1. Introduction

Optical waveguides consisting of dielectric and metallic parts may have very small geometry, a very high level of local field confinement, noticeable material dispersion, and substantial material losses [1–3]. If metals are used, the guiding capabilities are dominantly determined by the surface plasmon-polariton effect. Junctions of metallic parts with two different dielectrics, e.g., substrate and air, cause triple points which may lead to singularities in pure Maxwell models. Unlike wedge singularities, the triple-point singularities cannot be lifted by rounding the geometry. On one hand side, such singularities are attractive – for example in scanning nearfield optical microscopy (SNOM) – because they provide a strong field confinement and on the other hand side they cause heavy numerical problems and may lead to inaccurate simulations. In this paper, we consider a waveguide structure consisting of two metallic wires attached to a tiny dielectric fiber core which might be of interest at the very end of a SNOM tip. In practice, such a tip would be tapered. Full 3D modeling of a reasonably long section of a tapered SNOM tip is extremely demanding and it would not allow us to accurately calculate the triple point singularities with reasonable numerical costs. Therefore, we focus on the 2D analysis of waveguide modes on a cylindrical configuration. It should be mentioned that the same effects may also be observed for metallic waveguides mounted on a flat dielectric substrate, i.e., the findings in this paper are not only relevant in the context of SNOM.

Radiating modes with moderate radiation losses may become important when the material losses of strictly guided modes are too high. While the computation of guided modes on loss-free structures is well known and may be carried out by means of various commercial software packages, the analysis of lossy modes – especially when both material losses and radiation losses are present – is much more demanding and requires appropriate modifications of the numerical methods. When dispersive media, such as metals are present modes may “switch” from radiating to guided and vice versa. As a result, some modes may be strictly guiding only within a rather narrow frequency window. Beside the triple point problem, this is the second effect, which seems not to be well known. Since these effects need a very careful treatment of appropriate software, a big part of the paper focuses on numerical aspects.

The oldest and perhaps also the simplest method for the eigenvalue analysis of cylindrical waveguides solves the source-free Maxwell equations for a set of boundary conditions analytically, which yields a homogenous matrix equation M(e)X(e) = 0, where e denotes the chosen eigenvalue that may be either the frequency or propagation constant of a guided mode. To find the eigenvalues and eigenvectors of this system the transcendental equation $\det \left[M\left(e\right)\right]=0$ must be solved by using an iterative procedure such as the Bisection Method, Regula Falsi, etc [4]. In practice, this approach is limited to geometrically simple waveguides and the numerical solution of the transcendental equation is very difficult. Therefore, this analytic method is not widely used today.

For performing the eigenvalue analysis of optical waveguides with arbitrary complex geometries one of the available general methods for the numerical solution of Maxwell equations must be applied. The existing Maxwell solvers can be divided into the following two main classes: (a) domain methods which compute electromagnetic fields by approximating the unknown functions on a finite set of geometrically simple elements (sub-domains) obtained by meshing the entire computational domain, and (b) boundary methods that solve Maxwell equations by meshing the boundaries of the computational domain. Due to their relatively simple mathematical background, domain methods, such as the Finite Difference Method (FDM) [5] and the Finite Element Method (FEM) [6] in both time- and frequency-domain, are widely used in research and development. Boundary methods, such as the Boundary Element Method (BEM) [7] and the Multiple Multipole Program (MMP) [8–10] are almost exclusively used in frequency-domain. They have a higher level of accuracy compared to the domain methods but have also a more demanding mathematical background. Therefore they are mainly used in research and special development applications where a high level of accuracy is required.

For the eigenvalue analysis presented in this paper two essentially different Maxwell solvers were chosen, namely FEM and MMP. Using two fundamentally different methods and comparing the obtained results allows one to obtain high confidence in the results – provided that the observed differences are either small or may be well explained. This is of special importance if structures are analyzed, which require a high fabrication effort. It will be demonstrated that a good agreement of FEM and MMP results is obtained at least for non-radiating waveguide modes with moderate attenuation. In the case of radiating modes, FEM and MMP may provide differences caused by different modeling of the radiation terms.

2. Finite element method (FEM) and multiple multipole program (MMP)

As a general domain method for solving partial differential equations of second order, FEM is widely used in commercial simulation software for thermal, fluid flow, mechanical, and electromagnetic analysis. The efficiency and accuracy of FEM when applied to optical plasmonic nano-structures was reported in numerous publications (see, for example [11,12]).

