Sensing and Exploiting Static Femto-Newton Optical Forces by a

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Sensing and exploiting static femto-Newton optical forces by a nanofiber with white light interferometry Jianhui Yu, Liheng Chen, Huazhuo Dong, Xingyu Liu, Hankai Huang, Weiqia Qiu, Shiqing Huang, Wenguo Zhu, Huihui Lu, Jieyuan Tang, Yi Xiao, Yongchun Zhong, Yunhan Luo, Jun Zhang, and Zhe Chen ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00450 • Publication Date (Web): 30 May 2018 Downloaded from http://pubs.acs.org on May 31, 2018

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Communications; Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong Higher Education Institutes

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Article type: Article

Sensing and exploiting static femto-Newton optical forces by a nanofiber with whitelight interferometry

Jianhui Yu1,2, Liheng Chen1, Huazhuo Dong2, Xingyu Liu2, Hankai Huang2, Weiqia Qiu 2, Shiqing Huang3, Wenguo Zhu1,2*, Huihui Lu1,2, Jieyuan Tang2, Yi Xiao1,2, Yongchun Zhong1,2, Yunhan Luo1,2, Jun Zhang1,2, and Zhe Chen1,2,* *Corresponding Author: E-mail: [email protected], [email protected]

1

Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong

Higher Education Institutes, Jinan University, Guangzhou, 510632, China 2

Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Jinan

University, Guangzhou, 510632, China 3

MOE Key Lab of Disaster Forecast and Control in Engineering, Jinan University,

Guangzhou, 510632, China

Abstract Optical force determines the fundamental process of momentum exchange between light and matter. However, owing to the weak mechanical effect of the optical force and relatively large stiffness of optomechanical devices, pico-Newton (10-12 N) optical force is required to manipulate micro-/nanoparticles and the optical response of optical devices. It is still extremely challenging to sense static femto-Newton (fN) optical forces and exploit such forces to actuate micro-optical devices. Here, using a tapered nanofiber (TNF) with a high mechanical efficiency of 2.13 nm/fN, a sensitive and cost-effective scheme is demonstrated to generate, sense, and exploit fN optical force. Strong light coupling from the TNF to a glass substrate can result in a fN repulsive optical force, which can induce a TNF deformation of up to 425.6 nm. Such a large deformation allows white-light interferometry to detect a fN static 1 ACS Paragon Plus Environment

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optical force (5.2 fN). Moreover, the high optomechanical efficiency (15.6 nm/µW) allows us to all-optically control the signal power at values ranging from 0.09 µW to 17.1 µW with only micro-Watt pump power, which paves the way toward micro-Watt and fN-optical-force optomechanical devices. Keywords: femto-Newton optical force, all-optical control, linear momentum exchange, nanofiber, white-light interferometry, optomechanics

Introduction Optical forces determine a fundamental process of momentum exchange between light and matter.1-10 The two extremely controversial topics of optical momentum1,4,5 and optical force9 in dielectric media still need to be solved experimentally. Hence, sensing small optical forces is a critical technique for the experimental solution of the two controversies. In applications, optical forces have been widely and successfully exploited in manipulating micro/nanoparticles11-13 and cooling down atoms.14 Recently, many attempts have been made to enhance optical forces by the use of optical cavities,15-19 slow light,20 plasmons,21 and nanowaveguides.22-24 As a result, optical force has been successfully exploited in actuating micro-optical devices16,25-27 and all-optically manipulating their optical response. Furthermore, some interesting optomechanical phenomena induced by optical forces, such as electromagnetically

induced

transparency,18,28,29

back-action

effects

in

cavities,30

nonreciprocity31,32 and the mechanical Kerr effect,33 have been reported. However, due to the relatively large stiffness (or low optomechanical efficiency) of these devices,34 milli-Watt pump powers and pico-Newton optical forces are usually necessary to produce sufficient mechanical deformation to manipulate the optical response of these devices. Therefore, the low optomechanical efficiency of ~6.7 nm/mW of micro-optical devices16,27 impedes the exploitation of femto-Newton optical forces in optomechanical devices and thus the further reduction of operational pump power. On the other hand, sensing weak forces is the center of many fundamental studies in surface chemistry,35 the identification of individual atom species,36 and intermolecular and intramolecular interaction forces.37,38 The technique of atomic force microscopy (AFM) is usually used to sense the ultraweak force. However, due to the low efficiency of mechanical deformation (~10 nm/pN),16,27 high-Q mechanical resonators are necessarily exploited to detect femto-Newton forces through femtometer deformations.22-23 Despite the high 2 ACS Paragon Plus Environment

