Application of Generalized Mie Theory to EELS Calculations as a Tool for Optimization of Plasmonic Structures

Abstract

Technical applications of plasmonic nanostructures require a careful structural optimization with respect to the desired functionality. The success of such optimizations strongly depends on the applied method. We extend the generalized multiparticle Mie (GMM) computational electromagnetic method and use it to excite a system of plasmonic nanoparticles with an electron beam. This method is applied to EELS calculations of a gold dimer and compared to other methods. It is demonstrated that the GMM method is so efficient, that it can be used in the context of structural optimization by the application of genetic algorithms combined with a simplex algorithm. The scheme is applied to the design of plasmonic filters.

Appendix

The spherical vector wave functions (SVWF) can be constructed from the spherical wave functions (SWF). The SWFs are scalar solutions to the Helmholtz equation and can be written as

$$ u_{mn}^{1,3}(k \,\mathbf{r}) = z_{n}^{1,3}(k\,r) P_{n}^{|m|}(\cos \theta) \mathrm{e}^{\mathrm{i} m \varphi} ~, $$

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with n = 1,2,…, m = −n,…, n. (r, 𝜃, ϕ) are the spherical coordinates of the position vector r. Following the definition of [9], the radial part is the spherical Bessel function of first kind ${z_{n}^{1}}(k r) = j_{n}(k r)$ and the spherical Hankel function of first kind ${z_{n}^{3}}(k r) = h_{n}^{(1)}(k r)$ for incoming and outgoing waves, respectively. The polar part of the wave function is given by the associated Legendre polynomials, which depends on the magnitude of m.

Finally, the SVWFs can be constructed using the pilot vector r.

$$\begin{array}{@{}rcl@{}} \mathbf{M}_{mn}^{1,3}(k \,\mathbf{r}) &=& \frac{1}{\sqrt{2n(n+1)}} \nabla u_{mn}^{1,3}(k \,\mathbf{r}) \times \mathbf{r} \end{array} $$

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$$\begin{array}{@{}rcl@{}} \mathbf{N}_{mn}^{1,3}(k \,\mathbf{r}) &=& \frac{1}{k} \nabla \times \mathbf{M}_{mn}^{1,3}(k \,\mathbf{r}) \end{array} $$

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The final expressions used in this paper are

$$\begin{array}{@{}rcl@{}} \mathbf{M}_{mn}^{1,3}(k \,\mathbf{r}) &=& \frac{z_{n}^{1,3}(k \,r)}{\sqrt{2n(n+1)}} \left( \mathrm{i} m \pi_{n}^{|m|}(\theta) \mathbf{e}_{\theta} - \tau_{n}^{|m|}(\theta) \mathbf{e}_{\varphi}\right) \mathrm{e}^{\mathrm{i} m \varphi} \end{array} $$

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$$\begin{array}{@{}rcl@{}} \mathbf{N}_{mn}^{1,3}(k \,\mathbf{r}) &=& \frac{1}{\sqrt{2n(n+1)}} \left( n(n+1) \frac{z_{n}^{1,3}(k\,r)}{k\,r} P_{n}^{|m|}(\cos \theta) \mathbf{e}_{r} \right. \\ && \left.+ \frac{[k\,r z_{n}^{1,3}(k\,r)]'}{k\,r} \left( \tau_{n}^{|m|}(\theta) \mathbf{e}_{\theta} + \mathrm{i} m \pi_{n}^{|m|}(\theta) \mathbf{e}_{\varphi} \right) \right) \mathrm{e}^{\mathrm{i} m \varphi} \end{array} $$

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The angular functions τ and π are defined using the associated Legendre polynomials ${P_{n}^{m}}$ as

$$ {\tau_{n}^{m}}(\theta) = \frac{\mathrm{d}}{\mathrm{d} \theta} {P_{n}^{m}}(\cos \theta),\hspace*{20pt} {\pi_{n}^{m}}(\theta) = \frac{{P_{n}^{m}}(\cos \theta)}{\sin \theta}~. $$

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