies: (1) non-adiabatic directional coupling as illustrated in Fig. 1(f), and (2) adiabatic mode evolution as illustrated in Fig. 1 g. We then perform a rigorous theoretical comparison between these techniques to identify their respective benefits and/or limitations.

2. Design methodology

2. Non-adiabatic directional coupling

As detailed by Yariv [20] and later refined by Hardy and Streifer [21], the directional coupler allows for complete power transfer between any two waveguides, regardless of their cross-sectional design, so long as they can be phase-matched and placed in proximity to one another. Here, we employ coupled mode theory to design and predict the properties of a directional coupler formed by the evanescent interaction between a conventional strip waveguide and an ultra-low mode area V-groove waveguide (Fig. 1(f)). Although the appropriate choice of gap and coupler length naturally leads to a prospective design solution, the overall coupling performance must then be evaluated by also considering both fabrication tolerances and optical bandwidth.

Figure 2 presents the effective index vs. waveguide width for isolated silicon strip and V-groove waveguides, evaluated at $\lambda_{0}$ = 1550 nm. The design is targeted for fundamental mode quasi-transverse-electric (TE₀) coupling into a nominal silicon V-groove waveguide in 220 nm silicon-on-insulator (SOI), e.g. Figure 1(b), where w_out = 500 nm and h = 14 nm. The value of w_groove is parametrically defined alongside our choice of h according to w_groove = 2(220 nm – h)/tan(54.7°) since we consider a fixed 220 nm silicon thickness and a 54.7° groove angle consistent with wet etching of (100) silicon. For this V-groove waveguide, phase matching is achieved with a strip width w_in = 283 nm as indicated by the purple marker in Fig. 2(a) and 2(b).

 

Fig. 2. Effective indices as a function of waveguide width for: (a) the strip waveguide, (b) centered V-groove waveguide, and (c) off-centered V-groove waveguide (centered only at w_out = 600 nm). Purple markers indicate phase matching design points, while green/red markers indicate taper start/end points. Dashed lines indicate higher order modes.

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For a given choice of gap, g, the coupling coefficient $\kappa$ and required coupler length L_c can be derived from the effective indices n₁ and n₂ of the coupled waveguide system’s first two eigenmodes – i.e., the symmetric and anti-symmetric supermodes of the directional coupler. Where Δn is the difference n₁ - n₂ between the effective indices, at an operating wavelength λ₀, the fractional power coupling is described according to ${\kappa ^2} = \sin^2(z(\pi {\Delta }n)/{\lambda _0}).$ Complete power transfer and localization in one waveguide requires accumulating a π phase difference between the supermodes, which occurs at the cross-over length, $z = {L_c} = {\lambda _0}/2{\Delta }n.$ Choosing g = 300 nm alongside our nominal waveguide designs as an example, yields ${\kappa ^2} > \textrm{ }0.99$ for a coupler length L_c = 9.5 µm.

2.2 Adiabatic mode evolution

Operation of our adiabatic coupler based on mode evolution is distinct from phase matched directional coupling, in that it does not excite multiple supermodes but rather preserves light in the lowest order (highest effective index) supermode. Unlike adiabatic 2 × 2 couplers which rely on the same general concept [22], our device functions as a mode convertor which localizes light in one input (strip waveguide) and one output (V-groove waveguide). The design problem is then two-fold: (1) modulate the waveguide cross-sectional design along the optical axis to control where the lowest order eigenmode resides, e.g., employing tapering to localize the fundamental mode in either the strip or V-groove waveguide; and (2) ensure the tapering is sufficiently gradual such that high order modes and/or radiation modes are not excited. In our designs, we employ linear tapers on both the strip and V-groove waveguides. We further assume the dimensions of the V-groove to be constrained by wet etching and constant vs. length, so that we only taper the outer width of the V-groove waveguide. In principle, adiabatic mode evolution can be achieved by tapering solely the strip waveguide. Here, we prefer to employ dual tapering of both waveguides since it leads to reasonably compact design solution with a modest strip waveguide tip size, w_tip = 200 nm. The widths of the waveguide taper start and end points are illustrated in Fig. 2(a,c) via the green and red markers respectively. Our nominal adiabatic design utilizes: w_in = 350 nm, w_tip = 200 nm, w_s = 500 nm, and w_out = 600 nm.

