# Nondiffracting Supertoroidal Pulses: Optical “Kármán Vortex Streets”

Yijie Shen<EMAIL_ADDRESS>Nikitas Papasimakis  Optoelectronics Research
Centre & Centre for Photonic Metamaterials, University of Southampton,
Southampton SO17 1BJ, United Kingdom  Nikolay I. Zheludev  Optoelectronics
Research Centre & Centre for Photonic Metamaterials, University of
Southampton, Southampton SO17 1BJ, United Kingdom  Centre for Disruptive
Photonic Technologies, School of Physical and Mathematical Sciences and The
Photonics Institute, Nanyang Technological University, Singapore 637378,
Singapore

###### Abstract

Recently introduced supertoroidal light pulses [Nat. Commun. 15, 5891 (2021)]
are a family of space-time nonseparable freespace electromagnetic excitations
with unique topological properties including skyrmionic field configurations,
fractal-like patterns of singularities, and extended areas of energy backflow.
Here we report nondiffracting supertoroidal pulses (ND-STPs), propagation-
robust skyrmionic and vortex-ring field that endure the singular
configurations over arbitrary propagation distances. Intriguingly, the field
structure in of ND-STPs has a strong similarity with a von Kármán vortex
street, a pattern of swirling vortices observed in fluid and gas dynamics that
is responsible for the “singing” of suspended telephone lines in wind. We
argue that ND-STPs are of interest as directed energy channels for telecom
applications.

The topological properties of spatial light fields have been a subject of
fascination and intense research interest over the last half century
nye1974dislocations ; berry2000making ; dennis2009singular  with implications
for light-matter interactions bao2021light ; ozawa2019topological ;
malinauskas2016ultrafast , nonlinear physics luu2015extreme ;
krogen2017generation ; shen2017gain , spin-orbit coupling rego2019generation ;
dorney2019controlling ; shen2019optical , microscopy and imaging
davis2020ultrafast ; pu2021unlabeled ; pu2020label , metrology
yuan2019detecting , and information transfer xie2018ultra ; qiao2020multi ;
wan2021divergence . Light pulses can also be simultaneously structured in
space and time domains chong2020generation ; bliokh2021spatiotemporal ;
kondakci2017diffraction ; yessenov2019weaving ; zdagkas2021observation . Among
them, the toroidal pulses are of particular interest as they can engage
toroidal excitations in matter papasimakis2016electromagnetic ;
raybould2017exciting ; zdagkas2021observation  and exhibit space-time
nonseparability and isodiffraction zdagkas2020space ; shen2021measures ;
shen2022nonseparable . As a recent generalization of toroidal pulses, the
supertoroidal pulses (STPs) also exhibit striking topology including fractal-
like singularities, vortex rings, energy backflows, and optical skyrmionic
patterns zdagkas2019singularities ; shen2021supertoroidal .

In this Letter, we show that the transverse divergence of STPs can be tuned to
archive nondiffracting supertoroidal pulses (ND-STPs). Intriguingly, the
structure of an ND-STP resembles that of Kármán vortex street (KVS), staggered
vortex arrays observed in fluid and gas dynamics 1954aerodynamics , also
previously observed in CW structured light fields scheuer1999optical ;
molina2002observation ; molina2002singular ; shen2018vortex .

