Improving field enhancement of 2D hollow tapered waveguides via dielectric microcylinder coupling

We numerically study a novel scheme to improve the field enhancement of 2D hollow tapered waveguides (HTWs). A dielectric microcylinder is embedded into a metal–insulator–metal (MIM) HTW for resonant exciting gap surface plasmons (GSPs), which is different from the lowest propagating mode (TM0) excitation via the conventional fire-end coupling method. The physical mechanism of the field enhancement and the influence of critical parameters such as numerical aperture (NA) of the lens, permittivity of the microcylinder and the incident wavelength are discussed. The substantial improvement of the GSP excitation efficiency via dielectric microcylinder coupling shows potential in designing tapered MIM waveguides for nanofocusing and field enhancement.


Introduction
Dielectric microcylinder/-spheres, with the sizes of several light wavelengths, are attracting increasing interest because they can support high-quality whispering gallery modes (WGMs) [1] and provide nonresonant focusing of light into tiny spots termed photonic nanojets (PNJs) [2]. The WGM can be represented as a resonant electromagnetic wave recirculating in phase around the circumference of a dielectric microcylinder/-sphere due to the total internal reflection of the cylindrical/spherical surface. It is resonantly stimulated by evanescent coupling via a light waveguide and has become a powerful transduction mechanism for biosensing [3], solar cells [4], microcavity laser [5] and optical communications [6]. The PNJ is a non-evanescent propagating beam with subwavelength transverse dimensions, low divergence and high intensity, which makes it fruitful for applications in laser cleaning [7], nanopatterning [8], Raman spectroscopy [9], super-resolution imaging [10] and optical data storage [11]. Recent experiments [12] and simulation [13] show that the focal spot of the microcylinder/-sphere can also be near-field focal spot besides PNJ. The structural parameters of the microcylinder/-spheres and its adjacent object will modify the optical field distribution inside and outside of the microcylinder/-spheres, which is promising for the coupling from micro optical propagating to nanoscale focusing.
On the other hand, nanofocusing, i.e. effective delivering and concentrating the optical energy into nanoscale, has become an important issue in modern nanophotonics and nano-opitcs. Converting light into surface plasmons polaritons (SPPs) is deemed as one of the most promising approaches toward fulfilling this goal. Plasmonic nanofocusing is typically achieved using tapered metallic waveguides which can be divided into two categories: insulator-metal-insulator (IMI) structures such as tapered metal rods [14], nanopyramids [15], metal film tapers [16] and nanowedges [17] for slow surface plasmon (SSP) nanofocusing, and MIM structures such as tapered field lines [18], tapered gaps [19], tapered V-grooves [20] and nanocampanile (or 3D linear taper) [21,22] for gap surface plasmon (GSP) nanofocusing. Comparing to tapered IMI structures, tapered MIM structures can generate nanoscale confined light spots immune to the external background with higher field enhancement (FE) [23], which are qualified as aperture near-field scanning optical microscopy (NSOM) probes.
Efficient coupling of the electromagnetic radiation into tapered plasmonic waveguides is critical for realizing plasmonic nanofocusing. SPP excitation in an input entrance of larger scale and concentrating into a nanoscale region results in an extremely high FE in theory. However, for the tapered MIM structures, GSPs are generally excited by the TM 0 mode using the fire-end coupling method through bulk light or guided modes coupling. To reduce the reflection and scattering losses, the taper angles and the sizes of the input entrance of tapered MIM structures are generally smaller than the critical taper angle (usually less than 10 degrees to meet the adiabatic conditions) and the wavelength of the input beam, respectively [24,25], which is an obstacle for the design and application of tapered plasmonic waveguides.
In this paper, to the best of our knowledge, we report for the first time that FE in a 2D HTW of large taper angle and input entrance size can be remarkably improved via microcylinder coupling. A standard tapered MIM structure is chosen to study the physical phenomenon and the excitation mechanism of the GSPs. The influences of critical parameters such as the NA of the focusing lens (FL), the permittivity of the microcylinder and the incident wavelength are discussed.
The schematic diagram of a 2D HTW is illustrated in figure 1. The structure of the 2D HTW is assumed to be uniform and infinite in the y direction. Aluminum, which has smaller skin depth than noble metal and is commonly used in NSOM, is chosen as the metallic material for preventing optical leakage. Based on the 3D near-field probe used in our previous experiment [26], the metal thickness, entrance width D i , aperture width D o and taper angle of the HTW are 100 nm, 3 µm, 5 nm and 60°, respectively. The incident light is a linearly TM polarized Gaussian beam at λ = 500 nm with a 9 mm full width at half maximum (FWHM), which is further restricted by an aperture diaphragm (AD) of 9 mm diameter. The NA of the FL is 0.55 and the focal spot is assumed to be exactly located at the entrance of the 2D HTW. The dielectric microcylinder is tangent with the inner walls and its radius is chosen as 530 nm after optimization calculation.
The optical evolutions in the 2D HTWs are simulated by our self-developed finite difference time domain (FDTD) numerical model as in our previous work [27]. The grid cell is set as ∆x = ∆z = λ/500 = 1 nm for accurately describing the nanoscale waveguide structures. An anisotropic perfectly matched layer absorbing boundary condition is adopted for the truncation of FDTD lattices. The Lorentz-Drude model presented by Racik et al [28] is adopted to describe the permittivity of aluminum related to incident wavelength. The permittivities of the microcylinder, air and aluminum walls are taken as ε 1 = 2.2 (commonly used glass regardless of the dispersion), ε 2 = 1 and ε Al = −34.2 + 9.0i, respectively. Unless otherwise specified, the following calculations are based on the above parameter values. Figure 2 shows the time-averaged electric field intensity distributions in the 2D HTW. In absence of a microcylinder, most optical energy is reflected and absorbed by the metal wall due to the nonadiabatic conditions before evolving from TM 1 mode into TM 0 mode at the tip region, as shown in figure 2(a). The stimulated TM 0 mode is then reflected back along the internal walls or scattered back into the optical field. As a result, GSP excitation approaching the apex is not obvious and a FE of about 346 is obtained (the FE is defined as the ratio of the peak field intensity at the aperture to the average field intensity at the entrance). This GSP excitation, as a matter of fact, belongs to the usual fire-end coupling method.
