1. INTRODUCTION

Optomechanical forces provide a means of precisely controlling and manipulating micrometer- and nanometer-sized objects, and are commonly encountered in optical tweezers [1–5]. Driven by applications in biology and ultra-sensitive metrology, there is growing interest in trapping complex nonspherical nanostructured particles [6]. For smaller nanoscale objects, stable trapping is less efficient because the strength of the optical force scales approximately with the volume, with the consequence that Brownian motion is large enough to overwhelm the trapping [7,8]. Several approaches have been developed to enhance the optical force for smaller objects. For metallic nanoparticles and nanostructures, efficient trapping using milliwatt power has been achieved by means of plasmonic resonances [9–11]. New trapping schemes are required, however, for nanoscale dielectric nanoparticles because of the lack of optical resonances. A promising approach is so-called self-induced back-action, where the amplitude of the trapping force on a particle is enhanced by feedback from a planar photonic crystal cavity [12] or a nanoaperture in a metal film [13].

Here we report a novel system in which a fused silica “nanospike,” formed by tapering and etching a single-mode fiber (SMF), is used to launch light adiabatically into a hollow-core photonic crystal fiber (HC-PCF). The interplay and back-action between nanospike and HC-PCF modes causes optomechanical trapping of the nanospike at the core center, resulting in efficient and self-stabilized launching of light into the HC-PCF. The guided mode in the SMF spreads out into the surrounding space as it travels along the nanospike, adiabatically evolving into an eigenmode of “nanospike plus hollow core.” As a result the mechanically compliant nanospike “feels” the presence of the hollow core and becomes optomechanically trapped, with a restoring force that increases with optical power. The results are confirmed both by numerical simulations and by measurements of the optical spring effect at low pressure. Interestingly, optomechanical bistability is observed at higher radial displacements from the core center.

In addition to its intrinsic interest for studying optical forces and the effects of Brownian motion at low pressure, the system may provide a solution to the perennial challenge in single-mode fiber optics of maintaining stable launching of light into the core of the HC-PCF in the presence of external perturbations, especially at high power levels.

2. EXPERIMENTAL SETUP

A sketch of the setup is shown in Fig. 1(a). The nanospikes were formed by first thermally tapering SMF down to a diameter of $\sim 400\mathrm{nm}$, and then using HF etching to form a tip ($<1\mathrm{mm}$ long) of final diameter $\sim 150\mathrm{nm}$ (see Supplement 1). A scanning electron micrograph (SEM) of the nanospike was used to resolve its dimensions with resolution of 1 nm (inset of Fig. 1(a)). The HC-PCF had a seven-cell core with diameter 12.1 μm (Fig. 1(b), right-hand side).

 

Fig. 1. Optomechanically coupled silica nanospike and a HC-PCF. (a) 3D sketch of the experimental system. Inset: top, optical micrograph of a silica nanospike inserted into HC-PCF; bottom, SEM of the final section of the nanospike. (b) Left-hand plot: simulated adiabatic evolution of the nanospike mode ($z$ component of Poynting vector is plotted) over the 50 μm insertion length, with the nanospike placed at the core center. The gray-shaded area represents the core wall, and the dashed curves indicate the local MFD. Right-hand plot: SEM of the HC-PCF structure along with the measured near-field profile of the mode excited by the nanospike. (c) Left-hand plot: measured local taper angle versus diameter for the whole pretaper and nanospike (blue open-circles) before insertion into the HC-PCF. The solid black line represents the adiabaticity criterion in free space. Right-hand plot: local taper angle versus diameter close to the tip of the nanospike when it is inserted 50 μm into the HC-PCF. The solid curves show the adiabaticity criteria for the “nanospike plus hollow core” structure with different offsets $\delta$ of the nanospike from the core center. The joined circles show the actual taper angle versus diameter. Adiabaticity is violated for $\delta =3\mathrm{\mu m}$ at a taper diameter of $\sim 160\mathrm{nm}$.