Figure 1: An elementary toroidal pulse hellwarth1996focused  (a) and a ND-STP
(b): the isosurfaces with $\pm 0.15$ amplitude level and the amplitude
distributions on $x$-$z$ plane at various times $t=0,\pm q_{2}/(2c)$. Blue,
red, and yellow arrows mark optical cycle ($q_{1}$), Rayleigh range ($q_{2}$),
and transverse divergence ($q_{3}$), respectively. Parameters in simulation:
$q_{2}/q_{1}=40$ for the toroidal pulse; $q_{2}/q_{1}=100$ and $q_{3}/q_{1}=1$
for the ND-STP. Purple arrows mark the propagation profile. See Supplementary
Video 1 for the dynamic evolutions. Figure 2: (a) A characteristic example of
a transversely divergent pulse with $q_{3}/q_{1}=20$, which is represented by
electric field amplitude isosurfaces at levels $\pm 0.15$ of the maximum field
amplitude in three different moments in time $t=-q_{2}/(2c),0,q_{2}/(2c)$. The
presented case lies between the fundamental toroidal pulse [$q_{3}\to\infty$;
see Fig. Fig. 1(a)] and a ND-STP [$q_{3}=q_{1}$; see Fig. 1(b)]. The
transition from the former to the latter can be seen in Supplementary Video 1.
(b) The evolution of pulse width in the transverse upon propagation for
different $q_{3}$ values. The inset shows a 1D cross-section of the pulse
intensity in the transverse plane, where the pulse width is quantified by the
full width at half maximum (FWHM). c-f, The spatiotemporal evolution of
transversely divergent pulses with various $q_{3}$ values of 100, 20, 5, and
1, respectively. In each panel, the purple dashed lines mark the FWHM of the
corresponding pulses. Unit of length: $q_{1}$.

STPs and ND-STPs originate from the “electromagnetic directed-energy pulse
trains” (EDEPT) introduced by Ziolkowski ziolkowski1989localized . Following
Ziolkowski’s theory ziolkowski1989localized , the solutions are derived
through a scalar generating function $f(\mathbf{r},t)$ fulfilling the scalar
wave equation,
$(\nabla^{2}-\frac{1}{{{c}^{2}}}\frac{{{\partial}^{2}}}{\partial{{t}^{2}}})f\left(\mathbf{r},t\right)=0$,
where $\mathbf{r}=(r,\theta,z)$ represents spatial cylindrical coordinate, $t$
is time, $c=1/\sqrt{\varepsilon\mu}$ is the speed of light, and the
$\varepsilon$ and $\mu$ are the permittivity and permeability of medium. Here
$f(\mathbf{r},t)=f_{0}{e}^{-s/{q_{3}}}(q_{1}+i\tau)^{-1}(s+q_{2})^{-\alpha}$,
where $f_{0}$ is a normalizing constant, $s=r^{2}/(q_{1}+i\tau)-i\sigma$,
$\tau=z-ct$, $\sigma=z+ct$, $q_{1},q_{2},q_{3}$ are real positive parameters
with units of length, and the real dimensionless parameter $\alpha$ must
satisfy $\alpha\geq 1$, to ensure finite energy of the pulse
ziolkowski1989localized ; hellwarth1996focused . In order to retrieve the
electromagnetic fields, we construct the Hertz potentials
$\mathbf{\Pi}(\mathbf{r},t)=\mathbf{\hat{n}}f(\mathbf{r},t)$, where
$\mathbf{\hat{n}}$ can be arbitrary vector operator, and then the
electromagnetic fields of transverse electric (TE) mode can be obtained by
ziolkowski1989localized ; hellwarth1996focused :
$\mathbf{E}(\mathbf{r},t)=-{{\mu}_{0}}\frac{\partial}{\partial
t}\bm{\nabla}\times\mathbf{\Pi}$,
$\mathbf{H}(\mathbf{r},t)=\bm{\nabla}\times\left(\bm{\nabla}\times\mathbf{\Pi}\right)$.
If the Hertz potential is set as a transverse vortex vector field, i.e.
$\mathbf{\hat{n}}=\bm{\nabla}\times\mathbf{\hat{z}}$, the resulting
electromagnetic field is an azimuthally polarized toroidal pulse
hellwarth1996focused , as shown in Fig. 1(a). The transverse magnetic (TM)
mode can be obtained by exchanging electric and magnetic fields of the TE
mode. If the Hertz potential is defined as
$\mathbf{\hat{n}}=\mathbf{\hat{x}}$, the result is a linearly polarized
pancake-like pulse (polarized along $y$-axis) feng1999spatiotemporal . In
prior works, the conditions $q_{1}\ll q_{2}$, $\alpha=1$, and $q_{3}\to\infty$
were assumed to generate various focused structured pulses
hellwarth1996focused ; feng1999spatiotemporal ; feng1998gouy ;
lekner2004localized ; lekner2004helical , where $q_{1}$ and $q_{2}$ determin
the wavelength and the longitudinal divergence or Rayleigh length,
$z_{0}=q_{2}/2$ of the pulses, respectively, as marked in Fig. 1(a). The case
of $\alpha\geq 1$ and $q_{3}\to\infty$ was recently studied leading to the
introduction of supertoroidal pulses shen2021supertoroidal . In this work, we
explore STPs with finite $q_{3}$ (see Supplementary Material for detailed
derivation). A characteristic example of such a pulse with $q_{3}=q_{1}$ and
$\alpha=1$ is plotted in Fig. 1(b). Here, $q_{3}$ defines the degree of
transverse divergence: with decreasing value of $q_{3}$ the pulse envelope is
gradually squeezed into a dumbbell-like shape and eventually becomes
nondiffracting for $q_{3}=q_{1}$ and $\alpha=1$, fulfilling the definition of
nondiffraction $|E(r,\theta,z,t)|=|E(r,\theta,z+\Delta z,t+\Delta z/v)|$
hernandez2013non , where $\Delta z$ is a given propagation distance and $v$ is
group velocity of the wave here $v=c$, see Supplementary for the proof.

