**3. Frequency Sweeping Interferometry**

In the previous section it was mentioned that the intensity detected in a Michelson interferometer changes with the OPD change for a fixed laser frequency. If the frequency changes in time, the detected intensity will change not only due to the OPD change but also due to the frequency change.

FSI is based in the variation of the optical frequency ν, or angular frequency ω = 2πν, of a tunable laser in a Michelson interferometer. In this situation, the frequency no longer represents the periodical property of the wave, and in this case, the phase component is a non-linear function of time. Defining the instantaneous angular frequency as the derivative of the phase:

$$\cos(t) = \frac{d\phi(t)}{dt} \tag{1}$$

the phase is given by:

54 Modern Metrology Concerns

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

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

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

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

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

In the previous section it was mentioned that the intensity detected in a Michelson interferometer changes with the OPD change for a fixed laser frequency. If the frequency changes in time, the detected intensity will change not only due to the OPD change but also

FSI is based in the variation of the optical frequency ν, or angular frequency ω = 2πν, of a tunable laser in a Michelson interferometer. In this situation, the frequency no longer represents the periodical property of the wave, and in this case, the phase component is a non-linear function of time. Defining the instantaneous angular frequency as the derivative

> ( ) ( ) *d t <sup>t</sup> dt*

(1)

value defined by the total sweep range.

subsystem.

orders of magnitude.

due to the frequency change.

of the phase:

**3. Frequency Sweeping Interferometry** 

subsystem, normally based on a Fabry-Pérot interferometer (FP).

Michelson interferometer) will be determined without ambiguity.

$$\phi(t) = \int\_0^t \alpha(t)dt + \mathfrak{q}\_0 \tag{2}$$

where φ0 is the initial phase of the light source. The electric-field amplitude (not considering its spatial variation) is then described by:

$$E(t) = A(t)e^{i\psi(t)}\tag{3}$$

If two optical waves are derived from the same coherent light source, the angular frequency changing linearly in time (during the measurement period) but travel along different paths and are recombined at a point in space to interfere, the detected signal will change in time, unlike the case of a fixed frequency. In a Michelson interferometer, the angular frequency of the reference wave within the time interval *t* [0, Δ*t*] is given by:

$$
\alpha\_R(t) = \Omega \cdot t + \alpha\_0 \tag{4}
$$

where ω0 is the angular frequency at the beginning of the frequency sweep, and Ω is the angular frequency sweep rate. If the angular frequency variation has a linear behaviour, the value of the angular frequency sweep rate is given by:

$$
\Omega = 2\pi \frac{\Delta \mathbf{v}}{\Delta t} \tag{5}
$$

where Δν is the frequency sweep range. The phase of the reference wave is given by:

$$\text{op}\_R(t) = \frac{1}{2}\boldsymbol{\Omega} \cdot \boldsymbol{t}^2 + \boldsymbol{\alpha}\_0 \cdot \boldsymbol{t} + \boldsymbol{\varphi}\_0 \tag{6}$$

The wave function of the reference electric field is described by:

$$E\_{\mathcal{R}}\left(t\right) = A\_{\mathcal{R}} \cdot e^{i\left[\frac{1}{2}\Omega t^2 + \alpha\_0 t + \eta\_0\right]}\tag{7}$$

where *AR* is the amplitude of the reference wave (considered to be constant in time). The signal wave, travelling in the other arm of the interferometer, can be described similarly:

$$
\alpha\_S(t) = \boldsymbol{\Omega} \cdot \left(t + \boldsymbol{\tau}\right) + \alpha\_0 \tag{8}
$$

$$\log p\_R(t) = \frac{1}{2}\Omega \cdot \left(t + \tau\right)^2 + \alpha\_0 \cdot \left(t + \tau\right) + \eta\_0 \tag{9}$$

$$E\_S\left(t\right) = A\_S \cdot e^{i\left[\frac{1}{2}\Omega \left(t+\tau\right)^2 + \alpha\_0 \left(t+\tau\right) + \eta\_0\right]}\tag{10}$$