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A diagram of the measurement setup is given in Fig. 2(a). The nanospike was mounted on a precision piezo-driven $xyz$ stage (step resolution 50 nm) and pointed at the HC-PCF endface. A high-power laser beam at wavelength 1064 nm was then launched into the SMF. A $99:1$ coupler was used to monitor the input power (via photodiode PD1), and PD2 was used to monitor backreflections from the nanospike. A polarization controller (PC) was used to optimize for maximum transmission. At the output of the HC-PCF, the near-field profile was focused onto a CCD camera (CCD2) using a lens. A quadrant photodiode (QPD), with the function of differentiating the collected optical power between the quadrants, was used to follow the mechanical motion of the nanospike in the transverse plane. In the experiment, the displacement of the nanospike was recovered from the differential time-domain signal of the QPD using the following calibration procedure: the nanospike was displaced by a known value (via the piezo-controlled $xyz$ stage) and the corresponding QPD signal recorded on an oscilloscope. Given this conversion coefficient, the mechanical power spectrum of the nanospike could be obtained by Fourier-transforming the QPD signal. So as to control the ambient pressure, the nanospike and the HC-PCF (30 cm long, coiled with a diameter of $\sim 9\mathrm{cm}$) were placed inside a vacuum chamber. CCD1 along with a telescope system was used to monitor the position of the nanospike during the insertion process.

 

Fig. 2. Experimental setup. (a) Diagram of the measurement setup: PD, photodiode; PC, polarization controller; NS, nanospike; QPD, quadrant photodiode. (b) Typical Brownian motion spectrum at 10 μW input power, measured at 0.4 mbar and 0.32 μbar with the nanospike placed just outside the HC-PCF. (c) Measured spectral linewidth of thermal Brownian motion of the nanospike plotted against pressure. The dashed black line shows the linear fit to viscous damping.

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To explore the nanospike resonances in free space, a low power (10 μW) optical probe signal was launched into the SMF while placing the nanospike in front of the endface of the HC-PCF (i.e., without insertion). In this way optomechanical effects could be reduced to negligible levels while allowing the motion of the nanospike to be monitored using the HC-PCF. Under these conditions, Brownian motion and thermal vibrations drive the nanospike, producing the spectra plotted in Fig. 2(b) for pressures of 0.4 mbar (orange) and 0.32 μbar (blue). Both curves fit well to Lorentzian lineshapes. At the lower pressure the extremely high $Q$-factor (126,000) allowed the detection of two nondegenerate orthogonal nanospike resonances, separated by only 2.18 Hz, most likely caused by slight geometric irregularities. The increase in frequency of the resonant peak at the lower pressure (11 Hz) we attribute to the absence of air-related viscous drag ($\sim 10\mathrm{Hz}$ following the theory in [14]). Figure 2(c) plots the measured mechanical linewidths as a function of pressure. Above $\sim 1\mathrm{\mu bar}$ the damping is dominated by the viscosity of air, increasing linearly with pressure (dashed line), whereas below $\sim 1\mathrm{\mu bar}$ it is independent of pressure, indicating the contributions of intrinsic material and clamping losses.

3. PRINCIPLE OF OPERATION

As light progresses along the structure, it first evolves adiabatically [15] from the core mode of the SMF into the fundamental mode of the glass–air waveguide (Fig. 1(c), left-hand side). As the nanotaper gets slowly narrower and narrower, eventually reaching a diameter well below the laser wavelength (1064 nm), the mode spreads out strongly into surrounding air [16,17]. At this stage a second phase of “adiabatic following” takes place as the extended mode begins to feel the presence of the hollow core. Adiabatic following has previously been studied in conventional tapered fiber structures such as null couplers [18] and photonic lanterns [19]. The right-hand plot in Fig. 1(c) shows the length-scale adiabaticity criterion for the “nanospike plus hollow core” structure with the nanospike sitting at different displacements $\delta$ from core center. It may be seen that adiabaticity is maintained provided $\delta$ is less than 3 μm. The left-hand plot in Fig. 1(b) illustrates the adiabatic evolution of the nanospike mode over the 50 μm insertion length in the HC-HCF. Provided adiabatic following is satisfied, the mode-field diameter (MFD) at the tip of the nanospike is the same as that of a hollow core containing an axially uniform 150 nm diameter glass strand. Figure 3(c) shows the Poynting vector distributions for nanospike offsets $\delta$ of 1 and 3 μm from the core center. The presence of the HC-PCF strongly affects the field profiles, leading to the appearance of an optical restoring force when the nanospike is moved away from the axis.

