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The gap plasmon mode of an eccentric coaxial waveguide is analyzed by the effective index method. The results agree-well with fully-vectorial numerical calculations. In the eccentric structure, there is extreme subwavelength field localization around the narrowest gap due to the gap plasmon. Furthermore, the effective index of the lowest-order waveguide mode increases considerably, for example, to 3.7 in the structure considered with a 2 nm minimum gap. The nanostructure waveguide geometry and wavelength (4 μm) are comparable with recent experiments on coaxial structures, except that that position of the center island is shifted for the eccentric coaxial structure; therefore, the proposed structure is a good candidate for future fabrication and experiments. In the visible regime, the effective index increases to over 10 for the same structure. The influence of symmetry-breaking in the eccentric coaxial structure is discussed as a way to enhance the local field and improve optical coupling.
©2009 Optical Society of America
1. Introduction
Hole-shape influences the waveguide mode propagation inside a hole in a metal. The influence of hole-shape on the propagation and transmission of light through a single hole and hole-arrays in metals has been studied for many different shapes including cylindrical [1], square and rectangular [1–11], cylindrical coaxial [12–19], rectangular coaxial [20], rectangle-in-cylinder coaxial [21], eccentric-coaxial [22–26], elliptical [27–32], cruciform [33–36], C-, H-, and E-shaped [37–39], double-hole (overlapping [40,41] and separated [29]), triangular [42,43] and star-shaped [44] holes. These hole-shapes have been investigated throughout the electromagnetic spectrum; perfect electric conductors (PECs) approximate-well the behavior at long wavelengths, whereas the plasmonic response of the metal plays an important role at shorter wavelengths.
Gap plasmons have a strong influence on the waveguide mode. For example, the lowest order mode of a rectangular hole has a cut-off wavelength increases dramatically as the hole is made smaller along its shortest axis [7,9,45], as has been observed experimentally [46]. This effect is the result of the effective index of the gap plasmon increasing as the metal edges are brought closer together [7]. The waveguide modes of a cylindrical coaxial structure are similarly-dependent upon the gap plasmon [15–17]. The eccentric coaxial structure has a varying gap-width with angle, so the properties of the gap plasmon are of particular interest.
In this work, we study the gap-plasmon of an eccentric coaxial structure. The eccentric coaxial structure has been studied before for PECs using conformal mapping [22–26]. In those works, the cut-off wavelength was shown to be offset-dependent. Here, we focus on a different effect: the influence of the gap plasmon. The effective index method is used, which is adapted from the study of rectangular holes [7] to the cylindrical geometry. The structure analyzed is comparable to recent structures produced by interference lithography [18,19], which has the great advantage of allowing for mass production. The analytical results of the effective index model agree-well with comprehensive finite-difference mode-solver (FDMS) numerical calculations for the geometry considered for a wavelength of 4 μm. In the visible –near-IR region of the optical spectrum, the FDMS gives spurious results for smaller gaps; however, the simple effective index theory agrees quantitatively with a fully-vectorial finite-element method (FEM) in that regime.
2. Effective Index Model for Eccentric Coaxial Gap Plasmon
Past works on eccentric coaxial structures used conformal mapping to study the geometric influence on mode propagation for PECs. Here, the geometric influence is accounted for by the effective index variation from the gap plasmon, which is a good approximation for plasmonic materials when the effective index is determined predominantly by the gap variation. This approximation neglects conformal mapping effects that become important if the tangent angles to the inner and outer cylinders vary significantly – the approximation is suitable for relatively large inner island sizes or small offsets.
The effective index method is an approximation that assumes that separation of variables is allowed. We consider first consider the radial dependence of the field to determine an effective index as a function of angle – each angle has a different gap, which contributes to the change in the effective index. These effective index calculations assume that there is no angular dependence, so that the equations of a concentric structure can be used. The next step is to determine the angular dependence by using the effective index as a function of angle to replace the radial dependence. Therefore, the overall propagation constant of the mode can be determined from the one dimensional angular calculation.
Figure 1 illustrates the effective index method as applied to the eccentric coaxial structure, where the radial and angular co-ordinates are approximated to be separable. Figure 1(a) shows the structure under consideration – the metal cylinder island with radius a is offset from the center by d and the outer cylinder has radius b. Figure 1(b) shows the two extreme example angles where the rotationally symmetric structure is analyzed as if the island cylinder was coaxial and had the same radial extent as the eccentric cylinder at that angle. Figure 1(c) shows the equivalent structure given the effective index calculated in Fig.1(b), where the metal is replaced with a perfect electric conductor (PEC) and the air region is replaced with a dielectric. Figure 1(d) shows the angular dependence calculated with the same PEC-dielectric structure; however, the dielectric is modified as a function of angle, using the values from the part (c).
