1. INTRODUCTION

Understanding and controlling nonlinear optical processes in silicon [1–7] will be essential in low-cost, high-volume fabrication of integrated opto-electronic chips [8]. On-chip demonstrations of ${\chi }^{(3)}$ processes in silicon include soliton compression [3], supercontinuum generation [4], photon pair generation [5], Raman lasing [6], and third-harmonic generation [7], as well as applications in optical communications, interconnects, and switching [9–13]. The advantages of silicon for nonlinear photonic devices include its large Kerr nonlinearity and its capacity for subwavelength confinement of light. Combined with the potential for monolithically integrating photonics and micro-electronics in a single complementary metal-oxide-semiconductor (CMOS) platform [14], silicon continues to attract considerable attention from researchers around the world. It is clear that a deeper understanding of the physics of nonlinear pulse propagation in silicon is essential from both fundamental and practical points of view.

There are several optical nonlinearities present in silicon. In addition to the desirable ${\chi }^{(3)}$ Kerr effect ($n_2$), a major impairment to the development of nonlinear photonic devices in silicon is the presence of strong two-photon absorption (TPA) in the telecommunications band. The primary response of this nonlinear absorption is the reduction of pulse intensity $I$ and consequent clamping of the optical Kerr effect [15]. Recently we showed that cross-TPA in silicon affects photon pair sources even at very low power levels [16]. In addition, TPA also impacts the pulse dynamics through a second mechanism: the photogeneration of free carriers, which has been shown to greatly distort nonlinear pulse propagation in silicon [3,17].

Since the free-carrier density $N_c$ is generated by the intensity-dependent TPA mechanism, the resulting free-carrier refractive index change $\mathrm{\Delta }{n}_{\mathrm{FC}}$ is also nonlinear. Here $\mathrm{\Delta }{n}_{\mathrm{FC}}=n_{\mathrm{FC}}{N}_c(I)$, where $n_{\mathrm{FC}}$ is the free-carrier dispersion (FCD) coefficient, which can be derived from the Drude model [18]. Note that $n_{\mathrm{FC}}$ is a material parameter dependent on fundamental quantities and has a negative sign: $n_{\mathrm{FC}}=(-e^2/(2n_0{ϵ}_0{\omega }^2))(m_e^{*-1}+m_h^{*-1})$, where $e$ is the elementary charge, $n_0$ is the linear refractive index, ${ϵ}_0$ is the vacuum permittivity, $\omega$ is the angular frequency, and $m_{e/h}^{*}$ is the effective mass of the electrons/holes in the semiconductor. We recognize this as the general case of plasma dispersion, where the refractive index goes down with increasing carrier density. The total index of refraction in the silicon waveguide here is thus described by $n=n_o+n_{2}I+n_{\mathrm{FC}}{N}_c(I)$. In our investigation, we are in a regime in which both the Kerr and FCD effects play a role with FCD ultimately dominating the system. Associated with free-carrier dispersion (FCD) is free-carrier absorption (FCA), which has a relatively minor impact here.

Interplay between linear and nonlinear phase modulations leads to interesting dynamics not possible when either of these effects is present alone. A notable example is higher-order soliton formation, which requires both nonlinear self-phase modulation (SPM) and linear group-velocity dispersion (GVD) to observe temporal pulse narrowing [19,20]. While these effects are fundamentally dynamic interactions, to date the majority of investigations of nonlinear wave propagation in silicon have targeted the spectral domain [4–7,9,15,21]. This is likely due to the significant challenges in characterizing ultrashort pulses in the time domain, especially for the subpicojoule energies required in nanophotonic structures. Specifically, many temporal characterization techniques rely on nonlinear gating methods such as frequency-resolved optical gating, which require minimum power thresholds. Indeed, only last year free-carrier-induced temporal acceleration of picosecond optical pulses in semiconductor waveguides was identified [19]. Acceleration is defined as a reduced time-of-flight compared to linear propagation. These observations were enabled by direct time-domain measurements building on an earlier demonstration of soliton compression [20]. More recently we reported these effects in silicon [3].

