# Optical force balance for measurement in the nanonewton range

Simon Noviant, Colin Dartevelle, Zaccaria Silvestri, Alain Vissière

► To cite this version:

Simon Noviant, Colin Dartevelle, Zaccaria Silvestri, Alain Vissière. Optical force balance for measurement in the nanonewton range. Measurement - Journal of the International Measurement Confederation (IMEKO), 2026, 259, pp.119746. 10.1016/j.measurement.2025.119746. hal-05374727

HAL Id: hal-05374727

https://cnam.hal.science/hal-05374727v1

Submitted on 3 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Optical force balance for measurement in the nanonewton range☆

Simon Noviant* ID, Colin Dartelle ID, Zaccaria Silvestri ID, Alain Vissière

Laboratoire Commun de Métrologie LNE-CNAM, 1 Rue Gaston Boissier, Paris 75015, France,

## ARTICLE INFO

**Keywords:**

- Balance
- Comparator
- Magnetism
- Graphite pyrolytic
- High sensitivity
- Low mass
- Low force

## ABSTRACT

The redefinition of the kilogram in 2019, based on the Planck constant *h*, has led to new opportunities for innovation in metrology, particularly in the field of mass and related quantities. Responding to growing industrial and academic needs for the measurement of small forces and masses, several projects have been launched at LNE-CNAM. Among them, the 2023 project "Laser Power Measurement" (pPyC) led to the development of a traceable measurement system capable of generating an optical force with a relative standard uncertainty of 1 % in the nanonewton range.

Building on the expertise gained, a prototype of the first optical force balance has been developed in 2022, aimed at measuring very small forces (1 nN–100 nN). This prototype is based on the diamagnetic properties of pyrolytic graphite to create the conditions necessary for the levitation of a balance beam around a fixed axis. The creation of a contactless pivot enables very high sensitivity. The experimental results showed the balance’s sensitivity to be 13.5 µrad/nN, with a repeatability of 0.2 µrad/nN.

## 1. Introduction

Since the redefinition of the kilogram based on the Planck constant, the field of low-mass metrology and related quantities has opened up to numerous prospects for innovation. Several international strategic reports have highlighted the increasing demand for low-forces measurements below 1 N in both industrial and academic sectors [1].

Indeed, the demand for development in the field of small forces and masses continues to grow. These demands covering various fields such as biology [2,3] and biomechanics, where there is a need to characterize cells (stiffness [4], mass, and volume). Micro-electromechanical systems (MEMS) can enable this characterization, and traceable measurements of small forces and masses help to improve the reliability of these measurements [5,6].

In the field of Atomic Force Microscopy (AFM) instrumentation, measurement techniques are becoming increasingly widespread. For instance, in fundamental research, Casimir forces have been studied using AFM [7]. The force involved range from the piconewton to the micronewton scale. One of metrological traceability challenges is the determination of AFM cantilever stiffness. Between 2008 and 2010, national metrology institutes, including KRISS (South Korea), NIST (USA), NPL (UK), and PTB (Germany), participated in the first international comparison focused on micronewton-scale force measurements.

The objective was to measure the stiffness constant of five force sensors similar to AFM cantilevers [8]. Other application domains can also be cited, such as healthcare [9], where increasingly precise mass measurements are required, as well as micromanipulation [5,6] fields like MEMS assembly and robotic surgery.

Several national metrology institutes have already begun to innovate in the field of small force measurement (1 nN – 1 mN). For example, PTB has launched the Planck-Balance 2 (PB2) project [10] in the micronewton domain. In the nanonewton domain, laboratories such as KRISS [11], NIST [12], and PTB [13,14] have developed systems to calibrate AFM cantilevers using mass comparators with resolutions reaching down to 1 nN. In parallel, new methods for generating low forces below the micronewton level using laser radiation pressure are being developed, notably by NIST [15]. However, the field of low forces remains largely underexploited, with measurement uncertainties still too high. As of now, no national metrology institute has a Calibration and Measurement Capability (CMC) recognized by the BIPM for forces below 0.1 N.

