We report an experimental study of picosecond pulse propagation through a 4-mm-long Si nanophotonic wire with normal dispersion, at excitation wavelengths from 1775 to 2250 nm. This wavelength range crosses the mid-infrared two-photon absorption edge of Si at ~2200 nm. Significant reduction in nonlinear loss due to two-photon absorption is measured as excitation wavelengths approach 2200 nm. At high input power, self-phase modulation is clearly demonstrated by the development of power-dependant spectral fringes. Asymmetry and blue-shift in the appearance of the spectral fringes at 1775 nm versus 2200 nm is further shown to originate from a strong reduction in the intra-pulse density of two-photon absorption-generated free carriers and the associated free-carrier dispersion. Analysis of experimental data and comparison with numerical simulations illustrates that the two-photon absorption coefficient β TPA obtained here from nanophotonic wire measurements is in reasonable agreement with prior measurements of bulk silicon crystals, and that bulk Si values of the nonlinear refractive index n 2 can be confidently incorporated in the modeling of pulse propagation in deeply-scaled waveguide structures.
© 2011 OSA
Research into nonlinear optical propagation within Si nanophotonic wires has recently attracted much interest due to their strong optical confinement and large third-order nonlinear susceptibility χ(3) [1–11]. Particularly striking is the size of the effective nonlinearity parameter γ, as it can be up to five orders of magnitude larger than that of conventional silica glass fibers [12, 13]. This large nonlinearity facilitates the observation of many nonlinear processes and enables the realization of a wide variety of advanced chip-scale components for performing ultra-fast all-optical signal processing functions, e.g. wavelength conversion [4, 5, 14, 15], signal regeneration [16, 17], switching [18–20], and format conversion [21, 22].
Research efforts have recently considered nonlinear optical propagation in wavelength regions outside of the commercial telecommunication bands in order to mitigate the parasitic effects of nonlinear absorption. Within the telecommunication bands, the large χ(3) of silicon is accompanied by large nonlinear loss, via two-photon absorption (TPA) and the related effects of TPA-induced free-carrier absorption (TPA-FCA). This loss can limit certain all-optical applications requiring a strong pump beam, such as parametric amplification through four-wave mixing (FWM) . However, recent studies have shown that large nonlinear optical gain  and broadband wavelength conversion [24, 25] can be obtained with FWM by operating in the mid-infrared (mid-IR) near 2200 nm, corresponding to the wavelength spectrum beyond the onset of TPA.
These recent mid-IR demonstrations make it clear that an important direction for further research is to develop a full understanding of the behavior of the nonlinear refractive index n 2 and the two-photon absorption coefficient β TPA, specifically within the spectral region near and across the TPA edge. Previously, measurements of these quantities have been reported on bulk silicon crystals [26–28] (telecom-band and mid-IR), and Si nanophotonic wires [1, 8, 11] (telecom-band). However, only limited studies have been made of mid-IR pulse propagation in SOI waveguides [23, 24, 29]. In this paper we study the characteristics of mid-IR picosecond pulse propagation through Si nanophotonic wires, and specifically analyze the observed self-phase modulation (SPM) and nonlinear transmission. In so doing, we obtain valuable insight into the variation of the nonlinear coefficients n 2 and β TPA across the mid-IR TPA threshold near 2200 nm. In addition and in contrast to prior studies of mid-IR four-wave mixing effects in Si nanophotonic wires [23–25], we seek to avoid additional contributions from nonlinear Kerr-effect phenomena such as soliton generation [30, 31], self-steepening , and modulation instability , and thus use Si wires designed to have normal dispersion over the spectrum of interest.
2. Waveguide dispersion and effective nonlinearity
Our experiment employs a strip Si nanophotonic wire (Fig. 1(c) inset) with height h = 220 nm, width w = 600 nm and length L = 4 mm, patterned on a 2-μm-thick buried-oxide (BOX) layer. The waveguide is planarized with SiO2, and then clad with a 60 nm-thick silicon nitride (Si3N4) film and a thick SiO2 top layer. Optical coupling into and out-of the Si nanophotonic wire is via a SiOxNy mode-converter [34, 35]. All optical structures are fabricated on a standard CMOS line at the IBM T. J. Watson Research Center. Si nanophotonic wires with similar dimensions have been used extensively for nonlinear optical-signal-processing applications in the telecommunication band due to their large effective nonlinearity and strong modal confinement [4, 5, 9, 10, 14, 16–19, 21, 22].
