## Abstract

Nondiffracting and shape-preserving light beams have been extensively studied and were shown to exhibit intriguing wave phenomena, which led to applications including particle manipulation, curved plasma channel generation, and optical superresolution. However, these beams are generated by caustics, i.e., conical superposition of waves. As a result, they tend to propagate along a straight line or accelerate in a plane [(1 $+$ 1)D], and tailoring their propagation trajectories in a higher dimension [(2 $+$ 1]D] is challenging. Here we report both theoretically and experimentally a class of nondiffracting solutions to the paraxial wave equation perturbed by harmonic potential. We demonstrate that the initial wave packets of light can be engineered to accelerate along an arbitrary trajectory centered on an elliptic or a circular orbit in a (2 $+$ 1)D configuration, while maintaining their phase and polarization structures during propagation. Such particle-like features manifested by orbital movements can be attributed to the centripetal force of the underlying potential. We name such oscillating wave packets as pendulum-type beams. We suggest the concept can be generalized to other waves such as quantum waves, matter waves, and acoustic waves, opening possibilities for the study and applications of the pendulum-type wave packet in a wide range, e.g., it may be utilized in the field of laser scanning technology.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 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]

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]

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

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]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.

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.

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

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

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]:

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).

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.

## 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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