Study on the Evolutionary Characteristics of Airyprime Beams in Gaussian-Type PT Symmetric Optical Lattices

Abstract

The Airyprime beam, due to its adjustable focusing ability and controllable orbital angular momentum, has attracted significant attention in fields such as free-space optical communication and particle trapping. However, systematic studies on the propagation behavior of oscillating solitons in PT-symmetric optical lattices remain scarce, particularly regarding their formation mechanisms and self-accelerating characteristics. In this study, the propagation characteristics of Airyprime beams in PT symmetric optical lattices are numerically studied using the split-step Fourier method, and the generation mechanism and control factors of oscillating solitons are analyzed. The influence of lattice parameters (such as the modulation depth P, modulation frequency w, and gain/loss distribution coefficient W₀) and beam initial characteristics (such as the truncation coefficient a) on the dynamic behavior of the beam is revealed. The results show that the initial parameters determine the propagation characteristics of the beam and the stability of the soliton. This research provides theoretical support for beam shaping, optical path design, and nonlinear optical manipulation and has important application value.

1. Introduction

In recent years, Airyprime beams have become a research hotspot in optics due to their unique propagation characteristics. Early studies mainly focused on the beam’s transmission characteristics and optimization strategies in specific media. For example, in uniaxial crystals, the receiving field of the Airyprime beam exhibits a flat-top characteristic [1]; in a strong turbulent environment, symmetric Airyprime beams exhibit a reduced beam scintillation effect and bit error rate, which makes them more reliable in free-space optical communication [2]. Since then, people have begun to explore methods and the application potential of Airyprime beam regulation. For example, vortex-embedded circular Airyprime beams can achieve a tunable focusing capability and controllable orbital angular momentum [3]; the introduction of linear chirp enhances the self-focusing capability and extends the focal length of circular Airyprime beam arrays [4]. In addition, an array composed of centrosymmetric Airyprime beams [5] has been successfully demonstrated, which further expands the application possibilities of Airyprime beams. The current research focuses on revealing the mechanisms that regulate the self-focusing behavior of Airyprime beams. The off-axis vortex has a significant role in regulating the self-focusing behavior of circular Airyprime beams and its use is expected to show advantages in optical trapping [6]. Under the effect of linear chirp, the degree of change in the focal length of circular Airyprime beams is closely related to the exponential attenuation factor and the dimensionless radius of the main ring, which provides a theoretical basis for the precise tuning of the focal length of self-focusing beams [7]. In addition, in chiral media, circular Airyprime beams exhibit unique autofocusing properties, and the focal distance and peak intensity of their left and right rotationally polarized components can be flexibly adjusted [8]. In 2024, Liping Zhang et al. posited for the first time that circular Airyprime beams exhibit periodic evolutionary behavior in a parabolic potential medium and found that the focal distance of circular Airyprime beams is significantly affected by the parameters of their parabolic potential [9]. It was also found that the focal distance was significantly affected by the parameters of the parabolic potential. Based on Fourier spatial modulation technology, Xinqing Zheng’s team realized the adjustable self-focusing characteristics of circular Airyprime beams [10]. These research results provide new methods and ideas for the application of Airyprime beams in the fields of particle trapping and free-space optical communication.