Due to the fact that lossy dispersive optical waveguides support only hybrid propagation modes, i.e., modes with non-zero electric and magnetic field components in the propagation direction, the full-vector field formulation of the waveguide eigenvalue problem must be used even though the computational domain (waveguide cross-section) is 2D [6]. Consequently, the corresponding FEM discretization must be based on 2D vector elements [6,11].

The FEM discretization of the optical waveguide problem yields a large sparse linear system of equations with the matrix depending on the chosen eigenvalue. The role of eigenvalue is played by the propagation constant and the frequency is fixed prior to the eigenvalue extraction for the following two reasons: (a) material dispersion, i.e. frequency dependent material properties existing in the computational domain, and (b) frequency dependant absorbing boundary condition (ABC) terminating the computational domain.

The obtained FEM matrix is square and eigenvalue dependent. Thus, the required eigenvalues and eigenvectors can be extracted by solving the so-called generalized nonlinear eigenvalue problem [6,13].

The advantages of this FEM approach are its simplicity, the well understood mathematical foundation, the existing extensive algorithm bases, and the availability of several high quality commercial simulation tools. However, the drawbacks are also numerous: the accuracy is difficult to control, the accuracy is highly depending on the mesh quality, a rough guess for the eigenvalue range of interest is needed, which is a limiting constraint since the behavior of the eigenvalues in the corresponding complex search space is not known, etc.

MMP [8–10] is a boundary method based on expansions of analytical solutions of Maxwell equations for each sub-domain of the entire computational domain. The sub-domains are defined according to their distinctive material properties. The unknown coefficients of the field expansions are determined by imposing the well-known field continuity conditions and minimizing the weighted mismatching error along the boundaries, i.e., interfaces between the natural domains (air, fiber core, cladding in particular). This yields a system of linear equations that is usually characterized by a dense matrix that tends to become ill-conditioned. To avoid numerical problems caused by high condition numbers, our MMP approach employs the so-called generalized point matching technique which leads to an over-determined rectangular system of equations that is solved in the least square sense [13]. This drastically reduces condition problems but does not allow one to directly extract the eigenvalues [13]. Therefore, the latest MMP solver – contained in the OpenMaX [10] software package – includes several special techniques for solving eigenvalue problems by introducing an eigenvalue search function, for example [8–10]:

Where “Residual” may represent the norm of the weighted residual vector of the least square solution and “Amplitude” corresponds to a certain field quantity (usually the time-averaged Poynting vector through some cross section of the waveguide) computed at a given sensor point(s). Note that neither the definition of the residual and its weighting nor the definition of the amplitude is unique. An extensive efficiency analysis of various possible eigenvalue search functions was performed and reported [14]. As an outcome of this analysis, the latest MMP implementation includes nine different eigenvalue search functions with different tuning parameters and user-definable weighting and amplitude definition. Thus, applying the MMP eigenvalue solver is demanding for the user. Depending on the search function one may obtain “quick and dirty” results in relatively simple situations as well as more expensive but highly reliable results for complicated problems.

The advantages of MMP are its high accuracy, efficient accuracy control, robustness, and insensitivity to ill-conditioning. The drawbacks are its complex implementation, fully populated matrices, and experience required from the user for tricky cases.

3. Considered structures and numerical models

The examples considered in the following are inspired by Scanning Nearfield Optical Microscopy (SNOM) applications with an optical fiber feed. Along the circular fiber, a linearly polarized HE11 mode is propagating, whereas a radially polarized wire mode (TM01 type) is desirable at the tip. Standard, full metal coatings of the fiber do not allow linearly polarized modes to propagate without strong attenuation, especially when the cross section becomes small near the tip. By a partial cladding a two wire transmission line configuration, mounted on the fiber core, may be obtained. The analysis of such configurations shows that linearly polarized modes with rather low loss are present. These modes may couple well with HE11 fiber modes due to similar field configurations. Finally, symmetry breaking regions for converting the linearly polarized modes into radially polarized are required at the very end of the tip [15], [16]. For the analysis of the partially coated fiber, a 2D eigenvalue problem must be solved. In a second step, a full 3D analysis of the tip area is done and the most promising mode of the 2D analysis is used as excitation.

The 2D geometry of a dielectric optical waveguide with a partial metallic cladding is presented in Fig. 1a . A 50nm silica core is cladded with two symmetric silver strips. At the end of this waveguide it is possible to elongate one of the strips and to reduce its radius in order to achieve a metallic SNOM tip.

