1. Introduction

The development of nanoscale fabrication techniques have led to tremendous interest in designing nanosized waveguides. An important objective of nanosized waveguides is to couple light with deep subwavelengths; this is crucial in imaging systems [1], nanolithography [2], optical modulation [3], and smaller light sources [4]. However, it is difficult for conventional dielectric waveguides to achieve an ultrasmall light-spot size beyond the diffraction limit of light [5]. Almeida et al. was the first to present slot waveguides that allowed light confinement in a 50-nm-wide region by using a simple dielectric structure [6]. Although the strong electric field was localized to the nanoscale region by using the slot waveguide, the confinement was only in one dimension. Subsequently, based on the slot waveguide, some hollow core dielectric waveguides were proposed that could confine light in two dimensions [7,8]; however, the hollow core structure was difficult to fabricate, and the intensity in the core was not so highlighted.

It is well known that plasmonic waveguides can be used for nanoscale light confinement with high power-coupling efficiency, which is helpful for nano-spot realization [9,10]. The best known plasmonic waveguide is the optical probe with a metallic cladding; this device is used in scanning near-field optical microscopy [11–13]. However, a large loss is inevitable for surface plasmon polariton waves confined to the boundary between a metal and an insulator [14]; this large loss results in limited throughput, low damage threshold, and low brightness [15,16]. These drawbacks are not seen in dielectric waveguides. It is also difficult to fabricate nanosized waveguides from different types of materials. The major concern of this study is to design a simple all-dielectric waveguide to create a nanosized light spot with high intensity, which contributes to easy fabrication, usability, and tolerance for light power.

Shuren Hu et al. proposed an intriguing waveguide structure for the extreme concentration of light in photonic crystal cavities [17]. He showed that the bowtie-shaped dielectric structure is effective in squeezing the electric field energy for the transverse electric (TE) mode. This implies the possibility of having thin optical fibers with a nanosized mode field.

In this paper, we firstly propose a fiber model based on the bowtie-shaped core arrangement that looks similar to the infinity symbol (∞), and we analyze the structural parameters to improve the power confinement and the peak power density. We define the contrast ratio of the effective peak power density to the background density to show the degree of peak sharpness and the background suppression. These two factors will be useful to design an optical probe tip that is used for nanolithography as opposed to the conventional nanosized electron beam. Next, we analyze the loss spectrum of the optical fiber to compare the loss values among the conventional silicon (Si) waveguides having similar sizes. These analyses were performed by using finite-difference time-domain (FDTD) simulations [18].

2. Waveguide structure designing

Shuren Hu et al. proposed the bowtie-shaped dielectric structure for photonic cavities that progressively squeeze light into a point-like space because of the slot and anti-slot effect [17]. In this paper, we apply this bowtie-shaped structure to a thin optical fiber to improve its performance and make it behave as a light guide having the diameter of a nanosized mode field. Our proposed structure with bowtie-shaped cores (similar to the infinity symbol “∞”) is shown in Fig. 1. A pair of cores of the high-refractive-index Si is almost tangential to each other with a small gap of low-refractive-index cladding SiO₂; there is air in the background. In this study, we mainly focus on the gap distance and the bowtie angles approximated by the hyperbola curves to minimize a light spot.

 

Fig. 1 (a) Illustration of the bowtie-shaped slot core structure. (b) Tip of bowtie angle with a hyperbola-shaped profile.

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The geometric parameters of the optical-fiber model, that is, the diameter of the bowtie core d_H, the diameter of the cladding d_L, and the length L were 220, 1000, and 2000 nm, respectively. The gap width d_gap = 2a between the two tips was fixed at 5 nm, and the angle of the bowtie shape θ corresponding to the angle between the two asymptotes was 90°. The width of the bowtie core $w_H=d_H(1+\frac{1}{\sin \frac{\theta }{2}})$ was 531 nm for the case of a = b. For the light wavelength equal to 1550 nm, the refractive indices n_H of Si and n_L of SiO₂ were 3.48 and 1.46, respectively.