Figures 1(a) and 1(b) show the calculated electric field E x profiles of the quasi-TE00 mode using a finite-element (FEM) solver at wavelengths of 1550 nm and 2200 nm, respectively. This Si nanophotonic wire has a power confinement κ of 92% at 1550 nm, and, since the mode expands at longer wavelengths, the power confinement drops to 75% at 2200 nm. The wavelength-dependent second-order dispersion β 2 and effective nonlinearity parameter γ for the quasi-TE00 mode are shown in Fig. 1(c). The effective nonlinearity is derived using the expression γ = 3ωRe(Γ)/(4ε 0 A 0 v g 2), where ω is the angular frequency, ε 0 is the vacuum permittivity, A 0 is the area of the Si core, v g is the group velocity of the quasi-TE00 mode, and the quantity Γ is the complex effective waveguide susceptibility determined by the weighted integral of the third-order susceptibility χ(3) of bulk silicon over the waveguide mode [36, 37]. The effective nonlinearity of the Si nanophotonic wire (blue curve) decreases substantially as the wavelength increases, i.e. from ~300 W−1m−1 at 1750 nm to ~75 W−1m−1 at 2450 nm. This is largely due to decreasing mode confinement within the Si wire core, as seen in Figs. 1(a) and 1(b). The reduced effective nonlinearity could be mitigated by choosing slightly larger Si core dimensions for improved optical confinement, when optimizing Si nanophotonic wires for operation in the mid-IR near ~2200 nm. The waveguide dispersion information, calculated using a FEM mode solver with the same method as described in , is shown in the black curve of Fig, 1(c). Normal dispersion prevails over the wavelength range from 1775 to 2450 nm due to the reduced contribution of waveguide dispersion, again due to weaker optical confinement within the Si core. The worst-case dispersion (|β 2| < 20 ps2/m) yields a dispersion length of L D = T 0 2/β 2 > 200 mm, where T 0 (~2 ps) is the FWHM of the picosecond pulses used in this work. This dispersion length is much larger than the length of the Si nanophotonic wire (~4 mm), and therefore dispersion is expected to have minimal impact upon pulse propagation.
3.1 Measurement configuration
The experimental setup for the measurement of both the linear and nonlinear mid-IR propagation characteristics of the nanophotonic wires is shown in Fig. 2(a) . A pulse train of ~2 ps FWHM pulses is generated by a Ti:sapphire-pumped tunable optical parametric oscillator (OPO) system (Coherent Mira-OPO), and then fiber coupled using an objective lens. The pulse repetition rate is 76 MHz. The pump power coupled into the fiber is controlled using a neutral density (ND) filter. The center wavelength of the pump is tuned over the spectral range 1700-2300 nm by changing the cavity length of the OPO system, and is monitored along with its time-average power by a mid-IR optical spectrum analyzer (OSA; Yokogawa AQ6375). The pulse train is coupled into and out-of the Si nanophotonic wire chip via tapered lensed fibers, while the polarization of the pulses is aligned to excite the quasi-TE00 mode using an in-line fiber polarization controller. The transmitted pulse at the nanophotonic wire output is then directed to the OSA for analysis.
The peak pump power P P at the nanophotonic wire input is calculated using the expression P P = P avg/(Fτ). Here, P avg is the time-average input power integrated over the input pump spectrum, scaled by the wavelength-dependent fiber coupling loss (Fig. 2(b), explained further in section 3.2), F is the repetition rate of the OPO, and τ is the pulse duration.
