Introduction

The study of hyperbolic media and hyperbolic metamaterials have attracted significant attention in recent years due to their relatively simple geometry and many interesting properties, such as high density of states, all-angle negative refraction and hyperlens imaging beyond the diffraction limit1,2,3. Usually, both artificial4,5,6,7,8,9 and natural10,11 media with hyperbolic dispersion are uniaxial materials whose axial and tangential dielectric permittivities have opposite signs. In general, however, the propagation of electromagnetic waves inside a material is defined by both the dielectric permittivity and magnetic permeability tensors. Specifically, the electric response defines the dispersion for the transverse magnetic (TM) linearly polarized light, and the magnetic response defines the dispersion for transverse electric (TE) polarization (see details in Supplementary Note 1). Therefore, the ability to control both the electric permittivity and magnetic permeability gives a full flexibility for the dispersion engineering for any arbitrary polarization and nonpolarized light. This is of a major importance for the efficient interaction with randomly positioned emitters or thermal radiation. Moreover, a control over both electric and magnetic responses allows one to engineer the material dispersion and impedance independently, and, in particular, to achieve impedance matching between a hyperbolic material and the free space. Impedance matching prevents any light reflections at the interfaces and allows for efficient light coupling and extraction from the hyperbolic materials. Therefore, the development of magnetic hyperbolic materials with a simultaneous control over both dielectric permittivity and magnetic permeability tensors remains an important milestone. In particular, it is of a special interest to realize experimentally a magnetic hyperbolic material with the effective magnetic permeability tensor having principal components of the opposite signs1,12. Such a development would open new opportunities for super-resolution imaging, nanoscale optical cavities or control over the density of photon states and, in particular, the magnetic density of states for enhancing brightness of magnetic emitters13,14.

In recent years, we have seen the development of numerous structures with artificial magnetism. However, these structures are largely limited to planar metasurfaces of deeply subwavelength thickness. Importantly, many properties and functionalities of hyperbolic media rely on wave propagation inside them and therefore require essentially a three-dimensional design. For example, hyperlens super-resolution imaging relies on conversion of evanescent waves propagating in a bulk of hyperbolic media15,16. Nowadays, the realization of three-dimensional metamaterial structures17 is at the edge of technological possibilities associated with extreme fabrication difficulties and material constraints. To date, no photonic structures with hyperbolic dispersion in the magnetic response have been demonstrated, and such types of structures are only known for microwave systems18,19.

Here we demonstrate experimentally optical magnetic hyperbolic metamaterial with principal components of the magnetic permeability tensor having the opposite signs. We directly observe a topological transition between the elliptic and hyperbolic dispersions in metamaterials. In the hyperbolic regime, the length of wave vectors inside the metamaterial is diverging towards infinity. We reveal the effect of the hyperbolic dispersion on thermal emission, where the magnetic hyperbolic metamaterial demonstrates enhanced, directional, coherent and polarized thermal emission. Our experimental observations are supported by analytical calculations as well as full-wave numerical simulations.

Results

Sample fabrication

To realize a magnetic hyperbolic medium in optics, we employ multilayer fishnet metamaterials, known as the bulk-type metamaterials with negative refractive index at optical frequencies20. Multilayer fishnets were predicted theoretically to possess a magnetic hyperbolic dispersion21, but direct measurements of their dispersion remained out of reach. To test this, we fabricate a fishnet metamaterial by using focused ion beam milling through a stack of 20 alternating films of gold and magnesium fluoride (see details in Methods). The sample is fabricated on a 50-nm-thin silicon nitride membrane. A sketch and scanning electron microscopy image of the fabricated structure are shown in Fig. 1. The fishnets feature high optical transmission in the near-infrared spectral region, exhibiting a transmission maximum of 42% at ∼1,320 nm wavelength, as shown in Fig. 1b. We also measure the fishnet refractive index at normal incidence using spectrally and spatially resolved interferometry22. The fishnet’s refractive index shown in Fig. 1c is constantly decreasing with an increase of the wavelength exhibiting negative values at wavelengths above 1,410 nm.

