## Manipulating Optical Topology

Phase transitions in solid-state systems are often associated with a drastic change in the properties of that system. For example, metal-to-insulator transition or magnetic-to-nonmagnetic states find wide application in memory storage technology. An exotic electronic phase transition is the Lifshitz transition, whereby the Fermi surface undergoes a change in topology and a drastic change in the electronic density of states. **Krishnamoorthy et al.** (p. 205) now show that the notion of such a phase transition can be carried over to the optical regime by the suitable design of a metamaterial structure. This effect could be used to control the interaction between light and matter.

## Abstract

Light-matter interactions can be controlled by manipulating the photonic environment. We uncovered an optical topological transition in strongly anisotropic metamaterials that results in a dramatic increase in the photon density of states—an effect that can be used to engineer this interaction. We describe a transition in the topology of the iso-frequency surface from a closed ellipsoid to an open hyperboloid by use of artificially nanostructured metamaterials. We show that this topological transition manifests itself in increased rates of spontaneous emission of emitters positioned near the metamaterial. Altering the topology of the iso-frequency surface by using metamaterials provides a fundamentally new route to manipulating light-matter interactions.

Metamaterials are artificial media in which the subwavelength features of the designed unit cells and coupling between them governs the macroscopic electromagnetic properties (*1*). This control over material parameters has led to new applications (*2*–*4*) and also the ability to mimic and study physical processes, which is difficult by other methods (*5*–*7*). One specific design freedom afforded by metamaterials is the control over the iso-frequency surface, the surface of allowed wavevectors at constant frequency (*8*, *9*). The topology of this surface governs wave dynamics inside a medium.

The ideas of mathematical topology play an important role in many aspects of modern physics, from phase transitions to field theory to nonlinear dynamics (*10*, *11*). An important example of this is the Lifshitz transition (*12*), in which the transformation of the Fermi surface of a metal from a closed to an open geometry (because of, for example, external pressure) leads to a dramatic effect on the electron magneto-transport (*13*). In optics, the role of the Fermi surface is played by the optical iso-frequency surface , which can be engineered by tailoring the dielectric tensor, . We use this to demonstrate the optical equivalent of the “Lifshitz transition”—the optical topological transition (OTT) in which the very nature of the electromagnetic radiation in the metamaterial undergoes a drastic change. Effects on the kinetic and thermodynamic properties, such as the dynamics of propagating waves supported by the system and the electromagnetic energy density, respectively, are modified at the transition point and can be probed by following the light-metamaterial interaction using a quantum emitter.

We considered a metamaterial structure that has a uniaxial form of the dielectric tensor , where and . The iso-frequency surface for the extraordinary (TM-polarized) waves propagating in such a strongly anisotropic metamaterial is given by (1)Closed iso-frequency surfaces differing from a simple sphere (such as an ellipsoid) can occur in these metamaterials when and . On the other hand, an extreme modification of the iso-frequency surface into a hyperboloid occurs when the dielectric constants show opposite sign ( and ). We can design a metamaterial so that the dispersion of the dielectric constants leads to an OTT in the iso-frequency surface from an ellipsoid to a hyperboloid.

The photonic density of states (PDOS) in the metamaterial is related to the volume enclosed by the corresponding iso-frequency surface (*14*). Therefore, the topological transition from the closed (ellipsoid) iso-frequency surface for to an open (hyperboloid) iso-frequency surface for (Fig. 1) results in a nonintegrable singularity accompanied by a change in the density of states from a finite to an infinite value (in lossless effective medium limit). This optical analog of the Lifshitz transition in metamaterials is characterized by the appearance of additional electromagnetic states in the hyperbolic regime, which have wave vectors much larger than those allowed in vacuum. Light-matter interaction is enhanced because of the presence of these additional electromagnetic states, resulting in a strong effect on related quantum-optical phenomena, such as spontaneous emission.

