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The vector Kirchhoff integral theorem (VKI) is an important formula in electromagnetic (EM) theory,especially it is a basis of the optical diffraction theory.Recently,it has been found that there exist some flaws in the proofs presented in the literature.There are mainly two types of methods to prove the VKI.The first type of method is to employ the vector analysis to prove the VKI directly.Some flaws of this type of proof presented in the literature have been found and pointed out in this paper.The second type of method is to employ the scalar Kirchhoff Integral (SKI) to directly obtain the VKI. The SKI was first derived by Kirchhoff (1882).In spite of its mathematical inconsistency and its physical deficiencies, the SKI works remarkably well in the optical domain and has been the basis of most of the work on diffraction.However, the proofs for SKI usually need the scalar radiation conditions.The scalar radiation condition was first proposed by Sommerfeld to ensure the uniqueness of the solution of certain exterior boundary value problems in mathematical physics. But whether the scalar radiation conditions were suitable for the EM was not sure.In fact,for electromagnetic field,we have another vector radiation conditions which have been verified to be adaptable for all the radiation and scattering fields.It is difficult to obtain the scalar radiation conditions directly by just separating three Cartesian directions from the vector one,because the different scalar components are coupled together after the rotation and cross product operation.Actually,few strict proofs could be found to support the fact that EM satisfies the scalar radiation condition. So as the supplementary,the scalar radiation conditions will be derived in detail with far-field approximation method in this paper.To avoid using the scalar radiation condition which may bring some non-rigorousness,we perform a new strict proof for the VKI by using the vector analysis identities. The rest of this paper is organized as follows.In Section 2,the different proofs presented in the classical books will be analyzed in detail.The flaws existing in these proofs will be pointed out.After that,in Section 3,based on the Stratton-Chu formula,a new strict proof will be given with using the vector identities.In Section 4,a sensitivity analysis is numerically performed to confirm our demonstration.Finally,the conclusions are drawn from the present study in Section 5.The scalar radiation conditions will be discussed in the appendix.
[1] Jackson J D 1998 Classical Electrodynamics (3rd Ed.) (New York:Wiley-Interscience) pp479-482
[2] Born M, Wolf E 1986 Principles of Optics (6th Ed.) (New York:Pergamon Press Ltd) pp375-378
[3] Buchwald J Z, Yeang C P 2016 Arch. Hist. Exact Sci. 70 463
[4] Wang X F, Wang J Y 2011 Acta Phys. Sin. 60 025212 (in Chinese)[王晓方, 王晶宇2011 60 025212]
[5] Gordon W B 1975 IEEE Trans. Antennas Propagat. 23 590
[6] Umul Y Z 2013 Opt. Commun. 291 48
[7] Wang A, Prata A 1995 Opt. Soc. Am. A 12 1161
[8] Liu C X, Cheng C F, Ren X R, Liu M, Teng S Y, Xu Z Z 2004 Acta Phys. Sin. 53 427 (in Chinese)[刘春香, 程传福, 任晓荣, 刘曼, 滕树云, 徐至展2004 53 427]
[9] Sheng X Q 2016 Electromagnetic Theory, Computation, Application (Beijing:Higher Education Press) pp169-171(in Chinese)[盛新庆2016电磁理论、计算、应用(北京:高等教育出版社)第169–171页]
[10] Kong J A 1986 Electromagnetic Wave Theory (New York:Wiley-Interscience) pp381-383
[11] Ge D B 2009 Electromagnetic Wave Theory (Beijing:Science Press) pp334-337(in Chinese)[葛德彪2009电磁波理论(北京:科学出版社)第334–337页]
