Due to the sub-nm feature size of typical atomic or molecular structures, the sub-nm-confined optical field is fundamentally important for exploring light-matter interaction on the bottom and pushing the limit of optical technology ranging from super-resolution nanoscopy [1–3] and molecular spectroscopy [4–6] to atom/molecule manipulation [7–9]. Current available approaches to such fields mostly rely on plasmonic field localization, by which effective field confinement down to the 1- or sub-nm (i.e., ∼λ0/1000, where λ0 is the vacuum wavelength) level has been successfully realized [2,4,10–15]. However, due to the high optical loss and large wave vector of an extremely confined plasmonic mode, challenges such as thermal issues (e.g., thermal noise and damage caused by optical absorption [16,17]) and momentum mismatch between the confined and outside free-space fields [18] remain. Recently, sub-nm-confined optical fields generated in slit modes of coupled-nanowire pairs have been demonstrated [19,20]. Relying on coherently polarized bound electrons between two opposite hexagonal nanowire vertex edges, a point-like field with optical confinement down to sub-nm was obtained. While such fields have typically quasi-zero-dimensional cross-sections, an optical field having higher-dimensional cross-section is expected to offer higher flexibility and/or efficiency in some circumstances, as shown in the light sheet for far-field super-resolution optical microscopy [21,22].

In this work, we propose generating a nanoscale blade-like optical field with sub-nm thickness in a coupled nanofiber pair (CNP). Similar to that in the coupled nanowire pair [20], when two nanofibers are placed in parallel with a central slit down to the 1 nm level, a TE0-like nano-slit waveguiding mode with an extremely confined optical field can be obtained. However, unlike a hexagonal crystalline nanowire that typically has a 1-nm-scale diameter of the corner edge, a cylindrical glass nanofiber used here has a 100-nm-scale diameter, offering the slit an additional dimension (i.e., expanding the projection of the slit from a point-like to a line-like shape in the cross-section) for confining the central field to a blade-like profile in spatial distribution of the near-field intensity.

Previously, coupled modes in a CNP with a slit size down to 1 nm level have been studied under weak coupling conditions [23–26]. However, due to the strong mode coupling feature of a CNP in such cases (e.g., both the ratio of diameter to wavelength and the slit width are small), especially that a sub-nm-confined field can be obtained only in a slit waveguiding mode generated by strong mode coupling of the waveguiding modes of individual nanofibers, weak coupling approximation is not suitable in such cases.

Here, under the strong coupling condition, we investigate the slit waveguiding mode in a CNP and demonstrate a sub-nm thickness blade-like field around the slit region for the first time. The peak-to-background intensity ratio (PBR) of the confined field, fiber coupling scheme, modal field evolution, and waveguide dispersion are also studied.

The configuration of the CNP and the waveguiding nano-slit mode are schematically illustrated in Fig. 1(a). The CNP is formed by placing a pair of nanofibers closely in parallel. One end of the CNP having a flat end face is used as the output face, and the other end with one nanofiber connecting to a standard fiber (through a fiber taper) serves as the input port. The light input from the standard fiber is adiabatically transmitted through the fiber taper, coupled into the CNP waveguide, and finally converted into a TE0-like nano-silt mode as a result of strong mode coupling between fundamental modes of the two individual nanofibers (i.e., HE11 modes). Relying on the coupled oscillation of polarized bound electrons around both sides of the slit, the nano-slit mode can provide a blade-like optical field with a high PBR in the cross-section of the CNP [Fig. 1(b)].

Practically, benefitting from sub-nanometer roughness of a glass nanofiber surface [27,28] that is similar to that of melt formed glass surface [29,30], when two nanofibers are assembled in close contact in parallel, the CNP can naturally form a central slit ∼1nm in width. Meanwhile, material-rich nanofibers with high refractive indices and negligible absorption loss, such as As2S3 glass nanofiber from near- to mid-infrared bands and SiO2 glass nanofiber from ultraviolet to near-infrared bands, can be used to support the nano-slit mode in a wide spectral range. Also, it is worth mentioning that, in recent years, high-quality nanofibers have been reported for ultra-low-loss [31–33] and high-power optical waveguiding [34], making it possible to generate nanoscale blade-like optical fields with high peak power.

Generally, we use commercial software of Maxwell’s equations solver (Lumerical FDTD and COMSOL) to simulate mode evolutions in the CNP in three dimensions. Here, to obtain the spatial distribution of the nano-slit modes at the output end of the CNP with high precision, we mesh the area around the nano-slit with a minimum size of 0.01 nm. To avoid an intolerable amount of computation in the 3D simulation, we used a 2D-COMSOL simulation to calculate the field distribution in an As2S3 glass CNP with wavelength (λ)-dependent refractive index (n) of the As2S3 glass shown in Fig. 6 (Appendix A). As a more precise approximation, we assume a V-shaped linearly changing index profile across the slit (Fig. 7 in Appendix A), while providing typical results calculated with a step-index profile for reference (Figs. 7 and 8 in Appendix A).

