Abruptly autofocusing chirped ring Pearcey Gaussian vortex beams with caustics state in the nonlinear medium

Liping Zhang; Dongmei Deng; Xiangbo Yang; Guanghui Wang; Hongzhan Liu

doi:10.1364/OE.28.000425

1. Introduction

In 1999, Kaminski and Paris numerically evaluated and plotted zeroes for real values of the arguments of the Pearcey integral [1]. The complex amplitude distribution of the beam is described by the Pearcey function [2–4] and defined as an integral of the complex exponential function with a polynomial argument (similar to the definition of the Airy function). A closed-form expression has been derived for the Pearcey wave that simplifies to a paraxial Pearcey beam in the appropriate limit [4]. Pearcey beams generated by using a virtual source have been theoretically proposed [3] and demonstrated experimentally [3,5] by utilizing a spatial light modulator. Some outstanding properties of the Pearcey beam are not only similar to Airy beams but also similar to Gaussian beams and Bessel beams. These properties include inherent abruptly autofocusing (AAF), form-invariance on propagation and self-healing [6,7]. Kovalev et al. have considered the superposition of two paraxial two-dimensional half Pearcey laser beams in [8]. Recently, circular Pearcey beams (CPBs) [9–14] which are the circularly symmetric beams with Pearcey radial profile have received wide attention. Compared with the circular Airy beams, the CPBs can increase the normalized peak intensity, shorten the focus distance, especially, and eliminate the oscillation after the focal point [9]. In the meantime, the CPBs which maintain a low rotating speed before the focal point and rotate abruptly and quickly in the focal point [12] are new discovery for the family of rotating beams.

In nonlinear systems, solitary waves which are explored in many fields [15] reflect the balance between the linear dispersion and the nonlinear self-focusing effect [16]. New kinds of breathing soliton solutions of a (1+1)-dimensional nonlinear Schr$\ddot {o}$dinger (NLS) equation have been demonstrated in [17–19]. Such solitons stably propagate in the nonlinear medium and are stable against random perturbations [19]. Breathers which exist in systems with periodically modulated parameters (dispersion, gain/loss and nonlinearity coefficients) can be used for amplification and compression of solitons [20–22]. On plane wave background, a transformation between different types of solitons is discovered in [22]. The formation mechanisms of breather-like nonlinear waves are related to the energy conversion between two components resulting from four wave mixing [21]. On the basis of works of Refs. [23,24], a detailed account of mutual-focusing of two co-propagating cylindrical beams including the formation of spatial soliton pair and breather pair have been provided. Building on the concept of a collapsing caustic surface of revolution, the AAF wave family was broadened to include general power-law caustics that evolve from a sublinearly chirped input amplitude [25]. The optical bottle beam which is introduced describes a beam with a finite axial region of low (and even zero) intensity surrounded in all dimensions [26–28]. Some optical bottle beams have applications in optical tweezers for trapping low index microparticles [28–30]. In this paper, we explore the nonlinear dynamics of the chirped ring Pearcey Gaussian vortex (CRPGV) beams in the Kerr medium. We hereby study the features of the CRPGV beams whose rings are known to form paraboloids (optical bottle) in the Kerr medium. The new flexibility has enhanced focusing abruptness and larger intensity contrasts.

2. The theoretical model

We first consider the expression of the phase-modulated circularly symmetric wavefront of the initial CRPGV beams $\Psi _{0}$ in the cylindrical coordinates as follow

(1)$$\Psi_{0}(r, \theta, 0)=A_0Pe(r/p,\chi r/p+c)\exp(-\alpha^{2} r^2+i\beta_1 r+i\beta_2 r^2)(re^{i\theta}+r_se^{i\varphi})^m(re^{i\theta}-r_se^{i\varphi})^n,$$

where $Pe(\cdot )$ is the Pearcey integral [2,4–6] which is defined as an integral of the complex exponential function with a polynomial argument (similar to the definition of the Airy function) as follow

Pe(r/p,\chi r/p+c)=\int^{\infty}_{-\infty}\exp[i(s^4+(\chi r/p+c) s^2+rs/p)]ds.

