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# Hybridization of singular plasmons via transformation optics

Edited by John B. Pendry, Imperial College London, London, United Kingdom, and approved June 3, 2019 (received for review February 6, 2019)

## Significance

Surface plasmons, which are optical resonances supported by metallic nanostructures, are extremely useful for manipulating light on the nanoscale. However, designing novel plasmonic devices becomes quite challenging when the desired spectral response pattern is highly complex. Here we show that, by developing a theoretical model which combines plasmon hybridization theory with transformation optics, one can control both global and local features of the resonance spectrum in an intuitive way. As an application, we design a plasmonic metasurface whose absorption spectrum can be controlled over an extremely large class of complex patterns through only two structural parameters. Our theoretical model paves the way for the next generation of plasmonic devices, allowing for the efficient design of complex systems.

## Abstract

Surface plasmon resonances of metallic nanostructures offer great opportunities to guide and manipulate light on the nanoscale. In the design of novel plasmonic devices, a central topic is to clarify the intricate relationship between the resonance spectrum and the geometry of the nanostructure. Despite many advances, the design becomes quite challenging when the desired spectrum is highly complex. Here we develop a theoretical model for surface plasmons of interacting nanoparticles to reduce the complexity of the design process significantly. Our model is developed by combining plasmon hybridization theory with transformation optics, which yields an efficient way of simultaneously controlling both global and local features of the resonance spectrum. As an application, we propose a design of metasurface whose absorption spectrum can be controlled over a large class of complex patterns through only a few geometric parameters in an intuitive way. Our approach provides fundamental tools for the effective design of plasmonic metamaterials with on-demand functionality.

Metallic nanostructures have been extensively studied and used for subwavelength control of light due to their unique ability to support surface plasmon resonances, which are the collective oscillations of conduction electrons on metal–dielectric interfaces (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–22). The excitation of surface plasmon resonances leads to the concentration of light into nanometric volumes and extreme enhancement of the electromagnetic fields. These phenomena have important applications including optical nanocircuits (11), single-molecule sensing, spectroscopy, light harvesting (7), color nanotechnology (19), and nonlinear optics (13).

When designing plasmonic devices, one of the fundamental challenges is to find a geometry of the nanostructure which would yield the desired resonance spectrum. The plasmon hybridization (PH) model (8, 23, 24) has been successfully used to understand the spectral responses of various nanostructures in a simple and intuitive way. Since the PH model guides us to design overall features of the nanostructures with intuition, the specific characteristics can be optimized by tuning over small sets of structural parameters. For designing novel devices with custom-defined functionality, the desired spectra could be highly complex. In this case, continuing with the above direct design method, which is an intuition-based approach, encounters the challenge of increasing complexity. Inverse design methods (25) such as the adjoint method (26), evolution algorithms (27), or data-driven approaches based on machine learning (28⇓–30) could be used instead but these require significant computational effort. It would be preferable to use a deeper theoretical understanding to reduce the complexity of the direct design problem.

To mitigate this challenge, we propose a hybridization model for systems of strongly interacting particles by combining the PH model with transformation optics (TO) (14, 15, 31⇓–33), which is another theoretical method to understand surface plasmons. Our model greatly extends the applicability of the PH model so that it can describe a wide range of simple to complex spectrum patterns with only a few geometric parameters. In some sense, the TO approach captures the global behavior of the resonance spectrum, while the PH model captures the local spectral shift or splitting. Combining these two in an elegant way, our proposed model provides a deep physical insight into how to control the global and local features of the spectrum separately so that the range of achievable spectral patterns can be greatly extended. As a proof of concept, we propose a design of original metasurfaces whose absorption spectrum can be varied over a wide class, including simple and complex patterns, by changing only two geometric parameters in an intuitive manner. Our work shows the possibility of designing plasmonic metamaterials in an effective way even when the desired spectral response is highly complex.

## Results and Discussion

Before explaining our model, we briefly review the PH model and TO approach and discuss the related challenges. In the PH model, the plasmons of the whole interacting particle system are viewed as simple combinations of the individual particle plasmons. The PH model describes the spectral shifts, induced by the coupling between the particles, in a way analogous to molecular orbital theory, providing a general and powerful design principle (8, 23, 24). However, when the particles are extremely close to touching, the physical picture becomes complicated since a large number of uncoupled plasmons contribute to each hybridized plasmon.

