In this paper, we present the modulation of a tightly focused evanescent field by a nano-plasmonic waveguide, which consists of two silver nanorods lying on the interface of two dielectric media. Linearly polarized and radially polarized illuminating beams are investigated under the influence of localized surface plasmons effect. It is demonstrated that different polarization components of the tightly focused evanescent field can be modulated accordingly. The results obtained from the finite difference time domain simulation show that super-resolved focal spot can be achieved using the nano-plasmonic waveguide structure.
© 2009 OSA
Noble metals with nanostructure geometry have special optical properties because they can excite localized surface plasmons (LSPs) under the illumination of light field. LSPs are non-propagating excitations of the free electrons of metallic nanostructures coupled to the applied electromagnetic field . The optical properties of nanorods [2,3] and nanowires  have been studied, which proved that the excitation of LSPs is strongly dependent on the polarization mode of incident light, the geometry of metals and the size of the nanostructures . Due to the strong confinement of light field on the surface of metallic nanostructures, LSPs have been widely applied in the field of wave-guiding [6–8]. The tip forming metallic array of channel LSPs waveguides  was reported as a focusing launcher to form a 197 nm width focus under the illumination of Gaussian plane wave with 500 nm wavelength. A hybrid plasmonic waveguide formed by a high-permittivity dielectric cylindrical nanowire  had shown strong confinement of light field within approximately one tenth of the illuminating wavelength, and large propagation distance of LSPs ranging from 40 to 150 μm. The Kretschmann-Raether configuration based surface plasmon polaritons (SPPs) microscopy has proved that the transverse distribution of SPPs excited by the focused laser beam through a thin metallic film is influenced by the polarization of incident light . The lateral behaviour of the metallic thin film in SPPs microscopy also shows dependence on the defocus position along the optical axis .
A tightly focused evanescent field can be generated by a centrally obstructed high numerical aperture objective lens , and a super-resolved evanescent focal spot of λ/3 has been obtained . The enhancement of the electromagnetic (EM) field by tight focusing enables nano-lithography using evanescent field . It has been demonstrated that a tightly focused beam can be further modulated by a negative-refraction layer together with a nonlinear layer  or a saturable absorber  to approximately λ/4- λ/5 close vicinity of the focus. Considering the difficulty in realizing a negative-refraction layer in practice, here we introduce another mechanism of light modulation in the tightly focused region. Due to the use of high numerical aperture objective lens, the focused evanescent field is highly depolarized, which offers strong transverse and longitudinal polarization components. Therefore the deployment of nano-plasmonic structure, which is polarization sensitive, offers new mechanism to modulate the focused evanescent field.
In this paper we introduce a simple nano-plasmonic waveguide structure with two silver nanorods lying on the interface of two dielectrics. A new simulation model, incorporating three-dimensional (3D) finite difference time domain (FDTD) method  and high angle vectorial Debye theory , is developed to study the modulation of the focused evanescent field under the influence of LSPs effect, produced by the nano-plasmonic waveguide. The Drude model [20,21] is adopted to describe the frequency-dependent relative permittivity of dispersive material and the frequency- dependent finite-difference time-domain ((FD)2TD) method [22,23] is applied to simulate the optical properties of dispersive material. It has been proved that the Drude model fits the dispersive dielectric function for silver very well over a wide wavelength range .
2. Numerical simulation model
Our simulation configuration is shown in Fig. 1 . Two silver nanorods are lying on the interface of two dielectric media with the separation (D) of 120 nm centre-to-centre. The refractive index of lower medium (n1) is 1.78 and the refractive index of upper medium (n2) is 1.0. The size of the nanorods is 150 nm in length (l) and 50 nm in diameter (d). The numerical aperture of the objective is 1.65, and the pure focused evanescent field is generated by inserting a centrally placed obstruction with normalized radius εc = 0.606 , corresponding to the total internal refraction condition.
If the illuminating beam polarizes along the x direction, the electric field at the focal region, calculated with high angle vectorial Debye theory , is highly depolarized, and each polarization component can be expressed as:24]:
It should be pointed out that the electric fields under the radially polarized focal beam is circular symmetrical. The electric and magnetic fields are calculated at a plane one wavelength before the interface as the input source for the FDTD simulation. The total field/scatter field technique  is used to eliminate the light propagating to –z direction, so that the incident focal beam only propagates in forward direction.
In our simulation, the wavelength of incident focal beam is 532 nm. It should be noted that the annular beam illumination includes a wide range of incident angle (34.18°<θ<69.97°), which corresponds to a wide rage of surface plasmon resonance (SPR) wavelengths (340 nm<λ<1120 nm). Our simulations show that different wavelengths selected in this spectrum range make little difference for the excitation of LSPs by the nano-plasmonic waveguide. The grid sizes Δx, Δy and Δz for each dimension are set to 2.34 nm. According to the discretization in space domain, the discretization in time domain that satisfies the Courant stability condition  is adopted as:
The dispersive materials such as metals show strong frequency-dependent behaviour in visible range. The Drude model [20,21] has been developed and implemented in FDTD with the (FD)2TD method [22,23]. In frequency domain the relative permittivity of dispersive media described by Drude model is shown as:21].
