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Localized surface plasmon modes in a system of two interacting metallic cylinders

We study an optical response of a system of two parallel close metallic cylinders having nanoscale dimensions. Surface plasmon excitation in the gap between the cylinders are specifically analyzed. In particular, resonance frequencies and field enhancement were investigated as functions of geometrical characteristics of the system and Ohmic losses in the metal. The results of numerical simulations were systematically compared with the analytical theory, obtained in the quasi-static limit. The analytical method was generalized in order to take into account the retardation effects. We also present the physical qualitative picture of the plasmon modes, which is validated by numerical simulations and analytical theory.


Introduction
Localized surface plasmon excitations in metal-dielectric systems of subwavelength size is a topic under intensive study during the last decade. Fabrication techniques of wire-grid polarizers [1,2], nano-antennas [3][4][5][6][7], and arrays of metallic particles [8][9][10] are rapidly improving. Optical properties of aggregates of metallic grains are very different from those of the separate grains. Field refraction by such systems possesses typical features, in particular a strong field enhancement in the gaps between closely located metallic particles leading to an increase of scattering and absorption in comparison with those for single grains. The resonance frequency of surface plasmons in the systems depends both on particle sizes and inter-particle distances. It is red-shifted for modes with electric field polarization directed across the gaps between the grains and blue-shifted in the opposite case, see e.g. experimental works [5,6] and numerical investigations [11][12][13]. The electric field enhancement inside the gaps under the resonance conditions can reach sufficient values and one can use the effect to achieve Raman detection of single molecules placed into the gap [14,15].
One encounters difficulties trying to describe plasmon modes analytically since an exact solution is only possible for systems with very simple geometry. The geometry should allow to use an appropriate coordinate system for which separation of variables in the Helmholtz equation is possible [16]. Among the systems are spherical metal particles [17], surface plasmon propagation in plane metal films [18], dielectric gaps in metallic cladding [19] and along nanowires [20].
Although the solutions for these systems are quite simple, they yield basic understanding of the fundamentals of the surface plasmon physics at nanoscales.
The problem of two particles (in particular cylinders) in the external field can be approximately solved using dipole-dipole approximation if the inter-particle distance is much larger than their sizes. When the inter-particle distance gets smaller this approximation becomes inappropriate even for qualitative description of the system, and one should use multipole-multipole expansion technique [21]. There are some works that try to employ the multipole-multipole expansion technique in order to approach a system of two close metallic particles of more complex form [22], system of several particles [23] or metamaterials [24,25]. which is associated with the calculation of formally infinite series. The method has advantages for numerical simulation, whereas it does not allow to establish qualitative properties of the plasmon modes in the systems, in particular their scaling behavior on the geometry of the system. The problem can be simplified at scales less than the wavelength. In this case the Helmholtz equation is reduced to Laplace equation since retardation effects are negligible. This fact allows one to analytically solve the problem about surface plasmon mode structure for more complex systems such as two close spherical grains [26,27] and two close cylinders with circular cross section [28]. To analyze solutions it is reasonable first to consider a qualitative picture [26,28] which describes the plasmon modes in the systems. The qualitative picture can be constructed applying the solution for surface plasmon propagation in a thin dielectric gap [19] to the gap between two granules.
In the present work, we investigate scattering of light by a system of two close parallel metallic cylinders. We sequentially compare the results of numerical simulations of the electromagnetic near field distribution with predictions of the theory [28] developed for the quasi-static (long wavelength) limit. Following [28] we present the qualitative picture of the surface plasmon resonance in the system in more details, and show its agreement with both analytical (in quasi-static limit) and numerical solution of Maxwell's equations.
One of the main flaws of the method employed in [28] is that it does not account for retardation effects, which means that its applicability diminishes with the increase of the system size.
Moreover, it does not account for the radiation losses which could be significant even for the small system provided that the Ohmic losses are small enough. In the present paper, and it is its main point, we present the numerical results which on the one hand account for the retardation effects, thus are applicable for the systems of any size and material constants, and on the other hand extend the analytical methods of [28] to account for the radiation losses explicitly. We verify the accuracy of our numerical calculations by systematically comparing their results to analytical ones in the appropriate limit of small system size. We also present the results of numerical simulations for the silver cylinders in the experimentally interesting region of frequencies and for realistic sizes (up to 100 nm) which is slightly beyond the scope of the analytical quasi-stationary method which describes the picture only qualitatively. Silver is chosen since it is widely used in in experimental works in nanooptics (see e.g. [9,29]) due to its low Ohmic losses.

