## Abstract

We report time domain measurements of the group-velocity-dispersion-induced and nonlinearity-induced chirping of femtosecond pulses in subwavelength silicon-on-insulator waveguides. We observe that at a critical input power level, these two effects compensate each other leading to soliton formation. Formation of the fundamental optical soliton is observed at a peak power of a few Watts inside the waveguide. Interferometric cross-correlation traces reveal compression of the soliton pulses, while spectral measurements show pronounced dispersive (Cherenkov) waves emitted by solitons into the wavelength range of normal group velocity dispersion.

© 2010 Optical Society of America

Silicon photonics has attracted much attention recently due to its potential applications in photonic chips [1, 2, 3]. Compared with other possible materials for integrated photonics, an overwhelming advantage of silicon is its compatibility with silicon electronics. The large index contrast between silicon and other materials leads to a small footprint and to a straightforward opportunity to engineer dispersion properties using the waveguide geometry [3]. Silicon has strong third-order nonlinearities with the Kerr coefficient being more than 100 times larger than in silica glass [2]. All the above features can be used to realize a variety of on-chip nonlinear functions, including modulators, wavelength converters, light amplifiers and sources [1]. While spectral characterisation of nonlinear effects in subwavelength silicon-on-insulator SOI waveguides have been reported in numerous publications, see, e.g., [2, 3, 4], time domain or simultaneous spectral and time domain characterizations are required to reliably confirm such effects as soliton formation, pulse compression and more complex processes where short pulses mix with and generate dispersive waves [5].

Reference [6] and our previous work [7] have compared spectral measurements and numerical simulations to predict soliton generation in SOI waveguides. An important feature of these solitons is their record low peak powers of a few Watts for femtosecond pulse durations. In this paper, we investigate soliton formation in SOI waveguides by observing it in both the frequency and time domains. Applying a cross-correlation technique, we have been able to observe directly that the dispersion induced chirping of the low power femtosecond pulses is compensated by the opposing nonlinearity induced chirping at higher power levels, which provides a direct experimental verification of the main physical mechanism behind soliton formation.

In our experiments, we use 220nm thick, 380nm wide and 3.4 mm long waveguides fabricated using electron beam lithography (EBL) and inductively coupled plasma reactive ion etching [8]. The waveguide is surrounded by air on both sides and by a ≃ 100nm layer of resist on the top [8], with air above. The linear transmission loss is estimated to be 3.4dB/cm. In this study, we have used input pulses polarized parallel to the substrate. Using a free-space Mach-Zehnder interferometer, a supercontinuum source and a series of band-pass filters, we measure the group index at different wavelengths, as shown in Fig. 1(a). The time delay of the optical beam propagating along the waveguide, which is mounted in the signal arm of the interferometer, is measured by adjusting the optical length of the reference arm. Our group index measurements are so accurate that the error bars in Fig. 1(a) are smaller than the size of the symbols. Based on the measured group indices we use a 5th order polynomial to fit the experimental data. Introducing higher order fits is not necessary in this case, since the spectral range covered by the dispersion measurements and by the nonlinear spectral broadening is relatively narrow. Fig. 1b shows the group velocity dispersion across the wavelength region from 1.3 to 1.7*μ*m. The zero-dispersion wavelength is found to be around 1.6*μ*m. At the wavelength of 1.56*μ*m, which is the carrier wavelength of our pump pulses, the waveguide has anomalous group velocity dispersion (GVD), *β*_{2} = −3.01 ± 0.10ps^{2}/m, and 3rd order dispersion *β*_{3} = −0.070 ± 0.002ps^{3}/m.

