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As a new kind of electromagnetic pulses with finite energy, the flying electromagnetic toroid (FET), also called as the flying electromagnetic doughnut, has significant potential applications, such as the excitation of anapole non-radiation mode and the acceleration of charged particles. To show the propagation characteristic of FET, the spatial distribution and spectrum characteristic of the transverse and longitudinal components of FET and its topology evolution in the propagation process are discussed in this paper. Without loss of generality, we theoretically research the longitudinal field and transverse field of the transverse magnetic (TM) FET based on the real part of FET’s propagation equations. The field distribution, topology, and spectrum when the FET propagates to different positions can be calculated by assigning corresponding values to the time variable in FET’s propagation equations, therefore, the propagation characteristics of FET can be studied accurately in theory. The magnetic field of TM FET is distributed into rings in the plane vertical to the propagation direction and the electric field of TM FET is rotated around the magnetic field, which means the FET has a hypertorus topology. All the field components of FET are rotationally symmetric in the plane vertical to the propagation direction. The FET’s center is the maximum position of the longitudinal electric field component and the null position of the transverse electric and magnetic field components. Maximum values of FET’s longitudinal field are always located on the central line of FET’s propagation path and decrease gradually in the propagation process. Different from the longitudinal field, the maximum value of FET’s transverse field gradually moves away from FET’s center. The theoretical research indicates that the FET spreads quite slowly in its early propagation state and spreads linearly after propagating a long distance, which has the slowly spreading propagation characteristic even in the so-called focused range with stable toroidal topological structure. The further spectrum analysis shows that the high-frequency components spread less than the low-frequency components and the high-frequency components play a vital role in the topology retention of FET in the focused range, which may provide a basis for the generation and application of FET. At present, the theoretical research on FET’s characteristics is increasingly improved. To apply the attractive characteristics of FET in actual systems, it is necessary to actually generate FET. Therefore, the generation method of FET should become the next research emphasis.
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Keywords:
- flying electromagnetic toroid /
- toroidal structure /
- slowly spreading /
- propagation characteristic
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图 2 电磁飞环在t = 0时刻的场分布 (a) y = 0平面Ez; (b) y = 0平面Eρ; (c) y = 0平面Hθ; (d) z = 0平面Ez; (e) z = –q1平面Eρ; (f) z = –q1平面Hθ
Fig. 2. Field distribution of the FET when t = 0: (a) Ez on the y = 0 plane; (b) Eρ on the y = 0 plane; (c) Hθ on the y = 0 plane; (d) Ez on the z = 0 plane; (e) Eρ on the z = –q1 plane; (f) Hθ on the z = –q1 plane.
图 3 在t = 0时刻的z方向上的场分布 (a) 直线ρ = 0处; (b) 柱面ρ = 4.5q1上直线(x = 4.5q1, y = 0)处
Fig. 3. Field distribution along z direction when t = 0: (a) on the line ρ = 0; (b) on the line (x = 4.5q1, y = 0) of cylindrical surface ρ = 4.5q1.
图 4 电磁飞环在传播过程中场分布的演化 (a) 纵向电场Ez; (b)横向电场Eρ, 黑线表示电磁飞环传播到不同位置时横向电场最大值所在位置
Fig. 4. Evolution of the field distribution of FET: (a) Longitudinal electric field Ez; (b) transverse electric field Eρ, the black line indicates the position of maximum transverse electric field when the FET propagates to different positions.
图 5 电磁飞环传播到不同位置时横向电场最大值所在位置 (a) z < q2; (b) z > q2
Fig. 5. Position of maximum transverse electric field when the FET propagates to different positions: (a) z < q2; (b) z > q2.
图 6 电磁飞环经过z = 0平面时的归一化频谱分布(a)纵向电场Ez; (b)横向电场Eρ
Fig. 6. Normalized spectrum distribution when the FET propagates through the z = 0 plane: (a) Longitudinal electric field Ez; (b) transverse electric field Eρ.
图 7 横向电场Eρ各个频点最大值的位置曲线
Fig. 7. Position curves of maximum value of each frequency of the transverse electric field Eρ.
表 1 电磁飞环传输距离z与环半径ρ的扩散关系
Table 1. Relation between propagation distance and toroid radius of FET.
传输距离z/q1 环半径ρ/q1 扩散百分比/% 0 4.5 0 10 4.7 3 20 4.9 8 30 5.35 18 40 5.8 27 50 6.35 40 60 7 54 70 7.7 69 80 8.45 86 90 9.35 105 100 10.05 121 110 10.9 140 120 11.8 159 130 12.35 171 140 13.5 197 150 14.3 214 160 15.15 233 170 15.95 251 180 16.75 268 190 17.55 286 
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[1] McGloin D, Dholakia K 2005 Contemporary Phys. 46 15
Google Scholar
[2] Lu J, Greenleaf J 2002 IEEE Trans. Ultrasonics Ferroelectrics Freq. Control 39 19
[3] Saari P 2009 Laser Phys. 19 725
Google Scholar
[4] 闫孝鲁, 张晓萍, 李阳梅 2016 物理学报 65 138402
Google Scholar
Yan X L, Zhang X P, Li Y M 2016 Acta Phys. Sin. 65 138402
Google Scholar
[5] 韦永梅, 彭虎 2014 物理学报 63 198702
Google Scholar
Wei Y M, Peng H 2014 Acta Phys. Sin. 63 198702
Google Scholar
[6] Li H, Liu J, Bai L, Wu Z 2018 Appl. Opt. 57 7353
Google Scholar
[7] Ott P, Al Shakhs M H, Lezec H J, Chau K J 2014 Opt. Express 22 29340
Google Scholar
[8] Brittingham J N 1983 J. Appl. Phys. 54 1179
Google Scholar
[9] Ziolkowski R W 1985 J. Math. Phys. 26 861
Google Scholar
[10] Ziolkowski R W 1989 Phys. Rev. A 39 2005
Google Scholar
[11] Lekner J 2004 J. Opt. A Pure Appl. Op. 6 711
Google Scholar
[12] Hellwarth R W, Nouchi P 1996 Phys. Rev. E 54 889
Google Scholar
[13] Feng S, Winful H G, Hellwarth R W 1999 Phys. Rev. E 59 4630
[14] Papasimakis N, Fedotov V A, Savinov V, Raybould T A, Zheludev N I 2016 Nat. Mater. 15 263
Google Scholar
[15] Raybould T A, Fedotov V A, Papasimakis N, Youngs I J, Zheludev N I 2016 Opt. Express 24 3150
Google Scholar
[16] Zdagkas A, Papasimakis N, Savinov V, Dennis M R, Zheludev N I 2019 Nanophotonics 8 1379
Google Scholar
[17] Papasimakis N, Raybould T, Fedotov V A, Tsai D P, Zheludev N I 2018 Phys. Rev. B 97 201409
Google Scholar
[18] Raybould T, Fedotov V A, Papasimakis N, Youngs I, Zheludev N I 2017 Appl. Phys. Lett. 111 081104
Google Scholar
[19] Kaelberer T, Fedotov V A, Papasimakis N, Tsai D P, Zheludev N I 2010 Science 330 1510
Google Scholar
[20] Basharin A A, Chuguevsky V, Volsky N, Kafesaki M, Economou E N 2017 Phys. Rev. B 95 035104
Google Scholar
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