General properties of two-dimensional conformal transformations in electrostatics

Yong Zeng; Jinjie Liu; Douglas H. Werner

doi:10.1364/OE.19.020035

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 [1]. 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 [12], optical black holes [13] 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 [16]. 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 φ_n of the following generalized eigenproblem

\nabla \cdot [θ (r) \nabla φ_{n} (r)] = s_{n} \nabla^{2} φ_{n} (r),

where s_n represents the corresponding eigenvalues (see Appendix 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 [28]. Moreover, by defining a scalar product as

(ϕ | ψ) = \int θ \nabla ϕ * \cdot \nabla ψ d v,

we find that s_n of a non-trivial mode must be real and limited to the range 0 < s_n < 1, and the normalized eigenmodes φ_n can be used to expand an arbitrary function for points r that are inside the ɛ₁ material [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

\partial_{i} e_{i} \cdot (θ \partial_{j} φ_{n} e_{j}) = g_{i j} \partial_{i} (θ \partial_{j} φ_{n}) = g \partial_{i} (θ \partial_{i} φ_{n}),

where the Einstein summation convention is employed, and the metric tensor g_ij ≡ e_i · e_j = gδ_ij because of the conformal transformation. Similarly the right-hand side can be reformulated as

s_{n} \partial_{i} e_{i} \cdot (\partial_{j} φ_{n} e_{j}) = s_{n} g_{i j} \partial_{i} (\partial_{j} φ_{n}) = s_{n} g \partial_{i i} φ_{n} .

Evidently the transformed equation is identical to the previous one if we interpret the new equation as being in a right-handed Cartesian system and keep θ and φ_n unchanged.

The invariance of the non-trivial eigenmodes φ_n and their corresponding eigenvalue s_n 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 φ_t in terms of the associated eigenmodes,

\sum_{n} \frac{s (ω) (φ_{n} | φ_{0}) φ_{n}}{s (ω) - s_{n}} \equiv \sum_{n} β_{n} (φ_{n} | φ_{0}) φ_{n},

where k = ω/c, s(ω) = 1/(1 – ɛ₁) being the Bergman’s spectral parameter and φ₀ being the external potential. Notice that we do not include the radiation damping here. Consequently, once these eigenmodes of the original geometry are known, we can obtain the potential φ_t of any transformed or derivative geometry by simply calculating the expansion coefficients (φ_n|φ₀). In other words, transforming geometry is equivalent to transforming the external source, or more precisely, changing the expansion coefficient of each eigenmode. Furthermore, the time-averaged power P_a absorbed by the structure can be written as (See Appendix B)

P_{a} = ω ɛ_{0} Im (ɛ_{1}) I_{e} / 2,

P_{ex} = \frac{ω ɛ_{0}}{2} \sum_{n} {| (φ_{n} | φ_{0}) |}^{2} Im [(ɛ_{1} - 1) β_{n}] .

When we transform the external field φ₀ accordingly, i.e., φ₀(x, y) = φ′₀ (u, v) with u and v being the new coordinates, all the energy quantities, P_a, I_e and P_ex, are evidently invariant. On the other hand, if the external potential is fixed with respect to the w coordinate where w = u + iv, it follows that these energy quantities are proportional to a² when the conformal mapping has a form a × w(z) with z = x + iy and a is real. In other words, the larger the particle area, the stronger the absorption and extinction.

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) = s_n, or equivalently ɛ₁ = 1 – 1/s_n, when we do not include the radiation loss. Notice that the corresponding permittivity ɛ₁ of the particle should be real and negative since 0 < s_n < 1 [34]. 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 [36]. Furthermore, the complex frequency of the nth surface plasmonic resonance (SPR) is given by s(ω_n – iγ_n) = s_n with ω_n being real resonant frequency and γ_n being the relaxation rate [28,37]. Since the eigenvalues s_n 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 = e^z, its SPR hence can be determined by the nonretarded surface-plasmon condition of the metal-dielectric interface [38].

