We measure the transmission of ps-pulses through silicon-on-insulator submicron waveguides for excitation wavelengths between 1400 and 1650 nm and peak powers covering four orders of magnitude. Self-phase- modulation induced spectral broadening is found to be significant at coupled peak powers of even a few tens of mW. The nonlinear-index coefficient, extracted from the experimental data, is estimated as n2 ~5·10-18 m2/W at 1500 nm. The experimental results show good agreement with model calculations that take into account nonlinear phase shift, first- and second order dispersion, mode confinement, frequency dispersion of n2 , and dynamics of two-photon-absorption-generated free carriers. Comparison with theory indicates that an observed twofold increase of spectral broadening between 1400 and 1650 nm can be assigned to the dispersion of n2 as well as first order- rather than second-order dispersion effects. The analysis of pulse broadening, spectral shift and transmission saturation allows estimating a power threshold for nonlinearity-induced signal impairment in nanophotonic devices.
©2006 Optical Society of America
Silicon-on-insulator (SOI) is regarded as a potential material platform for ultra-dense on-chip integration of photonic and electronic circuitry due to its compatibility to CMOS fabrication and high refractive index contrast which results in a tight light confinement allowing efficient scaling of photonic components. In the last years, various photonic devices such as splitters, modulators and arrayed waveguide gratings have already been reported [1–7]. Moreover, the efficient light confinement entails high optical intensities in the waveguide and allows exploiting optical nonlinearities such as Raman amplification, Four-Wave mixing, continuum generation or 2R-data regeneration [8–17]. However, the efficient light confinement imposes severe limitations for devices operating in the linear regime. For instance, the spectral separation of channels in wavelength division multiplexing (WDM) is limited by self-phase modulation (SPM) and cross-phase modulation (XPM) induced crosstalk. Consequently, accurate knowledge of the waveguides properties such as nonlinear refractive index, free carrier density dependent losses or the onset of perceivable nonlinearity-induced spectral distortions is necessary in order to determine the power regime of impairment-free device operation. The efficiency of SPM not only depends on the optical intensities but also on the presence of free carriers generated by two-photon absorption (TPA) introducing free-carrier-absorption (FCA). TPA and FCA cause optical losses, which in turn lower the peak power inside the waveguide and therefore reduce the SPM-induced spectral broadening. In particular, the lifetime of generated carriers is a critical parameter. For instance, the switching speed of optical modulators is limited by the presence of absorbing carriers . Moreover, potential carrier accumulation increases the free carrier density even further. Accumulation is present, if e.g. the recombination time of carriers is longer or comparable with the repetition rate of the optical pulses.
In this paper, we report on experimental studies of SPM-induced spectral broadening of ps-pulses in sub-micron SOI photonic wires for pulse peak powers covering four orders of magnitude. We use a laser system with a repetition rate of only 1 kHz. To the best of our knowledge this is the first experimental study under near-single pulse condition, certainly excluding any potential interpulse carrier accumulation. We find noticeable SPM-induced spectral broadening even for peak powers as low as ~30 mW, and followed by saturation in transmission for higher powers levels. At 1500 nm, the nonlinear-index coefficient is measured as n2 =5±4·10-18 m2/W. In addition, a strong wavelength dependence of the SPM-induced spectral broadening is found. For the same peak power, the spectral broadening increases twofold from 1400 to 1650 nm. Our model calculations, taking into account propagation losses, phase shift and losses due to two photon absorption generated free carriers, as well as group velocity dispersion (GVD) and self-phase-modulation, are in good agreement with the experimental results.
2. Sample and experimental setup
The strip waveguides are processed on a standard CMOS line at the IBM Watson Research Center. The length of the uncladded (no oxide layer on top) SOI waveguides is 4 mm, with a width of w=470 nm and height h=226 nm, as determined from SEM images. At both ends of the waveguide, polymer-based spot-size converters with inverted taper geometry act as in-and output ports and insure efficient light coupling. Propagation losses at 1500 nm are measured to be 3.6 dB/cm. Further details on fabrication and characterization can be found elsewhere .