where *AS* is the amplitude of the signal wave (considered to be constant in time) and τ is the delay time of the signal wave with respect to the reference wave. The delay is related to the OPD in the interferometer by:

$$
\pi = \frac{n \cdot OPD}{c} \tag{11}
$$

where *n* is the refractive index of the optical propagation medium. When these two waves interfere, the intensity of the resulting electrical field can be written as:

$$\begin{aligned} I &= \left\langle \left| \mathbf{E}\_R \right|^2 \right\rangle + \left\langle \left| \mathbf{E}\_S \right|^2 \right\rangle + \left\langle \left| \mathbf{E}\_R \cdot \mathbf{E}\_S^\* \right| \right\rangle + \left\langle \left| \mathbf{E}\_R^\* \cdot \mathbf{E}\_S \right| \right\rangle \\ I(\tau, t) &= \left| A\_R \right|^2 + \left| A\_S \right|^2 + 2 \left( A\_R \cdot A\_S \right) \cos \left( \Omega \cdot \tau \cdot t + \alpha\_0 \cdot \tau + \frac{1}{2} \cdot \Omega \cdot \tau^2 \right) \end{aligned} \tag{12}$$

or equivalently by:

$$\begin{split} I(\boldsymbol{\tau}, t) &= I\_R + I\_S + 2\sqrt{I\_R I\_S} \cdot \cos\left(\boldsymbol{\Omega \cdot \boldsymbol{\tau} \cdot t + \alpha\_0 \cdot \boldsymbol{\tau} + \frac{1}{2} \cdot \boldsymbol{\Omega \cdot \boldsymbol{\tau}^2}}\right) \\ &= I\_0 \left[ 1 + V \cdot \cos\left(\boldsymbol{\Omega \cdot \boldsymbol{\tau} \cdot t + \alpha\_0 \cdot \boldsymbol{\tau} + \frac{1}{2} \cdot \boldsymbol{\Omega \cdot \boldsymbol{\tau}^2}\right) \right] \end{split} \tag{13}$$

where *IR* and *IS* are the intensities of the reference and signal wave, *I0* is the average intensity and *V* is the visibility (or contrast) of the signal:

$$V = \frac{2\sqrt{I\_R I\_S}}{I\_R + I\_S} \tag{14}$$

The relative importance of the different terms of the cosine argument in Eq.(13) depends on τ, Ω and ω0. The most relevant parameter is the delay τ. This value is limited by the coherence length of the tunable laser. For a coherence length of a few hundred metres, τ < 1 µs, the value of ½.Ω.τ2 will be several orders of magnitude lower than the other two terms (a typical value for Ω is 104 GHz/s). It is thus conceivable to neglect the term ½.Ω.τ2 in Eq.(13), and simplify the equation:

$$I(\tau, t) = I\_0 \left[ 1 + V \cdot \cos \left( \Omega \cdot \tau \cdot t + \alpha\_0 \cdot \tau \right) \right] \tag{15}$$

If we rewrite Eq.(15) considering the optical frequency and the OPD to describe the optical waves and the interference phenomena, we obtain (from Eq.(5) and (11)):

$$I\{OPD(t),t\} = I\_0 \left[1 + V \cdot \cos\left(2\pi \left(\frac{\Delta \mathbf{v}}{c} \cdot n \cdot OPD(t) \cdot \frac{t}{\Delta t} + \frac{n \cdot OPD(t)}{\lambda}\right)\right)\right] \tag{16}$$

where *OPD(t)* represents the OPD variation during the sweep and λ is the optical wavelength (in vacuum) at the beginning of the sweep. Defining a synthetic wavelength Λ, inversely proportional to the frequency sweep range, by:

$$
\Lambda = \frac{c}{\Delta \mathbf{v}} \tag{17}
$$

Eq.(16) becomes:

*n OPD c*

where *n* is the refractive index of the optical propagation medium. When these two waves

<sup>1</sup> , 2 cos

<sup>1</sup> , 2 cos

*II t*

where *IR* and *IS* are the intensities of the reference and signal wave, *I0* is the average intensity

