Hybrid electromagnetic toroidal vortices
Abstract
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.
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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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*matrix columns*: r=(-450:10:450)mm -
*matrix rows*: z=(50:10:1000)mm
Simulation
Sim_vortex_street.txt
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*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
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*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
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*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
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*matrix columns*: y=(-350:10:350)mm -
*matrix rows*: x=(-350:10:350)mm
FigS4(DEF).xlsx: phase responses of measured Er fields
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*matrix columns*: y=(-350:10:350)mm -
*matrix rows*: x=(-350:10:350)mm
FigS4(GHI).xlsx: amplitude responses of measured Ez fields
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*matrix columns*: y=(-350:10:350)mm -
*matrix rows*: x=(-350:10:350)mm
FigS4(JKL).xlsx: phase responses of measured Ez fields
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*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
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*matrix columns*: t=(-1.8:0.01:2.2)mm -
*matrix rows*: r=(0:1:350)mm
FigS5(DEF).xlsx
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
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*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
Theoretical, simulated, and measured results: calculated by matlab code, simulated by CST software, measured in anechoic chamber.