 

Fig. 1 Geometry of the considered optical fiber with metallic cladding (a) is presented. The core of the fiber is made of silica (ε_r = 2.25) and for the symmetric cladding two silver (ε_r = ε_r(λ)) strips are used. The measured permittivity data for silver modeling were used (b) [17].

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The corresponding 3D simulation model is presented in Fig. 2 . Note that this structure has not been optimized. In practice, the precise geometry would depend on the fabrication process, i.e., these simulations are only demonstrating that the concept of a partially cladded fiber with a metallic tip at the end would work as SNOM tip. From the computational point of view, the 2D eigenvalue is highly demanding for several reasons: 1) Both material losses (in the cladding) and radiation losses may be present, 2) sharp corners of the cladding lead to strong local field (which is indeed desirable at the end of the SNOM tip), and 3) triple points are created at the positions where air, metal, and core come together. Theoretically, field singularities are obtained in the triple points. These singularities cause severe numerical problems, require a very fine local discretization and may destroy fast convergence properties of the numerical method that is applied. Since the 2D eigenvalue analysis is numerically a very demanding and inappropriate modeling may lead to wrong results, two fundamentally different methods (FEM and MMP) were used for obtaining as much confidence in the results as possible. Furthermore, the MMP eigenvalue search is helpful for selecting the FEM eigenvalue areas.

 

Fig. 2 3D geometry of the considered SNOM tip based on the cladded optical waveguide presented in Fig. 1 is depicted. At the end of the fiber one silver strip is 40nm and the other one 90nm elongated beyond the core end in order to form a narrow field enhancement region.

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The MMP and FEM 2D modeling of the waveguide defined in Fig. 1 is illustrated in Fig. 3 . Only the natural boundaries, i.e., the interfaces between air and core (black line in Fig. 3a), cladding and core (red), and cladding and air (green) are discretized in the MMP model. The field in each region is then approximated by sets of multipole expansions distributed along the boundaries. The multipole locations are indicated by crosses in Fig. 3a. A multipole on one side of a boundary is assumed to illuminate the domain on the other side of the boundary as indicated by the arrows in Fig. 3a. Thus, all multipoles that expand the field in a certain domain are located outside that domain. As a consequence, the field in any domain has no singularity, which is reasonable in most cases and always in practice. However, the triple point singularities cause an exception here. In principle, it would be possible to place multipoles with appropriate singularity in the triple points. This would require a sophisticated analysis of the triple point singularity beforehand. Instead, a very fine discretization with multipoles located in the close vicinities of the triple points is used, which allows one to obtain sufficiently accurate results. Since the structure has two perpendicular symmetry axes, only the first quadrant needs to be discretized explicitly as shown in Fig. 3a. The resulting MMP matrix equation has 621 unknowns, i.e., degrees of freedom (DOFs). The MMP matrix is rather fully populated and has an overdetermination factor of approximately 2.

 

Fig. 3 MMP (a) and FEM (b) modeling of the waveguide shown in Fig. 1. The MMP field sources placed along different interfaces are marked by crosses. The FEM model also requires a termination of the surrounding air and the corresponding ABC. Both models use the same PEC and PMC boundary conditions along the symmetry planes.

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Since FEM is a domain method, it is necessary to truncate the surrounding air by using an absorbing boundary with imposed absorbing boundary conditions (ABC). MMP does not need such a boundary because the multipole expansions for the air domain expand the field anywhere.

These multipole expansions fulfill the Sommerfeld radiation conditions. It has already been mentioned that some modes may be radiating. For the analysis of radiating modes, the MMP model must include radiating multipoles, i.e., one must decide before starting the simulation whether radiating or strictly guided modes shall be analyzed. In the following MMP simulations, we assume that the Sommerfeld radiation condition is valid. This assumption is not made in the FEM model. The radiation of possible radiating modes would be absorbed by the ABC. The distance between the waveguide and truncating (ABC) boundary is chosen to be 500nm. A series of numerical tests were performed to verify that this distance is large enough not to influence the results. This resulted in the triangular FEM mesh presented in Fig. 3b. The mesh consists of 12’550 second order elements resulting in a FEM matrix with 88’215 DOFs.

