Free-carrier-induced soliton fission unveiled by in situ measurements in nanophotonic waveguides

Solitons are localized waves formed by a balance of focusing and defocusing effects. These nonlinear waves exist in diverse forms of matter yet exhibit similar properties including stability, periodic recurrence and particle-like trajectories. One important property is soliton fission, a process by which an energetic higher-order soliton breaks apart due to dispersive or nonlinear perturbations. Here we demonstrate through both experiment and theory that nonlinear photocarrier generation can induce soliton fission. Using near-field measurements, we directly observe the nonlinear spatial and temporal evolution of optical pulses in situ in a nanophotonic semiconductor waveguide. We develop an analytic formalism describing the free-carrier dispersion (FCD) perturbation and show the experiment exceeds the minimum threshold by an order of magnitude. We confirm these observations with a numerical nonlinear Schrödinger equation model. These results provide a fundamental explanation and physical scaling of optical pulse evolution in free-carrier media and could enable improved supercontinuum sources in gas based and integrated semiconductor waveguides.

Here we show the magnitudes of κ (2) F C (Eqn. 23), κ F C (Eqn. 7) and the soliton number N versus peak power P o . The markers indicate the minimum threshold from analytic modeling for the two effects.

Supplementary
Supplementary Note 1: Sample parameters Figure 1 summarises the properties of the GaInP photonic crystal waveguide (PhCWG) used in the experiment. Figure 1(a) is an electron micrograph showing the air-holes in the air-clad dielectric slab with a line-defect that forms the waveguide section. The lattice period is a = 471 nm, with slab thickness 190 nm, and bulk hole radius r = 0.23a. The dispersion is engineered following our method outlined in Ref. [1]. Specifically, we tune the width of the defect to W = 0.95 √ 3a, the innermost hole radii to r 1 = 0.20a and the second row radii to r 2 = 0.22a. The innermost holes are also shifted outwards by s 1 = 0.16 √ 3/2a. A twist parameter of t = 0.1a yields a flat group index band. The GaInP semiconductor has a refractive index of n 0 =3.17 and a band-gap of E g = 1.9 eV. Regarding nonlinear properties, the leading effect is a χ (3) electronic nonlinearity with a Kerr coefficient n 2 (intensity-dependent refractive index). At our working wavelength of 1553 nm (∼0.8 eV) and the intensities in our experiment, we also observe three-photon absorption (3PA, α 3 ) in the wide-gap material [2].
show the dispersion properties of the fabricated device we measured using a homebuilt setup similar to that of Ref. [3]. Figure 1(b) is a plot of the group index n g versus wavelength λ. Figure 1(c) shows the group-velocity dispersion (GVD, β 2 ) and third-order dispersion (TOD, β 3 ) coefficients of the device. Recall β n = d n β(ω)/dω n is the n th -order dispersion of the propagation constant β(ω) at frequency ω. The GVD is anomalous, as required for sustaining solitons, and on the order of ∼ps 2 /mm across the range of interest. The TOD term is small in the experiments, but included for completeness. Table I shows a summary of the experimental parameters described in the main text.

Supplementary Note 2: Spectral properties of the pulse transmission: experiment and model
Since soliton fission is fundamentally a time-domain phenomenon, we focused mostly on the experimentally measured temporal cross-correlations made with a brief discussion of spectral measurements. In this section we show the full measurement set of the spectral density of pulses transmitted through the PhCWG. Figure 2 shows the results of measurements (solid lines) made with an optical spectrum analyser. The vertical direction indicates increasing coupled peak power.
The separation of the smaller second soliton is clearly visible as we increase the coupled power.
As in the main text, we include the generalised nonlinear Schrödinger equation (GNLSE) model (dashed lines) with the solid agreement confirming the experimental results.