At LNE-CNAM, the pPyC project [16] based on torsion pendulum [17], enabled the measurement of laser power ranging from 100 mW to 5 W, corresponding to forces in the nanonewton range. Building on the outcomes of this project and considering the growing scientific and industrial needs, the design of the first optical force balance was initiated. Its objective is to measure small forces and masses (1 nN–100 nN) with a

<footnote>* This article is part of a special issue entitled: 'MEASUR_XXIV IMEKO World Congress' published in Measurement.</footnote>

<footnote>* Corresponding author.
E-mail address: simon.noviant@lecnam.net (S. Noviant).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Functional diagram of the NanoPyC balance with an optical load.</figcaption></figure>

standard uncertainty below 1 % [18]. Eventually, this system could
become a new reference in the field of low forces, particularly through the
establishment of new CMCs.

Section 2 of this article presents the first functional version of the project, followed by Section 3, which discusses the results obtained and outlines future developments and perspectives for the optical balance concept.

## 2. Optical force balance

This section will present the general principle of the optical balance
named NanoPyC, focusing on its two most important aspects: the crea-
tion of a magnetic pivot and the design of a reference force generator. It
will conclude by addressing the issue of system traceability.

### 2.1. *Theoretical principle of the balance*

The optical balance is based on the diamagnetic properties of pyro-
lytic graphite (PyC). These properties, combined with neodymium-iron-
boron (NdFeB) permanent magnets, enable the stable levitation of a

beam rotating around an axis. This configuration, theoretically free from
any restoring force, results in an almost zero stiffness around the axis
(see section 2.2). By using the high sensitivity of the revolute joint, it is
possible to design a balance system that acts as a comparator, with the
applied force to be measured on one side and an equivalent, traceable
force on the other side, with a relative standard uncertainty of less than
1 %.

In this article, we focus on a configuration of the optical balance
(Fig. 1) in which the load to be measured ($\vec{F}_a$) is applied using radiation
pressure. This force is generated by an optical force generator (see sec-
tion 2.3), which includes a laser source with a maximum power of 5 W.
The applied force can be modulated using an acousto-optic modulator,
while a power meter continuously measures it.

To simulate the application of a load, a laser beam corresponding to
the force $\vec{F}_a$ is directed onto a dielectric mirror attached to the balance
beam (placed approximately 50 mm from the centre of the revolute
joint). In doing so, it exerts a torque on the balance beam, inducing an
angular displacement around the $\vec{x}$ axis, denoted as $\alpha_x$.

To maintain the beam in its initial state ($\alpha_x = 0$ rad), that is, in its

<figure><img src="image_4.png" /><figcaption>Fig. 2. Kinematic diagram of the beam. The dashed part illustrates its behavior when a force $F_a$ is applied at point A.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Basic principle of the revolute joint around the x-axis.</figcaption></figure>

equilibrium position when no force $\vec{F}_a$ is applied, a reference force $\vec{F}_r$ is projected onto a second mirror attached to the balance beam. $\vec{F}_r$ is colinear with $\vec{F}_a$ but in the opposite direction. Since the amplitude of $\vec{F}_r$ is determined by a traceable optical measurement system (pPyC), it is possible to determine the applied load on the balance beam. Details on traceability will be discussed in section 2.4.

To ensure static equilibrium between the two forces, a feedback control of the reference force is required. The adjustment of $\vec{F}_r$ during the measurement is made possible by knowing the angular displacement $\alpha_x$ using an autocollimator system. The measured fluctuation of the reference force $F_r$ will allow the evaluation of the applied load $F_a$ during a test.

## 2.2. Principle of implementing a zero-stiffness revolute joint

The principle of this force comparator is based on the creation of a magnetic revolute joint with near-zero stiffness. To achieve this, the diamagnetic properties of PyC are used. This can be quantified by its magnetic susceptibility value, $\chi_m = 4 \times 10^{-4}$. By exploiting this property and using the magnetic force of NdFeB magnets, it is possible to design a contactless pivot with near-zero stiffness. We can therefore consider that

the beam is in equilibrium around a pivot joint, which will subsequently be regarded as perfect. The system is thus analogous to a simple heavy pendulum, where the restoring force is due to the lower position of the center of gravity relative to the pivot point. The kinematic diagram in Fig. 2 describes the operation of the beam when it is subjected to a force $F_a$.

By applying a force ($F_a$) with a lever arm $l$ on the balance beam, the balance beam is set into motion. Due to the offset between the balance beam's center of gravity and its axis of rotation, a restoring force appears in the direction opposite to the applied force, creating a new equilibrium state. A force analysis then yields the following equation:

$$F_a \cos(\alpha_x)l = mgsin(\alpha_x)d \quad (1)$$

With $m$ as the mass of the balance beam, $g$ as the gravitational constant, $l$ as the distance of the lever arm and $d$ as the distance between the center of rotation O and the center of gravity G. For calculation details, refer to the classical solution of a simple heavy pendulum.

