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Propagation dynamics of autofocusing circle Pearcey Gaussian vortex beams in a harmonic potential

Chao Sun, Dongmei Deng, Xiangbo Yang, and Guanghui Wang

Open Access Open Access

Abstract

We introduce the circle Pearcey Gaussian vortex (CPGV) beams in a harmonic potential for the first time and investigate their abruptly autofocusing properties by theoretical analysis and numerical simulations in this paper. By varying the spatial distribution factors, one can effectively control the propagating dynamics of the beams, including the position of the focus, the radius of the focal light spot and the intensity contrast. Meanwhile, the magnitude of topological charges and the position of the vortex can alter the focal pattern and the intensity contrast. Furthermore, the position of the focus can be flexibly controlled in a tiny range by adjusting the scaled parameter of the incident beam properly.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Abruptly autofocusing (AAF) beams have been studied theoretically [1] and experimentally [2,3]. Without invoking any lenses or nonlinearities, the AAF beams can abruptly focus their energy at the focal point in the free space while maintaining a low intensity profile up to that point [1]. This abruptly autofocusing property is quite useful in the fields of biomedical treatment [1] and generation of light bullet [4]. The AAF beams can also be used to improve the conventional optical tweezers, such as to enhance the attractive force or trap particles at different positions [5,6]. Besides, in order to better control the AAF properties of the beams, other kinds of AAF beams have also been reported [7–9].

A phase vortex with phase singularity or screw dislocation has a spiral phase front [10]. Due to their helical wavefronts and donut-like intensity distributions [11], optical vortices (OVs) play important roles in optical tweezers [12,13], opticial communications [14], high resolution imaging [15]. In recent years, the propagation properties of OVs embedded in various beams were extensively studied, such as an optical vortex imposed on an Airy beam [16], abruptly autofocusing vortex beams [17] and ring Airy Gaussian vortex beams [18].

On the other hand, the propagating dynamics of various beams in a harmonic potential has been studied, such as Airy Gaussian vortex beams [19], superposed Bessel-Gauss [20], Laguerre-Gauss [21], and circular Airy beams [20]. In addition, researchers also have investigated the propagation dynamics of beams in other types of potentials, such as localized potentials [22], dynamic linear potential [23], smooth-interface sigmoid-type potentials [22], parabolic potential [24] and higher-order power-law potentials [25].

Recently, Pearcey beams were theoretically introduced and experimentally generated in 2012 [26]. The Pearcey beam generation using a virtual source has also been theoretically reported [27]. After that, investigations of circle Pearcey beams have been reported in the free space [28]. However, to the best of our knowledge, the AAF properties of the circle Pearcey Gaussian vortex (CPGV) beams have not been explored in a harmonic potential. In this paper, the AAF properties of the CPGV beams are explored and investigated by numerical simulations. The propagating dynamics of the beams can be flexibly controlled by choosing the suitable parameters of the incident beams, including the position of the focus, the radius of the focal light spot, the focal pattern and the intensity contrast. Some interesting results are illustrated.

The organization of the paper is as follows. Firstly in Section 2, we introduce the theoretical model for the CPGV beams in a harmonic potential. Then in Section 3, the autofocusing property and propagating dynamics of the beams were discussed by theoretical analysis and numerical simulations. Finally, the main conclusions are drawn in Section 4.

2. The theoretical model

Within the framework of the paraxial approximation and the harmonic potential, the spatial beams propagating along the z-axis are governed by the dimensionless linear (2+1) dimensional Schrodinger equation [29]:

$${\nabla}^{2}_{ \bot }\mathbf{E}+2i \dfrac{\partial \mathbf{E}}{\partial z}-\alpha^{2}(x^2+y^2)\mathbf{E}=0,$$
where ${\nabla} ^{2}_{ \bot }=\dfrac {\partial ^{2}}{\partial x^{2}} +\dfrac {\partial ^{2}}{\partial y^{2}}$, $\mathbf {E}$ is the beam envelope, $\alpha$ stands for a parameter controlling the width of the potential, easily achieved, for example, in gradient-index media; x and y are the normalized transverse coordinates. In terms of the radially symmetrical CPGV beams solution, it is more convenient to describe it in polar coordinates, Eq. (1) can be rewritten as:
$$\dfrac{\partial^{2} \mathbf{E}}{\partial r^{2}}+r^{{-}1}\dfrac{\partial \mathbf{E}}{\partial r}+r^{{-}2}\dfrac{\partial^{2} \mathbf{E}}{\partial \varphi^{2}}+2i\dfrac{\partial \mathbf{E}}{\partial z}-\alpha^{2}r^{2}\mathbf{E} =0,$$
where $\varphi =arctan(y/x)$ is the azimuthal angle, the radial coordinate $r$ is normalized, scaled by an arbitrary transverse width $x_{0}$, and $z$ is the normalized propagation distance with the corresponding Rayleigh range $kx_{0}^{2}$. Here, $k=2 \pi n/\lambda$ is the wavenumber, n the ambient index of refraction, and $\lambda$ is the wavelength in the free space.

