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Dual-focusing behavior of a one-dimensional quadratically chirped Pearcey-Gaussian beam

Feng Zang, Lifeng Liu, Fusheng Deng, Yanhong Liu, Lijuan Dong, and Yunlong Shi

Open Access Open Access

Abstract

In this paper, we provide analytical solutions describing the dynamic behavior of the Pearcey-Gaussian beams propagating in free space. Based on the analytical solutions, explicit expressions governing the focusing distances of the Pearcey-Gaussian beams are found and verified by numerical simulations. For the linearly chirped Pearcey-Gaussian beam, it exhibits a uni-focusing behavior during propagation. Particularly, the focusing distance is independent on the linear chirp parameter and remains zf = 2 unchanged. Of particular interest is that the quadratically chirped Pearcey-Gaussian beam focuses twice when the quadratic chirp parameter β < 0. The first and the second focusing distances are determined by zf1 = 2/(1 − 4β) and zf2 = −1/(2β), respectively. Furthermore, we numerically investigate the peak powers at the different focusing positions and find that as β increases, the peak powers at zf1 and zf2 linearly decrease. It is expected that the characteristics can be used for manipulating the focusing distances and the peak powers to generate an optical beam with high peak power by adjusting the chirp parameter β.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Past few decades, special beams described by peculiar functions such as Airy beam [1–3], Gaussian beam [4,5] and Bessel beam [6,7] have attracted significant and increasing research interests due to their unique propagation characteristics [8,9], and provide a new possibility in fields of optical manipulation, Spatiotemporal light bullets and optical imaging [10,11]. Therefore, it becomes important to seek new special beams with different propagation properties. In 1946, another peculiar function, the Pearcey function, was first put forward in the study of the electromagnetic field structure near caustic [12]. Until 2012, the Pearcey beam was theoretically proposed and experimentally demonstrated [13]. The results show that the Pearcey beam exhibits some remarkable propagation characteristics such as form-invariance, self-focusing and self-healing. Subsequently, the dynamic properties of the circle Pearcey beams were theoretically investigated, the results show that the beams exhibit an abruptly autofocusing behavior. Compared with that of the circular Airy beams, the circle Pearcey beams increase the peak intensity contrast, shorten the focusing distance and, especially, eliminate the oscillation after the focal point [14]. Inspired by these results, a large number of works have been reported, such as nonparaxial propagation of the circular Pearcey Gaussian beams [15], abruptly autofocused and rotated circular chirp Pearcey Gaussian vortex beams [16], effects of the modulated vortex and second-order chirp on the propagation dynamics of ring Pearcey Gaussian beams [17], partially coherent Pearcey–Gauss beams [18], experimental demonstration of vortex circular Pearcey beams [19], flexible autofocusing properties of ring Pearcey beams with a cross phase [20] and so on. Also, the dynamics of the Pearcey beams in different systems including turbulent atmosphere [21], nonlinear medium [22–24], chiral medium [25] and quadratic-index medium [26–29] have been studied.

On the other hand, many different kinds of beams derived from the Pearcey integral including half Pearcey beams [30], dual Pearcey beams [31], odd-symmetric Pearcey beams [32] and symmetric Pearcey Gaussian beams [33] have been also intensively investigated. In fact, the Pearcey integral can be defined as $\operatorname *{Pe}\left ( x,y\right ) =\int _{-\infty }^{+\infty }ds\exp \left [ i\left ( s^{4} +s^{2}y+sx\right ) \right ]$. After setting the dimensionless transverse variable $y=0$, the Pearcey integral becomes into the one-dimensional case $\operatorname *{Pe}\left ( x,0\right ) =\int _{-\infty }^{+\infty }ds\exp \left [ i\left ( s^{4}+sx\right ) \right ]$. For this situation, its dynamics behavior has been studied [34,35], and the results show that the Pearcey beam evolves in the form of two-dimensional Pearcey function and exhibits a dual self-accelerating behavior [34]. By setting the other transverse coordinate $x=0$, the Pearcey function turns into $\operatorname *{Pe}\left ( x,0\right ) =\int _{-\infty }^{+\infty }ds\exp \left [ i\left ( s^{4}+s^{2}x\right ) \right ]$. However, for such Pearcey beam, its dynamics behavior is rarely involved in the literatures.