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sensitivity of this technique, extracting the femtometer deformation requires expensive instruments, and the force under detection should be modulated at the resonant frequency of the mechanical oscillator.22-23 Consequently, this technique cannot easily detect static fN optical forces, and thus, sensing static fN optical forces is a challenge. In addition, for optomechanical devices, static optical forces are more desirable and suitable for manipulating the optical response in a particular situation, since the optical response of optical devices is expected to be tuned and held for a period, as in the variable optical power controller shown in this work. However, the low mechanical efficiency of silicon-/SiN-based optical devices makes it very difficult to manipulate the optical response using static fN optical forces. Furthermore, it is also desirable for optomechanical devices to reach a large optomechanical deformation to manipulate the optical response.16 Most optomechanical devices are based on a gradient optical force that is proportional to the gradient of the intensity of the evanescent wave.34 The short range of the evanescent wave limits the range of the optical force and thus limits optomechanical deformation to only several tens of nanometers.16,27 Silica micro-/nanofiber (MNF) is an ideal candidate for optical force sensing and optomechanical devices because of its low loss, light weight, strong evanescent wave and outstanding flexibility.39 In recent years, MNF has become a versatile platform for various kinds of optical devices,39-45 ranging from microring resonators,46 Mach-Zehnder interferometers,47 and lasers48 to sensors.49 Here, a low-cost and simple scheme, namely, a mechanically sensitive tapered nanofiber (TNF) coupled with a wedge glass substrate, is reported to sense and exploit the static fN optical force. Benefitting from the outstanding flexibility of the TNF, a high mechanical efficiency of 2.13 nm/fN, almost 200 times larger than that reported before,14,25 is obtained, which allows us to achieve a high optomechanical efficiency of 15.7 nm/µW. In addition, the strongly evanescent coupling between the TNF and the substrate provides a new scheme to generate repulsive optical forces. It is shown that a large static optomechanical deformation of 425.6 nm, 20 times larger than that in two vertically coupled microdisk cavities,14 can be achieved with a static optical force of 198.9 fN (with a corresponding pump power of only 27.5 µW). Hence, the mature technique, whitelight interferometry with a microscope, is capable of sensing fN optical forces (5.2 fN in our scheme). In addition, the proposed scheme is also demonstrated to all-optically control the signal power using µW pump power. This work not only provides a novel way to sense fN optical forces but also extends the range of exploitable optical forces to the fN scale, which paves the way toward micro-Watt and fN-optical-force optomechanical devices. 3 ACS Paragon Plus Environment

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Theory and Model Repulsive optical force by linear momentum exchange. As shown schematically in Figure 1a, a TNF is used to guide the pump light, and the light will couple from the TNF to a glass substrate, thus generate a repulsive optical force. Figure 1b shows a snapshot of the continuous Ex field of the TE mode propagating in the TNF and coupling into the substrate, which is extracted from a simulation that used the three-dimensional finite difference time domain method (3D-FDTD).50 The generation of repulsive optical forces can be understood qualitatively as a coupling-induced momentum exchange between the nanofiber subsystem (red dashed rectangle in Figure 1c) and the substrate subsystem (blue dashed rectangle in Figure 1c). When a fundamental optical mode is propagating in TNF, the total optical momentum (TOM) carried by the optical wave has only a z component:

PzTNF

=neffUtot/c-

ngUdie/c, as reported previously.51 Here, c is the light speed in a vacuum; neff and ng are the effective and group refractive indexes, respectively; and Utot and Udie are the total energy and the energy stored in the dielectric of the nanofiber, respectively. However, the evanescent coupling from the TNF to the substrate, as shown in Figure 1b, will induce a negative y component of TOM in the substrate subsystem: Pysub = − PzTNF tan θ . To conserve the TOM for the whole system (comprising the substrate subsystem and nanofiber subsystem), the TNF subsystem will have a positive optical momentum along the y-axis, repulsive optical force

f yTNF

PzTNF tan θ ,

resulting in a

exerted on the TNF.

