1. Introduction

Recently, some peculiar functions have been utilized to describe manifold beams, one of which was the Pearcey function. The Pearcey function was first proposed in 1946, which can express a field structure at and near a line focus of a cylindrical electromagnetic wave train with any possible cylindrical aberration [1]. Until 2012, for the first time, the Pearcey beam was theoretically introduced and experimentally generated [2]. The propagation properties of the Pearcey beam were remarkable, including form-invariance, self-healing and noteworthy auto-focusing effect on propagation [2–4]. Since then, the Pearcey beam has become a subject of immense interest. Thus far, many derivative beams based on the Pearcey beam have been reported, such as circle Pearcey beam [5], half Pearcey beam [6], dual Pearcey beam [7], odd-Pearcey Gauss beam [8], and so on. Hence, some potential and possible applications are continuously expanded in optical imaging [9,10], optical trapping [11,12] and optical manipulation [13].

On the other hand, as an effective tool to modulate the dynamics of a laser beam, the photon potential was embedded in the refractive index of the medium. And the particular potential can be configured by the choice of the media [14,15]. The external potentials like linear potential [16,17] and parabolic potential [18,19] were tailored for achieving manipulations of the beam’s accelerating trajectory and altering the propagation characteristics of the beam. To portray the effect of the potential more lucidly, the potential was assimilated to a force that can stretch or suppress the beam during propagation [14,16,20,21]. Moreover, the dynamic potential which altered during evolution has been studied [15,20,22]. Consequently, the force which originated from the dynamic potential evolved into a more flexible one. The beam can follow a predesigned trajectory owing to the dynamic potential as a function of the propagation distance [22]. In other words, the accelerating trajectory was determined by the configuration of the refractive index of the medium.

The parabolic potential, one of the potential modes, has diverse impacts on the beam propagation properties, such as modulating a symmetric Gaussian [23,24] or a paraboloidal [25] intensity profile, being a Fourier transformer [26], and so on [27]. More importantly, the parabolic potential emerged provided that the strongly nonlocal nonlinearity existed in a certain form resembling that in Ref. [28], and furthermore, the boundary conditions of the strongly nonlocal media produced some prominent influences on the propagation dynamics of the beam [29]. To the best of our knowledge, the Pearcey derivative beam has been only considered in external parabolic potential [13,30]. For the sake of better manipulate the Pearcey beam’s propagation, the research on the Pearcey beam in a dynamic parabolic potential is required. Compared to the static parabolic potential, the dynamic one is generated by a more flexible configuration of the refractive index, and its potential varies with the beam-propagation distance. By definition, a specific dynamic parabolic potential can be formed by a correct choice of medium. Therefore, in this paper, we investigate the one-dimensional symmetric Pearcey Gaussian beam (SPGB$_1$) in a uniformly moving parabolic potential, which is one of the categories of the dynamic parabolic potential. The setup of the article is as follows. In Sec. 2, we introduce the theoretical model in detail. Then, we present the multivariate analytical solutions, numerical simulations and necessary discussions in Sec. 3. Eventually, we conclude the paper based on the results above in Sec. 4.

2. Theoretical model

In the paraxial optical system, a beam propagating in a dynamic parabolic potential satisfies the (1+1)D dimensionless Schrödinger equation [14]:

One figures out the solution of Eq. (1) step by step. By introducing a new variable $X = x - t(z)$ and utilizing the Fourier transform pair $\widetilde \varphi = \int _{ - \infty }^{ + \infty } \varphi \exp \left ( { - iKX} \right )dX$, $\varphi = \frac {1}{{2\pi }}\int _{ - \infty }^{ + \infty } {\widetilde \varphi } \exp \left ( {iKX} \right )dK$, now Eq. (1) can be written as [14]:

Next, we do a further coordinate transform operation through setting $\widetilde \varphi = \widetilde \phi \exp \left \{ {i\int {\frac {1}{2}{{\left [ {\frac {{dt\left ( Z \right )}}{{dZ}}} \right ]}^2}} dZ} \right \}$ and $K - \frac {{dt\left ( Z \right )}}{{dZ}} = \alpha \kappa$, in which case Eq. (2) is converted into another form:

Comparing Eqs. (1) and (3), it is not difficult to notice the resemblance in their forms. Equation (3) has many famous solutions, which can also be the solutions of Eq. (1) but expressed in different space and undergone a coordinate transform operation. We select one that fits for our study at hand, which can be written as [24,32]:

In terms of the form of Eq. (4), it is applied to the static parabolic potential. However, the fact that ${\widetilde \phi _0}\left ( {\xi ,0} \right )$ is constructed after the coordinate transform operations involving the dynamic function, allows Eq. (4) to be available for the dynamic parabolic potential. Conversely, the dynamic parabolic potential reduces to static one with $t\left ( z \right ) = 0$. Thence the versatility of the dynamic parabolic potential makes it more practical. The uniformly moving parabolic potential holds a specific form of the dynamic function $t\left ( z \right ) = \mu z + v$. This is a linear function of the beam-propagation distance, where $\mu$ is the slope parameter as and $v$ is the initial bias.

In another aspect, the Pearcey beam possesses the auto-focusing and periodic inversion behaviors during propagation, which can be described by the Pearcey integral $Pe\left ( {X',Y'} \right ) = \int _{ - \infty }^{ + \infty } {\exp \left [ {i\left ( {{s^4} + Y'{s^2} + X's} \right )} \right ]} ds$. The Pearcey integral can be calculated numerically using a contour rotation $s \to s{e^{i\pi /8}}$ in the complex $s$ plane, which guarantees the convergence of the integral when $s \to \pm \infty$ [1,33]. After setting the dimensionless transverse variable $Y' = 0$, the Pearcey integral evolves into the one-dimensional case $Pe\left ( {X',0} \right ) = \int _{ - \infty }^{ + \infty } {\exp \left [ {i\left ( {{s^4} + X's} \right )} \right ]} ds$, which serves the SPGB$_1$ scenario.

3. Analytical solutions and numerical simulations

3.1 One-dimensional symmetric Pearcey Gaussian beam with transverse displacement

Let us consider the first scenario in which case the SPGB$_1$ with transverse displacement ${\varphi _0}\left ( {x,z = 0} \right ) = Pe\left ( {x - {x_0},0} \right )\exp \left [ { - \sigma {{\left ( {x - {x_0}} \right )}^2}} \right ]$ is contemplated as an input. Here, $Pe\left ( \cdot \right )$ corresponds to the Pearcey integral, ${x_0}$ denotes a transverse displacement and $\sigma$ is a truncation coefficient which is related to the width of the Gaussian function. In the rest sections of the paper, we set $\sigma = 1/2500$.

In order to obtain the analytical expression of the beam in the uniformly moving parabolic potential, we first solve out the Fourier transform of the initial plane and do a coordinate transform operation. The result can be expressed as:

Substituting the result into Eq. (4), we obtain:

Find the corresponding integral formula:

After derivations, Eq. (6) can be written as:

Grounded on the specific model with respect to the parabolic potential, we indeed find that the SPGB$_1$ in the uniformly moving parabolic potential performs autofocus periodically [18,27] and the period is $T = 2\pi /\alpha$. The corresponding accelerating trajectory expression can be obtained from Eq. (11) as follows:

From the periodicity of the function in Eq. (12), thus the correctness of the period formula mentioned above is verified. According to Eq. (12), one can note that the evolution of the SPGB$_1$ in the uniformly moving parabolic potential demonstrates in three fundamental regimes as shown in Fig. 1:

 

Fig. 1. Three fundamental regimes of the propagation in a uniformly moving parabolic potential. The size pictured does not represent the actual one.

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Fig. 2. The propagation of the SPGB$_1$ with transverse displacement under various parameters settings: The schematics in the first column correspond to the normalized snapshots in the second column; the images in the same row share the same axes.

