1. Introduction

There is growing interest in ST wave packets, in which the spatial and temporal properties are mutually coupled [1,2]. Such packets can have unique propagation properties, including controllable group velocities [3], anomalous refractions [4], and other pathways for structuring optical fields [5–7].

Recently, another aspect of interest is the amount of OAM [8,9] that the generated ST wave packet carries [10–12]. In this regard, the LG modal basis is one form of spatial modes ($\ell \ne $ 0) carrying OAM. For LG modes, $\ell $ represents the number of $2\pi $ phase front changes along the azimuthal direction (azimuthal index), and p is related to the number of concentric rings (radial index). The LG modal basis forms a complete two-dimensional set and therefore can be used to decompose an arbitrary beam [13,14]. However, using a single frequency carrying either one OAM mode or multiple LG modes, the resulting monochromatic OAM beam has a static beam radius at each axial distance [5].

Beyond a beam that has a static amplitude and phase profile at some propagation distance, it is possible to create an ST wave packet that has a “dynamic” spatial profile [5,6,15]. To generate a dynamically varying ST wave packet, multiple spatial modes can be coherently superimposed at multiple frequencies. Previous reports have shown dynamic beam motions, such as (i) a Gaussian-like beam dot moving along one of the transverse axes [16], (ii) a light beam without carrying OAM rotating along its azimuthal direction or having in-and-out movement in the radial direction [17], and (iii) dynamic ST beams with two independent and controllable orbital-angular-momenta [15]. A laudable goal would be to generate ST wave packets with temporal changes of beam radius in time, as another type of dynamic motion, by exploiting the coherent combination of multiple spatial modes on multiple frequencies [18,19]. As a result, the temporally dynamic interference among the spatial modes could result in the generation of dynamic ST wave packets possessing time-dependent beam radii [15,20,21].

In this article, we demonstrate, through both simulation and experiment, OAM-carrying ST wave packets and having a controllable dynamically varying beam radius [18,19]. In our approach, each frequency line is spatially modulated with a combination of LG modes with the same $\ell $ value and different p values. The beam radius variations can be adjusted by manipulating the loaded modes on each frequency line. In the simulation, we investigate the effect of the number of frequency lines and LG modes, as well as that of the propagation distance on the radius of the generated OAM-carrying ST wave packet. The simulation results show that by coherently combining more spatial modes and frequency lines the dynamic range of oscillations along the radial axis can be increased. The simulated ST wave packet ($\ell $ = + 1 or +3) has an OAM purity of (i) > 95% when the propagation distance is 0, and (ii) ∼50% when the propagation distance is half the Rayleigh range. In the experiment, (i) six frequency lines from a Kerr frequency comb are spatially modulated with different spatial patterns to carry the coherent superposition of multiple LG modes, and (ii) the complex beam profiles (i.e., intensity and phase) at different time instants are captured by using off-axis digital holography and tuning the delay between the ST wave packet and a reference pulse. In the experiment, the generated ST wave packet ($\ell $ = + 1 or +3) has an OAM purity of (i) > 64% when the propagation distance is 0, and (ii) ∼20% when the propagation distance is half the Rayleigh range. The results indicate that the generated ST wave packet would attain larger beam radii at longer propagation distances.

2. Concept of generating OAM-carrying ST wave packets with a time-dependent beam radius

The concept for generating OAM-carrying ST wave packets that exhibit a time-dependent beam radius is depicted in Fig. 1. Using one OAM mode at a single frequency provides only a temporally fixed beam radius, as shown in Fig. 1(a). By combining multiple LG modes with the same beam waist but multiple higher p values, the resultant static OAM beam can be shaped or steered [Fig. 1(b)]. However, the relative phase difference ($\Delta {\varphi _p}$) between neighboring LG modes having different radial indices (i.e., $\Delta p\; \ne $ 0) will not be time-dependent. Thus, the beam remains static in time at a given distance and the oscillation rate of the intensity profile along the radial axis (${v_R}$) is zero when only one frequency is used [5,17].

