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Abstract
We investigate the propagation dynamics of the soliton-sinc, a kind of novel hybrid pulse, in the presence of higher-order effects with emphasis on the third-order dispersion (TOD) and Raman effects. At variance with the fundamental sech soliton, the traits of the band-limited soliton-sinc pulse can effectively manipulate the radiation process of dispersive waves (DWs) induced by the TOD. The energy enhancement and the radiated frequency tunability strongly depend on the band-limited parameter. A modified phase-matching condition is proposed for predicting the resonant frequency of the DWs emitted by soliton-sinc pulses, which is verified by the numerically calculated results. In addition, Raman-induced frequency shift (RIFS) of the soliton sinc pulse increases exponentially with a decrease of the band-limited parameter. Finally, we further discuss the simultaneous contribution of the Raman and TOD effects to the generation of the DWs emitted from the soliton-sinc pulses. The Raman effect can then either reduce or amplify the radiated DWs depending on the sign of the TOD. These results show that soliton-sinc optical pulses should be relevant for practical applications such as broadband supercontinuum spectra generation as well as nonlinear frequency conversion.
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1. Introduction
In fiber lasers, Gaussian-shaped and sech-shaped pulses are the universal optical pulses, and both occupy a large bandwidth in the return-to-zero optical transmission process. Therefore, they may not always be the best waveforms for improving spectral efficiency. Later, the Nyquist pulse [1,2] (also called sinc pulse) was proposed whose shape is given by the response function of the Nyquist filter. Its characteristic of the rectangular spectrum makes it possible to transmit information without inter-symbol interference in optical time division multiplexing (OTDM) systems. Moreover, the Nyquist pulse significantly improves dispersion tolerance due to the compact bandwidth. Using optical Nyquist pulses in OTDM systems allows us to attain ultra-high bit rates and spectral efficiency. However, nonlinear effects in optical fibers can destroy the ideal transmission characteristics of Nyquist pulses [3,4]. Recently, a new type of optical pulse combining the properties of the solitons and the Nyquist pulses, called the soliton-sinc pulse, has been proposed [5]. Such pulses retain an oscillation structure similar to the Nyquist pulse and are more robust to nonlinear signal distortions. Thus, the nonlinear propagation dynamics of the soliton-sinc pulse are fascinating. In addition, the experimental realization of the Nyquist-solitons (the soliton-sinc pulses) in the optical cavities has been reported [6]. Its emergence opens up many possibilities for ultra-high-speed optical communications and other nonlinear optical applications.
The optical solitons are of the perfect balance between the anomalous group velocity dispersion (GVD) and the Kerr-focusing nonlinear effects. The higher-order dispersions are typically regarded as perturbations that significantly affect the nonlinear propagation of soliton. The energy was transformed from the soliton frequency to the resonant frequency manifesting as a linear dispersive wave (DW). This spontaneous DW emission process, also known as resonant radiation or Cherenkov radiation, is fundamental in nonlinear optics [7–10]. The phase-matching condition determines the radiation frequency. This radiation has significant applications in the direction of supercontinuum generation [11–13], optical frequency comb [14–18], wavelength conversion [19–24], and suppression of beam collapse [25–27]. The DW emission is not limited to soliton-like pulses propagating in the anomalous dispersion region [28–31]. In addition, many studies on the interaction between the DWs and soliton systems have been reported [32–38]. On the other hand, it is necessary to consider the higher-order nonlinear effects to accurately describe the nonlinear propagation of femtosecond pulses. Stimulated Raman scattering is a critical high-order nonlinear effect in optical pulse propagation. Since the Raman response of the medium has a delayed characteristic, it causes a continuous redshift in the propagation of the pulse spectrum, called Raman-induced frequency shift (RIFS). The RIFS of soliton is particularly prevalent, called soliton self-frequency shift (SSFS). In 1986, SSFS was first discovered [39] and analyzed [40]. Furthermore, there is a corresponding theoretical analysis for the RIFS of non-soliton pulses [41,42]. Since its discovery, SSFS has been extensively investigated for different applications, including the development of optical analog-to-digital converters [43,44], tunable femtosecond laser sources [45], optical delayers [46], signal processing [47], and broadband supercontinuum generation [12].
