1. Introduction

For some metals, especially the gold and silver, the real part of the permittivity is negative within a certain frequency range of UV to NIR. The behavior is due to the collective motion of the free electrons in the metal, oscillating with the incident electromagnetic field. Because of that, there exists a unique surface electromagnetic wave propagating along a metal-dielectric interface, known as surface plasmon wave (SPW), or called surface plasmon polariton (SPP) [1]. The existence of SPW was first predicted by Ref. 2 in a transverse-magnetic (TM) mode, where the magnetic field is along the z-axis (out of the x-y plane) and the electric field is in the x-y plane. If the metal is a nonmagnetic medium, the dispersion relation of a SPW propagating along a planar metal-dielectric interface was derived as [1,2]

where k _sp is the wavenumber of the SPW in terms of the relative permittivity of the medium-1 (ε ₁) and medium-2 (ε ₂), ω is the angular frequency, and c is the light speed in vacuum. The medium-1 and medium-2 are the metal and dielectric, respectively. The characteristic of a SPW is that the amplitude of EM field decays exponentially as the distance from the interface. Recently, the characteristic equation of SPW propagating along a bend metal-dielectric interface was derived, and an asymptotic method was used to study the reflection and transmission coefficients of an incident SPW along a flat metal-dielectric interface encountering a bended corner [3, 4]. Moreover, SPW along a bended metal plate was studied [5], and FDTD method was used to show the wave propagation along the plate. The long-range SPP along a 3D finite-width strip and its bending loss were also investigated [6, 7]. In this paper, we will directly solve the characteristic equation numerically to identify the dispersion relation of SPW propagating along a curved interface quantitatively. Two types of SPW along curved metal-dielectric interfaces are studied by comparing their phase velocity and attenuation constant with that of SPW along a planar one; one is along the convex interface and the other along the concave. In order to identify the relation of the SPW’s dispersion relation with the radius of curvature of the metal-dielectric interface, we assume that the SPW propagates along the surface of a circular cylinder in the transverse-magnetic (TM) mode; i.e., the magnetic field is along the z-axis (out of the plane) and the electric field is in the x-y plane. For a convex case, a metallic circular cylinder with radius a embedded in a dielectric host is considered, and vice versa, a dielectric cylinder in a metal is considered for a concave one, as shown in Fig. 1, where the circular cylinder (medium-2) is embedded in the host (medium-1). Both media are assumed nonmagnetic materials. The relative permittivity of each medium is denoted by ε_j, and the relative permeability by µ_j, j=1, 2.

 

Fig. 1. The configurations of SPW propagating along (a) a convex and (b) a concave metal-dielectric interfaces with radius of curvature a.

Download Full Size | PDF

On the other hand, two SPWs were predicted [8] to be generated simultaneously to propagate along the circumference of a circular metallic cylinder of submicron-radius embedded in a dielectric host clockwise and counterclockwise, when the cylinder is irradiated by an incident plane wave. Therefore this model will be used to generate two SPWs propagating along a convex metal-dielectric interface. Utilizing a series solution of 2D Mie theory, this problem can be solved exactly, and calculated numerically to obtain the electromagnetic field distribution for each frequency. Furthermore, since the response of this scattering problem is time-harmonic steady-state, a dynamic animation of the propagation of SPW can be reconstructed. Using this technique, we can trace the migration of SPW, and its phase velocity will be measured to compare with the result of dispersion relation.

2. Characteristic equation of dispersion relation

Consider the electromagnetic field of a harmonic response for a two-dimensional (2D) TM-mode problem, where the time harmonic factor is exp(-iωt). In this paper, the characteristic equation [3] of a SPW propagating along a curved (convex or concave) metal-dielectric interface of radius of curvature a is rearranged as

where J_p is the Bessel function of first kind of order p and H ⁽¹⁾ _p is the Hankel function of first kind of order p in terms of z ₁=k ₁ a, z ₂=k ₂ a and the complex wavenumber $k_j=\frac{\omega \sqrt{{\epsilon }_j{\mu }_j}}{c},j=1,2$ . In Eq. (2), the primes of J_p and H ⁽¹⁾ _p denote the differentiation with respect to the argument. Here, the permeability of the medium-1 is the same with that of medium-2, µ ₂=µ ₁. The arguments z₁ and z₂ are given values (if a is given), and the order p is an unknown complex number, which is the root of Eq. (2). The magnetic fields of SPW in the two adjoining media are in the forms of H_z=AH ⁽¹⁾ _p(k ₁ r)e^ipθ and H_z=BJ_p(k ₂ r)e^ipθ, respectively, where A and B are the amplitudes. The term e^ipθ can be regarded as e ^i(p/a)aθ [9]. Physically, the meaning of p/a is the wavenumber of SPW propagating along the circumference of a circular cylinder with radius a, and aθ is the propagating distance of SPW. If the SPW creeps along the circumference of the circular cylinder counterclockwise, the real part of the complex wavenumber (p/a) should be positive because the phase shift should increase with the propagating direction, and the imaginary part should be positive, too, because of the attenuation of the amplitude; i.e., Re(p/a)>0, and Im(p/a)>0. Conversely, If the SPW creeps along the circumference clockwise, then Re(p/a)<0, and Im(p/a)<0. Since Eq. (2) is a transcendental equation, there is no analytical solution for the root, which is the complex order p. Therefore the numerical method is used for the calculation. When the radius of curvature, a, approaches infinite, the wavenumber (p/a) of SPW of the convex and concave cases will approach the value k _sp of a planar interface, as shown in Eq. (1).

