Dynamic spatiotemporal beams that combine two independent and controllable orbital-angular-momenta using multiple optical-frequency-comb lines

Novel forms of beam generation and propagation based on orbital angular momentum (OAM) have recently gained significant interest. In terms of changes in time, OAM can be manifest at a given distance in different forms, including: (1) a Gaussian-like beam dot that revolves around a central axis, and (2) a Laguerre-Gaussian (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LG_{\ell ,p}$$\end{document}LGℓ,p) beam with a helical phasefront rotating around its own beam center. Here we explore the generation of dynamic spatiotemporal beams that combine these two forms of orbital-angular-momenta by coherently adding multiple frequency comb lines. Each line carries a superposition of multiple \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LG_{\ell ,p}$$\end{document}LGℓ,p modes such that each line is composed of a different \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document}ℓ value and multiple p values. We simulate the generated beams and find that the following can be achieved: (a) mode purity up to 99%, and (b) control of the helical phasefront from 2π-6π and the revolving speed from 0.2–0.6 THz. This approach might be useful for generating spatiotemporal beams with even more sophisticated dynamic properties.

S tructured light has recently gained increased interest in that it can accommodate the production of uniquely propagating beams of light [1][2][3][4][5] . One particularly interesting aspect is the ability of a structured beam to carry orbital angular momentum (OAM) [6][7][8][9][10][11] . One form of momentum is a simple Gaussian beam dot that can rotate in a circular fashion as it propagates, illuminating a ring shape [12][13][14] ; this OAM is similar to revolution around a central axis. A second form of momentum is a subset of Laguerre-Gaussian (LG ';p ) beams in which the phasefront twists in the azimuthal direction as it propagates 15,16 . The amount of OAM (') is the number of 2π azimuthal phase changes, and p + 1 is the number of concentric rings on the intensity cross section for ' 6 ¼ 0. The beam rotates around its own beam center with a ring-like vortex intensity profile (Fig. 1d, h). This second type of OAM is similar to rotation. Indeed, the earth propagating around the sun exhibits both rotation around its own Earth center and revolution around a solar central axis 17 .
These two manifestations of momentum can occur in space during propagation, but yet the beam's intensity will appear static at any given point of propagation distance [18][19][20][21][22] . This scenario can be made more complex by enabling the generation and propagation of a dynamic spatiotemporal beam, such that the beam simultaneously revolves and rotates in the x-y plane in time at a given propagation distance z. Prior art has produced novel beams by combining different modes not only on the same frequency [18][19][20][21][22] but also on different frequencies [12][13][14][23][24][25][26][27] . This ability to produce different modes on different frequencies can be achieved by the use of optical-frequency combs, which have recently undergone much advancement 28 .
Specifically, it has been previously shown that a light beam can be created to exhibit unique dynamic features [12][13][14][23][24][25][26][27][29][30][31][32][33][34][35] , including the following examples: (a) a Gaussian-like beam dot that exhibits dynamic circular revolution at a given propagation distance by combining multiple frequency lines in which each line carries a different LG ';p mode (different ' but same p) [12][13][14] (Fig. 1e, i); (b) a Gaussian-like beam dot formed from multiple Hermite-Gaussian modes, each at a different frequency, such that the dot can dynamically move up-and-down in a linear fashion at a given propagation distance 23 ; (c) a light beam created by a pair of LG ';p modes with different ' and p values at two different

Frequency
LG 1,0 LG 2,0 LG 3 x y x Central axis Solar central axis

Dynamic rotation
Beam center Fig. 1 Generating a spatiotemporal beam exhibiting both dynamic rotation and revolution. a, b Illustration of a light beam that dynamically rotates around its beam center and revolves around another central axis; this is analogous to the earth orbiting around the sun, exhibiting both rotation around its Earth center and revolution around the solar central axis. c, g A Gaussian beam on a single frequency exhibits no dynamic rotation/revolution. d, h An LG 3,0 beam on a single frequency exbibits only dynamic rotation. e, i Using multiple frequency comb lines, in which each carries an LG ';p mode with a different ' value and the same p value, to generate a Gaussian-like beam dot exhibiting only revolution around a central axis. f, j Using multiple frequency comb lines, in which each carries a superposition of multiple LG ';p modes with one different ' value and multiple p values, to generate an LG 3,0 beam exhibiting both dynamic rotation and revolution.
