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Abstract
We investigate to the best of our knowledge the first time the interactions of in-phase and out-of-phase Airy beams in Kerr, saturable and nonlocal nonlinear media with fourth-order diffraction using split-step Fourier transform method. Directly numerical simulations show that normal and anomalous fourth-order diffractions have profound effects on the interactions of the Airy beams in Kerr and saturable nonlinear media. We demonstrate the dynamics of the interactions in detail. In nonlocal media with fourth-order diffraction, nonlocality induces a long-range attractive force between Airy beams, leading to the formation of stable bound states of both in-phase and out-of-phase breathing Airy soliton pairs which are always repulsive in local media. Our results have potential applications in all-optical devices for communication and optical interconnects, etc.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
After the pioneering work in quantum-mechanics [1], self-accelerating Airy beams and Airy pulses have been introduced in optics and drawn considerable attention during the past decade [2–4]. In particular, interactions of Airy beams in diversely nonlinear media were a hot topic [4]. In photorefractive crystal, interactions between two Airy beams [5,6] and temporal behaviors of the interactions [7] were studied theoretically. Waveguides induced by interactions of incoherent counter-propagating Airy beams [8–11] and soliton formation by decelerating interacting two-dimensional Airy beams [12] have been demonstrated experimentally in photorefractive media. In Kerr media, bound states of soliton pairs can be obtained by coherent [13–15] and incoherent [16] interactions of Airy beams. In optical fiber, interactions between Airy pulse [17–19], as well as generation of Airy pulses soliton pairs [20], were investigated extensively. Furthermore, interactions of Airy beams were discussed in quadratic nonlinear medium [21] and fractional Schrödinger equation [22–24]. Recently, interactions of Airy-Gaussian beams were also considered [25–28].
The dynamics of optical beams in nonlinear media depend crucially on the linear diffraction/dispersion and nonlinearity [29]. Generally, people only consider quadratic diffraction/dispersion effect. However, in realistic situations, high-order diffraction/dispersion should also be considered [30] for it affects the dynamics of solitons [30–34] and Kerr frequency comb generation [35] deeply. What is more, fourth-order diffraction/dispersion have drawn considerable attention in many nonlinear physical areas, such as kink dynamics [36], self-similar propagation of optical pulses [37], localized dissipative structures [38], spectra of polarization-modulational instability [39], beating between intrinsic frequencies [40], pure-quartic soliton laser [41], and self-trapping of novel soliton states [42–47].
Many works have shown that higher-order dispersion/diffraction, e.g., third-order dispersion, affects deeply the dynamics of Airy pulse [48,49]. As to fourth-order dispersion, the propagation of asymmetric and chirped Airy pulse [50], synchrotron resonant radiation [51], and solitons shedding from Airy pulses [52] in fiber have been investigated in detail.
In this paper, for the first time to our best knowledge, we investigate numerically the interactions of both in-phase and out-of-phase Airy beams in Kerr, saturable, and nonlocal nonlinear media with fourth-order diffraction. We demonstrate the dynamics of interactions using split-step Fourier transform method. In Kerr and saturable nonlinear media, normal fourth-order diffraction weaken the repulsion and anomalous fourth-order diffraction strengthen the repulsion of out-of-phase Airy beams. For in-phase Airy beams, normal diffraction weaken and anomalous diffraction strengthen the attraction in Kerr nonlinear media, which is opposite to the case in saturable nonlinear media. With the help of nonlocality, we can obtain stable bound states of breathing Airy soliton pairs in nonlocal nonlinear media which always repel in local nonlinear media. The theoretical works can be applied to optical switch and optical interconnects.