Using the 3D-FDTD method

47

and integrating the Maxwell stress tensor52 over the

surface of the TNF (see Method), we calculated the transverse optical force (TOF) exerted on the TNF. Figure 1d shows the change in TOFs with gaps g for the TM and TE modes. As shown in Figure 1d, the TOFs will be repulsive (+y direction), consistent with the prediction obtained from momentum analysis. In addition to the repulsive optical force, the TNF is exerted upon by the gradient optical force, which draws the TNF toward the substrate. The cooperation effect of these two kinds of forces results in a maximum TOF of 7.53 fN/µW (4.06 fN/µW) for TE (TM) mode at g=0.9 µm (0.8 µm). The TOF is in the +y direction for the gap in the range of 0.2 µm to 2.5 µm, indicating the large effective range of the repulsive optical force. In the FDTD calculations, the same parameters as in the experiment were used. The wavelength of the pump light is 1458 nm, the radius of the TNF is 239 nm, and the coupling length is 60 µm. Here, the TOF shown in Figure 1d is normalized by the input pump power. The insets in Figure 1d show the Ex field for the TE mode and the Ey field for the TM 4 ACS Paragon Plus Environment

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mode in the cross-section at the center of the TNF. As shown in the insets, the symmetry of the electric field guided by the TNF is broken due to the coupling with the substrate, thus leading to the transverse optical force. Femto-Newton force-sensing principle via white-light interferometry. As schematically shown in Figure 2, white-light interferometry is used to detect the fN optical force by measurement of the color pattern shift and the nanometer deformation of the TNF. A tilted angle ϕ of the substrate with respect to the TNF is introduced to form a color pattern. Therefore, the gap between the TNF and the substrate varies along the z-axis following g(z)=g0+z⋅tanϕ, where g0 is the initial gap between the TNF and the substrate at z=0. When a visible optical field Ein illuminates the TNF from the top, the fields of Er1 and Er2 will be reflected from the top surfaces of the TNF and the substrate, respectively, resulting in an interference pattern as shown in Figure 2b. The intensity of the interference pattern for a given wavelength λ can be written

 1 8π  4π  I r ( z, λ ) = ε 0c  Er21 + Er22 + 2Er1 Er 2 cos  ( g0 + z ⋅ tan ϕ ) + nr + (α1 + α 2 )   λ 2 λ  

(1)

Here, n and r denote the refractive index and radius of the nanofiber, respectively; and α1 and

α2 are additional phases induced by the reflections from the top surfaces of the nanofiber and the substrate, respectively. According to eq 1, for a particular component of color, its intensity will vary sinusoidally along the z direction with a spatial period of Tz=λ/2tanϕ. The red, green, and blue (RGB) components of the color pattern have different spatial periods Tz, owing to their differences in central wavelengths. Therefore, the ratio between RGB components will also vary periodically along the nanofiber, yielding a periodic color pattern, as clearly shown in Figure 2b. As schematically shown in Figure 2c and d, when a pump light is launched into the TNF, a repulsive optical force will be generated and push the TNF apart from the substrate, leading to an increase in the gap ∆g and thus a shift ∆z in the color pattern. Note that, in our experiments, the 25 mm-long suspended section of the TNF is 416 times longer than the 60 µm-long coupling region where the optical force is generated. Consequently, the gap change ∆g can be assumed to be a constant throughout the whole coupling region. From eq 1, the shift in color pattern is ∆z=-∆g/tanϕ. It is very interesting to note that a sufficiently small tilted angle ϕ can

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convert the vertical gap change in nanoscale into a horizontal shift in microscale, which can be directly measured by an optical microscope. To estimate the force exerted on the TNF, we theoretically analyze the mechanical deformation of the suspended TNF under an external force. As shown in Figure 3b, the TNF radius varies along the z-axis. According to the results measured by microscope, the TNF radius is fitted by an exponential function,53 r(z)=r0exp(γ|z|), where r0=239 nm is the waist radius of the TNF and γ is a shape factor. The fitting curve is in good agreement with the measured data, as shown by the inset of Figure 3b. The fitting curve allows γ=0.26954 mm-1 to be found. For a TNF with an exponential variation in radius, the analytical solution of the deformation induced by an external force exerted at the center of the TNF can be derived. The derivation of the analytical solution of the deformation and its verification by the finite element method and experimental measurements are presented in the supplementary information. The center of the TNF will undergo the largest deformation dopt(0). The deformation at the center is equal to the gap change in Figure 2c, dopt(0)=∆g. According to the analytical solution, the deformation is in the form of d opt (0) =

f yNF Ppump [ e-2γL + e2γL − 4L2 γ 2 − 2] 128EI 0 γ3 (e 2γL -1)

