Abstract
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 Gaussianlike beam dot that revolves around a central axis, and (2) a LaguerreGaussian (\(LG_{\ell ,p}\)) 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 orbitalangularmomenta by coherently adding multiple frequency comb lines. Each line carries a superposition of multiple \(LG_{\ell ,p}\) modes such that each line is composed of a different \(\ell\) 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 2π6π 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.
Introduction
Structured 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 (\(LG_{\ell ,p}\)) beams in which the phasefront twists in the azimuthal direction as it propagates^{15,16}. The amount of OAM (\(\ell\)) is the number of 2π azimuthal phase changes, and p + 1 is the number of concentric rings on the intensity cross section for \(\ell \,\neq \, 0\). The beam rotates around its own beam center with a ringlike 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 opticalfrequency 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 Gaussianlike beam dot that exhibits dynamic circular revolution at a given propagation distance by combining multiple frequency lines in which each line carries a different \(LG_{\ell ,p}\) mode (different \(\ell\) but same p)^{12,13,14} (Fig. 1e, i); (b) a Gaussianlike beam dot formed from multiple Hermite–Gaussian modes, each at a different frequency, such that the dot can dynamically move upanddown in a linear fashion at a given propagation distance^{23}; (c) a light beam created by a pair of \(LG_{\ell ,p}\) modes with different \(\ell\) and p values at two different 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 \(LG_{\ell ,p}\) mode with a different pair of indices (\(\ell ,p\)) and can produce a light beam that exhibits dynamic rotation around its center (azimuthal dimension) as well as inandout linear radial movement at a given propagation distance^{27}; and (e) a light beam, which is created by driving the highharmonicgeneration of two timedelayed pulses carrying different OAM values, can exhibit dynamic rotation around its center at a timedependent 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 opticalfrequency comb lines, with each line carrying a superposition of multiple \(LG_{\ell ,p}\) modes containing a different \(\ell\) 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 \(\bar \ell\) 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 \(LG_{\bar \ell ,\bar p}\) 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 selfrotating electric field at z = 0 (and more generally at a chosen distance), that also revolves around a central axis, O, distance R from its selfrotating 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) \(LG_{\bar \ell ,\bar p}\) beam at a speed of f_{rev} with a revolving radius of R, we can obtain a rotatingrevolving electric field (see Supplementary Note 6 for more details):
We refer to the beam with such a dynamic electric field as a rotatingrevolving \(LG_{\bar \ell ,\bar p}\) 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 \(LG_{\bar \ell ,\bar p}^{{\mathrm{Cartesian}}}\left( {x,y,z;\omega _0,w_0} \right)\) is the electric field in Cartesian coordinates of a conventional \(LG_{\bar \ell ,\bar p}\) beam. \(\varphi \left( t \right) = \omega _{{\mathrm{rev}}}t = 2\pi f_{{\mathrm{rev}}}t\) represents the revolving angular speed, and (\(x\cos \varphi \left( t \right)  y\sin \varphi \left( t \right)\) + \(R,x\sin \varphi \left( t \right)\) + \(y\cos \varphi \left( t \right),0;\omega _0,w_0\)) 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 rotatingrevolving \(LG_{\bar \ell ,\bar p}\) 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):