 

Fig. 1. (a) Using only one frequency line and one OAM mode generates a beam of a beam radius invariant over time. (b) Using a single frequency and multiple modes generates a continuous wave beam with a tailored radius. The relative phase difference of neighboring LG modes with $\Delta p \ne $ 0 ($\Delta {\varphi _p}$) would be time-independent, thereby leading to a static beam radius. (c) Combining frequency lines carrying multiple spatial modes generates an OAM-carrying ST wave packet with a time-dependent beam radius at a given propagation distance. The time-dependent $\Delta {\varphi _p}$ can cause the intensity of the generated ST wave packet to oscillate with the oscillation rate of ${v_R}(t )$. Also shown are the larger and smaller beam radii due to interference at times ${t_1}$ and ${t_2}$, respectively. Tx: transmitter aperture; Rx: receiver aperture.

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In general, an OAM-carrying ST wave packet can be generated when using multiple frequency lines, each carrying a designed spatial pattern. This coherent combination of multiple frequency lines and modes can be expressed as follows:

To create an OAM-carrying ST wave packet with a time-dependent beam radius, as shown in Fig. 1(c), each line is assigned to a designed spatial pattern. These patterns have (i) a Hermite distribution with the intensity along their radial axis identical to that used in [16], and (ii) a helical phase profile corresponding to the desired OAM order. Such a spatial pattern can be generated by a combination of LG modes. The complex coefficients ${\textrm{C}_\textrm{p}}$ are obtained by calculating the field overlap between the LG modes and the resultant spatial pattern. To tailor the beam radius at different times, the frequencies ${f_i}$ carrying different patterns are combined with coefficients ${\textrm{C}_\textrm{i}}$, which obey the Poisson formula [16]. The electrical field of an OAM-carrying ST wave packet with a time-dependent beam radius can be expressed as follows:

 

Fig. 2. (a) Intensity and (b) phase profiles of the OAM-carrying ST wave packets ($\ell $ = + 1 or −3) when using 1, 8, and 16 frequency lines with a frequency spacing of 0.2 THz, 4 mm as ${w_0}$, at $0.3\; {z_R}$, (see Visualization 1 for the real-time video of the intensity and phase profiles of the OAM-carrying ST wave packets).

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We note that the generated OAM-carrying ST wave packets with dynamically changing beam size are not diffraction-free, in contrast to previously demonstrated propagation-invariant ST wave packets [1–4]. This is due to the association of each spectral frequency with multiple spatial modes, breaking the one-to-one correspondence necessary for realizing diffraction-free propagation in free space.

It is also worth noting that our generated ST wave packet is different from the reported spatiotemporal OAM beams which are also referred to as spatiotemporal optical vortices (STOV) [23–28]. STOVs typically exhibit a topological charge transverse to the propagation direction (e.g., ($x,t$) domain and orthogonal to the direction of propagation) [2,23,29–31]. However, in our approach, one should capture the generated ST wave packet in the transverse spatial profile (i.e., ($x,y$) plane) to observe a spiral phase.

3. Simulation results

We use a simulation model with a pixel size of 0.3 mm and 400${\times} $400 pixels at the transverse plane and a time duration of 10 ps with a 50 fs time resolution. To analyze the resultant beam at a given distance, propagation distances are normalized to the Rayleigh range. According to the relation of ${z_R} = \pi w_0^2/\lambda $, the Rayleigh range equals 18.25 m (where ${w_0} = $ 3 mm and $\lambda = $ 1550.4 nm, corresponding to the center frequency line of 193.5 THz). Figures 2(a) and 2(b) show the intensity and phase profiles of the simulated ST wave packets at 0.3 ${z_R}$ when: (i) the frequency spacing $\Delta f$ is 0.2 THz, corresponding to an oscillation period of 5 ps; and (ii) each frequency line is assigned to different superpositions of multiple LG modes, all of which have the same $\ell $ value (+1 or −3) but different p values (with $\Delta p = $ 1). In Fig. 2, to observe a clear phase profile, we enlarged the beam waist to 4 mm and only depicted the phase profile in the area where the intensity is within the range of 1/5 of its peak value. As can be seen, there is no oscillation when using a single frequency. However, using multiple frequencies, the ST wave packet attains a dynamic change of its radius over time, which is lowest at half of the oscillation period (i.e., 2.5 ps). The beam radius can also be enlarged using a higher number of frequency lines and modes. In both the simulation and the experiment, the beam radius is defined as the position of the intensity peak along the radial axis (i.e., $ma{x_y}\; {|{E({x = 0,y,{z_0},{t_0}} )} |^2}\; $ at specific propagation distance of ${z_0}$ and time of ${t_0}$). However, the beam radius can also be defined by the second moment radius of the field or the hard aperture of the beams, in which these definitions may lead to different results [32–34]. Furthermore, the direction of the helical phase front rotation is reversed by negating the $\ell $ value, as shown in the fourth row of Figs. 2(a) and 2(b).