The eigenfields obtained from the 2D model are used to feed the 3D model presented in Fig. 4 . In order to reduce the number of elements of the 3D model, a special meshing technique, the so-called sweep meshing method was used. The surfaces of the input port of the straight 500nm long waveguide section were first meshed by using triangular planar elements. The 3D mesh of the straight waveguide section was then obtained by extruding the triangular elements in the perpendicular direction, simply producing prisms (Fig. 4, left). The number of prisms is much smaller than the number of tetrahedrons needed to cover the same volume by using a free volume mesh. For the tip region however, the sweep meshing method could not be applied as the geometry has no translational symmetry and a free tetrahedral volume meshing was applied (Fig. 4, right). The meshing process resulted in 710’863 first order vector edge elements which is translated into a FEM matrix with 950’482 DOFs.

 

Fig. 4 3D FEM modeling of the SNOM tip waveguide termination presented in Fig. 2. The straight section of the waveguide (500nm) was meshed by using a special meshing technique (sweep method) based on a triangular surface mesh and prisms generated in the perpendicular direction. By following the modeling from Fig. 3b the PMC boundary condition was used over the symmetry plane and the first order ABC was used over the cylindrical (the straight waveguide section) and spherical boundaries (the tips) air truncating boundaries.

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To mesh and solve the models presented in Figs. 3b and 4, the commercial FEM solver COMSOL was used [18]. A modern multicore high-performance PC (4 CPU cores, Intel Xeon 2.5GHz, 32GB RAM) is enough to perform the simulations. The 3D model has required around 20GB of RAM and has taken around 15 minutes of CPU time.

4. Numerical results

The first steps of our 2D FEM analysis were numerical studies of the required mesh size, distance of the ABC, etc. necessary to obtain a high level of accuracy. After this, the 2D eigenvalue analysis of the cladded waveguide revealed the results presented in Fig. 5 .

 

Fig. 5 2D FEM eigenvalue analysis results in the complex propagation constant plane (left) and the corresponding eigenfields (right) of the four guiding modes are depicted. Red circles in marked with M₁ – M₄ in the complex plane show the locations of the eigenmodes visualized. Since different modes cover different wavelength ranges and since the wavelength scale along the mode-traces is highly nonlinear, the wavelengths at the beginning and the end of each mode trace are given.

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Evidently, four different modes were found. They have complicated trajectories in the complex plane of the propagation constant (Fig. 5, left). Mode 2 performs even two almost complete loops. Modes with high imaginary part of the propagation constant (Mode 1 and Mode 3) have high losses and therefore have no practical importance. Within a certain wavelength range, Mode 2 and Mode 4 are very close to the real axis for certain frequency ranges, i.e. they have low losses. Hence they are attractive for practical applications. Note that Mode 2 has two relatively narrow frequency ranges with low loss close to Real(γ)/k₀ = 1, which indicates that the field is rather strong in the surrounding air domain. Mode 4 has low loss for wavelength longer than 500 nm, which is most promising, but its field most strongly confined near the triple point, which is rather problematic. Furthermore, the direction of the electric field outside the cladding is opposite to the direction the electric field inside the cladding, which is different from the field of a HE11 mode of fiber without cladding. Thus, it is expected that the HE11 mode couples much better to Mode 2 than to Mode 4.

As one can see in Fig. 5 (right) the field of Mode 1 and Mode 3 significantly penetrates into the silver of the stripes consequently causing high material losses. On the other hand the field of Mode 2 is almost completely in the air (as expected from its propagation constant) and the field of Mode 4 is confined in the narrow region of the triple point (air-silica-silver). Thus, it is not surprising that Modes 2 and Mode 4 have low losses.

The obtained FEM results were compared with highly accurate MMP results. This comparison is given in Fig. 6 . The obtained level of accuracy is very good, as the majority of the MMP and FEM curves almost overlap each other.

 

Fig. 6 Comparison of the 2D eigenvalue results between MMP and FEM. Evidently a very good agreement is obtained – except at areas where radiation loss may cause strong differences.

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A certain level of disagreement is observed where the losses and dispersion are very high (e.g. for some frequency range of Mode 1). In these areas, the MMP search function is very flat, which indicates that it is difficult to obtain accurate eigenvalue data. This is also physically justified because the complex propagation constant is very sensitive with respect to frequency and small modifications of the geometry in these areas. A severe disagreement is observed for the “curling” Mode 2. The normalized phase constant (Real(γ)/k₀) of this mode is sometimes below 1, which indicates that some radiation loss is present. When tracing this mode with MMP, the Sommerfeld radiation condition becomes invalid at certain points and then the MMP solution becomes invalid. Since the mode curls around in the complex plane, one would need to run different MMP models (with and without Sommerfeld radiation condition) and patch them together.