The employed light source is an optical parametric amplifier (OPA) generating 1.8 ps pulses at a repetition rate of 1 kHz with a tunable wavelength range from 1200 to 1700 nm. Its output power and polarization is controlled by reflective, metallic attenuators followed by a polarizer and λ/2-plates. Transmission spectra are all measured in TE polarization (in-plane). In order to eliminate nonlinear contributions in the transmission spectra, stemming from the experimental setup, no fiber-optics are used between light source and waveguide. A microscope objective (IR, 20x, NA=0.35) is used to focus the pulses on the input port of the waveguide. At the output port a micro-lensed tapered fiber collects and routes the transmitted signal to an optical spectrum analyzer (resolution 0.5 nm). We can exclude significant SPM contributions from the fiber, since no changes are observed in the transmission spectra for different lengths of the fiber (2 to 13.5 m). Frequency-Resolved-Optical-Gating technique (FROG) is used to characterize the spectral and temporal profile of the injected pulses .
The total signal losses in the photonic circuit are -27 dB, determined by measuring the transmission for very low peak powers (TPA, FCA and SPM are negligible). Taking into account intrinsic propagation losses of the 4 mm waveguides as well as coupling losses to the tapered output fiber , we determine the input coupling losses through the microscope objective as -24 dB. In the following, all given peak powers are normalized by -24 dB losses and correspond to the peak power inside the waveguide (coupled power).
3. Experimental results and discussion
3.1 Nonlinear-index coefficient n2
Figure 1 shows a series of TE-transmission spectra of the SOI waveguide for different coupled peak powers P at 1500 nm. The two lowest power spectra with P=1.8 and 8 mW, (corresponding to peak intensities of I ~0.003 and 0.014 GW/cm2, respectively) only differ in the transmitted intensity but not in spectral shape and position. The spectral full width at half maximum (FWHM) is identical to pulses measured without sample. Hence, the applied powers of the two low-power spectra belong to the regime where the waveguide is responding optically linear. However, by increasing the peak power to P=1.55 W (I≈2.6 GW/cm2), the spectral shape of the pulses broadens dramatically; now exhibiting several spectral side wings. The broadening continues for higher powers which can be seen in the spectrum with P=6.85 W (I≈11.6 GW/cm2).
The measured spectral pulse distortions can be explained by self-phase-modulation, which is known to cause an intensity dependent phase shift of the pulse carrier frequency. Within its own duration, the pulse experiences an intensity- and thus time-dependent refractive index. New frequency components are generated and the initial pulse spectrum broadens in an oscillatory manner while the temporal pulse shape remains unaffected. The degree of SPM-induced spectral broadening depends on the nonlinear refractive index n2 and the waveguide length L. Since the number of spectral oscillations in the transmission spectra is directly correlated with the nonlinear phase shift Φ, we can estimate n2 from the experiment by applying the formula n2 =(Φ·c·Aeff )/(P·Leff·ω) . Leff corresponds to an effective length of the waveguide that is smaller than L because of intrinsic propagation losses.
Assuming a homogeneous and isotropic medium with cubic optical nonlinearity the relation between nonlinear phase shift Φ and number of peaks N in the SPM-broadened spectra is given by Φ≈(N - 0.5)·π . However, in the spectra presented in Fig. 1, the peaks are not very pronounced indicating that the applied coupled powers do not cause odd multiples of 0.5 π phase shifts but rather intermediate values. Hence, the visibility of the SPM-induced fringes (self-interference) is reduced making the determination of N difficult. We have therefore applied large error bars allowing the interpretation of either one large or four small peaks (see green arrows in Fig. 1 for 1.55 W). The nonlinear phase shift for the two highest coupled powers then yields Φ=2±1.5 π and Φ=3±1.5 π (for 1.55 W and 6.85 W, respectively) resulting in an average nonlinear-index coefficient of about n2=(5±4)·10-18 m2/W. This estimate is in agreement with values previously reported from other groups, ranging from 4.5 to 14.5·10-18 m2/W [14, 16, 23–25]. From n2 we can derive the nonlinear length LNL which corresponds to an effective propagation distance at which a maximum phase shift of Φ=1 occurs. LNL is calculated by LNL =1/(P·γ) where γ is the nonlinear strength parameter defined as γ=(n2·ω)/(c·Aeff ). For the transmission spectra with the two highest peak powers (Fig. 1), the nonlinear length of the waveguide yields LNL =1.6±0.8 mm and 4.2±3 mm for 6.85 and 1.55 W, respectively.