2 *R S R S I I*

*I I*

The relative importance of the different terms of the cosine argument in Eq.(13) depends on τ, Ω and ω0. The most relevant parameter is the delay τ. This value is limited by the coherence length of the tunable laser. For a coherence length of a few hundred metres, τ < 1 µs, the value of ½.Ω.τ2 will be several orders of magnitude lower than the other two terms (a typical value for Ω is 104 GHz/s). It is thus conceivable to neglect the term ½.Ω.τ2 in

If we rewrite Eq.(15) considering the optical frequency and the OPD to describe the optical

where *OPD(t)* represents the OPD variation during the sweep and λ is the optical wavelength (in vacuum) at the beginning of the sweep. Defining a synthetic wavelength Λ,

> *<sup>c</sup>*

( ) ( ), 1 cos 2 ( ) *t n OPD t I OPD t t I V n OPD t*

waves and the interference phenomena, we obtain (from Eq.(5) and (11)):

2 2 2

interfere, the intensity of the resulting electrical field can be written as:

*R S RS*

*R S RS*

*I t A A AA t*

*R S RS RS*

0 0

*V*

*IV t*

<sup>1</sup> , 1 cos

**E E EE EE**

2 2

*ItI I*

*t*

and *V* is the visibility (or contrast) of the signal:

*I*

Eq.(13), and simplify the equation:

Eq.(16) becomes:

<sup>0</sup>

inversely proportional to the frequency sweep range, by:

*I*

or equivalently by:

(11)

0

0

*ItI V t* , 1 cos 0 0 (15)

*c t* 

2

2

2

2

2

(14)

(17)

(12)

(13)

(16)

$$I\left(OPD(\mathbf{t}),t\right) = I\_0 \left[1 + V \cdot \cos\left(2\pi \left(\frac{n \cdot OPD(\mathbf{t})}{\Lambda} \cdot \frac{t}{\Lambda t} + \frac{n \cdot OPD(\mathbf{t})}{\lambda}\right)\right)\right] \tag{18}$$

It is clear that the value of the OPD can contribute to the detected signal both in terms of the optical wavelength λ and/or in terms of the synthetic wavelength Λ generated by the frequency sweep.

To analyse Eq.(18) we consider two simple cases: 1 - no frequency sweep and, 2 - static OPD. In the first case, Δν = 0, Λ = ∞ and Eq.(18) becomes:

$$I\left(OPD(t), t; \Delta \mathbf{v} = 0\right) = I\_0 \left[1 + V \cdot \cos\left(2\pi \left(\frac{n \cdot OPD(t)}{\lambda}\right)\right)\right] \tag{19}$$

In this case, as the OPD changes in time, detector will sense a complete fringe for every λ/*n* variation in the OPD. This is the particular case of relative single wavelength interferometry.

In the second case, the OPD is constant during the frequency sweep, and Eq.(18) becomes:

$$I\left(OPD\_0, t\right) = I\_0 \left[1 + V \cdot \cos\left(2\pi \left(\frac{n \cdot OPD\_0}{\Lambda} \cdot \frac{t}{\Delta t} + \frac{n \cdot OPD\_0}{\lambda}\right)\right)\right] \tag{20}$$

The second term of the cosine argument represents only a constant phase and, as the frequency sweeps, the detector will sense a number of synthetic fringes that correspond to the absolute value of the OPD in Λ/*n* units. This is the ideal baseline for FSI. The absolute value of a distance *L* (half the OPD) in a FSI Michelson interferometer (Fig. 1) is obtained from the detected signal during the sweep interval Δ*t* by:

$$L = \frac{N}{2} \cdot \frac{\Lambda}{n} \tag{21}$$

where *N* is the number of synthetic fringes detected from the beginning to the end of the sweep (integer and fractional part).

However, a third more realist case must be considered when the two phenomena, frequency sweep and OPD variation, occur simultaneously. In this case, the contribution of the two phenomena is mixed up in the detected signal, and it is impossible to distinguish between the individual contributions from each one of them.