For an individual nanofiber, the fundamental mode is an HE11 mode. When the two nanofibers are placed in parallel and close contact, strong mode coupling occurs. Since the diameter of the nanofibers discussed here is relatively small, we plot only the lowest four eigenmodes. The spatial distribution of normalized electric field vectors and surface polarized bound charge density of the four lowest modes in an As2S3 CNP with a nanofiber diameter (D) of 200 nm and a slit width (W) of 1 nm at 640 nm wavelength are shown in Figs. 2(a)–2(d). It is worth noting that the field intensity confinement and intensity within the slit in TE0-like mode are much higher than those in other modes (here are TM0-, TE1-, and TM1-like modes), which can be attributed to the high polarized charge density with opposite signs gathered at a separation of only 1 nm [Fig. 2(a2)]. For comparison, the separations between oppositely polarized charges are much larger in the other three modes.

Benefitting from the broadband transparency of As2S3 glass (from 570 nm to 5 μm, shown in Fig. 6), the nano-slit mode in an As2S3 glass CNP can be obtained within a broad spectral range. Figures 2(e)–2(g) show D-dependent effective refractive index (neff) of the four lowest modes in the CNP at visible (640 nm), near-infrared (1550 nm), and mid-infrared (4.5 μm) wavelength. Due to the different cutoff diameter of each mode, it is possible to support TE0-like mode only and eliminate all other modes by selecting a proper D. However, in experiment, the inaccuracy in diameter measurement [35] and the varying refractive index of the material under different waveguiding power and temperature [36,37] make it difficult to precisely determine the cutoff diameter, although choosing a larger D is easier for micromanipulation in experiment. Fortunately, the polarizations of the TE0- and TM0-like modes are almost orthogonal, making it possible to selectively launch the TE0-like mode by controlling the polarization of the input fiber mode. Therefore, in the following text, we consider the CNP that supports the lowest two modes only (i.e., TE0- and TM0-like modes) and selectively launch the TE0-like mode by selecting the polarization of the input mode.

Figures 3(a)–3(c) show the field intensity distribution of a typical TE0-like mode in an As2S3 glass CNP with D=300nm and W=1nm at 1550 nm wavelength, in which a nanoscale blade-like optical field is clearly seen. In the central slit, the dominant peak offers an extremely tight field confinement of 0.28 nm (x axis) and 38.3 nm (y axis) in the full width at half-maximum (FWHM), with a peak intensity much higher (∼24dB) than the average intensity of the whole mode field. Besides the central dominant peak, there are two side peaks in the background field due to the dielectric noncontinuity at the edge of the CNP. For reference, Fig. 3(d) shows the two side peaks (peak2), which are 18.5 dB lower in intensity compared with the central peak (peak1). It is worth mentioning that, unlike the plasmon mode that can wholly break the diffraction limit, the TE0-like mode here as a whole is diffraction limited with an effective mode area of about 0.5μm2 [∼0.34(λ/neff)2, neff=1.28]. Despite its ultra-strong field intensity, the central peak contains only a very small fraction of the total mode power, agreeing well with the low momentum mismatch between the confined field and the free space. As shown in Figs. 3(e) and 3(f), the central peak region of the TE0-like mode accounts for 0.036% of the effective mode area and confines 0.88% of the total power. In addition, we have also calculated the dependence of fractional power confined in the central peak on D and W [Fig. 3(g)], showing that the fraction of the power confined in the central peak increases with D and W.

To investigate the behavior of PBR that is critical for practical applications (e.g., a large PBR is desired to obtain a high signal-to-noise ratio), we calculated the dependence of the PBR and FWHM of the dominant peak on W and D/λ (normalized diameter) in an As2S3 glass CNP. Figure 4(a) gives W-dependent PBR with D=300nm at 1550 nm wavelength. It shows that, when W increases from 1 to 40 nm, the PBR decreases monotonously from 23.8 dB to 15.9 dB, while the FWHM increases monotonously from 0.28 nm (in x axis) and 38.3 nm (in y axis) to 39.4 nm (in x axis) and 119.9 nm (in y axis). Figures 4(b) and 4(c) show D/λ-dependent PBRs and FWHMs with typical D values for visible (D=120nm) and near-infrared (D=300nm) bands, respectively. The results show that, within the two calculated spectral ranges, when D/λ increases, the PBRs decrease slightly and the FWHMs in y axis decrease evidently, while that in the x axis keeps almost a constant (e.g., 0.28 nm with D=300nm). The large difference in FWHM and its D/λ dependence in the x and y axes offers an opportunity to generate a nanoscale blade-like field with different geometries. For example, with a similar D/λ of about 0.19 and W=1nm, a blade-like field launched by 1550 nm wavelength light with D=300nm has a y axis width (38.3 nm) twice that of the field launched by 640 nm wavelength light with D=120nm [Fig. 4(d)], while the x axis thicknesses at y=0 (i.e., the minimum thickness), 0.25 nm and 0.28 nm for 640- and 1550-nm-wavelength light are very close.

