DOI: 10.5061/dryad.95x69p8vx
title: Hybrid electromagnetic toroidal vortices

description: The ubiquitous occurrence of toroidal vortices or vortex rings in
 fluid-dynamic scenarios in nature has garnered significant attention of
 scientific frontier, whilst, the electromagnetic counterparts of which
 were only proposed recently with two distinct manifestations: vector
 toroidal pulses and scalar phase toroidal vortices. Herein, we
 theoretically propose a novel form of electromagnetic toroidal vortex
 solutions, that uniquely integrate both scalar and vector characteristics,
 challenging the prevailing notion of their mutual exclusivity. We also
 present the experimental generation of the hybrid toroidal vortex pulses
 by a compact coaxial horn emitter augmented with a radial metasurface, and
 we experimentally observe the formation of an electromagnetic vortex
 street within these pulses. These pulses reveal the connection between
 transverse orbital angular momentum, topological skyrmion textures, and
 spatiotemporal field. These attributes introduce new dimensions in
 topologically complex structured waves, opening avenues for enhanced
 free-space information transmission, topologically nontrivial light-matter
 interaction and microscopy techniques.
description: Theoretical, simulated, and measured results: calculated by
 matlab code, simulated by CST software, measured in anechoic
 chamber.
description: # Hybrid electromagnetic toroidal vortices: Open source data — We have
 submitted the theoretical, simulated, and measured results corresponding
 to each figure (**data.zip**). Inside the submitted .zip file, data
 related to each figure are stored in individual figure files named by the
 figure number, e.g. **Fig2_unit_data**. Subfiles stored in each figure
 file are named based on the specific figure number and alphabet
 (**Fig2ABC.xlsx**). Codes related to plotting these results are also
 provided in the .m files. ## Descriptions —data for figures in the main
 text ### **Fig2\_unit\_data** This file contains three subfiles,
 **Fig.2ABC.xlsx**, **Fig.2DEF.xlsx**, and **Fig.2G.xlsx**, which present
 plots of the amplitude and phase distributions of units in the kx-w domain
 and the nodal line of the unit in the kx-w domain. The data for each
 figure are stored as a matrix in the file. #### **Fig.2ABC.xlsx** * ```
 *kx/k0 (column)*: range of kx/k0 corresponding to columns of the matrix
 ``` * ``` *f (row)*: range of frequency corresponding to rows of the
 matrix ``` * ``` *Fig2(a1)*: data matrix of Fig.2A ``` * ``` *Fig2(a2)*:
 data matrix of Fig.2B ``` * ``` *Fig2(a3)*: data matrix of Fig.2C ``` ####
 **Fig.2DEF.xlsx** * ``` *kx/k0 (column)*: range of kx/k0 corresponding to
 columns of the matrix ``` * ``` *f (row)*: range of frequency
 corresponding to rows of the matrix ``` * ``` *Fig2(b1)*: data matrix of
 Fig.2D ``` * ``` *Fig2(b2)*: data matrix of Fig.2E ``` * ``` *Fig2(b3)*:
 data matrix of Fig.F ``` #### **Fig.2G.xlsx** * ``` *nodal line 1*: data
 group for nodal line 1 ``` * ``` *nodal line 2*: data group for nodal line
 2 ``` * ``` *p*: period of unit simulated under Floquet boundary condition
 along y-axis ``` * ``` *kx/k0*: ratio of wavevector along x-axis and the
 wavevector ``` * ``` *f*: frequency [GHz] ``` ### **Fig3\_scalar\_data**
 This file contains three subfiles, **Measurement**, **Simulation**, and
 **Theoretical**, related to the scalar field results in Fig. 3. Scalar
 values on x-y plane at each moment are stored as a matrix in a specific
 .csv file with a number added on the file name to represent the time
 order. #### **Measurement** * ``` *matrix columns*: y=(-380:10:380)mm ```
