Open Access
Add to BibSonomy
Abstract
A new thin plane mirror with an Archimedes spiral structure (Archimedes-structure thin plane mirror - ATPM) that implements an elastic support boundary is proposed in this study. An optimal structure of ATPM is developed to achieve a linear displacement response with respect to optical forces. The displacement response of the optimized ATPM is analyzed by considering the combined effects of optical force and gravity. The distribution of the optical force density is calculated based on a tilted Gaussian laser beam. Experimental results demonstrate that the optimized ATPM can produce a steady-state displacement of 24.18 nm on average in a normal-gravity environment when subjected to an average optical force of 132.17 nN. When the optical force exceeds 133 nN, the nonlinearity of the displacement response of the optimized ATPM is less than 6.28%. An amplification of the optical force-induced displacement is achieved by more than 15 times compared with that for an unstructured mirror of the same size. The results of this study can assist the development of a miniaturized macroscale optical force platform based on an ATPM for practical applications including the in-situ laser power measurement and nN level force source in the atomic and close-to-atomic scale manufacturing.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
When an object is irradiated by a laser, it can be moved by multiple effects [1,2], including photothermal [3,4], photoacoustic [5], photoelastic [6], and optical force [7–9]. Optical force effects can provide precise and small forces ranging from pN to mN to drive objects, thus meeting the requirements of moving small and fragile objects. This has led to the development of technologies, such as optical cooling, trapping, and manipulation.
Optical cooling utilizes optical forces to generate coherent coupling between an optical field and atoms. This redistributes the optical field and decreases the thermal motion of atoms [10]. When the effective kinetic temperature is lowered to $\textrm{1}{\textrm{0}^{\textrm{ - 3}}}$ K, thermal motion of the atoms can be overcome, thus enabling the stable manipulation of atoms [11]. The objects of optical cooling are at the atomic scale, and their diameters range from 0.1 nm to 1 nm [12]. Optical tweezers and optical manipulation can move objects with diameters ranging from 0.1 nm to 100 µm [13]. These techniques are applied in cutting-edge fields [14], such as particle manipulation in meta-robot actuation [15]. Optical trapping can move objects with diameters up to 100 µm by varying the trapping position at high frequency by laser scanning [16,17]. Specially designed fibers can converge and modulate optical beams, thereby realizing low-cost flexible optical tweezers and standing-wave and dual-beam fiber traps. These are capable of moving objects with diameters up to 100 µm and have been applied to assembling multiple particles [18,19], thus achieving efficient micromanipulation. Holographic optical trapping and plasmonic tweezers can move objects with diameters ranging from 1 nm to 10 nm [20,21].
When the size of the irradiated object continues to increase and reaches the macroscopic scale, such as millimeters or even centimeters, the complexity of the optical force-based manipulation begins to increase. Because optical forces cannot overcome gravity and friction forces acting on macroscopic objects, the method of manipulating such objects has shifted from optical trapping to optical force propulsion and optical force-induced deformation, known as optical force-induced motion [22]. Optical force-induced motion has been applied in precision measurement research fields, such as the theory and verification of optical force [23], laser parameter measurement [24], and traceability of small forces [25]. However, owing to the extremely small scale of the optical force, it is difficult to measure the macroscopic optical force using traditional force measurement instruments or displacement transducers. For example, the displacement caused by a 532 nm laser pulse with a single pulse energy of 1 J incident on a fused silica substrate mirror with a thickness of 5 mm and a diameter of 23.5 mm is only tens to hundreds of picometers with fixed boundaries [7]. Therefore, the key issue in macroscopic optical force applications is the amplification of the optical force-induced motion.
Several methods have been used for the amplification of optical force-induced motion, including the enhancement of the laser source, use of multiple reflections, exploitation of reflective surface properties, and utilization of amplifying structures. Methods used to enhance the laser source include increasing its power density and the construction of resonant cavities. However, these methods may also amplify thermal effects and affect the distribution of the reflected laser intensity. The use of multiple reflections increases the optical path complexity. Exploiting reflective surface properties has higher requirements on the surface of objects as well as the laser source. The use of amplifying structures for amplification of optical force-induced motion offers advantages such as a simple optical path, good amplification capability, and miniaturization.
Amplifying structures for optical force-induced motions include suspensions, cantilevers, and elastic support structures. Jones et al. used a torsion balance suspension structure to hang a planar mirror by a 50 mm-long copper wire with a diameter of 0.04 mm and used an optical force from a 15 mW laser to drive it to deflect in the dispersive medium (the optical force-induced deflection angle was $\textrm{7} \times \textrm{1}{\textrm{0}^{\textrm{ - 5}}}\; \textrm{rad)}$ [26]. This was the first time that an optical force measurement experiment was performed in a medium [23]. However, the linear measurement range of the deflection angle of the torsional balance is small and is typically only near the zero position. The amplified deflection motion driven by the optical force may introduce nonlinear errors in angular measurements. This problem can be avoided by measuring the displacement induced by the optical force. When an object moves in a straight line subjected to an optical force, the displacement exhibits a linear relationship with the force or can be transferred to a linear relationship by experimental settings in a relative long range. Meanwhile, the optical force-induced displacement exhibits good traceability. Agatsuma et al. proposed a laser power measurement method by using optical forces to drive a mirror (with a diameter of 3 mm, which was suspended by 10 mm fibers) to perform an oscillating motion [27]. This yielded nanometer scale displacements with a 450 mW laser beam. The system used a double pendulum to ensure the mirror moved primarily along the direction of the optical force, and a Michelson interferometer was used to measure the mirror displacement, which reduced the uncertainty of the optical force measurement to values less than 1%.
Wilkinson et al. irradiated a cantilever with a 6.5 mW laser beam to achieve displacement amplitudes larger than 10 nm. The lever was approximately 11 mm long, 2 mm wide, and had an approximate thickness of 80 µm. The study indicates that optical force-induced displacement can be applied to the calibration of cantilever devices, such as AFM probes, demonstrating the potential application of optical force-induced displacement in metrology [28]. Ma et al. used 15 pN optical forces generated by a 9 mW laser beam to drive a 200 µm long, 5 µm wide silicon nitride microcantilever to produce a displacement of 10 nm [29]. The oscillation analysis used in the work can extract the optical force induced displacement from the oscillation induced by optical force coupled with thermal effects. This decoupling method provides a reference for the study of the light–matter interactions.
A vertically placed cantilever structure was designed in our previous study [30] using an ultrathin aluminum mirror coated with a high reflectance film. The mirror had a thickness of 100 µm, a diameter of 20 mm, and was fixed on a three-dimensional (3D) printed base made of polylactic acid. The structure generated an 80 nm optical force-induced displacement under a 29 W laser irradiation [31]. While this structure effectively amplifies the optical force-induced displacement, its design, which separates the mirror from the supporting base, requires adhesive bonding which reduces the reliability and introduces nonlinear errors within the amplification range of the optical force-induced displacement.
