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We numerically study the characteristics of optical rogue waves in the femtosecond supercontinuum (SC) generation and use the CW triggering mechanism to control the SC generation. Detailed simulation results show for the first time that a weak CW trigger can manipulate the behaviors of optical rogue waves in the femtosecond SC regime. For the proposed CW triggering technique which requires only wavelength tuning and is a handy approach for the active control of SC, the resultant spectrum can be greatly broadened, and the noise properties of the SC can be significantly improved in terms of both of the coherence and intensity stability.
© 2015 Optical Society of America
1. Introduction
Supercontinuum generation (SC) in photonic crystal fiber (PCF) provides an ultra-broadband high brightness spectrum [1] which makes it a ubiquitous “white light” source in diverse fields such as optical coherence tomography [2], frequency metrology [3], and fluorescence lifetime imaging [4]. The stability of light sources is a vital factor and SC noise properties remain as a subject of intense research recently. Solli et al. have shown that statistically rare “rogue” soliton events can be observed in the noise-driven SC generation with a picosecond pulse pump [5]. Subsequent research continued focusing on long-duration pulse pump (picosecond, nanosecond or even continuous wave) regime where modulation instability (MI) plays a key role in the spectrum broadening. The MI sidebands seeded from noise have been shown to underlie the initial SC and the subsequent SC is unstable [6]. Active control of rogue waves for stimulated SC generation was reported in [7] where a 200 fs pulse seed with 0.01% of the pump intensity can produce a SC with greatly enhanced coherence property. To harness the SC stability in the long-duration pulse pump regime, several effective control methods have also been proposed including a coherent pulse seed [8], a sliding frequency filter [9], the CW triggering mechanism [10, 11] or a solid-core photonic bandgap fiber [12].
The initial work on optical rogue waves was performed in the long-duration pulse pump including picosecond, nanosecond or even CW regime. However, in 2009, M. Erkintalo et al. demonstrated both numerically and experimentally that optical rogue waves can also been observed in the femtosecond SC generation where clear signature of soliton-related dynamics is observed [13]. A weak MI perturbation is only apparent in the initial propagation stage and even such a small perturbation has dramatic consequences on the generation of extreme amplitude wave [14]. For the femtosecond SC generation, there is a number of studies considering the SC coherence property [15]. give the comparison of SC coherence property with different input pulse durations in the femtosecond regime and different input powers. By changing the pump pulse length [16], the input wavelength [17], modulating the pump pulse shape [18], or adding a feedback [19], the phase stability and hence the SC coherence can be influenced to some degree [19]. numerically investigated the influence of feedback on SC generation in a microstructured fiber in the femtosecond regime, and the impact of delayed optical feedback on the SC noise properties is also investigated numerically and experimentally [20].
To the best of our knowledge [21], previous studies focus on the control of optical rogue waves for optical fiber SC generation in the picosecond regime [18, 21, 22], and no method has been applied to manipulate the optical rogue waves in the femtosecond SC generation. In this paper, we study the characteristics of optical rogue waves in the femtosecond SC generation using the generalized nonlinear Schrödinger equation (GNLSE). The statistical distribution of the time series which corresponds to the long-wavelength edge of the SC exhibits the L-shaped signature of rogue events. A weak CW trigger is used to control the behavior of the optical rogue waves in the femtosecond regime. One benefit of our CW-triggering technique is that it only requires wavelength tuning and does not rely on the time-delay tuning as required for picosecond seed pulses [11, 22]. Extensive simulation results show that adding a weak CW trigger can enhance not only the bandwidth but also the stability of SC. The stability is analyzed in terms of both of the coherence and intensity stability. By simply changing the position of the CW trigger, we can study the effect of CW trigger on SC generation in detail.
