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

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 Gaussian-like beam dot that revolves around a central axis, and (2) a Laguerre-Gaussian ($$LG_{\ell ,p}$$) 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 orbital-angular-momenta 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 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.

Introduction

Structured light has recently gained increased interest in that it can accommodate the production of uniquely propagating beams of light1,2,3,4,5. 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 shape12,13,14; 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 propagates15,16. 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 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 axis17.

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 distance18,19,20,21,22. 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 xy plane in time at a given propagation distance z. Prior art has produced novel beams by combining different modes not only on the same frequency18,19,20,21,22 but also on different frequencies12,13,14,23,24,25,26,27. This ability to produce different modes on different frequencies can be achieved by the use of optical-frequency combs, which have recently undergone much advancement28.

Specifically, it has been previously shown that a light beam can be created to exhibit unique dynamic features12,13,14,23,24,25,26,27,29,30,31,32,33,34,35, 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 $$LG_{\ell ,p}$$ mode (different $$\ell$$ 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 distance23; (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 distance26; (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 in-and-out linear radial movement at a given propagation distance27; 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 speed29. 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 $$LG_{\ell ,p}$$ 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 w0 = 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 fr = 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, two-dimensional (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 OAM11; however, such transverse OAM can be decomposed into different OAM components, which are related to the rotation around beam’s center15 and the revolution around another central axis24,25, 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 frev 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., self-rotating and not revolving) $$LG_{\bar \ell ,\bar p}$$ beam at a speed of frev with a revolving radius of R, we can obtain a rotating-revolving electric field (see Supplementary Note 6 for more details):

$$\begin{array}{c}E_1\left( {x,y,0,t} \right) = LG_{\bar \ell ,\bar p}^{{\mathrm{Cartesian}}}\left( {x\cos \varphi \left( t \right) - y\sin \varphi \left( t \right) + R,x\sin \varphi \left( t \right)} \right.\\ \left. { + \,y\cos \varphi \left( t \right),0;\omega _0,w_0} \right)\exp \left( {i\omega _0t} \right)\end{array}$$
(1)

We refer to the beam with such a dynamic electric field as a rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beam (i.e., a beam that exhibits dynamic rotation around its beam center as well as revolution around a central axis). In this equation, ω0 = 2πf0 is the angular frequency, w0 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; ω0, w0) in a reference frame rotating in the transverse plane at z = 0.

The electric field at z = 0 of such a rotating-revolving $$LG_{\bar \ell ,\bar p}$$ 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):

$$\begin{array}{c}E_1\left( {x,y,0,t} \right) = {\sum} {\mathop {\sum}\limits_{\ell ,p} {C_{\ell ,p}} } LG_{\ell ,p}\left( {r,\theta ,0;\omega _0 + \ell \omega _{{\mathrm{rev}}},w_0} \right)\\ {\mathrm{exp}}\left( {i\left( {\omega _0 + \ell \omega _{{\mathrm{rev}}}} \right)t} \right)\end{array}$$
(2)

$$LG_{\ell ,p}\left( {r,\theta ,0;\omega _0 + \ell \omega _{{\mathrm{rev}}},w_0} \right)$$ 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 rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beam.

In general, any spatial beam can be generated by a superposition of multiple modes from a complete spatial modal basis set36,37, 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 $$LG_{\bar \ell ,0}$$ 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 ring-like 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 non-zero azimuthal component. Because the Poynting vector indicates the propagation direction of light beams in free space, the phasefront of the above-generated beam dynamically rotates around its beam center, which is located at the central axis, in time at a given propagation distance16. 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 frequency38,39. 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 time-invariant, 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 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 $$LG_{\ell ,p}$$ 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 f0 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 = Δφ38,39,40. In our case, the time-variant 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 rotating-revolving LG beam

Here, we detail the method for generating a rotating-revolving $$LG_{\bar \ell ,0}$$ beam. As an illustrative example, we simulate the dynamic motion of an LG3,0 beam (beam waist w0 = 0.3 mm, center frequency f0 = 193.5 THz) revolving around a central axis with a radius of R = 0.75 mm at a speed of frev = 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 (equi-amplitude surface) structures of such a beam are simulated (see Fig. 3c), in which the mode purity of the generated rotating-revolving LG3,0 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 real-time video of rotating-revolving $$LG_{\bar \ell ,0}$$ beams with different rotating $$\bar \ell$$ values.)

