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Abstract
We present an in-depth investigation of optical rogue waves (ORWs) during picosecond supercontinuum generations in photonic crystal fibers with periodic landscapes of group velocity dispersion and nonlinearity, namely dispersion oscillating fibers (DOFs). Specifically, it is shown via ensembles of numerical simulations that during supercontinuum generations, the rogue nature of extreme and rare events formed in uniform fibers can be effectively manipulated in DOFs. This is also verified by comparing single evolution dynamics in different dispersion longitudinal profiles. For investigating the influence of slow dynamics of ORW generation in DOFs, we increase the propagating distance and find out MI gain is still the major factor that influences the generation of ORWs. In addition, analytical results associated with simulations indicate the rogue manipulations in DOFs are attributed to the adjustable modulation-instability-gain due to periodic dispersion variation along fiber length. Finally, unlike MI in uniform fiber, MI gain side lobes result from quasi-phase-matching (QPM) relation in DOFs provide additional degree of freedom to control generations of ORWs. We believe our results will provide not only a novel insight of understanding ORW dynamics in presence of dispersion modulations, but also a new way of harnessing rogue waves in oceanology.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Rogue waves describe extreme, rare and unpredictable events existing among sundry areas of nature [1–3]. In the context of nonlinear optics, they were first observed in the presence of picosecond supercontinuum generations in a nonlinear photonic crystal fiber (PCF), in which the red-shifted solitons with uncommon probability distributions were termed optical rogue waves [4–6]. Since then, substantial interests on ORWs were attracted to explore not only the underlying physical mechanisms per se [7–9] but also the possible practical manipulation approaches [10]. It has been proved that ORWs are dominated by the initial seeded noises, group-velocity dispersion (GVD), higher-order dispersions (HODs) as well as Raman effect [11–14]. Besides picosecond-pump ORWs, it is also showed that ORWs can form in femtosecond supercontinuum generations [10,15], continuous-wave (CW) supercontinuum generations [16], as well as the Akhmediev Breathers [17,18]. On the other hand, manipulating ORWs in optical fibers are paid great attentions as well, owing to their direct relationship with freak wave managements in oceans [8,19]. It is demonstrated that ORWs can be considerably controlled by incorporating an additional weak CW wave preferably located at maximal modulation (MI) or Raman gain frequencies [10,20]. Other approaches include shaping the pump pulses [21] or designing a fiber with two zero-dispersion wavelengths (ZDWs) [22], etc.
In fact, most of studies of ORWs are implemented in uniform fibers (UFs) within the frame of long-pulse supercontinuum generations. While recently, ORWs were also involved in nonuniform fibers in few earlier papers. For instance, Kudlinski et al. demonstrated that by using a tapered PCF, the pulse-to pulse fluctuations can be suppressed during CW supercontinuum generations [23]. Similar results can be found in Ref. [24]. Here we present a detailed numerical investigation of ORWs in dispersion oscillating fibers (DOFs) by recalling the unique property of DOFs for nonlinear fiber optics [25]. The picosecond-pump supercontinuum process is taken into consideration since MI process is dominated at beginning of propagation. Firstly, we show the overall MI gain could be manipulated by changing the oscillating period in DOFs, leading to efficient controllable soliton formations in the following propagation. This manipulation would result in a variable spectral bandwidth thus controlling ORWs eventually, which is verified by analyzing the statistical features of multiple simulations with distinct initial random noises in DOFs and UF, respectively. Secondly, we utilize several new DOFs with different longitudinal landscapes and amplitude of oscillating. Thirdly, considering the slowldynamics of ORW generation in DOFs compared with UF, we increase the fiber length for ORWs could be generated adequately. Results show that ORWs in DOFs have been restricted even though the propagating distance is long enough. Then we perform a single simulation for each fiber under an identical pump to compare the output spectra features. Simulation results exhibit that red-shifted Raman rogue solitons are indeed dominated therein. Finally, taking advantage of an additional special fiber structure, we verify that ORW manipulation in DOFs are induced by initial MI dominations.