 * ``` *matrix rows*: x=(-380:10:380)mm ``` * ``` *overall time*:
 t=(-2.5:0.01:2.5)ns ``` #### **Simulation** * ``` *matrix columns*:
 y=(-500:2:500)mm ``` * ``` *matrix rows*: x=(-500:2:500)mm ``` * ```
 *overall time*: t=(20.01:0.03:24)ns (NB: Time zero is not set to the time
 that vortex appeared in the original data.) ``` #### **Theoretical** * ```
 *matrix columns*: y=(-400:2:400)mm ``` * ``` *matrix rows*:
 x=(-400:2:400)mm ``` * ``` *overall time*: t=(-2:0.01:2)ns ``` ###
 **Fig4\_vector\_data** This file contains three subfiles, **Measurement**,
 **Simulation**, and **Theoretical**, related to the vector field results
 in Fig. 4. Specific data for figures in Fig.4 are stored in files inside
 each subfile named by the figure number. #### **Measurement** #####
 **Fig4(IJ).xlsx** * ``` *General description*: This file can be used to
 plot Fig. 4IJ. Er component of the measured electric field is in Sheet 1
 stored as a matrix on r-t plane, and Ez component of the measured electric
 field is in Sheet 2 stored as a matrix on r-t plane. ``` * ``` *matrix
 columns*: r=(0:10:400)mm ``` * ``` *matrix rows*: t=(-1.8:0.01:2.2)ns ```
 ##### **Fig4(KL).xlsx** * ``` *General description*: This file can be used
 to plot Fig. 4KL. Ex component of the measured electric field is in Sheet
 1; Ey component of the measured electric field is in Sheet 2; Ez component
 of the measured electric field is in Sheet 3. All Ex, Ey, and Ez are
 stored as a matrix on x-y plane. ``` * ``` *matrix columns*:
 x=(-180:10:180)mm ``` * ``` *matrix rows*: y=(-180:10:180)mm ``` ####
 **Simulation** ##### **Fig4(EF).xlsx** * ``` *General description*: This
 file can be used to plot Fig. 4EF. Er component of the simulated electric
 field is in Sheet 1 stored as a matrix on r-t plane, and Ez component of
 the measured electric field is in Sheet 2 stored as a matrix on r-t plane.
 ``` * ``` *matrix columns*: r=(0:18:400)mm ``` * ``` *matrix rows*:
 t=(-2:0.06:2)ns ``` ##### **Fig4(GH).txt** * ``` *General description*:
 This file can be used to plot Fig. 4GH. Space range and Ex, Ey, Ez
 components of the simulated electric field are stored in the file. ``` *
 ``` *x*: range of the field along x-axis [mm] ``` * ``` *y*: range of the
 field along y-axis [mm] ``` * ``` *z*: position of the field on z-axis
 [mm] ``` * ``` *Ex*: Ex component of the electric field [V/m] ``` * ```
 *Ey*: Ey component of the electric field [V/m] ``` * ``` *Ez*: Ez
 component of the electric field [V/m] ``` #### **Theoretical** #####
 **Fig4(AB).xlsx** * ``` *General description*: This file can be used to
 plot Fig. 4AB. Er component of the theoretical electric field is in Sheet
 1 stored as a matrix on r-t plane, and Ez component of the measured
 electric field is in Sheet 2 stored as a matrix on r-t plane. ``` * ```
 *matrix columns*: r=(0:0.5:400)mm ``` * ``` *matrix rows*: t=(-2:0.01:2)ns
 ``` ##### **Fig4(CD).xlsx** * ``` *General description*: This file can be
 used to plot Fig. 4CD. Ex component of the theoretical electric field is
 in Sheet 1; Ey component of the measured electric field is in Sheet 2; Ez
 component of the measured electric field is in Sheet 3. All Ex, Ey, and Ez
 are stored as a matrix on x-y plane. ``` * ``` *matrix columns*:
 x=(-200:0.5:200)mm ``` * ``` *matrix rows*: y=(-200:0.5:200)mm ``` ###
 **Fig5\_vortex\_street\_data** This file contains three subfiles,
 **Measurement**, **Simulation**, and **Theoretical**, related to vortex
 streets shown in Fig. 5. Specific data for figures in Fig.5 are stored in
 files inside each subfile named by the figure number. All data are stored
 in an individual matrix on r-z plane. #### **Measurement** #####
