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## Huygens’ principle derived by using momentum reciprocity theorem of electromagnetic field

Liu Guo-Qiang, Liu Jing
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• #### 摘要

经典电磁场互易定理(即洛伦兹互易定理)作为电磁学重要的理论之一, 被广泛应用于通信、天线信号传输和电磁成像等诸多领域, 它是一种“能量型”互易定理. 已有研究用微分形式扩展“Rumsey反应”的概念, 使其同时包含了洛伦兹力密度反应和功率密度反应项. 进一步有研究从麦克斯韦方程组导出了动量互易定理. 动量互易定理与洛伦兹互易定理一样, 既可以用于理论分析, 也可以解决实际应用问题. 因此利用洛伦兹互易定理可导出惠更斯原理, 本文利用动量互易定理导出惠更斯原理.

#### Abstract

The classical reciprocity theorem of electromagnetic field proposed by Lorentz H.A. in 1896 is one of the important theories of electromagnetics. The Lorentz reciprocity theorem is widely used in many fields such as communication, antenna signal transmission, and electromagnetic imaging. The Lorentz reciprocity theorem is an “energy-based” reciprocity theorem. Over the past hundred years, some new reciprocity theorems of electromagnetic field have been discovered, including reciprocity theorems in frequency domain and time domain. In 2020, Lindell et al. extended the concept of the 'Rumsey reaction' in a differential form to include both the Lorentz force density reaction term and the power density reaction term. In the same year, Liu et al. derived the momentum reciprocity theorem from the Maxwell's equations. The momentum reciprocity theorem, like the Lorentz reciprocity theorem, can be used for both theoretical analysis and practical applications. Huygens’ principle is usually derived by the Lorentz reciprocity theorem. In this paper, the momentum reciprocity theorem is employed to derive the Huygens’ principle.

#### 作者及机构信息

###### 通信作者: 刘国强, gqliu@mail.iee.ac.cn
• 基金项目: 国家自然科学基金重点项目(批准号: 51937010)和国家自然科学基金青年科学基金(批准号: 51907191)资助的课题.

#### Authors and contacts

###### Corresponding author: Liu Guo-Qiang, gqliu@mail.iee.ac.cn
• Funds: Project supported by the Key Program of the National Natural Science Foundation of China (Grant No. 51937010) and the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 51907191).

#### 参考文献

 [1] Lorentz H A 1896 Amsterdammer Akademie der Wetenschappen 4 176 [2] 陈云天, 王经纬, 陈伟锦, 徐竞 2020 物理学报 69 154206Google Scholar Chen Y T, Wang J W, Chen W J, Xu J 2020 Acta Phys. Sin. 69 154206Google Scholar [3] 符果行 1991 工程电磁理论方法 (北京: 人民邮电出版社) 第58, 59页 Fu G X 1991 Engineering Electromagnetic Theory and Methods (Beijing: People’s Post and Telecommunications Press) pp58, 59 (in Chinese) [4] 陈传升, 王秉中, 王任 2021 物理学报 70 070201Google Scholar Chen C S, Wang B Z, Wang R 2021 Acta Phys. Sin. 70 070201Google Scholar [5] Zhao Q, Li J, Liu S, Liu G Q, Liu J 2021 IEEE Trans. Instrum. Meas. 70 1Google Scholar [6] Feld Y N, Miller D G 1992 Soviet Phys. Doklady. 37 235 [7] Tai C T. 1992 IEEE Trans. Antennas. Prop. 40 675Google Scholar [8] Rumsey V H 1954 Phys. Rev. 94 1483Google Scholar [9] 赵双任 1987 电子学报 15 88Google Scholar Zhao S R 1987 Acta Electron. Sin. 15 88Google Scholar [10] Lindell I V, Sihvola A 2020 Prog. Electromagn. Res. Lett. 89 1Google Scholar [11] Liu G Q, Li Y Y, Liu J 2020 IEEE Antennas Wirel. Propag. Lett. 19 2159Google Scholar [12] 刘国强, 刘婧, 李元园 2020 电磁场广义互易定理 (北京: 科学出版社) 第38, 53, 95—101页 Liu G Q, Liu J, Li Y Y 2020 Generalized Reciprocity Theorem for Electromagnetic Fields (Beijing: Science Press) pp38, 53, 95–101 (in Chinese) [13] 全绍辉 2013 高等工程电磁场理论(北京: 北京航空航天大学出版社) 第149—151页 Quan S H 2013 Advanced Engineering Electromagnetic Field Theory (Beijing: Beijing University of Aeronautics and Astronautics Press) pp149–151 (in Chinese)

#### 施引文献

• 图 1  实际的电流源、磁流源、电荷源和磁荷源产生的场

Fig. 1.  Fields generated by the actual current, magnetic current, charge sources and magnetic charge sources.

图 2  惠更斯面电流源、面磁流源、面电荷源和磁荷源产生的场

Fig. 2.  Fields generated by the Huygens surface current, magnetic current, charge sources and magnetic charge sources.

•  [1] Lorentz H A 1896 Amsterdammer Akademie der Wetenschappen 4 176 [2] 陈云天, 王经纬, 陈伟锦, 徐竞 2020 物理学报 69 154206Google Scholar Chen Y T, Wang J W, Chen W J, Xu J 2020 Acta Phys. Sin. 69 154206Google Scholar [3] 符果行 1991 工程电磁理论方法 (北京: 人民邮电出版社) 第58, 59页 Fu G X 1991 Engineering Electromagnetic Theory and Methods (Beijing: People’s Post and Telecommunications Press) pp58, 59 (in Chinese) [4] 陈传升, 王秉中, 王任 2021 物理学报 70 070201Google Scholar Chen C S, Wang B Z, Wang R 2021 Acta Phys. Sin. 70 070201Google Scholar [5] Zhao Q, Li J, Liu S, Liu G Q, Liu J 2021 IEEE Trans. Instrum. Meas. 70 1Google Scholar [6] Feld Y N, Miller D G 1992 Soviet Phys. Doklady. 37 235 [7] Tai C T. 1992 IEEE Trans. Antennas. Prop. 40 675Google Scholar [8] Rumsey V H 1954 Phys. Rev. 94 1483Google Scholar [9] 赵双任 1987 电子学报 15 88Google Scholar Zhao S R 1987 Acta Electron. Sin. 15 88Google Scholar [10] Lindell I V, Sihvola A 2020 Prog. Electromagn. Res. Lett. 89 1Google Scholar [11] Liu G Q, Li Y Y, Liu J 2020 IEEE Antennas Wirel. Propag. Lett. 19 2159Google Scholar [12] 刘国强, 刘婧, 李元园 2020 电磁场广义互易定理 (北京: 科学出版社) 第38, 53, 95—101页 Liu G Q, Liu J, Li Y Y 2020 Generalized Reciprocity Theorem for Electromagnetic Fields (Beijing: Science Press) pp38, 53, 95–101 (in Chinese) [13] 全绍辉 2013 高等工程电磁场理论(北京: 北京航空航天大学出版社) 第149—151页 Quan S H 2013 Advanced Engineering Electromagnetic Field Theory (Beijing: Beijing University of Aeronautics and Astronautics Press) pp149–151 (in Chinese)
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##### 出版历程
• 收稿日期:  2021-12-17
• 修回日期:  2022-04-11
• 上网日期:  2022-07-09
• 刊出日期:  2022-07-20

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