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)

s_{k, \pm} = [1 \pm e^{- | k | (d_{2} - d_{1})}] / 2,

where k is a nonzero real valued parameter, and the corresponding eigenmodes α_kφ_k,_±e⁻^iky are given by

{\begin{array}{l} [e^{- 2 | k | d_{1}} \mp e^{- | k | (d_{1} + d_{2})}] e^{| k | x}, & x \leq d_{1} \\ [e^{- | k | x} \mp e^{- | k | (d_{1} + d_{2})}] e^{| k | x}, & d_{1} < x < d_{2} \\ (1 \mp e^{| k | (d_{2} - d_{1})}) e^{- | k | x}, & x \geq d_{2} \end{array}

where the normalization coefficient

α_{k}^{2} = 4 π | k | (e^{- 2 | k | d_{1}} - e^{- 2 | k | d_{2}})

. Evidently, in terms of the symmetry of the electric field, these eigenmodes can be cataloged into two groups. The subscript +(−) corresponds to an even (odd) mode respectively, while their associated eigenvalues are larger (smaller) than 0.5. Since the electric fields of the odd modes penetrate the metal weakly, they can propagate longer along the metallic surface than the even modes [38]. Furthermore, the resonant condition, approximately given by Re[s(ω_n)] = s_n 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 [27].

Fig. 1 A one-dimensional finite-thickness slab in xy coordinates (a) is transformed to an annulus (b) with a conformal transformation of w = e^z, a crescent (c) and two kissing cylinders (d) with a conformal transformation of w = 1/z, in the new uv coordinates.

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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) = p_uu + p_vv, which corresponds to a uniform electric field −p_ue_u – p_ve_v. The expansion coefficients in this case can be written as

(φ_{k, \pm} | φ_{0}) = \frac{2 k}{α_{k}} [ρ_{2} p F_{k} - ρ_{1} p * F_{- k}^{*}],

where

F_{k} = \int_{d_{1}}^{d_{2}} d x \int_{- \infty}^{\infty} d y g e^{k z^{*}}, p \equiv \frac{1}{2} (p_{u} - i p_{v}), g \equiv \frac{d w}{d z} .

In addition, ρ₁ = 1 and ρ₂ = ∓e^{–|k|(d₁+d₂)} for positive k, and they should be interchanged when k < 0. Here we use the Cauchy-Riemann equations of conformal mappings

\begin{array}{l} \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} = \frac{g + g *}{2}, \\ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} = \frac{g * - g}{2 i}, \end{array}

as well as

\frac{\partial φ_{0}}{\partial x} = g p + g * p *, \frac{\partial φ_{0}}{\partial y} = i (g p - g * p *) .

We first transform the slab to two coaxial cylinders (Figure 1(b)) with w = e^z, where the radii of the two cylinders are r₁ = e^d₁ and r₂ = e^d₂ respectively [39]. The expansion coefficients are found to be (See Appendix D)

(φ_{k, \pm} | φ_{0}) = \mp α_{k} p_{u} e^{d_{1} + d_{2}} δ (| k | - 1) / 4 .

Here p_v is assumed to be zero because of the structural symmetry. Evidently, only eigenmodes with k = ±1 are excited. We further use these coefficients to calculate the induced potential φ_i. It is found that σφ_i/[φ₀(1 – ɛ₁)], with

σ = r_{2}^{2} {(1 + ɛ_{1})}^{2} - r_{1}^{2} {(1 - ɛ_{1})}^{2}

, equals

{\begin{array}{l} (r_{1}^{2} - r_{2}^{2}) (1 - ɛ_{1}), & r \leq r_{1} \\ r_{1}^{2} (1 - ɛ_{1}) + r_{2}^{2} (1 + ɛ_{1}) - 2 r_{1}^{2} r_{2}^{2} / r^{2}, & r_{1} < r < r_{2} \\ r_{2}^{2} (r_{2}^{2} - r_{1}^{2}) (1 + ɛ_{1}) / r^{2}, & r \geq r_{2} \end{array}

Note that our results are identical to those obtained by expanding the potential in terms of Bessel functions [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 [22], 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)

(φ_{k, \pm} | φ_{0}) = α_{k} [p_{u} + i sgn (k) p_{v}] / 4,

which can be further used to achieve identical electric fields as those presented in Ref. [22] (See Appendix E). Evidently, each eigenmode is excited by the external potential. Furthermore, |(φ_k,_±|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 [22], a property which is not so obvious. Figure 2 shows the dependence of |(φ_k,_±|φ₀)|² on k achieved by setting p_u = p_v = 1, d₁ = d and d₁ = 2d with d being an arbitral real quantity. As mentioned, each eigenmode of the crescent is excited, consequently |(φ_k,_±|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.