Figure 3: The singular structures of electric field evolving with propagation
versus the time for a focused supertoroidal pulse (a) and a ND-STP (b): each
panel presents an isoline plot of the logarithm of the electric field of the
pulse in the $x$-$z$ plane at a given time. The singularities are highlighted
by bold black lines. Unit of length: $q_{1}$. See Supplementary Video 2 for
the dynamic evolutions of figures (a) and (b), and other general cases with
different parameters, respectively.

The evolution of the pulse from diffracting to nondiffracting is illustrated
in Fig. 2. An intermediate case of a weakly-diffracting supertoroidal pulse in
terms of $q_{1}$, $q_{2}$, and $q_{3}$, is presented Fig. 2(a). To illustrate
the dependence of the (non)diffracting nature of the pulse on $q_{3}$, we
calculate the $z$-dependent full width at half maximum (FWHM) radius of the
transverse intensity pattern of the STPs with $q_{2}=100q_{1}$ and $q_{3}$
varying from infinity to $q_{1}$, for propagation distances from focus to
$z=10^{3}q_{1}$, see Fig. 2(b). In the extreme case of $q_{3}\to\infty$
(fundamental toroidal light pulse), the FWHM of the pulse follows a hyperbolic
trajectory similar to a conventional focused Gaussian beam
hellwarth1996focused . With decreasing $q_{3}$, the divergence becomes weaker
and the pulse approaches a nondiffracting state for $q_{3}\leq 5q_{1}$.
Figures 2(c)-2(f) show the spatiotemporal evolution upon propagation for
toroidal pulses with different $q_{3}$ values. Here, decreasing $q_{3}$,
results in a faster spatiotemporal evolution of the cycle structure of the
pulse (see Supplementary Video 1), due to its increasingly complex shape. When
the $q_{3}$ value is decreased further to $q_{1}$, then the pulse becomes
X-shaped. Note that $q_{3}$ can be an arbitrarily small positive real number
and can take values smaller than $q_{1}$, in which case the pulse remains
nondiffracting and becomes further squeezed. Figure 1(b) plots the special
case of $q_{3}=q_{1}$, where the X-shape reveals the conical structure in
space, accommodating multiple cycles along the conical surface. Upon
propagation, the cycle structure keeps evolving on the conical surface akin to
a breather (see Supplementary Video 1). Note that similar X-type
nondiffracting pulses have been considered previously both theoretically and
experimentally and are typically termed Bessel-X pulses saari1997evidence ;
besieris20052+ ; bowlan2009measuring . However, previous works focused on
scalar long-pulses within the slowly-varying amplitude envelope approximation.
Here, our ND-STPs are few-cycle, space-time nonseparable with nontrivial
electromagnetic toroidal topology lacking in the Bessel-X pulses. A typical
example involves the intriguing skyrmions and KVS structures is discussed in
this work.