The decay rate near a half space of a metamaterial for a dipole-like emitter is given by (*15*–*20*) (2)where and are the decay rates due to propagating waves in vacuum and the surface plasmon polariton (SPP) modes supported by the metamaterial, respectively, and is the decay rate enhancement due to the high–wave vector states, which appear only beyond the OTT. In the near field of the metamaterial, when d << λ the decay rate is dominated by the contribution from the high–wave vector states (*21*): (3)where is the dipole moment of the perpendicularly oriented dipole, is the plane wave reflection coefficient of p-polarized light (*21*), and *d* is the distance of the dipole from the interface. In a hyperbolic metamaterial half space where (4)and for [], we have an elliptical dispersion with (assuming no losses). We thus introduce the topological transition parameter , which is proportional to the local density of electromagnetic modes and characterizes the emergence of high-k metamaterial states. The effect of dispersive and lossy effective medium dielectric constants on the topological transition parameter (proportional to the spontaneous emission rate) is shown in Fig. 2A. Although the losses reduce the sharp transition to a smooth crossover, the change in the iso-frequency surface topology still leads to an enhanced spontaneous emission rate.

Metamaterials with opposite signs of the dielectric constants can be realized by using metal-dielectric composites (*22*). We considered such a composite consisting of alternating layers of silver (9 nm) and titanium dioxide (TiO_{2}) (22 nm) corresponding to a 29% fill fraction of silver. Using the semiclassical theory of spontaneous emission (*23*), we calculated the lifetime for quantum dots (QDs) placed in the near field of this metamaterial. Our simulation takes into account nonidealities arising because of realistic losses, dispersion, finite thickness of layers, sample size, and the substrate. Even in a practical structure (Fig. 2B), a clear modification in lifetime of the emitter is expected as the system transitions from elliptical to the hyperbolic dispersion regime over the spectral range of interest. The transition occurs because of the particular dispersion of coupled plasmons, which contribute to lifetime decrease only on the hyperbolic side of the transition (*21*). Simulations carried out through effective medium theory (EMT) show good agreement with the prediction of numerical simulations (Fig. 2B).

To experimentally observe the signature of the predicted OTT manifested through enhancement in spontaneous emission rate, we investigated a metamaterial structure similar to that discussed above with multiple QD emitters positioned on its top surface (Fig. 3A) (*21*). The dielectric constants of the constituent thin films were extracted by using ellipsometry, and the effective medium parameters are shown in Fig. 3B. This structure is designed to have around 621 nm, which corresponds to the emission maximum of the CdSe/ZnS colloidal QDs used in the experiment. The photoluminescence (PL) from the QDs has a full width at half maximum (FWHM) of ~40 nm, which allows investigation of the phase space of both elliptical and hyperbolic dispersion regimes by use of the same sample. In order to isolate the effects of the nonradiative decay and SPP-based enhancement in the radiative rate due to the metal, we also measured the spontaneous emission rates of QDs on a control sample that consisted of one unit cell of the metamaterial (*21*).

Time-resolved PL measurements were carried out on the metamaterial sample, the control sample, and the glass substrate (Fig. 3C) at the anticipated transition wavelength (621nm) and at spectral positions on either side (605 and 635 nm). The large change in the spontaneous emission lifetime of the QDs on the metamaterial compared with the glass substrate is due to the excitation of the high-k metamaterial states as well as the nonradiative contribution and the SPP modes of the metamaterial. When compared with the control sample, the metamaterial shows an enhancement in the spontaneous emission rate by a factor of ~3 at the transition wavelength and ~4.3 deeper in the hyperbolic regime (635 nm). These enhancements are attributed to the high-k metamaterial states. The overall reduction in the lifetime of the QDs when compared with those on a glass substrate is ~11.

