## Abstract

By propagating femtosecond pulses inside submicron-cross-section Si photonic-wire waveguides with anomalous dispersion, we demonstrate that the pulse-propagation dynamics is strongly influenced by the combined action of optical nonlinearity and up to third-order dispersion with minimal carrier effects. Because of strong light confinement, a nonlinear phase shift of a few π due to self-phase modulation is observed at a pulse peak-power of just ~250 mW. We also observe soliton-emitted radiation, fully supported by theoretical analysis, from which we determine directly the third-order dispersion coefficient, *β*
_{3}=-0.73±0.05 ps^{3}/m at 1537 nm.

©2006 Optical Society of America

## 1. Introduction

In the last few years, there have been major advances in the fabrication, design, and device implementation of integrated Si photonics. A particularly exciting area has been focused on submicrometer-transverse-dimension Si-wire waveguides (Si-WWG) using the silicon-on-insulator (SOI) materials platform, where active functionalities such as Raman amplifiers and lasers, all-optical modulators, wavelength converters, and thermo-optic switches have been demonstrated [1–12]. Several of these functionalities rely on the fact that Si-wires possess substantial optical nonlinearities. Recent studies have demonstrated nonlinear effects such as self-phase modulation (SPM), cross-phase modulation (XPM), two-photon absorption (TPA), or four-wave mixing in Si-WWG [13–17]. These effects lead to functionalities such as pulse shaping, supercontinuum generation, and optical switching. On the other hand, they may also contribute impairments to Si-WWG-based optical systems such as spectral broadening with increasing peak power due to SPM or inter-channel crosstalk due to XPM.

An interesting property of deeply scaled Si-WWG is that their optical properties are governed chiefly by the waveguide dispersion [18–20]. By carefully choosing the transverse waveguide dimensions, one can *tailor* these dispersion properties. This characteristic raises the possibility of combining nonlinearities with dispersion to achieve additional optical-fiber-like functionalities, e.g., broadband optical parametric gain [21, 22], pulse compression in the normal dispersion regime [23], or soliton generation in the anomalous dispersion regime [24]. To observe the effects of dispersion and nonlinearity, the propagation length must be comparable to the dispersion and the nonlinear lengths. For optical fibers, the dispersion length is on the order of 10 m for ~100 fs pulses. However, for Si-WWG, the corresponding dispersion length is on the order of *only* 1 cm or less because of their large total dispersion. These dispersion-engineering and nonlinear optical properties provide the foundation for Si-wire-based waveguides as building blocks of a “fiber-on-a-chip” system.

Here we present an experimental and theoretical study of ultrashort laser pulse propagation in mm-long Si photonic wires. In this study, the waveguide length is comparable to its group-velocity dispersion (GVD), third-order dispersion (TOD), and nonlinear length. These three parameters determine correspondingly the length scales over which pulse broadening, pulse asymmetry, and nonlinearly induced self phase shift become appreciable, thereby allowing us to probe the interplay between GVD, TOD, TPA, and SPM. This is contrast to our previous ps study where SPM was the dominant effect [13]. In addition, compared to that of bulk Si, for the wavelengths used in this study, the waveguide GVD is ~ three times larger and has opposite sign, i.e., it is in the anomalous dispersion regime. We demonstrate the following: (1) saturation of the output power with input power, which arises only from bound-electron TPA; (2) SPM-induced spectral broadening that covers several tens of nanometers, namely from 1510 to 1590 nm and is strongly influenced by TOD; and (3) consistent with pulse propagation in the anomalous GVD regime, evidence of soliton-like behavior from which we extract the TOD coefficient. Furthermore, for the pulse width and pulse energy range used, the effect of free carriers generated by TPA is suppressed thereby allowing us to isolate the intrinsic effects of nonlinearity and dispersion. We finally discuss the implications of using Si-WWG as a switching device in terms of its figure of merit.

