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The electromagnetic helicity of the free electromagnetic field and the static magnetic helicity are shown to be two different embodiments of the same physical quantity, the total helicity. The total helicity is the sum of two terms: a term that measures the difference between the number of left-handed and right-handed photons of the free field, and another term that measures the screwiness of the static magnetization density in matter. Each term is the manifestation of the total helicity in different frequency regimes: $\ensuremath{\omega}g0$ and $\ensuremath{\omega}=0$, respectively. This unification establishes the theoretical basis for studying the conversion between the two embodiments of total helicity upon light-matter interaction.

Decomposing the field scattered by an object into vector spherical wave functions (VSWF) is a useful tool when discussing its optical properties on more analytical grounds. Thus far, it was frequently required in the decomposition that the scattered field is available on a spherical surface enclosing the scatterer. This requirement is adapted to the spatial dependency of the VSWFs but is rather incompatible with many numerical solvers. To mitigate this problem, we propose an orthogonal expression for the decomposition that holds for any surface that encloses the scatterer, independently of its shape. We also show that the orthogonal relations remain unchanged when the radiative VSWF used for the expansion of the scattered field are substituted by the regular VSWF used for the expansion of the incident illumination as test functions. This is a key factor for the numerical stability of our decomposition. As an example, we use a finite-element based solver to compute the multipole response of a nanorod illuminated by a plane wave and study its convergence properties.

Hierarchical self-assembled meta-atoms are made from a larger number of suitably arranged metallic nanoparticles. They constitute the basic building blocks for isotropic metamaterials. The properties of these meta-atoms are usually studied upon illumination with a plane wave and by analyzing the multipolar composition of the scattered field. This, however, does not always provide full information. The coupling between multiple meta-atoms is usually not considered, and a physical understanding for the cause of the response is often incomplete. Here we overcome these limitations by performing a spectral eigenvalue analysis of the transfer matrix of isolated and coupled self-assembled meta-atoms. Emphasis is put on using a transfer-matrix formulation in either a local or a global coordinate frame. We show that for the magnetic resonance, coupling to nearest neighbors is weak, suggesting the possibility to preserve the response of the isolated meta-atom upon tight packaging in a metamaterial.

Maxwell's equations in curved space time are invariant under electromagnetic duality transformations. We exploit this property to constrain the design parameters of metamaterials used for transformation electromagnetics. We show that a general transformation must be implemented using a dual-symmetric metamaterial. We also show that the spatial part of the coordinate transformation has the same action for both helicity components of the electromagnetic field, while the spatiotemporal part has a helicity dependent effect. Dual-symmetric metamaterials can be designed by constraining the polarizability tensors of their individual constituents, i.e., the meta atoms. We obtain explicit expressions for these constraints. Two families of realistically implementable dual-symmetric meta atoms are discussed, one that exhibits electric-magnetic cross polarizability and one that does not. In simple three-dimensional periodical arrangements of the meta atoms (Bravais lattices), the helicity dependent effect can only be achieved if the meta atoms exhibit nonzero electric-magnetic cross polarizabilities. In our derivations, we find that two dipoles located at the same point, one electric ($\mathbf{p}$) and one magnetic ($\mathbf{m}$), are needed to produce a total field with well-defined helicity equal to $+1$ or $\ensuremath{-}1$ and that they must be related as $\mathbf{p}=\frac{i}{c}\mathbf{m}$ or $\ensuremath{-}\frac{i}{c}\mathbf{m}$, respectively.