frequencies, such that it exhibits dynamic rotation around its center but no dynamic revolution around another axis at a given propagation distance 26 ; (d) a combination of multiple frequency lines, each of which carries one LG ';p mode with a different pair of indices ('; p) and can produce a light beam that exhibits dynamic rotation around its center (azimuthal dimension) as well as in-and-out linear radial movement at a given propagation distance 27 ; and (e) a light beam, which is created by driving the high-harmonic-generation of two time-delayed pulses carrying different OAM values, can exhibit dynamic rotation around its center at a time-dependent speed 29 . A laudable goal would be to produce a more sophisticated beam that can dynamically rotate and revolve at a tailorable speed and at a given propagation distance (Fig. 2).
In this paper, we explore the generation of spatiotemporal light beams that combine two independent and controllable orbital angular momenta. This scenario is enabled by using multiple optical-frequency comb lines, with each line carrying a superposition of multiple LG ';p modes containing a different ' value but multiple p values (Fig. 1f, j). As an example, we generate by simulation an LG 3;0 beam with a beam waist of w 0 = 0.3 mm, which exhibits dynamic rotation around its beam center as well as revolution around a central axis with a revolving radius of R = 0.75 mm at a speed of f r = 0.2 THz (Fig. 3). We show via simulation that we are able to control not only the spatiotemporal beam's helically twisting phasefront but also its dynamic, twodimensional (2D) motion of rotation and revolution at a given propagation distance. Specifically, we vary several parameters, including the rotating ' value, revolving speed, revolving radius, and beam waist of the generated spatiotemporal light beams.

Results
Introducing dynamic rotation and revolution. There are different types of dynamic optical beams that can exhibit simultaneously two forms of orbital angular momenta. One example of such beam propagation is an LG '; p beam rotating around its beam center while it also revolves around another central axis. We note that both such dynamic rotation and revolution can be described by the transverse OAM 11 ; however, such transverse OAM can be decomposed into different OAM components, which are related to the rotation around beam's center 15 and the revolution around another central axis 24,25 , respectively. We refer these two momenta related to the two types of motions as two forms of orbital angular momenta in order to distinguish them.
Our goal below is to generate a self-rotating electric field at z = 0 (and more generally at a chosen distance), that also revolves around a central axis, O, distance R from its self-rotating axis at a speed of f rev revolutions per second (or Hz) (i.e., the number of circles per second that the electric field revolves around O). Revolving the electric field at z = 0 of a conventional (i.e., selfrotating and not revolving) LG '; p beam at a speed of f rev with a revolving radius of R, we can obtain a rotating-revolving electric field (see Supplementary Note 6 for more details): We refer to the beam with such a dynamic electric field as a rotating-revolving LG '; p beam (i.e., a beam that exhibits dynamic rotation around its beam center as well as revolution around a central axis). In this equation, ω 0 = 2πf 0 is the angular frequency, w 0 is the beam waist, and LG Cartesian '; p x; y; z; ω 0 ; w 0 ð Þis the electric field in Cartesian coordinates of a conventional LG '; p beam. φ t ð Þ ¼ ω rev t ¼ 2πf rev t represents the revolving angular speed, and (x cos φ t ð Þ À y sin φ t ð Þ + R; x sin φ t ð Þ + y cos φ t ð Þ; 0; ω 0 ; w 0 ) is the coordinate transformation of (x, y, 0; ω 0 , w 0 ) in a reference frame rotating in the transverse plane at z = 0.
