We introduce the propagation properties of the circular Pearcey Gaussian (CPG) waves in Kerr medium for the first time. The breathers-like structure and breathers-like groups structure (filamentions) of CPG waves will form due to the interaction between the linear waves and the nonlinear medium. The focusing characteristics in Kerr medium can be adjusted by the deviation factor and the initial input power of the CPG waves. By choosing appropriate input power, the imaginary part of the CPG waves can split into some wavelets during the propagation. It is worth noting that the distinctive stepwise focusing of the imaginary part of the CPG waves which can be applied to wave modulation. Furthermore, the numerical experiment results show good agreement with the numerical simulation results.
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Since the abruptly autofocusing (AAF) waves were proposed in 2010 , people have paid great attention to them [2–4]. These AAF waves suddenly focus all their energy during propagation, while maintaining a relatively low intensity profile after the focus [1–5]. Some experimental observations [3,4] demonstrate that AAF waves could outperform standard Gaussian waves, especially in settings where high intensity contrasts and be delivered under conditions involving long focal-distance-to-aperture ratios. The AAF waves also can produce a greater gradient force on the particle in optical trapping , since the abruptly autofocusing property has a great intensity gradient in the focal region. As indicated in [1,6], the AAF feature is very useful in biomedical laser treatment, since the wave should only affect the intended area while leaving any preceding tissue intact . At the same time, AAF waves are observed and utilized for creating ablation spots , in particle manipulation  and for the controlled filament generation at particular spatial locations . Ring et al.  present a new solution of the paraxial equation based on the Pearcey function, which is related to the Airy function and describes diffraction about a cusp caustic. The Pearcey wave displays many properties including an inherent AAF effect, form-invariance on propagation and slef-healing [9, 10]. Then a new class of circle Pearcey beams (CPBs) with the AAF characteristics was introduced in [11–13]. The CPBs can increase the peak intensity contrast, shorten the focus distance, and especially eliminate the oscillation after the focal point in free space . An auto-focusing and auto-healing beam in linear media is investigated experimentally based on spatial light modulators and computer-generated holograms [14–16].
As far as we know, the optical nonlinearity  processes are caused by the anharmonicity of the motion equation of the bound charge. People devote much attention to the possibility of counteracting the dispersive spreading by focusing effects, due to medium nonlinearities e.g. the Kerr effect [18–21]. It is possible to control diffraction and self-focusing collapse by varying the waves geometry in Kerr medium . A kind solution of the nonlinear Schrödinger equation is represented by breathers (breathing solitons) , which periodically oscillate during the evolution. Breathing solitons present a state of self-sustained breathers which is supported by the balance between the dispersion and nonlinearity, concomitant with equilibrium between gain and loss that can be used for amplification and compression of solitons [23,24]. In 2018, Medina proved an existence theory of ring-profiled optical vortex solitons . The dynamics of filament formation in a Kerr medium shows that the number of filaments is expected to increase linearly with input power increasing . The propagation of various novel waves in Kerr medium has also been studied in many papers [27–34]. However, the circle Pearcey Gaussian (CPG) waves in Kerr medium have not yet been explored. In this article, we discuss the characteristics of the CPG waves propagating in Kerr medium.
The paper is organized as follows. In Sec. 2, we introduce the mathematical structure of the CPG waves under nonlinear medium. We extend the study in Sec. 3 by numerical simulation and discuss the focusing principle of the CPG waves. Further we study the propagation characteristics of real part and imaginary part of CPG waves, respectively. Finally we summarize our findings in Sec. 4.