Figure 1: Multilayer fishnet metamaterial.
figure 1

(a) Sketch of the structure. Thicknesses of MgF2 and Au layers are 45 and 30 nm, respectively. Thickness of Si3N4 membrane is 50 nm. Lattice period is 750 × 750 nm. Size of holes is 260 × 530 nm. (b) Experimentally measured transmission spectrum of the fishnet metamaterial. Inset shows a scanning electron microscopy image of the fabricated structure. (c) Effective refractive index of the fishnet metamaterial extracted for the normal incidence. The marked lines in b and c represent the wavelengths in the regions of elliptic dispersion (red), crossover optical topological transition (green) and hyperbolic dispersion (blue).

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Angular dispersion measurements

To reconstruct the dispersion isofrequency contours experimentally, we determine the length of the k-vectors of light inside the materials for a range of different directions. For this, we measure both amplitude and phase of the transmitted and reflected light, and find the k-vectors via the reverted Fresnel equations (see details in Supplementary Note 2).

For the phase measurements, we employ interferometry techniques. Specifically, for measuring a phase in transmission, we use a Mach–Zehnder-type interferometer23, while for the measurements of a phase in reflection we employ a Michelson–Morley interferometer23. To resolve transmission and reflection at different angles, we focus and collect the light using objective lens with high numerical aperture (Olympus LCPLN100XIR NA 0.85) and project the objective’s back-focal plane image onto an infrared camera (Xenics XS-1.7-320). The resulting image on the camera represents the k-space spectrum of the fishnet metamaterial with the central point of the image corresponding to the k-vectors normal to the fishnet’s surface. The edge of the image corresponds to k-vectors oblique to the fishnet at an angle ∼58°, limited by the numerical aperture of the objective. We note that for transmission measurements the numerical aperture is limited to 0.7 by the finite size of the silicon nitride membrane window etched into the supporting silicon handle wafer. This restriction also assures insignificant non-paraxial effects due to the sharp focusing. To obtain the phase information, we interfere the back-focal plane image with a reference beam. To reconstruct the phase information from the interference pattern, we employ off-axis digital holography technique24 (see details in Supplementary Note 3).

We measure the complex transmission and reflection for the following three different wavelengths: 1,310, 1,450 and 1,530 nm, marked in Fig. 1b,c with red, green and blue, respectively. We notice that for the first wavelength, the metamaterial exhibits positive refractive index for normal incidence of light; for the second wavelength, the refractive index is close to zero and for the latter wavelength, the refractive index is negative (Fig. 1c). The results of our angular measurements are presented in Fig. 2. We use a linearly polarized light source with the electric field polarized in the x direction. After the objective lens, the focusing beam has TE polarization along the ky axis and TM polarization along the kx axis. As a result, the back-focal plane images (Fig. 2) contain information about the optical response of the sample with respect to both TE- and TM-polarized light along the ky and kx axes, respectively. We notice that the material’s magnetic dispersion is measured along the ky axis, and the electric dispersion is measured along the kx axis.

Figure 2: Experimental results.
figure 2

Measured (a–c) transmission and (d–f) reflection amplitudes, (g–i) transmission and (j–l) reflection phase for three different wavelengths: (a,d,g,j) 1,310 nm, (b,e,h,k) 1,450 nm and (c,f,i,l) 1,530 nm. All the measurements are performed for the range of incident angles 0–60°and are plotted versus wave vector components kx and ky normalized by the length of the wave vector in air kair. Horizontal axes correspond to TM-polarized illumination. Vertical axes correspond to TE illumination. Square apertures for the transmission amplitude and phase measurements show the numerical aperture limited by the size of the windows in the supporting silicon wafer.

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From our measurements, we can see that for a specific range of angles of incidence at 1,310 nm and 1,450 nm wavelengths, the reflection becomes <4%, which, for comparison, is lower than the reflection of glass. This is a direct consequence of the impedance matching of the metamaterial to air.

Next, we analyse the phase accumulation of light inside the metamaterial (Fig. 2g–i). At 1,310 nm wavelength (Fig. 2g), the phase accumulation increases from the centre (normal incidence) to the edges (60° oblique incidence). This is similar to the response of a usual dielectric, where the phase accumulation increases with a growth of an optical path inside the material. At 1,450 nm wavelength, the phase accumulation along the ky axis remains nearly unchanged being close to zero for the entire range of incident angles. This corresponds to the case of ɛ-near-zero25 and μ-near-zero26 materials. Finally, at 1,530 nm wavelength, the phase accumulation is decreasing from the centre to the edges along the ky axis, while it is increasing along the kx axis.