At the same time, the concept of PT symmetric medium has attracted widespread attention in the field of optics and quantum mechanics. Previous studies have focused on constructing PT symmetric optical potential and exploring its basic properties. For example, the tunable PT symmetric optical potential constructed by a coherent four-level atomic medium opens up a new way to further explore related physical phenomena [11]. The study of the kink spectrum shows that the relative position of the kink center and the PT symmetry center will significantly affect the stability of the kink spectrum [12]. Under the action of PT symmetric potential, Chao-Qing Dai et al. found that the stability of the local solution of the power-law nonlinear Schrödinger equation is closely related to the imaginary part of the potential and the self-focusing characteristics of the medium [13]. At the same time, an anti-PT symmetric optical coupler was also proposed, and it was shown that it operates in the PT conjugate mode under Kerr nonlinearity [14]. With the deepening of research, people have begun to pay attention to the nonlinear effects, pulse propagation characteristics, and application in complex structures of PT symmetric media. For example, the application of the effective medium theory to PT symmetric multi-layer systems is discussed in depth, and the limitations of the traditional theory are clarified [15]; the phenomena of thresholdless PT symmetry breaking and unidirectional scattering [16] have been successfully realized. In Bragg geometry, the propagation of long and short pulses in quasi-PT symmetric media shows significant differences, indicating the influence of the pulse width on PT symmetric propagation [17]. D M Tsvetkov et al. found that the Bragg diffraction of chirped pulses can exhibit unique phenomena in dispersive quasi-PT symmetric photonic crystals [18]. At present, the research has begun to pay attention to the influence of the PT symmetric lattice on beam transmission characteristics. For example, Gang Yao, Yuhua Li, and Ruipin Chen studied the influence of PT symmetric lattices on the collapse of vortex beams in Kerr medium and found that PT symmetric lattices can cause asymmetric collapse, which provides a new idea for controlling beam collapse [19]. These research results provide a theoretical basis and experimental guidance for the application of PT symmetric media in the fields of optical device design, optical information processing, new optical materials, and optical fiber sensing.

The existing research on the use of bright solitons [20,21], Airy beams [22], and cosh-Airy beams [23] in PT symmetric lattices mainly focuses on the characteristics of shedding solitons. However, the research on the dynamic behavior of more complex beam structures (such as Airyprime beams) when propagating in PT symmetric lattices is still insufficient. In particular, the formation mechanism of oscillating solitons and the energy ratios of the remaining self-accelerating and self-restoring beams, including how they are controlled by the initial parameters of the lattice and the beam, still lack systematic discussion. This research introduces a novel type of mutation beam (Airyprime beams) that propagates in PT-symmetric optical lattices, during which it sheds oscillatory. This not only increases the diversity of solitons but also enhances the security of information transmission. Additionally, Airyprime beams demonstrate strong robustness in complex optical systems, maintaining stable propagation even under disturbances. This characteristic makes them less susceptible to external interference or manipulation in complex environments, and thereby enhances the confidentiality and security of information that is transmitted using Airyprime beams. Given the above shortcomings, this study aims to explore the transmission characteristics of Airyprime beams in PT symmetric optical lattices, focusing on the generation and control mechanism of oscillating solitons. By analyzing the effects of lattice parameters (such as the modulation depth and frequency) and initial beam characteristics (such as the truncation coefficient and peak intensity), this study attempts to reveal the internal physical mechanism of the dynamic behavior of oscillating solitons. The innovation of this study lies in determining a method for to precisely control the generation of oscillating solitons by optimizing the lattice configuration and beam control, which provides theoretical support and experimental guidance for beam shaping, optical path design, and nonlinear optical manipulation. It provides a new idea regarding advanced optical manipulation in complex optical environments and has important theoretical significance and application value.

2. Theoretical Model

The propagation of a beam in a PT-symmetric optical lattice is subject to medium loss and gain, which can be described by the normalized dimensionless nonlinear Schrödinger equation [22]:

$$i{\displaystyle \frac{\partial \phi }{\partial z}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2\phi }{\partial x^2}}+P\left[V\right(x)+iW(x\left)\right]\phi =0$$

(1)