The form of the cusp which underpins the intensity pattern satisfies [6] $Pe(r/p,f(r)),f(r)=-\frac {3}{2}(\frac {r}{p})^{\frac {2}{3}}$; $A_0$ is a normalized constant amplitude of the initial spatial field, $\alpha$ is a width factor making the beam feature tend to that of a chirped ring Pearcey vortex beam when it is a small value; or a chirped Gaussian vortex beam when it is a large value, $r$ is the polar distance normalized by $w_0$ ($w_0$ represents the initial width of the Gaussian beam), $p$ is a spatial distribution factor, $c$ is a constant [13] that varies the proportion of conventional Pearcey waves in the X- or Y-coordinate; $\chi$ is a scale factor of the initial intensity distributions [31]; the initial velocity ($\beta _1$) acts as an axicon [32,33], which usually affects the focal depth of the beams; the second-order chirp factor ($\beta _2$) can be used to adjust the positions of the focal point during the propagation progress; $r_s$ and $\varphi$ denote the radial location and the orientation of the vortices; $(m+n)$ is the topological charge.

The propagation of a CRPGV beam in the Kerr medium is described by a NLS equation [34–36] in the paraxial approximation

(2)$$2i\frac{\partial \Psi}{\partial Z}+\frac{\partial^2 \Psi}{\partial X^2}+\frac{\partial^2 \Psi}{\partial Y^2}+\frac{2n_{2}}{n_{0}\sigma^{2}}\mid{\Psi}\mid^2 \Psi=0,$$

where $\Psi$ is the amplitude of the CRPGV beams, $X=x/w_0, Y=y/w_0$ are the normalized transverse coordinates, $Z=z/2Z_R$ (the Rayleigh length $Z_R=\frac {kw_0^2}{2}$) means the normalized longitudinal propagation distance, $k=\frac {2\pi }{\lambda _0}$ ($\lambda _0$ is the center wavelength in the free space) being the wavenumber, $\sigma$ ($\sigma =\frac {1}{kw_0}$) is wave divergence angle of the Gaussian beam in the free space, $n_{0}$ represents the linear refractive index, $n_{2}$ is the nonlinear coefficient of the Kerr medium. In terms of the radially symmetrical beams (the CRPGV beams) solution, it is more convenient to describe it in cylindrical coordinates. Equation (3) can be rewritten as follow

(3)$$2i\frac{ \partial \Psi}{\partial Z}+\frac{\partial^2 \Psi}{\partial r^2}+r^{{-}1}\frac{\partial \Psi}{\partial r}+ r^{{-}2}\frac{\partial^2 \Psi}{\partial\theta^2}+\frac{2n_{2}}{n_{0}\sigma^{2}}\mid{\Psi}\mid^2 \Psi=0.$$

It is difficult to obtain the analytic expression for $\Psi (r, \theta , Z)$. Fortunately, we can use the split-step Fourier transform method to simulate [5,36] the propagation of the beams numerically. We also consider the situation that we can break the azimuthal symmetry by adding Gaussian perturbations (multiplication of the field by 1+$R\Upsilon (X,Y)$, where $R$ is the fluctuations percentage and $\Upsilon (X,Y)$ is the function of the Gaussian perturbations [37–39]). The coefficient for the Kerr nonlinearity $n_2 = 2.6\times 10^{-16}cm^{2}W^{-1}$ [35] leads to a critical power of collapse of a Gaussian beam for self-focusing $P_{cr}=\frac {3.77\lambda ^{2}}{8\pi n_0n_2}$ [40–44]. The connection between the initial input power of the CRPGV beams ($P_{in}$) and the critical power of collapse of a Gaussian wave ($P_{cr}$) is as follow [40]

(4)$$P_{in}=\frac{n_{2}k^2}{2\pi n_{0}}\int^{2\pi}_{0}\int^{\infty}_{0} P_{cr}|u(r, \theta,0)|^{2}rdrd\theta.$$

In the simulation, we set $\alpha$=1, $p= 0.1$, $\lambda = 0.532\times 10^{-6}$m, $w_0=1\times 10^{-3}$m, and $n_0=1.45$ throughout this paper.

