Propagation dynamics and radiation forces of autofocusing circle Bessel Gaussian vortex beams in a harmonic potential

Haonan Wang; Chao Sun; Jialong Tu; Weiming Zhen; Dongmei Deng

doi:10.1364/OE.435588

1. Introduction

In 1987, Durnin [1] used the scalar theory of nondiffracting beams for the first time to propose an accurate solution for the 0-order Bessel beam. The Bessel beam has good transmission characteristics of non-diffraction and self-healing [2,3], and also has the focusing characteristics of the small focal spot and sizeable focal depth [4–6]. These excellent characteristics have a wide range of potential applications in micro-lithography, high-resolution optical remote sensing, data storage, micro-manipulation [7], precision collimation, and biological imaging [8]. Efremidis et al. [9] theoretically proved that two-dimensional or three-dimensional beams could suddenly autofocus, and they proposed that this idea can be extended to other wave functions. Subsequently, the autofocusing ring-Airy beams with higher contrast than the Gaussian beams were observed in experiments [10]. It was proved theoretically and experimentally in 2013 that the ring-Airy beams gradually became a non-linear bullet-like wave packet with the increase of the transmission distance [11]. The circle Pearcey beams [12] and the annular Bessel beams [13,14] have also been explored sequentially. In a word, the above research demonstrates that the ring-shaped beams have a better autofocusing effect.

Inspired by the angular momentum of the circularly polarized light, it is proved in the spatial light modulator that the beam can also carry orbital angular momentum [15]. Due to the fact that an optical vortex has the following unique properties, such as spiral phase wavefront, phase singularity, multiple polarization states and infinite kinds of mutually orthogonal eigenstates, etc., it is used to trap large-scale medium objects [16], large-scale multiplexing communication [17,18], partially coherent vortex beams [19], and other fields have a wide range of applications. Recently, more and more scholars have also introduced optical vortices into the research of beams, such as the Airy Gaussian vortex beams [20], the circular Pearcey Gaussian vortex beams [21] and so on.

On the other hand, the photonic potential as a "potential" embedded in the refractive index of the medium is frequently used in linear optics and is extensively referenced in the literature. Mendlovic and his colleagues [22,23] noticed the phenomenon that the propagation of beams in a parabolic potential is equivalent to a fractional Fourier transform. And beyond that, Efremidis [24] found that the trajectory of Airy waves can be controlled by adjusting the index gradient in the refractive-index potential. Among various potentials, the harmonic potential is widely used in quantum mechanics as a potential field with an infinite number of completely equidistant energy levels [25], such as laser-plasma physics [26], Bose-Einstein condensates [27], ultracold atoms [28], ion-laser interaction [29], and optical lattices [30]. What’s more, the harmonic potential also has the advantages of the gradient refractive index and easy accesses in nature, which becomes a tool for effectively modulating the beam. Researchers have also analyzed the propagation of the Laguerre-Gauss beams [31], the circular Pearcey Gaussian vortex beams [21], and the Airy Gaussian vortex beams [20] in the harmonic potential, and they discovered some exciting phenomena. Lately, the propagation characteristics of the chirped annular Bessel Gaussian beams [13] in free space and the first-order annular Bessel Gaussian beams in uniaxial crystals [14] have been studied. Nevertheless, as far as we know, the topics covered in this paper have not been reported.

In this paper, based on the dimensionless linear (2+1)D Schrödinger equation, we mainly illuminate the transmission characteristics and the radiation force modulated by the vortex and Gaussian factors in the harmonic potential. We notice that the position, the intensity, the number of the focal points and the width of the transmitted beams can be flexibly adjusted by changing the parameters. At the same time, the number and the position of the vortex will significantly change the transmission form and the period of the beams. Additionally, we also analyze the radiation force of the beams.

The structure of the paper is as follows. In Sec. 2, we introduce the theoretical model of the circle Bessel Gaussian vortex beams (CBGVBs) under the harmonic potential. Then in Sec. 3, the beams’ autofocusing characteristics and the propagation dynamics are discussed under different vortex modulations through theoretical analysis and numerical simulation. Moreover, we analyze the radiation force in Sec. 4. Finally, the main conclusions are drawn in Sec. 5.