Recently, the propagation stability of the soliton-sinc pulse in the absence of higher-order effects has been investigated numerically [5]. It is shown that the soliton-sinc pulse has the advantage of keeping its spectral shape and compact bandwidth. The time structure also has less nonlinear distortion. In addition, Xue et al. generated experimentally the soliton-sinc pulse (called as Nyquist soliton) in a fiber ring cavity [6]. Hence, an open question is: how do the higher-order effects affect the propagation of soliton-sinc pulses? Here, we explore in detail the propagation dynamics of soliton-sinc pulses affected by the higher-order effects, which, to the best of our knowledge, have never been reported.
In this paper, we first investigate the DWs emission of the soliton-sinc pulses initiated by the third-order dispersion (TOD) effects. We examine the effect of finite bandwidth on radiating DW, such as DW radiation frequency and conversion efficiency. We propose a modified phase-matching condition to predict the resonant frequency of the soliton-sinc pulse emitting DW and verify its reliability numerically. Then, the RIFS simulated by the soliton-sinc pulse was also disclosed. In addition, we also discussed the propagation dynamics of the soliton-sinc pulse for the simultaneous presence of the TOD and Raman effects.
2. Propagation model
The propagation model of optical pulses in the anomalous dispersion region can be described by the nonlinear Schrödinger equation (NLSE), including TOD and Raman scattering terms. Its dimensionless form [48]:
The ideal soliton-sinc pulse can be obtained by filtering the soliton (sech) pulse with an ideal bandpass filter. The frequency domain expression for the soliton-sinc pulse is [5]
According to Eq. (4), we show the temporal intensity distribution of the soliton-sinc pulse as a function of $T_0$ and $B$, in Figs. 1(a) and 1(b), respectively. Here we use a logarithmic scale for the colormap to improve visibility. The black dashed lines mark the position of half the maximum intensity for the soliton-sinc pulse with specific $T_0$ and $B$; this gives us an overview of the variation of the full width at half of the maximum (FWHM). The pink dashed lines represent the intensity contour where $\left |A_{ss}\left ( t \right )\right |^2 = 0$. In Fig. 1(a), we fix $B = 3$. The FWHM of the soliton-sinc pulse increases with $T_0$, the peak intensity decreases, and the over-zero points gradually merge and disappear. As the soliton spectrum is continuously compressed during the increase of $T_0$, more soliton frequency components are included in the finite bandwidth, leading to the evolution of the soliton-sinc pulse to the fundamental soliton. Oppositely, when $T_0$ tends to 0, the soliton-sinc pulse evolves toward the pure sinc pulse. Similarly, in Fig. 1(b), we fix $T_0=1$. In the limiting case ($B=0$), the spectrum contains only fundamental frequency components, and the input time waveform is a continuous wave. As $B$ increases, the energy is transferred from the side lobes to the main lobe, the FWHM of the soliton-sinc pulse decreases, the peak intensity is enhanced, and the over-zero points merge and disappear, evolving continuously toward the fundamental soliton. In Figs. 1(c)–1(e) we compare the temporal waveforms of the sech soliton (black lines), the sinc (blue dashed lines), and the soliton-sinc (red lines); the frequency domain shapes are shown in the insets on the left. Both the sinc and the soliton-sinc pulse have decaying oscillation tails, the sinc pulse has periodic over-zero points while the soliton-sinc pulse does not. The first few over-zero intervals of the soliton-sinc pulse are narrower than those of the sinc pulse, and gradually increase and approach the constant period of the sinc pulse ($\Delta \tau =2\pi /B$) as the absolute value of $\tau$ increases. Therefore, we can manipulate the pulse width of the soliton-sinc pulse as well as the over-zero characteristics by adjusting $B$ and $T_0$.
