1. Introduction

The physics of nonlinear propagation effects in novel optical waveguides and photonic structures is a subject that continues to attract intense interest [1, 2]. Much of this recent work has been motivated by experiments in photonic crystal fibers, where the combination of engineered dispersion and elevated nonlinearity has led to the generation of broadband supercontinuum (SC) spectra that have found many important applications [3, 4]. The spectral broadening associated with supercontinuum generation is now well-understood, and arises from the combination of effects such as soliton fission, stimulated Raman scattering and four wave mixing [5]. In addition to these processes arising from the intensity-dependent Kerr nonlinearity, there has also been growing interest in the dynamics of third harmonic generation (THG) that depends on the full field-dependent χ⁽³⁾ response. A number of experiments have now reported THG generated in various higher-order modes of photonic crystal fibers, and these results have been well explained in terms of calculated multimode phasematching conditions [6-10]. Recent work has also studied non-phasematched THG observed in sub-cm segments of highly nonlinear fiber [11].

An additional degree of freedom in nonlinear waveguide design has been demonstrated through fiber tapering to sub-micron dimensions, and studies of nonlinear propagation in such “photonic nanowires” have demonstrated impressive results such as low threshold SC generation and pulse compression approaching the single-cycle regime [12–15]. Pulse propagation in nanowires has also been the subject of theoretical interest, and numerical simulations have successfully reproduced the characteristics of octave-spanning SC spectra seen in experiments [16]. Our objective here is to further extend this study of spectral broadening processes in nanowires through a detailed numerical study of the single mode propagation dynamics under conditions where the pump pulse spectral broadening is accompanied by simultaneous THG. Specifically, we show that appreciable spectral broadening due to THG is observed even under non-phasematched conditions in sub-mm length nanowires, and a particular result that we obtain is that the superposition of the THG with the spectrally-broadened pump pulse exhibits significant dependence on the input pump carrier-envelope offset (CEO) phase.

2. Simulations and evolution dynamics

Our numerical simulations use a unidirectional and single-mode generalized nonlinear envelope equation (GNEE) model where carrier amplitude and phase properties are integrated onto a generalized pulse envelope [17]. This propagation model builds on important previous studies by Blow and Wood [18], Brabec and Krausz [19] and others [20–22], and the explicit inclusion of the full χ⁽³⁾ nonlinearity including THG allows the accurate simulation of effects such as carrier shock generation and carrier-envelope offset (CEO) phase dependent propagation. Direct comparison of GNEE simulations with Maxwell equation solvers has shown quantitative agreement in both temporal and spectral characteristics for bandwidths spanning many times the optical carrier frequency [17, 23].

We model pulse propagation in a pure fused silica nanowire of diameter 600 nm. This yields near-optimal nonlinear response at the 1060 nm pump wavelength considered, and the nanowire has anomalous dispersion over the range 470 nm–720 nm with normal dispersion otherwise [12]. The pump lies in the normal dispersion regime and the nonlinear coefficient and group velocity dispersion at this wavelength are γ=0.347 W^-1m^-1 and β₂=1.030 ps²m^-1. We assume hyperbolic secant input pulses. The simulations included stimulated Raman scattering, the global dispersion characteristics and the wavelength dependence of the nonlinearity [24, 25]. We consider a propagation length of 0.5 mm which allows pump and THG interaction signatures to be clearly identified whilst allowing loss over the wavelength range of interest (230–1600 nm) to be neglected. Note that manipulation of fiber waveguides on mm and sub-mm length scales is now a practical technology [11, 26]. Noise was included via a phenomenological one photon per mode spectral density with random phase [5] and multiple simulations using different random noise seeds were used to generate an ensemble of results from which the coherence and stability properties could be determined.

Figure 1 shows numerical results illustrating the characteristic pulse propagation dynamics where both pump spectral broadening and THG are observed. The input pulses had 25 fs duration (FWHM) and 150 kW peak power, and we initially assume zero CEO phase such that the input carrier field corresponds to cosinusoidal oscillations under the envelope. The results show (a) the spectral intensity and (b) the temporal electric field at the distances indicated. We see that the initial evolution is associated with independent pump spectral broadening due to self-phase modulation and the generation of a distinct spectral component at the expected THG frequency. The non-phasematched THG broadens symmetrically as a result of pump-induced cross-phase modulation (XPM) up to ∼ 25 μm, before developing a more complex structure because of the pump-THG walk-off [27]. Indeed, the top subfigure of Fig. 1(b) shows the clear separation between the pump and THG at the fiber output. Despite the complex nature of these dynamics, multiple simulations performed in the presence of noise confirm that the spectral characteristics exhibit a very high degree of shot-to-shot stability. This is shown in the top subfigure of Fig. 1(a) that plots the (near-unity) spectral coherence function calculated from an ensemble of 100 simulations.

 

Fig. 1. (a) Spectral and (b) temporal characteristics at the distances shown for 25 fs input pulses and initial CEO phase ϕ_CEO=0. Spectral plots show both frequency and wavelength axes, and temporal plots use a co-moving frame. Frequency is normalized relative to the pump ω₀. The detailed views show: (a) the spectral coherence and (b) an expanded view of the output field.