Supplementary Note 3: Generalised Nonlinear Schrödinger Equation Model (GNLSE)
The nonlinear pulse propagation in the GaInP semiconductor waveguide can be described by a GNLSE based on a similar approach as in Refs. [4,5]. The following is the model we use throughout the main text. In addition to the dominant terms Kerr, GVD, and free-carrier dispersion (FCD, n F C ), we included higher-order terms in the full model in order to obtain a more accurate quantitative agreement between the GNLSE and the experimental data. These higher-order terms are: linear loss α, TOD, fourth-order dispersion (FOD, β 4 ), and self-steepening (SS, τ N L ).
The inclusion of the higher-order terms also serves a didactic purpose here in the Supplementary Information as these perturbations are known in the literature and serve as a reference point to understand our current system. We emphasise the basic physics could be explained only with the soliton and FCD terms as shown by Figs. 2(e), 2(f) and 3(d) in the main text. The full model is: On the left-hand side of the equation are the GVD and Kerr terms which compose the soliton while the right-hand side describes perturbations to the system. Here A(z, t) is the slowly varying amplitude of the pulse, with time t in the frame of the pulse. The input pulse peak power is P 0 = |A(0, 0)| 2 and k 0 is the vacuum wavenumber. The nonlinear self-phase modulation and 3PA terms are γ = (k 0 n 2 /A 3ef f ) and α T hP A = (α 3 /A 2 5ef f ), respectively, with the bulk coefficients for n 2 [6] and α 3 [7]. The cross-sectional mode areas for the χ (3) and χ (5) nonlinear effects are A 3ef f and A 5ef f , respectively [8].
The FCD coefficient is n F C with the free-carrier absorption cross-section (FCA, σ). Note that n F C is a material parameter dependent on fundamental quantities and has a negative sign: , where e is the elementary charge, 0 is the vacuum permittivity, ω is the angular frequency and m * e/h is the effective mass of the electrons/holes in the semiconductor [9]. The free-carrier density N c (z, t) is controlled by the generation and recombination of the 3PA-induced free carriers according to the following rate equation: where we have defined the power-normalised carrier generation rate ρ , with the photon energyhω 0 , and the effective free-carrier lifetime τ c .
The self-steepening term τ N L is related to the dispersion of γ(ω). Given the pulse duration T F W HM =2.2 ps in our experiments, it is surprising that we include this term at all as it is typically reserved for ultrafast pulses of tens of femtoseconds [10]. Nonetheless, it has recently been shown that the slow-light contribution to γ(ω) requires a new definition of τ N L and can add orders-of-magnitude to its strength [11]. In brief, Notice the traditional 1 ω dependence is not included since it is around one hundred times smaller. The Raman effect is completely negligible in our system due to both its narrowband (≈100 GHz) and far-detuned frequency peak in semiconductor systems (10-15 THz) [12,13].
As we conduct our experiments in a photonic crystal waveguide, we account for the reduced group velocity (increased group index n g ) and the associated slow-light enhancements to the path length and electric field by the slow down factor S = v p /v g = n g /n 0 [14]. The GNLSE describes our experiment when we replace the bulk parameters with the appropriate scaled values. For the free-carrier terms, n F C,ef f = S · n F C , σ ef f = S · σ, and for the nonlinear terms γ ef f = S 2 · γ [14] and α T hP A,ef f = S 3 · α T hP A [8]. This form is also applicable to channel waveguides with the appropriate S value.

Supplementary Note 4: Derivation of the free-carrier perturbation: three-photon absorption
It is common to non-dimensionalise the GNLSE (Eqn. 1) to analyse the relative strengths of physical effects governing the pulse evolution [15]. In the main text we introduced a new term κ (3) F C to describe the FCD perturbation using this procedure. With this non-dimensional parameter we analysed the strength of the perturbation in our experiments and found it to be an order-ofmagnitude above the minimum threshold. In this section, we show the derivation of the κ In our formulation we ignore the loss terms (α, α 3 , and σ) and FOD for clarity.
As we noted in the main text, these results are general to optical systems with nonlinear photo-carrier (photo-electron) generation including tunneling and multi-photon ionization regimes.
Applying the specific carrier generation mechanism to the formalism that follows is critical as the physical parameters governing the κ perturbation will scale differently.
We substitute the following dimensionless variables into (Eqn. 1): where ξ, τ , and U correspond to normalised propagation distance, time, and pulse envelope compared to the dispersion length L D , pulse duration T 0 , and peak power P 0 , respectively. This gives: In arriving at this result, we have used the well-known definitions for the dispersion lengths L D and L D , the nonlinear length L N L and the soliton number N : The non-dimensional parameters on the right-hand side govern the conditions to trigger soliton fission, amongst other effects [16,17]. The parameters related to TOD (δ 3 = β 3 6|β 2 |To ) [16] and SS (s= τ N L To ) [10] are well known. A higher-order soliton will break apart when the magnitude of any of these parameters exceeds a minimum threshold. Conversely, when the parameter is below the threshold, the higher-order soliton remains intact and recurrent behaviour is retained. The numerical value of the minimum threshold is different for each physical effect. Separate to this point, the minimum threshold to break a soliton decreases with increasing soliton number N . We will treat this topic in further detail below.
We now focus on the role of free carriers for which we have introduced the FCD perturbation with the characteristic FCD-3PA length: We have defined the free-carrier density generated from intrapulse 3PA with a peak amplitude In order to derive the L F CD we solved the free-carrier generation equation (Eqn. 2) assuming negligible intrapulse recombination, along with the condition n F C < 0 known from the Drude model [9]. The solution is N c (ξ, τ ) = ρ where the integral can be solved for exact pulse shapes (i.e. hyperbolic secant or Gaussian).

