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Abstract
We introduce controllable Laguerre Gaussian wave packets (LGWPs) with self-accelerating and self-focusing properties along their predesigned parabolic trajectory via phase modulation. Numerically and experimentally recorded intensity patterns of controllable LGWPs with topological charges are obtained, and it is obvious that they agree well with the theoretical model. Furthermore, spatiotemporally controllable LGWPs can propagate stably along predesigned trajectories for many Rayleigh lengths. This paper not only provides a theoretical propagation model for these multi-dimensional controllable LGWPs, but also promotes further development of the basic research into self-bending and autofocusing structured light fields.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, the study of novel multi-dimensional structured light fields has become a research hotspot in the physics field [1–3]. Berry and Balazs first proposed a solution to the Schrödinger equation using the Airy wave packet form [4]; then, Baumgartl et al. demonstrated that an Airy beam would drag a particle to the maximum strength of the main lobe and guide this particle along a parabolic trajectory [5]. By changing the vertical position of the phase diaphragm on the incident Gaussian beam and the space light modulator, Hu et al. regulated the height, distance, and the peak power position of the beam transmission orbit to achieve optimal control of the Airy beam trajectory transmission [6]. Goutsoulas et al. discussed precise control of the amplitude and trajectory of a self-accelerating circular Airy beam through amplitude and phase modulation [7]. Self-accelerating and quasi-nondiffracting wave packets are used widely in optical communication and light-matter interactions [1], which means that these results are highly important for applications involving beam precise control.
Additionally, abruptly autofocusing beams are special types of wave packets that can focus their power spontaneously [8]. Initially, the beam maintains a low intensity profile during propagation before the intensity increases abruptly by several orders of magnitude at the focal point; subsequently, the intensity decreases again. For example, a circular Airy beam [9], which is cylindrically symmetrical with an Airy radial profile, can realize abrupt autofocusing in linear media. Recently, spatiotemporal sharply autofocused dual-Airy-ring Airy Gaussian vortex wave packets were reported [10]. Circular Pearcey beams [11], which represent another type of circularly symmetrical beam with a Pearcey radial profile, have newly been proposed as abruptly autofocusing beams. Circular Pearcey Gaussian waves have been demonstrated numerically in a Kerr medium [12]; by varying the initial waves and the input power, both the peak intensity and the light filament can be controlled. Autofocusing beams of this type have potential applications including particle manipulation and controlled filament generation at specific spatial locations [12–15].
When an optical spatiotemporal wave packet propagates in a uniform medium, the wave packet will naturally expand in space and time because of the combined actions of diffraction and dispersion [16–20]. Over the past few years, researchers have used so-called "light bullets" to suppress diffusion of the packet’s energy [21–23]. Optical nonlinearity can compensate well for both dispersion and diffraction, thus allowing the shape of the pulse beam to be maintained over long propagation distances. A specific combination of parameters for the propagation medium and the optical wave packet is required when using nonlinear techniques, which cannot be tuned easily with respect to a wide range of parameters [1]. More recently, as a result of the mathematical similarity between the diffraction concept in the spatial domain and the influence of the second-order dispersion in the temporal domain, researchers have also considered the quasi-diffraction-free Airy function to be applicable to dispersion systems [24,25], thus providing a new opportunity to obtain spatiotemporal wave packets with self-accelerating properties. The spatiotemporal optical wave packets that are obtained from a self-accelerating Airy pulse beam [25] also provide an opportunity to obtain linear optical bullets with new features. This type of spatiotemporal wave packet has potential applications that include optical communication, highly localized optical detection, and spatiotemporally-resolved spectroscopy [1,26]. Therefore, acquisition of a controllable multi-dimensional (phase, amplitude, pulse width, and mode field) self-accelerating structured light field and discussion of the regulation and propagation of such a field have become hot research topics in optics.
Although self-accelerating and self-focusing wave families have previously been discussed rather independently, it would be interesting to combine the features of these waves to enable the design of new wave packet types [27]. One possibility would be to design wave packets carrying topological charges [28] with both self-acceleration and autofocusing properties that are capable of resistance to diffraction over many Rayleigh lengths [29]. Importantly, it would be desirable to control the propagation trajectory of these packets beyond the parabolic law. However, a few recent works have pointed toward this direction by proposing techniques to create Laguerre Gaussian wave packets (LGWPs) with predesigned trajectories. In addition, the research into controllable transmission of spatiotemporal light fields by adjustment of their optical parameters also needs to be expanded [30,31]. In this paper, LGWPs that are controllable along a predesigned trajectory in a linear medium using phase modulation are introduced.
