Vortex rings in fluid flows have long been a source of fascination: they are localized, axisymmetric disturbances which carry linear momentum through the medium . This behavior is understood by a combination of the topology of toroidal flow around a closed ring with the physics of fluid flow around a curved vortex, which guarantees the uniform forward motion of the ring. Here, we describe the topology underlying an electromagnetic analogue of fluid vortex rings, the so-called flying doughnut pulses (FD) . These are time-dependent, axially symmetric single-cycle toroidal disturbances propagating at lightspeed along the axis of the torus. The FD fields we study follow an envelope of a focused Gaussian beam; nevertheless, we show how the vectorial structure of the FD fields, consisting of the electric field, magnetic field and Poynting vector, can be understood in terms of their topology, in particular in terms of the instantaneous, real vector field zeros which form circulations, saddle points as well as sources and sinks in two-dimensional slices of the three-dimensional cylindrically symmetric fields. This description of the electromagnetic field topology is appropriate for FD fields which are pulses, rather than polarization singularities (e.g. C lines, L surfaces on which the polarization is circular and linear, respectively , , , , ) in monochromatic fields. Indeed, FD pulses are so short that polarization ellipses cannot be naturally defined in these fields, nor even can other, broadband descriptions of electric and magnetic field behavior such as Lissajous curves , . The vector field structures we study, nevertheless, are field zeros which form a topological skeleton providing a complete description of the anatomy of FD fields.
FDs are single-cycle pulses of toroidal topology, where the electric and magnetic fields are mainly confined on the surface and volume of a torus, respectively. In contrast to the typically encountered transverse waves, this field configuration leads to longitudinal field components that are oriented along the direction of pulse propagation (see Figure 1A). FD pulses belong to a wider family of exact solutions to Maxwell’s equations that were introduced by Brittingham and Ziolkowski , ,  in the context of non-diffracting waves (focus wave modes), in which the spatial and temporal dependence cannot be separated. However, FDs can be focused and diffract without distortion similarly to monochromatic Gaussian beams. FD pulses can be distinguished in transverse electric (TE) and transverse magnetic (TM). In cylindrical coordinates (ρ, θ, z), the electric and magnetic fields of the TE FD pulse coming to focus at t=0, z=0 are given by 
where σ=z+ct, τ=z−ct, and q1, q2 play the role of the effective wavelength and Rayleigh range, respectively. In particular, the value of the ratio q2/q1 indicates whether the pulse is collimated (q2/q1>>1) or strongly focused. The electric field is azimuthally polarized with no longitudinal or radial components, whereas the magnetic field is oriented along the radial and longitudinal directions with no azimuthal component (see Figure 1A). The TM FD pulse can be obtained by an exchange of electric and magnetic fields . From the complex electromagnetic fields of Eq. (1), two different pulses can be constructed corresponding to the real and imaginary parts of the fields, respectively. Both types of pulses are exact solutions to Maxwell’s equations. The real part corresponds to a pulse, which is single-cycle in the electric field and 1½-cycle in the magnetic field, whereas the imaginary part leads to a pulse, which is 1½-cycle in the electric field and single-cycle in the magnetic field. We refer to the real part as the single-cycle pulse and to the imaginary part as the 1½-cycle pulse. Upon propagation, single-cycle (1½-cycle) transforms to the 1½-cycle (single-cycle) pulse due to the Gouy phase shift .
The presence of both longitudinal and transverse field components has led to suggestions for applications, such as charged particle acceleration , whereas even simple properties of the pulse, such as interactions with dielectric and metallic interfaces, can be nontrivial . In particular, while the fields of the TE pulse retain their orientation relative to the propagation direction upon reflection, this is not true for the TM pulse. The latter acquires a longitudinal field antiparallel to its propagation direction when it is reflected from a perfect electric conductor or a dielectric interface. These field transformations are direct consequence of the boundary conditions at the interface in combination with the complex topology of the pulse. Importantly, the doughnut-like configuration of electric and magnetic fields is topologically similar to that of the recently discovered toroidal dipole excitations, and in fact it has been shown that FD pulses are promising candidates for exciting toroidal modes and non-radiating configurations in matter , . This has been demonstrated through the interaction of the FD pulse with spherical dielectric non-dispersive nanoparticles. The multipole expansion of the radiated field was used to show that a strong toroidal moment is induced despite the fact that the particle does not possess toroidal symmetry. In addition, the cross term in the scattered intensity between the toroidal and dipole moment showed that the two modes interfere destructively and overall lead to the excitation of a dynamic anapole, which manifests as a dip in the total scattering intensity at the corresponding resonance frequency. Recently, a scheme for the generation of FDs based on spatially and temporally dispersive metasurfaces was put forward, and efforts for experimental characterization of such pulses are underway , .
