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The electric field enhancing properties of the V-shaped optical resonant antenna are studied by using finite-difference time-domain method. Both dipolar and quadrupolar modes can be effectively excited and strong electric field enhancement in the gap of the V-shaped antenna is found. Compared with full-wave dipole antenna, the V-shaped antenna has a greater electric field enhancement, which can be attributed to the higher radiation directivity and the smaller curvature radius of the antenna arms. The more asymmetrical structure also contributes to the efficient quadrupolar excitation. The electric field enhancement of the V-shaped antenna has different dependences on the open angle of the V-shaped antenna for the dipolar and quadrupolar excitation. We obtained stronger electric field enhancing properties by using V-shaped bow-tie antennas, especially for the quadrupolar excitation. The V-shaped antenna and the bow-tie antenna can realize strongly localized and enhanced field and thus are well suitable for the use of near-field optics applications.
©2007 Optical Society of America
1. Introduction
The interconversion of propagating field and localized field becomes a core issue in many fields, such as optical characterization [1,2] and manipulation [3], optoelectronic devices [4], and quantum information processing [5,6]. Recently, the antenna operated in the optical frequencies was proposed [7–9] and realized experimentally [10, 11]. These antennas, including bow-tie antennas [11] and dipole antennas [10] concentrate the incident energy in a narrow gap and realize a localized and enhanced field. This enables the optical resonant antennas many applications such as nonlinear optics [10,12], surface modification [13,14] and scanning near-field optics applications [15–17].
Most of these optical antennas work for the dipolar resonant excitation [10,11], which has a high excitation efficiency and strongly enhanced field. Usually the size of the antenna is much smaller than the wavelength of the incident light and it is difficult to be fabricated. For the same resonant wavelength, the antenna length will be larger for the quadrupolar excitation than that for the dipolar excitation [18,19]. However, quadrupolar mode cannot be excited effectively because of the symmetry mismatching [18].
In this work, we present a V-shaped optical resonant antenna, in which both dipolar and quadrupolar modes can be effectively excited and strongly enhanced electric field is obtained. Its electric field enhancing properties is analyzed by using finite-difference time-domain (FDTD) technique. The influence of the open angle on the field enhancement is investigated. Compared with full-wave dipole (FWD) [8] antennas, the V-shaped antenna exhibits stronger electric field enhancement due to the higher radiation directivity and the smaller curvature radius of the two antenna arms. Moreover, its structure is more asymmetric and the quadrupolar mode can be excited effectively. Quadrupolar antennas have bigger antenna size [18,19] than that of the dipolar antennas and thus the fabrication process is easier.
To increase the field enhancement further, we change the antenna arms to triangular structure to form a bow-tie antenna and the electric field can be increased obviously, especially for the quadrupolar excitation. The strong field enhancement properties of the V-shaped antenna and the bow-tie antenna enable the promising applications in near-field optics.
2. Dipolar excitation of the V-shaped antenna
The basic geometry of a V-shaped antenna is shown in Figs. 1(a) and 1(b). Two golden nanostripes are located on the two opposite sides of a truncated pyramidal-shaped silica substrate with four-fold symmetry to form a V-shaped antenna. Both the width of the antenna arms w and the gap size of the antenna are kept 20 nm. The length of the antenna arms L varies between 70 nm and 520 nm and the open angle θ of the antenna changes from 60° to 120° in steps of 30°. The electric field calculations are performed based on FDTD technique using commercial software (REMCOM XFDTD 6.3.8.3 version). The Perfectly Matched Layer absorbing boundary conditions are implemented at the boundaries of the calculation region. To describe the gold permittivity, we use a modified Debye model which agrees well with experimental data in the spectral region between 600 nm and 1000 nm. To minimize numerical errors, we use a cell size of 2 nm for the antenna length smaller than 200 nm and 4nm for the antenna length larger than 200 nm. The V-shaped antenna is illuminated from the bottom by an 830 nm plane wave polarized along x axis as indicated in Fig. 1. In the following text the value of the incident field Einc is set as 1 V/m. The origin is the center point of the top surface of the V-shaped antenna. The field distribution on the top surface plane (z=0 plane) of the antenna is calculated and studied. In the practical cases, Ti:sapphire laser is a common light source and its typical wavelength is around 830 nm. In order to make our simulation practical, we varied the antenna length to keep the resonant wavelength at 830 nm.
