The proposed technique exploits the microscale light-thermal effect and macroscopic deformation-induced pulling force to taper down a fibre, which establishes a new method for MNF fabrication. To gain insight about the mechanism underlying the tapering process, next we trace back to the origins of “heat and pull” from a theoretical viewpoint.
Heat.– First, we calculated the light absorption of the plate–fibre system (the simulation model replicated the experimental geometry shown in Fig. 1b). The absorption spectrum displayed in Fig. S3A indicates the efficiency of heat generation inside the plate–fibre system in response to the supercontinuum light source used in the experiment. The gold plate and fibre in contact jointly function as a hybrid metal-dielectric waveguide. Given that the excitation of SPPs requires light fields carrying the electric component perpendicular to the metal-dielectric interface, the gold plate selectively absorbs more evanescent waves when the light field is guided in $ HE_{11}^{vertical} $ mode (see the modal profile in the inset of Fig. S3A)40-42. The total heat conversion efficiency depends on both the propagation loss of the hybrid waveguide and coupling efficiency between the fibre mode and SPP mode; it was calculated to be approximately between 2% (off-resonance) and 10% (on-resonance) for the $ HE_{11}^{vertical} $ mode. The total heat generation efficiency in both situations (on- and off-resonance) did not differ significantly owing to the finite fibre-plate contact length and propagation attenuation (the comparison is shown in Fig. S3D, E).
Fig. 3a displays the absorbed power profile inside the gold plate at the resonant wavelength ($ HE_{11}^{vertical} $ mode, 775 nm). As is one of the main characteristics of SPPs, the electromagnetic fields feature subwavelength-scale localisation around the contact line between the plate and fibre, with the light intensity quickly attenuating along the width and thickness directions, which suggests the significance of the contact length relative to other geometric parameters. As a result of the guide-mode interference, the specific E-field distribution would change as a function of the wavelength and geometric configuration (Fig. S3D-G). Those effects, however, would be smeared out by the heat transfer through the plate into the fibre.
Next, to elucidate the thermal response of the plate-fibre system to light absorption, heat transfer simulations were first performed in the static regime for the following reasons. The gold plate, referred to as the microsized heater in the context, generates heat via plasmonic effects and pumps it into the system, causing a considerable temperature rise. To thermally initiate the tapering process, groups of pulses should collectively and constructively contribute to accumulate heat, for which the very prerequisite is to set a high repetition rate of the light source (~100 kHz in our case). Once heat accumulation is assured, at an early stage, pulse-wise temperature rise is expected, as will be discussed later. However, the system will finally evolve into a thermally semi-stable state, where the system temperature is essentially invariant, with only slight periodic perturbations caused by the input light pulses and subsequent cooling. The latter stage can be approximated by static-state simulations, especially given that in the experiments, the time scale of the mechanical tapering process was measured in seconds, which is adequate for the balance to be reached in the fibre-plate system between the nanopulsed heat influx and outflux (see Fig. S4D for comparison. The response time with a 1-Hz modulated CW light source is within one second).
To examine the final profile of heat accumulation, we considered that a continuous 8-mW input power was injected into the system (the same simulation model used for electromagnetic calculation, based on Fig. 1b), corresponding to the average power of the pulsed light source. With the spatial distribution of the heat source inherited from that in Fig. 3a, the calculated temperature profile is displayed in Fig. 3b. Owing to the averaging effect of heat transfer over time, the resultant temperature profile is not traceable to the original E-field distribution, as shown in Fig. 3a. Instead, it is determined by the dynamic balance between heat influx and outflux, as demonstrated above. For simplicity and qualitative analysis, we chose to disregard the relaxation behaviour of the glass transition process. For the same reason, the phase change or laser ablation was not included in the heat transfer model. Specifically, a definite threshold temperature was set at 1473 K, beyond which the fibre could be sufficiently softened to be stretched and elongated. Bounded by this threshold condition $ T>Tg $ (indicated by arrows), the length of the effective heat zone surpasses the spatial range of the contact line (indicated by the gold dashed line) while still maintaining a relatively small dimension of a few tens of micrometres, which is consistent with the taper length obtained in the experiments.