The simulated intensity and phase profiles of the OAM-carrying ST wave packets with time-dependent beam radii are shown in Visualization 1. The results are obtained when (i) eight frequency lines with a frequency spacing of 0.2 THz are used; (ii) each line contains 30 LG modes ($\textrm{L}{\textrm{G}_{\ell ,0}}$ to $\textrm{L}{\textrm{G}_{\ell ,29}}$); and (iii) time duration equals 10 ps. This video includes different cases, such as (1) $\ell = \;$+1 and 0 ${z_R}$, (2) $\ell = $ −3 and 0 ${z_R}$, (3) $\ell = $ + 3 and 0 ${z_R}$, and (4) $\ell = $ + 3 and 2.5 ${z_R}$ for the OAM value and the propagation distance, respectively.

Now, we fix the number of spatial modes and frequency lines, while varying the frequency spacing. Figure 3(a) shows the radius amplitude at frequency spacings of 0.2, 0.5, and 0.8 THz. This trace is obtained by (i) using eight frequencies carrying a combination of 24 LG modes (e.g., ${\textrm{C}_i}\mathop \sum \nolimits_{p = 0}^{23} {\textrm{C}_p}\textrm{L}{\textrm{G}_{1,p}}$), and (ii) finding the oscillation path of the intensity peaks in a time duration of 10 ps and at $z = $ 0.3 ${z_R}$. In all cases, the beam oscillates around a fixed radial point equal to 32.5 mm (i.e., the beam radius when a single frequency is used). From the results, it can be inferred that the oscillation rate increases with frequency spacing. This is due to the relative phase delay of $2\pi \Delta ft$ between neighboring spatial patterns, which accompanies an oscillation rate of $\Delta f$ (i.e., 1/${\textrm{T}_\textrm{r}}$) [15].

 

Fig. 3. (a) The amplitude radius for the frequency spacings of 0.2, 0.5, and 0.8 THz when using eight frequencies, each of which carries 24 spatial modes ($\textrm{L}{\textrm{G}_{1,0}}$ to $\textrm{L}{\textrm{G}_{1,23}}$). (b) The dynamic range over different frequency lines and modes. (c) The radius when using (i) (8, 24), (ii) (12, 30), and (iii) (16, 40) frequency lines and LG modes, respectively. The results correspond to the simulation of an OAM-carrying ST wave packet ($\ell ={+} 1$) at $0.3\; {z_R}$ propagation distance.

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We also explore the effect of varying the number of frequency lines and modes in Fig. 3(b). The dynamic range of the beam radius, defined as the ratio of the maximum achievable radius to the minimum one, is used as the figure of merit. Here, as an example, we compare the dynamic range of the generated OAM-carrying ST wave packets with $\ell $ = + 1, which oscillate around the same point ${\rho _0}$ (∼ 32.5 mm) in the radial direction. As expected, for one frequency line, the dynamic range is one because there are no variations. Given the same number of modes, the dynamic range also increases when more frequency lines are used. Given the same number of frequency lines, decreasing the number of modes from 40 to 20 drops the dynamic range. This might be due to the limited spot size of the lower-order LG modes, which limits the ability to decompose the desired spatial patterns with larger radii, thereby reducing the dynamic range [14,32].

Figure 3(c) shows the radius of the simulated OAM-carrying ST wave packet with $\ell $ = + 1 in time. Here, the frequency spacing equals 0.2 THz, and the number of frequency lines and LG modes is (i) (8, 24), (ii) (12, 30), and (iii) (16, 40), respectively. The results indicate that the ST wave packet could attain higher dynamic ranges by using more frequency lines and LG modes in combination.