The field in the surrounding free-space is described by a multipole expansion based on Hankel function H_n(1) describing the guided and H_n(2) describing the radiated modes. Hence in MMP, to switch between the guided and radiated modes, it is necessary to change only the kind of Hankel function. However, in these settings radiating modes have increasing field function with respect to the distance and in infinity the field becomes also infinite. This is of course physically not correct.

Practically, waveguides are not infinite. The radiation lead to an increasing field function with respect to the distance up to a certain distance r = R_max. The value R_max depends on the propagation constant and on the distance to the beginning of the waveguide. For sufficiently large distances R_max is relatively large and the MMP model is accurate enough for describing the field in the vicinity of the waveguide.

However, it may be seen that a good agreement is obtained when Real(γ)/k₀>1 and when the mode dispersion is not very strong. A similar disagreement is also found for Mode 3 when Real(γ)/k₀ is close to 1. Here, the MMP search shows some discontinuous behavior. In fact, the mode is lost (because it disappears completely when the Sommerfeld radiation condition is assumed to be valid) and the smart eigenvalue search routine of OpenMaX the switches to another “evanescent” mode. In this context, it should be mentioned that a proper distinction of guided and evanescent modes is only possible for loss-fee waveguides, where the propagation constant is real for guided modes and imaginary for evanescent modes. When material andradiation losses are present, all modes are characterized by a complex propagation constant. Modes that follow mostly the imaginary axis may then be called “evanescent”, although this is not a strict definition.

It is worth mentioning that it is possible to extract only a few eigenvalues from a FEM matrix around a certain complex value of our interest which must be given prior to the eigenvalue extraction. Therefore, it is always necessary to have a good initial guess. This is where MMP has a big advantage over FEM, namely the possibility to visualize the search function over the complex propagation plane. The OpenMaX package even includes 9 different search functions for obtaining more information and confidence in the results. One of these search functions is presented in Fig. 7 . Having such plots at our disposal, the probability of not detecting some eigenvalues is very low.

 

Fig. 7 The behavior of a MMP search function (the residual norm of the MMP over-determined linear system of equations divided by the guided electromagnetic power as presented in [14]) in the complex propagation plane is shown. Sharp dips (dark areas) of this function correspond to the eigenvalues

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The main outcomes of our 2D eigenvalue analysis are that Mode 2 and Mode 4 have low losses and therefore could be very important for practical applications. Furthermore, Mode 2 has a very strange behavior in the gamma-plane. For this reason, it was very carefully analyzed. This analysis brings also more insight into the problem of radiating modes. The “curling” behavior of Mode 2 indicates that it is sometimes radiating and sometimes not – which makes the comparison with MMP difficult.

For a more detailed analysis the associated normalized radiation and material losses were computed. One has:guided power: $P_G=\frac{1}{2}{\displaystyle \underset{\left(S_{waveguide}\right)}{\iint }\overrightarrow{E}\times {\overrightarrow{H}}^{*}d\overrightarrow{S}}$,normalized radiation losses: $P_{RAD}=\frac{1}{2P_G}{\displaystyle \underset{\left(S_{ABC}\right)}{\iint }\overrightarrow{E}\times {\overrightarrow{H}}^{*}d\overrightarrow{S}}$,normalized material losses: $P_{RAD}=\frac{L}{2P_G}{\displaystyle \underset{\left(S_{waveguide}\right)}{\iint }\omega \cdot imag\left(\epsilon \left(\lambda \right)\right)\cdot {\overrightarrow{E}}^{2}dS}$.

The normalized radiation losses were computed for 1μm length of the waveguide. These results are presented in Fig. 8 (right).

 

Fig. 8 The behavior of Mode 2 in the γ-plane (left) and the associated normalized radiation and material losses (right).

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Evidently, Mode 2 makes two loops in the γ-plane over the wavelength range 300-900nm. The eigenvalue trace shown in Fig. 8 (left). It goes three times beyond the radiation limit ($real\left(\gamma \right)/k_0<1$). This is also translated into three distinctive parts of the blue radiation loss curve in Fig. 8 (right) with high values. Additionally, the two loops of Fig. 8 (left) have two upper parts with very high imaginary part. This corresponds very well to two distinctive parts of the red material loss curve depicted in Fig. 8 (right). The peak of the material loss curve that corresponds to lower wavelength is much higher than the peak at higher wavelength due to higher value of the imaginary part of the permittivity of silver at lower wavelengths presented in Fig. 1b.

