1. Introduction

Over the last few years, the drive to build integrated all-optical circuits on tiny chips has drawn attention to the problem of light propagation in single-mode silicon-on-insulator (SOI) photonic wire waveguides [1-4,16,17]. The nonlinear response of these waveguides is high as a result of the large nonlinear refractive index of silicon and is further enhanced by the very strong confinement possible in silicon waveguides with sub-wavelength dimensions. A large nonlinear response will be vital for applications such as switching, frequency conversion and parametric amplification – which are currently the subjects of intense exploration in SOI waveguides [1,2,5-12,16,17]. For waveguides with widths around 500nm and thicknesses of 200nm or 300nm, the group velocity dispersion (GVD) at 1.5µm wavelength is strongly anomalous because of the large waveguide dispersion [8,13,14]. Anomalous GVD combined with a strong focusing nonlinearity suggests that soliton dynamics will dominate the propagation of femtosecond pulses in long silicon photonic wires [4,5]. The effects of self-phase modulation, modulational instability and soliton formation, which in optical fibers are usually observed on length scales from tens of centimeters to hundreds of meters, can be expected after just a few millimeters of propagation in such extreme waveguiding environments [1,6,7,8,9,16,17]. Though nonlinear effects in silicon wires are influenced by strong two-photon absorption (TPA), many of the recent results are very encouraging. Previously, soliton features have been reported from 5mm long waveguides, corresponding to about one dispersion length of the 120fs pump pulses used [4]. Here we report femtosecond pulse propagation in an SOI photonic wire waveguide that has been specifically designed to span several dispersion lengths, so that one can more convincingly make the case that solitons have formed. Critically, we also present accurate measurements of the group index of the waveguide as a function of wavelength. The dispersion coefficients (up to the third order) derived from these measurements are used for our numerical modeling and for interpretation of the observed nonlinear output spectra.

2. Linear properties of the nanowires

Our photonic wire waveguide was fabricated on a silicon-on-insulator (SOI) wafer consisting of a 260nm thick, lightly p-doped silicon guiding layer on top of a 1µm thick silicon oxide layer, on a silicon substrate. The waveguide was patterned along the [110] crystallographic direction on the (100) wafer surface using electron beam lithography and inductively coupled plasma reactive ion etching. The top silicon layer was completely etched through, so that the rectangular waveguide cross sections were surrounded by air on both sides and by a residual ~100nm thick HSQ resist layer on the top, which was deposited before the etching process and acted as a hard mask. In order to improve the input/output coupling efficiency of the 480nm wide and 14.8mm long single-mode section of the waveguide, we fabricated short tapered sections expanding to 2.1 µm wide multimode waveguides at the edges of the chip (see Fig. 1). Untapered multimode waveguides of these dimensions were also fabricated on the same chip. Linear losses in the photonic wire waveguides were estimated to be α _dB=-3.4 dB/cm. In separate work, we have recently produced similar waveguide structures with propagation losses below 1 dB/cm [15].

Spectral group index measurements were carried out using a free-space Mach-Zehnder interferometer. An optical supercontinuum generated using a Q-switched microchip laser (1064nm) and a piece of photonic crystal fiber was used as a broadband light source. Wavelengths below 1220 nm were removed using a long-pass filter to avoid thermal effects and optical damage to the waveguide under study. The waveguide was placed in one arm of the interferometer. The delay of the reference arm could be scanned using a motorized control stage. The signal arm had an average power of 1mW, a repetition rate of 7 kHz, and a pulse length of 1ns. Using the estimated coupling coefficient in the interferometer (~-30dB), the peak power inside the waveguide was far too low to cause significant nonlinear response. At the output, the signal and reference beams were combined and passed through a band-pass filter, and then coupled into a single-mode collecting fiber. Interferograms were recorded using a lock-in amplifier. Fringe packets for the TE and TM modes were easily resolved. We also recorded the position of the fringes without the waveguide in the sample arm, to determine the actual values of the waveguide group indices. Coupled with our long waveguide length this method offers improved accuracy over previously reported measurements [13,14].

 

Fig. 1. (a) Schematics of the multi-mode and single-mode silicon-on-insulator waveguides. (b) & (c) Scanning electron micrographs of the single-mode SOI waveguide taken at the end (b) and in the middle (c) of the guide.

