Generation of static and dynamic small calibration forces and their measurements by electromagnetic force compensation balance


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ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 113 - 117 

 
 

GENERATION OF STATIC AND DYNAMIC SMALL CALIBRATION 

FORCES AND THEIR MEASUREMENTS BY ELECTROMAGNETIC 

FORCE COMPENSATION BALANCE 
 

S. Vasilyan1, N. Rogge1, E. Manske1, T. Fröhlich1 
1 Institute of Process Measurement and Sensor Technology, Ilmenau, Germany, suren.vasilyan@tu-ilmenau.de  

 

Abstract: 

The paper presents some of the results of the 

static and dynamic force measurements at 100 nN to 

sub-10 µN ranges which are generated due the 

photon-momentum. The force sensor with 

resolution about 20 nN and operating in differential 

measurement mode is developed by two 

electromagnetic force compensation balances. In 

order to generate these calibration forces, CW lasers 

with different operational modes, power levels, and 

wavelengths are used. Multi-reflection 

configuration of the laser beam inside the 

macroscopic cavity with highly reflective mirrors 

are used to test and variate the total amount of the 

forces. 

Keywords: photon momentum, static and 

dynamic small calibration forces, high power laser, 

highly reflective mirrors 

1. INTRODUCTION 

In recent years, the use of photon momentum to 

determine the optical power of lasers or to generate 

precision/calibration small forces [1, 2, 3] has made 

important progress, especially for the measurements 

of optical laser power at kilowatt levels [4]. The 

measuring principle is based on the measurement of 

the force which is exerted due to the transfer of the 

photon momentum upon reflection of the radiant 

power from a highly reflective mirror. A 

measurement device developed by Williams et 

al. [4] uses this measurement technique and 

achieves a relative expanded measurement 

uncertainty of 1.6 % for optical power levels 

between 1 kW and 50 kW. In the core of the device 

is a force sensor, consisting of a commercial off-the-

shelf electromagnetic force compensation (EMFC) 

balance and a mirror with high reflectivity 

(R = 0.999 8 %) attached to it. Vasilyan et al. [1, 2] 

carried out measurements with a device having 

similar components, however, here two force 

sensors adapted for differential measurements were 

used, by which the noise level has been reduced. 

Here, for the stability considerations a low power 

laser system (around 1 W) was used in multi-

reflection configuration to generate calibration 

forces at the currently existing lowest end of the 

small force standard typically referred to be from 

10 nN up to 10 µN. Despite the statistical error 

observed as an oversight between the measurements 

and simplified theoretical calculations, under the 

multi-reflection configuration the total measured 

force was amplified by at least an order of 

magnitude in comparison to the single reflection 

configuration. Furthermore, with the usage of 

single- and multi-reflection configurations, a 

possible standard for the force calibration routine, 

or reversed, standard for the optical (laser) power 

calibration routine with direct and more simplified 

traceable chain to the recently renewed SI base units 

[2, 5, 6] was already established. 

2. DESCRIPTION OF THE WORK  

In order to develop a proper methodology in 

using this effect of the photon 

momentum-generated small calibration forces, 

initially several test measurements have been 

carried out. The setup consists of two EMFC 

balances operating in synchronous measurement 

modes such that the difference signal can be 

obtained with substantially reduced noise level.  

 
Figure 1: Set-up for measuring the generated small 

calibration forces using EMFC balance, the supporting 

optical components and the reference optical power 

sensors. Inset the photograph showing the lower parts of 

the EMFC balances, mirrors and the reflection spots 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 114 

The EMFC balances are mounted independently 

from the common metrology frame and on each of 

them a highly reflective mirror is attached with total 

weight of about 70 g together with the adjustment 

mechanism (see scheme in Figure 1). A 

conventional off-the-shelf highly reflective mirror 

with reflectivity 𝑅 above 0.995, universally used for 
lasers with different wavelengths (see Table 1), has 

been used to create the macroscopic optical cavity 

where the multi-reflections of the laser beam are 

obtained. 

Table 1: Summary of the parameters of the laser 

(power 𝑃, power uncertainty 𝑈, wavelength 𝜆) and the 
mirror reflectivity coefficient 𝑅  used during different 
measurements 

 Laser 𝑹 
𝑷 / W 𝑼 / % 𝝀 / nm  

1 0.98 1 420 ± 10 0.999 

2 0.05 - 2 1 450 ± 7 0.999 

3 0.1 - 0.9 1 520 ± 2 0.995 

4 2 - 11 < 1 532 ± 5 0.995 

5 2.5 - 8.8 <1 532 ± 5 0.999 97 

 

Theoretically, at the single reflection of the laser 

beam with 1 W optical power from a perfectly 

reflecting mirror, a force of about 6.7 nN is 

generated. In the case of the multiple reflecting 

specular type of reflections, a force of about 220 nN 

can be created, see equations (1) and (2). 