\(LG_{\ell ,p}\left( {r,\theta ,0;\omega _0 + \ell \omega _{{\mathrm{rev}}},w_0} \right)\) is the electric field of an \(LG_{\ell ,p}\) mode in cylindrical coordinates, where \(r = \sqrt {x^2 + y^2}\) and θ = arctan(y/x), and (x, y, z, t) are the coordinate and time, respectively. For the \(\ell\)th frequency line carrying an \(LG_{\ell ,p}\) mode, \(C_{\ell ,p}\) is the complex coefficient and \(\omega _0 + \ell \omega _{{\mathrm{rev}}}\) is the angular 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 \(LG_{\bar \ell ,\bar p}\) 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 rotatingrevolving \(LG_{\bar \ell ,0}\) beam with a zero \(\bar p\) value. The generation of an \(LG_{\bar \ell ,0}\) 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 \(LG_{\ell ,p}\) modes. The generation can be understood by considering that a simple \(LG_{\bar \ell ,0}\) 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 \(LG_{\bar \ell ,0}\) mode can generate an \(LG_{\bar \ell ,0}\) beam with a ringlike intensity profile and a twisting phasefront of \({\mathrm{exp}}(i\bar \ell \theta )\) 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 nonzero 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 \(LG_{\ell ,p}\) 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 \(LG_{\ell ,p}\) modes on a single frequency produces a light beam with intensity and phase profiles as the same as those of an \(LG_{\bar \ell ,0}\) 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 \({\mathrm{exp}}(i\bar \ell \theta )\), 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 timeinvariant, thus the constructive interference produces an \(LG_{\bar \ell ,0}\) 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 timevariant relative phase delays among these modes. One possible approach is to combine different modes located on different frequencies. Here, we introduce a timevariant relative phase delay of Δφ = 2πΔft between the neighboring \(LG_{\ell ,p}\) modes (i.e., Δ\(\ell\) = 1) by combining multiple frequency comb lines. Each frequency line carries multiple \(LG_{\ell ,p}\) modes with a different \(\ell\) value and multiple p values, where f_{0} is the center frequency, Δf is the frequency spacing between neighboring frequency comb lines, and \(\omega _\ell = 2\pi (f_0 + \ell \Delta f)\) is the angular frequency of each \(LG_{\ell ,p}\) mode. In terms of the superposition of \(LG_{\ell ,p}\) modes on a single frequency, previous work has found that introducing a relative phase delay of Δφ between neighboring \(LG_{\ell ,p}\) modes will rotate the azimuthal location of the generated light beam by an angle of Δθ = Δφ^{38,39,40}. In our case, the timevariant relative phase delay will lead to dynamic constructive and destructive interferences, which produce an offset \(LG_{\bar \ell ,0}\) beam exhibiting not only dynamic rotation around its beam center but also dynamic revolution around a central axis (see Fig. 2c).
Generation of a rotatingrevolving LG beam
Here, we detail the method for generating a rotatingrevolving \(LG_{\bar \ell ,0}\) 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 \(LG_{\ell ,p}\) modes containing one unique \(\ell\) value and multiple p values, where p varies from 0 to 24. The electric field can be represented by \(\mathop {\sum}\nolimits_{\ell =  30}^{30} {\mathop {\sum}\nolimits_{p = 0}^{24} {C_{\ell ,p}} } LG_{\ell ,p}\left( {x,y,0,\omega _\ell } \right){\mathrm{exp}}(i\omega _\ell t)\) at distance z = 0, where \(\omega _\ell = 2\pi (f_0 + \ell \Delta f)\) is linearly dependent on the azimuthal mode index \(\ell\), and the frequency line at \(\omega _\ell\) carries a superposition of spatial patterns \(\mathop {\sum}\nolimits_{p = 0}^{24} {C_{\ell ,p}} LG_{\ell ,p}\left( {x,y,0,\omega _\ell } \right)\).
We characterize the beam’s spatial spectrum (i.e., spatial \(LG_{\ell ,p}\) mode distribution) using the amplitude and phase of its complex coefficients \(C_{\ell ,p}\) for each \(LG_{\ell ,p}\) mode (Fig. 3b). Moreover, we map the spatial spectrum onto the frequency spectrum based on the linear relationship between the mode index \(\ell\) and the angular frequency \(\omega _\ell\) (Fig. 3a). Specifically, we calculate the total power on each frequency comb line using the total power of the superposition of \(\mathop {\sum}\nolimits_{p = 0}^{24} {C_{\ell ,p}} LG_{\ell ,p}\left( {x,y,0,\omega _\ell } \right)\) (see Supplementary Fig. 1 for the spatial patterns on selected frequency lines). In addition, the phasefront and amplitude envelope (equiamplitude surface) structures of such a beam are simulated (see Fig. 3c), in which the mode purity of the generated rotatingrevolving 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. (See Supplementary Video 1 for a realtime video of rotatingrevolving \(LG_{\bar \ell ,0}\) beams with different rotating \(\bar \ell\) values.)