We used the TE-mode continuous wave with a Gaussian profile as the input light source in z = 2 μm, and we monitored the light amplitude and the intensity distributions at the output end of z = 0, where the fundamental guided mode dominated. We used monochromatic light with the wavelength of 1550 nm. The power monitor area was 1 μm × 1 μm. Material absorption was also considered in our simulations.

3. Simulation results and discussion

Figure 2(a) shows the contour map of the normalized E-field energy density distribution in the cross section. An ultrasmall bright spot can be seen around the gap region. The 3D plot of the normalized E-field energy density is also shown in Fig. 2(b). There exists a sharp peak in the center region. This structure was achieved to efficiently confine light in the x and y directions in a nanosized area. To compare with the conventional slot waveguide, we also evaluated the field distribution in an optical fiber with rectangular-solid cores, as shown in Fig. 3(a). The geometric parameters of side core width of 180 nm × 300 nm and the slot core width (gap) of 50 nm are typical sizes for efficient light confinement in Si–SiO₂–Si waveguides. Figure 3(b) shows the normalized E-field energy density distribution at z = 0. We can see a spreading elliptically deformed light-spot resulting from the 1D light confinement along x axis. This indicates the limitation of rectangular slot structure. The maximum E-field energy density in our proposed optical fiber is approximately 11 times higher than that in the optical fiber with the conventional slot structure.

 

Fig. 2 (a) Contour map of E-field energy density. (b) 3D surface plot of E-field energy density.

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Fig. 3 (a) Illustration of the conventional slot structure. (b) 3D surface plot of E-field energy density normalized to the maximum E-field energy density in Fig. 2.

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Next, we investigated the dependencies of the structural parameters on the spot size and energy density. Figure 4 shows the E-field energy density distribution for the bowtie-shaped slot core structure along x and y axes at z = 0 of the 3D surface plot given in Fig. 2(b). For simplicity, the optical spot area in the shape of a slightly deformed ellipse was expressed as two principle axes of δx_FWHM × δy_FWHM, where δx_FWHM and δy_FWHM are the full width at half maximum of the energy density distribution along x and y axes. Here, we introduce a new parameter, the energy density contrast ratio C_P, which is defined as the ratio of the average E-field energy density in the spot area to the average energy density over the surrounding area as follows: $C_P\equiv \frac{\frac{{\displaystyle {\iint }_{spotarea}E-fieldenergydensityds}}{\delta x_{FWHM}\times \delta y_{FWHM}}}{\frac{{\displaystyle {\iint }_{surroundingarea}E-fieldenergydensityds}}{surroundingarea}}$. When we focus on a central area around the peak, C_P reflects the effective peak area and the suppression of surroundings rather than the typical power confinement ratio of the peak power to the total input power. This is because light power mostly spreads thinly over the much broader cladding area than the spot. A high C_P improves the available power ranges for an optical nanosized beam used in nanofabrication or nanoscale sensing systems.

 

Fig. 4 The profile of E-field energy density at z = 0 in Fig. 2(b) for the bowtie-shaped slot core structure.

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Figures 5(a) and 5(b) show the optical spot size and energy density contrast ratio obtained in proposed optical fibers as a function of the angle θ of the bowtie shape; the shape changes are shown in Fig. 5(c). The gap width d_gap of 5 nm, and the diameter of the bowtie core d_H of 220 nm was fixed. The least value of the spot size is approximately 8 nm × 8 nm for the θ values of 75°–90°. This implies that δx_FWHM becomes larger than δy_FWHM when θ < 90° and vice versa when θ > 90° because of the anti-slot effect and slot effect [17], respectively. For instance, the confinement effect along the y axis due to the anti-slot waveguide structure becomes dominant for a small θ. Figure 5(b) illustrates the moderate variation of C_P against θand we can see that C_P takes a maximum in the 90°–105° region, which is approximately six times higher than the C_P of 19.34 for the conventional slot waveguide shown in Fig. 3. It can also be easily estimated that light confinement declines beyond θ = 120° from a geometric similarity to the conventional slot waveguide structure shown in Fig. 3(a). The dependence of the optical loss α on the angle θ is shown in Fig. 5(d). The optical loss estimated from the effective refractive index [19] is not sensitive to θ, and it is less than 0.7 dB/cm in the range 60° < θ < 90°. Although this loss value looks a bit larger than that in conventional slot waveguides in [20], it is still crucially small as compared with the nanosized optical spot transmission in plasmonic waveguides. From the three aspects of the optical spot size, the energy density contrast ratio, and the optical loss, we can conclude that the bowtie core structure with θ approximately equal to 90° is the best choice.