2. Theoretical model

We will theoretically analyze the propagation of the CBGVBs in a harmonic potential. The initial electric field intensity distribution of the CBGVBs is expressed as

(1)$$\begin{aligned} & U(r, \varphi, 0)= {C_{0}} J_{n}\left[({r_{s}-r})\beta\right] \exp \left[{-}b\left({r_{s}-r}\right)^ {2}\right] \\ & \times \left[r \exp (i \varphi)+r_{k}\exp \left(i \varphi_{k}\right)\right]^{m}\left[r \exp (i \varphi)-r_{k} \exp \left(i \varphi_ {k}\right)\right]^{l}, \end{aligned}$$

where $C_{0}$ is a constant, $J_{n}$ is the first kind of the n-order Bessel function, $r_{s}$ is the radius of the main ring when $n$=0 (the radius of the 0-order Bessel ring), $\beta$ is the horizontal spatial frequency and $b$ is the attenuation factor which controls the number of Bessel side rings in the initial field; $r$ and $\varphi$ represent the radial distance and the azimuth angle of the cylindrical coordinate system, respectively; $(r_{k},\varphi _{k})$ is related to the position of the vortex, where $r_{k}$ is the distance from the vortex to the center of the circle in the radial direction, $\varphi _{k}$ represents the counterclockwise rotation’s angle from the $x$-axis to the vortex in the argument direction; $m$ and $l$ represent the topological charges.

The CBGVBs propagate in a harmonic potential and satisfy the dimensionless linear (2+1) dimensional Schrödinger equation under paraxial conditions [32]

(2)$$\nabla_{{\perp}}^{2} {U(x,y,z)} + 2i \frac{\partial {U(x,y,z)}}{\partial z} - \alpha^{2}\left(x^{2}+y^{2}\right) {U(x,y,z)} = 0,$$

where ${U(x,y,z)}$ is the complex amplitude of the electric field at the propagation distance $z$, $\nabla _{\perp }^{2}=\frac {\partial ^{2}}{\partial x^{2}}+\frac {\partial ^{2}}{\partial y^{2}}$ is the transversal Laplace operator, $\alpha$ stands for a parameter controlling the width of the potential. In order to facilitate the description of the CBGVBs in cylindrical coordinates, we will rewrite Eq. (2) as

(3)$$\frac{\partial^{2} {U}}{\partial r^{2}}+r^{{-}1} \frac{\partial {U}}{\partial r}+r^{{-}2} \frac{\partial^{2} {U}}{\partial \varphi^{2}}+2 i \frac{\partial {U}}{\partial z}-\alpha^{2} r^{2} {U}=0,$$

where variables $r$ and $z$ are correspondingly the normalized transverse coordinates and propagation distances, scaled by some characteristic transverse width $x_{0}$ and corresponding Rayleigh range $Z_{r}=kx_{0}^{2}$. Here, $k=2 \pi n_{0} / \lambda$ is the wavenumber, $n_{0}$ is the ambient index of the refraction, and $\lambda$ is the wavelength in the free space.

The expression of the electric field propagating to $z$ can be obtained by the double integral of the two-dimensional Fourier transform of the initial field [31]

(4)$$\begin{aligned} U(x, y, z) & ={-}\frac{i}{2 \pi} N(x, y, z) \iint_{-\infty}^{\infty} U(\xi, \eta, 0) \exp \left[i M\left(\xi^{2}+\eta^{2}\right)\right] \\ & \quad\times \exp [{-}i K(x \xi+y \eta)] d \xi d \eta, \end{aligned}$$

where $N(x, y, z)=K\operatorname{exp}\left [iM\left (x^2+\ y^2\right )\right ], M=\alpha \cot (\alpha z) / 2$ and $K=\alpha / \sin (\alpha z)$. Obviously, $K x$ and $K y$ represent the spatial frequencies. Because of the ($r_{s}$ - $r$) term, it is not easy to obtain the integral result of Eq. (4). Nevertheless, we will obtain a numerical solution by the method of split-step Fourier.