The emergence of ND-STPs allows us to explore intriguing topological optical
effects. In our recent work, we showed that supertoroidal pulses exhibit a
complex topology (controlled by the parameter $\alpha$), including self-
similar fractal-like patterns, matryoshka-like singularity arrays, skyrmions,
and areas of energy backflow shen2021supertoroidal . Here, we show that such
topological structures are also present in nondiffracting supertoroidal
pulses. However, whereas in the former case, the self-similar topological
structures were only instantaneously observed at focus, in the ND-STP, they
can be observed over arbitrarily long distances [see example in Fig. 3(b)].
For instance, we have shown in our previous work that STPs exhibit a complex
topology that evolves rapidly upon pulse propagation with elements of self-
similarity consisting of concentric matryoshka-like spherical shells over
which the electric field vanishes [see Fig. 3(a)]. Similar propagation-robust
fractal-like topological structures also exist in the ND-STP [see Fig. 3(b)]
and persist upon pulse propagation (see also Supplementary Video 2).

Figure 4: Magnetic field (a) and Poynting vector (b) distributions of a ND-STP
with $q_{2}/q_{1}=100$ and $q_{3}/q_{1}=1$ at $t=0$. The contours show the
logarithm of the modulus of the corresponding vectors. (a1) A zoom-in of the
magnetic field with the arrow plot showing the vector distribution, the black
dots referring to the singularities, and surrounding red arrows marking the
type of the vector singularities (vortex and saddle) serving as guide to the
eye. (a2) Arrow plot showing a skyrmionic structure of magnetic field in the
transverse ($x$-$y$) plane at $z=10$ (indicated by the purple solid line in
(a1)). (b1) A zoom-in of the Poynting vector field, where the arrow plot
corresponds to the vector distribution, solid black lines and dots mark the
zeros of the Poynting vector, red and blue arrows indicate areas with forward
and backward energy flow, respectively. Unit of length: $q_{1}$.

The complex topology of the ND-STPs manifests also in the magnetic field
distribution. Figure 4(a) shows the magnetic field distribution of a ND-STP
with $q_{2}/q_{1}=100$ and $q_{3}/q_{1}=1$ at $t=0$. The magnetic field
includes both radial and longitudinal components resulting in vector
singularities of vortex and saddle types. The saddle-type singularities are
distributed along the propagation axis, while the vortices trace an off-axis
trajectory, see Fig. 4(a1). Such a singularity distribution induces multiple
electromagnetic skyrmions at transverse planes of the pulse, see Fig. 4(a2).
In contrast to the propagating electromagnetic skyrmions in prior works that
only exist around beam focus and collapse rapidly upon propagation
shen2021supertoroidal ; shen2021topological ; shen2022generation , here in the
ND-STP, the skyrmions persist upon propagation with their topological texture
exhibiting a periodic behaviour alternating between four different skyrmion
types (combination of opposite two polarities and two helical angles), see
Supplementary Video 3.

Figure 5: (a) Visualization of a KSV behind a cylinder in air, where the flow
is visualized by releasing glycerol vapour in the air near the cylinder. (b)
Simulation of a 2D KSV, where the right- and left-handed vortices were
highlighted by purple and cyan, respectively. (c) Artistic impressions of a
vortex-ring street analogue to the structure of magnetic field of a ND-STP.