## 2. Experiment

Our experiments employ a single-mode Si-WWG with cross-section *A*
_{0}=220nm×445nm and length *L*=4.7 mm, patterned on Unibond SOI with a 1-µm-thick oxide layer and aligned along the [110] crystallographic direction. The size uncertainty on either side is 1 nm as measured by scanning electron microscope. The devices were fabricated using the CMOS production line at the IBM T.J. Watson Research Center [25]. Each end of the waveguides has an inverse polymer mode-converter, which allows efficient in and out coupling. The measured intrinsic waveguide loss is *α*
_{in}=3.5 dB/cm for TE polarization near *λ*=1550 nm. The laser pulses are produced by a mode-locked fiber-laser with repetition rate of 37 MHz, pulse width of *T*
_{0}≈200 fs as measured by an autocorrelator, average power of 4 mW, center wavelength of 1537 nm, and spectral bandwidth of 16.5 nm. To prevent SPM from other optical elements prior to the waveguide, an objective lens was used to couple light into the waveguide; the output was then collected by a tapered fiber connected to a power meter or an optical spectrum analyzer (OSA). Using *α*
_{in} and the coupling efficiency between the tapered fiber and the waveguide (~3dB), we obtained an input coupling loss of ~30 dB.

#### 2.1 Optical limiting

The nonlinear response of the waveguide is demonstrated in Fig. 1. At relatively low input powers <50 mW, the output power scales linearly with the input power then saturates for higher input powers (here and henceforth, the input power *P*
_{0} refers to the in-coupled pulse *peak* power). Other groups have previously observed similar behavior using pulsewidths >1 ps but the saturation effect was attributed to TPA-generated free-carrier absorption (FCA). In this work, however, the maximum pulse energy used is 50 fJ, which is insufficient to create significant FCA, as discussed in detail below. Using the pulse-propagation model described in Section 3, we find from numerical simulations the onset of the saturation at *P*
_{0}≈50 mW, in agreement with experimental data.

The theoretical prediction denoted by the red solid line in Fig. 1, which includes all dispersion and nonlinear effects including FCA, was obtained assuming a sech-input pulse shape. The dashed green curve shows the result in the absence of FCA and overlaps with the red curve. This observation indicates that the optical-limiting behavior arises from TPA and not from the FCA, which was seen in longer pulse experiments [13, 16]. Using Gaussian input pulses (blue curve) yields a slightly different saturation level to that of sech pulses and better approximates the data, indicating the dependence of the output power on the input pulse shape. We consider input pulses with a sech shape since the output pulse of fiber lasers is known to have a temporal shape close to a sech profile. In addition, although the simulation using Gaussian pulses fits the experimental data better than for the sech pulses, our simulations show that sech pulses provide a much better fit of the spectral features as shown below in Section 3.3.

According to theory for low input powers, the output pulsewidth is slightly broader than its initial width by approximately 10% due to GVD and the output power is approximately 70% of the input power due to the intrinsic waveguide loss. At high powers, the temporal profile becomes strongly distorted due to the combined effects of GVD, SPM, and TOD. Due to this nontrivial pulse propagation behavior, we present in Fig. 1 the average power instead of the peak power. The simulated output power shown in Fig. 1 represents the average output power immediately before exiting the waveguide; the experimental data is adjusted from the measurements using the known coupling loss of the tapered fiber.

#### 2.2 Self-phase modulation

In addition to the saturation behavior, the output pulses show increasing spectral modulation, as shown in Fig. 2. As the input power is increased, the pulse spectrum broadens and then develops a multi-peak structure, a signature of SPM, which is the result of the phase interference of the pulse frequency components with a time-dependent SPM-induced frequency chirp. Similar results have recently been obtained for the case of ps pulses [13]. Assuming a transform-limited input pulse, we can estimate the maximum phase shift as *ϕ*
_{max}≈(*M*-1/2)π, where *M* is the number of peaks in the spectrum. At the highest input power (~250 mW) used, *ϕ*
_{max}≈2.5π. Because of the pulse distortion due to higher order dispersion acts to suppress features, i.e., “bumps”, in the spectra for higher powers, the 2.5π maximum phase value is believed to be an underestimate as far as using this approximate counting technique is concerned.