Elastic support structures can be employed to effectively amplify the optical force-induced displacement linearly. Ryger et al. proposed a micromachined silicon mirror with an elastic support structure. The mirror can convert the optical force-induced displacement into a change of the capacitance, which is detected by a bridge circuit for laser power measurements. The structure exhibits a response time less than 20 ms when subjected to modulated laser irradiation at 250 W [32]. Pinot et al. employed the diamagnetic properties of pyrolytic carbon to create a levitation system on which a mirror was placed. The levitation system acted as a virtual spring to support the mirror and balance gravity and resulted in a very small rotational stiffness in the horizontal direction. A horizontal displacement of 160 nm was induced by applying an optical force of 6.7 nN to the mirror [33]. Partanen et al. utilized a spring-suspended mirror to create a longitudinal oscillatory structure that effectively compensated the effects of gravity. They used a 1 W laser to drive a macroscopic object with a mass of 20 g and generated a maximum displacement of 100 nm [34].
This study aims to achieve linear amplification of optical force-induced displacement by an Archimedes-structure thin plane mirror (ATPM) with laser power at the watt level. It can produce a nanometer level linear displacement response driven by a nanonewton level optical force while presenting advantages such as low photothermal effects, miniaturization, and mass-production feasibility. A miniaturized macroscale optical force platform based on an ATPM can be developed for experimental and applied purposes. The structure of this paper is organized as follows:
Section 2.1 describes the ATPM structure and parameters. It employs an elastic support structure to resist gravity, thus ensuring that the central mirror undergoes linear motion when subjected to optical forces and maintains a linear relationship between the optical force and displacement response when subjected to gravity. In Section 2.2, the density distribution of the optical force on the ATPM is calculated based on the relationship between the laser power and the optical force. Structural mechanical steady-state displacement simulations are carried out to the ATPM based on the optical force load using the finite-element method (FEM) [35] to analyze the influences of the structural parameters on the optical force-induced displacement and the gravity-induced deformation. In Section 2.3, orthogonal simulations with three factors and three levels are designed and conducted to optimize the structural parameters of the ATPM with the objectives of maximizing the optical force-induced displacement and minimizing the gravity-induced deformation simultaneously. Section 3.1 analyzes the steady-state displacement response of the optimized ATPM structure and discusses the theoretical distribution of the optical force-induced displacement, internal stress, and linear displacement response range. In Section 3.2, the ATPM fabrication process is described wherein the internal stress of the fabricated sample is tested to confirm that no additional stress is introduced at the stress concentration positions during the fabrication process. Section 4.1 introduces the experimental setup and analyzes the uncertainty of the optical force in the experiment. Section 4.2 analyzes the measured steady-state displacement response of the ATPM in the experiment as well as the measurement uncertainty, and calculates the nonlinearity of the ATPM optical force-induced displacement. Finally, Section 5 summarizes the study and presents prospects for future research.
2. ATPM structural design and optimization
2.1 ATPM structural model
This study proposes a thin plane mirror with an Archimedes spiral structure, referred to as the ATPM, which can linearly convert the optical force into a measurable displacement with high precision in the macroscopic scale. As shown in Fig. 1, the mirror is positioned at the center of the ATPM, exposed to laser irradiation, and subjected to an optical force. The outermost ring of the structure is reserved as the clamping area and the mirror is connected to the clamping area using elastic support arms, as shown in Fig. 1(a).
Fig. 1. Archimedes-structure thin plane mirror (ATPM) structural model. (a) An angled view of ATPM, where the blue dashed line marks the mirror, and the outermost ring is the clamping area. The mirror is connected to the clamping area by elastic support arms. (b) Schematic of the ATPM inside the fixture subjected to the optical force and gravity. The direction of gravity is along the negative $y$-axis and the direction of the optical force is along the positive $z$-axis.
The laser incidence plane is the $xz$-plane with the $z$-axis oriented perpendicularly to the surface of the mirror. The laser beam obliquely irradiates the mirror at the center of the ATPM, is reflected and exerts an optical force on the mirror, as shown in Fig. 1(b). The optical force is along the positive $z$-axis. It pushes the mirror in the same direction. The optical force-induced displacement occurs from the elastic deformation of the three arms which together form a planar-spring structure. The support arms are designed in the shape of an Archimedes spiral, which increases the arm length and reduces the bending stiffness in the $z$-direction. This design is beneficial for amplifying the displacement response of the ATPM. The support arms rotate symmetrically around the geometric center of the ATPM, which cancels out the rotational torques around the $x$- and $y$-axes when the mirror is driven by the optical force. This ensures the mirror undergoes a linear displacement along the $z$-axis. Gravity acts on the ATPM along the negative $y$-axis, which is perpendicular to the direction of the optical force. To minimize the impact of gravity on the optical force-induced displacement, the ATPM is vertically mounted in the fixture along the $xy$-plane. The three arms form an angle of 120° with respect to each other, and the lines connecting the three connection points in the clamping area form an equilateral triangle. This design enhances the structural stability of the ATPM in the presence of gravity.
The parameters of the ATPM include the outer radius R, inner radius r, clamping area width L, support arm width W, thickness h, and Archimedes spiral curve parameter equation $C({x,y} )$, as shown in Fig. 2. The parametric equation $C({x,y} )$ is given in Eq. (1), where ${w_0}$ is the spiral radius and $\varphi $ is the offset constant, which is $\pi $, $- \pi /\textrm{3}$ and $\pi /\textrm{3}$ for the support arms, respectively.
The inner radius r of the ATPM should be larger than the radius of the laser irradiation area, and its value is fixed at 4 mm. The clamping area width L is set to 1 mm to ensure the stable fixation of the ATPM. To balance the efficiency and the yield of fabrication, the outer radius R is fixed at 10 mm, the minimum value of W is 1 mm, and the minimum value of the thickness h is 100 µm.
Fig. 2. Schematic of the ATPM and its parameters. (a) Front view of the ATPM, R is the outer radius, r is the inner radius, L is the clamping area width, W is the arm width, and h is the thickness. (b) Archimedes spiral curves in the ATPM (solid black curves). $C({x,y} )$ indicates a general point located on one of the spirals.
2.2 Optical force load and structural parameter sweep on ATPM
The distribution of the optical force density is derived from the laser power density based on electromagnetic theory. In this study, an optical force is generated by the transfer of linear momentum of the electromagnetic field. The momentum density carried by the laser beam is equal to the Poynting vector divided by the square of the speed of light as expressed by Eq. (2),
The parameters of the laser beam are shown in the Table 1. Introducing the parameters into Eq. (5) can derive the power density distribution $P({x,y} )$ with actual values, the result is shown in Fig. 3(a).