Figure 1 illustrates the influence of the vortex position and the number of topological charges on the initial plane ($z$=0) of the CBGVBs. Figures 1(a1)–1(b1) show the intensity of the CBGVBs without the vortex, which distributes uniformly on two rings. From Figs. 1(a2)–1(b2), whereas one can see that the intensity of the ring near the vortex decreases and the intensity increases far away from the vortex when the vortex is located at $r_{k}$=2.5 on the left $x$-axis. Simultaneously, the overall intensity is significantly improved compared with the case without the vortex (Fig. 1(a1)). In addition, Figs. 1(a4)–1(b4) establish that with the growth of the distance between the vortex and the center of the circle, the intensity is higher than that of Fig. 1(a2) when a vortex is at $r_{k}$=5 on the left $x$-axis. Similarly, it is interesting that the intensity distribution shows symmetry along the radius direction when a group of vortices symmetrical about the origin is placed on both sides of the $x$-axis at $r_{k}$=2.5. Furthermore, as the number of vortices increases, the integral intensity is also intensified compared to Fig. 1(a2). On the one side of the coin, for the phase evolution (Figs. 1(c2)–1(d2) and Figs. 1(c3)–1(d3)), the phase where the vortex is located will have obvious singularities and spirally distributed with a vortex in the main ring. Besides that, the rotation inside the main ring will also cause the rotation outside. As a whole, the rotation direction of the vortex in the main ring is the same as that of the outer.

Fig. 1. (a1)-(a4) The initial transverse intensity distributions of the CBGVBs ((a1) $m$=$l$=0; (a2) $m$=1, $l$=0, $r_{k}$=2.5; (a3) $m$=$l$=1, $r_{k}$=2.5; (a4) $m$=1, $l$=0, $r_{k}$=5); (b1)-(b4) The 3D intensity distributions corresponding to (a1)-(a4); (c1)-(c4) The 2D phase distributions corresponding to (a1)-(a4); (d1)-(d4) The 3D phase distributions corresponding to (a1)-(a4). If there is no other statement, these parameters shall remain constant ($\beta$=0.5, $\varphi _{k}$=0, $b$=0.2, $\alpha$=1, $n$=1).

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3. Results and analyses

3.1 Propagation dynamics of the CBGVBs in a harmonic potential

Figure 2(a) shows that the CBGVBs without vortex are periodically autofocusing along the $z$-axis due to the influence of a harmonic potential. Moreover, there will be twice focusing phenomena in a period, which is manifested explicitly as focusing first, then diverging, and then focusing. The focusing process from the initial plane to the focal plane presents the intensity of the outer ring gradually weakens, and the inner ring is increasing. Simultaneously, the radius of the outermost ring gradually decreases, and the intensity gradually converges toward the center. Finally, the intensity at the focal point (Fig. 2(b2)) is the strongest, where the radius of the outermost ring is the smallest. Figures 2(b2)–2(b4) are the divergence processes, the radius of the outermost ring gradually increases, and the number of rings increases. Furthermore, the intensity also transitions from the center to the outer ring. Besides, the radius of the outermost ring is the largest, and the intensity is the weakest when it reaches the position of Fig. 2(b4). What can see it from Figs. 2(c1)–2(c4) and Figs. 2(d1)–2(d4) that the intensity of the CBGVBs is uniform distributed on each ring.

Fig. 2. (a) The side view of the propagation with $m$=0, $l$=0, and $r_{k}$=0; (b1)-(b4) The transverse intensity distributions corresponding to the white dashed lines at planes 1-4 in (a); (c1)-(c4) The phase distributions corresponding to (b1)-(b4); (d1)-(d4) The 3D intensity distributions corresponding to (b1)-(b4).

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When an one-sided vortex is added to the $x$-axis, the periodically autofocusing phenomenon does not change in Fig. 3(a). However, the lower side of the middle spot is strengthened in the first and the second focusing processes, while the third and the fourth focusing processes are opposite within the propagation distance of 0-4.44$Z_{r}$. What’s more, the overall transmission trajectory is asymmetric about the $z$-axis, and the distance of one period is twice that of the one without vortex. Compared with the focus of the beam without vortex (Fig. 2(b2)), the spot at the focus (Fig. 3(b2)) is composed of a crescent-shaped bright spot and a weak dark spot. What’s interesting is that the crescent-shaped spot in the center will rotate counterclockwise during the focusing and the divergence processes (Figs. 3(b1)–3(b4) are the screenshot of 90$^\circ$ rotation each time). The details will be discussed in the center of mass and the energy flow section. In contrast to Fig. 2, it is clarified that the phase (Figs. 3(c1)–3(c4)) is distorted as a whole, the intensity (Figs. 3(d1)–3(d4)) is significantly improved, and the intensity of the ring-shaped uniform distribution is broken. In comparison with the intensity at the focal point (Fig. 3(d2)) and the initial field (Fig. 1(b2)), we can find that the intensity of the focal point is increased by about 4 times.

