1. Introduction

Limited by optical diffraction, freely propagating waves are hardly to be focused down to dimensions much smaller than about λ/2. This constrains many practical applications including microscopy, lithography, and sensing. However, light can be confined well below this limit in the immediate proximity of sub-wavelength structures. The first investigation of sub-wavelength structure is a circular type, which is used to improve the resolution of Near-Field Scanning Optical Microscopy (NSOM). However; transmission efficiencies of circular apertures suffer very low transmission efficiencies in agreement with predictions from Bethe which limits their use for practical applications [1–5]. Fortunately, not all sub-wavelength apertures are strictly limited by Bethe theory. For instance, bowtie apertures are capable of concentrating light to extreme low modal volume with transmission efficiency much higher than the regular shaped aperture commonly used [6–14]. These mainly own to their unique geometry design: open arms and the nanoscale gap. When illuminated by an incident light with proper polarization these two open arms provide longer cutoff wavelengths and the constrained light spot is related to the size of the gap. Comparing with other shaped apertures, the transmitted fields through the bowtie aperture is well f [15-20]. Plasmonic effects have previously been observed in bowtie apertures/waveguides in several studies [21-23]. However, hybridized plasmonic modes have not yet been pay much attention. In this letter, hybridized plasmonic modes and Fabry-Pérot effect of the nanoscale bowtie aperture waveguide is studied using modal analysis.

The outline of the bowtie aperture is constructed by width a, length b and two conducting ridges, as shown in Fig. 1. These two ridges both point to the center of bowtie aperture and a small gap between two ridges is characterized by a distance g. The four exterior corners are determined by the radius f and the tips of two ridges are rounded by a radius r. Thus, the simulation model is more close to the realistic apertures fabricated by focused ion beam (FIB), where the Gaussian profile ion beam will result in rounded ones. In addition, rounded corners can avoid infinitesimally sharp features, which may bring unwarranted field concentrations in the numerical calculation. The modal characteristics of bowtie aperture are studied by recognizing it as a typical optical waveguide. Starting with calculating the cutoff wavelengths of bowtie apertures with different sizes and gap sizes (g), the dispersion relation for transverse electric modes (TE mode) in perfect electrically conductor (PEC) and real metal are determined. Combined with dispersion model, we will find the spectral response of bowtie aperture waveguide can be explained and categorized into plasmonic and Fabry-Pérot resonance.

 

Fig. 1 Bowtie waveguide geometry.

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2. Mode analysis in PEC

Mode analysis and dispersion relations are carried out by using HFSS, which is a commercial frequency-based finite element method (FEM). The aperture is studied as Eigen-mode, freely supported PEC in air. Figure 2 shows the field distribution of five low-order modes inside a bowtie aperture design by a = b = 200 nm, f = 10 nm, r = 25 nm and t = 100nm. Note that all modes are happening to be transverse electric; the subscripts represent its relationship to the rectangular waveguide and do not come from any subjective solutions. TE₁₀ mode is always representing the fundamental mode. The TE₀₁ and TE₀₁* mode are symmetrically distributed on each side of the waveguide, this gap just separates the TE₀₁ mode into two modes. The electric fields of these two modes are symmetrically distributed and have almost the same cutoff wavelength. More detail can be available from the performance of electric field vector; this allows us to better understand the electromagnetic distribution.

 

Fig. 2 Lowest five modes of a bowtie waveguide modified by electric field vector .Bowtie waveguide in PEC defined by a = b = 200nm, f = 10 nm, and r = 25 nm. While λ_c represents cutoff wavelength.

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How well the field distribution of the mode couples to the incident wave determines whether a particular mode will be excited. Generally, whether a given mode can be excited depends on the following equation:

Although this integral term given by Eq. (2) can't be calculated to an exact value, we can still analyze it just by simple symmetry. For example, the orientation of field distribution for TE₁₀ and TE₃₀ modes is odd symmetry to the x-axis, so these modes cannot be excited if an x-polarized light is used (as $C(\theta )$ = 0). But the y-polarized wave can excite these modes because they are even symmetry to the y-axis ($C(\theta )$≠0).

From Eq. (2) and Fig. 2, neither the TE₀₁* or TE₂₀ modes can be excited by a normally incident light. And the TE₁₀ and TE₃₀ modes will couple to y-polarized light but not x-polarized light, while the TE₀₁ mode will only couple to x-polarized light and will not be excited by a plane wave polarized in the y-direction. So the useful modes of the waveguide depend on whether the light is polarized. Multi-mode propagation will occur if the light is not polarized or if more than one mode is not cutoff.

As the frequency increases, the dispersion curve for the waveguide crosses “the light line” (“The light line” is simply the dispersion line for light in free space). The waveguide changes from traditional photonic to surface plasmonic. Then the curves will intersect within a range at high frequency and these transverse modes become hybridized with surface plasmon polaritons (SPPs), wedge plasmon polaritons (WPPs) and channel plasmon polaritions (CPPs) (shown in Fig. 3).

 

Fig. 3 Dispersion curves for bowtie waveguide defined in PEC.