3. Propagation dynamics

We first show some typical initial beams in Fig. 1. We know that the CRPGV beams can be regarded as the Pearcey beams revolve around the axis with the locations in X-Y plane as depicted by the dotted lines (different $\chi$ and $c$) in Fig. 1(a) [13,31]. For better visualization, the intensities in Figs. 1(a1)–1(a4), 1(b)–1(c), 1(b2)–1(c2) and 1(b3)–1(c3) have all been scaled to the same spatial scale [see Fig. 1(a2)]. From Figs. 1(a1)–1(a4), we can see that the intensity distributions of the CRPGV beams with different $\chi$ (can be regarded as the slope of the lines) correspond to different scales of the initial intensity. In the meantime, the parameter $c$ [see Figs. 1(a4) and 1(c)] varies the proportion of conventional Pearcey waves in the X- or Y-coordinate with the same $\chi$. The form of the cusp which underpins the intensity pattern is shown as a dashed line marked (b) in Fig. 1(a) [6]. Next, we simulate two representative cases (Figs. 1(b) and 1(c)) of the initial CRPGV beams propagation in the Kerr medium within $Z=0.2$. When various initial CRPGV beams are appropriately weighted by different chirp conditions, the relative phases of the beams can be adjusted for achieving the perfectly mirror-symmetric optical bottle which can be used as an optical trap. By rotating the cusp dotted line (curve) of the Pearcey beams, we can obtain the initial CRPGV beams with the maximum intensity in the center and the intensity decreasing outward gradually [see Fig. 1(b)]. The transverse intensity distribution of Fig. 1(b) shows that the initial input beam has a circularly symmetric amplitude oscillating outward from a bright nucleus. The corresponding propagation progress is illustrated in Fig. 1(b1) as $\beta _1=5, \beta _2=0$, one optical bottle appears after the beam traveling a distance in a relatively focused state, then the beams focus into breathers-like states ultimately during the transmission. As the beam’s cross-sectional area gradually decreases, the maximum intensity over the transverse plane increases, focuses at the center. In the view of the relation between the chirp modulation and the nonlinearity on the beams, the optical bottle phenomenon of the beams during the propagation progress can be explained. The “defocusing-to-focusing" effect of the beams creates an optical bottle with paraboloid intensity boundaries and radial dependence. This diffraction is inhibited by the effect of the chirped modulation and nonlinearity so that the beam finally gathers in foci. At this moment, the effect of the chirped factor on the beam is just like a focusing lens. We focus on the transverse intensity distributions of the beams in this chirped condition. As the beams propagate a distance ($Z=0.02$), the transverse intensity distribution of the CRPGV beams is shown as a hollow focused ring [see Fig. 1(b2)], which means the intensity of the center point of the beams is not the maximum. When $Z=0.08$, the hollow transverse distribution of the paraboloid optical bottle creates an optical potential well in dark regions [see Fig. 1(b3)]. To show the diversity of “defocusing-to-focusing" effect, we carry out the simulation on the evolution of Fig. 1(c) as $\beta _1=20, \beta _2=-20$, showing that the optical bottle of the CRPGV beams appears at the beginning of the transmission. In this case, the chirped modulation, the nonlinear effect and the AAF feature act on the beams synergistically, and the chirped modulation is the dominant component of the light field at the beginning of the transmission. This phenomenon can be understood that various initial beams with different energy distributions lead the magnitude of effect of various factors on the light field is different. From Fig. 1(c2), the transverse intensity distributions of the CRPGV beams are shown as the concentric rings with the maximum intensity mainly focuses on the center and other energy concentrates on one outer ring. In the meantime, the optical bottle transverse distribution is shown in Fig. 1(c3). Next, the CRPGV beams also reach a breathers-like balance as the paraboloid state of the CRPGV beams ends. However, the breathers-like balance states are also different with various initial energy distributions and various factors.