Note κ (3)
F C depends on both the power and temporal duration of the input optical pulse, with the material contributing via constants. The FCD length is proportional to peak power cubed due to 3PA carrier generation. The pulse duration dependence comes from the accumulation of free carriers across the pulse, as represented by the integral in the carrier equation. Once again, different nonlinear carrier generation mechanisms will have different scaling. In contrast, the characteristic scale of the Raman effect depends only on the native material response τ R [18]. More generally, the definition of L F CD depends on the specific carrier-generation mechanism for N c . For example, a related derivation was shown for materials limited by two-photon absorption (TPA, α 2 ) in [19] though the normalised perturbation was not analysed in that case. We show more details on this below due to the relevance in silicon waveguides. Now that we have our definitions in place, we treat these effects quantitatively to show that FCD is the dominant mechanism in our experiments.

Supplementary Note 5:
Analytic estimate of the minimum free-carrier perturbation strength to cause soliton fission In the main text we presented both analytical and numerical methods to estimate a quantitative threshold κ (3) F C,min for FCD to cause soliton fission. Here we show the analytic formalism and compare this with known parameters.
Using the method of moments, we can estimate the threshold for soliton fission to occur. This analysis is based on the general evolution equations described in Ref. [20]. The starting point is to consider the equations for the blueshift Ω and the temporal shift T c due to FCD driven by 3PA: with the pulse energy E and our other previously defined variables. The field envelope A(z, t) is such that the peak power is P = |A| 2 . Neglecting recombination in the carrier generation equation (Eqn. 2) and inserting into Eqn. 9, we get the blueshift evolution: This equation will change depending on the specific carrier generation mechanism as noted earlier. We now use the hyperbolic secant pulse: where τ is the pulse duration and C is the chirp. The blueshift evolution follows: Noting that for a hyperbolic secant pulse with peak power P the energy is E = 2P τ , and that the chirp C is zero for solitons, we integrate over z assuming constant power to get: The assumption of constant power is taken as a first approximation. Indeed, in our earlier work with the moments method we examined specific cases where one can solve for Ω and T c analytically when power varies with distance due to, for example, dispersion, linear loss, or nonlinear loss. Critically, we found the input power P o remains the key parameter and that any variation in the power along the waveguide becomes a modest scale factor without loss of generality to the main physics.
We can now evaluate the temporal shift due to the FCD blueshift. Substituting our result for Ω into Eqn. 10 and integrating in z assuming constant power gives: We can use this expression for T c to estimate the threshold for FCD to cause soliton fission.
In order to proceed, we must first recall theory describing the amplitudes of the constituent solitons composing a more energetic higher-order soliton. The peak power P corresponds to a single constituent soliton fissioning out of the bound state [21]. We define the relative power η = P/P 0 , where P 0 is the input peak power. For a N = 2 soliton, the constituent fundamental solitons have relative powers η = 9/4 and η = 1/4 [22]. For higher-order solitons, one must consult these relations for the appropriate values. Considering a single constituent P = η P o : We normalise the temporal shift to the input pulse duration T 0 and use the definitions for L (3) F CD and L D above to get: At this point we employ the characteristic scales of solitons for length (z o ) and pulse duration (T o ) as in the main text. Namely, the shift must occur within a soliton period z 0 = (π/2)L D . We also recall κ F CD and the soliton relation L D = N 2 L N L to simplify to: Additionally, we set the fission threshold such that after one soliton period the main pulse has advanced by T 0 , i.e. T c /T 0 = 1. Inverting this relation we get the main result: Importantly, κ F C,min (analytic) ∝ 1 N 2 , which explains why solitons of higher order N are easier to break apart.
Considering the highest power constituent of a N = 2 soliton has an amplitude of η = 9/4, we get the quantitative threshold for the N =2 soliton: F C,min (analytic) > 0.039.
At the maximum power in our experiments we had κ F C,max =0.35, which is an order-of-magnitude above the minimum threshold κ F C,min predicted by both analytic theory and simulation. This value is about 50% larger than the value obtained from numerical simulations κ F C,min (GNLSE) = 0.029. The soliton recoil from momentum conservation probably accounts for most of this deviation [21]. This is expected since our analytic formalism treats the constituents independently and neglects soliton interactions. Indeed, for a fission of a N = 2 soliton caused by the Raman effect, the discrepancy between the threshold predicted by analytic theory and simulations was about 2.5, however after a much longer propagation distance of ≈6z 0 . The smaller soliton has amplitude η=1/4 and its temporal shift is negligible.