2. Numerical model
The GNLSE is used to model the SC generation [1]:
where is the field envelop, βk and are dispersion and nonlinear coefficients . The second and higher-order dispersion coefficients (truncated at β10) at the pump wavelength are respectively: 6.8264 × 10−2 ps2/km, 6.7868 × 10−2 ps3/km, −9.8974 × 10−7 ps4/km, 2.4921 × 10−7 ps5/km, −7.7253 × 10−10 ps6/km, 3.4112 × 10−12 ps7/km, −1.5464 × 10−14 ps8/km, 4.7042 × 10−17 ps9/km, −6.5618 × 10−20 ps10/km. The nonlinear coefficient γ = 11 W−1/km. where is pump frequency. The Raman response function in defined as where is 0.18 and is determined from the experimental fused silica Raman cross-section [1]. We consider the propagation of a chirp-free hyperbolic secant pulse with 200 fs full width at half maximum (FWHM), 15 kW peak power in a 4 m PCF with zero dispersion wavelength (ZDW) at 1029 nm. The pump wavelength is 1028 nm which is slightly in the normal dispersion regime. Input pulse noise is modeled semiclassically by adding one photon per mode noise with random phase on each spectral discretization bin [9]. An ideal CW trigger () is included propagating with the pump where is the coefficient of the CW trigger intensity compared with the pump power and is assumed to be 10−6. is the normalized frequency of the CW trigger, and the position of the CW trigger can be easily adjusted by changing .3. Numerical results
We first study the effect of a minute CW trigger on the femtosecond SC generation. The CW trigger wavelength is selected to be 1130 nm which is of the best performance for the improvement of SC coherence property in the wavelength range of 1030 and 1210 nm. Figure 1 shows the comparison between the untriggered and CW triggered SC. Figure 1(a) and 1(e) explicitly plot the output spectra of the untriggered and CW triggered SC. Gray lines represent an ensemble of 1000 simulations with different initial random noise seed (for clarity only 100 simulations are plotted) and the black line represents the average spectrum. The bandwidth of the untriggered SC in Fig. 1(a) spans 846 nm which is from 707 nm to 1553 nm at −50 dB level. With the use of a CW trigger, the bandwidth in Fig. 1(e) extends to 981.1 nm (from 669.9 nm to 1651 nm) at −50 dB level, which is about 135 nm broader than that in the untriggered case. The expanded view of the long wavelength above 1450 nm which is given in Fig. 1(b) clearly shows a small number of rogue wave events with increased red frequency shift. Compared to significant shot-to-shot fluctuations in the Fig. 1(b), the Raman solitons in Fig. 1(f), which is the expanded view of Fig. 1(e) for the long wavelength above 1450 nm, appear in a more deterministic manner. Figure 1(c) and 1(g) show the first-order temporal coherence |g12(λ)| of the untriggered and CW triggered SC to characterize the phase fluctuations over SC. |g12(λ)| is a value in the interval [0,1] and 1 means perfect coherence [16]. In order to quantify the overall coherence across SC spectrum, it is convenient to introduce a spectrally averaged overall coherence [1]. The coherence property of the CW-triggered SC in Fig. 1(g) is greatly improved, while only a narrow band near the pump wavelength in Fig. 1(c) has a relatively good coherence. The overall coherence of the untriggered SC in the 600-1650 nm wavelength range is only 0.040, while the triggered SC coherence is improved in Fig. 1(g) and the overall coherence has increased to 0.248. Figure 1(d) and 1(h) show the histograms of the peak power probability distribution of the untriggered and triggered SC. In order to isolate the shot to shot fluctuation in the long wavelength edge of SC, a long pass filter is used to select the spectrum components above certain wavelength. A long-pass filter at 1430 nm is used here. A time series can be extracted from the inverse Fourier Transforms of the selected spectrum components. Figure 1(d) shows a typical long tail characteristic of extreme-value events which indicate optical rogue waves exist in the femtosecond regime. Here only the pulses with the peak power larger than 30 W and the temporal separation larger than 1 ps are counted. After adding the CW trigger, the time series exhibits a better intensity stability and the histogram shows that the peak power are clustering in a narrow range (4350 W- 4500 W) as given in Fig. 1(h).
Fig. 1 Comparisons between (a, b, c, d) untriggered SC and (e, f, g, h) CW triggered SC where the CW trigger is at 1130 nm. Spectra of (a) untriggered and (e) CW triggered SC where the individual spectrum from an ensemble of 1000 simulations is shown as the gray line (for clarity only 100 simulations are plotted) and the calculated average spectrum is shown as the black line; (b) and (f) expanded view of (a) and (e) above 1450 nm; The first-order temporal coherence of (c) untriggered and (g) CW triggered SC; Histogram of the peak power probability distribution of (d) untriggered and (h) CW triggered SC after a long-pass filter (>1430 nm) using 180 W bins.
Figure 2(a) and 2(b) further illustrate the temporal pulses after the long-pass filter at 1430 nm. For the untriggered SC in Fig. 2(a), pulses with different intensity randomly scatter from 22 ps to 39 ps. Where CW trigger wavelength is at 1130 nm, the ensemble of pulses is of a far more uniformed intensity and scatter within 3 ps (from 56 to 59 ps). For clarity, only 50 simulations are plotted. The standard deviation of intensities in Fig. 2(b) is as low as 0.002 P0 while the standard deviation in Fig. 2(a) is 0.106 P0. The intensity stability can be quantified by the signal-to-noise ratio (SNR) which is defined as the ratio of the mean to the standard deviation [8]. Figure 2(c) and 2(d) show the comparison of SNR for both the untriggered and CW triggered SC generation. Compared to a low flat SNR across the SC spectrum in the untriggered case in Fig. 2(c), the CW trigger at 1130 nm pull the SNR high in long-wavelength range around 1570 nm dramatically (Fig. 2(d)). In Fig. 2(d), the short-wavelength range of spectrum around 790 nm also shows a better intensity stability, which is conincide with the result in Fig. 1(g). Similar to the overall coherence, we also define the overall SNR as given in [8] to quantify overall intensity stability of an SC from 600nm to 1650 nm. The overall SNR of the CW triggered SC is 3.472, while the overall SNR is only 0.959 for the untriggeed SC.