2. The theoretical model
We will consider the CPG waves propagating along the Z axis. In the 3D paraxial optical system, the wavepacket evolution is described by the nonlinear Schrödinger equation in the Kerr medium, which can be written in the normalized form [2, 28]:Equation (1) can be rewritten as
The electric field of the initial CPG waves in cylindrical coordinates can be expressed into two forms9,35,36],
It is difficult to obtain the analytic expression for u(r, θ, Z). Fortunately, we can use the split-step Fourier transform method to numerically simulate [28,37] the propagation of the waves. In the simulation, we set that λ = 0.532× 10−6m, α=0.05, and n0 = 1.45 throughout this paper. The coefficient for the Kerr nonlinearity n2 = 2.6 × 10−16cm2W−1  leads to a critical power of collapse of a Gaussian wave for self-focusing [30,39–42]. The connection between the initial input power of the CPG waves (Pin) and the critical power of collapse of a Gaussian wave (Pcr) is as follow [30,38]
3. Numerical analysis
3.1. Demonstration of simulated phenomena
We first discuss different cases of the initial fields of the CPG waves. In Fig. 1, we show the intensity distribution of the rings for different values of c. Figure 1(a) is the intensity distribution of the conventional Pearcey waves based on the Pearcey function of catastrophe optics. In catastrophe optics, the Pearcey function represents the diffraction about a focus, which experiences spherical aberration (the rays underlying the diffraction caustic form a cusp either below or above the geometrical focus depending on the sign of the spherical aberration) . Its transverse profile resembles a diffraction cusp whose light intensity automatically focuses to a small spot centered in an hourglass-shaped distribution during propagation . The CPG waves can be regarded as the Pearcey waves revolve around the axis with the location in X-Y plane given by the c parameter as depicted by the dotted lines in Fig. 1. The parameter c also varies the proportion of conventional Pearcey waves in the X- or Y-coordinate and affects the position of intensity maximum of the CPG waves radially. Then, different c affects the positions of the main intensity of the CPG waves. The dotted lines marked (1), (2), (3) correspond to Figs. 1(a1)–1(a3) which relate to u1 function. Besides, the dotted lines marked (4), (5), (6) correspond to Figs. 1(a4)–1(a6) which relate to u2 function. We first observe Figs. 1(a1)–1(a3): When c = 0, the CPG waves have a circularly symmetric profile that develops outward of a dark disk and oscillates radially, forming a series of concentric intensity rings with decreasing width. At this moment, the maximum intensity position is close to the center of rings. As c increases, the ring with maximum light intensity moves toward the outer edge of the CPG waves disk. The distribution shows that energy gradually disperses outward along rings forming a disk, and the central light intensity is weak. The case simulated by u2 function is shown as Figs. 1(a4)–1(a6), the distribution of initial intensity also varies in different c. When c = 0, the CPG wave is a circular light spot with the wave intensity concentrating in the center. When c = 2, the CPG waves are formed by rotation under most concentrating light intensity state with few rings. With the further increase c, the ring numbers of the CPG waves become more, and the maximum intensity position moves to outside of the rings. Different c can make the initial waves take on different states, which shows that the initial waves can be adjusted in a more realistic way.
Comparing with other cylindrical waves [14,30], the initial circular Pearcey Gaussian beam can be adjusted by choosing different parameters, and the maximum intensity at the focus and the focal length can be controlled. We study the transmission characteristics within normalized propagation distance 10ZR in Fig. 2. For an input energy of 0.7Pcr in Fig. 2(a), we find that the focusing of the CPG waves has already formed but no breathers-like structure emerges. This abrupt increase of the intensity is mainly due to the autofocusing effect of the CPG waves. Rings have different velocities toward the center, and energy rushes toward the focus. After the focusing, the rings begin to separate and the light intensity starts to decrease. This decrease is rapid and monotonous, which just means that the linear effect dominates, and the CPG waves still act as the propagation rules in free space. While the input energy is big enough in Fig. 2(b) (Pin =8.8Pcr), we can observe that two strong foci occur in the front part of the transmission, and then small breathers-like structure is generated in Kerr medium which is a special state of the interaction between the linear waves and nonlinear medium. The formation of the breathers-like structure can be described by the action of the optical Kerr effect acting against diffraction. Typically multiple filamentations in the Kerr medium for a ring Airy Gaussian waves are not expected to occur unless the input power is greater than 10Pcr in Ref. . When the input power is 8.8Pcr, the CPG waves are expected to occur filamentation. Because