 

Fig. 5 (a) Optical spot size. (b) Energy density contrast ratio. (c) The bowtie core changes with the variation of θ. (d) Optical loss for θ from 60$°$ to 120$°$ with a step of 15$°$.

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We investigated the dependence on the gap width d_gap for θ = 90° and d_H = 220 nm. As shown in Fig. 6(a), the optical spot size almost linearly increases with d_gap, whereas the C_P quickly declines with the increase of d_gap. The bowtie-shaped core squeezes light into a point-like space in both x and y directions; therefore, to decrease d_gap, the equivalent effect is to put another smaller bowtie-shaped core in the central area. This would further squeeze light into a tiny space. Figure 6(b) shows this iterative process in such nested structure. Considering the calculation time restriction, we used the minimum grid size of 1 nm, which is insufficient for the case of d_gap = 0 nm. Therefore, the spot size at d_gap = 0 nm is for reference. Actually, it seems unrealistic to fabricate an ultimate shape of d_gap = 0 and θ = 90°. In this study, we achieved a minimum optical spot of 5 nm × 5 nm with C_P = 167.41 at d_gap = 2 nm. To our knowledge, this is the smallest optical spot with a high energy contrast ratio in fully dielectric waveguides. Figure 6(c) shows the optical loss α versus d_gap. As shown in the angle θ dependency in Fig. 5(d), α is also not sensitive to d_gap and equals 0.58 dB/cm at d_gap = 2 nm.

 

Fig. 6 (a) Spot size and the energy density contrast ratio. (b) The schematic diagram of the tip profile changing with the gap width d_gap. The yellow circle roughly represents the optical spot; a darker color indicates a higher intensity. (c) Optical loss over different d_gap.

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Finally, the wavelength dependence on the optical spot size, the energy density contrast ratio C_P, and the optical loss α were estimated for θ = 90$°$ and d_gap = 5 nm, as shown in Fig. 7. From Fig. 7(a), we can see that the spot size is almost independent of the wavelength ranging from 1000 nm to 1700 nm. This reflects the slot waveguide property that the mode field diameter is mainly determined by the slot width rather than the wavelength, unlike conventional slab waveguides. In Fig. 7(b), large optical losses appeared in shorter wavelength region because of Si absorption. The energy density contrast ratio C_P remained larger than 100 at wavelengths ranging from 1200 nm to 1600 nm, but it tended to decrease in shorter wavelength region. This was mainly because of the increase in the optical loss. In other words, we can expect that a similar ultrasmall optical spot could be obtained in the visible or the shorter wavelength region also for other dielectric materials with low loss and high refractive indices, such as SiC.

 

Fig. 7 Wavelength dependence of (a) the spot size, (b) the energy density contrast ratio and the optical loss.

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4. Conclusions

In this study, we have presented a fiber model based on a dielectric bowtie-shaped core. This simple structure shows the possibility of generating a high-intensity nanosized optical spot by using all-dielectric waveguides with low propagation losses of less than 1 dB/cm at wavelengths ranging from 1200 nm to 1700 nm. A 5 nm × 5 nm optical spot was achieved with the energy density contrast ratio C_P = 167.41 and an optical loss of α = 0.58 dB/cm by using FDTD simulation. Because of the fibrous structure, such as polarization-maintaining fibers, this model will be easier to fabricate as compared with conventional nanosized waveguides. Our simulation calculations show that the proposed fiber has the potential to be applied in nanolithography with high-power lasers and tip devices of near-field microscopes.