Automatic unit for measuring refractive index of air based on Ciddor equation and its verication using direct interferometric measurement method

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Automatic unit for measuring refractive index of air based on

Ciddor equation and its veriﬁcation using direct

interferometric measurement method

V. Hucl, M. Cizek, J. Hrabina, B. Mikel, S. Rerucha, Z. buchta, P. Jedlicka, A. Lesundak, J.

Oulehla, L. Mrna, M. Sarbort, R. Smid, J. Lazar, O. Cip

Institute of Scientiﬁc Instruments of the Czech Academy of Sciences (ISI),

Kr´alovopolsk´a 147, 612 64 Brno, Czech Republic

ABSTRACT

In scanning probe microscopy laser interferometers are usually used for measuring the position of the probe

tip with a metrological traceability. As the most of the AFM setups are designed to work under standard

atmospheric conditions the changes of the refractive index of air have an inﬂuence to measured values of the

length with 1.0×10−4relatively. In order to achieve better accuracies the refractive index of air has to be

monitored continuously and its instantaneous value has to be used for compensating the lengths measured by all

of the interferometric axes. In the presented work we developed a new concept of an electronic unit which is able

to monitor the refractive index of air on basis of measurement of ambient atmospheric conditions: temperature,

humidity, pressure of the air and the CO2 concentration. The data processing is based on Ciddor equation for

calculating the refractive index of air. The important advantage of the unit is a very low power consumption of

the electronics so the unit causes only negligible temperature eﬀects to the measured environment. The accuracy

of the indirect measuring method employed by the unit was veriﬁed. We tested the accuracy in comparison

with a direct method of measuring refractive index of air based on an evacuatable cell placed at the measuring

arm of a laser interferometer. An experimental setup used for veriﬁcation is presented together with a set of

measurements describing the performance. The resulting accuracy of the electronic unit falls to the 4.1×10−7

relatively.

Keywords: Refractive index of air, precise measurement, laser interferometer

DOI: 10.1117/12.2020756

Note: This is a preprint rendition of a conference paper V. Hucl et al, Automatic unit for measuring refractive

index of air based on Ciddor equation and its veriﬁcation using direct interferometric measurement method, Proc.

SPIE, vol. 8788, pp. 878837, 2013.

1. INTRODUCTION

The precise measurement of lengths in the industrial or laboratory conditions by laser interferometers relies

very much on the knowledge of a value of the refractive index of air. The particles and the air itself make

the measuring optical path longer than the geometrical is. Therefore the determination of the index value is

necessary to measure in real-time when the measurement by lasers is carried-out. For laboratory measurement

is very useful a direct measurement of the index by a laser refractometer. This measuring setup works as an

interferometer where ﬁxed known geometrical length is detected by laser interferometer. The ﬂuctuation of the

index is then visible as a change of the optical path length at the output of the interferometer [1]. On the other

hand the direct method needs very complicated optical setup. In this case there is also problem with placing of

the refractometer setup close to the measuring interferometer used for the desired distance detection.

Further author information: please send correspondence to e-mail address treak@isibrno.cz (V. Hucl). arXiv rendition

and further development by S. Rerucha res@isibrno.cz

arXiv:2210.13223v1 [physics.ins-det] 18 Oct 2022

2. METHODOLOGY

For simple and relatively precise measurement of the refractive index of air the non-direct measurement is

very useful. This principle is based on measurement of set of quantities of the ambient air which cover the

measuring area: temperature, relative humidity and air pressure. From known values of these quantities we

determine this index on basis of one of these empiric formulas: Edlen [2], Bonsch and Potulski [3], or Fira [4]. In

general all of above mentioned formulas came from fundamental equation founded by Edlen [2]. The empirical

measurement and comparison of calculated values with direct measurement of the index by interferometric way,

i.e. laser refractometers [5,6] led to additional improvement. This modiﬁed Edlen formula can be expressed by

the following procedure:

A= 8342.54, B = 2406147, C = 15998, D = 96095.43, E = 0.601, F = 0.00972, G = 0.003661

S=1

λ2

ns= 1 + 10−8A+B

130 −s+C

38.9−S

X=1 + 10−8(E−F t)p

1 + Gt

ntp = 1 + p(ns−1)X

ns=ntp −10−10 292.75

t+ 273.15(3.7345 −0.0401S)pv

where t is the temperature of air [degC], pis the pressure [Pa], pvis the water vapor partial pressure, λis

the wavelength [µm]. Water vapor partial pressure we calculate using pv=RH

100 psv(t) expression, where RH is

the relative humidity [%] and psv is the saturation vapor pressure [Pa].

On basis of the research of the relation between the refractive index and a carbon dioxide concentration the

Ciddor formula is nowadays considered as a very reliable equation [7]. When using the Ciddor formula it is

necessary to express humidity as a mole fraction:

α= 1.00062, β = 3.14e−8, γ = 5.60e−7

f(p, t) = α+βp +γt2

xv=RH

100 f(p, t)psv

For calculation the refractive index of air we deﬁne ﬁrst following constants:

w0= 295.235 w1= 2.6422 w2=−0.03238 w3= 0.004028

k0= 238.0185 k1= 5792105 k2= 57.362 k3= 167917

a0= 1.58123e−6a1=−2.9331e−8a2= 1.1043e−10

b0= 5.707e−6b1=−2.051e−8

c0= 1.9898e−4c1=−2.376e−6d= 1.83e−11 e=−0.765e−8

pR1= 101325 TR1= 288.15 Za= 0.9995922115

ρvs = 0.00985938 R= 8.314472 Mv= 0.018015

Then we insert measured atmospheric parameters to equations:

ras = 1e−8k1

k0−S+k3

k2−S

rvs = 1.022 −8(w0+w1S+w2S2+w3S3)

Ma= 0.0289635 + 1.2011e−8(xCO2−400)M

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摘要：

AutomaticunitformeasuringrefractiveindexofairbasedonCiddorequationanditsvericationusingdirectinterferometricmeasurementmethodV.Hucl,M.Cizek,J.Hrabina,B.Mikel,S.Rerucha,Z.buchta,P.Jedlicka,A.Lesundak,J.Oulehla,L.Mrna,M.Sarbort,R.Smid,J.Lazar,O.CipInstituteofScienticInstrumentsoftheCzechAcademyofSci...

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Automatic unit for measuring refractive index of air based on Ciddor equation and its verication using direct interferometric measurement method

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