Fig. 1. (a): The initial intensity distribution of the Pearcey beam; (a1)-(a4),(b) and (c): the initial intensity distribution of the CRPGV beams; (b1) and (c1): the propagation process of the CRPGV beams in the nonlinear medium within $Z=0.2$; (b2)-(b3) and (c2)-(c3): the intensity distributions of the CRPGV beams at various locations [as the dotted lines in (b1) and (c1)]. The parameters are chosen as $\chi =1,c=0$ in (a1), $\chi =0,c=0$ in (a2), $\chi =-1,c=0$ in (a3), $\chi$ doesn’t exist, $c_2$=0 in (a4), $\chi$ doesn’t exist, $c_2$=−2 in (c), (b) corresponding to the cusp function integral, and other parameters, $P_{in}=20.1P_{cr}, m=n=0, R=0.$

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To further reveal the focusing properties of the nonlinear field, we then model the propagation progress of the beams corresponding to two cases (the initial beams same as Figs. 1(a2) and 1(a4)) of functions with the same chirp condition in Fig. 2. The propagation dynamics in Fig. 2(a) is from the situation 1 ($\chi =-1, c=0$), and the propagation dynamics in Fig. 2(b) is from the situation 2 ($\chi$ doesn’t exist, $c=-2$). For Fig. 2(a), as the CRPGV beams propagate toward the center, a paraboloid caustic surface is formed that "collapses" on axis thus leading to a large intensity buildup right before the focus. Next two continuous paraboloids of the CRPGV beams appear at the beginning of the propagation progress. The transverse intensity distribution of the first parabola [see Fig. 2(a1)] is shown as some hollow rings, while the transverse intensity distribution of the second parabola ($Z=0.03$) is one ring formed by some light points [see Fig. 2(a2)]. This phenomenon (the CRPGV beams gradually focus into a series of symmetrical ring beads) can be explained by the fact that the rotating state of the CRPGV beams is disturbed by focusing effect. The result is similar with the vortex multiple-filamentation observed in [37] achieving more perfect symmetry here. As the distance increases, the beam will focus into a breathers-like oscillations, and the focusing point is shown in Fig. 2(a3). From Fig. 2(a0), we can get the result that the intensity of the first focal point is the biggest. The intensity at the first focus comparing with the maximum input intensity is higher several orders of magnitude. We now turn to consider the case of the initial beam (case 2) whose energy mainly focuses on the center [see Fig. 2(b)] than that of dispersing on the outer ring. At this moment, the beam comes into a stable breather-like state after defocusing for some distances. When $Z=0.01$, the transverse intensity distribution of the CRPGV beams is shown in Fig. 2(b1), as the distance increases, the distribution becomes a hollow focused ring in Fig. 2(b2). When the beams come into breathers-like oscillations, the focusing point will not keep the hollow state and the intensity mainly concentrates on the center. The normalized intensity from Fig. 2(b0) is smaller than that of Fig. 2(a0). Comparing those two cases of the CRPGV beams, we find that the case 1, has stronger focusing ability even though its initial energy disperses on rings outside the center. This phenomenon can be explained that the case 1, with many rings contains more energy than case 2, so the initial focusing intensity is bigger. In other words, the focusing progress of the CRPGV beam is that the beams concentrate all the energy to a point with high energy density.