[12] Zhang S J 2009 Engineering Electromagnetics (Beijing:Science Press) pp638-640(in Chinese)[张善杰2009工程电磁场(北京:科学出版社)第638–640页]
[13] Yang R G 2008 Advanced Electromagnetic Theory (Beijing:Higher Education Press) pp175-177(in Chinese)[杨儒贵2008高等电磁理论(北京:高等教育出版社)第175–177页]
[14] Gong Z L 2010 Modern Electromagnetic Theory (2nd Ed.) (Beijing:Peking University Press) pp288-291(in Chinese)[龚中麟2010近代电磁理论第2版(北京:北京大学出版社)第288–291页]
[15] Schot S H 1992 Hist. Math. 19 385
[16] Sommerfeld A 1949 Partial Differential Equations in Physics (New York:Academic Press) pp188-193
[17] Ji J R 2007 Advanced Optical Tutorial (Beijing:Science Press) pp166-168(in Chinese)[季家镕2007高等光学教程(北京:科学出版社)第166–168页]
[18] Goodman J W 1996 Introduction to Fourier Optics (2nd Ed.) (New York:McGraw-Hill) pp42-44
[19] Huang K Z 2009 Tensor Analysis (2nd Ed.) (Beijing:Tsinghua University Press) pp139-149(in Chinese)[黄克智2009张量分析第2版(北京:清华大学出版)第139–149页]
[20] Sheng X Q 2008 A Brief Treatise on Computational Electromagnetics (2nd Ed.) (Hefei:Press of University of Science and Technology of China) pp42-43(in Chinese)[盛新庆2008计算电磁学要论第2版(合肥:中国科学技术大学出版社)第42–43页]
[21] Tai C T 1997 Generalized Vector and Dyadic Analysis (2nd Ed.) (New York:Wiley-Interscience) pp124-127
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[1] Jackson J D 1998 Classical Electrodynamics (3rd Ed.) (New York:Wiley-Interscience) pp479-482
[2] Born M, Wolf E 1986 Principles of Optics (6th Ed.) (New York:Pergamon Press Ltd) pp375-378
[3] Buchwald J Z, Yeang C P 2016 Arch. Hist. Exact Sci. 70 463
[4] Wang X F, Wang J Y 2011 Acta Phys. Sin. 60 025212 (in Chinese)[王晓方, 王晶宇2011 60 025212]
[5] Gordon W B 1975 IEEE Trans. Antennas Propagat. 23 590
[6] Umul Y Z 2013 Opt. Commun. 291 48
[7] Wang A, Prata A 1995 Opt. Soc. Am. A 12 1161
[8] Liu C X, Cheng C F, Ren X R, Liu M, Teng S Y, Xu Z Z 2004 Acta Phys. Sin. 53 427 (in Chinese)[刘春香, 程传福, 任晓荣, 刘曼, 滕树云, 徐至展2004 53 427]
[9] Sheng X Q 2016 Electromagnetic Theory, Computation, Application (Beijing:Higher Education Press) pp169-171(in Chinese)[盛新庆2016电磁理论、计算、应用(北京:高等教育出版社)第169–171页]
[10] Kong J A 1986 Electromagnetic Wave Theory (New York:Wiley-Interscience) pp381-383
[11] Ge D B 2009 Electromagnetic Wave Theory (Beijing:Science Press) pp334-337(in Chinese)[葛德彪2009电磁波理论(北京:科学出版社)第334–337页]
[12] Zhang S J 2009 Engineering Electromagnetics (Beijing:Science Press) pp638-640(in Chinese)[张善杰2009工程电磁场(北京:科学出版社)第638–640页]
[13] Yang R G 2008 Advanced Electromagnetic Theory (Beijing:Higher Education Press) pp175-177(in Chinese)[杨儒贵2008高等电磁理论(北京:高等教育出版社)第175–177页]
[14] Gong Z L 2010 Modern Electromagnetic Theory (2nd Ed.) (Beijing:Peking University Press) pp288-291(in Chinese)[龚中麟2010近代电磁理论第2版(北京:北京大学出版社)第288–291页]
[15] Schot S H 1992 Hist. Math. 19 385
[16] Sommerfeld A 1949 Partial Differential Equations in Physics (New York:Academic Press) pp188-193
[17] Ji J R 2007 Advanced Optical Tutorial (Beijing:Science Press) pp166-168(in Chinese)[季家镕2007高等光学教程(北京:科学出版社)第166–168页]
[18] Goodman J W 1996 Introduction to Fourier Optics (2nd Ed.) (New York:McGraw-Hill) pp42-44
[19] Huang K Z 2009 Tensor Analysis (2nd Ed.) (Beijing:Tsinghua University Press) pp139-149(in Chinese)[黄克智2009张量分析第2版(北京:清华大学出版)第139–149页]
[20] Sheng X Q 2008 A Brief Treatise on Computational Electromagnetics (2nd Ed.) (Hefei:Press of University of Science and Technology of China) pp42-43(in Chinese)[盛新庆2008计算电磁学要论第2版(合肥:中国科学技术大学出版社)第42–43页]
[21] Tai C T 1997 Generalized Vector and Dyadic Analysis (2nd Ed.) (New York:Wiley-Interscience) pp124-127
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