The electric field at z = 0 of such a rotating-revolving LG '; p beam can also be described as a superposition of multiple frequency comb lines with each line carrying a unique spatial pattern. It can be written in the form (see the "Methods" section and Supplementary Note 6 for more details): LG ';p r; θ; 0; ω 0 þ 'ω rev ; w 0 ð Þ is the electric field of an LG ';p mode in cylindrical coordinates, where r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x 2 þ y 2 p and θ = arctan(y/x), and (x, y, z, t) are the coordinate and time, respectively. For the '-th frequency line carrying an LG ';p mode, C ';p is the complex coefficient and ω 0 þ 'ω rev is the angular  Simulation results of a spatiotemporal beam exhibiting both rotation and revolution at a given distance. For the generated rotating-revolving LG 3,0 beam revolving at 0.2 THz, we simulate its a frequency spectrum; b spatial LG ';p mode distribution, namely the amplitude and phase of the complex coefficients C ';p of all the LG ';p modes used for superposition; c envelope structure (i.e., the iso-surface with an amplitude of 1/10 of the peak value), where the top cap represents the helically twisting phasefront; d power distribution on light beams with different rotating ' values; and e dynamically rotating and revolving intensity/phase profiles. Scale bar, 1 mm. The spatiotemporal beam consists of multiple frequency comb lines, in which each line is a superposition of multiple LG ';p modes (same beam waist of 0.3 mm) with one ' value but multiple p values. The dynamic helical phasefront and envelope indicate that the beam not only rotates around its beam center but also revolves around another central axis 0.75 mm away from its center. freq.: frequency.
frequency. Clearly, the expansion of Eq. (2) requires infinite number of modes to be a perfectly accurate one, but we will show below that a reasonable number of a few tens provides an acceptable accuracy, as testified by the purity of the rotatingrevolving LG '; p beam.
In general, any spatial beam can be generated by a superposition of multiple modes from a complete spatial modal basis set 36,37 , and the dynamic revolution and rotation motions in this paper can be realized by the judicial selection of spatial modes and frequencies with appropriate complex coefficients, as described in Eq. (2). As an illustrative example, we first consider the principle of generating a rotating-revolving LG ';0 beam with a zero p value. The generation of an LG ';0 beam that dynamically rotates and revolves in time at a given distance can be explained by the coherent interference among all the frequency comb lines that each line carries a superposition of multiple LG ';p modes. The generation can be understood by considering that a simple LG ';0 beam that rotates around its beam center should be firstly offset from the central axis and then be made to revolve around the original central axis, as the following two steps: In the first step, we introduce the dynamic rotation with a spatial offset. A single frequency carrying an LG ';0 mode can generate an LG ';0 beam with a ring-like intensity profile and a twisting phasefront of expði 'θÞ in a circle around its beam center, which is located at the central axis (Fig. 2a). Such a phasefront leads to a Poynting vector with a non-zero azimuthal component. Because the Poynting vector indicates the propagation direction of light beams in free space, the phasefront of the abovegenerated beam dynamically rotates around its beam center, which is located at the central axis, in time at a given propagation distance 16 . Such a structured beam can keep its intensity and phase profiles, and be made offset by combining several modes from a complete LG ';p modal basis set on a single frequency 38,39 . The approach is to choose an appropriate complex coefficient for each mode. As shown in Fig. 2b, the constructive interference of multiple LG ';p modes on a single frequency produces a light beam with intensity and phase profiles as the same as those of an LG ';0 beam, whose beam center is made radially offset from the central axis by a certain distance. Because the light beam still has a twisting phasefront of expði 'θÞ, the phasefront dynamically rotates around its beam center, which is offset from the central axis, in time at a given distance.
In the second step, we introduce the dynamic revolution. For the above superposition, the relative phase delays among all the modes are time-invariant, thus the constructive interference produces an LG ';0 beam whose intensity profiles appears static at any given point of propagation distance. An additional dynamic revolution around the central axis could be introduced by choosing appropriate time-variant relative phase delays among these modes. One possible approach is to combine different modes located on different frequencies. Here, we introduce a time-variant relative phase delay of Δφ = 2πΔft between the neighboring LG ';p modes (i.e., Δ' = 1) by combining multiple frequency comb lines. Each frequency line carries multiple LG ';p modes with a different ' value and multiple p values, where f 0 is the center frequency, Δf is the frequency spacing between neighboring frequency comb lines, and ω ' ¼ 2πðf 0 þ 'Δf Þ is the angular frequency of each LG ';p mode. In terms of the superposition of LG ';p modes on a single frequency, previous work has found that introducing a relative phase delay of Δφ between neighboring LG ';p modes will rotate the azimuthal location of the generated light beam by an angle of Δθ = Δφ 38-40 . In our case, the time-variant relative phase delay will lead to dynamic constructive and destructive interferences, which produce an offset LG ';0 beam exhibiting not only dynamic rotation around its beam center but also dynamic revolution around a central axis (see Fig. 2c).