Fig. 2. (a), (b): The optical transmission diagrams of the CRPGV beams relating two cases functions within $Z=0.1$. (a0), (b0) the normalized intensity diagrams; (a1)-(a3) and (b1)-(b3): the intensity distributions of the CRPGV beams at normalized propagation distances; other parameters as follows, $m=n=0, \beta _1=5, \beta _2=0, R=0.$

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In Figs. 3(a) and 3(b), we investigate the impact of the initial input powers for the propagation dynamics of the CRPGV beams in the Kerr medium. To this end, we choose $P_{in}=9.1P_{cr}$ and $P_{in}=23.2P_{cr}$ in the simulation, respectively. We know that the CRPGV beams maintain a low rotating speed before the focus, and rotate abruptly and quickly in the focal point [8]. We first focus on the propagation diagrams of Figs. 3(a) and 3(b). After the first focus point, the outside rings of the CRPGV beams will form into the paraboloid, while the intensity of the center is not equal to zero because of the off-axis vortices pair. It is clear that breathers-like states of the beams are changing with various $P_{in}$ which represents the size of the nonlinear effect on the beams. When the nonlinearity is bigger, the intensity contrast of the CRPGV beams decreases and the stability of the breathers-like states becomes poor [see Figs. 3(a0) and 3(b0)]. From the propagation plots, we can see that the focus length and focal positions are nearly invariant when we take the various initial input powers. However, the transverse intensity distributions of Figs. 3(a1)–3(a3) in the course of the propagation have great differences with those of Figs. 3(b1)–3(b3). Comparing Figs. 3(a1) and 3(b1), it can be said that higher input power provides excellent performance improvement on the intensity contrast of the CRPGV beams in the Kerr medium. As the beams continue to transmit, the transverse intensity distributions of the beams are shown as a little whirlwind with two gears of Fig. 3(a2) and a small ring enclosing two smaller rings of Fig. 3(b2). When $Z=0.022$, the rotating state of the CRPGV beams is shown as Figs. 3(a3) and 3(b3). As shown in Figs. 3(a3) and 3(b3), a lot of side lobe rings for the CRPGV beams happen with $P_{in}=23.2P_{cr}$, and the side lobe rings of the beams for $P_{in}=9.1P_{cr}$ can be ignored. From this point, we can say that nonlinear effect of Kerr medium increases as the input power growing.

Fig. 3. (a), (b): Optical transmission diagrams with different initial input power of the CRPGV beam in the nonlinear medium within $Z=0.08$. (a0), (b0): the three-dimensional normalized intensity; (a1)-(a3) and (b1)-(b3): the transverse intensity distributions of the CRPGV beams at $Z=0.01, 0.015, 0.022$; other parameters as follows, $c=0, \chi$ doesn’t exist, $\beta _1=1, \beta _2=-35, m=n=1, r_s=0.5, R=0$.

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We simulate the propagation of the CRPGV beams producing breathers-like state in the Kerr medium with various chirp factors within $Z=0.07$ as Fig. 4. The intensities in Figs. 4(a2)–4(c2) have all been scaled to the same spatial scales [see Fig. 4(b2)]. Figs. 4(a1)–4(b1) show the propagation process of the CRPGV beams with different $\beta _1$. After the first focus, the CRPGV beams will not disperse completely, some rings of the beams keep a relative convergence state during the propagation progress. Comparing with Fig. 4(a1), the two focal positions of Fig. 4(b1) appear earlier. The transverse intensity distributions ($Z=0.002$) [see Figs. 4(a2)–4(c2)] show interesting phenomena: when $\beta _1>0$, a dark spot appears in the center of CRPGV beams; when $\beta _1<0$, the intensity of the CRPGV beams focuses toward the center. This phenomenon will continue until the beams focus. It can be explained that the action of $\beta _1$ as an axicon with different parameters affects the propagation state of the CRPGV beams in the Kerr medium. Figure 4(d1) shows the relation between the two intensity peaks at the ends of paraboloid and parameters $\beta _1$, $\beta _2$. The distance between the ends of paraboloid increases as $\beta _1$ increases and the absolute value of $\beta _2$ ($\beta _2<0$) negative decreases. When $\beta _2=-50$ [see Fig. 4(b1)], the focusing length between two foci of the CRPGV beams becomes shorter comparing with that of Fig. 4(a1). After two foci, the CRPGV beams will focus into multifarious breathers-like oscillations in the Kerr medium. By increasing the absolute value of the second-order chirp factor, the focusing points of the CRPGV beams are shifting to a more front position. This can be understood that various chirped conditions are equivalent to focusing lens with different focusing lengths and locations. Therefore, one can alter Pearcey beams’ optical structure, as well as their shape, focusing location, dark region by changing the parameter of the beam. In the meantime, the controllability of their optical structure is expected to make Pearcey beams more useful in some research fields.