3.2 Asymmetric spectral broadening
As can be seen from Fig. 1, the spectral broadening of the transmission spectra is strongly asymmetric. Since SPM alone is known to yield a symmetric spectral distribution around the injected laser frequency, the asymmetry must stem from other factors such as chirped injected laser pulses, self steepening, GVD or changes in the free carrier density by TPA.
Recently, Modotto et al. reported asymmetric SPM broadening in AlGaAs waveguides . It was argued and confirmed by calculations that the asymmetry is caused by a chirp imposed on the injected pulses. This phenomenon is well known for optical fibers  but does not hold for our experimental conditions. The inset in Fig. 1 shows a spectrally- and time-resolved trace of the injected pulses, measured with FROG. The analysis does not reveal any significant distortions (temporal phase deviation is below 0.1 radians), meaning an almost chirp-free pulse is launched into the waveguide.
Self-steepening can also be ruled out. First, within the narrow spectral bandwidth of the ps-pulses a wavelength dependence of n2 is negligible. Second, self-steepening induces a spectral red-shift [22, 26] which is opposite to the experimental findings in Fig. 1.
Group velocity dispersion should be of minor influence as well. We have recently demonstrated that, although the GVD in such photonic wires is very high, yielding 4400 ps/nm·km, the dispersion length LD is ~1 meter for ~2 ps pulses . Compared to this and as determined above, the nonlinear length is in the range of a few millimeters, nearly three orders of magnitude shorter.
Similar asymmetry of SPM-induced spectral broadening was reported and explained by two-photon absorption generated free carriers (FC) [9, 24]. Since the FC-density follows the integral of the temporal pulse shape, the pulse experiences a rapidly changing FC-density. Moreover, since the FC-induced phase shift is proportional to the FC-density, a more efficient phase shift occurs in the trailing edge than in the leading edge of the pulse, causing an asymmetric broadening of the spectra. This description agrees well with the observed experimental spectra shown in Fig. 1. We therefore conclude that carrier-induced changes of the refractive index are the most likely source for the asymmetric spectrum.
3.3 FCA induced saturation in transmission
To investigate the impact of optically generated free-carriers in more detail, we measure the transmission through the photonic wire as a function of very low peak powers (0.8–400 mW). Figure 2 plots coupled peak power against output peak power. The output peak power is obtained by measuring the spectrally integrated average output power and normalizing it to the pulse duration and repetition rate. Since the impact of GVD can be neglected in these waveguides , SPM will not affect the temporal shape of the pulses . Therefore the method of output power normalization we used in this experiment is acceptable. Within this series, a clear transition between linear and nonlinear optical response of the waveguide is observed. For low input powers the output signal shows a fast increase. The double-logarithmic scale, shown in the inset of Fig. 2, demonstrates the strictly linear response for peak powers up to ~30 mW, with a slope of m ~1.05±0.15. With increasing coupled power the transmitted output power reveals a self-limiting behavior and remains nearly constant for peak powers larger than 100 mW. Given that SPM is an energy-conserving process it cannot explain the saturation in transmission. The behavior is rather due to optical losses caused by intrapulse TPA and FCA. We want to stress that interpulse carrier accumulation is impossible with the used laser system since the pulse distance is nearly four orders of magnitude larger than reported carrier lifetimes.
From the experimental saturation curve we can conclude that for peak powers above a threshold of 30 mW (I~50 MW/cm2) the absorption due to generated FCs is not negligible anymore. This threshold corresponds to a FC-density of N=5.3·1016 cm-3 exceeding the initial carrier density in the unexcited waveguide more than tenfold (p-doped, N~1·1015 cm-3).
3.4 Power threshold for signal impairment
In Subsection 3.3 we have seen that for higher peak powers TPA-induced FCA leads to saturation in transmission. But simultaneously, the optical pulses also undergo a spectral broadening due to SPM. This imposes a power limitation for SOI waveguides if incorporated into photonic devices. For instance, in dense wavelength division multiplexing (DWDM) a crosstalk must be suppressed to at least below 10–15 dB between adjacent channels separated by Δλ=0.8 nm in order to maintain data integrity.