In the formula, the parameter φ represents the field envelope distribution of the beam, the parameter z = Z/Z₀ represents the normalized longitudinal coordinates along the propagation direction, the parameter x = X/X₀ represents the normalized transverse coordinates, and the parameter P represents the modulation depth of the medium. In our study, X₀ represents the initial width of the beam, which is typically related to the transverse characteristic scale of the beam. As for Z₀, it is a normalized scaling length, defined as $Z_0=\sqrt{Z_d\cdot Z_{nl}}$, where Z_d is the dispersion length and Z_nl is the nonlinear length. This length affects the strength of the complex potential. V(x) and W(x) are the real and imaginary parts of the dimensionless complex refractive index of the PT symmetric optical lattice, respectively. To satisfy the PT symmetry condition, the complex refractive index requires that the refractive index distribution of the optical lattice is an even function of the transverse spatial index, and that the gain/loss curve distribution is an odd function. That is, the real part and the imaginary part of the refractive index have parity symmetry. In this paper, the Gaussian PT symmetric structure optical lattice is used, the expression for which is as follows:

$$V\left(x\right)=\exp \left(-{\left(\omega x\right)}^2\right)$$

(2)

$$W\left(x\right)=W_0\left(\omega x\right)\exp (-{\left(\omega x\right)}^2)$$

(3)

In the above formula, w is the modulation frequency of the complex potential, and W₀ is the gain/loss distribution coefficient. The greater the modulation depth, the greater the intensity of the complex potential, and the stronger the modulation effect on the Airy beam. The modulation frequency determines the width and modulation range of the complex potential. The larger the modulation frequency, the narrower the complex potential and the smaller the modulation range of the beam. When the gain/loss distribution coefficient W₀ > 0, the PT optical lattice generates gain in the x < 0 region, and the energy propagates to the gain region, resulting in the transverse flow of the beam to the right first. When W₀ < 0, the gain is generated in the x < 0 regions, and the loss is generated in the x > 0 regions, resulting in the transverse flow of the beam to the left first. As the absolute value of W₀ increases, the peak power and period of the soliton increase, which affects the propagation characteristics of the beam, including the generation, moving direction, transmission trajectory, and self-acceleration characteristics of the soliton. The initial input for the Airyprime beam can be written as follows:

$$\phi (z=0,x)=F\left(a\right)A{i}^{\prime }(x)\exp (ax)$$

(4)

In the formula, F(a) is introduced to keep the peak intensity of the beam at 1, 0 < a < 1 is the truncation coefficient, and Ai′(x) is the first derivative of the Airy function. The expression of the Airy function is:

$$Airy(x)={\displaystyle \frac{1}{2\pi }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\exp [i(u^3/3+xu)]du}$$

(5)

The initial waveform of an Airyprime beam is an obvious asymmetric structure. The main peak is located in the second peak on the right side, and the first peak on the right side decays rapidly. There are multiple oscillation peaks on the left side of the main peak, and the oscillation amplitude gradually decreases, forming a long trailing structure.

3. Numerical Calculations and Analysis

We first discuss the potential well characteristics of the PT symmetric optical lattice to clarify the characteristics of its optical modulation effect. Then, the influence of the initial parameters of the PT symmetric optical lattice (such as the modulation depth P, modulation frequency w, and gain/loss distribution coefficient W₀) and the initial characteristics of the beam (such as the truncation coefficient a) on the transmission characteristics is analyzed, and the influence of each parameter on the peak power and period of the shedding soliton is statistically analyzed. Finally, the similarities and differences between the evolution characteristics of Airyprime beams with trailing in front and trailing behind in PT symmetric media are studied.

3.1. Profile of an Optical Lattice Potential

Figure 1 shows the real and imaginary parts of the Gaussian PT symmetric optical lattice. The real part represents the refractive index, which is symmetrical and is the highest at x = 0, which restricts the beam. The imaginary part represents the gain and loss, and exhibits odd symmetry. The maximum absolute value occurs near x = 0 (approximately at ±0.7). As the distance from the center increases, the imaginary part gradually approaches 0. In the region where x < 0 and near the center of the waveguide, loss is observed, while in the region where x > 0, gain is observed. As the distance from the center increases, both the gain and loss gradually approach 0. When the modulation frequency w is large (Figure 1a), the modulation range of the potential function is narrow, only covering the main lobe of the Airyprime beam, and the beam is less affected. When the modulation frequency w is small (Figure 1b), the modulation range of the potential function is wide, meaning that it can cover the entire Airy beam, and the imaginary part shows gain and loss in different directions. By comparing Figure 1a with Figure 1b, it can be seen that increasing the absolute value of w will narrow the modulation range of the PT symmetric optical lattice, with this effect mainly acting on the central region of the beam; reducing the absolute value of w will expand the modulation range, and the influence range of the beam will increase. In addition, when w is negative, the symmetry of the gain and loss of the imaginary part is reversed, which affects the direction of the gain distribution of the medium. That is to say, whether the modulation frequency w is positive or negative not only affects the range of modulation but also changes the distribution mode of the gain and loss.