Fig. 2 Comparisons between (a, c) untriggered SC and (b, d) CW triggered SC where the CW wavelength is at 1130 nm. The temporal pulses corresponding to the long-wavelength edge of (a) untriggered and (b) CW triggered SC (for clarity, only 50 simulations are plotted); The SNR of (c) untriggered SC and (d) CW triggered SC.
Figure 3 presents the spectrum evolution of untriggered and CW triggered SC. The parameters are as same as Fig. 1. In order to isolate the effect of noise, initial random noise seed is fixed. Although the pump lies in the slightly normal dispersion regime, soliton dynamics process started to play an important role after the spectrum was broaden across the ZDW and expanded in the anomalous regime. For given parameters, the calculated characteristic soliton fission length (~LD / N, 1.069 m) is comparable to the calculated MI length (~16 LNL, 0.097 m). Figures 3(a) and 3(b) show the spectral evolution during the initial stage of propagation (1 m) for both the untriggered and 1130 nm CW triggered SC. Figures 3(c) and 3(d) plot spectra at selected distance of 0.025 m, 0.05 m, 0.075 m, 0.1 m, 0.5 m, 1 m (from bottom to top). As shown in Fig. 3(c), the spectrum broadening occurs in the initial stage and is almost symmetric. When the spectrum broadens to 1130 nm at about 0.05 m, MI starts to play a role. MI perturbation is apparent at about 0.075 m, but its amplitude is quite lower compared with pump spectrum component. Even such a weak MI perturbation has dramatic consequences on the rogue events that undergo the extreme frequency shifts. When the CW trigger is placed at 1130 nm, a clear soliton is ejected at around 0.6 m and experiences continuous shift to long wavelength due to Raman self-frequency shift. Meanwhile, the dispersive wave on the blue-wavelength side trapped by soliton becomes apparent and broadens its spectrum. Since the initial MI process is seeded from the CW trigger rather than the random noise, the soliton appears in a similar manner even if the initial random noise seed is changed. Consequently, the long wavelength range of the triggered SC has a better intensity stability.
Fig. 3 False color representation of (a) untriggered SC and (b) CW triggered SC spectral evolution during the initial stage of propagation. The CW trigger wavelength is at 1130 nm. Initial random noise seed is fixed to isolate the effect of noise. (c) Untriggered SC and (d) CW triggered SC spectrum at selected distances of 0.025 m, 0.05 m, 0.075 m, 0.1 m, 0.5 m, 1 m (from bottom to top).
By changing the position of the CW triggers, we carefully investigate the influence of CW triggers on SC characteristics. The CW triggers are placed in the anomalous dispersion regime and their wavelengths change every 20 nm from 1030 nm to 1210 nm. We stopped at 1210 nm since the CW triggering effect is not obvious when the CW trigger is above 1210 nm. Here, 200 simulations were carried out for each trigger positon. The output spectra and coherence property are presented in Figs. 4(a) and 4(b) respectively, where the result of 1130 nm is shown as the black line and results of rest wavelengths are shown as blue lines. In Fig. 4(a), the bandwidth of the spectrum is broadened and achieves its maximum value 981.1 nm when the CW trigger is at 1130 nm. Such enhancement in the bandwidth is caused by the Raman soliton in the long-wavelength and the dispersive-waves blue-shifts induced [13]. Figure 4(b) shows the dependence of coherence on CW trigger wavelength. For 1130 nm CW trigger, not only the wavelength around the pump but also two sidebands in the short-wavelength range has a high coherence. The coherence degradation occurs when the trigger is deviated from 1130 nm. Figures 4(c) and 4(d) are the overall coherence and SNR versus CW trigger position. The overall coherence is almost at the same level as the untriggered case when the CW trigger is below 1070 nm,and the value rapidly increases after 1070 nm and achieves its maximum value (0.2478) when the CW trigger is at 1130 nm. After 1130 nm, the overall coherence drops rapidly. The CW triggers with wavelength with 1 nm interval from 1125 nm to 1135 nm have been investigated specifically and the results are shown in the inset of Fig. 4(c). The results confirmed that 1130 nm trigger is the optimal result in the anomalous dispersion regime. The overall SNR, which is the measurement of the intensity stability of SC, achieved its maximum value 2.978 at 1134 nm (plotted in the inset). The 1130 nm trigger also has an obvious effect on SNR. The value of SNR at 1130 nm is 2.613 compared to 0.097 in the untriggered case. The triggering effect is not obvious if the CW trigger is too close (below 1070 nm) or too far (above 1150 nm) from the pump since the CW trigger experiences high MI gain when the wavelength is among the range from 1070 nm to 1130 nm.