 

Fig. 3. Optical trapping force calculation and optical spring effect. (a) Simulated optical trapping force for 1 W of power plotted against the nanospike offset from the core center, calculated by integrating the Maxwell stress tensor (blue open-circles) and using response theory (blue dashed line). The red solid curve shows the trapping force when a focused Gaussian beam, with waist matching the MFD of the HC-PCF, is used. The orange dashed line plots the calculated effective mode index of the fundamental mode of a hollow core with a 150 nm glass strand placed inside. (b) Scaling of measured and simulated values of $f_R^2-f_m^2$ versus power for different values of base offset $\mathrm{\Delta }$ from fiber axis. (c) Simulated Poynting vector distributions of the supermode at $\delta =1$ and 3 μm for a hollow core containing a 150 nm glass strand. The bottom figures show the zoom-in around the strand. (d), (e) Measured Brownian motion spectra at (d) 0.4 mbar and (e) 0.32 μbar for several different power levels. The solid curves are fits to Lorentzian lineshapes.

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A. Optical Force Calculation

Starting with the local mode approximation, we calculated the Maxwell stress tensor at each point along the nanospike so as to obtain the local force per unit length. We then found the total restoring force $F_{\text{opt}}$ by integrating over the insertion length. The resulting dependence of $F_{\text{opt}}$ on the offset $\delta$ from the core center is plotted in Fig. 3(a) (open circles).

$F_{\text{opt}}$ may also be calculated by considering the gradient of the free energy (proportional to the gradient in effective index) of the supermode with respect to nanospike position $\delta$ [20]. The orange dashed curve in Fig. 3(a) plots the calculated value of $n_{\text{eff}}$ (via finite element modeling) versus $\delta$ for a tip of diameter 150 nm. It can be seen that $\partial n_{\text{eff}}/\partial \delta$ peaks at a radius of $\sim 2.8\mathrm{\mu m}$. Under the local mode approximation, the total force acting on the nanospike is then given by

B. Optical Spring Effect Inside HC-PCF

The optical force $F_{\text{opt}}$ will create an optical spring effect and thus modify the resonant frequency of the taper. If the shape of the pretaper and nanospike are accurately known, the resonant frequency $f_R$ of the nanospike can be calculated using finite element modeling (see Supplement 1). As in any mechanical resonator, $f_R$ is proportional to the square root of the stiffness, which in this case is ($k_m+k_{\text{opt}}$), where $k_m$ and $k_{\text{opt}}=-\partial F_{\text{opt}}/\partial \delta$ are the mechanical and optical spring constants. Since $k_{\text{opt}}$ is proportional to the optical power $P$, we can write $f_R^2-f_m^2\propto P$, where $f_m$ is the resonant frequency at zero optical power.

To explore the optical spring effect, the nanospike was inserted $\sim 50\mathrm{\mu m}$ into the hollow core and attention was focused on one of the orthogonal mechanical eigen-resonances of the taper. Figure 3(d) shows the measured resonant power spectra, driven by Brownian motion, at 0.4 mbar pressure for several values of input power from 100 μW to 54 mW when the nanospike was placed at the center of the core. The resonant frequency changes from 1 to 1.42 kHz, indicating that there is a strong optical spring effect. The green squares in Fig. 3(b) plot the measured values of $f_R^2-f_m^2$ at several values of input power, while the green dashed line follows the theory presented in Supplement 1 with no free parameters. The error bars reflect uncertainties in the Lorentzian fits to the measured spectra. The excellent agreement between experiment and simulation confirms the accuracy of the spring effect model used to plot Fig. 3(a).

C. Effect of Displacement from Core Center

The system further allows the offset of the nanospike base from the core center to be precisely adjusted using the $xyz$ stage, which means that the optical stiffness can be directly measured. The orange and pink symbols in Fig. 3(b) show the measured values of $f_R^2-f_m^2$ for base offsets ($\mathrm{\Delta }$) of 1 and 2 μm, and the corresponding dashed lines show fits calculated following the theory in Supplement 1. At relatively low power levels ($<\sim 20\mathrm{mW}$), the measured data follow the theoretical predictions quite well, while at higher power levels the optical force begins to cancel out the mechanical force, causing the nanospike to shift toward the core center and increasing the value of $f_R^2-f_m^2$. (This is apparent for the 1 and 2 μm traces in Fig. 3(b)).

