Abstract
We numerically investigate and statistically analyze the impact of medium parameters (modulation depth P, modulation factor ω, and gain/loss strength W _{0}) and beam parameters (truncation coefficient a and distribution factor χ _{0}) on the propagation characteristics of a coshAiry beam in the Gaussian paritytime (PT)symmetric potential. It is demonstrated that the main lobe of a coshAiry beam is captured as a soliton, which varies periodically during propagation. The residual beam selfaccelerates along a parabolic trajectory due to the selfhealing property. With increment in P, the period of a trapped soliton decreases almost monotonically, while the peak power of a trapped soliton increases monotonically. With the increase in ω or decrease in the absolute value of W _{0}, the period and peak power of a trapped soliton decrease rapidly and then almost remain unchanged. Moreover, it is indicated that the period of a trapped soliton remains basically unchanged no matter a and χ _{0} increase or decrease. The peak power of a trapped soliton increases with increment of a, but the peak power of a trapped soliton stays relatively constant irrespective of variation in χ _{0}.
1 Introduction
In 1979, Berry and Balazs deduced a nondiffracting Airy wave packet solution in Schrödinger equation [1]. In 2007, Christodoulides et al. first reported the generation of a finiteenergy Airy beam in experiment. Then, the Airy beam has attracted significant attention due to its unique properties, such as nondiffraction, selfacceleration, and selfhealing [1,2,3,4,5,6]. Because of these properties, an Airy beam has important applications in plasma waveguides [7], optical micromanipulation [8], light bullet [9,10], plasmonic energy routing [11], and so on. An Airy beam propagating in different kinds of medium has been investigated in recent years [12,13,14,15,16,17,18,19,20,21,22]. Combing Airy beams with other shaped beams, some novel beams have been proposed and studied, including Airy–Gaussian beams [23], Airy–Bessel beams [24], Airy–Vortex beams [25], Airy–Laguerre–Gaussian beams [26], Airy–Ince–Gaussian beams [27], and Airy–Hermite–Gaussian beams [28]. Recently, a coshAiry beam was proposed by superposition of two Airy beams with different truncation coefficients [29]. The beam propagation factor of a coshAiry beam has been studied by the secondorder moments [30]. The propagation characteristics of the coshAiry beam is similar to that of an Airy beam, except that the coshAiry beam has more manipulation degrees of freedom [29]. Additionally, the selfhealing ability of the coshAiry beam is higher than that of an Airy beam [31]. Moreover, few works reported on the propagation of the coshAiry beam in uniaxial crystals orthogonal to the optical axis [32], quadraticindex inhomogeneous medium [33], and parabolic potential [34].
With the rapid development of material technology, lots of new nonlinear materials have been discovered, such as the paritytime (PT)symmetric medium [35]. PTsymmetric medium is a kind of typical inhomogeneous medium. PTsymmetric demonstrates that the real and imaginary part of a complex potential must be an even function and an odd function, respectively [36]. Christodoulides et al. creatively introduced the PTsymmetric into optics field [37]. Adding the periodic gain/loss to an optical lattice, the distribution function of refractive index is an even function and the distribution function of gain/loss is an odd function, so a PTsymmetric optical lattice is constructed [38]. The PTsymmetric medium with some unique properties supports an optical soliton generation. Therefore, generation and propagation of a variety of PT soliton in different types of PTsymmetric potentials have been investigated extensively, for example periodic potentials [39], hyperbolic potentials [40], Bessel potentials [41], parabolic potentials [42], and Gaussian potentials [43].
The Gaussian PTsymmetric profile has an explicit expression, the real part is refractive index modulation, and the imaginary part plays a role in phase modulation. Therefore, research on the Gaussian PTsymmetric medium is particularly important. To the best of our knowledge, there are few research reports about coshAiry beams; however, the previous research works have not investigated the influence of PTsymmetric medium with Gaussian potential on the propagation properties of a coshAiry beam. In this work, we study the propagation characteristics of a coshAiry beam in an inhomogeneous medium with Gaussian PTsymmetric potential. Our simulation result illustrates that a trapped soliton is generated from the main lobe of a coshAiry beam, the residual part can also selfaccelerate along a parabolic trajectory. The propagation characteristics of a coshAiry beam are controlled by the Gaussian PTsymmetric potential parameters and the coshAiry beam parameters. This work is organized as follows. In Section 2, the model describing a coshAiry beam propagating in the Gaussian PTsymmetric potential is displayed. In Section 3, the influence of modulation depth, modulation factor, gain/loss strength, truncation coefficient, and distribution factor on the propagation properties of a coshAiry beam is analyzed and discussed in details. Conclusion is presented in Section 4.