Fig. 4 (a) Spectra and (b) coherence for CW triggers at different wavelengths where the specific CW wavelength at 1130 nm is plotted in black and the CW trigger wavelengths change every 20 nm in anomalous dispersion regime from 1030 nm to 1210 nm (from bottom to top); (c) Overall coherence and (d) overall SNR versus CW trigger wavelength where the specific CW wavelength at 1130 nm is marked by the cross symbol and specific view of CW triggers wavelengths from 1125 nm to 1135 nm (1 nm as interval) are shown in the inset.
We also investigate the influence of the CW trigger when the trigger is placed in the normal dispersion regime in Fig. 5. Considering the wavelength range from 830 nm to 1010 nm, the trigger placed at 943 nm is shown to have the best performance in terms of coherence property. The mentioned 943 nm here corresponds to a same frequency shift towards to the center frequency as the 1130 nm. The blue lines correspond to spectrum/coherence when CW trigger changes every 20 nm from 830 nm to 1010 nm, and the black line corresponds to the result for 943 nm. In Fig. 5(a), the spectrum in the 943 nm case is featured by the soliton in the long-wavelength and the dispersive wave on the short-wavelength range which is similar to the 1130 nm CW triggered SC spectrum. The coherence property is shown in Fig. 5(b). For the 943 nm trigger case which is the black line in Fig. 5(b), multi-bands have a high coherence, and coherence degradation happens when triggers move away from 943 nm. Figures 5(c) and 5(d) show the dependence of overall coherence and overall SNR on CW trigger wavelength where the specific CW wavelength at 943 nm is marked by the cross symbol and specific view of CW triggers wavelengths from 937 nm to 945 nm using 1 nm as interval are shown in the inset. The 943 nm trigger improves the SC coherence property while the overall coherence remains low and is almost of the same level as that in the untriggered case for most of CW triggers. In the inset of Fig. 5(c), the 943 nm trigger shows the best improvement for the SC coherence property, and the overall coherence increases from 0.040 (untriggered case) to 0.223. As shown in Fig. 5(d), the overall SNR is greatly improved when the trigger wavelength is between 910 nm to 950 nm, while no obvious improvement of SNR occurs when CW trigger wavelength is below 900 nm or above 950 nm. For the inset of Fig. 5(d), the best SNR enhancement is achieved at 945 nm, which is very close to the position with the best coherence property 943 nm. The value of overall SNR is 4.020 at 943 nm and 4.497 at 945 nm respectively.
Fig. 5 (a) Spectra and (b) coherence for CW triggers at different wavelengths where the specific CW wavelength at 943 nm is plotted in black and the CW trigger wavelengths change every 20 nm in anomalous dispersion regime from 830 nm to 1010 nm (from bottom to top); (c) Overall coherence and (d) overall SNR versus CW trigger wavelength where the specific CW wavelength at 943 nm is marked by the cross symbol and specific view of CW triggers wavelengths from 937 nm to 945 nm (1 nm as interval) are shown in the inset.
4. Conclusions
We have numerically shown that optical rogue waves can be observed with high power femtosecond pulses and normal dispersion pumping regime, confirming the presence of L-shaped statistics in SC generation using femtosecond pulses other than long pulse excitation. In addition to our previous work [11], which showed the possibility of controlling optical rogue waves in the picosecond SC generation by using a weak CW trigger, the CW triggering technique has been demonstrated as an effective method in controlling behaviors of optical rogue waves in the femtosecond SC regime. This is the first time showing control of optical rogue waves in the femtosecond SC generation. The optimized CW trigger has been shown to lead to a remarkable enhancement of the spectrum bandwidth and the stability for both the anomalous and normal dispersion regime. We believe the results presented here can give a deep insight on the impact of a weak CW trigger on SC generation with femtosecond or picosecond pulse pump. This could pave the way to provide SC generation with outstanding high stability.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. Project 61307051), Shenzhen Technology and Innovation Council (Project JCYJ20140419131807789, KQCX20140521150127440)
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