of the effect of nonlinearity is stronger with the increasing input power, the auto-focusing point and the nonlinear focusing point of the wave become different. On the other hand, the nonlinear effect is strongest at the centre of the wave, where the intensity is highest, and most rings will start to compress. The action of Kerr medium causes different velocities of rings toward the center means that the focusing is not synchronous. After the first focusing, the rings soon reach another stronger focus and achieve the maximum energy convergence. It is interesting that the small multiple refocusing can continually replenish the energy loss in the core. The breathers-like structure of the CPG waves can sustain a long range due to the Kerr nonlinearity, the rings will no longer completely spread out, and the energy concentrating in the center. Figures 2(a1)–2(a4) and Figs. 2(b1)–2(b4) show the transverse intensity distribution of the CPG waves at different transmission distances. At the input plane, the intensity distributions of the CPG waves are shown in Figs. 2(a1) and 2(b1). When Pin=0.7Pcr, the point where all rings reach the center at the same time, which is called the focal point in Fig. 2(a3) (Z = 3.9). By comparing Figs. 2(a1)–2(a4) and Figs. 2(b1)–2(b4), we find that the initial input power can affect the focusing of the CPG waves in Kerr medium. The breathers-like structures of CPG waves are formed after the focusing, and the transmitting state of the waves relates to the interaction between the linear waves and the nonlinear medium. We can get the result that the nonlinear effect of Kerr medium changes as the input power growing. Figures 3(a1)–3(b1) are the intensity distribution of CPG waves about the function u1. Figures 3(a1)–3(a2) show that the CPG waves with c = 0 develop to the relatively stable breathers-like structure after two main focusing, which is similar to Fig. 2(b). When the main intensity of the rings moves outward as shown in Fig. 3(b1), the focus distance becomes longer than that in Fig. 3(a2) during the propagation. Here (Pin=28.5Pcr), the stable breathing-like subgroups (oscillations) are formed during the propagation, which needs stronger initial input power than the case for c = 0. At this time, these stable states are different with other waves in Kerr medium and the physical explanation is that the nonlinearity effect makes the focusing of rings with big input power be inconsistent with that of rings with small input power. The groups are stable, and once they are formed they survive as the propagation length or energy is increased. Next, we study Figs. 3(c1)–3(d1) which show the initial intensity distribution from the function u2. The most concentrated state of the CPG waves shown as Fig. 3(c1) (c = 2), its propagation with few rings and the energy mainly concentrates on the center of the light spot. At this time, a small focus of the CPG waves occurs first, then a stable breathers-like group structure is gradually formed during the propagation. When c = 5, the rings with maximum light intensity move outward, and its propagation progress is similar with Fig. 3(c2). However, the rings far from the center with less energy are not convergence to the axis, means that the nonlinearity has small influence for those rings. In the process, the steady-state breathers-like subgroups of the CPG waves can be achieved in the center means that higher intensity rings still are influenced by nonlinearity. Meantime, we can get the result that the focal length of the CPG waves is affected by the deviation factors.
3.2. Exhibition of numerical experiments
We show the numerical experiment  results of the CPG waves for studying their evolution properties. It is worth emphasizing that the main characteristic of the numerical experiment is the calculated hologram results of the CPG waves. Figures 4(a1)–4(a2) are the intensity patterns of the initial input of the CPG waves. Figures 4(a2)–4(b2) are the phase patterns of the initial input of the CPG waves. We find that the phase distribution that is a circularly symmetric profile forming a series of concentric intensity rings which is similar with the intensity distribution. By calculating the off-axis interference patterns between the complex amplitude profiles of the CPG waves at the Z = 0 plane and a plane wave, we can get the holograms as shown in Figs. 4(a3)–4(b3). An initial wave is launched to reconstruct the off-axis computer-generated holograms as shown in Figs. 4(a4)–4(b4) of the desired beam profiles in generation of the numerical experiments. We obtain the transverse average intensity patterns at the input plane as Figs. 4(a5)–4(b5). The results show that the shapes of interference average intensity diagrams are consistent with the initial input intensity. Here, we take different c for obtaining the varied initial waves. Figure 4(c) shows the numerical experiment progress of the the CPG waves in Kerr medium. We can see from the Fig. 4(c) that the focusing process and phenomena (stable breathers-like structures) are also consistent with Fig. 3(a2). We can get the conclusion that our numerical experiments are consistent with the numerical simulation.