Here we investigate the topological properties of FD pulses in terms of the singular points, that is, the real vector field zeros of the pulse, where either the electric field or magnetic field, and the Poynting vector vanish. We demonstrate that in contrast to the relatively simple toroidal shape presented by the high intensity areas of the pulse (see Figure 1A), a complex structure of electric and magnetic field vortices, as well as energy backflow, is revealed at the lower energy scales (Figure 1B and C). Our study focuses on the single-cycle (real) and 1½-cycle (imaginary) TE pulse; however, our conclusions hold for the TM case as well (by exchanging electric and magnetic fields). Moreover, in the following we choose a small value of the Rayleigh range q2=20q1, which places our study in the non-paraxial regime and allows us to investigate all singular points of the pulse in a relatively small region.
2 Electric field singularities
The instantaneous electric field for the single-cycle TE FD pulse is presented in Figure 2 for t=0, when the pulse comes into focus. Owing to its azimuthal polarization, the electric field vanishes along the propagation axis (ρ=0) similarly to the phase singularities observed in vector polarized monochromatic beams  (see the vertical solid black line in Figure 2A). The electric field also vanishes in the z=0 plane (see the horizontal solid black line in Figure 2B) due to the single-cycle nature of the pulse, which implies odd symmetry for the electric field upon reflection across the x-axis (at t=0). In contrast to the simple two-lobe picture presented in Figure 1A, the electric field presents a complex behavior, where it vanishes on two spherical shells (indicated by the solid circles in the left panel of Figure 2) on the positive and negative z-axis, respectively. This behavior can be more clearly observed in Figure 2A–C, where cross-sections normal to the propagation axis are presented. In accordance to Figure 1A, the electric field close to and away from the center of the pulse (cross-sections A and C) rotates counter-clockwise forming a vortex around the singularity along the propagation axis. However, at a distance of ~5q1 from its center (z=0), the electric field vanishes on a circular boundary, which corresponds to a spherical region inside which the electric field is oriented along the clockwise direction, whereas outside this region the electric field remains oriented in the opposite (counter-clockwise) direction (see cross-section B).
A similar behavior is displayed by the 1½-cycle pulse (see Figure 3), where the electric field also vanishes along the propagation axis. However, in contrast to the single-cycle pulse of Figure 2, the electric field here does not vanish on the z=0 plane; rather it is at a maximum in this region as a result of its 1½-cycle nature, which leads to even symmetry of the electric field at t=0 (see the left panel in Figure 3). As in the case of the single-cycle pulse, the electric field is zero on two spherical shells on positive and negative z, respectively, which separate regions with opposite directions of the electric field. Indeed, the electric field vortex around the propagation axis changes the direction of its rotation, from counter-clockwise at z=0 to clockwise inside the spherical regions within the circular singularities, to counterclockwise again at large values of z (see Figure 3A–C). Similarly, the electric field rotates counterclockwise away from the propagation axis.
3 Magnetic field singularities
The magnetic field of the FD pulses presents an even more complex topological behavior than the electric field. In Figure 4, we show the singularities of the modulus of the magnetic field of the (real) single-cycle pulse, defined as
In the case of the 1½-cycle (imaginary) FD pulse, the magnetic field presents a simpler topological structure with a smaller number of singularities (see Figure 5): one saddle point singularity at the origin (ρ=0, z=0) and four vortices away from the propagation (ρ=0) axis. A zoom-in of the latter singularities in the x>0 half-space is shown in Figure 5B. Here, the formation of magnetic field vortices around the singularities can be observed clearly with the vortex at z>0 (z<0) rotating counter-clockwise (clockwise). The field structure around the singularity at ρ=0, z=0 reveals a saddle point with the radial component of the magnetic field pointing away from the singularity, while the longitudinal component points toward the singularity (see Figure 5C). In contrast to the (real) single-cycle pulse, here the topological structure of the magnetic field is confined at a region close to the pulse center, while due to the even number of vortex-like regions the on-axis “isolated” singularities are absent.
