## Abstract

We propose a novel silicon waveguide that exhibits four zero-dispersion wavelengths for the first time, to the best of our knowledge, with a flattened dispersion over a 670-nm bandwidth. This holds a great potential for exploration of new nonlinear effects and achievement of ultra-broadband signal processing on a silicon chip. As an example, we show that an octave-spanning supercontinuum assisted by dispersive wave generation can be obtained in silicon, over a wavelength range from 1217 to 2451 nm, almost from bandgap wavelength to half-bandgap wavelength. Input pulse is greatly compressed to 10 fs.

©2012 Optical Society of America

## 1. Introduction

Chromatic dispersion is one of essential properties of an optical waveguide. It not only shapes the temporal waveform of an optical pulse traveling along the waveguide, but also acts together with nonlinearity to affect pulse spectral broadening [1]. In particular, zero-dispersion wavelength (ZDW) plays a critical role in determining phase matching conditions and conversion efficiency in nonlinear parametric processes [1]. Silica step-index optical fibers usually exhibit one ZDW at near infrared wavelengths of interest, while photonic crystal fibers (PCFs) provide a great opportunity to tailor dispersion profile, being able to produce three [2,3] or four [4,5] ZDWs and greatly tune their spectral positions. Such a spectral profile of the dispersion strongly influences nonlinear processes in fibers (e.g., supercontinuum generation) and generates rich and unprecedented phenomena [6–10].

Integrated waveguides with high nonlinearity have generated much excitement recently [11–15]. In particular, flat and low dispersion over a wide wavelength range becomes desirable for femtosecond pulses or octave-spanning spectral broadening. Dispersion tailoring with two ZDWs in high-index-contrast waveguides was presented [16–19]. More recently, a dispersion flattening technique was proposed, producing 20 times better dispersion flatness and three ZDWs [20]. Dispersion tailoring assists octave-spanning super-continuum generation in integrated chalcogenide and silicon nitride waveguides [21–23], where index contrast is lower than in silicon waveguides and waveguide dispersion becomes smaller. Nevertheless, such supercontinuum generation requires a high peak power of input femtosecond pulses, which is feasible only in a material platform with two-photon absorption (TPA) absent or extremely low. In silicon waveguides, strong dispersion and considerable TPA make it quite challenging to achieve octave-spanning supercontinua [24–27].

We propose a silicon slot waveguide that exhibits four ZDWs in a wavelength range from 1461 and 2074 nm, for the first time to the best of our knowledge. Dispersion values and ZDW locations can be varied, which provides a great opportunity to explore rich nonlinear effects such as solitons propagation, modulation instability, and supercontinuum and frequency comb generation. We show by simulation that the unique dispersion assists octave-spanning supercontinuum generation from 1217 to 2451 nm on a silicon chip. In this process, the input pulse is greatly compressed to 10 fs.

## 2. Dispersion-flattened silicon slot waveguide with four ZDWs

The waveguide has a horizontal silica slot formed between two silicon layers, as shown in Fig. 1
. The waveguide parameters are: width *W* = 610 nm, upper height *H _{u}* = 136 nm, lower height

*H*= 344 nm, and slot height

_{l}*H*= 40 nm. The buried oxide substrate is 2-μm thick. To calculate the dispersion of quasi-TM mode (vertically polarized), we obtain its effective index of refraction as a function of wavelength [20], using a full-vector mode solver, COMSOL, with material dispersions of Si and SiO

_{s}_{2}taken into account.

Figure 2
shows silicon material dispersion and the flattened dispersion of the quasi-TM mode, calculated as D = -(c/λ)∙(d^{2}n_{eff}/dλ^{2}). Dispersion varies between −22 and + 20 ps/(nm·km) over a 667-nm bandwidth, from 1435 to 2102 nm. Four ZDWs are found at 1461, 1618, 1889, and 2074 nm, respectively, as shown in Fig. 2(b). The group delay, defined as τ = (1/c)∙(n_{eff}-λ∙dn_{eff}/dλ), has a small variation of 40 fs/cm from 1403 to 2146 nm, which produces ultra-broadband group-velocity match for femtosecond pulse interactions.