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With regard to the previous settings, the parabolic potential depth $\alpha$, the transverse displacement ${x_0}$ and the slope parameter $\mu$ play a key role in the trajectory manipulation, as does the initial bias $\nu$. From Fig. 2, all these patterns reveal the periodicity of the SPGB$_1$. The parabolic potential depth $\alpha$ alters the period of the SPGB$_1$. As the value of the parabolic potential depth $\alpha$ goes up, the period of the SPGB$_1$ shrinks. For instance, when the parabolic potential depth $\alpha$ doubles, the period shrinks to half of the original, which can be seen from Figs. 2(c1)–2(c3). Both the period calculation formula $T = 2\pi /\alpha$ and the accelerating trajectory expression can be verified. One notices that the slope of the accelerating trajectory is associated with the slope parameter $\mu$. In other words, the slope parameter $\mu$ controls the propagation orientation of the SPGB$_1$. Let us observe the patterns in the second row in Fig. 2. It is apparent that the transverse amplitude of the accelerating trajectory is controlled by ${x_0}$, and the initial bias is controlled by $\nu$. The above phenomenon can be properly elucidated by Eq. (12).

It is worth mentioning that, with respect to the positions $z = \left ( {2m + 1} \right )T/4$, Eq. (4) becomes into inverse Fourier transform of the initial plane distribution whose coordinate transform operation has been done. After analyzing Eq. (9), every time in these positions, the automatic Fourier transforms of the initial plane distribution perform particularly in the static parabolic potential [19,31], while in the dynamic one, the imperfect automatic Fourier transforms perform but accompanied by some newly generated phase-modulation items.

3.2 One-dimensional linearly chirped symmetric Pearcey Gaussian beam

Prompted by the impact of a chirp on the Pearcey beam’s propagation properties [34,35], we are inquisitive about the behavior of the chirped SPGB$_1$ in the uniformly moving parabolic potential. Scrutinizing Eq. (4) and the entire derivation process, we note that the initial beam has already involved a quadratic chirp in Fourier space. From another perspective, we embark on investigating the initial beam combined with a chirp in spatial space. Subsequently the SPGB$_1$ with two typical chirps will be delved into in Sec. 3.2 and 3.3, respectively.

The initial plane of the SPGB$_1$ with a linear chirp can be written as:

Compared with Eq. (12) in the former case, this accelerating trajectory expression appends an additional term originated from the linear chirp. Though the modification of the accelerating trajectory occurs, the linearly chirped SPGB$_1$ still possesses the similar properties as before. For instance, the period of the linearly chirped SPGB$_1$ in the uniformly moving parabolic potential is still $T = 2\pi /\alpha$. Moreover, as can be seen from Eq. (15), it is possible for the beam to propagate in three other modes which relate to the linearly chirped factor ${\beta _1}$: a) a controllable straight accelerating trajectory; b) a ladder-like accelerating trajectory with the "platforms" in different positions; c) a simple harmonic motion accelerating trajectory with amplitude controlled by the linearly chirped factor ${\beta _1}$.