The electric fields of the initial CPGV beams can be expressed as:

$$\begin{aligned} E(r, \varphi, 0) & =A_{0}Pe(-\frac{r}{p_{s}}, -\frac{r}{\xi})\exp{({-}r^{2})}q(r)\\ & \times(re^{i \varphi}+r_{k}e^{i \varphi_{k}})^{m}(re^{i \varphi}-r_{k}e^{i \varphi_{k}})^{l}, \end{aligned}$$
where $A_{0}$ is the constant amplitude of the electric field, $Pe(\cdot )$ corresponds to the Pearcey integral [30], $p_{s}$ is the spatial distribution factor which can be used to control the intensity distribution of the input beams, at the same time, $p_{s}$ also determines the position of the focal plane (i.e., the focal length $L_{s}=2p_{s}^{2}$), $\xi$ is the scaled parameter, $(r_{k},\varphi _{k})$ denotes the location of the OVs, $(m,l)$ represents the topological charges of the OVs, $q(r)$ can be expressed as:
$$q(r)=\left\{ \begin{aligned} & Ce^{a r ^{\beta} }, &r<r_{0}\\ & 0, & r \geq r_{0} \end{aligned}, \right.$$
where $C$ is a constant, $a$ and $\beta$ are used to adjust the amplitude distribution of the initial electric field at $r<r_{0}$, and they also control the intensity of the focal point. Similar to the Airy beam and the Bessel beam, a pure Pearcey beam has infinite energy, Eq. (4) can ensure that the total conveyed power $\int ^{2 \pi }_{0}\int ^{\infty }_{0}E(r, \varphi , 0)r d r d \varphi$ is finite, which does not significantly change the beams’ properties.

For a given initial electric field Eq. (3), the solution to Eq. (2) can be expressed in terms of the Fresnel integral:

$$\mathbf{E}(r,\varphi,z)=\int^{2 \pi}_{0}\int^{\infty}_{0}\dfrac{\mathbf{E}(\rho,\theta,0)}{2 \pi i z}e^{i \dfrac{\rho^{2}+r^{2}-2\rho r cos(\varphi-\theta)}{2z}}\rho d \rho d \theta.$$
It is difficult to obtain the analytical solution for Eq. (5). Fortunately, numerical simulations are carried out by using the fast Fourier transform method [31,32]. In the following simulation, the parameters are chosen as $A_{0}=1$, $p_{s}=0.1$, $\xi =2$, $m=0$, $l=0$, $x_{0}=100\mu m$, $\lambda =533nm$, $C=1$, $a=0.1$, $\beta =1$, $r_{0}=1.5$. Hereafter, the parameters are the same as those aforesaid unless stated otherwise.

3. Discussions and analyses

The propagation properties of the CPGV beams are discussed in Fig. 1. The transverse intensity profiles of the CPGV beams at different propagation distances are elucidated in Figs. 1(a)–1(c). It is clear to see from Fig. 1(a) that the transverse intensity profiles are formed by a circular light spot in the center and a series of concentric symmetric intensity rings at the initial plane, and most of the energy is concentrated in the central circular light spot area. As the propagation distance increases, the concentric intensity rings shrink toward the center and the intensity of the circular light region increases, which is manifested in Fig. 1(b). In the focal plane, all concentric intensity rings shrink sharply toward the center simultaneously, as observed in Fig. 1(c), all the energy is concentrated in a very small area. The intensity of the CPGV beams at the focal plane is $80$ times higher than that of the initial plane. In order to more intuitively demonstrate the evolution properties of the CPGV beams, the amplitude of the CPGV beams is depicted as a function of z in Figs. 1(d) and 1(e). It is easy to find from Fig. 1(d) that CPGV beams can abruptly focus their energy at the focal point while maintaining a low intensity profile up to that point [1]. After the focal plane, all concentric intensity rings begin to diffuse and the maximum peak intensity drops sharply, which well proves that the abrupt increase of the intensity at the focal plane is due to the abruptly autofocusing properties of the CPGV beams. Comparing Figs. 1(d) and 1(e), one can find that the evolution behavior of the CPGV beams amplitude does not change, only the magnitude of the absolute amplitude is scaled accordingly. As the value of a increases, the location of the focus point does not change, but the peak intensity of the focus increases dramatically, the peak intensity of the focal plane is controlled by adjusting the parameter a. The phase distribution of the CPGV beams is closely aligned with the concentric ring shape in Fig. 1(f).

 figure: Fig. 1.