Here we report the first description, to the best of our knowledge, of temporal broadening due to dynamic interaction between nonlinear-induced FCD and linear GVD in nanostructured waveguides made of silicon. While these effects have been studied separately, the observations presented here arise uniquely from their simultaneous interaction in the system. In addition, we unveil the role of the third-order dispersion (TOD) interacting with FCD, which, to our knowledge, is also not described in the present literature. We further demonstrate that TOD can either counteract or support the FCD-GVD effect. Critically, the ability to independently tune the dispersion and the nonlinearity in the photonic crystal waveguide (PhC-wg) enables our examination into distinct nonlinear regimes simply by changing the input wavelength of the pulses. This allows us to understand the basic physics in silicon waveguides for a wide range of parameters. We measure the dynamics of electric field evolution of the subpicojoule pulses with an ultrasensitive linear frequency-resolved electrical gating (FREG) [22] technique. The FREG measurements are supported by strong agreement with a generalized nonlinear Schrödinger equation (GNLSE) model. We further demonstrate that dispersion engineering is a powerful control mechanism for free-carrier temporal effects. In addition to our main goal of elucidating the fundamental light–matter interaction of FCD-GVD dynamics in silicon waveguides, these results provide direct insight into fundamental nonlinear optical processes and means to control free-carrier effects in silicon waveguides for potential applications in optical interconnects [23].

2. DISPERSION ENGINEERED SILICON WAVEGUIDES AND DESCRIPTION OF THE EXPERIMENT

The sample is a dispersion engineered slow-light silicon PhC-wg of length $L=396\mathrm{\mu m}$ [24]. The measured group index ($n_g$), the GVD (${\beta }_2$), and the TOD (${\beta }_3$) vary with wavelength as shown in Fig. 1(a). In addition to many degrees of freedom for dispersion engineering, the photonic crystal also significantly enhances the nonlinear optical interactions due to a reduction in the group velocity in the structure known as slow light. The reduced group velocity $v_g=c/n_g$ yields a larger electric field for a given power, as well as a longer effective optical path length [25]. The exact slow-light scaling of the nonlinear parameters is given in Section 3 and discussed in detail in Supplement 1. We note that the slow-light-enhanced nonlinearity discussed here is derived from the structure. In contrast, material slow light from atomic and related systems does not exhibit enhanced nonlinear effects [26].

 

Fig. 1. Sample parameters and temporal characterization setup. (a) Measured group index (blue line), and second-order (red line) and third-order (green line) dispersion of the photonic crystal waveguide. The markers represent the values for the measured wavelengths. (b) Experimental FREG setup, consisting of the following: MML, mode-locked laser; WS, pulse shaper; PhC-wg, photonic crystal waveguide; $\tau$ variable delay; PD, fast photodiode; MZM, Mach–Zehnder modulator; OSA, optical spectrum analyzer. The inset shows a scanning electron micrograph of our silicon PhC-wg device.

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In order to obtain a complete temporal and spectral characterization of the ultrafast pulses propagating through the PhC-wg, we have built a FREG apparatus [22] in a cross-correlation configuration as shown in Fig. 1(b). The FREG technique is a subclass of spectrographic techniques including frequency-resolved optical gating [27]. With the FREG we measure a series of spectrograms $S(\omega ,\tau )$, gated optical power versus frequency $\omega$ and delay $\tau$, for varying input powers. The spectrogram $S(\omega ,\tau )$ is defined as

3. MODELING THE INTERACTION OF FREE-CARRIER EFFECTS AND DISPERSION

In order to understand the physics behind the observed temporal dynamics, we employ a model based on a GNLSE. In addition to dispersion and SPM, this incorporates TPA, FCD, and FCA present in silicon [17]:

The free-carrier density is $N_c(z,t)$, and the generation of the TPA-induced free carriers obeys the following rate equation:

We have defined the power-normalized carrier generation rate ${\rho }_{\mathrm{FC}}={\alpha }_{\mathrm{TPA}}/(2h{\nu }_0{A}_{\text{eff}}^2)$, with the photon energy $h{\nu }_0$, and ${\tau }_c\sim 0.5\mathrm{ns}$ is the carrier lifetime [15].