If we assume the small-angle approximation ($\alpha_x < 200 \mu rad$), then the equation becomes:

$$F_a l = mg\alpha_x d \quad (2)$$

We can then consider that the restoring force is proportional to $\alpha_x$,

<figure><img src="image_6.png" /><figcaption>Fig. 4. Reference optical force generator.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5. Optical force generator calibration.</figcaption></figure>

which means that the system's sensitivity, defined by Eq. (3), is constant. The restoring force is also proportional to the length of the beam's lever arm and inversely proportional to its mass and to the distance of its center of gravity from the pivot point.

$$s_x = \alpha_x/F_a \quad (3)$$

The assumption of constant sensitivity will be verified during the experimental tests in section 3.2.

Fig. 3 details the main components enabling the creation of this

revolute joint. The lifting magnets exert a magnetic force on smaller magnets (floating magnets) attached to the balance beam; this force maintains the balance beam in levitation by compensating for its weight. PyC is naturally repelled by external magnetic fields, which allows the formation of magnetic potential wells resulting in the appearance of stable equilibrium points for the levitating system [19].

The structure of the balance beam allows forces to be applied with a defined lever arm and its motion to be characterised through the implementation of mirrors.

<figure><img src="image_8.png" /><figcaption>Fig. 6. Calibration curve of the ratio $R_k$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 7. Traceability diagram of the laser power measurement system developed at LNE-CNAM.</figcaption></figure>

## 2.3. Reference optical force generator

Another important component of the optical force balance is its reference optical force generator, which enables the injection of a compensating force that is potentially traceable. The idea is to split the laser beam into two parts: one that will be continuously measured and the other that will apply the force. To ensure the traceability of this system, it is important that it has a constant and stable splitting ratio over time. A study on its stability is presented below. Fig. 4 details its main components.

This system consists of a stabilised laser source (KEOPSYS laser, model CYFL-TERA-10-LP-1064) with a wavelength of 1064 nm, capable of generating a continuous laser beam with a maximum power of 5 W. The laser output is fiber-coupled and directly connected to an acousto-optic modulator (AOM from AA Opto-electronic, model MT80-IR60-Fio-PM5-J1-A-VSF), which adjusts the output power via a voltage control signal ranging from 0 to 5 V, with a response time of less than 1 µs. The beam at the output of the AOM is collimated to a diameter of approximately 2 mm. It is then injected into the polarizer to fix the polarization at the output. The beam is split by a 90/10 beam splitter cube. The weaker portion (10 %) is directed to the reference power meter (Thorlabs S144C, referred to as PM<sub>Thr</sub>), while the stronger portion (90 %) passes through a focusing lens and is reflected off the balance beam's mirror. The focusing lens reduces the beam diameter from 2 mm to 0.5 mm at the mirror, which is necessary to minimize warming of elements near the mirror (such as the beam structure in Polylactic acid (PLA), PyC, etc.). By intentionally creating an angle of incidence (θ) at the beam balance mirror (see Fig. 4), we ensure that the reflected beam is directed into an external laser trap outside the balance (not shown in the diagram). This configuration allows us to significantly reduce heating caused by the laser power.

The power at the output of the beam splitter (10 % side), measured with the power meter PM<sub>Thr</sub>, is denoted P<sub>Thr.d</sub>. As a reminder, the power injected onto the beam, denoted P<sub>r</sub>, corresponds to the power at the 90 % output of the optical force generator. The relationship between P<sub>r</sub> and P<sub>Thr.d</sub> is defined by the following equation:

$$P_r = R_k \times P_{Thr.d} \quad (4)$$

where R<sub>k</sub> is the splitting ratio measured at both outputs of the optical force generator during its calibration, and P<sub>Thr.d</sub> is the power measured on the 10 % side by the power meter during a measurement.

The advantage of this setup is that once the splitting ratio R<sub>k</sub> is calibrated, we are no longer dependent on the uncertainty of the beam splitter cube ratio or the power loss caused by the multiple interface (AOM, polarizer and focusing lens). Indeed, it is sufficient for the entire system to have a constant and stable splitting ratio R<sub>k</sub> over time in order to continuously know the power P<sub>r</sub>. By calibrating the ratio R<sub>k</sub> with a laser power measurement reference traceable to the SI, it is thus possible

to have an optical force generator that is itself traceable (see section 2.4).