Generation of a rotating-revolving LG beam. Here, we detail the method for generating a rotating-revolving LG ';0 beam. As an illustrative example, we simulate the dynamic motion of an LG 3,0 beam (beam waist w 0 = 0.3 mm, center frequency f 0 = 193.5 THz) revolving around a central axis with a radius of R = 0.75 mm at a speed of f rev = 0.2 THz. We use 61 frequency comb lines with a frequency spacing Δf of 0.2 THz. Each line is a superposition of multiple LG ';p modes containing one unique ' value and multiple p values, where p varies from 0 to 24. The electric field can be represented by Δf Þ is linearly dependent on the azimuthal mode index ', and the frequency line at ω ' carries a superposition of spatial patterns P 24 p¼0 C ';p LG ';p x; y; 0; ω ' ð Þ . We characterize the beam's spatial spectrum (i.e., spatial LG ';p mode distribution) using the amplitude and phase of its complex coefficients C ';p for each LG ';p mode ( Fig. 3b). Moreover, we map the spatial spectrum onto the frequency spectrum based on the linear relationship between the mode index ' and the angular frequency ω ' (Fig. 3a). Specifically, we calculate the total power on each frequency comb line using the total power of the superposition of P 24 p¼0 C ';p LG ';p x; y; 0; ω ' ð Þ(see Supplementary  Fig. 1 for the spatial patterns on selected frequency lines). In addition, the phasefront and amplitude envelope (equi-amplitude surface) structures of such a beam are simulated (see Fig. 3c), in which the mode purity of the generated rotating-revolving LG 3,0 beam is obtained to be~99% (see Fig. 3d). As shown in Fig. 3e, the dynamic helical phasefront and amplitude profiles indicate that the beam exhibits both dynamic rotation and revolution in time at a given distance. LG 3,0 beam. With further propagation, such as at a distance of 70z R , the interferogram of a conventional LG 3,0 beam remains as a twisting shape, while the interferogram of the rotating-revolving LG 3,0 beam is distorted. The mode purity of the rotatingrevolving LG 3,0 beam is >90% from 0 to 20z R ; it decreases to 38% at 70z R (see the blue curve in Fig. 4e). Figure 4e also shows that the propagation distance of the rotating-revolving LG 3,0 beam with mode purity of >90% increases from 2z R to more than 100z R , when the revolving speed f r decreases from 2 to 0.02 THz.
The difference between the diffraction effects of an offset conventional LG 3,0 beam and a rotating-revolving LG 3,0 beam can be understood in the following manner: (a) The electric field (z = 0) of an offset conventional LG 3,0 beam and a rotating-revolving LG 3,0 beam can be expressed as a superposition of multiple LG ';p modes with the same mode distribution but different frequency spectra (i.e., one frequency line at ω 0 or multiple frequency lines at ω 0 þ 'ω rev ); (b) When the frequency difference 'ω rev is ( ω 0 and the beam is within the Rayleigh range, the diffraction effects of the LG ';p mode carried by the frequency line at ω 0 are almost the same as those of the same mode carried by the frequency line at ω 0 þ 'ω rev . Thus, the two superpositions with the same mode distribution but different frequency spectra are similar to each other in the near-field (see Supplementary Notes 5 and 7 for more analysis details); and  (c) However, such diffraction effects tend to differ with each other (i) with further propagation at far-field and (ii) as the frequency difference 'ω rev increases. The diffraction difference might introduce different spatial amplitude and phase distortions to the same mode carried by the frequency lines on ω 0 and ω 0 þ 'ω rev . As a result, the superposition of multiple modes carried by a single frequency line remains as an LG 3,0 beam, while the superposition of multiple modes carried by multiple frequency lines is distorted; thus, the mode purity decreases.
