Pendulum-type light beams

Junhui Jia; Junhui Jia; Haolin Lin; Haolin Lin; Yixuan Liao; Yixuan Liao; Zhen Li; Zhen Li; Zhen Li; Zhen Li; Zhenqiang Chen; Zhenqiang Chen; Zhenqiang Chen; Shenhe Fu; Shenhe Fu; Shenhe Fu

doi:10.1364/OPTICA.477076

1. INTRODUCTION

The diffracting property of light beams is a fundamental feature that gives rise to spatial broadening of localized beams during their propagation. The Gaussian light beam and its higher-order counterparts (exact solutions to the free-space Helmholtz wave equation in the paraxial approximation) are typical examples of such diffracting light beams [1]. In addition to the diffracting solutions, the paraxial Helmholtz wave equation admits solutions for nondiffracting light beams such as Bessel beams in cylindrical coordinates [2–4] and cosine beams in Cartesian coordinates [5]. Contrary to the Gaussian beam, Bessel and cosine beams exhibit interesting nondiffracting features and can preserve their beam profiles while propagating along a distance. The first realization of such nondiffracting light beams in experiment were reported by Durnin et al. in 1987, by conical superposition of infinite plane wave components [2]. This nondiffracting beam was shown to self-heal when it was partially blocked by a small obstacle [4]. Note that the nondiffracting solutions have been recently applied to low-dimension wave systems such as surface plasmonic waves [6] and surface gravity water waves [7–10]. Apart from these nondiffracting light beams that propagate along a straight line, a peculiar class of nondiffracting solutions of the wave equation is beams able to self-accelerate along a spatial trajectory. The acceleration is mainly due to a highly asymmetric transverse profile of the light beams. For example, the paraxial Airy beam, first predicted in quantum mechanics [11] and subsequently realized in optics [12], accelerates along a parabolic trajectory while maintaining its Airy beam shape during propagation. Because of the unique properties, the study of nondiffracting and self-accelerating light beams has drawn significant attention [13–18], and other nonparaxial solutions for accelerating and nondiffracting wave packets have been found, including Mathieu beams [19,20] and Weber beams [21]. These beams have also led to many interesting applications including particle manipulation [22,23], curved plasma channel generation [24], superresolution imaging [25], optical routing [26].

It is worth noting that although the aforementioned diffraction-free beams process different wave forms, they actually share common characteristics, i.e., they are resulted from an appropriate conical superposition of infinite plane-wave components [2,3]. A direct outcome of such a coherent effect is that the generated wave packet is propagation invariant, while its center of gravity remains invariant (a result of Ehrenfest’s theorem) during propagation. Despite the acceleration property of the local main lobe, the propagation trajectory of previously reported nondiffracting and accelerating wave packets is still limited to a plane [(1 $+$ 1)D] configuration, and engineering the propagation trajectory in a higher dimension [(2 $+$ 1)D] is difficult to achieve [27,28]. This is because the realization of such a complex trajectory of light is at the expense of destroying its self-similar wave configurations. As a result, the beam tends to diffract in the course of propagation.

In this paper, we demonstrate theoretically and experimentally a new class of nondiffracting solutions to the paraxial wave equation perturbed by harmonic potential. We show that, unlike all the aforementioned nondiffracting solutions whose gravity centers are on a straight line, the presented new solutions support nondiffracting wave packets that can accelerate along an arbitrary trajectory centered on an elliptic or a circular orbit in the (2 $+$ 1)D configuration. The acceleration is due to the centripetal “force” of the underlying potential, while the associated trajectory is closely related to linear momentum initially added to the wave packet. Furthermore, we demonstrate that any considered propagation trajectory can support a complete set of general nondiffracting solutions, in both scalar and vector regimes. This enables generating complex states of light, e.g., a state that simultaneously carries intrinsic and extrinsic orbital angular momenta. We reveal that the motion of the wave packets of light described by the Schrödinger-like equation is analogous to the motion of a classical particle described by the Newton equation. Based on this similarity, we name such nondiffracting and accelerating wave packets as pendulum-type beams. Finally, we expect that the presented pendulum-type beams may be used in the field of laser scanning technology.