However, for this study, we only used a commercial power meter (Ophir 3A-IS-IRG, referred to as PM<sub>Oph</sub>) as a reference. During calibration, the reference power meter is placed at the output of the system, replacing the balance beam's mirror (see Fig. 5).

The characterisation of R<sub>k</sub> (Fig. 6) is performed using the AOM (control from 1 V to 5 V) over a power range from 0.1 W to 1.8 W (on the 90 % side), corresponding to a force F<sub>a</sub> ranging from 1 nN to 12 nN. We can then determine a value of R<sub>k</sub> as the slope of the line given by the equation:

$$P_{Oph} = R_k \times P_{Thr.s} \quad (5)$$

To differentiate the power read by the power meter PM<sub>Thr</sub> during calibration from the power measured during the actual measurement, we denote P<sub>Thr.s</sub> as the power measured by the power meter PM<sub>Thr</sub> during calibration.

Based on a series of three calibrations conducted over three days, the value of R<sub>k</sub> is finally given as 118.0 ± 6.5 (k = 1), with the uncertainty of R<sub>k</sub> taking into account the uncertainty in the accuracy of the two power meters (u<sub>k.just</sub> = 6.5) and the uncertainty related to the repeatability of the ratio R<sub>k</sub> (i.e., u<sub>k.rep</sub> = 0.1). The uncertainty on the ratio R<sub>k</sub> is relatively high (> 5 %) due to the use of commercial power meters with relative standards uncertainties of 5 %. However, it is noted that the uncertainty related to the repeatability of the system is very good (< 1 %). It will therefore be possible to significantly reduce the uncertainty associated with the generated force through calibration of the commercial power meters. This validates the architecture of the optical force generator.

The calibration of the ratio R<sub>k</sub> thus enables the continuous determination of the power P<sub>r</sub> by reading the value from PM<sub>Thr</sub>.

The output power of the generator induces radiation pressure on the balance mirror. The resulting force F<sub>r</sub> can be considered as the resultant of this pressure [17] and is defined by the following equation:

$$F_r = \frac{2\cos(\theta)}{c} P_r \quad (6)$$

where θ is the laser beam injection angle in radian, P<sub>r</sub> is the power of the loading laser at the output of the optical force generator (in watt), and c is the speed of light in vacuum.

Subsequently, we define:

$$c_e = \frac{2\cos(\theta)}{c} \quad (7)$$

with c<sub>e</sub> = 6.67 nN/W, which is considered a constant given the small range of measured angles (±150 µrad).
<figure><img src="image_10.png" /><figcaption>Fig. 8. Experimental set-up of the initial version (V1) of the NanoPyc balance.</figcaption></figure>

## 2.4. The traceability of the balance

The development of the reference optical force generator with very high repeatability ensures the traceability of the balance by referencing a traceable laser power measurement system. This is why the entire generator was designed to be both portable and mountable onto the pPyC laser power measurement system [20].

This system, developed at LNE-CNAM, is based on a torsion pendulum and provides a traceable method for measuring laser powers ranging from 100 mW to 5 W, with a relative standard uncertainty of less than 1 %. It can therefore be considered a traceable measurement instrument over a range centred around one watt, which corresponds to approximately 0.1 nN to 500 nN.

Fig. 7 shows the traceability scheme of the pPyC instrument and its outputs. The reference force generator can thus be considered a transfer standard, which through comparison with $F_r$ makes it possible to render the measurement of the applied force $F_a$ traceable to the SI units (kilogram, meter, and second).

This article does not cover the calibration of the optical force generator using the pPyC system, as this part of the project is still ongoing.

# 3. Experimental set-up and sensitivity study

In the previous section, we discussed the general principles of the optical force balance. This section provides a detailed description of the experimental setup used to obtain the first measurements related to the system's sensitivity. We can note that several aspects of the setup differ from the general version. As the project is still under development, some aspects have not yet been developed. We should also note that the experimental measurement range is narrower (4 nN–12 nN) for the same reasons.

