I -

INTRODUCTION

laser interferometr, allows one to realize dimensional measurement with a low uncertainty the 10 level can be reached when the measurements are made under Vacuum) In an however an additional uncertainty arises from the unknowledge of the refractive index of the medium n

To Determine n one needs to measure the temperature. the pressure, the carbon dioxide concentrate and the relatie humiditve with the help at calibrated sensors and the olden formula Neverthless the relative uncertainty is still about 1 part in 15 which limits the accuracy of dimentional measurement

In this paper we describe an interferometer which enables one to measure distances between 0.4 and 8 m in air and in vacuum with a relative uncertainty close to 1 part in 10⁸

2

PRINCIPLE [June 97] [likes 92] [Dänd 88][Gill 83]

Fig 1 shows the principle of the apparatus It is based on a double channel interferometer illuminated by the beam of a frequency tunable laser diode around 1.55 μm of frequency v₁. The beam splitter-compensatot assembly is composed of 3 identical presnel parallelepipeds optically adhered together and forming an optical block. One of the parallelepipeds has a semireflecting coatting on the internal face Only one of the arms of the interferometer has a totally reflecting glass-air on vaccum interface the glass part of the other arm behaves as a compensating plate The first arm on the interferometer is ended with a corner cube reflector mounted on a piezoelectric transducer The second one is ended with an identical reflector situated a distance D further from the optical block.

Any incident beam whose polarizsation is onented at 45° to the plane of incidence (equivalent to the plane of Fig 1 ), can be split into two orthogonal polarizations, p and s. respectively parallel and perpendicular to the plane of incidence We can consider that these two orthogonally polanzed output beams interfere independently inside the interferometer After a simple total reflection for each direction of propagation, these beams acquire a phase difference Δϕ derived from the Fresnel formulae

from which we deduce

where I’ts the beam angle ofincidence on the total reflecting surface and ng is the refractive index of the medium An angle of incidence i=55° is necessary to obtain a 90° phase shift at λ=633nm with an optical block made from Schott BK7 glass (n_g= l.515)

At the output of the optical block. a polarizing beam splitting cube separates the p and s components of the polarization The two output intensities are

where I_o is proportional to the intensity The quantity is the output phase of the interferometer, c is the speed of light m vacuum. n and v_t are defined above. D is the distance to be measured. k is the integer part of the interference order and ε is the fractional part From the output beam of the interferometer and using two perpendicular polarizers set m front of detectors. we obtain the two signals tn phase quadrature given in (2 3)

Fig 1

Interferometric set-up OB optical block CC comercube . P polarizer . D, and D, cosine and sine detectors . M and DM multiplexer and demultiplexer. D distance to measure . v_a and V_a' source A frequencies . v_b source B frequency

After an appropriate electronic treatment (subtraction of DC levels and intensity normalization). these two signals are reduced to

For control purpose, these two quadrature signals can be visualised bysending. them to the inputs of an oscilloscope set to the XY configuration The position of the spot gives directly a value of φ(Fig 2) Practical values of φ are obtained by adequate computer treatment of these signals

Fig 2

Lissajou figure

After computing, I_o and I, are send to an A/D converter and to a sine and cosine inputs of a reversible counter This counter determines the integer number of half fringes during the scan of the laser frequency

To determine the fractional part, we have to

Such that we have finally

The value of φis given bv φ=tan^-1(Y/X) from which we extract ε=φ/2π (where 0<ε<1 ) with an uncertainty dε of order 10^-3

3.

MEASUREMENT UNDER VACUUM

3 - 1

Description

The frequency v_L of the tuneable laser diode is scanned from V_a to V_a’, reference frequencies corresponding to two optical transitions near l.5μm of the acetylene molecule The absolute distance D can be determined for each frequency

where k₁ is the integer part of the interference order and ε_i is the fractional part

D is also given by

Where The ratio c/Δv is often called the synthetic wavelength In our case is of order 292 Ghz

In this step, Δk is known unambiguously and fractional parts at the beginning and end of the scan are determined by the method explained above From equation (3 2) we calculate a first value of D with an uncertainity around 1.53μm derived from the uncertainity of the frequency difference Δv (97 kHz in our case) and from the uncertainity of Δε (1.4 10^-3) with this uncertainity it is not possible however to obtain a better accuracy in the measurement of D by using equation (3 1) directly since the determination of k remains ambiguous

The ambiguity can be solved by using a third frequency standard v _bsuch that v_b-v_a (900 GHz in our case) is three times larger than va-va’. This allows one to reduce the uncertainity of D to 0.5μm For this latter case, since it is not possible to scan continuously the lase frequency from v_a to v_b, the value k_b-k_a cannot be measured directly. It is however calculated without ambiguity from the values determined using the first step of the measurement

Equation (3 1) constitutes the last step of the measurement since now the uncenaintv of D obtained by step - permits a determination of k_a The final uncertainty m D is obtained using equation ( 3 1) and becomes 2.5nm ( 10^-9 relative uncenainty for D=3m)

3 - 2

Experimental results

For the moment, we realised the first step of the measure. which consists in

• - detemnning ε_a

• - counting the number of fringes during the scan of the laser diode frequence from v_a to v_a’