(2)

where Young’s modulus E=73.1 GPa;39 the inertia momentum at the TNF waist I0=πr04/4; the length of the suspended TNF L=25 mm; and the shape factor of the TNF γ=0.26954 mm-1. With these parameters, the TNF has a high mechanical sensitivity of 2.13 nm/fN and a high stiffness of kTNF=0.467 fN/nm. The TNF stiffness can be measured based on energy equipartition theory. As shown in the supplementary information, the measured stiffness is in good agreement with the theoretical prediction. Additional stresses may be acquired in the TNF during the pulling procedure. However, these stresses will not affect the estimated optical force, since the measured TNF stiffness contains a residual small additional stress. Note that the high mechanical sensitivity combined with the large range of the optical force allow a large TNF deformation of 425.8 nm to be achieved in the experiment when a 198.9 fN optical force is exerted on the TNF. The large static deformation is 20 times larger than that induced by the gradient optical force in the vertically coupled cavity,16 while the required repulsive optical force is ~3.7×104 times smaller. Therefore, by measuring the deformation (∆g) of the TNF with a white-light interferometer, the optical force or other kinds of forces acting on the TNF can be obtained via eq 2. Thus, 6 ACS Paragon Plus Environment

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our scheme is a competitive method for fN-force sensing with high-sensitivity and costefficiency features. Experimental Details As shown in Figure 3a, a TNF is horizontally double-clamped on a U-shaped holder. The waist radius of the TNF was measured to be 239 nm under scanning electronic microscopy, as shown in Figure 3c. The U-shaped holder has two 25 mm-spaced arms, allowing the suspension of a 25 mm-long TNF, as shown in Figure 3a. The position where the TNF is fixed by UV adhesive is marked by two black arrows in Figure 3b. A wedge glass substrate beneath the suspended TNF is mounted on and moved by the piezoelectric translator (PZT) with 1 nm relocation precision. An XYZθxθy five-dimensional translation/rotation stage, together with the PZT, can adjust the wedge glass substrate to the desired gap and tilted angle. A 1458 nm pump laser with tunable output power is divided into two beams by a 1:9 fiber coupler. The first beam is used to monitor the input power, while the second one is launched into the TNF to generate the repulsive optical force. Before the pump laser is launched into the TNF, a polarization controller (PC) is used to select the TE mode of the TNF. A photodetector (PD) (Thorlbs model 8088) is used to monitor the transmission pump power after the pump laser travels through the TNF. A computer equipped with home-developed software can acquire the transmission pump power using a prior-calibrated linear relationship between the photon voltage and optical power. At the same time, the computer can also precisely control the vertical movement of the piezoelectric translator with 1 nm resolution. A microscope with a 20X objective (NA=0.4) and a 16 bit-color charge-coupled device (CCD) is used to record the shift in the color pattern on the TNF. The integration time of the CCD is set to be 29 ms during the measurement. Microscope photographs of the color pattern along the TNF and its corresponding RGB components along the TNF are shown in Figure 3d and 3e, respectively. It can be seen from Figure 3e that the oscillations of RGB components will decay along the TNF due to the short coherent length of the illumination source. Here, an LED light source with a coherent length of 1.7 µm (estimated from the spectrum of the LED) is used as a white-light illumination source. The luminous flux of the LED source is 92.7 Lux. Results and Discussions Linear relationship between color pattern shift and gap change. Before measuring the deformation induced by the optical force, we measure the linear relationship between the gap 7 ACS Paragon Plus Environment