Thus, in this article we investigate the propagation dynamics of one-dimensional Pearcey-Gaussian beams in free space. For normal and linearly chirped Pearcey-Gaussian beams, they all focus once during propagation, and the focusing distances are $z_{f}=2$. When the Pearcey-Gaussian beam carries a quadratic chirp, two situations are considered. For $0\leq \beta <0.25$, the Pearcey-Gaussian beam also focuses once during propagation. The focusing distance and the peak power can be manipulated by changing the quadratic chirp parameter. For $\beta <0$, the difference from the cases above is that the Pearcey-Gaussian beam exhibits a dual-focusing behavior. The focusing distances and the peak powers are dependent on the parameter $\beta$.

The rest of this paper is structured as follows. In Sec. 2, we introduce the theoretical model and discuss the characteristics of the Pearcey function. In the next section, we provide an analytical solution describing the evolution of Pearcey-Gaussian beam without chirp. The analytical description for the evolution of the Pearcey-Gaussian beam with a linear chirp is presented in Sec. 4. The analytical result is also presented when the Pearcey-Gaussian beam carries a quadratic chirp. Finally, the main results of the paper are summarized in Sec. 6.

2. Theoretical model and the characteristics of the Pearcey function

We consider a Pearcey beam propagating in free space. Under this case, the evolution of optical beam along propagation direction obeys the normalized potential-free Schrödinger equation:

$$i\frac{\partial\psi}{\partial z}+\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}\psi\left( x,z\right) =0,$$
where $\psi \left ( x,z\right )$ is the slowly varying amplitude of optical beam. $x$ and $z$ account for normalized dimensionless transverse coordinate and propagation distance, scaled by some transverse width $x_{0}$ and the corresponding Rayleigh range $kx_{0}^{2}$. $k=2\pi n/\lambda _{0}$ represents the wavenumber, $n$ is the linear refraction index, $\lambda _{0}$ is the wavelength. The general solution of Eq. (1) in real space can be written as
$$\psi(x,z)=\frac{1}{2\pi} {\displaystyle\int_{-\infty}^{+\infty}} \widehat{\psi}(k,0)\exp\left({-}i\frac{1}{2}k^{2}z\right) e^{ikx}dk.$$
Here $\widehat {\psi }(k,0)$ is the Fourier spectrum of the initial beam $\psi (x,0)$. By employing the convolution property of Fourier transform for Eq. (2), we have
$$\begin{aligned} \psi(x,z) & =\psi(x,0)\ast F^{{-}1}\left( e^{{-}i\frac{1}{2}k^{2}z}\right)\\ & =\frac{\left( 1-i\right) }{2\sqrt{\pi z}}{\int\nolimits_{-\infty }^{+\infty}}\psi(\xi,0)\exp\left[ \frac{i\left( x-\xi\right) ^{2}} {2z}\right] d\xi. \end{aligned}$$
From expression (3), one can see that for a given initial beam $\psi (x,0)$, we can get the evolution solution of Eq. (1) by computing the integral in Eq. (3).

In this paragraph, we will discuss the characteristics of Pearcey function. The one-dimensional Pearcey function is defined as