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Fig. 2. (a) Group index of the single-mode (solid squares) and multi-mode (open squares) SOI waveguides in the TE mode as a function of frequency. The colored lines (black, red and green) are the 3^rd, 4^th, and 5^th order polynomial fits respectively. (b) Derived 2^nd- and 3^rd- order dispersion coefficients vs. frequency.

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Taking account of various sources of error and uncertainty in our measurements we have established an uncertainty in our value of group index of 0.037. Most of this error is independent of wavelength, and the wavelength-dependent uncertainty of our group index measurements (which is the only part which impacts on our GVD estimates) is below 0.001.

Figure 2(a) shows the group indices of the multi-mode (open squares) and single-mode (filled squares) SOI waveguides in the TE polarization state across the wavelength region from 1.24 to 1.64µm. The data have been processed to remove the effects of the multimode input and output and the tapers by assuming that in the multimode region only the fundamental TE mode is excited, and by using the known lengths (Fig. 1(a)) of the different sections together with the measured group index curves. The effect of the strong waveguide dispersion on the group index and its slope when the size of the waveguide is decreased is very apparent. The results presented are those derived from the measured data for the 480 nm×260 nm cross-section waveguide. Figure 2(a) also shows fits for different orders of polynomial to the group indices of the single-mode SOI waveguide. Figure 2(b) shows the second and third order dispersion coefficients derived from the polynomial fits. The values obtained at a wavelength of 1.5 µm are β₂=-2.31±0.04 ps²/m and β₃=0.0119±0.0009 ps³/m respectively. The uncertainties quoted here are the result of the variation obtained when fitting to different order polynomials since relative to this variation the uncertainties due to the measurement error of the group index can be ignored. Our dispersion values are in the order of magnitude agreement with the previously reported ones [13,14].

3. Nonlinear measurements

Our nonlinear experiments were carried out using the experimental setup shown in Fig. 3. An optical parametric amplifier (OPA) provided ultrashort pulses centered at ~1.5µm with a full width at half maximum (FWHM) pulse length of ~100fs and a repetition rate of 250kHz. The first polarizing cube after the OPA was used to remove the idler beam. By rotating the second polarizer we could continuously adjust the power going through the third polarizer, which was set to excite the TE mode in the waveguide. A singlet coated objective lens was then used to couple the pulses into the SOI waveguide. This no-fiber arrangement minimized the nonlinear and dispersive degradation of the input pulses. At the output end, the transmitted pulses were collected through an AR-coated high NA (numerical aperture, 0.65) objective lens followed by a piece of single mode fiber, and analysed using an optical spectrum analyzer (OSA) or a photodetector.

 

Fig. 3. Schematic of the experimental configuration for nonlinear measurements.

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As indicated in Fig. 3, a continuous wave (cw) laser at ~1.55µm was launched into the experimental setup using a flipper mirror, in order to characterize the linear loss caused by coupling and propagation. A thermal power meter and a photodetector were cross calibrated and then used to measure the power before/after the SOI waveguide and after the collecting single-mode fiber. The results obtained combined with the known Fresnel reflection loss between silicon and air (-1.56 dB) and the propagation loss along the waveguide (α_dB=-3.4 dB/cm) indicated that the mode-matching to the SOI waveguide caused a loss of ~-23dB. The light collection optics at the output caused losses ~-9dB. Using these values of linear loss, if an OPA pulse train with an average power of 100µW were launched into the experimental setup, the nonlinear length in the single-mode collecting fiber would be greater than 1km. Thus the nonlinear response in this element can be neglected. We have also found that the spectra measured using this setup are modulated, which can be removed by placing a polarizer (Pol. 4 in Fig. 3) before the collecting fiber. These evenly-spaced peaks appeared at the same wavelength independent of the power level. We believe that these features are likely to be caused by polarization mode conversion in the experimental setup and the resulting mode beating following propagation through the (non-polarization-maintaining) collecting fiber. Although this mode conversion should be very weak, the two photon absorption (TPA) induced nonlinear attenuation could make the intensities of the TE mode and the TM mode comparable at the output. The measurements presented below were obtained with the polarizer 4 set for the TE polarization.

