Supertoroidal light pulses as electromagnetic skyrmions propagating in free space

Abstract

Topological complex transient electromagnetic fields give access to nontrivial light-matter interactions and provide additional degrees of freedom for information transfer. An important example of such electromagnetic excitations are space-time non-separable single-cycle pulses of toroidal topology, the exact solutions of Maxwell's equations described by Hellwarth and Nouchi in 1996 and recently observed experimentally. Here we introduce an extended family of electromagnetic excitation, the supertoroidal electromagnetic pulses, in which the Hellwarth-Nouchi pulse is just the simplest member. The supertoroidal pulses exhibit skyrmionic structure of the electromagnetic fields, multiple singularities in the Poynting vector maps and fractal-like distributions of energy backflow. They are of interest for transient light-matter interactions, ultrafast optics, spectroscopy, and toroidal electrodynamics.

Fig. 1. From toroidal to supertoroidal light pulses.

a, b Isosurfaces for the electric fields of a the fundamental TLP Re[E_θ(r, t)], and b a STLP $\text{Re}\left[E_{\theta }^{\left(\alpha \right)}\left(\mathbf{r},t\right)\right]$ of α = 5, at amplitude levels of E = ±0.1 and the Rayleigh range of q₂ = 100q₁, at different times of t = 0, ±q₂/(4c), and ±q₂/(2c). x–z cross-sections of the instantaneous electric field at y = 0. The insets in (a) and (b) are schematics of spatial topological structures of magnetic vector fields at focus (t = 0) for the fundamental TLP and STLP, respectively. The gray dots and rings mark the distribution of singularities (saddle points and vortex rings) in magnetic field, large pink arrows mark selective magnetic vector directions, and the smaller colored arrows show the skyrmionic structures in magnetic field.

Fig. 2. Electric fields of toroidal and supertoroidal light pulses.

a, b The isoline plots of the electric field in the x–z plane for a the fundamental TLP, Re[E_θ(r, t = 0)], and b the STLP of α = 5, $\text{Re}\left[E_{\theta }^{\left(\alpha =5\right)}\left(\mathbf{r},t=0\right)\right]$, in logarithmic scale. The bold black lines represent the zero-value singular lines. The dashed purple lines represent the positions of propagation distance corresponding to the transverse plots at right side. Panels a1–a3 and b1–b3 show the electric field distributions in the transverse planes. The field magnitude is plotted as contours (in logarithmic scale), while the field orientation is presented by arrow plots. Electric field zeros are marked by the black solid bold lines and black dots. Blue and red arrows represent the two opposite azimuthal directions of the electric fields. Unit for coordinates: q₁.

Fig. 3. Magnetic fields of toroidal and supertoroidal light pulses.

a, b Isoline and arrow plots of the magnetic fields in the x–z plane for a the fundamental TLP and b the STLP of α = 5, in logarithmic scale. Magnetic field singularities are marked by black dots with red arrows correspondingly marking the saddle or vortex style of the vector singularities. Panels a1 and b1 present the zoom-in plots corresponding to regions of blue boxes in (a) and (b), respectively. Panels a2 and b2 present transverse distributions of magnetic amplitude (in logarithmic scale) and normalized magnetic vectors at z = 0 planes, the positions marked by the black dashed lines in (a) and (b), respectively, where the magnetic fields vanish along the circular solid black lines with red arrows marking the styles of singularities (vortex for a2 and saddle for b2). Skyrmionic structures in magnetic fields of toroidal and supertoroidal light pulses: c Various textures of Néel-type skyrmionic structure observed at various transverse planes (see dashed purple lines in a and b) for the fundamental TLP (c1–c2) and the STLP of α = 5 (c3–c6), which are demonstrated by the arrows with color-labeled longitudinal component value of magnetic field. The up-right insert of each panel shows the basic texture of the skyrmionic structure. Unit for coordinates: q₁.

Fig. 4. Poynting vector fields of toroidal and supertoroidal light pulses.

a, b Contour and arrow plots of the Poynting vector fields in the x–z plane, in logarithmic scale, for a) the fundamental TLP and b the STLP of α = 5. Panels a1 and b1 present the zoom-in plots of the regions of blue boxes in (a) and (b), respectively. Poynting vector field zeros are marked by the black solid lines and black dots. Red and blue bold arrows highlight the regions of energy forward flow and backflow, respectively. Unit for coordinates: q₁.

Fig. 5. Fractal-like patterns in electromagnetic fields of supertoroidal pulses.

a The isoline plot of the electric field in the x–z plane for the STLP of α = 20, $\text{Re}\left[E_{\theta }^{\left(\alpha =20\right)}\left(\mathbf{r},t=0\right)\right]$, in logarithmic scale. Electric field zeros are marked by the black solid lines and black dots. Panel a1 presents the zoom-in plot of region highlighted by blue box in (a). b The isoline plot of magnetic field amplitude (in logarithmic scale) and arrow plot of normalized magnetic vectors in the x–z plane for the STLP of α = 20. Magnetic field singularities are marked by black dots with red arrows correspondingly marking the saddle or vortex style of the vector singularities. Panel b1 presents the zoom-in plot of region highlighted in (b). Subwavelength features of skyrmionic structures: c1–c4 The skyrmionic distributions of magnetic field at several transverse planes marked by dashed lines marked by “c1–c4” in (b1). d1–d4 The distribution of normalized magnetic field and its absolute value versus x for the skyrmionic structures in c1–c4. Insets illustrate the Subwavelength features at the regions highlighted by gray bands. Unit for coordinates: q₁.

Fig. 6. Evolution of on-axis singularity distribution versus supertoroidal order.

a, b The numbers (a) and positions (b) of on-axis saddle-singularities of the magnetic field of the STLP (q₂ = 20q₁) within the range of z ∈ [0, 80q₁], versus α. The blue dashed line marks where the number of singularities decreases.