3.2. The Influence of Optical Lattice Parameters

Figure 2 illustrates the spatial evolution of the Airyprime beam under different modulation depths in a PT symmetric optical lattice (Figure 2a–c), as well as the variations in the spatial and centroid positions concerning their propagation distance (Figure 2d,e). The modulation depth significantly influences the existence, stability, and transmission characteristics of the soliton. From the spatial evolution diagrams (Figure 2a–c), it can be seen that, due to the modulation of the PT potential, the main peak of the Airyprime beam is compressed in the initial stage. After propagating for a certain distance, a soliton with oscillatory transmission, where both its energy and width vary periodically, is shed.

A periodic oscillation occurs, causing the energy and width of the soliton to vary periodically. The rest of the soliton can recover its shape and self-acceleration along the parabolic trajectory due to its self-healing characteristics. Under the action of PT potential, the shedding soliton first moves to the right, and then reflects to the center after colliding with the boundary. This behavior indicates that the PT potential exerts an attractive force on the soliton, pulling it from the center towards the sides. However, upon reaching the boundary, an elastic collision occurs, altering the soliton’s direction of motion. After this collision, the soliton moves toward the center, and upon reaching the center, it continues in its original direction. However, due to the modulation range of the PT potential primarily affecting the central region, the velocity of the soliton decreases as it moves further away from the center. Once the velocity reaches zero, the soliton reverses its direction and moves back toward the center. This repetitive motion mirrors the harmonic oscillation of a spring oscillator. Moreover, as the modulation depth increases, the transmission characteristics of the Airyprime beam undergo significant changes. As the modulation depth P increases (from 1.5 to 2.5), the beam’s oscillation frequency accelerates, and the oscillation period decreases notably. Figure 2d and Figure 2e, respectively, show the spatial variations in the peak intensity and the centroid position of the beam at different modulation depths. It can be observed that, with an increasing modulation depth, the soliton exhibits higher peak power and stronger transverse oscillations. These results indicate that the modulation depth P has a significant impact on the transmission characteristics of the Airyprime beam.

Figure 3 illustrates the propagation evolution of Airyprime beams in a PT symmetric optical lattice under different gain/loss distribution coefficients, which are denoted as W₀. Figure 3a–c show the spatial evolution of the Airyprime beam at different transmission distances when W₀ is −0.6, 0, and 0.6, respectively. It is evident that the evolution behavior of the Airyprime beam undergoes significant changes with variations in the gain/loss distribution coefficient W₀. When W₀ = −0.6 (Figure 3a), the beam propagation is somewhat suppressed, and there is a reduced lateral energy flow. In contrast, when W₀ = 0 (Figure 3b), the beam exhibits a more uniform energy distribution, and its propagation characteristics remain relatively stable. However, when W₀ = 0.6 (Figure 3c), the enhanced gain distribution results in a stronger lateral energy flow, leading to more pronounced oscillatory behavior. Figure 3d,e illustrate the variation in the peak intensity and centroid position of the Airyprime beam, when the propagation distance is at a fixed propagation distance of Z = 30, for different gain/loss distribution coefficients W₀. It can be observed that, as W₀ increases, the peak intensity of the beam gradually rises, while the centroid position undergoes periodic fluctuations. Notably, when W₀ is positive (Figure 3c), the gain causes the beam’s energy to propagate outward along the gain region. However, when W₀ is negative (Figure 3a), the beam’s energy tends to flow towards the loss region, resulting in backward propagation. Overall, the sign of W₀ not only affects the beam’s propagation direction but also determines its energy distribution and propagation stability. The larger the absolute value of W₀, the more pronounced the propagation effects are, with increasing lateral shifts and fluctuations.

