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Abstract
We investigate the propagation dynamics of the Pearcey-Gaussian (PG) pulses in the presence of time-dependent potentials in a linear medium both theoretically and numerically. We demonstrate that the combination of the linear potential and the initial chirp of PG pulses can flexibly control the propagation trajectory and inherent focusing properties of the PG pulses. When the parabolic potential is taken into account, the chirped PG pulses are periodically focused and reversed. By adjusting the parabolic potential and the pulse chirp, the characteristics of the focal points, such as position, intensity, and spacing between focal points, can be manipulated effectively. The interaction of two temporally separated PG pulses still shows a periodic evolution with controllable focusing characteristics. These results can broaden the application range of PG pulses and provide some inspiration for the control of PG pulses under nonlinear conditions.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Past few decades, different types of caustic beams based on catastrophe theory, such as Airy beam [1–3], Pearcey beam [4–6] and swallowtail beam [7–9], have stimulated great theoretical and experimental enthusiasm due to their special structure and transmission characteristics, as well as in various applications such as optical manipulation, micro-Nano processing, light bullets and imaging [10]. The Pearcey beam, described by the Pearcey function, was theoretically and experimentally introduced into optics in 2012 by superimposing the exponential function on the Pearcey function to obtain a beam with finite energy [4]. Actually, the Pearcey function was first proposed in 1946 to describe the diffraction of cusp caustics [11]. Since then, the Pearcey beam has attracted the attention of many researchers due to its unique properties, including inversion, self-focusing, and self-healing. Until now, various derivative beams based on Pearcey beam obtained by different methods have been reported [12–15]. For example, a dual-Pearce beam with a stable transmission structure can be generated by Fresnel diffraction [12], a strongly focused symmetrical Pearcey-Gaussian (PG) beam can be obtained based on two quaternary spectral phases [6] and partially coherent PG beams with controllable intensity distribution can be obtained based on phase modulation techniques [14]. Encouraged by these studies, the linear and nonlinear propagation dynamics of Pearcey and its derivative beams have been extensively studied [16]. In addition, the effect of the chirp on the evolution of the PG beam in free space is also investigated [17]. The results show that the quadratic chirp makes the PG beam focus twice, and the linear chirp cannot change the inherent focusing characteristics of the PG beam. Inspired by time-space duality, the temporal PG pulses carried the similar characteristics of the PG beams have been proposed [18]. It has been demonstrated that the PG pulse shows remarkable advantages in dispersive wave radiation and supercontinuum generation, because it has asymmetric structure both in the time and frequency domains [19,20].
Controllable light propagation has always been an active research area, and researchers have focused on two directions: light-medium interactions and light-light interactions [21–23]. Among these numerous investigations, light manipulation by changing the properties of the medium is the most shining, and the effective method is the external potentials (such as linear potentials, parabolic potentials, and dynamic potentials) related to the refractive index of the medium. It has been proved that the linear potential can manipulate the propagation trajectory and alter the properties of the light beam by changing the speed of the photon [24–26]. The parabolic potential makes the photons resonate and oscillate, so the beam is periodically reversed and focused during the transmission process and even generate a self-induced Fourier transform [27–29]. In fact, the effect of strong nonlocal nonlinearity is equivalent to the parabolic potential, so its modulation can be easily studied by reducing the nonlinear problem to a linear one [30]. For the dynamic potential, the beam can be periodically focused and inverted along a certain acceleration trajectory, which can be understood as the linear superposition of the linear potential and the parabolic potential [31,32]. Recently, the temporal potential resulting from the nonlinear interaction of weak signals and strong pumps has received special attention due to its advantages in the manipulation of temporal or spectral signals [33,34].