D. Comparison with Focused Gaussian Beam

For comparison we calculated the optical force exerted on the nanospike by a fixed Gaussian beam with waist matched to that of the fundamental mode of the HC-PCF (solid red curve in Fig. 3(a); see Supplement 1). The results show that the maximum trapping force and trap stiffness in the HC-PCF are an order of magnitude stronger than in conventional optical tweezers. This is because the trapping mechanism is fundamentally different, being strongly affected by changes in modal distribution with displacement (Fig. 3(c)), which enhances the optical force. In Fig. 3(b) the calculated value of $f_R^2-f_m^2$ for the Gaussian beam is plotted (solid red curve), showing a much smaller frequency shift than in the hollow core. A further advantage of the nanospike is that it is mechanically connected to its base, making exploration of optical forces at μbar pressures much easier than for levitated particles (which tend to become untrapped at very low pressures [21–24]). Figure 3(e) shows the measured optical spring effect at 0.32 μbar, when the high $Q$-factor (126,000) makes it possible to measure a shift in resonant frequency as small as 10 mHz at extremely low trapping power levels ($<210\mathrm{\mu W}$). This frequency resolution corresponds to a minimum detectable stiffness of $\sim 8$ attoN/μm.

4. SELF-ALIGNED COUPLING INTO HC-PCF

Self-alignment of the nanospike provides an elegant means of coupling light from SMF to HC-PCF with high efficiency, while providing automatic stabilization against external perturbations. For a small enough nanospike end-diameter, Fresnel reflections are practically eliminated without need for an anti-reflection coating (the effective modal index is very slightly less than 1.0), and the usual requirement for MFD matching is avoided by the adiabatic nature of the transition. This is illustrated in Fig. 4(a), where the calculated fundamental MFD for a uniform glass strand, placed in free space (blue curve) and inside the HC-PCF (orange curve), is plotted against the strand diameter. The results show that, at a nanospike end-diameter of $\sim 190\mathrm{nm}$, the free-space mode already has an MFD equal to the MFD of the HC-PCF, so that light will be captured and guided by the HC-PCF when the nanospike is inserted into it. Below this critical end-diameter, the MFD of the “nanospike plus HC-PCF” mode starts to approach the MFD of the HC-PCF, ensuring perfect mode matching and near-unity coupling efficiency (purple dot-dashed curve obtained by calculating the overlap integral, with the purple points corresponding to measured results from different samples). Efficient coupling into the HC-PCF also enhances the optical spring effect, with the result that the nanospike can be robustly stabilized at the core center without the need for physical contact with the HC-PCF.

 

Fig. 4. Efficient coupling from SMF to HC-PCF. (a) Simulated mode-field diameter for a nanospike in free space (blue) and in the center of the hollow core (orange) for a core diameter of 12.1 μm. For tip diameters below $\sim 190\mathrm{nm}$, the fundamental mode is guided mainly by the HC-PCF (gray-shaded area). The purple curve shows how the coupling efficiency varies with the final tip diameter when the nanospike is centered in the core. The solid purple dots show the coupling efficiencies measured for nanospikes with different final tip diameters. (b) Self-alignment and self-stabilization measured at atmospheric pressure, when the base of the nanospike is offset from the center by 0, 1, 2, 3, and 4 μm. Inset: transmitted signal recorded over 200 s at 450 mW (dark green) and 1 mW (bright green) when the nanospike is at the core center. The gray curve is the laser output monitored simultaneously (via PD1, scaled for comparison) at 450 mW, showing that the noise on the transmitted signal is caused by the laser.

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A. Launch Efficiency and Self-Alignment

The launch efficiency was tested at atmospheric pressure with the nanospike sitting at the core center. For 450 mW launched power the measured total transmission from the SMF to the output end of the HC-PCF was 81%, corresponding to a launch efficiency of 87.8% (0.57 dB insertion loss) after taking into account the transmission loss of the taper (0.31 dB) and the HC-PCF (0.13 dB/m). This value is quite close to the theoretical prediction (88.3%) in Fig. 4(a). The right-hand side of Fig. 1(b) shows that the measured mode intensity distribution at the output end of the HC-PCF has a clean single-lobed shape. The total transmission was found to drop by $\sim 10\%$ when launching into the orthogonal polarization state, perhaps because of polarization-dependent loss in the taper and the HC-PCF. The backreflection was measured to be $\sim 0.05\%$, which is much less than the 4% Fresnel reflection from a silica–air interface. This new coupling method is advantageous compared to techniques such as butt-coupling to a standard SMF, where there is a strong Fresnel reflection unless the fiber end is anti-reflection coated. This is especially interesting at high power levels.

The blue curve in Fig. 4(b) plots the measured total transmission as a function of input power with the nanospike placed at the core center. The error bars show the standard deviation, normalized to input power, of the time-domain signal collected over 200 s. The transmission becomes more stable with increasing power due to the stronger optical trapping force. This self-stabilization effect is more noticeable when time-domain traces at low (1 mW) and high (450 mW) power are compared (inset of Fig. 4(b)). Indeed, at power levels higher than $\sim 100\mathrm{mW}$, the standard deviation of the time-domain signal is dominated by power fluctuations in the laser. Note that at atmospheric pressure any mechanical oscillations of the nanospike are strongly damped and so are not observed in the system.

