Abstract
Toroidal vortices, also known as vortex rings, are whirling, closedloop disturbances that form a characteristic ring shape in liquids and gases and propagate in a direction that is perpendicular to the plane of the ring. They are wellstudied structures and commonly found in various fluid and gas flow scenarios in nature, for example in the human heart, underwater air bubbles and volcanic eruptions^{1,2,3}. Here we report the experimental observation of a photonic toroidal vortex as a new solution to Maxwell’s equations, generated by the use of conformal mapping^{4,5,6,7}. The resulting light field has a helical phase that twists around a closed loop, leading to an azimuthal local orbital angular momentum density. The preparation of such an intriguing state of light may offer insights for exploring the behaviour of toroidal vortices in other disciplines and find important applications in light–matter interactions, optical manipulation, photonic symmetry and topology, and quantum information^{8,9,10,11,12,13,14,15,16,17}.
Main
Toroidal vortices, also known as vortex rings, are intriguing propagating ringshaped structures with whirling disturbances rotating about the ring. Toroidal vortices are not uncommon in nature and science. Indeed, aquarium visitors are often amazed at how dolphins master creating and playing with bubble rings, which are airfilled vortex rings propagating in water. The scientific investigation of vortex rings dates back to 1867, when Lord Kelvin proposed the vortex atom model^{18}. After one and a half centuries, vortex rings are still being actively investigated in a variety of disciplines. For example, in meteorology, vortex rings of wind, rain and hail are tightly related to the development of microbursts, which pose a great threat to aviation safety^{19}. In cardiology, asymmetric redirection of blood flow through the heart resembles a toroidal vortex^{1}. In magnetics, the experimental observation of vortex rings in a bulk magnet has only just been accomplished, opening up possibilities for studying complex threedimensional (3D) solitons in bulk magnets^{20}. In photonics and light science, there have been studies on toroidal dipole and multipole excitation in metamaterials^{12,13,14,15,16,17}. However, the theory and experimental demonstrations of a propagating photonic toroidal vortex remain elusive. In this Article we fill this gap, based on conformal mapping. Conformal mapping is an anglepreserving transformation that has been utilized to bend optical rays with metamaterials in mapped directions to circumvent hidden objects so as to achieve optical cloaking^{21,22}. Here we exploit conformal mapping to reshape a photonic vortex tube into a toroidal vortex. The photonic toroidal vortex is an approximate solution to Maxwell’s equations and can propagate without distortion in a uniform medium with anomalous group velocity dispersion (GVD). The observation of a photonic toroidal vortex, like its counterparts in other disciplines, will open up a wealth of physical mechanisms to explore, such as toroidal electrodynamics, toroidal plasma physics, complex symmetry and topology for light confinement, sensing and manipulation, and interaction of light with metamaterials. The discovery of photonic toroidal vortices may also open up possibilities for the development of novel laser designs, as well as energy and information transfer methods.
The electric field of an optical wave packet is expressed as the product of the carrier at the central frequency and the envelope function. Under a scalar, paraxial and narrow bandwidth approximation, the envelope function that describes the dimensionless light field in a uniform medium with anomalous GVD is given by^{23}
where Ψ is the scalar wavefunction, x and y are the normalized transverse coordinates, τ is the normalized retarded time, and z is the propagation distance. The reason why the envelope function is expressed in a dimensionless way is to conveniently unify the spatial and temporal coordinates and derive a stable solution in the spatiotemporal domain. Anomalous GVD is chosen so as to obtain the same sign for the spatial and temporal second derivatives. As demonstrated by Supplementary equation (14), a spatiotemporal Laguerre–Gauss tube with the vortex line directed in the y direction is a solution to equation (1). The isointensity surface of a spatiotemporal Laguerre–Gauss tube is labelled at position 0 in Fig. 1. According to conformal mapping theory, the complex exponential function corresponds to a logpolartoCartesian coordinate transformation. This conformal mapping transforms a line to a circle in two dimensions and performs a tubetotoroidal mapping in 3D space. Here we adopt an afocal system that consists of two computergenerated phase masks to perform optical conformal mapping \({{\left( {x,\,y} \right)} \mapsto {\left( {u,\,v} \right)}}\), where (x, y) and (u, v) are the Cartesian coordinates in the planes at which the two phase elements are located. The logpolartoCartesian coordinate transformation requires \({u} = {b\exp }{\left( {  \frac{x}{a}} \right)}\cos {\left( {\frac{y}{a}} \right)}\) and \({v} = {{b\exp }{\left( {  \frac{x}{a}} \right)}\sin {\left( {\frac{y}{a}} \right)}}\). The first phase mask performs the mapping from a tube to a toroid, and its phase profile can be derived through stationary phase approximation as^{4,5}
where k is the wave number, d is the distance between the two phase masks, and a and b are used to adjust the beam size and position in the (u, v) plane. The simulated evolution process for the mapping from a tube to a vortex ring in freespace propagation is labelled at positions from 1 to 5 in Fig. 1. The second phase mask recollimates the light. In the timereversal direction, the second phase mask performs a Cartesiantologpolar coordinate transformation. Similarly, its phase profile can be derived as
After travelling through the two phase masks, the spatiotemporal vortex tube will be mapped into a photonic toroidal vortex.
The toroidal vortex as a solution to equation (1) in the remapped coordinates can be approximately expressed as
where \({r_\bot} = {\sqrt {{u^2} + {v^2}}}\), r_{0} is the radius of the circular vortex line, l is an integer, and z_{0} is a constant. Figure 2 shows the isointensity surface of a photonic toroidal vortex in different perspective views. The photonic toroidal vortex possesses a 3D phase structure that rotates around a closed loop, forming a ringshaped vortex line while advancing in the direction perpendicular to the ring orifice at the speed of light. In Fig. 2, the rotating spatiotemporal spiral phase is represented by colours and the direction of local orbital angular momentum (OAM) density is marked with arrows.