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The physical origin of the spectral structure is seen more clearly in Fig. 2 which presents spectrograms of the output field after 500 μm. To distinguish features arising from the intensity dependent refractive index and the THG, we show the two cases where THG is (a) neglected and (b) included in the simulations. The normal and anomalous dispersion regions are identified by the dashed lines, and the caption provides a link to an animation of Fig. 2(b). The results show how the transfer of energy from the normal into the anomalous dispersion regime leads to the formation of a soliton component (SOL), and a more detailed analysis shows that the pulse attains a soliton order N≈1 after a distance of around ∼100 μm. Significantly, the anomalous dispersion regime where the soliton forms is located between two regimes of normal dispersion, and thus the subsequent propagation dynamics can include both short-wavelength and long-wavelength dispersive waves, as well as soliton-dispersive wave interactions mediated by cross-phase modulation and Raman scattering [28–33].

From our simulation results, we see significant dispersive wave (DW) transfer towards short wavelengths, and a distinct DW component is generated at ω/ω₀≈3.7 after a propagation distance of ∼150 μm. The localization and structure of this DW arises from the interaction with the soliton pulse through cross-phase modulation and Raman scattering [33], and we have confirmed that simulations with and without the inclusion of Raman gain yield differences in the detailed DW structure. On the other hand, although the pump pulse in the normal dispersion regime undergoes significant temporal broadening, its structure is not significantly affected by dispersive wave transfer from the soliton back towards longer wavelengths. This has been confirmed by carrying out simulations without any higher-order dispersion terms such that the pump pulse evolves alone independently of any soliton or dispersive wave generation. Nonetheless, the generation of multiple dispersive waves may well influence the dynamics with different pump and fiber parameters and would be expected to further modify the observed spectral structure. In this context, we also note that the soliton pulse at around 500 nm does not stimulate any short-wavelength THG itself because such radiation lies well within the fused silica absorption bands and is attenuated. However, such soliton-induced THG may also be observable in a different parameter regime, possibly providing an additional mechanism for short wavelength SC extension [9].

Figure 2(b) shows the novel features associated with THG. The corresponding animation shows the initial growth of THG at 3ω₀ and its subsequent XPM-induced broadening. This latter effect has important implications for increasing the SC short wavelength extension [27]. Fig. 2(b) also shows the spectral overlap between the THG and the pump spectral tail and, in fact, the spectral interference between these components is sensitive to the CEO phase of the input pulses. This is shown in Fig. 2(c) where ∼8 dB difference between spectra generated with ϕ_CEO=0 (solid line) and ϕ_CEO=π/2 (dashed line) is seen at ω/ω₀≈2.8. This difference becomes less apparent for higher frequencies where the DW component appears more strongly. Such CEO-phase sensitivity in few cycle pulse propagation has been previously observed in harmonic generation and dispersionless pulse propagation [17, 34], and the results here demonstrate that it is also expected in SC generation. Significantly, multiple simulations using different random quantum noise seeds show that both spectra (generated with ϕ_CEO=0 and ϕ_CEO=π/2) remain essentially identical from shot-to-shot such that the averaged spectrum over multiple shots is visually indistinguishable from the single shot spectra shown in Fig. 2. This suggests that the detection of such CEO-phase sensitivity in the SC spectral structure should indeed be experimentally possible.

 

Fig. 2. Simulated spectrograms of fiber output for 25 fs input pulses: (a) neglecting and (b) including THG for an initial CEO phase ϕ_CEO=0. An animation for (b) is available (Movie 1.8 MB) [Media 1]. The gate function had duration 25 fs, and the plots show intensity on a logarithmic scale as shown. (c) Detail for initial CEO phase of ϕ_CEO=0 (solid) and ϕ_CEO=π/2 (dashed).

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We have carried out simulations to investigate these dynamics over a wider range of peak powers and pulse durations, and the results show that similar qualitative characteristics are observed for pulses in the range 10–150 fs and for peak powers in the range 100–150 kW. For this parameter regime, the intensity in the nanowire remains below the fused silica ionization threshold of ∼10¹⁴ Wcm^-2, and thus these parameters correspond to the regime where damage-free pulse propagation can be expected [35].

At a peak power of 150 kW, Fig. 3 shows spectrograms after 500 μm for the durations indicated. The initial CEO phase for all cases was ϕ_CEO=0 and all other parameters were as above. It is instructive to view the corresponding animations over the full propagation distance and links are given in the caption. The qualitative form of the spectrogram evolution is similar, and pump spectral broadening, soliton dynamics, dispersive wave generation and XPM modification of the THG are observed in all cases. The major quantitative difference is that the distance over which the localized soliton formation and dispersive wave radiation is observed increases with the pulse duration. An additional feature to note is the weak pulse duration dependence of the short wavelength extension. Although perhaps surprising, this can be understood when considering the physical mechanisms leading to the spectral broadening. Specifically, the blue edge of the spectrum is generated from pump-THG interaction through XPM and, although the XPM-induced frequency shift is larger for shorter pulses, it is mitigated by the shorter walk-off distance. Thus the short-wavelength extension of the SC spectrum is primarily determined by the initial peak power.