Fig. 3. (a) The side view of propagation with $m$=1, $l$=0, $r_{k}$=2.5, and $\varphi _{k}$=0; (b1)-(b4) The transverse intensity distributions corresponding to the white dashed lines at planes 1-4 in (a); (c1)-(c4) The phase distributions corresponding to (b1)-(b4); (d1)-(d4) The 3D intensity distributions corresponding to (b1)-(b4).

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As illustrated in Fig. 4(a), the overall propagation trajectory is symmetrical about the $z$-axis because a pair of off-axis vortices that are symmetrical about the origin is added to the $x$-axis. In addition, compared with the propagation trajectory of a single off-axis vortex beam (Fig. 3(a)), the focal spots become two with the two off-axis vortices beams in the side view, and the side lobes close to them have also been significantly enhanced. On the whole, the focusing and the divergence processes of two off-axis vortices beams are similar to those of the one without vortex. The difference is that the $x$-$y$ transverse distributions (Figs. 4(b1)–4(b4)) become a crescent-shaped distribution symmetrical about the $x$-axis, and the focal point is composed of two crescent-shaped bright spots and two dark spots. Interestingly, a pair of off-axis vortices on the $x$-axis will cause the beam to focus significantly on the $y$-axis (focusing on the vortices’ vertical direction). Inspired by this, we can realize the focus of the CBGVBs with a pair of symmetric off-axis vortices in any direction on the $x$-$y$ section by changing $\varphi _{k}$ and $r_{k}$, which will also change the transmission trajectory of the overall beam (Fig. 5 is an example). Comparing the intensity of the focal point of the beam without vortex with the two off-axis vortices beams (Fig. 2(d2) and Fig. 4(d2)), the side lobes of the crescent-shaped spot at the focal point are significantly improved. There are two apparent singularities in the phase distribution (Figs. 4(c1)–4(c4)), and the overall intensity (Figs. 4(d1)–4(d4)) is higher than that of a single vortex off-axis beam.

Fig. 4. (a) The side view of propagation with $m$=1, $l$=1, and $r_{k}$=2.5; (b1)-(b4) The transverse intensity distributions corresponding to the white dashed lines at planes 1-4 in (a); (c1)-(c4) The phase distributions corresponding to (b1)-(b4); (d1)-(d4) The 3D intensity distributions corresponding to (b1)-(b4).

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Fig. 5. (a1)-(a5) The transverse intensity distributions at different positions along the $z$-axis ($m$=1, $l$=1, $r_{k}$=2.5, and $\varphi _{k}$=$\frac {\pi }{3}$).

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Next, we discuss the envelope and the tangential intensity of the above three types of beam propagation trajectories. It can be observed that the envelope of the beam without vortex is composed of many rings (Fig. 6(a1)), and the intensity distributions of the $y$-$z$ (Fig. 6(b1)) and $x$-$z$ sections (Fig. 6(c1)) are the same. The location of a single off-axis vortex forms a hollow area in the envelope (Fig. 6(a2)), and the crescent-shaped spot rotates continuously with the propagation distance. Moreover, the focusing intensity in the $y$-axis direction has a tremendous increase if the vortex is on the left side of the $x$-axis by observing the intensity distribution of the $y$-$z$ plane and the $x$-$z$ plane of a single vortex beam(Figs. 6(b2)–6(c2)). By observing the $x$-$z$ section of (Fig. 6(c2)), we find that the focusing phenomenon of the $x$-axis presents two spots with the asymmetric intensity. The intensity of the spot far away from the vortex is greater than that of the spot near the vortex, and their position will alternate periodically with the transmission distance increasing. The overall envelope of the beams of the two off-axis vortices presents a double crescent distribution (Fig. 6(a3)). The vortices on the $x$-axis enhance the focusing spot intensity of the $y$-axis more than that of the $x$-axis one, which is similar to a single off-axis vortex beam.

Fig. 6. (a1)-(a3) The full-wavepacket diagrams under the parameters (a1) $m$=$l$=0; (a2) $m$=1, $l$=0, $r_{k}$=2.5; and (a3) $m$=$l$=1, $r_{k}$=2.5. (b1)-(b3) and (c1)-(c3) are the half-wavepacket diagrams of the $y$-$z$ section and $x$-$z$ section corresponding to (a1)-(a3).