Fig. 1. The intensity profiles of the CPGV beams propagating at different propagation distances. The transverse intensity profiles and the corresponding cross lines of the CPGV beams (a) at the initial plane ($z=0$), (b) before the focal plane ($z=L_{s}/2$), (c) at the focal plane ($z=L_{s}$), the detailed intensity distribution of the propagation dynamics for (d) $a=0.1$ and (e) $a=0.5$, (f) the phase pattern at the initial plane.

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The numerically simulated side-view intensity profiles of the CPGV beams at different propagation distances are depicted in Fig. 2(a). One can see that the CPGV beams are abruptly autofocusing at the focus position, and the intensity profiles are centrosymmetric distribution. From Figs. 2(b1)–2(b6), the transverse intensity patterns are taken at the planes marked by the dashed lines in Fig. 2(a). One can see clearly that the center of the transverse intensity profiles is always hollow, and a hollow channel is formed due to the vortex on axis along the propagation. As the propagation distance increases, the phase distribution of the CPGV beams carrying the on-axis vortex is distorted and exhibits a scroll shape in Figs. 2(c1)–2(c6).

 figure: Fig. 2.

Fig. 2. The intensity profiles of CPGV beams propagating at different propagation distances. (a) The numerically simulated side-view propagation of the CPGV beams, (b1) - (b6) snapshots of the transverse intensity patterns are taken at the planes marked by the dashed lines in (a); (c1) - (c6) the corresponding phase distributions at different planes marked in (a). Other parameters are the same as those in Fig. 1 except $m=1$, $r_{k}=0$ and $\varphi _{k}=0$.

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Figure 3 shows the evolutions of the intensity distributions of the CPGV beams with the different spatial distribution factor $p_{s}$. One can see from Fig. 3(a1) that many of the hollow cylinder structures are arranged in a stepped pattern on the initial plane, and their strength decreases step by step. However, at the focal point, only a hollow cylinder structure with high peak intensity appears at the center of the intensity distribution in Fig. 3(b1). For a larger distribution factor $p_{s}$, the number of hollow cylinder structures decreases and other hollow cylindrical intensity values near the main ring increase in Fig. 3(a2), whereas the radius of the hollow cylinder in the focal plane increases slightly and its maximum peak strength decreases significantly in Fig. 3(b2). As the distribution factor $p_{s}$ increases further, only three hollow cylinders are visible intuitively in the initial plane, the radius of the hollow cylinder in the focal plane is further increasing, as is depicted in Fig. 3(a3). In order to more intuitively demonstrate the evolution behavior of the CPGV beams, the corresponding cross lines of the intensity distribution are depicted in Figs. 3(c1)–3(c3). As $p_{s}$ increases, only the number of peaks decreases, and the maximum peak intensity is nearly equivalent in the initial plane, however, the distance between the two peaks increases in the focal plane, which means the radius of the ring increases, meanwhile, the peak intensity decreases in the focal plane.

 figure: Fig. 3.

Fig. 3. The evolutions of the intensity distributions of the CPGV beams with the different spatial distribution factor (a1) and (b1) $p_{s}=0.1$; (a2) and (b2) $p_{s}=0.13$, (a3) and (b3) $p_{s}=0.15$. The corresponding cross line of the intensity distribution on $y=0$ (c1) at the initial plane, (c2) at the focal plane. (c3) The peak intensity distribution of the CPGV beams versus the propagation distance. Other parameters are the same as those in Fig. 2 except $a=0.6$.

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The magnitude of topological charges and the position of the vortex have a tremendous influence on the propagation characteristics of the CPGV beams. When the on-axis vortices are imposed, one can see that the hollow region becomes bigger with the increase of topological charges at the initial plane, compared to Fig. 2(b1). But in the focal plane, not only the ring radius increases but also the intensity decreases in Fig. 4(b2). When an off-axis vortex is imposed in Figs. 4(a1)–4(a3), it is obvious from Fig. 4(a1) that most of the energy is concentrated in the upper right corner of the main ring, and the energy of the secondary ring is very small. In Fig. 4(a2), the intensity pattern of the CPGV beam is a pea-shaped light spot due to the influence of the off-axis vortex in the focal plane, and the intensity pattern rotates 90 degrees counterclockwise compared to Fig. 4(a1). When two off-axis vortices are imposed at the same position in Figs. 4(c1)–4(c3), it is clear that most of the energy is concentrated in the upper right corner of the secondary ring in the initial plane, but the main ring has very little energy. In the focal plane, the shape of the light spot remains the same, but the intensity of the light spot increases. It is clear to see that the intensity contrast (defined as the ratio between the peak light intensity in propagating and the initial peak intensity) decreases with $m$ increasing when the CPGV beams have on-axis vortices in Fig. 4(d1), whereas, when the CPGV beams have off-axis vortices in Fig. 4(d2), the magnitude of topological charges has little effect on the intensity contrast.

 figure: Fig. 4.