One striking observation in Fig. 2 is that the output spectrum becomes more asymmetric with increasing input power. In optical fibers, SPM-induced spectral broadening is normally symmetric around the center frequency. For Si-WWG excited with ps or longer pulses, or larger waveguides that tend to have longer carrier lifetimes excited with fs pulses, similar asymmetry has been observed and was attributed to FCA [13, 15]. While FCA may become dominant for ps or longer pulses, it is significantly reduced for fs pulses since the total pulse energy is greatly reduced. Consequently, as we demonstrate below, TOD becomes the dominant factor responsible for the spectral asymmetry in Fig. 2.

## 3. Theoretical study and simulation

#### 3.1 Dispersion properties

To describe the dynamics of the pulse propagation, we first determined the waveguide dispersion properties, viz. effective index (*n*
_{eff}), group index (*n*
_{g}), and GVD coefficient (*β*
_{2}). We calculated *n*
_{eff} using the RSoft BeamPROP software based on a full vectorial beam propagation method, which was crosschecked with a finite-element algorithm and experimental data [18]. All results agree within 0.1% of each other. We fitted the values of *n*
_{eff} with a 7^{th}-order polynomial and took numerical derivatives of this polynomial to obtain *n*_{g}
and *β*
_{2}, which are defined by *n*
_{g}=*β*
_{1}
*c* and *β*_{m}
=*d*^{m}*β*
_{0}/d*ω*^{m}
, where *β*
_{0}=*n*
_{eff}(*ω*)*ω*/*c* and *ω* is the carrier frequency. Figure 3 shows the dependence of these parameters on the wavelength *λ* (left panel), and the resulting major and minor modes, i.e., the *E*_{x}
and *E*_{y}
electric field components, respectively (right panels). Notice that for the wavelength range used in our experiments, our waveguide exhibits anomalous dispersion. For comparison, the GVD coefficient *β*
_{2}=-3.97 ps^{2}/m (*D*=d*β*
_{1}/d*λ*=3.2 ps/nm-m) at 1537 nm is comparable to *β*
_{2}=-2.60 ps^{2}/m (*D*=2.1 ps/nm-m) of a Si-WWG of slightly different dimensions (525×226 nm^{2}) [19].

#### 3.2 Simulation of pulse propagation

The pulse dynamics are governed by the interplay of SPM and dispersion whose relative strengths can be determined by several characteristic lengths, namely the GVD and TOD lengths, defined as *L*_{D}
=${T}_{0}^{2}$/|*β*
_{2}| and *L*_{D}
′=${T}_{0}^{3}$/|*β*
_{3}|, respectively, and the nonlinear length, defined as *L*_{NL}
=*ε*
_{0}
*A*
_{0}/3*ω*
${\beta}_{1}^{2}$
*P*
_{0}Γ′. The nonlinear effects described by our model, i.e., SPM and TPA, are characterized by a complex effective 3^{rd}-order nonlinear coefficient of the Si-WWG, Γ=Γ′+*i*Γ″, which is defined as the overlap integral of the bulk 3^{rd}-order susceptibility tensor χ^{(3)} of silicon and the waveguide mode. The quantities Γ′ and Γ″ are directly related to the *effective* Kerr nonlinear refractive index, *n*
_{2}, and the TPA coefficient, *β*, respectively, according to *n*
_{2}=3Γ′/(4*ε*
_{0}
*cn*
^{2}) and *β*=3*ω*Γ″/(2*n*
^{2}
*c*
^{2}
*ε*
_{0}). In addition, we take into account the FCA and free-carrier-induced dispersion. We also introduce the normalized time, *τ*=(*t*-*β*
_{1}
*z*)/*T*
_{0}, the normalized carrier lifetime *τ*_{c}
=*t*_{c}
/*T*
_{0}, and the normalized length, *ξ*=*z*/*L*_{D}
. In terms of these parameters, the dynamics of pulse propagation in the silicon wire, described by the slowly varying envelope *ψ*(*z*, *t*) of the laser pulse, is governed by the following system of coupled nonlinear differential equations [18]:

where Δ*N* is the free-carrier density, *δ*=-sgn(*β*
_{2})*L*_{NL}
/*L*_{D}
, *δ*′=-sgn(*β*
_{3})*L*_{NL}
/*L*_{D}
′, *θ*=*cβ*
_{1}
*κL*_{NL}
/2*n*, *η*=2*θω*/*c*, *γ*=Γ/Γ′, and *ρ*=3*T*
_{0}
${\beta}_{1}^{2}$Γ″/*ε*
_{0}*ħ*${A}_{0}^{2}$. The other parameters are *κ*, a coefficient related to the effective area of the waveguide mode, *α*
_{FC} is the FCA coefficient, *δn*
_{FC} is the FC-induced change of the refractive index, and *ħ* is the reduced Planck constant.

For ps or longer pulses with *P*
_{0}=5 mW or larger, we have *δ*=*L*_{NL}
/*L*_{D}
≪1 and *δ*′=*L*_{NL}
/*L*_{D}
′≪1. In this case, the second and the third terms on the LHS of Eq. (1) may be ignored and SPM dominates the pulse evolution inside the waveguide. If, instead, the pulse width is in the fs regime, *L*_{D}
, *L*_{D}
′, and *L*_{NL}
are comparable for mW-level powers. For *T*
_{0}=200 fs, used in our experiment, *L*_{D}
≈10 mm and *L*_{D}
′≈11 mm. Here we point out that *β*
_{3}, which yields *L*_{D}
′, is extracted from the data as described below. The length *L*_{NL}
depends on power, e.g., if *P*
_{0}=5 mW, *L*_{NL}
≈9 mm so that *δ*≈*δ*′≈1. Consequently, near or above *P*
_{0}≈5 mW, the GVD, TOD, and SPM all become relevant to the overall pulse dynamics.

#### 3.3 Soliton-emitted radiation and third-order dispersion

Experimentally, we find that SPM is evident in the spectra at high pump powers (Fig. 2). In addition as the input power increases, an incipient spectral feature develops near 1590 nm, shown in detail in Fig. 4(a) (brown curve) at *P*
_{0}=200 mW. Such a spectral feature is consistent with soliton-emitted radiation as might be expected, considering that the soliton number is *N*
_{soliton}=6.6, i.e., the pulse propagation is in the soliton regime [26, 27]. Furthermore, this observation is consistent with pulse propagation in the anomalous GVD regime, a key requirement for soliton propagation. From the position of this peak we can extract an *estimated* value of *β*
_{3} according to the relation *β*
_{3}=3|*β*
_{2}|*T*
_{0}/*ω*_{r}
, where *ω*_{r}
is normalized angular frequency separation between the center frequency and the soliton feature [26, 27]. We then solve Eqs. (1) and (2) numerically while varying value of *β*
_{3} until the various features of the spectrum shown in Fig. 4(a), such as the peaks and dips are reproduced. This yields *β*
_{3}=-0.73±0.05 ps^{3}/m. Beyond this range of *β*
_{3}, the spectral features are no longer reproduced. Note that *β*
_{3} is the only free parameter used in our simulations. This method provides a direct determination of *β*
_{3}, which must otherwise be obtained via sequential differentiation of a fitted function to *n*
_{eff} (i.e., *β*
_{3}=d^{3}
*β*
_{0}/d*ω*
^{3}), a procedure prone to error since any initial error in the fitting of *n*
_{eff} will be significantly magnified after three differentiation stages, especially when the absolute value of *β*
_{3} is small. The nonlinear coefficient, Γ=(2.35×10^{4}+*i*4.94×10^{3}) pm^{2}V^{-2} [18], is calculated using the experimental value of the bulk Kerr coefficient *n*
_{2} and TPA absorption coefficient [28].