## 3.1. *Experimental set-up*

The experimental setup differs slightly from that shown in Fig. 1. All

<figure><img src="image_11.png" /><figcaption>Fig. 9. Functional diagram of the experimental setup version of NanoPyc balance.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 10. Angular displacement of the balance beam $\alpha_x$ during a 60 s loading corresponding to an applied force of 11 nN.</figcaption></figure>

components involved in the creation of the revolute joint remain present: namely the support magnets, PyC, the beam assembly, and the floating magnets. The angular measurement device (autocollimator system) also remains. In this configuration, the reference force and the applied load are combined, and no feedback control system is implemented. Thus, only the reference optical generator, previously detailed in Fig. 4, remains in the setup. Additionally, two enclosures not mentioned in the previous section are now introduced:

* An acoustic enclosure, which isolates the entire setup from external disturbances (not shown in Fig. 8);

* An aluminum enclosure, which isolates the beam from internal air currents and holds the two PyC discs in place.

The picture in Fig. 9 illustrates the experimental setup featuring. Fig. 1 shown above can be adapted as follows in Fig. 9.

In this configuration, the measurement of the applied force $F_a$ is performed differently. Since the applied load and the reference force are combined, it becomes necessary to characterise the system's sensitivity. Once this sensitivity is known, a load can be applied to the system, and the resulting angular variation between the two equilibrium states (with and without the load) can be observed. Based on this angular displacement and the previously determined sensitivity, the applied force can then be calculated. This configuration has the advantage of being simpler to implement, as it does not require a control system. However, it also has several disadvantages: since the measurement depends on sensitivity, it is necessary to characterize it before each test, as it may vary over time. The variation in sensitivity is on the order of 0.1 µrad/nN on several days. This variation is likely due to the construction of the beam from PLA and the use of adhesive putty for balancing. All these

elements can easily deform, which implies a change in the weight distribution of the beam and therefore in the sensitivity. Furthermore, using a single load may introduce unwanted force components at the revolute joint of the balance beam.

The sensitivity $s$ of the system is defined by the following equation:

$$s = \frac{\Delta\alpha}{F_a} \qquad (8)$$

where:

$$\Delta\alpha = \alpha_{x,moy1} - \alpha_{x,moy2} \qquad (9)$$

with $\alpha_{x,moy1}$ representing the average angular position of the balance beam in its first equilibrium state (without applied force), and $\alpha_{x,moy2}$ the average angular position in its second equilibrium state (with an applied load). Thus, $\Delta\alpha$ is the angular displacement of the balance beam between the two positions around the x-axis.

Angular measurement is therefore essential for determining the system's sensitivity. It must be calibrated by comparison with a reference autocollimator (Newport Conex LDS 1000 with a resolution of 0.01 µrad). By using a reference autocollimator, the uncertainty $u_{\alpha_{x,ref}}$ related to angular measurement becomes negligible ($u_{\alpha_{x,ref}} < 0.2\%$) compared to the uncertainties arising from the external sources of noise of the beam (see Section 3.3). The PSD used for angular measurement comes from Thorlabs, model PDP90A.

By characterizing the repeatability and accuracy of the system's sensitivity, it becomes theoretically possible to determine a load applied to the balance beam. An initial series of tests was therefore carried out to characterize this sensitivity, serving as a basis to demonstrate the feasibility of force measurements.
<figure><img src="image_13.png" /><figcaption>Fig. 11. Angular position difference of the beam ($\alpha_x$) during a series of 20 loadings using a 4 V AOM control signal (equivalent to approximately 10.5 nN). Angular position calculated between interval I1 and I2.</figcaption></figure>

To determine the sensitivity, it is necessary to characterize the system's key parameters: the angular displacement and the applied load.

In the following sections, we will discuss the determination of the system's sensitivity, the uncertainty associated with it, and its accuracy. Moreover, due to technical constraints on the maximum injection power and experimental limitations related to the system's stability, the measurement range was reduced from (0.1 nN–100 nN) to (5 nN – 12 nN).

## 3.2. Experimental results

To determine the sensitivity during our measurements, we operate with the balance, using only a single laser: the load laser (see Figs. 8 and 9). The measurements are carried out in two experimental tests:

1. Repetition of 20 load with an identical AOM command, each lasting 60 s. The applied force values range between 5 nN and 12 nN.

2. Randomized loading cycle involving forces between 5 nN and 12 nN, with 20 load applications per series.

The objective of these two experimental tests is to define the repeatability and accuracy of the balance, considering its operational mode. Indeed, it is possible to calibrate the sensitivity of the balance just before a measurement series using a fixed load (close to the target force), or to calibrate the sensitivity across the entire range to measure several forces within the desired measurement range. During both phases, to optimize results, the balance is manually adjusted daily before each measurement series.