Figure 3(a) shows a series of transmission spectra of optical pulses, centered at 1500 nm, as a function of coupled peak power. For spectra with lowest coupled power (8 and 16 mW), the pulses show only negligible spectral distortions and demonstrate again the primarily linear optical response of the photonic wire, as was shown in the inset of Fig. 2. With increasing coupled powers (62 to 378 mW), the spectral width increases significantly. Moreover, at the same time the pulses undergo a blueshift but with nearly unchanged peak intensity (e.g. for 170 and 378 mW). For P=378 mW (I~0.65 GW/cm2) the pulse has already shifted by more than 1 nm. The power threshold from whereon a peak shift is detectable (~100 mW, see inset of Fig. 3a) agrees well with the onset of saturation of the integrated output power (Fig. 2). While the pulse broadening can be assigned to SPM (note the rise of SPM-typical spectral side wings in the spectra), the shift of the peak position is caused by generated FC known to lower the refractive index due to the plasma effect .
Significant crosstalk penalty in DWDM eventually occurs when SPM-induced spectral broadening of the pulses exceeds channel separation. It can be estimated by measuring the spectral broadening of the pulses at FWHM (-3 dB, blue triangles in Fig. 3(b)). However, this can only give an upper estimate of the power threshold since it is known that SPM commences in the spectral wings of a pulse. Figure 3(b) tracks the spectral width of the pulses at 4 different intensity levels: -3, -5, -10, and -15 dB. As a reference width the spectrum with P=8 mW is taken. As one can see, at a -3 dB level the coupled power needs to be higher than 200 mW to keep the spectral broadening below 0.8 nm (black dotted line). However, at a -15 dB level, the power threshold is already lowered by one order of magnitude to only 20 mW (black triangles). Thus severe power limitations for DWDM components on a photonic wire platform can be envisioned.
The method we used is oversimplified and can give only an order of magnitude estimate of crosstalk penalties in real communications systems as XPM , four-wave mixing , stimulated Raman  and waveguide dispersion  are not considered. Moreover, for data rates higher than 1 Gb/s, interpulse as opposed to intrapulse carrier accumulation should be taken into account. In principle, this might decrease SPM-induced broadening thus increasing the power threshold. In addition, the strong dependence of SPM on the spectral shape of the optical pulses might also alter the power threshold. However, considering the very low crosstalk levels required for real communication channels, e.g. far below -20 dB, the expected power threshold might become even lower than estimated above.
3.5 Wavelength dependent SPM
The results in Figs. 1–3 demonstrate that nonlinearities such as SPM and TPA-induced FCA strongly perturb the spectral shape of optical pulses at 1500 nm. However, telecommunication bands cover spectral ranges between 1490 nm and 1612 nm (S, C, and L-band). It is therefore of importance to have knowledge about the SPM wavelength dependence.
In order to study this in detail we have measured the spectral broadening as a function of wavelength (1400–1650 nm) while keeping coupled power and pulse width constant. Figure 4 compares transmission spectra at 1400 nm and 1650 nm with the spectrum at 1500 nm (same as in Fig. 1). The peak power is set to 6.85 W. At all excitation wavelengths the output spectra are broadened and asymmetric. The significant difference is that SPM-induced broadening becomes more efficient for longer wavelengths. The spectral distance between the short-wavelength SPM oscillation and the originally injected laser wavelength increases from 8 to 16 nm when the OPA is tuned from 1400 to 1650 nm. As will be discussed in the next section, this enhancement for longer wavelengths is probably due to an increase of group index ng and nonlinear refractive index n2 .
