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

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.

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,

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

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

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

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

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

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

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

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

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

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

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