Electrostatic properties of two-dimensional nanosystems can be completely described by their non-trivial geometry modes. In this paper we prove that these modes as well as the corresponding eigenvalues are invariant under any conformal transformation. This invariance suggests a new way to study electrostatic conformal transformations, while also providing an in-depth interpretation of the behavior exhibited by singular plasmonic nanoparticles.
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An intriguing property of Maxwell’s equations is their form invariance under arbitrary coordinate transformations, assuming the field quantities and the material properties are transformed accordingly . This invariance leads to a powerful design tool, referred to as transformation optics or transformation electromagnetics [1–7]. Since the introduction of transformation optics in 2006, a wealth of novel and unique devices have been theorized and demonstrated [7,8]. Some examples include: electromagnetic cloaks [9–11], event cloaks , optical black holes  and field splitters [14, 15].
When the electromagnetic wavelength is far longer than the characteristic size of the structure, quasi-static approximations can be employed. As a result, the interactions between light and a medium can be approximately described by Laplace’s equation . As with the full-wave Maxwell equations, the two-dimensional Laplacian equation is also invariant under conformal transformations. Additionally, the electrostatic transformation does not alter the properties of the constituent material, in direct contrast with the full-wave coordinate transformations which generally result in a physical medium with complex or exotic properties. This unique property of the Laplacian equation has been used to design near-field perfect lenses and reflectors [17–21]. Recently, it has been employed to study two-dimensional plasmonic particles. Wherein, the transformed geometry can be studied analytically by mapping a complicated two-dimensional structure into a simple one-dimensional configuration, such as a metallic slab of finite-thickness [22–27]. It was further found that these singular plasmonic nanoparticles, including crescent and touching or non-touching cylindrical dimers, can exhibit broadband properties and significantly enhance the local electric fields.
In this paper, we will study electrostatic conformal transformations by taking advantage of the invariance of geometry modes and their associated eigenvalues. We will show that transformation of a geometry can be equivalently explained by transforming the excitation source and modifying the amplitudes of the associated geometry modes. A new approach is developed that exclusively depends on the structural geometry. This method is then used to interpret two important features of singular plasmonic structures, i.e., broadband response and divergent electric fields around the singularities.
To begin, we will invoke the spectral Bergman-Milton theory [28, 29], which states that the electrostatic behavior of a nanosystem consisting of two different mediums, with permittivities of ɛ1(ω) and ɛ2(ω) respectively, can be described by a set of non-trivial eigenmodes φn of the following generalized eigenproblemAppendix A). The function θ(r) characterizes the geometry of the composite: θ(r) = 1 when r is inside the medium with dielectric constant ɛ1 and equals 0 elsewhere. To simplify the following discussion, ɛ2(ω) is assumed to be 1. Since this equation depends exclusively on the geometry, but not on the material composition, the resultant eigenmodes are therefore referred to as geometry modes . Moreover, by defining a scalar product as 30–33].
At this point it is important to notice that, for a two-dimensional system, Eq. (1) remains invariant under any conformal coordinate transformation. To prove this fact, we rewrite its left-hand side as
The invariance of the non-trivial eigenmodes φn and their corresponding eigenvalue sn immediately suggest a new way to interpret and study the electrostatic response of the transformed structure. This new approach depends exclusively on the geometry, as described by the function θ(r), and does not need any information regarding the material parameters ɛ(ω) or the external potential. For instance, we can expand the total potential φt in terms of the associated eigenmodes,Appendix B)
It should be mentioned that the geometry modes actually are the surface modes of the particle [34, 35], and the corresponding resonance condition for the nth mode is strictly s(ω) = sn, or equivalently ɛ1 = 1 – 1/sn, when we do not include the radiation loss. Notice that the corresponding permittivity ɛ1 of the particle should be real and negative since 0 < sn < 1 . The most striking property of the surface mode is that the resultant total electrostatic energy of the entire system, including the free space and the particle, is exactly zero . Furthermore, the complex frequency of the nth surface plasmonic resonance (SPR) is given by s(ωn – iγn) = sn with ωn being real resonant frequency and γn being the relaxation rate [28,37]. Since the eigenvalues sn are conserved under any conformal mapping, the transformed structure will have the same SPRs as the original structure at the identical resonant frequencies. For instance, since a metallic cylinder can be transformed from a metal-dielectric interface by w = ez, its SPR hence can be determined by the nonretarded surface-plasmon condition of the metal-dielectric interface .
As a proof of principle, we consider a one-dimensional dielectric slab with finite thickness (Figure 1(a)) and its derivative systems obtained through different conformal transformations. The eigenvalues of the slab are given by (See Appendix C)38]. Furthermore, the resonant condition, approximately given by Re[s(ωn)] = sn when the relaxation rate is weak, suggests that only the odd modes will be excited resonantly when Re(ɛ1) is smaller than −1. Moreover, these eigenmodes and eigenvalues can be directly applied to the complementary structure of the finite slab, i.e. a free-space gap sandwiched by two semi-infinite metallic spaces. Since these two structures have identical geometry, they therefore share the same eigenvalues and eigenmodes. To study the complementary structure, all we need to do is interchange ɛ1 and ɛ2. As a direct consequence, when Re(ɛ1) is smaller than −1, only the even modes will be resonantly excited in the semi-infinite structure .