 **Fig5(EF).xlsx** * ``` *matrix columns*: r=(-450:10:450)mm ``` * ```
 *matrix rows*: z=(50:10:1000)mm ``` #### **Simulation** #####
 **Sim\_vortex\_street.txt** * ``` *x*: range of the field along x-axis
 [mm] ``` * ``` *y*: range of the field along y-axis [mm] ``` * ``` *z*:
 range of the field along z-axis [mm] ``` * ``` *Ex*: Ex component of the
 electric field [V/m] ``` * ``` *Ey*: Ey component of the electric field
 [V/m] ``` * ``` *Ez*: Ez component of the electric field [V/m] ``` ####
 **Theoretical** ##### **Fig5(AB).xlsx** * ``` *matrix columns*:
 r=(-450:2:450)mm ``` * ``` *matrix rows*: z=(0:3:999)mm ``` —data for
 figures in the supplementary materials ### **FigS1\_data** This file
 contains three subfiles, **FigS1(FGH).xlsx**, and **FigS1(IJK).xlsx**,
 which present plots of the amplitude and phase distributions of units with
 different asymmetric levels in the kx-w domain. Data for each figure are
 stored as a matrix in the file. #### **FigS1(FGH).xlsx** * ``` *kx/k0
 (column)*: range of kx/k0 corresponding to columns of the matrix ``` * ```
 *f (row)*: range of frequency corresponding to rows of the matrix ``` *
 ``` *FigS1(d1)*: data matrix of Fig. S1F ``` * ``` *FigS1(d2)*: data
 matrix of Fig. S1G ``` * ``` *FigS1(d3)*: data matrix of Fig. S1H ``` ####
 **FigS1(IJK).xlsx** * ``` *kx/k0 (column)*: range of kx/k0 corresponding
 to columns of the matrix ``` * ``` *f (row)*: range of frequency
 corresponding to rows of the matrix ``` * ``` *FigS1(e1)*: data matrix of
 Fig. S1I ``` * ``` *FigS1(e2)*: data matrix of Fig. S1J ``` * ```
 *FigS1(e3)*: data matrix of Fig. S1K ``` ### **FigS4\_data** This file
 contains four subfiles, **FigS4(ABC).xlsx**, **FigS4(DEF).xlsx**,
 **Fig4(GHI).xlsx**, **FigS4(JKL).xlsx**, which can be used to plot the
 amplitude and phase responses of measured Er and Ez electric components at
 2.15GHz, 2.2GHz, and 2.25GHz in Fig. S4. All these results are stored as
 an individual matrix in the subfiles (Sheet1: 2.15GHz, Sheet2: 2.2GHz, and
 Sheet3: 2.25GHz) with the same range on x-y plane. ####
 **FigS4(ABC).xlsx**: amplitude responses of measured Er fields * ```
 *matrix columns*: y=(-350:10:350)mm ``` * ``` *matrix rows*:
 x=(-350:10:350)mm ``` #### **FigS4(DEF).xlsx**: phase responses of
 measured Er fields * ``` *matrix columns*: y=(-350:10:350)mm ``` * ```
 *matrix rows*: x=(-350:10:350)mm ``` #### **FigS4(GHI).xlsx**: amplitude
 responses of measured Ez fields * ``` *matrix columns*: y=(-350:10:350)mm
 ``` * ``` *matrix rows*: x=(-350:10:350)mm ``` #### **FigS4(JKL).xlsx**:
 phase responses of measured Ez fields * ``` *matrix columns*:
 y=(-350:10:350)mm ``` * ``` *matrix rows*: x=(-350:10:350)mm ``` ###
 **FigS5\_data** This file contains two subfiles, **FigS5(ABC).xlsx** and
 **FigS5(DEF).xlsx**, which can be used to plot the measured scalar and
 vector spatiotemporal fields under different radial slices in Fig. S5.
 Data for the three chosen slices are stored in Sheet1, Sheet2, and Sheet3
 in the subfiles in **FigS5(ABC).xlsx** for scalar results and in
 **FigS5(DEF).xlsx** for vector results. All these results are stored as an
 individual matrix with the same range on r-t plane in the subfiles. ####
 **FigS5(ABC).xlsx** * ``` *matrix columns*: t=(-1.8:0.01:2.2)mm ``` * ```
 *matrix rows*: r=(0:1:350)mm ``` #### **FigS5(DEF).xlsx** * ``` *matrix
 columns*: t=(-1.8:0.01:2.2)mm ``` * ``` *matrix rows*: r=(0:1:350)mm ```
 ### **FigS6\_Er\_data** This file contains three subfiles,
 **FigS6(AB).xlsx**, **FigS6(CD).xlsx**, and **FigS6(EF).xlsx**, which can
 be used to plot the theoretical, simulated, and experimental results of
 electric fields on spatiotemporal planes (r-t plane and z-t plane) in Fig.