It must be emphasised that, for a typical ADM application, the absolute value of the length to be measured is always several orders of magnitude larger than the length variations during the sweep (hereafter designated by drift). When drift occurs during the frequency sweep, two different types of fringes will be generated:


As it is impossible to distinguish these two types of fringes, λ-fringes will be misinterpreted as Λ-fringes, causing the noise error (drift) to be multiplied by a large amplification factor Λ/λ.

The effect of drift in the FSI measurement can be separated in two different cases. If the drift signal (movement direction) changes during the sweep duration (usually less than a second), we are in presence of a low amplitude drift typically with frequencies identical or higher than the measurement frequency. In this case, the small amplitudes will have a smaller influence and, if it is not negligible, due to its random nature, it can be reduced applying a mean filter to the measurements.

In a second case, if the drift signal does not change during the sweep, we are in the presence of a low frequency drift with considerable amplitude even within the small time duration of the sweep (bearing in mind that this effect will be amplified). In this case, the drift must be compensated. Independent relative metrology and control can always reduce the drift by measuring it and by actuating a delay line during the sweep, or take it into account in the distance calculations. Nevertheless, this option would increase the complexity of the sensor, jeopardizing one of the FSI advantages.

If the OPD is written as:

$$OPD(t) = \mathcal{Z} \cdot \left(L\_0 + l(t)\right) \tag{22}$$

where *L0* is the distance at the beginning of the frequency sweep and *l*(*t*) the relative distance variation in time, the drift (the factor of 2 is due to the round trip in the measuring arm of the interferometer), the detected intensity will be given by:

$$I(t) = I\_0 \left[ 1 + V \cdot \cos \left( 2 \pi \left( \frac{n \cdot 2 \cdot \left( L\_0 + l(t) \right)}{\Lambda} \cdot \frac{t}{\Lambda t} + \frac{n \cdot 2 \cdot \left( L\_0 + l(t) \right)}{\lambda} \right) \right) \right] \tag{23}$$

The absolute distance measurement is based on the fringe counting from the beginning (*t* = 0) to the end of the frequency sweep (*t* = Δ*t*). As long as the drift signal does not change during the sweep, it is not important to know how the fringes change in time but only to measure the total number of detected fringes from the start to the end of the sweep. For this purpose, we need only to describe the drift as:

$$l(t) = \begin{cases} 0 & \text{for } \ t = 0 \\ \Delta L & \text{for } \ t = \Delta t \end{cases} \tag{24}$$

From Eq.(23) and (24) we can determine the number of detected fringes by subtracting the initial from the final phase of the cosine argument (divided by 2π):

$$N = \left[\frac{n \cdot 2 \cdot \left(L\_0 + \Delta L\right)}{\Lambda} + \frac{n \cdot 2 \cdot \left(L\_0 + \Delta L\right)}{\lambda}\right]\_{t=\Delta t} - \left[\frac{n \cdot 2 \cdot \left(L\_0\right)}{\lambda}\right]\_{t=0} \tag{25}$$

resulting in:

$$N = \frac{2 \cdot n}{\Lambda} \left( L\_0 + \left( 1 + \frac{\Lambda}{\lambda} \right) \cdot \Delta L \right) \tag{26}$$

From Eq.(26) it is possible to determine the absolute distance at the end of the sweep:

The effect of drift in the FSI measurement can be separated in two different cases. If the drift signal (movement direction) changes during the sweep duration (usually less than a second), we are in presence of a low amplitude drift typically with frequencies identical or higher than the measurement frequency. In this case, the small amplitudes will have a smaller influence and, if it is not negligible, due to its random nature, it can be reduced

In a second case, if the drift signal does not change during the sweep, we are in the presence of a low frequency drift with considerable amplitude even within the small time duration of the sweep (bearing in mind that this effect will be amplified). In this case, the drift must be compensated. Independent relative metrology and control can always reduce the drift by measuring it and by actuating a delay line during the sweep, or take it into account in the distance calculations. Nevertheless, this option would increase the complexity of the sensor,

where *L0* is the distance at the beginning of the frequency sweep and *l*(*t*) the relative distance variation in time, the drift (the factor of 2 is due to the round trip in the measuring