The Helmholtz equation for the axial electric field, Ez (θ), is used since this field component is present in the gap-plasmon and is continuous at the boundaries:
where εz (r,θ) is the relative permittivity of dielectric material in the annular region, ω is the angular frequency and c is the speed of light in vacuum.
For the concentric cylindrical coaxial structure (i.e., no angular variation), the effective index of the mode is found by equating the tangential electric and magnetic fields at the boundary of island (radius a) and outer circle (radius b) to gives the solution [47]:
where
and Io, Ko ,I 1,K 1 are modified Bessel functions of order zero and one,
Eqs. (2–6) are solved for varying inner-diameter, a′, of concentric coaxial cylindrical structures to determine the angle-dependent effective index, n eff (θ). The angle-dependent effective index found in each case is used to solve the angular dependence of electric field in the eccentric structure, and the propagation constant along the axial direction, β, of the mode using:
where we set r = b to solve the field at the rotationally invariant outer radius. It should be noted that n eff (θ) is the effective index in the “effective index method”. It is only a function of θ and not the same as εz, which is a function of both r and θ. The parameters d,a,b are contained within the calculation for n eff (θ), so they do not appear explicitly in Eq. 7.
While Eq.7 is valid for concentric structures only, it is used here to approximate the behavior of the eccentric structure within the approximation of the effective index method. In Cartesian co-ordinates, the effective index method takes a similar approximation by replacing the real problem of non-matching boundaries by one with a uniform effective index. In cylindrical co-ordinates, we use a similar approach. For the radial electric field component, this can be pictured as a thin slice surrounded by a PEC boundary (as shown in Fig. 1).
Approximate approaches to solve for the propagation constant may be used, such as the Wentzel–Kramers–Brillouin method; however, due to the simplicity of this one-dimensional differential equation, we solve it numerically using discretization in 1 degree angular steps and the “eig” function in Matlab to find largest eigenvalue to give β and the corresponding eigenvector for Ez (θ) of the lowest order mode, where the radial dependence considered previously is assumed constant in the effective index method. The lowest order mode is the one that has phase change in the radial electric field in the gap region.
Fig. 1. (a) Schematic of eccentric cylindrical coaxial waveguide in gold with air gap. (b) Equivalent structures to calculate radial contribution to effective index assuming at each angle that the structure is rotationally symmetric. (c) Effective index of the rotationally symmetric structure is equivalent to a dielectric inside a coaxial perfect electric conductor (PEC). (d) Angular dependence, using effective index values calculated from the radial dependence at each angle.
3. Effective Index Increase and Field Localization for Eccentric Coaxial Structure
3.1 Infrared example comparable to recent experiments and FDMS calculations
We consider a coaxial structure with inner island radius of 224 nm, and outer radius of 286 nm, in gold for the free-space wavelength of 4 μm. These values were chosen to be similar with experiments on structures created by interference lithography [18]. At this wavelength the relative permittivity of gold is −600 + 130i. Fig. 2 shows the comparison between effective index (using the common definition βc/ω – not n eff (θ) as defined above) of the eccentric coaxial structure as calculated by the effective index method, as outlined in the previous section, and by a commercially available FDMS. Good agreement is seen between the two methods. The commercial FDMS incorporates the loss of the material from an imaginary part of the relative permittivity (not shown). In principle, this can be incorporated in the effective index method; however, since the loss is small for this example, and it complicates the analysis by adding complex roots, the imaginary part of the relative permittivity is ignored.
Fig. 2. Comparison of lowest order mode effective index (βc/ω) calculated by the effective index method (line) and calculated by a comprehensive vectorial FDMS (crosses). The structure chosen is gold, with an air gap, an outer cylinder radius of 286 nm, and an inner island radius of 224 nm. The inner island is offset to produce different narrowest gap values.
Figure 3 shows the square of the electric field amplitude of the lowest order mode calculated by the effective index method. As described above, Ez in Fig. 3 is determined from the eigenvector of Eq. (7).For a 60 nm shift in the center island, the field is strongly localized to a FWHM of 56° in a 2 nm gap. As shown in Fig. 4, a similar localization is seen for the comprehensive FDMS calculations (~60°), which also contain a radial dependence.
