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We present a concept of suppression of the influence of variations of the refractive index of air in displacement measuring interferometry. The principle is based on referencing of wavelength of the coherent laser source in atmospheric conditions instead of traditional stabilization of the optical frequency and indirect evaluation of the refractive index of air. The key advantage is in identical beam paths of the position measuring interferometers and the interferometer used for the wavelength stabilization. Design of the optical arrangement presented here to verify the concept is suitable for real interferometric position sensing in technical practice especially where a high resolution measurement within some limited range in atmospheric conditions is needed, e.g. in nanometrology.
©2012 Optical Society of America
1. Introduction
Interferometric measuring techniques with highly coherent and stable laser sources represent a key technique in dimensional metrology. As according to the definition of the unit of length the fundamental physical constant – the speed of light in vacuum – converts time interval into a certain distance, an optical frequency of a coherent light source can be converted into wavelength. This principle is the basis of interferometric measuring techniques. A highly stable laser (with stable optical frequency) is a technical representation of a length standard.
To get to the real length measurement, the elementary length quanta – the discrete wavelengths – are counted over the measured displacement through laser interferometry. Resolutions deep below one wavelength have been achieved thanks to interpolation of one interference fringe through advanced signal processing techniques. In vacuum conditions the conversion from optical frequency to wavelength does not introduce any additional source of uncertainty. The relative stability of the present laser standards go really down to remarkable values. Traditional He-Ne lasers stabilized to the active Doppler-broadened line in Ne can operate with relative frequency stability on the level 10−8 – 10−9, He-Ne laser stabilized through subdoppler spectroscopy in iodine on the 10−11 – 10−12 level and the potential of iodine stabilized lasers based on frequency doubled Nd:YAG is very close to the 10−14 level [1]. The reproducibility of their absolute frequencies is another goal in metrology and is limited to 2.1 x 10−11, resp. 9 x 10−12 [2]. A length measuring interferometric setup is a subject to a host of other sources of uncertainty limiting the real length measurement precision much more than the fundamental laser standard.
When the measurement is done on air the refractive index of this environment becomes the key limiting factor of the overall uncertainty. In case of commercial interferometric systems the only reasonably simple approach is to measure the refractive index indirectly and to use the value for calculation of the wavelength correction. This indirect technique is based on measurement of air temperature, pressure, and humidity and when better precision is needed also the content of carbon dioxide. The value of refractive index is extracted by evaluation of the empirical Edlen formula [3]. This fundamental formula was further tested and a set of improvements followed [4–7]. All measurements of the refractive index of air performed by refractometers or by evaluation of the Edlen formula suffer from one principal limitation namely the fluctuations of air along and around the laser beam axis.
Precise interferometric measurements in fundamental metrology in laboratory conditions are accompanied by measurement of refractive index of air with refractometers. The principle of refractometry of air is an inverse one compared to interferometry, change of optical length is measured during evacuation of a cell of known and fixed mechanical length. Tracking refractometer converting the refractive index variations into laser optical frequency has been presented in [8]. In this case the concept relies on coherent and broadly tunable laser sources [9,10]. Refractometers serve also as a tool for further improvements of the Edlen formula. Under precisely controlled laboratory conditions uncertainties close to the 10−9 [11, 12] have been achieved. The limiting factor is definitely the homogeneity and stability of the atmosphere along the beam path. The practical limit in evaluation of the refractive index of air is determined by effects such as thermal gradients and air flow. They cannot be completely avoided; they depend on particular application and measurement configuration.
To get the information about the distribution of the refractive index along the measuring beam path, the sensors, especially temperature ones should be placed close to the beam. Still, the indirect measurement relies on a set of sensors with relatively slow response time unable to follow the changing nature of the air refractive index. The idea of combining refractometer with displacement measuring interferometer has been proposed [13]. Authors suggest using a set of two identical interferometers where one is fixed in the length and serves as a reference for the laser wavelength.
2. Stabilization of wavelength
Our proposal represents an interferometric system similar to a concept of combined interferometer and refractometer. Tracking refractometer is able to monitor the variations and drift of the refractive index of air. This value can be on-line used for calculation of the wavelength on air. More, our optical design unifies the laser beam path of the tracking refractometer with displacement measuring interferometer reducing thus significantly the uncertainty associated with inhomogeneity of air and air flow. In this setup the laser is not stabilized to optical frequency reference but its atmospheric wavelength is locked to a mechanical reference of length. This length is also the measuring range of the positioning. A He-Ne laser for interferometry locked to the active line in Ne can offer a relative stability of the optical frequency in the order of 10−8. This is on the same level as the coefficient of thermal expansion of highly stable materials such as Zerodur from Schott or ULE from Corning. It seems to be feasible to use mechanical standards made from these materials as a reference for stabilization of the wavelength under conditions of varying refractive index. Next to relying on the low thermal expansion of a frame the reference could be derived from a vacuum path monitored by an independent interferometer.