Diffraction of a rotatingrevolving LG beam
To characterize the quality of the rotatingrevolving LG_{3,0} beam at various propagation distances, we analyze the freespace diffraction effects in the near and farfield. The examples in Fig. 4a–d show the comparison of the freespace propagation between an offset conventional \(LG_{3,0}\) beam (beam center at (x, y) = (−0.75 mm, 0)) and the abovegenerated rotatingrevolving LG_{3,0} beam (revolving speed f_{rev} = 0.2 THz, revolving radius R = 0.75 mm). The Rayleigh range is z_{R}(ω_{0}, w_{0}) = 45.6 mm for the center frequency line. Figure 4c, d shows that the spatiotemporal beam counterclockwise revolves around the central axis as a function of z for t = 0 (see Supplementary Notes 5 and 7 for the analysis). Within the Rayleigh range, the shapes of the intensity profiles of the rotatingrevolving LG_{3,0} beam and its interferograms with Gaussian beams are almost the same as those of a conventional 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 rotatingrevolving 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 rotatingrevolving 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 rotatingrevolving 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 rotatingrevolving LG_{3,0} beam can be expressed as a superposition of multiple \(LG_{\ell ,p}\) modes with the same mode distribution but different frequency spectra (i.e., one frequency line at ω_{0} or multiple frequency lines at \(\omega _0 + \ell \omega _{{\mathrm{rev}}}\));

(b)
When the frequency difference \(\ell \omega _{{\mathrm{rev}}}\) is \(\ll \omega _0\) and the beam is within the Rayleigh range, the diffraction effects of the \(LG_{\ell ,p}\) 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 \(\omega _0 + \ell \omega _{{\mathrm{rev}}}\). Thus, the two superpositions with the same mode distribution but different frequency spectra are similar to each other in the nearfield (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 farfield and (ii) as the frequency difference \(\ell \omega _{{\mathrm{rev}}}\) 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 \(\omega _0 + \ell \omega _{{\mathrm{rev}}}\). 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 \(\bar \ell\) values and the revolving speed of different rotatingrevolving \(LG_{\bar \ell ,0}\) beams. These two momenta are associated with the dynamic rotation and revolution, respectively (Fig. 5). Specifically, we investigate the cases for a rotatingrevolving \(LG_{\bar \ell ,0}\) beam (i) revolving clockwise at a speed of 0.2 THz and carrying a rotating \(\bar \ell\) value varying from 1 to 3 (Fig. 5a–c), or (ii) carrying the same rotating \(\bar \ell = 3\) 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 \(\bar \ell\) value of the rotatingrevolving \(LG_{\bar \ell ,0}\) beam can be controlled by changing the spatial \(LG_{\ell ,p}\) mode distribution carried by each frequency line (see Supplementary Fig. 2 for details). Second, the \(LG_{\bar \ell ,0}\) beam revolves at a speed equal to the frequency spacing Δf. This is because that the dynamic revolution is related to the timevariant relative phase delay between the neighboring \(LG_{\ell ,p}\) mode for superposition, and its value is Δφ = 2πΔft. Moreover, it is possible to change the direction of revolution of a rotatingrevolving \(LG_{\bar \ell ,0}\) beam by reassigning each modal combination \(\mathop {\sum}\nolimits_p {C_{\ell ,p}} LG_{\ell ,p}\left( {x,y,0,\omega _\ell } \right)\), which is originally carried by a frequency line on \(\omega _\ell = \omega _0 + \ell \omega _{{\mathrm{rev}}}\), to be carried by the one on \(\omega _0  \ell \omega _{{\mathrm{rev}}}\)^{12,13}. Therefore, the total amount of orbital angular momenta associated with these two motions could be independently controlled by changing the spatial \(LG_{\ell ,p}\) mode distribution and frequency spectrum, respectively. (See Supplementary Fig. 3 for the cases of flipping the sign of the rotating \(\bar \ell\) 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 \(LG_{\ell ,p}\) 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 \(\bar \ell\) 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 \(\bar \ell \, \ne \, 3\) 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 Fig. 6c that combining ~30 frequency lines could generate a rotatingrevolving \(LG_{\bar \ell ,0}\) beam, where \(\bar \ell = 0,1,2,3\), with mode purity of >90%. For the cases where a limited number of frequency lines, such as 20, are used, the mode purity of the generated rotatingrevolving \(LG_{\bar \ell ,0}\) beam is higher for smaller rotating \(\bar \ell\) values. Figure 6d–f shows the number of frequency comb lines within the 10dB bandwidth of the frequency spectra for generating rotatingrevolving \(LG_{\bar \ell ,0}\) beams with different revolving radii or beam waists. The simulation results show that a larger number of frequency comb lines would generate a rotatingrevolving \(LG_{\bar \ell ,0}\) beam with a (i) larger revolving radius, (ii) smaller beam waist, or (iii) higher rotating \(\bar \ell\) 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 \(LG_{\ell ,p}\) modes with different azimuthal index \(\ell\) values^{38}. As the light beam’s (i) revolving radius increases, (ii) beam waist decreases, or (iii) rotating \(\bar \ell\) 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 rotatingrevolving \(LG_{\bar \ell ,0}\) beams with a revolving speed of subTHz, 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 subTHz by changing the frequency spacing of the frequency comb. Besides, if frequency lines with nonconstant frequency spacing are coherently combined, the generated light beam might exhibit dynamic motions with timevariant speed.