Fig. 4. The transverse intensity profiles of the CPGV beams (a1)–(c1) at the initial plane (z = 0), (a2)–(c2) at the the focal plane (z = Ls), (a3)–(c3) the phase pattern at the initial plane. Comparison of $(I/I_{0})_{max}$ of the CPGV beams with different m as a function of z for (d1) on-axis vortices and (d2) off-axis vortices. Other parameters are the same as those in Fig. 3(a1), except $r_{k}=0.5$ and $\varphi _{k}=\pi /3$ in (a1)–(c1), m=2 in (a2)–(c2), and m=2, $r_{k}=0.5$ and $\varphi _{k}=\pi /3$ in (a3)–(c3).

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From Fig. 5, we can see that when an off-axis vortex pair is imposed, the CPGV beams will exhibit different propagation dynamics and initial phase patterns. Interestingly, the off-axis vortex pair contracts toward the center and also rotates relative to the axis of propagation with the increase of the propagation distance. Comparing Figs. 5(a1)–5(a4) to Figs. 5(b1)–5(b4), one can see that, obviously, the intensity of main ring light increases at the initial plane, however, the light spots are completely separated and two hollow areas can be clearly observed in Fig. 5(a3), and only one light spot and the intensity increases in Fig. 5(b3) at the focal plane, this phenomenon is mainly due to the increase of $r_{k}$. Comparing Figs. 5(a1)–5(a4) to Figs. 5(c1)–5(c4), it can be clearly seen that the number of intensity rings decreases in the initial plane, but in the focal plane, the light spot shape is unchanged, the radius of the light spot becomes larger, and the intensity of the light spot decreases due to the increase of $p_{s}$. The phase pattren of the off-axis vortex pair is shown in Figs. 5(a5)–5(c5). In Fig. 5(d), we plot the intensity contrast of the RPCPV beams versus z. It is not difficult for us to perceive that when $p_{s}$ is a constant, the larger $r_{k}$ is, the larger the intensity contrast is. The position of the off-axis vortex pair has little effect on the intensity contrast of the CPGV beams. However, when the off-axis vortex pair is in the same position, the intensity contrast goes down with $p_{s}$ increasing. The AAF properties of the CPGV beams with a small $p_{s}$ can remarkably improve, as shown in Figs. 5(d) and 3(c3).

 figure: Fig. 5.

Fig. 5. The transverse intensity profiles and the phase patterns of the CPGV beams with an off-axis vortex pair at different propagation distances. Other parameters are the same as those in Fig. 4(a1), except l=1 in (a1)–(a5); l=1 and $r_{k}=0.7$ in (b1)–(b5); l$=1$ and $p_{s}=0.13$ in (c1)–(c5).

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In Fig. 6, we plot the peak intensity contrast as a function of z. It is easy for us to find that changing the value of $\xi$ can slightly tune the position of the focus in Fig. 6(a), and the position of the focus moves slightly forward and the intensity contrast insignificantly increases with $\xi$ increasing. In Fig. 6(b), the peak intensity contrast of the CPGV beams increases with the increase of $a$.

 figure: Fig. 6.

Fig. 6. (a) The peak intensity contrast of the CPG beams versus the propagation distance for (a) different $\xi$, (b) different $a$. Other parameters are the same as those in Fig. 1.

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4. Conclusion

In summary, we have demonstrated the evolution of the abruptly autofocused CPGV beams in a harmonic potential by means of numerical calculations. In the focal plane, the beams can be intensively focused to a very small area due to all energy shrink sharply toward the center simultaneously, the light intensity reaches their maximum. Furthermore, by varying the value of $p_{s}$, one can effectively control the propagating dynamics of the beams, including the position of the focus, the radius of the focal light spot and the intensity contrast. When $p_{s}=0.1$, the abruptly autofocusing property of the beams can achieve optimal result. Meanwhile, the magnitude of topological charges and the position of the vortex can alter the focal pattern and the intensity contrast. Interestingly, the intensity contrast decreases with $m$ increasing when the OVs of the CPGV beams are on-axis, however, when the OVs the CPGV beams are off-axis, the magnitude of topological charges and the position of the vortex have little effect on the intensity contrast. Finally, the intensity pattern of the beams in the focal plane rotates 90 degrees counterclockwise compared to the initial plane, and the position of the focus can be flexibly controlled in a tiny range through an appropriate $\xi$, and the peak intensity contrast can also be controlled by adjusting the parameter $a$. Our results may be useful for potential applications in biomedical treatment, optical transport, optical trapping and optical manipulation of microparticle.