Figure 2 shows the experimental setup used for nonlinear measurements. An Erbium-doped fiber ring laser produces ultrashort pulse trains at 1.56*μ*m with a repetition rate of 33MHz. This pulse train is passed through a band-pass filter to remove spectral sidebands. After filtering, the ultrashort pulses are sent into an Erbium-doped fiber amplifier (EDFA). In order to minimize the nonlinearity inside the fiber amplifier, a stretched-pulse scheme is adopted, so that, along the whole length of the EDFA, the pulse is first dispersed and then re-compressed [9]. Using an auto-correlator at the exit of the EDFA, see the insert in Fig. 2, we measure the full width of the pulse at half maximum of its intensity (FWHM) to be *T _{fwhm}* = 175fs. The chirp of this sech-like pulse is small, since its time-bandwidth product equals 0.39, and is not far from the transform-limited value of 0.3148. In our setup, two optical isolators are used to block back-reflection, several polarizer cubes are used to constrain the polarization to be parallel to the substrate (TE mode), and a 5% plate beam splitter is used to construct the reference arm of the cross-correlator. After this beam splitter, the pulse train is coupled into the SOI waveguide through an AR-coated singlet lens (numerical aperture, NA = 0.65) with the estimated coupling coefficient of 12dB. At the waveguide output the optical pulses are characterized using either an optical spectral analyzer or a cross-correlator.

To demonstrate the soliton formation process, we use a cross-correlation technique based on two photon absorption in a silicon detector induced by the sum of the signal field *E _{s}* =

*A*(

_{s}*t*)

*e*

^{−iωst}+

*c.c*. (pulse after the waveguide) and the reference field

*E*(

_{r}*t*) =

*A*(

_{r}*t*)

*e*

^{−iωrt}+

*c.c*. (input pulse). The corresponding cross-correlation integral is

*I*(

_{c}*τ*) =

*R*∫ (

*E*(

_{r}*t*)+

*E*(

_{s}*t*+

*τ*))

^{4}

*dt*, where

*τ*is the time delay between two pulses and

*R*is the repetition rate [10]. We measure weak signals by exploiting a strong reference pulse in the cross-correlator. We do not adopt a further amplification scheme, since it would unavoidably distort the signal. For the power levels and spectral broadening reported in this work it is safe to approximate ${I}_{c}=\mathit{const}+R\frac{3}{4}\left[{e}^{-i{\omega}_{s}\tau}\int |{A}_{r}{|}^{2}{A}_{r}^{*}{A}_{s}{e}^{i({\omega}_{r}-{\omega}_{s})t}dt+c.c.\right]+R\frac{3}{2}\int |{A}_{r}{|}^{2}|{A}_{s}|dt+\dots $. Using relatively high repetition rates helps to improve the ratio between the signal-carrying 2nd and 3rd terms with respect to the large ( $\sim {E}_{r}^{4}$) constant term in

*I*, which has motivated our choice of a fiber ring laser having

_{c}*R*= 33MHz. It should be noted that we use moderate values of the repetition rate to avoid the effects of inter-pulse free-carrier absorption in waveguides.

At low input powers GVD is expected to be the dominant factor determining the frequency chirp. The GVD length *L _{d}* of a sech input pulse,

*sech*(

*t*/

*T*

_{0}) is defined as ${L}_{d}={T}_{0}^{2}/|{\beta}_{2}|=3.3\text{mm}$, where ${T}_{\mathit{fwhm}}=2\text{ln}\left(1+\sqrt{2}\right){T}_{0}$. Thus

*L*of the input pulse is comparable to the waveguide length, 3.4mm and temporal broadening and GVD induced chirping of the transmitted pulses in the linear regime should be noticeable. Figure 3a shows measured interferometric cross-correlation traces with varying input power and Fig. 3b shows how the FWHM of

_{d}*I*varies with power. The fast oscillations of

_{c}*I*provide direct information about the distribution of the signal frequency across the pulse profile. We equally divide every trace into the leading, central, and trailing sections and determine the averaged carrier wavelength by measuring the distance corresponding to 16 continuous oscillations in every section (Fig. 3c). Thereby we are able to observe the pulse chirp directly. For small powers the leading part of the pulse is bluer than the trailing part (see Fig. 3c), which is consistent with the anomalous GVD. Comparing Figs. 3b and 3c one can see that the GVD induced chirping is directly associated with the pulse broadening in the time domain.