4 Energy backflow and Poynting vector singularities
The singularities of the electric and magnetic field of the FD pulse lead to a complex topological behavior for the energy flow as represented by the instantaneous Poynting vector (see Figure 6). A striking observation for both the single-cycle (Figure 6A and B) and the 1½-cycle (Figure 6C and D) pulse is the presence of energy backflow (white arrows in Figure 6), a behavior similar to that predicted by Michael Berry for superposition of waves , as is the case of superoscillatory light fields . Indeed, the Poynting vector at regions of high intensity is oriented along the propagation direction (parallel to the positive z-axis; see purple arrows in Figure 6), while at regions of relatively low intensity extended energy backflow can be observed (white arrows in Figure 6). In the case of the single-cycle pulse (Figure 6A and B), the Poynting vector vanishes at the positions, where the respective electric and/or magnetic fields are zero. This leads to the vanishing Poynting vector on the xy plane at z=0 and along the propagation axis (ρ=0). In addition, the Poynting vector vanishes on the two spherical shells around the z-axis, where the electric field is zero (marked by the circular regions in Figure 6A). The Poynting vector also presents four singular points on the z-axis, where both the electric and magnetic fields vanish. All of the on-axis singular points are saddle-like with forward (backward) energy flow close to (away from) the origin. Moreover, the singularities of the Poynting vector include two additional circular regions (indicated as points in Figure 6B) that act as sources and sinks. We would like to note that in contrast to the case of continuous waves, here we are considering the instantaneous energy flow in the pulse, and hence source and sink points have no implications for energy conservation. The Poynting vector for the 1½-cycle pulse presents a very similar topological structure to that of the single-cycle pulse. Indeed, the Poynting vector here also vanishes on the propagation axis due to the azimuthally polarized electric field and on two spherical shells around the z-axis. The main difference compared to the single-cycle pulse is the absence of the singularity plane at z=0 and of the saddle point singularities on the z-axis. This is a direct result of the simpler magnetic field topology of the 1½-cycle FD, which in turn is linked to the temporal shape of the pulse.
In summary, we have presented a study of the topological structure of the FD pulse. The availability of analytical expressions for the pulse allows us to examine closely its fine-scale features revealing a large number of singularities in the electric and magnetic fields of various types, including saddle points, vortex lines and rings, planes and spherical surfaces. The field singularities lead to an even more complex topological structure for the instantaneous Poynting vector. Indeed, our results indicate that the FD pulse exhibits extended areas of energy backflow, with the Poynting vector oriented antiparallel to the pulse propagation direction. Nevertheless, this energy backflow is instantaneous and corresponds to areas of very low energy density; hence, the total energy of the pulse still propagates forward. Our analysis has focused on the TE FD pulse, which is azimuthally polarized, but the findings can be readily extended to the radially polarized TM FD by simply exchanging electric and magnetic fields. Although only a single ratio of q2/q1 has been considered here, we have observed that by moving closer to the paraxial regime (larger q2/q1 ratios) only the location of the singularities changes appreciably but not their type or number. On the other hand, the topology of the pulse experiences dramatic changes upon propagation related to the Gouy phase shift. An in-depth study of the link between the FD topology and propagation dynamics will be the focus of a future study.
Our findings put forward the FD pulse as an ideal platform to study the instantaneous field singularities of ultrafast vector polarized pulses in both the paraxial and strongly focused regime without resorting to approximations. We anticipate that our results will be relevant to the experimental and theoretical efforts involving the generation, propagation properties and light matter interactions , , , , , , , . We note that although a part of the fine topological structure in the FD pulse occurs at spatial regions of very low energy, a number of intriguing features manifest close to regions of relatively high energy (e.g. the multiple magnetic field vortices in Figure 4), which can be observed experimentally. Finally, the results presented here suggest that such ultrashort pulses can be employed to study transient phenomena in matter , , , .
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