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Abstract
The Talbot-like effect of symmetric Pearcey beams (SPBs) is presented numerically and experimentally in the free space. Owing to the Talbot-like effect, the SPBs have the property of periodic, multiple autofocusing and self-healing. Meanwhile, the focusing positions and focusing times of SPBs are controlled by the beam shift factor and the distribution factors. Furthermore, the beam shift factor can also affect the Talbot-like effect and the Talbot period. It is believed that the results can diversify the application of the Talbot effect.
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1. Introduction
The Talbot effect, first discovered by Henry Fox Talbot [1] in 1836, was also referred to as lens-less imaging or self-imaging. About half a century later, Lord Rayleigh explained this phenomenon as the interference of the diffracted beams and performed the analytical calculations for the plane waves illumination [2]. He found the exact reconstructed images of the grating repeated with the longitudinal period, Talbot period $Z = 2m\frac {d^{2}}{\lambda }$, where $d$ is the grating constant, $\lambda$ denotes the wavelength, and $m$ is a positive integer. In 1996, Montgomery demonstrated that the lateral periodicity of the object is a sufficient condition to perform its reconstruction [3]. This means that the periodic reconstruction of the wave can be obtained without the use of optical lenses when the Montgomery’s condition is fulfilled. Therefore, Kyva1sk pointed out in Ref. [4], the longitudinal periodicity of the wave fields can be presented as a superposition of nondiffracting beams or modes. Then in Ref. [5], similar conclusions are drawn in the frequency and the time domains, respectively. Many scientists had studied the Talbot effect and generalized it to practical applications, such as image processing and synthesis [6], photolithography [7], optical testing [8], optical metrology [9], etc. This lens-less imaging phenomenon has also been observed in various areas of research, such as atomic optics [10], nonlinear optics [11], quantum optics [12], waveguide arrays [13], plasmon [14], photonic lattices [15], metamaterials [16] and the fractional Schrödinger systems [17], etc. Some corresponding conceptual extensions of the Talbot effect have been proposed, including the quantum Talbot effect [12], nonlinear Talbot effect [11], parity-time symmetric Talbot effect [15], plasmon Talbot effect [14], Talbot-Lau effect [18], and Airy-Talbot effect [5], etc. On the other hand, autofocusing beams have recently attracted a lot of interest, and various autofocusing beams have been produced. [19–21]. We know that symmetric Pearcey beams (SPBs) have excellent properties such as autofocusing, nondiffraction, vortex-guiding [21] and Pearcey beams have periodic focus and dispersion in the parabolic refractive index [22]. Meanwhile, the Talbot effect of certain autofocusing beams has been studied [5,23–25] and the Talbot effect has a wide range of applications in optics [26–28]. What will happen if we combine SPBs with the Talbot effect, making SPBs periodic in free space? Therefore, we study the self-imaging of the SPBs by taking the spectrum of SPBs as a periodic function, which generates a Talbot-like field [29], and we call it the symmetric Pearcey Talbot-like effect. We add the Gaussian term to SPTBs, which can be implemented in an actual experiment. Furthermore, we add a cubic phase to observe the change of the optical field more clearly. At last, we explore the influence of parameter setting on the optical field of the Talbot-like effect.