The experimental apparatus begins with a dispersionmanaged modelocked fibre laser that emits a chirped pulse (Fig. 3). The light from the source splits into a reference pulse and a signal pulse. The reference pulse is dechirped to a transformlimited pulse through a grating pair. A reflective mirror is mounted on a precision stage after the grating pair so that the path length difference between the reference pulse and the signal pulse can be controlled accurately. The signal pulse propagates through a 2D pulse shaper that consists of a grating, a cylindrical lens and a reflective programmable spatial light modulator (SLM 1). SLM 1 is situated in the spatial frequency–temporal frequency plane and applies a controlled spiral phase to the signal pulse. After a spatiotemporal Fourier transform, the signal pulse is converted to a wave packet carrying a spatiotemporal vortex^{24,25,26,27}. The spatiotemporal vortex pulse then travels through an afocal cylindrical beam expander and stretches in the direction of the vortex line. SLM 2 and SLM 3 form an afocal mapping system. They are programmed with the phase profiles as given by equation (2) and equation (3), respectively. Parameters a and b are chosen so that the light occupies a great portion of the liquidcrystal area. The stretched spatiotemporal vortex pulse transforms into a toroidal vortex pulse through the conformal mapping system.
To characterize the 3D field information of the toroidal vortex pulse, the transformlimited reference pulse is combined with the toroidal vortex pulse by a beamsplitter (BS 4) and the interference patterns are recorded by a chargedcoupled device (CCD) camera. The reference pulse is considerably shorter (~90 fs) than the toroidal vortex pulse (~3 ps), so, by the use of a precision stage and a movable mirror, the reference pulse interferes with each temporal slice of the toroidal vortex pulse. By utilizing the characterization method^{28,29}, the amplitude and phase information can be retrieved from the interference patterns. Consequently, the 3D isointensity profile of the toroidal vortex pulse can be reconstructed as shown in Fig. 4. Semitransparency is applied so that the ring orifice and the hidden ringshaped vortex core are clearly shown. The outer layer of the isosurface is coloured in purple and the vortex core surface is coloured in red to make the vortex core stand out. Because the phase gradient is dominant in the radial–temporal plane, the local OAM density direction is at a tangent to the vortex line, as indicated by green arrows. To present the spiral phase rotating about the vortex core, three slices are taken in the radial direction of the ring, as marked numerically (1–3). The phase profiles of the three slices in local coordinates are reconstructed and shown in Fig. 4c. All three images show a spiral phase of topological charge 1. The intensity and phase information demonstrate that the generated pulse is indeed a toroidal vortex. The experiment is carried out in air of very weak positive GVD. The topological charge of the photonic toroidal vortex in the radial–temporal plane is 1, and propagation of the photonic toroidal vortex in air over a short distance will introduce only small distortions. More details on the method and data are provided in Supplementary Figs. 3–7.
In conclusion, we have experimentally demonstrated the generation of a photonic toroidal vortex as a new solution to Maxwell’s equations through optical conformal mapping. The preparation of photonic vortex rings is of great importance for studying toroidal electrodynamics, light–matter interactions and photonic symmetry and topology. The concept and conformal mapping method can be readily extended to other spectra, such as electron beams, Xrays and even mechanical waves such as acoustics, hydrodynamics and aerodynamics, opening up new opportunities for studying vortex rings in a wide range of disciplines.
Methods
The laser source is a dispersionmanaged modelocked fibre laser with a central wavelength of 1,030 nm. As shown in Fig. 3, a chirped pulse emitted from the laser source is split by a beamsplitter (BS 1). One split pulse passes through a 2D reflective pulse shaper that consists of a grating, a cylindrical lens and a SLM positioned in a 4f configuration. The SLM is a GAEA2NIR069 made by Holoeye. A spiral phase pattern is displayed on the SLM. The phase singularity point of the spiral phase pattern is aligned with the centre of the incident pulse. The 2D pulse shaper turns a Gaussian pulse into a STOV that has a spiral phase in the x–t plane. The STOV is elongated along the vortex line direction (y direction) and becomes a spatiotemporal tube. SLM 2 and SLM 3 are loaded with the phase patterns described by equations (2) and (3), respectively. The two SLMs perform the conformal mapping and transform the spatiotemporal tube into a toroidal vortex. The other pulse split from the laser source is dechirped by a grating pair mounted on an electrically controlled precision stage (Zolix MAR2065) with a scan step of 5 μm. The transformlimited pulse is utilized as the probe pulse to interfere with each temporal slice of the toroidal vortex. The intensity and phase information of each temporal slice is retrieved from the interference fringe pattern. The 3D isosurface plot is reconstructed through stitching of the 2D temporal slice information.
Data availability
Extra data for this study are available in the Supplementary Information.
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Acknowledgements
We acknowledge support from the National Natural Science Foundation of China (NSFC; nos. 92050202 (to Q.Z.), 61875245 (to C.W.) and 61805142 (to J.C.)), Shanghai Science and Technology Committee (no. 19060502500, to Q.Z.) and Wuhan Science and Technology Bureau (no. 2020010601012169, to C.W.).
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C.W., A.C. and Q.Z. proposed the original idea and performed the theoretical analysis. C.W. performed all experiments and data analysis. Q.C. and J.C. contributed to developing the measurement method. Q.Z. guided the theoretical analysis and supervised the project. All authors contributed to writing the manuscript.
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Wan, C., Cao, Q., Chen, J. et al. Toroidal vortices of light. Nat. Photon. 16, 519–522 (2022). https://doi.org/10.1038/s4156602201013y
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DOI: https://doi.org/10.1038/s4156602201013y
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