Since OAM-carrying ST wave packets are not diffraction-free, in Fig. 4, we characterize the free-space diffraction of the generated ST wave packets leading to beam divergence. In Fig. 4(a), the oscillation of intensity peaks is found when: (i) $\ell $ = + 1, (ii) using eight frequency lines, each carrying 30 $\textrm{L}{\textrm{G}_{1,p}}$ modes, and (iii) at 10 ps. Different propagation distances in the range of {0, 0.2, 0.5, 0.8, and 1}×${z_R}$ and {1.5, 2, and 2.5}×${z_R}$ are considered to analyze the near-field and the far-field propagations, respectively. The intensity and phase profiles of the generated ST wave packets at time $t = 0$ are also shown in Fig. 4(b). In the near-field, the beam maintains its dynamic range; however, the instant of occurrence of the max beam intensity changes. Furthermore, at longer propagation distances, the dynamic range of beam radius is higher than its near-field counterpart. However, the OAM purity, which will be explained later, could be lower compared to the case of using a higher number of frequencies and modes. The results could be explained by: (i) In the near-field, the same LG mode carried by different frequency lines might experience similar diffraction. Thus, the generated ST wave packet can be approximated by an offset ST wave packet in time; and (ii) In the far-field, likely due to the increased divergence of the higher-order LG modes used in the synthesis and higher difference of diffraction effect for the same LG modes at different frequencies, the generated ST wave packet could be distorted [14,32,35].

 

Fig. 4. The simulated OAM-carrying ST wave packet’s (a) beam radius over a time duration of 10 ps and at different propagation distances of {0, 0.2, 0.5, 0.8, 1, 1.5, 2, and 2.5}×${z_R}$ when $\ell $ = + 1, and (b) intensity and phase profiles at {0, 1, and 2.5} ×${z_R}$ when $\ell $ = + 1. (c) OAM purity of the generated OAM-carrying ST wave packets with $\ell $ = + 1 and +3 in a logarithmic scale for different propagation distances and frequency spacings (FSs) of 0.2 and 2 THz. The results correspond to the coherent combination of eight frequencies each carrying 30 LG modes.

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The effect of free-space propagation and frequency spacing on the purity of the generated ST wave packet is also investigated as shown in Fig. 4(c). To evaluate the OAM purity, (1) we calculate the normalized power weight coefficients ${|{{C_{\ell ,p}}} |^2} = {\left|{\mathrm{\int\!\!\!\int }{E_{\textrm{ST}}}({x,y,z} )\;E_{\textrm{LG}}^\mathrm{\ast }({x,y,z} )\;dxdy} \right|^2}$, where ${E_{ST}}({x,y} )$ is the normalized electrical field of the simulated OAM-carrying ST wave packet at $t$ = 0 ps, ${E_{\textrm{LG}}}({x,y} )$ is the normalized electrical field of a LG beam at the same propagation distance, and * denotes the complex conjugate operation; and (2) subsequently the modal weights ${|{{C_{\ell ,p}}} |^2}$ are summed up over different radial indices (e.g., ${|{{C_1}} |^2} = \mathop \sum \nolimits_{p = 0}^{29} {|{{C_{1,p}}} |^2}$ when $\ell $ = + 1), which is taken to represent the OAM purity. The results show that OAM purity decreases when the frequency spacing increases. Additionally, the OAM purity is ∼50% around $2.5{z_R}$. The reduction in OAM purity when increasing the (i) frequency spacing or (ii) propagation distance might be blamed for the increasing differential divergence among the different frequency lines carrying spatial patterns [15,32,36]. These effects lead to different distortion effects in their phase and amplitude when synthesizing the modes, thereby lowering OAM purity [14,15,36].