2. Theory model
We know that the Pearcey integral is defined as [30]: ${\rm {Pe}} \left ( {X} , {Y}\right ) = \int _{+\infty }^{-\infty } \exp [i \left (s^{4} + {Y}s^{2} + {X}s \right )]\, ds$, which can be calculated numerically using a contour rotation in the complex plane [31]. Under the paraxial system, we can obtain the spatial spectrum of SPBs [21]:
A superposition of nondiffracting beams will exhibit the longitudinal periodicity of the Talbot-like effect, the effect of which is equivalent to the propagation of nondiffracting beams through the grating [4,29]. There are two methods to demonstrate the Talbot-like effect for the one dimensional paraxial wave equation in the free space. The first method is the moving frame approach, which derives from the free space [23]. The second method is to look at a field whose spatial spectrum is a periodic function. The two methods are equivalent, and both prove that the superposition of several single beams with constant beam shift can form the Talbot-like effect of beams [5]. We choose the second method, modifying the spatial spectrum of SPBs. A two dimensional periodic spatial spectrum $f\left ( k_x , k_y \right )$ with a period of $\frac {2\pi }{\triangle }$ can be defined as [5]
Similarly, we superimpose on the SPGTBs a cubic phase factor ${\rm exp}[i{\beta }\frac {(x-\triangle n)^{3}+(y-\triangle n)^{3}}{{w_0}^{3}}]$, giving SPGTBs a parabolic trajectory. The initial field of the cubic symmetric Pearcey Gaussian Talbot-like beams (CSPGTBs) can be expressed as:
3. Pearcey Talbot-like effect
This section describes the propagation properties of the SPGTBs and CSPGTBs in the free space by simulation and experiment. In previous research [32], for Gaussian beams illumination, the Talbot distance is expressed by $Z_G = 2m\frac {d^{2}}{\lambda }[1+(\frac {\lambda {z}}{\pi {w_0}})^{2}]$, where $w_0$ is the Gaussian waist, $d$ is the grating spacing, $z$ is the propagation distance and $m$ is a positive integer. Hence the period is $Z_G = 2m\frac {d^{2}}{\lambda } * 1.16$, and we approximate it to the Talbot distance of the plane wave for convenience ( $Z_G \approx Z_T = 2m\frac {d^{2}}{\lambda }$). We know that $\frac {1}{\triangle }$ is the fundamental spatial spectrum, so $\triangle$ denotes the grating constant $d$ in the free space [4]. Therefore, the Talbot distance can be expressed as $Z_T = 2m\frac {{\triangle }^{2}}{\lambda }$. When $m = 1$, we define that $Z_F = \frac {2{\triangle }^{2}}{\lambda } = 0.074{Z_R}$ ($Z_R$ is the Rayleigh distance $\frac {\pi {w_0}^{2}}{\lambda }$) represents one Talbot period, whose position is the primary Talbot image plane and $Z_H = Z_F / 2$ represents half of the Talbot period, whose position is the secondary Talbot image plane.
3.1 Symmetric Pearcey Gaussian Talbot-like beams in the free space
Figure 1 shows the numerical simulation and experimental implementation of the SPGTBs. We have set the wave packet of SPGTBs to comprise a superposition of 11 SPGBs, shifted along the z-axis with $\triangle$, and each multiplied by the same coefficient 1. Due to the interference between the wavefronts of 11 SPGBs, the intensity distributions of SPGTBs are redistributed in the initial and later planes, manifesting as a matrix distribution of several bright spots, shown in Fig. 1. As shown in Figs. 1(a1)–1(a4), the intensity distributions of each Talbot period are similar, uniform matrix distribution of small bright spots. There are 10, 10, 8, 4 bright spots in each row, and other spots are diverged, corresponding to the Figs. 1(a1)–1(a4), respectively. Because of the propagation property of the SPGBs, the number of the small bright spots decreases as the propagation distance increases, which will explain in Fig. 3. Similarly, the attenuation of the small bright spots on both sides can be seen in the intensity curves [white lines at the bottom of the Figs. 1(a1)–1(a4)]. Although the intensities of the small bright spots around the overall spot at each Talbot period planes decrease as the propagation distance increases, the relative distributions of the small bright spots still have a certain similarity. Therefore, we claim that the SPGTBs have the Talbot-like Effect in the free space. Meanwhile, we utilize the holographic principle generating the SPGTBs experimentally, with the experimental configuration shown in Fig. 2. Collimated and expanded by a beam expander, the linearly polarized Gaussian beams (GBs) emitted with a He-Ne laser, is reflected by a reflector and propagates to the reflective spatial light modulator (rSLM). A computer-generated hologram (CGH) is loaded onto the rSLM, from which the GBs will obtain the amplitude and phase information of the Talbot arrays. After loaded the information of the Talbot arrays, the beams are filtered by a $4f$ filter system (consist of L1, L2 and AP) and then incident into CCD. Moving the CCD, we can observe the transverse patterns of the Talbot arrays in different propagation distances. Figures 1(b1)–1(b3) are the experiment results corresponding to Figs. 1(a1)–1(a3) and the experimental patterns agree with the theoretical ones.
Fig. 1. Numerical simulation and experimental implementation of the SPGTBs with $(b_1, b_2, p_1, p_2) = (0.072, 0.072, 0, 0)$; (a1)-(a4) numerical snapshots of normalized intensity distributions at Talbot period planes each, and the white lines at the bottom of the snapshots are the normalized intensity curves; (b1)-(b3) experimental snapshots corresponding to (a1)–(a3).