$$\operatorname*{Pe}\left( x/b,a\right) =\int_{-\infty}^{+\infty}ds\exp\left[ i\left( as^{4}+s^{2}x/b\right) \right] .$$
Here, $a$ represents the coefficient of $s^{4}$ in integral term, and $b$ is an arbitrary scaling factor. Figures 1(a) and 1(b) present the profiles and the phase distributions of the Pearcey function with different $a$ and $b$. From Fig. 1(a), one can see that the profile of the Pearcey function is asymmetry about $x=0$, and there are many small side lobes next to the main lobe, which is similar to that of the Airy function. From Fig. 1(b), one can find that the phase of the Pearcey function differs from that of Airy function, it continuously varies from $-\pi$ to $\pi$. We numerically calculated a larger number of graphs of the Pearcey functions with different $a$ and $b$, Figs. 1(a) and 1(b) only show part of them. By comparing the graphs of the Pearcey function with different $a$ and $b$, we can find that for a fixed $b$, when the parameter $a$ changes the symbol, the profile and the phase of Pearcey function invert, while the inversion of the phase is more complicated. In other words, the profiles of Pearcey function with $a$ and $-a$ are symmetry about $x=0$. According to the continuity of the function for variable, we can find that $a=0$ is a transition point of the spatial inversion. For a fixed $a$, one can find that when the parameter $b$ changes the symbol, the profile and the phase of Pearcey function invert. By the same logic, $b=0$ is a transition point of the spatial inversion ($b=0$ is a singularity). Thus, we conclude that the parameters $a$ and $b$ affect the profile and the phase of the Pearcey function, when $a$ ($b$) changes the symbol, the profile of Pearcey function inverts.

 figure: Fig. 1.

Fig. 1. (a) The profiles and (b) the phase distributions of Pearcey functions with different $a$ and $b$.

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3. Self-inverting and self-focusing dynamic behavior of Pearcey-Gaussian beams

To study the dynamic behavior of the Pearcey beam, the input beam is taken as

$$\psi(x,0)=\operatorname*{Pe}\left( x,0\right) \exp\left( -\sigma x^{2}\right) .$$
$\operatorname *{Pe}\left ( x,0\right ) =\int _{-\infty }^{+\infty }ds\exp \left [ i\left ( s^{4}+s^{2}x\right ) \right ]$ denotes a Pearcey function. Gaussian function $\exp \left ( -\sigma x^{2}\right )$ is used to ensure that the energy of Pearcey beam is finite. $\sigma$ is a parameter related to the width of Gaussian function. To maintain the propagation characteristics of the Pearcey beam as much as possible, the Gaussian function width must be very large, that is, $\sigma$ must be very small. In the following discussions, the parameter $\sigma =1/400$ is fixed. Figures 2(a) and 2(b) show the profile and the phase distribution of the input Pearcey-Gaussian beam (5 ). We simulate the Pearcey-Gaussian beam propagating in free space by using the split-step Fourier method. Figures 2(c) and 2(d) present the numerical evolutions of the intensity and the phase of the Pearcey-Gaussian beam in free space, respectively. From them, one can see that the Pearcey-Gaussian beam inverts and focuses during propagation, and the focusing position, i.e., the inverting point, is located at $z_{f}=2$.

 figure: Fig. 2.

Fig. 2. (a) The profile and (b) the phase distribution of the input Pearcey-Gaussian beam (5). Numerical evolutions of (c) the intensity and (d) the phase of the Pearcey-Gaussian beam obtained by solving numerically Eq. (1). Analytical evolutions of (e) the intensity and (f) the phase of Pearcey-Gaussian beam governed by Eq. (6).

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In the following, we try to explain the self-focusing dynamic behavior of the Pearcey-Gaussian beam by an analytical method. Substituting Eq. (5) into Eq. (3) and using the definition of Pearcey function, the analytical dynamic solution for Eq. (1) with initial condition (5) can be written as

$$\begin{aligned} \psi(x,z) & =\sqrt{\frac{1}{\eta\left( z\right) }}\exp\left[ -\frac{\sigma x^{2}}{\eta\left( z\right) }\right]\\ & \times {\displaystyle\int_{-\infty}^{+\infty}} ds\exp\left\{ is^{4}\left[ 1-\frac{z}{2\eta\left( z\right) }\right] +is^{2}\frac{x}{\eta\left( z\right) }\right\} , \end{aligned}$$
where $\eta \left ( z\right ) =1+i2\sigma z$. According to solution (6), the evolutions of the intensity and the phase of the Pearcey-Gaussian beam (5) are depicted in Figs. 2(e) and 2(f), respectively. From them, one can see that the Pearcey-Gaussian beam does invert and does focus during propagation, which match very well with the numerical results given by Figs. 2(c) and 2(d). Comparing solution (6) with the Pearcey function (4), we can find that $a=1-z/\left ( 2\eta \right )$ and $b=\eta$. When $\sigma$ is indefinitely small, the imaginary part of $\eta \left ( z\right ) =1+i2\sigma z$ can be regarded as a small quantity and be neglected, so the coefficient $a$ and the scaling factor $b$ can be approximated as $1-z/2$ and $1$. According to the previous analysis of the Pearcey function, when $1-z/2=0$, i.e., $z_{f}=2$, the Pearcey function inverts. Therefore the analytical result suggests the focusing distance of the Pearcey-Gaussian beam is $z_{f}=2$, which is consistent with the numerical result.