Fig. 2 The dependence of the expansion coefficient on the mode index k, when it is assumed that p_u = p_v = 1. Notice that two eigenmodes with identical k while different symmetry have identical amplitudes of the expansion coefficients, mode symmetry therefore is not specified. Here d is an arbitrary real quantity.

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Another example consists of the kissing cylinders suggested in [27], 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)

(φ_{k, \pm} | φ_{0}) = \frac{α_{k}}{4} [p_{u} \frac{e^{| k | (d_{2} + d_{1})} \pm 1}{e^{| k | (d_{2} - d_{1})} \pm 1} + i sgn (k) p_{v} \frac{e^{| k | (d_{2} + d_{1})} \mp 1}{e^{| k | (d_{2} - d_{1})} \mp 1}] .

Again all the eigenmodes are excited by the external source. Note that when d₁ = −d₂ the cylinders have the same radius. The kissing cylinders then possess both u and v mirror symmetry. Hence, the even mode φ_k,₊ or the odd mode φ_k,₋ can be excited by a u- or v- polarized electric field respectively. An example dependence of |(φ_k,_±|φ₀)|² on k is plotted in Figure 2, with p_u = p_v = 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 ω_n, the resonant frequency of the n-th surface mode. The total potential of Eq. (5) can therefore be approximated as β_n(φ_n|φ₀)φ_n. Since only φ_n is position dependent, the amplitude of the corresponding electric field is proportional to |∇_wφ_n| = |dz/dw||∇_zφ_n|. 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/e^x 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, I_e, 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)

It is useful to reformulate the generalized eigenproblem, Eq. (1), as an integral equation [33]

s_{n} φ_{n} (r) = \hat{G} φ_{n} (r) \equiv \int d v^{'} θ (r^{'}) \nabla^{'} G (r, r^{'}) \cdot \nabla^{'} φ_{n} (r^{'}),

with the help of the Green’s function G(r,r′) which satisfies the following Laplace’s equation

\nabla^{2} G (r, r^{'}) = - δ (r - r^{'}) .

Notice that both φ_n and G should satisfy the homogeneous Dirichlet-Neumann boundary conditions. If a scalar product of two functions further is defined as

(ϕ | ψ) = \int θ (r) \nabla ϕ * \cdot \nabla ψ d v,

the operator Ĝ is then Hermitian. To prove this fact, we can show that

\begin{array}{l} (ϕ | \hat{G} ψ) & = & \int θ (r) \nabla ϕ * \cdot \nabla (\hat{G} ψ) dv \\ = & \int dv θ (r) \nabla ϕ * \cdot \nabla [\int d v^{'} θ (r^{'}) \nabla^{'} G (r, r^{'}) \cdot \nabla^{'} ψ] \\ = & \int dv \int d v^{'} θ (r) θ (r^{'}) \nabla ϕ * \cdot \nabla [\nabla^{'} G (r, r^{'}) \cdot \nabla^{'} ψ], \end{array}

as well as

\begin{array}{l} (\hat{G} ϕ | ψ) & = & \int θ (r) \nabla (\hat{G} ϕ *) \cdot \nabla ψ dv \\ = & \int dv θ (r) \nabla ψ \cdot \nabla [\int d v^{'} θ (r^{'}) \nabla^{'} G (r, r^{'}) \cdot \nabla^{'} ϕ *] \\ = & \int d v \int d v^{'} θ (r) θ (r^{'}) \nabla ψ \cdot \nabla [\nabla^{'} G (r, r^{'}) \cdot \nabla^{'} ϕ *] \\ = & \int dv \int d v^{'} θ (r) θ (r^{'}) \nabla^{'} ψ \cdot \nabla^{'} [\nabla G (r^{'}, r) \cdot \nabla ϕ *] \\ = & \int dv \int d v^{'} θ (r) θ (r^{'}) \nabla^{'} ψ \cdot \nabla^{'} [\nabla G (r, r^{'}) \cdot \nabla ϕ *], \end{array}

where the symmetry of the Green’s function has been employed. Further notice that

