To calculate the formation of SPPs, we imported the field distribution of focus,
57
generated by vectorial diffraction theory, into commercial FDTD software, SIM 3D_MAX, by MMResearch. The two-dimensional field distributions of |Ez| component are identical to the field distribution of excited SPPs. Figure 4.4 illustrates the field distribution of excited SPPs for single excitation. To improve the visualization of the outer ring, the results were selected to illustrate the amplitude of its field distribution rather than intensity. With a single excitation, we modulated the size and position of a single TM-polarized sector to observe the corresponding SPPs fields. As shown in Fig. 4.4(a) and (b), the center of the angle of TM-polarized sector, 0, determined the direction of propagation of the plasmonic waves.
The size of the TM-polarized sector, δ, corresponded reciprocally to the excited area. Such plasmonic manipulations show potential as a new scheme, for achieving high throughput and coupling efficiency for the planar plasmonic waveguides or devices.
Fig. 4. 4 The calculated field distribution of SPPs when the SIP beam was focused on the Au/Air interface. Subfigures (a) to (h) display single excitations with different ratios of TM-polarization (indicated with black arrows along the radial direction, indicated by the white background) at the pupil entrance, where (a) 0 = 202.5°
and δ°, (b) 0 = 22.5° and δ°, (c) 0 = 90° and δ°, (d) 0 = 67.5°
and δ°, (e) 0 = 90° and δ°, (f) 0 = 112.5° and δ°, (g) 0 = 135°
58
and δ°, (h) 0 = 157.5° and δ°.
The field distribution of excited surface plasmon waves along the radial direction was expressed as ESP~E0exp(ikspr)exp(-LSP/r), where LSP was the propagation length with 1/e attenuation of the SPPs amplitude and r represented the radial propagation distance from the position of excitation. The propagation constant and length of SPPs were calculated by taking the real part on the field distribution of |Ez|. In the case of 0 = 22.5° and δ°
[Fig. 4.4(b)], the numerical result of the length of propagation and resonant wavelength were LSP = 0.98 um and SPP = 601nm, respectively. The numerical results agreed closely with the theoretical prediction under SPP = 2Re[k0()1/2 = 598 nm, whereas the theoretical length of propagation (6.24 um) was longer than that of the numerical results. This was because the field distribution of SPPs was formed by constructive interference induced from partially in-phase angular vector ksp. This intensity peak was much higher than that of any ray excited surface plasmon wave.
When the size of the TM-polarized sector δ was increased from 90o to 315o, the field distribution of excited SPPs revealed a gradual tendency toward local concentration, as shown in Figs. 4.4(c) to (h). In addition to the concentration of energy, the propagating SPPs were steered counterclockwise as δ increased. At the same time, a series of interference ripples along the azimuthal plane became noticeable, yielding side lobes in the shape of discontinued arcs, due to the consequences of omni-directional SPPs propagation. As TM-polarized light occupied the entire pupil of illumination, the Bessel field distribution became excited creating a tiny spot at the center dressing the side lobe with a concentric ring. This was consistent with results in previous studies [73, 74].
Figure 4.5 shows two animations of SPPs with different formations generated by the double excitation scheme. The polarization distribution of the SIP beam was designed to aid
59
in the investigation of the interference behavior of the two propagated surface plasmons. In Fig. 4.5(a), the TM-polarized sector was divided into two equal parts where δ 45o and varied in Δ. The inserted TE-polarized sector acted as a barrier to isolate two generated plasmon waves with an angular distance Δ. As Δ changed from 15 to 135 degree, the interferometric patterns of the two oblique plasmonic waves gained additional outer edges with corresponding sway. The angular distance between the two edges was identical to Δ.
In addition, a constructive bright spot was observed at the center, which had been created by in-phase counter-propagating vectors, but was irrelevant to the change of angular distance.
Fig. 4. 5 Double excited SPPs generated by purposely designed SIP beams, with the point of observation lying on the plane of focus. (a) the TM-polarized sector is divided into two part with equal δ but varied in Δ. As Δ changed from 15° to 135°, the interferometric patterns of two oblique plasmonic waves show additional outer swayed edges. (b) the polarization distribution of SIP beam consists of double TM-polarized sectors which was arranged on the opposite side with variations in the size of δ. A clear plasmonic interference pattern spreading along vertical direction can be observed due to the counterpropagation of the SPPs.
60
Figure 4.5(b) (right side) depicts the normalized field distribution of SPPs observed on the focal plane illuminated by an SIP beam featuring two counter TM-polarized sectors with varied δ. A clear plasmonic interference pattern extending vertically was observed, due to the counter-propagation of the SPPs. The modulation of the interference pattern implied a recipitical relation between the size of sector and the lateral elongation of the interference lines. When the size of the sector δ shrank to a narrow slit on each side, the interference pattern of SPPs resembled that of two counter-propagating plane waves. This approach provided an easy, but effective way for scientists to investigate the interference of SPPs without the need for complicated nano-structures in the near-field.
Figure 4.6 shows additional methods for the manipulation of the SPPs, via scanning the observation plane through the geometrical focus. As mentioned before, the cross-sectional points between every TM-polarized ray and dielectric/metal interface comprised a virtual annular ring referring to the initiation points of the SPPs. When metallic film was placed below (Z < 0) or over the focus (Z > 0), excited plasmonic waves propagated either toward the center or away from the virtual annular ring, yielding obvious or obscure individual interference patterns. One point of note was that the excitation position of SPPs shifted with additional extension or reduction, depending on the axial position of the focus, as shown in Fig. 4.6(a). This behavior was consistent with predictions from proposed phenomenological model. The radius of the virtual annular ring (the parameter of L in Eq. 4.3) was largely dependent on defocus.
As we split the TM-polarized sector into three sections and scanned the metallic film through the focus, several unusual patterns were observed. These had been created either by interference between counter-propagating plasmons at the center, or three independent propagating SPPs, as shown in Fig. 5(b). It is vital to note that the resulting interference patterns for such triple excitations was a group of 150-nm-radius (in magnitude of 1/e decay) bright spots in a 300-nm-period hexagonal arrangement caused by many in-phase SPPs. The
61
separation distance and spot radii followed the basic concept of interference in which the period and the size are close to half the effective wavelength of SPPs. The spatial distribution of subwavelength spots could be manipulated by varying the size of individual TM-polarized sectors and the angular distance between each sector. On the other hand, when the focus moved above the interface, three propagated SPPs were simultaneously launched. This yielded a field distribution in the shape of a shamrock. Such multiple excitations could be applied to future’s applications in planner optics.
Fig. 4. 6 Dual excited SPPs, generated by purposely designed SIP beams with the observation plane scanning through the focus. (a) and (b) shows the field distribution of SPPs under specific SIP illuminating with dual and triple TM-polarized sectors (with equal δ), respectively.