For our silicon PhC-wg the linear loss is ${\alpha }_l=60\mathrm{dB}/\mathrm{cm}$ [24]. The dispersion values ${\beta }_n$ and group indices $n_g$ at the chosen wavelengths are given in Table 1. The effective area $A_{\text{eff}}$ is calculated using the MPB package [30], being 0.28, 0.34, and $0.44{\mathrm{\mu m}}^2$ for 1541, 1543, and 1545 nm, respectively. The slow-light enhancement of nonlinearity is included in the GNLSE model by scaling the relevant parameters by the slow-down factor $S=v_p/v_g=n_g/n_0$, where $v_p$ is the phase velocity and $n_0$ is the bulk refractive index (for silicon we use $n_0=3.46$) [15]. As detailed in Supplement 1, the GNLSE describes our experiment when we replace the nonlinear bulk parameters with the following slow-light-scaled values [31]: $n_2=S^2·6·{10}^{-18}{\mathrm{m}}^2/\mathrm{W}$, ${\alpha }_{\mathrm{TPA}}=S^2·10·{10}^{-12}\mathrm{m}/\mathrm{W}$, $n_{\mathrm{FC}}=S·-6·{10}^{-27}{\mathrm{m}}^3$, and $\sigma =S·1.45·{10}^{-21}{\mathrm{m}}^2$ [18,29]. The FCD parameter used here differs slightly from our previous experiment due to the fact that $n_{\mathrm{FC}}$ depends on the exact free-carrier concentration and distribution, which vary somewhat with the exact powers and pulse durations used [3,18]. We emphasize that the physical effects described by this equation are general to all silicon waveguides.

Table 1. Measured Parameters and Ratios for the Wavelengths Studied^a

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4. TIME-RESOLVED MEASUREMENTS AND PHYSICAL LENGTH SCALES IN SILICON WAVEGUIDES

We first characterized the input pulses with the FREG by bypassing the sample. Figure 2(a) shows a measured spectrogram (top), which we take before the pulse is input into the PhC-wg. Then we inject 1.3 ps pulses at 1541, 1543, and 1545 nm into the PhC-wg, gradually increasing the input power and recording their output spectrograms using our FREG arrangement. The pulse duration is defined for Gaussian pulses at full width-half-maximum $T_{\mathrm{FWHM}}$. In the spectrograms shown in Fig. 2(a), increasing blue shift and acceleration (diminished time of flight) are observed with increasing input power for the 1543 nm input wavelength. Similar blue-shift and acceleration trends were observed for the other two input wavelengths. However, Fig. 2(b), showing the $T_{\mathrm{FWHM}}$ of the retrieved output pulses, clearly indicates distinct pulse temporal dynamics for each measured wavelength. In Fig. 2(b) we highlight three key regimes dominated by linear dispersion (red), Kerr and dispersion (yellow), as well as FCD and dispersion (blue). In this paper we describe the physical origin of these dynamics.

 

Fig. 2. Frequency-resolved electrical gating (FREG) measurements. (a) Experimental spectrograms at the input (waveguide bypassed) and output for three different coupled peak powers at 1543 nm. The cross-correlation delay is $\tau$. (b) Measured pulse durations as a function of coupled peak power for three different wavelengths. Color-shaded regions indicate the dominant effect: dispersive broadening (GVD, red), Kerr-induced narrowing (SPM, yellow), and free-carrier dispersion broadening (FCD, blue).

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To elucidate the driving effects behind the temporal regimes observed in Fig. 2, we quantify the strengths of the various linear and nonlinear effects (GVD, TOD, Kerr, FCD) using the associated length scales ($L_D$, $L_D^{\prime }$, $L_{\mathrm{NL}}$, $L_{\mathrm{FCD}}$). Devices longer than these length scales will exhibit the properties described by these effects. For example, dispersive temporal broadening of $\sqrt{2}{T}_{\mathrm{FWHM}}$ occurs when $L=L_D$ for Gaussian pulses. The dispersion length $L_D=T_0^2/|{\beta }_2|$ and TOD length $L_D^{\prime }=T_0^3/|{\beta }_3|$ are linear effects with $T_0=T_{\mathrm{FWHM}}/1.665$ being the pulse duration for Gaussian pulses. The nonlinear length $L_{\mathrm{NL}}=1/(\gamma P_0)$ describes the strength of SPM [32]. The characteristic FCD length,

