Hybridization of singular plasmons via transformation optics

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. 1A, 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. 1B). 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.

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Fig. 1.

Schematic description of our proposed model. (A) Comparison with the standard hybridization model. (B) A more general 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 ϵ=1−ωp2/ω2, where ωp is the bulk plasma frequency and the background permittivity is ϵ0=1. We also adopt the quasi-static approximation by assuming the system to be small compared with the wavelength of the incident light. We remark that the radiation reaction can be incorporated to go beyond the quasi-static limit, as described in refs. 14 and 55.

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. 2A). 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 ωnTO are given byωnTO=ωp/e−ns⁡sinh(ns), n=1,2,3,…,[1]with the parameter s satisfying sinh2⁡s=(δ/R)(1+δ/4R). We denote their associated gap plasmons by ωnTO. When the gap distance δ gets smaller, as shown in Fig. 2B, the frequencies ωnTO are red shifted singularly and the spectrum becomes denser. Thus, the TO description captures the singular behavior of gap plasmons (see SI Appendix, section 2 for more details).

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Fig. 2.

Dimer plasmons in TO approach. (A) Transformation of a dimer into a concentric shell. Red solid lines represent the oscillations of a dimer gap plasmon. (B) The singular red shift of the spectrum for a dimer. We set R=20 nm and ωp=8 eV.

We now turn to our SPH model, taking a trimer as an example (Fig. 3A). The trimer plasmon is specified as a superposition of the gap plasmon of the pair (B1,B2) and that of the pair (B2,B3). We let (an,bn) represent the following linear combination of the gap plasmons: anωnTO(B1,B2)+bnωnTO(B2,B3). Their hybridization is characterized by the following coupled-mode equations:(ωnTO)2ΔnΔn(ωnTO)2anbn=ω2anbn.[2]Here, Δn represents the coupling between the two gap plasmons. As the bonding angle θ between the two gap plasmons decreases, the coupling strength Δn increases, which is to be expected since the two gaps get closer. This coupled-mode system is derived using the spectral theory of the Neumann–Poincaré operator (56⇓⇓⇓⇓–61) and TO (see SI Appendix, section 5 for details). The coupling term Δn is calculated without any assumption or fitting procedure. We emphasize that the above equation is a simplified version of our theory. Although we require additional TO gap plasmons for improved accuracy, we shall see that this simplified version can already capture the physics. Solving the equation, we obtain the hybridized plasmons for the trimerωn±≈12ωnTO(B1,B2)∓ωnTO(B2,B3),n=1,2,…,[3]and their resonant frequenciesωn±≈ωnTO±Δn, n=1,2,⋯ .[4]So our theory predicts that the spectrum consists of a family of pairs (ωn−,ωn+) of resonant frequencies which are split from the dimer resonant frequencies ωnTO. The dimer part ωnTO is singularly shifted as the gap distance δ gets smaller, while the splitting part Δn remains moderate.

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Fig. 3.

Trimer plasmons in the SPH model. (A) Geometry of the trimer. The pairs (B1,B2) and (B2,B3) are close to touching while B1 and B3 are well separated. (B) Hybridization of trimer singular plasmons. (C and D) Absorption cross-sections when the gap distance is δ=1.25 nm. The open circles (resp. solid lines) represent the numerical simulation results (resp. theoretical results from the SPH model), respectively. The red and blue colors represent the x-polarized and y-polarized incident light cases, respectively. (E and F) The same as C and D but with the gap distance δ=0.25 nm. We set R=20 nm, ωp=3.85 eV, and γ=0.1 eV.

We call ωn− and ωn+ the bonding trimer plasmon and antibonding trimer plasmon, respectively (Fig. 3B). 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 ωn± for the trimer consists of two parts: the singularly shifted part ωnTO and the regular splitting part Δn. The singular part ωnTO is determined by the small gap distance δ, which is a “local” feature of the geometry. On the other hand, the regular part Δn is determined by the bonding angle θ, which is a “global” feature of the geometry. In other words, the small gap distance δ affects the “overall” behavior of the spectrum while the bonding angle θ controls the “detailed” splitting of the spectrum. This shows an interesting relation between the spectrum and the geometry: Local (and global) features of the geometry can determine the global (and local) behavior of the spectrum, respectively. This relation provides us with a design principle: One should manipulate the local geometric singularity (such as interparticle gap distances) to control the overall behavior of the spectrum, while manipulating the global geometry to achieve the desired detailed splitting of the spectrum. Our SPH model provides a systematic way of achieving such a design using gap plasmons as basic building blocks. Our approach is valid for general systems consisting of an arbitrary number of interacting particles, with arbitrary positions and different radii.