Fig. 3. Optical force load on the ATPM. (a) Laser power density distribution $P({x,y} )$ and (b) optical force distribution ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over G} _z}({x,y} )$ (the red dashed arrow indicates the direction of the optical force).
Table 1. Parameters of the Laser Beam
The optical force distribution can be obtained by substituting $P({x,y} )$ into Eq. (4), where ${R_f} = $ 99.90% and $n = $1.00029. All the parameters listed above are experimentally measured. The calculation results for ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over G} _z}({x,y} )$ exerted on the ATPM are shown in Fig. 3(b). ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over G} _z}({x,y} )$ exhibits a Gaussian distribution with the largest value at the center and gradually decreases toward the periphery of the mirror. The direction of the optical force is along the positive $z$-axis. The area integration of the optical force density distribution in Fig. 3(b) shows that the average value of the optical force on the mirror under the experimental conditions is 132.17 nN.
The geometry of the ATPM support arms, which determines the mechanical response of the ATPM, depends on the values of the three parameters ${w_0}$, W and h. Based on the ATPM model, this study employs the optical force-induced displacement $u({x,y} )$ of the mirror along the positive $z$-axis, evaluated at the center of the mirror $[{x,y} ]= [{\textrm{0},\; \textrm{0}} ]$, to characterize the amplification of the ATPM displacement response to the optical force. Similarly, the deformation $v({x,y} )$ along the gravitational direction (negative $y$-axis direction), evaluated at the center of the mirror $[{x,y} ]= [{\textrm{0},\; \textrm{0}} ]$, is used to characterize the effect of gravity on the ATPM. A parameter scan analysis is conducted on ${w_0}$, W, and h using the FEM. The clamping area width L, outer circle radius R, and inner circle radius r are fixed owing to conditional limitations. The ATPM’s optical force-induced displacement u and gravity-induced deformation v under the action of a specific optical force load presented in Fig. 3(b) are shown in Fig. 4. The blue and orange curves respectively depict the relationships between u or v and the structural parameters.
Fig. 4. Parametric study of the steady-state displacement response as a function of the ATPM’s (a) spiral radius ${w_0}$, (b) support arm width W, and (c) thickness h. Optical force-induced displacement $u({\textrm{0},\textrm{0}} )$ for the optical force load shown in Fig. 3(b) is depicted in blue and the deformation $v({\textrm{0},\textrm{0}} )$. due to gravity is depicted in orange.
Figure 4 shows a negative correlation between ${w_0}$, W, h, and u (or $v$). Reducing the numerical values of the three parameters of the support arms can effectively amplify the displacement response of the ATPM to the optical force. However, it can also introduce gravitational deformation along the negative $y$-axis, which affects the elastic support arm geometry, thereby reducing the linearity of the ATPM displacement response. Increasing the numerical values of these parameters can increase the bending stiffness of the ATPM in the $x$- and $y$-directions to reduce the deformation under gravity, but can also increase the bending stiffness in the $z$-direction, which decreases the ATPM displacement response to the optical force. Therefore, optimizing the design of the support arm structure is necessary to ensure linear amplification of the ATPM displacement response to optical forces.
2.3 Structural optimization
Nine groups of parameters for ${w_0}$, W and h are obtained by using ${\textrm{L}_\textrm{9}}\textrm{(}{\textrm{3}^\textrm{3}}\textrm{)}$ three-factor (A, B, C), three-level (1, 2, 3) orthogonal experimental design. The steady-state displacement responses of the ATPM are calculated with different parameter combinations by using FEM. Table 2 summarizes the results. In the structural optimization analysis, the optical force load and gravity acceleration used in all the simulations remain the same. Gravity is not applied when the optical force-induced displacement u is calculated and the optical force is not applied when the gravity-induced deformation v is calculated, thus ensuring the orthogonality of the experimental design.
Table 2. Three-factor, Three-level Orthogonal Experimental Design for the Analysis of Optical Force-induced Displacement ${\boldsymbol u}$ and Gravity-induced Deformation ${\boldsymbol v}$
In Table 2, the larger the value of the optical force-induced displacement u is, the better the amplification of the ATPM displacement response to the optical force will be for this parameter combination. But at the same time, the value of the deformation v induced by gravity is larger which incidents the greater impact of gravity to the ATPM. Larger gravitational deformation causes the ATPM to rotate around the $x$-axis (horizontal axis) applied to the optical force, thus resulting in an increase in the nonlinear error of the displacement response. This reduces the linear amplification of the optical force-induced displacement of the ATPM.
The parameter combination with the best amplification of optical force-induced displacement is A1B1C1 whose u is 84.99 nm. However, the value of v in this set is 46.37 nm, which is the maximum among all combinations. The combination with the smallest deformation induced by gravity is A3B3C3 whose v is 0.10 nm. But the optical force-induced displacement is 0.10 nm, which does not amplify the optical force-induced displacement. The other combinations in Table 2 exhibit an insufficient amplification for the optical force-induced displacement u. The design of the ATPM can be optimized based on the appropriate combination of ${w_0}$, W and h. A range analysis of the results in Table 2 is conducted to establish Tables 3 and 4 to determine the sensitivities of u and v to ${w_0}$, W and h.
Table 3. Range Analysis of Orthogonal Experiments on Optical Force-induced Displacements
Table 4. Range Analysis of Orthogonal Experiments on Gravity-induced Deformations
The sensitivities of the optical force-induced displacement u to ${w_0}$, W and h are listed in Table 3 based on the range analysis. The average K-value is obtained by dividing the sum of all values at a certain level for one factor by three which is the number of data used in the sum process. A larger average K-value in the table indicates that the corresponding structural parameter has a greater impact on the optical force-induced displacement for the given level. In the case where the level equals one, the average K-value of ${w_0}$ is the largest, thus indicating that ${w_0}$ has the greatest influence on u among all three structural parameters. h has the greatest impact when the level equals two, and W has the greatest influence when the level equals three. The average K-value for h decreases with increasing levels, thus indicating that u decreases at increasing h values. For ${w_0}$ and W, the average K-value first decreases and then increases at increasing levels, thus indicating that ${w_0}$ and W have the smallest impact on u when the level equals two. By sorting the average K-values in each column, the combination that maximizes the optical force-induced displacement u is ${w_0} = 1\; \textrm{mm}$, $W = 1\; \textrm{mm}$, and $h = 100\;\ \mathrm{\mu} \textrm{m}$. The range value in the last row is obtained by making a difference between the maximum and minimum values of the average K-value for each factor. It quantifies the sensitivity of u to the structural parameters. Therefore, the ability of the three parameters to enhance the optical force-induced displacement u can be ranked in the order of the decreasing range value as $h$> ${w_0}$> W.
