Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

Min Hyung Cho; Wei Cai

doi:10.1007/s10915-016-0341-3

Journal of Scientific Computing

June 2017, Volume 71, Issue 3, pp 1319–1350 | Cite as

Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

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Authors and affiliations

Min Hyung ChoEmail author
Wei Cai

Article

First Online: 29 December 2016

Received: 21 October 2016

Accepted: 15 December 2016

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2 Citations

Abstract

Concise and explicit formulas for dyadic Green’s functions, representing the electric and magnetic fields due to a dipole source placed in layered media, are derived in this paper. First, the electric and magnetic fields in the spectral domain for the half space are expressed using Fresnel reflection and transmission coefficients. Each component of electric field in the spectral domain constitutes the spectral Green’s function in layered media. The Green’s function in the spatial domain is then recovered involving Sommerfeld integrals for each component in the spectral domain. By using Bessel identities, the number of Sommerfeld integrals are reduced, resulting in much simpler and more efficient formulas for numerical implementation compared with previous results. This approach is extended to the three-layer Green’s function. In addition, the singular part of the Green’s function is naturally separated out so that integral equation methods developed for free space Green’s functions can be used with minimal modification. Numerical results are included to show efficiency and accuracy of the derived formulas.

Keywords

Maxwell’s equations Dyadic Green’s functions Sommerfeld integrals Layered media

Mathematics Subject Classification

65Z05 45H05 78A25

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Notes

Acknowledgements

The authors also like to thank Dr. William Beck from Army Research Laboratory for helpful discussions during this work.

Appendix: Bessel Identities

A derivation of the dyadic Green’s function in multi-layered media is very tedious but it is required for a proper implementation. The Bessel identities play a key role in the derivation and they are based on a integral representation of the Bessel function and recurrence relation (See Ref. [1]), namely,

$$\begin{aligned} J_{n}(z) e^{in\theta }= & {} \frac{1}{2\pi } \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}+in\phi - in\frac{\pi }{2}} d\phi , \end{aligned}$$

(219)

$$\begin{aligned} J_{n+2}(z)= & {} \frac{n+2}{z}J_{n+1}(z)-J_{n}(z), \end{aligned}$$

(220)

respectively. For convenience, the most often used identities are listed in the following

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}~d\phi&= 2\pi J_{0}(z), \end{aligned}$$

(221)

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}\cos {\phi }~d\phi&= 2\pi i J_{1}(z)\cos {\theta },\end{aligned}$$

(222)

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}\sin {\phi }~d\phi&= 2\pi i J_{1}(z)\sin {\theta },\end{aligned}$$

(223)

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}\cos {2\phi }~d\phi&= -2\pi J_{2}(z)\cos {2\theta },\end{aligned}$$

(224)

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}\sin {2\phi }~d\phi&= -2\pi J_{2}(z)\sin {2\theta },\end{aligned}$$

(225)

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}\cos ^{2}{\phi }~d\phi&= \pi J_{0}(z)-\pi J_{2}(z)\cos {2\theta },\end{aligned}$$

(226)

$$\begin{aligned} \int _{0}^{2\pi } e^{iz \cos {(\phi -\theta )}}\sin ^{2}{\phi }~d\phi&= \pi J_{0}(z)+\pi J_{2}(z)\cos {2\theta }. \end{aligned}$$

(227)

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Authors and Affiliations

Min Hyung Cho
- 1
Email author
Wei Cai
- 2

1.Department of Mathematical SciencesUniversity of Massachusetts LowellLowellUSA
2.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA

Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgements

References

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