\nabla ϕ * \cdot \nabla [\nabla^{'} G (r, r^{'}) \cdot \nabla^{'} ψ] = \nabla^{'} ψ \cdot \nabla^{'} [\nabla G (r, r^{'}) \cdot \nabla ϕ *],

and hence the operator Ĝ is Hermitian, such that its eigenvalues s_n are all real. Moreover,

\int {| \nabla φ_{m} |}^{2} dv = \frac{1}{s_{m}} \int θ (r) {| \nabla φ_{m} |}^{2} dv,

which suggests that the eigenvalues lie between 0 and 1. As pointed out in Reference [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

(φ_{m} | φ_{n}) = δ_{mn} .

B. Absorption and extinction

The normalized non-trivial eigenmodes φ_n are a complete orthogonal set [33], and the external potential, inside the ɛ₁ material, can be expanded in terms of these modes

φ_{0} = \sum_{n} (φ_{n} | φ_{0}) φ_{n} .

Using the fact that

\nabla \cdot (θ \nabla φ_{0}) = s (ω) \nabla^{2} φ_{i} - \nabla \cdot (θ \nabla φ_{i}),

where φ_i being the induced potential, we achieve the following

φ_{i} = \sum_{n} \frac{s_{n} φ_{n}}{s (ω) - s_{n}} (φ_{n} | φ_{0}) .

The total potential hence can be given as

φ_{t} = φ_{i} + φ_{0} = \sum_{n} \frac{s (ω) φ_{n}}{s (ω) - s_{n}} (φ_{n} | φ_{0}) \equiv \sum_{n} β_{n} (φ_{n} | φ_{0}) φ_{n} .

Moreover, the corresponding electric field is given by E = −∇φ. Here the radiation loss is not included.

In electromagnetics, the extinction of a particle illuminated by an external field is given by [16]

P_{ex} = \frac{ω ɛ_{0}}{2} Im \int (ɛ_{1} - 1) E_{0}^{*} \cdot E_{t} dv .

Here the integration is performed over the particle, and the surrounding material is assumed to be air. Substituting Eq. (B.1) and (B.4) into the above equation yields

P_{ex} = \frac{ω ɛ_{0}}{2} \sum_{n} {| (φ_{n} | φ_{0}) |}^{2} Im [β_{n} (ɛ_{1} - 1)] .

Furthermore, the electrostatic energy inside the metallic region is expressed as

I_{e} = \int_{v} θ E_{t}^{*} \cdot E_{t} dv = \sum_{n} {| β_{n} (φ_{n} | φ_{0}) |}^{2} .

In a similar way, the absorption may be represented as

p_{a} = \frac{ω ɛ_{0}}{2} Im (ɛ_{1}) \int θ E_{t}^{*} \cdot E_{t} dv = \frac{ω ɛ_{0}}{2} Im (ɛ_{1}) I_{e} .

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 e^iky for x < d₁, ce^–|k|x e^iky for x > d₂, and b₁ e^|k|x e^iky + b₂e^−|k|x e^iky in the middle. The boundary conditions at x = d₁ and x = d₂ yield

\begin{array}{l} (a - b_{1}) m^{2} = b_{2}, a m^{2} = \frac{(s - 1)}{s} (b_{1} m^{2} - b_{2}) \\ c - b_{2} = b_{1} n^{2}, c = \frac{(1 - s)}{s} (b_{1} n^{2} - b_{2}), \end{array}

where m = e^|k|d₁ and n = e^|k|d₂. Two sets of eigenmodes can be obtained. The eigenvalues of the odd modes are given by

s_{k, +} = \frac{1 + e^{- | k | (d_{2} - d_{1})}}{2},

and the associated eigenmodes are

α_{k} φ_{k, +} e^{- iky} = {\begin{array}{l} [e^{- 2 | k | d_{1}} - e^{- | k | (d_{1} + d_{2})}] e^{| k | x}, & x \leq d_{1} \\ - e^{- | k | (d_{1} + d_{2})} e^{| k | x} + e^{- | k | x}, & d_{1} < x < d_{2} \\ (1 - e^{| k | (d_{2} - d_{1})}) e^{- | k | x}, & x \geq d_{2} . \end{array}