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change ∆g and the shift ∆z of the color pattern. Starting from an initial gap of g0=0.6 µm (see “locating initial gap” in the supplementary information), we vertically moved the wedge glass substrate away from the TNF in 20 nm steps using the piezoelectric translational stage and recorded microscopic images of the color pattern. Figure 4a shows the microscopic photographs of the color pattern. Figure 4b shows the distributions of the RGB components along the TNF, as extracted from the microscopic images. As indicated by the black arrows in Figure 4b, the red, green, and blue scattered points in Figure 4c are obtained by tracing the dips in the red, green, and blue intensity distributions, respectively. The red, green, and blue solid lines in Figure 4c are their corresponding linear fitting curves. The slopes of ∆z/∆g are fitted to be -66.5, -66.3, and -68.2 for the red, green, and blue lines, respectively, with corresponding linear correlation coefficients of 99.2%, 99.1%, and 99.7%. Since the minimum resolvable distance of the microscope with an objective lens of NA=0.4 is approximately 0.9 µm according to Abbe’s limit, a slope of -66.5 allows us to accurately measure a gap change ∆g of down to 13.6 nm by only using the microscope. From the relationship between the gap change and pattern shift, tanφ=∆g/∆z, the tilted angle of the substrate is ϕ=0.86°. To find the optimized tilted angle, the same experiments were carried out at other tilted angles. Measurement of femto-Newton optical force. Figure 5a shows the microscopic images of the color pattern along the TNF (z-axis) as the pump power increased from 0 µW to 27.5 µW. Here, the pump power guided in the TNF is calibrated by multiplying the input pump power with the prior-measured transmittance (see the Method section). The initial gap between the TNF and the glass substrate was chosen to be g0=0.6 µm for the three tilted angles of ϕ=0.37°, 0.86°, and 1.4°. Extracting the RGB component along the TNF from the microscopic images in Figure 5a allows us to obtain the RGB intensity distribution as shown in Figure 5b. The dependence of the pattern shift ∆z on the pump power for the three tilted angles ϕ=0.37°, 0.86°, and 1.4° is shown in Figure 5c. Using the prior-calibrated slopes of ∆z/∆g, we can derive the gap change ∆g due to the optical force and its dependence on the pump power. As shown in Figure 5d, the static TOF and its dependence on the pump power can be estimated for the three tilted angles by multiplying the measured deformation (∆g) with the stiffness. The fitting curves shown by the solid lines in Figure 5d yield normalized TOF values of fyTNF=5.48 fN/µW, 7.22 fN/µW, and 6.286 fN/µW, respectively, for ϕ=0.37°, 0.86°, and 1.4°. Among the three tilted angles, ϕ=0.86° is the optimized angle that yields the largest 8 ACS Paragon Plus Environment

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normalized optical force of 7.31 fN/µW, which agrees well with the FDTD-calculated value of 7.22 fN/µW for g0=0.6 µm, as presented in Figure 1d. The small difference of 0.09 fN/µW between the measured optical force and the FDTD calculation may be due to the inaccuracy of the tilted angles. Using the minimum measurable gap change of 13.6 nm as estimated above, we conclude that the minimum static optical force of 6.4 fN can be sensed by use of the current optical microscopy. As shown in Figure 5d, the minimum detectable optical force is 5.2 fN for ϕ=0.86°, which is in good agreement with the estimated value of 6.4 fN. An objective lens with a higher NA could further reduce the minimum detectable optical force. Using eq 2 and a similar method to that shown in Figure 5, the linear dependence of the total TOF on pump power is measured. Here, a newly fabricated TNF is used to measure the total optical force. For the TNF, the shape factor is 0.231 mm-1 and the waist diameter is 600 nm, allowing the mechanical sensitivity of 1.352 nm/fN (stiffness of 0.740 fN/nm) to be calculated through eq 2. Figure 6a shows the linear dependence of the total TOF on the pump power for different initial gap g0 values ranging from 0.25 µm to 0.91 µm. From Figure 6a, the dependence of the optical force on g0 is obtained and shown by the blue spheres in Figure 6b. As shown in Figure 6b, the measured result is in good agreement with the FDTDsimulated results, which confirms the validity of the technique for fN optical force measurement. The repeatability of the fN optical force is investigated. A new TNF is fabricated with a stiffness of kTNF=0.540 fN/nm. The gap changes of the TNF are measured for pump power changes in the range of 0 to 100 µW with 20 µW steps. In addition, the optical forces are estimated after five measurements (detailed in the supplementary information). The optical force can be repeatably controlled by the pump power. When the pump power is 1.2 µW, a 5.2 fN average optical force is obtained with a standard deviation of 2.5 fN. Therefore, the fN optical measurements are repeatable within the range of error. Thermal contributions to the displacement of the TNF are considered as a thermal buckling process in the supplementary information. As estimated in the supplementary information, the thermal displacement of the TNF arising from thermal buckling is only five orders of magnitude smaller than that induced by the optical force. Therefore, with sufficient accuracy, we can ignore the thermal displacement of the TNF in our above estimation of the optical force.