In fact, the propagation trajectories and self-focusing properties of various derivative beams of the Pearcey beam controlled by external potentials have been reported [35–37]. To the best of our knowledge, the effect of external time-dependent potentials on the evolution dynamics of chirped PG pulses has not been explored. In this work, we investigated the propagation dynamics of the chirped PG pulses in external time-dependent potentials. The results show that the participation of the initial linear or quadratic chirp of the PG pulse provides a higher degree of freedom for controlling the inherent focusing properties of the PG pulse while the linear potential manipulates the PG pulse transmission trajectory. When the parabolic potential is exploited, the PG pulse is periodically focused and reversed, and appropriate chirp parameters can adjust the position and intensity of the focusing points. In addition, the interaction between the dual PG pulses also exhibits periodic evolution with controllable focusing properties. Furthermore, we solve the evolution dynamics expression of the chirped PG pulse and verify the result by numerical calculation. We believe that the obtained results can provide an efficient method for the control of PG pulses and enhance the freedom degree of the PG pulse control for their applications.
2. Propagation model
Here, we consider the chirped PG pulses propagating in the presence of time-dependent potentials in a linear medium. It was shown by Han et al. in 2019 that, strong pump pulses can create the time-dependent potentials on the slowly varying envelope of weak signal light through cross-phase modulation [33]. The modified pulse propagation equation can be simplified to a dimensionless standard form, which makes it more universal in the study of time-dependent potentials. In this case, the propagation evolution of the pulse follows the following normalized dimensionless linear Schrödinger-like equation [33,38]:
Here the slowly varying amplitude $\varphi (T,Z)$ of the pump pulse is defined as one. $T = {t / {{T_0}}}$ and $\textrm{ }Z = {z / {{L_D}}}$ are dimensionless using the initial pulse width ${T_0}$ and the dispersion length (${L_D} = {{{T_0}^2} / {|{{\beta_2}} |}}$) related to second-order dispersion coefficient ${\beta _2}$, respectively. $V(T )$ represents a time-varying potential, $V(T )= \alpha T$ is a linear potential, and $V(T )= 0.5{\beta ^2}{T^2}$ is a parabolic potential. Here $\alpha $ and $\beta $ represent the depth of potentials, respectively.
The envelope of the PG pulses can be described by $PG(T )= Pe(T )\exp ({ - A{T^2}} )$, where $Pe(T )= \int_{ - \infty }^{ + \infty } {ds} \exp ({i{s^4} + i{s^2}T} )$ is the one-dimensional form of Pearcey function with the truncation coefficient A. In order to ensure the consistency of analysis results, here we choose $A = 0.01$. In this paper we mainly consider the evolution of PG pulses with initial linear and quadratic chirps. Therefore, the PG pulse carrying the initial chirps has the following form:
According to $\tilde{\varphi }({\omega ,0} )= \int_{ - \infty }^{ + \infty } {\varphi ({T,0} )\exp ({i\omega T} )} dT$, the Fourier transforms of Eq. (2) is:
3. Results and discussions
3.1. PG pulse propagation in linear potential
Here, we consider the propagation dynamics of the chirped PG pulses under the linear potential. The general solution of the Eq. (1) where $V(T )= \alpha T$ in the Fourier domain is found to be:
In order to deepen the understanding of the focusing characteristics of the chirped PG pulse, we carefully analyze Eq. (6) and find that the focusing point can only appear at the coefficient of ${s^4}$ where the coefficient is 0 and the coefficient of ${s^2}$ at the singular point. That is to say, the focus position of the chirped PG pulse evolution process appears at $1 - {Z / {2M}} = 0$ and $M = 0$, and the truncation coefficient A is small and neglected to solve:
To verify the theoretical results based on Eq. (6), we numerically solve the Eq. (1) with the linear potential by using the split-step Fourier method, and the results are shown in Fig. 1. Figures 1(a)–1(c) show the temporal evolutions of the PG pulses with different linear potentials. Figure 1(b) shows the temporal evolution of the PG pulses in the absence of linear potential $\alpha = 0$, exhibiting the inherent focusing and inversion properties of the PG pulses. When the linear potentials are considered, the PG pulses are deflected in the time domain. Specifically, the PG pulses deviates toward its leading edge for $\alpha < 0$, and while moves toward its trailing edge for $\alpha > 0$. It is worth mentioning that the linear potential do not affect the uni-focusing phenomenon inherent in the PG pulses. To reveal the relationship between the focusing properties and the linear potentials, the corresponding evolution of the maximum intensity of the PG pulse under different $\alpha $ is shown in Fig. 1(d). The results show that the linear potential does not cause the peak intensities of the pulses evolution and the position of the focus point (the main lobe width is the narrowest but the intensity is the largest). The focus distance ${Z_f} = 2$ is consistent with the analytical result of Eq. (8). Subsequently, the variations of the temporal position of the main lobe during pulse propagation for different linear potential parameters are shown in Fig. 1(e). It can be found that the larger the absolute value of the linear potential depth $|\alpha |$, the bigger the deviation of the main lobe of the PG pulse with respect to the case of $\alpha = 0$. It is worth mentioning that when the absolute values of the linear potential depths $|\alpha |$ are the same, the time domain absolute offsets are also the same, which can be clearly explained from Eq. (7). In addition, the rhombuses obtained by Eq. (7) in Fig. 1(e) are highly consistent with the numerical results plotted as solid lines, which also confirm the correctness of the analytical results.
Fig. 1. Temporal evolutions of PG pulse in a linear potential with different $\alpha $: (a) $\alpha ={-} 1$, (b) $\alpha = 0$, and (c) $\alpha = 1$. (d) The peak intensity of the PG pulses as a function of propagation distance Z and potential depth $\alpha $. (e) The main lobe position variations of the PG pulses during propagation for different $\alpha $. The curves and the diamonds represent numerical and analytical results, respectively.
Since the effect of spatial chirp on the propagation dynamics of the Pearcey beam is fascinating, we consider the effect of the initial pulse chirp on the evolution dynamics of the PG pulse. The role of the initial linear chirp is first considered, and the corresponding results are shown in Fig. 2. Figures 2(a) and 2(b) plot the temporal evolutions of the PG pulses with different linear chirps when the linear potential is $\alpha ={-} 1$. The results show that the linear chirp does not destroy the uni-focusing process of the PG pulses, and under the premise of following the direction of the deviate caused by the linear potential, the time domain deviates can be enhanced or weakened. Furthermore, the analytical results obtained from Eq. (7) (yellow diamonds) are in agreement with the simulations (color density maps). Figure 2(c) shows the evolution of the maximum intensity of the PG pulses for different initial linear chirp C, revealing the relationship between the properties of the focal point and the linear chirp. It is shown that the focus distance is remains unchanged at ${Z_f} = 2$, which is independent of the initial linear chirp. This behavior coincided exactly with the analytical predictions from Eq. (8). Comparing Figs. 2(a) and 2(b), we can find that although the focal distance does not change, the temporal position (TP) of the focusing point depends strongly on the linear chirp parameter. According to the TP formula (${T_{focus}} = 2({C + \alpha } )$) of the focusing point, Fig. 2(d) shows the relationship between the TP of the focusing point and the initial linear chirp under different linear potentials. The results show that the TP of the focusing point increases linearly with the increase of the linear chirp, and also maintains a linear relationship with the linear potential depth. More interestingly, when the values of the linear potential and the linear chirp keep the relationship of $C + \alpha = 0$, the TP of the focus point can be fixed to 0. To confirm the above analytical results, our numerical results are shown as diamonds in Fig. 2(d). It is clearly seen from Fig. 2(d) that they agree with each other very well. By properly choosing the values of the linear potential and the linear chirp, we can significantly control the TP of the PG pulse focusing point. This is useful for broadening the perspective for its application.