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It is unnecessary to talk about TE₀₁* and TE₂₀ modes because they are both hard to be excited by a linearly polarized light. Modes TE₁₀, the fundamental mode, act more similar to the CPPs propagating along the waveguide In this case a groove in the aperture can guide channel plasmon polaritons (CPPs) and the electric fields under the groove come from the hybridization of TE₁₀ mode with CPPs which propagates through the groove in the corners to the exit. And for modes TE₀₁, the WPPs waveguides (usually combined by two wedges close to each other and running along the edges of the structure) suit well. But if we turn to mode TE₃₀, we will find surface plasmon Polaritons (SPPs) play a significant role. The SPPs propagate along the surface. After that it couples to the waveguide by its four corners.

3. Analysis in real material as a Fabry-Perot cavity

While the apertures are defined in real metals, it can be seen that the situation is much more complex. Degiron and Smith’s iterative approach is used to identify the real and imaginary parts of the propagation constant as well as the field distribution. Also, this case varies from metal to metal. The difference is that the cutoff wavelength is slightly longer for a given bowtie aperture due to field penetration into the metal. The dominant mode is the one that has the lowest cut-off frequency, which is to say, TE₁₀ mode in the bowtie waveguide. TE₁₀ is widely used for it concentrates almost all the energy of the electromagnetic field in the waveguide.

When we considered the aperture as a Fabry-Perot cavity, multiple reflections from within the cavity will separate each mode to a series of standing waves along the axial direction of the cavity (longitudinal direction). These generate different longitudinal modes TE_10-1, TE_10-2, TE_10-3 and so on (all based on the TE₁₀ aperture mode) and the number of nodes of the standing wave are determined by the thickness, as Fig. 4 shown (the transmission efficiency represents relative intensity to the incident wave. Arbitrary Unit (A.U.) is used). The first FP modes resonant wavelength is independent of the thickness of the film, always occurring near the cutoff wavelength for the waveguide similar to the PEC case. The higher FP modes can be seen in thicker film and occur at longer wavelengths than they do in the PEC system because the propagation constant in the waveguide is much higher than that in the free space.

 

Fig. 4 Transmission spectra for a bowtie aperture defined in an Ag film Eigen-mode analysis resolving first four Fabry-Pérot modes of TE₁₀.

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Figure 5 shows transmission through Au and Al films. Interestingly these two materials get different characteristics at their own transmission performance. The Au results got higher order FP modes and the higher order FP modes disappear due to a higher attenuation coefficient; similar situations also appear in Ag. Al behaves more like PEC at optical frequencies. The Al film has higher transmission through the TE_10-2 mode than TE_10-1 mode, primarily due to the inter-band absorption at this wavelength (with a skin depth about 830nm). This absorption shift down the transmission curve around a specific wavelength, affects Al by mode TE_10-1 and Au by higher order FP modes. These interesting phenomenon guides us to choose different materials for different exposure conditions.

 

Fig. 5 Transmission spectra for a 200 nm bowtie waveguide defined in free standing 200 nm thick Au /Al films supported in the air.

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4. Verification experiment

In order to discuss the characteristics between measured transmission spectra and simulated results, experiments were carried out to verify our numerical analysis. Silver was chosen for its low dissipation at visible frequencies. Then e-beam evaporation is used to deposit 200nm thick silver on a fused quartz wafer. After that, Focused Ion Beam (FIB, FEI Helios) is used to mill the bowtie apertures with a gap size less than 20nm under the condition of 30kV, 7.7 pA in this silver film.

As shown in Fig. 6, µPAX-2 Pulsed Xenon Light Source is used for our system. These Xenon lamps generate light over a continuous spectrum from UV to infrared (220-2000nm) and have stability less than 1% CV. Corresponding to ultraviolet part, low-loss quartz optical fibers with SMA905 connectors guide the light into the system. Then the collimated light is split into two beams. The transmitted light will be focused on the mask to spread through the Nano-apertures. By collecting and measuring this part of energy, comparing the spectrum with the reference light before, we can get the transmission efficiency of the bowtie aperture. Considering the optical loss and the interference of stray light, measurement at nanoscale will not be easy. But the following relationship is still trustworthy:

 

Fig. 6 Measured system for bowtie apertures (a = b = 200 nm) formed on a 200 nm thick silver film, with a gap about 15 nm.

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Comparing Fig. 7 with Fig. 4, three FP modes of the TE₁₀ can be found: TE_10-1, TE_10-2 and TE_10-3, and higher-order modes just disappear completely because they are too weak. The resonance wavelength of each mode is 806nm, 578nm, and 408nm respectively. For simulation results, they are 853nm, 621nm, and 444nm. So there seems to be an error around 6%-8%. The corresponding error is caused by nano-fabrication error during the fabrication process. Experimental results further verified this phenomenon qualitatively and qualitatively to a certain extent.

 

Fig. 7 Measured transmission spectra for bowtie apertures with a = b = 200 nm g = 15 nm formed on 200 nm thick silver film.

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5. Conclusion

In summary, nanoscale bowtie aperture waveguide changes from traditional photonic to surface plasmonic and almost all these allowed transverse modes become hybridized with SPPs, WPPs, and CPPs. Fabry-Perot effect also plays a significant role. Dramatic enhancements are possible by optimizing the design to be resonant at the desired wavelength. This work could be useful in designing optimum structure to obtain sub-wavelength spots for the application in nanolithography, data storage, and many other areas where high optical resolution is required.