# Part I. Metamaterials

## Electromagnetic crystals and negative index materials.

The characteristic feature of meta-materials is the coupling of macroscopic fields through structurally generated sub-wavelength resonance. Well known meta-materials include stained glass and colloidal suspensions of gold and silver nanoparticles (Bohren and Huffman, 2004). For artificial magnetism generated in bulk meta-materials this is experimentally accomplished at microwave frequencies using split ring resonators made from perfect conductors as in (Pendry et al., 1999). One then adds a lattice of thin conducting wires to the resonators to generate simultaneously negative dielectric permittivity and magnetic permeability (Smith et al.,2000). Left handed media are responsible for supporting novel wave propagation phenomena including waves with phase velocity and power flow in different directions, inverse Doppler effect, and an inverted Snell's law (Veselago, 1968).

Unfortunately at optical frequencies metallic resonators no longer perform well due to resistive loss and new strategies are required for generating local sub wavelength resonances. Current state of the art applies phenomenological formulas describing negative index materials in the optical regime (Huang, Povinelli, and Joannopoulos, 2004), (Jelinek and Marquez, 2010), (Lepetit and Akmansoy, 2008), (O'Brien and Pendry, 2002), and (Shvets and Urzhumov, 2004). However this approach is insufficient for the systematic and rational design of metamaterials

In this research we work at near infrared and optical frequencies and focus on creating novel modes for wave propagation inside heterogeneous materials using frequency dependent dielectric media. Here the heterogeneous material is a patterned inclusion geometry surrounded by a connected phase of a different material. The period length is strictly below that of the wavelength of radiation that is to be controlled. These materials are known as meta-materials and control electromagnetic radiation at longer wavelengths through sub-wavelength resonances excited by the patterned dielectric.

We focus on understanding wave propagation inside a bulk meta-material and we assume an infinite periodic patterned material. In the sub-wavelength regime the period size $d$ is smaller than the wavelength of light $\lambda$. We are interested in frequencies $\omega$ available for wave propagation inside the material for a given wavelength. The available frequencies are written $\omega(\lambda)$ and this is the dispersion relation for the meta-material.