We validate our model with numerical examples for the trimer. We set the radius of the particles to be R=20 nm. We consider the two cases when the interparticle gap distances are (i) δ=1.25 nm and (ii) δ=0.25 nm. Note that the ratio δ/R is very small so that the particles are close to touching. We assume the Drude model ϵ=1−ωp2/(ω(ω+iγ)) with ωp=3.85 eV and γ=0.1 eV. In Fig. 3 C and D, we plot the absorption cross-section (red and blue circles) for the trimer with the gap distance δ=1.25 nm when the bonding angle is θ=150○ (weak coupling) and θ=85○ (strong coupling), respectively. In the latter case, the coupling strength between the gap plasmons is stronger since the gaps are closer to each other. The solid lines (and open circles) represent the theoretical results (and numerical simulation results) performed by our SPH model (and the fully retarded version of multipole expansion method), respectively. The red and blue colors correspond to the 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 δ=0.25 nm. We emphasize that we have used a complete version of our SPH model for theoretical results (see SI Appendix, section 5 for details). We also plot the values of the resonant frequencies ωn− and ωn+ (red and blue dashed, vertical lines) computed by the SPH model. Their corresponding plasmons are dominated by bonding and antibonding combinations of gap plasmons, respectively. As expected, the resonance peaks of the absorption are located near the bonding (and antibonding) plasmon frequencies ωn− (and ωn+) for the 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 ωnTO computed using the TO approach. As the gap-distance δ gets smaller, the overall spectrum is significantly red shifted in conjunction with the singular shift of ωnTO. The green arrows indicate how much the trimer frequencies ωn± have split from the dimer frequency ωnTO. It is clearly shown that the splitting ωn±−ωnTO is moderate regardless of the interparticle gap distance. In the strong coupling case (smaller bonding angle) the splitting is more pronounced. Further, as the bonding angle θ decreases the absorption of the 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. 4A, is a 1D periodic array consisting of two particles with different radii. This array is a combination of two subarrays, B1# which consists of the largest particles and B2# which consists of the smallest ones.

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Fig. 4.

(A–D) Metasurface geometry (A), singular gap plasmons of the metasurface (B), and the absorption spectrum patterns (C and D) for various gap distances and bonding angles. We set R1=30 nm, R2=15 nm, ωp=3.5 eV, and γ=0.05 eV.

To explain the motivation behind our metasurface design, we begin by considering the array of larger particles B1#. As the particles composing B1# become closer, the spectrum becomes more dense. To generate various spectral patterns, we introduce the array B2# and position it close to B1#. This leads to the formation of three kinds of singular gap-plasmon arrays in each gap, as in Fig. 4B, 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 θ# decreases. We verify this prediction with numerics. We set the radii to be R1=30 nm, R2=15 nm. We assume that the two gap distances are the same, δ1=δ2=δ, and consider two cases, (i) δ=3 nm and (ii) δ=0.3 nm. For each gap distance, we also consider four cases of different bonding angles from θ#=90○ (weak coupling) to θ#=50○ (strong coupling). We plot the absorption in Fig. 4 C and D, respectively, assuming a normal incidence to the metasurface. Assuming the Drude model ϵ=1−ωp2/(ω(ω+iγ)) with ωp=3.5 eV and γ=0.05 eV, we compute the absorption spectra using our SPH model. See 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 B1# (gray dotted lines). In the weak coupling regime, the introduction of the array B2# results in a somewhat minor alteration of the spectrum. In the strong coupling regime, new resonance peaks appear due to the strong hybridization of singular plasmons (the new peaks are marked with gray triangles). Clearly, this shows that a variety of patterns of the plasmon spectrum, ranging from simple to highly complex ones, are generated by adjusting only two geometric parameters: the gap distance δ and the bonding angle θ#. This is possible because δ controls the global spectral shift while θ# controls the local spectral splitting. The diversity of the generated spectrum comes from a combination of these two geometric parameters. This could have applications in plasmonic color engineering (12, 19). We should also mention that these various spectral patterns and their resonance peaks can be explained using our SPH model, although we do not present the detailed analysis for the hybridized modes here. More general spectral patterns could be realized by considering more complex particle configurations.

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.