The spectrum of our OPA pulses is shown in the lower panel in Fig. 4(a), and the full width at half maximum (FWHM) of the spectrum was measured to be 5.22THz. Assuming a Gaussian shape, the FWHM pulse length was measured to be 100fs using an autocorrelator. Hence, the time-bandwidth product of the input pulses was 0.522, implying a modest chirp. Figure 4(a) shows the measured output spectra from our SOI waveguide as the input power was increased. At low powers (average powers 1-2µW) the output spectra are narrowed relative to the input spectrum. Spectral broadening dominates the spectral evolution in the input power range from 4µW to 40µW. Above 40µW the spectral evolution begins to saturate, with further power increases (up to 90µW) resulting in only minor changes to the output spectrum. Note that in order to convert the given measured average powers to find the peak powers in the waveguide, the repetition rate of 250kHz, pulse length of 100fs and coupling coefficient of -23dB must be taken into account. Numerical simulations are plotted in Fig. 4(b) and will be described in the next section. We have used a calibrated photodetector to measure the transmitted average power through the collection fiber. Figure 5 plots this transmission curve as a function of the input power. Complete saturation of the transmission is evident at input powers above 40µW. In the quasi-linear regime we measured a total linear attenuation of -40.6 dB, very close to the loss (-39.2 dB) measured using the cw laser at 1.55µm.

 

Fig. 4. (a) Output spectra (linear scale) experimentally recorded at various input powers plotted. (b) The same spectra modeled numerically. These spectra drawn for TPA only (dotted blue line), TPA with free carrier effects (dashed red line) and TPA together with 3PA (solid green line). 1µW of the averaged input power corresponds to ~0.18W of the peak power inside the waveguide.

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4. Analysis and modelling

In order to reproduce our experimental measurements numerically and to draw conclusions about the nonlinear dynamics of pulses inside the waveguide, we have used a generalized Nonlinear Schrodinger equation (NLS). This equation included the effects of the second and third order dispersion, Kerr nonlinearity, multi-photon absorption, free-carrier generation and linear loss. We have found that the effects of Raman scattering and dispersion of the nonlinearity are negligible, the former is mainly due to slow time response of phonons relative to the pulse duration and smallness of the Raman coefficient [10], and we have excluded them from our model. The system was modeled using the equation

 

Fig. 5. Measured (black line with squares) average output power as a function of average input power. The other lines are numerical calculations. These are drawn for TPA only (dotted blue line), TPA with free carrier effects (dashed red line) and TPA with 3PA (solid green line).

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Here E is the slowly varying envelope of the electric field, τ is time in the reference frame moving with the input pulse group velocity and ξ is distance along the waveguide. These dimensionless units are related to real distance and time by t=T ₀τ and z=ξL_D, where L_D=T ₀ ²/|β ₂|=1.4mm is the dispersion length and T ₀ is the input pulse duration. The shape of the input pulse is assumed to be sech(t/T ₀), because precise information about the chirp on our input pulses is not available. T ₀=57fs relates to the FWHM as T_FWHM=2ln(1+√2)T ₀. Dispersion is accounted for by the experimentally-determined coefficients β ₂ and β ₃. The scaled third-order-dispersion parameter is p ₃=β ₃/(6|β ₂|T ₀). The power is scaled such that the P ₀|E|² gives physical power, where P ₀=|β ₂|/(γT ₀ ²)=0.36W is the soliton threshold for the ideal NLS and γ is the nonlinear parameter of the waveguide. One should note that it is difficult to estimate γ reliably from the analytical formulae, because the longitudinal polarization component is strong and the field variations close to the waveguide boundaries are steep so that many standard approximations are violated. In the existing literature one can find various approaches to calculation of the nonlinear coefficients in photonic wire waveguides, which give typical values from hundreds to a few thousands of (Wm)^-1. See, e.g. [8,12]. We use γ as a fitting parameter, and have found that reasonable agreement of our modeling results with spectral measurements is achieved for γ=2000/W/m. Taking into account the overall coupling loss before the photonic wire of -23dB, we find that P ₀ corresponds to ~2µW of the averaged laser power measured before the waveguide.