Figure 4(b) shows the Poynting vector field distribution of the ND-STP, which
possesses layered energy forward-flow and backflow structures, see Fig. 4(b1).
Interestingly, energy transport is mediated by the vortex arrays of the
magnetic field. Vortices on the front half of the pulse act as energy sources,
whereas the vortices in the rear half behave as sinks. In between sources and
sinks, we observe areas of extended backflow. In areas of high intensity,
energy flows primarily forward, ensuring the propagation of the pulse.

Moreover, we observe that the vortex arrays exhibited by the ND-STPs form a
striking trail of two-vortex clusters (with opposite circulations, namely a
vortex dipole) propagating in a periodic staggered manner, evocative of the
KVS structure [Fig. 5(a)]. In fluid dynamics, a KVS is such a classic pattern
of swirling vortices caused by a nonlinear process of vortex shedding, which
refers to the unsteady separation of flow of a fluid around blunt bodies.
Vortex-street-like optical fields were reported previously in stationary
optical fields exhibiting phase vortex patterns scheuer1999optical ;
molina2002observation ; molina2002singular ; shen2018vortex . In contrast,
here we observe KVS-like structures in propagating electromagnetic
spatiotemporal pulses in linear regime. In fluid dynamics, a KVS is a pattern
of repeating swirling vortices constructed in the flow velocity field, whilst
in our optical analogy, the KVS pattern is constructed by the electric or
magnetic field of ND-STPs. Moreover, whereas KVS in fluid dynamics is a
nonlinear phenomenon, here we show that similar effects can be observed in a
linear system. The analogy between KVS in fluid flows and ND-STPs can be drawn
further by considering for instance the motion of electrons along the vortex
streets of a TM ND-STP or the propagation of supertoroidal pulses in nonlinear
media. Finally, note that while the KVS is usually demonstrated in 2D [Fig.
5(b)], here the KVS in ND-STP corresponds to a 3D set of vortex rings, due to
the cylindrical symmetry [Fig. 5(c)].

ND-STPs are vector polarized, space-time nonseparable pulses and as such they
can be generated by spatially gradient metasurfaces, similarly to the toroidal
pulses discovered by Helwarth and Nouchi hellwarth1996focused ;
zdagkas2021observation . Of particular importance here is the relation between
the size of the generating metasurface (aperture) and the distance over which
the generated pulses propagate without diffraction. We have shown that in
contrast to the well known Bessel beams, the field of the ND-STPs is
exponentially confined (see Supplementary). We argue that the latter is less
sensitive to the metasurface size and thus practical implementations of the
latter remain closer to their ideal form.

In conclusion, we demonstrate nondiffracting toroidal pulses. Importantly, the
sophisticated vector field configuration of ND-STPs induce robust topological
structures including fractal-like singularities, skyrmions, vortex rings, and
energy backflows, all of which can stably propagate at an infinitely long
distance. Due to its propagation-robust topology, the ND-STP acts as a
spectacular display of striking trail of staggered electromagnetic vortex
dipole arrays with stable propagation, evocative of the classic KVS. The ND-
STPs and optical KVSs unveil intriguing analogies between fluid transport and
flow of energy in structured light. In particular, the robust topological
structure of ND-STPS that remains invariant upon propagation could be used for
long-distance information transfer encoded in the topological features of the
pulses. Finally, we anticipate that ND-STPs will inspire potential
applications such as light-matter interactions, superresolution microscopy,
telecommunications, remote sensing and lidar.

Acknowledgments. The authors acknowledge the supports of the MOE Singapore
(MOE2016-T3-1-006), the UKs Engineering and Physical Sciences Research Council
(grant EP/M009122/1, Funder Id: http://dx.doi.org/10.13039/501100000266), the
European Research Council (Advanced grant FLEET-786851, Funder Id:
http://dx.doi.org/10.13039/501100000781), and the Defense Advanced Research
Projects Agency (DARPA) under the Nascent Light Matter Interactions program.

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