In order to examine which physical mechanism is dominant we compute ratios of the key effects at the wavelengths under investigation. Figure 3 shows the scaling of these ratios as a function of power for the 1543 nm case. Similar results occur for the other wavelengths. We observe that the first ratio to exceed the threshold is $L/L_{\mathrm{FCD}}$. The ratio $L/L_{\mathrm{FCD}}$ indicates the relative importance of nonlinear FCD over the propagation length $L$ with pulse conditions of $L/L_{\mathrm{FCD}}>1$ exhibiting significant FCD effects such as frequency blue shift [15].

 

Fig. 3. Ratios of key physical length scales in silicon waveguides for the 1543 nm case. As $L_{\mathrm{FCD}}$ and $L_{\mathrm{NL}}$ are power dependent, we show the relative balance of the characteristic length scales as a function of coupled power. The other wavelengths exhibit similar behavior.

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The next ratio to come into play is $L_{\mathrm{NL}}/L_{\mathrm{FCD}}$, which represents the importance of SPM in relation to FCD. When $L_{\mathrm{NL}}/L_{\mathrm{FCD}}>1$, the instantaneous phase shift induced by the free carriers is larger than the Kerr-induced phase shift. Note that $L_{\mathrm{FCD}}$ scales inversely with $P_o^2$ for free carriers generated by TPA, while $L_{\mathrm{NL}}$ scales with $P_o$. We expect SPM to be the dominant nonlinearity at lower powers, while FCD will dominate at high powers due to its power-squared dependence as shown in the plot. As dispersion is quite strong in this system from both the large ${\beta }_2$ and short pulse duration, dispersion is the last effect overcome by nonlinear FCD. This is shown by the ratio $L_D/L_{\mathrm{FCD}}>1$, which indicates the importance of FCD compared to GVD. The temporal broadening mechanism is tied directly to this ratio as we describe below. It is clear that the nonlinear FCD effect ultimately dominates in the system as we increase the input power.

Table 1 summarizes these values for the other wavelengths we investigated. To illustrate the dynamics, we choose an intermediate power point (3 W), as higher power levels are all dominated by FCD. Here we see that $L/L_{\mathrm{FCD}}$ is well above a value of 1 in all cases, showing that FCD effects are quite strong even at these modest power levels. Notice that at this intermediate power, the free-carrier effects ($L_{\mathrm{FCD}}$) are stronger than SPM ($L_{\mathrm{NL}}$) and on the order of $L_D$ for all wavelengths. The last ratio in the table, $L_D/L_D^{\prime }$, reveals the significance of the second- and third-order dispersion terms over the pulse propagation. Despite being relatively small compared to $L_D$, later we show that $L_D^{\prime }$ plays a role in the temporal dynamics.

The largest point of difference between the photonic crystal employed here and nanowire (channel) waveguides is the relative strength of the dispersive and nonlinear effects. More specifically, the dispersion and nonlinear values in nanowire waveguides are smaller than in PhC-wgs [4,33]. Despite these differences, all of the ratios will eventually be dominated by FCD, similar to the PhC-wg. The main difference is the order in which the phenomena are overcome and the required pulse durations and powers. Detailed analysis can be found in Supplement 1. For the case in [33], we expect that significant FCD temporal dynamics will be visible at peak powers slightly beyond those reported there, which could be achieved with improved insertion loss. Given that shorter pulse durations of hundreds of femtoseconds are often used in nanowires, the pulse energies are the same order of magnitude as those used here for PhC-wgs.