Recently, the TO approach (31⇓–33) was applied to two close-to-touching particles and other geometrically singular structures (14, 15, 34⇓–36). We also refer to refs. 37⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–48 for related works. TO shows that, as the two particles get closer, the discrete plasmon spectrum becomes more dense and eventually converges to a continuous spectrum at the touching limit. In the TO approach, conformal mappings are used to transform the singular structure to one with the same spectrum but having better symmetry, thereby providing a unique physical insight into the origin of the broadband light harvesting as well as analytic solutions (*SI Appendix*, sections 1–3). Moreover, for two close-to-touching particles, the TO approach is more accurate and efficient than the standard hybridization method (see *SI Appendix*, section 4 and Fig. S2 for their comparison). Nevertheless, TO alone cannot be applied to systems featuring three (or more) particles.

We should mention that, in our previous work (48), we considered a similar problem and developed an efficient numerical method for a system of close-to-touching plasmonic particles. But this numerical method, like FEM or FDTD, does not provide deep physical insights into the hybridization of surface plasmons. We emphasize that our present work’s focus is on advances in physical understanding of the plasmonic interaction.

As mentioned previously, our proposed model combines the advantages of both the PH and TO approaches to deal with an arbitrary number of close-to-touching particles. We consider the system of close-to-touching particles as a prototypical example. Our approach can more generally be applied to other singular systems consisting of crescents with corners, eccentric shells with small gaps, or their mixtures.

We should mention that the nonlocal effect, which has a quantum origin, is an important issue when the gap distance between the particles is extremely small (below 0.25 nm) (49⇓⇓⇓⇓–54). Our focus is not on modeling the nonlocal effect but on understanding the strong interaction between the particles. We assume a local model for the metal permittivity. The nonlocal effect can be accounted for by using the approach of refs. 14, 52, and 53.

We now explain our proposed model which we call the singular plasmon hybridization (SPH) model. In the standard hybridization model, a plasmon of the system is a combination of plasmons of individual particles. On the contrary, in our approach, the basic building blocks are the gap plasmons of a pair of particles whose singular behavior is captured using the TO approach. This simple conceptual change is the key to solving the aforementioned challenges. In Fig. 1*A*, we show a schematic comparison for a trimer, as it is the simplest example for our model (we emphasize that our model can be applied to a general configuration of particles, as shown in Fig. 1*B*). The trimer plasmon is now treated as a combination of two gap plasmons. In our picture, the new plasmons are formed by the hybridization of these gap plasmons. The gap plasmons are strongly confined in their respective gaps and all of the gaps are well separated, meaning that the gap plasmons do not overlap significantly. Hence, the spectral shifts due to their hybridization should be moderate and we can expect to find a simple picture even in the close-to-touching case.

To gain a better understanding, we develop a coupled-mode theory for the hybridization of singular gap plasmons. For simplicity, we consider only 2D structures (however, our theory can be extended to the 3D case). We assume the Drude model for the metal permittivity

We begin with the TO description (14, 15, 34⇓–36) of gap plasmons which are the basic building blocks of our proposed model. Consider a dimer of cylinders of radius R separated by a distance δ. By an inversion conformal mapping, the dimer is transformed to a concentric shell which is an analytically solvable case (Fig. 2*A*). Then TO reveals that, when the two cylinders get closer, the wavelength of their plasmon near the gap becomes smaller and energy accumulation occurs in the gap region. This gives rise to an extreme field enhancement. TO can also describe the singular spectral shift of gap plasmons. Let us consider the gap plasmons whose dipole moment is aligned parallel to the dimer axis since these plasmons contribute to the optical response significantly. Their resonant frequencies *B*, the frequencies *SI Appendix*, section 2 for more details).

We now turn to our SPH model, taking a trimer as an example (Fig. 3*A*). The trimer plasmon is specified as a superposition of the gap plasmon of the pair *SI Appendix*, section 5 for details). The coupling term

We call *B*). The bonding plasmon has a net dipole moment pointing in the *x* direction so that it can be excited by the *x*-polarized light. Similarly, the antibonding plasmon can be excited by the *y*-polarized light. These plasmons are very different from the bonding plasmon and the antibonding plasmon of a dimer in the standard hybridization model. They are trimer plasmons and are capable of capturing the close-to-touching interaction via TO. We emphasize that, contrary to the standard hybridization approach, these “simple” combinations of gap plasmons remain effective for describing the hybridized plasmons even when the particles are close to touching. In other words, the required number of uncoupled gap plasmons does not increase as the gap distance δ gets smaller and hence our model gives a simple picture in the close-to-touching case. We also mention that our physical picture for the trimer is qualitatively different from the standard hybridization one given in refs. 62 and 63.