Fig. 1. (a) Pulse intensity distribution as a function of $T_0$ with fixed parameter $B = 3$; (b) Pulse intensity distribution as a function of $B$ with fixed parameter $T_0 = 1$. The pink dashed lines represent the intensity contour where $\left |A_{ss}\left ( t \right )\right |^2 = 0$. The black dashed lines mark the FWHM of the soliton-sinc pulse. Temporal waveforms of the sech soliton (black lines), the sinc (blue dashed lines), and the soliton-sinc (red lines) with (c) $B = 3$, (d) 2, and (e) 1. The insets show the frequency domain shapes. Here $T_0 = 1$.
3. Numerical results
3.1 Effect of TOD on the soliton-sinc pulse
In this subsection, we investigate the bandwidth-limited soliton-sinc pulse propagation, subject to the combined effects of TOD and self-phase modulation (SPM). According to Eq. (4), for a constant $T_0$, the peak intensity of the pulse decreases with $B$. To fairly compare the dispersive waves generated by different incident pulses, we normalized their peak intensity $P_0=1$, to ensure that they all have the same nonlinear intensity. Figure 2 compares the DWs generated by the fundamental soliton and three different bandwidth soliton-sinc pulses with $B$ values of 3, 2, and 1. The common parameters are set as $\delta _3 = -0.08$, $N = 1$, and $T_0 = 1$. Figures 2(a) and 2(f) show the temporal and spectral intensity of different incident pulses at $\xi =10$, respectively. There are significant disparities between the soliton-sinc pulse and the fundamental soliton. A clear formation of solitons and DWs can be seen from the temporal evolution in Figs. 2(b)–2(e). Furthermore, the spectral evolution in Figs. 2(g)–2(j) shows the generation of DWs spectral peaks. For the fundamental soliton, which is equivalent to a soliton-sinc pulse with infinite bandwidth, we can see from Figs. 2(b) and 2(g) that the intensity and frequency shift of the generated DW are weak. When we limit the bandwidth of the soliton-sinc pulse ($B = 3$), the intensity and frequency shift of the generated DW are increased, as shown in Figs. 2(c) and 2(h). At the beginning of propagation, the main lobe of the soliton-sinc pulse is narrowed by nonlinear compression due to the combined effects of GVD and SPM. A portion of the energy is shed from the main lobe to form a soliton, which continues to exchange energy with the side lobes in further propagation. Continuing to reduce $B$, the soliton-sinc pulse transforms toward the sinc pulse with higher dispersion tolerance, and the spectral broadening and nonlinear compression are enhanced, eventually intensifying the generation of DWs. The SPM effects is dominant for smaller $B$ values. Additionally, as $B$ decreases, the temporal oscillation period increases and the intensity of the side lobes enhances, which is consistent with the pulse features in Fig. 1. Meanwhile, the main lobe of the soliton-sinc pulse becomes wider as $B$ decreases, and the propagation distance required for pulse compression increases. Since the spectrum of the soliton-sinc pulse is truncated at the bandwidth boundaries, the spectral intensity at the boundaries is discontinuous, as shown in Figs. 2(h)–2(j). This unique phenomenon can be understood as follows. Due to the SPM-induced frequency chirp at the beginning of pulse propagation, the in-band spectral phase precedes the out-of-band one. And this phase difference will continue for a while in further propagation. Throughout the generation of DWs, the entire energy of the DWs comes from the in-band pumping energy because the frequency component outside the initial bandwidth of the soliton-sinc pulse is zero.
Fig. 2. (a) Temporal and (f) spectral shapes for different incident pulses at $\xi = 10$. Temporal (b)-(e) and spectral (g)-(j) evolution with propagation distance for different incident pulses: (b), (g) sech soliton, and the soliton-sinc with (c), (h) $B = 3$, (d), (i) 2, and (e), (j) 1. The black dashed lines represent the bandwidth boundaries. The colored dashed lines represent the calculated theoretical phase-matching frequencies. Here $\delta _3 = -0.08$, and $\tau _R = 0$.
