Vortex rings in fluid flows have long been a source of fascination: they are localized, axisymmetric disturbances which carry linear momentum through the medium [1]. 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) [2]. 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 [3], [4], [5], [6], [7]) 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 [8], [9]. 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 [10], [11], [12] 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 [2]

Figure 1: Large- and fine-scale topology of the single-cycle TE FD pulse.

(A) Isosurface of the electric field consisting of two doughnut-shaped regions. Red and blue colors indicate the two half-cycles of the pulse, respectively. Green lines show the (radially and longitudinally oriented) magnetic field winding around the (azimuthally oriented) electric field, while red arrows mark the direction of the latter. The solid green arrow indicates the direction of pulse propagation along the positive *z*-axis. (B, C) Isosurface plots of the logarithm of the normalized electric (B) and magnetic (C) field modulus at different field values. Purple color marks the strong field central regions of the pulse presented in (A), while light blue indicates the location of the zeros of the fields. In order to reveal the complex field structure, only the half of the pulse that lies in the negative *x*-space is presented in (B) and (C). In all panels, the pulse is considered at focus (*t*=0), while *q*_{2}=10*q*_{1}.

$$\begin{array}{l}\overrightarrow{E}={E}_{\theta}\widehat{\theta}=-4i{f}_{0}\sqrt{\frac{{\mu}_{0}}{{\epsilon}_{0}}}\frac{\rho \mathrm{(}{q}_{1}+{q}_{2}-2ict\mathrm{)}}{{[{\rho}^{2}+\mathrm{(}{q}_{1}+i\tau \mathrm{)}\text{\hspace{0.17em}}\mathrm{(}{q}_{2}-i\sigma \mathrm{)}]}^{3}}\widehat{\theta}\\ \overrightarrow{H}={H}_{\rho}\widehat{\rho}+{H}_{z}\widehat{z}=4{f}_{0}\hspace{0.17em}[i\frac{\rho \mathrm{(}{q}_{2}-{q}_{1}-2iz\mathrm{)}}{{[{\rho}^{2}+\mathrm{(}{q}_{1}+i\tau \mathrm{)}\text{\hspace{0.17em}}\mathrm{(}{q}_{2}-i\sigma \mathrm{)}]}^{3}}\widehat{\rho}\\ \text{}-\frac{{\rho}^{2}-\mathrm{(}{q}_{1}+i\tau \mathrm{)}\text{\hspace{0.17em}(}{q}_{2}-i\sigma \text{)}}{{[{\rho}^{2}+\mathrm{(}{q}_{1}+i\tau \mathrm{)}\text{\hspace{0.17em}}\mathrm{(}{q}_{2}-i\sigma \mathrm{)}]}^{3}}\widehat{z}],\end{array}$$(1)

where *σ*=*z*+*ct*, *τ*=*z*−*ct*, and *q*_{1}, *q*_{2} play the role of the effective wavelength and Rayleigh range, respectively. In particular, the value of the ratio *q*_{2}/*q*_{1} indicates whether the pulse is collimated (*q*_{2}/*q*_{1}>>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 [2]. 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 [13].

The presence of both longitudinal and transverse field components has led to suggestions for applications, such as charged particle acceleration [2], whereas even simple properties of the pulse, such as interactions with dielectric and metallic interfaces, can be nontrivial [14]. 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 [15], [16]. 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 [17], [18].

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 *q*_{2}=20*q*_{1}, 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.

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