Fig. 2. Experimental configuration for generating the Talbot array of SPGBs and CSPGBs. BE, beam expander ($\times 8$); rSLM, reflective spatial light modulator (Santec SLM-200, 1900$\times$1200 pixels); BS, beam splitting cube; R, reflector; L1, L2, lens; AP, aperture; CCD, charge-coupled device camera (BeamPro 11.11). L1, L2 and AP form a $4f$ filter system that filters just the positive first-order.
Fig. 3. Propagation of the SPGTBs; side view when (a) $x = y$ and (b) $y = 0$; (c) side view of the SPGBs; (d1)-(d6) numerical snapshots of normalized intensity distributions of the SPGTBs at every $Z_H, Z_F$ marked in dash lines in (a) and (b), and the white crosses in the center mean the axis origin $(0,0)$; (e1)-(e6) numerical snapshots of normalized intensity distributions of the SGTBs at planes 1-6 marked in dash lines in (c) corresponding to (d1)-(d6); (f1)-(f6) maximum normalized intensity curves along x-axis corresponding to (d1)-(d6), (e1)-(e6). The parameters are the same as those in Fig. 1.
Figure 3 shows the propagation of the SPGTBs. From [21], we know that the SPGBs have a long focal distance, and its pattern is a circular main lobe with four side lobes that will split into four off-axis main lobes as the propagation distance increases (about $0.051Z_R$), and finally the lobes will diverge. For the whole distribution of SPGTBs, the transverse intensities distributions of SPGTBs are similar to that of the SPGBs during propagation. From Figs. 3(d1)–3(d4), the small bright spots arranged by the matrix have the strongest intensity in the center of the matrix, then the intensity weakens slightly from the center to the outward, and divergent spots are around the matrix, like Figs. 3(e1)–3(e4). Excluding the bright spots caused by the constructive interference between beams, we can observe that the intensity distributions of Figs. 3(d5), 3(d6) and Figs. 3(e5), 3(e6) are similar, according to the distribution of bright spots and divergence spots. As we can see from Figs. 3(f1)-(f6) the red curves have two small protrusions on either side of the zero point, representing the side lobes of Figs. 3(e1)–3(e6), corresponding to the black curves having small protrusions on either side of the edge. Figures 3(f1)–3(f6) also show some similarities between the envelope intensity of SPGTBs and the intensity profile of SPGBs (red curves). Every single beam of SPGTBs itself is a SPGB, so the whole distribution of SPGTBs is similar to SPGBs, the small bright spots around the SPGTBs will attenuate. That is why the number of the small bright spots will decrease [Figs. 1(a1)–1(a4)]. In addition, we know that the SPGTBs have multiple autofocusing property [Figs. 3(a) and 3(b)]. Although every single beam of SPGTBs splits into four off-axis main lobes at every $Z_F$ plane, SPGTBs still present autofocusing because of the constructive interference of the 11 SPGBs. While SPGTBs’ autofocusing at every $Z_H$ plane is caused by each beam of SPGTBs’ autofocusing and periodic reconstruction of the wavefront by the Talbot Effect.
We know that the Pearcey beams have the self-healing property, so do the Talbot arrays of the SPGBs have the same property? Can one observe the effect of autofocusing self-Imaging of SPGTBs when the beam is partially obscured. In the following, we introduce a Gaussian soft well function as a barrier, shown in Fig. 4(a) and 4(b).
Fig. 4. Propagation of the SPGTBs with a Gaussian soft well($a = 0.15 \rm {mm}$); side view when (a) $x = y$ and (b) $y = 0$; (c1)-(c8) numerical snapshots of normalized intensity distributions of the SPGTBs at planes 1-8 marked in red in (a) and (b); (d1)-(d8) numerical snapshots of normalized intensity distributions of the SPGTBs at the same position in Fig. 3(a) and 3(b). The green arrows denote poynting vectors of the SPGTB and $\rm {SPGTB}_{well}$. The small figures in the corner of (c1)-(c8) and (d1)-(d8) are the partial detail enlargement, and the white crosses in the center mean the axis origin $(0,0)$. The parameters are the same as those in Fig. 1.