To reveal how the frequency of DW excited by the soliton-sinc pulse varies with the bandwidth $B$, we investigated its theoretical phase-matching condition. Equation (4) states that for the specified values of $B$ and $T_0$, the incident pulse $A_{ss}(t)$ will reach its maximum value at the center, $t = 0$. The equation for the maximum value is:
It is important to note that the incident pulse is not a standard fundamental soliton, and only part of the energy is converted to shed solitons. Since the incident pulse is multiplied by a normalization factor $A_{ssMAX}^{-2}$, which allows the total incident pulse energy to be compensated accordingly, the peak intensity of the shedding soliton $P_s$ can be approximated. The approximate peak intensity of the shedding soliton is $P_s \approx P_0/A_{ssMAX}^{2}$. Now we obtain the nonlinear phase $\phi _{NL} \approx P_s\xi$ and linear phase $\phi _{L} = \left (\delta _3 \omega ^3+0.5 \omega ^2\right )\xi$, then make them equal to finally get the phase-matching condition:
where $\omega =2\pi (\nu -\nu _0)T_0$ is the frequency shift of the DW relative to the soliton. By solving the roots of Eq. (6) we can obtain the frequency of DWs. When $B = 3$, 2, and 1, the DW frequencies of soliton-sinc pulses are calculated to be -1.03, -1.3, and -1.5, respectively. The calculated results are marked with dashed lines of different colors in Figs. 2(g)–2(j), which greatly agree with the spectral evolution. During the radiation process, the energy in the band is shifted from the central frequency $\omega _0$ to the resonant radiation frequency $\omega _{DW}$. Here $\delta _3<0$, which corresponds to generating the redshift DWs.3.1.1 Effect of TOD on the DWs
To better investigate the effect of TOD on DWs, we give the output spectra of different incident pulses, each at $\xi = 10$ as a function of $\delta _3$ in Fig. 3. The remaining parameters are the same as in Fig. 2. The black dotted lines are the theoretical phase-matching curves. The black dashed lines represent the bandwidth boundaries. The results show that $\delta _3$ has a crucial role in controlling the generation of DWs, and the theoretical phase-matching curves are highly consistent with each output spectra. Here $\delta _3<0$ makes DWs appear only on the red side of the spectrum. As $\left |\delta _3\right |$ increases, the phase-matching position $\omega _{DW}$ moves from the red side to the center and the peak intensity of the DWs enhances. In Fig. 3(a), the incident pulse is the fundamental soliton. When $\left |\delta _3\right |$ is small, it is hard to observe the production of DWs until $\left |\delta _3\right |$ grows to a particular value ($\delta _3 \approx -0.065$). The minimum $\left |\delta _3\right |$ threshold corresponding to the emergence of a noticeable resonance peak varies for different incident pulses. Decreasing the bandwidth $B$ yields a lower $\left |\delta _3\right |$ threshold, as shown in Figs. 3(b)–3(d). $\delta _3$ thresholds for the soliton-sinc pulses with $B=3$, 2, and 1 are -0.055, -0.04, and -0.028, respectively. In addition, when $\left |\delta _3\right |$ is small, the soliton-sinc pulse output spectra have an approximately symmetric spreading with respect to the initial bandwidth. The phase-matching condition is satisfied near the threshold point, and the red side energy is rapidly shifted to the resonant frequency. The results show that the soliton-sinc pulse with a small bandwidth $B$ facilitates the generation of DWs, and we can precisely control the frequency of DWs by manipulating $\delta _3$.