Fig. 2. Temporal evolutions of PG pulse for the same linear potential ($\alpha ={-} 1$) and different linear pulse chirps: (a) $C ={-} 1$ and (b) $C = 1$. The diamonds represent analytical results. (c) The evolution of the maximum intensity of the PG pulses with distance for different linear pulse chirps. (d) The temporal position (TP) of the focusing point versus the chirp C for different potential depths. The solid curves and the diamonds represent numerical and analytical results, respectively.
Next, we focus on the effect of the initial quadratic chirp on the propagation dynamics of the PG pulse in presence of linear potential, and the corresponding results are shown in Fig. 3. Figures 3(a) and 3(b) plot the temporal evolutions of PG pulses with different quadratic chirps when the linear potential is $\alpha ={-} 1$, and the curved dashed black lines are the trajectories of the main lobe of the PG pulses without quadratic chirp. After comparison, it is found that under strict compliance with the curved trajectory caused by the linear potential, the PG pulse focused twice when $Q < 0$, and the uni-focusing distance is delayed when $0 \le Q < 0.25$. It is worth noting that for $Q > 0.25$, the inherent uni-focusing characteristic of the PG pulse disappears. To verify these, the maximum intensity (${I_{Max}}$) versus distance for the PG pulses carrying different initial quadratic chirps are plotted in Fig. 3(c). The results show that with the increase of Q with the range of $0 \le Q < 0.25$, the focusing distance increases, and the intensity of the focusing point decreases. For the case of $Q < 0$, as Q decreases, the focusing distance shorter, and the intensity of the focusing point increases. The diamonds in Fig. 3(c) show the focal position calculated according to Eq. (8), which is consistent with the maximum point of the numerically calculated ${I_{Max}}$ curve. In addition, according to Eq. (8), we can calculate the interval between two focusing point distances as:
Fig. 3. Temporal evolutions of PG pulse for the same linear potential ($\alpha ={-} 1$) and different linear pulses chirps: (a) $Q ={-} 0.2$ and (b) $Q = 0.1$. (c) The evolution of the maximum intensity of PG pulses with distance for different quadratic chirps. The diamonds represent the analytical focal point. (d) The interval between two focal points and the corresponding intensity ratio as a function of the parameter Q. The black solid curves and the diamonds represent numerical and analytical results, respectively.
Such analytical results are shown as the black curve in Fig. 3(d), and the numerical results were plotted as the diamonds for comparison. It can be found that the interval between two focusing points grows exponentially with the quadratic chirps and is very sensitive to the large value of Q. The intensity relationship between the two focal points is depicted by the red curve in Fig. 3(d). It can be seen that with the increase of Q, the intensity ratio (${{{I_2}} / {{I_1}}}$) at ${Z_{f1}}$ and ${Z_{f2}}$ decreases continuously, and when $Q ={-} 0.6$, the intensities at ${Z_{f1}}$ and ${Z_{f2}}$ are equal. This means that the intensity drops faster at ${Z_{f2}}$ than at ${Z_{f1}}$. In addition, the effect of the truncation coefficient A is also analyzed. The results show that the focus distance remains constant and the focus intensity decreases with the increase of A. It is worth noting that when $A > 0.05$, there is only one side lobe of the PG pulse whose intensity is much smaller than that of the main lobe. This indicates that it is difficult for the PG pulse to perform secondary focusing when A is large. The controllability of the focal point intensity and position of the PG pulse can facilitate its application in particle manipulation.
3.2. PG pulse propagation in parabolic potential
Next, we consider the propagation dynamics of the chirped PG pulse under the parabolic potential. For any incident pulse $\varphi ({T,0} )$, the general solution of the Eq. (1) with $V(T )={-} 0.5{\beta ^2}{T^2}$ is [27]:
According to Eq. (10), the evolution of the chirped PG pulses can be described by:
Obviously, the chirped PG pulse evolves continuously with the period ${{2\pi } / \beta }$ when the parabolic potential exists, and its period is only determined by the potential parameter. Consistent with the analysis method of Eq. (8), by setting $1 - {1 / {4M}} = 0$ and ${{4M} / {({\beta \csc ({\beta Z} )} )}} = 0$ to find the two focal point positions as:
Therefore, the combination of parabolic potential and quadratic chirp can change the distance of the focal point, while the linear chirp parameter can only change the temporal position of the focal point.
