1. Introduction

Optical nonlinearity and dispersions constitute critical parameters in the design of advanced nonlinear devices such as optical modulators, switches, supercontinuum sources, parametric amplifiers, and wavelength converters [1,2]. To increase the mode confinement (and thus nonlinearity) and also to tailor their dispersion properties, different forms of waveguides have been suggested [3–6]. Slot waveguides, providing additional design freedom as a multi-layer structure [7], for the above reasons have been the foci of recent waveguide research, for applications which require higher nonlinearity [7–11] as well as tailored dispersions [12–17]. Unconventional characteristics, which are difficult to get using channel waveguides, such as wideband-flat dispersion, have also been achieved with a strip-slot hybrid structure [16,17]. Nonetheless of the success, past efforts for slot waveguides design have been limited to the infrared range, and also to treating only a specific set of materials and geometric parameters. A generalized analysis and design guideline clarifying the role of waveguide geometric parameters (cross-section area and fill factor) for slot waveguide dispersion has been carried out only recently by Mas [13], identifying material- and waveguide-dispersion dominant regimes and also comparing the result to that of a channel waveguide.

Another generic parameter, not yet explored for the slot waveguide design, is the index contrast, which has been widely employed in the study of double clad fiber [18,19]. In this paper, inspired by the structural similarity of the multi-layer construction of the slot waveguide and double clad fiber, we focus on the variable of index contrast Δn_slot-clad and Δn_clad-cover to approach/examine the problem of slot waveguide design, and then also explore the missing spectral region (<1 μm) in slot waveguide studies. Within the boundary of practically accessible refractive indices of materials, results show that the waveguide dispersion property and modal confinement of a slot waveguide are mainly determined by: 1) Δn_slot-core and Δn_core-clad, each for short (0.6–1.2 μm) and long wavelength (1–1.7μm) regimes; and 2) index contrast Δn rather than index n, especially in the short-wavelength region. Limiting the problem to short-wavelength slot of reasonable intensity confinement (I_slot > I_core), we also identify waveguide parameters appropriate for zero and flattened dispersion curves. The feasibility of low-threshold power super-continuum sources (either of wide bandwidth or of high coherence) for short wavelength application is demonstrated, assuming a TiO₂/Ta₂O₅/SiO₂ slot driven by 0.8μm, 100 fs input pulse (pulse energy 0.1 nJ).

2. Structure and analysis methods

Figure 1(a) illustrates the cross section of the horizontal slot waveguide [8–10]. A slot layer of width w and thickness w_s is placed between two core layers of thickness h, which is then surrounded by clad layers. The fill factor or normalized slot thickness δ is defined as w_s/(2h), adopted from [13,18]. Figure 1(b) shows the core-index-normalized index n/n_core profile of the slot. The core index n_core is normalized to unity, and the normalized index contrasts Δñ_sc and Δñ_cc are defined as (n_core – n_slot)/n_core and (n_core – n_clad)/n_core respectively. By changing the material or dopants (e.g., high Kerr silicon nanocrystal), the absolute index of the slot or core layer could be adjusted. To note, considering the minor contribution of absolute index to slot characteristics (in section 3), here we use the value of n_core = 3, unless stated otherwise.

 

Fig. 1 (a) Cross section of horizontal slot waveguides (b) Profile of normalized refractive index for the investigated slot waveguide. Δñ_sc = (n_core – n_slot)/n_core, and Δñ_cc = (n_core – n_clad)/n_core.

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The mode effective index n_eff and effective area A_eff of quasi-TM mode in slot waveguides were calculated by full-vectorial 2D finite element method using a FEM solver, COMSOL multiphysics [20]. The dispersion curve of the slot waveguide was obtained using D = –(λ/c₀)·(d²n_eff/d λ ²) [21], in excellent agreement with previous arts [12,13]. Field intensities in the slot and core layers were also compared to study their field confinement contributions.

3. Results and discussions

Without critical loss of generality, we start our discussion with conventional slot geometric parameters taken from literatures (w = 300 nm, h = 200 nm, w_s = 40 nm) [8–14]. By considering the refractive index of applicable materials (Table 1 ), we then investigated the behavior of waveguide dispersion D_w, normalized intensity I, and effective area A_eff, for six representative index profiles (combinations of Δñ_sc = 0.2, 0.35, 0.5 and Δñ_cc = 0.5, 0.65; Fig. 2(a) ).