4. Experimental results

The experimental setup for the generation and detection of such ST wave packets is illustrated in Fig. 5(a). A single-soliton Kerr frequency comb with a frequency spacing of ∼192 GHz is used to generate multiple frequency lines by coupling a continuous-wave pump to an integrated microresonator [37,38]. A waveshaper is used to (i) select six comb lines, and (ii) adjust the amplitude and phase of the frequency lines to minimize the pulse width of the output pulse. The input and output optical spectra of the waveshaper are shown in Fig. 5(b) and (c), respectively. To create a reference pulse, which is used to measure the complex profiles of the generated ST wave packets at different times [11,39,40], the output of the collimator is split into two paths using a beam splitter (BS). One path is used to generate ST wave packets in which an optical grating diffracts each comb line to a different location on the first spatial light modulator (SLM). Six different phase patterns are loaded on these different locations on SLM1 according to the designed complex-weighted combinations of LG modes on each frequency. The location of each phase pattern is tuned to align the spatial location of each frequency such that a relatively high mode purity is achieved. The accuracy of the alignment could be related to the pixel sizes of the SLMs used in the experiment [41]. Each location imparts a prescribed phase pattern that creates a superposition of seven LG modes, which share the same $\ell $ index but have different p indices (from 0 to 6), on each frequency line. For instance, the phase patterns loaded on SLM1 corresponding to the fifth and sixth frequency lines for generating the ST wave packet with $\ell ={+} 1$ are shown in Fig. 5(d). The loaded phase pattern on the SLM1 for each frequency is a combination of (i) the beam phase profile and (ii) a grating pattern whose diffraction efficiency is designed at different positions according to the beam amplitude profile [42]. Therefore, the phase pattern appears to be a fringe pattern. The grating period of each phase pattern on SLM1 is tuned such that the output frequencies are directed to the same position on SLM2. The beam waist of LG modes is ∼0.3 mm. Note that by (i) reducing the beam waist, the SLM pixel size may limit the generated mode purity, and (ii) increasing the beam waist, the higher-order LG modes would be larger than the Gaussian input beam, which can be challenging to modulate and combine them. After spatial modulation, the comb lines are coherently combined using SLM2. In our experiment, the phase pattern on SLM2 is designed based on a combination of six grating patterns. The grating periods of these six patterns are tuned such that the output directions of the frequencies after SLM2 are the same.

 

Fig. 5. (a) Experimental setup for the generation and detection of ST OAM-carrying wave packets. The optical spectrum at the (b) input and (c) waveshaper, respectively. (d) Phase patterns loaded on SLM1 correspond to the fifth and sixth frequency lines. AFG: arbitrary function generator, ECL: external cavity laser, FS: frequency spacing, LF: lensed fiber, EDFA: erbium-doped fiber amplifier, PC: polarization controller, Col.: collimator, BS: beam splitter, SLM: spatial light modulator.

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To capture the complex profiles (i.e., amplitude and phase) of the generated ST wave packets in the transverse plane, the off-axis holography technique is used [39,40]. We adjust the delay stage to change the delay between the reference pulse and the generated ST wave packet for recording the profiles at different time instants. After the delay stage, an off-axis Gaussian pulsed beam (i.e., the reference pulse that contains the same six generated frequency comb lines) interferes with the generated ST wave packet after being combined at another BS. This interference is recorded on the infrared camera. We apply digital processing to retrieve the complex profiles [39,40].

Figures 6 show the intensity and phase profiles, the estimated beam radius, and the OAM purity of experimentally generated ST wave packets, respectively. The three cases considered for the azimuthal index and the propagation distance are as follows: (a) { $\ell $ = + 1, $z\; \sim $ 0${z_R}$}; (b) {$\ell $ = + 1, $z\; \sim $ 0.5${z_R}$}; and (c) { $\ell $ = + 3, $z\; \sim $ 0${z_R}$}. The Rayleigh range equals ∼18.1 cm for the center frequency line. The intensity profiles indicate that the radius oscillates with an oscillation period of ∼5 ps, which corresponds to the frequency spacing of the comb source (i.e., ∼192 GHz). Moreover, the helical phase rotation follows the OAM order ($\ell $), as shown in Figs. 6(a1,b1,c1). We note that in the simulation results in Figs. 2(b), ring-shaped apertures are manually applied to the phase profiles according to the corresponding intensity profiles. In the experimental results in Figs. 6(a1,b1,c1), circular apertures with a fixed size corresponding to the maximum beam radius are applied to the phase profiles as the corresponding intensity profiles have some undesired rings due to imperfections. The experimentally measured intensity profiles of the OAM-carrying ST wave packets tend to have one main ring having most of the power as shown in the simulation and other rings having some residual powers. This might be due to the experimental artifacts and various nonidealities such as that (i) the effective area on the SLM modulating the input beam is limited, which might result in a limited aperture size on the output beam and consequently, modal coupling to other p modes [35,36] and (ii) the complex coefficients loaded on each frequency comb lines when synthesizing LG modes to experimentally generate such OAM-carrying ST wave packet might be different from the designed values in the simulation [43,44]. These effects could induce an imperfect combining of spatial modes, which could lead to undesired intensity rings at some positions [44].