𝐹 =
𝑃

𝑐
(1 + 𝑅) (1) 

∑ 𝐹𝑗 =

𝑛

𝑗=1

(1 + 𝑅)

𝑐
∑ 𝑃𝑗

𝑛

𝑗=1

 (2) 

where 𝐹  is the force in N, 𝑃  is the average laser 
power in W, 𝑐 is the speed of light in m/s, 𝑅 is the 
reflectivity coefficient of the mirror, and 𝑗  is an 
integer assigned to each of the 𝑛 reflections. 

During the measurements the photon 

momentum-generated forces due to specular type of 

multi- reflection are described in equation (3). 

𝐹total = 𝐹1 − 𝐹2 = 

= [𝐹P1 + 𝐹Err1] − [−𝐹P2 + 𝐹Err2] 
(3) 

where 𝐹P1  and 𝐹P2  are the photon momentum-
generated forces acting on the EMFC balances 

attached left and right mirrors from 2nd, 4th, 6th… 

and 1st, 3rd, 5th … reflections respectively, 𝐹Err1 
and 𝐹Err2  are the noise measured by each EMFC 
balance which are assumed to be the same (see 

Figure 1). 

The parameters listed in Table 1 were used 

during different measurement campaigns. The 

averaged laser power has been pre-set using varying 

methods. We have separated dynamically changing 

and statically applied laser powers in order to 

characterise the force measurement signals in 

relation to them, more specifically in the time or 

frequency domains. In the case of the static signals 

have been used either a shutter system or completely 

turning the lasers off. In the case of the dynamic 

signals the power of the laser has been varied by 

varying the gain current of the laser’s control unit 

using equation (4). 

𝑃 ~ 𝐼𝑖 (𝑓𝑖 ) = ∆𝐼 sin(2𝜋𝑓𝑖  ∆𝑡𝑖 ) = 

= ∆𝐼 sin(2𝜋[𝑓0 + 𝑖 ∙ ∆𝑓] ∆𝑡𝑖 ) 
(4) 

where ∆𝐼  is the peak-to-peak amplitude of the 
applied gain current, within the frequency range 𝑓0 
to 𝑓end, with ∆𝑓 separation between the grid points, 
and ∆𝑡𝑖  is the signal length. Any other function type 
could be used, e.g. a square wave (see Figure 2). 

 
Figure 2: Example of the measured forces as a) sine and 

b) square wave functions [2] 

3. UNCERTAINTY ASSESSMENT 

The photon momentum-based method, as 

mentioned, may directly relate the measure of the 

force, which is produced by the momentum transfer 

of the absorbed and re-emitted photons from a 

highly reflective mirror, with the measure of the 

optical power of a laser beam. The following 

simplified numerical example, based on equating 

the photon momentum-generated forces with the 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 115 

gravitational force acting on a standard mass piece, 

gives an understanding of the advantages 

underlying this method. We should note here 

beforehand that as it is known from recent 

developments the mass pieces mainly at 1 kg and 

500 g are already calibrated directly traceable to the 

quantum-based natural constants, more specifically 

it is traceable to the numerical value of the Planck 

constant, however there are still ongoing 

developments to extend that calibration method of 

mass pieces for the range of smaller mass pieces 

(down to 1 mg) [6]. Using the Kibble balance 

apparatus, one may even provide calibration of the 

arbitrary values of mass pieces, therefore on the 

account of measuring the gravitational force acting 

to this standard mass pieces one may also provide 

static force calibration values with similar accuracy 

and precision. Here we will rather provide our 

analysis on the basis of existing basic specifications 

of the measurement components that are well 

known/established and at the same time assumes the 

future speculations related to possible approaches of 

calibrating forces at smaller force levels. 

It is known that the optical power measurements 

at 1 kW level are typically made using reference 

standard thermal detectors with several percent of 

uncertainty, at the same time becoming a non-trivial 

technological task to implement since their 

measurement capability and accuracy strongly 

depend on the absorbance and heat capacity of the 

cavity used as sensor. In comparison to this, the 

1 kW laser power after first reflection from the 

ultra-high reflective mirror with 0.999 95 

reflectivity exerts a force of about 6.673 116 µN, 

whereas after 33 reflections it is 220.036 76 µN. 