In order to compare the phase velocity and the attenuation constant of a SPW along a curved interface with those of a planar one, the relative wavenumber α and the relative attenuation constant β are defined as

where Re is the real part and Im is the imaginary part. The relative wavenumber α is the ratio of the wavenumber of SPW along a curved interface to that of a planar one, and the relative attenuation constant β is the ratio of the attenuation constant of SPW along a curved interface to that of a planar one at the same frequency. Here the relative wavenumber α is also regarded as the ratio of the phase velocity of SPW along a planar interface to that of a curved one.

3. Numerical Results and Discussion

The terms in the left-hand side of Eq. (2) is the residue corresponding to any complex order p for the given z₁ and z₂. The roots of the complex order p, which enable the real and imaginary parts of the residue of Eq. (2) to be zero, represent the SPW modes. Using a mathematic package (Maple) to calculate the Bessel and Hankel functions of complex order with complex argument, the root of Eq. (2), the complex order p, can be solved numerically. For simplicity, we only consider the case of counterclockwise SPW, so that the root of Eq. (2) in the first quadrant of the complex domain p is searched. For example, consider a SPW propagating along an Ag circular cylinder of radius 400nm in air at 2.88eV (λ=430.6 nm). The relative permittivity of Ag is (-6.059844, 0.19696) [10] for this frequency. The distribution of the absolute value of the residue of Eq. (2) in the first quadrant of p is plotted in Fig. 2. Figure 2 indicates that there are several discrete zero-points of Eq. (2), where the first zero with the minimum imaginary part of p represents the fundamental mode (the first mode) of SPW, and the others are the higher-order modes of SPW [3]. For this typical case, the complex order of the fundamental mode is p=(7.1936, 0.6431), where the real part of p/a (the wavenumber of SPW along a convex interface) is 1.79835×10^-7 (1/m); i.e., the wavelength of SPW is λ_spw=349.4 nm. The relative wavenumber of the fundamental mode is α=1.12604404, and the relative attenuation constant is β=31.3828. These higher-order modes of SPW decay very fast due to their larger attenuation constants, so that the fundamental mode becomes relatively essential, because it can propagate a longer distance. Therefore, in the following calculations, only the fundamental mode of SPW is searched.

 

Fig. 2. The distribution of the absolute value of the residue in the first quadrant of the order p.

Download Full Size | PDF

For a SPW along a convex or a concave Ag-air interface, the dispersion curves of the relative wavenumber α and the relative attenuation constant β are plotted in Fig. 3 for different radii a (200, 400, and 1000 nm). The frequency-dependent permittivity of silver is cited from Ref. 10. The curves of α and β versus radius of curvature a are also plotted in Fig. 4 for different frequencies (2.13, 2.63, and 3 eV). These curves show that the relative wavenumber α of a convex case is always larger than one, and that of a concave case is always less than one. This is to say that the phase velocity of a convex case is always slower than that of a planar case, and the phase velocity of a convex case is always faster than that of a planar case. Generally, the larger the radius of curvature, the less the value of α for the convex case but the larger the relative wavenumber α for the concave case. In addition the higher the frequency is, the less the relative wavenumber α will be for the convex case but the larger the value of α for the concave case. On the other hand, the relative attenuation constant β for both cases (convex and concave) is always larger than one; i.e., the attenuation constant of SPW along a curved interface is always larger than that of a planar one for all frequencies and radii of curvature. This is because that the radial radiation of the electromagnetic energy into the host accompanies the propagation of SPW along a curved interface, except the dissipation in the metal. In addition, the larger the radius of curvature is, the less the relative attenuation constant β will be for both cases; the attenuation constant of SPW along a curved interface approaches the value of Eq. (1), as the radius of curvature increases. Moreover, when the frequency increases, the value of β also becomes less for both cases.

 

Fig. 3. The curves of (a) the relative wavenumber α vs. frequency, and (b) the relative attenuation constant β vs. frequency of SPW at Ag-air interface of different radii a (200, 400, 1000 nm), where the solid points: convex, and the void points: concave.

Download Full Size | PDF

 

Fig. 4. The curves of (a) the relative wavenumber α vs. radius a, and (b) the relative attenuation constant β vs. radius a of SPW at Ag-air interface for different frequencies (2.13, 2.63, 3 eV), where the solid points: convex, and the void points: concave.