Control two orbital angular momenta. Based on our simulations, the two momenta can be independently and separately controlled by tuning the rotating ' values and the revolving speed of different rotating-revolving LG ';0 beams. These two momenta are associated with the dynamic rotation and revolution, respectively (Fig. 5). Specifically, we investigate the cases for a rotating-revolving LG ';0 beam (i) revolving clockwise at a speed of 0.2 THz and carrying a rotating ' value varying from 1 to 3 (Fig. 5a-c), or (ii) carrying the same rotating ' ¼ 3 value and revolving at a speed varying from 0.2 to 0.6 THz (Fig. 5d-f). Two phenomena can be discerned from Fig. 5. First, the rotating ' value of the rotating-revolving LG ';0 beam can be controlled by changing the spatial LG ';p mode distribution carried by each frequency line (see Supplementary Fig. 2 for details). Second, the LG ';0 beam revolves at a speed equal to the frequency spacing Δf. This is because that the dynamic revolution is related to the time-variant relative phase delay between the neighboring LG ';p mode for superposition, and its value is Δφ = 2πΔft. Moreover, it is possible to change the direction of revolution of a rotating-revolving LG ';0 beam by reassigning each modal combination P p C ';p LG ';p x; y; 0; ω ' ð Þ , which is originally carried by a frequency line on ω ' ¼ ω 0 þ 'ω rev , to be carried by the one on ω 0 À 'ω rev 12,13 . Therefore, the total amount of orbital angular momenta associated with these two motions could be independently controlled by changing the spatial LG ';p mode distribution and frequency spectrum, respectively. (See Supplementary Fig. 3 for the cases of flipping the sign of the rotating ' value and/or the revolving direction.) Furthermore, we investigate the quality of the dynamic spatiotemporal beam with respect to the frequency spectrum. Here, all the frequency comb lines carry multiple LG ';p modes with the same beam waist of 0.3 mm. Figure 6a-c shows the relationship between the power distribution on light beams with different rotating ' values and the number of selected frequency comb lines. Figure 6b shows that when the number of comb lines is selected to be <10, the power coupling (the difference between the blue curve and other curves) to the light beams with the undesired rotating ' ≠ 3 value is >−5 dB and the mode purity of the generated rotatingrevolving LG 3,0 beam is <25%; while when the number of comb lines is >40, the power coupling is <−20 dB and the mode purity is >95%. We can see from   Supplementary Fig. 2 for details of the spatial LG ';p mode distributions used for superposition. d The corresponding profiles/structures of the generated rotating-revolving LG ';0 beams with the same ' = 3 value but a different revolving speed. e Examples of an LG 3,0 beam revolving clockwise at different speeds from 0.2 to 0.6 THz; and f an LG 3,0 beam revolving counterclockwise at a speed of 0.2 THz. Except for the varied parameters and the spatial/frequency spectra, all the other parameters are the same as those in Fig. 2. NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17805-1 ARTICLE NATURE COMMUNICATIONS | (2020) 11:4099 | https://doi.org/10.1038/s41467-020-17805-1 | www.nature.com/naturecommunications generated rotating-revolving LG ';0 beam is higher for smaller rotating ' values. Figure 6d-f shows the number of frequency comb lines within the 10-dB bandwidth of the frequency spectra for generating rotating-revolving LG ';0 beams with different revolving radii or beam waists. The simulation results show that a larger number of frequency comb lines would generate a rotatingrevolving LG ';0 beam with a (i) larger revolving radius, (ii) smaller beam waist, or (iii) higher rotating ' value. These relationships can be understood by referring to a Fourier transformation; by looking at the dynamic azimuthal mode (the generated spatiotemporal beam) at a given time, the beam can be described as a superposition of multiple LG ';p modes with different azimuthal index ' values 38 . As the light beam's (i) revolving radius increases, (ii) beam waist decreases, or (iii) rotating ' value increases, the azimuthal mode will be spatially distributed within a smaller azimuthal range; thus the number of comb lines increases after applying a Fourier transformation from the azimuthal spatial domain to the frequency domain 38 .

Discussion
We have explored the generation of a spatiotemporal light beam containing two independent orbital angular momenta using multiple frequency comb lines. Although our examples only focus on the generation of rotating-revolving LG ';0 beams with a revolving speed of sub-THz, it might be possible to generate spatiotemporal light beams with different speeds and more sophisticated structures.
The speed of the dynamic motion could be controlled by tuning the frequency spacing between the frequency lines. It is thus possible to vary the revolving speed from several MHz to sub-THz by changing the frequency spacing of the frequency comb. Besides, if frequency lines with non-constant frequency spacing are coherently combined, the generated light beam might exhibit dynamic motions with time-variant speed.