0 0

The absolute distance measurement is based on the fringe counting from the beginning (*t* = 0) to the end of the frequency sweep (*t* = Δ*t*). As long as the drift signal does not change during the sweep, it is not important to know how the fringes change in time but only to measure the total number of detected fringes from the start to the end of the sweep. For this

*n L lt n L lt <sup>t</sup> It I V*

( ) for

From Eq.(23) and (24) we can determine the number of detected fringes by subtracting the

22 2

0

From Eq.(26) it is possible to determine the absolute distance at the end of the sweep:

*n L Ln L L n L*

 

*l t*

initial from the final phase of the cosine argument (divided by 2π):

2 () 2 () 1 cos 2

0 for 0

*L tt* 

00 0

*t*

 

*t*

arm of the interferometer), the detected intensity will be given by:

*OPD t L l t* () 2 () <sup>0</sup> (22)

(24)

*t t t*

<sup>2</sup> <sup>1</sup> *<sup>n</sup> NL L* (26)

0

(23)

(25)

applying a mean filter to the measurements.

jeopardizing one of the FSI advantages.

0

purpose, we need only to describe the drift as:

*N*

resulting in:

If the OPD is written as:

$$L = N \frac{\Lambda}{2 \cdot n} - \frac{\Lambda}{\lambda} \cdot \Delta L \tag{27}$$

Comparing Eq.(27) and (21), it is clear that the drift introduces an error equal to the drift amplitude multiplied by an amplification factor given by the ratio between synthetic and optical wavelengths.

When drift is neglected, compensation is not performed and the second term of Eq.(27) corresponds to an additional error in the measurement. As an example, for a sweep range of 50 GHz (Λ ≈ 6 mm) in the visible (λ ≈ 600 nm), the amplification factor is ≈ 10 000; a drift with an amplitude of 1 nm would introduce an error of 10 µm in the measurement. Depending on requirements, such an error may or may not be negligible.

It is thus very important to work with the lowest possible amplification factor either by using a longer laser wavelength (near IR) or a smaller synthetic wavelength by increasing the sweep range (without increasing the sweep duration otherwise the drift would also increase). Nevertheless, special care must be taken when the synthetic wavelength decreases, because the number of synthetic fringes will increase (which may also increase the complexity of the electronics counting subsystem).

#### **3.1 Synthetic wavelength measurement and distance calculation**

To calculate the absolute distance it is necessary to know not only the number of fringes but also the value of the synthetic wavelength, by measuring the value of the frequency sweep range.

In the experimental examples that will be discussed, the frequency sweep range measurement subsystem is based on a Fabry-Pérot interferometer (FP). The sweep range measurement is obtained by multiplying the FP free spectral range (FSR) by the number of resonances detected while the laser frequency sweeps. The beginning of the sweep is determined by a particular cavity mode; the laser sweeps from mode to mode and resonances, separated by the FP FSR, are detected and counted (Fig. 2). The frequency sweep range is then given by:

$$
\Delta \mathbf{v} = \mathbf{r} \cdot \text{FSR} \tag{28}
$$

where *r* is the number of detected *FSR* (number of resonances minus one).

From Eq.(17) and (28) we obtain:

$$
\Lambda = \frac{c}{r \cdot FSR} \tag{29}
$$

The measured length is, considering Eq.(27) and (29), given by:

$$\Delta L = N \frac{\mathcal{L}}{2 \cdot n \cdot r \cdot FSR} - \frac{\mathcal{L}}{\lambda \cdot r \cdot FSR} \cdot \Delta L \tag{30}$$

Without drift, or when its influence is smaller than the required uncertainty, we are in a Static Mode and the absolute distance can be calculated simply from:

$$L = N \frac{c}{2 \cdot n \cdot r \cdot FSR} \tag{31}$$

where *c*, *n* and *FSR* are constants and *N* and *r* are the variables to be measured.