Fig. 1. (a). Top view and (b). side view of the V-shaped antenna; (c). The geometrical model of a full-wave dipole antenna; (d). The geometry model of a modified full-wave dipole antenna.
When a V-shaped antenna is resonantly excited, the electric field is localized in the antenna gap. Figure 2 plots the electric field distribution on the top surface plane (z=0 plane) of a V-shaped antenna with an open angle of 120°. From the figure we can see that the field on the antenna arms near the gap is greater than other areas, which indicates that a coupling between the two antenna arms occurs. Although stronger coupling and more intense field can be obtained by reducing the gap size [18,20], we use a gap size of 20 nm through our simulation due to the practical nanofabrication ability.
Figure 2(b) shows the electric field distributions along the line y=0 and y=10 nm in the z=0 plane (i.e. along the edge of the antenna arms and the central line of the antenna gap). The edge of the antenna arms has strong field enhancement and the biggest field enhancement exists at the corner of the antenna arms, which is indicated by Ecorner [see Fig. 2(a)]. The value of Ecorner depends on the small curvature radius and the meshing result at the corner of the structure. We did not make particular treatment on the mesh at the edge and the corner of the antenna arms during the simulation. For the present nanofabrication methods, the sample usually can not have such sharp edge and corner. In the following discussion, we only study the electric field at the center of the antenna gap Ecenter. In Fig. 2(b), the curve is not symmetrical about the zero point. This is because the meshing result of the two antenna arms is not symmetrical during the auto-meshing process. However, this asymmetry should not influence the properties of the Ecenter in physics.
Fig. 2. (a). Electric field distribution in the z=0 plane for the V-shaped antenna with an open angle of 120°; (b). Electric field distribution along the line y=0 nm and y=10 nm in the z=0 plane. The FWHMs in (b) are 25 nm and 28 nm respectively. (c). Near-zone field scattering spectra for the V-shaped antennas with an open angle of 120°. The antenna lengths L are 70 nm (red curve) and 230 nm (black curve) respectively.
The length of the V-shaped antenna is varied to keep the resonance wavelength at 830 nm. We show the near-zone field scattering spectra in Fig. 2(c), which is obtained by the transform method [21,22]. A V-shaped antenna with an open angle of 120° and a length of 70 nm has a dipolar resonant wavelength at 830 nm and a quadrupolar resonant wavelength at 616 nm. When we increase the length to 230 nm, the quadrupolar resonant wavelength shifts to 830 nm.
First we study the influence of the open angle on the electric field enhancing properties of the antenna and show the results in Table 1. For the dipolar excitation, the electric field enhancement increases with the open angle of the antenna. A V-shaped antenna with an open angle of 120° has an electric field value of 48.4 V/m while one with an open angle of 60° has an electric field value of 29.8 V/m. There are two reasons accounting for this phenomenon. A greater open angle results in a bigger projection area of the antenna arms and higher excitation efficiency. Second, the free charge tends to accumulate at the place or point where the curvature radius is small. For the V-shaped antenna, the edge of the antenna arms near the gap is sharp and the curvature radius reduces with the increment of the open angle.
The full width at half maximum (FWHM) of the electric field distribution along the line y=0 on the top surface of the antenna (z=0 plane) is 25 nm and hardly vary with the open angle. The small FWHM value means high spatial resolution. The V-shaped antenna realizes a strongly localized and enhanced field and thus can be used in the near-field optics applications.
Table 1. Simulation results of the dipolar excitation of the V-shaped antenna at different open angles.