Fig. 3. Output spectra at $\xi = 10$ as a function of $\delta _3$ for different incident pulses: (a) sech soliton, and the soliton-sinc with (c) $B = 3$, (b) 2, and (d) 1. The black dashed lines represent the bandwidth boundaries. The black dotted lines represent the calculated theoretical phase-matching frequencies.
3.1.2 Effect of bandwidth B on the DWs
To more comprehensively demonstrate the effect of bandwidth $B$ on the DWs emission. In Fig. 4(a), we show the variation of the soliton-sinc pulse output spectra with $B$. The black dashed and dotted lines represent the initial bandwidths and the theoretical phase-matching curves, respectively. When $B \geq 3$, there is little difference in the output spectrum because the soliton-sinc pulse is similar to the fundamental soliton. As $B=3$ continues to reduce $B$, the spectral spreading enhances significantly, and the redshift amount and peak intensity of the DWs also increase. Interestingly, as $B$ decreases below 2, multiple weaker spectral peaks emerge in the gap between the center of the spectrum and the red side spectral peak. This is all caused by the collision between the shedding soliton and the side lobes, which generates new transmitted and reflected components. It is worth noting that the correspondence between the theoretical phase-matching curve and the spectrum deteriorates near the small bandwidth because the decrease in $B$ causes less and less of the fundamental soliton spectrum to be retained. The soliton-sinc pulse diverges from the fundamental soliton, which leads to the phase-matching condition no longer being applicable.
Fig. 4. Output spectra at $\xi = 10$ as a function of $B$ with (a) $\tau _R=0$, and (b) 0.04. The black dotted lines represent the calculated theoretical phase-matching frequencies. The black dashed lines represent the bandwidth boundaries. Energy conversion efficiencies of DWs vary with $\xi$ for different incident pulses, (c) $\tau _R=0$, and (d) 0.04.
Thus far, the phase-matching condition allows us to calculate the DW emission frequency. The energy conversion efficiency can depict the energy transfer process, which is defined as $\mu = (E_{DW}/E_{total}) \times 100\%$. At $\delta _3=-0.08$, the variation curves of the energy conversion efficiency with $\xi$ are shown in Fig. 4(c). As the pulse propagates, the energy conversion efficiency rises steeply near the maximum compression distance, reaching saturation at $\xi = 6$. To better understand the DWs emission of the soliton-sinc pulse, we added the sinc pulse with the same bandwidth. We can see that the soliton-sinc pulse and the sinc pulse have higher energy conversion efficiency compared to the fundamental soliton. With the same $B$, the energy of the soliton-sinc pulse is concentrated at the spectral center, while the energy of the sinc pulse is uniformly distributed in the bandwidth range $\left [ -B/2, B/2 \right ]$, so the former has a higher energy conversion efficiency. The energy conversion efficiency of the soliton-sinc pulse at $\xi =10$ has a great improvement as $B$ decreases and the DWs emission is delayed. For specific values, the energy conversion efficiency of the fundamental soliton excitation DWs is only 0.09%; while 1.31%, 4.8%, and 9.63% for soliton-sinc pulses with $B=3$, 2, and 1, respectively. The results show that the bandwidth-limited soliton-sinc pulse can significantly improve the energy conversion efficiency of DWs. This bandwidth-limiting method can also be used for other pulses (such as the Gaussian pulse). The case of considering the Raman effects would be discussed further in Section 3.3.
