According to waveguide theories presented in Chapter 2, propagation modes supported by a waveguide depend on the wavelength of incident light and the geometry and dimensions of the waveguide. Therefore, prior to the study of the hybrid effect, the dependence of the transmission on the dimensions of a C‐shaped aperture is studied and the dimensions are optimized to obtain the maximum transmission for the incident wavelength of 633 nm.
3.2.1 Rectangular Apertures
The power throughput and spot sizes at 50nm away from a rectangular aperture as a function of the side length are calculated as a comparison base for following study. The simulation result in Fig. 3‐1 shows that the PT through a square aperture in a finite‐conducting metal film is approximated as a fourth power function of the side length. The behavior generally obeys the trend predicted by Bethe’s theory. However, the exact value can only be obtained by numerical methods because of the difference between real metal materials and perfect conductors. It also implies that no propagation mode exists in a square aperture at a wavelength of 633 nm.
Spot sizes of the square apertures along X and Y axis are listed in Table 3‐1.
The spot sizes in the X direction generally are larger and also have higher variations in the change of aperture sizes than that in the Y direction. This is because X‐polarized light is used to illuminate the aperture. Two edges of the aperture perpendicular to the polarization result in two local intensity maximums and thus broaden the spot size. The location of the two local
maximums is determined by the aperture edges so the spot size in the X direction is almost linearly proportional to the aperture size. In contrast, the spot size in the Y direction slightly decreases with the reduction of the aperture size. Since spot sizes are the dominate factor to determine the spatial resolution, we chose the 60‐nm aperture whose PT is 6.4x10‐5 as a comparison base.
Fig. 3‐2 The PT as a function of the side length of a square aperture
Table 3‐1 Spot sizes at 50 nm away from square apertures
3.2.2 Optimization of Ridge Part
The first step to optimize the C‐shaped aperture is to fix the aperture
dimensional variables a and b as 210 nm and 84 nm and analyze the dependence of transmission on the ridge length c and gap g. The simulation result in Fig. 3‐3 shows that gap g is a dominate parameter to determine the power throughput through the waveguide. The PT gradually increases with the decrease of the gap and reaches a peak of 1.8 when the gap is 38 nm. As the gap is less than 38 nm, the PT drops dramatically. The drop as the gap less than a specific value indicates a cut‐off for the incident wavelength regardless of ridge length c. This characteristic behaves like a waveguide. The cut‐off wavelength of a specific propagation mode is determined by the geometry and dimensions of the waveguide. When the incident wavelength is above the cut‐off wavelength, the corresponding propagation mode can not be supported within this waveguide.
Fig. 3‐3 The PT as a function of gap g with various ridge lengths c
In contrast to the gap, the ridge length c has a minor influence on the transmission. Instead, the ridge length alters the boundary conditions that determine the spot size along Y axis. Therefore, the spot size along Y axis increase with the increase of the ridge length.
3.2.3 Optimization of Aperture Part
Fig. 3‐4 The PT as a function of aspect ratio AR with various aperture lengths a as ridge width d is 46 nm
Based on the results in the previous section, the second step is to analyze the influence of aperture length a and aspect ratio AR, defined as the ratio of aperture width b to aperture length a, on the PT as ridge length c and ridge width d are kept 46 nm and 86 nm, respectively. The PT reaches a peak at aspect ratio of ~0.4 regardless of various aperture lengths, as shown in Fig. 3‐4. This cut‐off characteristic exhibits that a C‐shape aperture behaves like a single ridge waveguide. If the ridge area is relatively small compared to the aperture size, the ridge waveguide effect vanishes. The PT of the aperture will have similar tendency with that of a rectangular aperture. This explains a reduction in the maximum PT with an increase of the aperture size and a fixed ridge part. In contrast, if the aperture shrinks, the ratio of the ridge area to the aperture increases and consequently the gap is reduced. The PT of the aperture will behave like a nano slit so the PT has a sharp peak as the aperture area decreases.
Since the gap g, which is equivalent to (a x AR – d), is a dominant factor to determine propagation modes of a waveguide, we calculate the PT as a function of the aspect ratio with a different aperture width d. Our calculation, as illustrated in Fig. 3‐5, shows a similar result as that in Fig. 3‐3 but yields a higher peak value of PT. For a curve with a fixed aperture length, the change in the aspect ratio indicates the change in the ratio of the slit part and also the gap.
Therefore, for a specific ridge width, there is a corresponding gap that determines corresponding propagation modes and consequently a peak PT.
Fig. 3‐5 The PT as a function of aspect ratio AR with various aperture lengths a as ridge width d is 34 nm
In summary, according to our simulation results, the maximum PT exhibits an optimized design. The dimensions of the optimal C‐shaped aperture has an aperture length a of 210 nm, an aperture width b of 84 nm, a ridge length c of 86 nm, and a ridge width d of 34 nm. Since a C‐shaped aperture function like a waveguide, we will study the coupling effect between SPP modes and propagation modes of the aperture by employing this optimal design.