2 Theoretical model
When a beam propagates in a onedimensional PTsymmetric optical lattice, the refractive index has the following form:
where φ is the amplitude of a beam, n _{0} is the linear refractive index, and n _{2} is the nonlinear index coefficient. n _{2} is positive or negative, which represents the selffocusing or selfdefocusing nonlinear effect, respectively. n _{R}(X) and n _{I}(X) are the real and imaginary parts of the complex refractive index, representing the refractive index distribution and gain/loss distribution of an optical lattice, respectively. To satisfy the PTsymmetry condition, the complex refractive index is written as follows:
According to the Eq. (2), it is demonstrated that the refractive index distribution must be an even function, and the gain/loss distribution must be an odd function. In other words, the real and imaginary parts of the complex refractive index must be even and odd symmetric, respectively. Without Kerr nonlinearity, we consider the (1 + 1)D normalized nonlinear Schrodinger equation for a coshAiry beam propagating in a Gaussian PTsymmetric potential [43].
where φ(x, z) is the field envelop, and z represents the normalized longitudinal coordinate. x = X/X _{0} is the dimensionless transverse coordinate, and X _{0} is the beam width. V(x) and W(x) represent the real and imaginary parts of a PTsymmetric potential, respectively. P is the modulation depth of a PTsymmetric potential. The Gaussian PTsymmetric potential is presented as follows [43]:
where ω is a modulation factor of the Gaussian PTsymmetric potential. W _{0} is the gain/loss strength. For the complex Gaussian PTsymmetric potential, all eigenvalues are real when the refractive index strength is stronger than the gain/loss strength, otherwise, the eigenvalues are mixed [43]. The initial field distribution of a coshAiry beam is taken as
where a is a truncation coefficient that must satisfy the condition 0 < a < 1 to ensure the physical realization of a finite energy coshAiry beam [1]. The side lobes of the coshAiry beam become more and more obvious when a is close to 0. The coshAiry beam becomes a Gaussian beam if a is close to 1. χ _{0} is a distribution factor, Eq. (6) stands for an Airy beam for χ _{0} = 0. Ai(·) represents an Airy function.
The coshAiry beam is regarded as the result of superposition of two Airy beams with different truncation coefficients [29], so Eq. (6) is expressed as
where
When the modulation depth and the gain/loss strength are set as P = 1 and W _{0} = 1, Figure 1 displays the refractive index distribution function V(x) and the gain/loss distribution function W(x) for various values of ω and the initial coshAiry curve with different χ _{0}. In Figure 1(a), it is presented that real part V(x) is even symmetry and imaginary part W(x) is odd symmetry, which satisfies the PTsymmetric condition. From the real part V(x), it is illustrated that the value of V(x) is positive and is maximum at x = 0. The value of V(x) is gradually close to 0 as x increases or decreases. The center refractive index of a Gaussian PTsymmetric potential is the largest, while the refractive index gradually decreases to 0 away from the center position. From the imaginary part W(x), it is shown that the value of V(x) is positive for x > 0, V(x) is negative for x < 0, and the maximum absolute value of W(x) occurs around x = 0. The value of W(x) is gradually close to 0 as x increases or decreases. The Gaussian PTsymmetric potential generates loss and gain for x < 0 and x > 0, respectively. In Figure 1(b), it is shown that the side lobe intensity of a coshAiry beam with truncation coefficient a = 0.1 gradually increases with increment of χ _{0;} however, the main lobe intensity varies little.
From Figure 1, it is also found that the coshAiry beam will be constrained due to the highest refractive index in the Gaussian PTsymmetry center. Meanwhile, the power of a coshAiry beam will generate oscillation because the gain/loss of imaginary part are inconsistent near the Gaussian PTsymmetry center. The modulation range is wide enough to control the entire coshAiry beam for ω = 0.1 (blue cure). However, the potential width or modulation range is much narrower that merely covers the main lobe of a coshAiry beam for ω = 1.0 (red curve). A more complicated inhomogeneous medium can be composed by superposition and combination of these different unique properties medium. Therefore, the influence of a Gaussian PT potential on the propagation characteristics of a coshAiry beam may be different due to different modulation ranges.