When the nanospike was initially offset from the core center, the transmission improved with increasing power (Fig. 4(b)), indicating that the nanospike is pulled toward the core center by the optical forces. At around 200 mW the transmission saturated to a value some 3% lower than in the perfectly aligned system, most probably because of the slight bending of the entire taper.

Self-alignment also operates at low pressures, when the system is able to suppress the effects of Brownian motion, especially at high powers. It is additionally feasible to further cool residual vibrations in the taper by parametric feedback to the trapping beam.

B. Bistability and Hysteresis

To explore the robustness of coupling, the $xyz$ stage supporting the nanospike base (some 10 mm away from the tip) was deliberately offset by $\mathrm{\Delta }$ from the core center while keeping the power constant. Figures 5(a)–5(d) show the measured transmission at different power levels when scanning the base away from (blue) and back toward (orange) the axis. At 450 mW (Fig. 5(a)), high transmission (and thus optical trapping) is maintained even for $\mathrm{\Delta }>12\mathrm{\mu m}$, i.e., well beyond the core edge (at 6 μm). At 98.2 and 52.1 mW (Figs. 5(b) and 5(c)), high transmission is maintained up to a certain offset, but beyond this point it drops sharply to a lower value, accompanied by a jump in the position of the nanospike (monitored via CCD1) and a change in the transmitted mode profile from a single lobe to a mixture of single- and double-lobed shapes (inset of Fig. 5(c)). When the nanospike is scanned back toward the fiber axis, the system switches to its original state but at a lower value of offset, exhibiting hysteresis.

 

Fig. 5. Bistability and hysteresis of the transmission at atmospheric pressure. (a)–(d) Measured transmission at different input powers when the base of the nanospike is scanned from the central axis of the HC toward the core wall (blue) and then back (orange). The dashed arrows in (b) and (c) indicate the theoretical predicted switching points of the nanospike. Inset in (c): measured near-field mode profile for a value of $\mathrm{\Delta }$ beyond the switching point. (e) Illustration of the bistability and hysteresis behavior due to the excitation of the ${\mathrm{LP}}_{11}$ mode of the “nanospike plus hollow core” structure.

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This behavior is a consequence of the breakdown in adiabatic following at higher offset values of the nanospike from the fiber axis ($>3\mathrm{\mu m}$; see right-hand plot in Fig. 1(c)), which causes a mixture of ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes to be excited in the HC-PCF, as illustrated in Fig. 5(e). Excitation of the ${\mathrm{LP}}_{11}$ mode creates a second local minimum in total potential energy at which the nanospike can also be trapped. The equilibrium position of the nanospike therefore switches sharply between the two local minima at a certain threshold value of $\mathrm{\Delta }$, which takes a different value when scanning toward and away from the core center. When the (lossy) ${\mathrm{LP}}_{11}$ mode is excited, the result is a drop in transmission. The predicted switching values of $\mathrm{\Delta }$ (dashed arrows in Figs. 5(b) and 5(c)), which were obtained numerically by requiring static equilibrium (see Supplement 1), agree reasonably with the observations. If the input power is further reduced (Fig. 5(d)), the hysteresis disappears because the optical stiffness is not strong enough to compete with the mechanical stiffness. Note that if the switching point is too far from the core center (represented by the yellow-shaded areas), the nanospike adheres to the core wall and cannot be detached by optical forces.

5. CONCLUSIONS AND PROSPECTS

The use of an SMF nanospike allows robust, self-aligning, self-stabilized, and efficient launching of high power laser light into the HC-PCF without the need for electromechanical stabilization. In addition, because the nanospike is held rigidly at one end (while maintaining a high $Q$-factor), the system may be used for systematic studies of optomechanical trapping under high vacuum, when tweezered nanoparticles (even though they have higher $Q$-factors) tend to become unstable under the effects of Brownian motion, rapidly escaping from their traps [21–24]. Stable optical trapping in air, especially at low pressure, also provides exciting opportunities to test fundamental laws of physics, e.g., the fluctuation-dissipation theorem [25,26] and the contributions of equilibrium and nonequilibrium Brownian motion [21,27]. Also, by applying direct or parametric feedback to the trapping beam, it should be possible to efficiently cool the mechanical motion of the nanospike to a lower center-of-mass temperature.