3.3. Exploration of focusing principle
As opposed to the self-focusing effects mediated by Kerr nonlinearities, this autofocusing behavior of the CPG waves is purely linear in origin and is a result of the optical field structure itself . In order to explore the reasons for focusing of the CPG waves in theory, we simulate the propagation of the waves in Kerr medium with small input power. The propagation process of the real part, the imaginary part of the CPG waves is shown in Fig. 5. Figures 5(a2)–5(c2) are intensity distribution located at Z = 2. In this case of Fig. 5 (a1), different rings move toward the center, and energy rushes in an accelerated fashion toward the focusing point. For longer propagation distances, all rings begin to separate, and the peak intensity starts to decrease. The results show that all rings reach the focal point at the same time. For the real part and imaginary part of the CPG waves, some small foci appear before the main focusing occurs. Figures 5(b1s) and 5(c1s) are the enlarged parts of dotted circle of Figs. 5(b) and 5(c). It is clear from the propagation progress as shown in Figs. 5(b) and 5(c) that rings with different energies focus synchronously. Interestingly, the imaginary part of the CPG waves has many foci in the front part of the transmission until the waves converge on the main focus ultimately. The physical interpretation of this distinctive stepwise focusing of halo is that those rings with big energy and closing to the center of the waves are greatly affected by the nonlinearity and focus earlier. Meantime, the AAF feature of the CPG waves also has bigger impact on the area where the intensity is big. During the propagation, the small foci make the transverse light intensity distribution more dispersed at the location Z = 2 in Fig. 5(c2). Comparatively speaking, only one small obvious focus appears before reaching the main focus of the real part of the CPG waves in Fig. 5(b1s). We can conclude that the imaginary part of the CPG waves focuses spontaneously, and the characteristic can be applied in theory to modulate some other waves. This trait is helpful for optical trapping, light manipulation etc.
Next, the propagation of the CPG waves with larger input power is studied. The first case (u1) is shown as Fig. 6, the focusing position of the real part of the CPG waves can change as the initial input power changes. By comparing Figs. 6(a1) and 6(a2), we find that the focal distance of CPG waves is shorter as the input power increases. After the main autofocusing, the wave can propagate a distance in the state of light filament, and gradually weaken as the distance increases. Even the input power is larger, some rings with small energy will not focus on the central axis, and travel with the CPG waves all the way. That is to say, the nonlinearity has little effect on rings which have small energy and farther away from the center. When the CPG waves transmit to Z = 6 as shown in Figs. 6(a2)–6(d2), we can see that the number of rings increases with the input power growing. It is worth noting that the imaginary part of the CPG waves splits into three wavelets in an accelerated fashion away from the central axis in Fig. 6(d1), after the main focusing, and the strongest intensity mainly distributes in the middle wavelet as shown in Fig. 6(d2). Same as the real part of the CPG waves, the imaginary part of the CPG waves has different focusing positions with different input power in Kerr medium. This phenomenon explains the significant variations of the CPG waves in focusing position as the input power increases. Clearly, we can conclude that the effect of Kerr medium makes the CPG waves propagation progress different to that in free space.
The next case (u2) is shown as Fig. 7, the focusing position in real part also changes as the initial input power. Compared with Fig. 6, the energy of the CPG waves mainly concentrates on the central axis during the propagation. We can find that the focusing position of the CPG waves appears earlier and the formation of the breathers-like structure needs bigger initial power. Meantime the focusing position about the imaginary part of the CPG waves in Kerr medium also changes with different input power. The breathers-like groups structure is more intensive with bigger input power in Fig. 7(d1). The imaginary part of the CPG waves has more rings with small energy during the transmission, and the waves can split out as the distance and initial power increase. In a word, the waves from u2 function have the similar propagating phenomena with the waves from u1 except the shorter focal distance and more intensive breathers-like groups structure.
In conclusion, we provide the first numerical demonstration of the CPG waves in Kerr medium and identify many interesting behaviors. Motivated by the potential applications of filaments, we are able to control the peak intensity and the light filament by changing the initial waves and the input power. Different c can make the initial waves take on various states, which provides us a more realistic way to adjust the waves. We observe that a unique form of the CPG waves in Kerr medium which appears to be the breathers-like structure and breathers-like groups structure. We also present some numerical experiment results of the CPG waves which are in accordance well with simulation results. Then a special analysis of the CPG waves is made about the propagation of the real part and imaginary part. When the input power Pin =0.7Pcr, the propagation characteristics of the wave are nearly same as those in free space. Interestingly, we find that the imaginary part of the CPG waves has many foci with energy gradually increasing until it converges on main focus ultimately. However, the nonlinearity action has little impact for the rings with small energy, and as the initial input power increases, the nonlinearity has a great influence on the energy-intensive rings. The stepwise focusing of the imaginary part of the CPG waves can be applied in theory to modulate some other waves. This trait is helpful for optical trapping, light manipulation etc.
National Natural Science Foundation of China (NSFC) (11775083, 11374108, 11674107, 61875057); Innovation Project of Graduate School of South China Normal University (2018LKXM012).
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