2. Modulation instability analysis in UFs and DOFs
MI is a universal phenomenon occurring in optical fibers [26,27]. Periodic pulses can be excited in presence of a CW pump undergoing a MI stage, creating double new sidebands located in the vicinity of the pump frequency. Usually MI occurs in anomalous GVD regime of optical fibers. However, it is confirmed as well that normal GVD MI is capable of being realized thanks to the negative value of the fourth-order dispersion [28]. The well-known maximal MI gain in the anomalous GVD region of a nonlinear fiber can be written as $g_{max}=2\gamma P_{0}$. $\gamma$ is the nonlinear coefficient, $P_{0}$ is the pump power. We then consider a DOF counterpart with the following dispersion landscape:
3. Fiber properties and simulation models
Before presenting simulation results, we first introduce the fiber properties and simulation models in the paper. Several dispersion-oscillating PCFs (we call them DOFs hereafter) of 30 meters are employed with periodic variations of the pitch along fiber lengths following sine shapes ($\Lambda$ and pitch scale depends on the fiber). The ratio between pitch and air-hole diameter (0.52) of the DOFs maintains constant along fibers. The generated GVD described by Eq. (1) is $\beta _{2}^{0}=-1.476 ps^{2}/km$, $\beta _{2}^{1}=0.24 ps^{2}/km$ of DOF#1-4, $\beta _{2}^{1}=0.45 ps^{2}/km$ of DOF#5 and $\beta _{2}^{1}=0.69 ps^{2}/km$ of DOF#6. In fact, the HODs as well as $\gamma$ also change along fiber lengths, and in simulations we consider these variations. For comparison,we use a uniform PCF (UF) with a constant pitch (3.55 $\mu$m) equivalent to the average pitch of the DOFs. The ZDW of the DOF oscillates from 1049.4 nm to 1053.4 nm, while the ZDW of the UF is 1051 nm. Similar GVD characteristics of the DOF have been fabricated in [39]. Table 1 is used to mark DOFs with different oscillating periods and show the detailed DOF properties.
Table 1. Structure parameters of the DOFs.
Regarding the simulation model, we utilize the generalized nonlinear Schrödinger equation (GNLSE), including the GVD, the HODs up to 10-th order, the frequency-dependent nonlinear parameter, as well as Kerr (including the shock term) and Raman nonlinearities [40]. In addition, for the seeded pump noise, we employ a realistic model by incorporating a small amplitude (0.01% of the peak power of initial pulses) random noise within a limited spectral bandwidth (40 nm) directly to the initial pump pulses [4]. Actually, earlier results have proven that specific noise models are not crucial for ORW statistical features but influence ORW growth rates during supercontinuum generations [4,10].
4. Simulation results
MI spectra in DOFs with different oscillating periods have been depicted in Fig. 1. Figure 1(a) shows the resultant pitch landscapes of DOF#1-3. Figure 1(b) corresponds to the spontaneous MI spectrum obtained by pumping DOF#1-4 with CW at center wavelength equals to 1064nm and small amplitude random noise. Zero-frequency shift means the center wavelength where a CW with average power of 20 W is pumped. The frequency feature of the noise is a limited spectral bandwidth from 1044 to 1084nm and the amplitude feature of the noise is less than 0.01% of the pump power. The MI gain of UF(blue line) is used as a comparison. From the result of Fig. 1(b), the phenomena that MI gain spectra would be suppressed with the increase of oscillating period of DOFs has been revealed. Figure 1 indicates that MI gain profiles can be manually designed to either cancel or improve specifical sidebands by tailoring longitudinal dispersion landscapes [37].
Fig. 1. (a) Outer diameter of DOF#1-3 versus fiber length. (b) MI gain spectra at the output of DOFs with different oscillating periods and UF.