The sensitivities of the gravity-induced deformations v to ${w_0}$, W and h are listed in Table 4. A larger average K-value indicates that the corresponding structural parameter has a greater impact on the gravity-induced deformation v at that level. For the level equals to one, the average K-value of ${w_0}$ is the largest, thus indicating that ${w_0}$ has the greatest impact on v. h has the greatest impact when the level equals two, and W has the greatest impact when the level equals three. The average K-values for ${w_0}$ and h decrease as ${w_0}$ and h increase, thus indicating that larger ${w_0}$ or h values lead to a smaller v value. For W, the average K-value first decreases and then increases as the level increases, thus indicating that W has the smallest impact on v when the level is two. Sorting the average K-values in each column reveals that the combinations with the smallest v are ${w_0} = \textrm{3}\; \textrm{mm}$, $W = \textrm{2}\; \textrm{mm}$, and $h = \textrm{300}\; \mathrm{\mu m}$. According to the range values in the last row, the influence of the three parameters on deformation v can be ranked in the order of the decreasing range values as ${w_0}$> $W$> h.
The optimal combination of the ATPM structural parameters is obtained and is presented in Table 5. Groups 1 and 2 represent the best combinations for optimizing separately the values of u or v based on the range analyses in Tables 3 and 4. The optimal group is a combination obtained by weighted averaging the parameter values of Groups 1 and 2. The weight values of ${w_0}$, W and h are the corresponding range values in the Tables 3 and 4. There are two integer combinations near the optimal combination ${w_0} = \textrm{1}.92\; \textrm{mm}$, $W = \textrm{1}.36\; \textrm{mm}$, $h = \textrm{1}.58\; \textrm{mm}$, which are shown in the Groups 3 and 4. The requirement of the integer is due to the resolution of the fabrication system. Although Group 3 has less gravity-induced deformation, the amplification of the optical force-induced displacement is also negatively affected. Therefore, the optimal combination of structural parameters (Group 4) is determined as ${w_0} = \textrm{2}$ mm, $W = \textrm{1}$ mm, and $h = \textrm{100}\; $µm. For this parameter combination, u is 25.43 nm, and v is 4.25 nm. The optimized ATPM can reduce the gravity-induced deformation while ensuring an amplification of the optical force-induced displacement.
Table 5. Optimized Parameters of ATPM and the Corresponding Simulated Values of ${\boldsymbol u}$ and ${\boldsymbol v}$
3. ATPM performance simulation and fabrication
3.1 ATPM performance simulation
Table 6 shows the optimized structural parameters of the ATPM. The steady-state displacement field $u({x,y} )$ of the ATPM is calculated with the optical force load shown in Fig. 3(b), and the results are shown in Fig. 5.
Fig. 5. Steady-state displacement field $u({x,y} )$ of the ATPM subjected to the optical force. (a) Schematic of the ATPM deformation (a scale factor of $\textrm{5} \times \textrm{1}{\textrm{0}^\textrm{4}}$). The deformation is exaggerated with a scale factor to clearly show the deformed shape. (b) ATPM displacement field driven by the optical force. (c) the displacement field of the central mirror of the ATPM, where r is 4 mm.
Table 6. Parameters of the Optimized ATPM
As shown in Fig. 5(a), when subjected to an optical force, the clamping area of the ATPM remains stationary, the central mirror moves linearly along the $z$-axis, and the three support arms undergo bending deformation. A scalar factor is used to exaggerate the plotted deformation to clearly show the deformed shape. The optical force-induced displacement at the center position $u({\textrm{0},\; \textrm{0}} )= $ 25.43 nm is shown in Fig. 5(b). The shape of the center mirror subjected to the optical force is shown in Fig. 5(c), with a peak-to-valley value (PV value) < 0.5 nm, which maintains good flatness.
The high flatness of the mirror subjected to the optical force is advantageous for improving the stability of the optical force magnitude and direction during laser irradiation. If the irradiated area undergoes significant deformation and produces a lens effect, it can alter the laser beam reflection consequently leading to a reduction in the amplification and linearity of the optical force-induced displacement. A small PV value of the mirror is also suitable for optical interferometric displacement measurements and avoids the geometric errors caused by the deformation.
Figure 6 shows the internal stress distribution of the ATPM when it reaches the maximum steady-state displacement subjected to the optical force shown in the Fig. 3. The stress is mainly distributed on the support arms, which are the main regions where the deformation occurs. The stress in the shear deformation Region A concentrates in a small area and its value surges around the corner reaching $\textrm{6}\textrm{.98} \times \textrm{1}{\textrm{0}^\textrm{3}}\; \textrm{N/}{\textrm{m}^\textrm{2}}$. The stress in the bending deformation Region B is 1718 $\textrm{N/}{\textrm{m}^\textrm{2}}$ on average. The internal stress within the irradiated area of the center mirror is less than 100 $\textrm{N/}{\textrm{m}^\textrm{2}}$, therefore, the mirror does not deform.
Fig. 6. Stress distribution inside the ATPM during deformation. (a) Angled view and (b) stress distribution under the steady-state displacement owing to the optical force (regions A and B indicate the locations with ccentrated shear and bending stresses, respectively).
The displacement response of the center point of the mirror subjected to the combined action of the optical force and gravity is shown in Fig. 7. The curves are as a function of the total applied optical force. For comparison, the curves are normalized by the respective maximum displacement values. The gray dashed line represents the ideal linear optical force-displacement response curve. Smaller difference between the curves and the gray dashed line means the better linearity of the displacement response driven by the optical force. The blue curve in Fig. 7 is obtained by using the parameters from Group 1 in Table 5. It describes the structure before optimization. For this set of parameters, the optical force-induced displacement is the largest, but the linearity of the displacement response is reduced owing to the large deformation induced by gravity. The maximum nonlinear deviation observed at zero applied optical force is 23.25% of the maximum value of the optical force-induced displacement. When the optical force exceeds 260 nN, the blue curve approximately coincides with the ideal linear displacement response curve (gray dashed line). The orange curve represents the displacement response of the optimized ATPM. Its linearity is significantly improved and coincides with gray dashed line when the optical force is greater than 80 nN. The maximal nonlinear deviation reduces to 7.31% at the zero applied optical force. When the optical force is greater than 15 nN, the nonlinear deviation of the orange curve is less than 3.83%. When the optical force is greater than 50 nN, the nonlinear deviation is less than 1.00%. The optimized ATPM performs better in linearly amplification when the optical force is greater than 50 nN. Additionally, the optimal design of the ATPM can resist the deformation induced by gravity at any rotation angle about the z-axis. For the reflectance over 99.90%, the thermal effect will not interfere the optical force-induced displacement in this study.
Fig. 7. ATPM displacement driven by the optical force in the presence of gravity. (a) The blue curve is the ATPM’s displacement response to the applied optical force before optimization, the orange curve is optimized, and the gray dashed line is the ideal linear relationship between the optical force and the induced displacement. (b) Relative non-linear error as a function of the applied optical force. Relative non-linear error is calculated as (actual response - ideal response)/ideal response at the applied optical force.