4. Comparison with theory and discussion
4.1 Model description
To analyze the experimental data we have used a recently developed theoretical model to describe the pulse dynamics in Si photonic wires. It accounts for GVD, parametric and nonparametric nonlinear effects such as SPM, XPM, TPA, or Raman interaction; free carrier-induced effects, such as FCA and free carriers dynamics . In the case studied here, of single-frequency pulse propagation, a simplified version of the model, excluding XPM and Raman terms, is used. The dynamics of pulse propagation in silicon photonic wires is governed by the following system of coupled nonlinear differential equations
where ψ(z,t) is the slowly varying envelope of the pulse, measured in W1/2, βi (i=1, 2) are the dispersion coefficients of the i th order, c the speed of light, n the refractive index of silicon, ω the carrier frequency, ε0 the vacuum permittivity, A0 the waveguide transverse area, P the peak power of the pulse, κa correction factor that takes into account the fraction of the optical mode confined within the waveguide and hence generating FC, αin and αFC the intrinsic and FC-induced losses, respectively, δnFC the FC-induced change of the refractive index, ΔN the FC density, tc the carrier lifetime and ħ the Plank’s constant. Γ is the complex effective third-order nonlinear coefficient of the silicon photonic wire describing the contribution of nonlinear third order susceptibility χ (3). Details of the derivation of Eqs. (1)–(2) can be found in Ref. . The effective third-order nonlinear coefficient Γ=Γ′+iΓ″ determines both the SPM and the TPA coefficients in the photonic wire. The real part describes changes in the refractive index whereas the imaginary part corresponds to the TPA coefficient α2 . Assuming a homogeneous and isotropic medium with cubic optical nonlinearity one can define an effective nonlinear-index coefficient n2 and an effective TPA coefficient α2 .
However, the waveguide geometry plays a crucial role; one of the consequences of this fact is that the nonlinear strength parameter γ commonly used for optical fibers, γ=n2ω/cA0  is replaced by γ̄=3Γ′/ε0A0  in the model description of the photonic wire and accounts for the wavelength dependence of the group index ng=β1·c .
4.2 Input parameters
The intrinsic material dispersion of Si is calculated by using the Sellmeier equation  and is included in the calculations of the mode propagation constant βI , which is obtained by using the Eigen-Mode-Expansion (EME) method . The modal dispersion therefore contains both material and waveguide contributions. The EME method is chosen due to its good agreement with experimental measurements of the group index ng =β1·c for such photonic wires.
The complex effective third-order nonlinear coefficient Γ=Γ′+iΓ″ of the photonic wire is given by an overlap integral of the χ (3) tensor of bulk silicon and the calculated waveguide modes inside the wire . To take into account the frequency dispersion of the nonlinear susceptibility χ (3) of the bulk silicon we have used experimental values of the n2 coefficient at λ=1.54 µm for the  and  directions, measured by Dinu et al. , and determined the ratio between the only independent components of the χ(3) tensor as χ1122/χ1111=0.25. Since n2 was measured for only one wavelength for two different crystal directions, we used the wavelength dependence of the cubic susceptibility χ (3)~λ6  and determined the nonlinear susceptibility χ (3) that leads to the best fit of the experimental data. The obtained values at 1550 nm are =6.95·10-19 m2/V2 and =1.77·10-19 m2/V2.
The free carrier absorption αFC and the FC-induced change of the refractive index δnFC are calculated by applying the Drude model for free carriers . Finally, κ is calculated as the fraction of the mode contained within the waveguide . For the carrier lifetime we use tc =0.5 ns, as previously measured by optical pump-probe experiments . Furthermore, the following experimental input parameters are used for the calculation: intrinsic propagation losses with αin =3.6 dB/cm  and hyperbolic secant shaped input pulses with a FWHM of 1.8 ps. The waveguide transverse area A0 is calculated from the waveguide width and height, as determined from SEM images.
4.3. Saturation in transmission
In Fig. 2 we compare the measured saturation in transmission (squares) with model calculation (blue line), as described in Section 4.1. The output pulse peak power is shown as a function of the coupled pulse peak power. As can be seen from the inset in Fig. 2, the low-power regime where the waveguide responds optically linear is well reproduced by the model (blue line, the dotted line is a guide to the eye). As already described in Section 2, input coupling losses are taken into account for the experimental data. Hence, the difference between input and output power is therefore solely due to intrinsic propagation losses and coupling losses to the output fiber, leading to a 3 dB attenuation of the signal intensity. With increasing peak power, TPA induced FCA starts to contribute and eventually dominates the overall transmission intensity. Also for this nonlinear high-power regime, the calculated saturation in transmission is in good agreement with the experiment. We want to stress that the free carrier recombination time of tc =0.5 ns used in the calculations is a non-critical parameter. The optical pulse length is 1.8 ps, hence orders of magnitude shorter, while the pulse repetition rate is 1 ms, orders of magnitude larger than tc .