The result in Fig. 1 hints the variable in MI gain due to the difference of oscillating periods in DOFs will influence generation of ORWs. So in Fig. 2 the manipulation of ORW in DOF#1 compared with UF has been illustrated. For each of the fiber, an ensemble of 300 simulations is carried out by launching Gaussian pulses with 5-ps temporal width (FWHM) and 80W peak power, together with initial random noises with 0.01% of pulse. The pump wavelength is 1064 nm so that pumps are located in anomalous GVD regime for both fibers. The soliton number at the fiber input is about 20. Simulation results are depicted in Fig. 2(a) and 2(b). It is evident that ORWs appear in both fibers by viewing the long wavelength edge (beyond 1175nm), which allows us to see small number of ORWs having a large redshift. However, one also can identify that the mean spectrum bandwidth of DOF#1 is considerably compressed than that of the UF in both short- and long-wavelength regimes. A natural idea is that ORWs of extreme and rare events formed in long-wavelength end are suppressed compared with ORWs of UF. To check our conjecture, we calculate the histograms of the total power beyond 1175 nm [green lines in Fig. 2(a) and 2(b)] for the 300 output spectra (i.e. the power of the Raman rogue solitons) in both fibers. Results shown in Fig. 2(c) verify ORW suppressions in DOF#1. Specifically, the power distribution of UF follows the L-shape probability distribution [10]. Yet, most of the spectral components lie below 1175 nm thus the long tail of the power histogram disappears. It is noteworthy that ORW degenerations in DOFs rely on spectral bandwidth compressions, the coherence property of ORWs in DOFs remains low, hence if we analyse the statistics of output spectra by extracting the total power from the wavelength far below 1175 nm, the resultant histogram of DOF#1 will have a L-shape distribution as well. Briefly speaking, ORWs in DOFs can be effectively inhibited rather than totally disappearing.
Fig. 2. Simulations of ORWs in DOF#1 and the UF. (a),(b) 300 final output spectra (gray lines) in presence of distinct initial noises in DOF#1 and the UF, respectively. The blue and red lines correspond the output mean spectra. The light yellow line in (a) is ZDW (1064nm) and the light yellow area in (b) are ZDW ranges (from 1049.4 to 1053.4nm due to variable dispersion in DOF#1). (c) Histograms of the total power beyond 1175 nm [green lines in (a) and (b)] of the output spectra in DOF#1 (red marks) and the UF (blue marks). (d) The part histogram distributions of (c) for clarity.
To further exhibit the relation between ORWs and the distinct types of DOFs, we employ four DOFs (DOF#1-4) with different oscillating periods in Fig. 3(a) and four DOFs with different initial phase of fiber oscillation in Fig. 3(b). Similarly, we carry out an ensemble of 300 simulations for each fiber using the parameters mentioned above, and then we can obtain the final output mean spectra. Also, mean spectra of the UF are shown for sake of comparisons in Fig. 3(a).
Fig. 3. (a) Output mean spectra of the UF (blue line), DOF#1 (red line), DOF#2 (green line), DOF#3 (black line) as well as DOF#4 (yellow line). DOF#1-4 with different oscillating periods. (b) Output mean spectra of DOF#2 (green line), DOF#2 with initial phase 0.5$\pi$ (green dash line), DOF#4 (yellow line), DOF#2 with initial phase 1.5$\pi$ (yellow dash line).DOF#4 is DOF#2 with initial phase $\pi$.
First, it is straightforward that the long-wavelength spectral components are all compressed in the DOFs compared with that in the UF. This feature indicates that ORWs are degenerated in all DOFs. Second, with lower oscillating period in DOFs, broader the mean spectra in the long and short-wavelength side, indicating that the difference of MI gain in DOFs influences the bandwidth of mean spectra of ORWs. Finally, ORWs with other value of phase difference (0.5$\pi$, $\pi$, 1.5$\pi$) are shown in Fig. 3(b). The oscillating period and amplitude of DOFs are fixed and $\beta _{2}^{0}$, $\beta _{2}^{1}$ in Eq. (1) are constants, which indicates that MI gains in DOFs with different phase are nearly overlaped, so that the output mean spectra do not change too much.