3 Results and discussion
Based on Eq. (3), we numerically simulate the propagation process of a coshAiry beam in the Gaussian PTsymmetric potential by considering the effect of modulation depth P, modulation factor ω, gain/loss strength W _{0}, truncation coefficient a, and distribution factor χ _{0}.
At ω = 1, W _{0} = 0.9, a = 0.1, and χ _{0} = 0.02, Figure 2 presents the propagation properties of a coshAiry beam in the Gaussian PTsymmetric potential for P = 1.6, 2.0, 2.5, and 3.5, respectively. In Figure 2(a)–(d), it is illustrated that the main lobe of a coshAiry beam is captured as a soliton, i.e., trapped soliton, because the refractive index in the PTsymmetric center is higher than that of others. The trapped soliton varies periodically during propagation. The remaining part of the coshAiry beam is still able to selfaccelerate along a parabolic trajectory due to the selfhealing property. The refractive index of Gaussian PTsymmetric potential in the center position increases gradually with increment in modulation depth P, which shows that the restraint ability of a beam increases gradually. On increasing the modulation depth P, the peak power of a trapped soliton increases and the period of a trapped soliton becomes short in Figure 2(a)–(e).
When we set P = 2.5, W _{0} = 0.9, a = 0.1, and χ _{0} = 0.02, Figure 3 shows the propagation properties of the coshAiry beam in Gaussian PTsymmetric potential for ω = 0.8, 1.1, 1.3, and 1.5, respectively. The potential width is much narrower for ω = 1 that is equal to the main lobe width of the coshAiry beam (Figure 1(a)). With increasing modulation factor ω, the potential width decreases gradually that is narrower than the main lobe width of the coshAiry beam. It is indicated that the restraint ability of a beam decreases gradually with increment in ω. Since the refractive index in the center of a Gaussian PTsymmetric potential is the highest, the main lobe of the coshAiry beam is also captured as a soliton in Figure 3(a)–(d). The trapped soliton still varies periodically in the process of propagation. The remaining part of the coshAiry beam is still able to selfaccelerate along a parabolic trajectory due to the selfhealing properties (Figure 3(a)–(d)). In Figure 3(a)–(e), we find that the peak power of a trapped soliton decreases and the period of a trapped soliton becomes short with increase in ω.
The propagation characteristics of the coshAiry beam in Gaussian PTsymmetric potential with different gain/loss strength W _{0} are shown in Figure 4 for P = 2.5, ω = 1, a = 0.1, and χ _{0} = 0.02. For W _{0} > 0, the Gaussian PTsymmetric potential generates loss at x < 0, while the gain appears at x > 0 (Figure 1(a)). However, for W _{0} < 0, the Gaussian PTsymmetric potential generates gain at x < 0, and the loss appears at x > 0. It is presented that a soliton sheds from the main lobe of the coshAiry beam because of the effect of Gaussian PTsymmetric potential and propagates periodically in Figure 4. The remaining part of the coshAiry beam is also able to selfaccelerate along a parabolic trajectory due to the selfhealing properties. Comparing Figure 4(a1–d1 and a2–d2) shows that the deflection direction of a trapped soliton is opposite. On increasing the absolute value of W _{0}, the peak power of a trapped soliton increases and the period of a trapped soliton becomes long in Figure 4(e1) and (e2).
The effect of Gaussian PTsymmetric potential parameters on the propagation properties of a trapped soliton is statistically analyzed. Variation in the peak power and period of a trapped soliton with modulation depth P, modulation factor ω, and gain/loss strength W _{0} is depicted in Figure 5. It is demonstrated that the period of a trapped soliton decreases almost monotonically with increase in P, while the peak power of a trapped soliton increases monotonically with increment of P (Figure 5(a)). On increasing ω or decreasing the absolute value of W _{0}, the period and peak power of a trapped soliton decrease rapidly and then almost tend to be stable (Figure 5(b) and (c)). Thus, we found that a trapped soliton shedding from a coshAiry beam can be manipulated by changing the Gaussian PTsymmetric potential parameters.