The idea of generating four ZDWs is the following. First, material dispersion in silicon is always negative at wavelengths of interest (see Fig. 2(a)). Forming a waveguide, we see that, at short wavelengths close to the Si bandgap wavelength (~1100 nm), material dispersion is dominant, so overall dispersion is negative. At long wavelengths where the guided mode approaches cut-off, the dispersion is dominated by waveguide dispersion, which is negative [18]. Thus, as long as one tailors waveguide dimensions so that, in the middle wavelength range, waveguide dispersion is positive and stronger than material dispersion, a positive overall dispersion can be obtained with two ZDWs [18]. Then, we use the anti-crossing effect caused by mode transition, that is, the guided mode evolves from being more strip-mode-like to being more slot-mode-like as wavelength increases (see Fig. 1). This induces additional negative waveguide dispersion [20] and generates another two ZDWs in the middle.

One can change the structural parameters in the waveguide around the values given above to tailor dispersion value and slope. We individually decrease the upper height and increase the lower height, keeping the others the same. The dispersion profile is moved from normal to anomalous dispersion regime in Fig. 3 , with a dispersion value change of 18.5 and 15.9 ps/(nm·km) per nm, respectively. No significant change in dispersion slope is observed.

Dispersion slope can be greatly tailored by varying the slot height from 32 to 48 nm, as shown in Fig. 4(a) . On the other hand, ZDWs are moved with a spacing change among the ZDWs as the waveguide width is increased, which causes a slight rotation of the dispersion profile. The dispersion tailorability by varying the structural parameters gives us a valuable space for device design, although the dispersion properties in the high-index-contrast waveguides could be sensitive to fabrication imperfections.

## 3. Octave-spanning supercontinuum generation in Si waveguides

The flattened dispersion profile with multiple ZDWs can be used to generate rich nonlinear effects, and one of advantages of using it is to generate octave-spanning supercontinua. We note that supercontinuum generation in microstructured fiber [28,29] critically relies on its engineered dispersion. In normal dispersion regime, self-phase modulation (SPM) is mainly responsible for pulse spectral broadening, while higher-order soliton fission and dispersive wave generation are identified to be the main reasons for supercontinuum generation in anomalous dispersion regime [28]. Recently, we showed that, with nearly zero dispersion over a wide wavelength band, the self-steepening effect in nonlinear pulse propagation can greatly enhance spectral broadening to produce a two-octave supercontinuum on a chip [22]. To generate a supercontinuum in silicon, the approaches based on self-steepening and SPM tend to require very high peak power, which becomes impractical due to TPA in silicon [30]. Then, operating in the anomalous dispersion regime, one either used femtosecond pulses to reduce free carrier absorption [25,26] or moved to mid-infrared wavelengths to mitigate TPA [27], but, without sufficient capability to engineer the dispersion of high-index-contrast Si waveguides, octave-spanning supercontinuum generation still remains challenging.

Tailoring the dispersion to produce the saddle-shaped dispersion profile (four ZDWs) is substantially different from what was proposed in [20], in which three ZDWs were obtained in low-dispersion bandwidth, in terms of the ability to broaden pulse spectrum. This is because a dispersion profile with three ZDWs has strong anomalous dispersion at the long-wavelength end of the low-dispersion bandwidth, where dispersive wave can hardly be generated [28]. In contrast, being able to have the saddle-shaped dispersion, one not only achieves flat dispersion over an even wider bandwidth but also, more importantly, produces normal dispersion at both ends of the bandwidth. Assisted by the dispersive wave generation at the both ends, an octave-spanning supercontinuum is finally obtained in Si.

Here we choose width *W* = 610 nm, upper height *H _{u}* = 132 nm, lower height

*H*= 344 nm, and slot height

_{l}*H*= 40 nm. The flattened part of the dispersion profile is intentionally moved into anomalous dispersion regime, as shown in Fig. 3(a). Two ZDWs are located at 1418 and 2108 nm. Two local maxima of dispersion are 80 and 79 ps/(nm·km) at 1540 nm and 1965 nm wavelengths, respectively, and a local minimum of dispersion is 53.7 ps/(nm·km) at 1750 nm. To model supercontinuum generation, we use the following nonlinear envelope equation, with high-order dispersion of nonlinearity parameters including nonlinear index n