The reduced size of the microheater played a major role in narrowing down the heat zone. To illustrate this size effect, we considered a fixed rate of heat deposition (adopting the same absorbed optical power as in Fig. 3b, which is a practical value considering both the allowable light power used in the experiments and the simulated heat conversion efficiency) assigned to heat sources with various dimensions. Given that the major concern here was the dimension of the heat source, for the purpose of variable control, the heat source was assumed to be uniform, regardless of the E-field distribution. Fig. 3c shows the resultant temperature profile along the fibre axis. Note that the dependence of the temperature uniformity on the heat-source dimension is plainly revealed. As the heat source dimension decreases below 5 $ \mu $m, the corresponding temperature profile of the fibre converges, approaching that of a point source. In the experiments, although the size of the microheaters was in the range of a few tens of micrometres, the exact contact length could be properly adjusted to a few micrometres, as is the case exhibited in Fig. 1b. Therefore, it is inferred that the microheaters can generate a point-source-like hot zone in which the temperature gradient is significant. The nonuniform temperature profile in the hot zone further renders a similar gradient distribution of the viscosity of the softened fibre, thus causing inhomogeneous stretching of the fibre (that is, a less viscous, hotter region would be stretched faster than a more viscous, colder region). This explains why the resultant fibre taper showed a nonuniform waist (Fig. 1c). In comparison, when the heat source dimension is in the submillimetre regime, the corresponding temperature profiles tend to flatten, which is associated with a drop in the peak temperature as a compensation for a larger heating area. This latter observation indirectly suggests that, although the heat conversion efficiency generally becomes lower when the size of the plasmonic heater decreases, its salient peak temperature nevertheless reduces the burden of the input light power required to soften the fibre. Furthermore, given that the basis of the hot-zone theory is the assumption of an equally heated hot region, it only applies in cases where the heat source is above the millimetre scale, e.g., the flame burner26. For a highly localised heat source, modifications are needed to account for the temperature nonuniformity, and waist-less tapers are to be expected.
To verify the aforementioned insight obtained from simulation, that is, the fact that the microheater could generate sufficient heat to soften the fibre material, we transferred gold plates to standalone straight fibre probes (to exclude the impact of macrobending) and coupled light into them. Both the pulsed light source and CW laser were used in the experiment. The average input power of both sources was a few milliwatts. With light leaking out from the thinned region near the probe tips, the gold plates began to heat the fibres till the point at which the silica material visibly shrunk and exhibited liquid-like behaviour (Fig. 4 and Movie S8), suggesting effective heating and softening of the fibre induced by light-thermal effects. Further increasing the light power led to the formation of microbumps on the tips of the probes, similar to the reported phenomena in which $ CO_2 $ lasers with a few watts of input power were used to generate microcavities in silica fibres43. For fibre material away from the heated region, the solid-state geometry was still well preserved.
The early stage of the heat accumulation scheme was also considered, during which the heating effects of individual pulses are not yet overwhelmed by the accumulated temperature background. In nanosecond laser heating of microobjects, electrons were found to be in equilibrium with the lattice in previous studies44,45. Therefore, the classical heat transfer picture still holds, and the time-dependent Fourier heat conduction model was adopted in this study with slightly modified thermal properties for micro and nanoscale materials45. Gaussian pulses with 4-ns pulse width and an integrated pulse energy of 10 nJ were set up in the simulation based on the experimental measurements. Under single-pulse illumination, Fig. 5a illustrates the temperature evolution probed within the plasmonic hot region (far end of the plate–fibre contact line). Here, we can see that while the temperature almost synchronises with the light pulse at the rising edge, the cooling process evolves slowly and exhibits a long trailing end. Moreover, Fig. 5b shows the temperature evolution probed at positions away from the plasmonic hot region (probe locations are indicated in the inset of Fig. 5d), which is in stark comparison with Fig. 5a. The widened and time-delayed temporal envelopes suggest a retarded heating process of the fibre due to slow heat conduction (heat convection can be ignored in high vacuum and radiation contributes little in the low-temperature regime). The transient process of heat transfer can be further visualised in Fig. 5c, which shows the temperature distribution of the fibre-plate system in four snapshots taken from an extended period of 1 ms. At 10 ns, the temperature profile still resembles that of the absorption profile. As the heat transfer proceeds, the input energy is distributed more evenly across the fibre–plate system. Given that the thermal diffusivity of the gold plate and fused silica differ by two orders of magnitude ($ \alpha_{Au} \sim 10^{-4}\; m^2 $/$ s $, $ \alpha_{fibre} \sim 10^{-6} \;m^2 $/$ s $), while the gold plate is quick in reaching temperature uniformity, the fibre appears more lagging in dissipating the incoming heat flux from the heat source.