 

Fig. 6. (a1-c1) Intensity and phase profiles, (a2-c2) the estimated beam radius along with the simulated radius, and (a3-c3) the OAM purity of the generated OAM-carrying ST wave packets. In (a) $\ell = $ + 1 and at 0 ${z_R}$, (b) $\ell = $ + 1 and at 0.5 ${z_R}$, and (c) $\ell = $ + 3 and at 0 ${z_R}$, respectively. All results are measured in a time duration of 5 ps.

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In Figs. 6(a2,b2,c2), the time dependence of the beam radius is investigated both experimentally and computationally. In this Section, the beam waist is chosen as 0.3 mm in the experiment as well as the simulation. The beam radius oscillates from ∼0.24 mm to ∼0.68 mm. By comparing Figs. 6(a2) and 6(b2), we conclude that the beam radius increases with distance, as explained earlier. The deviation of the measured beam radius with respect to the simulation values is likely due to the imperfect mode generation by the SLM1. The beam radii of the generated ST wave packet could have some error due to the power and phase variations of the six frequency comb lines. The measurement error could be related to the (i) temperature drifts and mechanical vibrations of the fiber, which could result in varying phases and polarizations of different frequency lines [45]; (ii) the relative phase noise between the generated frequency lines of the comb source, which could result in a random phase shift between the frequency lines [46]; and (iii) the temperature drift of the microresonator and the frequency drift of the pump laser, which could change the state of the generated comb [47].

OAM purity values are plotted in Figs. 6(a3,b3,c3) using the approach illustrated in Sec. 3. The variations in measured OAM purity might be due to the phase variation at given times, which are shown in the second and fourth columns of Fig. 2, and this phase distortion reduces OAM purity. OAM purity values are: (i) ∼64% to ∼86% for $\ell = $ + 1 and +3 at z ∼ 0, and (ii) ∼20% to ∼84% for $\ell = $ + 1 at ∼ 0.5${z_R}$. OAM purity values are lower: (i) at larger distances, and (ii) compared to the simulation results in Fig. 4. The relatively low OAM mode purity might originate from the limited pixel resolution of both SLM1 and SLM2, which affects the quality of the synthesized spatial modes [48,49]. The OAM purity can be potentially increased by synthesizing a larger number of LG modes including higher-order p modes.

OAM-carrying ST wave packets with a time-dependent beam radius are not limited to LG modes [12]. Other types of spatial modes (e.g., Bessel-Gaussian $\textrm{B}{\textrm{G}_{\ell ,{k_r}}}$ modes with $|\ell |\ge \;$1) can carry OAM as well [50–53]. Therefore, our approach could potentially be used to generate such wave packets with similar time-dependent properties by combining other spatial modes and frequencies.

We note that in our demonstration, the topological charge of the OAM-carrying ST wave packet is $\ell $ = + 1 or +3. We could also choose other topological charges (e.g., $\ell $= + 2 instead of $\ell $ = + 3). However, choosing higher-order topological charges (e.g., $\ell $ > + 3) for OAM-carrying ST wave packets may lead to a larger beam size due to increased divergence of the LG modes [35,36].

5. Summary

We have demonstrated the generation of an OAM-carrying ST wave packet with a time-dependent beam radius using both simulations and experiments. In this approach, the interference resulting from the coherent combination of LG modes, which are superimposed on multiple frequency lines, can lead to beam-radius oscillation in the radial dimension. Increasing the number of frequency lines and LG modes can accommodate achieving oscillations with a higher dynamic range. The OAM purity of the generated OAM-carrying ST wave packet drops with propagation distances. This paper demonstrates the ability to synthesize OAM-carrying ST wave packets such that the structured beam radius can independently be tailored to dynamically change at a given propagation distance. In terms of dynamic beam control, this represents another capability that is in addition to previously published control of specific characteristics of dynamic ST wave packets [12,15,43]. According to Refs. [12,54,55], we note that structured beams of different radii can have various applications, e.g., sensing, particle manipulation, and lithography. Although our paper demonstrates tailorable control of structured beam radii, we imagine that there are various applications that have yet to be identified.

The number of modes and frequency lines used in the experimental demonstration limits the ability to generate OAM-carrying ST wave packets with higher OAM purity and dynamic ranges of the beam-radius oscillations. However, the OAM purity and this dynamic range can be potentially improved by using SLMs having a (i) higher resolution, which is related to the quality of generated LG modes [49]; and (ii) larger aperture size to modulate more frequency lines and extend the number of LG modes synthesized on each frequency line [56].