3.2 Symmetric Pearcey Gaussian Talbot-like effect with a cubic phase in the free space
The cubic phase makes the beam have Airy distribution and parabolic trajectory, which helps us observe the transverse intensity distribution of the beam better. We obtain the symmetric cubic Pearcey Gaussian beams (CSPGBs) by imposing the cubic phase in the SPGBs. In the same way, we obtain the CSPGTBs by superimposing CSPGBs. Figure 5 shows the numerical simulation and experimental implementation of the CSPGTBs. Like the SPGTBs, the transverse intensity distributions of CSPGBs is redistributed in the initial and later planes. Analogously, the intensity distributions of the CSPGTBs of each Talbot period are similar [Figs. 5(a1)–5(a4)]. During the propagation, the intensity will be concentrated more on the upper right and radiate to the right and up, whereas the bright spots on the lower and left sides will diverge. As we predetermined, the spots of CSPGTBs emerge Airy distribution because of the cubic phase of CSPGBs. Similar to the SPGTBs, we claim that the CSPGTBs also have the Talbot-like Effect in the free space because the relative distributions of the small bright spots on each Talbot plane still have a certain similarity. The experiment results are shown in Figs. 5(b1)–5(b4), which are in good agreement with Figs. 5(a1)–5(a4).
Fig. 5. Numerical simulation and experimental implementation of the CSPGTBs with $\beta = 7$, $(b,p) = (0.072,0)$; (a1)-(a4) numerical snapshots of normalized intensity distributions at Talbot period planes each, and the white lines at the bottom of the snapshots are the normalized intensity curves of Talbot period planes each; (b1)-(b4) experimental snapshots corresponding to (a1)–(a4).
Figure 6 shows the propagation of the CSPGTBs. We know that the CSPGTBs also have multiple autofocusing property [Figs. 6(a) and 6(b)], whose causes are the same as those of SPGTBs. However, the focus plane is not in the Talbot period planes or the secondary Talbot period planes. So the analysis of the period is different from section 3.1. If we want to change the position of focus, we can adjust the distribution factor $b$ or $p$ . Due to the cubic phase, the propagation trajectory of the CSPGTBs is parabolic. From Figs. 6(d1)–6(d6), the small bright spots in the upper right corner have stronger intensity, which is consistent with the case of the CSPGBs [Figs. 6(e1)–6(e6)]. The positions of the dash lines in the Figs. 6(e1), 6(e2), 6(e6) have no intensities, and correspondingly, the intensities at the positions of the dash lines in the Figs. 6(d1), 6(d2), 6(d6) are relatively low. As the propagation distance increases, both the CSPGTBs and the CSPGBs present Airy distribution [Figs. 6(d4)–6(d6) and Figs. 6(e4)–6(e6)]. Meanwhile, Figs. 3(f1)–3(f6) also show some similarities between the envelope intensity of CSPGTBs and the intensity profile of CSPGBs (red curves).Therefore, the transverse intensity distribution of the CSPGTBs presents that of CSPGBs during propagation. In conclusion, the transverse intensity distributions of the superimposed beams will present that of the single beam, and the superimposed beams have the property of multiply autofocusing attributed to the focusing of the Talbot-like effect and the autofocusing of Pearcey beams. The CSGTBs are derived from SPGTBs, which also have the Talbot-like effect, so we can modify the SPGTBs to obtain the desired beam with the Talbot-like effect according to the requirements.
Fig. 6. Propagation of the CSPGTBs; side view of the CSPGTBs when (a) $x = y$ and (b) $y = 0$; (c) side view of the CSPGBs when $y = 0$; (d1)-(d6) numerical snapshots of normalized intensity distributions every $Z_H ,Z_F$ marked in dash lines in (a) and (b), and the white crosses in the center mean the axis origin $(0,0)$; (e1)-(e12) normalized intensity distributions of the CSPGBs at planes 1-6 marked in dash lines in (c) corresponding to (d1)-(d6); (f1)-(f6) maximum normalized intensity curves along x-axis corresponding to (d1)-(d6), (e1)-(e6). The parameters are the same as those in Fig. 5.