The retarded heat transfer allows heat to accumulate in pulses, as shown in Fig. 5d. As the repetition rate of the light source increases, the long tail of the cooling edge of the current heat pulse gradually overlaps with the next incoming temperature rise. Therefore, the starting temperature of each pulse is based on the accumulated effect of previous pulses, guaranteeing that the system can be heated constantly46,47. Another remarkable aspect is the smearing effect of the pulse-wise temperature evolution. An external observer with millisecond temporal resolution (common for heat-driven dynamic processes) could hardly resolve the dynamic changes occurring within a few hundred nanoseconds. A smoothed temperature envelope would be detected instead, resembling that of steady-state heating, and the effect would be the same if the observer were located far from the pulsed thermal source48(see the two curves at the bottom right of Fig. 5d). The temporal response of the plate-fibre system to the modulated CW laser features a pattern of quasi-stationary heat transfer, the details of which are provided in Section S1 of the Supplementary Information.
Pull.– While heat takes effect within a localised region at microscale, the other element of the proposed fibre tapering, namely the pulling force, is implanted in the system at macroscale. Specifically, this force is determined by the equilibrium of the bending moment in the fibre deformed in a designed pattern. As sketched in Fig. S6A (see also Fig. S1D), the deformed fibre is composed of a transversely suspended region, two vertical holders, and two bending regions connecting the former two parts, which combine to form a "door-shaped" structure. The bending regions are associated with bending moments inversely proportional to the radius of curvature. The equilibrium condition requires that the suspended region, though barely bent, should possess a specific stress distribution to generate a counter-moment offsetting the one at the bending region (see the left panel in Fig. 1a). Consequently, the suspended region where the microheater is placed is bound to be subject to a pure pulling force; a detailed discussion on this point is provided below (see also Supplementary Section S3).
The door-shaped deformation pattern was quantified by extracting the radius of curvature of a symmetrically deformed fibre, as shown in Fig. S6B. Note that the smallest bending radius (largest bending curvature) is located within the two bending connectors, and it increases along the fibre axis towards the centre of the suspended region, where the fibre becomes nearly straight with a bending radius greater than 1 m. By scanning the view of the optical microscope along the deformed fibre axis, the part of the fibre with the smallest bending radius was found to be located within the transition region of the pre-tapered fibre, the scale of which was typically a few millimetres. Fig. 6a exhibits an optical image of this example. The average bending radius of the displayed region is 3 mm.
To reproduce the manually applied deformation computationally, solid mechanics simulation was performed, in which inward rotations were prescribed for the two outermost regions of a cylindrical fibre (Movie S9). The stress distribution in the resultant door-shaped structure is shown in Fig. 6b. Owing to the large deformation in the simulation model, the geometric nonlinearity was taken into consideration by using the Green–Lagrange strain, and the second Piola-Kirchhoff stress along the deformed axial direction was selected as a quantifier for the normal stress.