Traditional approach in displacement measuring interferometry is based on a laser source with highly stable optical frequency. Lasers for metrology are often stabilized by locking to a suitable frequency reference, such as atomic or molecular transitions. Definition of a length unit – the metre relies on a unit of time and a speed of light (in vacuum). Conversion of a stable optical frequency into stable wavelength suitable for measuring of distances is done through the speed of light which in air includes the refractive index (Eq. (1)). This has to be acquired independently.
where λa is wavelength in atmosphere, na is refractive index, c speed of light in vacuum, νstab stabilized, constant optical frequency. The refractive index is calculated as a function of the parameters of atmosphere, temperature υ, pressure p, relative humidity RH and partial pressure of CO2 pCO2 [3–7].The concept of referencing of λ, i.e. stabilization of wavelength in air keeps the wavelength constant, derived from a mechanical reference. The length of the mechanical reference, here the length LC is monitored with an interferometer. Its output is used to control the laser optical frequency which varies with the fluctuating refractive index of air na within the length LC (Eq. (3)).The control loop keeps constant N, the number of wavelengths within the length LC (Eq. (4)).This principle can be seen as moving with the carriage over a fixed scale locked to the length of the measuring range which is considered highly stable thanks to the for example low expansion material.When the sources of uncertainty in interferometry are considered, next to the refractive index, one of the dominating is always the mechanical stability of the whole arrangement. Close to the nanometer level the mechanics becomes critical. The approach we present here combines the mechanical referencing of the interferometer itself with referencing of the laser wavelength. The mechanical referencing simply cannot be avoided so we at least link one source of variations and uncertainty (refractive index) to another (mechanics). We first proposed a concept with an over-determined counter-measuring interferometric displacement measuring setup [14, 15] where the length in one axis was measured by two interferometers with their reference points fixed to a highly stable mechanical reference – frame. In this case the reference relied on a material with thermal stability low enough to overcome the uncertainty caused by fluctuations of the refractive index of air. We used “0”–grade Zerodur ceramics from Schott, with stability at 10−8/K level for a wide range of temperatures from 0°C to 50°C. In a smaller range the coefficient of thermal expansion should have a plateau with even smaller thermal expansion.
In our first arrangement the wavelength of the laser source was fixed by a control loop to a sum value of the two interferometers representing a principle of stabilization of wavelength. This lead to a design with a sort of standing wave interferometry [16–18]. In this contribution we present a new version of this concept focused on a design applicable in real displacement measurements. The setup consists of three interferometers where the overall length is not a sum value of two but an independently measured value (Fig. 1 ).
Fig. 1 Configuration with corner-cube reflectors measuring directly the overall length and two particluar displacements. CC: corner-cube reflector, PBS: polarizing beamsplitter, NP: non-polarizing plane, λ/2: half-wave plate, F: fiber-optic light delivery, OA, OB, OC outputs, Lc. La, Lb: particular lengths determining the position of the moving carriage.
The system consists of three independent interferometers where each measures the specified part of the overall length (A, B, C, see Fig. 1). The left polarizing beamsplitter with a corner-cube reflector serves as a reference arm for the interferometer measuring the distance between the left reference point and the moving carriage (A) as well as for the interferometer measuring the overall length (C). The moving carriage carries another beamsplitter with corner-cube reflector generating a reference arm for the interferometer measuring the distance between the moving carriage (B) and the right reference point. The beam of the interferometer C only passes through the beamsplitter on the moving carriage. Beam paths on air of the interferometers A and B are identical with proportional parts of the beam path of the interferometer C.
Quite complex optical setup driven by the effort to unify the beam paths results in a beam being a measuring beam in one interferometer and reference beam in another, etc. This raises the danger of crosstalk between the interferometers. Only the interferometer B is free from this effect. In case of A and C, tilt of the two beamsplitters with respect to each other may cause this problem. In a regime of stabilization of wavelength referenced to the output from C the phase in C is kept constant reducing its crosstalk into A. In our arrangement with proper alignment of the two beamsplitters the effect of crosstalk was suppressed deep below the level of fluctuations of the refractive index of air.
