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Abstract
Based on the dispersion management technology, the transmission characteristics of Airy pulses in optical fiber systems are studied theoretically and numerically. The results show that the group velocity dispersion and third-order dispersion that change periodically along the transmission direction of the optical fiber can prolong the transmission of pulses. Under the action of periodically varying group velocity dispersion, the symmetrical reversal of the Airy pulse can be realized which the shape of the pulse keeps invariable and the tail converses after the inversion. When the periodic third-order dispersion effect is also considered, the periodically symmetrical reversal of the Airy pulse happens and the pulse can be regenerated periodically at the certain transmission distance. Adjusting the parameters of the third-order dispersion, the inversion position and the period can be manipulated. In addition, it is found that the tight focusing of the Airy pulse is also controllable.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The Airy beam is a non-diffracting wave packet predicted by Berry and Balazs in the context of quantum mechanics [1]. Since the Airy beam has infinite energy, it could not be achieved experimentally for practical applications. In 2007, the concept of finite energy Airy beam (FEAB) was proposed theoretically and then realized experimentally [2,3]. It is found that FEABs exhibit the unique characteristics of quasi-diffraction-free, self-acceleration and self-healing [4–8], which attracted great research interest. Due to their potential applications, more and more intriguing results have been reported [9–19].
Because temporal dispersion and spatial diffraction are isomorphic, the concept of finite energy Airy pulses (FEAP) was introduced. The FEAP are self-accelerating light wave packets distributed over time, which have self-accelerating properties in dispersive media. Recently years, Airy pulses have been widely studied by researchers. In particular, the influence of Airy pulses in the nonlinear domain has received extensive attention, such as Airy pulses form solitons during propagation under the action of Kerr nonlinearity [20], the evolution of FEAPs in optical fibers with cubic-quintic nonlinearity [21], dynamic propagation of FEAP and the manipulation of Raman-induced frequency shift in the presence of higher-order effects [22,23], controllable transmission of Airy pulses in nonlinear dissipative system [24]. Meanwhile, the interaction of pulses has also been extensively studied, such as interactions of two truncated Airy pulses produces diverse breathing solitons in optical fibers [25], the dynamic propagation of symmetric Airy pulses with initial chirp in optical fibers [26], controlling soliton self-frequency shift via Airy-soliton interactions [27] etc.
Further, some initiatives have been introduced recently where the time varying optical potentials are used to manipulate the trajectory of FEAP. In presence of the linear potential, the acceleration of Airy pulse can be counteracted with the appropriate value of linear potential [28]. The Airy pulse trajectory can be engineered in the presence of external harmonic potential [29]. Especially, when the parabolic potential is considered, the FEAP exhibits the periodic oscillation trajectory [30]. In addition, dispersion management (DM) techniques can also be used to manipulate the trajectory of Airy pulse [31]. Recently, a scheme based on DM was proposed for the increase of Airy pulse longevity [32–34]. When the pulse width is shorter, the third-order dispersion (TOD) should be considered. The existence of TOD will cause the Airy pulse inverse but the peak power decrease after inversion [35–37]. How to keep the inversion pulse invariant is an issue which is not indicated in other literatures as far as we know. In this paper, based on dispersion management technology, the symmetrical reversal and regeneration of Airy pulse are investigated and the requirement for symmetrical reversal is analytically deduced. The results show that under the joint action of periodic GVD and TOD, the periodic inversion and regeneration can be realized. And adjusting the parameters of TOD, the reversion position and period can be controlled.
The paper is organized as follows. In the next section, the theoretical model is introduced and analysis is performed. In Section 3, the transmission characteristics of Airy pulses under different GVDs and TODs are discussed theoretically and numerically. In the end, the main results are summarized in Section 4.