Note that the bending and suspended areas (circled in blue and red, respectively, in Fig. 6b) behave differently: the normal stress inside the bending area has opposite signs on the convex and concave sides, suggesting a transition from tensile stress to compressive stress across the cross-section and the existence of a neutral layer in between. The stress distribution in the bending region is similar to that of a cantilever beam subjected to pure bending. In comparison, the suspended region only experiences tension, thereby echoing the moment analysis described above. We attribute this axial evolution of the normal stress to the geometrical evolution of the deformed fibre, particularly to the axial evolution of the bending curvature. In contrast, a circumferentially deformed fibre possesses a similar curvature through the bending connectors towards the suspended region, all of which feature the cantilever-like stress distribution (Fig. S6C, D). Indeed, the generation of pure tension can only be guaranteed by the law of moment equilibrium in the specified deformation manner, i.e., the door-shaped structure. For a pre-tapered fibre deformed in the door-shaped manner with an axially varying diameter, as in the experiment, the tensile stress within the suspended region can be several orders of magnitude larger than the maximum stress at the bending connectors owing to the reduced cross-section (Fig. S8). The calculated normal stress is below the rupture stress of tapered fibres measured experimentally49.
After the fibre is softened with heat input, the internal stress distributed in the deformed fibre can be released through a self-stretching process, as schematically illustrated in Fig. 6c. First, since the localised hot zone is always in the suspended region where the microheater is placed, the softened fibre material is subjected to pure tensile stress, and will be tapered and drawn thin, as predicted in the hot-zone theory. The tapering process inside the hot zone leads to an increase in the fibre’s total length, which then responds to the tendency of the bending connector to release its internal stress, such that the elongated side (convex side) is tightened and the compressed side (concave side) is extended, that is, the bending radius increases. Therefore, the coordinated effect of the displacements in the suspended region and bending connector would be an axial stretch inside the hot zone, accompanied by gradual loosening of the bending connector that causes the "forward" motion of the suspended region. As the self-stretching event progresses, the internal stress of the deformed fibre can be released accordingly. In this way, the self-modulation effect is introduced such that, at a later stage, a thinner fibre is subject to a smaller pulling force as a result of the gradually loosened bending, preventing the fibre from experiencing sudden tensile failure36. Nevertheless, given that the pulling force is preset in the fibre and can not be removed, a continuous input of heat may break the fibre (complete relaxation of the internal stress); alternatively, when the taper is drawn sufficiently thin (below 800 nm), the inner stress could surpass the rupture strength of fused silica as a brittle material and cause mechanical failure. To prevent such events, the dynamic tapering process must be terminated in due course by switching off the light source.
In the conducted experiments, the self-stretching event was manifested by both the formation of the nonadiabatic taper and a shift of the fibre image in the direction vertical to the suspended fibre, which can be seen in multiple video recordings in the Supplementary (Movies S2, S5-S7). The simulation results in Fig. 6d are consistent with these observations where a gradual drop in Young’s modulus was prescribed within a specified hot zone to mimic the heat-induced larger deformability. The total length of the 1-cm fibre in the simulation model increases by 24 $ \mu $m after the softening event in the hot zone; this is associated with a shift in the fibre position in both the suspended and bending regions. More precise results can be obtained by treating the whole fibre as a viscoelastic material and enforcing a drop in viscosity as the temperature increases.
The asymmetry of the fabricated taper mainly arises from the asymmetry in the bending connector, given that the asymmetry in the heat source can be obliterated by the heat transfer. Specifically, the two bending connectors might be bent with different bending angles and radii, which causes the normal stress to distribute and evolve differently along the two arms towards the position of the heat source, as shown in Fig. S9. On the other hand, even if the pre-tapered fibre is symmetrically deformed, there is no guarantee that the microheater can be placed right at the centre of the suspended region. A deviation of the microheater from the axis of symmetry would make the softened hot zone be subjected to unequal pulling velocities in opposite directions, which subsequently leads to asymmetry in the geometry of the taper. To make the effect of asymmetry more evident, we used a microfibre (4.73 $ \mu $m in diameter) to form a loop and placed a gold plate to the right of its axis of symmetry. The previous discussion suggests that the circumferentially deformed structure possesses a cantilever-like stress distribution, behaving poorly when it comes to providing the pulling force. As expected, no self-stretching event was observed after the softening of the silica material. Instead, the fibre loop shifted towards the right, which favoured the stress release of the silica material closer to the microheater. While the curved fibre on the right side straightened, it was at the cost of increased curvature of the fibre material on the left. Thereupon, asymmetry in the initial configuration could manifest itself in the fabricated taper samples.