3.2 ATPM fabrication and internal stress testing
There are two main procedures in the fabrication of the ATPM. The first procedure prepares the ATPM substrate by laser cutting. The second procedure is then coating a highly reflective multilayer on the ATPM substrate. In the first laser fabrication procedure, the equipment we used is a custom femtosecond laser processing platform. The processing laser source is an infrared fiber femtosecond laser (central wavelength 1030 nm, repetition frequency 5–500 kHz, pulse width 285 fs) produced by BWT Laser Tianjin Ltd. The workpiece is placed on a 3-axis displacement stage with a resolution of 500 µm. With the femtosecond laser processing platform [36,37], the ATPM is cut from a single piece of thin planar soda-lime glass. We describe the soda-lime glass as the workpiece in the following content. There are four steps in the laser cutting procedure. The first step of laser cutting is to clean the workpiece to avoid the dust causing damages during cutting. The second step is to place the workpiece on the starting point of the cutting processing trajectory on the displacement stage. The third step is to turn on the femtosecond laser to start the cutting. The displacement stage will move the workpiece along the preset processing trajectory. There are two kinds of processing trajectories. One is used for the complete ATPM’s structure, as shown in Fig. 2(b). The other one is a single circle with a diameter of 20 mm which is used for the unstructured mirror in the control experiment in the Raman spectra test (Fig. 9) and the optical force experiment (Fig. 12). After the cutting, the fourth step is to retrieve the ATPM substrate from the workpiece and settle it in the drying cabinet for 24 hours. This step is to release the residual stress introduced in the laser cutting before the coating procedure.
In the second procedure, the ATPM substrate is coated by the Ion-Beam Assisted Deposition (IBAD). The coating is a multilayer Bragg reflective coating (IDTE, Tianjin, China; custom made). It is composed of SiO2 and ZrO2 with refractive indices of ${n_s} = $1.46099 and ${n_z} = $1.92638 at the wavelength of 527 nm. The top layer of the coating is a 180.36 nm thick SiO2 layer followed by alternating layers of 68.39 nm thick ZrO2 layers and 90.18 nm thick SiO2 layers. The total thickness of the coated layers is 3193.21 nm. The machine is programmed to coat automatically according to the preset parameters. After coating, the ATPM is complete and needed to be settled in the drying cabinet for 48 hours to release the residual stress introduced in the coating procedure.
The ATPM substrate cut from a soda-lime glass with the optimized structural parameters is shown in Fig. 8(a). The coated ATPM is shown in Fig. 8(b). Figure 8(c) illustrates the ATPM mounted on the fixture described in Section 2.1.
Fig. 8. ATPM fabrication. (a) ATPM substrate, (b) coated ATPM, (c) and the ATPM mounted in the fixture.
To minimize the subsurface damage and ensure the surface integrity, it is important to avoid introducing additional stress to the support arms during fabrication. It reduces the nonlinear effects of the ATPM displacement response to the applied optical force, decreases elastic hysteresis, and ultimately increases the lifespan of the ATPM.
The residual stress introduced during fabrication may affect the internal stress distribution of the ATPM subjected to an optical force, which in turn affects the amplification and linearity of the optical force-induced displacement. The residual stress causes changes in the lattice of the sample, thereby influencing the spectrum of the scattered light measured by Raman spectrometer. By measuring the shift of the Raman spectra, we can characterize the residual stress in the sample [38]. Raman spectral intensity testing is conducted at the location of the stress concentration in the ATPM in depth of 50 µm before and after processing, as shown in Fig. 9. Figure 9(a) shows the Raman spectral intensity at the center of the uncoated ATPM before fabrication, which is farthest from the processing trajectory and serves as the reference curve for the internal stress. Figures 9(b) and 9(d) show the test results at positions #1 and #3, respectively. These two positions are the areas of internal stress concentration of the ATPM subjected to the optical force, as shown in Region A of Fig. 6. No Raman peak wavelength shifts occur at positions #1 and #3 where the mirror and support arms are joined, thus indicating the cutting and coating procedures do not introduce additional stress to the shear deformation area responding to the optical force. The Raman spectral intensity at the middle position (position #2) of the coated ATPM support arm after fabrication remains unchanged compared with the baseline, thus indicating no additional stress introduces to the bending area responding to the optical force.
Fig. 9. Comparison of internal stress within the elastic deformation region of the ATPM support arm (a) before and (b) - (d) after fabrication. (e) is a photo of the Raman spectra test of the ATPM
4. Experimental setup and results
4.1 Experimental setup
Experimental setup for measuring the optical force-induced displacement of the ATPM is shown in Fig. 10. A high-power laser system A (model AWAVE-527-20W, Advanced Optowave, US) is used to generate the laser beam, which is then directed through the Z-type alignment optical path B. The beam passes through aperture C and half-wave plate D before irradiating the ATPM mounted in the fixture F. The reflected light is absorbed by the absorber E, and the optical force-induced displacement of the ATPM is measured by a laser interferometer G with a wavelength of 632 nm.
Fig. 10. Schematic of the optical path in the experiment. (A) Laser source, (B) Z-type alignment optical path, (C) aperture, (D) 1/2 wave plate, (E) absorber, (F) ATPM with a fixture, (G) laser interferometer, and (H) five-axis displacement stage.
Optical path B is used to adjust the direction of the incident laser beam, and aperture C blocks the stray light and ensures that the laser irradiates only the center position of the ATPM to prevent thermal expansion of the fixture. The half-wave plate D is used to precisely adjust the polarization direction of the laser beam incident on the ATPM. For the laser irradiation with an incident angle of $\textrm{7}\mathrm{.5^\circ }$ in this experiment, the average value of the optical force generated by the laser in different polarization directions is considered to be the same within the same time frame (continuous laser or pulsed laser with a pulse width larger than nano second) [39] and thus, the polarization direction of the incident laser beam does not affect the optical force-induced displacement, which was experimentally verified. The five-axis displacement stage H can accurately adjust the posture of the ATPM to resist the effect of gravity based on the optimized design of the ATPM. The optical force-induced displacement of the ATPM is measured by a laser interferometer (model SIOS SP2000, Germany), which aims at the center position of the mirror in the ATPM. The laser interferometer is calibrated by the Physikalisch-Technische Bundesanstalt (PTB) to ensure the traceability of the displacement measurement.