2. RESULTS AND DISCUSSION

Our idea to generate pendulum-type light beams is inspired by the concept of a mechanical pendulum driven by the gravitational potential. Let us remind that a gravitational pendulum comprises a particle of mass $m$ suspended by a massless string of length $L$ in a space where the gravitational field is uniform. The potential of the particle depends quadratically on its equilibrium position $x$, written as $V(x) = mg{x^2}/(2L)$, where $g$ is the gravitational acceleration. If the pendulum particle is pulled away from the equilibrium and then released, it moves in a harmonic oscillatory manner in a vertical plane, which appears as $\ddot x + {\omega ^2}x = 0$, where $\omega = (g/L{)^{1/2}}$ denotes the harmonic angular frequency. Such a classical pendulum can be extended to a quantum pendulum governed by a parabolic potential well [29]. On the other hand, we note that the time evolution of a wave packet in quantum mechanics is analogous to spatial propagation of a wave beam in the classical regime [30]. The parallels between wave equations in quantum mechanics and optics allow us to study pendulum-type structured light beams in the classical domain. Under slowly varying approximation, the paraxial Schrödinger equation is written as [31]

(1)$$\left[{i\frac{\partial}{{\partial z}} + \frac{1}{{2{n_0}{k_0}}}\nabla _ \bot ^2 + V(x,y)} \right]\psi ({x,y,z} ) = 0,$$

where $\psi (x,y,z)$ denotes the wave packet of light propagating along $z$, and ${k_0} = 2\pi /\lambda$ is its wave number in vacuum, with $\lambda$ being the carrier wavelength. $\nabla _ \bot ^2 = \partial _x^2 + \partial _y^2$ represents a transverse Laplacian operator in $x$ and $y$ coordinates. Analogous to the form of a gravitational potential, we consider a two-dimensional optical potential in the form of $V(x,y) = - 0.5{n_0}{k_0}{\alpha ^2}({x^2} + {y^2})$, where ${n_0}$ represents the refractive index of the background, and $\alpha$ is an arbitrary constant used to adjust the harmonic potential. In this manner, the broadening effect of light is arrested by the potential.

General solutions to the wave equation [Eq. (1)] have already been found, e.g., see [32–34], among others. Here we utilize these theoretical frameworks and introduce a new pendulum-type beam by considering an initial wave packet that contains a tunable transverse momentum. Specifically, we consider the scaled variables $u = \sqrt {{n_0}{k_0}} x$, $v = \sqrt {{n_0}{k_0}} y$, within which Eq. (1) can be decomposed accordingly into two independent ones. Their solutions are found as [34,35]

(2)$$\begin{split}{{\psi _u}(u,z) }=&{f({u,z} )\int {\psi _u}(u^\prime ,0)\exp \big({i\beta {{u^\prime}^2} - i\gamma u^\prime} \big)\text{d}u^\prime ,}\\{{\psi _v}(v,z) }=&{f({v,z} )\int {\psi _v}(v^\prime ,0)\exp \big({i\beta {{v^\prime}^2} - i\gamma v^\prime} \big)\text{d}v^\prime ,}\end{split}$$

respectively, where $f(\tau ,z) = \exp (i\beta {\tau ^2})[- i\gamma /(2\pi \tau {)]^{1/2}}$ (here $\tau = u$ or $v$). $\beta = \alpha \cot ({\alpha z})/2$ and $\gamma = \alpha \tau \csc ({\alpha z})$ are two parameters associated with the harmonic potential and distance, respectively. Because of the symmetric distribution of the harmonic potential $V(x,y)$, the two components of the wave packets exhibit equivalent behaviors in the course of wave propagation. The combination of them leads to an analytic solution: $\psi = {\psi _u} \cdot {\psi _v}$ [34,35]. For a fixed $\alpha$, the values of $\beta$ and $\gamma$ vary periodically with $z$, causing a periodic phase transition of the wave packets at the distance that satisfies $\sin (\alpha z) = 0$. This brings about harmonic oscillation of the wave packets. The oscillation period is associated with the potential parameter $\alpha$.