This paper is organized as follows. First, in section 2, we investigate the theoretical model of controllable LGWPs that propagate along predesigned trajectories in both the spatial and temporal dimensional configurations. Then, in section 3, the dynamics of these controllable LGWPs, which have both self-accelerating and self-focusing properties along their predesigned trajectories, are discussed in detail. Using computer-generated holograms, numerical and experimental results are obtained for the controllable LGWPs. The changes in the energy flow that occur during propagation are shown. The quadratic parameters that affect self-focusing for the different vortex orders of the wave packets are discussed. Additionally, spatiotemporally controllable LGWPs can propagate stably along predesigned trajectories for many Rayleigh lengths are shown. Finally, we present our conclusions about the work in section 4. This research reveals the relationship between the modulated phase and the propagation dynamics of a novel structured optical field.
2. Theoretical model of controllable LGWPs
The propagation of spatiotemporal wave packets in linear medium can be described by the (3+1)D free-space Schrödinger equation [18], we equalize the diffraction and dispersion effects to simplify the analysis [24], in the dimensionless coordinates, it can be written as
The initial spatiotemporal LGWPs can be expressed as $U(X_0,Y_0,T_0)=M(X_0,Y_0)A(T_0)$. We consider the initial spatial field distribution of phase-modulated LGWPs [29,32] in the input plane as
Without loss of generality, we consider the initial temporal field distribution of phase-modulated zeroth order Laguerre Gaussian pulse in the input plane as
Similar to the spatial part, we set the $\tau$-direction predesigned parabolic trajectories as: $h(Z)= Z^2/\alpha _{\tau }$, $\alpha _{\tau }$ is the temporal quadratic term ratio of a parabolic. Refer to the spatial part processing method, the temporal part results can be obtained by reducing one dimension. Now, we focus on the propagation dynamic characteristics of the controllable LGWPs.3. Discussion
3.1 $2D$ controllable LGWPs along predesigned trajectories
In general, a single-ring Laguerre-Gaussian beam is a vortex beam with the simplest possible optical field structure that has zero light intensity in its central area. First, we theoretically simulated the transmission of single-ring LGWPs carrying two topological charge numbers with $m=2$, $n=0$. In Fig. 1, the properties of the LGWPs that are modulated by the special phase $Q_{s}(X,Y)$ are shown. Figure 1(a) depicts that the spatial phase distribution is axisymmetric. On the one hand, in Fig. 1(b), the evolution of the overall propagation trajectory of the modulated LGWPs follows a parabolic trajectory, indicating self-bending (self-accelerating), which is unlike the linear trajectory of traditional Laguerre Gaussian beams when propagating in free space. On the other hand, unlike the self-focusing effects mediated by the Kerr nonlinearities, the autofocusing behavior shown in Fig. 1(b) is purely linear in origin and is a result of the optical field structure itself. The corresponding transverse intensity distributions shown in Figs. 1(c1)–(c4) are indicated by the dotted lines in Fig. 1(b). In the initial plane, as shown in Fig. 1(c1), there is a ring modulated by the phase $Q_{s}(X,Y)$ with equiphase surfaces that are single-ring type. Figures 1(c2)–(c4) show that the light field is composed of a central vortex ring that surrounds some side-lobe rings with increasing $Z$, similar to the distributions of higher-order Bessel functions, but also show that the field is gradually bending. The vortex ring accelerates along the parabolic orbit toward the $X$ direction. During this acceleration, the central region of the LGWPs retains their quasi-nondiffracting characteristics.
Fig. 1. Numerical simulation of controllable LGWPs ($n=0$, $m=2$) propagating along a parabolic trajectory by using fast-Fourier-transform method. (a) The spatial phase at the initial plane. (b) The side-view propagation on $y=0$. (c1)–(c4) The corresponding transverse intensity distributions taken by the white dotted lines in (b), respectively. $\alpha _{s}=10$.
We investigate the propagation properties of these controllable LGWPs experimentally using the setup presented in Fig. 2. A Gaussian beam with $\lambda =532 nm$ propagates toward a reflective spatial light modulator (SLM). This SLM then transforms the Gaussian beam into the complex field expressed by Eq. (5) using a computer-generated hologram with the desired phase information encoded into it. This transformed beam is then imaged on a charge-coupled device (CCD) camera after passing through a $4f$ system to show the controllable LGWP patterns.
Fig. 2. Experimental setup for generating the controllable LGWPs. SLM, spatial light modulator (Santec SLM-200, 1900$\times$1200 pixels); CA, circular aperture; CCD, charge coupled device camera (BeamPro 11.11).