We now calculate the resultant expansion coefficients of a structure transformed from the finite slab through a conformal mapping of w = w(z). It is further assumed that the external potential φ0(u, v) = puu + pvv, which corresponds to a uniform electric field −pueu – pvev. The expansion coefficients in this case can be written as
We first transform the slab to two coaxial cylinders (Figure 1(b)) with w = ez, where the radii of the two cylinders are r1 = ed1 and r2 = ed2 respectively . The expansion coefficients are found to be (See Appendix D)34].
Since the eigenvalues of the finite slab cover a wide region (0,1) (note that the eigenvalues of Eq. (1) are limited to between 0 and 1), any negative Re(ɛ1) will excite a surface mode as long as its expansion coefficient is not zero. Consequently the transformed system will present broadband response in principle. One example is the crescent studied in , which can be obtained by using w = 1/z and d1 > 0 (Figure 1(c)). The expansion coefficients are found to be (See Appendix D)22] (See Appendix E). Evidently, each eigenmode is excited by the external potential. Furthermore, |(φk,±|φ0)| depends exclusively on the amplitude of the external electric field. The energy quantities such as the absorption hence do not depend on the direction of the incident field , a property which is not so obvious. Figure 2 shows the dependence of |(φk,±|φ0)|2 on k achieved by setting pu = pv = 1, d1 = d and d1 = 2d with d being an arbitral real quantity. As mentioned, each eigenmode of the crescent is excited, consequently |(φk,±|φ0)|2 is always positive. Furthermore, in this case these expansion coefficients are strongly localized in the regime 0.3 ≤ |k|d ≤ 1.6 with a maximum around |k|d = 0.72.
Another example consists of the kissing cylinders suggested in , which is obtained by transforming two semi-infinite slabs, with d1 < 0 and d2 > 0, using the conformal mapping given by w = 1/z (Figure 1(d)). The corresponding expansion coefficients are calculated as (See Appendix D)Figure 2, with pu = pv = 1 as well as d1 = −d and d1 = 2d. Similar to the curve associated with the crescent, the eigenmodes with 0.2 ≤ |k|d ≤ 1.6 contribute significantly to the absorption cross section, which guarantees that the kissing cylinders will exhibit broadband behavior.
Considerable field enhancement and confinement has been observed around the singularity of the metallic crescent or cylinder dimers under a uniform illumination [22–27]. To interpret this phenomenon, we assume the frequency of the external excitation coincides with ωn, the resonant frequency of the n-th surface mode. The total potential of Eq. (5) can therefore be approximated as βn(φn|φ0)φn. Since only φn is position dependent, the amplitude of the corresponding electric field is proportional to |∇wφn| = |dz/dw||∇zφn|. Taking the crescent as an example, |dz/dw| = x2 + y2 since w = 1/z. Consequently, when we map the point (a,∞) in xy coordinates to the singular point of the crescent (the origin of the uv coordinates), the corresponding electric field tends to infinity. On the other hand, |dz/dw| = 1/ex for the two coaxial cylinders we studied above, and the resultant electric field for this case only depends on r and does not exhibit singular behavior. Note that |dz/dw|2 is actually the stretching factor for area when we transform the xy coordinates to the uv coordinates. The field enhancements because of |dz/dw| hence are purely induced by the coordinate transformations. Furthermore, we want to point out that, although the electric field at a few specific points can be infinite, Ie, the integration of the electric field intensity over the system, is finite and invariant under any conformal transformation.
In conclusion, we proved that the electrostatic eigenmodes and eigenvalues of two-dimensional nanosystems with proper boundary conditions are invariant under any conformal transformation. Based on this property, we suggested a new approach to studying the electrostatic responses of the transformed structures. Namely, transforming a geometry is equivalent to transforming the external potential or the expansion coefficients of the invariant eigenmodes. This method is then used to interpret two important features of singular plasmonic structures, i.e., broadband response and divergent electric fields in the neighborhood of the singularities.
A. Properties of Eq. (1)33], the eigenmodes corresponding to the eigenvalues of 0 and 1 are unimportant because their scalar product with any function vanishes. We therefore only consider the non-trivial modes and normalize them in the following way
B. Absorption and extinction
The normalized non-trivial eigenmodes φn are a complete orthogonal set , and the external potential, inside the ɛ1 material, can be expanded in terms of these modes
In electromagnetics, the extinction of a particle illuminated by an external field is given by Eq. (B.1) and (B.4) into the above equation yields
C. Geometry modes of one-dimensional finite slab
We consider a one-dimensional dielectric slab with finite thickness (See Figure (1a)). The slab sits between [d1, d2]. The possible solution has a form ae|k|x eiky for x < d1, ce–|k|x eiky for x > d2, and b1 e|k|x eiky + b2e−|k|x eiky in the middle. The boundary conditions at x = d1 and x = d2 yield
D. Expansion coefficients of different structures
Kissing cylinder dimer: A kissing dimer can be transformed from a slab with d1 < 0 and d2 > 0, by using a conformal mapping of w = 1/z.
We first calculate Fk,
E. Electric field of the crescent
The total electric field, including both incident and scattered fields, can be expressed asEq. (8), βk,± can be reformulated as 22] as well as setting g = 1, the above equation is found to be identical to Eq. (28) of Ref. .
This work was supported in part by the Penn State MRSEC under NSF grant no. DMR 0213623.
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