There are four principal sources of damping in our model. The first is linear loss, as represented by ε ₁=T ₀ ²|α|/(8.7|β ₂|)=0.055. Two-photon absorption (TPA) and three-photon absorption (3PA) ε_tpa and ε_3pa are scaled to the nonlinear refractive index of silicon. For TPA we have used a standard value of ε_tpa=0.1 [10]. Three-photon absorption is usually not included in the modeling of pulse propagation in silicon photonic wires but we have found that in order to match the measured dependence of the output vs input power relationship (see Fig. 5) simply accounting for TPA and free-carrier absorption is insufficient. Phenomenological inclusion of three-photon absorption improves agreement between the modeling and measurements. We have selected ε_3pa=0.05. The scaled free-carrier charge (FCC) density n evolves as ∂_τn=|E|⁴-T ₀ n/τ_c. The first term on the right-hand side corresponds to carrier excitation due to TPA. The second term corresponds to the carrier recombination, which can be ignored because the carrier lifetime τ_c (which is of the order 10ns) is both much larger than the pulse duration, and much smaller than the inter-pulse interval. Therefore, we can assume that the carrier density starts from zero at the beginning of each pulse, with negligible intra-pulse decay. The physical density is N ₀ n, where N ₀=β_tpaT ₀ P ₀ ²/(2ħω ₀ A)=7×10¹⁸ m ^-3. Here β_tpa=5×10^-12 m ^-1 W ^-1 is the TPA coefficient, ω ₀ is the pump frequency and A is the waveguide cross-sectional area. The FCC absorption coefficient ε_fcc=σL_DN ₀(1+iµ)=1.6×10^-5×(1+i7.5), where the absorption cross-section is σ=1.45×10^-21 m ².

We have solved Eq. (1) numerically for the case of 100 fs pump pulses, neglecting propagation in the multimode section of the waveguide. We present here the results for three cases for comparison with experimental measurements and to clarify the role of the different physical mechanisms influencing nonlinear propagation in the photonic wires. The results presented take account of both the input and the output losses for direct comparison with experiments.

In generating the first set of numerical data we have ignored both the coupling of the light field to the free carrier equation and 3PA, as shown using dotted blue lines in Figs. 4(b) and 5. The slight shift at 4µW in the spectral peak towards longer wavelengths can be seen in both modeling and experiment, see Fig. 4. We have confirmed that this asymmetry appears due to third order dispersion and can be reversed by changing its sign. With increasing power the discrepancy between the model and experiment starts to develop. One can see that the width of the modeled spectrum and the number of spectral peaks for powers above 20µW quickly exceeds the corresponding experimental values. On the output vs. input power plots (see, Fig. 5) the experimental curve also saturates faster, in agreement with the narrower measured spectra.

The second set of data shown by the solid green lines in Figs. 4(b) and 5 addresses the problem of weak saturation by including the 3PA term. One can see that the high power part of the numerically generated plot in Fig. 5 is now much flatter and fits the experimental data better. The spectral width generated numerically for high powers also agrees well with measurements. In addition for small powers the correct asymmetry of the spectral peak is retained.

The third set of data marked with the dashed red line disregards the 3PA term, but introduces the generation of free carriers into the model. One can see that the spectral width generated for high powers diverges well beyond the experimental data and for small powers, the spectral asymmetry introduced by the free carriers is opposite to that seen in experiments. Indeed, free carriers generated at the front edge of the pulse start to affect its trailing edge. However, anomalous GVD implies that shorter wavelengths in the pulse spectrum propagate faster, than the longer ones, implying that the shorter wavelengths are almost unaffected by the free carriers, while the longer wavelengths are attenuated. For the parameters chosen in the modeling this effect overcomes the spectral asymmetry introduced by the third order dispersion and results in noticeable disagreement with experimental observations at 4µW. Thus, for the conditions of our experiment the best agreement across the relevant power range is observed when the third order dispersion, linear loss, and instantaneous TPA and 3PA are added to the ideal NLS model.