4. Dynamic behavior of linearly chirped Pearcey-Gaussian beams

In this section, we consider the evolution of the Pearcey-Gaussian beam with a linear chirp. Thus, we take the initial beam as a Pearcey-Gaussian beam with a linear chirp

$$\psi(x,0)=\operatorname*{Pe}\left( x,0\right) \exp\left( -\sigma x^{2}\right) \exp\left( iCx\right) .$$
Where, $C$ is linear chirp parameter related to the incident angle of the optical beam. Substituting Eq. (7) into Eq. (3), after some algebra, the solution for Eq. (1) with initial input (7) can be written as
$$\begin{aligned} \psi(x,z) & =\sqrt{\frac{1}{\eta\left( z\right) }}\exp\left[ -\frac{\sigma x^{2}}{\eta\left( z\right) }\right] \exp\left[ i\frac{C\left( x-Cz/2\right) }{\eta\left( z\right) }\right]\\ & \times {\displaystyle\int_{-\infty}^{+\infty}} ds\exp\left\{ is^{4}\left[ 1-\frac{z}{2\eta\left( z\right) }\right] +\frac{is^{2}\left( x-Cz\right) }{\eta\left( z\right) }\right\} , \end{aligned}$$
where $\eta \left ( z\right ) =1+i2\sigma z$. Comparing solution (8) with the Pearcey function (4), one can find that when $\sigma$ is very small, $a$ and $b$ can be approximated as $1-z/2$ and $1$, respectively. Therefore, the analytical result predicts when $1-z/2=0$, i.e., $z_{f}=2$, the linearly Pearcey-Gaussian beam undergoes a spatial inversion and focuses during propagation, and the parameter $C$ does not affect the focusing distance of the Pearcey-Gaussian beam. To verify the analytical result, Figs. 3(a) and 3(b) display the numerical evolutions of the Pearcey-Gaussian beam for the different parameters $C$, the corresponding evolutions of the maximum intensity are shown in Fig. 3(c). From them, it can be easily seen that for the different parameters $C$, the Pearcey-Gaussian beam exhibits a uni-focusing behavior, and the focusing distance hardly changes. Furthermore, the dependence of the focusing distance $z_{f}$ on the chirp parameter $C$ is also calculated numerically, as shown by the red points in Fig. 3(d). One can see that with the increasing of $C$, the focusing distance remains unchanged, which agrees with the analytical result $z_{f}=2$ shown by the black curves in Fig. 3(d).

 figure: Fig. 3.

Fig. 3. Numerical evolutions of the Pearcey-Gaussian beam with different parameters $C$, (a) $C=-2$, (b) $C=1$. (c) The evolutions of the maximum intensity of the Pearcey beam for different parameter $C$. (d) The focusing distance $z_{f}$ versus the chirp parameter $C$.