Equation (2) generates nontrivial propagation dynamics if we take into account the initial condition

(3)$$\psi ({u,v,0} ) = \phi (u - {u_0},v - {v_0})\exp ({i{k_u}u + i{k_v}v} ),$$

where $\phi (u,v)$ represents an arbitrary wave form of light beams, which is initially launched onto the input end of the potential, at a transverse position of ${{\bf r}_0} = ({u_0},{v_0})$ (note that ${u_0} = \sqrt {{n_0}{k_0}} {x_0}$ and ${v_0} = \sqrt {{n_0}{k_0}} {y_0}$). The initial wave packet $\phi (u,v)$ is endowed with a transverse wavefront, characterized by a transverse wave vector (${k_u},{k_v}$); hence, it contains a linear momentum ${{\bf p}_0} \propto ({k_u},{k_v})$ with direction transverse to the propagation direction $z$. To reveal wave propagation phenomena, we adopt the concept of center of mass for the wave packet, whose spatial position can be expressed as [34]

(4)$$\tilde u(z) = \frac{{\int u{{| {{\psi _u}({u,z} )} |}^2}\text{d}u}}{{\int {{| {{\psi _u}({u,z} )} |}^2}\text{d}u}},\quad \tilde v(z) = \frac{{\int v{{| {{\psi _v}({v,z} )} |}^2}\text{d}v}}{{\int {{| {{\psi _v}({v,z} )} |}^2}\text{d}v}}.$$

Integrating Eq. (2) with Eqs. (3) and (4) immediately yields an important result:

(5)$$\begin{split}{\tilde u(z)}&={ - \frac{{{k_u}}}{\alpha}\sin (\alpha z) + {u_0}\cos (\alpha z),}\\{\tilde v(z)}&={ - \frac{{{k_v}}}{\alpha}\sin (\alpha z) + {v_0}\cos (\alpha z).}\end{split}$$

Equation (5) displays propagation trajectory of the center of mass of the wave packet, which exhibits a harmonic oscillation with $z$. Eliminating the propagation variable $z$ gives rise to an equation of an ellipse, expressed as

(6)$$\frac{{{{\tilde u}^2}}}{{{A^2}}} + \frac{{{{\tilde v}^2}}}{{{B^2}}} - 2\frac{{\tilde u\tilde v}}{{AB}}\cos (\Delta \theta) = \mathop {\sin}\nolimits^2 (\Delta \theta),$$

where $A = (k_u^2/{\alpha ^2} + u_0^2{)^{1/2}}$ and $B = (k_v^2/{\alpha ^2} + v_0^2{)^{1/2}}$ determine the major and minor axes of the ellipse, respectively, and $\Delta \theta = {\theta _u} - {\theta _v}$ is defined as a phase mismatch between $\tilde u(z)$ and $\tilde v(z)$, where ${\theta _u} = \text{arctan} (- {u_0}\alpha /{k_u})$ and ${\theta _v} = \text{arctan} (- {v_0}\alpha /{k_v})$. It is the phase mismatch $\Delta \theta$ that leads to an intriguing class of pendulum-type light beams. The wave packets in the potential exhibit nondiffracting properties with nonuniform accelerations in propagation. At any distance, the locus of the propagating light field is an ellipse in general. These intriguing motions of wave packets revealed by Eq. (6) are analogous to those theoretical results reported in the context of quantum coherent wave states [36] and Bose–Einstein condensates [37] whose evolution trajectories are described by the Lissajous figure. We experimetnally realize these pendulum-type wave phenomena with structured light, described by the elliptical Lissajous pattern. To this end, it is required to engineer the initial wave packets of light with tunable linear momentum ${{\bf p}_0}$. Since Eqs. (5) and (6) are independent of the wave form, such wave phenomena can be further generalized to complex wave forms in the higher-order regime, which has not been reported before.