One may use standard practices known from 

mass/force metrology to perform the force 

measurements at these levels. In the case that one 

uses a commercially available precision balance 

with 3 µg reproducibility (standard deviation) in 

measurements of the weight pieces, the equivalent 

force calculated with equation (2) is to be 29.4 nN 

assuming the value for the local gravitational 

acceleration of 9.812 503 m·s-2 at the site of the 

measurements. Thus, in this case we can measure 

the optical power generated forces after first 

reflection with 0.440 % relative standard deviation 

and with 0.013 % after 33 reflections. However, 

since force standards at the nanonewton level have 

not yet been established, the implementation of a 

traceable optical power measurement using the 

photon momentum approach can only be verified in 

connection to the measurement capabilities of a 

certain class of apparatuses, e.g. with an 

electromagnetic force compensation - EMFC 

weighing balances down to 10 µN (with about 

0.3 % relative measurement uncertainty) traceable 

from there down to about 10 nN only accounting for 

the reproducibility or in general the Type B 

uncertainty evaluation (an alternative to be the 

development of an uniquely-designed custom-made 

instrument with well characterised and SI-traceable 

calibration certificates whose performance could be 

verified by an independent party). 

Generally, the balances are calibrated by 

employing the deadweight effect exerted on 

standard weight pieces in accordance with the 

equation 𝐹 = 𝑚𝑔. On the other hand, the optical 
power can be determined by combining this with 

equations (1) and (2) for single or multiple 

reflection cases yielding the simplified form given 

in equation (5). 

from which, by using the standard uncertainty 

propagation, we obtain equation (6). 

The value of the gravitational acceleration can 

typically be obtained to better than 0.2 ppm. The 

values of the reflectivity for the ultra-high reflective 

mirrors to our best knowledge can be estimated only 

indirectly by the measurement of the losses 

typically given as the optical transmission curve. In 

accordance to most of the datasheets provided by 

manufacturers it can be determined with 70 ppm 

uncertainty [9]. The uncertainty values of the mass 

vary dependent from the nominally used discrete set 

of standard mass piece (ref. Figure 3) against which 

the nominally applied magnitude of the optical 

power should be compared. For this, one should 

follow at least the recommendations given in [8]. 

 
Figure 3: Expanded relative uncertainties and maximum 

permissible errors associated with weights of 

OIML R 111-1 class E1 [8] 

Thus, if the measurements would be conducted 

at the 1 kW optical power level for the single 

reflection case the combination of uncertainty 

𝑃 = 𝑐
𝑚𝑔

1 +  𝑅𝐿
 (5) 

𝑢(𝑃)

𝑃
 = √(

𝑢(𝑚)

𝑚
)

2

+ (
𝑢(𝑔)

𝑔
)

2

+ (
𝑢(𝑅𝐿 )

𝑅𝐿
)

2

 (6) 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 116 

contributions using equation (6) leads to an 

estimated standard uncertainty of the optical power 

measurements of about 𝑢(𝑃)/𝑃  = 1 002 ppm; for 
the multi-reflection configuration with 

33 reflections the value is 86 ppm. In Figure 4 the 

full estimation curves of relative measurement 

uncertainties in the optical power obtained by 

equation (6) are presented. Here the relative 

standard uncertainty is presented as a function of the 

nominal magnitude of the applied optical power in 

four different (1, 3, 21, 33) reflection configuration 

cases with interpolated data as an extended estimate 

for the power levels below 1 kW. The blue dashed 

line presents the lower limiting factor as a result of 

the reflectivity coefficient of the mirrors, the solid 

black line with 0.05 % notation highlights the 

typical limit of the relative standard uncertainties in 

measuring laser power at 10 W and above with 

conventional power meters [7]. 

 
Figure 4: Estimated relative standard uncertainty of the 

optical power realisation by equations (5) and (6) 

This method of establishing the traceability of 

the optical power measurements through the use of 

measurements of the photon momentum-generated 

forces and their further intercomparisons with the 

standards and procedures known from mass/force 

metrology is theoretically more direct and simplistic 

which may further improve and lower the 

measurement uncertainties at least in continuous 

wave optical power measurement. 

4. RESULTS 

Currently, we perform deliberately planned trial 

measurements in collaboration with Division 4 

(optics) PTB, Braunschweig for establishing the 

traceability of the optical power measurements 

through the use of photon momentum-based force 

measurements and we compare the results with 

those performed by using a calibrated reference 

standard detector, Thermopile and Si-diode 

traceable to PTB’s primary standard for optical 

power (the cryogenic radiometer) as shown in 

schematics in given in Figure 1. 

The optical power values were used to calculate 

theoretically the expected photon momentum-

generated forces for comparison with the data from 

the measurements. In this configuration we can 

directly compare the reference for the force 

measurements and the reference for the optical 

power measurements, towards the SI-based 

traceable comparison (actually in future to be in 

reference to the Planck constant) of the force/mass 

and laser power references. 

In Figure 5 we show one typical set of static 

forces and the corresponding laser power 

measurements. 

