Ridge-based Aperture

Various geometric apertures are applied to maintain the strong transmission and small spot size at the same time. I and C shaped apertures are reformed from the slit, as presented in Fig. 3-10. The corresponding cutoff wavelengths of the fundamental propagation mode are 420nm, 577nm, and 600nm for slit, C-, and I- shaped aperture, respectively [26]. No propagation mode could be sustained in our discussion having 633nm incident wavelength.

y

x

a a a

b b

d

d d

s s

Fig. 3-10 The structures of (a) slit-, (b) C-, and (c) I-aperture, the corresponding lengths are a=210 nm, b=84 nm, d=38 nm, and s=86 nm.

When the Ex polarization illuminates the slit aperture in 200nm thick silver film, the entrance Ex-field distribution is depicted as Fig. 3-11(a). The accumulated charges along the long axis because of the discontinuity of the incident polarization result in the narrow Ex filed profile at horizontal cross section at y=0 and the other broad one at x=0, as shown in Fig. 3-11(b).

Fig. 3-11 (a)Ex distribution and (b) Ex cross section profile at the entrance of the slit aperture (38nm×210nm) in 200nm thick Ag film.

Other improved design, such as C- and I- shaped aperture, maintain the ridge part and add two extra arms to confine the charges distribution, such as electrical dipole concentrated around the central part of the aperture .The cross section profiles in Fig.

3-12 show the centralized diploes perform stronger filed intensities almost two times than uniform diploes in the slit.

Fig. 3-12 The Ex-filed distribution (a)(c)and Ex cross section profile (b)(d) at the entrance of the C- and I-shaped aperture in 200nm thick Ag film.

Even though the ridge-based aperture excites more intense LSP field intensity at specific cross section, the geometry restricts the LSP distributions over all the open aperture area. The cross section profile in Fig. 3-13 expresses that the two extra arms of the ridge-based aperture reduce the full width of half maximum (FWHM) narrower than the slit in the y direction. It explains that the slit excites the LSP most efficiently along its edges. The slit owns better photon capture ability than the ridge-based aperture.

Fig. 3-13 The Ex-field cross section profile at the x=0 for the slit, C- and I- shaped aperture

After the discussion on the LSP excitation at the entrance plane, the energy transportation issue will be presented next. However, the Bethe’s rule is limited in the

infinite thin film. After passing through the thick film, the beam profile at the output is not the same as the entrance filed distribution. Furthermore, the fundamental propagating mode disappears at the sub-wavelength scale on the waveguide theorem.

Fortunately, Prof. M. Mansuripur [27] supports the “radiation pressure” concept to solve this energy transportation problem. When the Ex polarized plane-wave illuminates the narrow gap of the width δ << between two dielectric hosts with λ_o refractive index no as Fig. 3-14. The field amplitudes inside the medium are (Ex,Hy) = (Eo,Ho) = (Eo,noEo/Zo).And the electromagnetic field inside the gap is the superposition of two evanescent plane-waves, each of which must satisfy the constrain k• k=k02, k E=0, and (k/k• o)E=ZoH imposed by Maxwell’s equations.

Fig. 3-14 A linearly polarized plane-wave propagates along the z-axis in a dielectric host of refractive index no. A narrow gap of width δ << is assumed to exist in this λ_o

medium; the plane of the gap is yz in (a) and xz in (b).

The fields of these evanescent waves must satisfy the boundary conditions of the both walls of the gap. In case (a), the continuity is required of the perpendicular D-field,D_x =ε_on_o²E_o, and the tangential H-field,H_y =H_o.The k-vector and field amplitudes are in the following forms.:

z The momentum density p in the gap is derived from the Poynting vector component Sz along the propagation direction, namely.

2

The general expressions for dielectric material are not the same as our metal structure. The metal will screen the fields’ penetration and no fields will appear inside the metal. However, we adapt the Drude model to take the surface charges within skin depth into accounts. The accumulated charges resulting from the incident polarization which is perpendicular to the slit can still produce stronger radiation pressure at the entrance of the thick metal film, as shown in Fig. 3-15. The phenomenon matches the theoretical analysis on the dielectric gap.

Fig. 3-15 The energy transportation inside the slit for (a) TM and (b) TE incidence

When the accumulated charges radiate the energy through the aperture, the aperture depth influences the energy transportation. The power throughput oscillated with the aperture depth as Fig. 3-16 (a). For the slit with x-polarized incidence, the oscillating period is almost half the incident wavelength. And the C- and I- shaped aperture perform shorter oscillating period physically. The enhancement inside the sub-wavelength aperture results from the Fabry Perot-like resonance effect, as depicted in Fig. 3-16 (c).

Furthermore, PT decays sharper in the slit than C- and I-aperture under the cut-off condition with y-polarized incidence, such as in Fig. 3-16 (b). Because the two extra arms of the C- and I- shaped apertures excites LSP under y-polarized illumination, which wouldn’t occur for the slit aperture. However, the y-polarization which induced weak LSP at entrance and faint driving radiation pressure inside the aperture causes PT for these structures all less unity.

Fig. 3-16 Power throughput comparison of slit-, C- and I- aperture versus the depth of aperture under (a) x- and (b)y-polarized illumination.(c) illustrates the Fabry Perot-like

resonance for X-polarized excitation corresponding to 250nm and 600nm film thickness.

We discuss the spot size by measuring the FWHM of Poynting vector at 50nm away from the aperture. For the C-shaped aperture, it is not right-left symmetrical at the line x=0 as the slit and I-shaped aperture. The two extra arms shift the energy distribution of the C-shaped aperture away from the center at horizontal cross section, in Fig. 3-17 (a). It is also the anti-symmetry avoids LSP to locate at the ends of the long axis and keeps the concentrated output performance in the vertical cross section in Fig. 3-17 (b).

Fig. 3-17 The spot size x-(at y=0) and y-(at x=0) cross section Poynting vector profile for the slit (black solid), C-(red dash), and I-shaped (blue dash) aperture at 50nm away

from the exit plane.

Generally speaking, the simple geometry of the slit captures the photon most efficiently. And the wide-distributed LSP for the slit raises the enhancement by sacrificing the energy confinement (spot size). The improved ridge-based aperture re-distributes the charges accumulation. It confines the stronger LSP excitation into the smaller area. The reformed apertures compress the light into smaller spot size by sacrificing the photon capture ability by the two extra arms at the entrance surface.

Although the C- and I-shaped apertures both own the better performance in PTD than the slit in Tab. 3-1, the asymmetric (along x axis) C-shaped aperture produces more concentrated spot size.

Aperture Area (λ²) PT (a.u.) Spot size (λ²) PTD (μm^-2)

Slit 0.020 1.30 0.049 66.1

C 0.032 1.27 0.038 89.5

I 0.032 1.27 0.038 89.5

Tab. 3-1 Output performance comparison between the slit, C-and I-shaped aperture. 

The “area” denotes the size of opening at entrance plane.