_{s}_{2}, TPA coefficient β

_{TPA}, and effective mode area A

_{eff}:

*A = A(z,t)*is the complex amplitude of an optical pulse, and γ

_{n}is the

*n*th-order dispersion coefficient of nonlinearity, γ

_{n}= ω

_{0}·∂

^{n}[γ(ω)/ω]/∂ω

^{n}. We include up to the 6th-order here. Other terms are defined the same way in [22], and Raman terms are not needed for the TM-mode [12]. Detailed derivation on the above equation is given in [31]. It was confirmed that the simulation of pulse propagations using nonlinear envelope equations is quite accurate [32]. The propagation loss is set to be 7 dB/cm, which may be lower in practice [33], since more light is confined in lower crystal silicon layer in our case. All order dispersion terms and carrier dynamics are included as shown in [25], and the free-carrier effects play a negligible role in the nonlinear processes. We estimate the dispersion of modal distribution by calculating the mode overlap factor given in [12] for the widely separated wavelengths at 1230, 1810, and 2395 nm. The overlap factor equals 0.745 between 1230 and 1810 nm and 0.971 between 1810 and 2395 nm. We believe that the major results presented here would hold well when the mode overlap is included. In our simulations, the time window length is 50 ps (Δf = 20 GHz), and the whole bandwidth in the frequency domain is 1000 THz.

We launch a chirp-free hyperbolic secant pulse into the dispersion-flattened silicon slot waveguide. The pulse has the center wavelength at 1810 nm and a full width at half-maximum (FWHM) T_{0} of 120 fs. Its peak power is 62 W. At 1810 nm, we find nonlinear index n_{2}=7.2×10^{−18} m^{2}/W and TPA coefficient β_{TPA}=5.3×10^{−12} m/W, based on the measurements in [34,35]. The nonlinear coefficient γ of the waveguide is (102+10.8j) /(m·W) using a full-vector model [36]. The shock time τ equals (1.16+0.33j) fs. At 1810 nm, the 2^{nd}-order dispersion coefficient β_{2} = −0.999 ps^{2}/km. Therefore, characteristic dispersion and nonlinearity lengths L_{D} = T_{0}^{2}/|β_{2}| = 14.4 cm and L_{N} = 1/real(γ)P = 0.16 mm, which gives a soliton number of 30.

Figure 5 shows pulse spectral broadening along the waveguide. At the beginning of the propagation, SPM causes obvious spectral broadening. Due to a relatively low dispersion over a wide band and thus small walk-off of frequency components, the pulse is significantly compressed, which further enhances the spectral broadening. In addition, the pulse transfers energy to high and low spectral components located around 1230 and 2395 nm, where dispersion is normal. This is likely to be dispersive wave generation, and group delay matches for 1300, 1810 and 2240 nm. At a propagation distance of 3.5 mm, the total spectrum bandwidth at −25 dB level is 1234 nm, from 1217 to 2451 nm, more than one octave. Accordingly, the compressed pulse has a FWHM of 12.2 fs, as shown in Fig. 5(b). The pulse waveform has beating patterns at rising and falling edges, which correspond to the dispersive waves at long and short wavelengths, respectively. With an input pulse FWHM of 60 fs, we obtain a supercontinuum of 1250 nm at −17 dB level, from 1200 to 2450 nm, when the pulse propagates 1.6 mm, exhibiting a FWHM reduced to 10.1 fs, as shown in Fig. 5(d). In these two cases, the pulse root-mean-square width is 155 and 73.6 fs, respectively, due to the pulse pedestals caused by dispersive waves. No soliton fission is observed, and we expect high spectral coherence.

## 4. Conclusion

We present a dispersion tailoring technique that produces four ZDWs in a silicon slot waveguide. With a flattened dispersion over a 670-nm bandwidth and normal dispersion at both short- and long-wavelength ends of this bandwidth, one can generate octave-spanning supercontinua in near-infrared wavelength region using the silicon waveguides, which would enable numerous chip-scale sub-systems for frequency metrology, sensing, optical coherence tomography, pulse compression, microscopy and spectroscopy, and telecommunication.

## Acknowledgments

This work is supported by HP Labs.

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