We now discuss how our SPH model gives additional physical insights into the relationship between geometry and plasmons. The power of our model comes from its ability to decompose the plasmon spectrum into a singular part, which depends on the local geometry, and a regular part, which depends on the global geometry. The resonant frequency

We validate our model with numerical examples for the trimer. We set the radius of the particles to be *i*) *ii*) *C* and *D*, we plot the absorption cross-section (red and blue circles) for the trimer with the gap distance *x*-polarized and *y*-polarized incident light cases, respectively. Similarly, in Fig. 3 *E* and *F*, we plot the absorption cross-section in the case of the smaller gap distance *SI Appendix*, section 5 for details). We also plot the values of the resonant frequencies *x*-polarized (and *y*-polarized) incident field, respectively. See *SI Appendix*, section 6 and Figs. S3–S6 for the contour plots of potential distributions at these resonant frequencies. The gray dots represent the dimer frequency *y*-polarized incident field becomes stronger since the net dipole moment of the antibonding mode increases. Hence, the numerical results are consistent with the prediction of our proposed SPH model.

We now move on to consider the design of metasurfaces, which is a key application of this work. Recently, Pendry *et al*. (36) proposed a broadband absorbing metasurface consisting of a grating with points of vanishingly small thickness (which are geometric singularities). Remarkably, they interpreted its broadband spectral response as a realization of compacted dimensions. They used a geometric singularity to control the global behavior of the spectrum: The discrete spectrum converges to the continuous one as the small thickness goes to zero. We refer to refs. 64⇓–66 for related works on the metasurfaces. In this work, we propose a metasurface which can generate a large class of simple to complex spectral patterns through only two geometric parameters. This is achieved by controlling both global and local spectral shifts. Our metasurface geometry, shown in Fig. 4*A*, is a 1D periodic array consisting of two particles with different radii. This array is a combination of two subarrays,

To explain the motivation behind our metasurface design, we begin by considering the array of larger particles *B*, the hybridization of which results in new resonance peaks. As in the case of a single trimer, it is natural to expect that the hybridization becomes stronger as the bonding angle *i*) *ii*) *C* and *D*, respectively, assuming a normal incidence to the metasurface. Assuming the Drude model *SI Appendix*, section 8 for the SPH theory of the metasurface. Our theoretical results, which are based on the quasi-static approximation, are in excellent agreement with the fully retarded simulation results (performed by using the multipole expansion method). This is because the period of the structure is small compared with the wavelength. We also compare the results with the absorption of a metasurface consisting of only the largest particles

Let us discuss the advantages the SPH model gives us in the design of the metasurface. First, the proposed metasurface has “only a few” geometric parameters but can generate a large class of spectral responses. Second, the SPH model yields (semi)“analytic” solutions for the hybridized plasmons of the metasurface, as described in *SI Appendix*, section 8. These two features allow us to easily optimize the metasurface’s parameters for the targeted application.

We finally mention that the extension of the SPH model to the 3D case is nontrivial. A spherical dimer supports three branches of gap plasmons (35): the odd modes, normal even modes, and anomalous even modes. The anomalous ones are not confined entirely within the gap but spread over the whole particle surface. Hence their hybridization cannot be described by the present form of the SPH model and should be considered carefully.

## Conclusions

We have proposed the singular plasmon hybridization model for plasmons of strongly interacting particles which gives a simple and intuitive physical picture when the particles are close to touching. The proposed model demonstrates an elegant interplay between the plasmon hybridization model and transformation optics, clarifying a deep geometric dependence of the plasmon spectrum. It enables us to design a plasmonic device whose spectral characteristics can be controlled over a large class of patterns through only a few geometric parameters. We believe that our model can have a significant impact on the design of a variety of complex plasmonic devices.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: sanghyeon.yu{at}sam.math.ethz.ch.

Author contributions: S.Y. designed research; S.Y. and H.A. performed research; and S.Y. and H.A. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1902194116/-/DCSupplemental.

Published under the PNAS license.

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