The structure of the generated spatiotemporal light beam could be tuned by changing the spatial \(LG_{\ell ,p}\) mode distribution. For example, it is possible to extend our method to generate rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beams with nonzero \(\bar p\) values. Moreover, if each frequency comb line carries a superposition of multiple \(LG_{\ell ,p}\) modes containing both multiple \(\ell\) values and multiple p values, it might be possible to simultaneously generate multiple rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beams with different parameters, such as different nonzero \(\bar p\) 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 the same dynamic properties at different distances. It might be possible to generate a nondiffraction rotatingrevolving 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 rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beam that also carries spin angular momentum. One potential method could be realized in three steps: (i) generating two rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beams on x and ypolarizations, (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 \(LG_{\ell ,p}\) modes in order to generate a rotatingrevolving \(LG_{\bar \ell ,\bar p}\) 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 \(LG_{\ell ,p}\) 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 \(LG_{\ell ,p}\) beams with high mode purity.
Methods
Simulation details
The scalar electric field of an \(LG_{\ell ,p}\) mode in cylindrical coordinates can be described by^{16}:
where U(r, z; ω, w_{0}) is the complex electric field independent with θ, \(L_p^{\left \ell \right}\) are the generalized Laguerre polynomials, and \(C_{\ell ,p}^{{\mathrm{LG}}}\) are the required normalization constants, \(w\left( {z,\omega } \right) = w_0\sqrt {1 + \left( {z/z_{\mathrm{{R}}}(\omega ,w_0)} \right)^2}\) is the beam waist, and \(R\left( {z,\omega } \right) = z(1 + (z_{\mathrm{R}}(\omega ,w_0)/z)^2)\), where \(z_{\mathrm{R}}\left( {\omega ,w_0} \right) = \omega w_0^2{\mathrm{/}}2c\) is the Rayleigh range in free space, k is the wave number; and ψ(z) is the Gouy phase and equals \(\left( {\left \ell \right + 2p + 1} \right)\arctan \left( {z/z_{\mathrm{R}}(\omega )} \right)\). 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 \(LG_{\bar \ell ,0}\) beam centered at (x, y) = (−R, 0) by decomposing its electric field into an \(LG_{\ell ,p}\) 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 \(\ell\) value but different p values to obtain the spatial pattern of the frequency line at \(\omega _\ell = 2\pi (f_0 + \ell \Delta f)\), where Δf is the revolving speed; (iii) we calculate the electric field of the rotatingrevolving \(LG_{\bar \ell ,0}\) 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 6μm pixel size in the (x, y) plane, and 400 pixels with a 12.5fs pixel size in time.
Mode purity calculation
Considering that the observed intensity and phase profiles of the generated rotatingrevolving \(LG_{\bar \ell ,0}\) beams remain relatively invariant if an observer moves dynamically with the rotatingrevolving 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 \(C_\ell ^2 = \left {{\iint} {E_1\left( {x,y} \right)} E_2^ \ast \left( {x,y} \right)dxdy} \right^2\)^{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 \(LG_{\ell ,0}\) 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, \(\left { {\iint}{E_i\left( {x,y} \right)} E_i^ \ast \left( {x,y} \right)dxdy} \right^2 = 1\), where i = 1 or 2. We calculate the \(C_\ell ^2\) using the integral, (i) over the whole transverse plane when the beam is within the Rayleigh range, and (ii) over a ringshape 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 \(\bar \ell\) value and the total power of the generated beam.