Table 1. List of materials and their index values, within the scope of this study [22–26]

View Table

 

Fig. 2 (a) Waveguide dispersion (D_w), (b) Normalized intensities in slot (I_slot) and core (I_core) layers, and (c) effective area (A_eff) plotted for six different sets of index contrasts, Δñ_sc and Δñ_cc. (d) Normalized modal power of quasi-TM mode (λ = 0.65–1.85μm) for slot design of index contrast Δñ_sc = 0.2: (d)-1, ~5 and Δñ_sc = 0.5: (d)-6, ~10 (Δñ_cc = 0.5 in solid line, Δñ_cc = 0.65 in dotted line). For all the plots, n_core = 3.0, w = 300 nm, h = 200 nm, w_s = 40 nm.

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Figure 2(a) shows the calculated D_w over the frequency range of 0.65 to 1.85 μm, for the three by two (Δñ_sc x Δñ_cc) combinations of index profiles: Δñ_sc = 0.2 (red), 0.35 (black), 0.5 (blue) and Δñ_cc = 0.5 (solid line), 0.65 (dashed line). For the waveguide dispersion, two notable features were observed. First, in the short wavelength regime, the D_w curves sharing the same Δñ_sc (color code) converged with only minor dependence on Δñ_cc. On the other hand in the long wavelength regime, the effect of Δñ_cc was more pronounced, and D_w curves of the same Δñ_cc (line type) showed similar behaviors, with minor dependence on Δñ_sc.

It’s worth noting, this phenomena is in line with the modal intensity spectra in Fig. 2(b) and modal shape shown in Fig. 2(d). For the short wavelength, as the mode is mostly confined in the slot-core layer (Fig. 2(d)-1 and 2(d)-6), its waveguide dispersion becomes more dependent on Δñ_slot-core than Δñ_core-clad. In the short-wavelength region, for increased Δñ_sc (as in Fig. 2(d)-6 vs. Figure 2(d)-1, or the blue line compared to the red line in Fig. 2(b)), the faster decrease of I_slot causes a steeper change in its dispersion slope dD_w/dλ (Fig. 2(a)). In contrast, for the long wavelength, the mode extends from core to the clad layer (Fig. 2(d)-2(d)-5, or Fig. 2(d)-10) and thus by changing Δñ_cc it becomes possible to control the D_w in this regime. These observations imply that the shape of the waveguide dispersion is mostly governed by the change of index difference Δñ_sc for the short-wavelength regime, and on the other hand Δñ_cc for the long-wavelength regime. Considering both dispersion flatness and strong slot confinement, it was found that too large Δñ_sc or Δñ_cc are not suitable for short wavelength applications.

The effects of absolute refractive index and waveguide geometry on D_w curve tuning also have been carried out. Considering the numerous combinations one can assume for various waveguide parameters, here we keep the modal confinement within the core region (~n_core × h) to a constant value, thus to reduce the redundancies in parameter adjustments. For two representative slot designs appropriate for short-wavelength application (of flattened D_w and small A_eff, in Fig. 2; set of Δñ_sc = 0.2, Δñ_cc = 0.5 and Δñ_sc = 0.35, Δñ_cc = 0.5), the effect of absolute refractive index n_core and fill factor δ = w_s/(2h) was examined. Figure 3 shows the results of waveguide dispersion (a1–c1), normalized intensity (a2–c2) and effective area (a3–c3) for the changes of slot width (δ = 0.15, 0.10, and 0.05) and core refractive index (n_core = 3.5, 3.0, and 2.5) respectively. The waveguide width was set to w = 300 nm, and core thickness h was adjusted to keep n_core × h at a constant value.

 

Fig. 3 (a1–c1) Waveguide dispersion (D_w). (a2–c2) Normalized intensity (I_slot and I_core), and (a3–c3) Effective area (A_eff) for the slot design of; (Δñ_sc, Δñ_cc) = (0.2, 0.5); red curves and (Δñ_sc, Δñ_cc) = (0.35, 0.5); black curves, for different values of n_core = 3.5 (solid circle), 3.0 (line), and 2.5 (open circle). Core thickness h was adjusted to keep n_core × h constant. Slot thickness was changed from δ = 0.15, 0.10, and 0.05 for (a), (b), and (c) respectively.

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By decreasing δ (or equivalently w_s) or increasing n_core, the mode confinement in the slot layer was increased, with the associated increase in the D_w values. Yet worth noting is the behavior of dispersion and slot confinement, which shows a convergent behavior in the short wavelength regime irrespectively of the values of absolute n_core; indicating the dominance of index contrast and slot width when compared to the absolute refractive index, in the design of short-wavelength slot waveguides. Also worth mentioning, by decreasing core thickness h at fixed n_core, zero dispersion at even shorter-wavelength (<0.65μm) was possible.