4. Influence of parameters on the symmetric Pearcey Gaussian Talbot-like effect
Inquisitive about the behavior of the symmetric Pearcey Talbot-like effect with different parameters, we observe the SPGTBs under various parameters setting in Fig. 7. Only when we select suitable beam shift factor $\triangle$, the number of superimposed beams $\rm {N}$ and distribution factors $b_1, b_2, p_1, p_2$, can the beams show the Talbot-like Effect. Therefore, we select the parameters setting shown in Figs. 7(a1)–7(d3). The $\triangle$ affects not only the focusing positions and times but also the intensity distributions [Figs. 7(a1)–7(a3)]. Moreover, the overall spot size gets large [Fig. 7 (a5)] and the distance between each beam increases with the increase of $\triangle$. According to Fig. 7(a4), SPGTBs exhibit better periodicity and higher intensity only $\triangle$ is appropriate. When $\triangle$ is as small as 0.4, the periodicity of SPTGBs’ intensity is not very apparent. In addition, the number of superimposed beams $\rm {N}$ also affects the Talbot effect of SPGBs. The more beams are superimposed, the farther SPGTBs can propagate and the more times SPGTBs will focus [Figs. 7(b1)–7(b3)]. From Fig. 7(b4), although the number of superimposed beams $\rm {N}$ increases, the intensity of the beams along the z-axis remains almost constant in the first Talbot period plane. Meanwhile, the relative intensity of the beams along the x-axis remains almost constant, except for the expansion of the beam to the sides [Fig. 7(b5)].
Fig. 7. Propagation of SPGTBs under various parameters setting: (a1)-(a3), (b1)-(b3), (c1)-(c3) and (d1)-(d3) side views of SPGTBs when $x = y$ with $\triangle = 0.4, 0.5, 0.6$, $N = 5, 7, 9$, $b_1 = b_2 = 0.06, 0.07, 0.08$, $p_1 = p_2 = 0, 1.5, 3$, respectively, and the white dash lines represent the first Talbot period planes; (a4)-(d4)maximum intensity curves along z-axis while $x=y$ in (a1)-(a1), (b1)-(b3), (c1)-(c3), (d1)-(d3); (a5)-(d5) normalized intensity curves in the red dash lines (the second Talbot half period ) in (a1)-(a1), (b1)-(b3), (c1)-(c3), (d1)-(d3). Other parameters are the same as those in Fig. 1.
In the following analysis, $b_1, b_2$ and $p_1, p_2$ are collectively referred to as $b$ and $p$, respectively. The increase of $b$ not only affects the focusing times but also influences the focusing positions. For every single beam, the focal length will be longer with the increase of $b$, and the maximum intensity will move backwards [21]. According to the previous conclusion, the superimposed beams will present the single beam’s transverse intensity distributions. Therefore, the SPGTBs will have more focuses and the maximum intensity will shift back, when $b$ increase [The red asterisks in Fig. 7(c4) represent the maximum intensity]. Due to the backward offset of the maximum intensity, the focusing positions move backwards. [Figs. 7(c1)–7(c3)]. The value of $b$ also needs to be controlled, when $b$ are close to 0.08, although the number of focusing times (self-imaging planes) becomes more, but compared with $b_1,=b_2=0.06$, $b_1,=b_2=0.07$ the overall intensity of the spot becomes lower and the periodicity becomes poor. Next, we analyze the influence of the parameter $p$ on the Talbot-like Effect. The $p$ also affects the focusing positions and times like $b$. For every single beam, $p$ can also do what the $b$ does, having a larger adjustable range than $b$, so $p$ can roughly adjust the focusing positions and times, while $b$ can fine tune the focusing positions and times [Figs. 7(c1)–7(c3), 7(d1)–7(d3)]. In like manner, the larger $p$ is, the more focusing times are, the farther the focusing positions are. However, the larger $p$ will decrease the intensity, so we should choose $b$ and $p$ reasonably. In addition, the parameters $b$ and $p$ have a fascinating influence on SPGTBs. Under appropriate range, the larger $b$ and $p$ are, the less likely SPGTBs are to diverge and the better the focusing effect is. Comparing three figures in Figs. 7 (c1)–7(c3), 7(c5) and Figs. 7 (d1)–7(d3), 7(d5), respectively, the larger the $b$ and $p$, the more energy is concentrated in the center and farther, but the overall intensity of the beam reduces [Figs. 7(c4) and 7(d4)]. In the traditional application of the Talbot effect, a Gaussian beam passing through a finite periodic object, such as a two-dimensional grating, will generate the Gaussian Talbot array [32]. But the array will diverge with the increase of the propagation distance. However, the Talbot-like effect generated by symmetric Pearcey beams in this paper has a focusing effect attributing to $b$ and $p$. In the case of the Talbot effect in lithography, etc, if the position of the image plane (Talbot plane) needs to be changed the only method is changing the period of the grating [27,32]. Because the periodic structure of the objects decides the position of the Talbot plane. However, we can change the $b$ and $p$ to change the position of the Talbot plane without changing the periodic structure of the objects.