The elliptical trajectory degenerates to special forms for a certain quantity of the phase mismatch. For example, for $\Delta \theta = 0$ or $\pi$ as shown in Figs. 1(a) and 1(e), Eq. (6) is considered as a straight line equation with zero intercept. It corresponds to a harmonic oscillation along only one direction, tantamount to a classical simple pendulum described by the Newton equation; for $\Delta \theta = \pi /2$ and $3\pi /2$, the cross-product term $\tilde u\tilde v$ in Eq. (6) disappears. As a result, the ellipse becomes standard, as shown in Figs. 1(c) and 1(g), respectively. Particularly, together with the condition $A = B$, Eq. (6) degenerates to an equation of a circle, in which the state of light in the potential exhibits invariant extrinsic orbital angular momentum. In this particular scenario, the oscillation of the wave packet is analogous to a conical pendulum. For other values of $\Delta \theta$, the wave packets propagate along the trajectories centered on the elliptic orbits, as shown in Figs. 1(b), 1(d) and 1(f), 1(h). Note that the light beam moves on a clockwise or anticlockwise orbit, depending on the sign of the phase mismatch, as illustrated in Fig. 1.

Fig. 1. Propagation trajectory of center of mass of the pendulum-type light beam in the transverse plane ($\tilde u$, $\tilde v$). The beam moves on a clockwise (a)–(d) or anticlockwise (e)–(h) orbit, relying on the value of phase mismatch $\Delta \theta$, defined by Eq. (6): (a) $\Delta \theta = 0$; (b) $0 \lt \Delta \theta \lt \frac{\pi}{2}$; (c) $\Delta \theta = \frac{\pi}{2}$; (d) $\frac{\pi}{2} \lt \Delta \theta \lt \pi$; (e) $\Delta \theta = \pi$; (f) $\pi \lt \Delta \theta \lt \frac{3}{2}\pi$; (g) $\Delta \theta = \frac{3}{2}\pi$; (h) $\frac{3}{2}\pi \lt \Delta \theta \lt 2\pi$.

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We performed experiments to verify these findings. Initial wave packets with linear transverse momentum were prepared by a spatial light modulator (SLM) illuminated by a linearly polarized He–Ne laser beam ($\lambda = 632.8\;\text{nm} $); see the experimental setup in Fig. 2. In experiments, we used the encoding technique by Bolduc [38,39] to generate the phase-only mask. The reflected beam from the SLM was then modulated by the phase mask that encoded both the amplitude and phase information of the initial wave function. After telescopic lenses, the light beam was then Fourier transformed by a lens. The generated wave packet was normally coupled into a graded-refractive-index fiber ($\alpha = 6.06\;{\text{cm}^{- 1}}$ and ${n_0} = 1.611$), which plays the role of the harmonic potential. The transverse size of the potential is $1\,\,\text{cm} \times 1\;\text{cm} $. To observe the propagation dynamics of the wave packet, 16 fibers with selected propagation lengths $z = mp/8$ (where $p = 2\pi /\alpha$ is the period of the harmonic oscillation, and $m = 1,2,\ldots 16$) were prepared. These samples were left aligned and mounted onto a stage such that they could be moved along vertical direction. In this manner, we can monitor dynamical behaviors of the light states in the fiber. An imaging system with calibrated magnification times ($\times 20$) was utilized to record the light state at the output faces of the fibers.

Fig. 2. Experimental setup. The He–Ne laser working at wavelength of $\lambda = 632.8\;\text{nm} $ is divided into two beams by the beam splitter (BS), after passing through the beam expander (L1 and L2). One of the split beams is normally injected onto the phase-only spatial light modulator (SLM). The modulated wavefront, after telescopic lenses (L3 and L4), is Fourier transformed by an objective lens (OL1). This prepared wave packet is then injected into the sample. A microscopy system consisting of an objective lens (OL2), tube lens (TL), and charge coupled device (CCD) is used to monitor the propagation dynamics of light. We use another split beam to interfere with the pendulum-type beam to detect the wavefront of the output light from the sample. M, mirror.