4.4 Wavelength dependent SPM
Figure 5 summarizes the spectral broadening, shown in frequency units, as a function of the center wavelength of the input pulses. The experimental results are deduced from transmission spectra (e.g. Fig. 4) and correspond to the spectral position of the most intense, shorter wavelength SPM oscillation. The experiment was repeated several times for different waveguides owning the same structural parameters and coupled peak power of 6.85 W (asterisks). Within the investigated wavelength regime of 250 nm, the experimentally found SPM-induced spectral broadening increases nearly twofold (~1 THz) with the wavelength.
The calculated wavelength dependence of the SPM-induced spectral broadening shows reasonable agreement with the experiment as it also predicts an increase with longer wavelengths (solid lines). This dependence might be caused by several factors. First, the group index ng =β1·c increases with the wavelength . Although this change is in the order of only 10 %, the quadratic dependence of the SPM coefficient on β1 enhances the SPM effect for longer wavelengths (as described in Sec. 4.1). Second and more important, the frequency dispersion of the Kerr nonlinearity, namely the increase of the cubic susceptibility χ (3) for longer wavelengths also contributes to an increased SPM-induced broadening . This is in agreement with the experimental results from Dinu et al , reporting a nearly twofold increase of n2 for wavelengths from 1270–1550 nm.
The comparison of experiment and calculation (Fig. 5) also indicates that GVD just barely influences SPM-induced spectral broadening. The calculated curves, without GVD (red line) and including GVD (black line) are almost identical. This is in very good agreement with recent measurements, where we have shown that the dispersion length LD of such photonic wires for ps pulses is ~1 m , orders of magnitude larger than the waveguides’ nonlinear length of some millimeters, as determined in Section 3.1. Consequently, for the waveguides and for wavelengths sufficiently separated from the zero GVD point (where third order dispersion becomes important), GVD should not have a significant influence on SPM.
The inset of Fig. 5 compares the spectral broadening of optical pulses at 1500 nm with model calculations, now as a function of the input peak power. Note that the applied powers are very high (up to 6.85 W). Hence, this plot can be seen as the high power continuation of the low power series of Fig. 3 (up to 0.38 W). The experimentally measured SPM peak positions (blue and red dots) show the same power dependence as the model calculations (solid lines). The observed saturation, starting already at low peak powers, is in good agreement with the saturation in the transmitted output power, as shown in Fig. 2 and again is caused by optical losses, originating from TPA induced FCA. We can conclude that FCA is the main limitation with regard to nonlinear spectral broadening of optical pulses in the photonic wires.
In summary, we have measured SPM-induced spectral broadening of 2-ps pulses in SOI photonic wires as a function of excitation wavelength and coupled power. The nonlinear response of the 4 mm long and 226×470nm waveguides is extremely efficient and the spectral shape of the injected pulses is significantly distorted already for very low input peak powers of some tens of mW. The nonlinear-index coefficient is estimated as n2≈5·10-18 m2/W at 1500 nm. For coupled peak powers of a few W, the nonlinear length of the waveguide is reduced to some millimeters, nearly three orders of magnitudes shorter than the dispersion length. The results demonstrate that even extremely high GVD, characteristic for such photonic wires, has only marginal influence on the SPM-broadened ps pulses. The onset of nonlinearities is found to increase with the wavelength and the SPM-induced spectral broadening doubles between 1400 and 1650 nm caused by an increase of ng and n2 . For high coupled peak powers, the spectra saturate in transmission and show strong spectral asymmetry, both caused by intrapulse free carrier generation. The FC induced optical losses also lead to saturation in SPM-induced spectral broadening. The experimental findings demonstrate that already at very low coupled peak powers of a few tens of mW, the SPM-induced spectral broadening, measured at -15 dB intensity level, is approaching 1 nm. This might result in severe crosstalk penalties for designing DWDM components based on a SOI photonic wire platform.
The authors gratefully acknowledge the contributions of Dr. Sharee McNab and Dr. Fengnian Xia (IBM T. J. Watson Research Center). This work was supported in part by the DARPA Slow Light project (J. Lowell, DSO), under grant N00014-04-C-0455. The Columbia part of this work has been financially supported by Air Force Office of Scientific Research through contracts FA9550-05-1-0428 and FA9550-04-C-0022. We are grateful to Dr. Jerry Dadap and Andy Hsieh for many useful discussions.
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