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3.2 Peak intensity/beam width/the center of mass

In order to demonstrate the focusing characteristics of the CBGVBs in detail, we discuss the parameters that affect the maximum light intensity of the CBGVBs in the propagation process as follows. Figure 7(a) analyzes the influence of the main ring radius ($r_{s}$) on the peak intensity. We find that increasing the radius of the main ring will gradually bring the focus position closer to the middle, and the intensity will also be improved. In Fig. 7(b), when $\beta$ is small, the increase of $\beta$ does not significantly change the position of the focus, but the increase of the intensity is more obvious. Therefore, inspired by the above results, the position and the intensity of the focus can be effectively adjusted through the combination of $r_{s}$ and $\beta$ parameters.

Fig. 7. The peak intensity is a function of $z/Z_{r}$ for three groups of different parameters.

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Finally, we study the influence of the width $\alpha$ of the harmonic potential medium on the peak intensity. It can be found that the number of periods in the same range will also increase by the exact multiple when the $\alpha$ with a two-fold increase. In addition, the peak intensity will also be significantly increased, and the weaker focus will also be formed at the junction of the two periods, as shown in Fig. 7(c) $(\alpha =4)$. Consequently, by controlling the $\alpha$, what can effectively adjust the number and the intensity of the focus in the same period.

Next, we introduce the beam width in the $x$ and $y$ directions to discuss the propagation characteristics of the CBGVBs. The beam width in the $x$-$y$ direction is defined as follows [33]

(5)$$w_{x}=\sqrt{\frac{\iint_{-\infty}^{\infty} x^{2} |U(x, y, z)|^{2} d x d y}{\iint_{-\infty}^{\infty} |U(x, y, z)|^{2} d x d y}}, w_{y}=\sqrt{\frac{\iint_{-\infty}^{\infty} y^{2} |U(x, y, z)|^{2} d x d y}{\iint_{-\infty}^{\infty} |U(x, y, z)|^{2} d x d y}}.$$

Since the CBGVBs are symmetric in the $x$-$y$ section without vortex, we only select $w_{x}$ as the research object. Figure 8 shows the beam width changes with each parameter. On the whole, the beam width is the smallest at the highest intensity. Besides, the beam width will also increase with the increase of $\beta$. We find that $r_{s}$ has a significant modulation on the beam width and the transmission trajectory of the initial light field, but it does not change the maximum beam width too much. Similarly, the modulation of $\alpha$ to the beam width is similar to the peak intensity.

Fig. 8. The beam width is a function of $z/Z_{r}$ for three groups of different parameters.

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To further study the propagation trajectory of the CBGVBs, it is indispensable to observe the change of the centroid of the intensity. The definition of the centroid is as follows [32]

(6)$$x_{c}=\frac{\iint_{-\infty}^{\infty} x|U(x, y, z)|^{2} d x d y}{\iint_{-\infty}^{\infty}|U(x, y, z)|^{2} d x d y}, y_{c}=\frac{\iint_{-\infty}^{\infty} y|U(x, y, z)|^{2} d x d y}{\iint_{-\infty}^{\infty}|U(x, y, z)|^{2} d x d y}.$$

According to Eq. (6), we discuss the influence of the topological charge, the $r_s$ and the $\beta$ parameters on the center of mass. Since the intensity distributions of without vortex and those two off-axis vortices with symmetrical positions are symmetrical about the origin, it can be observed from Fig. 9(a) that the trajectory of the center of mass will move along the optical axis. Interestingly, the center of mass will rotate along the optical axis with an off-axis vortex. Moreover, the projection on the $y$-$z$ plane is a sine function, and the projection on the $x$-$z$ plane is a cosine function. Figure 9(b) shows that as $r_{k}$ increases, the center of mass will rotate around the $z$-axis along a larger radius, but it does not change the overall contour. In Fig. 9(c), we find that when $r_{s}$ is small, the $x$-$y$ projection of the centroid trajectory is a longitudinal ellipse, and as $r_{s}$ increases, the centroid trajectory gradually transitions to the horizontal ellipse.

Fig. 9. The CBGVBs centroid change locus propagating in the harmonic potential along the $z$-axis.