Download Full Size | PDF

Furthermore, the curves of the relative wavenumber α and the relative attenuation constant β versus the relative permittivity of the dielectric medium are plotted in Fig. 5 for different radii a (200, 400, and 1000 nm) at 2.63 eV, where the metal is silver. These curves show that the larger the permittivity of the dielectric medium, the less the relative wavenumber α for the convex case and the larger the value of α for the concave case. However, these phase velocities of the convex and concave cases will approach the value of a planar case by increasing the permittivity of the dielectric medium. Moreover, the larger the permittivity of the dielectric medium is, the less the relative attenuation constant β will be for both the convex and concave cases, if the frequency and the radius of curvature are fixed.

 

Fig. 5. The curves of (a) the relative wavenumber α, and (b) the relative attenuation constant β vs. the relative permittivity of the dielectric medium for different radii a (200, 400, and 1000 nm) at 2.63 eV, where the solid points: convex, and the void points: concave.

Download Full Size | PDF

4. Numerical Experiment

Consider the previous case, SPW propagating along an Ag circular cylinder of radius 400nm in air at 2.88eV. From the characteristic equation, the value of p is p=(7.1936, 0.6431); i.e., the wavelength of SPW is λ_spw=349.4 nm. From the other aspect, when the Ag circular cylinder is irradiated by an incident plane wave of p-polarization, two SPWs are predicted to be generated simultaneously; one propagates along the circumference clockwise and the other counterclockwise [8]. The phenomenon is very similar to the SPP along a ring resonator [12]. Moreover, a standing wave will be formed on the backside of the cylinder due to the interference of the two opposite-direction SPWs [13]. According to this phenomenon, we can estimate the wavelength of SPW along a convex metal-dielectric interface of a specific radius of curvature for a specific frequency by measuring the arc length of two adjacent nodal points of the standing wave. Utilizing an analytic solution of 2D Mie theory in series form, this problem can be solved exactly and calculated numerically to obtain the distributions of the electric and magnetic fields for each frequency. Since the field values we obtain are complex values of the time-harmonic responses of steady state in terms of the phase difference, they can be reconstructed in time domain by using the formulation,

where the field function G(x) can be the magnetic field H_z and the electric fields E_x, E_y. We divide the motion of a period T into 64 frames. The time response of each frame at t=nT/64, n=0, 1, 2…63, can be calculated by using Eq. (3), and then an animation of the steady-state wave propagating can be obtained by playing these frames in sequence. In Figs. 6(a) and 6(b), only the results of the magnetic field of t=5T/64 and the electric field of t=13T/64 at 2.88eV are shown respectively, in which the incident wave propagates from the left-hand side to the right-hand side. Obviously, there is a standing wave on the backside of the Ag-cylinder, no matter from the electric field or the magnetic field. Since the attenuation of SPW is associated with the propagation due to the existence of the imaginary part of the wavenumber, the amplitudes of the two SPW decay as they propagate along the circumference. The attenuation makes the pattern of the standing wave blurred. However on the right backside, which is around the region of θ=0°, the amplitudes of both SPWs are almost the same, because their propagating distance are almost identical. Therefore, we only measure the angle between the two marked nodal points, as shown in Fig. 6(b). The angle is 50°, which covers a wavelength, so that the estimated wavelength is λ_spw=349 nm, which is in agreement with the theoretical value of λ_spw=349.4 nm obtained from the characteristic equation. Using this method, the wavelengths of the other frequencies are also checked, and they are all consistent with the theoretical ones.

 

Fig. 6. (a). The distribution of the magnetic field at t=5T/64. (b) The distribution of the absolute of the electric field at t=13T/64 of Ag cylinder of a=400nm irradiated by a plane wave at 2.88eV. All the values are normalized with the amplitudes of the incident fields.

Download Full Size | PDF

5. Conclusion

The characteristic equation of a SPW creeping along a convex or concave Ag-dielectric interface was solved numerically to obtain its dispersion relation. The results indicate that the phase velocity of SPW propagating along a convex interface is always less than that along a planar one, and will approach the value of the planar one as the radius of curvature of the convex interface, the permittivity of the dielectric host, or the frequency increases. In contrast, the phase velocity of SPW propagating along a concave interface is always faster than that along a planar one, and will close to the planar one as the radius of curvature of the concave interface, the permittivity of the dielectric cylinder, or the frequency increases. In addition, the attenuation constants of SPW of the convex and concave cases are always larger than that of a planar one, due to the radial radiation of the electromagnetic energy into the host, except the dissipation in the metal. These dispersion relations will be useful to interpret the bend-induced loss. Using a model of an incident plane wave irradiating an Ag circular cylinder in air for generating two SPWs, the wavelength of SPW along the convex interface can be obtained from the nodal points of the standing wave. The wavelength of this model is in agreement with that of the dispersion relation.