We investigate the evolution of asymmetric self-accelerating finite energy Airy pulses (FEAP) in optical fibers with emphasis on the role of Raman scattering. We show that the Raman-induced frequency shift (RIFS) of soliton initiated by an asymmetric self-accelerating FEAP depends not only on the launched peak power but also on the truncation coefficient imposed on the asymmetric self-accelerating FEAP. We find that the RIFS of asymmetric self-accelerating FEAP increases with a decrease in the truncation coefficient, while the peak power and spectrum width of the outermost red shift of the shedding soliton spectrum are almost unchanged. The time and frequency shifts of the shedding soliton are found to be sensitive to the truncation coefficient when the truncation coefficient is in the range of 0 to 0.1. These excellent features would lead to the realization of a RIFS-based tunable light source by launching self-accelerating FEAP with different truncation coefficient into an optical fiber.
© 2014 Optical Society of America
The original Airy wave packet was predicted by Berry and Balazs in the context of quantum mechanics as a solution to the potential-free Schrödinger equation . However, such Airy wave packet is physically unrealizable as it carries infinite energy. In 2007, the concept of finite-energy Airy beam (FEAB) was introduced theoretically  and demonstrated experimentally  within the area of spatial optics. Since then, FEAB have been attracting considerable interest [4–8], due to their intriguing characteristics of transverse acceleration, quasi-diffraction-free and self-reconstruction [2, 3, 9].
More recently, finite energy Airy pulses (FEAP), known as pulsed version of FEAB, were introduced from spatial to temporal domain by exploiting the analogy between “diffraction in space” and “dispersion in time” . In fact, the generation of FEAP has been reported as early as 1979 as a result of the propagation of Gaussian pulse through a fiber with only third-order dispersion [10, 11]. FEAP can also be produced by imparting a cubic spectral phase on an incident pulse by other pulse shaping techniques . The resulting FEAP can propagate further without broadening in the quadratic dispersive (normal or anomalous) media, leading to spatially and temporally confined pulses or light bullets [13, 14]. The other linear propagation dynamics of FEAP was studied as well [15–17]. Alongside the linear dynamic properties of FEAP, nonlinear propagation of high intensity FEAP has also attracted attentions [18–23]. The effect of Kerr nonlinearity on the evolution of FEAP in optical fiber with normal or anomalous group velocity dispersion (GVD) was analyzed [18, 19]. Zhang et al. reported soliton pair was generated and engineered from the interaction of Airy pulse . Ament et al. has shown both experimentally and numerically that, multi-peak dispersive waves are emitted when a femtosecond FEAP propagates in a highly nonlinear photonic crystal fiber with a single zero-dispersion wavelength . It has been reported that the high-order linear and nonlinear effects play an important role in the propagation of self-decelerating FEAP, indicating that the deceleration of the main lobe is enhanced by Raman scattering effects . In addition, Zhang et al. investigated the impact of truncation coefficient on the modulation instability of FEAP .
To accurately describe the femtosecond pulse nonlinear propagation, higher-order liner and nonlinear effects should be considered . Of all the higher-order nonlinear effects, stimulated Raman scattering plays a crucial role in light pulse propagation . As the pulse propagates in the medium, a continuous red-shift of the pulse spectrum is induced-a feature that is referred to as the Raman-induced frequency shift (RIFS), where low-frequency photons of the pulse are amplified by its high-frequency photons through Raman scattering from of the optical phonons. RIFS was first discovered and analyzed in 1986 [26, 27] and has been investigated extensively in the various optical fibers and metamaterials since then [28, 29]. It has found a number of practical applications, including supercontinuum generation , tunable femtosecond pulse sources , analog-to-digital conversion , and signal processing . Hence, it is very important to realize the RIFS manipulation under different conditions. For this purpose, several techniques have been proposed to control the RIFS, including RIFS suppression by bandwidth-limited amplification [34, 35], cross-phase modulation , upshifted filtering , negative dispersion slope  and self-steepening  or RIFS enhancement by reducing the initial pulse duration , and by optimization of the photonic crystal fibers [41, 42].