With the microcylinder, the situation is quite different. Pure SPPs are primitively excited by the evanescent fields existing in the microcylinder-wall gaps close to the microcylinder-wall contact points (gap width between the two walls is too large to support the GSP modes). The excited SPPs propagate forward along the walls and are coupled into GSPs gradually in the tip region when the gap width of the two walls is small enough to support the GSP modes. Most of the GSPs are reflected back by the aperture edge and convert back into SPPs along the metal walls. The superposition of the forward and backward SPP (GSP) waves constructs strong standing waves, which results in a FE of about 1834 at the 5 nm aperture (compared with the situation without microcylinder, the light field coupling efficiency is improved by 5.3 times). In this case, the GSP excitation mainly comes from the near-field coupling of the microcylinder-wall gaps, which is completely different from the fire-end coupling method without microcylinder. The reflection coefficients for To view insight into the physical phenomenon of the GSP resonant excitation, the relationship between the FE and the microcylinder radius is investigated for various NAs of the FL, permittivities of the microcylinder and incident wavelengths, respectively.
Firstly, in order to excite a pure SPP on an inner metal wall, the component of the wave vector of the incident light parallel to the metal surface must equal the wavenumber of the pure SPP (1) The evanescent fields existing in the microcylinder-wall gaps close to the contact points present abundant wave vectors larger than the wave vector of the incident wave, so the wave vector of the SPPs readily matches with the wave vector of the evanescent fields by itself and pure SPPs are primitively excited on the inner metal walls close to the contact points. The contact points and the corners of the aperture edges compose a Fabry-Perot kind of resonator, so the SPPs (GSPs) are resonantly excited and magnified by the multiple reflections of the counter-propagating SPPs (GSPs). For resonant excitation of the SPPs (GSPs), the distance L between the contact points and the corners along the walls must comply with the standing wave condition where β SPP(GSP) is the propagation constant of the SPPs (GSPs). It is a constant when the gap width is large enough (SPPs) and increases with the decrease of the gap width in the tip region (GSPs), l is the spacing between the contact points and the corners along the wall and φ R is the total reflection phase. Figure 3 shows the FE at the 5 nm aperture as a function of the microcylinder radius for FL with various NAs (the other parameters are the same as those stated in figure 1). The microcylinder radius ranging from 150 to 850 nm can be readily achieved in the real-world process [29,30]. For NA = 0.15, the GSP excitation and FE is unconspicuous. For NA ≥ 0.35, the FE of the 2D HTW can be improved as the microcylinder radius is large than 180 nm. In addition, the FE curves fluctuate upon the microcylinder radius ranging from 150 to 850 nm and resonance peaks become more and more highlighted with the increase of the NA values. The maximum peak values appear when the microcylinder radii are 530 and 630 nm with small and large NA respectively. This mainly depends on two factors: the standing-wave conditions and optical energy (as much as possible) contained by the near field in the microcylinder-wall gap. It is found that larger NA is beneficial to the FE and a FE of about 3702 is obtained as the microcylinder radius is 630 nm for NA = 0.95. On the one hand, more optical energy can be coupled into the HTW with the FL of a larger NA. On the other hand, a larger NA results in a smaller focal spot and the higher divergence of the focused optical field can transfer more optical energy into the microcylinder-wall gaps. Figure 4 shows the FE at the 5 nm aperture as a function of the microcylinder radius for various permittivities of the microcylinder (the other parameters are the same as those  stated in figure 1). By increasing permittivity from ε 1 = 2.0 to ε 1 = 2.6, the whole FE curve moves up and the microcylinder radii corresponding to the resonant peaks decrease slightly. This illustrates that a microcylinder with larger permittivity can transfer more optical energy into the near field in the microcylinder-wall gaps and results in a slightly larger GSP propagation constant in the microcylinder-wall gaps. Figure 4 also shows that when ε 1 further increases from 2.6 to 2.8, the maximum resonant peak of the FE curve decreases abnormally. It indicates that the permittivity cannot be too large; otherwise, the optical reflection on the incident cylindrical surface is large enough to decrease the GSP excitation efficiency in return. Figure 5 shows the FE at the 5 nm aperture as a function of the microcylinder radius for various incident wavelengths (the other parameters are the same as those stated in figure 1). It indicates that the shorter the incident wavelength is, the larger the FE will be. With the decrease of the incident wavelength, the resonant peaks become clearer and more highlighted and move toward the left entirety. On the one side, a shorter-wavelength light field is more conducive for focusing via coupling lens. On the other side, a shorter-wavelength light field excitation results in a relatively larger GSP propagation constant which is advantageous to the stronger field localization of the GSPs and the locations of the nodes and antinodes move left accordingly.
In summary, we find that the field intensity of a 2D HTW in nonadiabatic conditions can be greatly enhanced via embedding a dielectric microcylinder. The efficient excitation and compression of GSPs in the 2D HTW result in the high FE, which is different from the TM 0 mode excitation using the common fire-end coupling method. The results are envisioned to be important for the design of tapered MIM waveguides of large taper angle and micrometer-size input entrance for nanofocusing and FE.