Magnetic hyperbolic optical metamaterials

Strongly anisotropic media where the principal components of electric permittivity or magnetic permeability tensors have opposite signs are termed as hyperbolic media. Such media support propagating electromagnetic waves with extremely large wave vectors exhibiting unique optical properties. However, in all artificial and natural optical materials studied to date, the hyperbolic dispersion originates solely from the electric response. This restricts material functionality to one polarization of light and inhibits free-space impedance matching. Such restrictions can be overcome in media having components of opposite signs for both electric and magnetic tensors. Here we present the experimental demonstration of the magnetic hyperbolic dispersion in three-dimensional metamaterials. We measure metamaterial isofrequency contours and reveal the topological phase transition between the elliptic and hyperbolic dispersion. In the hyperbolic regime, we demonstrate the strong enhancement of thermal emission, which becomes directional, coherent and polarized. Our findings show the possibilities for realizing efficient impedance-matched hyperbolic media for unpolarized light.

Magnetic hyperbolic dispersion. Light is an electromagnetic wave, i.e. harmonic oscillations of electric and magnetic fields periodic in time and in space. Periodicity of light wave in time is defined by its frequency ω. And its periodicity in space is defined by the wave-vector, or k-vector. Time and space periodicity of light are connected together with the dispersion relation. Importantly, the dispersion is largely defined by properties of a given optical medium. We will first consider a special case of local media. This assumes that the electric displacement vector D and the magnetic field H at a given point in space can be written in terms of averaged electric fields E and average induction field B. For the local media the electromagnetic properties are defined by the tensors of electric permittivity ε and magnetic permeability μ (1) With this definition of the tensors we leave out of consideration media exhibiting gyrotropy or magnetoelectric coupling.
For a given frequency ω all the k-vectors belong to a certain three-dimensional surface, called isofrequency surface. And the shape of the isofrequency surface depends on the material parametersε andμ. To analyze the possible shapes of these surfaces we write explicitly a set of two equations for two principal linear polarizations: TE and TM. Without lack of generality we assume that for TE-polarization electric component of wave is pointing in x -direction, and for TM-polarization magnetic component is pointing in y-direction. The resulting dispersion relations take form: These equations describe the two types of isofrequency contours: either elliptic or hyperbolic depending on the relative signs of the components ofε andμ tensors. Essentially, in the media with hyperbolic dispersion, the tensorsε orμ have diagonal components of opposite signs. If the tensor of electric permittivityε has both positive and negative components, this results in hyperbolic dispersion for the T M -polarization, or electric hyperbolic dispersion. Similarly, components with opposite signs in magnetic permeability µ tensor result in hyperbolic dispersion for the T E-polarization, or magnetic hyperbolic dispersion.

Supplementary Note 2
Determination of k-vector from complex transmission and reflection coefficients. The normal component of the vector k z can be found as [1][2][3][4][5]: where t and r are complex transmittance and reflectance, h is the thickness of the material slab and m is an integer number. Complex transmission and reflection coefficients carry the information about both the amplitude and the phase of light. The two tangential components k x and k y remain continuous at the interface between the media according to the boundary conditions.

Supplementary Note 3
Phase retrieval technique. We experimentally detect an interference pattern of the sample wave and reference wave coming at an angle with respect to each other. We perform a Fourier-transform of the image of the interference pattern and in Fourier-image we filterout all the spatial frequencies except the frequencies corresponding to a single maximum of the first order. After the filtering, we perform the inverse Fourier-transform, which gives us a two-dimensional distribution (image) of a complex field, where the phase of the field represents a phase difference between the sample and reference beams. We normalize our transmission measurements (both amplitude and phase) to the transmission through an empty space, and normalize the reflection measurements to the reflection from a golden mirror.
Spatial dispersion. In some cases approximation of local medium is not sufficient to describe the optical properties of media and corresponding dispersion. In this case the theory of spatial dispersion needs to be employed [6,7]. Within this theory components of electric permittivity tensor of the medium are considered to be a function of a k-vector ε = ε(k).
This can be expanded into Taylor series by k. Here we will consider centra-symmetric media.
In such media even terms of Taylor expansion vanish. Thus, the expression for ε in the case of spatial dispersion takes form: The tensor Σ ij ∂ 2 ε ∂k i ∂k j is also called quadrupole susceptibility tensor. We further take the quadrupole susceptibility into consideration and neglect higher-order terms. The quadrupole tensor may have up to six independent components in non-symmetric structures, while symmetries reduce this number [8,9].
We further focus on spatial dispersion of multilayer fishnet metamaterials. We assume our electric field to be in the x − z plane, thus we consider ε x and ε z components of the electric permittivity tensor.
Our dispersion relation for TM-polarization take form: Here we take into account that quadrupole susceptibility cross-components ∂ 2 ε i ∂kx∂kz = ∂ 2 ε i ∂kz∂kx = 0 due to the C 2 point symmetry of the structure. The dispersion isofrequency contours are correspondingly the fourth-order curves.
And the dispersion isofrequency contours are the second-order curves (such as ellipses or hyperbolas).
To sum up, the isofrequency dispersion of fishnet metamaterials for TE-polarization can be described by two independent parameters: either (ε loc x , µ loc y ), or (ε loc x , ∂ 2 εx ∂k 2 z ). For TMpolarization the total number of parameters is six. However we note, that two of them are same as for TE-polarization. Therefore, the dispersion for TM-polarization can be described by extra four independent parameters: (ε loc z , ∂ 2 εx .

Supplementary Note 5
Numerical simulations of complex dispersion. In addition, we perform numerical calculations of the dispersion of the metamaterial using CST Microwave Studio commercial software. In the calculations we use realistic material parameters of gold, magnesium fluoride and silicon nitride [11][12][13][14]. Theoretically calculated and experimentally measured isofrequency contours are in a good agreement with each other. absorption and thermal emission to have the same pattern. We observe that, while the resolution of directionality diagrams for thermal emission is lower due to increased material loss with the increase of the temperature as well as experimental limitations, the diagrams share the same patterns. The fact that in the magnetic hyperbolic regime the thermal emission is directional implies that the emission is spatially coherent.

Supplementary Note 7
Numerical calculations of thermal emission. We provide full-wave numerical simulations of the thermal emission directionality and spectra. We rely on the fact that absorption equals emission, and therefore we find emission by calculating absorption. In our calculations we take into account change of resistivity of gold with temperature by approx. Theoretical spectrum shows the emission averaged over directions within a 45 degree cone.
This resembles experimental condition of emission collection with 0.7 NA objective lens.