Figure 3 depicts the effective indices and mode profiles of the three lowest order supermodes in the adiabatic taper design. At the start of the taper the fundamental supermode confines light to the strip waveguide, whereas at the end of the taper light is localized in the ultra-small mode area V-groove waveguide. For a sufficiently long or adiabatic taper, light will be preserved in the fundamental supermode and higher order supermodes should not be excited to any significant degree. Interestingly, the higher order supermodes exhibit polarization rotation, an effect which is aided by the broken vertical symmetry of the V-groove structure. This suggests the V-groove geometry could prove useful in polarization diverse applications, such as in the design of on-chip polarization rotators [23,24]. This is however not the focus of the present study, as we are focused on efficiently coupling light into the fundamental quasi-TE mode of the V-groove waveguide from the fundamental quasi-TE mode of a strip waveguide.

 

Fig. 3. Visualization of the supermodes’ electric field intensity and effective indices in the adiabatic mode evolution design, where: w_in = 350 nm, w_tip = 200 nm, w_s = 500 nm, w_out = 600 nm, and g = 200 nm. The strip waveguide supermode evolves into a V-groove waveguide supermode as the taper passes through an anticrossing. Higher order supermodes exhibit polarization rotation effects.

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The coupling efficiency was evaluated vs. taper length for various gaps, using a commercially available 3D eigenmode expansion (EME) solver (Lumerical Inc.) with results depicted in Fig. 4. The coupling efficiency is defined as P_out/P_in × 100% which describes the ratio of power in the desired output V-groove mode, P_out, normalized to the input power launched into the strip waveguide, P_in. For gaps g = 200 nm or 300 nm, >99% efficiency is observed for L_c > 100 µm and 200 µm respectively. To provide manufacturing margin and ensure broadband operation, we select lengths L_c = 124 µm and 250 µm for our respective g = 200 nm and 300 nm adiabatic coupler designs. We’ve also independently quantified the worst case substrate leakage loss [25] for two buried oxide (BOX) thicknesses. For a BOX thickness of 1 µm the substrate leakage is ∼5.5 dB/cm at the mid-point of the taper; whereas a BOX thickness of 2 µm is adequate to suppress this value to <0.001 dB/cm, making it is negligible compared to typical propagation losses.

 

Fig. 4. Evaluation of the adiabatic mode evolution based design: coupling efficiency vs. coupler length for varying gaps.

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3. Results and discussion

Figure 5 presents a visualization of the simulated electric field |E| profile for both the directional coupler and adiabatic mode evolution based designs, where light is injected into the strip waveguide before the coupling region. Both structures successfully achieve high >99% coupling efficiency from a silicon strip waveguide into an ultra-small mode area V-groove waveguide. At a glance, it might appear as though both devices perform equivalently, except for the more compact footprint of the directional coupler. These results, however, are for the nominally ideal case which assumes zero fabrication errors and operation at exactly 1550 nm. In practice, these devices must achieve high coupling efficiency amidst non-zero errors in critical dimensions (CD), and ideally over a broad wavelength window.

 

Fig. 5. Simulated electric field |E| for: (a) directional coupler and (b) adiabatic mode evolution based example designs. The field profiles are taken at the plane y = 15 nm above the SiO₂/Si interface.

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Next, we investigate the fabrication tolerance of both designs by skewing three critical dimensions: (1) silicon device layer thickness (nominal = 220 nm), (2) global variations in waveguide width Δw, and (3) variations in the V-groove dimension. Here we assume the V-groove angle is held constant at 54.7° and consider nanoscale errors in the silicon bridge height Δh remaining beneath the groove (see Fig. 1(c)). Given the strong field concentration near the tip of the V-groove, the effective index is especially sensitive to both the silicon device layer thickness and the V-groove etch depth which control h [14]. As observed in Fig. 6(a), the directional coupler design is very sensitive to device layer thickness, falling to ∼50% efficiency for a 10 nm thickness deviation. Meanwhile the adiabatic design shows negligible impact from silicon thickness variations.

 

Fig. 6. Simulated coupling efficiencies of adiabatic and directional coupler (DC) type devices vs. three key parameters: (a) silicon device layer thickness, (b) error in waveguide width Δw and (b) error in V-groove bridge height Δh (as depicted in Fig. 1(c)).