Fig. 4. (a1)-(c1): The propagation progress of the CRPGV beam for $P_{in}=10.5P_{cr}$ in the Kerr medium with different chirp factors $\beta _1,\beta _2$; (a2)-(c2): the three-dimensional transverse distributions; (d) the relationship presentation between the normalized distance of two intensity peaks and $\beta _1$, $\beta _2$; the other parameters as follows, $c=0, \chi$ doesn’t exist, $m=n=1, r_s=0.5, R=0$.

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From practical considerations, one needs to take the stochastic type perturbations into account. We combine multiple laser beams with uncorrelated phases into a diffraction-limited beam using strong self-focusing in a nonlinear medium as [39]. There is a significant variation of the distance to the collapse and sensitivity to phase distribution of the laser beams at the entrance to the waveguide. Figures 5(a) and 5(b) show the intensity distributions of the CRPGV beams in the Kerr medium with various fluctuations. Under our initial perturbations conditions ($R=0.01$), the position of the filaments undergoing collapse does not significantly change with propagation. Then the impact of the stochastic perturbations is also almost negligible. However, when $R=0.5$, the transverse intensity of the beams is deeply influenced by the perturbations. As the beams approach the point of collapse, breaking the azimuthal symmetry by adding high-frequency amplitude and phase perturbations. The propagation state is no longer symmetrical in Fig. 5(b1), and an obvious deflection angle $\alpha _1$ appears during the propagation. The breakup of the beam suggests that phase modulation instability occurs, similar to that observed in the case of other ring profiles [37,38].

Fig. 5. (a), (b): Initial intensity distributions of the CRPGV beam with different $R$; (a), (b): optical transmission progress; (a2), (b2): the normalized intensity diagrams; other parameters as follows, $P_{in}=10.5P_{cr}$, $c=0, \chi$ doesn’t exist, $\beta _1=1, \beta _2=-35, m=n=1, r_s=0.5$.

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4. Summary

We study in detail the CRPGV beams propagating in the Kerr medium and identify many interesting behaviors. Firstly, we theoretically show the optical bottles and breathers-like oscillations of the CRPGV beams in the Kerr medium. When the various initial CRPGV beams are appropriately weighted by the different chirped modulations, the relative phases of the beams can be adjusted for achieving the different caustics. From the propagation progress of the CRPGV beam, we can conclude that the focusing progress of the CRPGV beam is that the beams concentrate all the energy to a point with high energy density. We obtain the chirped Pearcey Gaussian beams with an off-axis vortex pair following a paraboloid in the nonlinear medium. The sign of $\beta _1$ decides the light intensity of the beams whether concentrating in the center. Two peak intensity positions of the paraboloid can be controlled by $\beta _1$ and $\beta _2$. For practical considerations, we take the stochastic type perturbations into account. The result shows that the perturbations break the phase symmetry influencing the beam symmetrical structure. One can alter CRPGV beams’ optical structure, as well as their shape, focusing location, dark region by changing the parameter of the beam in the nonlinear medium.

Funding

National Natural Science Foundation of China (11374108, 11674107, 11775083, 61875057).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Abruptly autofocusing chirped ring Pearcey Gaussian vortex beams with caustics state in the nonlinear medium

Abstract

1. Introduction

2. The theoretical model

3. Propagation dynamics

4. Summary

Funding

Disclosures

References

Cited By

Figures (5)

Equations (5)

Optics Express