3.2 RIFS of the soliton-sinc pulse
Interpulse Raman scattering (IRS) is the dominant of the higher-order nonlinear effects that make the pulse spectrum energy continuously shift to longer wavelengths in propagation. In this section, we focus on the influence of IRS on the nonlinear propagation of the soliton-sinc pulse, setting $\delta _3=0$ and $\tau _R=0.04$. Figure 5 shows the temporal and spectral evolution for the different incident pulses in the presence of the Raman effect. Due to the Raman effects, the fundamental soliton is decelerated and presents a bending trajectory in the time domain, as shown in Fig. 5(b). Accordingly, its spectrum is redshifted, as shown in Fig. 5(g). Since the fundamental soliton can propagate stably and maintain the parameter N=1, the Raman effects play a relatively minor role here. The temporal and spectral shifts are small. In Figs. 5(c)–5(e), for the soliton-sinc pulse, a stronger shedding soliton is formed by the main lobe narrowing at an early stage. With decreasing $B$, the intensity of the shedding solitons increases and the pulse width becomes shorter, as shown in Fig. 5(a). These contribute to the enhancement of RIFS. The spectral energy of the soliton-sinc pulse jumps when it crosses the bandwidth boundaries, as shown in Figs. 5(h)–5(j). When $B = 3$ and 2, the spectral evolutions show that the spectral energy in the bandwidth is shifted to the red side with a slower rate. For $B=1$, the spectrum broadens rapidly at the initial stage of propagation, and then a portion of the energy forms a smooth redshift spectrum at the red side. In addition, the residual pumping energy near the bandwidth shows a multi-peak structure due to the SPM.
Fig. 5. (a) Temporal and (f) spectral shapes for different incident pulses at $\xi = 10$. Temporal (b)-(e) and spectral (g)-(j) evolution with propagation distance for different incident pulses: (b), (g) sech soliton, and the soliton-sinc with (c), (h) $B = 3$, (d), (i) 2, and (e), (j) 1. Here $\delta _3 = 0$, and $\tau _R = 0.04$.
Figure 6 visualizes the variation of temporal and spectral intensity distribution at $\xi =10$ with $B$. In Fig. 6(a), with decreasing $B$, the shedding soliton is stronger and shorter, being further decelerated. So the time shift increases. Correspondingly, Fig. 6(b) shows that the spectral shift also increases. When the value of $B$ is between 1.8 and 2.8, there is a significant discontinuity in the spectrum crossing the bandwidth boundaries. This is due to the phase lead in the band, which reaches the bandwidth boundaries faster under the Raman effects. Continuing to reduce $B$ creates a smooth redshift spectrum at the red side, and some residual spectrum near the frequency band. Overall, the RIFS of soliton-sinc pulses shows an exponential increase with the decrease of $B$. The results show that we can manipulate the RIFS of the soliton-sinc pulse by changing $B$.
3.3 Impact of Raman effect on the DWs
So far, we have investigated the influence of the TOD and Raman effects on the propagation dynamics of the soliton-sinc pulse when acting alone, respectively. In this section we discuss the role of TOD and Raman effects combined on soliton-sinc pulse propagation dynamics. Figure 4(b) shows the variation of the output spectrum at $\xi =10$ with $B$. Here $\tau _R = 0.04$. Compared with Fig. 4(a), the presence of Raman effects makes more blueside spectrum is shifted to the redside and the spectrum is smoother. The spectral peaks of the redshifted DWs slightly shift toward the center of the spectrum with enhanced peak intensity. These are due to the RIFS, which causes a slight change in the frequency of DWs and provides pumping energy for the redshifted DWs throughout the propagation process. Figure 4(d) shows the energy conversion efficiency curve in the presence of Raman effects; compared to the case without Raman effect, the energy conversion efficiency of the soliton-sinc pulse with $B=1$ is enhanced from 9.63% to 19.66%. Figures 7(a) and 7(b) show more clearly the frequency shift and the peak intensity of the DWs as a function of $B$. The results correspond at $\xi = 6$, where the radiation of the first DWs has just finished. At $B=5$, the Raman effect reduces the redshift of DWs by almost 0.2, and the peak intensity of DWs is enhanced by more than 6 dB. Interestingly, there is no significant frequency shift in the spectral peaks of the DWs, which indicates that the DWs are generated immediately after the soliton shedding process. The frequencies and amplitudes of the DWs are less affected by the subsequent energy transfer.
Fig. 7. (a) Frequency shift and (b) relative peak intensity of the DW peak at $\xi =6$ plotted as a function of $B$, where $\tau _R=0$ (blue lines), and 0.04 (red lines). Here $\delta _3=-0.08$.