To verify the characteristics shown in Fig. 1, we compare the numerical and experimental results for holographic generation of the modulated LGWPs, which are presented in Fig. 3. We obtain the computer-generated hologram in Fig. 3(a) by computing the interference pattern between the complex amplitude profiles of the beams at the initial plane and a plane wave, as shown in Fig. 3(b), which will then be used to simulate the reflection to enable numerical reconstruction of the information from the encoded beams using a typical $4f$ system with numerical spatial filtering. Subsequently, the trajectory is shown in Fig. 3(c), where the side-view propagation image is in good agreement with that shown in Fig. 1(c). The corresponding numerically (Figs. 3(d1)–(g1)) and experimentally (Figs. 3(d2)–(g2)) recorded transverse intensity patterns of the controllable LGWPs are then obtained; these patterns are obviously in accordance with the findings presented in Figs. 1(c1)–(c4). Therefore, the validity of the analytical model has been demonstrated convincingly.
Fig. 3. By using computer hologram, the numerical and experimental results of controllable LGWPs are obtained. (a) Computer-generated hologram. (b) Interference intensity of the initial generated beam and a plane wave. (c) Numerical recorded normalized transverse intensity trajectory. (d1)–(g1) Numerical and (d2)–(g2) experimental recorded corresponding transverse intensity patterns taken by the white dotted lines in (c).
In addition, to determine more of the propagation properties of these wave packets, the energy flow must also be discussed. The propagation properties of electromagnetic fields are closely related to their local energy flow [33,34], which can generally be expressed in terms of the Poynting vector. Given the polarized vector potential $M(x,y,0,z)e^{-ikz}{\bf {e}}_{x}$, where ${\bf {e}}_{x}$ is the unit vector oriented along the $X$ direction, the time-averaged Poynting vector for a linear polarized field can then be written as $\langle {\bf {S}}\rangle =\frac {c_{0}}{4\pi }\langle \bf {E}\times \bf {B}\rangle$, where $c_{0}$ is the speed of light in free space, $\bf {E}$ and $\bf {B}$ are the electric and magnetic fields, respectively. The values in Fig. 4 were computed numerically; the Poynting vector has a magnitude that is expressed as the energy per unit area (or per unit time), and a direction that represents the energy flow direction at any point in the field. The directions and magnitudes of the arrows (shown as blue solid lines) correspond to the directions and magnitudes of the Poynting vector in the transverse plane. By comparing the energy flow distributions in Figs. 4(a)–(c) with the intensities shown in Figs. 1(c2)–(c4), we find that the rotation of the energy flow agrees well with the rotation of the wave packet distributions.
Fig. 4. Poynting vector of the controllable LGWPs at different positions. Other parameters are the same as those in Figs. 1(c2)–(c4).
As a trajectory control parameter, $\alpha _{s}$ plays an important role in changing the bending curvature of the parabolic trajectory and the sharp intensity enhancement at the focal point. To gain deeper insight into the effects of $\alpha _{s}$, we plot the peak intensity distributions of the spatially controllable LGWPs as shown in Fig. 5. Under different vortex order ($m$) conditions, we can see the complete propagation characteristics of the spatial LGWPs for various values of $\alpha _{s}$. Notably, the wave packets are focused abruptly at the focal point, where the intensity is enhanced suddenly. Additionally, larger values of $m$ result in longer autofocus distances. When the controllable LGWPs are carrying vortexes, their energy is split, causing the phenomenon where the peak value is diminished in Figs. 5(b) and 5(c) when compared with that shown in Fig. 5(a).
Fig. 5. The peak intensity distributions of controllable LGWPs along different predesigned parabolic trajectories with (a) $m=0$ (Gaussian), (b) $m=1$ and (c) $m=2$.
3.2 Spatiotemporal controllable LGWPs along predesigned trajectories
From self-accelerating pulse, beam to spatiotemporal wave packets, researchers are paying increasing attention to optical wave packets with novel properties [1]. Controllable LGWPs can also be used in combination with their temporal and spatial field configurations to describe stable multi-dimensional finite energy wave packets.