Although RIFS has been extensively studied in optical fibers since the discovery of the phenomenon, the overwhelming majority of prior studies use an optical pulse with compact and symmetric temporal profiles such as sech, Gaussian, or super-Gaussian. FEAP can exhibit self-acceleration or self-deceleration depending on its tail behind or in front of its main lobe. The influence of Raman scattering on the propagation of self-decelerating FEAP is presented in Reference . However, the actual group velocity of self-decelerating FEAP is distinctively different from that of self-accelerating FEAP. Spectrogram (time frequency space) of the self-decelerating FEAP is the total opposite of the one by self-accelerating FEAP. It is therefore still unknown how exactly the Raman scattering executes its role in the propagation dynamics of self-accelerating FEAP. In addition, self-accelerating FEAP also has the ability to resist dispersion and their intensity peaks accelerate and self-reconstruct during the linear propagation. Do these unique linear properties make the RIFS of self-accelerating EFAP different from that of symmetric pulses or self-decelerating FEAP? In this paper, we focus our interest on the studies of self-accelerating FEAP propagating in an optical fiber with the inclusion of Raman scattering effects.
This paper is organized as follows. Section 2 presents the theoretical model, while Section 3 presents numerical results of the self-accelerating FEAP propagating in an optical fiber in the presence of Raman scattering effects, and a comparison of the RIFS of self-accelerating FEAP and that of symmetric pulse. Then, a brief summary of the results and conclusions are given in Section 4.
2. Theoretical model
To isolate the effects of Raman scattering, we neglect the impact of high-order dispersion, self-steepening and fiber losses on the pulse propagation along optical fiber. Therefore, a modified nonlinear Schrödinger equation (NLSE) with an added Raman term can be written in the form of Eq. (1) :Equation (1) with the simple model of Raman scattering can be used to reasonably describe the propagation of picosecond pulse but is not applicable for pulses whose lengths are close to or shorter than 100 femtosecond . For the latter, a comprehensive model of Raman Scattering and other higher-order linear and nonlinear effects must be considered in the propagation equation.
By substitution of parameters we haveEq. (1) can be reduced to its normalized form
The initial self-accelerating FEAP can be written asEq. (1) for the propagation dynamics of self-accelerating FEAP in an optical fiber with emphasis on the role of Raman scattering.
3. Numerical results
In the absence of Raman scattering and the presence of anomalous dispersion (), Eq. (3) predicts that, when , an input pulse with the amplitude propagates as the fundamental soliton with the general solution in the form of . Clearly, the shape and width of such an input pulse does not change perfectly during propagation. FEAP are a solution to Eq. (3) with so that the self-phase modulation (SPM) is absent. The linear propagation of self-accelerating FEAP is displayed in Figs. 1(a) and 1(b).
Figures 1(a) and 1(b) show the temporal and spectral evolutions of the self-accelerating FEAP with the truncation coefficient and as a function of propagation distance in the linear regime. It is indicated by Fig. 1(a) that the self-accelerating FEAP displays all the remarkable properties of the ideal self-accelerating Airy pulses in a finite propagation distance. The shape of self-accelerating FEAP remains quasi-invariant over several dispersion lengths while again the intensity features tend to “freely accelerate” until dispersion dominates. Compared to the linear propagation of Gaussian pulse, whose pulse duration is same as that of the main lobe of self-accelerating FEAP [Figs. 1(c) and 1(d)], the temporal broadening rate of the self-accelerating FEAP is bigger than that of Gaussian pulse. The physical reason behind such difference can be understood by noting that self-accelerating FEAP have broader spectrum compared with that of Gaussian pulse. We carried out a series of numerical experiments to disclose the impact of combined effects of anomalous GVD and SPM on the propagation dynamics of the self-accelerating FEAP. It is found that its propagation dynamics is similar to that of self-decelerating FEAP .