The value and its uncertainty of the average optical force exerting on the ATPM in this experiment are calculated using Eq. (4). The uncertainty of the optical force is composed of the measurement uncertainties of the laser beam power P, incident angle $\theta $, air refractive index n, and reflectance ${R_f}$. The laser system generates linearly polarized laser with a wavelength of 527 nm, and the geometric parameters of the laser beam is shown in the Table 1 in the Section 2.1. The maximum average power P is $\textrm{20} \pm \textrm{0}.2\; \textrm{W}$. The measurement of the incident angle $\theta $ is converted into a length measurement using the inverse trigonometric function relationship by the equation of $\textrm{2}\theta = \textrm{arccos}({LR/LH} )$, where $LR{\; }$ is the path length of the incident beam (as the right-angle side of the right triangle) and $LH$ is that of the reflected beam (as the hypotenuse of the right triangle) on the same horizontal plane. The length measurement results are $LR = \textrm{521}\textrm{.5} \pm \textrm{0}.5\; \textrm{mm}$, $LH = \textrm{539}\textrm{.9} \pm \textrm{0}.5\; \textrm{mm}$, and $\theta $ is calculated to be $\textrm{7}\textrm{.50} \pm \textrm{0}\mathrm{.21^\circ }$. The air refractive index n is measured with the laser interferometer’s environmental compensation unit, the result is $n = \textrm{1}\textrm{.00029} \pm \textrm{0}\textrm{.00010}$.
The reflectance, ${R_f}$ is measured for the $\textrm{7}\mathrm{.5^\circ }$ incident angle of laser at the wavelength from 400 nm to 700 nm using a spectrophotometer (model MSP-100B, INOVIA, Turkey), and the results are shown in Fig. 11. Some of the measurement values at points between 500 nm and 570 nm exceed 100% because the reflectance of the ATPM coating is higher than that of the standard mirror (with a nominal reflectance of 99.90%) used in instrument calibration. The ATPM coating has a designed reflectance of 99.99% at the wavelength of 527 nm with an incident angle of $\textrm{7}\mathrm{.5^\circ }$. In this study, the measured reflectance of ATPM at 527 nm is considered to be $\textrm{99}\textrm{.90\%} \pm \textrm{0}\textrm{.10\%}$, which belongs to Type B uncertainty estimation. The experimental temperature is $\textrm{20} \pm \textrm{0}\textrm{.5}\mathrm{\circ{C}}$, and the humidity is $\textrm{50\%} \pm \textrm{1\%}$.
According to Table 7, the calculation of the average optical force and evaluation of the standard uncertainty are performed [40]. The average value of the optical force in the experiment is 132.17 nN, standard uncertainty is 2.21 nN, expanded uncertainty is 4.42 nN with a confidence factor of two, and the confidence interval is 95.45%. Therefore, the maximal optical force in the experimental setup is $\textrm{132}\textrm{.17} \pm \textrm{4}.42\; \textrm{nN}$.
Table 7. Elements Used to Evaluate the Uncertainty of Optical Force (Type B Uncertainty)
4.2 Experimental results
An experimental comparison is conducted between two structures, namely an unstructured thin plane mirror and the ATPM, to verify the optical force-induced displacement amplification effect of the ATPM. The two structures have the same outer radius R, thickness h, substrate material, and coating. Figure 12 shows the displacement measurement results for the two structures subjected to an optical force of $\textrm{137}\textrm{.17} \pm \textrm{4}.42\; \textrm{nN}$.
Fig. 12. Measured displacements of the unstructured thin plane mirror and the ATPM subjected to the optical force with the density distribution shown in Fig. 3(b). The gray curves represent the raw data, while the black curves represent the data after moving average processing. (a) Measured displacement during and after laser irradiation on the unstructured thin plane mirror. (b) Measured displacement of the ATPM during and after laser irradiation. The light-blue dotted line represents the time-varying profile of the incident laser power. (c) Distribution of errors introduced by environmental noise in the interferometric measurement of ATPM displacement in the absence of laser irradiation. (d) Unstructured thin plane mirror (left) and ATPM (right).
Figure 12(a) shows the measured displacement results for an unstructured thin plane mirror subjected to laser irradiation for 0.5 s (segment of the displacement trace between the vertical red dashed lines). The sampling frequency is 1000 Hz. The raw data (gray curve) reveals that no unidirectional movement exceeds the noise during continuous laser irradiation on the unstructured mirror. The low-pass-filtered data (black curve) reveals the vibrations with amplitudes less than 2 nm. The maximum vibration value is 1.64 nm, which is consistent with the optical force-induced displacements reported in the literature for thin plane mirrors of similar sizes [30].
Figure 12(b) shows the results of the ATPM’s optical force-induced displacement experiment. The raw data (gray curve) indicates that the mirror in the ATPM undergoes a unidirectional displacement greater than the noise level during laser irradiation. According to the experimental setup shown in Fig. 10, when the mirror moves along the positive $z$-axis, the distance between the mirror and the interferometer decreases, and the displacement data recorded by the interferometer are negative. The measured displacement results in Fig. 12(b) verify that the ATPM’s optical force-induced displacement direction is consistent with the optical force direction. To obtain the steady-state displacement response, the laser power is set to gradually increase linearly to 20 W and is sustained until the ATPM’s displacement response no longer increases. The time profile of the incident laser power is indicated by the light-blue dotted line in Fig. 12(b). The low-pass filtered data (black curve) in Fig. 12(b) indicates that the maximal steady-state displacement of the ATPM under a laser beam power of 20 W is 24.90 nm.
The displacement measurement uncertainty comes from the performance of the interferometer (nominal ${\pm} 20\; \textrm{pm}$), stability of the experimental optical path, and environmental noise. In this study, the uncertainty of the displacement measurement is determined by environmental noise, which belongs to Type A uncertainty. Figure 12(c) shows the distribution of the measured environmental noise results obtained from the displacement measurement of the ATPM in a stationary state without laser irradiation with a sample size of 1500. The standard deviation of the environmental noise data is 1.21 nm. As shown in Fig. 12(c), the environmental noise data approximately follows a normal distribution. A confidence factor of two and a confidence level of 95.45% are chosen to evaluate the dispersion of the noise data. The expanded uncertainty of displacement measurement introduced by environmental noise is ${\pm} \textrm{2}.42\; \textrm{nm}$. More information about the uncertainty analysis for displacement sensors can be found in the Ref. [41]. Therefore, the steady-state displacement of the ATPM subjected to an optical force of $\textrm{132}\textrm{.17} \pm \textrm{4}.42\; \textrm{nN}$ is $\textrm{24}\textrm{.90} \pm \textrm{2}.42\; \textrm{nm}$.
By comparing the displacement measurement results of the unstructured thin plane mirror and the ATPM subjected to the same optical force, it demonstrates that the ATPM can effectively amplify the displacement response subjected to an optical force of 132.17 nN with a maximum amplification factor of 15.18.
Figure 13(a) shows that the ten repeated measured displacements (red data points) and the average of the measured displacement (blue data points) exhibit a linear relationship with the magnitude of the optical force, which is consistent with the simulation results within the range of displacement measurement uncertainty. Figure 13(b) shows a scatter plot of the residuals versus the optical force, where the residuals indicate the deviation of the measured displacement from the ideal linear relationship between the optical force and displacement.