Supplementary Note 6: Comparison with known perturbation mechanisms
As mentioned above, the TOD perturbation δ 3 is well known. It has been shown numerically that a value of δ 3,min = 0.022 can trigger fission for a soliton of order N = 2 [16]. For our parameters δ 3 = 0.016, which is below the threshold. Figure 3(c) of the main text shows the higher-order soliton under the influence of TOD only. The N =2 soliton evolves periodically with no visible temporal separation. In contrast, Fig. 3(d) of the main text shows clear temporal separation of the solitons under the influence of FCD, highlighting that TOD is much weaker than FCD for our experimental conditions.
For SS, we use the formalism in Ref. [11] which includes contributions from the waveguide dispersion, slow light, and modulation in the mode area. Since we could not find a minimum threshold s min in the literature, we conducted simulations and found the threshold to be s min =0.17.
Using the GNLSE we found our experimental value of s = τ N L T 0 =0.09 (including the contributions from dispersion and area) is not sufficient to trigger fission and gives qualitatively similar results to Fig. 3(c) in the main text. This is not surprising as it is a factor of two smaller than the threshold.
In summary, the FCD perturbation κ (3) F C in our experiments was ten times larger than the minimum required threshold κ (3) F C,min and is moreover the strongest perturbation to the higherorder soliton propagation. We thus conclude the FCD is the dominant fission mechanism in our experiments. Supplementary Note 7: Analytic treatment of fission from free carriers generated by TPA Given the strong interest in the optical properties of silicon (E g =1.1 eV) at 1550 nm, we also quote the equivalent threshold for soliton fission caused by free carriers generated from two-photon absorption. Following a similar procedure as above we find: with the appropriate FCD length from TPA as in Ref. [19]: Once again we define the power-normalised free-carrier generation rate ρ F C = α 2 /(2hω 0 A 2 3ef f ). Note that the absolute magnitude of κ Using the moments method treatment above and the results for Ω and T c in Ref. [20] we find the minimum threshold: This value is about a factor of two larger than the 3PA parameter. Importantly, the nondimensional threshold still scales as 1 N 2 . Figure 3 shows a comparison of κ (2) F C and κ F C for identical waveguide parameters used in this study. The TPA coefficient for silicon at 1550 nm is taken as its experimentally measured value of α 2 = 1 cm/GW [23]. Note the TPA perturbation κ (2) F C grows both linearly in power and much more rapidly than the 3PA perturbation. We also highlight the minimum thresholds with markers.
The TPA material clearly exceeds the threshold at a peak power that is smaller by an order-ofmagnitude, highlighting the stronger relative impact of free carriers in the TPA system.
In the main text, we highlighted the importance of the FCD fission mechanism in explaining recent supercontinuum (SC) generation experiments in silicon [24]. Recall that it has been shown that the larger the soliton order N , the smaller the required perturbation to cause soliton fission [16]. In. Ref. [24], they have N =19, and κ (2) F C,exp ≈ 2.4, which exceeds the N =2 threshold by more than a factor of thirty, and at least two orders of magnitude larger than the fission threshold for a N =19 soliton. It is clear that FCD played a significant role in that experiment. Though not measured, it is likely the fissioned solitons would have advanced in time in the semiconductor medium, and the broadening on the blue side attributed to blue-shifting solitons. We suggest all past, present, and future supercontinuum demonstrations in semiconductor waveguides and ionized gases consider the role of FCD in order to create improved on-chip integrated SC sources with broader and brighter spectra.