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Let us first consider the pendulum-type light beam having the Gaussian wave function expressed as

(7)$$\phi (u,v) = {\exp} \left[{- \frac{{{{(u - {u_0})}^2} + {{(v - {v_0})}^2}}}{{({n_0}{k_0}\sigma _0^2)}}} \right],$$

where ${\sigma _0}$ denotes the Gaussian duration. To observe pendulum-type phenomena, the light width should be much smaller than the size of the potential. Hence, in experiments, we set it as ${\sigma _0} = 14\,\,{\unicode{x00B5}\text{m}}$. Without loss of generality, we examine the propagation trajectory having a standard ellipse. This requires to satisfy the condition of $\cos (\Delta \theta) = 0$, from which we obtain the relation ${k_u}{k_v} + {u_0}{v_0}\alpha = 0$. According to this relation, a typical set of parameters was chosen as ${u_0} = \sqrt {{n_0}{k_0}} {x_0}$, ${v_0} = \sqrt {{n_0}{k_0}} {y_0}$ (here ${x_0} = 100\,\,{\unicode{x00B5}\text{m}}$ and ${y_0} = 0$), and ${k_u} = 0$, ${k_v} = 121.27\;{\text{m}^{- 1/2}}$. Figure 3(a) illustrates the experimental measurements at different propagation distances, which show good agreement with the analytical results [Fig. 3(b)]. It can be recognized that the generated light beam approximately preserves its beam width (nondiffracting) and accelerates along a standard elliptic orbit, which emulates the motion of a particle in the conical pendulum.

Fig. 3. Generation of the pendulum-type Gaussian beam. Panels show experimentally (a), (c) and theoretically (b), (d) the intensity distributions of the Gaussian beam at different propagation distances: ${z_j} = jp/8$ ($j = 1,{2},{\ldots}\;{8}$). Two propagation trajectories are presented: (a), (b) trajectory centered on an elliptic orbit and (c), (d) trajectory centered on a circular orbit. Parameters for these trajectories are chosen as ${\sigma _0} = 14\,\,{\unicode{x00B5}\text{m}}$, ${k_u} = 0$, ${k_v} = 121.27\;{\text{m}^{- 1/2}}$, and (a), (b) ${x_0} = 100\,\,{\unicode{x00B5}\text{m}}$, ${y_0} = 0$; (c), (d) ${x_0} = 50\,\,{\unicode{x00B5}\text{m}}$, ${y_0} = 0$. Arrows denote direction of motion of light. Scale bars: (a), (b) $50\,\,{\unicode{x00B5}\text{m}}$; (c), (d) 40 µm.

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We further study the case where the propagation trajectory of the pendulum-type Gaussian beam becomes a circle. This is more involved because a circular orbit of motion requires to simultaneously satisfy the conditions $\cos (\Delta \theta) = 0$ and $A = B$. Experimentally, we achieve these two conditions by gradually reducing the value of ${x_0}$. There should be a critical point (${x_0} = 50\,\,{\unicode{x00B5}\text{m}}$) that satisfies the two conditions simultaneously. Hence, we modify the value of ${x_0}$, while keeping other parameters unchanged. In this case, we display the intensity distributions of the light beams at different distances, both experimentally [Fig. 3(c)] and analytically [Fig. 3(d)]. Again, the experiments agree well with the analytical outcomes. It is shown that the Gaussian wave packet indeed accelerates along a circular orbit, without changing its intensity profile. Since the beam centered on a circular orbit is nondiffracting and shape-preserving during propagation, the beam is particle-like and carries an invariant extrinsic orbital angular momentum whose magnitude depends on the angular frequency of orbital motion as well as the radius of the circular orbit.

In addition to the Gaussian wave function that is a zeroth-order solution of Eq. (1), Eqs. (5) and (6) suggest other wave functions that exhibit pendulum-type characteristics. We study higher-order wave functions including vortex beams and vector beams. These beams are spatially structured in phase or polarization, causing intriguing wave phenomena [40–44].

For the pendulum-type vortex beam, the incident wave form is replaced by the Laguerre–Gaussian function coupled with a helical phase. It is written as

(8)$$\phi (u,v) = \frac{r}{{\sqrt {{n_0}{k_0}} {\sigma _0}}}\exp \left({- \frac{{{r^2}}}{{{n_0}{k_0}\sigma _0^2}}} \right)\exp (il\theta),$$

