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Clarification and extension of the optical reciprocity theorem

J. Math. Phys. 50, 072901 (2009); doi:10.1063/1.3162201

Published 10 July 2009

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H. Y. Xie (謝懷毅), P. T. Leung (梁培德), and D. P. Tsai (蔡定平)
Department of Physics, National Taiwan University, Taipei 10617, Taiwan, Republic of China
Clarifications on the optical reciprocity theorem are provided by explicitly proving the equivalence between the Lorentz lemma and the symmetry of the Green dyadic for the electromagnetic wave equation. This is achieved by explicitly including the surface term in the former so that different boundary conditions can be considered as required in the formulation of the latter. In addition, we shall also extend the theorem to include anisotropic magnetic materials with a nonlocal response, leading to a result which will be useful for the study of materials possessing such properties such as certain types of metamaterials. ©2009 American Institute of Physics
History: Received 18 December 2008; accepted 3 June 2009; published 10 July 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/072901/1
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KEYWORDS and PACS

Keywords
PACS
  • 41.20.Jb
    Electromagnetic wave propagation; radiowave propagation
  • 42.70.-a
    Optical materials
  • YEAR: 2009

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ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (21)
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    13. Note that this paper adopted a quantum mechanical formulation and the proof of the Green dyadic symmetry in Appendix A of this paper contains several errors. In particular, Eq. (A3) should read as

    [dformula H[sub ij] = {[partial-derivative][sub i][sup r] kappa([bold r],omega)[partial-derivative][sub j][sup r] - [[partial-derivative][sub l][sup r] kappa([bold r],omega)[partial-derivative][sub l][sup r]+((omega[sup 2])/c[sup 2])epsilon([bold r],omega)]delta[sub ij]}delta([bold r] - [bold r][sup [prime]]).]

    Thus we believe the proof in this paper is questionable.

  13. See, e.g., D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003)
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  16. Note that the condition introduced in Eq. (17) is a generalization of that used in Ref. 6 with the explicit inclusion of the magnetic permeability tensor.
  17. Note that the proof of the equivalence between the two versions of the reciprocity principle in Sec. II remains valid for the case with nonlocal response, with Eq. (12) generalized to the following form:

    [dformula -((4 pi i omega p)/c){[[bold G][sub e]([bold r][sup [double-prime]],[bold r][sup [prime]])][sub ij] - [[bold G][sub e]([bold r][sup [prime]],[bold r][sup [double-prime]])][sub ji]} = ((omega p)/i)[contour-integral][sub S]da[integral]d[sup 3]x[sub 1]([bold mu ][sup -1]([bold r],[bold r][sub 1])[bullet][bold [del]][sub 1] x [bold G][sub ej]([bold r][sub 1],[bold r][sup [prime]]))[bullet][[bold n] x [bold G][sub ei]([bold r],[bold r][sup [double-prime]])]-((omega p)/i)[contour-integral][sub S]da[integral]d[sup 3]x[sub 1]([bold mu ][sup -1]([bold r],[bold r][sub 1])[bullet][bold [del]][sub 1] x [bold G][sub ei]([bold r][sub 1],[bold r][sup [double-prime]]))[bullet][[bold n] x [bold G][sub ej]([bold r],[bold r][sup [prime]])],]

    and the Neumann condition in Eq. (17) in the nonlocal case has also to be generalized to the following form: [integral]n×[µ−1(r,r1)•[del]1×Ge(r1,r[prime])]d3x1=0.

  18. We thank an anonymous referee for the suggestion to include this clarification for the symmetry condition of the dielectric and permeability tensors.
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  20. J. A. Kong, Proc. IEEE 60, 1036 (1972).
  21. C. T. Tai, Generalized Vector and Dyadic Analysis, 2nd ed. (IEEE, New York, 1997), Chap. 7.

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