In order to know whether the small curvature radius of the antenna arms is the only reason of the strong field enhancement of the V-shaped antenna, we excite the dipolar mode of a FWD antenna and a modified FWD antenna [sees Figs. 1(c) and 1(d)]. The antenna arm of the modified FWD antenna is a diamond with an inner angle φ of 30°, which has a similar corner shape with the V-shaped antenna with an open angle θ of 120°. The antenna size of the FWD antenna is 80 nm long, 20 nm wide and 20 nm thick while it is 72 nm long, 20 nm wide and 20 nm thick for the modified FWD antenna. The gap size of the two antennas is also kept 20 nm, which is the same as that of the V-shaped antenna. When the dipolar mode is resonantly excited, the Ecenter is 15.3 V/m for the FWD antenna and 20.0 V/m for the modified FWD antenna. However, the field enhancement of the V-shaped antenna with an open angle of 120° is much stronger than that of the modified FWD antenna. The V-shaped antenna has a higher directivity than the dipolar antenna [8], which means a larger maximum effective aperture, i.e., receiving cross section [23]. Therefore, the power received by the V-shaped antenna is more than that received by the dipolar antenna, which results in a stronger field enhancement.
3. Quadrupolar excitation of the V-shaped antenna
For dipole antennas composed of nanorods parallel to the incident polarization, the excitation efficiency of the quadrupolar mode is fairly low [18] because of the symmetry mismatching [19]. However, for the V-shaped antenna, the antenna arms are not parallel to the polarization of the incident beam and there is retardation along the antenna arms. Quadrupolar mode is expected to be excited effectively. The resonant wavelength is still kept at 830 nm and the results are shown in Table 2. High field enhancement is also obtained and the peak value of 30.8 V/m occurs at an open angle of 90°. Because the light comes from the bottom of the antenna, the phase varies when it reaches different parts of the antenna and the electric field can be antiparallel for different parts of the antenna. The results agree with the experiment of Au nanowires with an obliquely incident light [19]. By comparing the values in Table 1 and 2, we can find that for all the three open angles calculated the dipolar mode has bigger field enhancement than the quadrupolar mode. This case will change by using a bowtie antenna in section 4.
Table 2. Simulation results of the quadrupolar excitation of the V-shaped antenna at different open angles.
We also calculate the quadrupolar mode of the FWD antenna and the modified FWD antenna to study the influence of the curvature radius of antenna arms on the field enhancement. Their sizes are 210 nm×20 nm×20 nm and 265 nm×20 nm×20 nm (L×w×d), respectively, for a resonant wavelength of 830 nm. The modified FWD antenna has an inner angle φ of 45° [see Fig. 1(d)], which has the same curvature radius at the corner as the V-shaped antenna with an open angle of 90°. The Ecenter values are 5.63 V/m and 11.4 V/m for the FWD and the modified FWD antenna, respectively. Both the small curvature radius and more asymmetry of the modified FWD antenna contribute to the field enhancement. However, the Ecenter of the modified FWD antenna is still much smaller than that of the V-shaped antenna with an open angle of 90°. Besides the high directivity and a small curvature radius, the oblique angle to the incident polarization, which results in a more asymmetrical charge distribution, is a main reason of the high field enhancement of the V-shaped antenna for the quadrupolar excitation.
According to the equation λeff×j/2=L, where j is an integer and λeff represents the effective wavelength of the light in the metal antenna arms [24], if the effective wavelength λeff is kept the same, the antenna length L increases with the mode number j. High order multipolar excitation means long antenna arms, which makes the micro-fabrication process of the antenna easy.
4. Bow-tie antenna
In order to increase the electric field enhancement further, we consider increasing the structure asymmetry of the V-shaped antenna. As discussed above, a nanorod is symmetrical in its axis direction. Nevertheless, a triangle nanoparticle is asymmetrical and the quadrupolar mode can be excited [25]. Bow-tie antenna, which is composed of two triangle nanoparticles, has higher asymmetry in the axis direction than the V-shaped antenna composed of two nanorods. Therefore, the charge and current density distributions are more asymmetrical [26] and the electric field enhancing properties are expected to be better.