The eigenvalues of the even modes are

s_{k, -} = \frac{1 - e^{- | k | (d_{2} - d_{1})}}{2},

and the corresponding eigenmodes are

α_{k} φ_{k, -} e^{- iky} = {\begin{array}{l} [e^{- 2 | k | d_{1}} + e^{- | k | (d_{1} + d_{2})}] e^{| k | x}, & x \leq d_{1} \\ e^{- | k | (d_{1} + d_{2})} e^{| k | x} + e^{- | k | x}, & d_{1} < x < d_{2} \\ (1 + e^{| k | (d_{2} - d_{1})}) e^{- | k | x}, & x \geq d_{2} . \end{array}

The normalization coefficient

α_{k}^{2} = 4 π | k | (e^{- 2 | k | d_{1}} - e^{- 2 | k | d_{2}})

.

D. Expansion coefficients of different structures

Two coaxial cylinders: The conformal mapping used here is w = e^z. To obtain the expansion coefficients, we use Eqs. (10) and (11). It is found that

F_{k} = \int_{d_{1}}^{d_{2}} d x \int_{- \infty}^{\infty} d y e^{z} e^{k z^{*}} = π (e^{2 d_{2}} - e^{2 d_{1}}) δ (k - 1),

as well as

F_{- k}^{*} = \int_{d_{1}}^{d_{2}} d x \int_{- \infty}^{\infty} d y e^{z^{*}} e^{- k z} = π (e^{2 d_{2}} - e^{2 d_{1}}) δ (k + 1),

where use has been made of the fact that

\int_{- \infty}^{\infty} e^{- iky} dy = 2 π δ (k) .

Because of the structural symmetry we assume that p_v = 0, and the resultant expansion coefficients are

(φ_{k, \pm} | φ_{0}) = \mp \frac{α_{k}}{4} p_{u} e^{d_{1} + d_{2}} δ (| k | - 1) .

Evidently, only eigenmodes with k = ±1 are excited. Making use of these coefficients, the induced potential φ_i = Σ_n(β_n – 1)(φ_n|φ₀)φ_n can be expressed as

φ_{i} (u, v) = \frac{p_{u} r_{1} r_{2}}{4} [(β_{1, -} - 1) (α_{1} φ_{1, -} + α_{- 1} φ_{- 1, -}) - (β_{1, +} - 1) (α_{1} φ_{1, +} + α_{- 1} φ_{- 1, +}],

with r₁ = e^d₁, r₂ = e^d₂, and

β_{1, \pm} - 1 = \frac{(1 - ɛ_{1}) (r_{2} \pm r_{1})}{(1 + ɛ_{1}) r_{2} \mp (1 - ɛ_{1}) r_{1}} .

Finally we achieve, by using the fact that φ₀ = p_uu = p_ue^x cos y = p_ur cosθ, the following result:

\frac{σ φ_{i}}{φ_{0}} = {\begin{array}{l} (r_{1}^{2} - r_{2}^{2}) {(1 - ɛ_{1})}^{2}, & r \leq r_{1} \\ r_{1}^{2} {(1 - ɛ_{1})}^{2} + r_{2}^{2} (1 - ɛ_{1}^{2}) - 2 (1 - ɛ_{1}) r_{1}^{2} r_{2}^{2} / r^{2}, & r_{1} < r < r_{2} \\ r_{2}^{2} (r_{2}^{2} - r_{1}^{2}) (1 - ɛ_{1}^{2}) / r^{2}, & r \geq r_{2} \end{array}

with

σ = r_{2}^{2} {(1 + ɛ_{1})}^{2} - r_{1}^{2} {(1 - ɛ_{1})}^{2}

.

Crescent: It is known that a slab, with d₁ > 0, can be transformed to a crescent by a conformal mapping of w = 1/z. Consequently g = dw/dz = −1/z². By using Eqs. (10) and (11), it is found that

F_{k} = \int_{d_{1}}^{d_{2}} d x \int_{- \infty}^{\infty} dy \frac{- 1}{{(x + i y)}^{2}} e^{k (x - iy)} = \int_{d_{1}}^{d_{2}} e^{kx} dx \int_{- \infty}^{\infty} dy \frac{- 1}{{(x + i y)}^{2}} e^{- iky} .