Fig. 2. FSI Setup. While the laser sweeps the frequency, the interferometer detector acquires the fringes and a FP measures the sweep range by counting the resonances of the cavity.

#### **3.2 Drift compensation method for dynamic mode measurements**

When it is not realistic to assume that the distance under measurement is constant during the sweep, the drift effect on the measurement has to be compensated. If we can assume that during the sweep there is no acceleration, i.e. the drift speed is constant, it is possible to remove the drift influence from two consecutive measurements with different values of sweep duration or/and sweep range. For a constant drift speed *Sdrift*, the drift amplitude is described by:

$$
\Delta L = S\_{drift} \cdot \Delta t \tag{32}
$$

and Eq.(27) can be written as:

$$L = N \frac{\Lambda}{2 \cdot n} - \text{sign}(\Delta \mathbf{v}) \cdot \frac{\Lambda}{\lambda} \cdot S\_{drift} \cdot \Delta t \tag{33}$$

where sign(Δν) is the sign of the frequency variation (+ for an increasing and - for a decreasing frequency sweep). For two consecutive measurements, 1 and 2, the following system of equations holds:

*n r FSR* (31)

*LS t drift* (32)

(33)

2 *<sup>c</sup> L N*

where *c*, *n* and *FSR* are constants and *N* and *r* are the variables to be measured.

Fig. 2. FSI Setup. While the laser sweeps the frequency, the interferometer detector acquires the fringes and a FP measures the sweep range by counting the resonances of the

When it is not realistic to assume that the distance under measurement is constant during the sweep, the drift effect on the measurement has to be compensated. If we can assume that during the sweep there is no acceleration, i.e. the drift speed is constant, it is possible to remove the drift influence from two consecutive measurements with different values of sweep duration or/and sweep range. For a constant drift speed *Sdrift*, the drift amplitude is

> sign( ) <sup>2</sup> *L N S t drift <sup>n</sup>*

where sign(Δν) is the sign of the frequency variation (+ for an increasing and - for a decreasing frequency sweep). For two consecutive measurements, 1 and 2, the following

**3.2 Drift compensation method for dynamic mode measurements** 

cavity.

described by:

and Eq.(27) can be written as:

system of equations holds:

$$\begin{cases} L\_1 = N\_1 \frac{\Lambda\_1}{2 \cdot n} - \text{sign}(\Delta \mathbf{v}\_1) \cdot \frac{\Lambda\_1}{\lambda} \cdot S\_{drift} \cdot \Delta t\_1 \\\\ L\_2 = N\_2 \frac{\Lambda\_2}{2 \cdot n} - \text{sign}(\Delta \mathbf{v}\_2) \cdot \frac{\Lambda\_2}{\lambda} \cdot S\_{drift} \cdot \Delta t\_2 \\\\ L\_2 = L\_1 + S\_{drift} \cdot \Delta t\_3 \end{cases} \tag{34}$$

where Δ*t3* represents the time interval between the end of the first and the end of the second measurements. Drift speed and the corrected absolute distance for the second measurement are recovered by inverting Eq.(34) and using Eq.(29):

$$\begin{cases} S\_{drift} = \frac{\lambda}{2 \cdot n} \cdot \frac{\left(\frac{N\_2}{r\_2} - \frac{N\_1}{r\_1}\right)}{\left(\Delta t\_3 \cdot \frac{FSR}{c} \cdot \lambda\right) + \left(\text{sign}(\Delta \mathbf{v}\_2) \frac{\Delta t\_2}{r\_2} - \text{sign}(\Delta \mathbf{v}\_1) \frac{\Delta t\_1}{r\_1}\right)}\\ L\_2 = \frac{c}{r\_2 \cdot FSR} \cdot \left(\frac{N\_2}{2 \cdot n} \cdot \text{sign}(\Delta \mathbf{v}\_2) \cdot S\_{drift} \cdot \frac{\Delta t\_2}{\lambda}\right) \end{cases} \tag{35}$$

These equations are exact as long as drift speed is constant for the time span of the two measurements. Note that the frequency of the dynamic mode measurement can still be the same as it would be in the static mode single measurement as the first measurement can always be the same as the second measurement of the previous pair of sweeps. Thus, every new measurement enables another dynamic mode measurement (in this case, the characteristics of the two sweeps would swap from measurement to measurement).