The lifetime of the QDs increases as a function of wavelength on both the glass substrate and the control sample (Fig. 3D). This is due to the size distribution of QDs and the dependence of the oscillator strength on the energy (*24*–*26*). On the contrary, the metamaterial sample shows a decrease in the lifetime as a function of wavelength and the shortest lifetime owing to coupling to the high-k metamaterial states (Fig. 3D). The coupling of the emission from the QDs into the metamaterial states was also verified by using steady-state PL measurements (fig. S6) in which a reduction in the PL intensity emitted in the direction away from the metamaterial sample was observed (*21*, *27*).

To demonstrate the OTT, we studied the radiative lifetimes in three samples with differing volume ratio of metal to dielectric, which correspond to different transition wavelengths. We compared the lifetimes of the sample with 29% fill fraction of silver to one with 23% fill fraction, which lies deep in the elliptical phase, and to another sample that lies deeper in the hyperbolic phase, with fill fraction of 33% (*21*). To make any conclusions regarding the effect of the OTT on the radiative lifetimes from the measurement of the total decay rates, the contribution of the high-k metamaterial states has to be distinguished from nonradiative decay and the SPP-assisted decay of the unit cell. Because the spacer and first-layer environment of the QDs is the same in the control and the metamaterial sample, we expected similar quantum yield dependence on wavelength owing to near-field interaction of the QDs and the metallic structure. Thus, to account for purely the contribution from the high-k metamaterial states to the overall lifetime change, we normalized the QD lifetime on the metamaterials with that on the control samples (Fig. 4). The controls for the three samples are different and correspond to the unit cell of each metamaterial sample. To compare these three samples, which have OTTs at different wavelengths, we normalized the wavelength with respect to the transition wavelength, with positive and negative Δλ corresponding to the hyperbolic and elliptical dispersion regimes, respectively.

We clearly observed a sharp reduction in the normalized lifetime of the samples with 29 and 33% fill fractions that cross the transition wavelength, whereas the sample with 23% fill fraction, which lies in the elliptical regime, did not show this reduction. Whereas the combination of metal losses and finite thickness of layers leads to a smooth crossover, the signature of the transition is clear from the reduction in the normalized lifetime in the hyperbolic regime. The difference in the absolute value of the normalized lifetimes in the two samples that show the transitions is due to the differing fill fractions (29 versus 33%) and associated difference in the dielectric constants. Thus, the changes in the lifetime observed experimentally in these metamaterial structures can be attributed to the increase in the photonic density of states that manifests when the system goes through the topological transition in its iso-frequency surface from an ellipsoid to a hyperboloid, which is in good agreement with the theoretical prediction of Fig. 2B.

We have established the occurrence of OTT in two metamaterial structures using spontaneous emission from a quantum emitter as the probe. Absence of OTT in a structure that lies solely in the elliptical dispersion phase has also been demonstrated. A host of interesting effects can transpire at the transition wavelength, such as the sudden appearance of resonance cones, which are characteristic of hyperbolic metamaterials (*28*), enhanced nonlinear effects, and abrupt changes in the electromagnetic energy density. We expect the OTT to be the basis for a number of applications of both fundamental and technological importance through use of metamaterial-based control of light-matter interaction.

## Supplementary Materials

## References and Notes

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**Acknowledgments:**V.M.M. and H.N.S.K. acknowledge partial support through the Materials Research Science and Engineering Center program of the National Science Foundation through grant DMR 1120923 and the Army Research Office (ARO) grant W911NF0710397. V.M.M. and I.K. acknowledge support through Round 14 of the CUNY Collaborative Incentive Research Grant Program. E.N. was partially supported by ARO–Multidisciplinary University Research Initiative grants 50342-PH-MUR and W911NF-09-1-0539. Z.J. was partially supported by Natural Sciences and Engineering Research Council of CanadaDiscovery grant 402792 and Canadian School of Energy and Environment Proof of Principle Grant. The ellipsometric measurements were carried out at the Center for Functional Nanomaterials at Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract DE-AC02-98CH10886.