 S6. There are two sheets in each subfile in which Sheet1 is for electric
 field on r-t plane and Sheet2 is for electric field on z-t plane. Data in
 each subfile are stored as a matrix. #### **FigS6(AB).xlsx** * ```
 *Sheet1*: theoretical electric field on r-t plane ``` * ``` *matrix
 columns*: r=(0:2:1000)mm ``` * ``` *matrix rows*: t=(-12.5:0.01:12.5)ns
 ``` * ``` *Sheet2*: theoretical electric field on z-t plane ``` * ```
 *matrix columns*: z=(0:3:3000)mm ``` * ``` *matrix rows*:
 t=(-12.5:0.01:12.5)ns ``` #### **FigS6(CD).xlsx** * ``` *Sheet1*:
 simulated electric field on r-t plane ``` * ``` *matrix columns*:
 r=(0:2:1000)mm ``` * ``` *matrix rows*: t=(-12.1:0.03:12.9)ns ``` * ```
 *Sheet2*: simulated electric field on z-t plane ``` * ``` *matrix
 columns*: z=(0:2:3000)mm ``` * ``` *matrix rows*: t=(-14.9:0.03:11.1)ns
 ``` #### **FigS6(EF).xlsx** * ``` *Sheet1*: experimental electric field on
 r-t plane ``` * ``` *matrix columns*: r=(0:10:1000)mm ``` * ``` *matrix
 rows*: t=(-10:0.01:15)ns ``` * ``` *Sheet2*: experimental electric field
 on z-t plane ``` * ``` *matrix columns*: z=(50:10:1000)mm ``` * ```
 *matrix rows*: t=(-12.7:0.01:12.3)ns ``` ### **FigS7\_data** This file
 contains two subfiles, **FigS7(AB).xlsx**, and **FigS7(CD).xlsx**, which
 can be used to plot the theoretical, and simulated results of magnetic
 fields on spatiotemporal planes (r-t plane and z-t plane) in Fig. S7.
 There are two sheets in each subfile in which Sheet1 is for magnetic field
 on r-t plane and Sheet2 is for magnetic field on z-t plane. Data in each
 subfile are stored as a matrix. #### **FigS7(AB).xlsx** * ``` *Sheet1*:
 theoretical magnetic field on r-t plane ``` * ``` *matrix columns*:
 r=(0:2:1000)mm ``` * ``` *matrix rows*: t=(-12.5:0.01:12.5)ns ``` * ```
 *Sheet2*: theoretical electric field on z-t plane ``` * ``` *matrix
 columns*: z=(0:3:3000)mm ``` * ``` *matrix rows*: t=(-12.5:0.01:12.5)ns
 ``` #### **FigS7(CD).xlsx** * ``` *Sheet1*: simulated magnetic field on
 r-t plane ``` * ``` *matrix columns*: r=(0:2:1000)mm ``` * ``` *matrix
 rows*: t=(-12.1:0.03:12.9)ns ``` * ``` *Sheet2*: simulated electric field
 on z-t plane ``` * ``` *matrix columns*: z=(0:2:3000)mm ``` * ``` *matrix
 rows*: t=(-14.9:0.03:11.1)ns ``` ### **FigS8\_data** This file contains
 three subfiles, **FigS8(A).xlsx**, **FigS8(B).xlsx**, and
 **FigS8(C).xlsx**, which can be used to plot spectrum distribution of
 radially polarized electric field at different propagation distances in
 Fig. S8 for experimental, simulated, and theoretical results. There are
 three sheets in each subfile to store the electric field spectrum at
 z=300mm, z=600mm, and z=900mm. Data in each subfile are stored as a matrix
 on the frequency-radius (f-r) plane. #### **FigS8(A).xlsx** * ```
 *Sheet1*: spectrum distribution of radially polarized electric field at
 z=300mm (experimental) ``` * ``` *Sheet2*: spectrum distribution of
 radially polarized electric field at z=600mm (experimental) ``` * ```
 *Sheet3*: spectrum distribution of radially polarized electric field at
 z=900mm (experimental) ``` * ``` *matrix columns*: f=(2.2:0.05:2.4)GHz ```
 * ``` *matrix rows*: r=(0:10:1000)mm ``` #### **FigS8(B).xlsx** * ```
 *Sheet1*: spectrum distribution of radially polarized electric field at
 z=300mm (simulated) ``` * ``` *Sheet2*: spectrum distribution of radially