The most simple implementation of this compensation method is to use two consecutive measurements with different signs on the sweep range (Δν) and equal durations, corresponding to a symmetrical triangular shaped frequency sweep (Δ*t2* = Δ*t1* = Δ*t* and Δν*2* = -Δν*1* = Δν *r2* = *r1* = *r*). In this case, we obtain:

$$\begin{cases} S\_{drift} = \frac{\lambda}{2 \cdot n} \cdot \frac{N\_2 - N\_1}{\left(\Delta t\_3 \cdot \frac{r \cdot FSR}{c} \cdot \lambda\right) - 2\Delta t} \\\\ L\_2 = \frac{c}{r \cdot FSR} \cdot \left(\frac{N\_2}{2 \cdot n} + S\_{drift} \cdot \frac{\Delta t}{\lambda}\right) \end{cases} \tag{36}$$

With this approach, we are not only correcting the length measurement but also providing a measurement of the drift speed.

#### **3.3 Measurement performances**

FSI sensor performance does not depend on the stability of the absolute value of the frequency but on the uncertainty in the frequency sweep range: there is no need to calibrate the system for absolute frequencies.

The system final uncertainty has two major contributors:

	- uncertainties in the *FSR* value and on
	- determination of the number of detected FSR (*r*).

Without drift (static mode), the final uncertainty, corresponding to a coverage probability of approximately 95% (IOS, 1995), can be obtained from (31):

$$\begin{aligned} \delta L &= \sqrt{\sum\_{i} \left( \frac{\partial L(X\_{i})}{\partial X\_{i}} \cdot \delta X\_{i} \right)^{2}} \\\\ \delta L &= \sqrt{\frac{c}{2 \cdot n \cdot FSR \cdot r} \cdot \delta N} \Bigg( + \\ \delta L &= \sqrt{\frac{c \cdot N}{2 \cdot n \cdot FSR^{2} \cdot r} \cdot \delta FSR}^{2} + \left( \frac{c \cdot N}{2 \cdot n \cdot FSR \cdot r^{2}} \cdot \delta r \right)^{2} + \\ &\left( \frac{c \cdot N}{2 \cdot n^{2} \cdot FSR \cdot r} \cdot \delta n \right)^{2} \end{aligned} \tag{37}$$

The uncertainty component in *N* is only the fringe phase uncertainty, δ*N*, because the integer number of fringes can be considered to be measured exactly. In the determination of the integer part, if a fringe is missed or over counted, the effect is easily noticed and can be corrected using simple outlier removal procedures. This uncertainty does not change with range, and depends only on the value of the synthetic wavelength. This component is dominant for small distances because it does not increase with *L*.

The next two components, in *r* and in *FSR*, determine the uncertainty in the sweep range measurement and are related to the FP performances.

The uncertainty in *r* depends on FP finesse and on the capability of the signal processing to localize resonance maxima. In order to locate resonance maxima in time unambiguously, the maxima of the signal generated by the FP should be clearly discriminated. The higher the finesse, the easier it is to locate accurately each resonance maxima in time and, therefore, reduce derived sensor errors.

The value of the *FSR* uncertainty is determined by the FP calibration and stability. The stability depends on length and optical variations induced by temperature and misalignments. For high resolution (low *FSR* uncertainty), thermal stabilization and lowthermal expansion materials may be required.

In contrast to the uncertainty in the synthetic fringe interpolation, the uncertainty components related to the sweep range measurement increase with range. These components become dominant for large distances.