3.2 Linear propagation loss
The propagation loss of the quasi-TE00 mode for the Si nanophotonic wire is measured using the cut-back method (comparing relative transmission through 4 mm-long and 20 mm-long wires) at several discrete input wavelengths across the TPA edge of Si. To avoid nonlinear loss, measurements are performed with the input peak power attenuated to less than 0.25 mW. The measured loss is shown as a black dashed line through square black data points in Fig. 2(b), and ranges from 4.5 dB/cm to 7 dB/cm. For comparison, the propagation loss across the telecommunication bands is also shown in Fig. 2(b) (solid black curve, measured with 1500-1700 nm broadband LED source). This data shows that within the mid-IR wavelength region < 2100 nm the linear propagation loss of this wire is significantly less than that in the telecommunication bands. The propagation loss decreases with increasing wavelength, approximately following the expected behavior for Rayleigh scattering in such a waveguide . However, this loss also shows a perceptible increase when the wavelength approaches 2200 nm, most likely due to a) absorption loss from the outer cladding of the nanophotonic wire, i.e., SiO2 , or b) mode leakage into the Si substrate. Such losses could be mitigated by using low-loss conformal spacer layers, e.g. air , Al2O3 , Si3N4 [39, 41], etc., and/or by increasing the BOX layer thickness [23, 39]
Also shown in Fig. 2(b) is the wavelength-dependent facet coupling loss between lensed fiber and the SiOxNy mode-converter, as illustrated by the blue dashed line through blue circle data points. While the loss is measured to be ~5 dB/facet at 1800 nm, the loss becomes larger than 12 dB for wavelengths ≥ 2200 nm. Facet coupling loss is larger at long wavelengths because the SiOxNy mode-converter used here was of a design previously optimized for operation at 1550 nm [34, 35].
3.3 Self-phase modulation versus input wavelength
As discussed in the introduction, it is important to characterize the optical parameters describing nonlinear wave propagation in Si nanophotonic wires. Here we utilize a series of SPM experiments to assess both the nonlinear refractive index n 2 and the two-photon absorption coefficient β TPA as a function of wavelength. The behavior of SPM is characterized at four different pump wavelengths. The measured power-dependent transmission spectra at the center wavelengths 1775 nm, 1988 nm, 2200 nm, and 2250 nm are shown in Figs. 3(a) –3(d), respectively. For each input wavelength, output transmission spectra are recorded at several values of input peak power; these wavelengths are designated by the color-coding shown in the upper-right corner of each panel.
Figure 3 illustrates the development of an increasing number of sharp spectral fringes as the peak pump power increases for all input wavelengths, both above and below the TPA edge near 2200 nm. The observed spectra exhibit clear indications of self-phase modulation (SPM) of the input picosecond pulse, via the strong Kerr nonlinearity of the Si nanophotonic wires. The increasing coupling loss between the lensed fibers and the SiOxNy mode-converters (Fig. 2(b)) limits the maximum peak power level achievable at the waveguide input, and accordingly the total number of fringes observed across the pump spectrum decreases with increasing wavelength. Moreover, the number of fringes and their related total nonlinear phase shift vary sub-linearly with peak power due to the nonlinear loss in the waveguide , particularly at wavelengths less than 2200 nm where the effects of TPA are most pronounced.
Note that except for when the pump is centered at 1775 nm with a peak power of 33.5 W, the mid-IR experimental spectra here do not exhibit the significant spectral broadening observed in previous telecom-band SPM experiments with Si nanophotonic wires [1, 8, 9]. This general lack of broadening is a consequence of the fact that the input pump pulse is not transform-limited, but has a time-bandwidth product of ~1.5 for all four input wavelengths, as suggested by separate autocorrelation and pump spectrum measurements. Note that a similar effect has previously been observed in optical fibers , and attributed to the fact that the input pump pulse was negatively chirped. The effective “positive” chirp introduced by SPM  is largest at the highest experimental peak power ~33.5 W, which is obtained only at a wavelength of 1775 nm due to small on-chip coupling loss (Fig. 2(b)). In this case alone, the positive chirp induced via SPM is sufficient to overcome the pump’s initial negative chirp, and thus substantial spectral broadening is observed. The negative chirp of the input pulse also explains why the SPM-induced spectral fringes consistently emerge from the red and blue “shoulders” of the pulse spectrum for all input wavelengths, rather than from the spectral peak, as typically occurs for an un-chirped pulse [42, 43].