With the physical length scales in mind, we return to Fig. 2(b) and elucidate the dominant mechanism in each region. In the linear regime shaded red in Fig. 2(b), the GVD dominates and the pulse experiences dispersive broadening from its initial pulse duration of 1.3 ps to an output duration scaling with ${\beta }_2$. This significant broadening is due to the fact that the dispersion lengths, $L_D$ (65, 62, and 86 μm for each of the three wavelengths, respectively), are well below the length of the 396 μm PhC-wg. Coherently with the dispersion curve shown in Fig. 1(a), the wavelength with the largest GVD parameter, i.e., 1543 nm with ${\beta }_2=-10.1{\mathrm{ps}}^2/\mathrm{mm}$, broadens the most, while 1545 nm, with ${\beta }_2=-8.4{\mathrm{ps}}^2/\mathrm{mm}$, broadens the least in the linear regime.

As the peak power coupled into the PhC-wg increases, the pulse tends to narrow in time from its initially dispersion-broadened state, as indicated by the yellow-shaded region in Fig. 2(b). The temporal narrowing in this region is governed by SPM-induced chirp interplaying with the anomalous GVD-induced chirp. Note, however, that the observations here are not higher-order soliton compression, as opposed to our previous work [3]. In this case, the SPM is only beginning to cancel the dispersive chirp.

For all wavelengths, there is a certain peak power threshold from which the pulse experiences a significant temporal broadening, as shown by the blue-shaded region of Fig. 2(b). The power threshold relates to the relative significance of the free carriers with respect to the GVD and sample length and occurs at the point where both $L_D/L_{\mathrm{FCD}}>1$ and $L/L_{\mathrm{FCD}}>1$. The power threshold is different with each wavelength due to the different sample parameters. Physically this nonlinear effect is a direct consequence of the interplay of the anomalous GVD and new blue components generated by FCD, as we will discuss in detail in the next section.

5. TEMPORAL PULSE DYNAMICS DOMINATED BY FREE-CARRIER NONLINEARITY IN SILICON

Figures 4(a) and 4(b) show the normalized intensity for several input power levels in both the time and the frequency domains for the 1543 nm case and the 1545 nm case. The pulse retrieved from the input spectrogram in Fig. 2(a) is used as the simulation input. The red lines represent the retrieved experimental FREG pulses (time) or OSA traces (frequency). Our GNLSE model, indicated by the blue lines, shows excellent agreement with the spectral and temporal profiles of the optical pulses. For all the wavelengths we examined, the pulse consistently experiences a spectral blue shift and accelerates, i.e., diminishes its time of flight, with increasing powers, as clearly indicated in Figs. 4(a) and 4(b). The three previously identified regimes of temporal propagation: dispersive broadening, Kerr-effect narrowing, and free-carrier broadening are noticeable in Figs. 4(a) and 4(b) using the same color code as in Fig. 2(b).

 

Fig. 4. Temporal and spectral pulse profiles: experiment and model. (a),(b) Normalized intensity in the time and frequency domains for (a) 1543 nm and (b) 1545 nm. The red lines represent the experiment and the solid blue lines the GNLSE simulations. Color-shaded regions indicate the dominant effect: dispersive broadening (GVD, red), Kerr-induced narrowing (SPM, yellow), and free-carrier dispersion broadening (FCD, blue).

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As is known from past experiments, the spectral blue shift is a direct consequence of FCD alone [15–17]. This results from the time-dependent phase modulation induced by TPA-generated free carriers: $\delta {\omega }_{\mathrm{FC}}=-\frac{d{\varphi }_{\mathrm{FC}}}{dt}$, where ${\varphi }_{\mathrm{FC}}=k_o{n}_{\mathrm{FC}}{N}_c(z,t)$. The inclusion of GVD induces only a small perturbation to this trend in the frequency domain. In contrast, the temporal properties are dominated by the dynamic interaction of FCD with dispersion. For example, the acceleration of the pulse stems from the free-carrier generated blue components traveling faster in time due to the strong anomalous dispersion of the waveguide [3]. In this work we demonstrate pulse temporal advances up to more than six times the initial pulse duration $T_{\mathrm{FWHM}}$, notably larger than any prior reported acceleration [34,35]. Apart from those known effects, here we report a significant nonlinear pulse broadening associated with the FCD-GVD interaction, which we now examine in detail.