where $r = \sqrt {{{(u - {u_0})}^2} + {{(v - {v_0})}^2}}$, and $\theta = \text{arctan} [(v - {v_0})/\def\LDeqbreak{}(u - {u_0})]$. $l$ is the topological charge of the vortex beam. This beam can be well generated by a computer-generated hologram based on the SLM; see the experimental setup in Fig. 2. We consider the same propagation trajectory, i.e., the trajectory centered on a standard ellipse, as illustrated in Figs. 3(a) and 3(b). Therefore, the experimental parameters for generating the pendulum-type vortex beam such as ${\sigma _0}$, (${u_0}$, ${v_0}$) and (${k_u}$, ${k_v}$) are set the same as those shown in Fig. 3. Despite the complex wave function, the higher-order vortex light beam still follows the given trajectory during its propagation in the fiber. As an illustration, Fig. 4(a) presents the experimental results, showing the intensity distributions of the first-order ($l = 1$) vortex beam at different propagation distances. The beam maintains its doughnut shape and beam width during propagation. Its center of mass follows an elliptical trajectory. To verify the helical phase profile, the pendulum-type vortex beam is superimposed with a plane wave; see the setup in Fig. 2. The resultant interference patterns at the corresponding distances are presented in Fig. 4(b). As clearly shown, all the patterns exhibit an identical dislocation in the fringes, indicating a helical wavefront with a topological charge of $l = 1$. This new state of light in the potential is intriguing since it contains both intrinsic and extrinsic orbital angular momenta.

Fig. 4. Generation of pendulum-type structured light beams, including a vortex beam (a), (b) and vector beam (c), (d). (a), (c) Measured intensity distributions of the vortex beam and vector beam, respectively, at different propagation distances: ${z_j} = jp/8$ ($j = 1,{2},{\ldots}\;{8}$). The two structured beams follow the propagation trajectories as those shown in Figs. 3(a) and 3(c); hence, parameters for these observations are set as ${\sigma _0} = 14\,\,{\unicode{x00B5}\text{m}}$, ${k_u} = 0$, ${k_v} = 121.27\;{\text{m}^{- 1/2}}$, and (a) ${x_0} = 100\,\,{\unicode{x00B5}\text{m}}$, ${y_0} = 0$; (c) ${x_0} = 50\,\,{\unicode{x00B5}\text{m}}$, ${y_0} = 0$. (b) Interference patterns of (a); (d) vertical polarization component of (c), at the corresponding distance. Arrows in (a), (c) denote direction of motion of light. Scale bars: (a) $50\,\,{\unicode{x00B5}\text{m}}$; (c) $35\,\,{\unicode{x00B5}\text{m}}$.

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For the vector beam, the pendulum-type dynamics of the vectorial wave packet $\vec \psi (x,y,z)$ still can be governed by Eq. (1). The initial wave packet is denoted as $\vec \psi (z = 0) = \vec \phi \exp (i{k_u}u + i{k_v}v)$, with $\vec \phi$ taking the following form [40]:

(9)$$\vec \phi (u,v) = \frac{r}{{\sqrt {{n_0}{k_0}} {\sigma _0}}}\exp \left({- \frac{{{r^2}}}{{{n_0}{k_0}\sigma _0^2}}} \right)[\cos (\theta + \varphi)\vec u + \sin (\theta + \varphi)\vec v],$$

where $\vec u$ and $\vec v$ are unit vectors associated with the scaled coordinates $u$ and $v$, respectively, and $\varphi$ is a constant denoting the polarization angle. For instance, $\varphi = 0$ and $\varphi = \pi$ correspond to radially and azimuthally polarized vector beams, respectively. Here we consider the case of $\varphi = \pi /2$, corresponding to a spirally polarized vector beam, as shown in Figs. 4(c) and 4(d). To show the pendulum-type vector beam, we initially modulate the incident laser beam with a q-plate (topological charge is 0.5) inserted into the setup. By rotating the q-plate appropriately, we can achieve an output from the wave plate as the spirally polarized vector beam. Again, we consider the same propagation trajectory as that shown in Fig. 3(c). With these conditions, we find interestingly that the vector beam propagates along the given circular trajectory [Fig. 4(c)] and preserves its polarization topology, which can be confirmed by the vertical polarization component of the vector beam [Fig. 4(d)]. The results in Figs. 4(c) and 4(d), together with the results in Figs. 4(a) and 4(b), suggest that any phase- and polarization-structured light beam, in addition to the studied cases, could preserve its topological property when it oscillates in the harmonic potential. This provides an opportunity to generate the complex propagating states of light in the higher (2 $+$ 1) dimension, which is difficult to achieve by conical superposition of waves.