To ensure that any observed nonlinearity originates solely from transmission through the Si nanophotonic wire, reference measurements with the nanophotonic wire removed from the optical path are performed, by directly coupling the tips of the tapered input/output fibers. Even at the highest input power levels used across all wavelengths in the above measurements, there is no observable SPM or nonlinear absorption. Therefore, the nonlinear contribution from all passive fiber components within the setup is found to be negligible.
4.1 Self-limiting of transmission and mid-IR nonlinear loss
The power-dependent transmission spectra in Fig. 3 are analyzed in greater detail to characterize the Si nanophotonic wire’s nonlinear loss as a function of input wavelength. For each set of input conditions, the output peak power is calculated as described in section 3.3 above, using the integrated time-averaged power within the Si nanophotonic wire output spectrum. Figure 4(a) shows the nonlinear transmission response of the Si nanophotonic wire for wavelengths between 1775 and 2250 nm, obtained by plotting the output peak power versus the input peak power. The error bars in the figure reflect the power fluctuations (~1 dB) measured at the waveguide output. Note that the effects of inter-pulse free-carrier accumulation are negligible, since the 13.1 ns period between pump pulses is much larger than the free-carrier recombination lifetime (approximately 0.5-1 ns) in our sub-micrometer Si nanophotonic wires [44, 45].
At low input peak power (< 2 W), the output power shows a linear dependence upon input power for all four wavelengths. However, in the case of the two shortest wavelengths of 1775 nm and 1988 nm, the output power begins to saturate at input peak powers of ~2 W and ~2.5 W, respectively. In addition, the data in Fig. 4(a) clearly demonstrate an increase in the saturated output power level as the wavelength shifts toward the mid-IR TPA threshold. For example, the saturated output peak power increases from ~1 W at 1775 nm to ~2.3 W at 1988 nm. Despite significant coupling loss at the two longer wavelengths, high peak-power levels can still be obtained at both 2200 nm (~6 W) and 2250 nm (~3 W). At these power levels, the transmission behavior is still within the linear regime for 2200 nm and 2250 nm, in contrast to the strongly saturating transmission observed for 1775 nm and 1988 nm at comparable input power. These results are anticipated due to the reduced TPA coefficient for wavelengths ≥ 2200 nm, and are in fact consistent with previously published experimental measurements on bulk silicon [26–28, 46]. The value of the TPA coefficient β TPA can be estimated from the slope of the reciprocal transmission (1/T or P in/P out) versus input peak power [11, 47], as shown in Fig. 4(b). Following the numerical approach described in  and using the slopes obtained from linear fits (dashed lines in Fig. 4(b)), the β TPA values are extracted from the experimental data and summarized in Table 1 . In order to ensure that the extracted β TPA values can be compared against data generated through measurements of bulk silicon [27, 28], our extraction of β TPA incorporates an additional factor which normalizes out the effect of optical modal overlap  with the silicon nanophotonic wire core. The β TPA values are estimated to be 1.4±0.2 cm/GW and 0.6±0.1 cm/GW for wavelengths of 1775 nm and 1988 nm, respectively. For the wavelength of 2200 nm, the estimated β TPA value is significantly lower at 0.03±0.1 cm/GW. Note that this non-zero value of β TPA occurs because the picosecond input pulse train has a significant spectral bandwidth centered at 2200 nm, and therefore samples an average of β TPA coefficients from both above and below the theoretical TPA threshold at 2200 nm. This low value of the TPA coefficient facilitates efficient nonlinear processes, as has been illustrated by a recent report of greater than 25 dB on-chip parametric gain in a Si nanophotonic wire pumped at a wavelength of ~2200 nm . At 2250 nm, the β TPA value is estimated to be negative, at −0.06±0.2 cm/GW. However, the restricted range of accessible input power levels in our 2250 nm experiments produces a relatively large error bar as well as uncertainly in the mean value of β TPA. Additional measurements with longer Si nanophotonic wires having lower facet coupling losses would be required in order to yield more accurate values of β TPA, particularly at wavelengths near and immediately beyond silicon’s TPA threshold.