In order to verify the above theoretical results, we numerically solve the Eq. (1) with parabolic potential, and the results are shown in Fig. 4. Figures 4(a)–4(c) show the temporal evolutions of the PG pulses with different linear and quadratic chirps when the parabolic potential parameter is $\beta = 0.3$. The results show that PG pulses exhibit a periodic evolution pattern, which is consistent with the reported dynamics of beams evolution under parabolic potential [35–37]. However, due to the inherent uni-focusing and inversion properties of the PG pulses, the focusing phenomenon will occur multiple times during the period of evolution. In order to clearly disclose the characteristics of the focal points, we only show the evolution dynamics of one period in the early stage of the PG pulse propagation in Figs. 4(a)–4(c). There are four focusing points in a single evolution period and the focus appears at $T = 0$, which can be found in the temporal evolution of the chirp-free PG pulse under parabolic potential in Fig. 4(b). When a chirped PG pulse is incident, the linear chirp keeps the focal length constant, while the temporal position of the focal point changes. This can be obtained by directly combining Eqs. (12) and (13). For the quadratic chirp case, the focus distance changes while the temporal position of the focus remains unchanged. This is consistent with the analytical result of Eq. (11). To visualize the effect of parabolic potential depth on the two focal points, the distance of the two focal points and the corresponding intensity ratio as a function of $\beta $ are plotted in Figs. 4(d) and 4(e). The results show that the focusing distance becomes shorter, while the interval decreases with the increase of the parabolic potential depth. The trend of focusing distance versus parabolic potential based on Eq. (13) (black and blue curves) is exactly the same as the numerical simulation (diamonds). When only the linear chirp is considered, it can also be seen from the red curve in Fig. 4(d) that with the increase of $\beta $, the intensity ratio increases continuously. In particular, the intensities of the two focal points are the same when $\beta = 0.5$. For the quadratic chirp, the intensity ratio decreases as $\beta $ increases, and the second focus is always stronger than the first one. Furthermore, the truncation coefficient reduces the intensity of the focusing point without affecting the focus distance.
Fig. 4. Temporal evolutions of the PG pulses in a parabolic potential ($\beta = 0.3$) for different linear and quadratic chirps: (a) ${C = 1,}\,{Q = 0}$, (b) ${C = 0,}\,{Q = 0}$, and (c) ${C = 0,}\,{Q = 0.2}$. The position and intensity ratio of the two focal points as a function of the parabolic potential for different chirps: (d) ${C = 1,}\,{Q = 0}$ and (e) ${C = 0,}\,{Q = 0.2}$. The solid curves and the diamonds shown in (d) and (e) represent numerical and analytical results, respectively.
Since the quadratic chirp shows strong talent for controlling the focusing properties of the PG pulse, we discuss the effect of the quadratic chirp on the two focal points in the early stage of evolution in detail, and the results are shown in Fig. 5. The maximum intensity of the PG pulses as a function of quadratic chirps for a fixed propagation distance, shown in Fig. 5(a). The results show that the focusing distance of the PG pulse increases with the quadratic chirp. They are identical to the analytical results shown by the black dashed line in Fig. 5(a). Specifically, the relationship of the two focal points such as maximum intensity, intensity ratio and interval as a function of quadratic chirp are shown in Figs. 5(b)–5(d). The results show that with the increase of Q, the maximum intensity of the PG pulse decreases and then increases. The physical reason behind this phenomenon can be understood as the spectrum of a quadratic chirped PG pulse has oscillation tails either behind or in front of the main peak, depending on whether the $\beta $ is negative or positive. The dispersion-induced chirp in the anomalous dispersion region is negative, and the sign of the initial quadratic chirp determines whether they are superimposed or canceled. Through the sin-type change of the two focal intensity ratio curves, the intensity of the second focal point first decreases and then increases, even three times the intensity of the first focal point. From Eq. (13), it can be known that the interval between two focal points in the early stage of PG pulse evolution is:
Fig. 5. (a) The maximum intensity of the PG pulses as a function of quadratic chirps for a fixed propagation distance, where ${\beta = 0.3,}\,{C = 0}$ and the dashed curves represent analytical results. The maximum intensity (b), intensity ratio (c), and the interval (d) of the two focal points as a function of the quadratic chirp Q.