As illustrated in Fig. 1(a), to simplify the launching structure, we propose using a nanofiber with one side connected to a standard fiber via a fiber taper (i.e., the nanofiber is naturally tapered down from the standard fiber). To obtain a high coupling and converting efficiency, we propose a waveguiding launching scheme including an adiabatic mode transition and side coupling with a matched effective refractive index (neff). Figure 5(a) gives a typical example of such a launching structure for 1550-nm-wavelength light with D=300nm. The input mode from a single-mode fiber (i.e., LP01 mode) with horizontal polarization is first converted into the nanofiber mode (i.e., HE11 mode) via a fiber taper with a tapering angle of ∼3° (to ensure an adiabatical mode transition [38]) and then evanescently coupled into the CNP when it reaches the overlapping area. As neff of an individual nanofiber is evidently smaller than that of the CNP, to match the neff in the evanescent coupling process, the overlapping of the second nanofiber starts from the taper region of the launching fiber [a 2-μm-length taper is long enough for ensuring an intersection point between neff of the launching fiber taper and the CNP, as shown in Fig. 5(b)]. Owing to the small diameter of the nanofiber, the slight bending of the second nanofiber can be realized by either elastic [39] or plastic [27] bending.

The light coupled into the CNP will evolve into TE0-like nano-slit mode while propagating along the CNP. Our calculation shows that, after waveguiding through the propagation area (∼7.63μm in length) in CNP, the mode is output from the end of the CNP as a high-purity nano-slit mode. Owing to a tapering profile and thus a wide-range neff of the input fiber taper, the waveguiding scheme can be operated within a broad spectral range. For example, within about 800 nm bandwidth (from 1016 to 1870 nm wavelength), the mode purity of the nano-slit mode is higher than 90%, with a maximum of 98.3%. From 1.1 μm to 1.8 μm wavelength, the coupling efficiency of the excited nano-slit mode is higher than 90%, with a maximum efficiency approaching 98.1%, as shown in Fig. 5(c). The slight impurity may come from the forward scattering fields (due to the breakage of the symmetry of the waveguiding structure) and/or a very weak TM0-like mode (due to the non-strictly orthogonal polarization with TE-polarized mode) that may be excited during the mode coupling and evolution processes.

Also, we investigated the group velocity (Vg) and dispersion of the TE0-like mode around 1550 nm wavelength, as shown in Fig. 5(d). Due to the relatively small nanofiber diameter (i.e., D·n/λ<1), the increasing wavelength leads to the increasing fractional power in the air and thus increasing Vg with negative dispersion. Although the dispersion of the nano-slit mode is orders of magnitude larger than those of conventional waveguides (e.g., ∼0.6fs·nm−1·mm−1 at 1320 nm for single-mode fiber [40,41]), the short length of the CNP used for in-coupling and mode evolution (e.g., <10μm) makes ultrafast pulsed operation possible. For example, after waveguiding through a 10-μm-length As2S3 CNP (D=300nm, W=1nm), a 100 fs pulse with a central wavelength of 1550 nm and a bandwidth of 30 nm will be broadened ∼1.4fs in pulse width.

In conclusion, we propose a blade-like field with sub-nanometer thickness in a waveguiding CNP. Compared with other ultra-tightly confined optical fields, such a field significantly expands its width while maintaining an extreme optical confinement in the thickness direction (i.e., 0.28nm×38nm, with an aspect ratio ∼100). Using a fiber taper-assisted launching scheme, the waveguiding mode from a single-mode optical fiber can be coupled into the CNP and converted into a high-purity (up to 98.3%) TE0-like nano-slit mode with high efficiency (up to 98.1%), within a propagation length less than 10 μm at 1550 nm wavelength. In the same CNP, the TE0-like nano-slit mode can be operated within a broad spectral range with relatively low group velocity dispersion, making it possible for ultrafast pulsed operation. Moreover, by using nanofibers with other materials (e.g., silica with lower surface roughness and shorter operation wavelength), the nano-slit can be operated with tighter confinement (see Fig. 9 in Appendix B) and/or at shorter wavelength (e.g., ultraviolet spectral range). For reference, using a CNP consisting of two 50-nm-diameter silica nanofibers, it is possible to obtain an optical confinement down to 0.15 nm at 200 nm wavelength (see Fig. 9). In addition, as the CNP reported here is a fully dielectric structure, typical issues (e.g., fluorescence quenching [42,43], Ohmic heating [17], and electron tunneling [44]) that exist in plasmonic structures will be alleviated or avoided in this case. Considering its flexibility in field generation and manipulation, such a nanoscale “optical blade” may find applications in atomic-level light-matter interaction and ultra-high-resolution optical technologies ranging from optical spectroscopy and super-resolution optical nanoscopy to chemical bond manipulation.