The structure of the generated spatiotemporal light beam could be tuned by changing the spatial LG ';p mode distribution. For example, it is possible to extend our method to generate rotatingrevolving LG '; p beams with non-zero p values. Moreover, if each frequency comb line carries a superposition of multiple LG ';p modes containing both multiple ' values and multiple p values, it might be possible to simultaneously generate multiple rotatingrevolving LG '; p beams with different parameters, such as different non-zero p values, or revolving radii. In addition, a spatiotemporal light beam would experience spatial beam diffraction when propagating in free space. As a result, it might not maintain LG , beam, varying from 0 to +3 0 LG , beam, varying from 0 to +3 0 # of frequency comb lines Revolving radius (R/a) Beam waist (w 0 /a) Fig. 6 The relationship between the quality of the rotating-revolving LG--';0 beams and the frequency spectrum. a We first calculate the spatiotemporal spectra of the rotating-revolving beam using the approach described in Methods, and then we select a certain number of frequency lines/modes from the calculated spectra for power/mode purity calculation in (b) and (c). b The power distribution on light beams with different rotating ' values for generating a rotating-revolving LG 3,0 beam, when the number of selected frequency comb lines is varied. c The mode purity of a generated rotating-revolving LG ';0 beam ( ' varies from 0 to +3), when the number of selected frequency comb lines is varied. d We calculate the spatiotemporal spectra of different rotatingrevolving beams and count the number of frequency lines in the 10-dB bandwidth of the frequency spectra; panels (e) and (f) show the number of frequency comb lines in the 10-dB bandwidth for generating a rotating-revolving LG ';0 beam with (i) the same beam waist of w 0 = 0.2 mm and a revolving radius of R varied from 0.75 to 1.5 mm; and (ii) the same revolving radius of R = 1.5 mm and a beam waist of w 0 varied from 0.15 to 0.5 mm, respectively. a = 0.3 mm in (e) and (f). the same dynamic properties at different distances. It might be possible to generate a non-diffraction rotating-revolving Bessel beam through combining multiple frequency comb lines with each carrying multiple modes in the Bessel modal basis 41 .
We note that our analysis does not include the beam polarization since we are trying to isolate the effects of orbital angular momenta without considering spin angular momentum. However, we believe that it might be possible to generate a rotating-revolving LG '; p beam that also carries spin angular momentum. One potential method could be realized in three steps: (i) generating two rotatingrevolving LG '; p beams on xand y-polarizations, (ii) subsequently adding a phase delay of π/2 to one of the beams 24 , and (iii) finally coherently combining the two beams.
We also note that our results indicate that we might need to combine a large number of frequency comb lines with each carrying a large number of LG ';p modes in order to generate a rotatingrevolving LG '; p beam with high mode purity. Although these large numbers are difficult to achieve at present, there have been reports of generating such large numbers of modes and frequency lines that could potentially be used for spatiotemporal light shaping. For example, reports have shown the generation and combination of (i) 210 LG ';p modes 42 , and (ii)~90 frequency lines with each carrying different modes 43 . We believe those techniques indicate the potential to handle the experimental feasibility of the rotatingrevolving LG ';p beams with high mode purity.

Methods
Simulation details. The scalar electric field of an LG ';p mode in cylindrical coordinates can be described by 16 : LG ';p r; θ; z; ω; w 0 ð Þ ¼ U r; z; ω; w 0 ð Þ expði'θÞ where U(r, z; ω, w 0 ) is the complex electric field independent with θ, L ' j j p are the generalized Laguerre polynomials, and C LG ';p are the required normalization constants, w z; ω ð Þ¼w 0 is the beam waist, and R z; ω ð Þ¼zð1 þ ðz R ðω; w 0 Þ=zÞ 2 Þ, where z R ω; w 0 ð Þ¼ωw 2 0 =2c is the Rayleigh range in free space, k is the wave number; and ψ(z) is the Gouy phase and equals ' j j þ 2p þ 1 ð Þ arctan z=z R ðωÞ ð Þ . The parameters (r, θ, z; ω, w 0 ) have the same definitions as in the "Results".