In general, the β TPA values reported in Table 1 are in reasonable agreement with those measured previously in bulk Si crystals using the z-scan technique [27, 28]. Quantitatively, these β TPA values are approximately a factor of 1.5 × larger than those reported in , and about a factor of 5 × larger than reported in . One potential source of discrepancies with comparison to bulk Si may originate from the fact that the fabrication processes used to define the Si nanophotonic wires (i.e., reactive ion etching, oxidation, deposition of cladding films by plasma-enhanced chemical vapor deposition, etc.) can create crystal damage at the surfaces and within the bulk of the wire. It is known that surface states  and bulk states from vacancies and/or interstitials  in silicon are capable of producing linear absorption at wavelengths longer than the bandedge wavelength of silicon. It is therefore possible to envision that such damage-induced bulk and surface states present in our Si nanophotonic wires may also contribute to nonlinear absorption at wavelengths near 2200 nm which would otherwise be free of TPA in bulk silicon crystals. Note that even in the case of the C-band, where the β TPA and n 2 nonlinear coefficients have been studied more extensively [1, 26–28, 50], experimental uncertainties make definitive statements about the sources of different behavior between bulk Si and Si nanophotonic wires difficult without further study. Despite measurement uncertainties, the values of β TPA in Table 1 provide useful experimental quantities for the design and modeling of future Si nanophotonic wire-based mid-IR nonlinear optical devices. In analyzing our measured value of the β TPA coefficient, note that it is not possible to relate this value via the Kramers-Kronig relation to the n 2 coefficient, due to the presence of other nearby-lying nonlinear processes , and the fact that our measurements extend over only a limited frequency range.
It is worthwhile to note that three photon absorption (3PA) has recently been revealed as the leading-order source of nonlinear loss for sufficiently high peak power (tens of Watts) pulse propagation experiments in Si near λ = 2200 nm [23, 51]. However, given the magnitude of the 3PA coefficient (γ3PA ~0.025 cm3/GW2 at λ = 2170 nm), 3PA is not expected to contribute significantly to nonlinear loss at peak input power levels < 6 W and wavelengths > 2200 nm in the present experiments. An upper limit of nonlinear loss due to 3PA (in dB units) can be estimated by α = 10·log(exp(γ 3PA(κP P/A 0)2 L)), where γ 3PA is the 3PA coefficient, κ is power confinement factor, P P is peak power in the Si nanophotonic wire, A 0 is the area of the silicon core, and L is the length of the Si nanophotonic wire. This estimation predicts a maximum nonlinear loss of 0.5 dB from 3PA within the 4 mm-long Si nanophotonic wire.
4.2 Free-carrier dispersion-induced spectral asymmetry
The SPM spectra in Fig. 3 also reveal additional qualitative characteristics of nonlinear pulse transmission through the nanophotonic wire as the input wavelength is scanned across the TPA threshold of silicon. For example, at the lower values of input power shown in Fig. 3(a)–3(d), the spectral fringes appear symmetrically on both sides of the pump spectrum for all input wavelengths. At higher input powers, the total number of fringes increases, as expected due to the increasing nonlinear phase accumulated across the pump spectrum. However, the fringes become distributed asymmetrically, i.e. the fringes on the red side of the spectrum become spaced more closely together than those on the blue side. This effect is most particularly pronounced for input wavelengths below the TPA edge of silicon, i.e. 1775 nm and 1988 nm. Moreover, these same spectra show a significant spectral blue-shift at high input peak power levels, in contrast to the spectra at 2200 nm and 2250 nm.
These wavelength-dependent SPM fringe asymmetries and spectral blue-shift characteristics are highlighted in Fig. 5(a) and 5(b), through a comparison of the experimental output spectra at wavelengths of 1775 nm (P p = 15.2 W) and 2200 nm (P p = 5.7 W). The notable difference in SPM spectral evolution correlates with the input wavelength being situated far below or near the TPA threshold of silicon. In fact, the observed fringe asymmetry and spectral blue-shift are consistent with the presence of intra-pulse free-carrier dispersion (FCD) effects [1, 8, 9], originating from TPA-generated free carriers.