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5. Dynamic behavior of quadratically chirped Pearcey-Gaussian beams

Now, we turn to discuss the dynamic behavior of the Pearcey-Gaussian beam with a quadratic chirp. The initial beam is chosen as a Pearcey-Gaussian beam with a quadratic chirp

$$\psi(x,0)=\operatorname*{Pe}\left( x,0\right) \exp\left( -\sigma x^{2}\right) \exp\left( i\beta x^{2}\right)$$
with $\beta$ being the quadratic chirp parameter related to the ratio of the Rayleigh length to the focal length of a spherical lens. Substituting Eq. (9) into Eq. (3), we can readily obtain the propagation solution of Eq. (1) corresponding to the initial condition (9):
$$\begin{aligned} \psi(x,z) & =\sqrt{\frac{1}{\eta\left( z\right) }}\exp\left[ -\frac{\sigma x^{2}}{\eta\left( z\right) }\right] \exp\left[ i\frac{\beta x^{2}}{\eta\left( z\right) }\right]\\ & \times {\displaystyle\int_{-\infty}^{+\infty}} ds\exp\left\{ is^{4}\left[ 1-\frac{z}{2\eta\left( z\right) }\right] +\frac{is^{2}x}{\eta\left( z\right) }\right\} , \end{aligned}$$
where $\eta \left ( z\right ) =1+i2\sigma z+2\beta z$. From solution (10), we can find that the focusing distance of the Pearcey-Gaussian beam is of the form
$$z_{f}=\frac{2}{1-4\beta}.$$
To ensure that $z_{f}$ is a positive number, the quadratic chirp parameter $\beta$ must satisfy $\beta <0.25$, i.e., when $\beta \geq 0.25$, the Pearcey-Gaussian beam does not focus. According to the range of the chirp parameter $\beta$, two situations, $0\leq \beta <0.25$ and $\beta <0$, need to be discussed in detail.

For $0\leq \beta <0.25$, when $\sigma$ is very small, the scaling factor $b=\eta \left ( z\right )$ can be approximated as $1+2z\beta$. Due to $1+2z\beta \neq 0$, only a focusing position, i.e., the transition point appears. Therefore, the analytical result indicates the Pearcey-Gaussian beam focuses once during propagation, and the focusing distance can be described by expression (11). Figures 4(a) and 4(b) show the numerical evolutions of the Pearcey-Gaussian beam for different $\beta$. Furthermore, the corresponding evolutions of the maximum intensity are depicted in Fig. 4(c). From them, one can see that for different $\beta$, the Pearcey beam does exhibit a uni-focusing behavior, and the parameter $\beta$ plays an important role on the focusing distance $z_{f}$ and the peak power at $z_{f}$. Moreover, Figs. 4(d) and 4(e) present the dependences of the focusing distance $z_{f}$ and the peak power on $\beta$, respectively. From them, one can see that with the increasing of $\beta$, $z_{f1}$ increases, which matches very well with the analytical result (11) shown by the black curve in Fig. 4(d); while the peak power linearly decreases [see Fig. 4(e)].

 figure: Fig. 4.

Fig. 4. Numerical evolutions of the Pearcey-Gaussian beam for the different parameters $\beta$. (a) $\beta =0.05$, (b) $\beta =0.1$. (c) The evolutions of the maximum intensity of the Pearcey beam for different parameters $\beta$. (d) The focusing distance $z_{f}$ versus the chirp parameter $\beta$. (e) The peak power at $z_{f}$ versus the parameter $\beta$.

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For $\beta <0$, we simulate the evolutions of the Pearcey-Gaussian beams for the different parameters $\beta$, the corresponding results present in Fig. 5(a) and 5(b). From them, one can obviously see that Pearcey-Gaussian beam exhibits a dual-focusing behavior during propagation. Moreover, Fig. 5(c) displays the evolutions of the maximum intensity for different negative parameters $\beta$. From it, one can see that for different negative parameters $\beta$, the Pearcey beam does focus twice during propagation, and the focusing distances and the peak powers rely on the chirp parameter $\beta$. To explain this dual-focusing behavior of the Pearcey-Gaussian beam, we go back to solution (10). When $\sigma$ is very small, the coefficient $a$ and the scaling factor $b$ can be approximated as $1-z/\left [ 2\left ( 1+2\beta z\right ) \right ]$ and $1+2\beta z$, respectively. According to the previous analysis for the Pearcey function, when $1-z/\left [ 2\left ( 1+2\beta z\right ) \right ] =0$ and $1+2\beta z=0$, the Pearcey-Gaussian beam exhibits a dual-focusing behavior, and two focusing distances can be deduced from two formulas above