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3.3 Poynting vector and the orbital angular momentum

After analyzing the transmission and the focusing characteristics of the CBGVBs in the harmonic potential, in order to understand the evolution process of the CBGVBs, we decide to study the Poynting vector and the orbital angular momentum. As we all know, the Poynting vector represents the energy passing through a vertical unit area per unit time, the Poynting vector can be defined as

(7)$$\langle\vec{S}\rangle=\frac{c}{4 \pi}\langle\vec{E} \times \vec{B}\rangle, \quad|\vec{S}|=\sqrt{S_{x}^{2}+S_{y}^{2}+S_{z}^{2}}.$$

Figure 10 depicts the Poynting vector of the CBGVBs propagating in a harmonic potential. During the focusing process (Figs. 10(a1)–10(a2)), it can be found that the energy in the $x$-direction gradually flows to the center, while the energy in the $y$-direction is far away from the center of the circle. Nonetheless, the energy flow in the $x$ and the $y$ directions is opposite to the focusing process in the divergence process (Figs. 10(a3)–10(a4)). From Figs. 10(b1)–10(b8), it can be clarified that the energy will flow clockwise from the strong side to the weak side with an off-axis vortex. Moreover, the spot at the focal point is no longer a solid round spot but a crescent-shaped round spot with a hollow center. As the propagation distance increases, the central crescent-shaped round spot rotates counterclockwise (Figs. 10(b1)–10(b8) is the cross-section of 90$^\circ$ rotation each time), and the energy will flow clockwise. Figures 10(c1)–10(c4) show the CBGVBs with two vortexes, and the energy flow is similar to Figs. 10(a1)–10(a4). The difference is that two hollow dark spots are formed at the focal point due to the vortex.

Fig. 10. The transverse Poynting vector of the CBGVBs at different positions along the $z$-axis. (a1)-(a4) $m$=$l$=0; (b1)-(b8) $m$=1, $l$=0, $r_{k}$=2.5; (c1)-(c4) $m$=$l$=1, $r_{k}$=2.5;

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The time-average angular momentum density can be obtained by the cross product of the position vector and the Poynting vector [34],

(8)$$\langle\vec{J}\rangle=\vec{r} \times\langle\vec{E} \times \vec{B}\rangle, \quad|\vec{J}|=\sqrt{J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}.$$

The background is the angular momentum value, and the blue arrow indicates the direction of the $x$-$y$ section angular momentum. The transverse normalized angular momentum density of CBGVBs with different propagation distances is demonstrated in Fig. 11. The angular momentum is distributed on the ring-shaped spot or crescent-shaped spot as a whole, and they are more concentrated on high intensity. What is more, it is interesting that all angular momentum’s directions are clockwise.

Fig. 11. The CBGVBs’ transverse angular momentum at different positions along the $z$-axis. (a1)-(a4) $m$=$l$=0; (b1)-(b8) $m$=1, $l$=0, $r_{k}$=2.5; (c1)-(c4) $m$=$l$=1, $r_{k}$=2.5.

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4. Radiation force

To further understand the electromagnetic momentum, we will explore the radiation force of the CBGVBs in a harmonic potential in this section. As an excellent structured light, the Bessel beams also have a good performance in capturing particles [35]. For example, the annular structure of the Bessel beams make it possible to capture both low and high refractive index particles at the same time [36], and the self-reconstruction property dramatically improves the possibility of capturing spatial-separated particles [37]. Few people know that the reason why the trajectory of the comet’s tail is far away from the sun because of the effect of the solar radiation. Maxwell proved that light could exert force in the theory of electromagnetics. This force is the scattering force (it can be understood as the result of the momentum transferred by scattered photons). The direction is along the direction of light propagation, and the size is proportional to the light intensity, which can be expressed as [38]

(9)$$\vec{F}_{scat}(r)=\frac{8{\pi}n_{2}}{3c} \left(k \delta\right)^{4} \delta^{2}\left(\frac{q^{2}–1}{q^{2}+2}\right)^{2} I(r) \vec{e}_{z},$$

where $n_{2}$ is the refractive index of the surrounding medium, $\delta$ represents the particle radius and $n_{1}$ is the particle refractive index, $q=n_{1}/n_{2}$ is the relative refractive index, $\epsilon _{0}$ is the permittivity of vacuum and $I(r)=\frac {n_{2} \varepsilon _{0} c}{2}|U(r)|^{2}$.