Optical arrangement of the triple interferometer presented here was designed for homodyne detection where the measuring beam leaving the first beamsplitter is polarization rotated 45° to be split again into measuring and reference beams of the interferometer B. To operate in a heterodyne detection scheme, this setup as it is would be able to measure the overall length LC and carriage position LA. Still this would mean positioning and displacement measurement over a range with stabilized wavelength. To get the output from the interferometer B, the setup would have to be redesigned. Linearity of the scale in a homodyne interferometer – the cyclic error has been compensated in our experiment. We applied our method developed previously [19].
The principle combines one-axis interferometric measurement with Michelson type interferometer and tracking refractometer that is able to follow the variations of the refractive index just in the beam path of the measuring interferometer. This is the key improvement over other concepts of stabilization of wavelength in air [20, 21]. Our arrangement includes two interferometers measuring the displacement in a counter-measuring setup and a third one that gives the information of the overall optical length changes. Considering the physical length of the interferometer C constant or constant with precision overwhelming the precision of the refractive index evaluation the output of the interferometer C serves a reference for the atmospheric wavelength stabilization. Average value of wavelength in the range given by interferometer C is kept constant and the carriage moves within.
The carriage position can be seen in our arrangement as overdetermined, it is measured from both sides, referred here as A and B. The carriage displacement may be referenced either to the left or right end of the measuring range. Still the identity of the displacement measuring beam path (on air) and the beam path of the tracking refractometer is limited by the ratio given by the carriage position. The value of the refractive index may differ in the left and right part (A, resp. B) of the setup. The best approximation of the resulting carriage position should be thus a value calculated from both A and B positions.
3. Tracking of the refractive index drift with laser optical frequency
Testing of the performance of the interferometric system in the regime of stabilization of wavelength was done in a double-wall glass box with the walls filled with water circulating within and outside through Peltier heater/cooler. Thermal control of the environment inside reduced temperature gradients thanks to even distribution of temperature on the walls. Temperature control of the air inside allowed to let the air be heated or cooled gradually so the refractive index of air would vary within some range. To monitor the atmosphere inside, we added temperature, pressure, humidity sensors together with a sensor monitoring the content of CO2. Refractive index of air was instantly calculated and recorded from these measurements to be compared the drift of the laser optical frequency following the stabilization of atmospheric wavelength. We used the following set of sensors: precision thermometer Isotech F100 with Pt100 sense with response time approx. 30 s, pressure sensor Freescale semiconductor MPXH6101A with response time 15 ms, relative humidity module Measurement Specialties HTM25X0LF with response time 10 s, and carbon dioxide probe Ahlborn FYA 600 CO2 with response time 1 min.
To investigate the level of relevance of this concept we recorded the variations of refractive index monitored by the interferometer outputs in a steady-state regime with the carriage in approx. central position within the measuring range. The recording in Fig. 2 shows outputs from the three interferometers (A, B, and C) with the temperature control box closed and open (in laboratory conditions). A sum value of interferometers A and B is added. In both cases there is a small mechanical drift of the carriage but the recordings show well the level, frequency and overall nature of these fluctuations. The comparison between the sum of A and B and the interferometer C output shows how successful the stabilization of wavelength may be when used for compensation of the refractive index variations in the measuring axis. The differences between A + B and C should be interpreted in comparison with the sole variations of A, resp. B. The air path of the whole measuring range (monitored by the interferometer C) was 195 mm. The mean value of these differences is in closed box 8 nm which means relative uncertainty 4 x 10−8. Correlation coefficients (sometimes referred to as cross-correlation) may be expressed to show the similarity of recorded random processes. Here, when applied to the A + B and C recordings, R = 0.9531 for the closed box case (Fig. 2 left), resp. R = 0.9817 for the open laboratory environment (Fig. 2 right) show the advantage of unified beam paths for displacement measurement and stabilization of wavelength.
Fig. 2 Recording of the variations of the interferometers A (red line), B (yellow line), and overall length measuring C (blue line) together with the sum of A and B (green line) over time in a closed thermal box (left) and opened (right).
4. Experimental verification of the system performance
The interferometric setup was subject to a slow drift of the refractive index of air induced through heating of the air within the thermal control box. The aim was to compare the recording of the varying refractive index evaluated through Edlen formula with tracking of these variations through the laser optical frequency in the regime of stabilization of wavelength. The laser was locked through a servo control loop to the constant output of the interferometer C, at the beginning of the experiment reset to zero value. In Fig. 3 and 4 we show the refractive index drift and the corresponding drift of the laser optical frequency. The proof of the concept was based on comparison of these two recordings.