Fig. 13. Measured steady-state displacements of the ATPM subjected to varying optical force. (a) Repeatedly measured individual responses (red data points), average measured response (blue data points), FEM simulated response (red dotted line), and ideal linear response (gray dashed line) (b) Residuals between the average measured response and the ideal linear response.
Based on the nonlinearity of ATPM displacement response to the applied optical force, as shown in Table 8, it can be observed that as the optical force-induced displacement increases, the nonlinearity gradually decreases. This outcome is consistent with the trend of nonlinearity observed in the simulation results shown in Fig. 7(b). In the experiment, when the optical force is greater than 100.07 nN, the nonlinearity of the optical force-induced displacement response is less than 8.87%, and when the optical force is greater than 132.17 nN, the nonlinearity is less than 6.28%. The effective compliance k can be calculated as 5.47 N/m based on the optical force of 132.17 nN and the average displacement response of 24.18 nm.
Table 8. Experimental Results and the Residuals of the ATPM Displacement Response
5. Conclusion and prospects
ATPM is a novel type of optical force-sensitive macroscopic mirror based on the principle of elastic support. It can effectively amplify the optical force-induced displacement compared to the unstructured mirror in normal gravity environments. The conclusions of this study are as follows.
- 1) The ATPM adopts a rotationally symmetric planar spring structure. The three support arms are 120° apart. This configuration increases the structural stability of the ATPM in the presence of gravity. The mirror in the ATPM moves in a nearly straight line subjected to an optical force, thus ensuring the good linearity of the optical force-induced displacement response of the ATPM.
- 2) The optimal design of the ATPM structural parameters is conducted using orthogonal simulation experiments and range analysis. The optimized ATPM ensures a good amplification and linearity of the optical force-induced displacement. An experimental platform for the measurement of steady-state displacement is established to generate optical forces and verify the performance of the ATPM. Uncertainty analysis reveals that the average value and uncertainty of the optical force are 132.17 nN and 4.42 nN, respectively.
- 3) The measured displacement response of the ATPM subjected to the optical force from an incident laser beam power of 20 W is $\textrm{24}\textrm{.90} \pm \textrm{2}.42\; \textrm{nm}$. The average value of ten repeated optical-induced steady-state displacement measurements is 24.18 nm for the maximal laser power of 20 W. These results are consistent with the simulation result of 25.43 nm within the uncertainty range. The experimental results show that the nonlinear deviation of the optical force-induced displacement of the ATPM is less than 6.28% when the optical force is greater than 133 nN.
The ATPM has a diameter of 20 mm, it is smaller than the previously reported devices [30,31,33,34] and facilitates the miniaturization of the macroscale optical force platform. The fabrication of ATPM contains two procedures commonly used in the optical device fabrication which indicates the feasibility of mass production and cost-effective advantages. The ATPM can serve as a sensitive element for optical force sensors applied in the laser-matter interaction research and applications. The sensitivity of the physical parameters can refer to the sensitivity coefficients shown in the Table 7. The ATPM can be employed in the in-situ laser power measurement in the high-power laser assisted machining system [42]. It is smaller than the conventionally used optical power meter in size and weight which can be integrated in the system and reduced the total mass of the laser assisted machining tools. The ATPM measures the laser power based on the optical force generated from the high reflectance which incidents the fast response and free of thermal degradation. ATPM will be further applied to provide small force source in the atomic and close-to-atomic scale manufacturing [43]. The flexibility of the structure parameters of the ATPM (as shown in Table 2) enriches the possibilities of extension applications.
Funding
National Natural Science Foundation of China (52035009); Ministry of Education of the People's Republic of China (B07014).
Acknowledgments
Thanks extend to Jinshi Wang, Peng Lyu, Kaiyuan You, Kun Zhang, Bing Dong, Yunxuan Li, Yuxing Niu, Shufan Chen and Yukun Yuan for the discussion of the ATPM structure model and fabrication method, and to Jinyan Lyu, Changkun Shi and Bing Dong for the support to the experiments.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding author upon a reasonable request.
References
1. D. Gao, W. Ding, M. Nieto-Vesperinas, X. Ding, M. Rahman, T. Zhang, C. Lim, and C.-W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light: Sci. Appl. 6(9), e17039 (2017). [CrossRef]
2. P. Zemánek, G. Volpe, A. Jonáš, and O. Brzobohatỳ, “Perspective on light-induced transport of particles: from optical forces to phoretic motion,” Adv. Opt. Photonics 11(3), 577–678 (2019). [CrossRef]
3. D. Kiracofe, K. Kobayashi, A. Labuda, A. Raman, and H. Yamada, “High efficiency laser photothermal excitation of microcantilever vibrations in air and liquids,” Rev. Sci. Instrum. 82(1), 013702 (2011). [CrossRef]
4. V. Pini, B. Tiribilli, C. M. C. Gambi, and M. Vassalli, “Dynamical characterization of vibrating AFM cantilevers forced by photothermal excitation,” Phys. Rev. B 81(5), 054302 (2010). [CrossRef]
5. V. Koskinen, J. Fonsen, K. Roth, and J. Kauppinen, “Progress in cantilever enhanced photoacoustic spectroscopy,” Vib. Spectrosc. 48(1), 16–21 (2008). [CrossRef]
6. W. Holzapfel, S. Neuschaefer-Rube, and M. Kobusch, “High-resolution, very broadband force measurements by solid-state laser transducers,” Measurement 28(4), 277–291 (2000). [CrossRef]
7. T. Požar, J. Laloš, A. Babnik, R. Petkovšek, M. Bethune-Waddell, K. J. Chau, G. V. Lukasievicz, and N. G. Astrath, “Isolated detection of elastic waves driven by the momentum of light,” Nat. Commun. 9(1), 3340 (2018). [CrossRef]