### 3.2.1. Test 1

Fig. 10 shows an example of a loading cycle using a 4 V control signal on the AOM, corresponding to a force of 11 nN, applied over a 60 s plateau. Three distinct stages can be observed: the first, where the

balance beam is at rest (its equilibrium state in the absence of any applied force), the second, where the force $F_a$ is applied and the balance beam reaches a new equilibrium state and the third, where the balance beam returns to its initial equilibrium position.

It should be noted that the data are filtered (red curve) using a bandstop filter between 0.2 Hz and 0.4 Hz. All results presented in the remainder of this report will be based on the filtered data. Furthermore, to avoid including transient regimes in the analysis, we define three 15-s intervals (I1, I2 and I3) within which the measured data will be processed.

Based on the 20 load applications, it is therefore possible to calculate 20 sensitivity values for a single AOM control input, corresponding to a power variation of approximately 0.25 W. This confirms the need, at this stage, to use a monitoring power meter.

Fig. 11 shows all calculated angular position differences $\Delta\alpha_x$ between two intervals as a function of the power injected during those same intervals. As a reminder, the injected laser power is measured using the power meter PM<sub>Thr</sub> and the ratio $R_k$. The angular position difference is computed between the mean angular position of the first interval ($\alpha_{x,moy1}$) and that of the second interval ($\alpha_{x,moy2}$): I1 – I2. It would also be possible to calculate the angular difference between the second and third intervals ($\alpha_{x,moy3}$): I2 – I3.

For this given example, the mean sensitivity $\bar{s}$ is 13.5 µrad/nN, with a repeatability $u_{s,rep}$ of 0.2 µrad/nN (equation (10)).

$$u_{s,rep} = \sqrt{\frac{\sum (s_i - \bar{s})^2}{n}} \times \frac{1}{\sqrt{n}} \quad (10)$$

With n the numbers of series.

### 3.2.2. Test 2

The second phase shown on Fig. 12 covers a larger measurement
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Angular position difference of the beam ($\alpha_x$) during a series of 20 loadings using randomly 2.5 V – 4 V AOM control signal (equivalent respectively to approximately 5 nN and 12 nN). Angular position calculated between interval I1 and I2.</figcaption></figure>

<table><caption>**Table 1**

Error sources and metrological characteristics.</caption><thead><tr><th>Error source</th><th>Metrological characteristic</th></tr></thead><tbody><tr><td>Background noise:<br>vibration and air movement</td><td>Repeatability, Resolution</td></tr><tr><td>Temperature variation</td><td>Reproducibility, Drift (OoM: 20 µrad/°C)<br>Sensitivity (stiffness of the revolute joint)</td></tr><tr><td><ul><li>Magnetic inhomogeneities of the magnets</li><li>Dimensions and position of the beam's center of gravity</li><li>Spacing between the pyrolytic graphite plates</li></ul></td><td></td></tr><tr><td>Experimental protocol: Measurement method and sampling</td><td>Repeatability</td></tr><tr><td>Model parameters</td><td>Trueness</td></tr></tbody></table>

range, from 5 nN to 12 nN, with a randomized order of load applications.

In this case, the mean sensitivity $\bar{s}$ is 13.2 µrad/nN, with a repeatability of 0.2 µrad/nN. This test clearly shows that the sensitivity can be considered constant when the angular displacement is less than 200 µrad, which validates our hypothesis (section 2.2). Therefore, in this balance configuration, it is necessary to characterize the sensitivity before each test. With this method, the uncertainty related to repeatability remains of the same order.

### 3.3. Uncertainty budget

Table 1 presents a list of error sources that affect the metrological characteristics of the device.

This section aims to establish a preliminary uncertainty budget for the sensitivity $\bar{s}$ of the optical force balance. We apply the classical GUM method [21] to propagate uncertainties based on the equations shown in Fig. 13.

<figure><img src="image_15.png" /><figcaption>**Fig. 13.** Overview of the equations used in the balance model.</figcaption></figure>
<table><caption>**Table 2**
Uncertainty budget of the sensitivity *s* for a load of approximately 11 nN.</caption><thead><tr><th>Component</th><th>Value</th><th>Standard uncertainty</th><th>Unit</th></tr></thead><tbody><tr><td><i>s</i></td><td>13.4</td><td>1.2</td><td>μrad/nN</td></tr><tr><td>Δα</td><td>152.9</td><td>6.1</td><td>μrad</td></tr><tr><td>Fa</td><td>11.4</td><td>0.8</td><td>nN</td></tr><tr><td>ce</td><td>6.67</td><td>0</td><td>nN/W</td></tr><tr><td>Rk</td><td>118.0</td><td>6.5</td><td></td></tr><tr><td>PThr,d</td><td>13.0</td><td>0.7</td><td>W</td></tr></tbody></table>

Table 2 presents the uncertainty budget leading to the standard uncertainty on the sensitivity (as well as its contributions) presented for a load application of approximately 11 nN (corresponding to an AOM command of 4 V).