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All-optical control with a micro-Watt pump. Owing to the high optomechanical efficiency of the TNF, the above simple scheme can not only be exploited to sense fN optical force but also provides a way to all-optically control the signal power using only a micro-Watt pump. As shown in Figure 7a, a signal at 974 nm and a pump at 1458 nm are launched into a standard single-mode fiber and merged into a PC through a wavelength-division multiplexing coupler (WDMC). After TE polarization is selected by the PC, the beams are launched into the TNF, which is enclosed by a chamber, and then split back into signal and pump beams by another WDMC. The spectrum of the signal beam is measured with an optical spectrum analyzer (OSA), and the optical power of the pump beam is monitored with a power meter. The transmitted powers of the signal and pump beams are recorded while the glass substrate is moved closer to the TNF in 5 nm steps. By this method, we obtain the transmission as a function of the gap g for the 974 nm (black line) and 1458 nm (red line) beams, as shown in Figure 7b. Owing to the wavelength difference between the signal and pump, the transmission increases from 0.0052 to 1 for the signal and from 0.09 to 0.17 for the pump when the gap increases from 0 µm to 0.767 µm. On the other hand, for a fixed initial gap of g0=0.1 µm, the gap can be changed linearly by the pump, with a power ranging from 0 µW to 146.5 µW, as shown by the blue spheres in Figure 7d. The linear fitting yields an optomechanical efficiency of 5.34 nm/µW. The total TOF values are therefore estimated with a stiffness of 0.740 fN/nm and are shown by the red spheres in Figure 7d. The linear fitting of the TOF to the pump power yields an optical force of 3.95 fN/µW. Therefore, a 544 fN TOF can be generated by only a 136.8 µW pump, which induces a 736 nm gap change. Such a gap change allows the signal power to be all-optically tuned from 0.09 µW (-41.0 dBm) to 17.1 µW (22.5 dBm), as shown in Figure 7c. The measured power relationship between the signal and the pump is shown by the blue spheres in Figure 7e. The theoretical power relationship between the signal and the pump, as shown by the red line in Figure 7e, is calculated using the transmission in Figure 7b and the gap change ∆g in Figure 7d. One can conclude from Figure 7e that the theoretical relationship is in good agreement with the measured data. The transient response of our design is measured by using two photodetectors, which monitor the transmission power of the optical signal and pump. The optical pump is modulated by a chopper with a period of 8 s before being input into the TNF. The dynamics of the transmission power of the pump and signal are shown in Figure 8. The response times of the rises and falls in power are found to be 2.3 s and 0.4 s, respectively. To explain the relatively low response time, we simplify the TNF as a cylinder with a uniform radius of 1 µm 10 ACS Paragon Plus Environment

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and a length of 25 mm. According to Ref. [54], a double-clamped cylinder possesses different flexural vibration modes in an undamped system. The vibration frequency of the lowest vibration mode is inversely proportional to L2 and is estimated as ~0.52 Hz for L=25 mm by using double-clamped cylinder model [54]. When pumped by the 1458 nm laser, the TNF undergoes damped motion in an air environment. For an underdamped system, the response time is mainly determined by the vibration frequency of the undamped system. Thus, the response time is about 0.96 s, which is comparable with the experimental results. The response time should be speeded up by improving the production quality of the TNF and optimizing the structural parameters. Conclusions In summary, a simple and cost-effective scheme, which consists of a mechanically sensitive TNF evanescently coupled with a glass substrate, is demonstrated to sense and exploit fN static optical forces. Owing to the high mechanical efficiency of the TNF (2.13 nm/fN), the mature technique, white-light interferometry, is capable of sensing femto-Newton optical forces through large deformation. A weak optical force of 5.2 fN is detectable in our scheme, which can be improved further by using a higher-NA objective lens. In contrast to the gradient optical force (a short range of several tens of nanometers), the repulsive optical forces generated from the momentum exchange between the TNF and the substrate can have a larger range of up to ~2 µm, allowing a larger static deformation of up to 736 nm. Owing to the high optomechanical efficiency (15.7 nm/µW), our scheme can be used to all-optically tune the power of the signal at a 974 nm wavelength from 0.09 µW to 17.1 µW by a 146.5 µW pump at a 1458 nm wavelength. The proposed scheme not only provides a useful method to sense the fN static force but also opens up opportunities for micro-Watt and fN-force optomechanical devices.