We numerically generate the spatiotemporal beam in three steps: (i) we first calculate the complex spatial mode distribution of a conventional LG ';0 beam centered at (x, y) = (−R, 0) by decomposing its electric field into an LG ';p mode basis centered at (x, y) = (0, 0). The frequency is f 0 , the revolving radius is R, z = 0, and t = 0; (ii) we then coherently combine all the spatial modes with the same ' value but different p values to obtain the spatial pattern of the frequency line at where Δf is the revolving speed; (iii) we calculate the electric field of the rotating-revolving LG ';0 beam by coherently combining the electric fields of all the frequency comb lines. We only consider the cases in which the frequency separation Δf is a constant and the center frequency is 193.5 THz. In our simulation model, there are 500 × 500 pixels with a 6-μm pixel size in the (x, y) plane, and 400 pixels with a 12.5-fs pixel size in time.
Mode purity calculation. Considering that the observed intensity and phase profiles of the generated rotating-revolving LG ';0 beams remain relatively invariant if an observer moves dynamically with the rotating-revolving beams (Fig. 2f), we calculate the mode purity as the normalized power weight coefficient of the generated spatiotemporal beam at time t = 0 and distance z = 0 using jC ' j 2 ¼ RR E 1 x; y ð ÞE * 2 x; y ð Þdxdy 2 9 , where E 1 (x, y) is the generated electric field of the generated spatiotemporal beam, and E 2 (x, y) is the electric field of a conventional LG ';0 beam with center overlapping with the generated beam, the operator * denotes the conjugation calculation. Both E 1 (x, y) and E 2 (x, y) are normalized, namely, RR E i x; y ð ÞE * i x; y ð Þdxdy 2 ¼ 1, where i = 1 or 2. We calculate the jC ' j 2 using the integral, (i) over the whole transverse plane when the beam is within the Rayleigh range, and (ii) over a ring-shape area with a radius from 0.9R max to 1.1R max (where R max is the distance from the intensity peak to the beam center) outside the Rayleigh range. Here, the calculated mode purity represents the ratio between the power on the spatiotemporal beam with the desired rotating ' value and the total power of the generated beam.
Generalization for the generation of rotating-revolving LG beams. We have shown in the Results some special cases as illustrative examples of the generation of rotating-revolving LG '; p beams. However, it is interesting to consider the generalization of our generation method to a broader range so that it can generate a rotating-revolving LG '; p beams with other ð '; pÞ values (e.g., ' values of >10 or non-zero p values). A rotating-revolving LG '; p beam can be generated by offsetting a conventional LG '; p beam to have an electric field of ψ x; y; 0 ð Þ exp iω 0 t ð Þ at z = 0 and subsequently dynamically revolving the beam around a central axis. According to the modal decomposition method 36,37 , any offset conventional LG '; p beam with arbitrary ð '; pÞ values can be represented by a combination of multiple LG ';p modes, namely, E 0 x; y; 0; t ð Þ= ψ x; y; 0 ð Þ exp iω 0 t ð Þ = PP ';p C ';p LG ';p r; θ; 0; ω 0 ; w 0 ð Þ exp iω 0 t ð Þ. When the beam revolves clockwise around the origin at a speed of f r revolutions per second, the revolution motion introduces a frequency shift of 'ω r to each LG ';p mode so that ω 0 shifts to ω 0 þ 'ω rev 13,44 . Such a frequency shift transforms the electric field of a single frequency line carrying multiple modes into the form which is the electric field at z = 0 of a rotating-revolving LG '; p beam with arbitrary ð '; pÞ values (see Supplementary Note 6 for details). Equation (4) indicates that a rotating-revolving LG '; p beam can be generated by combining multiple frequency comb lines with each carrying multiple LG ';p modes. This method could be generalized to generate rotating-revolving LG '; p beams with any ð '; pÞ values, by judiciously selecting the coefficient C ';p to be an integral RR ψ x; y; 0 ð Þ ð LG ';p r; θ; 0; ω 0 ; w 0 ð Þ Þ * dxdy 9,36,37 . However, when the ð '; pÞ values increase, the coefficient C ';p might still have nonnegligible values for LG ';p modes with modal indices out of the ranges shown in the Article. Thus, in this case, we believe that a higher number of LG ';p modes should be utilized for the combination to generate a rotating-revolving LG '; p beam.

Data availability
All data, theory details, simulation details that support the findings of this study are available from the corresponding authors on reasonable request.