Figure 3 illustrates the propagation of the linearly chirped SPGB$_1$ in diverse modes. Based on Eqs. (14) and (15), one notices the distinctive relation between the slope parameter $\mu$ and the linearly chirped factor ${\beta _1}$. Hence, we envision two separate scenarios. One is that the linearly chirped factor is $2$ times larger than the slope parameter (${\beta _1} = 2\mu$), the other is that two parameters are equal (${\beta _1} = \mu$). From the patterns in the first row of Fig. 3, the slope parameter $\mu$ is still the key to manipulate the orientation of the beam propagation. The linearly chirped factor ${\beta _1}$ does not impact on the slope of the trajectory but the amplitude of the sinusoidal wave due to the additional term $- {\beta _1}\sin \left ( {\alpha z} \right )/\alpha$ in Eq. (15). When considering the case ${\beta _1} = 2\mu$, we convert $- U\sin \left ( {\alpha z} \right )/\alpha$ into a form $\mu \sin \left ( {\alpha z} \right )/\alpha = - \mu \sin \left ( {\alpha z + T/2} \right )/\alpha$ which makes the "platforms" shift by a half of a period. Otherwise, the patterns in the second row of Fig. 3 reveal the linearly chirped SPGB$_1$ propagating in an alterable-incline straight line when the two parameters are equal ${\beta _1} = \mu$. In retrospect, the SPGB$_1$ without a chirp only can propagate along a straight line with unalterable and horizontal angle when all the parameters set to zero but the parabolic potential depth. After setting the condition ${\beta _1} = \mu$ and ${x_0} = 0$, the accelerating trajectory expression transforms into $x = \nu + \mu z$, which is the interpretation of this phenomenon. In addition, from the patterns in the third row of Fig. 3, one can see that the simple harmonic motion arises and its amplitude is controlled by twice value of the linearly chirped factor $\beta _1$. The introduction of a linear chirp does not change the autofocusing positions in the beam-propagation direction, because the specific autofucusing positions only depend on the parabolic potential depth $\alpha$. The parabolic potential depth $\alpha$ is fixed, so are the autofocusing positions in the beam-propagation direction. Meanwhile, the accelerating trajectory undergoes a shift by a quarter of a period which makes the autofocusing positions coincide with the maximum transverse movement positions of the accelerating trajectory when the slope parameter $\mu$ and the transverse displacement ${x_0}$ set to zero. With respect to the relative magnitude between the linearly chirped factor $\beta _1$ and the slope parameter $\mu$, the whole patterns in Fig. 3 indicate that it determines the amplitude of the sine term in Eq. (15). Thus, this is the key to manipulate the linearly chirped SPGB$_1$.

 

Fig. 3. The propagation of the linearly chirped SPGB$_1$ under various parameters settings but the parabolic potential depth $\alpha = 0.5$: The schematics in the first column correspond to the normalized snapshots in the second column; the images in the same row share the same axes.

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3.3 One-dimensional quadratically chirped symmetric Pearcey Gaussian beam

Commence with the last case in which the SPGB$_1$ with a quadratic chirp can be given in the form:

Distinguished from Eqs. (12) and (15), this accelerating trajectory expression involves an exceptional term which is relevant to the quadratically chirped factor $\beta _2$ and the transverse displacement ${x_0}$. One means that the manipulation effect from the quadratic chirp exists only when the transverse displacement ${x_0}$ is non-zero. Hence the same accelerating trajectory manipulation as before can be achieved when the transverse displacement ${x_0}$ sets to zero. Besides, the periodicity does not change.

Figure 4 exhibits the propagation of the quadratically chirped SPGB$_1$ with diverse selections of the parameters (Figs. 4(a1)–4(a3) have been processed by the high-pass filtering for better manifestation). Next, we discuss about two particular accelerating trajectory scenarios based on the assumption ${x_0} \ne 0$. Scenario one seen in the first row of Fig. 4: in the static parabolic potential, the introduction of a chirp makes it possible to adjust the amplitude as well as the phase shift of the simple harmonic motion accelerating trajectory. The accelerating trajectory expressions with linear or quadratic chirp in the static parabolic potential are respectively rewritten as:

Scenario two seen in the second row of Fig. 4: like the introduction of a linear chirp, the introduction of a quadratic chirp can also achieve a ladder-like accelerating trajectory, when the condition ${\alpha ^2}{x_0} + 4{\beta _2}^2{x_0} - 4\mu {\beta _2} = 0$ is met. Because of the limitation of aforementioned condition, the phase shift of the simple harmonic motion superimposed on the oblique straight line is restricted, which in turn restrains the positions of the "platforms" of the "ladder". In fact, whether it is the introduction of a linear chirp or a quadratic chirp, it makes the position of the "platform" difficult to control. Without a chirp, the positions of the "platforms" $z = mT$ are only determined by the parabolic potential depth $\alpha$. In other cases seen in the third row of Fig. 4, that is, the general cases in which the accelerating trajectories retain its shape of Mode 3 in Fig. 1.