3. Simulation results and discussion
3.1 Linearly polarized focal beam
The intensity distribution of a focused evanescent field under the linearly polarized illumination is shown in Fig. 2 and agrees well with previous theory [13,25]. Under the conditions described in Section 2, the intensity of Ey component is one order smaller than either Ex or Ez, so the overall impact of Ey on the intensity distribution is less significant. Due to the depolarization effect, a strong longitudinal Ez component appears, with its strength comparable to the illuminating polarization component Ex, i.e., |Ez|2/|Ex|2≈0.85. As a result, the intensity distribution of the total field is splitted to two lobes shown in Fig. 2 (a).
The modulation of the focused evanescent field by a pair of silver nanorods is demonstrated in Fig. 3 , where the intensity distributions in planes of different distances from the interface are illustrated. In the left column, when there is no nanorod on the interface, the focal spot splits into two lobes at different distances above the interface. In the middle column, the nanorods are lying along x direction, parallel to the dominant transverse polarization component Ex. In the right column, the nanorods are lying along y axis, perpendicular to the dominant transverse polarization component Ex. In Fig. 3 (a), it is noted that at the xy plane 25 nm above the interface, the electric field is significantly enhanced by the LSPs between the nanorods, if the nanorods are lying in the y direction, resulting in a strong localized field between the two nanorods. While the nanorods are lying in the x direction, the electric field shows less enhancement and localization. At the plane 60 nm above the interface (10 nm above the nanorods, Fig. 3 (b)), it is observed that the longitudinal electric field component Ez displays a strong enhancement at both ends of each nanorod, forming four strong intensity lobes. When the nanorods are lying along x direction, the dominant transverse electric field component Ex is not significantly enhanced, so the electric field is dominated by the longitudinal component Ez, which shows four strong intensity lobes. However when the nanorods lie in the y direction, due to the significant enhancement of Ex component between the nanorods, the four strong lobes become less evident. At the horizontal planes that further away from the nanorods, i.e. at the planes 75nm (Fig. 3 (c)) and 100nm (Fig. 3 (d)) above the interface, super-resolved focal spots are demonstrated. In particular, with the significant enhancement of dominant transverse electric field component Ex, the focal spot shows a narrower distribution and a stronger strength, when the nanorods lie in the direction perpendicular to the Ex component.
The detailed analysis of the LSPs effect on each polarization component at the plane 100nm above the interface is illustrated in Fig. 4 . When the nanorods are lying along x direction (Fig. 4(b)), the enhancement of Ex component is insignificant compared with the case that the nanorods lie in the y direction. Nevertheless, it still produces a narrower focal spot with 46.6% reduction in the full width at half maximum (FWHM) and 61% increase in strength. When the nanorods are lying in y direction (Fig. 4(c)), the dominant transverse polarization Ex is not only significantly enhanced in strength, e.g. by a factor of 2.71, but also becomes more localized, e.g. the FWHM is reduced from 388 nm to 149 nm. This phenomenon indicates the LSPs excited by the Ex component couple between the nanorods and re-distribute the energy of electromagnetic field. The longitudinal component Ez is also enhanced by a factor of 1.4, and becomes narrower, e.g. the distances between two intensity peaks reduced from 266 nm to 127 nm. As a result, a super-resolution focal spot can be formed outside the waveguide. The above analysis shows that the nano-plasmonic waveguide provides strongest enhancement to the transverse polarization component perpendicular to the nanorods, followed by the longitudinal polarization component, and the enhancement for the transverse polarization component parallel to the nanorods is least significant.
3.2. Radially polarized focal beam
It is well known that the focal spot for radially polarized beam is circularly symmetrical [14,22]. According to the analysis demonstrated in the previous section, it is expected that the circular symmetry would be broken due to the LSPs effect which is polarization sensitive. Figure 5 shows the intensity distributions of evanescent radially polarized focal beam at xy planes of different distances above the interface for three cases, including without nanorods, nanorods lying in the x and y directions, respectively. The focal spot is circularly symmetrical and decay exponentially further from the interface without the presence of the nanorods, as shown in the left column of Fig. 5. With strong LSPs effect from the nano-plasmonic waveguide, the circular symmetry is broken, and the focal spot shows a strong intensity lobe at each of the four ends of the two nanorods (Figs. 5 (b-d)). It is noted that the intensity of the longitudinal component Ez is approximately one order of magnitude stronger than that of the transverse components. As a result, the intensity distribution at the focal region shows four strong intensity lobes produced by LSPs effect excited by the longitudinal polarization component Ez.
In this paper we demonstrate a new method to modulate highly focused evanescent field with a nano-plasmonic waveguide. The modulation of focus is based on the mechanism that the LSPs are polarization sensitive and the focus is strongly depolarized by a high numerical aperture objective. For a simple nano-plasmonic waveguide that consists of two silver nanorods lying on the interface between two dielectrics, LSPs effect is strongest for the polarization component perpendicular to the nano-plasmonic waveguide. A super-resolved focal spot with significantly enhanced strength can be achieved, when the nanorods are lying perpendicular to the dominant polarization component. The design of the nano-plasmonic waveguide structure gives rise to a new approach to further improve the tightly focused evanescent field to achieve the resolution beyond diffraction limit, and thus facilitates potential applications in nano-trapping and nano-lithography.
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