Obviously, the curve of the interval between the two focal points of the Gaussian-type change with different quadratic chirp parameters, it can be known that the interval between the two focal points increases first and then decreases. Through the above comparison, one can find that the focusing characteristic of the PG pulse under the parabolic potential has a strong dependence on the quadratic chirp, and is more sensitive to quadratic chirp $|Q |< 0.5$.
3.3. Dual PG pulse propagation in parabolic potential
Next, we study the interaction of two in-phase PG pulses in parabolic potential controlled by Eq. (1). Therefore, the incident pulse based on Eq. (2) can be written as:
where B is the parameter controlling the pulses initial interval.By using the same analysis method as in Section 3.2, Figs. 6(a)–6(c) present the evolution dynamics of PG pulses with different initial chirps when the initial interval $B = 0$. The results show that the dual PG pulses also exhibit periodic focusing and defocusing behavior, and this phenomenon is consistent with other in-phase pulse interactions such as Airy pulses and solitons [43,44]. It is worth noting that the evolution trajectories of the dual PG pulses and the focusing distance are consistent with the single one, following the analytical results of Eqs. (12) and (13). The existence of the linear chirp only bends the evolution trajectory of the main lobe without changing its focusing distance, which is the same as the case of single-pulse incidence. Figure 6(d) shows the evolution of the maximum intensity of the dual PG pulse at different initial linear chirp C. Comparing Figs. 6(a) and 6(d), it can be found that neither the position nor the intensity of the two focal points changes with the linear chirp C. After the quadratic chirp is taken into account, as shown in Fig. 6(e), the situation is completely different, the position and intensity of the focal point vary with Q. We can find that the center of the dual PG pulse is $T = 0$, consistent with the chirp-free case. Furthermore, through extensive statistical analysis, the evolution properties of the dual PG pulses, such as the manipulation trends of the maximum intensity, intensity ratio and interval two focal points, are the same as the single one. However, the dual pulses case can achieve high focusing intensity, about twice of the single pulse incident. In addition, the effect of the parabolic potential depth on the focal point of the dual PG pulse is also considered, and the results are similar to those obtained by the single pulse incidence.
Fig. 6. Temporal evolutions of dual PG pulse in a parabolic potential ($\beta = 0.3$) for different linear and quadratic chirps: (a) ${B = 0,}\,{C = 1,}\,{Q = 0}$, (b) ${B = 0,}\,{C = 0,}\,{Q = 0}$, and (c) ${B = 0,}\,{C = 0,}\,{Q = 0.2}$. The evolution of the maximum intensity of dual PG pulses as a function of linear (d) and quadratic (e) chirps, where the dashed curves represent analytical results.