Apart from the particle-like property of pendulum-type light beams, we also examine their wave-like property in the same setting, hence confirming the famous wave–particle duality. We note that wave–particle duality has been studied in both classical and the quantum regimes. In the classical regime, one has considered using a soliton, which is a localized wave packet in space or time and behaves as a particle, to demonstrate such wave–particle duality [45–50]. However, the generation of solitons strongly depends on the nonlinear effect, which requires intense light interacting with a nonlinear medium. We consider using the optical pendulum to observe the particle and wave properties simultaneously. Without loss of generality, we consider two pendulum-type vortex beams to demonstrate wave–particle duality, with outcomes shown in Fig. 5. While the particle-like property is manifested by orbital motions (periodic oscillations), the wave nature is manifested by the interference effect between two pendulum-type vortex beams. To this end, we experimentally generate two pendulum-type vortex beams in the potential, which are initially separated by a distance of $\Delta {x_0} = 60\,\,{\unicode{x00B5}\text{m}}$. The phase mismatch quantity for the two vortex beams is set to $\Delta \theta = 0$, in which manner the two wave packets are oscillating along the $x$ axis during propagation. Figure 5(a) illustrates their propagation trajectories in the $x - z$ plane. It is shown that the two vortex beams having an identical topological charge of $l = 1$ follow their respective trajectories during propagation, featuring two “particles” as demonstrated in Fig. 5(b). Interference occurs when they come close to each other, giving rise to interference fringes; see Fig. 5(c). The fringes contain two dislocations, indicating a helical wavefront $\exp (i2l\theta)$. This is a result of superposition of the two identical vortex beams, featuring the wave property. Noteworthily, with the increase of propagation distance, the fringes disappear and the two vortex beams revive, retaining their topological charges. Therefore, we observe simultaneously both the particle-like and wave-like properties of the pendulum-type light beams. Furthermore, two vortex beams having opposite charges ($l = 1$ and $l = - 1$) are also considered to reveal the duality; see Figs. 5(e)–5(g). Owing to the wave-like property, the superposition of them gives rise to a non-helical wavefront, as indicated by the regular fringes without any dislocation; see Fig. 5(f). However, owing to the particle-like property, the wavefront becomes helical again when they move away from each other, which is suggested from the doughnut-shaped intensity distribution in Fig. 5(g).

Fig. 5. Observation of wave–particle duality of the pendulum-type vortex beam. (a) Propagation trajectories of the two initially separated vortex beams. We consider two different scenarios of incident beams having identical topological charge [(b)–(d) $l = 1$], and opposite topological charges [(e)–(g) $l = 1$ and $l = - 1$]. (b)–(g) Intensity distributions of light beams at different propagation distances: (b), (e) ${z_1} = 0$; (c), (f) ${z_2} = p/4$; (d), (g) ${z_3} = p/2$. Panels in (b)–(g) share the same scale. Scale bars: $50\,\,{\unicode{x00B5}\text{m}}$.