Figures 5(a)–5(c) depict the experimental and GNLSE modeled output values of $T_{\mathrm{FWHM}}$ as a function of the coupled peak power for the three different wavelengths. The pulse broadens in time at least six times the initial $T_{\mathrm{FWHM}}$ for all cases. The strong agreement between the full model (black solid lines) and experiments (markers) across the three cases allows us to switch on and off each effect independently to explain the origins of the observed pulse propagation behavior. We focus on the 1545 nm case in Fig. 5(a), as it gives the clearest presentation of these effects.

 

Fig. 5. Nonlinear temporal dynamics in the silicon waveguide FCD-GVD regime. (a)–(c) Pulse duration as a function of coupled peak power for (a) 1545 nm, (b) 1541 nm, and (c) 1543 nm cases. The color markers represent the experimental FREG measurements, the solid black lines represent the full GNLSE simulation, the dashed black lines neglect the free-carrier effects, the purple dotted lines neglect TOD, the red dotted–dashed line neglects GVD and TOD, and the blue dotted line neglects all effects except for SPM.

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First, we focus our attention to the origin of temporal broadening. When we turn the free-carrier effects off in the model there is no broadening, but rather a continued narrowing as shown by the black dashed lines. This is the expected interaction for a pulse undergoing SPM with anomalous GVD. Importantly, we confirm that temporal broadening requires both FCD and GVD. We confirm this by turning off only the GVD and the TOD (red dotted–dashed line) and observing that the pulse does not experience significant broadening at high powers. The marginal change in $T_{\mathrm{FWHM}}$ is due to TPA. We further verified that FCA has only a marginal impact for our conditions by turning this off in the model. As this effect is negligible we exclude this case on the plot for clarity. Switching off TPA (not shown here for simplicity) would result in a narrower pulse, since the SPM-induced chirp would not be capped by this nonlinear absorption mechanism [17,19]. Turning off all effects except SPM, we recover the initial pulse with no temporal changes (blue dotted line).

Examining these results in further detail, we observed physics particular to free carriers interacting with the TOD ${\beta }_3$. To our knowledge, this process has not been reported in the literature. The purple dotted lines in Figs. 5(a)–5(c) represent the results of the pulse duration using our GNLSE model neglecting the effect of TOD (${\beta }_3=0$). For positive ${\beta }_3$ ($+1.1{\mathrm{ps}}^3/\mathrm{mm}$) in the case of 1541 nm, the absence of TOD broadens the output pulse. Stated equivalently, the positive ${\beta }_3$ acts counter to the anomalous FCD-GVD-induced broadening. In the case of a large negative ${\beta }_3$ ($-2.57{\mathrm{ps}}^3/\mathrm{mm}$), as in the 1545 nm case, the absence of TOD results in a narrower pulse, thereby demonstrating negative ${\beta }_3$ reinforces the FCD-GVD broadening. The pulse shape is relatively unchanged for the 1543 nm case.

We have also verified that ${\beta }_3$ has a non-negligible impact on the pulse temporal acceleration. Figure 6 shows the measured (green dots) and simulated acceleration for 1545 nm. Notice that the pulse acceleration is stronger with (solid black line) than without TOD (dotted purpled line), from which we conclude that negative ${\beta }_3$ causes larger values of free-carrier-induced acceleration, once again supporting the combined effect of FCD and negative ${\beta }_2$. Two effects can potentially contribute to this: intrapulse TOD broadening and the effective dispersion at the blue-shifted frequency. Intrapulse TOD broadening plays a minor role here due to the relatively long pulses. The effective dispersion ${\beta }_2^{\prime }$ can be understood by recognizing that TOD corresponds to the dispersion slope. The blue-shifted pulses thus experience the effective dispersion ${\beta }_2^{\prime }={\beta }_2+{\beta }_3\mathrm{\Delta }{\omega }_{\mathrm{FC}}$ [32]. At 1545 nm and $P_0=6\mathrm{W}$, the ${\beta }_3\mathrm{\Delta }{\omega }_{\mathrm{FC}}$ product increases by up to $-5.5{\mathrm{ps}}^2/\mathrm{mm}$ for the maximum blue shift of 0.34 THz. This effect contributes additional pulse acceleration as it is the same order and sign as ${\beta }_2$. It can be seen from ${\beta }_2^{\prime }$ that if the sign of TOD was opposite to GVD, it would instead reduce the effective dispersion and thus acceleration.