Finally, we consider the effect of the initial interval of the dual PG pulse on the focusing properties. Figures 7(a) and 7(b) show the temporal evolutions for the linear chirp $C = 1$ and the quadratic chirp $Q = 0.2$ for the initial interval $B = 2$ and the parabolic potential $\beta = 0.3$, respectively. It can be found that the pulse at the leading edge in the initial stages undergoes the same process as in Fig. 4, while the trailing edge pulse does the opposite. That is to say, the leading PG pulse must first follow its inherent focusing characteristics and then interact with the trailing one. This is due to the opposite effects of the same chirping parameters on the leading and trailing PG pulses. Figures 7(c) and 7(d) are the curves of the maximum intensity of dual PG pulses with different initial linear and quadratic chirps as a function of the initial interval. Obviously, the increase of the initial interval does not cause a change in the positions of the two focal points, even for $B < 0$. This is due to the fact that the first focal point is generated by the uni-focusing of the PG located at the leading edge, and the second focal point is brought by the evolution of the parabolic potential. When the initial interval is large, the leading edge of PG pulse completes the focusing process before the two pulses interact. This is also confirmed by the fact that the intensity of the first focal point is almost unchanged with increasing spacing. Therefore, it can also be found that the intensity of the second focal point increasing continuously as the distance increases. In particular, the intensity of the second focal point can be increased to more than 40 times the incident intensity by adjusting the combination of parameters, which has been confirmed by calculations. Through simple pulse temporal superposition, a focusing intensity dozens of times stronger than itself can be obtained, which has a wide range of applications in micro-nano processing and rogue wave generation. In addition, we can use a synchronized multi-wavelength fiber laser [45], and then the output pulse is transformed into a PG wave by a pulse shaper. Afterward, the PG pulse is coupled into a waveguide with a wedge-shaped dielectric-supported plasmonic structure or a wedge-shaped metal-dielectric-metal structure [24], and its propagation at a linear potential is experimentally investigated. Furthermore, by coupling the PG pulses into a medium whose refractive index changes along a parabolic trend, the parabolic potential strength can be experimentally controlled. The effect of strong nonlocal nonlinearity is equivalent to the parabolic potential [30]. Therefore coupling the PG pulses into a strong nonlocal medium such as a nematic liquid crystal can experimentally investigate the effect of the parabolic potential.
Fig. 7. Temporal evolutions of dual PG pulses in the parabolic potential ($\beta = 0.3$) with the initial interval ($B = 2$) for different linear (a) and quadratic (b) chirps. The evolution of the maximum intensity of dual PG pulses for linear (c) and quadratic (d) chirp as a function of initial interval. Insets in (c) show the zoomed region for improving visibility.
4. Conclusion
In conclusion, we have theoretically and numerically investigated the propagation dynamics of the chirped PG pulses and their interactions in a linear media with external linear or parabolic time-dependent potentials. We find that under the combined action of the linear potential and the linear chirp of the PG pulse, the transmission trajectory of the PG pulse can be effectively controlled without affecting the inherent uni-focusing distance of the PG pulse. Following the transmission trajectory induced by the linear potential, the PG pulse maintains a uni-focusing behavior when the quadratic chirp parameter is $0 \le Q < 0.25$, and the PG pulse exhibits a double focusing phenomenon when $Q < 0$. When the parabolic potential is taken into account, the chirped PG pulses are periodically focused and reversed. For linear chirp, the PG pulse evolution trajectory can be changed without affecting the focus distance and focus intensity. For quadratic chirp, the characteristics of the focal points of the PG pulse, such as position, intensity, and spacing between focal points can be manipulated. In addition, dual PG pulses also exhibit periodic evolution with controllable focusing properties. By adjusting the interval of the incident pulses, the intensity of the second focal point can be increased by more than ten times while the intensity of the first focal point is basically unchanged. Furthermore, the truncation coefficient reduces the intensity of the focus without affecting the focus distance. The analytical expression of the focal distance of the PG pulse is derived and verified by numerical simulation. These results provide some inspiration for the regulation of PG pulses and broaden the application scenarios of the PG pulses and time-dependent potentials.
Funding
National Natural Science Foundation of China (61975130); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515010084); The Special Funding of Guiyang Science and Technology Bureau and Guiyang University (GYU-KY-[2021]).
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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