Diffraction of a rotating-revolving LG beam

To characterize the quality of the rotating-revolving LG3,0 beam at various propagation distances, we analyze the free-space diffraction effects in the near- and far-field. The examples in Fig. 4a–d show the comparison of the free-space propagation between an offset conventional $$LG_{3,0}$$ beam (beam center at (x, y) = (−0.75 mm, 0)) and the above-generated rotating-revolving LG3,0 beam (revolving speed frev = 0.2 THz, revolving radius R = 0.75 mm). The Rayleigh range is zR(ω0, w0) = 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 rotating-revolving LG3,0 beam and its interferograms with Gaussian beams are almost the same as those of a conventional LG3,0 beam. With further propagation, such as at a distance of 70zR, the interferogram of a conventional LG3,0 beam remains as a twisting shape, while the interferogram of the rotating-revolving LG3,0 beam is distorted. The mode purity of the rotating-revolving LG3,0 beam is >90% from 0 to 20zR; it decreases to 38% at 70zR (see the blue curve in Fig. 4e). Figure 4e also shows that the propagation distance of the rotating-revolving LG3,0 beam with mode purity of >90% increases from 2zR to more than 100zR, when the revolving speed fr decreases from 2 to 0.02 THz.

The difference between the diffraction effects of an offset conventional LG3,0 beam and a rotating-revolving LG3,0 beam can be understood in the following manner:

1. (a)

The electric field (z = 0) of an offset conventional LG3,0 beam and a rotating-revolving LG3,0 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 ω0 or multiple frequency lines at $$\omega _0 + \ell \omega _{{\mathrm{rev}}}$$);

2. (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 ω0 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 near-field (see Supplementary Notes 5 and 7 for more analysis details); and

3. (c)

However, such diffraction effects tend to differ with each other (i) with further propagation at far-field 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 ω0 and $$\omega _0 + \ell \omega _{{\mathrm{rev}}}$$. As a result, the superposition of multiple modes carried by a single frequency line remains as an LG3,0 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 rotating-revolving $$LG_{\bar \ell ,0}$$ beams. These two momenta are associated with the dynamic rotation and revolution, respectively (Fig. 5). Specifically, we investigate the cases for a rotating-revolving $$LG_{\bar \ell ,0}$$ 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 rotating-revolving $$LG_{\bar \ell ,0}$$ 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 time-variant 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 rotating-revolving $$LG_{\bar \ell ,0}$$ 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 rotating-revolving LG3,0 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 rotating-revolving $$LG_{\bar \ell ,0}$$ 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 rotating-revolving $$LG_{\bar \ell ,0}$$ beam is higher for smaller rotating $$\bar \ell$$ values. Figure 6d–f shows the number of frequency comb lines within the 10-dB bandwidth of the frequency spectra for generating rotating-revolving $$LG_{\bar \ell ,0}$$ beams with different revolving radii or beam waists. The simulation results show that a larger number of frequency comb lines would generate a rotating-revolving $$LG_{\bar \ell ,0}$$ 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 $$\ell$$ values38. 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 domain38.

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 $$LG_{\bar \ell ,0}$$ 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 $$LG_{\ell ,p}$$ mode distribution. For example, it is possible to extend our method to generate rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beams with non-zero $$\bar p$$ 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 rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beams with different parameters, such as different non-zero $$\bar p$$ 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 non-diffraction rotating-revolving Bessel beam through combining multiple frequency comb lines with each carrying multiple modes in the Bessel modal basis41.

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 $$LG_{\bar \ell ,\bar p}$$ beam that also carries spin angular momentum. One potential method could be realized in three steps: (i) generating two rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beams on x- and y-polarizations, (ii) subsequently adding a phase delay of π/2 to one of the beams24, 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 rotating-revolving $$LG_{\bar \ell ,\bar p}$$ 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 $$LG_{\ell ,p}$$ modes42, and (ii) ~90 frequency lines with each carrying different modes43. We believe those techniques indicate the potential to handle the experimental feasibility of the rotating-revolving $$LG_{\ell ,p}$$ beams with high mode purity.