Fig. 3 Recording of a slow refractive index drift evaluated from measurement of air temperature, pressure, humidity and CO2 content.
Fig. 4 Recording of the optical frequency tuning of the laser following stabilized wavelength over the measuring range.
The gradual drop of the refractive index recorded over approx. 20 min recorded in Fig. 3 performs a smooth decrease due to slow response of the sensors for measuring of the parameters of the atmosphere insensitive to faster variations seen in Fig. 2. The recording of the optical frequency shows the sensitivity of the system following the small-scale variations similar in nature and scale to those in Fig. 2 and thus the ability of this system to compensate for them.
To follow the principle of referencing to high-stability mechanical frame, it should include the central beamsplitter on the moving carriage to be made out such material as well, at least quartz glass. Thermal expansion of this component inserted into the beam paths contributes to the temperature stability of the overall length. In our case we used SF-14 glass for technology reasons and its slow gradual heating together with high thermal expansion coefficient and high refractive index consequently acted against the course of the drift. To evaluate the agreement between these two recordings we considered only this limited time period of the slope before the expansion of the glass showed up. The agreement for this period can be expressed as on the level 2 x 10−8 difference between the relative change of refractive index (Fig. 3) and relative change of laser optical frequency (Fig. 4).
Recording of the optical frequency drift in Fig. 4 was derived from a control voltage tuning the laser via PZT. We used the calibrated PZT voltage not only for technical simplicity. We wanted to show the nature of the variations of the optical frequency including faster changes. Precise optical frequency monitoring through counting of beat frequency with a reference stabilized laser would result in only slow frequency sampling. Nonlinearity in PZT response was in our case surely below the level of agreement we were able to achieve mentioned above.
The whole experiment was controlled by software written in LabView environment. The servo loop locking the laser optical frequency to the constant value of the refractive index of air included proportional-integral controller, digital-to-analog converter and high-voltage amplifier driving the PZT.
5. Discussion and conclusions
Performance of this system can be judged on the basis how the laser locked to the constant wavelength is able to follow the fluctuations of the refractive index of air and how relevant the displacements measured by the interferometers A and B are within the measuring range set by the interferometer C with the stable wavelength. Agreement between the values measured by all three interferometers can be assumed from the Fig. 2. Agreement between the A and B value together with their sum compared with C shows good coincidence more than an order of magnitude smaller compared to the amplitude of fluctuations themselves. Understandable disagreement between A and B (their beam paths are not identical) results in a suggestion to evaluate the measured displacement of the carriage as an average from A and B of the overdetermined system. This would refer to the constant value of wavelength kept within the whole distance of C (see good agreement A + B vs. C). Thanks to constant wavelength within the measuring range, the interferometers can operate in their incremental regime without the need to know absolute (air) lengths of A, B or C.
The recordings in Fig. 2 as well as 4 show the nature of the refractive index variations. In case of the interferometer C in Fig. 2, 8 nm corresponds to 4 x 10−8 of the refractive index change. In our experiment both the interferometer detection chain as well as the frequency response of the optical frequency servo loop were well over the frequency bandwidth of this process. This may be seen in a sharp contrast with the slow response of the indirect evaluation of the refractive index through Edlen formula (Fig. 3).
The ability of the servo loop controlling the laser optical frequency to follow the fluctuations of the refractive index of air is limited by the continuous tuning range of the laser. The relative frequency tuning corresponds to the relative change of the refractive index. It is independent from the optical length of the beams in air. Considering the parameters of atmosphere, for example the drift of temperature over 1 K needs tuning of green, frequency doubled Nd:YAG laser over approx. 1 GHz.
The concept presented here is well able to follow the fluctuations of the refractive index of air and effectively compensate for them. Recording of the laser optical frequency in the locked regime shows very similar nature to the fluctuations of the refractive index. It is not able to measure the value of the refractive index absolutely. The initial value has to be measured a traditional way or with the help of a refractometer.
Acknowledgments
The authors wish to express thanks for support to the grant projects from Grant Agency of the Czech Republic, projects: GA102/09/1276, GAP102/11/P820, Technology Agency of the Czech Republic, projects: TA02010711, TE01020233, European Commission and Ministry of Education, Youth, and Sports of the Czech Republic, project: CZ.1.05/2.1.00/01.0017, and RVO: 68081731.
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