2. Theoretical model and analysis
When GVD and TOD effects dominate in linear dispersion-managed fibers, the transmission of the pulse is described by the normalized variable coefficient linear Schrödinger equation [31]:
The initial incident is an Airy pulse truncated exponentially:
where ${A_0}$ is the amplitude of the input pulse; a is the truncation parameter and set to be 0.05.By solving Eq. (1) with the initial input (2), the transmission dynamics of Airy pulse can be expressed as:
WhereIt can be seen from Eq. (6) that when $\eta ({{Z_n}} )= 2$, there will be a singularity in Eq. (3). The location of the singularity is ${Z_n} = {\eta ^{ - 1}}(2 )$ (n = 0,1,2…), where Airy pulse is nullified and exhibits a Gaussian envelope. The corresponding expression is:
It can be noted from Eq. (8) that for a small truncation coefficient a, the approximate expression of the FWHM (full width at half maximum) of the Gaussian pulse at the reversal position is:
3. Numerical analysis and discussion
3.1 Symmetrical reversal transmission of Airy pulses
In this section, the transmission characteristics of Airy pulse under periodic varying GVD and constant coefficient of TOD are discussed with $\Gamma (Z )= 0$. Set ${\delta _3}(Z )= 1$, according to Eq. (6) in which $\eta (Z )= 2$, there will be a singularity ${Z_0}$. The pulse is reversed at the singularity [31]. Equation (3) clearly describes the expression of the transmission trajectory of the Airy pulse along the fiber:
When the corresponding T at the peak position is replaced by the transmission trajectory (11), the peak power evolution of the pulse is approximately expressed as: where ${\rm K} = RaT(Z )+ \frac{{a{R^2}}}{\textrm{2}}\rho {(Z )^2} - \frac{{{a^\textrm{3}}}}{3}({1 + 2{R^2} - 3R} )$, ${P_0}$ is the initial peak power.According to the analysis of Eq. (11), when the truncation parameter a is small and $\rho {(Z )^2}$ is a symmetry function about ${Z_0} = 2$, the approximate expression of the pulse transmission trajectory can be obtained: $T({ - Z + {Z_0}} )\approx{-} T({Z + {Z_0}} )$. At this time, it can be found that the transmission trajectory of the pulse is oddly symmetrical with respect to the reversal point. In the same way, according to Eq. (12) we can approximate the expression of $P({ - Z + {Z_0}} )\approx P({Z + {Z_0}} )$, the peak power of the pulse is even symmetrical about the reversal point. The theoretical analysis indicates that the Airy pulse undergoes symmetrical inversion with appropriate form of GVD and the reversal recovery can be realized. In addition, by setting the TOD, it can adjust the position of reversal point.
To confirm the propagation dynamics of Airy pulses obtained from the analytical analysis, Eq. (1) is numerically simulated using the split -step Fourier method. We set the GVD is longitudinally periodical to be ${\delta _2}(Z )= h\cos ({b\pi Z} )$, h and b represent the oscillation intensity and period respectively, in which $b = {k / {({2{Z_0}} )}}$ (k = 1,2,3…) is to satisfy the condition that $\rho {(Z )^2}$ is symmetrical about the reversal point. Figure 1 shows the dynamic evolution of Airy pulse with different h and b. It can be clearly seen that the pulse reverses at ${Z_0} = 2$ and the output (red dotted line) restores its original shape (black solid line) with the direction opposite, which is indicated in Figs. 1(a)-(d). (I) When k is an odd number, there will be an inversion region in the pulse transmission, and the area of the inversion region gradually increases with the increase of h; When h is the same, the area of the inversion region gradually decreases with the increase of k, as shown in Figs. 1 (e) (i) and (g) (k). (II) When k is an even number, the pulse will be tightly focused at the inversion, and the trajectory oscillation will become more obvious as h increases, as shown in Figs. 1 (f) (j) and (h) (l). Therefore, by adjusting the parameters of periodic varying GVD, it can not only control the pulse to switch between the reverse and the tight focus, but also adjust the size of the reversal region and the transmission trajectory.
To further illustrate the above results, Fig. 2 shows the evolution of the peak power of the Airy pulse with different k and h. It can be seen that the peak power is symmetrical about the reversal point. When k is an odd number, it corresponds to the case (I). With the increase of h, the peak power decreases and the corresponding inversion region becomes wider, in which the peak power closes to stable, as shown in Figs. 2(a) and (c). When k is even, it corresponds to the case of tight focus (II). With the increase of h, the peak power decreases, but the peak power changes rapidly near the tight focus and a sharp of peak power appears. With the increase of k there will be multiple sub-peaks on both sides, and the sub-peaks become more obvious when h increase, as shown the curves in Figs. 2(b) and (d).
Fig. 2. Peak power of Airy pulse for different oscillation intensity h. (a) k = 1; (b) k = 2; (c) k = 3; (d) k = 4.
3.2 Periodic reversion and regeneration of Airy pulses
As mentioned above, with the action of the periodic varying GVD and the constant TOD, the pulse undergoes the symmetrical reverse but reverse only once. In this section, both of the periodic GVD and TOD are considered with ${\delta _2}(Z )= h\cos ({b\pi Z} )$ and ${\delta _3}(Z )= {h_1}\sin ({{b_1}\pi Z} )$, in which ${h_1}$ and ${b_1}$ represent the oscillation intensity and period of TOD respectively. According to Eq. (6), it can be deduced that $\eta (Z )\textrm{ = }\frac{{{h_1}}}{{{b_1}\pi }}\textrm{ - }\frac{{{h_1}}}{{{b_1}\pi }}\cos ({{b_1}\pi Z} )$ and when ${{{h_1}} / {({{b_1}\pi } )}} = 2$, it exists multiple singularities in Eq. (3) and the occurrence of each singularity has a certain periodicity which is:
${Z_n}$ represents the position where the nth singularity occurs. It can be known from Eq. (13) that the reversal position of the pulse is related to the TOD period ${b_1}$, so the position of the reversal point and the reversal period can be controlled. Figure 3 shows the dynamic evolution of Airy pulse with different k, where ${h_1} = \pi $, ${b_1} = 0.5$, h = 2. Obviously, when ${R^{ - 1}} = 0$, the singularity exists and the location is:${Z_n} = 1 + 2n$. It can be seen that the pulse trajectory exhibits periodical inversion and the period is 4 with the selected parameters. Similarly, when k is odd, it displays a finite area at the inversion; when k is an even number, tight focusing occurs at the reversal position in propagation, as shown in Figs. 3(b) and (c). The pulse profiles under different propagation distances are presented in Fig. 3(d). It reveals the symmetrical reverse and regeneration periodically of Airy pulse and exhibits a Gaussian envelope at the reverse points during the transmission within two cycles. The numerical simulation, black solid line is well consistence with the analytical solution, the red dotted line.Figure 4 further discusses the dynamics of Airy pulse with different ${b_1}$. It can be seen with the decrease of ${b_1}$ and satisfying the condition ${{{h_1}} / {({{b_1}\pi } )}} = 2$, the reversal period increases. Therefore, under the joint action of periodic GVD and TOD, not only the periodic symmetrical inversion transmission of Airy pulses can be realized, but also the inversion period is controllable.