 polarized electric field at z=600mm (simulated) ``` * ``` *Sheet3*:
 spectrum distribution of radially polarized electric field at z=900mm
 (simulated) ``` * ``` *matrix columns*: f=(2:0.05:2.2)GHz ``` * ```
 *matrix rows*: r=(0:10:1000)mm ``` #### **FigS8(C).xlsx** * ``` *Sheet1*:
 spectrum distribution of radially polarized electric field at z=300mm
 (theoretical) ``` * ``` *Sheet2*: spectrum distribution of radially
 polarized electric field at z=600mm (theoretical) ``` * ``` *Sheet3*:
 spectrum distribution of radially polarized electric field at z=900mm
 (theoretical) ``` * ``` *matrix columns*: f=(2:0.05:2.2)GHz ``` * ```
 *matrix rows*: r=(0:10:1000)mm ``` ### **FigS9\_data** This file contains
 one subfile, **FigS9.xlsx**, which can be used to plot the simulated
 vector field of the coaxial horn emitter in Fig. S9. #### **FigS9.xlsx** *
 ``` *z_reference*: the referred range along z-axis transferred from time
 [mm] (NB: *z_reference* is actually the time range shown by c*t in this
 referred form.) ``` * ``` *r*: radius of selected field under cylindrical
 coordinate [mm] ``` * ``` *Ez*: longitudinally polarized electric field
 [V/m] ``` * ``` *Er*: radially polarized electric field [V/m] ``` ###
 **FigS10\_data** This file contains one subfile, **FigS10.xlsx**, which
 can be used to plot the loss of the microwave metasurface under different
 substrate loss tangents in Fig. S10. #### **FigS10.xlsx** * ``` *f*:
 frequency range [GHz] ``` * ``` *tanδ=0.05*: material loss of microwave
 metasurface under loss tangent of 0.05 ``` * ``` *tanδ=0.001*: material
 loss of microwave metasurface under loss tangent of 0.001 ``` ###
 **FigS11\_data** This file contains one subfile, **FigS11.xlsx**, which
 can be used to plot the field distribution on a radial cross-section when
 the loss tangent of the metasurface substrate is 0.05 in Fig. S11. There
 are two sheets in the subfile. Sheet1 is the radially polarized electric
 field, and sheet2 is the longitudinally polarized field. All data are
 stored as a matrix with the same range on r-t plane. #### **FigS11.xlsx**
 * ``` *matrix columns*: t=(29.01:0.03:36.99)ns (NB: Time zero is not set
 to the time that vortex appeared in the original data.) ``` * ``` *matrix
 rows*: r=(0:2:500)mm ``` ## Code/Software Matlab 2020 is required to run
 the provided .m codes. Annotations (%explanation) are provided to separate
 the codes for corresponding figures. The specific part of the code to plot
 the results in each file are listed in the following: ###
 **theoretical.m** This file can be used to calculate the theoretical
 results. ### **scalar\_field.m** This file can be used to plot all scalar
 results. * *%3(a)*: code to plot the 3D isosurface in Fig. 3ADG * *%3(b)*:
 code to plot phase in spatiotemporal domain in Fig. 3BEF * *%3(c)*: code
 to plot radially polarized scalar electric field for all scalar field on
 r-t plane and z-t plane ### **vector\_field.m** This file can be used to
 plot all vector results except the vortex street. * *%code for
 Fig.4(a)(b)*: code to plot the vector field on r-t plane * *%code for
 Fig.4(c)*: code to plot the vector field on x-y plane * *%code for Fig.
 4(d)*: code to plot the vector field on a sphere ### **vortex\_street.m**
 This file can be used to plot the vortex street. * *%code for Fig5*: code
 to plot the vortex street