Isofrequency analysis

Next, we reconstruct the isofrequency contours out of the measured transmission and reflection data (see details in Supplementary Note 2). For the three wavelengths and two polarizations, these dispersion contours are shown in Fig. 3b–d,f–h.

Figure 3: Experimental observation of a transition from elliptic to hyperbolic dispersion.
figure 3

(a) Sketch of the TE illumination geometry showing relative orientations of the sample, wave vector k and electric field E. (b–d) Isofrequency dispersion contours for the TE polarization at wavelengths 1,310, 1,450 and 1,530 nm, respectively. Wave vector components ky and kz are normalized by the length of the wave vector in air kair. (e) Sketch of the TM illumination geometry. (f–h) Isofrequency dispersion contours for the TM polarization at wavelengths 1,310, 1,450 and 1,530 nm, respectively. Dots mark experimental data, and lines correspond to analytical results. Grey circles correspond to the isofrequency contour of light in vacuum.

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We use an analytical approach to reveal the shapes of isofrequency surfaces obtained experimentally. We write explicitly a set of two equations for two principal linear polarizations: TE and TM. Without a lack of generality, we assume that for TE polarization the electric field component is pointing in the x direction, and for TM polarization the magnetic field component is pointing in the y direction. The resulting dispersion relations take the form:

Thus, the material response is described by a set of parameters ɛx, ɛz, μy and μz. For the case of purely real values of parameters (for example, for materials with no loss or gain), the dispersion equations describe two types of isofrequency contours: either elliptic or hyperbolic depending on the relative signs of the parameters. In particular, opposite signs of the electric permittivity components ɛx and ɛz lead to hyperbolic isofrequency contours for TM polarization, while the opposite signs of the magnetic permeability components μy and μz lead to hyperbolic isofrequency contours for the TE polarization. Here we take into account the absorption of light in the metamaterial and consider the parameters ɛx, ɛz and μy as complex numbers with the imaginary parts representing losses. We assume μz=1 for all the cases, as we do not expect artificial magnetic response from the structure in the z direction. Table 1 summarizes the values of ɛx, ɛz and μy used in our analytical model to describe the experimental data.

Table 1 Effective parameters of the metamaterial dispersion.
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As we observe, at 1,310 nm wavelength (Fig. 3b,f), all the material parameters have positive real parts, thus representing the cases of the elliptic dispersion. Interestingly, in both cases, the shapes of the isofrequency contours deviate from elliptical. This effect comes from the imaginary parts of ɛx, ɛz and μy. It is known that finite material losses lead to a hybridization of propagating and evanescent modes27. Importantly, in our case for the TM polarization (see Fig. 3f), the hybridization leads to a new class of topology of isofrequency contours that is different from either elliptic or hyperbolic.

At 1,530 nm wavelength and TE polarization, the permeability coefficient μy has a negative real part, which is opposite to the sign of the permeability coefficient μz=1. Therefore, in this spectral region, the material dispersion becomes magnetic hyperbolic (see Fig. 3d). The branches of the hyperbola go beyond the isofrequency contour of light in air. We notice that the k-vectors with tangential components larger than |kair| are not accessible experimentally when the metamaterial is illuminated from free space. However, an analytical extrapolation of the experimental curves supports the existence of propagating waves with large wave vectors. This is a key to achieve extraordinary optical properties of hyperbolic media, such as super-resolution imaging, nanoscale optical cavities and control over the density of photon states. For the other TM polarization, however, all three material parameters ɛx, ɛz and μy are simultaneously negative resulting in elliptic dispersion with a complex topology of isofrequency contours due to the presence of losses (see Fig. 3h).

