Here we examine the waveguide dispersion property of slot waveguides, approaching/analyzing the given problem with respect to the normalized index contrast, Δnslot-core/ncore and Δncore-clad/ncore between adjacent layers. For two index contrasts of concern, it is found that their contributions to slot waveguide dispersions are substantially different, with Δnslot-core and Δncore-clad each acting preferentially on short- and long-wavelength regions. Additional degrees of freedom in the waveguide design, such as the effect of absolute refractive index and waveguide geometry are also investigated to enable flexible tuning of the waveguide dispersion. Focusing on the unexplored regime of slot waveguides design in short wavelength (<1 μm), we also study the feasibility of low-threshold super-continuum sources using a Ta2O5/TiO2/silica slot, either of two-octave spectral width (0.467–1.581 μm), or of one-octave, near unity coherence |g12(1)| = 1.
©2012 Optical Society of America
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 , 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 , 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 Δnslot-clad and Δnclad-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) Δnslot-core and Δncore-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 (Islot > Icore), 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 ws 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 ws/(2h), adopted from [13,18]. Figure 1(b) shows the core-index-normalized index n/ncore profile of the slot. The core index ncore is normalized to unity, and the normalized index contrasts Δñsc and Δñcc are defined as (ncore – nslot)/ncore and (ncore – nclad)/ncore 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 ncore = 3, unless stated otherwise.
The mode effective index neff and effective area Aeff of quasi-TM mode in slot waveguides were calculated by full-vectorial 2D finite element method using a FEM solver, COMSOL multiphysics . The dispersion curve of the slot waveguide was obtained using D = –(λ/c0)·(d2neff/d λ 2) , 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, ws = 40 nm) [8–14]. By considering the refractive index of applicable materials (Table 1 ), we then investigated the behavior of waveguide dispersion Dw, normalized intensity I, and effective area Aeff, for six representative index profiles (combinations of Δñsc = 0.2, 0.35, 0.5 and Δñcc = 0.5, 0.65; Fig. 2(a) ).
Figure 2(a) shows the calculated Dw over the frequency range of 0.65 to 1.85 μm, for the three by two (Δñsc x Δñcc) combinations of index profiles: Δñsc = 0.2 (red), 0.35 (black), 0.5 (blue) and Δñcc = 0.5 (solid line), 0.65 (dashed line). For the waveguide dispersion, two notable features were observed. First, in the short wavelength regime, the Dw curves sharing the same Δñsc (color code) converged with only minor dependence on Δñcc. On the other hand in the long wavelength regime, the effect of Δñcc was more pronounced, and Dw curves of the same Δñcc (line type) showed similar behaviors, with minor dependence on Δñsc.
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 Δñslot-core than Δñcore-clad. In the short-wavelength region, for increased Δñsc (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 Islot causes a steeper change in its dispersion slope dDw/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 Δñcc it becomes possible to control the Dw in this regime. These observations imply that the shape of the waveguide dispersion is mostly governed by the change of index difference Δñsc for the short-wavelength regime, and on the other hand Δñcc for the long-wavelength regime. Considering both dispersion flatness and strong slot confinement, it was found that too large Δñsc or Δñcc are not suitable for short wavelength applications.
The effects of absolute refractive index and waveguide geometry on Dw 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 (~ncore × 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 Dw and small Aeff, in Fig. 2; set of Δñsc = 0.2, Δñcc = 0.5 and Δñsc = 0.35, Δñcc = 0.5), the effect of absolute refractive index ncore and fill factor δ = ws/(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 (ncore = 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 ncore × h at a constant value.
By decreasing δ (or equivalently ws) or increasing ncore, the mode confinement in the slot layer was increased, with the associated increase in the Dw 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 ncore; 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 ncore, 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 —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  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  and TiO2  each for slot and core, considering their index contrast (Δñsc = 0.11 and Δñcc = 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 (Dw and Dm); 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)/cAeff(ω0) and dispersion coefficients βm = (dmβ/dωm)ω = ω0 (up to the 6th-order  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  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)| .
Figure 4(a) shows the output spectrum and coherence, pulse evolution in the slot, and chromatic dispersion curves Dch = Dm + Dw, for the slot design of slot thickness δ = 0.08. Operating in the anomalous Dch 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 Dch (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, Aeff = 0.445μm2, orders larger than the γ ~ 1.6/W/m of the Ta2O5 rib waveguide  or the highly nonlinear PCF 0.11/W/m ), the required threshold pulse power for short-wavelength SC operation was considerably smaller than previous reports.
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 Δnslot-clad and Δnclad-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.
This work was supported by the National Research Foundation; GRL, K20815000003 and Center for Subwavelength Optics, SRC 2012-0000606, and in part by Global Frontier Program (2011-0031561) all funded by the Korean government.
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