Funding

National Natural Science Foundation of China (11374108, 11775083); Supported by the Innovation Project of Graduate School of South China Normal University (2018LKXM043).

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Figures (6)
Fig. 1.
Fig. 1. The intensity profiles of the CPGV beams propagating at different propagation distances. The transverse intensity profiles and the corresponding cross lines of the CPGV beams (a) at the initial plane ( $z=0$ ), (b) before the focal plane ( $z=L_{s}/2$ ), (c) at the focal plane ( $z=L_{s}$ ), the detailed intensity distribution of the propagation dynamics for (d) $a=0.1$ and (e) $a=0.5$ , (f) the phase pattern at the initial plane.

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Fig. 2.
Fig. 2. The intensity profiles of CPGV beams propagating at different propagation distances. (a) The numerically simulated side-view propagation of the CPGV beams, (b1) - (b6) snapshots of the transverse intensity patterns are taken at the planes marked by the dashed lines in (a); (c1) - (c6) the corresponding phase distributions at different planes marked in (a). Other parameters are the same as those in Fig. 1 except $m=1$ , $r_{k}=0$ and $\varphi _{k}=0$ .

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Fig. 3.
Fig. 3. The evolutions of the intensity distributions of the CPGV beams with the different spatial distribution factor (a1) and (b1) $p_{s}=0.1$ ; (a2) and (b2) $p_{s}=0.13$ , (a3) and (b3) $p_{s}=0.15$ . The corresponding cross line of the intensity distribution on $y=0$ (c1) at the initial plane, (c2) at the focal plane. (c3) The peak intensity distribution of the CPGV beams versus the propagation distance. Other parameters are the same as those in Fig. 2 except $a=0.6$ .

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Fig. 4.
Fig. 4. The transverse intensity profiles of the CPGV beams (a1)–(c1) at the initial plane (z = 0), (a2)–(c2) at the the focal plane (z = Ls), (a3)–(c3) the phase pattern at the initial plane. Comparison of $(I/I_{0})_{max}$ of the CPGV beams with different m as a function of z for (d1) on-axis vortices and (d2) off-axis vortices. Other parameters are the same as those in Fig. 3(a1), except $r_{k}=0.5$ and $\varphi _{k}=\pi /3$ in (a1)–(c1), m=2 in (a2)–(c2), and m=2, $r_{k}=0.5$ and $\varphi _{k}=\pi /3$ in (a3)–(c3).

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Fig. 5.
Fig. 5. The transverse intensity profiles and the phase patterns of the CPGV beams with an off-axis vortex pair at different propagation distances. Other parameters are the same as those in Fig. 4(a1), except l=1 in (a1)–(a5); l=1 and $r_{k}=0.7$ in (b1)–(b5); l $=1$ and $p_{s}=0.13$ in (c1)–(c5).

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Fig. 6.
Fig. 6. (a) The peak intensity contrast of the CPG beams versus the propagation distance for (a) different $\xi$ , (b) different $a$ . Other parameters are the same as those in Fig. 1.

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Equations (5)

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∇ ⊥ 2 E + 2 i ∂ E ∂ z − α 2 ( x 2 + y 2 ) E = 0 ,
∂ 2 E ∂ r 2 + r − 1 ∂ E ∂ r + r − 2 ∂ 2 E ∂ φ 2 + 2 i ∂ E ∂ z − α 2 r 2 E = 0 ,
E ( r , φ , 0 ) = A 0 P e ( − r p s , − r ξ ) exp ⁡ ( − r 2 ) q ( r ) × ( r e i φ + r k e i φ k ) m ( r e i φ − r k e i φ k ) l ,
q ( r ) = { C e a r β , r < r 0 0 , r ≥ r 0 ,
E ( r , φ , z ) = ∫ 0 2 π ∫ 0 ∞ E ( ρ , θ , 0 ) 2 π i z e i ρ 2 + r 2 − 2 ρ r c o s ( φ − θ ) 2 z ρ d ρ d θ .
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