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Finally, we discuss possibility of using pendulum-type light beams in the field of laser scanning technology. Although laser scanning has been widely used in many different fields such as optical microscopy imaging [51,52], switching [53], and sensing [54], it is still subjected to the constraint of the trade-off between the laser scanning area and beam diffraction. Because of the optical diffraction, it is difficult to achieve a large scanning area and deflecting angle [55,56]. The presented pendulum-type light beams might provide a new mechanism for scanning technology, overcoming the constraint mentioned above. To address this issue, we set the phase mismatch between the two components ${\psi _u}$ and ${\psi _v}$ to be $\Delta \theta = 0$. In this case, Eq. (6) reduces to a straight line equation, expressed as $\tilde v = A\tilde u/B$. We further simplify the linear equation as $\tilde v = {k_v}\tilde u/{k_u}$, by considering the conditions ${u_0} = 0$ and ${v_0} = 0$ (i.e., the laser is launched into the center of the fiber). It is clear that the straight line is zero-cross, and the slope is determined by the value ${k_v}/{k_u}$. Therefore, for a fixed fiber length, the light beam is deflected to different positions according to the values of ${k_u}$ and ${k_v}$. We performed experiments to confirm this phenomenon, using the same setup in Fig. 2. We swept the values of ${k_u}$ and ${k_v}$, respectively, from ${-}242.5$ to ${+}242.5\;{\text{m}^{- 1/2}}$, with an interval of $2 \times 242.5/7\;{\text{m}^{- 1/2}}$. This generates 49 points in the reciprocal space (${k_u}$, ${k_v}$), corresponding to 49 kinds of initial light states. These states were injected into the fiber center (a typical fiber length was chosen as $0.75p$). The output beams were then deflected to different positions, as shown in Fig. 6. The experiment agrees approximately with the theory. With these conditions, we achieved a scanning area of $200\,\,{\unicode{x00B5}\text{m}} \times 200\,\,{\unicode{x00B5}\text{m}}$, by adjusting ${k_u}$ and ${k_v}$. We can increase the scanning spatial resolution by reducing the interval of ${k_u}$ and ${k_v}$. Note that the resolution is limited by the pixel size of the SLM used in the experiment. Since the values of ${k_u}$ and ${k_v}$ are programmably controlled, we can achieve a large-area dynamical scanning technology, which is not subjected to the diffraction-induced constraint.

Fig. 6. Theoretical (blue) and experimental (red) results for light beam deflection. To obtain these results, we sweep the values of ${k_u}$ and ${k_v}$ from ${-}242.5\;{\text{m}^{- 1/2}}$ to ${+}242.5\;{\text{m}^{- 1/2}}$, with an interval of $2 \times 242.5/7\;{\text{m}^{- 1/2}}$, hence generating 49 kinds of initial states of light. The output beam is deflected from the center to different positions, covering an area of $200\,\,{\unicode{x00B5}\text{m}} \times 200\,\,{\unicode{x00B5}\text{m}}$. The associated parameters in the experiment are chosen as ${\sigma _0} = 14\,\,{\unicode{x00B5}\text{m}}$, ${x_0} = 0$, and ${y_0} = 0$.

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3. CONCLUSION

In conclusion, we have presented for the first time pendulum-type structured light beams that are nondiffracting and shape-preserving during propagation along trajectories that can be arbitrarily controlled in (2 $+$ 1)D configurations. We realized this new type of beam based on the paraxial Schrödinger wave equation, which is perturbed by the harmonic potential. A general theoretical solution for the pendulum-type light beam was presented and confirmed by experiments. We have shown that any initially structured light beam (e.g., Gaussian beam, vortex beam, or vector beam) with appropriate linear momentum can accelerate along a certain trajectory governed by a general equation of an ellipse. According to this formulism, we have revealed intriguing propagation dynamics of pendulum-type light beams including nondiffracting and shape-preserving propagation, and orbital motions. Apart from these particle-like properties, we also revealed their wave-like properties, which are manifested by the interference effects between two pendulum-type beams. We emphasize that there is a strong interest in investigations of nondiffracting and accelerating beams. Regarding this aspect, we believe that the results presented here bring to attention a new class of interesting light beams. It hence opens new possibilities for generating complex states of nondiffracting beams such as a state that simultaneously contains intrinsic and extrinsic orbital angular momenta. Potential applications of pendulum-type beams are expected, e.g., in the field of laser scanning. We suggest that, owing to the similarities between wave equations in quantum mechanics, optics, acoustics, etc., [30,57], the concept of pendulum-type beams can be readily extended to other waves such as quantum waves [58], matter waves [59], and acoustic waves [60], providing possibilities for the study and potential applications of pendulum-type wave packets in a wide range of fields.

Funding

Guangdong Provincial Pearl River Talents Program (2017GC010280); National Natural Science Foundation of China (11974146, 62175091); Guangzhou Science and Technology Program Key Projects (202201020061).

Acknowledgment

The authors thank Dr. Yaoyu Cao from Jinan University for help with experiments.

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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Pendulum-type light beams

Abstract

1. INTRODUCTION

2. RESULTS AND DISCUSSION

3. CONCLUSION

Funding

Acknowledgment

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (6)

Equations (9)

Optica