The basic model of a bow-tie antenna is shown in Fig. 3(a), which is similar to the V-shaped antenna with a truncated pyramidal-shaped silica substrate with four-fold symmetry. Two opposite sides of the substrate are coated with a layer of Au film. The metal film thickness d and the gap size are kept 20 nm. The width w of the Au film on the top surface plane of the antenna is 20 nm. The length of the antenna L is varied to let the resonance wavelength at 830 nm. The incident light is polarized along the gap direction and illuminates the antenna from the bottom. When the open angle ranges from 30° to 120° in steps of 30°, the lengths of the antenna arms are 115 nm, 120 nm, 110 nm and 84 nm, respectively, for the dipolar excitation and 310 nm, 318 nm, 300 nm and 262 nm, respectively, for the quadrupolar excitation. Figure 3(b) shows the near-zone field scattering spectra of a bow-tie antenna with an open angle of 120°. For the resonant quadrupolar excitation at 830 nm, the antenna length is 262 nm and the corresponding dipolar resonant wavelength shifts to 1584 nm.
Fig. 3. (a). Geometrical model of the bow-tie antenna; (b). Near-zone field scattering spectra for the bow-tie antennas with an open angle of 120°. The antenna lengths L are 84 nm (black curve) and 262 nm (red curve), respectively. (c) and (d) show the electric field distributions of the y=0 plane for the dipolar excitation and quadrupolar excitation, respectively. The value 0 dB equals a field value of 76.3 V/m.
The electric field distributions of the y=0 plane are shown in Figs. 3(c) and 3(d) for the dipolar and quadrupolar excitation, respectively, for an open angle of 120°. In Fig. 3(d), there is a node at the center of the antenna arm. The vector plot of the instantaneous electric field in y=0 plane (not shown here) indicates that the fields have inverse directions at the two sides of the node. These facts demonstrate that the mode excited is really a quadrupolar mode. The field enhancement of the bow-tie antenna is plotted in Fig. 4, which is larger than that of the V-shaped antenna with the same open angle. When the open angle is bigger than 60°, the open angle does not influence the field enhancement too much and the values of Ecenter are between 43 V/m and 52 V/m. The FWHM values, which are 24 nm and 28 nm corresponding to the dipolar excitation and the quadrupolar excitation, respectively, hardly change with the open angles.
An encouraging result is that for the quadrupolar excitation the Ecenter of the bow-tie antenna with an open angle of 90° is increased to 49.6 V/m, which is even greater than that for the dipolar excitation with the same open angle. The triangle geometry of the bow-tie antenna enables peak current density to be largest at the apex of the triangle [26]. From Fig. 3(d), we can see that the excitation efficiency of the upper part of the antenna arms (the part close to the gap) is much higher than that of the bottom part. Compared to the V-shaped antenna, the charge density distribution is more asymmetrical along the antenna arms, which enhances the even order multipole excitation. The big field enhancement for the quadrupolar excitation shows the advantage of the bow-tie antenna.
The V-shaped antenna and the bow-tie antenna have strong field enhancement and high spatial resolution, which make them particularly suitable for near-field optics applications, such as single molecule fluorescence detecting [17], surface ablation [14], nonlinear optics [10,12] and lasing [4]. Besides, the V-shaped antenna and the bow-tie antenna are three dimensional and can be integrated to scanning near-field probes. Because of the strong excitation efficiency of the quadrupolar mode, we can use it to obtain high field enhancement while keeping a long resonant antenna length and therefore make its fabrication process easier than that of the dipole antenna [10,15,17].
5. Conclusion
In conclusion, we have investigated the electric field enhancing properties of the V-shaped optical antenna by using FDTD simulations. High enhancement factors have been obtained and both the dipolar and quadrupolar modes can be excited effectively as compared with the FWD antenna. The large enhancement is attributed to the high radiation directivity and the small curvature radius of the V-shaped antenna. The more asymmetrical structure also contributes to the efficient excitation of the quadrupole. The influence of the open angle on the field enhancement is discussed. Furthermore, the bow-tie antenna is studied and exhibits stronger field enhancement, especially for the quadrupolar excitation. The results demonstrate that the V-shaped antenna and the bow-tie antenna have potential applications for the near-field optics.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under grant Nos 10434020 and 10521002, and the National Basic Research Program of China under grant No 2007CB307001.
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