Problem formulation
We examine scattering of an electromagnetic plane wave by two close metallic cylinders under the conditions of surface plasmonic resonance. The size of the cylinders in cross-plane is assumed to be of the order or less than the wavelength. of both cylinders are equal to a and the width of the gap between the cylinders is δ. We consider a particular case of TM-wave when the electric field of the incident wave is polarized along the line connecting the cylinders axes. Such choice is made because surface plasmon modes with redshifted resonance frequencies and high local enhancement of the electric field in the gap between the cylinders [28] is achieved for this polarization only. The red shift corresponds to high negative value of the dielectric constants contrast (permittivity ratio) ε = ε m /ε d , where ε m and ε d are the permittivities of the metal and the surrounding dielectric respectively.
The complete system of Maxwell's equations can be reduced to wave equation on the only nonzero magnetic field Z-component H, where ω is the frequency of the incident wave. We assume the magnetic permeability µ to be unity both in metal and in surrounded space. The electric field is orthogonal to OZ-axis, thus the problem becomes effectively two-dimensional. The electric field distribution can be restored as where η αβ is two-dimensional antisymmetric tensor, η xy = 1, and α, β runs values x, y.
The numerical simulations were performed with commercially available software JCMsuite [30]. The software solves Maxwell's equations based on a finite element method (FEM). It gives high advantages in simulations of structures with small curved elements. In particular, the package JCMsuite showed very good results in a benchmark simulation of plasmonic nano antennas [31], which are similar to our structure. The simulations are challenging because of the narrow gap between cylinders. The studied system is under resonance condition and exhibits large field  enhancement, so field distribution based adaptive meshing should be applied. An optimal number of refinement steps was found as well as a number of points on circuits in the vicinity of the gap for the manual mesh specification. A solution convergence based on posteriori error estimation was examined in the same way as in [30] and [32]. Setting permittivity of surrounding medium ε d = 1 in numerical simulations, we suppose that permittivity ratio is equal to complex metal permittivity and can be expressed as ε = ε + iε , i.e.
ε is the real part of the permittivity and ε is the imaginary part.
In the following we use the term 'model' metal. By this we mean that we can arbitrarily assign any values to its permittivity at any wavelength in order to illustrate the dependence of certain quantities (eg. resonance conditions, field enhancement factor) on the parameters of the system.
Having understood the general properties of plasmon modes in such systems we turn to investigation of the optical properties on silver cylinders. For silver, both the real ε and the imaginary ε parts of the dielectric permittivity are functions of the wavelength λ, which dependencies can be extracted from [35].

Analytic solution
If the cylinders radius a is much less than the wavelength in the outer space, √ ε d ka 1, and the skin layer in the metal, |ε m |ka 1, with k being the free space wavenumber, it is possible to develop an analytical solution. One can neglect retardation effects and assume ω = 0 in Eq. (1), thus describing the system in the quasi-static approximation as a pair of cylinders in a homogenous external electric field. The solution of such problem depends on frequency implicitly via metal permittivity ε(ω). This description is valid at distances close to the system, r λ/ √ ε d .  It is more convenient to use the electric field potential Φ instead of the magnetic field H in this approximation. The electric field is E = − grad Φ in the case. The reason is that the qualitative   a=10nm, δ=1nm a=15nm, δ=1.5nm a=30nm, δ=3nm a=100nm, δ=10nm a=200nm, δ=20nm a=15nm, δ=1nm In order to solve the above stated problem we use the so-called bipolar coordinates system: two dimensionless coordinates ξ and η are related to the Cartesian coordinates as follows: where h = a sinh ξ 0 /(cosh ξ − cos η) is the scaling function.
The partial solutions of Laplace equation can be written as e ±nξ cos nη or e ±nξ sin nη in separated variables. In order to solve the problem of two cylinders in the external field one has to expand the potential in terms of these partial solutions. Using the symmetry we can write: A n e −nξ cos nη; ξ > 0 in the first resonance. Wavelength is λ = 2 µm, radius of the cylinders is a = 50 nm, gap width is δ = 1 nm. Dotted line corresponds to law (15) at resonance in pure quasi-static limit, dashed line accounts for radiative losses which are given by (17).
where Φ in is the potential inside the cylinders and Φ out,ind is the induced part of the potential outside the cylinders. Coefficients A n and B n are to be found from the boundary conditions. Now let us expand the potential of the external field Φ 0 ≡ −E 0 y in terms of partial solutions of Laplace equation. Simple but cumbersome calculations give: 2e −n|ξ| cos nη + 1 .  The solution of the problem can be written explicitly (n ≥ 1) A n = − 2 sinh ξ 0 e nξ 0 sinh nξ 0 (ε + coth nξ 0 ) .
Expression (6) allows one to calculate the amplitude of the electric field. Explicit expression for electric field components reads see (16).
In what follows, we use these expressions to evaluate the electric field spatial distribution in the vicinity of the cylinders.

