**2. Absolute Distance Measurement by optical interferometry**

Absolute Distance Measurements (ADM) can be performed by several optical methods. Generally, one has non-interferometric methods for large ranges with moderate accuracies and interferometric methods for high accuracies and moderate ranges, limited by the coherence length of the source. ADM can also be performed on the basis of low-coherence or white light interferometry but, in this case, the measuring range is limited to a few millimetres. Lately, the measuring range of interferometric methods increased due to the increase of the coherence length and the development on new coherent sources.

ADM interferometric techniques are typically based on the use of multiple or time varying wavelength sources. Single wavelength interferometry is a powerful tool in displacement measurement when high accuracy is required, however, as a major drawback, is only applicable to relative measurements. To measure an absolute distance it is necessary to use incremental interferometry: to measure a displacement it is necessary to fix a reference position and to carefully displace a suitable reflector from that position to the final one, and basically counting each interferometer fringe event. This must be done avoiding any optical misalignment that would result in a loss of the interference signal and therefore of the measure.

Multiple Wavelength Interferometry (MWI) is a technique were the use of (at least) two different wavelengths allows the generation of a synthetic wavelength much longer than the two individual optical wavelengths, actually increasing the non-ambiguity range (NAR) for interferometry (Dändliker et al., 1988; Dändliker et al., 1995). In MWI, each synthetic wavelength (one or several) is generated using two different and very accurately known optical wavelengths. By selecting a small synthetic wavelength it is possible to achieve high resolution (at the micrometre level or even smaller). However the intrinsic ambiguity of the measurement limits the measurement range. To overcome the ambiguity, a chain of increasing synthetic wavelengths is generated (or a combination with another sensor) to cover the required range (Salvadé et al., 2000). For large ranges, several wavelengths are required and MWI can become indeed a very complex solution.

In a first approach, FSI is equivalent to MWI. In FSI, the generation of the synthetic wavelength is based on frequency sweeping the laser source within a given sweep range. The technique is not new, has strong similarities to radar, dating back to the 80's (Kikuta et al., 1986), but it was not studied extensively until the development of tunable lasers and the emergence of External Cavity Diode Lasers (ECDL) (Hecht, 2001). As frequency sweeps, detection electronics counts synthetic wavelength fringes (temporal "synthetic fringes") without ambiguity, thus making it particularly interesting for large measurement ranges (Thiel et al., 1995; Stone, 1999; Edwards et al., 2000; Coe et al., 2004; Cabral & Rebordão, 2005; Swinkels et al., 2005). While in Double Wave interferometry (DWI) – which is a particular case of MWI - we have a fixed synthetic wavelength, while in FSI, as the frequency is being swept, the value of the synthetic wavelength is decreasing down to a value defined by the total sweep range.

In contrast to MWI, FSI does not require independent stabilized and well known laser sources and relies only on a tunable laser and a frequency sweep range measurement subsystem, normally based on a Fabry-Pérot interferometer (FP).

As in DWI, the synthetic wavelength Λ is inversely proportional to the frequency sweep range Δν (see next section). While in MWI we measure only the fractional part of the synthetic wavelength fringe, in FSI both the fractional and integer number of synthetic fringes can be measured. Thus, the absolute value of the OPD (between the two arms of a Michelson interferometer) will be determined without ambiguity.

It is not possible to elaborate an objective comparison between different techniques without an a priori definition of the sensor requirements: maximum range, accuracy, technological complexity, reliability, cost, etc. Nevertheless, a trade-off between the two basic methods, FSI and MWI, can be done in a qualitative way, considering the three main parameters: measurement distance, measurement uncertainty and overall complexity of the sensor subsystem.

FSI main advantage is the capability to perform large range measurements due to the nonambiguity nature of the technique, limited only by the coherence length of the tunable laser (a few hundred metres). FSI main drawback is the sensitivity to drift that, even with a compensation method, will limit the accuracy at the micrometre level. With an affordable complexity in the frequency sweeping (i.e. synthetic wavelength) measurement sub-system, for large distances, FSI can achieve a relative accuracy around 10-5. With a high degree of complexity (locking the laser to a resonant cavity) accuracies can be improved by one or two orders of magnitude.