 
Figure 5: Example of the static force measurements with 

a periodically applied laser power (10 s) in the case of 33 

reflections. Top - measured force signal from each 

balance. Middle - the difference signal (a feed backward 

averaging filter was chosen for the last 15 bins to filter 

raw data). Bottom - input and output power 

From the raw data of the force measurements 

during the 33 reflections configuration (wavelength 

532 nm, mirror reflectivity 0.999 95, input power 

8.5 W, output power 8.3 W), a mean value of 

1 873 nN is obtained with a relative combined 

standard deviation of 1.71 % (32 nN). As a 

comparison, the standard uncertainty of the optical 

power measurements using the calibrated reference 

standard detector was 0.3 %. Furthermore, in this 

particular case, the mean value of the measured 

forces in comparison to the value calculated 

theoretically differs by 1.07 %, from all 

measurements this difference is within 3 %. 

With these measurements at 10 W level using 

well established methods from mass/force 

metrology we demonstrate that upon the increase of 

the input power entering to the optical cavity and the 

number of reflections, the resulting uncertainties of 

generated small force values or vice versa the 

measurements of the input optical power can 

markedly be reduced. Thus, by collecting more 

experimental evidence at the edge of the small force 

measurements in static and dynamic regimes, 

improving the accuracy of the measurements, and 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 117 

eliminating further the systematic errors associated 

with the measurements, then the photon 

momentum-based force generation can serve more 

evidently as a superior method for obtaining 

accurate measurement of small (calibration) forces 

below 10 µN through the traceable optical power 

measurements. Thereafter, by virtue of the same 

relation, the photon momentum-based force 

measurements can be used to develop viable, 

accurate and absolute optical power detectors / 

sensors / power meters at higher ranges with direct 

traceability to SI units. In accordance with the 

existing technical limits and instrumentation 

capabilities as well as the theoretical obtained 

relations, we can still improve the accuracy and the 

precision of the optical power and small force 

measurements by at least several orders of 

magnitude (ref. Figure 4). 

5. SUMMARY 

In this work we present photon momentum-

generated precision small force measurements in 

static and dynamic modes, in the range of forces 

below 10 μN and above 100 nN, using a force 

sensor with about 20 nN resolution. Static 

(considered to be below 0.2 Hz) and dynamic 

(ranging from 0.2 Hz to 10 Hz) modes of 

measurements are considered for different 

excitation signals with sine and square waveforms 

whose peak-to-peak amplitudes range from several 

hundred milliwatts up to several watts. We compare 

the measured forces in reference to the calculated 

values. The calculations of the theoretical data are 

made based on reference measurements of the 

optical power by calibrated reference standard 

detector, Thermopile and Si-diode traceable to 

PTB’s primary standard for optical power (the 

cryogenic radiometer). 

6. ACKNOWLEDGEMENT 

This work is funded by DFG Projektnummer 

409476492. The authors are grateful to colleagues 

from Institut für Prozessmess- und Sensortechnik 

and collaborators from PTB Division 4 for all 

fruitful discussions. 

7. REFERENCES  

[1] S. Vasilyan, T. Fröhlich, E. Manske, “Total 
momentum transfer produced by the photons of a 

multi-pass laser beam as an evident avenue for 

optical and mass metrology,” Optics Express, 

vol. 25, pp. 20798-20816, 2017. 

[2] E. Manske, T. Fröhlich, S. Vasilyan, “Photon 
momentum induced precision small forces: a static 

and dynamic check”, Meas. Sci. Technol. vol. 30, 

105004, 2019. 

[3] G. A. Shaw, J. Stirling, J. Kramar et al., 
“Comparison of electrostatic and photon pressure 

force references at the nanonewton level”, 

Metrologia, vol. 56, 025002, 2019. 

[4] P. Williams et al., “Portable, high-accuracy, non-
absorbing laser power measurement at kilowatt 

levels by means of radiation pressure”, Opt. 

Express, vol. 25, pp. 4382–92, 2017. 

[5] P. A. Williams et al., “Meta-study of laser power 
calibrations ranging 20 orders of magnitude with 

traceability to the kilogram”, Metrologia, vol. 57, 

015001, 2020. 

[6] C. Rothleitner, J. Schleichert, L. Günther et al., 
“The Planck-Balance – Using a fixed value of the 

Planck constant to calibrate E1/E2-weights,” Meas. 

Sci. Technol., vol. 29, 074003, 2018. 

[7] https://www.ptb.de/cms/ptb/fachabteilungen/abt4/
kalib.html, PDF Kalibrier-und Messmöglichkeiten 

der Abt. 4. 

[8] OIML R 111-1: Weights of classes E1, E2, F1, F2, 
M1, M1-2, M2, M2-3 and M3 – Part 1: Metrological 

and technical requirements, International 

Organization of Legal Metrology, 2004. 

[9] Super Mirror, 30 mm diameter, 532 nm 
wavelength, TFHSM-50C08-532, SIGMAKOKI 

Group. 

 
 

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