Generalization for the generation of rotatingrevolving LG beams
We have shown in the Results some special cases as illustrative examples of the generation of rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beams. However, it is interesting to consider the generalization of our generation method to a broader range so that it can generate a rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beams with other \((\bar \ell ,\bar p)\) values (e.g., \(\bar \ell\) values of >10 or nonzero \(\bar p\) values). A rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beam can be generated by offsetting a conventional \(LG_{\bar \ell ,\bar p}\) beam to have an electric field of \(\psi \left( {x,y,0} \right){\mathrm{exp}}\left( {i\omega _0t} \right)\) at z = 0 and subsequently dynamically revolving the beam around a central axis. According to the modal decomposition method^{36,37}, any offset conventional \(LG_{\bar \ell ,\bar p}\) beam with arbitrary \((\bar \ell ,\bar p)\) values can be represented by a combination of multiple \(LG_{\ell ,p}\) modes, namely, \(E_0\left( {x,y,0,t} \right)\) = \(\psi \left( {x,y,0} \right){\mathrm{exp}}\left( {i\omega _0t} \right)\) = \({\sum} {\mathop {\sum}\nolimits_{\ell ,p} {C_{\ell ,p}} } LG_{\ell ,p}\left( {r,\theta ,0;\omega _0,w_0} \right){\mathrm{exp}}\left( {i\omega _0t} \right)\). 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 \(\ell \omega _r\) to each \(LG_{\ell ,p}\) mode so that ω_{0} shifts to \(\omega _0 + \ell \omega _{{\mathrm{rev}}}\)^{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 rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beam with arbitrary \((\bar \ell ,\bar p)\) values (see Supplementary Note 6 for details). Equation (4) indicates that a rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beam can be generated by combining multiple frequency comb lines with each carrying multiple \(LG_{\ell ,p}\) modes. This method could be generalized to generate rotatingrevolving \(LG_{\bar \ell ,\bar p}\) beams with any \((\bar \ell ,\bar p)\) values, by judiciously selecting the coefficient \(C_{\ell ,p}\) to be an integral \({\iint} \psi \left( {x,y,0} \right)( {LG_{\ell ,p}\left( {r,\theta ,0;\omega _0,w_0} \right)} )^ \ast dxdy\)^{9,36,37}. However, when the \((\bar \ell ,\bar p)\) values increase, the coefficient \(C_{\ell ,p}\) might still have nonnegligible values for \(LG_{\ell ,p}\) modes with modal indices out of the ranges shown in the Article. Thus, in this case, we believe that a higher number of \(LG_{\ell ,p}\) modes should be utilized for the combination to generate a rotatingrevolving \(LG_{\bar \ell ,\bar p}\) 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.
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Acknowledgements
We thank Dr. Jing Du for helpful discussions. This work is supported by Vannevar Bush Faculty Fellowship sponsored by the Basic Research Office of the Assistant Secretary of Defense (ASD) for Research and Engineering (R&E) and funded by the Office of Naval Research (ONR) (N000141612813); and Office of Naval Research through a MURI subaward from the University of Central Florida.
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All the authors contributed to the interpretation of the results and manuscript writing. Z.Z. conceived the idea. Z.Z. and Hao S. developed the simulation model. Z.Z., Hao S., and R.Z. with the help of K.P., C.L., Haoqian S., A.A., K.M., and H.Z. performed the simulation and data analysis. B.L., R.W.B., M.T., and A.E.W. provided the technical support in the simulation and data analysis. The project was supervised by A.E.W.
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Zhao, Z., Song, H., Zhang, R. et al. Dynamic spatiotemporal beams that combine two independent and controllable orbitalangularmomenta using multiple opticalfrequencycomb lines. Nat Commun 11, 4099 (2020). https://doi.org/10.1038/s41467020178051
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