Methods

Simulation details

The scalar electric field of an $$LG_{\ell ,p}$$ mode in cylindrical coordinates can be described by16:

$$LG_{\ell ,p}\left( {r,\theta ,z;\omega ,w_0} \right) = U\left( {r,z;\omega ,w_0} \right){\mathrm{exp}}(i\ell \theta )\\ = \frac{{C_{\ell ,p}^{{\mathrm{LG}}}}}{{w\left( {z,\omega } \right)}}\left( {\frac{{r\sqrt 2 }}{{w\left( {z,\omega } \right)}}} \right)^{\left| \ell \right|}\exp \left( { - \frac{{r^2}}{{w^2\left( {z,\omega } \right)}}} \right)L_p^{\left| \ell \right|}\left( {\frac{{2r^2}}{{w^2\left( {z,\omega } \right)}}} \right)\\ \quad {\mathrm{exp}}\left( { - i\left( {k\frac{{r^2}}{{2R\left( {z,\omega } \right)}} + kz - \ell \theta - \psi \left( {z,\omega } \right)} \right)} \right)$$
(3)

where U(r, z; ω, w0) 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, w0) 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 f0, 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 rotating-revolving $$LG_{\bar \ell ,0}$$ 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 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 $$LG_{\bar \ell ,0}$$ 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 $$|C_\ell |^2 = \left| {{\iint} {E_1\left( {x,y} \right)} E_2^ \ast \left( {x,y} \right)dxdy} \right|^2$$9, where E1(x, y) is the generated electric field of the generated spatiotemporal beam, and E2(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 E1(x, y) and E2(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 ring-shape area with a radius from 0.9Rmax to 1.1Rmax (where Rmax 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 rotating-revolving LG beams

We have shown in the Results some special cases as illustrative examples of the generation of rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beams. However, it is interesting to consider the generalization of our generation method to a broader range so that it can generate a rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beams with other $$(\bar \ell ,\bar p)$$ values (e.g., $$\bar \ell$$ values of >10 or non-zero $$\bar p$$ values). A rotating-revolving $$LG_{\bar \ell ,\bar p}$$ 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 method36,37, 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 fr revolutions per second, the revolution motion introduces a frequency shift of $$\ell \omega _r$$ to each $$LG_{\ell ,p}$$ mode so that ω0 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

$$E_1\left( {x,y,0,t} \right) = {\sum} {\mathop {\sum}\limits_{\ell ,p} {} C_{\ell ,p}} LG_{\ell ,p}\left( {r,\theta ,0;\omega _0 + \ell \omega _{{\mathrm{rev}}},w_0} \right){\mathrm{exp}}\left( {i\left( {\omega _0 + \ell \omega _{{\mathrm{rev}}}} \right)t} \right)$$
(4)

which is the electric field at z = 0 of a rotating-revolving $$LG_{\bar \ell ,\bar p}$$ beam with arbitrary $$(\bar \ell ,\bar p)$$ values (see Supplementary Note 6 for details). Equation (4) indicates that a rotating-revolving $$LG_{\bar \ell ,\bar p}$$ 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 rotating-revolving $$LG_{\bar \ell ,\bar p}$$ 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 rotating-revolving $$LG_{\bar \ell ,\bar p}$$ 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) (N00014-16-1-2813); and Office of Naval Research through a MURI subaward from the University of Central Florida.

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Authors

Contributions

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.

Corresponding authors

Correspondence to Zhe Zhao or Alan E. Willner.

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Zhao, Z., Song, H., Zhang, R. et al. Dynamic spatiotemporal beams that combine two independent and controllable orbital-angular-momenta using multiple optical-frequency-comb lines. Nat Commun 11, 4099 (2020). https://doi.org/10.1038/s41467-020-17805-1

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• Generation and Detection of Structured Light: A Review

• Jian Wang
•  & Yize Liang

Frontiers in Physics (2021)