$$\begin{aligned} z_{f1} & =\frac{2}{1-4\beta},\\ z_{f2} & ={-}\frac{1}{2\beta}. \end{aligned}$$
According to expression (12), we plot the dependences of $z_{f1}$ and $z_{f2}$ on $\beta$ in Fig. 5(d), respectively. From it, we can see that with the increasing of $\beta$, $z_{f1}$ and $z_{f2}$ increase, and the larger the parameter $\beta$ is, the faster the increasing trend is, which are in agreement with the numerical results shown by the points in Fig. 5(d). Figure 5(e) depicts the dependences of the peak powers at $z_{f1}$ and $z_{f2}$ on $\beta$, respectively. One can see that as $\beta$ increases, the peak powers at $z_{f1}$ and $z_{f2}$ linearly decrease, and the peak power at $z_{f2}$ reduces faster than that at $z_{f1}$. When $\beta =-0.6$, the peak powers at $z_{f1}$ and $z_{f2}$ are equal. These properties can be used for manipulating the focusing distances and the peak powers to generate optical beam with high peak power by adjusting the chirp parameter $\beta$. When $\beta \geq 0.25$, due to the propagation distance $z\geq 0$, neither $a=1-z/\left [ 2\left ( 1+2\beta z\right ) \right ]$ nor $b=1+2\beta z$ are equal to $0$ during propagation, and with the increasing of $z$, $b$ increases. So the analytical results predict that the Pearcey-Gaussian beam does not focus, and the width of the beam becomes larger and larger.

 figure: Fig. 5.

Fig. 5. (a) Dual-focusing behavior of the Pearcey-Gaussian beam for the different parameters $\beta$ (a) $\beta =-0.25$, (b) $\beta =-0.2$. (c) The evolutions of the maximum intensity of the Pearcey beam for different parameters $\beta$. (d) The first focusing distance $z_{f1}$ and the second focusing distance $z_{f2}$ versus the chirp parameter $\beta$, respectively. The curves and points correspond to the analytical results and the numerical results, respectively. (e) The peak powers at $z_{f1}$ and $z_{f2}$ versus the parameter $\beta$, respectively.

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6. Conclusions

In summary, we have analytically investigated the self-focusing behavior of the Pearcey-Gaussian beams in free space. Moreover, we have got analytical expressions governing the focusing distances of the Pearcey-Gaussian beams, and the results have been verified by numerical simulations. For unchirped Pearcey-Gaussian beam, it focuses once during propagation, and the focusing distance is $z_{f}=2$. When the Pearcey-Gaussian beam carries a linear chirp, it also focuses once during evolution. The focusing distance is not impacted by the chirp parameter, and remains $z_{f}=2$ unchanged. For the Pearcey-Gaussian beam with a quadratic chirp, two situations are considered. For $0\leq \beta <0.25$, the Pearcey-Gaussian beam also exhibits a uni-focusing behavior during propagation, and with the increasing of $\beta$, the focusing distance increases, while the peak power linearly decreases. Differing from the cases above, the Pearcey-Gaussian beam focuses twice during propagation for $\beta <0$. The first and the second focusing distances can be described by $z_{f1}=2/(1-4\beta )$ and $z_{f2}=-1/(2\beta )$, respectively. Furthermore, the peak powers have been numerically investigated. The results have shown that as $\beta$ increases, the peak powers in $z_{f1}$ and $z_{f2}$ linearly decrease.

Funding

Science and Technology Innovation Team in Shanxi Province (201805D131006); Shanxi Provincial Key Research and Development Project (201923D121026, 201923D121071); National Natural Science Foundation of China (11874245).

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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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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Figures (5)
Fig. 1.
Fig. 1. (a) The profiles and (b) the phase distributions of Pearcey functions with different $a$ and $b$ .