Figure 4 indicates the influence of different amplitude of diameter variable in DOFs (DOF#2, DOF#5-6) on ORWs. MI spectra in DOFs with different amplitude have been revealed in Fig. 4(a) and output mean spectra based on 300 simulations for DOFs(DOF#2, DOF#5-6) have been shown in Fig. 4(b). Although the amplitude changes linearly from DOF#2 to DOF#6, MI gain spectra in Fig. 4(a) are nearly unchanged. MI gain is the conclusive factor to influence the generation of ORWs, so the spectra of DOF#2 to 6 don’t have much more differences. Especially the spectra of DOF#2 and 5 almost overlap with each other.
Fig. 4. (a) MI gain spectra at the output of DOFs with different amplitude of diameter variable and UF. (b) Output mean spectra of the UF (blue line), DOF#2 (green line), DOF#5 (purple line) and DOF#6 (brown line). DOF#2, DOF#5-6 with different ammplitude of pitch.
Although output mean spectra in DOFs and UF may reveal that ORWs in DOFs are suppressed compared with UF, we don’t consider there is a possibility that the suppression of ORWs in DOFs is induced due to slower dynamics in DOFs before, which means that after propagating enough distance in DOFs, the difference of ORWs between UF and DOF#1 may be ignored. Figure 5 further researches the generation of ORWs as propagating longer fiber length. Fiber length is increased from 30m to 120m in UF and DOF#1 and other values don’t change for ORWs could be generated adequately. Mean spectra of UF and DOF#1 at different fiber length has been shown in Fig. 5(a) and (b). The bandwidths of spectra are all broadened as the fiber length increases in UF and DOF#1. However, benefiting from Raman effect, the energy of output mean spectrum at the wavelength range from 1175 to 1400nm in UF is much larger than DOF#1, as shown in Fig. 5(c). The energy ratio of ORW means the ratio between the energy of mean spectrum beyond 1175nm and the total energy in fiber. When fiber length beyond 30m, the energy ratio in UF increases from 3% to 53%. As a comparison, the energy ratio in DOF#1 at 120m just reaches 14%. We can get a conclusion that as the fiber length increasing, the difference of the ORW energy in UF and DOF#1 is enlarged.
Fig. 5. (a) output mean spectra of the UF with the increase of fiber length. (b) output mean spectra of DOF#1 with the increase of fiber length. (c) energy transfer efficiency of spectra beyond 1175nm in UF and DOF#1
Typically, statistical analyses of ORWs are employed through simulating many roundtrips under different pump noises. As a contrast, single-roundtrip simulation is prompt and straightforward for investigating one specifical evolution detail. Here we utilize this method to further verify ORW dynamics in DOFs. Adopting the same parameters as before, we simulated 300 times with distinct random noises in the UF and DOF#1 and chose one of the largest energies of ORWs respectively, as shown in Fig. 6(a) (output spectrum of UF) and (b) (output spectrum of DOF#1). Figure 6(c)and(d) are the evolution dynamics of ORWs in UF and DOF#1.
Fig. 6. single-roundtrip simulation of orws scenarios in UF and DOF#1. (a) output spectra of the UF. The energy ratio of ORW is 29.13% at 1343.7nm. (b) output spectra of DOF#1. The energy ratio of ORW is 18.29% at 1281.3nm.(c)(d) corresponding detailed spectral evolutions dynamics in UF and DOF#1 along the fiber lengths.