Here, we discuss the properties of $3D$ wave packets to gain a physical insight into the spatiotemporally controllable LGWPs, which are demonstrated in terms of their two spatially transverse coordinates and the temporal coordinate. Figures 6(a)–(c) provide some examples of spatiotemporally controllable LGWPs ($m=0$) at different propagation distances, where the hole shape remains good for several Rayleigh lengths. Figure 6(d) shows the intensity plot of the $1D$ temporal distribution for different propagation distances. Unlike the Gaussian distribution at $Z=0$ (black dotted line in Fig. 6(d)), the wave packets are self-accelerating in the positive $T$ direction with the main lobe in the middle and the side lobes distributed on two sides, with the intensity clearly being enhanced from $Z=0$ to $Z=4$. Figure 6(e) shows the side-view propagation of the spatial part with $m=0$, where the parabolic trajectory $(f(Z),g(Z))=( Z^2/5,0)$ continues to bend in the $X$ direction. The spatial peak intensity distribution can be obtained as indicated in Fig. 5(a) by the red solid line.
Fig. 6. Spatiotemporal controllable LGWPs ($m=0$) at (a) $Z=2$, (b) $Z=3$ and (c) $Z=4$. $\alpha _{s}=5$, $\alpha _{\tau }=5$. (d) Intensity plot of $1D$ temporal distribution at different propagation distances with $\alpha _{\tau }=5$. The black arrow represents the wave packet self-accelerating along the positive $T$ direction. (e) The side-view propagation with $\alpha _{s}=5$.
In addition, a comparison of the corresponding results for spatiotemporally controllable LGWPs with $m=1$ is depicted in Fig. 7. In Figs. 7(a)–(c), the entire spatiotemporal controllable wave packet propagates smoothly in each case while basically maintaining its profile, with the vortex ring’s main lobes located at the center and the ring side lobes distributed outside the main lobes. The self-accelerating feature of the wave packets in the positive $T$ direction with $\alpha _{\tau }=10$ is shown in Fig. 7(d). The side-view propagation of the spatial part with the vortex directed along a parabolic trajectory is shown in Fig. 7(e). The spatial peak intensity distribution can be obtained as indicated in Fig. 5(b) by the blue dotted line. Although they are not strictly no expanding, these spatiotemporally controllable LGWPs with vortexes could propagate stably for many Rayleigh lengths and could thus play an essential role in multi-dimensional optical communications.
Fig. 7. Spatiotemporal controllable LGWPs with vortex ($m=1$) at (a) $Z=3$, (b) $Z=4$ and (c) $Z=5$. $\alpha _{s}=40$, $\alpha _{\tau }=10$. (d) Intensity plot of $1D$ temporal distribution at different propagation distances with $\alpha _{\tau }=10$. The black arrow indicates the wave packet self-accelerating along the positive $T$ direction. (e) The side-view propagation with $\alpha _{s}=40$.
4. Conclusion
In conclusion, we have investigated the dynamics of controllable LGWPs, which are studied in both the spatial and temporal dimensional configurations. Through phase modulation of conventional LGWPs, a class of self-accelerating wave packets with controllable transmission trajectories and autofocusing intensities has been constructed. Using fast-Fourier-transform method and computer-generated holograms, respectively, the numerical and experimental results are shown to be consistent with the theoretical model, thus providing a convincing demonstration of the validity of the proposed model. The energy flow changes during propagation, with the energy flow rotation agreeing with the distributions of the controllable LGWPs. In addition, the quadratic parameter affects the self-focusing behavior of the different vortex orders of the controllable LGWPs. Using larger $m$ values leads to the increase of the autofocus distance. In the temporal distribution, it is notable that the initial Gaussian envelope gradually turns into the multi-lobe distributions, with the wave pckets self-accelerating along the positive $T$ direction in each case. The trajectory control parameters $\alpha _{s}$ and $\alpha _{\tau }$ play important roles in changing the bending curvature of the parabolic trajectory. This approach provides a way to construct multi-dimensional wave packets along a specific trajectory in which the desired self-accelerating and self-focusing features are combined. Furthermore, different types of spatiotemporally controllable LGWPs propagating along predesigned trajectories are shown. Clearly, these spatiotemporally controllable LGWPs could propagate stably for many Rayleigh lengths with their main lobes being located at the center, while maintaining the external distribution of their ring side lobes.
Our results not only enhance the current understanding of spatiotemporally self-accelerating and self-focusing wave packets in linear media, but also broaden the potential applications of these wave packets. Furthermore, the unique characteristics and the application value of the novel spatiotemporal wave packets presented here provide a further indication of the necessity to explore new types of multi-dimensional wave packets.
Funding
National Natural Science Foundation of China (12004081, 11775083, 11874321, 11947103, 12174122, 12174338, 62175042); Education Department Foundation of Guangdong Province (2018KQNCX136, 2019KTSCX083); Postdoctoral Research Program of Zhejiang Province (ZJ2021034); Talent Introduction Project Foundation of Guangdong Polytechnic Normal University (2021SDKYA142).
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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