The modulation frequency w is another factor that has a strong influence on the transmission characteristics of the soliton. At the modulation frequency w = 0.8 (Figure 4a), the shedding soliton has strong localization, the Airyprime beam is relatively concentrated and has small transverse oscillations which show relatively strong contraction during transmission, and the intensity of the shedding soliton is also larger. As the modulation frequency increases (Figure 4b,c), the peak power of the shed soliton decreases significantly. This is mainly because the attraction of the PT symmetry potential to the shedding soliton is weaker at this time, while the influence of the diffraction effect is relatively larger, resulting in the inability of the shedding soliton to maintain a stable shape. With increases in the modulation frequency w, the attraction of the PT symmetric potential to the shedding soliton is gradually enhanced, which leads to the gradual concentration of the energy of the shedding soliton and a decrease in its oscillation amplitude. Therefore, the modulation frequency directly affects the distribution of the falling soliton in the spatial domain. A lower modulation frequency (e.g., w = 0.8) results in a stronger localization of the beam, leading to a smaller propagation range. In contrast, a higher modulation frequency (e.g., w = 1.2) causes the beam to spread more, weakening its localization and increasing its extent. Figure 4d,e illustrate the spatial variation in the peak intensity and the centroid position of the Airyprime beam under different modulation depths. It can be observed that, as the modulation frequency increases, the falling soliton exhibits a lower peak power and smaller lateral oscillations. These results indicate that the modulation frequency w has a significant impact on the transmission characteristics of the Airyprime beam.

3.3. The Influence of Airyprime Beam Parameters

Figure 5 demonstrates the effect of the modulation of parameters of the PT symmetric optical lattice on the peak power and period of the shedding soliton of the Airyprime beam. Figure 5a shows that the peak power of the shedding soliton monotonically increases with increases in the modulation depth P, while the period decreases with increases in P. The peak power of the soliton oscillates faster than that of the modulation depth P, and the period decreases with increases in P. This implies that, the larger P is, the faster the soliton oscillates and the higher its energy. Figure 5b shows that the peak power and period of the shedding soliton show a U-shaped variation with the absolute value of the gain/loss distribution coefficient W₀, and that the minimum value occurs at W₀ = 0. This means that, the larger W₀ is, the faster the soliton oscillates and the higher its energy. Figure 5c demonstrates that the peak power and period of the shedding soliton decrease monotonically with increases in the modulation frequency w, indicating that, the higher w is, the weaker the soliton oscillation state is. By adjusting these parameters, precise control of the soliton properties can be realized, allowing adjustment of the speed, energy, and oscillatory state of the soliton. This is of great significance in studying the properties and applications of optical solitons.

The truncation coefficient a is a degree of freedom that controls the waveform of the Airyprime beam. A numerical study was carried out to investigate the effect of the truncation coefficient on the propagation of the Airyprime beam in a PT symmetric optical lattice. The Airyprime beam exhibits an asymmetric tail-dragging oscillatory structure at the initial position, with the main peak having the highest energy. When propagating in a PT symmetric optical lattice, the main peak sheds an oscillatory soliton and forms a long trailing tail due to its self-recovery property (Figure 6a). The Airyprime beam exhibits an asymmetric multi-peak structure, with the truncation coefficient aaa controlling the level of side-lobe energy. As aaa increases, the side-lobe energy of the oscillatory tail decreases rapidly. Since the total energy remains constant, the energy from the tail side-lobes is transferred to the main peak (Figure 6b). When the truncation coefficient is increased to 0.4, the paraboloids of the beam gradually disappear, the beam almost evolves into a Gaussian shape, the main peak energy reaches a maximum, and the transverse self-accelerating property almost disappears (Figure 6c). Figure 6d,e show the variation in the peak intensity and the position of the center of gravity of the shedding soliton, respectively. The results show that, the larger the truncation coefficient a is, the higher the peak intensity of the shedding soliton is, and that the oscillation amplitude of the center of gravity position gradually increases with a, indicating that the truncation coefficient a has a significant modulation effect on the energy distribution of the Airyprime beam as well as the peak intensity and center of gravity position of the shedding soliton.