We consider inclusions of two types (Chen and Lipton, 2013a), (Chen and Lipton, 2013b) and (Chen and Lipton, 2013c), imbedded inside a third connected phase and design the dispersion relation $\omega(\lambda)$ by changing the shape and geometric arrangement of the inclusions within the period cell. Mathematically the problem for two dimensional electromagnetic crystals and three dimensional acoustic media is of the form $$-\nabla\cdot(a(\omega,x)\nabla h(x)) =\omega^2 h(x) ,\hbox{ $x\in\mathbb{R}^n$ $\,n=2,3$}$$ where $\lambda$ is fixed and we seek to find $\omega$. Here $\omega$ appears as an eigenvalue $\omega^2$ and also in the coefficient of the PDE. The eigenvalue problem is novel but is present in the applications and demands creative approaches. Interestingly this problem does not appear to been attempted before. We take as our small parameter the ratio of period to wavelength $\eta=d2\pi/\lambda$ and seek a power series solution $$\omega(\lambda)=\sum_{m=0}^\infty\eta^m\omega_m(\lambda)\qquad h=\sum_{m=0}^\infty\eta^m\omega_m.$$ When the particles are sufficiently regular (no cusps or reentrant corners) we prove that the power series converge in the $\mathbb{R}\times H^1$ norm and provide a weak solution to the eigenvalue problem, this is done in recent work with former graduate student Yue Chen in (Chen and Lipton, 2013c). Here convergence is established using majorizing series. The first term $\omega_0(\lambda)$ gives the dispersion relation to leading order. From this we can observe the novel wave propagation features of the meta-material. We find that it is possible to have double negative wave propagation certain frequencies (phase velocity and group velocity in opposite direction). The main mathematical challenge behind our approach is to appropriately handle the underlying PDEs which loose ellipticity in the classic sense. At every order in the power series development a divergence form PDE must be solved on the unit period cell $Y$. It is of the following type: $$-\nabla\cdot(a(\omega,x)\nabla h(x)) =f(x) ,\hbox{ x\in Y.}$$ Here the differential operator is defined on periodic mean zero functions in $H^1(Y)$. Inside the inclusion wth frequency dependent dilectric the coefficient depends on frequency, i.e. $a(\omega,x)=z(\omega)$, inside the high dielectric inclusion $a(\omega,x)=\epsilon_r/d^2$ and outside the inclusions $a(\omega,x)=1$ in the connected phase. We allow for a wide range of frequency dependence hence we relax the ellipticity requirement and suppose that the dielectric coefficient inside the inclusion phase can take any value $z\in\mathbb{C}$. Because of this we do not use the classic Lax Milgram theory and instead find new criteria on particle geometry that determines the set of $S\subset \mathbb{C}$ that $z=z(\omega)$ can belong to so that the operator $-\nabla\cdot(a(\omega,x)\nabla$ is invertible. We solve this problem and construct a complete orthonormal basis for the space for $H^1(Y)$ on which the operator $-\nabla\cdot(a(\omega,x)\nabla$ and its inverse can be expressed explicitly as a function of $z$, this work has appeared in (Chen and Lipton, 2013c). We show that the orthonormal basis developed here is intimately related to the spectra of the well known Poincare Neumann operator (Khavinson, Putinar, and Shapero, 2007) and is associated with the geometry of the patterned dielectric. We then develop our understanding of the Neumann Poincare spectra and its relation to geometry and use this to design patterned dielectric geometries for introducing negative wave propagation in frequency bands at given frequencies. We find new criteria on particle geometry that explicitly determines the set $S\subset \mathbb{C}$ which $z=z(\omega)$ can belong to such that the operator $-\nabla\cdot(a(\omega,x)\nabla$ is invertible. These results together with dispersion relations computed for the metamaterial are given in (Chen and Lipton, 2013a), (Chen and Lipton, 2013b), and Chen and Lipton, 2013c These results are applied in the time domain to illustrate negative index refraction of signals (Chen and Lipton, 2018).

Recent work (with Ph.D. student Abiti Adili) extends these results to fully three dimensional electromagnetic crystals via the vector wave equation $$\nabla\times\left(a(\omega,x)\nabla\times H(x))\right) =\omega^2 H(x) ,\hbox{ \nabla\cdot {H}=0, x\in\mathbb{R}^3.}$$ For this case we establish existence of the solution operator $$\left[\nabla\times a(\omega,x)\nabla\times \right]^{-1},$$ in an analogous way using making use of the structural spectra associated with the magnetic dipole operator.

The work discussed here has appeared in
Archive for Rational Mechanics and Analysis 209 (2013) pp. 835--868, Multiscale Modeling and Simulation, SIAM, 11:192--212, 2013, Multiscale Methods for Engineering Double Negative Metamaterials. Photonics and Nanostructures - Fundamentals and Applications, 11 (2013) 442-452., New Journal of Physics, 12, 2010, 083010, and Y. Chen and R. Lipton. Controlling refraction using sub-wavelength resonators, Appl. Sci. 2018, 8(10), 1942; doi:10.3390/app8101942.
This work is supported by NSF Grant DMS-1211066 and Air Force Research Office through award FA9950-12-1-0489

## Effective Maxwell's Equations and Negative Permittivity Materials in 3d.

We analyze the time harmonic Maxwell's equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size $\eta$ of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeability of the medium can be large and/or negative. Together with Ben Schweizer we provide the first rigorous proof of negative effective magnetic permeability from periodic metal split ring resonators, see (Lipton and Schweizer, 2018). Here the metal is modeled as a perfect conductor given by the boundary condition $\mathbf{E}\times \vec{n}=0$ on the surface of the metal. This is the the correct behavior of metals at microwave frequencies where the split ring geometry is used in the applications. It turns out that the proof of this is straight forward when modeling the metal using the boundary condition $\mathbf{E}\times \vec{n}=0$. This work has appeared in Archive for Rational Mechanics and Analysis, see (Lipton and Schweizer, 2018), Published March 22, 2018.

This work supported by the Air Force Research Office through award FA9950-12-1-0489

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