Methods Measurement of the pump power guided in the tapered nanofiber. The pump power guided in the tapered nanofiber (TNF) can be derived from the input power Pin and T. Here, T is the transmittance through two sections: one section of single-mode fiber connected with the TNF, and the tapered section where the fiber radius decreases sharply from 125 µm to 0.5 µm. Accordingly, the pump power guided in the tapered nanofiber Ppump=Pin⋅T. As shown in Figure 3a, the input power Pin can be monitored and calculated by the power meter. To obtain 11 ACS Paragon Plus Environment

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the transmittance, first we set the input pump power Pin=1 mW and move the substrate far from the TNF, and then we cut the TNF at the center, where the coupling with the substrate occurs. Finally, the output power from the cut end face of the TNF is measured to be 11.75 µW, yielding a transmittance T=0.01175 (corresponding to a loss of 19.3 dB). Therefore, the pump power guided in the TNF can be calculated as Ppump=0.01175*Pin. FDTD calculation of optical force. The light coupling from the TNF to the glass substrate is simulated using the three-dimensional finite different time domain method.44 In the simulation, the computational space is set to be 9 µm×9 µm×65 µm with additional perfect match layers to avoid leaving out the evanescent wave of the TNF. As the radius of the tapered nanofiber changes very slowly along the TNF, we let the TNF have a uniform diameter of 478 nm and a length of 65 µm in the simulation. The TNF is located at the center of the computational space along the z-axis, while the glass substrate, a rectangle having a length of 60 µm and width of 9 µm, is located parallelly beneath the TNF with a gap. Here, the grid size is set to be 20 nm in the xy direction and 50 nm in the z direction. For a given gap, the continuous-wave TE and TM fundamentals are launched into the TNF. After simulation for a sufficiently long time, the electric and magnetic fields are extracted from the FDTD simulation and used to calculate the repulsive optical force, as shown in Figure 1c. The transverse optical force exerted on the TNF is calculated by integrating the Maxwell stress tensor (MST)46 over the surface of the TNF. For a continuous wave, the time-averaged optical force is derived by46

F NF =

1 T ⋅ ds 2 ∫S

(3)

where s is the surface in the air that encloses the coupling section of the TNF, and the timeaveraged MST is written as 46

T = ε 0 EE * + µ 0 HH * −

(

1t 2 I ε 0 E + µ0 H 2

2

)

(4)

t I Here, is the unitary matrix. The calculated transverse optical force is normalized by the input power monitored in a 9 µm×9 µm cross-section. The cross-section for the power monitor is placed in front of the glass substrate.

Supporting Information 12 ACS Paragon Plus Environment

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Additional supporting information may be found in the online version of this article at the publisher’s website. Choose of initial gap between the TNF and the substrate, mechanical deformation of the double-clamped tapered nanofiber, measurement of the stiffness, repeatability of fN optical force measurement, thermal mechanical effect, and dynamics of temperature rise of TNF. Acknowledgments The authors would like to thank Prof. Jingang Zhong for his helpful discussion. The authors would like to acknowledge the support from National Natural Science Foundation of China (61675092, 61705086, 61475066); Natural Science Foundation of Guangdong Province (2017A010102006, 2017A030313375, 2016TQ03X962; 2016A030311019), and the Science and Technology Projects of Guangdong Province (2016B010111003, 2016A030313079, 201704030105). Science & Technology Project of Guangzhou（201707010396）

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Figure 1. Repulsive optical force by linear momentum exchange. (a) Schematic for the generation of optical force in the TNF-glass substrate system. (b) Snapshot of the CW Ex field of the TE mode propagating in the TNF and coupling into the substrate, which is simulated by the 3D-FDTD method; (c) Schematic illustration of the transverse momentum exchange between the TNF and substrate subsystems; (d) Variation in normalized TOF with a gap g. Insets show the electric field Ex for the TE mode and the electric field Ey for the TM mode when coupling occurs. The coupling-induced asymmetry of the electric field results in the TOF.