2. Physical model
Considering an Airy beam in a bulk nonlinear self-focusing media with fourth-order diffraction, the envelope $\psi (x,z)$ is described by the following normalized nonlinear Schrödinger equation [43,44],
Here, we only consider coherent interactions [53] of Airy beams, and two shifted counter propagating Airy beams can be represented as [13,14],
here $A$ and $B$ are amplitude and beam intervals. $\rho =0$ and $\rho =1$ describe in-phase and out-of-phase Airy beams, respectively [13,14]. $a>0$ is the decaying factor [2]. In this paper, $a=0.2$ is assumed. In Fig. (1), we show the intensities of Airy beams with different values $B$ when $\rho =0$ and $\rho =1$, respectively.3. Kerr nonlinear media
Firstly, interactions of Airy beams will be investigate numerically in Kerr medium. We integrate directly Eq. (1) with split-step Fourier transform method. Here $\delta n(I)$ is described as
Firstly, we consider the interactions of Airy beams with $\rho =0$. We show in Fig. (2) the numerical results with different $B$ and fourth-order diffraction coupling constant $\beta$. For well separated Airy beam, e.g., $B=4$ and $B=-3$, interactions are weak and two parallel solitons can be generated [13]. Thus here we only focus on the interactions with smaller beam intervals ($B=0, 1, 2, 3$). We also assume the amplitude is $A=3$.
When $\beta =0$ (without fourth-order diffraction), breathing solitons pairs can be obtained with different breathing periods [Figs. 2(a2-d2)] [13,14]. The attraction become stronger and the breathing period decrease when the beam interval $B$ decreases [13,14]. When $B = 1$, the attraction is strongest with the smallest breathing period [Fig 2(b2)] due to the incident beam has the maximum peak intensity [Fig. 1(a)] [14].
Fig. 1. Intensity distribution of the incident beams with different $B$. Solid lines describe out of phase ($\rho =1$) and dashed lines represent in-phase ($\rho =0$) Airy beams, respectively. The parameters are $A=3$ and $a=0.2$.
Fig. 2. Intensity trajectories of interactions of Airy beams with $\rho =0$ in Kerr medium. The beam intervals are B=0 (a1-a3), B=1 (b1-b3), B=2 (c1-c3), and B=3 (d1-d3). The fourth-order diffraction coupling constants are $\beta >0$ (normal diffraction), $\beta =0$ (without fourth-order diffraction), and $\beta <0$ (anomalous diffraction) for (a1-d1), (a2-d2), and (a3-d3), respectively. The amplitudes are $A=3$.
When $\beta >0$ (normal diffraction), as shown in Figs. 2(a1-d1), bound breathing solitons form initially, and eventually repulsions emerge for all beam intervals. However, for $\beta <0$ (anomalous diffraction), situations become different. From Figs. 2(a3-d3), it is obviously that attractions between in-phase Airy beams are enhanced obviously. The breathing period of the bound state become smaller for all $B$. Generally, the bound states are balances between the effects of diffraction and nonlinearity. The diffraction effect of the beams will be strengthen (weaken) for $\beta >0$ ($\beta <0$). Thus nonlinearity can not overcome the normal fourth-order diffraction and interactions of Airy beams with $\rho =0$ exhibit repulsive behaviors for $\beta >0$. Otherwise, nonlinearity will reinforce the interactions for anomalous fourth-order diffraction ($\beta <0$).
Besides the results obtained above, we also find some anomalous interactions of the in-phase Airy beams. When $B=0$ and $\beta =-0.001$, we can see repulsion appears in the case of anomalous fourth-order diffraction. The similar phenomena and corresponding physical mechanism have been studied in previously works [14,54].
Another anomalous interaction happens when $B=3$ and $\beta =0.001$. From Figs. 3(b) and 2(d2), it is clearly that the attraction of the Airy beams increases obviously under the condition of normal diffraction. From Fig. 1(c), we can see another intensity peak is located at the middle of two main lobes of the in-phase Airy beam. Although the linear diffraction is enhanced with $\beta >0$, the third intensity peak helps the bound state to attract better [54].
Fig. 3. Intensity trajectories of interactions of Airy beams with $\rho =0$ in Kerr medium. The beam intervals are B=0 (a) and B=3 (b). The fourth-order diffraction coupling constants are $\beta =-0.001$ (a) and $\beta =0.001$ (b). The amplitudes are $A=3$.