This example shows that the effective index method provides a reasonable approximation for the field localization and effective index increase of the lowest order waveguide mode in an eccentric-coaxial structure of a plasmonic metal. In the following section, we will discuss the specific benefits of the eccentric coaxial structure.
Fig. 3. Amplitude squared of electric field of lowest order mode calculated using calculated by the effective index method for eccentric coaxial structure, described in Fig. 2, with offset of 0 nm, 45 nm, and 60 nm (black, red, blue). The 60 nm offset has a 2 nm narrowest gap, which leads to strong field localization.
Fig. 4. Axial component of the electric field intensity for the same structures as in Fig. 3, with offsets of d= 0 nm, 45 nm, and 60 nm (left to right). Normalized color scale: red − 1, blue − 0.
3.2 Extension to visible – near-IR region and comparison with FEM calculations
The values presented in Sec. 3.1 were chosen to be comparable to recent experiments. We also investigated the extension of these results to the visible and near-IR regions of the optical spectrum and for another metal (silver). We use the same structure, but vary the wavelength and relative permittivity values appropriately [48]. In this region, there is significant penetration of the electric field into the metal, which is not well-captured by the FDMS for d > 40 nm; even for grid sizes of 0.2 nm, beyond which the calculation did not converge. (We attribute the inaccuracy of the FDMS method to the poor representation of the curved geometry using a Cartesian grid and linear interpolation).
We found better agreement for the effective index model calculations with a commercially available FEM solver. As shown in Fig. 5, we compare with the effective index method model for d = 55 nm and 60 nm, with good agreement between effective index method and the FEM solver. That figure also shows the calculated FDMS values. It is noteworthy that the FDMS calculated index actually reduces when the gap is made narrower from 7 nm to 2 nm, which contrary to the usual behavior of gap modes and is believed to be a spurious result.
Fig. 5. Effective index calculations for (a) gold and (b) silver in the visible – near-IR region. EIM: effective index method; FEM: finite element method; FDMS: finite difference mode solver.
4. Discussion
While FDMS and FEM methods can be used to study the eccentric structure, they are entirely numerical and general. As a result, they do not contain any special insight into the physics of strong field localization. Using the effective index method, it is clear that the localization is physically the result of the gap mode having an increase in the local index. Furthermore, since the behavior of the gap mode can be well-approximated by a simple parametric expression, the effective index method has the possibility of providing fully-analytical information about the behavior of the eccentric structure (and other cylindrical geometry structures). This will allow for rapid design and optimization of such structures. For example, all of the effective index method calculations in this manuscript can be completed in less than a minute using Matlab, whereas the FDMS and FEM methods take several minutes for each case.
As could be seen from the last section, the eccentric coaxial structure allows for strong field localization in the region of the narrowest gap. This provides even greater field localization than the concentric coaxial waveguide, which has already attracted great interest from the plasmonics community.
An additional benefit from the eccentric coaxial structure comes from symmetry-breaking, which introduces linear polarization to the lowest order mode. The lowest order mode of the corresponding concentric structure is radially polarized. As a result, linearly polarized light can be used to excite the lowest order mode of the eccentric structure, but not the concentric structure. This is important to practical application, when it comes to actually exciting these modes in real structures.
Past works have used focused-ion beam milling (e.g. Ref. 21) and interference lithography (e.g., Ref. 18) to produce coaxial structures. Those approaches can be readily extended to the eccentric structure described here; however, to obtain narrow gaps of only a few nanometers, the interference lithography approach seems to be especially interesting. In particular, it is possible to use adapt that method by using angled-evaporation (in step 6a of Fig. 1, Ref. 18) to produce the central island with an offset, and at least in principle, the center island can be made arbitrarily close to the outer cylinder by this method.
5. Conclusion
We have demonstrated that the gap plasmon mode of an eccentric coaxial structure allows for an increased effective index, strong field localization and linear polarized field to benefit coupling to linear polarized light. Each of these properties becomes more prominent as the narrowest gap is reduced by shifting the center island, to make the structure more eccentric. It is expected that proposed structures may be fabricated by focused-ion beam milling or interference lithography methods. Several potential applications arise from the strong field localization of this structure including: nonlinear optics [42,44,49–51], surface-enhanced Raman scattering [52–56], and optical trapping [57].
This paper also showed an example of using the effective index method for a cylindrical geometry to rapidly produce results that are in good agreement with more-cumbersome comprehensive numerical calculations. This method may also be applied to other concentric structures.
Acknowledgment
The authors acknowledge financial support from the NSERC (Canada) Discovery Grant.
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