According to the 2D eigenvalue analysis, the waveguide has several modes with moderate loss. In fact, there are also other modes available, which do not have the assumed symmetry properties (PEC and PMC walls in Fig. 3), but these modes are considered to be less promising because of the desired coupling with the HE11 fiber mode. Mode 1 and Mode 3 have very high losses due to the fact that their eigenfields affect a lot the silver strips. Mode 4 has low loss above 500nm but it is based on the field confinement around the triple point and is expected to have a weak coupling with a HE11 fiber mode. Therefore, we have chosen Mode 2 for feeding the 3D structure with the SNOM tip.

To analyze different situations and to improve the understanding of this structure, three different wavelengths of Mode 2 were chosen, as shown in Fig. 9 . At the lowest wavelength of 329nm Mode 2 is beyond the radiation limit (Fig. 9a) and therefore has large radiation losses (Fig. 9b). Since its eigenfield is also substantial inside the silver strips (Fig. 9c) its material losses are high (Fig. 9b). At the remaining two wavelengths 378nm and 396nm Mode 2 has very different field patterns, namely the field is confined in the core or in the air outside, respectively. Additionally, at both wavelengths Mode 2 is above the radiation limit. Hence the radiation and material losses are very small and near these two wavelengths we expect a good performance of the structure.

 

Fig. 9 Three different wavelengths of Mode 2 were chosen (a), the corresponding radiation and material losses computed (b), and the corresponding eigenfields depicted (c). The eigensolutions are used for feeding the 3D structure (d).

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Having the eigensolutions (see Fig. 9c), it is possible to use them as a field source of the 3D structure. This is illustrated in Fig. 9d. The chosen eigenfield is applied over the surface of the so-called input port of the structure. Thus, the attached eigenfield will propagate along the waveguide and at its end it will encounter the SNOM tip.

This technique for feeding the 3D structure is very favorable as it allows one to model rather short straight waveguide sections before the waveguide discontinuity. However, this method has also some drawbacks. The input port with an eigenfield applied will work efficiently only if there is no possibility that a mode conversion occurs at the waveguide end (waveguide discontinuity) and that some other converted mode is reflected to the input port. In our case this method of feeding could be applied without numerical problems as the other possible modes are very lossy at the chosen three wavelengths. Even though they are partially excited, they are so weak at the input port that they may be neglected.

The solutions of the 3D analysis are shown in Fig. 10 . The electric field (Fig. 10a) confirms the predictions of the 2D eigenvalue analysis. At 329nm Mode 2 has high material and radiation losses. Its energy is dissipated and radiated much before reaching the tip. At 378nm and 396nm the mode has a very efficient transmission and the wave reaches the tip resulting in a narrow field enhancement region. To quantify the effect, the electric field 5nm away from both tips is given in Fig. 10b. Evidently the longer tip exhibits the highest field enhancement factor at 396nm.

 

Fig. 10 The results of the 3D analysis are presented. The absolute value of the electric field is depicted (a) and the electric field values 5nm away from the silver tips are compared (b). The field values are normalized according to the highest field value obtained.

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5. Conclusions

The algorithms and results of the MMP and FEM eigenvalue analysis of an optical waveguide - based on a dielectric fiber with partial metallic cladding - were presented. The results of the eigenvalue analysis revealed several guided modes. One of them exhibits a reasonably low level of material and radiation loss within some frequency range and has a field pattern similar to the one of HE11 modes on uncladded fibers. Thus, it is expected that the mode couples well to HE11 modes. This mode has a very strange behavior, switching back and forth from radiating to non-radiating and from low to high material loss. Therefore, it was very carefully analyzed. In the second step, a SNOM tip was attached by extending the silver strips of the partial coating beyond the dielectric core. The corresponding 3D model was fed by the waveguide mode with the most efficient guiding and analyzed at three different wavelengths. The obtained 3D results confirmed the 2D predictions.

The 2D eigenvalue analysis of the waveguide with a relatively simple geometry yielded the modes with very complicated behavior in the complex γ-plane. Since a good initial guess is always required for the FEM eigenvalue analysis, it was very difficult to find all the eigensolutions by only using FEM. Therefore, the corresponding MMP analysis was very helpful. By plotting the various MMP eigenvalue search function in the γ-plane and by checking the MMP mismatching errors, one may obtain a very high confidence in the results. However, after the parameters of the FEM modeling such as the mesh size and distance to the ABCs were optimized and after rough initial guesses for the guided modes were available, a high level of agreement between FEM and MMP results was achieved – except in areas of very high dispersion and in areas of substantial radiation loss, where the Sommerfeld radiation condition (imposed in the MMP ansatz) is violated.