Figure 7 illustrates the effects of different truncation coefficients on the Airyprime beam. Figure 7a shows the spatial intensity distribution of the Airyprime beam at different truncation coefficients. It can be seen that the initial waveform of the Airyprime beam has an obvious asymmetric structure. The main peak is located in the second peak on the right side, and the first peak on the right side decays rapidly, while there are multiple oscillatory peaks on the left side of the main peak. The amplitude of the oscillations decreases gradually, forming a long trailing structure. It is worth noting that the intensity of the Airyprime beam gradually becomes more concentrated as the truncation factor increases, which causes the intensity of the main peak to gradually increase and the energy of the side flaps to decrease significantly. Figure 7b shows the effect of the truncation coefficient on the peak intensity of the shedding soliton, which shows a monotonically increasing trend with increases in a. The peak intensity of the shedding soliton is also shown in Figure 7b, which shows the effect of the truncation coefficient a on the peak intensity of the shedding soliton. Overall, the larger the truncation coefficient a is, the more concentrated the energy of the Airyprime beam is, and the higher the peak intensity of the shedding soliton is.

3.4. Comparison of Trailing Leading and Trailing Lagging Airyprime Beams

Figure 8 shows evolution diagrams of Airyprime beams with trailing leading (Figure 8a) and trailing lagging (Figure 8b) in the PT symmetric optical lattice. The shedding solitons are identical for both beams, except for a symmetry change in the direction of self-acceleration of the remaining Airyprime beam. The Airyprime beam with the trailing tail leading accelerates and deflects to the right, while the Airyprime beam with the trailing tail lagging accelerates and deflects to the left. These evolutions reveal a significant effect of the gain/loss of the PT symmetric optical lattice on the propagation behavior of the Airyprime beam, especially in the trailing and shedding solitons. The shedding soliton propagation properties remain unchanged, while the self-deflection pattern of the remaining Airyprime beam changes with the leading and lagging tails. This result indicates the symmetry and consistency of the action of the PT symmetry potential on Airyprime beams with leading and lagging trailing tails.

4. Conclusions

In this study, the evolutionary properties of Airyprime beams in Gaussian-type PT symmetric optical lattices are systematically explored. Numerical simulations reveal the significant effects of lattice parameters (e.g., modulation depth, modulation frequency, gain/loss distribution coefficients) and the initial parameters of the beam (truncation coefficients) on the propagation characteristics of Airyprime beams. It is shown that an increase in the modulation depth leads to a faster oscillation frequency and a shorter oscillation period of the beam, as well as an enhanced transverse oscillation of the beam. Changes in the strength of the gain/loss distribution coefficients affect the energy flow and propagation stability of the beam, which manifests itself in different gain and loss effects, which in turn affect the generation of solitons and the trajectory of the motion. The adjustment of the cutoff coefficient plays an important role in regulating the energy distribution, peak intensity, and generation of shedding solitons in Airyprime beams, especially in controlling the stability and self-accelerating properties of solitons. It is also found that the PT symmetric optical lattice has a highly symmetric effect on the beam, especially in the case of symmetric variation of the trailing position, which affects the beam’s self-deflection and propagation behavior. This study provides an important theoretical foundation and experimental guidance for the application of Airyprime beams in PT symmetric lattices, which has a wide range of application prospects, especially in the fields of beam shaping, optical path design, and nonlinear optical manipulation.