Effect of index contrasts in the wide spectral-range control of slot waveguide dispersion

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 shortand longwavelength 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 twooctave spectral width (0.467–1.581 μm), or of one-octave, near unity coherence |g12| = 1. ©2012 Optical Society of America OCIS codes: (190.4390) Nonlinear optics, integrated optics; (230.3120) Integrated optics devices; (000.4430) Numerical approximation and analysis; (190.5530) Pulse propagation and temporal solitons; (320.7110) Ultrafast nonlinear optics. References and links 1. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Quantum Electron. 12(6), 1678–1687 (2006). 2. B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24(12), 4600–4615 (2006). 3. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006). 4. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25(19), 1415–1417 (2000). 5. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31(9), 1295–1297 (2006). 6. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14(10), 4357–4362 (2006). 7. V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). 8. P. Sanchis, J. Blasco, A. Martinez, and J. Marti, “Design of silicon based slot waveguide configurations for optimum nonlinear performance,” J. Lightwave Technol. 25(5), 1298–1305 (2007). 9. F. Dell’Olio and V. M. N. Passaro, “Optical sensing by optimized silicon slot waveguides,” Opt. Express 15(8), 4977–4993 (2007). 10. P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17(11), 9282–9287 (2009). 11. C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for alloptical signal processing,” Opt. Express 15(10), 5976–5990 (2007). 12. L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18(12), 13187–13193 (2010). 13. S. Mas, J. Caraquitena, J. V. Galán, P. Sanchis, and J. Martí, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18(20), 20839–20844 (2010). 14. A. E. Willner, L. Zhang, and Y. Yue, “Tailoring of dispersion and nonlinear properties of integrated silicon waveguides for signal processing applications,” Semicond. Sci. Technol. 26(1), 014044 (2011). 15. L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express 17(9), 7095–7101 (2009). #166171 $15.00 USD Received 5 Apr 2012; revised 15 May 2012; accepted 15 May 2012; published 25 May 2012 (C) 2012 OSA 4 June 2012 / Vol. 20, No. 12 / OPTICS EXPRESS 13189 16. L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18(19), 20529–20534 (2010). 17. L. Zhang, Y. Yan, Y. Yue, Q. Lin, O. Painter, R. G. Beausoleil, and A. E. Willner, “On-chip two-octave supercontinuum generation by enhancing self-steepening of optical pulses,” Opt. Express 19(12), 11584–11590 (2011). 18. S. KAWAKAMI and S. NISHIDA, “Characteristics of a doubly clad optical fiber with a low index inner cladding,” IEEE J. Quantum Electron. QE-10, 879–887 (1974). 19. L. G. Cohen, W. L. Mammel, and S. Lumish, “Tailoring the shapes of dispersion spectra to control bandwidths in single-mode fibers,” Opt. Lett. 7(4), 183–185 (1982). 20. ComsolMultiphysics by COMSOL, © ver. 3.3 (2006). 21. G. P. Agrawal, Nonliner Fiber Optics (Academic Press, 2007). 22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998) 23. R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As2Se3 chalcogenide fibers,” J. Opt. Soc. Am. B 21(6), 1146–1155 (2004). 24. E. V. Stryland, Handbook of Optics (McGraw–Hill, 2009). 25. G. A. Al-Jumaily and S. M. Edlou, “Optical properties of tantalum pentoxide coatings deposited using ion beam processes,” Thin Solid Films 209(2), 223–229 (1992). 26. R. Spano, J. V. Galan, P. Sanchis, A. Martinez, J. Marti, and L. Pavesi, “Group velocity dispersion in horizontal slot waveguide filled by Si nanocrystals,” in 2008 5th IEEE International Conference on Group IV Photonics (IEEE, 2008), pp. 314–316. 27. J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010). 28. C. Y. Tai, J. Wilkinson, N. Perney, M. Netti, F. Cattaneo, C. Finlayson, and J. Baumberg, “Determination of nonlinear refractive index in a Ta2O5 rib waveguide using self-phase modulation,” Opt. Express 12(21), 5110– 5116 (2004). 29. V. Dimitrov and S. Sakka, “Linear and nonlinear optical properties of simple oxides. II,” J. Appl. Phys. 79(3), 1741–1745 (1996). 30. J. Jasapara, A. V. V. Nampoothiri, W. Rudolph, D. Ristau, and K. Starke, “Femtosecond laser pulse induced breakdown in dielectric thin films,” Phys. Rev. B 63(4), 045117 (2001). 31. J. Yao, Z. Fan, Y. Jin, Y. Zhao, H. He, and J. Shao, “Investigation of damage threshold to TiO2 coatings at different laser wavelength and pulse duration,” Thin Solid Films 516(6), 1237–1241 (2008).


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][4][5][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][8][9][10][11] as well as tailored dispersions [12][13][14][15][16][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 (either of wide bandwidth or of high coherence) for short wavelength application is demonstrated, assuming a TiO 2 /Ta 2 O 5 /SiO 2 slot driven by 0.8μm, 100 fs input pulse (pulse energy 0.1 nJ).

Structure and analysis methods
Figure 1(a) illustrates the cross section of the horizontal slot waveguide [8][9][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.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 [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.
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.
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.

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 Ta 2 O 5 [25] and TiO 2 [24] each for slot and core, considering their index contrast (Δñ sc = 0.11 and Δñ cc = 0.38 with silica clad), Kerr coefficients (n 2 = 7.2x10 −19 m 2 /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 = ω 0 n 2 (ω 0 )/cA eff (ω 0 ) and dispersion coefficients β m = (d m β/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 |g 12 (1) | [3].degradation in its coherence from the soliton fission.With the slot design of all-normal D ch (δ = 0.16), much improved coherence |g 12 (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 = 0.445μm 2 , orders larger than the γ ~ 1.6/W/m of the Ta 2 O 5 rib waveguide [28] or the highly nonlinear PCF 0.11/W/m [3]), the required threshold pulse power for shortwavelength SC operation was considerably smaller than previous reports.

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.

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 #166171 -$15.00USD Received 5 Apr 2012; revised 15 May 2012; accepted 15 May 2012; published 25 May 2012 (C) 2012 OSA