_{c}As the input power increases, nonlinearity induced chirp starts to balance the GVD induced chirping. The two chirping rates become equal at around 1mW of the averaged input power, which corresponds to the soliton formation threshold in our system. We relate the average *P _{av}* shown in Fig. 3 to the pulse peak power

*P*at the end of the waveguide as

*P*=

*P*/[

_{av}*T*]/

_{fwhm}R*q*, where

*q*accounts for the input-coupling loss

*q*= 12

_{r}*dB*and the linear and nonlinear power loss during propagation in the waveguide

*q*= 3

_{w}*dB: q*= 10

^{(qr +qw)/10}≃ 31. To estimate the nonlinear part of

*q*we have assumed that the 2 photon absorption rate is 0.2

_{w}*γ*, where

*γ*is the nonlinear waveguide parameter [11]. Calculating the later according to Ref. [2], we find

*γ*= 240[Wm]

^{−1}. Thus we can estimate the peak power of the pulses with the balanced linear and nonlinear chirping rates to be

*P*≃ 5W. Equating the dispersion

*L*and nonlinear

_{d}*L*= 1/[

_{nl}*γP*] lengths, we find the output pulse duration to be

*T*

_{0}= 50fs, which agrees well with predictions of the numerical modeling of the generalised nonlinear Schrödinger equation previously used by us in Ref. [7]. Increasing the input power further, so that the output power goes above the soliton threshold, we observe that the chirping of the pulse is determined by the Kerr nonlinearity, i.e., the leading edge of the pulse is red shifted with respect to the trailing one, see Fig. 3c. For these power levels the output pulses remain compressed below their input duration, see Fig. 3b.

Figure 4 shows the output spectrum as a function of the input power. When the power is above the soliton threshold the spectrum develops pronounced asymmetry due to the peak growing at around 1670nm in the range of normal GVD. We attribute this peak to the Cherenkov radiation emitted by the solitons [5, 12, 13, 14]. This effect has been well studied in photonic crystal fibers and is known to be closely associated with supercontinuum generation [5]. Matching the propagation constants of the linear dispersive wave to the propagation constant of a soliton gives the wavelength of the Cherenkov resonance [5]. It is known in the context of fiber optics that the complex interaction between the radiation and solitons leads to the spectral drift (recoil) of the latter. Note, that Fig. 3c indicates that the average pulse wavelength drifts towards shorter values. Assuming the soliton carrier wavelength to be in the range from 1.5 to 1.56*μ*m, we find that the Cherenkov radiation should appear between 1.625 and 1.690*μ*m (see Fig. 1b), which agrees well with our measurements.

Figure 5 provides a detailed view of the cross-correlation traces measured for three different power levels and the associated spectra. At the soliton threshold and below the spectrum is symmetric. At 4 times above threshold, the spectrum develops two strong peaks offset roughly by *δλ* ≃ 34nm and a much weaker Cherenkov peak at 1670nm. Both of the stronger peaks are located in the anomalous GVD range. The cross-correlation trace is also split into two features, one is well localised and the other is broad. One can loosely estimate the temporal separation between the two to be *δτ* ≃ 0.23ps. The product of the frequency interval *cδλ*/*λ*^{2} ≃ 4.25THz, associated with *δλ*, and *δτ* is close to one, and is in agreement with the uncertainty principle relating the frequency and time domain signals. Thus our data indicate that in the multisoliton regime (4 times above the single soliton threshold), only the fittest soliton survives in our system. Cherenkov radiation is not strong enough to be seen within *I _{c}*(

*τ*).

The consistency of the spectral and temporal descriptions is further supported by plotting the input-output transmission curves, see Fig. 6. Integration of the output spectra across the acquisition wavelength range gives the transmission curve with a saturation occurring at the input powers > 2mW, Figure 6a. Two-photon absorption, multi-photon absorption, and free-carrier absorption are factors which contribute to this nonlinear damping [7]. The averaged output power indicated includes significant out-coupling losses. Fig. 6b shows ∫*I _{c}*(

*τ*)

*dτ*, where the onset of the saturation is derived from the time domain measurements and agrees well with Fig. 6a. Note, that the soliton appears at 1mW input, i.e. before the onset of strong saturation.

## Acknowledgments

This work is supported by UK EPSRC project EP/G044163. We acknowledge the support of the James Watt Nanofabrication Centre at Glasgow University.

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