Next, we turn our attention to the question of whether our spectral measurements and modeling allow us to draw any conclusions about the formation of quasi-soliton pulses in silicon photonic wires. To answer this question we focus on the regime of relatively low input powers (<5µW) below the powers corresponding to the onset of the spectral splitting. In this power regime all three models considered above give approximately the same results. Increasing the peak intensity of the pump pulse should induce increasing nonlinear absorption and the quasi-soliton pulses, if formed, are expected to spread temporally with a simultaneous drop of their amplitude. If, however, such behavior happens in the adiabatic regime, when the drop in the amplitude and increase in the length develop in a way that their product remains constant, or almost constant, then one can speak about formation of adiabatically evolving solitons. Indeed, the soliton solution of the ideal dimensionless NLS model is, $E=\sqrt{2q}\mathrm{sech}\left(\tau \sqrt{2q}\right)e^{\mathit{iq}\xi }$ . If the soliton parameter q is assumed to evolve in ξ, then the amplitude-width product of this adiabatically evolving soliton still remains a ξ - independent constant. Fig. 6(a) shows the temporal length (FWHM) normalized to the input pulse width, which is always greater than one and hence the pulse spreads in time. However in the power interval from ~1µW up to ~3µW the product of the amplitude and the pulse FWHM (shown in Fig. 6(c)) remains quasi-constant and approximately corresponding to the value found for the ideal NLS soliton (see dashed horizontal line). Another test that can indirectly verify the formation of solitons is measurement of the spectral width of the signal [9]. If an adiabatically dispersing soliton is formed, then the time-domain expansion should be simultaneously accompanied by spectral narrowing. This test has been used in [9] as a criterion for soliton formation in silicon photonic wires. Figure 6(b) shows the ratio of the output to the input spectral width. One can see that the spectral narrowing begins to occur at input powers well below the point where the amplitude-width products becomes quasiconstant, thereby showing the limitations of such an approach.

 

Fig. 6. (a) Output pulse length normalized to the input pulse length, plotted versus averaged input power. (b) The same as (a), but for normalized spectral width. (c) Pulse amplitude multiplied by the pulse length (FWHM) vs input power. The line colors and style are the same as in Figs. 4 and 5. The horizontal line in (c) shows the value for the soliton in the ideal NLS model.

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Numerically computed evolution of a single quasi-soliton pulse along the waveguide (using the model including 3PA) is shown in Fig. 7(a). For powers >3µW the pulse spreading becomes dominant and then, at 6µW and above, the pulse splitting begins to develop. The spectral splitting of the pulse corresponds to splitting in the time domain (see Fig. 7(b)) and can be characterized as fission of higher order quasi-solitons strongly perturbed by the linear and nonlinear absorption. Figure 8 shows the evolution of the FWHM times root-peak-power parameter along the waveguide for several input powers corresponding to the single peak pulses. One can see that in the 1 and 4µW cases the parameter changes sufficiently rapidly demonstrating no evidence for the adiabatic soliton evolution. At the same time, we have found that in the interval from 1.5 to 3µW (see 1.7 and 2µW curves), variations of the FWHM times root-peak-power parameter along the waveguide length can be considered insignificant and this product stays sufficiently close to the value found for the ideal NLS, thereby confirming adiabatic evolution of the quasi-solitons in the presence of strong linear and nonlinear absorption effects.

 

Fig. 7. Time domain evolution of the pulse amplitude along the waveguide. (a) corresponds to the 2µW input and (b) to 8µW.

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Fig. 8. Evolution of the pulse amplitude multiplied by the pulse length (FWHM) along the waveguide. The horizontal dashed line shows the value for the soliton in the ideal NLS model.

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5. Summary

We have carried out experimental and numerical studies of the propagation of 100fs pulses in 15mm long silicon-on-insulator photonic wire waveguides in the regime of strongly anomalous dispersion. The second- and third-order dispersion values used in our modeling have been reliably estimated from direct experimental measurements of the group index. Reasonable agreement between experiment and numerical simulation for both low and high powers has allowed us to conclude that there is a range of input powers for which quasisoliton pulses with peak powers below and around 1W are formed and propagate within the photonic wires. Our numerical analysis of the soliton propagation regimes and use of the long 15mm waveguides provide more evidence for the existence of quasi-solitons in the SOI photonic wires in the presence of strong linear and nonlinear absorption, than has been previously reported. In the case of the single-soliton femtosecond pump accounting for the two photon absorption and linear losses suffices for adequate description of the experimental observations. However, the best modelling to experiment fit for the high energy multi-soliton pump pulses is achieved, when the phenomenological three photon absorption term is added into the model, thus calling for further work on first principles calculations of the ultrafast nonlinear behavior of silicon.