Since x ≥ d₁ > 0, the second integration has a pole at y = ix in the upper half complex plane. Consequently for negative k, we have

F_{k < 0} = 2 π k \int_{d_{1}}^{d_{2}} e^{2 kx} dx = π (e^{2 k d_{2}} - e^{2 k d_{1}}) .

Similar procedures can be applied to calculate

F_{- k}^{*}

for positive k, which yield

F_{- k}^{*} = \int_{d_{1}}^{d_{2}} dx \int_{- \infty}^{\infty} dy \frac{- 1}{{(x - iy)}^{2}} e^{- k (x + iy)} = π (e^{- 2 k d_{2}} - e^{- 2 k d_{1}}),

because of the fact that

\int_{- \infty}^{\infty} dy \frac{- 1}{{(x - iy)}^{2}} e^{- iky} = - 2 π k e^{- k x} .

Here the integration has a pole at y = −ix in the lower half complex plane. Hence the resultant expansion coefficients are given by

(φ_{k, \pm} | φ_{0}) = \frac{α_{k}}{4} [p_{u} + i sgn (k) p_{v}] .

Notice that for same k, the even mode and the odd mode result in identical expansion coefficients.

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 F_k,

F_{k} = \int_{d_{1}}^{d_{2}} dx \int_{- \infty}^{\infty} dy \frac{- 1}{{(x + i y)}^{2}} e^{k (x - iy)} = \int_{d_{1}}^{d_{2}} e^{kx} dx \int_{- \infty}^{\infty} dy \frac{- 1}{{(x + i y)}^{2}} e^{- iky} .

To evaluate the second integration, we notice that its integrand has a pole at y = ix, which is located in the upper half plane for positive x while in the lower half plane for negative x. By carefully choosing the integral contour, it is found that

F_{k} = π (e^{2 k d_{2}} - 1), F_{- k}^{*} = π (e^{- 2 k d_{1}} - 1),

for negative k, and

F_{k} = π (e^{2 k d_{1}} - 1), F_{- k}^{*} = π (e^{- 2 k d_{2}} - 1),

for positive k. The corresponding expansion coefficients are then given by

(φ_{k, \pm} | φ_{0}) = \frac{α_{k}}{4} [p_{u} \frac{e^{| k | (d_{2} + d_{1})} \pm 1}{e^{| k | (d_{2} - d_{1})} \pm 1} + i sgn (k) p_{v} \frac{e^{| k | (d_{2} + d_{1})} \mp 1}{e^{| k | (d_{2} - d_{1})} \mp 1}] .

E. Electric field of the crescent

The total electric field, including both incident and scattered fields, can be expressed as

E = - \nabla_{w} φ_{t} = - \sum_{n} β_{n} (φ_{n} | φ_{0}) \nabla_{w} φ_{n},

which can be further manipulated into the form

\begin{array}{l} E_{u} = - \sum_{n} β_{n} (φ_{n} | φ_{0}) [\frac{\partial φ_{n}}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial φ_{n}}{\partial y} \frac{\partial y}{\partial u}], \\ E_{v} = - \sum_{n} β_{n} (φ_{n} | φ_{0}) [\frac{\partial φ_{n}}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial φ_{n}}{\partial y} \frac{\partial y}{\partial v}], \end{array}

with

\frac{\partial x}{\partial u} = \frac{\partial y}{\partial v} = \frac{z_{w} + z_{w}^{*}}{2}, \frac{\partial x}{\partial v} = - \frac{\partial y}{\partial u} = i \frac{z_{w} - z_{w}^{*}}{2},

as well as z_w = dz/dw. By representing the geometrical mode φ_k,_± as

φ_{k, \pm} (x, y) = \frac{1}{α_{k}} e^{iky} (a_{k, \pm} e^{| k | x} + b_{k, \pm} e^{- | k | x}),