• - determining ε_a’

Because the measure of ε_a and ε_a’ is not simultaneous. we have to take into account any vibration or thermal expansion of our interferometer during the measuring time of around 30s So that we use a second laser source which is a He-Ne laser at λ₁=633nm It works as a classical interferometer At last the absolute distance D is given by

The quantity d_t is the correction determined from the calculation of ε_r and ε_r’ at the beginning and the end of the scan

Distance measurements are made under vacuum ( 10^-5mbar and 0.1mbar) at about 0.41 and 3m The reference hollow corner cube in then moved from about 77μm by mean of the piezoelectric transducer The panel below summarizes the results obtained by 100 measurements each time The mean value, the experimental standard deviation (esd). the repeatabilit. the reproducibility and the global uncertaintv are calculated

	10-5mbar	0.1 mbar
3m	0.41m	3m	0.41m
D(m)	2.956901	0.411941	2.956900	0.411936
esd (μm)	1.5	2.1	1.1	0.43
repeatability (μm)	1.2	1	1	0.43
Reproducibility (μm)	2.8	1.7	2.5	0.43
uncertainty (μm)	1.4	1.1	1.1	0,43
Corner cube displacement (μm)	around
D(m)	-	-	2.956821	0.411857
esd (μm)	-		0.9	0.8
repeatability (μm)	1 -	-	l	0.4
reproducibility (μm)	-	-	l	0.72
uncertainty (μm)	---	-	1	0.5

The uncertainty obtained here is of the same order than theoretical uncertainty calculated into 3 - J The uncertainty is greater at 10^-3mbar than 0.1mbar because we worked with the vacuum pump on we suppose that it is the cause of interference vibrations

The displacement of the comer cube measured by the interferometer is 79μm

We also examined the effect of the external temperature T (the variation is about 2°C) on the absolute distance D and we observe a linear relation between D and T The factor is 12.5μm/°C at 3m (Fig 3)

Fig 3

a)Temperarure and absolute distance vs time b) Absolute distance vs temperature

We submitted the laser source at temperatures of 10. 20 and 30^°C and we made absolute distance measurements at 3m as described above The panel below shows the results

	10° C	20°C	30°C
D (m)	2,956904	2.956906	2,956905
uncertainty (μm)	2.6	1	1.5

We also examined the correlation between the absolute distance measured as described above and the displacement determined by the visible source working as a classical interferometer (Fig 4)

Fig 4

Absolute distance and visible measurement vs time

We can see that the two curves have the same behaviour. the visible measurement is 1.44μm and the infrared one is 1.61μm

We prove that we can measure an absolute distance with an uncertainty of 2 μm We show that a second source is necessary to make a correct measurement of D We show too that our svstem can detect any variation of D around a face value, 0.41 or 3 m

If the method is applied to distance measurement in an the relative uncertainty will be increased to 10^-7 because of the unknowledge of the refractive index n

4-

MEASUREMENT IN AIR OR ANY GASEOUS MEDIUM

In order to make distance measurement in air (or another gaseous medium). a ne w type of source is used with the interferometer described above The source in question is an air wavelength standard developed at the BNM-INM (Bureau National de Metrologie-Institut National de Metrologie) lts

relative uncenaintv is about l part in 10^x and it is insensitive to the refractive index of the A

medium. generally air Fig. 5 shows the principle of this wavelength standard

Fig 5

Air wavelength standard

The source is based on a plane-plane Fabrv Perot cavin with a zerodur spacer to which the silica, minors are opticallv adhered The gold coated mirrors haw a reflectivity of 97% at 633nm and even h igher in the infrared ( 1550nm)

The design of this svstem allows one to determine unambiguously the interference order k of the transmission peak to which the frequency V₁ of a laser diode is locked after locking. the frequency of the laser source tracks in real time the refractive index fluctuations such that nv_t =constant The

wavelength of the source given bv remains unaltered This wavelength is determined by

measunng e.under vacuum. with the help of an optical standard and the method of exact fractions U sing this system. the three preceding reference frequencies v_a, v_a’. and v_b are replaced bv the reference wavelength men by a laser diode locked on different transmission peaks of known interference orders The distance D is determined as descnbed above

The measurement uncenaintv is limned by the wavelength standard (about 10^-8) and is free of refractive index corrections

5.

CONCLUSION AND PERSPECTIVES

In this paper we have outlined the pirnciples of distance measurements using a device based on a fringe counting sigmameter and the concept of synthetic wavelengths A prototype interferometer has been developed at CSO Mesure in order to prove the feasibility of the first step of the measurement We prove that we can measure an absolute distance up to 3m with an uncertainty of 2μm m using one svnthetic wavelength and the continuous scan between two frequencies It will allow us to link two discrete frequencies synthetic wavelength about 310μm) thus reach an uncertainty of 0.1 μm Finall y we will use the basic wavelength (1.5-μm) to measure the absolute distance with an absolute uncerfaint y about 3nm

The next step consists in associating our interferometer and the air-wavelength standard and to sho w that we can make distance measurement in air without any problem