At 1,450 nm wavelength, the parameters ɛx, ɛz and μy are vanishing simultaneously, representing the regime of optical topological transition2. Around the topological transition, ɛx, ɛz and μy change their signs due to the resonant nature of the metamaterial’s response. This results in an increase of the phase velocity of light towards infinity inside the structure. Importantly, at this wavelength, the structure supports propagating waves with k-vectors substantially smaller than the k-vectors in air, while all conventional optical materials support k-vectors larger than those in air. As local material parameters become close to zero, we expect to see strong contributions from nonlocal response of the metamaterials28. This implies that the permittivity coefficients ɛx and ɛz become functions of the wave vector k (see details in Supplementary Note 4). We find that for the magnetic (TE) dispersion of the fishnet metamaterials, both local and nonlocal models result in the same dispersion relation. Therefore, we consider the electric (TM) dispersion equation only, which in the nonlocal case takes the form:

We notice that (ref. 28). We further neglect the nonlocal parameter , as we find it to be of a minor importance28. Thus, for the case of 1,450 nm wavelength, near the point of the optical topological transition of the metamaterial, we introduce two extra spatially dispersive terms and to describe the experimental dispersion. The values of the material parameters for the 1,450 nm wavelength are also given in Table 1. Our results suggest that a wide range of nontrivial isofrequency dispersion contours can be realized by an appropriate tuning of material’s loss, gain and spatial dispersion. While all possible types of isofrequency contours for local media without loss/gain are limited to the second-order geometrical curves29 (such as an ellipse or hyperbola), the presence of loss, gain and spatial dispersion extends the possible cases of isofrequency contours to the fourth-order curves. This leads to new topologies of the metamaterial dispersion.

In addition, we use full-wave numerical simulations to calculate material’s isofrequency contours (see details in Supplementary Note 5 and Supplementary Fig. 1). We find that numerical results are in a good agreement with both experimental measurements and analytical calculations.

Manipulation of thermal emission from fishnet metamaterials

Next, we study the effect of hyperbolic dispersion on far-field thermal emission. In our experiments, we heat the sample up to 400 °C with a ceramic heater. At this temperature, the metamaterial gives relatively bright thermal emission in the spectral region of interest, and it remains undamaged by heating. We collect the thermal emission of the fishnet metamaterial sample by an objective lens with 10 mm working distance and 0.7 numerical aperture. In our experiments, we ensure that only thermal emission from the metamaterial sample is collected by the objective lens. We then direct it onto an infrared spectrometer and measure the thermal emission spectra. We take a reference measurement of the thermal emission from a silicon sample next to the fishnet metamaterial The reference measurement allows us to find emissivity of fishnets (radiation of fishnets normalized by the black body radiation) using the known emissivity of silicon30 and, in particular, characterize the degree of polarization of the emitted light (see details in Supplementary Note 6 and Supplementary Figs 2 and 3). We then measure the polarization states of the emissivity by employing Stokes vector formalism (see details in Supplementary Note 6 and Supplementary Figs 4 and 5). We find that in the spectral region with the magnetic elliptic dispersion, the thermal emission remains largely unpolarized. However, the degree of polarized light grows rapidly as we approach the point of the optical topological transition. In the spectral region of the magnetic hyperbolic dispersion, the thermal emission becomes partially linearly polarized. Figure 4a shows the emissivity spectrum of our sample. We notice that the unpolarized fraction of the emissivity remains almost unchanged over the measurement spectral range. The polarized part of the emission, however, increases at around the topological transition region and in the region with hyperbolic dispersion. These phenomena can be explained only by the enhanced density of photon states due to the magnetic hyperbolic dispersion. In addition, we argue that our far-field results suggest that the near-field thermal radiation can be characterized as super-Planckian, that is, exceeding the black body limit31.

Figure 4: Thermal emission from bulk magnetic hyperbolic metamaterials.
figure 4

(a) Spectrum of thermal emission normalized by the black body spectrum (emissivity). The unpolarized portion of emission is shown in grey, whereas the total emissivity is shown in orange above the unpolarized part. Line represents the corresponding theoretical calculation for the total emissivity. (b–d) Experimentally measured directionality of thermal emissivity at wavelengths 1,310, 1,450 and 1,530 nm, respectively, plotted versus wave vector components kx and ky normalized by the length of the wave vector in air kair. (e–g) Theoretically calculated directionality for the same three wavelengths, respectively. Images (c–g) have the same coordinate system as b.