we finally arrive at

\begin{array}{l} E_{u} & = & - \sum_{k, \pm} \frac{| k | e^{iky}}{α_{k}} β_{k, \pm} (φ_{k, \pm} | φ_{0}) [(a_{k, \pm} e^{| k | x} z_{w} - b_{k, \pm} e^{- | k | x} z_{w}^{*}) ϑ (k) + (a_{k, \pm} e^{| k | x} z_{w}^{*} - b_{k, \pm} e^{- | k | x} z_{w}) ϑ (- k)], \\ E_{v} & = & - i \sum_{k, \pm} \frac{k e^{iky}}{α_{k}} β_{k, \pm} (φ_{k, \pm} | φ_{0}) [(a_{k, \pm} e^{| k | x} z_{w} + b_{k, \pm} e^{- | k | x} z_{w}^{*}) ϑ (k) + (a_{k, \pm} e^{| k | x} z_{w}^{*} + b_{k, \pm} e^{- | k | x} z_{w}) ϑ (- k)], \end{array}

where ϑ(x) = 1 for positive x while ϑ(x) = 0 when x < 0. Furthermore, using Eq. (8), β_k,_± can be reformulated as

β_{k, \pm} = \frac{s (ω)}{s (ω) - s_{k, \pm}} = \frac{2}{ɛ_{1} + 1} \frac{1}{1 \pm e^{γ - | k | t}},

where t = d₂ – d₁ being the slab thickness, and eγ = (ɛ₁ – 1)/(ɛ₁ + 1). Moreover, since k is continuous, the electric field, taking the u component as an example, may be expressed as

\begin{array}{l} - E_{u} & = & \frac{p_{u} + i p_{v}}{2 (ɛ_{1} + 1)} \int_{0}^{\infty} dkk e^{iky} (\frac{a_{k, +} e^{| k | x} z_{w} - b_{k, +} e^{- | k | x} z_{w}^{*}}{1 + e^{γ - kt}} + \frac{a_{k, -} e^{| k | x} z_{w} - b_{k, -} e^{- | k | x} z_{w}^{*}}{1 - e^{γ - k t}}) \\ - \frac{p_{u} - i p_{v}}{2 (ɛ_{1} + 1)} \int_{- \infty}^{0} dkk e^{iky} (\frac{a_{k, +} e^{| k | x} z_{w}^{*} - b_{k, +} e^{- | k | x} z_{w}}{1 + e^{γ + kt}} + \frac{a_{k, -} e^{| k | x} z_{w}^{*} - b_{k, -} e^{- | k | x} z_{w}}{1 - e^{γ + kt}}) \end{array} .

We now assume that, in a certain frequency region, Re(ɛ₁) < −1 as well as the Im(ɛ₁) is positive and close to zero. Consequently e^γ is located in the first quadrant of the complex plane. Further assuming that only the poles of the integrand, given by k = ±γ/t, contribute to the integration, we finally obtain

- E_{u} = i \frac{π γ (p_{u} + i p_{v})}{(ɛ_{1} + 1) t^{2}} (a_{-} e^{γ x / t} e^{i γ y / t} z_{w} - b_{-} e^{- γ x / t} e^{i γ y / t} z_{w}^{*})

for positive y, and

- E_{u} = i \frac{π γ (p_{u} - i p_{v})}{(ɛ_{1} + 1) t^{2}} (a_{-} e^{γ x / t} e^{- i γ y / t} z_{w}^{*} - b_{-} e^{- γ x / t} e^{- i γ y / t} z_{w})

for negative y. The electric field in the x < d₁ region, where a₋ = (1+e^−γ)e^−2γd₁/t and b₋ = 0, can then be expressed as

E_{u} = \frac{π γ (i p_{u} - sgn (y) p_{v})}{(ɛ_{1} + 1) t^{2}} (1 + e^{- γ}) {(x + i | y |)}^{2} e^{γ (x + i | y | - 2 d_{1}) / t} .

By using the same notation as Ref. [22] as well as setting g = 1, the above equation is found to be identical to Eq. (28) of Ref. [22].

Acknowledgments

This work was supported in part by the Penn State MRSEC under NSF grant no. DMR 0213623.

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General properties of two-dimensional conformal transformations in electrostatics

Abstract

A. Properties of Eq. (1)

B. Absorption and extinction

C. Geometry modes of one-dimensional finite slab

D. Expansion coefficients of different structures

E. Electric field of the crescent

Acknowledgments

References and links

Cited By

Figures (2)

Equations (64)

Optics Express