3.1 Influence of Raman scattering on the propagation of self-accelerating FEAP
Figure 2 shows the temporal and spectral evolution of the self-accelerating FEAP with truncation coefficient and different launched peak powers. Its temporal evolution is apparently different from that of self-decelerating FEAP . A detailed comparison can be investigated and will be reported elsewhere. Figures 2(b)–2(d) show that, the shedding soliton slows down due to the Raman scattering effects, and it experiences successive collisions with self-accelerating side lobes. This is manifested as the bending trajectory in time domain and the spectral red-shift in frequency domain. For self-decelerating FEAP, the deceleration of their main lobe is enhanced . The temporal deceleration rate is directly proportional to parameter N. The larger the value of parameter N, the more the time delay becomes evident, if assuming a fixed propagation distance [Fig. 2(a)]. The reason for this behavior can be understood as follows. When Raman scattering effects is not taken into consideration, the primary lobe of self-accelerating FEAP with high launched peak power N has enough energy to form soliton within a short propagation distance, and then shed it with a short pulse width. Short pulse duration is very helpful for enhancing the RIFS .
In the frequency domain [right column of Fig. 2], the RIFS of self-accelerating FEAP was observed in pulse spectrum toward long wavelengths along the fiber. When a self-accelerating FEAP for which the width and peak power are chosen such that , is launched inside an ideal lossless fiber, the peak intensity of shedding soliton is smaller than 1, indicating that its pulse width is most broad [Fig. 2(a)]. It means that the Raman effects play a relatively minor role here. So the RIFS is not obvious. With increasing launched peak power, the peak intensity of shedding soliton is increased above 1, and its pulse width becomes narrower. As a result, the spectrum shifts significantly towards longer wavelength due to the contribution of RIFS, while the blue-shifted spectra almost remain the same. At the early stage, the Raman-induced red-shifting frequency will interfere with the residual frequency, leading to discrete multi-peaks. As the propagation distance is increased, the IRS effects gradually start to dominate and shift soliton to the back of pulse. The larger the value of the parameter N, the greater and smoother the outermost red-shifted spectra will be obtained. Clearly, when and 1.2, the spectral evolutions exhibit multi-peaks structure as a result of such interference [Figs. 2(f) and 2(g)]. For , such multi-peaks structure is still seen in the initial stage of propagation. This unique behavior can be understood as follows. The self-accelerating FEAP has a broader spectrum. Because of the SPM-induced frequency chirp, at distinct temporal points, the pulse may have the same instantaneous frequency but different phases that may interfere destructively and constructively relying on their relative phase difference. In addition, Raman effects become weak in the case of low peak power. The pulse spectra show multiple peaks structure as a result of such interference.
3.2 Influence of truncation coefficient on the RIFS of self-accelerating FEAP
The truncation coefficient is a very important parameter of self-accelerating FEAP since its unique features depend considerably on the truncation degree. When the self-accelerating FEAP have a large value of truncation coefficient, their unique features disappear rapidly. In this section, we are devoted to disclose how the truncation degree affects the RIFS of self-accelerating FEAP. We set the parameter N to 1.6 while the truncation coefficient is limited to the range of 0.03 to 0.15.
Figure 3 shows the temporal and spectral evolution of self-accelerating FEAP with three different values of truncation coefficient over 15 dispersion lengths by numerically solving Eq. (3) with . It is clearly seen that the impact of truncation coefficient on the RIFS of self-accelerating FEAP. The smaller the truncation coefficient, the larger the temporal shift of the shedding soliton position. Figures 3(a) and 3(e) present, for comparison, the output pulse shapes and spectra at propagation distance for the truncation coefficient to 0.05, 0.10, and 0.15. It is indicated that the peak power and the temporal shift of shedding soliton increase with the decreasing truncation coefficient; as a result, the RIFS decreases with an increase in the truncation coefficient. But the peak power and spectral width of the outermost red-shifted spectra are almost the same for different truncation coefficients. The reason is that, for small truncation coefficient, the self-accelerating FEAP contains more energy owing to much more side lobes, leading to the formation of a soliton with high peak power and short pulse width. The shorter pulse with higher peak power is capable of generating a larger RIFS, because RIFS scales quadratically with the parameter , or with the width as . Moreover, the truncation coefficient can be used to manipulate the RIFS for realizing a broad tunable light source. It should be pointed out that the increasing number of side lobes is not clearly visible in Figs. 3(b)–3(e) when the truncation coefficient decreases, because their intensities are much smaller compared to that of the main lobe. On the other hand, the time window covers a wide range from −20 to 210, resulting in loss of visual quality. If the evolution of absolute amplitude , rather than of the intensity , and a smaller time window are displayed, a better visualization of the increasing number of the low-intensity side lobes will be obtained.