In particular, from Fig. 6(a)(b), it is easy to identify that the output spectra of DOF#1 is compressed than that of the UF for the long-wavelength part. Although the center wavelength of ORW in DOF#1 reaches 1281.3nm after 120m propagation, the center wavelength of ORW in UF also increases to 1343.7nm. The energy ratios of ORW in UF and DOF#1 still have huge difference. The energy ratio of ORW in the UF reaches 29.13% but in DOF#1 the ratio is just 18.29%. Figure 6(c)(d) further exhibit the evolution dynamics of ORWs in UF and DOF#1. After propagating about 30m fiber length, ORW in UF and DOF#1 generate. ORW in UF will still has red-shifted after 120m. As a comparison, ORW in DOF#1 oscillates because of the variable dispersion along propagating direction and finally after 105m propagation, ORW doesn’t show an obvious red-shift anymore. For the short-wavelength DWs, it is straightforward to observe from Fig. 6(c)(d) that soliton-DW trapping leads to the deep blue extensions of the spectra. On the other hand, the DW spectra reshapes are also obvious if we compare the evolution dynamics in the UF and DOF#1, where soliton-DW collisions are able to generate new spectral components in the presence of the dispersion variation along a fiber [41]. Simulation results in Fig. 6 proof that even if the propagating distance in DOF#1 is long enough, the best result of ORW in DOF#1 have restrictions on Raman redshift and energy transfer efficiency compared with ORW in UF. The simulation results in Fig. 5 and Fig. 6 also show that although the dynamics of ORWs in UF are more advanced compared with DOFs, the difference of MI gain spectrum between the UF and DOFs is still the major factor that influences the generation of ORWs.
Based on the discussion about the ORW manipulation above, we now consider ways ORW promoting induced by MI gain side lobes [42]. Figure 7(a) shows MI gain at the output of DOF#1 and UF scaled from 1090nm to 1160nm (center wavelength at 1064nm). Here, the initial input CW with average power of 20W imposes on a small amplitude random noise (0.01% of power of CW). Compared with MI gain of UF, dispersion oscillating fiber (DOF#1) provides additional gain, especially in the scale between 1090nm to 1120nm. Numerical results in Fig. 7(b) illustrate the effect of MI gain side lobes on the rogue wave dynamics. Here, a small amplitude (0.01% of the average power of CW) random noise within a limited spectral bandwidth (40nm) is imposed on input CW with average power of 20W (center wavelength equals to 1072nm) and simulations of propagating evolutions in 30 meters UF and DOF#1 correspond to blue and red line. As a comparison, a simulation result with unlimited spectral bandwidth input random noise in 30 meters DOF#1 is shown in orange line. The simulation results with different input random noise spectral bandwidth clearly reveal that MI gain side lobes in dispersion oscillating fiber promote the generation of optical rogue wave if the spectral bandwidth of the noise contain MI gain side lobes. Especially when the frequency beyond 1100nm, there are about 15dB difference between orange line (noise with unlimited spectral bandwidth) and red line (noise with spectral bandwidth from 1090nm to 1160nm). This result illustrates that MI gain side lobes at mid-infrared frequency band accelerate the generation of optical rogue wave based on Raman soliton evolution in dispersion oscillating fiber.
Fig. 7. (a) MI gain spectra at the output of DOF#1 and UF scaled from 1090nm to 1160nm. (b) Output mean spectra of the UF (blue line), DOF#1 (red line) and DOF#1 with unlimited spectral bandwidth noise (orange line).
5. Conclusion
In conclusion, we investigate the ORW dynamics in PCFs with periodic landscapes of dispersions and nonlinearities, relying on the platform of noise-seeded picosecond supercontinuum generations. MI analysis in DOFs reveals that ORW regulation is feasible in DOFs since the MI-dominated soliton generation rates are controllable through tailoring the oscillating period of DOFs. Simulations results verified the ORW suppressions in DOFs, giving rise to the compressed bandwidths of the long-wavelength spectra thus the rogue statistical feature disappeared for the most red-shifted solitons. In addition, by increasing the fiber length of UF and DOF#1, the difference of ORWs in UF and DOF#1 is enlarged. Even though the number of ORWs in DOF#1 will increase, the energy and wavelength of ORWs have huge differences with UF. What’s more, by means of the single-roundtrip simulations in each of the fibers (i.e. UF and DOF#1), we confirm ORWs in DOFs could be regulated by the underlying MI gain manipulation during the initial propagation. We believe the results presented here will provide a novel avenue for future exploring more approaches of harnessing ORWs. Finally, the phenomena that the MI gain side lobes promoting ORWs in DOFs have been revealed, which hints that the bandwidth of noise is a new factor to control the generation of ORWs in DOFs.
Funding
National Natural Science Foundation of China (91750108).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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