 

Fig. 4. The propagation of the quadratically chirped SPGB$_1$ under various parameters settings but the initial bias $\nu = 0$: The schematics in the first column correspond to the normalized snapshots in the second column; the images in the same row share the same axes.

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In addition, we observe that there is drastic alteration of the energy distribution of the beam. Note that neither of two beams in Sections 3.1 and 3.2 can adjust the intensity distribution of the beam’s propagation without altering the parabolic potential depth $\alpha$. Exclusively, the quadratic chirp impinges upon the beam’s energy distribution. It is meaningful for the quadratically chirped factor $\beta _2$ to adjust the relative intensity of the focal point. Therefore, the beam’s energy can be redistributed according to actual requirements.

3.4 Extension into two-dimensional scenario

We consider an input in the form of a superposition of the one-dimensional symmetric Pearcey Gaussian beams [36]:

The superimposed beam, the so-called two-dimensional symmetric Pearcey Gaussian beam (SPGB$_2$) should be centrosymmetric in theory. Still in paraxial approximation, the SPGB$_2$ evolves in the uniformly moving parabolic potential according to the (2+1)D dimensionless Schrödinger equation, namely,

The specific uniformly moving parabolic potential in this subsection is configured into the case of Mode 1 in Fig. 1, which will cause the SPGB$_2$ to propagate along the ladder-like accelerating trajectory. As expected, the SPGB$_2$ will undergo a ladder-like accelerating trajectory during evolution, as shown in Fig. 5. The results in Fig. 5 show the SPGB$_2$ does own the characteristic of central symmetry. Near the focal point, the energy of the SPGB$_2$ gradually as well as uniformly disperses from the center to the four corners, leading to energy gathering at the four corners and thus a square-shaped pattern. In conformity with the accelerating trajectory expression in Eq. (12), the light spot will offset uniformly from the axis origin along the $x$ and $y$ directions by a distance

As an example, the snapshot in the position $z = 0.9\pi$ seen from Fig. 5(b4) represents a near-focus state, in which the SPGB$_2$ shifts from the axis origin by a distance ${d_{shift}} = 2.72$. After the derivation and stimulation, the one-dimensional case and the two-dimensional case are in good agreement.

 

Fig. 5. The propagation of the SPGB$_2$ in the configured uniformly moving parabolic potential; (a) the side view; (b1)-(b6) the numerical snapshots of the normalized intensity distributions at planes 1-6 marked in (a). The white crosses in the center mean the axis origin ($x=0$, $y=0$).

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4. Conclusion

In summary, based on the dimensionless Schrödinger equation, we have theoretically and numerically investigated the SPGB$_1$ in a uniformly moving parabolic potential. Meanwhile, we also present the analytical solutions of two other scenarios in which the SPGB$_1$ attaches a linear or quadratic chirp, respectively. The accelerating trajectory can be controlled by the parabolic potential which resembles a force to stretch or suppress the beam. After setting a specific configuration of the parabolic potential, the SPGB$_1$ propagates along the predesigned trajectory. Generally speaking, the accelerating trajectory of the displaced and chirped SPGB$_1$ in the uniformly moving parabolic potential maintains a linear superposition form of an oblique straight line with a controllable slope and a simple harmonic motion with the controllable amplitude and phase shift, including two particular scenarios: a) a ladder-like accelerating trajectory with the adjustable "platforms"; b) an alterable-incline straight line. When the dynamic parabolic potential degenerates into a static one, i.e. dynamic function $t\left ( z \right ) = 0$, the accelerating trajectory does not receive the manipulation impact from the slope term, thus appearing in a simple harmonic motion form with the controllable amplitude and phase shift so as to manipulate the specific positions of the focal points. It is worth mentioning that the quadratic chirp impinges up the SPGB$_1$’s energy distribution. Last but not least, we propose an extension into the two-dimensional scenario. We believe that the results shown in the paper would contribute to the accelerating trajectory manipulation of the Pearcey beam, and broaden more applications from the dynamic parabolic potential.