The other force is the gradient force caused by the fluctuating electric dipole when light passes through a transparent object. In other words, the gradient force is produced by the Lorentz force and can be expressed as [39]

(10)$$\vec{F}_{grad}(r)=\frac{2 \pi n_{2} \delta^{3}}{c}\left(\frac{q^{2}–1}{q^{2}+2}\right) \nabla I(r),$$

among them $\nabla$ is gradient calculation.

An essential prerequisite for the particle capture is that the gradient force is greater than the scattering force, so we must ensure that this condition is met when selecting the particle radius. Figure 12(a) shows the variation of the maximum scattering force and the maximum gradient force with the particle radius during the propagation process. It is found that both of them will enhance with the increase of the particle radius, and they are equal to about 2nm. Under capture conditions, we select 1.7nm particles for research.

Fig. 12. (a) The relationship between the maximum amplitude of the radiation force and the particle radius in the propagation process; (b) Longitudinal gradient force, scattering force, and their combined force with a radius of 1.7nm. (a)-(b) $m$=$l$=0, $\alpha$=4, $n_{1}$= 1.755 (i.e., dense flint glass nanoparticle).

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Figure 12(b) demonstrates the longitudinal gradient force, the scattering force, and their combined force in a period. The distribution of the scattering force is similar to the peak intensity during transmission, and the direction of the force will not change. As we all know, when the electrolyte sphere is in the light gradient field, the resultant force of all light before the force balance tends to push the sphere to the area with more vigorous light intensity [40]. However, due to the scattering force, the particles initially trapped at the focal points (C, D) are pushed out of the focal point for a certain distance and finally form traps at points (A, B) and are stably captured. We explore the lateral gradient force of points (A, B) and its direction and find that the gradient force of the points forming a stable trap is the most dramatic change in Figs. 13(a1)–13(a2).

Fig. 13. (a1)-(a2) The gradient force in the $x$ direction on the capture plane varies with $x$; (b1)-(b2) The transverse gradient force of the capture plane, and the background represents the transverse strength.

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By comparing the magnitude of the gradient force in Figs. 13(a1)–13(a2) with the slope of the curve at the two points in Fig. 12(b), we find that the greater the slope of the combined force of the scattering force and the gradient force, the greater the corresponding lateral gradient force, and the easier it is to capture a particle. In addition, the intensity of the point B is slightly greater than that of the point A, and the direction of the transverse gradient force of them both points to the center in Figs. 13(b1)–13(b2).

5. Conclusion

In summary, this paper investigates the transmission characteristics and the radiation force of the CBGVBs in a harmonic potential. For the transmission part, we numerically simulate the transmission of the periodic autofocusing CBGVBs with no vortex, a single off-axis vortex, and two off-axis vortices. We find that the number, the position, and the intensity of the focal spot can be adjusted. The beam’s transmission trajectory can also be changed by controlling the number and the position of the vortex. Discussing the parameters that may affect the light intensity, the beam width, and the center of mass, it is found that the position, the intensity, and the number of the focus can be changed through the joint adjustment of the parameters. It is interesting to observe that the trajectory of the center of mass of a single vortex is spiral. In addition, we discuss the Poynting vector and the angular momentum and find that the focusing and the diverging energy flow in opposite directions. For the angular momentum, the whole is distributed on the ring, and the direction is clockwise. In the radiation force part, we study the scattering force in the transmission direction, the gradient force, and their resultant force under the former embodiment that the gradient force is greater than the scattering force. Moreover, we discuss the possible capture points and find that the point where the resultant force changes steeply is easier to capture particles. We believe that the results of our investigation may not only provide effective and diverse methods for linear manipulation of circle Bessel Gaussian vortex beams, but also have enlightening ideas of such beams.

Funding

Science and Technology Program of Guangzhou (2019050001); National Natural Science Foundation of China (11374108, 11775083).

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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Propagation dynamics and radiation forces of autofocusing circle Bessel Gaussian vortex beams in a harmonic potential

Abstract

1. Introduction

2. Theoretical model

3. Results and analyses

3.1 Propagation dynamics of the CBGVBs in a harmonic potential

3.2 Peak intensity/beam width/the center of mass

3.3 Poynting vector and the orbital angular momentum

4. Radiation force

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Equations (10)

Optics Express