Secondly, we focus on interactions of Airy beams with $\rho =1$. When $\beta =0$ (without fourth-order diffraction), as shown in Figs. 4(a2-d2), soliton pairs are generated and repel each other [13,14]. The repulsion become stronger when the beam interval decrease [13]. When $B=0$, repulsion of the soliton pairs is strongest [13,14]. Normal fourth-order diffraction ($\beta >0$) always weaken the repulsion [Figs. 4(a1-d1)], whereas, anomalous fourth-order diffraction ($\beta <0$) always enhance the repulsion [Figs. 4(a3-d3)]. In fact, for interactions of out-of-phase beams, the stronger (weaker) the nonlinearity, the stronger (weaker) the repulsion. Similar with the case of in-phase Airy beams, normal (anomalous) fourth-order diffraction will strengthen (weaken) the diffractive effect, hence the nonlinear effect is weaken (strengthen) accordingly, result in the decrease (increase) of the repulsion when $\rho =1$.
Fig. 4. Intensity trajectories of interactions of Airy beams with $\rho =1$ in Kerr medium. The beam intervals are B=0 (a1-a3), B=1 (b1-b3), B=2 (c1-c3), and B=3 (d1-d3). The fourth-order diffraction coupling constants are $\beta >0$, $\beta =0$, and $\beta <0$ for (a1-d1), (a2-d2), and (a3-d3), respectively. The amplitudes are $A=3$.
4. Saturable nonlinear media
Subsequently, we investigate numerically interactions of Airy beams in saturable medium with fourth-order diffraction. Thus $\delta n(I)$ is represented as
As shown in Fig. (5), for smaller beam intervals, the interaction of in-phase ($\rho =0$) Airy beams in saturable medium can only generate one central individual solitons [Figs. 5(a1-a3)], whereas, it can generate one central individual solitons and soliton pairs for larger beam intervals [Figs. 5(b1-b3)] [13,14]. Fourth-order diffraction also affects the interactions of Airy beams with $\rho =0$, however, different with dynamics in Kerr nonlinear media. When beam interval is larger $B=3$, from Figs. 5(b1-b3), we can see that the attraction of the soliton pair is strengthened for normal diffraction ($\beta >0$) and weakened for anomalous diffraction ($\beta <0$). However, normal (anomalous) fourth-order diffraction always weaken (strengthen) the attractions of in-phase Airy beams in Kerr medium [Fig. (2)]. As shown in Figs. 5(a1-a3), when the beam interval $B$ is smaller, the width of the individual central solitons becomes narrower (wider) with normal (anomalous) fourth-order diffraction, which also indicate the nonlinear effect become stronger (weaker) when the coupling constant $\beta$ is positive (negative).
Fig. 5. Intensity trajectories of interactions of Airy beams with $\rho =0$ in saturable medium. The beam intervals are B=2 (a1-a3) and B=3 (b1-b3). The fourth-order diffraction coupling constants are $\beta >0$, $\beta =0$, and $\beta <0$ for (a1, b1), (a2, b2), and (a3, b3), respectively. The amplitudes are $A=3$.
The interactions of Airy beams with $\rho =1$ in saturable medium can only generate repulsive soliton pairs without central individual solitons [13,14], as shown in Figs. 6(a2-d2). When fourth-order diffractions are took into account, dynamics of the interactions are similar with the situation in Kerr medium. Normal fourth-order diffraction ($\beta >0$) will weaken the repulsion [Figs. 6(a1-d1)], e.g., parallel soliton pair without repulsion can even be generated for $B=3$ and $\beta =0.1$ [Fig. 6(d1)]. It is also true that anomalous fourth-order diffraction ($\beta <0$) will enhance the repulsion [Figs. 6(a3-d3)]. The physical reason of interactions of Airy beams with $\rho =1$ in saturable medium has been explained in Kerr medium.
Fig. 6. Intensity trajectories of interactions of Airy beams with $\rho =1$ in saturable medium. The beam intervals are B=-1 (a1-a3), B=1 (b1-b3), B=2 (c1-c3), and B=3 (d1-d3). The fourth-order diffraction coupling constants are $\beta >0$, $\beta =0$, and $\beta <0$ for (a1-d1), (a2-d2), and (a3-d3), respectively. The amplitudes are $A=3$.