8. S. M. Kharlamov, V. V. Guzanov, A. V. Bobylev, S. V. Alekseenko, and D. M. Markovich, “The transition from two-dimensional to three-dimensional waves in falling liquid films: Wave patterns and transverse redistribution of local flow rates,” Phys. Fluids 27(11), 114106 (2015). [CrossRef]
9. J. Lu, Q. Li, C.-W. Qiu, Y. Hong, P. Ghosh, and M. Qiu, “Nanoscale Lamb wave–driven motors in nonliquid environments,” Sci. Adv. 5(3), eaau8271 (2019). [CrossRef]
10. S. Chu, “Laser Manipulation of Atoms and Particles,” Science 253(5022), 861–866 (1991). [CrossRef]
11. A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature 400(6747), 849–852 (1999). [CrossRef]
12. T. A. Nieminen, “Trapping ions,” Nat. Photonics 4(11), 737–738 (2010). [CrossRef]
13. Y. Shi, Q. Song, I. Toftul, T. Zhu, Y. Yu, W. Zhu, D. P. Tsai, Y. Kivshar, and A. Q. Liu, “Optical manipulation with metamaterial structures,” Appl. Phys. Rev. 9(3), 031303 (2022). [CrossRef]
14. Y. Ren, Q. Chen, M. He, X. Zhang, H. Qi, and Y. Yan, “Plasmonic Optical Tweezers for Particle Manipulation: Principles, Methods, and Applications,” ACS Nano 15(4), 6105–6128 (2021). [CrossRef]
15. Y. Shi, T. Zhu, T. Zhang, A. Mazzulla, D. P. Tsai, W. Ding, A. Q. Liu, G. Cipparrone, J. J. Sáenz, and C.-W. Qiu, “Chirality-assisted lateral momentum transfer for bidirectional enantioselective separation,” Light: Sci. Appl. 9(1), 62 (2020). [CrossRef]
16. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Laser-Scanning Micromanipulation and Spatial Patterning of Fine Particles,” Jpn. J. Appl. Phys. 30(Part 2, No. 5B), L907–L909 (1991). [CrossRef]
17. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett. 16(19), 1463 (1991). [CrossRef]
18. G. Thalhammer, R. Steiger, S. Bernet, and M. Ritsch-Marte, “Optical macro-tweezers: trapping of highly motile micro-organisms,” J. Opt. 13(4), 044024 (2011). [CrossRef]
19. H. Xin, R. Xu, and B. Li, “Optical formation and manipulation of particle and cell patterns using a tapered optical fiber: Optical formation and manipulation of particle/cell patterns,” Laser Photonics Rev. 7(5), 801–809 (2013). [CrossRef]
20. D. G. Grier and Y. Roichman, “Holographic optical trapping,” Appl. Opt. 45(5), 880–887 (2006). [CrossRef]
21. P. R. Huft, J. D. Kolbow, J. T. Thweatt, and N. C. Lindquist, “Holographic Plasmonic Nanotweezers for Dynamic Trapping and Manipulation,” Nano Lett. 17(12), 7920–7925 (2017). [CrossRef]
22. Y. Yuan, C. Gu, S. Huang, L. Song, and F. Fang, “Advances on studying optical forces: optical manipulation, optical cooling and light induced dynamics,” J. Phys. Appl. Phys. 53(28), 283001 (2020). [CrossRef]
23. M. Mansuripur, “Resolution of the Abraham–Minkowski controversy,” Opt. Commun. 283(10), 1997–2005 (2010). [CrossRef]
24. P. Williams, B. Simonds, J. Sowards, and J. Hadler, “Measuring laser power as a force: A new paradigm to accurately monitor optical power during laser-based machining operations,” Proc. SPIE 9741, 97410L (2016). [CrossRef]
25. E. Manske, T. Fröhlich, and S. Vasilyan, “Photon momentum induced precision small forces: a static and dynamic check,” Meas. Sci. Technol. 30(10), 105004 (2019). [CrossRef]
26. R. V. Jones and B. Leslie, “The measurement of optical radiation pressure in dispersive media,” Proc. R. Soc. Lond. A 360(1702), 347–363 (1978). [CrossRef]
27. K. Agatsuma, D. Friedrich, S. Ballmer, G. DeSalvo, S. Sakata, E. Nishida, and S. Kawamura, “Precise measurement of laser power using an optomechanical system,” Opt. Express 22(2), 2013 (2014). [CrossRef]
28. P. R. Wilkinson, G. A. Shaw, and J. R. Pratt, “Determination of a cantilever’s mechanical impedance using photon momentum,” Appl. Phys. Lett. 102(18), 184103 (2013). [CrossRef]
29. D. Ma, J. L. Garrett, and J. N. Munday, “Quantitative measurement of radiation pressure on a microcantilever in ambient environment,” Appl. Phys. Lett. 106(9), 091107 (2015). [CrossRef]
30. Y. Yuan, C. Gu, S. Huang, L. Song, G. Yan, Z. Li, and F. Fang, “Study of laser induced dynamics on macroscopic mirror,” J. Opt. 22, 035402 (2020). [CrossRef]
31. Y. Yuan, C. Gu, Y. Cao, S. Wang, and F. Fang, “Observation of radiation pressure induced deformation of high-reflective reflector,” J. Phys. Commun. 1(5), 055031 (2017). [CrossRef]
32. I. Ryger, A. B. Artusio-Glimpse, P. Williams, N. Tomlin, M. Stephens, K. Rogers, M. Spidell, and J. Lehman, “Micromachined Force Scale for Optical Power Measurement by Radiation Pressure Sensing,” IEEE Sens. J. 18(19), 7941–7948 (2018). [CrossRef]
33. P. Pinot and Z. Silvestri, “Optical power meter using radiation pressure measurement,” Measurement 131, 109–119 (2019). [CrossRef]
34. M. Partanen, H. Lee, and K. Oh, “Radiation pressure measurement using a macroscopic oscillator in an ambient environment,” Sci. Rep. 10(1), 20419 (2020). [CrossRef]
35. “COMSOL Multiphysics,” retrieved https://www.comsol.com.
36. K. Zhang, Z. Xu, B. Dong, and S. Zhang, “Process exploration of β-Ga2O3 blind hole processing by water-assisted femtosecond laser technology,” J. Alloys Compd. 939, 168769 (2023). [CrossRef]
37. B. Dong, Z. Xu, C. Shi, K. Zhang, Y. Zhang, R. Hua, W. Zhao, and J. Wang, “High-quality micro/nano structures of 4H-SiC patterning by vector femtosecond laser,” Opt. Laser Technol. 163, 109338 (2023). [CrossRef]
38. Y. Kang, Y. Qiu, Z. Lei, and M. Hu, “An application of Raman spectroscopy on the measurement of residual stress in porous silicon,” Opt. Lasers Eng. 43(8), 847–855 (2005). [CrossRef]
39. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12(22), 5375 (2004). [CrossRef]
40. “Guide to the Expression of Uncertainty in Measurement,” ISO/IEC 98:1993(E).
41. A. J. Fleming, “A review of nanometer resolution position sensors: Operation and performance,” Sens. Actuators, A 190, 106–126 (2013). [CrossRef]
42. K. You and F. Fang, “High effective laser assisted diamond turning of binderless tungsten carbide,” J. Mater. Process. Technol. 302, 117505 (2022). [CrossRef]
43. J. Wang, F. Fang, H. An, S. Wu, H. Qi, Y. Cai, and G. Guo, “Laser machining fundamentals: micro, nano, atomic and close-to-atomic scales,” Int. J. Extreme Manuf. 5(1), 012005 (2023). [CrossRef]