4. Applications to short-wavelength super-continuum sources

Application of slot waveguides for super-continuum (SC) sources have been stated in [13,14], but their feasibility has been studied only recently by Zhang [17]—assuming a mid-IR (2.2 μm) pulse fed into a strip/slot hybrid structure composed of silica and silicon nitride. Meanwhile there is good interest and applications for SC sources in the spectral region around 0.8μm, and while short-wavelength SC sources constructed of nonlinear fiber [27] and Ti:Sapphire oscillator exist, still, there are no reports of exploiting the slot waveguide advantage for short-wavelength SC sources. From the knowledge gained in the previous sections for the short-wavelength slot, here we examine the feasibility of slot waveguide base, short-wavelength SC sources. For the material, we adopted Ta₂O₅ [25] and TiO₂ [24] each for slot and core, considering their index contrast (Δñ_sc = 0.11 and Δñ_cc = 0.38 with silica clad), Kerr coefficients (n₂ = 7.2x10⁻¹⁹m²/W [28,29]), and laser-induced damage threshold [30,31]. Near the wavelength of input pulse (0.8 μm), we tested two different designs of slot, considering both waveguide- and material-dispersion (D_w and D_m); one of anomalous and another of normal, by using slot thickness of δ = 0.08 and 0.16 respectively (for w = 300 nm, and h = 200 nm). After calculating the nonlinearity γ = ω₀n₂(ω₀)/cA_eff(ω₀) and dispersion coefficients β_m = (d^mβ/dω^m)_{ω = ω0} (up to the 6th-order [21] using Sellmeier equation [22,24,25]), a pump pulse (hyperbolic secant, Δτ = 100 fs and energy of 0.1nJ) was input to solve the nonlinear Schrodinger equation [27] to obtain the pulse evolution. The coherence of the pulse is calculated by using the modulus of the complex degree of first-order coherence |g₁₂⁽¹⁾| [3].

Figure 4(a) shows the output spectrum and coherence, pulse evolution in the slot, and chromatic dispersion curves D_ch = D_m + D_w, for the slot design of slot thickness δ = 0.08. Operating in the anomalous D_ch regime, a soliton-induced spectral broadening over two octaves (0.467–1.581 μm, after 5 mm slot propagation) was observed, yet with significant degradation in its coherence from the soliton fission. With the slot design of all-normal D_ch (δ = 0.16), much improved coherence |g₁₂⁽¹⁾| = 1 was observed over a one-octave spectral range (0.523 to 1.226 μm, Fig. 4(b)). For our slot waveguide having extremely high nonlinearity (γ = 12.76/W/m, A_eff = 0.445μm², orders larger than the γ ~ 1.6/W/m of the Ta₂O₅ rib waveguide [28] or the highly nonlinear PCF 0.11/W/m [3]), the required threshold pulse power for short-wavelength SC operation was considerably smaller than previous reports.

 

Fig. 4 (a-1,b-1) Pulse evolution and spectrum after 5mm propagation, (a-2,b-2) degree of coherence, and (a-3,b-3) chromatic dispersion (D_ch = D_w + D_m, in black); for slots of normalized slot thickness (a) δ = 0.08 and (b) δ = 0.16. w = 300 nm, h = 200 nm.

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6. Conclusion

Motivated by the present lack of generalized analysis for the slot waveguide design, and also noting the multi-layer structural similarity between the slot and double-clad fibers, we investigated the problem of slot waveguide dispersion and modal confinement by using the variable of index contrast—with special attention to the short-wavelength applications. It was found that Δn_slot-clad and Δn_clad-cover regulate the slot waveguide dispersion curve—each preferentially for the short- and long-wavelength regions. The observed phenomena were explained in terms of the modal intensity shift, from core to clad, and then to cover layers. Focusing on the unexplored regime of the short-wavelength slot, design guidelines were provided to realize both flattened near-zero dispersion and high confinement factor. The effect of geometric parameters and absolute refractive index were also explained. Application example of the learning was given with two types of short-wavelength super-continuum sources, one having a two-octave broadband spectrum and the other having unity coherence over one-octave, both enjoying the benefit of low-threshold power from the tight mode confinement of slot. Our study would work as guidelines in the selection of slot materials/dopants or in tailoring the dimensions of slot, depending on the required dispersion/confinement/nonlinearity at a target wavelength.