It is noteworthy that only the Raman effects are considered in Fig. 5, producing a strong RIFS. However, this phenomenon does not seem to be reflected in the output spectrum with the addition of the TOD effect, as shown in Fig. 4(b). To reveal why the RIFS is suppressed, in Figs. 8(a) and 8(c), we show the temporal and spectral evolution of the soliton-sinc pulse with $B=1$ when the propagation distance reaches $\xi =40$, respectively. The black dashed line represents the zero-dispersion (ZD) frequency. An interesting feature seen in Fig. 8(a) is the unusual trajectory of the Raman soliton. While the soliton is initially accelerated (a negative delay), after a certain distance it starts to slow down and eventually moves at a constant speed. Correspondingly, in Fig. 8(c), the RIFS is canceled when the Raman soliton approaches the ZD point attributed to the spectral recoil [8,49]. As the energy of the soliton shifts progressively, the DWs of the subsequent radiations weaken and its spectral peaks show a multi-peak structure and expand toward the center of the spectrum. In addition, the bending time trajectory leads to generate a refractive index potential well. As a result, the temporal reflection and refraction is observed [50]. This can generate new frequency components located the spectral gap between the DWs and the input pulse. Next, we changed the sign of TOD so that $\delta _3=0.08$ and show the influence of Raman effects on blueshifted DWs in comparison in Figs. 8(b) and 8(d). Figure 8(b) shows the spectral evolution of only the TOD, presented as the mirror image of Fig. 2(j), with DWs generated on the blue side. Figure 8(d) shows the spectral evolution in the presence of Raman effects. We can see the blueshifted DWs are suppressed, its intensity was reduced and its frequency was increased, together with a significant soliton spectral redshift. So the spectral recoil does not occur at the positive TOD. In summary, we can obtain the redshift DWs are enhanced when the negative TOD ($\delta _3 < 0$) and Raman effects are combined to influence. If $\delta _3 > 0$, the blueshifted DWs will be suppressed by the Raman effect.
Fig. 8. Temporal (a) and spectral (c) evolution with propagation distance for the soliton-sinc with B = 1. Here $\delta_3=-0.08, \tau_R=0.04$, and the maximum propagation distance $\xi = 40$. The black dashed line represents the ZD frequency. spectral evolution with propagation distance for the soliton-sinc with B = 1, (b) $\tau_R =0$, and (d) 0.04. Here $\delta_3 = 0.08$, and the maximum propagation distance $\xi = 10$.
4. Conclusion
In conclusion, we have numerically studied the dynamics of the soliton-sinc pulse propagation in optical fibers, considering the TOD and Raman effects. First, we have found that the band-limited parameter can effectively control the conversion efficiency and resonant frequency of the DWs generated by the TOD effects. Unlike the fundamental soliton, the soliton-sinc pulses have a stronger ability to modulate the DW emission. The energy conversion efficiency of the DWs induced by the soliton-sinc pulses is much higher than that for the fundamental soliton. The improved phase-matching condition accurately predicts the DW radiation frequency of soliton-sinc pulses, which shows that a decreasing $B$ effectively reduces the values of $\vert \delta _3\vert$ threshold required for the DWs generation. For a soliton-sinc pulse with $B=1$, $\vert \delta _3\vert$ is only 0.028, while it increases to 0.065 for the fundamental soliton. In addition, the RIFS of the soliton-sinc pulses exhibits exponential increases with a decreasing $B$. Finally, the combined impact of the TOD and Raman effects on the soliton-sinc pulse propagation has revealed as well. The Raman effects can enhance the redshifted DWs or weaken the blueshifted DWs. These results show that soliton-sinc pulse can offer possibilities in scenarios such as broadband supercontinuum generation, frequency combs, and other spectroscopic applications.
Funding
National Natural Science Foundation of China (61975130); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515010084);.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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