 

Fig. 3. Spectrograms of fiber output and evolution animations for pulse durations: (a) 15 fs (Movie 1.8 MB) [Media 2] (b) 50 fs (Movie 1.8 MB) [Media 3]; (d) 100 fs (Movie 1.9 MB) [Media 4]. Initial CEO phase ϕ_CEO=0 in all cases. The gate function has duration 25 fs in all cases and we use the same intensity scale as Fig. 2.

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For all these cases, additional simulations in the presence of noise have confirmed that the output SC retains spectral coherence as shown in Fig. 1. For this range of pulse durations, we have therefore also studied the CEO phase dependence of the spectral structure, examining in particular the variation in spectral structure as a function of propagation distance for different values initial CEO phase. In fact, a strong dependence on propagation distance would be expected because the degree of overlap between the THG radiation and the components generated by the broadening pump varies significantly with propagation distance.

Typical results are shown in Fig. 4 for pulse durations of (a) 25 fs and (b) 100 fs. The left-hand figures in each case show the spectral evolution by plotting spectral slices at selected propagation distances as indicated for the particular value of CEO phase ϕ_CEO=0. These results complement the spectrogram animations shown in Fig. 4 in highlighting the variation in the detailed propagation dynamics with pulse duration. In addition, for selected propagation distances as indicated, the right-hand figures also show the detailed spectral structure in the overlap region between the broadened pump and the THG. This allows us to examine the difference in spectral amplitude for two values of CEO phase. As would be expected, the CEO phase dependence of the spectral amplitude is strongly dependent on propagation distance, only becoming significant when the broadening pump and THG components are superposed. Because initial pump spectral broadening (due to self phase modulation for example) is greater for shorter input pulses, we see CEO-phase dependent spectral amplitude at shorter propagation distances for the 25 fs pulse compared to the 100 fs pulse.

 

Fig. 4. Evolution of spectral characteristics for (a) 25 fs and (b) 100 fs input pulses at the distances shown for initial CEO phase ϕ_CEO=0. The subplots on the right illustrate the differences in spectral characteristics at selected distances for an initial CEO phase of ϕ_CEO=0 (solid line) and ϕ_CEO=π/2 (dashed line).

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A potential application of CEO phase dependent spectral broadening is in the stabilization of ultrashort pulse oscillator phase using spectral amplitude variations as a convenient feedback signal. Indeed, the use of spectral amplitude stabilization based on SC generation could represent a considerable simplification when compared to f to 2f interference techniques. In this context, however, the results above indicate that obtaining the largest variation in CEO-phase dependent spectral structure would require careful matching of the waveguide length and pulse duration, and Fig. 5 shows results of additional simulations where we study this in more detail.

 

Fig. 5. For the pulse durations and propagation distances indicated, the figure shows spectra obtained for initial CEO phase of ϕ_CEO=0 (solid line) and ϕ_CEO=π/2 (dashed line).

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Specifically, the figure shows a series of spectra for two values of initial CEO phase at particular propagation distances selected such that the integrated spectral difference for orthogonal spectral phases is maximized over the normalized frequency range 2.5–3.5 ω/ω₀. At these distances we observe CEO phase-dependent spectral variations exceeding 20 dB even for the longest pulse duration of 100 fs. From these results, we can establish practical guidelines to estimate the adequate nanowire length in order to induce detectable CEO spectral dependence. On the one hand, the length should allow for the pump to expand down to the high-frequency normal dispersion region so as to overlap with the THG spectrum and thus produce CEO-dependent spectral interference, yet on the other hand the distance should be limited to avoid excessive generation of a localized DW spectral component [32] that would reduce the signal to noise ratio in detecting any CEO phase dependent structure.

4. Discussion and outlook

The dynamics of SC generation in highly nonlinear nanowires in the presence of THG clearly leads to several novel features, and there are several conclusions to be drawn from these results. Firstly, assuming short propagation lengths and material transparency, the generation of SC bandwidths spanning 230–1600 nm should be possible at nanojoule pulse energies. This may well have important applications for generating extended short wavelength SC spectra. Secondly, both dispersive wave radiation and THG have both been shown to contribute to the short wavelength spectral structure, and thus quantitative comparison between theory and experiment for blue-extended SC spectra will require the use of numerical modelling based on the full χ⁽³⁾ susceptibility and not only the intensity-dependent refractive index. Finally, for SC generation in the regime where THG effects are important, our results have shown a significant dependence of the spectral intensity on the initial CEO phase of input pulses. Such a CEO phase dependence should be easily experimentally detectable with appropriate matching of input pulse duration and waveguide length, and could have impact in the development of compact devices for frequency comb stabilization. Finally we note that, although we have considered such effects only in the case of non phasematched harmonic generation, our results may stimulate further studies into possible CEO dependent propagation effects in multimode phasematched THG.