Next, we will concentrate on the distribution factors $b_1$ and $b_2$. The dashed rectangles delineate the scope of the main lobes of SPGTBs. We know that when $b_1 = b_2$ the focusing positions shift back with $b_1, b_2$ increasing. Similarly, when $b_1 \neq b_2$, the focusing positions shift back too [Figs. 8(a), (b), (d), (e)]. Figures 8(f1)-(f3) depict the intensity along the z-axis, which is associated with the larger of the two factors. When $b_1 = b_2$, there are side lobes before and after the main lobes, and the intensities of the side lobes are the same, symmetrical about the half Talbot period plane [Fig. 8(c)]. It is interesting to note the situation when $b_1 \neq b_2$. When $b_2 > b_1$ [Figs. 8(a), (e)], the intensity will be more concentrated in the front side lobes, while the rear side lobes are virtually no intensity (defined propagation direction positive direction for the front). The situation with $b_1 > b_2$ is opposite [Figs. 8(b), (d)]. Therefore, by adjusting the ratio of $b_1$ and $b_2$, it is possible to adjust the intensity ratio between the main lobes and the side lobes. The red asterisks in Figs. 8(f1) and (f2) denote the position of the maximum intensity of the main lobes. When $b_1 = b_2$, seeing from Figs. 8(c1)-(c5), the spot is square, and the side lobes appear and change in both $x$ and $y$ directions simultaneously in one Talbot period [the enlargements are on the corner of Figs. 8(c1),(c2)]. When $b_1 \neq b_2$, seeing from Figs. 8(a1)-(a5) and Figs. 8(d1)-(d5) (in which both $b_2 = 0.07$), the side lobes exist first to the left and right of the main lobes in the x-direction and then to the top and bottom of the main lobes in the y-direction [the enlargements are on the corner of Figs. 8(a1),(a2) and Figs. 8(d1),(d2)]. This variation alternates in one Talbot period, and finally in the fourth Talbot period plane, the bright spots show a rectangular distribution. The case with $b_1 = 0.07$ is the opposite of the case with $b_2 = 0.07$, seeing from Figs. 8(b1)-(b5) and Figs. 8(e1)-(e5). This indicates that the side lobes will diverge in the direction whose distribution factor $b$ larger. It is worth mentioning that $p_1$ and $p_2$ can also be divided into two unequal values in the $x$ and $y$ directions. We know that $p_1$ and $p_2$ are the shift factors for the Pearcey integral Pe($\cdot$), and $b_1$ and $b_2$ are the scaling factors [21]. To sum up, SPGTBs have two kinds of distribution factors to control the shape of the intensity distributions, focusing position, focusing times and focal length, which shows SPGTBs also have a high-operable autofocusing ability like SPGBs.
Fig. 8. Numerical intensity distributions of SPGTBs with different distribution factors $(b_1,b_2,p_1,p_2)$: (a)-(e) side view of the SPGTBs; (a1)-(e5) numerical snapshots of normalized intensity distributions marked in red in (a)-(e), respectively; (f1)-(f3) maximum intensity curves along z-axis. The parameter settings for each row from top to bottom in the schematic are $(0.06,0.07,0,0)$, $(0.07,0.06,0,0)$, $(0.07,0.07,0,0)$, $(0.08,0.07,0,0)$, $(0.07,0.08,0,0)$, respectively, and the white dash lines represent the first Talbot period planes. Other parameters are the same as those in Fig. 1.
5. Conclusion
In conclusion, we have demonstrated the symmetric Pearcey Talbot-like Effect numerically and experimentally by the superposition of the symmetric Pearcey beams. In the symmetric Pearcey Talbot-like effect, the wavefront of the superimposed beams is redistributed periodically because of interference. The superimposed beams will present that of the single beam. Therefore, the symmetric Pearcey Talbot-like effect can make the symmetric Pearcey beams have the property of multiple autofocusing and self-healing. Additionally, by adjusting the beam shift factor $\triangle$, the distribution factors $b$ and $p$, we can also control the intensity distribution, the focusing positions and times of the symmetric Pearcey beams. Our results further connect Talbot effect and symmetric Pearcey beams. This symmetric Pearcey Talbot-like effect not only provides a deeper insight into the self-imaging beams, but also provide some potential applications in Talbot effect’s optical applications such as optical imaging, optical traps, lithography, etc.
Funding
Guangdong provincial Natural Science Foundation of China (2022A1515011482); Science and Technology Program of Guangzhou (2019050001); National Natural Science Foundation of China (12174122, 11775083, 11374108).
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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