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Fig. 2.
Fig. 2. (a) The profile and (b) the phase distribution of the input Pearcey-Gaussian beam (5). Numerical evolutions of (c) the intensity and (d) the phase of the Pearcey-Gaussian beam obtained by solving numerically Eq. (1). Analytical evolutions of (e) the intensity and (f) the phase of Pearcey-Gaussian beam governed by Eq. (6).

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Fig. 3.
Fig. 3. Numerical evolutions of the Pearcey-Gaussian beam with different parameters $C$ , (a) $C=-2$ , (b) $C=1$ . (c) The evolutions of the maximum intensity of the Pearcey beam for different parameter $C$ . (d) The focusing distance $z_{f}$ versus the chirp parameter $C$ .

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Fig. 4.
Fig. 4. Numerical evolutions of the Pearcey-Gaussian beam for the different parameters $\beta$ . (a) $\beta =0.05$ , (b) $\beta =0.1$ . (c) The evolutions of the maximum intensity of the Pearcey beam for different parameters $\beta$ . (d) The focusing distance $z_{f}$ versus the chirp parameter $\beta$ . (e) The peak power at $z_{f}$ versus the parameter $\beta$ .

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Fig. 5.
Fig. 5. (a) Dual-focusing behavior of the Pearcey-Gaussian beam for the different parameters $\beta$ (a) $\beta =-0.25$ , (b) $\beta =-0.2$ . (c) The evolutions of the maximum intensity of the Pearcey beam for different parameters $\beta$ . (d) The first focusing distance $z_{f1}$ and the second focusing distance $z_{f2}$ versus the chirp parameter $\beta$ , respectively. The curves and points correspond to the analytical results and the numerical results, respectively. (e) The peak powers at $z_{f1}$ and $z_{f2}$ versus the parameter $\beta$ , respectively.

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Equations (12)

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i ∂ ψ ∂ z + 1 2 ∂ 2 ∂ x 2 ψ ( x , z ) = 0 ,
ψ ( x , z ) = 1 2 π ∫ − ∞ + ∞ ψ ^ ( k , 0 ) exp ⁡ ( − i 1 2 k 2 z ) e i k x d k .
ψ ( x , z ) = ψ ( x , 0 ) ∗ F − 1 ( e − i 1 2 k 2 z ) = ( 1 − i ) 2 π z ∫ − ∞ + ∞ ψ ( ξ , 0 ) exp ⁡ [ i ( x − ξ ) 2 2 z ] d ξ .
Pe ⁡ ( x / b , a ) = ∫ − ∞ + ∞ d s exp ⁡ [ i ( a s 4 + s 2 x / b ) ] .
ψ ( x , 0 ) = Pe ⁡ ( x , 0 ) exp ⁡ ( − σ x 2 ) .
ψ ( x , z ) = 1 η ( z ) exp ⁡ [ − σ x 2 η ( z ) ] × ∫ − ∞ + ∞ d s exp ⁡ { i s 4 [ 1 − z 2 η ( z ) ] + i s 2 x η ( z ) } ,
ψ ( x , 0 ) = Pe ⁡ ( x , 0 ) exp ⁡ ( − σ x 2 ) exp ⁡ ( i C x ) .
ψ ( x , z ) = 1 η ( z ) exp ⁡ [ − σ x 2 η ( z ) ] exp ⁡ [ i C ( x − C z / 2 ) η ( z ) ] × ∫ − ∞ + ∞ d s exp ⁡ { i s 4 [ 1 − z 2 η ( z ) ] + i s 2 ( x − C z ) η ( z ) } ,
ψ ( x , 0 ) = Pe ⁡ ( x , 0 ) exp ⁡ ( − σ x 2 ) exp ⁡ ( i β x 2 )
ψ ( x , z ) = 1 η ( z ) exp ⁡ [ − σ x 2 η ( z ) ] exp ⁡ [ i β x 2 η ( z ) ] × ∫ − ∞ + ∞ d s exp ⁡ { i s 4 [ 1 − z 2 η ( z ) ] + i s 2 x η ( z ) } ,
z f = 2 1 − 4 β .
z f 1 = 2 1 − 4 β , z f 2 = − 1 2 β .
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