5. Nonlocal nonlinear media
At last, interactions of Airy beam will be studied in nonlocal medium [55] with fourth-order diffraction. In the area of optics, nonlocal nonlinearity means that the light-induced refractive index change of a material at a particular location is determined by the light intensity in a certain neighborhood of this location [55]. Nonlocality affects propagation [56–63] and interactions [54,64–69] of Airy beams in unique manners. In nonlocal Kerr medium, $\delta n(I)$ is represented as [55]
where $R(x)$ is nonlocal response functions [55]. The profile of nonlocal response function $R(x)$ of realistic medium, e.g., nematic liquid crystals [70,71] and lead glass [72], is in the exponential profile where $\sigma$ is the degree of the nonlocality [55].Without fourth-order diffraction ($\beta =0$), interactions of Airy beams with $\rho =0$ [Figs. 2(a2-d2)] and $\rho =1$ [Figs. 4(a2-d2)] in nonlocal medium have been discussed previously in detail [65]. Here we only concentrate on the interactions in nonlocal media with fourth-order diffraction. Nonlocality helps to form stable bound states of Airy beams [65]. Figure (7) displays interactions of in-phase ($\rho =0$) Airy beams with both normal and anomalous fourth-order diffraction in nonlocal Kerr nonlinear media with different $\sigma$. For $\beta >0$, we can see nonlocality can balance the repulsion induced by normal diffraction, result in bound states of soliton pairs [Figs. 7(a1-d1)], which repel in local Kerr medium [Figs. 2(a1-d1)]. For $\beta <0$, nonlocality also helps to reduce the breathing periods of the bounding soliton pairs [Figs. 7(b2-d2)], and overcome the repulsion [Fig. 7(a2)] observed in Fig. 3(a).
Fig. 7. Intensity trajectories of interactions of Airy beams with $\rho =0$ in nonlocal medium with different $\sigma$. The parameters are B=0 (a1, a2), B=1 (b1, b2), B=2 (c1, c2), and B=3 (d1, d2). The fourth-order diffraction coupling constants are $\beta >0$ and $\beta <0$ for (a1-d1) and (a2-d2), respectively. The amplitudes are $A=3$.
In nonlocal media, stable bound states of Airy beams with $\rho =1$ always require some larger amplitudes [65]. The interactions of out-of-phase Airy beams in nonlocal medium with both normal [Figs. 8(a1-d1)] and anomalous [Figs. 8(a2-d2)] fourth-order diffraction are displayed in Fig. (8). The parameters of beam intervals $B$ and coupling constants $\beta$ are same with Fig. (4), but with different amplitudes $A$. With appropriate $\sigma$, bound states of out-of-phase Airy beams with both normal and anomalous fourth-order diffraction can be formed, which are repulsive in local Kerr medium. For strong repulsion with smaller beam intervals, it always require a larger $\sigma$ to balance the repulsion [65]. For a given beam interval $B$, the amplitude $A$ with anomalous diffraction should be larger than that with normal diffraction [65].
Fig. 8. Intensity trajectories of interactions of Airy beams with $\rho =1$ in nonlocal medium with different $\sigma$. The parameters are B=0 (a1, a2), B=1 (b1, b2), B=2 (c1, c2), and B=3 (d1, d2). The fourth-order diffraction coupling constants are $\beta >0$ and $\beta <0$ for (a1-d1) and (a2-d2). The amplitudes are $A=12, 16$ for (a1, a2), $A=8, 10$ for (b1, b2), $A=5, 6$ for (c1, c2), and $A=4, 5$ for (d1, d2), respectively.
6. Conclusion
In conclusion, we have investigated the interactions of Airy beams in local Kerr, saturable, and nonlocal Kerr nonlinear media with fourth-order diffraction. We demonstrated the effects of normal and anomalous diffraction on the interactions of in-phase and out-of-phase Airy beams by directly numerical simulations using split step Fourier transform method. In nonlocal nonlinear media, with the help of nonlocality, stable bound states of breathing Airy solitons can be obtained in the presence of fourth-order diffraction which are repulsive in local nonlinear media. These numerical simulations are helpful to study the interactions of Airy pulses in optical fiber with fourth-order dispersion in the future.
Funding
Science and Technology Commission of Shanghai Municipality (19ZR1417900); National Natural Science Foundation of China (61975109).
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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