Fig. 3. (a) Periodic singularity occurs. (b)(c) Transmission characteristics of Airy pulses with different k. (d) Pulses profiles at k = 1 under different transmission positions. Here ${h_1} = \pi $, ${b_1} = 0.5$, h = 2.
Fig. 4. Transmission characteristics of Airy pulses with different TOD parameters. Here $b = {k / {({2{Z_0}} )}}$, k = 1, h = 4.
Figure 5 discusses the effect of the truncation coefficient on the evolution of the pulse peak power and the pulse shape at the reversal position. Figure 5(a) shows the evolution of the peak power of Airy pulse transmission when the truncation coefficient takes different values. The results show that the peak power of the pulse decreases with the increase of the truncation parameter, and the area of the inversion region increases slightly. The change of the truncation coefficient does not change the position of the reversal point and when the pulse reaches the inversion point, the truncation coefficient has little effect on the peak power of the Gaussian pulse,as shown in Fig. 5(b). At the same time, when the truncation coefficient increases gradually, the pulse width of the Gaussian pulse at the inversion becomes smaller, which can be verified from the expression of the Gaussian pulse in Eq. (10).
Fig. 5. (a) Peak power variation of Airy pulse versus propagation distance with different truncation coefficient a; (b) Airy pulses at reversal point Z0 for different a. Here h = 2, k = 1, ${h_1} = \pi $, ${b_1} = 0.5$.
3.3 Periodic tight focusing transmission of Airy pulses
From the previous sections, tight focusing occurs at the reversal position when the TOD coefficient is much larger than the GVD coefficient. This section focuses on controlling where tight focus occurs through related parameters. With the selected periodic GVD parameters, when $b = {1 / {[{{Z_0}({2{k_1} - 1} )} ]}}$ (${k_1}$=1,2,3…), the reversal position ${Z_{{k_1} + ({2{k_1} - 1} )n}}$ (n = 0,1,2…) satisfies $\rho ({{Z_{{k_1} + ({2{k_1} - 1} )n}}} )= 0$. It means that the periodic tight focusing can be achieved selectively and the position where the tight focus occurs can be regulated by controlling the value of the period GVD parameter.
Figure 6 shows the transmission dynamics of Airy pulses under tight focusing with different periods, and the values of k are 2, 3, 4, and 5. When ${k_1} = 2$, the Airy pulse produces the first tight focus at the reversal position ${Z_2}$ and recycles every two inversion regions. It can be seen that the position where the tight focusing first appears in the pulse transmission is ${Z_{{k_1}}}$, and the period of the tight focusing is $2{k_1} - 2$, which indicates that the position and period of tight focusing are controllable.
Fig. 6. Transmission evolutions of Airy pulses with different GVD parameter ${k_1}$. (a) ${k_1} = 2$; (b) ${k_1} = 3$; (c) ${k_1} = 4$; (d) ${k_1} = 5$.
4. Conclusions
In this paper, we investigate the propagation dynamics of temporal Airy pulses in inhomogeneous fibers through theoretical analysis and numerical simulation. The effects of periodic GVD and TOD on the transmission characteristics of Airy pulses are studied in detail. The results show that the symmetrical inversion of the pulse can be achieved under the action of periodic GVD, and the peak power is symmetrical about the inversion position. When the dispersion period k is odd, there is a finite area at the reversal position and dispersion intensity h determines the size of reversal region. When the coefficient k is even, tight focusing occurs at the reversal position. Under the joint action of periodic GVD and TOD, the periodic inversion can be realized and in each inversion period, the pulse shape is regenerated with the certain transmission period. Adjusting the parameters of TOD, the reversion position and period can be manipulated. In addition, it is found that the tight focusing of the Airy pulse is also controllable. The results may lead to the potential applications in pulse remodeling and signal processing systems.
Funding
the Shanxi province Postgraduate Innovation Project (2021Y156); National Natural Science Foundation of China (11705108).
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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