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Figure 2. Schematic illustration of white-light interferometry for detecting the optical force in terms of color pattern shift. (a) Schematic of the formation of color patterns along the TNF due to white-light interference; (b) Color and its RGB distribution along the TNF; (c) Deformation of the TNF induced by the repulsive optical force when a pump light is launched into the TNF; (d) Shift in color pattern toward the –z direction when the gap is increased by the repulsive optical force.


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Figure 3. Implementation of experiment for generating and sensing fN optical forces. (a) Schematic diagram of the experimental setup; (b) Fiber radius variation along the TNF and its zoomed-in profile near the TNF waist; (c) Image of the TNF waist taken by scanning electronic microscope, from which the diameter of the TNF waist is measured to be 478 nm; (d) Microscope image of the color pattern along the TNF; (e) Intensity of the red, green, and blue components extracted from the color pattern along the TNF.


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Figure 4. Calibration of the linear relationship between the pattern shift ∆z and gap change ∆g. (a) Microscopic images of shifts of color patterns when the wedge substrate is vertically moved away from the TNF in 20 nm steps by the piezoelectric translational stage with an initial gap of g0=0.6 µm; (b) Distributions of the RGB components extracted from the microscopic images; (c) Linear relationship between the gap change ∆g and color pattern shift ∆z for the three RGB components.

Figure 5. Measurement of pump-induced optical force. (a) Microscopic images of color patterns along the TNF for the tilted angle ϕ=0.860 as the pump power increases from 0 to 27.5 µW; (b) Intensity distributions of the RGB components extracted from the microscopic images; the blue arrows indicate the pattern shift ∆z for the red component; (c) Linear 19 ACS Paragon Plus Environment

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dependence of the shift ∆z on pump power for the tilted angles ϕ=0.37°, 0.86°, and 1.4°; (d) Measured dependence of the total TOF on the pump power.

Figure 6. Measurement of optical force with different initial gap g0 values using a 0.6 µm waist diameter TNF. (a) Total TOF values with different pump powers measured for different initial gaps g0, where a TNF with a mechanical sensitivity of 1.352 nm/fN (stiffness of 0.740 fN/nm) is used. (b) Dependence of measured and FDTD-calculated optical forces on g.


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Figure 7. All-optical control of signal (974 nm) optical power using a micro-Watt continuous pump (1458 nm). (a) Schematic of the experimental setup of all-optical signal power control. (b) Transmissions of the TNF as a function of gap g for the pump at 974 nm (black) and signal at 1458 nm (red). (c) Spectra of the signal controlled by pump power from 0 µW to 146.5 µW with an initial gap of g0=0.1 µm. (d) Linear dependence of gap change and total TOF on the pump power for g0=0.1 µm. (e) All-optical dependence of the output signal power on the pump power. The experimental results, as shown by blue spheres, are measured with an optical spectrum analyzer (OSA). The theoretical results, as shown with a red solid line, are estimated using the transmission curve (e) and gap dependence (d).


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0.02 0.3

0.2 0.01 0.1

0.0 0

5

10

15

20

Time (s) Figure 8. Transient response of the all-optical control experiment.


25

PD Voltage (V)

1458nm 980 nm

0.4 PD Voltage (V)

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For Table of Contents Use Only Sensing and exploiting static femto-Newton optical forces by a nanofiber with whitelight interferometry Jianhui Yu, Liheng Chen, Huazhuo Dong, Xingyu Liu, Hankai Huang, Weiqia Qiu , Shiqing Huang, Wenguo Zhu, Huihui Lu, Jieyuan Tang, Yi Xiao , Yongchun Zhong, Yunhan Luo, Jun Zhang, and Zhe Chen A suspended optical tapered nonafiber (TNF) is pumped by a 1458nm laser. The linear momentum transformation from the TNF to the wedge glass substrate leads to the repulsive optical force exerted on the TNF. The force deforms the TNF, which induces an interference pattern shift in CCD. With the measured pattern shift and TNF stiffness, the repulsive optical force can be derived. Experimental results show that this simple scheme is sensitive to fN optical force (5.2 fN). The application of fN optical force in all-optical control by a microWatt pump is demonstrated.


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Sensing and Exploiting Static Femto-Newton Optical Forces by a

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