Results for the limit of close cylinders
In this section we analyze the limit of two close cylinders, when the width δ of the gap between the cylinders is small compared to the cylinder radius a, i.e. δ a. The condition means that the dimensionless parameter ξ 0 ≈ δ/a 1 in (3).
The resonance condition (9) becomes with the limit n a/δ. The result can be qualitatively explained as follows. The resonance condition enables the existence of standing surface plasmon waves in the flat region of the gap which has the approximate length ∼ √ aδ [19]. This is a general point, which is valid in the case of two close metal spheres as well [26,27]. Expression (12)  We performed numerical simulations for silver cylinders using experimental values of permittivity [35] for silver. We chose the size of the system to be close and slightly below experimentally achieved (see, e.g. [3]). Resonance permittivity (its real part) ratio as the function of geometrical c /E 2 0 for large values of the cylinders' radii due to radiation losses, the relative importance of which decreases with the wavelength λ (see (17) below). Now let us examine in details the enhancement of the electric field in the gap when the real part of permittivity contrast ε is close to the first resonance position ε 1 = − a/δ. One can see that in the vicinity of the resonance the contribution from the first harmonics in (6) is much larger than from all the rest. It follows from the symmetry of the problem that electric field on the OXZ-plane is directed normal to the plane and is equal to where we have taken into account the fact that δ a. This expression is valid provided kx 1.
Electric field changes its sign at x = l, where and this length should be interpreted as the mode size, i.e. wavelength of plasmon in flat dielectric gap with thickness δ between two bulk metal media [19].
Numerical simulation results for the mode size as a function of geometrical parameters of the system are shown in FIG. 6. The expression (14) is consistent with our qualitative explanation (12) and is valid under the same condition √ ε m k √ aδ 1.
It follows from (13) that the field at the center of the gap is The first representation in (15) is written for general case, whereas the second one is valid for the limit of closely located cylinders. At the resonance, i.e. real part ε = ε 1 , Eq. (15) is reduced to which means that the radiation losses are small compared to the Ohmic ones. For silver cylinders the deviation of the enhancement factor from law (15) is substantial. This is due to the fact that wavelength at the resonance frequency is not well above the size of the system.
At scales x √ aδ expression (13) gives quadratic decay law of the electric field as a function of coordinate The qualitative explanation of this law for scales Decaying law (16) for larger scales, x a, corresponds to field of dipole d = −2a 2 E 0 /(ε−ε 1 ), the asymptotics is valid for any value of the ratio δ/a. Note, that the dipole moment is formed at scales x ∼ a, and the minor part of the total surface charge is involved in the process. In fact, the dipole moment can be estimated as the integral over the region l x a, d ∼ dx∆Φ. The integral is formed at distances x ∼ a, where only the small part δ/a of surface charge is located.
Inner region does not give feasible contribution in d due to the small charge separation length.
One can improve Eq. (15) by taking into account the radiation losses. As a result, the resonance value of dielectric permittivity ε 1 (9) achieves imaginary negative contribution iε 1 , ε 1 < 0. The correction can be extracted from the condition, that dipole radiation is equal to power supplied to the system from external electric field provided Ohmic losses are zero, I = W . The intensity of the radiation is I = (πck 3 /4)|d 2 |, and the power supplied to the system is where E 0 is assumed to be real and Im[d] is the imaginary part of the dipole moment induced in the system. At resonance conditions, dipole moment is pure imaginary, thus we find d = (2i/π)E 0 /k 2 .
where ε 1 should be determined from (9). The correction is valid if (ka) 2 1 and is applicable for arbitrary ratio of cylinders' radii and the inter-cylinder distance, including the case when δ ∼ a. In   FIG. 7 and 9 theoretical curves which take into account the radiation correction (17) are presented.
The curves fit numerical data quite good, thus the deviation of the data curve from the theoretical prediction (15) with pure real ε 1 should be assigned just to unaccounted radiation losses.

Conclusion
We have numerically investigated the distribution of electro-magnetic field induced by the incident plane wave near a system of two closed parallel metallic cylinders of nanoscale dimensions modeled by Maxwell's equations. The comparison of the results of the numerical simulation with the analytical solution for the plasmon modes in the system governed by Maxwell's equations in the quasistatic limit showed agreement between these two approaches.
The position of the resonance is determined by the geometrical characteristics of the gap between the cylinders. Resonance corresponds to existence of standing wave inside the almost flat part of the gap, which can be thought of as a plane metal-dielectric-metal structure. Since the wavelength of the plasmon depends on its frequency (through permittivity of metal), there exist a set of resonance frequencies corresponding to the set of standing modes in the gap. Comparison with numerical simulations gives a good agreement, since the dimension of the gap is smaller than the size of the whole metallic granules, and the quasi-static approximation used in the analytical theory has high accuracy.
The analytically obtained [28] field enhancement factor in the center of the gap for the first resonance has also good agreement with that obtained numerically. It is determined both by Ohmic losses in the metal and radiative losses, the relative importance of the radiative losses rises with the dimension of the system. We assumed the radii of the cylinders to be of the order of 100 nm, which is implemented in experiments [3]. The skin layer depth in metal is of the same order thus the quasi-static approximation has some deviations from the exact solution of the Maxwell's equations.
Finally, we studied the dependence of the field enhancement for the case of silver cylinders with permittivity taken from experimental data. Our numerical simulations for such systems showed that only the Ohmic losses are reasonable, and only the first two resonances can be really observed.