The normalized simulation results presented in Fig. 4(b) are for input pulses with either a hyperbolic secant or a Gaussian-pulse shape. The output spectrum obtained with the hyperbolic-secant pulse shape agrees with the experimental data in Fig. 4(a), particularly with regard to the output spectral shape, the spectral shift of the split peaks and the position of the soliton-emitted radiation peak at 1590 nm. The output spectrum corresponding to the Gaussian input contains less spectral features, but the imbalance between the left and right peaks is closer to the experimental result. This demonstrates the strong sensitivity of the output spectrum to input pulse shape. Hence, the discrepancy between the experiment and simulation can be attributed chiefly to the input pulse shape as can be seen in the low power data and simulation in Fig. 4 (blue curves). This discrepancy is also manifested in the optical limiting measurements of Fig. 1.

We can directly estimate the carrier effects by considering the steady state solution of Eq. (2). Assuming *t*_{c}
=0.5 ns, typical for Si-WWG [29], we obtain Δ*N*=4.1×10^{16} cm^{-3} for *P*
_{0}=200 mW. This yields *δn*
_{FC}/*n*=2.3×10^{-5} and the peak carrier-induced loss *α*
_{FC, max}=0.3 dB/cm, i.e., carrier effects are negligible. These maximum values correspond to the conditions just after the pulse enters into the waveguide. As the pulse propagates along the waveguide, its power decreases exponentially due to linear losses, and therefore the generated FC concentration will have an even smaller influence on the pulse propagation. Moreover, carrier accumulation can be neglected since the temporal separation of adjacent pulses is about 50 times longer than the carrier lifetime. These estimates show that for fs-pulse propagation in Si-WWG, TOD instead of FCA, is the main cause of the spectral asymmetry as observed in Fig. 2. In addition, by fixing all parameters and varying the input power, we reproduce the output-power response of the Si-WWG, shown in Fig 1. We thereby conclude that TPA is the cause of saturation observed in Fig. 1.

#### 3.4 Figure of Merit

For telecommunication applications, the nonlinear switching properties of devices may be described in terms of the figure of merit, *FOM*=*n*
_{2}/*βλ*, which is used to quantify the nonlinear phase shift achieved over an effective absorption-limited length [29, 30]. The *FOM* value is relatively independent of the dimensions of the waveguide. The operational value of *FOM* is device-specific, i.e., depending on the switching mechanism, the required nonlinear phase shift varies from π to 4π, which corresponds *FOM* values of 0.5 to 2. For bulk Si, the *FOM*=0.37, which suggests that silicon-wires may not be ideal for certain forms of nonlinear optical switching components. For the Si wire that we considered in this paper, *n*
_{2} and *β* are strongly enhanced by the waveguide confinement properties as shown in Section 3.2. Because the enhancement to *n*
_{2} and *β* are comparable, the FOM does not change significantly from their bulk value. However, the extremely low pulse energy required for Si wire devices is itself interesting and may justify utilizing the low *FOM* of value of silicon despite its associated performance penalty. We note also that since in the case XPM-based switches, the required *FOM* is decreased to half of the value corresponding to the SPM case, the limitation encountered here can be overcome [17].

## 4. Conclusion

We have presented an experimental and theoretical study of optical nonlinearity and dispersion of ultrafast pulses in Si wires possessing anomalous GVD under a regime where the waveguide, the characteristic dispersion and nonlinear lengths are comparable to each other. Thus, there is a complex interplay among all these effects, which ultimately determines the pulse dynamics. We have observed optical limiting due to TPA and spectral broadening and splitting arising from SPM. Our simulations show that TOD is the main factor that leads to the spectral asymmetry and that FC absorption and dispersion have minimal effect due to the very low pulse energy. Furthermore, we show the use of soliton-emitted radiation from Si wires to determine via its spectral position the waveguide TOD coefficient, *β*
_{3}=-0.73±0.05 ps^{3}/m. Because of all these factors, careful theoretical analysis is critical in the design of high transmission rate intra-chip optical communication systems based on silicon wires.

## Acknowledgments

This work was supported in part by the DoD STTR Contract No. FA9550-05-C-1954, and by the AFOSR Grant FA9550-05-1-0428. The IBM part of this work was supported by Grant No. N00014-07-C-0105 ONR/DARPA. We thank Julian Tauber for his help with the computational work and Professor B. J. Eggleton for useful critical comments on Si-wire devices.

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