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Further, we study directionality of thermal emission at the three wavelengths of 1,310 nm, 1,450 and 1,530 nm. For this, we translate the back-focal plane image of the collecting objective onto the infrared camera through a corresponding band-pass filter. We again employ Stokes formalism to characterize the polarization states of directionality diagrams (details of the back-focal plane polarimetry method can be found in ref. 32). With this method, we retrieve the polarized portion of the thermal emission and plot its directionality diagrams for the three wavelengths in Fig. 4b–d. We notice that the directionality of emission at 1,310 nm (elliptic dispersion) is not pronounced, while the emission at 1,530 nm has a noticeable north–south directionality. Importantly, the directions of high thermal radiation correspond to the directions with large k-vectors on the magnetic hyperbolic dispersion curve in Fig. 3d. Emission at the point of topological transition exhibits noticeable directionality as well; in particular, the emission in the direction normal to the sample is suppressed (the centre of the image in Fig. 4c). This corresponds to the region with near-zero k-vectors. The fact that in the regime of the magnetic hyperbolic dispersion, the thermal emission is directional implies that it exhibits a high degree of spatial coherence.

We further calculate the spectra and directionalities of thermal emission theoretically (see details in Supplementary Note 7). The results of our calculations are sown in Fig. 4a with a line for the spectral density and Fig. 4e–g for the directionality. The calculated spectra and directionality diagrams show an excellent qualitative agreement with our experimental measurements.

In addition, we compare experimental thermal emission directionalities at 400 °C with experimentally measured absorption directionalities at room temperature (see details in Supplementary Note 6 and Supplementary Fig. 6). The directionalities look similar, while resembling some differences in details associated with the change of material properties with temperature.

Discussion

We have demonstrated experimentally optical magnetic hyperbolic metamaterial with the principal components of the magnetic permeability tensor having the opposite signs. We have developed an experimental method for direct measurements of isofrequency dispersion contours of three-dimensional metamaterials and directly observed a topological transition between the elliptic and hyperbolic dispersions in metamaterials. In the hyperbolic regime, the length of wave vectors inside the metamaterial is diverging towards infinity.

We have applied an analytical theory that takes into account losses and spatial dispersion to describe the measured isofrequency contours, and demonstrated the importance of nonlocal contributions33 in the regime of optical topological transitions, associated with vanishing local parameters. A control of loss, gain and spatial nonlocalities in metamaterials opens up new opportunities for engineering isofrequency dispersion contours beyond elliptic or hyperbolic, with nontrivial geometry and topology. The magnetic hyperbolic dispersion of metamaterials together with their electric response enables impedance matching between hyperbolic media and air, resulting in an efficient light transfer through interfaces. Our results suggest that other three-dimensional metamaterials assembled from magnetically polarizable or chiral elements34,35,36 may possess magnetic hyperbolic dispersion as well.

In addition, we have studied the effect of the hyperbolic dispersion on thermal emission of matematerials, and revealed that the magnetic hyperbolic metamaterial demonstrates enhanced, directional, coherent and polarized thermal emission. These results suggest an advanced thermal management that can find applications in thermophotovoltaics37, scanning thermal microscopy38, coherent thermal sources39 and other thermal devices.

Methods

Nanofabrication

The bulk fishnet metamaterial is fabricated on a suspended 50-nm low-stress silicon nitride (Si3N4) membrane made from standard microelectromechanical systems fabrication technologies. The metal–dielectric stack is then deposited onto the Si3N4 membrane using layer-by-layer electron beam evaporation technique at pressure ∼1 × 10(−6) Torr without vacuum break. The chamber temperature is cooled down on each layer of evaporation to avoid buildup of excessive heating and stress. Essentially 10 repeating layers of gold (Au, 30 nm) and magnesium fluoride (MgF2, 45 nm) are deposited. Next, the sample is turned upside down and mounted on a special stage holder, which has a matching trench that prevents any mechanical contact with the fragile multilayer structure sitting on the membrane. The nanostructures are milled by gallium (Ga) focused ion beam from the membrane side. Milling from the membrane side prevents the implantation of Ga ions into the metal layers at the unpatterned areas that reduces optical losses and improves the overall quality. This is essential to mask the implantation of Ga ions into the metal layers at the unpatterned areas. The final structure has a slight sidewall angle along the thickness direction, but is found to have minor influence on the optical properties. Another important advantage of focused ion beam fabrication of the structure on a thin membrane compared with conventional bulk substrates is the ability for Ga ions to enter the free space (that is, no substrate for Ga ions to accumulate and cause undesired contamination and absorption).

Additional information

How to cite this article: Kruk, S. S. et al. Magnetic hyperbolic optical metamaterials. Nat. Commun. 7:11329 doi: 10.1038/ncomms11329 (2016).