The refractive index of air contributes to the measurement uncertainty due to the instabilities in the optical medium caused by the variations of the air temperature, pressure

uncertainty in the measured number of fringes *N* (the synthetic fringe interpolation

Without drift (static mode), the final uncertainty, corresponding to a coverage probability of

2

2

The uncertainty component in *N* is only the fringe phase uncertainty, δ*N*, because the integer number of fringes can be considered to be measured exactly. In the determination of the integer part, if a fringe is missed or over counted, the effect is easily noticed and can be corrected using simple outlier removal procedures. This uncertainty does not change with range, and depends only on the value of the synthetic wavelength. This component is

The next two components, in *r* and in *FSR*, determine the uncertainty in the sweep range

The uncertainty in *r* depends on FP finesse and on the capability of the signal processing to localize resonance maxima. In order to locate resonance maxima in time unambiguously, the maxima of the signal generated by the FP should be clearly discriminated. The higher the finesse, the easier it is to locate accurately each resonance maxima in time and, therefore,

The value of the *FSR* uncertainty is determined by the FP calibration and stability. The stability depends on length and optical variations induced by temperature and misalignments. For high resolution (low *FSR* uncertainty), thermal stabilization and low-

In contrast to the uncertainty in the synthetic fringe interpolation, the uncertainty components related to the sweep range measurement increase with range. These

The refractive index of air contributes to the measurement uncertainty due to the instabilities in the optical medium caused by the variations of the air temperature, pressure

*n*

*c N c N <sup>L</sup> FSR <sup>r</sup> n FSR r n FSR r*

2 2

2 2

2 2

(37)

uncertainty in the sweep range measurement Δν. The later depends on:

<sup>2</sup>

*<sup>c</sup> <sup>N</sup> n FSR r*

 

*i*

*i*

*L X L X X*

uncertainties in the *FSR* value and on

determination of the number of detected FSR (*r*).

*i i*

2

dominant for small distances because it does not increase with *L*.

*c N*

*n FSR r*

 

approximately 95% (IOS, 1995), can be obtained from (31):

2

2

measurement and are related to the FP performances.

thermal expansion materials may be required.

components become dominant for large distances.

reduce derived sensor errors.

uncertainty);

and humidity. The value of the refractive index of air can be determined by the Edlén equation (Stone & Zimmerman, 2004). The different parameters contribute differently to the index uncertainty. As an example, for normal laboratory conditions (p = 100,4 kPa; T = 20 ºC; RH = 50%), the contribution given by a change in the humidity of 50% is equivalent to a change of 0,2 kPa or 0,5 ºC. The influence of the humidity is thus irrelevant compared to the influence of pressure and temperature. In order to neglect the contribution of the laboratory environment the refractive index contribution should be kept at least around 10-6. This can be achieved by measuring the pressure with an uncertainty around ±0,2 kPa (±2 mbar) and by measuring (and controlling) the temperature within ±0,5 ºC. Note that for a space application this factor in null.

Fig. 3 shows an example of the system performances for these three components as a function of distance, considering typical parameters (presented in Table 1) used in the experimental results to shown later.

The value of the PF FSR should be selected in order to optimize its contribution to the final uncertainty. For the same sweep range, the number of resonances decreases when the FSR increases. Thus, if the δ*FSR* contribution is larger than the δ*r* contribution, the value of the FSR should increase and vice-versa. It must be noted that a low FSR is limited by the maximum available etalon length (the length of the FP increases when the FSR decreases) and by the FSR uncertainty (as the FSR becomes smaller, the number of resonances for the same sweep range increases and the contribution of the FSR uncertainty increases as well).

The previous analysis corresponds to a driftless static mode. In the dynamic mode with a constant drift speed, the same analysis can be performed starting with Eq. (35) or (36), leading in this case to a more complex solution. Nevertheless, the conclusions are basically the same.

Fig. 3. Example of the FSI model, considering typical parameters presented in Table 1. (δ*N* = 1/360 of a fringe, δ*FSR* = 5.10-6, δ*r* = 3.10-6, δ*n* = 1.10-6).


Table 1. FSI parameters used in the simulation presented in Fig. 3.