In order to further confirm our understanding of the characteristic spectral differences observed across the TPA threshold, we use a perturbed nonlinear Schrödinger equation (NSE) to numerically simulate picosecond pulse propagation in the Si nanophotonic wires [9, 13]. The numerical model captures the negative linear chirp of the input pump pulses, the value of which is determined by fitting to the measured input pump spectral width. The wavelength-dependent dispersion and effective nonlinearity parameter in Fig. 1(c) are incorporated into the model, as well as the values of β TPA determined from the self-limiting measurements (Section 4.1 and Table 1). Finally, the model also uses the values of n 2 taken from measurements on bulk silicon [27, 28]. The important parameters used in the simulations are all summarized in Table 2 .
Figures 5(c) and 5(d) contain the simulated output transmission spectra calculated using the same values of input peak power as in the experimental data of Fig. 5(a) and 5(b) (solid lines). As illustrated by comparison with Figs. 5(a) and 5(b), the simulated spectra for both 1775 nm and 2200 nm input wavelengths agree qualitatively well with the spectra obtained from experiment. Moreover, although the maximum experimentally accessible peak pump powers were lower at 2200 nm as compared to 1775 nm, a simulated spectrum for which the pump power is increased to 15.2 W (black dashed curve in Fig. 5(d)) confirms that spectral symmetry (i.e., symmetric SPM fringes and absence of blue-shift) is preserved at 2200 nm for pump power conditions comparable with those at 1775 nm. This result is consistent with the negligible impact of TPA-induced free-carrier generation and dispersion expected near and above the ~2200 nm TPA threshold. In fact, our model shows that the peak intra-pulse free-carrier density at the beginning of the Si nanophotonic wire is reduced by a factor of 60, from 6x1018 cm−3 at the wavelength 1775 nm to 1x1017 cm−3 at the wavelength of 2200 nm, when using the same coupled peak power value of 15.2 W.
The measurements and simulations of pulse propagation and SPM presented here do not permit reliable extraction of silicon’s nonlinear parameter n 2, primarily due to uncertainty in the input pulse’s chirp, which is itself a fitting parameter as stated above. Nevertheless the agreement between the experiments and simulations in Fig. 5 gives confidence that the values of the nonlinear parameters n 2 (based upon bulk Si measurements) and β TPA (extracted from self-limiting transmission experiments above) used as simulation inputs, in fact provide an accurate description of the mid-IR nonlinear characteristics of the Si nanophotonic wires. Moreover, we can also infer that the numerical tools used to compute the spatial mode distribution and waveguide dispersion are appropriate for precise mid-IR modeling.
In this paper we report an experimental and numerical study of mid-IR linear and nonlinear picosecond pulse propagation through a normally dispersive 4 mm-long Si nanophotonic wire. Using a set of input wavelengths which cross the silicon TPA threshold at ~2200 nm, we demonstrate characteristics of SPM and self-limiting transmission having markedly different behavior at longer versus shorter mid-IR wavelengths. We interpret both quantitative and qualitative aspects of these experimental characteristics in the context of fundamental TPA and TPA-induced free-carrier generation/dispersion. Through analysis of the experimental data as well as comparison with numerical simulations, we illustrate that the nonlinear refractive index n 2 and the two-photon absorption coefficient β TPA derived from measurements of bulk silicon can be used to model the mid-IR nonlinear transmission characteristics of Si nanophotonic wires with reasonably good accuracy. This appears to be the case even for deeply scaled waveguide structures in which the (non-ideal, damaged) patterned silicon surfaces are in close interacting proximity with the highly-confined optical mode.
The authors are grateful for the assistance of the staff at the IBM Microelectronics Research Laboratory where the Si nanophotonic wires are fabricated. The authors also wish to acknowledge IBM Summer Internship support of Xiaoping Liu, as well as partial support under the Columbia University portion of DARPA program #CU08-7741.
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