To get a better understanding of the influence of truncation coefficient on the RIFS, we computed the time centroid and frequency centroid which defines that as a function of propagation distance. Figure 4 shows the time centroids () and frequency centroids () over 15 dispersion lengths for different values of truncation coefficients. It is clearly seen from Fig. 4(b) that the frequency centroids are almost the same over the first 5 dispersion lengths. While numerical results shown in Fig. 4(a) indicate that the time centroids bears small deviations over the first 5 dispersion lengths for different truncation coefficient because the time centroid of initial FEAP depends on their truncation coefficient. With further increment of the propagation distance, they are quite sensitive to the truncation coefficients with its value in the range from 0 to 0.1. This is very useful for spectral tuning based on the RIFS of the self-accelerating FEAP.
3.3 Comparison of the RIFS of self-accelerating FEAP and that of symmetric pulses
The RIFS of light pulses with symmetric profiles such as Gauss and sech have been intensively investigated. A comparison shown in Fig. 5 reveals difference between the RIFS of the self-accelerating FEAP and that of symmetric Gauss and sech pulses. Figures 5(a) and 5(d) show the output pulse intensity and spectra at in the anomalous dispersion regime under the action of SPM () and Raman scattering ()effects. The time and frequency centroids (, ), , and TP of are also shown in Fig. 5. A careful comparison shows that, under the action of Raman effects, the outermost red-shifted spectra with larger intensity and much larger shift will be obtained by using the self-accelerating FEAP pump. It can be seen from Fig. 5(e) that the frequency centroid of self-accelerating FEAP is almost identical to that of soliton in first 9 dispersion lengths. With further propagation, the frequency shifting rate of self-accelerating FEAP is faster than that of solitons. The RIFS of Gaussian pulse is not obvious compared with that of the other input pulses. We compare the maximum intensity evolutions for three cases in Fig. 5(c): at the first 5.5 dispersion lengths, the maximum intensity of soliton shed from the main lobe of FEAP is comparable to that of sech pulse and is larger than that of Gauss pulse; but it becomes significantly larger as propagation distance is increased. The reason is that, at the initial propagation stage, the soliton is formed out of the main lobe of FEAP and experiences successive collisions with side lobes; the shedding soliton then interacts with the dispersion background, leading to energy exchange in the propagation process . As a result, the maximum intensity of shedding soliton increases gradually and exhibits multi-peaks structure. The larger the peak intensity, the bigger the RIFS of soliton shows.
In summary, we have numerically investigated the nonlinear propagation of self-accelerating FEAP in optical fibers with emphasis on the Raman scattering effects. It is shown that soliton is shed from the main lobe of self-accelerating FEAP and then slows down, as is apparent from its bent trajectory. At the same time, the shedding soliton interact with secondary lobes and dispersive background, resulting in its maximum intensity being periodically manifested. Furthermore, we find that the RIFS of shedding soliton can be manipulated by varying the truncation coefficient imposed on the self-accelerating FEAP. It is shown that the smaller the truncation coefficient, the bigger the amount of the RIFS. In addition, we also present a comparison of the nonlinear propagation of self-accelerating FEAP with that of pulse having the same width of the main lobe of self-accelerating FEAP as well as symmetric profiles (such as Gauss, sech), showing that the asymmetric self-accelerating FEAP with truncation coefficient exhibits the largest RIFS. The numerical results obtained demonstrate that a tunable light source based on RIFS of the self-accelerating FEAP can be achieved for various potential applications.
This work was supported by the Natural Science Foundation of SZU (Grant Nos. 201449, 201450) and the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ4108).
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