It should be noted that the uncertainty related to the angular position difference Δα is not computed using a mathematical model, as it includes external sources of noise of the system that cannot be modeled at this stage. To account for all sources of uncertainty, we rely on the observed physical motion of the beam. This uncertainty is therefore calculated as the quadratic sum of the uncertainties associated with the standard deviation of the angular displacement αₓ over the defined intervals (I1 and I2).

This uncertainty thus includes both the uncertainty from the angular measurement system, i.e. the uncertainty of the autocollimator described above, and the uncertainty associated with external sources of noise of the system (such as vibrations, air movement, temperature variations, etc.). To validate this method, it is required to verify that the uncertainty due to the measurement device is negligible compared to the uncertainty caused by the system's external sources of noise:

$u_{α_{x,ref}} \ll u_{Δα_x}$ with $u_{α_{x,ref}}$, the relative standard uncertainty of the angular device measurement.

Finally, the standard uncertainty obtained for the sensitivity of the balance revolute joint around the x-axis is.

$s = (13.4 \pm 1.2) \mu\text{rad/nN} (k = 1)$ where $Δα_x$ and $F_a$ contribute in relative respectively to 30.9 % and 69.1 % to the uncertainty of the sensitivity. The main contributions to the uncertainty in sensitivity, originate from the system's external sources of noise and from the power meters used during the measurements. Both sources of uncertainty can be improved. During these tests, neither the feedback control of the balance beam around a reference force nor the force standard (pPyC) were implemented: the feedback control should significantly reduce the system's external source of noise, although at this stage it remains difficult to estimate the extent of this improvement. The effect of using pPyC as a reference for the optical force is more easily quantifiable, based on the results from the internal report [16], which indicate a relative uncertainty of 1 % for powers between 100 mW and 5 W.

## 4. Conclusions

In this article, we presented an innovative optical force balance concept. The experimental results allowed us to confirm the theoretical principle of our balance by characterizing the sensitivity of our system over a force range from (5 nN–12 nN), with a repeatability uncertainty of less than 2 %. This balance concept has the advantage of potentially being traceable to the SI, by referencing a system already in place at LNE-CNAM.

There are numerous prospects for improvement around this novel measurement concept. Regarding the uncertainties (< 2 %) in the targeted measurement range in next version of the balance (0.1 nN–100 nN), which is currently scarcely or not at all covered by LNE-CNAM or other major international laboratories. Moreover, although this aspect was not explored during this article, the balance concept could also enable measurements in the domain of small masses (10 ng – 1 µg).

Applications in this area are increasingly numerous, and this system could potentially be adapted to meet those needs. For example, one

future objective is the characterization of AFM cantilevers.

## Funding statement

This work has been partly funded by the French National Research Agency (ANR) under the project ANR-23-CE42-0022–01 and French Metrology.

## CRediT authorship contribution statement

**Simon Noviant:** Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. **Colin Dartelvé:** Writing – review & editing, Investigation. **Zaccaria Silvestri:** Writing – review & editing, Validation, Supervision. **Alain Vissière:** Writing – review & editing, Investigation, Conceptualization.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: [Simon Noviant reports financial support was provided by French National Research Agency. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper].

## Data availability

No data was used for the research described in the article.

## References

[1] CCM, "Consultative Committee for Mass and related quantities." [Online]. Available: http://www.bipm.org/utilies/common/pdf/KCDB_CMCs.pdf.

[2] K. Park, et al., Measurement of adherent cell mass and growth, Proc. Natl. Acad. Sci. U. S. A. 107 (48) (2010) 20691–20696, https://doi.org/10.1073/ pnas.1011365107.

[3] D. Martínez-Martín, et al., Inertial picobalance reveals fast mass fluctuations in mammalian cells, Nature 550 (7677) (2017) 500–505, https://doi.org/10.1038/ nature24288.

[4] E.Piat, "Traçabilité des faibles forces et caractéristique métrologique d'un dispositif biomédical innovant pour la PMA - TRAFALDA," ANR. [Online]. Available: https://anr.fr/Projet-ANR-23-CE42-0022.