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Figure 6(b) reports the coupling efficiency as a function of the CD error in waveguide width, Δw. Here the nominal designs are biased according to w_actual = w_design + Δw to mimic the effect of CD bias errors that may occur after lithography and/or etching. We observe the directional coupler based design to suffer a strong coupling loss penalty as a function of Δw, whereas the adiabatic design shows no measurable impact to coupling efficiency. The observed penalty in the directional coupler case arises due to two closely related effects. First, the perfectly phase matched condition of the ideal structure breaks in the presence of width bias because, unlike a conventional directional coupler formed from an equivalent waveguide pair, the strip waveguide and V-groove waveguide are not symmetric structures. As visible in curves of Fig. 2(a,b), the strip waveguide and V-groove waveguide exhibit different effective index sensitivities to waveguide width, dn_eff/dw, resulting in a phase mismatch which grows with CD error. Secondly, for non-zero Δw the beat length deviates from the nominal design value, as in a traditional directional coupler, and therefore the maximum coupling occurs at a different design point. Neither of these two challenges exist in the adiabatic coupler design and thus it can perform equally well for any CD error Δw considered in the range from −30 nm to +30 nm.

Figure 6(c) reports the coupling efficiency as a function of the CD error Δh, where we assume h_actual = h_design + Δh and evaluate device performance for the case h_design = 14 nm. As was found for CD errors in waveguide width, we observe that the directional coupler based design suffers a significant coupling loss penalty with increasing CD error Δh. We further explore how this penalty is affected by the choice of coupler gap. Notably, we observe that larger gap designs suffer a stronger penalty vs. CD error, with the coupling efficiency falling to ∼40% for the g = 400 nm design and ∼82% for the g = 200 nm design for a CD error Δh = +10 nm. The adiabatic design again shows robust tolerance to CD errors. In this case, the adiabatic design benefits from the fact that the strip waveguide taper spans a large range of n_eff from ∼1.5 to ∼2.1, while the corner case effective indices of the V-groove waveguide are always contained well within this range. Hence, the CD errors considered here are not large enough to impact the localization of the lowest order eigenmode or to substantially alter the device efficiency.

Lastly, we evaluated the wavelength dependence of both coupler types over the range λ₀ = 1550 nm ± 100 nm. Consistent with expectations, we find the adiabatic mode evolution based design exhibits superior bandwidth as compared to the directional coupler based design as illustrated in Fig. 7. Specifically, these two design types are found to exhibit >95% coupling efficiency over wavelength ranges of >200 nm and ∼50 nm respectively. To provide additional insight into the CD error penalties associated with the directional coupler in particular, we also illustrate wavelength dependence of devices where h_actual is set to 10 nm or 19 nm, equivalent to Δh = - 4 nm and + 5 nm respectively. A clear shift in wavelength is seen for changing heights. Here, these small nanoscale variations in bridge height shift the peak coupling wavelength of the directional coupler based design by ± 50 nm. This highlights the difficult task of operating with a directional coupler, especially when the coupled waveguides are dissimilar in nature and asymmetrically sensitive to fabrication variations.

 

Fig. 7. Simulated wavelength dependence of nominal adiabatic and directional coupler (DC) type devices, and two illustrative examples of DCs with CD errors.

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4. Conclusion

In this work, we theoretically investigated the coupling performance of a strip to V-groove waveguide coupler in both adiabatic and non-adiabatic regimes. We showed that both design types are theoretically capable of achieving low loss coupling into ultra-small mode area silicon V-groove waveguides. However, the adiabatic mode evolution based design provides better overall performance amidst realistic fabrication non-idealities alongside a wider operating bandwidth in comparison to the non-adiabatic design. Despite requiring a longer device length in general, the adiabatic coupler is favored since it is more robust yet remains simple to design. We expect many of the design principles utilized here can equally apply to other types of waveguide systems, particularly those which interface between conventional modes and dissimilar modes or those exhibiting strong subdiffraction character. These results illustrate a clear path to efficiently interfacing with novel types of all-dielectric subdiffraction waveguides, thus making them accessible in future experimental works that seek to harness their strong subwavelength field enhancement.