Truncation coefficient a is an important parameter used to manipulate the waveform of a coshAiry beam. Figure 6 presents the propagation characteristics of a coshAiry beam in a Gaussian PTsymmetric potential for three truncation coefficients under the condition of P = 1.5, ω = 1, W _{0} = 0.9, and χ _{0} = 0.01. In Figure 6(a), the initial spatial shape of the coshAiry beam is asymmetric oscillation structure with multipeak. When the truncation coefficient is smaller (a = 0.05), a trapped soliton generates from the main lobe of the coshAiry beam due to the effect of Gaussian PTsymmetric potential and propagates periodically. The remaining part of the coshAiry beam is also able to selfaccelerate along a parabolic trajectory because of the selfhealing properties. In Figure 6(b), the side lobes of the coshAiry beam decreases rapidly and the main lobe is also captured as a soliton with increase in a to 0.25. In Figure 6(c), when the truncation coefficient becomes much large (a = 0.35), the side lobes disappear slowly and the coshAiry beam evolves almost into a Gaussian beam, so the property of transverse selfacceleration almost loses. In addition, a trapped soliton also generates from the coshAiry beam and propagates periodically. As the truncation coefficient increases, the peak power of the trapped soliton increases, but the period the trapped soliton almost remains unchanged (Figure 6(a)–(d)).
Distribution factor χ _{0} is another important parameter to control the waveform of the coshAiry beam. For P = 1.5, ω = 1, W _{0} = 0.9, and a = 0.3, the propagation properties of the coshAiry beam in a Gaussian PTsymmetric potential with three distribution factors is displayed in Figure 7. Distribution factor can be used to control the side lobe intensity of the coshAiry beam, but has little effect on the main lobe intensity (Figure 1(b)). In Figure 7(a)–(c), when the distribution factor χ _{0} increases gradually, it is illustrated that a soliton sheds from the main lobe of the coshAiry beam and propagates periodically, meanwhile the property of transverse selfacceleration recovers gradually due to the increase in the side lobe energy. In Figure 7(d), it is found that the peak power and period of a trapped soliton almost remains unchanged with increment of χ _{0}.
The influence of beam parameters on the propagation characteristics of a trapped soliton is statistically analyzed. Figure 8 shows the peak power and period of a trapped soliton varying with truncation coefficient a and distribution factor χ _{0}. It is illustrated that the period of a trapped soliton remains basically unchanged irrespective of increase or decrease in a and χ _{0} (Figure 8(a) and (b)). The peak power of a trapped soliton increases with increment of a; however, the peak power of a trapped soliton stays relatively constant in spite of variations in χ _{0} (Figure 8(a) and (b)). Consequently, we found that a trapped soliton shedding from a coshAiry beam can be controlled by changing the beam parameter a, while another beam parameter χ _{0} has little effect on the propagation properties of the trapped soliton.
4 Conclusion
In conclusion, the propagation characteristics of a coshAiry beam in the inhomogeneous nonlinear medium with Gaussian PTsymmetric potential is numerically investigated in detail. It is found that a trapped soliton sheds from the main lobe of a coshAiry beam and propagates periodically, while the residual part can also selfaccelerate along a parabolic trajectory because of the selfhealing property. Furthermore, the influence of Gaussian PTsymmetric potential parameters and beam parameters on the propagation properties of a coshAiry beam is statistically analyzed. It is presented that the period of a trapped soliton decreases almost monotonically with increase in the modulation depth P, while the peak power of a trapped soliton increases monotonically with increment in the modulation depth P. On increasing the modulation factor ω or decreasing the absolute value of gain/loss strength W _{0}, the period and peak power of a trapped soliton decrease rapidly and then almost stays relatively constant. Moreover, it is illustrated that the period of a trapped soliton remains basically unchanged irrespective of variation in truncation coefficient a and distribution factor χ _{0}. The peak power of a trapped soliton increases with the increment in the truncation coefficient a; however, the peak power of a trapped soliton remains unchanged in spite of increase or decrease in the distribution factor χ _{0}. Hence, we found that the propagation properties of a trapped soliton shedding from a coshAiry beam can be manipulated by appropriately choosing the Gaussian PTsymmetric potential parameters and beam parameters.

Funding information: This work was partially supported by the National Natural Science Foundation of China (No. 11947088), the Hunan Provincial Natural Science Foundation of China (No. 2022JJ50276 and 2021JJ40020), and the Scientific Research Fund of Hunan Provincial Education Department (No. 21A0499, 20B107, and 19B098).

Author contributions: Y.B. Deng: conceptualization, writing, review and editing; B. Wen: data curation and methodology; L.Z. Chen: conceptualization and supervision; S.W. Zhang: validation; G.F. Zhang: methodology; C.X. Xiong: investigation and software; X.L. Leng: data curation and software. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
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