[5] M.C. Carrozza, A. Eisinberg, A. Menciassi, D. Campolo, S. Micera, P. Dario, Towards a force-controlled microgripper for assembling biomedical microdevices, J. Micromech. Microeng. 10 (2) (2000) 271–276, https://doi.org/10.1088/0960- 1317/10/2/328.

[6] M. Boudaoud, S. Regnier, An overview on gripping force measurement at the micro and nano-scales using two-fingered microrobotic systems, Int. J. Adv. Robot. Syst. 11 (1) (2014), https://doi.org/10.5772/57571.

[7] S.K. Lamoreaux, Comment on 'precision measurement of the casimir force from 0.1 to 0.9 µm., Phys. Rev. Lett. 83 (16) (1999) 3340, https://doi.org/10.1103/ PhysRevLett.83.3340.

[8] M.S. Kim, J.R. Pratt, U. Brand, C.W. Jones, Report on the first international comparison of small force facilities: a pilot study at the micronewton level, Metrologia 49 (1) (2012) 70–81, https://doi.org/10.1088/0026-1394/49/1/011.

[9] "EURAMET TC-M Strategy 2024," 2024.

[10] S. Vasilyan, et al., The progress in development of the Planck-Balance 2 (PB2): a tabletop Kibble balance for the mass calibration of E2 class weights, Tech. Mess. 88 (12) (2021) 731–756, https://doi.org/10.1515/teme-2021-0101.

[11] M.S. Kim, J.H. Choi, Y.K. Park, J.H. Kim, Atomic force microscope cantilever calibration device for quantified force metrology at micro- or nano-scale regime: the nano force calibrator (NFC), Metrologia 43 (5) (2006) 389–395, https://doi. org/10.1088/0026-1394/43/5/008.

[12] R.S. Gates, J.R. Pratt, Prototype cantilevers for SI-traceable nanonewton force calibration, Meas. Sci. Technol. 17 (10) (2006) 2852–2860, https://doi.org/ 10.1088/0957-0233/17/10/041.

[13] V. Cherkasova and T. Fröhlich, "Capacitive Calibration Capabilities in an Emfc Balance," 24th IMEKO TC3 Conf. Meas. Force, Mass Torque 2022, Together with 14th TC5 Conf. Meas. Hardness, 6th TC16 Conf. Press. Vac. Meas. 5th TC22 Conf. Vib. Meas., no. October, pp. 1–6, 2022, doi: 10.21014/tc3-2022.111.

[14] V. Nesterov, et al., SI-traceable determination of the spring constant of a soft cantilever using the nanonewton force facility based on electrostatic methods, Metrologia 53 (4) (2016) 1031–1044, https://doi.org/10.1088/0026-1394/53/4/ 1031.

[15] S. Vasilyan, et al., Revisiting the limits of photon momentum based optical power measurement method, employing the case of multi-reflected laser beam, Metrologia 58 (1) (2021), https://doi.org/10.1088/1681-7575/abc86e.
[16] Z. Silvestri and A. Vissière, "Applications métrologiques du graphite pyrolytique - Mesure de la puissance d'un Laser - mPyC - pPyC Zaccaria Silvestri Rapport final," pp. 1-19, 2023.

[17] P. PINOT, Z. SILVESTRI, and A. VISSIERE, "Mesure de la puissance laser - Techniques de mesure basées sur la pression de radiation," *Mes. mécaniques Dimens.*, vol. 33, no. 0, 2021, doi: 10.51257/a-v1-r6739.

[18] S. Noviant, A. Vissière, Z. Silvestri, Development of an optical force balance for low mass at LNE-CNAM, Meas. Sensors (2024) 1-5, https://doi.org/10.1016/j.measen.2024.101362.

[19] Q.H. Gao, et al., Magnetic levitation using diamagnetism: mechanism, applications and prospects, Sci. China Technol. Sci. 64 (1) (2021) 44-58, https://doi.org/10.1007/s11431-020-1550-1.

[20] P. Pinot, Z. Silvestri, Optical power meter using radiation pressure measurement, Meas. J. Int. Meas. Confed. 131 (2019) 109-119, https://doi.org/10.1016/j.measurement.2018.07.087.

[21] Joint Committee for Guides in Metrology, JCGM 100:2008(F) GUM 1995 évaluation des données de mesure — guide pour l’expression de l’incertitude de mesure, JCGM 100 (2008) 2008.