 

Fig. 6. Pulse acceleration for the 1545 nm case. We show the experimental FREG measurements (colored markers), the full GNLSE simulation (solid black line), free-carrier effects off (dashed black line), TOD off (purple dotted line), and positive ${\beta }_2$ with the remaining parameters from the full simulation (turquoise dot-dashed line).

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Though not experimentally demonstrated here, FCD coupled with normal dispersion (${\beta }_2>0$) would yield pulse delay and pure temporal broadening of $T_{\mathrm{FWHM}}$ while retaining a spectral blue shift. This case (still with negative ${\beta }_3$) is shown as the turquoise dot-dashed line in Fig. 6 with evident pulse delay.

6. FUTURE DIRECTIONS

In this section, we consider the implications of these results from a practical perspective. Silicon waveguides are expected to play a key role as optical interconnects in future integrated opto-electronic chips [36]. Concretely, signal processing of dense optical channels confined to subwavelength waveguides will inevitably trigger nonlinear effects and require mitigation akin to the nonlinear Shannon limit in fiber optic telecommunications networks [37]. The nonlinear temporal broadening mechanism due to FCD-GVD presented here will certainly need to be taken into account in future on-chip photonic networks.

While TPA is intrinsically linked to the band gap of silicon ($E_g=1.1\mathrm{eV}$), and therefore unavoidable, there are a number of proposed strategies for controlling the free carriers. Some examples include the use of reversed p-i-n junctions to sweep the carriers out of the waveguide [38], the modification of the waveguide fabrication process to minimize the carrier lifetime [39], or the use of short (femtosecond) pulses [40]. Notably, other demonstrations harness p-i-n junctions to inject carriers into the optical mode for modulation [36,41,42], regeneration [43], and detection [44].

These experiments reveal a new approach to control the free-carrier effects with dispersion engineering. In particular the combination of the group index ($n_g$) and the GVD parameter (${\beta }_2$) determines the power threshold at which the FC effects start dominating the temporal dynamics. Further, TOD (${\beta }_3$) becomes an instrument to counteract or support these FC effects in silicon waveguides. This is in analogy with the work presented in [45], where dispersive radiation was used to cancel the most significant higher-order nonlinearity in glass, the Raman effect. Here we control the most significant higher-order nonlinearity in semiconductors, the free-carrier effects. More complex temporal dynamics ($T_{\mathrm{FWHM}}$ and acceleration) are possible when dispersion profiles with similar amplitudes of ${\beta }_2$ and ${\beta }_3$ ($L_D/L_D^{\prime }\sim 1$) and mixed signs are considered. The new insight presented here provides a powerful control mechanism for free-carrier temporal effects in silicon devices simply by designing the waveguide.

7. CONCLUSION

By making use of an ultrasensitive FREG technique we examined the nonlinear pulse dynamics of subpicojoule pulses in nanostructured silicon waveguides directly in the time domain. Importantly, the unique ability to independently tune the dispersion and nonlinearity in our PhC-wg enabled our systematic investigation into different strengths of linear and nonlinear effects, which would be challenging in other media. In particular, we experimentally demonstrated free-carrier-induced temporal broadening of picosecond pulses. The temporal broadening stems from the combination of nonlinear-induced FCD combined with GVD. We further examined the role of ${\beta }_3$ with the conclusion that positive values of the TOD parameter, ${\beta }_3>0$, counteract the effects of free carriers in anomalous dispersion media, whereas ${\beta }_3<0$ reinforces them. These results were supported by strong agreement with a numerical GNLSE model. The physics unveiled in this paper provide further insight into the role of free carriers in nonlinear pulse propagation in silicon and means to control their effects. These realizations could be key to the development of optical interconnects and data processing in integrated CMOS opto-electronic chips.